Physics Reports vol.316

D. Youm/Physics Reports 316 (1999) 1}232 BLACK HOLES AND SOLITONS IN STRING THEORY Donam YOUM School of Natural Scienc...

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D. Youm/Physics Reports 316 (1999) 1}232

BLACK HOLES AND SOLITONS IN STRING THEORY

Donam YOUM School of Natural Sciences, Institute for Advanced Study Olden Lane, Princeton, NJ 08540, USA

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Physics Reports 316 (1999) 1}232

Black holes and solitons in string theory Donam Youm School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA Received October 1997; editor: A. Schwimmer Contents 1. Outline of the review 2. Soliton and BPS state 2.1. Physical parameters of solitons 2.2. Supermultiplets of extended supersymmetry 2.3. Positive energy theorem and Nester's formalism 3. Duality symmetries 3.1. Electric-magnetic duality 3.2. Target space and strong-weak coupling dualities of heterotic string on a torus 3.3. String}string duality in six dimensions 3.4. ;-duality and eleven-dimensional supergravity 3.5. S-duality of type-IIB string 3.6. ¹-duality of toroidally compacti"ed strings 3.7. M-theory 4. Black holes in heterotic string on tori 4.1. Solution generating procedure 4.2. Static, spherically symmetric solutions in four dimensions 4.3. Rotating black holes in four dimensions 4.4. General rotating "ve-dimensional solution 4.5. Rotating black holes in higher dimensions

4 5 7 11 20 24 26

37 43 47 49 50 56 62 62 65 80 83 89

5. Black holes in N"2 supergravity theories 5.1. N"2 supergravity theory 5.2. Supersymmetric attractor and black hole entropy 5.3. Explicit solutions 5.4. Principle of a minimal central charge 5.5. Double extreme black holes 5.6. Quantum aspects of N"2 black holes 6. p-branes 6.1. Single-charged p-branes 6.2. Multi-charged p-branes 6.3. Dimensional reduction and higher dimensional embeddings 7. Entropy of black holes and perturbative string states 7.1. Black holes as string states 7.2. BPS, purely electric black holes and perturbative string states 7.3. Near-extreme black holes as string states 7.4. Black holes and fundamental strings 7.5. Dyonic black holes and chiral null model 8. D-branes and entropy of black holes 8.1. Introduction to D-branes 8.2. D-brane as black holes 8.3. D-brane counting argument Acknowledgements References

E-mail address: [email protected] (D. Youm) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 3 7 - X

92 92 93 95 96 103 111 119 119 123 149 165 167 170 174 177 182 191 192 197 204 212 213

D. Youm / Physics Reports 316 (1999) 1}232

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Abstract We review various aspects of classical solutions in string theories. Emphasis is placed on their supersymmetry properties, their special roles in string dualities and microscopic interpretations. Topics include black hole solutions in string theories on tori and N"2 supergravity theories; p-branes; microscopic interpretation of black hole entropy. We also review aspects of dualities and BPS states. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.27.#d

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1. Outline of the review It is a purpose of this review to discuss recent development in black hole and soliton physics in string theories. Recent rapid and exciting development in string dualities over the last couple of years changed our view on string theories. Namely, branes and other types of classical solutions that were previously regarded as irrelevant to string theories are now understood as playing important roles in non-perturbative aspects of string theories; these solutions are required to exist within string spectrum by recently conjectured string dualities. Particularly, D-branes which are identi"ed as non-perturbative string states that carry charges in R-R sector have classical p-brane solutions in string e!ective "eld theories as their long-distance limit description. p-branes, other types of classical stringy solutions and fundamental strings are interrelated via web of recently conjectured string dualities. Much of progress has been made in constructing various p-brane and other classical solutions in string theories in an attempt to understand conjectured (non-perturbative) string dualities. We review such progress in this paper. In particular, we discuss black hole solutions in string e!ective "eld theories in details. Recent years have been active period for constructing black hole solutions in string theories. Construction of black hole solutions in heterotic string on tori with the most general charge con"gurations is close to completion. (As for rotating black holes in heterotic string on ¹, one charge degree of freedom is missing for describing the most general charge con"guration.) Also, signi"cant work has been done on a special class of black holes in N"2 supergravity theories. These solutions, called double extreme black holes, are characterized by constant scalars and correspond to the minimum energy con"gurations among extreme solutions. Among other things, study of black holes and other classical solutions in string theories is of particular interest since these allow to address long-standing problems in quantum gravity such as microscopic interpretation of black hole thermodynamics within the framework of superstring theory. In this review, we concentrate on recent remarkable progress in understanding microscopic origin of black hole entropy. Such exciting developments were prompted by construction of general class of solutions in string theories and realization that non-perturbative R-R charges are carried by D-branes. Within subset of solutions with restricted range of parameters, the Bekenstein} Hawking entropy has been successfully reproduced by stringy microscopic calculations. Since the subject reviewed in this paper is broad and rapidly developing, it would be a di$cult task to survey every aspect given limited time and space. The author made an e!ort to cover as many aspects as possible, especially emphasizing aspects of supergravity solutions, but there are still many issues missing in this paper such as stringy microscopic interpretation of black hole radiation, M(atrix) theory description of black holes and the most recent developments in N"2 black holes and p-branes. The author hopes that some of missing issues will be covered by other forthcoming review paper by Maldacena [470]. The review is organized as follows. Sections 2 and 3 are introductory sections where we discuss basic facts on solitons and string dualities which are necessary for understanding the remaining sections. In these two chapters, we especially illuminate relations between BPS solutions and string dualities. In Section 4, we summarize recently constructed general class of black hole solutions in heterotic string on tori. We show explicit generating solutions in each spacetime dimensions and discuss their properties. In Section 5, we review aspects of black holes in N"2 supergravity theories. We discuss principle of a minimal central charge, double extreme solutions and quantum corrections. In Section 6, we


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summarize recent development in p-branes. Here, we show how p-branes and other related solutions "t into string spectrum and discuss their symmetry properties under string dualities. We systematically study various single-charged p-branes and multi-charged p-branes (dyonic p-branes and intersecting p-branes) in di!erent spacetime dimensions. We also discuss black holes in type-II string on tori as special cases and their embedding to p-branes in higher dimensions. In Sections 7 and 8, we summarize the recent exciting development in microscopic interpretation of black hole entropy within the framework of string theories. We discuss Sen's calculation of statistical entropy of electrically charged black holes, Tseytlin's work on statistical entropy of dyonic black holes within chiral null model and D-brane interpretation of black hole entropy.

2. Soliton and BPS state Solitons are de"ned as time-independent, non-singular, localized solutions of classical equations of motion with "nite energy (density) in a "eld theory [106,512]. Such solutions in D spacetime dimensions are alternatively called p-branes [228,234] if they are localized in D!1!p spatial coordinates and independent of the other p spatial coordinates, where p(D!1. For example, the p"0 case (0-brane) has a characteristic of point particles and is also called a black hole; p"1 case is called a string; p"2 case is a membrane. The main concern of this paper is on the p"0 case, but we discuss the extended objects (p51) in higher dimensions as embeddings of black holes and in relations to string dualities. As non-perturbative solutions of "eld theories, solitons have properties di!erent from perturbative solutions in "eld theories. First, the mass of solitons is inversely proportional to some powers of dimensionless coupling constants in "eld theories. So, in the regime where the perturbative approximations are valid (i.e. weak-coupling limit), the mass of solitons is arbitrarily large and the soliton states decouple from the low energy e!ective theories. So, their contributions to quantum e!ects are negligible. Their contribution to full dynamics becomes signi"cant in the strong coupling regime. Second, solitons are characterized by `topological chargesa, rather than by `Noether chargesa. Whereas the Noether charges are associated with the conservation laws associated with continuous symmetry of the theory, the topological conservation laws are consequence of topological properties of the space of non-singular "nite-energy solutions. The space of non-singular "nite energy solutions is divided into several disconnected parts. It takes in"nite amount of energy to make a transition from one sector to another, i.e. it is not possible to make a transition to the other sector through continuous deformation. Third, the solitons with "xed topological charges are additionally parameterized by a "nite set of numbers called `modulia. Moduli or alternatively called collective coordinates are parameters labeling di!erent degenerate solutions with the same energy. The space of solutions of "xed energy is called moduli space. The moduli of solitons are associated with symmetries of the solutions. For example, due to the translational invariance of the Yang}Mills}Higgs Lagrangian, the monopole solution sitting at the origin has the same energy as the one at an arbitrary point in R; the associated collective coordinates are the center of mass coordinates of the monopole. In addition, there are collective coordinates associated with the gauge invariance of the theory. Note, monopole carries charge of the ;(1) gauge group which is broken from the non-Abelian (S;(2)) one at in"nity (where the Higgs "eld takes its value at the gauge symmetry breaking vacuum). Thus, only relevant gauge

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transformations of non-Abelian gauge group that relate di!erent points in moduli space are those that do not approach identity at in"nity, i.e. those that reduce to non-trivial ;(1) gauge transformations at in"nity. Another important characteristic of solitons is that they are the minimum energy con"gurations for given topological charges, i.e. the energy of solitons saturates the Bogomol'nyi bound [94,510]. The lower bound is determined by the topological charges, e.g. the winding number for strings and ;(1) gauge charge carried by black holes. The original calculation [94] of the energy bound for a soliton in #at spacetime involves taking complete square of the energy density ¹ ; the minimum RR energy is saturated if the complete square terms are zero. Solitons therefore satisfy the "rst-order di!erential equations (`complete square termsa"0), the so-called Bogomol'nyi or self-dual equations. An example is the (anti) self-dual condition F "$夹F for Yang}Mills instantons IJ IJ [68]. Another example is the magnetic monopoles [507,592] in an S;(2) Yang}Mills theory, which satisfy the "rst-order di!erential equation BG"$DGU relating the magnetic "eld BG to the Higgs "eld U. Here, the Higgs "eld takes its values at the minimum of the potential B

1 1 d"x(!g F . S " BU 16pG" 2(d#1)! B> , We omit the dilaton factor in the kinetic term for the sake of the argument.

(9)

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Note, in this kinetic term, G" is absorbed in the action in contrast to the form of the matter term in , (1). The "eld equations and Bianchi identities of A are B (10) d夹F "2i (!)B夹J , dF "0 , B B> B> " where J is the rank d source current and 夹 denotes the Hodge-dual transformation in D spacetime B dimensions, i.e. 1 (夹A )I2I"\B, eI2I"(A ) "\B>2 " with e2"\"1. I BI B d! The soliton that carries the `Noethera electric charge Q under A is an elementary extended B B object with d-dimensional worldvolume, called (d!1)-brane, and has the electric source J coming from the p-model action of the (d!1)-brane. The `topologicala magnetic charge P I of A is carried B B by a solitonic (i.e. singularity and source free) object with dI -dimensional worldvolume, called (dI !1)-brane, where dI ,D!d!2. The `Noethera electric and the `topologicala magnetic charges of A are de"ned as B 1 (!)B夹J " 夹F , Q ,(2i " M B (2i "\B\ B> B "\B 1 " 1 PI, F . (11) B (2i B> B> 1 " These charges obey the Dirac quantization condition [482,591]:

Q PI n B B" , n3Z . 4p 2

(12)

The electric and magnetic charges of A have dimensions [Q ]"¸\"\B\ and B B [P I ]"¸"\B\, respectively. Electric/magnetic charges are dimensionless when D"2(d#1). B Examples are point-like particles (d"1) in D"4, strings (d"2) in D"6 [311,542] and membranes (d"3) in D"8 [311,388]. From (11), one sees that the AnsaK tze for F for the soliton that carries electric or magnetic B> charge of A are respectively given by B , (13) 夹F "(2i Q e I /X I , F "(2i P I e /X B> " B B> B> " B B> B> B> where e denotes the volume form on SL, and the electric and magnetic charges of A are de"ned L B from the asymptotic behaviors: Q PI X u B , F & B> B , (14) A& B B> (2i rB> B (2i r"\B\ " " where r is the transverse distance from the (d!1)-brane, u is the volume form for the (d!1)B brane worldvolume and X is the volume form of SB> surrounding the brane. B> From the elementary (d!1)-brane, one "nds that the electric charge Q is related to the tension B ¹ of the (d!1)-brane in the following way: B Q "(2i ¹ (!)"\BB> . (15) B " B


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Here, ¹ has dimensions [¹ ]"M¸B\ in the unit c"1 and therefore is interpreted as mass per B B unit (d!1)-brane volume. In particular, for a 0-brane (d"1) the tension ¹ is the mass. The Dirac quantization condition (12), together with (15), yields the following form of the magnetic charge P I of A : B B 2pn PI" (!)"\BB>, n3Z . (16) B (2i ¹ " B We comment on the ADM mass of (d!1)-branes. Note, in deriving (8) we assumed that the metric g depends on all the spatial coordinates. So, (8) applies only to the 0-brane type soliton (or IJ black holes). The (d!1)-branes do not depend on the (d!1) longitudinal coordinates internal to the (d!1)-brane and therefore the Laplacian in (4) is replaced by the #at (D!d!1)-dimensional one. As a consequence, in particular, the (t, t)-component of the metric has the asymptotic behavior:

16pG" M 1 , B g &! 1! . (17) RR (D!2)X < r"\B\ "\B\ B\ Here, the ADM mass M of the (d!1)-brane is de"ned as M ,¹ d"\x"< ¹ d"\Bx, B B B\ where < is the volume of the (d!1)-dimensional space internal to the (d!1)-brane. So, for B\ (d!1)-branes it is the ADM mass `densitya

M o , B " ¹ d"\Bx B < B\ that has the well-de"ned meaning. As an example, we consider the elementary BPS (d!1)-brane in D dimensions. The leading order asymptotic behavior of the (t, t)-component of the metric of (d!1)-brane carrying one unit of the d-form electric charge is

D!d!2 c" B g &! 1! , dI '0 , RR D!2 r"\B\

(18)

where c",2i ¹ /dI X I is the unit (d!1)-brane electric charge and r,(x#2#x ) B " B B> "\B\ is the radial coordinate of the transverse space. For (d!1)-branes carrying m units of the basic electric charge, c" in (18) is replaced by mc". From (17) and (18), one obtains the following ADM B B mass density of the (d!1)-brane carrying one unit of electric charge: o "¹ "(1/(2i ) "Q " . " B B B

(19)

2.2. Supermultiplets of extended supersymmetry 2.2.1. Spinors in various dimensions Before we discuss the BPS states in extended supersymmetry theories, we summarize the basic properties of spinors for each spacetime dimensions D. More details can be found, for example, in

When dI "0, e.g. a string in D"4, the metric is asymptotically logarithmically divergent. In this case, the ADM mass density is determined from volume integral of the (t, t)-component of the gravitational energy-momentum pseudo-tensor [440].

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[438,575,628,629]. We assume that there is only one time-like coordinate. The types of superPoincareH algebra satis"ed by supercharges depend on D. The superalgebra is classi"ed according to the fundamental spinor representations of the homogeneous group SO(1, D!1) and the vector representation of the automorphism group that supercharges belong to. The pattern of superalgebra repeats with D mod 8. In even D, one can de"ne c-like matrix c,gcc2c"\ which anticommutes with cI and has the property c"1 (implying g"(!1)"\), required for constructing a projection operator. So, the 2 " complex component Dirac spinor t, which is de"ned to transform as dt"!e RIJt (RIJ,![cI, cJ]) under the in"nitesimal Lorentz transformation, in even D IJ is decomposed into 2 inequivalent Weyl spinors t "(1#c)t and t "(1!c)t with \ > 2"\ complex components each. We discuss the reality properties of spinors. One can always "nd a matrix B satisfying RIJH"BRIJB\. B de"nes the charge conjugation operation: tPtA"Ct,B\tH .

(20)

By de"nition, the charge conjugation operator C commutes with the Lorentz generators RIJ, implying that t and tA have the same Lorentz transformation properties. If C"1, or equivalently BBH"1, the Dirac spinor t can be reduced to a pair of Majorana spinors (i.e. eigenstates of C) t "(1#C)t and t "(1!C)t. This is possible in D"2, 3, 4, 8, 9 mod 8. First, in odd D, Majorana spinors are necessarily self-conjugate under C and are always real. So, the Dirac spinors in odd D are real [pseudoreal] in D"1, 3 mod 8 [D"5, 7 mod 8]. In even D, Majorana spinors can be either complex or real depending on whether t and tA have the same or opposite helicity. Namely, since Cc"(!1)"\cC, t and tA have the same (opposite) helicity for even (odd) (D!2)/2, i.e. D"2 mod 8 [D"4, 8 mod 8]. So, in even D, the Dirac spinors are real (complex) or pseudoreal for D"2 mod 8 (D"4, 8 mod 8) or D"6 mod 8, respectively. In particular, in D"2 mod 8, both the Weyl and the Majorana conditions are satis"ed, and therefore in this case the Dirac spinor t is called Majorana}Weyl. We saw that supercharges Q (i"1,2, N), transforming as spinors under SO(1, D!1), have G di!erent chirality and reality properties depending on D. The set +Q ,, furthermore, transforms as a G vector under an automorphism group, with i acting as a vector index. The automorphism group depends on the reality properties of +Q ,. The automorphism group is SO(N), ;Sp(N) or S;(N);;(1) for G real, pseudoreal or complex case, respectively. In D"2 mod 8 and D"6 mod 8, the pair of Weyl spinors with opposite chiralities are not related via C and therefore are independent: the automorphism groups are SO(N );SO(N ) and ;Sp(N );;Sp(N ) in D"2 mod 8 and D"6 mod 8, > > > > respectively, where N #N "N. The central charge Z'( transforms as a rank 2 tensor under the > \ automorphism group with (I, J) acting as tensor indices. In D"0, 1, 7 mod 8 [D"3, 4, 5 mod 8], the central charge has the symmetry property Z'("Z(' [Z'("!Z(']. The number N of supercharges Q' in each D is restricted by the physical requirement that particle helicities should not exceed 2 when compacti"ed to D"4 [17,69,169,274,479]. This limits the maximum D with 1 time-like coordinate and consistent supersymmetric theory to be 11 with N"1 supersymmetry, i.e. 32 supercharge degrees of freedom. This corresponds to N"8 supersymmetry in D"4 when compacti"ed on ¹. In D(11, the number of spinor degrees of freedom cannot exceed that of N"1, D"11 theory. For the pseudoreal cases, i.e. D"5, 6, 7 mod 8, only even N are possible.


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2.2.2. Central charges and super-PoincareH algebra We discuss types of central charge Z'( one would expect in the super PoincareH algebra. According to a theorem by Haag et al. [330], within a unitary theory of point-like particle interactions in D"4 the central charge can be only Lorentz scalar. However, in the presence of p-branes (p51), central charges Z'(2 N transforming as Lorentz tensors can be present in the I I superalgebra without violating unitarity of interactions [201]. In fact, as will be shown, it is the Lorentz tensor type central charges in higher dimensions that are responsible for the missing central charge degrees of freedom in lower dimensions when the higher-dimensional superalgebra is compacti"ed with an assumption that no Lorentz tensor type central charges are present [603]. The Lorentz tensor type central charges appear in the supersymmetry algebra schematically in the form: (21) +Q', Q(,"d'((CcI) P # (CcI2IN) Z'(2 N , ?@ I I ? @ ?@ I N2 where P is D-dimensional momentum, I, J"1,2, N label supersymmetries and a, b are spinor I indices in D dimensions. Here, (CcI) in (21) is replaced by (CcIP!) for positive or negative chiral ?@ ?@ Majorana spinors Q' (e.g. type-IIB theory), where P projects on the positive or negative ! ! chirality subspace, and also similarly for (CcI2IN) . Note, Z'(2 N commute with Q' and P , but ?@ I I ? I transform as second rank tensors under the Lorentz transformation, and therefore are central with respect to supertranslation algebra, only. The number of central charge degrees of freedom is determined by the number of all the possible (I, J) in (21) [41}44]. In the sum term in (21), one has to take into account the overcounting due to the Hodge-duality between p and D!p forms (Z'(2 N&Z'(2 "\N). When p"D!p, Z'(2 N are I I I I I I self-dual or anti-self dual. (For this case, the degrees of freedom are halved.) (I, J) on Z'(2 N are I I de"ned to have the same permutation symmetry as (a, b) in cI2IN so that cI2INZ'(2 N is symmetric ?@ ?@ I I under the simultaneous exchanges of indices in the pairs (I, J) and (a, b) so that they have the same symmetry property (under the exchange of the indices) as the left-hand side of (21). Namely, only terms associated with cI or cI2IN that are either symmetric or antisymmetric under the exchange ?@ ?@ of a and b can be present on the right-hand side of (21). 2.2.3. Central charges and i-symmetry The p-form central charge Z'(2 N in (21) arises from the surface term of the Wess}Zumino (WZ) I I term in the p-brane worldvolume action [201]. Before we discuss this point, we summarize how WZ term emerges in the p-brane worldvolume action [599,600]. In the Green}Schwarz (GS) formalism [86,313,315,351] of the supersymmetric p-brane worldvolume action, one achieves manifest spacetime supersymmetry by generalizing spacetime with bosonic coordinates XI (k"0, 1,2, D!1) and global Lorentz symmetry to superspace R with coordinates Z+"(XI, h?) and super-PoincareH invariance. Here, a is a Ddimensional spacetime spinor index and the spacetime spinor h? takes an additional index I (I"1,2, N) for N-extended supersymmetry theories, i.e. h'?. Fields in the GS action are

A Majorana spinor Q is de"ned as QM "Q2C, where the bar denotes the Dirac conjugate. The positive or negative chiral spinor Q is de"ned as cQ "$Q . ! ! !

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regarded as maps from the worldvolume = to R. The worldvolume = of a p-brane has coordinates mG"(q, p ,2, p ) with worldvolume vector index i"0, 1,2, p. We denote an immersion from N = to R as : =PR. The pullback H of a form in R by induces a form in =. To generalize the bosonic p-brane worldvolume Lagrangian density L "¹ [!det(R XI(m)R XJ(m)g )] to be N G H IJ invariant under the supersymmetry transformation as well as local reparameterization and global PoincareH transformations, one introduces a supertranslation invariant D-vector-valued 1-form PI,dXI!ihM cI dh. This corresponds to the spacetime component of the left-invariant 1-form P"(PI, P?"dh?) on R. The simplest and straightforward supersymmetric generalization of the bosonic worldvolume p-model action for a p-brane is [2,85}87,201,313,315,351,377] S "¹ N

dN>m(!det(PI(m)PJ(m)g ) , (22) G H IJ 5 where ¹ is the p-brane tension and PI is the mG-component of the pullback of the 1-form PI in R by N G

, i.e. ( H P)(m)"P(m) dmG with PI"R XI!ihM cIR h and P?"R h? (R ,R/RmG). (22) is manifestly G G G G G G G invariant under the global super-PoincareH and local reparameterization transformations, but is not invariant under a local fermionic symmetry, called ì-symmetrya [3,88,131,313,377], which is essential for equivalence of the GS and NSR formalisms of the worldvolume action. To make (22) invariant under the i-symmetry, one introduces an additional term S , called `Wess}Zumino 58 (WZ) actiona, into (22). To construct the Wess}Zumino (WZ) action for a p-brane, one introduces the super-PoincareH invariant closed (p#2)-form h on R. Such closed (p#2)-forms exist only N> for restricted values of D and p. The complete listing of the values of (D, p) are found in [2]. The maximum values of allowed D and p are D "11 and p "5, which can also be determined

by the worldvolume bose-fermionic degrees of freedom matching condition discussed in the next paragraph. The super-PoincareH invariant closed (p#2)-form, in general, has the form is closed, one can locally write h in terms of h "PI2PIN> dhM c 2 N> dh. Since h I I N> N> N> a (p#1)-form b (on R) as h " db . Then, a super-PoincareH invariant WZ action for N> N> N> over = [2,85}87,201,313,315,351,377]: a p-brane is obtained by integrating b N>

. S "¹ dN>m H b N> 58 N

(23)

Note, whereas S and S are individually invariant under the local reparameterization and global 58 super-PoincareH transformations, the i-symmetry is preserved only in the complete action S "S #S . The i-symmetry gauges way the half of the degrees of freedom of the spinor h, N 58 thereby only 1/2 of spacetime supersymmetry is linearly realized as worldvolume supersymmetry [378]. To summarize, the invariance under the i-symmetry necessitates the introduction of b (on R) via the WZ term; b couples to the worldvolume of the p-branes and becomes the N> N> origin of the central charge term in the supersymmetry algebra. We comment on the allowed values of p and the number N of spacetime supersymmetry for each D. This is determined [241] by matching the worldvolume bosonic degrees N and fermionic degrees N of freedom. First, we consider the case where the worldvolume theory corresponds to $ scalar supermultiplet (with components given by scalars and spinors). By choosing the static gauge (de"ned by XI(m)"(XG(m),>K(m))"(mG,>K(m)), with i"0, 1,2, p and m"p#1,2, D!1), one "nds that the number of on-shell bosonic degrees of freedom is N "D!p!1. We denote the


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number of supersymmetries and the number of real components of the minimal spinor in D-dimensional spacetime [(p#1)-dimensional worldvolume] as N and M [n and m], respectively. Then, since the i-symmetry and the on-shell condition each halves the number of fermionic degrees of freedom, the number of on-shell fermionic degrees of freedom is N "mn"MN. The allowed $ values of N and p for each D is determined by the worldvolume supersymmetry condition N "N , $ i.e. D!p!1"mn"MN. The complete listing of values of N and p are found in [234,241]. The maximum number of D in which this condition can be satis"ed is D "11 (p"2) with M"32

and N"1. So, for other cases (D(11), MN432. Similarly, the maximum value of p for which this condition can be satis"ed is p "5. The `fundamentala super p-branes [2] that satisfy

this condition are (D, p)"(11, 2), (10, 5), (6, 3), (4, 2). The 4 sets of p-branes obtained from these `fundamentala super p-branes through double-dimensional reduction are named the octonionic, quaternionic, complex and real sequences. Note, in addition to scalars and spinors, there are also higher spin "elds on the worldvolume [241]: vectors or antisymmetric tensors. First, we consider vector supermultiplets. Since a worldvolume vector has (p!1) degrees of freedom, the worldvolume supersymmetry condition N "N becomes D!2"mn"MN. This condition intro $ duces additional points in the brane-scan. Vector supermultiplets exist only for 34p49 and the bose}fermi matching condition can be satis"ed in D"4, 6, 10, only. Second, we consider tensor worldvolume supermultiplets. In p#1"6 worldvolume dimensions, there exists a chiral (n , n )"(2, 0) tensor supermultiplet (B\ , j', '( ), I, J"1,2, 4, with a self-dual 3-form "eld > \ IJ strength, corresponding to the D"11 5-brane. The decomposition of this (2, 0) supermultiplet under (1, 0) into a tensor multiplet with 1 scalar and a hypermultiplet with 4 scalars, followed by truncation to just the tensor multiplet, leads to worldvolume theory of 5-brane in D"7. 2.2.4. Central charges and topological charges We illustrate how Lorentz tensor type central charges (associated with p-branes) arise in the supersymmetry algebra [201]. Since the action S has the manifest super-PoincareH invariance, N one can construct supercharges QG from the conserved Noether currents j associated with ? ? super-PoincareH symmetry. Whereas (22) is invariant under the super-PoincareH variation, i.e. d S "0, the integrand of the WZ action (23) is only quasi-invariant. Namely, since d b "dD 1 1 N> N for some p-form D , the integrand of S transforms by total spatial derivative: N 58 ¹ d ( H b )"d( H D ),R DG dqdp2 dpN. It is D that induces `topological N 1 N> N G N N chargea which becomes central charge in the super-PoincareH algebra. Generally, when a Lagrangian density L is quasi-invariant under some transformation, i.e. d L"R DG , the associated G Noether current jG contains an ànomalousa term DG : RL !DG . jG "d Z+ R(R Z+) G Such an anomalous term modi"es the algebra of the conserved charges Q"dNp jO to include a topological (or central) terms A . For a p-brane, the WZ action (23) gives rise to the central term in the supersymmetry algebra of the form:

A "¹ (CcI2IN) dNp jO I2IN , ?@ 2 ?@ N

(24)

16


where jO I2IN is the (worldvolume) time component of the topological current density 2 jG I2IN"eGH2HNR XI(m)2R NXIN(m). So, p-form central charges in supersymmetry algebra (21) has 2 H H the following general form [44] given by the surface integral of a (p#1)-form local current J'( 2 N(x) over a space-like surface embedded in D-dimensional spacetime: II I

Z'(2 N" d"\RI J'( 2 N(x) . II I I I

(25)

Here, the (p#1)-form local current J'( 2 N(x) has contributions from individual p-branes with the II I coordinates X? (q, p ,2, p ) and charges z'( (index a labeling each p-brane): I N ?

J'( 2 N(x)" dq dp 2 dp z'(d"(x!X?(q, p ,2, p )) N ? N II I ? (26) ;R X? 2R NX? N (q, p ,2, p ) . N N I

O I This (p#1)-form current is coupled to a (p#1)-form gauge potential A'( 2 N(x) of the low energy II I e!ective supergravity in the following way:

S& d"x AII2IN(x)J'( 2 N(x) '( II I

(27) " dq dp 2 dp AII2IN(X?)R X? 2R NX? N z'( , O I N '( N I ? ? where z'( are the charges of A'( 2 N(x) carried by the ath p-brane with the coordinates X? . The ? II I I "eld equation for A'( 2 N(x) is II I (28) RHR A'( 2 N (x)"J'( 2 N(x) . II I H I I I

So, one can think of Z'(2 N as being related to charges of A'( 2 N(x) with the charge source given II I I I by p-branes with their worldvolumes coupled to A'( 2 N(x). There is a one-to-one correspondence II I between A'( 2 N(x) in the e!ective supergravity theory and Z'( 2 N in the superalgebra, i.e. there II I II I are as many central extensions as form "elds in the e!ective supergravity. 2.2.5. S-theory The maximally extended superalgebra has 32 real degrees of freedom in the set +Q', of ? supercharges, i.e. N"1 supersymmetry in D"11 or N"8 supersymmetry in D"4. So, the right-hand side of (21) has at most (32;33)/2"528 degrees of freedom; the sum of D degrees of freedom of the momentum operator PI and the degrees of freedom of central charges Z'(2 N in (21) I I has to be 528. This is the main reason for the necessity of existence of p-branes in higher dimensions [603]; N"1 supersymmetry in D"11 without central charge has only 11 degrees of freedom on the right-hand side of (21).

In the supersymmetry algebra (21), the p-brane tension ¹ is set equal to 1. N So, a p-form central charge is related to boundaries of the p-brane. For example for a string ( p"1), Z & I X (0)!X (p). I I


17

The extended superalgebra (21) can be derived by compactifying either a type-A superalgebra in D"2#10 or a type-B superalgebra in D"3#10 (the so-called `S-theoriesa) [41}44], with the Lorentz symmetries SO(10, 2) and SO(9, 1)SO(2, 1), respectively. The superalgebra of the typeIIA(B) theory is obtained by compactifying the type-A(B) algebra. First, the type-A algebra has the form: (29) +Q , Q ,"c++Z #c+2+Z>2 , ?@ ++ ?@ + + ? @ where M "0, 0, 1,2, 10 is a D"12 vector index with 2 time-like indices 0, 0. Note, in D"12 G with 2 time-like coordinates, only gamma matrices which are (anti)symmetric are c++ and c+2+, with c+2+ being self-dual. So, one has terms involving 2-form central charge Z and ++ self-dual 6-form central charge Z>2 , without momentum operator P+ term, in (29). Generally, + + in the D(12 supersymmetry algebra compacti"ed from the type-A algebra, the spinor indices a and a of 32 spinors Q? are regarded as those of SO(c#1, 1) and SO(D!1, 1) Lorentz groups, ? respectively, where c is the number of compacti"ed dimensions from the point of view of D"10 string theory. Second, the type-B algebra has the form: (30) +Q? , Q@M M ,"cI M (cq )? @M PG #cI M IIc? @M > #cI M 2I(cq )? @M XG 2 , ? @ ?@ ?@ ?@ G I III G I I where the indices are divided into the D"10 ones a, bM "1, 2,2,16 and k"0, 1,2, 9 with the Lorentz group SO(1, 9), and the D"3 ones a , bM "1, 2 and i"0, 1, 2 with the Lorentz group SO(1, 2). (The barred [unbarred] indices are spinor [spacetime vector] indices.) Here, cI [q ] are G gamma matrices of the SO(1, 9) [SO(1, 2)] Cli!ord algebra and c? @M "ip? @M "e? @M . We discuss the maximal extended superalgebras that follow from the type-A algebra. First, the following N"1, D"11 superalgebra is obtained from (29) by compactifying the 0-coordinate: +Q , Q ,"(CIC) P #(CIIC) Z #(CI2IC) X 2 , ?@ I I ?@ I I ? @ ?@ I where each term on the right-hand side emerges from the terms in (29) as

(31)

Z PP Z 66"11#55 ++ I II (32) 462"462 . Z>2 PX 2 I I + + The central charges Z and Z 2 are associated with the M2 and M5 branes, respectively. The I I II maximal extended superalgebras in D(11 are obtained by compactifying the D"11 supertranslation algebra (31) on tori. The central charge degrees of freedom in lower dimensions are counted by adding the contribution from the internal momentum P (m"1,2, 11!D) and the number of K ways of wrapping M2 and M5 branes on cycles of ¹\" in obtaining various p-branes in lower dimensions. Schematically, decompositions of the terms on the right-hand side of (31) are P "P P , Z "Z ZKZKK , II I I I K II (33) X 2 "X 2 XK2 XKK XKKKXK2KXK2K . I I III II I I I I I The N(N!1) Lorentz scalar central charges of the (maximal) N-extended D(11 superalgebras originate from the Lorentz scalar type terms (under the SO(D!1, 1) group) on the right-hand sides of (33), i.e. P , ZKK and XK2K. In this consideration, one has to take into account equivalence K under the Hodge duality in D dimensions. N(N!1)"56 Lorentz scalar central charges of N"8 superalgebra in D"4 originate from the 7 components of P , (7;6)/(2;1)"21 terms in ZKK, K

18


(7;6;5;6;5)/(5;4;3;2;1)"21 terms in XK2K and 7 terms in the Hodge dual of XK2 . I I The similar argument regarding the supertranslational algebras of type-IIB and heterotic theories can be made, and details are found, for example, in [44,603]. We saw that one has to take into account the Lorentz tensor type central charges in higher dimensions to trace the higher-dimensional origin of N(N!1) 0-form central charge degrees of freedom in N-extended supersymmetry in D(11. This supports the idea that in order for the conjectured string dualities (which mix all the electric and magnetic charges associated with N(N!1) 0-form central charges in D(10 among themselves) to be valid, one has to include not only perturbative string states but also the non-perturbative branes within the full string spectrum. In lower dimensions, these central charges are carried by 0-branes (black holes). It is a purpose of Section 4 to construct the most general black hole solutions in string theories carrying all of 0-form central charges. In Section 6.3.3, we identify the (intersecting) p-branes in higher dimensions which reduce to these black holes after dimensional reduction.

2.2.6. Central charges and moduli xelds We comment on relation of central charges to ;(1) charges and moduli "elds [7,8]. Except for special cases of D"4, N"1, 2 and D"5, N"2, scalar kinetic terms in supergravity theories are described by p-model with target space manifold given by coset space G/H. Here, a non-compact continuous group G is the duality group that acts linearly on "eld strengths HK2 N> and is on-shell I I and/or o!-shell symmetry of the action. The isotropy subgroup HLG is decomposed into the automorphism group H of the superalgebra and the group H related to the matter

multiplets: H"H H . (Note, the matter multiplets do not exist for N'4 in D"4, 5 and in

maximally extended supergravities.) The properties of supergravity theories are "xed by the coset representatives ¸ of G/H. ¸ is a function of the coordinates of G/H (i.e. scalars) and is decomposed as: ¸\"(¸ K, ¸ K) , (34) ¸"(¸KR)"(¸K , ¸K), ' ' where (A, B) and I respectively correspond to 2-fold tensor representation of H and the fundamental representation of H . Here, K runs over the dimensions of a representation of G.

The (p#2)-form strength HK kinetic terms are given in terms of the following kinetic matrix determined by ¸: . (35) NKR"¸ K¸ R !¸ K¸' ' R So, the `physicala "eld strengths of (p#1)-form potentials in supergravity theories are `dresseda with scalars through the coset representative and the (p#2)-form "eld strengths appear in the supersymmetry transformation laws dressed with the scalars. The central charges of extended superalgebra, which is encoded in the supergravity transformations rules, are expressed in terms of electric QK, "\N\GK and magnetic PK, N>HK charges of (p#2)-form "eld strengths 1 1 HK"dAK (GK"RL/RHK) and the asymptotic values of scalars in the form of the coset representative, manifesting the geometric structure of moduli space. These central charges satisfy the di!erential equations that follow from the Maurer}Cartan equations satis"ed by the coset representative. One of the consequences of these di!erential equations is that the vanishing of a subset of central charges (resulting from the requirement of


19

supersymmetry preserving bosonic background) forces the covariant derivative of some other central charges to vanish, i.e. `principle of minimal central chargea [258,259]. Furthermore, from de"ning relations of the kinetic matrix of (p#2)-form "eld strengths and the symmetry properties of the symplectic section under the group G, one obtains the sum rules satis"ed by the central and matter charges. For other cases, in which the scalar manifold cannot be expressed as coset space, one can apply the similar analysis as above by using techniques of special geometry [132,270,576]. For this case, the roles of the coset representative and the Maurer}Cartan equations are played respectively by the symplectic sections and the Picard}Fuchs equations [134]. 2.2.7. BPS supermultiplets We discuss massive representations of extended superalgebras with non-zero central charges [264,330,485,489,575], i.e. the BPS states. It is convenient to go to the rest frame of states de"ned by P "(M, 0,2, 0), where M is the rest mass of the state. The little group, de"ned as a set of I transformations that leave this P invariant, consists of SO(D!1), the automorphism group and I the supertranslations. Since central charges Z'( transform as a second rank tensor under the automorphism group, only the subset of automorphism group that leaves Z'( invariant should be included in the little group. Central charges inactivate some of supercharges, reducing the size of supermultiplets. In the following, we illustrate properties of the BPS states for the D"4 case with an arbitrary number N of supersymmetries. In the Majorana representation, the central charges ;'( and , N C N"(¸2)\N¸\, F> C F>"aF>#bgNF>, N C N"(dN#cg)(bgN#a)\ .

(84)

Note, O(m, n) [S¸(2, R)] is a perturbative (non-perturbative) symmetry, since N is not inverted (gets inverted). S¸(2, R) is the symmetry of the equations of motion only, since electric and magnetic charges get mixed, and since this corresponds to the transformations (70) and (71) with B O0OC . R R

32


3.1.3. Target space manifolds of N"2 theories Contrary to D"4, N53 theories, the scalar manifolds of N"2 theories are not necessarily expressed as homogeneous symmetric coset manifolds. Scalar manifold of the D"4, N"2 theory has the generic form M "SM HM , (85) L K where SM is a special KaK hler manifold of the complex dimension n" `the number of the vector L supermultipletsa, and the manifold HM spanned by the scalars in the hypermultiplets has the K dimension 4m, where m" `the number of the hypermultipletsa. So, the metric g ( ) of M has '( the form (86) g ( )d 'd ("g *dz?dz @*#h dqSdqT . ST '( ?@ Here, g * [h ] is the special KaK hler metric on SM [the quaternionic metric on HM ]. ST L L ?@ 3.1.3.1. Special KaK hler manifolds. N"2 super-Yang-Mills theory is described by a chiral super"eld U, which is de"ned by DM ? G U"0 (like chiral super"eld in N"1 theory), with an additional constraint: (87) D? D@ Ue "e e lDM ? IDM l@Q UM e Q , ?@ G H ?@ GI H where i"1, 2 labels supercharges of N"2 theory, a, a"1, 2 are indices of chiral spinors, and DM ? is a covariant chiral superspace derivative. The component "elds of a N"2 chiral super"eld U are a scalar X, spinors jG, ;(1) gauge "eld strength F , and auxiliary scalars > satisfying a reality IJ GH constraint > "e e l>M Il (due to the constraint (87)). The action of N"2 chiral super"elds U is GH GI H determined by an arbitrary holomorphic function F(U) of U as dxdh F(U)#c.c., and is given, in terms of the component "elds, by (88) L"g M R XRIXM #g M jM cIR j M #Im(F F\F\ IJ)#2 , IJ I G I where the dots denote the interaction terms involving fermions, g M "R R M K is a KaK hler metric, and F ,R R F. Note, this action is a special case of the most general coupling of N"1 chiral super"elds to N"1 Abelian vector super"elds in which the KaK hler potential K and the holomorphic kinetic term function F take the following special forms [283,572]: K(X, XM )"i[FM (XM )X!F (X)XM ] (F ,R F), F "R R F . (89) The KaK hler manifold with the KaK hler potential K(X, XM ) determined by the prepotential F(X) [161,211,212] through (89) is called the special Ka( hler manifold [113,128,198,210,212,222,576]. But there is a subclass of homogeneous special manifolds, which are classi"ed in [166]. These are S;(1,1) S;(1,n ) S;(1,1) SO(2,n ) Sp(6,R) S;(3,3) T T , , , , , S;(n );;(1) ;(1) SO(2);SO(n ) S;(3);;(1) S;(3);S;(3) ;(1) T T SOH(12) E \ , and S;(6);;(1) E ;SO(2) with the corresponding symplectic groups Sp(2n #2) respectively given by Sp(4), Sp(2n #2), Sp(2n #4), Sp(14), Sp(20), T T T Sp(32), and Sp(56).


33

When N"2 chiral "elds UK (K"0, 1,2, n ) are coupled to the Weyl multiplet (with compoT nents given by vierbein, 2 gravitinos and auxiliary "elds) [210,212], the invariance under the dilatation requires F(X) to be a homogeneous function of degree 2 (so that F(X) has Weyl weight 2) [210,212]. Furthermore, the requirement of canonical gravitino kinetic term imposes one constraint on the set of scalars XK as i(XM KFK!FM RXR)"1 ,

(90)

leading to gauge "xing for dilations and the special KaK hler manifold of the dimension n . Note, the T extra chiral super"eld U is introduced to "x the dilatation gauge, to break the S-supersymmetry, and to introduce the physical ;(1) gauge "eld in the N"2 supergravity multiplet (the scalar and the spinor components of the super"eld U do not become additional physical particles). The "nal form of bosonic action describing n numbers of N"2 vector multiplets coupled to N"2 T supergravity is (91) e\L"!R#g *R z?RIz @*!Im(NKR(z, z )F>KF>RIJ) , IJ ?@ I where z? (a"1,2, n ) are the coordinates of a KaK hler space spanned by the scalars T XK (K"0, 1,2, n ) which satisfy one constraint (90) (therefore, the manifold spanned by XK has T n complex dimensions). A convenient choice for z? is the inhomogeneous coordinates called the T special coordinates: z?"X?(z)/X(z), a"1,2, n . (Note, X?(z)"z? in special coordinates in which T R(X?/X)/Rz@"d? [113,128,134,576].) Here, K and NKR (cf. the scalar matrix N in (71)) are @ determined by F(X) to be of the forms [130,161,198,211,212]: e\)XX "iZM K(z )FK(Z(z))!iZR(z)FM R(ZM (z )) , Im(FK )Im(FRP)X XP , (92) Im(F P)X XP where ZK(z)"e\)XK and ZM K(z )"e\)XM K (K"0, 1,2, n ) are holomorphic sections of the projective space PC L> [128,129,198], and FKR,RKFR(X). We give some examples of the holomorphic function F(X) of N"2 theories and the corresponding special KaK hler manifold target spaces: NKR"FM KR#2i

F"iXX,

S;(1, 1) , ;(1)

F"(X)/X,

S;(1, 1) , ;(1)

F"!4(X(X),

S;(1, 1) , ;(1)

S;(1, n) F"iX gKRX , , S;(n);(1) K

R

dK RXKX XR F" , X

Calabi}Yau .

XQ L SO(2, 1) SO(2, n) F"!i (X)! (X?) , ; . X SO(2) SO(2);SO(n) ?

(93)

34


So far, we de"ned the special Ka( hler manifold as the KaK hler manifold with the special form of the KaK hler metric given by (92), which depends on the holomorphic prepotential F. Now, we discuss the symplectic formalism of the special KaK hler manifolds of N"2 supergravity coupled to n vector supermultiplets. For the symplectic formalism [132,133,214] of the special KaK lher manifold M, one considers the tensor bundle of the type H"SVL. Here, SVPM denotes a holomorphic #at vector bundle of rank 2n #2 with structural group Sp(2n #2, R), and LPM denotes the complex T T line bundle whose "rst Chern class equals the KaK hler form of the n -dimensional Hodge}KaK hler T manifold M. A holomorphic section of the bundle H has the form [113}115,128,134,198]:

XK

X"

K, R"0, 1,2, n , T

FR

(94)

which is de"ned for each coordinate patch ; LM of the (Hodge}KaK hler) manifold M and G transforms as a vector under the symplectic transformation Sp(2n #2, R). The Hodge}KaK hler T manifold M with a bundle H described above is called special Ka( hler, if the KaK hler potential is expressed in terms of the holomorphic section X as K"!log(i1X"XM 2)"!log[i(XM KFK!FM RXR)] ,

(95)

where

1X"XM 2,!XR

0

I

!I 0

X

denotes a symplectic inner product. One further introduces the symplectic section of the bundle H according to

¸K

N, eIJMNR a"(!ge\EgINgJHgMOH , KL N NHO where H is the "eld strength of B . Then, from the above real scalars we de"ne the following IJM IJ complex scalars [215] S"S #iS ,a#ie\E , ¹"¹ #i¹ ,b#ie\N , ;"; #i; ,c#ie\M ,

(150)

46


which (within the framework of the heterotic string) are respectively the dilaton}axion "eld, the KaK hler structure and the complex structure. Then, the "nal form of the D"4 e!ective action is [236]:

1 (!g R !S gIIYgJJYF2 (M M )F #gIJ Tr(R M LR M L) L" E IJ 2 3 IYJY I 2 J 2 16pG 1 gIJR SR SM , (151) # gIJTr(R M LR M L)! I 3 J 3 I I 2(S ) where M , M 3S¸(2, R) are de"ned as 2 3 ¹ ; 1 1 1 1 , , M , (152) M , 3 ; ; ";" 2 ¹ ¹ "¹" and the ;(1) gauge "elds AG (i"1,2, 4) are given by A"B , A"B , A"A, A"A. I I I I I I I I I Here, L is an S¸(2, Z) invariant metric. The action is manifestly invariant under the S¸(2, R);S¸(2, R) T-duality:

M Pu2 M u , M Pu2 M u , F P(u\u\)F , (153) 2 2 2 2 3 3 3 3 IJ 2 3 IJ where u 3S¸(2, R) and the rest of the "elds are inert. In addition, the theory has an on-shell 23 S-duality symmetry:

FG FG a b IJ Pu\ IJ ; u" , (154) 1 夹FG 夹FG c d IJ IJ where a, b, c, d3Z satisfy ad!bc"1, and 夹FG is the Hodge-dual (de"ned from the action (151)) of IJ the "eld strength FG . IJ We denote the theory described by (151) as H , meaning the heterotic theory with the 123 dilaton-axion "eld, the KaK hler structure and the complex structure given respectively by S, ¹, ; de"ned in (150). Under the Mirror symmetry, the KaK hler structure and the complex structure are interchanged, and therefore we obtain H theory, i.e. the heterotic string with the KaK hler 132 structure and the complex structure now respectively given by ; and ¹ de"ned in (150); the e!ective action is (151) with ¹ and ; "elds interchanged. We call H and H as the S-strings, 123 132 meaning the string theories with the dilaton}axion "eld given by S de"ned in (150). Under the D"6 string}string duality (144), the H is transformed to A . Under the Mirror 123 213 symmetry, A is transformed to B . So, the A and B are the T-strings. 213 231 213 231 We apply string}string duality to B to obtain B . Under the Mirror symmetry, B is 231 321 321 transformed to A . Therefore, we have the ;-strings given by B and A . 312 321 312 We comment on relation of the D"4 S-duality to the D"6 string}string duality. Since e!ect of the D"6 string}string duality on the D"4 theory is to interchange the complex structure and the dilaton}axion "eld, the S¸(2, Z) subset T-duality of one theory accounts for the S-duality of the string}string duality transformed theory [226,233]. Namely, the large}small radius T-duality (RPa/R) of one theory corresponds to the strong}weak coupling duality (g/2pP2p/g) of the dual theory. In terms of transformation of ;(1) gauge "elds, one can understand this as follows [236]. Under the string}string duality, electric [magnetic] charges of 2-form ;(1) gauge "elds of aS#b SP , cS#d


47

one theory is transformed to magnetic [electric] charges of 2-form ;(1) gauge "elds of the dual theory, while those of KK ;(1) "elds remain inert. Since the T-duality interchanges KK ;(1) gauge "elds and 2-form ;(1) gauge "elds (associated with the same internal coordinates), under the combined action of the D"6 string}string duality and the D"4 T-duality, electric [magnetic] charges of KK ;(1) gauge "elds and magnetic [electric] charges of 2-form ;(1) gauge "elds are exchanged, which is exactly the D"4 S-duality. Thus, since the T-duality is proven to hold order by order in string coupling, proof of the conjectured D"4 S-duality amounts to proof of the D"6 string}string duality, and vice versa. Since the string}string duality interchanges the dilaton}axion "eld with the KaK hler structure, the string coupling g/2p of one theory is transformed to the worldsheet coupling a/R of the dual theory [226]. So, string quantum corrections controlled by the string coupling in one theory correspond to the stringy classical corrections (a corrections) controlled by the worldsheet coupling in the dual theory; the a [quantum] corrections in one theory can be understood in terms of the quantum [a] corrections of the dual theory. 3.4. U-duality and eleven-dimensional supergravity Hull and Townsend [381] conjectured that the type-II superstring theories on a torus has full symmetry of low-energy e!ective "eld theory, which is larger than the direct product of the D"10 S¸(2, Z) S-duality and the O(10!d, 10!d, Z) T-duality in D"d(10 [306]. For example, the e!ective action of type-II string on ¹, which is D"4, N"8 supergravity [159,160], has an on-shell E symmetry [160], which contains S¸(2, R);O(6, 6) as the maximal subgroup. Hull and Townsend [381] conjectured that the subgroup E (Z) (broken due to the DSZW charge quantiz ations) extends to the full string theory as a new uni"ed duality group, called U-duality. U-duality uni"es the S and ¹ dualities and mixes p-model and string coupling constants. The discrete subgroup E (Z) is the intersection of the continuous E group and the discrete symplectic Sp(28, Z) group, which transforms 28 ;(1) gauge "elds of the e!ective theory linearly: E (Z)"E 5 Sp(28, Z) [381]. Under U-duality, a set of 28;2 electric and magnetic charges transform as a vector, and all the scalars in the theory mix among themselves. Unlike other types of duality, which assigns a special role to the dilaton, under U-duality the dilaton is on the same footing as moduli. Thus, unlike T-duality which is perturbative in nature, U-duality, like S-duality [560], is non-perturbative in nature, so cannot be tested within perturbative spectrum of string theories. We list the conjectured U-duality groups in various dimensions [381]. Note, the duality group in higher dimensions is a subgroup of lower dimensional duality group, since it survives compacti"cation (cf. The duality group does not act on the Einstein-frame metric). The SO(10!d, 10!d, Z) T-duality in D"d(10 and the S¸(2, Z) S-duality in D"10 are uni"ed to U-duality for d(8. The U-duality groups are, in the descending order starting from D"7: SO(5, 5, Z), S¸(5, Z), E (Z), E (Z), E (Z), E (Z), E (Z). These U-duality groups in D"d are generated by the T-duality in D"d and n numbers of U-duality groups in D"d#1 [381], where n is the possible numbers of ways in which one can compactify from D"10 to D"d#1 and then down to D"d. In comparison, the heterotic string on ¹\B with d'3 maintains the duality group in the form (T-duality group);(S-duality group), i.e. SO(10!d, 26!d, Z);Z for 105d55 or

48


SO(10!d, 26!d, Z);S¸(2, Z) for d"4. For d"3 [561] and d"2 [563], the T- and S-dualities are uni"ed to U-duality given by SO(8, 24, Z) and SO(8, 24)(Z), respectively. As we will see in Section 7.1, BPS electric states are within perturbative spectrum of heterotic string [248]; all the 28 electric charges in the heterotic string on ¹ are related through the `perturbativea O(6, 22, Z) T-duality. Also, there is non-perturbative spectrum carrying the remaining 28 magnetic charges, related to perturbative spectrum via the Z subset of the `non-pertur bativea S¸(2, Z) S-duality [486,544,558]. These magnetic charges are carried by solitons. The mass of a state in heterotic string on ¹ carrying electric [magnetic] charges of the ;(1) gauge group behaves as &1 [&1/g] in the string frame, as expected for a fundamental string [a soliton]. Q For the type-II string, only 12 of the 28 electric charges couple to perturbative string states, since the remaining 16 electric charges are R-R charges, which cannot be coupled to perturbative string states. The Z subgroup of the S¸(2, Z) S-duality group relates these perturbative states to solitonic states carrying 12 magnetic charges of the same 12 ;(1) gauge "elds. The remaining 16 electric and 16 magnetic charges of the ;(1) gauge group are carried by another type of non-perturbative states, whose mass behaves as &1/g . Thus, under the (T-duality);(S-duality) subgroup, i.e. Q SO(6, 6, Z);S¸(2, Z)LE (Z), the fundamental representation 56 (representing 28 electric and 28 magnetic charges of the D"4 ;(1) gauge group) is decomposed as 56"(12, 2);(32, 1). Here, the "rst factor (12, 2) corresponds to the 12 numbers of S¸(2, Z) doublets of perturbative and solitonic states in the NS-NS sector and the second factor (32, 1) denotes the remaining non-perturbative states, which are singlets under the S¸(2, Z) group and carries 16 electric and 16 magnetic charges of 16 ;(1) gauge "elds in the R-R sector. Since the O(6, 6, Z) T-duality group of the type-II string on ¹ mixes only NS-NS charges among themselves, it is not inconsistent that string states carry only NS-NS charges. However, it is not consistent with the conjectured U-duality, since U-duality puts all the 28;2 electric and magnetic charges of D"4 ;(1) gauge group on the same putting. In addition, as we saw in the decomposition of the 56 representation, the U-duality requires existence of additional 16#16 electric and magnetic charges in the RR-sector that transform irreducibly under the O(6, 6, Z) T-duality group. Hence, one leads to the conclusion that all the RR charges, which cannot be carried by perturbative string states or solitons, should be carried by another type of nonperturbative states. The low-energy or long-distance description of these non-perturbative string states is R-R p-branes or black holes. In the original work by Hull and Townsend [381], they show that all the R-R charged black holes can arise from extreme p-branes of the D"10 e!ective supergravity via dimensional reduction. In [498], Polchinski shows that the states carrying R-R charges can be realized within string theories without introducing exotic extended objects like p-branes. Such objects are D-branes [193,445], boundaries to which the ends of open strings (with Dirichlet boundary condition) are attached. D-branes carry one unit of R-R charges. D-branes are dynamic objects and the open string states describe their #uctuations. In the strong coupling limit, D-branes become black holes. Such identi"cation made it possible to give precise statistical explanation of black hole entropy. In Section 8, we will summarize aspects of D-branes and the recent development in D-brane calculation of black hole entropy. In the following sections, we discuss the S-duality of the type-IIB string and the T-duality of type-II string on a torus. The U-duality of type-II string on a torus is generated by these two dualities. In particular, starting from generating black holes of type-II theories on a torus, one


49

obtains black holes with the general charge con"guration by applying the S-duality and the T-duality transformations. For p-branes in type-II theories, one can generate p-branes of the general charge con"guration by "rst imposing SO(1, 1) boost on a charge neutral solution to induce KK electric charge and then applying the ¹- and the S-duality transformations and/or another SO(1, 1) boosts sequentially. 3.5. S-duality of type-IIB string The type-IIB string [312,536] has the S¸(2, Z) symmetry [381]. The Z LS¸(2, Z) trans formation exchanges the NS-NS 2-form potential BK (coupled to a perturbative string state) and the R-R 2-form potential BK (coupled to a non-perturbative D-brane) while changing the sign of the dilaton (or inverting the string coupling). So, the S¸(2, Z) symmetry is a strong}weak coupling duality. It is well-known that a covariant e!ective action for type-IIB string does not exist, while only the "eld equations [376,536] exist. The only problem with construction of the covariant e!ective action is the R-R 4-form potential DK (with the self-dual 5-form "eld strength FK ), whose equation of motion FK "夹FK cannot be derived from the covariant action. So, to construct the covariant e!ective action for type-IIB string, one is forced to set DK to zero. However, it is found in [72] that one can construct the type-IIB e!ective action which gives rise to the correct "eld equations and compacti"es to the correct action for the dimensionally reduced type-IIB theories without setting DK to zero. In this approach, one keeps FK di!erent from zero in the e!ective action but eliminates the self-duality constraint. After the "eld equations are obtained from this e!ective action, the self-duality constraint is imposed in order to get the correct "eld equations for the type-IIB theory. In the string-frame, such e!ective action for type-IIB string has the form [72]:

3 1 S " dx (!GK e\U !RK #4(RU)! (HK ) '' 4 2

1 3 5 1 ! (Rs)! (HK !sHK )! FK ! seGHeDK HK GHK H , 2 4 6 96(!G

(155)

The "eld strengths of the 2-form potentials BK G and the 3-form potential DK , and their gauge transformation rules are HK G"RBK G,

dBK G"RRK G ,

FK "RDK #eGHBK GBK H, dDK "RKK !eGHRRK GBK H ,

(156)

where RK G and KK are in"nitesimal gauge transformation parameters. The S¸(2, Z) symmetry of the type-IIB theory is manifest in the e!ective action in the Einsteinframe. To go to the Einstein-frame, one Weyl-rescales the metric GK "e\UGK . +, +,

50


The resulting Einstein-frame action has the form:

1 1 3 S# " dx (!GK !RK # Tr(R MK R+MK \)! HK GMK HK H '' + GH 2 4 4

5 1 ! FK ! eGHeDK HK GHK H , 6 96(!GK

(157)

where MK is a 2;2 real matrix formed by the complex scalar jK "s#ie\U:

"jK " !Re jK 1 . MK " Im jK !Re jK 1

(158)

(157) is manifestly invariant under the S¸(2, R) transformation [376,536]:

HK HK

Pu

HK HK

, MK P(u\)2MK u\,

u3S¸(2, R) .

(159)

The S¸(2, R) transformation on jK has the usual fractional-linear form jK P(ajK #b)/(cjK #d). When the DSZW type charge quantization is taken into account, the S¸(2, R) symmetry breaks down to the S¸(2, Z) subset. 3.6. ¹-duality of toroidally compactixed strings Closed strings in D dimensions in target space background with d commuting isometries have O(d, d, Z) T-duality symmetry [304,305,308,570]. The O(d, d,Z) symmetry is a perturbative symmetry proven to hold order by order in string coupling. For heterotic string, the symmetry is enlarged to O(d, d#16, Z) due to the additional rank 16 background gauge "elds. Under the Z subset that inverts the radius of S (i.e. R Pa/R ) and interchanges winding and momentum G G modes (i.e. m n ), the type-IIA and the type-IIB theories are interchanged if odd number of circles G G are acted on by the Z transformations, while heterotic string transforms to itself. The Z transformation between the type-IIA and the type-IIB strings at the e!ective "eld theory level is understood as "eld rede"nition between type-IIA and type-IIB theories, since the compacti"cation of the type-IIA and the type-IIB supergravities leads to the same supergravity theory. We consider the bosonic string worldsheet p-model with only NS-NS sector "elds (target space metric G (X), 2-form potential B (X) and dilaton U(X)), which are common to both type-II and IJ IJ heterotic strings, turned on. The action with the (curved) D-dimensional target space has form

1 dz[(G (X)#B (X))RXIRXJ! (X)R] . S" IJ IJ 2p

(160)

Let us assume that (160) is invariant under the d commuting, compact Abelian isometries [309] dXI"ekI (i"1,2, d) along the XG-direction, where [k , k ]"0. Then, the background "elds G G H become independent of XG. First, we consider the case where the background "elds have only one isometry (d"1) along, say, the direction h"X. The T-dual pair p-model actions are obtained by gauging the Abelian isometry. Following the procedures discussed in [80,82,100}102], one obtains the action dual to


51

(160) with the dual background "elds (with primes) related to the original ones (without primes) as 1 B G G #B B ? @, G " , G " ? , G "G ! ? @ G ? G ?@ ?@ G ?@ (161) G B #B G ? @, U"U#log G , B "G G , B "B ! ? @ ? ? ?@ ?@ G where XI"(h, X?) (a"1,2, D!1). This is the curved background generalization of the RPa/R T-duality of closed strings on S. Alternatively, the dual pair p-models are obtained by the method of chiral currents [516]. One starts with a (D#d)-dimensional p-model with d Abelian (left-handed) chiral currents JG and (right-handed) anti-chiral currents JM G [309]: S "S #S #S[X] , ">B ? 1 S " dz[RhG RM hG #RhG RM hG #2R (X)RhG RM hH #C* (X)RX?RM hG #C0 (X)RhG RM X?] , 2p * * 0 0 GH 0 * ?G * G? 0

1 dz[RhG RM hG !RhG RM hG ] , S" * 0 0 * ? 2p

(162)

1 dz[C (X)RX?RM X@!U(X)R] , S[X]" ?@ 2p where i, j"1,2, d and a, b"d#1,2, D. Here, chiral and anti-chiral currents, corresponding to the ;(1)B ;;(1)B a$ne symmetries dhG "aG (z), are * 0 *0 *0 JM G"RM hG #R hM H #C0 RM X? . (163) JG"RhG #R hH #C* RX?, 0 GH * G? * HG 0 ?G By gauging either a vector or an axial subgroup of ;(1)B ;;(1)B , one has the following * 0 D-dimensional dual pair p-model actions:

1 1 S!" dz E! (X?)RXI RM XJ ! !(X?)R " 2p IJ ! ! 4

1 " dz E!(X?)RhG RM hH #F0!(X?)RhG RM X?#F*!(X?)RX?RM hG GH ! ! G? ! ?G ! 2p

1 #F!(X?)RX?RM X@! !(X?)R , ?@ 4

(164)

where (XI )"(hG , X?) with k, l"1,2, D, i"1,2, d, a"d#1,2, D. Here, hG ,hG $hG ! ! ! 0 * and the upper (lower) signs in $ and G correspond to the axial (vector) gauged p-model. The

More general transformations with non-zero R-R "elds are derived in [81].

52


background "elds are

E! F0! G@ , !"U#log det(I$R) , E! "G! #B! " GH IJ IJ IJ F*! F! ?H ?@ 1 E!"(I$R) (IGR)\, F!"C $ C* (IGR)\C0 , GH GI IH ?@ ?@ 2 ?G GH H@

(165)

F0!"(IGR)\C0 , F*!"$C* (I$R)\ , G? GH H? ?G ?H HG with G! [B!] denoting the symmetric (the antisymmetric) part of E!. The action S! has an " isometry under the translation in the hG -direction. 8 The dual pair actions (164) are the most general p-model with d commuting compact Abelian symmetries. S> and S\ are related under the combined operations of the sign reversal of R and C*, " " and the coordinate transformation hG P!hG (i.e. hG hG ), implying equivalence of two gaugings * * > \ at least locally. To achieve global equivalence of the two gaugings, one has to impose the same periodicity conditions on both hG and hG , i.e. hG ,hG #2p. Then, one establishes the equivalence > \ ! ! of vector and axial gauged p-model actions S! (the so-called `vector-axial dualitya [426]). " One can relate S! to the bosonic string p-model action S in (160) by identifying XI"XI . Then, " ! the background "elds in S! are related to those in S in the following way: " G "(E!#E!), B "(E!!E!) , HG GH GH HG GH GH G "(F0!#F*!), B "(F0!!F*!) , ?G G? G? ?G G? G? G "(F!#F!), B "(F!!F!) . @? ?@ ?@ @? ?@ ?@

(166)

From this, transformation rule of background "elds under T-duality that relates the dual pair bosonic string p-model actions (one related to S> and the other related to S\) follows. When " " S! have isometry along only one coordinate direction (d"1), one recovers the factorized duality " transformation (161). We discuss transformations [305,309] that relate the di!erent backgrounds (of the same action) describing the equivalent conformal "eld theory. S! have the manifest invariance under the " following O(d, d, Z) transformation E!PE!"(a( E!#bK )(c( E!#dK )\

E!

(a!E!c)F0!

"

F*!(cE!#d)\ F!!F*!(cE!#d)\cF0!

det G! 1 ,

!P !" !# log det G! Y 2

g"

a b c

d

3O(d, d, Z) ,

,

(167)

where D;D blocks a( , bK , c( and dK of the O(D, D, Z) matrix are

a( "

a

0

0 I "\B

,

bK "

b 0 0 0

, c( "

c

0

0 0

,

dK "

d

0

0 I "\B

.

(168)


53

Here, I is the n;n identity matrix. The d;d matrices E! transform fractional linearly under L O(d, d, Z): E!PE!"(aE!#b)(cE!#d)\ .

(169)

The constant E! case corresponds to the transformation in the toroidal background. The O(d, d, Z) transformation is generated by the following transformations. E Integer `Ha-parameter shift of E, i.e. E PE #H (H "!H ): GH GH GH GH HG a b I H " B s.t. H"!H2 . c d 0 I B E Homogeneous transformations of E!, F*!, F0! under G¸(d, Z), F*!PA2F*!, F0!PF0!A (A3GL(2, Z)):

a b

A2

0

0

A\

"

c

d

s.t. A3GL(2, Z) .

(170) i.e. E!PA2E!A,

(171)

E Factorized dualities D , corresponding to the inversion of the radius R of the ith circle, i.e. G G R P1/R : G G G\ B\G\ CDE CDE a b I!e e G G " s.t. e "diag(0, 2, 0, 1, 0, 2, 0). (172) G c d e I!e GF F FHF F FI G G

The maximal compact subgroup of O(d, d, Z) is O(d, Z);O(d, Z) with a group element having the form:

hG PO hG , * * *

as ">B h O #O O !O h \ P1 * 0 * 0 \ , 2 O !O O #O h h > * 0 * 0 > C0PO C0, CPC , 0

a b

1 O*#O0 O*!O0 " s.t. O , O 3O(d,Z) . * 0 2 O !O O #O c d * 0 * 0 The O(d, Z);O(d, Z) transformation naturally acts on the action S hG PO hG , 0 0 0

(173)

RPO RO2, C*PC*O2, (174) 0 * * meanwhile on the actions S! as " E!PE!"[(O #O )E!#(O !O )][(O !O )E!#(O #O )]\ , * 0 * 0 * 0 * 0 F*!PF*!"2F*![(O !O )E!#(O #O )]\ , * 0 * 0 (175) F0!PF0! "[(O #O )!E! (O !O )]F0! , * 0 * 0 F!PF! "F!!F*![(O !O )E!#(O #O )]\(O !O )F0! . * 0 * 0 * 0 We now specialize to strings in #at background (i.e. toroidal compacti"cation) to understand properties of perturbative string spectrum under T-duality. The relevant part of the worldsheet

54


action is the toroidal (¹B) part:

1 1 p (176) S" dp dq (gg?@G R XGR XH#e?@B R XGR XH! (g R , GH ? @ GH ? @ 2 4p where XG&XG#2pmG and i, j"1,2, d. Here, mG is a string `winding numbera along the XGdirection. G and B can be collected into the `background matrixa E"G#B. The matrix E is GH GH a special case of E! in (165) where E! do not depend on X?. The lattice KB, which de"nes ¹B"RB/(pKB), is spanned by basis vectors e satisfying B e?e?"2G . ? G H GH G The mode expansions of XG and conjugate momenta 2pP "G XQ H#B XH are G GH GH i 1 XG(p, q)"xG#mGp#qGGH(p !B mI)# [aG (E)e\ O\N#aG (E)e\ O>N] , H HI L (2 L$ n L 1 [E2 aH (E)e\ O\N#E aH (E)e\ O>N] , (177) 2pP (p, q)"p # GH L GH L G G (2 L$ where we made analytic continuation qP!iq and the momentum zero modes p take integer G values, i.e. p "n 3Z. The equal-time canonical commutation relations [XG(p, 0), P (p, 0)]" G G H idG d(p!p) lead to commutation relations among the oscillator modes: H [xG, p ]"idG , [aG (E), aH (E)]"[aG (E), aH (E)]"mGGHd . (178) H H L K L K K>L The Hamiltonian takes the form

1 1 p dp(P#P)" Z2M(E)Z#N#NI , H"¸ #¸I " * 0 4p 2 G!BG\B BG\ m , Z" ? , M(E)" !G\B G\ n ? where P are the left- and the right-moving momenta de"ned as *0 P "[2pP #(G!B) XHY]eGH, P "[2pP !(G#B) XHY]eGH , *? G GH ? 0? G GH ? and the number operators of the left- and the right-moving modes are

(179)

(180)

N " aG (E)G aH (E), N " aG (E)G aH (E) . (181) * \L GH L 0 \L GH L L L Here, eGH are dual basis vectors, satisfying B e?eHH"dH and B eGHeHH"(G\)GH. The left- and ? G ? G ? ? ? the right-moving momenta zero modes p "[n2#m2(B!G)]eH, p "[n2#m2(B#G)]eH , (182) 0 * transforming as a vector under O(d, d, R), form an even self-dual Lorentzian lattice CBB [480,481], i.e. p!p"2mGn 32Z. * 0 G While CBB is preserved under O(d, d, R), the Hamiltonian zero mode H "(p#p) is invariant 0 * only under its maximal compact subgroup O(d, R);O(d, R). So, the zero-mode spectrum is


55

unchanged under the O(d, R);O(d, R) subgroup, only. Note, from (182) one sees that (p , p ) and * 0 hence CBB are in one-to-one correspondence with a particular background E"G#B. Thus, the moduli space is isomorphic to O(d, d, R)/[O(d, R);O(d, R)]. Under the O(d, d, Z) transformation (169) [439], M(E)PgM(E)g2 ,

(183) a (E)P(d!cE2)\a (E), a (E)P(d#cE)\a (E) . L L L L So, N are manifestly invariant under O(d, d, Z); the spectrum is O(d, d, Z) invariant. The O(d, d, Z) *0 transformation is generated [304,305,308,570] by integer H-parameter shift of E, the G¸(2, Z) transformation and the factorized duality D , as discussed above. Particularly, under G¸(d, Z), G which changes the basis of KB, EPAEA2 and (m, n)P(A2m, A\n) (A3G¸(d, Z)). In addition, the spectrum is invariant under the worldsheet parity [305] pP!p, which acts on E as BP!B. The e!ect of the worldsheet parity on the spectrum is to interchange the left-handed and the right-handed modes: p p and a a . The above transformations generate the full spectrum * 0 L L preserving symmetry group G . B A particular element g of O(d, d, Z) with a"d"0 and b"c"I, i.e. EPE\ [308,570], corresponds to vector-axial duality symmetry (165). Under this transformation, n m and the Hamiltonian (179) is manifestly invariant. When B"0, the transformation becomes GPG\, i.e. the volume inversion of ¹B. A signi"cance of EPE\ is that the gauge symmetry is enhanced to the a$ne algebra S;(2)B ;S;(2)B at a single "xed point G"I and B"0. The gauge symmetry is * 0 maximally enhanced [308] at "xed points under EPE\ modulo S¸(d, Z) and H(Z) transformations, i.e. E such that E\"M2(E#H)M (M3S¸(d, Z)). Hence, an enhanced symmetry point corresponds to an orbifold singularity point [220,221] in the moduli space under some non-trivial O(d, d, Z) transformation. At the "xed point, E takes the following form in terms of the Cartan matrix C of the rank d, semi-simple, simply laced symmetry group [252]: GH E "0 (i(j) . (184) E "C (i'j), E "C , GH GH GH GG GG Non-maximally enhanced symmetry points correspond to "xed points under factorized dualities D instead of the full inversion EPE\. A simplest but non-trivial example is the d"2 case G (i.e. compacti"cation on ¹), which we discuss in Section 4.2.2. At the "xed point E"I under EPE\, the gauge symmetry is enhanced to (S;(2);S;(2)) ;(S;(2);S;(2)) . The gauge * 0 symmetry is maximally enhanced to S;(3) ;S;(3) at the point * 0 1 1 . E" 0 1

These "xed points correspond to orbifold singularities [570] in the fundamental domain of the moduli space (parameterized by two complex coordinates of the moduli space S¸(2, R)/;(1); S¸(2, R)/;(1)).

Note, O(2, 2, R) S¸(2, R);S¸(2, R). Therefore, E, which parameterizes the moduli space O(2, 2, R)/[O(2, R) ;O(2, R)], is reparameterized by the complex coordinates o and q, each parameterizing S¸(2, R)/;(1).

56


3.7. M-theory In this section, we discuss some aspects of M-theory. We illustrate how di!erent superstring theories emerge from di!erent moduli space of compacti"ed M-theory and discuss the M-theory origin of string dualities. In this picture, each of "ve di!erent string theories represents a perturbative expansion about di!erent points in moduli space of the compacti"ed M-theory. Namely, 5 perturbative string theories and uncompacti"ed M-theory are located at di!erent subsets of moduli space, and it is dualities that map one subset of moduli space to another, thereby making transition between di!erent theories. In the following, we illustrate this idea by showing how di!erent theories are achieved by taking di!erent limits of parameters of moduli space and how dualities are realized as transformations of parameters of moduli space. First, we discuss the connection between the type-II theories and M-theory. Type-IIA theory is obtained from M-theory by compactifying the extra 1 spatial coordinate on S of radius R [112,232,383,601,635]. The type-IIA and the type-IIB theories are related via T-duality [193,216]. Namely, the type-IIA theory on S of radius R is perturbatively equivalent (under T-duality) to the type-IIB theory on S of radius R "1/R . So, one can think of the S-compacti"ed type-IIB theory as ¹ compacti"ed M-theory. To understand the connection between type-II theories and M-theory, one has to compactify M-theory on ¹"S;S (with the radii of each circle given by R and R from the D"11 point of view), and compactify the type-IIA and the type-IIB theories on circles of radii R and R , respectively. Here, the radius R [R ] is measured with D"10 string frame metric of the type-IIA [the type-IIB] theory. We "rst relate parameters of the type-II theories (i.e. the radii R and the string couplings g in the type-IIA/B theories) to parameters R and R of ¹ moduli space before we Q understand the various limits in the moduli space. R is related to g as R "(g). As for the Q Q second circle of ¹"S;S, which is also the circle upon which the type-IIA theory is compacti"ed, the radius is measured di!erently depending on the dimensionality of spacetime. Note, we denoted the radius measured in D"11 [in D"10 by the type-IIA string-frame metric] as R [R ]. Namely, since the string-frame metric g'' (k, l"0, 1,2, 9) of the type-IIA theory is IJ related to the D"11 metric G (M, N"0, 1,2, 10) as G&e\(g'', where is the IJ +, IJ dilaton of the type-IIA theory, one sees that R "R /(g). Furthermore, one can relate the Q string coupling g of the type-IIB theory to R and R as follows. Under the T-duality between Q the type-IIA and the type-IIB theories, the string couplings are related as g "g/R . By using Q Q other relations among parameters, one "nds that g "R /R . Q

Note, due to the no-go theorem for KK compacti"cation of the D"11 supergravity [633], it might be impossible to obtain a chiral theory like type-IIB supergravity through dimensional reduction. This no-go theorem can be circumvented to obtain the (chiral) type-IIB theory by compactifying on orbifolds (rather than manifolds) [197,358,568,639]. In the case of compacti"cation of M-theory on ¹ (which is relevant to our discussion), when the size of ¹ goes to zero at the "xed shape, one obtains `chirala type-IIB theory, due to additional massive &wrapping' modes (of membrane) which become massless [22,81,540,541]. The string coupling g is de"ned as g "e6(7. Q Q


57

We discuss the various limits in the M-theory moduli space of ¹ in terms of R and R [22]: M-theory and the type-IIA, B theories are located at various limiting points in the ¹-moduli space. First, M-theory is located at (R , R )"(R,R) (i.e. the decompacti"cation limit), which is also the strong coupling limit (g"(R )PR) of the type-IIA theory [601,635]. The (uncomQ pacti"ed) type-IIA theory, de"ned as R PR and "nite string coupling g, is located at Q (R , R )"(R, xnite), i.e. M-theory on S of radius R (R. The (uncompacti"ed) type-IIB theory can be de"ned as the limit R PR, R P0 and "nite string coupling g . In this limit, Q R "R /(g)"R /(R g )"R(g )\"R\(g )\P0. So, in terms of para Q Q Q meters of ¹, the (uncompacti"ed) type-IIB theory corresponds to the limit in which (R , R )"(0, 0) while keeping the ratio g "R /R "nite. The value of g depends on how Q Q the limit (R , R )P(0, 0) is taken and, therefore, (R , R )"(0, 0) is not really a point in the moduli space. Note, when 2 circles in ¹"S;S are exchanged (i.e. R R ), g "R /R is inverted: Q g P1/g . Such interchange of 2 circles is a subset of more general SL(2, Z) reparameterization of Q Q ¹, which acts on the complex modulus q of ¹ fractional linearly. Thus, the reparameterization symmetry of ¹, upon which M-theory is compacti"ed, manifests in the type-IIB theory as the S¸(2, Z) S-duality [381], which acts on the complex scalar o"s#ie\( (formed by 0-form s and the dilaton ) fractional linearly. Next, we discuss string theories with N"1 supersymmetry, i.e. the E ;E and SO(32) heterotic strings and type-I string. To understand the connection between M-theory and these N"1 string theories, one has to consider the moduli space of M-theory compacti"ed on S/Z ;S, i.e. a cylinder of length ¸ and radius R. We "rst comment on the relation of type-I string theory to M-theory. One can think of the type-I theory as an &orientifold' of the type-IIB theory, namely a theory of unoriented closed string (gauged under the worldsheet parity transformation X [355,356,509]) and open string with SO(32) Chan-Paton factor [314]. To see the direct relation to M-theory, it is convenient to "rst compactify one spatial coordinate, which we call >, of the type-I theory on S and then T-dualize along the S-direction, inverting the radius of S. We call such theory as the type-I theory [193,505]. Since the dual coordinate >I is pseudo-scalar under the worldsheet parity transformation (i.e. X[>I ](q, p)"!>I (q,!p)), S is mapped under this T-duality to the orbifold S/Z with "xed points at >I "0, p. So, the type-I theory is e!ectively described by the type-IIA theory on S/Z ; closed strings wrapped around S/Z look like open string stretched between two 8-plane boundaries located at "xed points of S/Z . (These (parallel) boundaries corresponds to D 8 branes.) Second, the E ;E heterotic string theory is obtained by compactifying M-theory on the orbifold S/Z [357,358]. Namely, M-theory on S/Z of length ¸ gives rise to spacetime with two D"10 faces (the so-called ènd-of-the-world 9-branesa) which are separated by a distance ¸. Each of the two faces carries an E gauge "eld of the E ;E heterotic string [357,358]. In this picture, a fundamental string of the E ;E theory is interpreted as a cylindrical M 2-brane attached between the two faces. (So, the intersection of the cylindrical M 2-brane with the faces is a circle.) The string coupling is g#"¸ and, therefore, in the strong coupling limit (g# G G> G G\ G !2(l !K )cos h sin h cos t 2 cos t cos t sin t dh dt G> G> G\ G G G !2 (l!K )cos h cos t 2 cos t H H G\ GH ;cos t sin t 2 cos t cos t sin t dt dt G G H\ H H G H 4l l kkN k 4l kN # G [(r#l)D#2lkN] d ! G G dt d # G H G H d d , G G G G G G H D D D GH

(187)

where for E Even dimensions: "\ "\ D,a (r#l)#r k(r#l)2(r#l ) G G G\ G G ;(r#l )2(r#l ), G> "\ K ,l sin t #2#l cos t 2 cos t sin t , G G> G "\ G "\ "\ N"mr ,

(188)

While the SO(1, 1) boosts in SO(22, 1);SO(6, 1)LSO(22, 2);SO(6, 2) induce electric charges of the D"4 ;(1) gauge group, the remaining SO(1, 1) boosts in SO(22, 2);SO(6, 2)!SO(26!D, 1);SO(10!D, 1) induce magnetic charges of the ;(1) gauge group.

64


and k ,sin h, k ,cos h sin t ,2 , k ,cos h cos t 2 cos t sin t , "\ "\ "\ a,cos h cos t 2 cos t , "\

(189)

E Odd dimensions: "\ D,r k(r#l)2(r#l )(r#l )2(r#l ), G G\ G> "\ G K ,l sin t #2#l cos t 2cos t sin t G G> G "\ G "\ "\ #l cos t 2cos t , "\ G "\ N"mr ,

(190)

and k ,sin h, k ,cos h sin t ,2 , k ,cos h cos t 2 cos t sin t , (191) "\ "\ "\ k ,cos h cos t 2 cos t . "\ "\ Here, the repeated indices are summed over: i, j in t [ ] run from 1 to ["\] [from 1 to ["\]]. The ADM mass and the angular momenta J are G X 2 (D!2)X "\m, J " "\ml " Ml , (192) M" G 4pG G D!2 G 8pG " " where G is the D-dimensional Newton's constant and " 2p"\ X " "\ C("\) is the area of S"\. When compacti"ed to D!1 dimensions [3 dimensions (for the 4-dimensional Kerr solution)], the transformation that generates inequivalent solutions from the Kerr solution is (SO(26!D, 1);SO(10!D, 1))/(SO(26!D);SO(10!D)) [(SO(22, 2);SO(6, 22))/(SO(22);SO(6) ;SO(2))], which has (9!D)#(25!D) [2;28#1] parameters; these parameters are interpreted as (9!D)#(25!D) electric charge degrees of freedom [2;28 electric and magnetic charge degrees of freedom plus unphysical Taub-NUT charge] introduced to the Kerr solution [562]. For the D"5 case, an additional charge associated with the NS-NS 2-form "eld is generated by an SO(1, 1) boost in O(8, 24) [182]. After the generating solutions are constructed from the SO(1, 1) boosts, the remaining charge degrees of freedom are induced (without changing the Einstein-frame spacetime) from subsets of D-dimensional (continuous) duality transformations that generate new charge con"gurations from the generating ones while keeping the canonical scalar asymptotic values intact. This is


65

[SO(10!D);SO(26!D)]/[SO(9!D);SO(25!D)], which introduces (9!D)#(25!D) new charge degrees of freedom, and, for the D"4 case, SO(1)LS¸(2, R), which introduces one more charge degree of freedom. Subsequently, to obtain solutions with arbitrary scalar asymptotic values, one has to undo the transformations (185) and (186). In the following sections, we discuss the generating solutions in each dimensions. Note, due to the conjectured string}string duality between heterotic string on ¹ and type-II string on K3, these also correspond to the generating solutions of general black holes in type-II strings on K3;¹L for n"6!D"0, 1, 2. For this case, some of charges of the generating solutions can be dualized to R-R charges, rendering interpretation in terms of D-branes. It turns out that by applying ;-dualities of type-II string on tori to such generating solutions, one can generate the general class of solutions of the e!ective type-II string on tori as well [171] (see Section 6.3.2 for discussions). 4.2. Static, spherically symmetric solutions in four dimensions 4.2.1. Supersymmetric solutions In this section, we derive a general BPS spherically symmetric solution with a diagonal moduli [178]. Such a solution, after subsets of O(6, 22) and S¸(2, R) transformations are applied, satis"es one ;(1) charge constraint, missing one parameter to describe the most general BPS solutions. The solution generalizes the previously known black hole solutions in heterotic string on tori as special cases, and are shown to be exact to all orders in expansions of a [173]. At particular points in moduli space, such a solution becomes massless, enhancing not only gauge symmetry but also supersymmetry [179]. 4.2.1.1. Generating solutions. A general BPS non-rotating black hole solution with a diagonal moduli matrix is obtained by solving the Killing spinor equations. With spherically symmetric AnsaK tze for "elds and a diagonal form of moduli M, the Killing spinor equations dt "0, dj"0 + and ds'"0 (cf. (119)) are satis"ed by restricted charge con"gurations (see [178] for details on allowed charge con"gurations), which we choose without loss of generality to be P, P, Q, Q. The explicit BPS non-rotating solution with such a charge con"guration has the form [178] j"r/[(r!g P)(r!g P)(r!g Q)(r!g Q)] , . . / / R"[(r!g P)(r!g P)(r!g Q)(r!g Q)] , . . / / (r!g P)(r!g P) . . , eP" (r!g Q)(r!g Q) / /

r!g P . , g " r!g P .

r!g Q / , g " r!g Q /

(193)

g "1 (mO1, 2) , KK

where j and R are components of the metric g dxI dxJ"!j dt#j\ dr# IJ R(dh#sin h d ), g "$1 and the radial coordinate is chosen so that the horizon is at r"0. ./ The requirement that the ADM mass saturates the Bogomol'nyi bound restricts choice of g ./ such that g sign(P#P)"!1 and g sign(Q#Q)"!1, thus yielding non-negative . /

66


ADM mass of the form ""P#P"#"Q#Q" . (194) .1 Note, the relative signs for the pairs (Q, Q) and (P, P) are not restricted but in this section we consider the case where both of pairs have the same relative signs so that the solution (193) has regular horizon. The Killing spinor e of the above BPS solution satis"es the following constraints: M

E PO0 and/or PO0: CK CK ?e"ig e, . E QO0 and/or QO0: CK CK ?e"g e. / From these constraints, one sees that purely electric (or magnetic) solutions preserve 1/2, while dyonic solutions preserve 1/4 of N"4 supersymmetry. The former and the latter con"gurations fall into vector- and hyper-supermultiplets, respectively. Since the Killing spinor equations are invariant under the O(6, 22) and S¸(2, R) transformations, one can generate new BPS solutions by applying the O(6, 22) and S¸(2, R) transformations to a known BPS solution. The [SO(6)/SO(4)];[SO(22)/SO(20)] transformation with 6 ) 5!4 ) 3 22 ) 21!20 ) 19 # "50 2 2 parameters to the solution (193) leads to a general solution with zero axion and 4#50" 54"56!2 charges. 28 electric Q and 28 magnetic P charges of such a solution satisfy the two constraints P2M Q"0 (M ,(¸M¸) $¸) . (195) ! ! The subsequent SO(2)LS¸(2, R) transformation introduces one more parameter (along with a non-trivial axion), which replaces the two constraints (195) with the following one S¸(2, R) and O(6, 22) invariant constraint on charges: P2M Q [Q2M Q!P2M P]!(#!)"0 . (196) \ > > Thus, general solution in this class has 4#50#1"55"2 ) 28!1 charge degrees of freedom. By applying the O(6, 22) and S¸(2, R) transformations to (194), one obtains the following ADM mass for general solutions preserving 1/4 of supersymmetry: M "e\P+P2M P#Q2M Q#2[(P2M P)(Q2M Q)!(P2M Q)], . (197) .1 > > > > > This agrees with the expression (213) obtained [236] by the Nester's procedure. When magnetic P and electric Q charges are parallel in the SO(6, 22) sense, i.e. P2M Q"0, (197) becomes the > ADM mass of con"gurations preserving 1/2 of N"4 supersymmetry [34,299,337, 420}422,495,560,564]: M "e\((P2M P#Q2M Q) , > > .1 whose corresponding generating solution is purely electric or magnetic subset of (193).

(198)


67

4.2.1.2. General supersymmetric solution with xve charges. The BPS solution (193) has a charge con"guration satisfying one constraint (196) when further acted on by the [SO(6)/SO(4)]; [SO(22)/SO(20)] and SO(2) transformations. So, to construct the generating solution for the most general BPS solution, which conforms to the conjectured classical `no-haira theorem, one has to introduce one more charge degree of freedom into (193). Such a generating solution was constructed in [174] by using the chiral null model approach, and has the following charge con"guration: (Q, P)"(q, P ), (Q, P)"(Q , 0) , (Q, P)"(!q, P ), (Q, P)"(Q , 0) . Explicitly, the solution has the form r j" , [(r#Q )(r#Q )(r#P )(r#P )!q[r#(P #P )]] (r#P )(r#P ) , eP" [(r#Q )(r#Q )(r#P )(r#P )!q[r#(P #P )]] q(P !P ) W" , 2(r#P )(r#P ) r#P r#Q q[r#(P #P )] , . , G " G " G "!B " r#P r#Q (r#Q )(r#P ) For this solution to have a regular horizon, the charges have to satisfy the constraints P '0, P '0, Q '0, Q '0 , Q Q !q'0, (Q Q !q)P P !q(P !P )'0 . The ADM mass has the same form as that of the 4-parameter solution (193):

(199)

(200)

(201)

M

"Q #Q #P #P , (202) "+ independent of the additional parameter q. Meanwhile, the horizon area, i.e. A,4p(j\r) , is P modi"ed in the following way due to q: A"4p[(Q Q !q)P P !q(P !P )] . (203) The following ADM mass and horizon area of BPS non-rotating black hole with general charge con"guration are obtained by applying the [SO(6)/SO(4)];[SO(22)/SO(20)] and SO(2)LS¸(2, R) transformations to (202) and (203): M "a 2k a #e\Pb2k b#e\P[(b2k b)(a2k a)!(b2k a)] , "+ > > > > > A"peP[(b2¸b)(a2¸a)!(b2¸a)] ,

(204)

where the charge lattice vectors a and b live on the even self-dual Lorentzian lattice K with signature (6, 22), a ,a#W b and k ,M $¸. Here, a and b are related to the physical ;(1) !

68


charges Q and P as: (2P "¸ b , (205) (2Q "ePM (a #W b ), G GH H H G GH H where we assumed that a"2, G "aeP"eP. , The ADM mass and horizon area in (204) can be put in the S¸(2, Z) S-duality, as well as the O(6,22) T-duality, invariant forms by expressing them with new charge lattice vector *2"(*2, *2),(a2, b2) and by introducing the following S¸(2, R) invariant matrices:

M"eP

W

1

W W#e\P

, L"

0

1

!1 0

.

(206)

The "nal forms are M "(8G )\(M (*?2k *@)#[2L L (*?2k *@)(*A2k *B)]) , "+ , ?@ > ?A @B > > (207) 1 A . "p L L (*?2¸*@)(*A2¸*B) S" 2 ?A @B 4G , These are manifestly S¸(2, R) invariant, since M and * transform under SL(2, R) as [560]

MPuMu2,

*PLuL2*,

u3S¸(2, R) .

(208)

An important observation is that while for "xed values of a and b, mass changes under the variation of moduli and string coupling, entropy remains the same as one moves in the moduli and coupling space [258}260]. The fact that entropy is independent of coupling constants and moduli is consistent with the expectation that degeneracy of BPS states is a topological quantity which is independent of vacuum scalar expectation values and the fact that entropy measures the number of generate microscopic states, which should be independent of continuous parameters. 4.2.1.3. Bogomol'nyi bound. We derive the Bogomol'nyi bound on the ADM mass of asymptotically #at con"gurations within the e!ective theory of heterotic string on ¹ [236]. For this purpose, we introduce the Nester-like 2-form [483]: EK ,e cIJMd tI , (209) IJ C M where d tI is the supersymmetry transformation of physical gravitino in D"4. Given supersymC I metry transformations (119) of fermionic "elds expressed in terms of D"4 "elds [178], the Nester's 2-form reduces to the form: 1 EK IJ"e IJMd e# e\Pe (< ¸(F!icF I )IJ)?C?e#2 , M 0 2(2

(210)

where < is a vielbein de"ned in (122) and ¸ is an invariant metric of O(6, 22) given in (127). Derivation of the Bogomol'nyi bound consists of evaluating the surface integral of the Nester's 2-form (210), which is related through the Stokes theorem to the volume integral of its covariant derivative. The surface integral yields

1 1 dS eEK IJ"e P"+cI# e\P+< ¸(Q!icP),?C? e , IJ I 0 4pG R 2(2G .

(211)


69

where P"+ is the ADM 4-momentum [20] of the con"guration, and Q and P are physical electric I and magnetic charges of ;(1) gauge group. The integrand of the volume integral is a positive semide"nite operator, provided spinors e satisfy the (modi"ed) Witten's condition [631] n ) K e"0 (n is the 4-vector normal to a space-like hypersurface R). Thus, the bilinear form on the right-hand side of (211) is positive semide"nite, which requires that the ADM mass M has to be greater than or equal to the largest of the following 2 eigenvalues of central charge matrix: 1 e\P[Q #P $2+Q P !(Q2 P ),] , "Z "" 0 0 0 0 0 0 (4G )

(212)

where Q ,(2(< ¸Q) and similarly for P . This yields the Bogomol'nyi bound: 0 0 0 1 M 5 e\P[Q #P #2+Q P !(Q2 P ),] . "+ (4G ) 0 0 0 0 0 0

(213)

This bound is saturated i! supersymmetric variations of fermionic "elds are zero, i.e. BPS con"gurations. The Bogomol'nyi bound (213) can be expressed explicitly in terms of electric Q and magnetic P charges, and asymptotic values M and u of scalars, by using the identity: ¸A ?\A E L "+q"o\,: a"k "$(0, 1, 0,!1) contributing to ;(1) ;;(1) ;;(1) ;S;(2) ! ? A @>B @\B E L "+q"o!i,: a"k "$(1, 1,!1, 0) contributing to ;(1) ;;(1) ;;(1) ; ! B ?>A ?\@\A S;(2) ?>@\A E L "+q"o/(io#1),: a"k "$(1, 0,!1, 1) contributing to ;(1) ;;(1) ;;(1) ! @ ?>A ?\A\B ;S;(2) . ?\A>B At points where the lines L intersect [120,122], there are additional massless states, resulting in G the maximal enhancements of gauge symmetries: E L 5L "+q"o"1,: a"k or k contributing to ;(1) ;;(1) ;S;(2) ;S;(2) ! ! ?>A @>B ?\A @\B E L 5L 5L "+q"o"e p,: a"k or k or k contributing to ;(1) ;;(1) ; ! ! ! ?>A ?\@\A\B S;(3) . @\B?>@\A>B Along with the above perturbative massless states, there are accompanying in"nite massless dyonic states, so-called S-orbit pa( "sbK , related via S¸(2, Z) S duality. G G 1 Second, we consider the intermediate multiplets, i.e. the BPS multiplets with "Z "O"Z ". They have the highest spin 3/2 states and preserve 1/4 of the N"4 supersymmetry. In this case, a( and bK are not proportional, i.e. a( bK !a( bK O0. The requirement that the ADM mass is zero, i.e. G H H G "Z ""0 and *Z""Z "!"Z ""0, leads to the relations [119]: a!aqo#iao!ia"0, b!bqo#ibo!ib"0 . (225) These relations are satis"ed by the following "xed points [119,141]: E q"o"i: (a, b)"(k , k ), ! ! E q"o"e p: (a, b)"(k ,k ), 24i(j44. ! H! In addition to the above massless dyonic states, there are in"nite number of S¸(2, Z) related 1 dyonic states. Since these additional massless states belong to the highest spin supermultiplet, supersymmetry as well as gauge symmetry are enhanced [179]. 4.2.2.3. Properties of massless black holes. When both of the pairs (Q, Q) and (P, P) have the same relative signs [178], the singularity of the solution (214) is always behind or located at the event horizon at r"0, corresponding to the Reissner-NordstroK m-type horizon or null singularity,


73

respectively. However, the moment at least one of the pairs has the opposite relative signs [48,179,406], there is a singularity outside of the event horizon, i.e. naked singularity r '0. Explicitly, the curvature singularity is at r"r ,max+min["P ", "P "], min["Q ", "Q "],'0. These singular solutions have an unusual property of repelling massive test particles [406]. (Note, the BPS black holes need not be massless to be able to repel massive test particles [179].) There is a stable gravitational equilibrium point for a test particle at r"r where the graviA tational force is attractive for r'r and repulsive for r(r [406]. This can also be seen by A A calculating the traversal time of the geodesic motion for a test particle with energy E, mass m and zero angular momentum along the radial coordinate r, as measured by an asymptotic observer [179]:

t(r)"

P

E dr

P j(r)(E!mj(r)

.

(226)

The minimum radius that can be reached by a test particle corresponds to r 'r for which

j(r"r )"E/m, since it takes in"nite amount of time to go beyond r"r . Here, r"r is

the singularity. Massive test particles cannot reach the singularity of singular black holes in "nite time and are re#ected back. On the other hand, classical massless particles with zero angular momentum do not feel the repulsive gravitational potential due to increasing j(r), and they reach the singularity in a "nite time. Note, for regular solutions, studied in Section 4.2.1, j41, while for singular solutions studied in this section, j51 for r small enough. Thus, for regular solutions, particles are always attracted toward the singularity. When only one charge is non-zero, the regular solution has a naked singularity at r"0; t(r"r "0) is "nite. 4.2.3. Non-extreme solutions The following non-extreme generalization [180] of the BPS solution (193) is obtained by solving the Einstein and Euler}Lagrange equations: j"r(r#2b)/[(r#P)(r#P)(r#Q)(r#Q)] , R(r)"[(r#P)(r#P)(r#Q)(r#Q)] ,

eP"

(r#P)(r#P) , (r#Q)(r#Q)

r#P , g " r#P

W"0 ,

r#Q , g " r#Q

g "B "0 (mOn), KL KL

(227)

g "1 (mO1, 2) , KK

a' "0 , K

where b'0 measures deviation from the corresponding BPS solution and P, b$((P)#b, etc. The ADM mass is M"P#P#Q#Q!4b .

(228)

74


The signs $ in the expressions for P, etc. should be chosen so that MPM as bP0. To .1 have a regular horizon, one has to choose the relative signs of both pairs (Q, Q) and (P, P) to be the same [178]. In this case, the non-extreme solution is (227) with positive signs in the expressions for P, etc. and have the ADM mass M"((P)#b#((P)#b#((Q)#b#((Q)#b , which is always compatible with the Bogomol'nyi bound:

(229)

""P"#"P"#"Q"#"Q" . .1 Such solutions always have nonzero mass.

(230)

M

4.2.3.1. Space}time structure and thermal properties. We now study spacetime properties [178,180] of the regular D"4 4-charged black hole discussed in the previous subsections. There is a spacetime singularity, i.e. the Ricci scalar R blows up, at the point r"r where R"0. The event horizon, de"ned as a location where the r"constant surface is null, is at r"r & where gPP"j"0. The horizon(s) forms, provided r 'r (time-like singularity). In some cases, & singularity and the event horizon coincide: r "r . In this case, the singularity is (i) naked & (space-like singularity) when the singularity is reachable by an outside observer (at r"r 'r ) in & a "nite a$ne time q"P dr(EPP"P& dr j\(r) and (ii) also an event horizon (null singularity) ERR P P when q"R. Thermal properties of the solution (193) are speci"ed by spacetime at the event horizon. The Hawking temperature [345,346] is de"ned by the surface gravity i at the event horizon: ¹ "i/(2p)""R j(r"r )"/(4p) . & P & Entropy S is given by the Bekenstein's formula [38,64,65,343,347]:

(231)

S";(the surface area of the event horizon)"pR(r"r ) . (232) & We classify thermal and spacetime properties according to the number of non-zero charges: E All the 4 charges non-zero: There are 2 horizons at r"0,!2b and a time-like singularity is hidden behind the inner horizon, i.e. the global spacetime is that of the non-extreme Reissner} NordstroK m black hole. The Hawking temperature is ¹ "b/(p(PPQQ) and the & entropy is S"p(PPQ. When bP0, spacetime is that of extreme Reissner} ,MPBQRPMK m black holes. E 3 non-zero charges: A space-like singularity is located at the inner horizon (r"!2b). For example when P"0, ¹ "b/(p(2PQQ) and S"p(2bPQQ. When bP0, & the singularity coincides with the horizon at r"0. E 2 non-zero charges: A space-like singularity is at r"!2b. For example when PO0OP, ¹ "1/(2p(PP) and S"p(4bPP. As bP0, the singularity co & incides with the horizon at r"0. E 1 non-zero charge: A space-like singularity is at r"!2b. For example when PO0, ¹ "1/(2p(2bP) and S"p(8bP. As bP0, the singularity becomes naked. &


75

4.2.4. General static spherically symmetric black holes in heterotic string on a six-torus In Section 4.2.1, we saw that a general solution obtained by applying subsets of O(6,22) and S¸(2,R) transformations on the 4-charged solution has 1 charge degree of freedom missing for describing non-rotating black holes with the most general charge con"guration. It is a purpose of this section to introduce such 1 missing charge degree of freedom by applying 2 SO(1,1) boosts (in the D"3 O(8,24) duality group) along a ¹ direction (associated with 4 non-zero charges) with a zero-Taub-NUT constraint to construct the `generating solutiona for non-extreme, nonrotating black holes in heterotic string on ¹ with the most general charge con"guration of ;(1) gauge group [181]. (See [390] for an another attempt.) So, the generating solution is parameterized by the non-extremality parameter (or the ADM mass) and 6 ;(1) charges with one zero-TaubNUT constraint. 4.2.4.1. Explicit form of the generating solution. For the purpose of constructing the generating solution in a simplest possible form, it is convenient to "rst generate the 4-charged non-extreme solution (227) with the following non-zero charges by applying 4 SO(1, 1) boosts to the Schwarzschield solution: P"2m sinh d cosh d ,P , P"2m sinh d cosh d ,P , N N N N Q"2m cosh d sinh d ,Q , Q"2m cosh d sinh d ,Q . O O O O

(233)

Only non-extreme solutions compatible with the Bogomol'nyi bound and, therefore, within spectrum of states, are those with the same relative signs for both pairs (Q , Q ) and (P , P ). For this case, PK ,2m cosh d !m"$((P )#m, etc. are given with plus signs. N As the next step, one introduces one missing charge degree of freedom by applying 2 SO(1, 1)LO(8, 24) boosts with parameters d satisfying the zero Taub-NUT condition: P tanh d !Q tan d "0 .

(234)

Assuming, without loss of generality, that Q 5P , one has, from (234), d in terms of the other parameters: cosh d "Q cosh d /D,

sinh d "P sinh d /D ,

(235)

where D,sign(Q )((Q ) cosh d !(P ) sinh d .

An additional SO(1, 1) boost along a ¹ direction on the 4-charged black hole solution necessarily induces Taub-NUT term, since the metric components g get mixed with the -component of the ;(1) gauge potential, which is IJ singular [562]. One can induce any 2 of the remaining charges in the ;(1);;(1);;(1);;(1) gauge group. But we here choose to induce P and Q. For the case Q 4P , the role of d and d are interchanged.

76


The "nal form of the generating solution [181] (with zero Taub-NUT charge) is = (r#m)(r!m) , R"(X>!Z), eP" , j" X>!Z (X>!Z) 1 R W" [D(P Q #P Q )#P Q [(P )(r#QK )!(Q )(r#PK )] P D=

;

with

P P (r!QK ) sinh d #Q Q (r#PK ) cosh d sinh d cosh d , X>!Z

X > G " , G " , (r#PK )(r#QK ) (r#PK )(r#QK ) Z G "! , (r#PK )(r#QK ) [(Q )(r#PK )!(P )(r#QK )] cosh d sinh d , B "! D(r#PK )(r#QK ) G "d , B "0 (i, jO1, 2), a' "0 GH GH GH K

(236)

X"r#[(PK #QK ) cosh d #(QK !PK ) sinh d ]r#(PK QK sinh d #QK PK cosh d ) , 1 >"r# [(P )(PK !QK ) sinh d #(Q )(PK #QK ) cosh d ]r D 1 # [(P )QK PK sinh d #(Q )QK PK cosh d ] , D 1 Z" [(P P #Q Q )r#(PK Q Q #QK P P )] cosh d sinh d , D

(237)

1 ="r# [(Q )(PK #PK ) cosh d #(P )(QK !QK ) sinh d ]r D 1 # [(Q )PK PK cosh d #(P )QK QK sinh d ] . D For the sake of simpli"cation, the coordinate is chosen so that the outer horizon is at r"m. This solution has the following non-zero charges: P"P Q /D, Q"(PK !PK !QK !QK ) cosh d sinh d , P"0, Q"(Q Q cosh d #P P sinh d )/D , P"(Q P cosh d #Q P sinh d )/D, Q"0 , P"P Q (Q !Q !P !P ) sinh d cosh d /D, Q"D ,

(238)

The BPS limit (m"0 and d PR) of this solution is related to the solution (200) via subsets of SO(2);SO(2)LO(2, 2) (¹ ¹-duality) and SO(2)LS¸(2, R) transformations.


77

and the ADM mass, compatible with the BPS bound [178,236], is M

1 " [(P )(PK !QK ) sinh d #(Q )(PK #QK ) cosh d ] "+ D #(PK #QK ) cosh d #(QK !PK ) sinh d .

(239)

4.2.4.2. S- and T-duality transformations. The additional 51 charge degrees of freedom needed for parameterizing the most general ;(1) charge con"guration are introduced by [O(6);O(22)]/ [O(4);O(20)] and SO(2) transformations. The resulting general solution has the charge con"guration ; (e !e ) ; (m !m ) 1 S B S B , P"( U2 , (240) Q" U2 e #e m #m S B S B (2 ; ; 0 0

CDE e2,(Q cos c#P sin c, Q cos c#P sin c, 0, 2, 0) , S

where

CDE e2,(P sin c, Q cos c, 0, 2, 0) , B

(241)

CDE m2,(P cos c, Q sin c, 0, 2, 0) , S CDE m2,(P cos c#Q sin c, P cos c#Q sin c, 0, 2, 0) , B

c is the SO(2)LS¸(2, R) rotational angle, ; 3SO(6), ; 3SO(22), 0 is a (16;1)-matrix with zero entries and U3O(6, 22, R) brings to the basis where the O(6, 22) invariant metric (127) is diagonal. And the complex scalar S and the moduli M transform to (W cos c!sin c)#ie\P cos c S" , (W sin c#cos c)#ie\P sin c

(242) ;2 0 UMU2 U, ; 0 ;2 where W, e\P and M are the axion, the dilaton and the moduli of the generating solution (236). The Èinstein-framea metric g in (236) remains unchanged, but the `string-framea metric g is IJ IJ transformed to the most general form g "g /Im (S). IJ IJ

; M"U2 0

0

4.2.4.3. Special cases of the general solution. The generating solution (236), when supplemented by appropriate subsets of S- and T-dualities, reproduces all the previously known spherically symmetric black holes in heterotic string on ¹. Here, we give some examples.

78


E Non-rotating black holes in Einstein}Maxwell-dilaton system with the gauge kinetic term e\?PF FIJ [245,281,298,354]: IJ (1) P "P "Q "Q O0 case: the Reissner}NordstroK m black hole, i.e. a"0 (2) any of 3 charges non-zero and equal: the a"1/(3 case [236] (3) only 2 magnetic (or electric) charges non-zero and equal: the a"1 case [409] (4) only 1 charge non-zero: the a"(3 case, which contains in the extreme limit the followings (i) P O0 case: KK monopole [506,299,325,574], and (ii) P O0 case: H-monopole [57,286,420}422]. E P "P and Q "Q solution with subsets of S- and T-dualities applied becomes general axion}dilaton black holes found in [83,404,410,487,486]. E The solution with Q O0OQ , when supplemented by S- and T-dualities, is the general electric black hole in heterotic string [562,564]. The S-dual counterpart is the general magnetic solution [58]. E The non-BPS extreme solution (i.e. mP0, P "Q "0, "Q "!"P "P0 and d PR, while keeping meB and ("Q "!"P ")eB as "nite constants) is related by S- and T-dualities to the non-BPS extreme KK black hole studied in [301]. 4.2.4.4. Global space-time structure and thermal properties. We classify all the possible spacetime and thermal properties of non-rotating black holes in heterotic string on ¹. These properties are determined by the 6 parameters P , Q , d and m of the generating solution (236), since the D"4 T- and S-dualities, which introduce the remaining charge degrees of freedom, do not a!ect the Èinstein-framea spacetime. We separate the solutions into non-extreme (m'0) and extreme (m"0) ones. Within each class, we analyze their properties according to the range of the other 5 parameters P , Q and d . 4.2.4.4.1. Global space-time structure. There is a spacetime singularity at r"r where R(r)"0. The event horizon(s) is located at r"r where j"0, provided r 5r . ! ! (A) Non-extreme solutions (m'0). By analyzing roots of X>!Z, one sees that a singularity is always at r 4!m. Thus, global spacetime is either that of non-extreme Reissner}NordstroK m black hole when r (!m (case with 2 horizons at r"$m) or that of Schwarzschield black hole when r "!m. X>!Z has a single root at r "!m, in which case a singularity and the inner horizon coincide at r"!m, when (a) d O0 and P "0, or (b) d "0 and at least one of P and Q is zero. X>!Z has a double root at r "!m, in which case the inner horizon disappears and a singularity forms at r"!m, when (a) d O0 and only Q is non-zero, or (b) d "0 and at least 2 of P , Q are zero.

In the following analysis, it is understood that Q O0 when d O0. When Q "0, P "0 due to initial assumption "Q "5"P ". Then, d are not constrained by (234). In this case, we have a non-extreme 4-charged solution with charges P, P, Q and Q. Such a solution is related to (236) through subsets of SO(2);SO(2)LO(2, 2)LO(6, 22) and SO(2)LS¸(2, R) transformations.


79

(B) Extreme solutions (mP0). When d is "nite, the ADM mass of the generating solution always saturates the Bogomol'nyi bound as mP0, i.e. becomes BPS extreme solution. When both pairs (P , P ) and (Q , Q ) have the same relative signs, the singularity is always at r 40. Global spacetime is, therefore, that of the extreme Reissner}NordstroK m black hole when r (0 (time-like singularity), or the singularity and the horizon coincide (null singularity) when r "r "0. The latter case happens when at least one out of P , Q (and Q ) is zero with ! d O0 (with d "0). The horizon at r "0 disappears (naked-singularity) when (i) only Q is non-zero with d O0, or (ii) only one out of P , Q is non-zero with d "0. When at least one of the pairs (P , P ) and (Q , Q ) has the opposite relative sign, the singularity is outside of the horizon, i.e. r '0 (singularity is naked) [48,179,400,406]. In the case of non-BPS extreme solutions [177,181], the singularity is always behind the event horizon (r (r "0), i.e. the global spacetime of the extreme Reissner}NordstroK m black hole (time-like singularity). 4.2.4.4.2. Thermal properties. Thermal properties are speci"ed by spacetime at the (outer) horizon. So, we consider only regular solutions, which include non-extreme solutions compatible with the Bogomol'nyi bound and extreme solutions with the same relative sign for both pairs (P , P ) and (Q , Q ). The entropy S is of the form p S" "[(QK #m)(PK #m) cosh d #(PK #m)(QK !m) sinh d ] "D" ;[(Q )(QK #m)(PK #m) cosh d #(P )(QK #m)(PK !m) sinh d ] ![P P (QK #m)#Q Q (PK #m)] cosh d sinh d " ,

(243)

where PK "#(P#m, etc. Entropy increases with d , approaching in"nity ["nite value] as d PR [non-BPS extreme limit is reached]. For BPS extreme solutions, entropy is (a) non-zero and "nite, approaching in"nity as d PR, when P and Q are non-zero, and (b) always zero when at least one of P , Q (and Q ) is zero with d O0 (with d "0). The Hawking temperature ¹ ""R j(r"m)"/4p is & P m S\ . ¹ " & (2

(244)

As d increases, ¹ decreases, approaching zero. In the BPS extreme limit with at least 3 of & P ,Q non-zero, ¹ is always zero. With 2 of them non-zero, ¹ is non-zero and "nite, & & approaching zero as d PR. When only one of them (only Q ) is non-zero (for the case d O0), ¹ becomes in"nite. In the non-BPS extreme limit, ¹ is zero. & & 4.2.4.5. Duality invariant entropy. We discuss the duality invariant form of entropy of nearextreme, non-rotating black hole in heterotic string on ¹ [170].

80


The (t,t)-component of metric for general N"4 spherically symmetric solutions has the form g "p(r#m)(r!m)S\(r) with S(r) given by RR S(r)"p ((r#j ) . G G

(245)

Entropy S of non-extreme solutions is given by S(r) at the outer horizon, i.e. S,S(m). Generally, j are functions of 28#28 electric and magnetic charges (240) and m (through m), G and their duality invariant forms are hard to obtain. However, for the near-extreme case, in which j are expressed to leading order in m around their BPS values j, one can obtain the T- and G G S-duality invariant entropy expression, which reads p S"p (j # m (1/j j j j #O(m) . G G H I G 2 G GHI G

(246)

Here, the T- and S-duality invariants are j ,S /p"F(¸, C)F(¸,!C) , .1 G G j ,M "(F(M , C)#(F(M ,!C) , G .1 > > G 1 j j " (Q2¸Q#P2¸P)#(F(M ,C)F(M ,!C) , > > G H 2g Q GH 1 j j j " +M (Q2¸Q#P2¸P)!(Q2M Q!P2M P) .1 > > G H I 4gM Q .1 GHI ;(Q2¸Q!P2¸P)!4(Q2¸M ¸P)(Q2M P), , >

(247)

where 1 F(M ,$C)" (Q2M Q#P2M P$C(M )) > > > > 2g Q C(M )"(4(P2M Q)#(Q2M Q!P2M P) . > > > >

(248)

4.3. Rotating black holes in four dimensions We generalize the 4-charged non-extreme solution (193) to include an angular momentum [183]. (For an another attempt, see [389]. But this solution has only 3 charge degrees of freedom and is a special case of a general solution to be discussed in this section.)


81

4.3.1. Explicit solution By applying the solution generating technique discussed in the beginning of this section, one obtains the following D"4, non-extreme, rotating black hole solution [183]: (r#2m sinh d )(r#2m sinh d )#l cos h N C , g " (r#2m sinh d )(r#2m sinh d )#l cos h N C 2ml cos h( sinh d cosh d sinh d cosh d ! cosh d sinh d cosh d sinh d ) N N C C N N C C , g " (r#2m sinh d )(r#2m sinh d )#l cos h N C (r#2m sinh d )(r#2m sinh d )#l cos h N C , g " (r#2m sinh d )(r#2m sinh d )#l cos h N C 2ml cos h( sinh d cosh d cosh d sinh d ! cosh d sinh d sinh d cosh d ) N N C C N N C C , B "! (r#2m sinh d )(r#2m sinhd )#l cos h N C (r#2m sinh d )(r#2m sinh d )#l cos h N N eP" , D

r!2mr#l cos h dr ds"D ! dt# #dh # D r!2mr#l #

sin h +(r#2m sinh d )(r#2m sinh d )(r#2m sinh d ) N N C D

(r#2m sinh d )#l(1# cos h)r#=#2mlr sin h,d C 4ml ! +( cosh d cosh d cosh d cosh d ! sinh d sinh d sinh d sinh d )r N N C C N N C C D

#2m sinh d sinh d sinh d sinh d , sin h dt d , N N C C

(249)

where D,(r#2m sinh d )(r#2m sinh d )(r#2m sinh d )(r#2m sinh d ) N N C C #(2lr#=) cos h , =,2ml( sinh d # sinh d # sinh d # sinh d )r N N C C #4ml(2 cosh d cosh d cosh d cosh d sinh d sinh d sinh d sinh d N N C C N N C C !2 sinh d sinh d sinh d sinh d ! sinh d sinh d sinh d N N C C N C C !sinh d sinh d sinh d ! sinh d sinh d sinh d N C C N N C !sinh d sinh d sinh d )#l cos h . (250) N N C The axion W also varies with spatial coordinates, but since its expression turns out to be cumbersome, we shall not write here explicitly.

82


The ADM mass, ;(1) charges, and angular momentum are M"2m(cosh 2d # cosh 2d #cosh 2d #cosh 2d ) , C C N N Q"2m sinh 2d , Q"2m sinh 2d , C C (251) P"2m sinh 2d , P"2m sinh 2d , N N J"8lm(cosh d cosh d cosh d cosh d !sinh d sinh d sinh d sinh d ) , C C N N C C N N where G"" and the convention of [478] is followed. , When Q"Q"P"P, all the scalars are constant, and thus the solution becomes the Kerr}Newman solution. The d "d "0 case is the generating solution of a general electric N N rotating solution [562]. The case with Q"P and Q"P is constructed in [389]. The solution (249) has the inner r and the outer r horizons at \ > (252) r "m$(m!l , ! provided m5"l". In this case, the solution has the global spacetime of the Kerr}Newman black hole with the ring singularity at r"min+Q, Q, P, P, and h"p/2. The extreme solution (r "r ) is obtained by taking the limit mP"l">. In this case, the global > \ spacetime is that of the extreme Kerr}Newman solution. The BPS limit is reached by taking mP0 and d PR while keeping meBCCNN as "nite CCNN constants so that the charges remain non-zero. When J is non-zero, i.e. lO0, the singularity is naked since the condition m5"l" for existence of regular horizon (252) is not satis"ed. To have a BPS solution with regular horizon, one has to take lP0, leading to a solution with J"0. Thus, the only regular BPS solution in D"4 is the non-rotating solution, with global spacetime of the extreme Reissner}NordstroK m black hole. This is in contrast with the D"5 3-charged solution [98,182], where one can take l to zero (so that the BPS solution has regular horizon) but the angular momenta J can be non-zero. For D'5, the regular BPS limit with non-zero angular momentum is achieved without taking l to zero if only one angular momentum is non-zero [366]. G 4.3.2. Entropy of general solution The thermal entropy of the solution (249) is [183]

1 A"16p m cosh d # sinh d #m(m!l cosh d ! sinh d S" G G G G 4G , G G G G "16p m cosh d # sinh d # m cosh d ! sinh d !J , (253) G G G G G G G G where d ,d and A" dh d (g g " > is the outer-horizon area. CCNN FF (( PP Note, the thermal entropy has the form which is sum of `left-movinga and `right-movinga contributions. Each term is symmetric in d , i.e. in the 4 charges, manifesting U-duality symmetry G [381]. On the other hand, (253) is asymmetric in J: only the right-moving term has J, which reduces

See [610] for the same result from the conformal p-model perspective.


83

the right-moving contribution to the entropy. This re#ects right-moving worldsheet supersymmetry of the corresponding p-model. When J"0, the entropy becomes [180]: S"32pm cosh d , G G

(254)

which again has U-duality symmetry under the exchange of 4 charges. In the regular BPS limit as well as the extreme limit, the `right-movinga term in (253) becomes zero, however entropy has di!erent form in each case. In the regular BPS limit (J"0) [178]: S"32pm cosh d "2p(PPQQ , G G

(255)

while in the extreme limit:

S"16pm cosh d # sinh d "2p(J#PPQQ . G G G G

(256)

Entropy of a black hole with general charge con"guration in the class and with arbitrary scalar asymptotic values is independent of scalar asymptotic values when expressed in terms of the charge lattice vectors a and b, and has the S- and T-duality invariant form [183]: S"2p(J#+(a2¸a)(b2¸b)!(a2¸b), .

(257)

4.4. General rotating xve-dimensional solution We construct the most general rotating black hole in heterotic string on ¹ [182]. In D"5, black holes carry only electric charges of ;(1) gauge "elds. Since the NS-NS 3-form "eld strength H is Hodge-dual to a 2-form "eld strength in D"5 in the following way: IJM eP HIJM"! eIJMHNF , HN 2!(!g

(258)

where F is the "eld strength of a new ;(1) gauge "eld A , black holes in D"5 carry an additional IJ I charge associated with the NS-NS 2-form "eld B as well as 26 electric charges of the ;(1) gauge IJ group. Thus, the most general black hole in heterotic string on ¹, compatible with the conjectured `no-hair theorema [125,343,344,385,386], is parameterized by 27 electric charges, 2 angular momenta and the non-extremality parameter. 4.4.1. Generating solution We choose to parameterize the `generating solutiona in terms of electric charges Q, Q and Q associated with H , A and A , respectively. These charges are induced through solution IJM I I generating procedure described in Section 4.1.

84


The "nal form of the generating solution is [182] r#2m sinh d #l cos h#l sin h C g " , r#2m sinh d #l cos h#l sin h C (r#2m sinh d #l cos h#l sin h) C eP" , (r#2m sinh d #l cosh#l sinh) CG G m cosh d sinh d C C A" , R r#2m sinh d #l cos h#l sin h C l sinh d sinh d cosh d !l cosh d cosh d sinhd C C C C C C, A "m sin h ( r#2m sinh d #l cos h#l sin h C l cosh d sinh d sinh d !l sinh d cosh d cosh d C C C C C C, A "m cos h ( r#2m sinh d #l cos h#l sin h C m cosh d sinh d C C A" , R r#2m sinh d #l cos h#l sin h C l cosh d sinh d coshd !l sinh d cosh d sinh d C C C C C C, A "m sin h ( r#2m sinh d #l cos h#l sin h C l sinh d cosh d sinh d !l cosh d sinh d cosh d C C C C C C, A "m cos h ( r#2m sinh d #l cos h#l sin h C l sinh d sinh d cosh d !l cosh d cosh d sinh d C C C C C C, BK "!2m sin h R( r#l cos h#l sin h#2m sinh d C l sinh d sinh d cosh d !l cosh d cosh d sinh d C C C C C C, "!2m cos h BK R( r#l cos h#l sin h#2m sinh d C 2m cosh d sinh d cos h(r#l#2m cosh d ) C C C , BK "! (( r#l cos h#l sin h#2m sinh d C (r#l cos h#l sin h)(r#l cos h#l sin h!2m) ds"DM ! dt # DM

r 4m cosh sinh # dr#dh# [l l +(r#l cosh#l sinh) (r#l)(r#l)!2mr DM !2m( sinh d sinh d # sinh d sinh d #sinh d sinh d ), C C C C C C #2m+(l#l) cosh d cosh d cosh d sinh d sinh d sinh d C C C C C C !2l l sinh d sinh d sinh d ,]d d

C C C 4m sin h [(r#l cos h#l sin h)(l cosh d cosh d cosh d ! C C C DM


85

!l sinh d sinh d sinh d )#2ml sinh d sinh d sinh d ]d dt C C C C C C 4m cos h ! [(r#l cos h#l sin h)(l cosh d cosh d cosh d C C C DM !l sinh d sinh d sinh d )#2ml sinh d sinh d sinh d ] d dt C C C C C C sin h # (r#2m sinh d #l) (r#2m sinh d #l cos h#l sin h) C CG DM G #2m sin h+(l cosh d !l sinh d )(r#l cos h#l sinh) C C #4ml l cosh d cosh d cosh d sinh d sinh d sinh d C C C C C C !2m sinh d sinh d (l cosh d #l sinhd ) C C C C

!2ml sinh d (sinh d #sinh d ), d C C C

cos h # (r#2m sinh d #l) (r#2m sinh d #l cos h#l sin h) C CG DM G #2m cos h+(l cosh d !l sinh d )(r#l cos h#l sin h) C C #4ml l cosh d cosh d cosh d sinh d sinh d sinh d C C C C C C !2m sinh d sinh d (l sinh d #l cosh d ) C C C C

!2ml sinh d (sinhd #sinh d ), d , C C C

(259)

where DM ,(r#2m sinh d #l cos h#l sin h)(r#2m sinh d #l cos h#l sin h) C C ;(r#2m sinh d #l cos h#l sin h) , (260) C and the subscript E in the line element denotes the Einstein-frame. The ;(1) charges, the ADM mass and the angular momenta of the generating solution (259) (with G""p/4) are , Q"m sinh 2d , Q"m sinh 2d , Q"m sinh 2d , C C C M"m(cosh 2d #cosh 2d #cosh 2d ) C C C (261) "(m#(Q)#(m#(Q)#(m#Q , J "4m(l cosh d cosh d cosh d !l sinh d sinh d sinh d ) , C C C C C C J "4m(l cosh d cosh d cosh d !l sinh d sinh d sinh d ) . C C C C C C The solution has the outer and inner horizons at: 1 1 1 r "m! l! l$ ((l!l)#4m(m!l!l) , ! 2 2 2 provided m5("l "#"l ").

(262)

86


When Q"Q"Q, the generating solution becomes the D"5 Kerr}Newman solution, since g and u become constant. The generating solution with Q"Q corresponds to the case where the D"6 dilaton u "u# log det g is constant. In this case, with a subsequent rescaling of scalar asymptotic values one obtains the static solution of [368] and rotating solution of [95]. The BPS limit with J O0 and regular event horizon is de"ned as the limit in which mP0, l P0 and d PR while keeping meBC"Q, meBC"Q, meBC"Q, CCC l /m"¸ and l /m"¸ constant. In this limit, the generating solution becomes Q A" , R r#Q

J sin h J cos h A " , A " , ( r#Q ( r#Q

Q A" , R r#Q

J sin h J cos h A " , A " , ( r#Q ( r#Q

J sin h J cos h BK "! , BK " , BK "!Q cos h , R( R( (( r#Q r#Q r#Q (r#Q) , eP" g " , r#Q [(r#Q)(r#Q)]

r dr J cos h sin h ds"DM ! dt# #dh# d d

# DM r 2DM 2Jr cos h 2Jr sin h dt d # dt d

! DM DM

1 sin h (r#Q)(r#Q)(r#Q)! J sin h d # 4 DM

1 cos h (r#Q)(r#Q)(r#Q)! J cos h d , # 4 DM

(263)

where DM ,(r#Q)(r#Q)(r#Q) . The solution is speci"ed by 3 charges and only 1 angular momentum J:

(264)

J "!J ,J"(2QQQ)(¸ !¸ ) , while its ADM mass saturates the Bogomol'nyi bound:

(265)

M

"Q#Q#Q . .1

When 1 or 3 boost parameters are negative, one has the BPS limit with J "J .

(266)


87

4.4.2. ¹-duality transformation The remaining 27 electric charges (needed for parameterizing the most general charge con"guration) are introduced by the [SO(5);SO(21)]/[SO(4);SO(20)] transformation on the generating solution (259). The "nal expression for electric charges is

; (e !e ) S B Q" U2 e #e S B (2 ; 0 1

,

(267)

where

CDE CDE e2,(Q, 0, 2, 0), e2,(Q, 0, 2, 0) , B S

(268)

; 3SO(5), ; 3SO(21), 0 is a (16;1)-matrix with zero entries and U3O(5, 21, R) brings to the basis where the O(5, 21) invariant metric ¸ (127) is diagonal. And the charge Q associated with B remains unchanged. The moduli M is transformed to IJ ; 0 ;2 0 M"U2 UMU2 U, (269) 0 ; 0 ;2 where M is the moduli of the generating solution (259). The subsequent O(5, 21);SO(1, 1) transformation leads to the solution with arbitrary asymptotic values M and u .

4.4.3. Entropy of general solution The thermal entropy of the generating solution (259) is [183]

1 A"4p m+2m!(l !l ), cosh d # sinh d S" G G 4G , G G #m+2m!(l #l ), cosh d ! sinh d G G G G 1 "4p 2m cosh d # sinh d ! (J !J ) G G 16 G G 1 # 2m cosh d ! sinh d ! (J #J ) , (270) G G 16 G G where d ,d , G "p/4 and the outer horizon area is de"ned as A" CCC , dh d d (g (g g !g )" >. ( ( PP FF ( ( ( ( Note, each term is symmetric under the permutation of d (i.e. 3 charges), manifesting the G conjectured U-duality symmetry [381]. Again, as in the D"4 case, the entropy (270) is cast in the form as sum of `left-movinga and `right-movinga contributions, hinting at the possibility of statistical interpretation of each term as left- and right-moving (D-brane worldvolume) contributions to microscopic degrees of freedom. Each term now carries left- or right-moving angular momentum that could be interpreted as left- or right-moving ;(1) charge [35,36] of the N"4

88


superconformal "eld theory when the generating solution (259) is transformed to a solution of type-IIA string on K3;S through the conjectured string}string duality in D"6 [635]. When J "0, the entropy rearranges itself as a single term [183,362]: G (271) S"8(2pm cosh d , G G which again has manifest symmetry under permutation of charges. We discuss duality invariant forms of the entropy and the ADM mass of non-extreme, rotating black hole with general charge con"guration (267). The entropy and the ADM mass are expressed in terms of the following T-duality invariants (obtained by applying T-duality to charges of the generating solution): QPX"(Q2M Q#(Q2M Q , > \ (272) QP>"(Q2M Q!(Q2M Q , > \ while Q remains intact under T-duality. From these 3 T-duality invariant `coordinatesa X, >, Q, one de"nes the following duality invariant `non-extreme hatteda quantities XK : G XK ,(X#m, X "(X, >, Q) . (273) G G G Duality invariant forms of the entropy and the ADM mass are

S"2p

XK #m XK #( (XK !m)!(J !J ) G G G G G G

# XK #m XK !( (XK !m)!(J #J ) , G G G G G G M"XK #>K #QK . When J "0, the duality invariant expression for entropy is G S"4p((XK #m)(>K #m)(QK #m) .

(274)

(275)

4.4.3.1. BPS limit. In the regular BPS limit, the event horizon area (270) becomes [182] (276) "4p[(QQQ)(1!(¸ !¸ ))]"4p[QQQ!J] . .1 Entropy of BPS black hole with general charge con"guration (267) and with arbitrary scalar asymptotic values depends only on (quantized) charge lattice vectors a and b [182], being a statistical quantity [174,178,258}260,441,579]: A

"4p(b(a2¸a)!J . .1 Here, a and b are related to the physical charges Q and Q as S

(277)

Q"ePM a, Q"e\Pb .

(278)


89

In the regular BPS limit, the ADM mass (261) becomes M

"Q#Q#Q . (279) "+ A subset of the SO(5, 21) T-duality transformation on (279) leads to the following T-duality invariant expression for ADM mass of the general D"5 black hole: "eP[a2(M #¸)a]#e\Pb , "+ which has dependence on scalar asymptotic values as well as charge lattice vectors. M

(280)

4.4.3.2. Near-extreme limit. In"nitesimal deviation from the BPS limit is achieved by taking the limit in which m and l are very close to zero, and d's are very large such that charges and l /m"¸ remain as "nite, non-zero constants, and then keeping only the leading order terms in m [182]. To the leading order in m, the inner and the outer horizons are located at r +m(1!(¸#¸)$([2!(¸ #¸ )][2!(¸ !¸ )]) . ! The outer horizon area to the leading order in m is [95,182] A+4p[(QQQ)(1!(¸ !¸ )#m(QQ#QQ#QQ) ;([1!(¸ #¸ )]] . J and J are no longer equal in magnitude and opposite in sign anymore: J,(J !J )"(2QQQ)(¸ !¸ )#O(m) , 1 1 1 *J,(J #J )"m(2QQQ) # # (¸ #¸ )#O(m) , Q Q Q while the ADM mass still has the form

(281)

(282)

(283)

M"(m#(Q)#(m#(Q)#(m#Q . (284) Note, when one of the charges is taken small, e.g. QP0, as in study of the microscopic entropy near the BPS limit [95,368], the ADM mass is M"M #O(m), while the area is .1 A"A #O(m). However, when all the charges are non-zero, the deviation from the BPS limit .1 is of the forms M"M #O(m) and A"A #O(m). .1 .1 4.5. Rotating black holes in higher dimensions We discuss rotating black holes in heterotic string on ¹\" (44D49) with general ;(1)\" electric charge con"gurations [184,449]. The generating solution is parameterized by the ADM mass M (or alternatively the non-extremality parameter m), ["\] angular momenta & J (i"1,2, ["\]), and 2 electric charges of the KK and the 2-form ;(1) gauge "elds associated G with the same compacti"ed direction, which we choose without loss of generality to be Q and Q, i.e. those associated with the "rst compacti"ed direction, as well as asymptotic values of a toroidal modulus G and the dilaton u .

90


The non-trivial "elds of the generating solution are [184] N sinh d cosh d N sinh d cosh d , , A" A" R R 2N sinh d #D 2N sinh d #D Nl k sinh d cosh d Nl k sinh d cosh d , A " G G , A " G G (G (G 2N sinh d #D 2N sinh d #D 2N sinh d #D D eP" , G " , 2N sinh d #D = 2Nl k sinh d sinh d [m(sinh d #sinh d )r#D] G G B G"! , R( = B G H"!4Nl l kk sinh d sinh d cosh d cosh d GH G H (( ;[N(sinh d #sinh d )#D][2N sinh d sinh d #ND(sinh d # sinh d !1)#D]/[(D!2N)=] , D!2N dr dt# ds"D"\"\="\ ! = "\ (r#l)!2N G G r#l cos h#K sin h # dh D

cos h cos t 2cos t G\ (r#l cos t #K sin t )dt # G> G G> G G D l!K H cos h cos t 2cos t !2 H G\ D GH ;cos t sin t 2cos t cos t sin t dt dt G G H\ H H G H l !K G> cos h sin h cos t 2cos t cos t sin t dh dt !2 G> G\ G G G D k # G [(r#l)(2N sinh d #D)(2N sinh d #D) G D= 2Nl k cosh d cosh d G G dt d

#2lN(D!2N sinh d sinh d )] d ! G G G = # GH

4Nl l kk(D!2N sinh d sinh d ) GH G H d d , G H D=

(285)

where =,(2N sinh d #D)(2N sinh d #D) and D, K , N, k , a are de"ned separately for even and odd D in (188)}(191). G G

(286)


91

The ADM mass, angular momenta and electric charges of the generating solution are X m M " "\ [(D!3)(cosh 2d #cosh 2d )#2] , & 16pG " X J " "\ml cosh d cosh d , G G 4pG " (287) X Q" "\ (D!3)m sinh 2d , 16pG " X Q" "\ (D!3)m sinh 2d . 16pG " For the canonical choice of asymptotic values G "d , i.e. compacti"cation on (10!D) self-dual GH GH circles with radius R"(a, the D-dimensional gravitational constant is G "G /(2p(a)\". " Also, the KK and the 2-form "eld ;(1) charges Q and Q are quantized as p/(a and q/(a, respectively, where p, q3Z. The outer horizon area of the generating solution is [184,366] A "2mr X cosh d cosh d , (288) " > "\ where the outer horizon r is determined by > "\

"0 . (289) (r#l)!2N G PP> G The surface gravity i at the (outer) event horizon is de"ned as i"lim > j Ij, where PP I mIm ,!j and m,R/Rt#X R/R . Here, X is the angular velocity at the (outer) horizon and is I G G G de"ned by the condition that m is null on the (outer) horizon. The surface gravity and angular velocity at the outer-horizon of the generating solution are

1 1 R (P!2N) l P G i" , X" , (290) G cosh d cosh d r #l cosh d cosh d 4N PP> > G where P, "\ (r#l). G G The generating solution has a ring-like singularity at (r, h)"(0, p/2) and thus spacetime is that of the Kerr solution. The BPS limit of (285), where the ADM mass M saturates the Bogomol'nyi bound & M 5"Q#Q" , (291) & is de"ned as the limits mP0 and d PR such that Q remain as "nite constants. For D56 with only one of l non-zero, the BPS limit is also the extreme limit [366], i.e. all the horizons G collapse to r"0 as mP0. However, with more than one l non-zero, the singularity at r"0 G becomes naked, i.e. horizons disappear.

We use the convention of [478], keeping in mind that matter Lagrangian in (124) has 1/(16pG ) prefactor. "

92


5. Black holes in Nⴝ2 supergravity theories 5.1. N"2 supergravity theory 5.1.1. General matter coupled N"2 supergravity We consider the general N"2 supergravity [5,6,128,198,211,271] coupled to n vector mul tiplets and n hypermultiplets. The "eld contents are as follows. The N"2 supergravity multiplet & contains the graviton, the S;(2) doublet of gravitinos tG (the S;(2) index i"1, 2 labels two I supercharges of N"2 supergravity and k"0, 1, 2, 3 is a spacetime vector index), and the graviphoton. The N"2 vector multiplets contain ;(1) gauge "elds, doublets of gauginos j? and G scalars z? (a"1,2, n ), which span the n -dimensional special KaK hler manifold. The hypermultip lets consist of hyperinos f , f? (a"1,2, 2n ) with left and right chiralities and real scalars ? & qS (u"1,2, 4n ), which span the 4n -dimensional quaternionic manifold. The general form of the & & bosonic action is [5] L,"(!g[!R#g H(z, z ) Iz? z @H#h (q) IqS qT ?@ I ST I K R K R #i(N M KRF\ F\ IJ!NKRF> F> IJ)] , (292) IJ IJ where g H"R R HK(z, z ) is the KaK hler metric, h (q) is the quaternionic metric, ?@ ? @ ST F!K,(FK $(i/2)eIJMNFK ) are the (anti-)self-dual parts of the "eld strengths II IJ MN FK "R AK!R AK#gf KR ARA of the ;(1) gauge "elds AK (K"0, 1,2, n ) in the N"2 IJ I I J I I J I supergravity and N"2 vector multiplets, and g is the gauge coupling. Here, the gauge covariant di!erentials on the scalars are de"ned as:

z?,R z?#gAKk?K(z) , I I I

z ?H,R z ?H#gAKk?H (z ) , (293) I I I K

qS,R qS#gAKkSK(q) , I I I where k?K(z) [kSK(q)] are the holomorphic [triholomorphic] Killing vectors of the KaK hler [quaternionic] manifold (cf. see (65)). We introduce a symplectic vector of the anti-self-dual "eld strengths:

Z\,

F\K G\ R

,

(294)

where G\ M KRF\R [212]. The symplectic vector Z> of self-dual "eld strengths is the complex K ,N conjugation of (294). It is convenient to rede"ne "eld strengths FK as [93,132] ¹\,1 and G>"(C/2)F e> (e> obeys GI ' ' IJ "ie> and is normalized to give 2p after being integrated over S) substituted, one obtains the *e> IJ IJ following solution for the metric: e\3"1#(CCM /4r .

(307)

This solution has the surface area given by p A" CCM . 4

(308)

5.4. Principle of a minimal central charge At the "xed attractor point in phase space, the central charge eigenvalue is extremized with respect to moduli "elds, so-called `principle of a minimal central chargea [254,258,259,414,415]. For N'2 theories, the largest eigenvalue is extremized and the smaller central charges become zero [259] at the "xed point. Since scalar asymptotic values are expressed only in terms of ;(1) charges at the "xed point, the extremized (largest) central charge depends only on ;(1) charges, thereby becoming a candidate for describing black hole entropy. It turns out that entropy of extreme black holes for each dimension has the following universal dependence on the extremal value Z of the (largest) central charge eigenvalue regardless of the number N of supersymmetries [258,259]: A S" "p"Z "? , 4

(309)

where a"2[3/2] for D"4 [D"5]. As an example, we consider the BPS dilatonic dyon [292,409] in D"4 with the mass: ""Z""(e\P"p"#eP"q") . (310) .1 The minimum of the central charge "Z" is located at g "eP""p/q", which leads to the following correct expression for the entropy which is independent of dilaton asymptotic value: M

S"A/4"p"Z ""p"pq" . (311) One can prove the principle of a minimal central charge as follows. We consider the ungauged N"2 supergravity coupled to Abelian vector multiplets and hypermultiplets, de"ned by the Lagrangian (292) and the supersymmetry transformations (296) with g"0. Since we are interested in con"gurations at the "xed point, the derivatives of scalars are zero, i.e. R z?"0 and R qS"0, at I I These are obtained by solving dj?G"0. Thus, for N54 theories, one can determine moduli "elds (at the "xed point) in terms of ;(1) charges, by minimizing the largest eigenvalue and setting the remaining eigenvalues equal to zero.


97

the horizon. So, from dj?G"0, one has F\?"0. Note, the covariant derivative of the central IJ charge de"ned in (300) is 1 Z , Z"! ? ? 2

g HF>@H"(QK ¸K!PR MR)"1; "Q2 . ?@ ? ? ?

(312) 1 Since F\?"0 is equivalent to F>?"0, the central charge Z is covariantly constant at the "xed point of the moduli space:

Z " Z"1; "Q2"0 . (313) ? ? ? It can be shown [258] that the condition (313) is equivalent to the statement that the central charge takes extremum value at the "xed point: R "Z""0 . (314) ? Thus, within ungauged general Abelian N"2 supergravity we establish that central charge is minimized at the xxed point of geodesic motion of moduli evolving with o"1/r. At the "xed point in the moduli space, the central charge is expressed in terms of the symplectic ;(1) charge vector Q and the moduli as [258]: "Z""!Q2 ) M(N) ) Q , Im N#(Re N)(Im N)\(Re N) !(Re N)(Im N)\ , M(N), !(Im N)\(Re N) (Im N)\

(315) (316)

with the moduli in the matrix M(N) taking values at the "xed attractor point. The central charge minimization condition (313) "xes the asymptotic values of the moduli in terms of Q. By using the relations 1; "
QR"2 Im(ZM MR) .

(318)

Eq. (318) can be solved to express the moduli (at the "xed point) in terms of ;(1) charges:

1 , >M ),!i1PM "P2!1PM #P"Q2 ,

(321)

98


where

P,

>K

FR(>)

;

>K,ZM XK .

(322)

Namely, at the minimum of VQ(>,>M ), relation (317) for the minimal central charge, which can be expressed in terms of P as P!PM "iQ, is satis"ed. In particular, the entropy for D"4 black holes is rewritten as S ""Z ""i1PM "P2"">" exp[!K(z, z )]" . p

(323)

Many black holes are uplifted to intersecting p-branes. In this case, energy of black holes is sum of energies of the constituent p-branes. The minimal energy of p-branes corresponds to the ADM mass of the corresponding double-extreme black holes in lower dimensions. In taking variation of moduli to "nd the minimum energy con"guration, one has to keep the gravitational constant of lower dimensions as constant. The minimum energy of p-brane is achieved when energy contributions from each constituent p-brane are equal [403]. 5.4.1. Generalization to rotating black holes Generally, rotating black holes have naked singularity in the BPS limit. D"5 rotating black holes with 3 charges has regular BPS limit (thereby the horizon area can be de"ned), if 2 angular momenta have the same absolute values [98,182]. We discuss generalization of the principle of minimal central charge to the rotating black hole case [414]. We consider the following truncated theory of 11-dimensional supergravity compacti"ed on a Calabi}Yau three-fold [103,328,329,491]: L"(!g[!R!ePF FIJ!e\PG GIJ#(R u)] IJ IJ I 1 eIJMNHF F B . ! IJ MN H 4(2

(324)

This corresponds to the N"2 theory with F"CKR XKXRX . The supersymmetry transforma tions of the gravitino t and the gaugino s in the bosonic background are I 1 1 e\PG )e , dt " e# (KMN!4dMCN)(ePF ! I MN (2 MN I I 12 I (325) 1 1 CIR ue# CMN(ePF #(2e\PG )e . ds"! I MN MN 4(3 2(3

Note, lower-dimensional gravitational constant is expressed in terms of the D"10 gravitational constant and the volume of the internal space, i.e. a modulus. It is argued in [288] that singular D"4 heterotic BPS rotating black holes can be described by regular D"5 BPS rotating black holes which are compacti"ed through generalized dimensional reduction including massive Kaluza}Klein modes.


99

The model corresponds to the N"2 supergravity with the graviphoton (ePF ! MN (1/(2)e\uG ) coupled to one vector multiplet with the vector "eld component MN (ePF #(2e\PG ). MN MN We consider the following D"5 BPS rotating black hole solution [98,182,609] (cf. (263)) to the above theory:

r 4J sinh 4J cosh dt! d # dt ds" 1! r p(r!r) p(r!r) r \ dr!r(dh#sin h d #cos h dt) . ! 1! r

(326)

For this solution, the scalar u is constant everywhere (double-extreme): eP"j. So, from ds"0, one sees that the vector "eld in the vector multiplet vanishes, i.e. B"!(j/(2)A, from which one can express u in terms of ;(1) charges as

(2 (2 8Q e\P夹G, Q , eP夹F . (327) eP" $ ,j; Q , & 4p $ 16p pQ 1 1 & Furthermore, the entropy is still expressed in terms of the central charge Z at the "xed point, but modi"ed by the non-zero angular momentum J: S"p(Z !J . (328) The argument can be extended to more general rotating solutions in the N"2 supergravity coupled to n vector multiplets with the gaugino supersymmetry transformations 1 3 i (329) dj "! g (u)CIR u@ e# tK CIJFK e . ? IJ I ? 4 4 2 ?@

From dj "0, one sees that at the "xed point (R u?"0) FK "0. So, the central charge is ? I IJ extremized at the "xed point: R Z"0. ? We discuss the enhancement of supersymmetry near the horizon [138,412]. Since the vector "eld in the vector multiplet is zero for (326), the solution is e!ectively described by the pure N"2 supergravity [265] with the graviphoton FI ,((3/2)jF. The supersymmetry transformation for the gravitino is 1 dt " K e" e# (CMNC #2CMdN)FI e . I I I I I MN 4(3 The integrability condition for the Killing spinor equation dt "0 is I [ K , K ] e"RK e"0 , ? @ ?@ where the super-curvature RK for solution (326) is de"ned as ?@ (r!r) X (1#C) . RK " ?@ ?@ r

(330)

(331)

(332)

100


Here, the explicit forms of matrices X , which can be found in [414], are unimportant for our ?@ purpose. At the horizon (r"r ) and at in"nity (rPR), RK "0, thereby (331) does not constraint ?@ the spinor e, i.e. supersymmetry is not broken. However, for "nite values of r outside of the event horizon, for which RK O0, e is constrained by the relation: ?@ (1#C)e"0 , (333) indicating that 1/2 of supersymmetry is broken. 5.4.2. Generalization to N'2 case The principle of minimal central charge is generalized to the N"4, 8 cases by reducing N"4, 8 theories to N"2 theories, and then by applying the formalism of N"2 theories [258]. For N54 theories, there are more than 1 central charge eigenvalues Z (i"1,2, [N/2]). The ADM mass of G the BPS con"guration is given by the max+"Z ",. When the principle of minimal central charge is G applied to this eigenvalue, the smaller eigenvalues vanish and all the scalar asymptotic values are expressed in terms of ;(1) charges, only [259]. So, the extremum of the largest central charge continues to depend on integer-valued ;(1) charges, only. The entropy of extreme black holes in each D has the universal dependence on the extreme value of the largest central charge eigenvalue: S"A/4"p"Z "?, where a"2 [3/2] for D"4 [D"5], regardless of the number N of super symmetry. 5.4.2.1. Pure N"4 supergravity. Pure N"4 theory can be regarded as N"2 supergravity coupled to one N"2 vector multiplet. This can also be regarded as either S;(2);SO(4) or S;(2); S;(4) invariant truncation of N"8 theory. The former corresponds to the N"2 theory with F(X)"!iXX and the latter has no prepotential. These two theories are related by the symplectic transformation [136]: XK "X, FK "F , XK "!F , FK "X , (334) where the hat denotes the S;(4) model [165]. The SO(4) version [162,164,194] of the D"4, N"4 supergravity action without axion is

1 dx(!g[!R#2R uRIu!(e\PFIJF #ePGI IJGI )] , I" I IJ IJ 16p

(335)

where the "eld strength GI of the SO(4) theory is related to that G of the S;(4) theory as IJ IJ i 1 GI IJ" e\PeIJMHG . MH 2 (!g The dilatino supersymmetry transformation is 1 1 dK "!cIR e # pIJ(e\(F a !e(GI b )\e( . I ' (2 IJ '( IJ '( 2 '

(336)

At the "xed point (R "0), the Killing spinor equations dK "0 "x in terms of electric and I ' magnetic charges: e\( ""q"/"p". Then, writing dK "0 at the "xed point in the form (Z ) e("0, ' '(


101

one learns [409] that E pq'0 case: e non-vanishing, Z "0 and M ""Z ". "+ E pq(0 case: e non-vanishing, Z "0 and M ""Z ". "+ So, smaller eigenvalues, which correspond to broken supersymmetries, vanish and entropy is given by the largest eigenvalue at the "xed point. 5.4.2.2. N"4 supergravity coupled to n vector multiplets. The target space manifold of N"4 supergravity coupled to n vector multiplets is O(6, n ) S;(1, 1) ; O(6);O(n ) ;(1) with the "rst factor parameterized by the axion-dilaton "eld S and the second factor by the coset representatives ¸K "(¸GHK , ¸?K) (i, j"1, 2, 3, 4, K"1,2, 6#n and a"1,2, n ) [84]. The central charge is Z "e)[¸K qK!S¸ KpK] , GH GH GH where K"!ln i(S!SM ) is the KaK hler potential for S. At the "xed point, the gaugino Killing spinor equations dj?"0 require that G K K S¸?Kp !¸? qK"0 .

(337)

(338)

The dilatino Killing spinor equation dsG"0 requires the following smaller of the central charge eigenvalues to vanish: (339) "Z ""(Z ZM GH!((Z ZM GH)!"eGHIJZ Z ") , GH GH IJ GH which "xes S at the "xed point. At the "xed point, the di!erence between two eigenvalues "Z "!"Z ""((Z ZM GH)!"eGHIJZ Z " GH GH IJ becomes independent of scalars and gives rise to the horizon area [178,236] A"4p(M ) "4p"Z ""2p(qp!(q ) p) . "+

(340)

(341)

5.4.2.3. N"8 supergravity. The consistent truncation of N"8 down to N"2 is achieved by choosing HLS;(8) such that 2 residual supersymmetries are H-singlet. Such theory corresponds to N"2 supergravity couple to 15 vector multiplets (n "15) and 10 hypermultiplets (n "10). & (This is the upper limit on the number of matter multiplets that can be coupled to N"2 supergravity.) Under N"2 reduction of N"8, S;(8) group breaks down to S;(2);S;(6), leading to the decomposition of 26 central charges Z of N"8 into (1, 1)#(2, 6)#(1, 15) under S;(2);S;(6). The S;(2) invariant part (1, 1)#(1, 15) is (Z, D Z), where Z is the N"2 central G charge. So, the horizon area is again A&"Z". For example, for type-IIA theory on ¹ /Z truncated so that only 2 electric and 2 magnetic charges are non-zero, the central charge at the "xed point is product of ;(1) charges, which is black hole horizon area.

102


We consider the following truncation of N"8 supergravity:

1 1 S" dx(!g R! [(Rg)#(Rp)#(Ro)] 16pG 2

!

eE [eN>M(F )#eN\M(F )#e\N\M(F )#e\N>M(F )] . 4

(342)

This is a special case of S¹; model [236] with the real parts of complex scalars zero e\E"Im S,s,

e\N"Im ¹,t, e\M"Im ;,u .

(343)

The following black hole solution to this model is reparameterization [511] of the general solution obtained in [178]: ds"!e3dt#e\3 dx , e3"t t s s , t t t s t s e\E" , e\N" , e\M" , s s s t t s F "$dt dt, FI "$ds dt , F "$dt dt, FI "$ds dt , where "q " \ "p " \ t " eE>N>M# , s " e\E\N>M# , r r

"q " \ "p " \ t " eE\N\M# , s " e\E>N\M# , r r

(344)

(345)

and s are magnetic potentials related to FI "eE!N\M夹F . The ADM mass of (344) is 1 s u t M " stu"q "# "q "# "P "# "p " . (346) "+ 4 tu st su

By minimizing (346) with respect to s, t, u, one obtains the ADM mass at the "xed point [178]: (M ) ""q p q p " , (347) "+ and "nds that the smaller central charges are zero at the "xed point. This result is proven in general setting as follows. We consider the N"8 supersymmetry transformations [160] of gravitinos W and fermions s at the "xed point: I ! dW "D e #Z cJe , ds "Z pIJe , (348) I I IJ ! IJ !

where A"1,2,8 labels supercharges of N"8 theory. We truncate the Killing spinors as e "0, e "+e , e O0, e "e "0, , ? G

(349)

This model also corresponds to ¹ part of type-IIA theory on K ;¹ or heterotic theory on ¹;¹. See Section 3.2.2 for the explicit action.


103

where 6 supersymmetries are projected onto null states. Here, we splitted the index A as A"(i, a) in accordance with the breaking of S;(8) to S;(4);S;(4). By bringing Z to a block diagonal form [264] through S;(8) transformation (See Section 2.2, for details.), one "nds that the supersymmetry variations of t , s and s vanish due to (349) and the block diagonal I? ?@A ?GH choice of Z . From ds "0, one "nds that Z "0 (i.e. Z "Z "0) and from ds "0, one G?@ ?@ GHI "nds that Z "0. So, we proved within the class of con"gurations characterized by truncation (349) that the condition for unbroken supersymmetry requires the smaller central charges to vanish. And the largest central charge at the "xed point gives the ADM mass and the horizon area. 5.4.2.4. Five-dimensional theories. N"1, N"2 and N"4 theories in D"5 [11,27,103,213, 328,329,491] have 1, 2 and 3 central charges, respectively. At the "xed point, the largest central charge is minimized and the smaller central charges vanish. The horizon area is given in terms of the central charge at the "xed point by A"4pZ. The general expressions for the (largest) central charge at the "xed point for each case are as follows: E N"1 theory: Z "(d (q)\q q , where d (q)\ is the inverse of d "d t!(z) evaluated ! at the "xed point [258]. E N"2 theory: Z "(Q Q), where Q is a charge of the 2-form potential and Q is the & $ & $ Lorentzian (5, n ) norm of other 5#n charges [580]. E N"4 theory: Z "(q XHJq XKLq XNG), where q is 27 quantized charges transforming GH JK LN GH under E (Z) and XGH3Sp(8) is traceless. 5.5. Double extreme black holes We discuss the most general extreme spherically symmetric black holes in N"2 theories in which all the scalars are frozen to be constant all the way from the horizon (r"0) to in"nity (rPR) [415], called double-extreme black holes. For this case, the ADM mass (or the largest central charge) takes the minimum value (related to the horizon area) and, therefore, is equal to the Bertotti}Robinson mass. Whereas all the scalars are restricted to take values determined by ;(1) charges, all the ;(1) charges can take on arbitrary values. Double-extreme black holes are also of interests since they are the minimum-energy extreme con"gurations in a moduli space for given charges. The general double-extreme solution is obtained by starting from the spherically symmetric Ansatz for metric ds"e3 dt!e\3 dx, ;(r)P0,

as rPR ,

(350)

After the "rst draft of this section is "nished, more general class of N"2 supergravity black hole solutions [60,61,521,522,520], which include general rotating black holes and Eguchi}Hanson instantons, are constructed. These solutions are entirely determined in terms of the Kahler potential and the Kahler connection of the underlying special geometry, where also the holomorphic sections are expressed in terms of harmonic functions. Such general class of solutions turns out to be very important in addressing questions related to the conifold transitions in type II superstrings on Calabi-Yau spaces, when they become massless [60].

104


and assuming that all the scalars are constant everywhere (R zG"0 and R qS"0) and that I I consistency condition F\G"0 for unbroken supersymmetry is satis"ed. Since all the scalars are constant, the spacetime is that of extreme Reissner}Nordstrom solution: e\3"1#M/r .

(351)

By solving the equations of motion following from (292), one obtains FK"e3

2QK 2PK dtdr! r dhr sin h d . r r

(352)

From equations of motion along with (350)}(352), one obtains the ADM mass M in terms of the electric QK and magnetic PR charges of FK: M"!2 Im NKR(QKQR#PKPR) .

(353)

The ;(1) charges (PR, QK) are related to the symplectic charges Q"(qK , q )"(FK, GR) as: R qK 2PK " . (354) 2(Re NKR)PK!2(Im NKR)QK q R

), M is expressed as In terms of (qK , q R (Im N#Re N Im N\Re N)KR (!Re N ImN\)KR 1 M"! (qK , q K ) 2 (!Im N\ Re N)KR (Im N\)KR

qR q R

""Z"#" Z" . (355) ? From the consistency condition F\?"0 for unbroken supersymmetry, one has Z"0, which is ? with equivalent to R Z"0. So, the ADM mass of double extreme black holes is M""Z" ? ?

8 scalars constrained to take values de"ned by Z"0. By solving Z"0, one obtains the ? ? following relation between (q, q ) and the holomorphic section (¸K, MR): qK 2iZM ¸K "Re , (356) 2iZM MR q R

which can be solved to express (¸, M) in terms of (q, q ). Since the ADM mass M for double-extreme solutions obeys the stabilization equations (356), the entropy is related to the ADM mass as: S"pM? ,

(357)

where a"2[3/2] for the D"4 [D"5] solutions. 5.5.1. Moduli space and critical points We have seen that the BPS condition requires scalars at the event horizon take their "xed point values expressed in terms of quantized electric/magnetic charges and, thereby, the (largest) central

The other sum rule for Z and Z is "Z"!"Z ""!Q2M(F)Q. ? ?


105

charge at the event horizon is related to the black hole entropy. In this subsection, we point out that such property of extreme black holes at the "xed point can be derived from bosonic "eld equations and regularity requirement of con"gurations near the event horizon without using supersymmetry [254]. For non-extreme con"gurations, the horizon area has non-trivial dependence on (continuous) scalar asymptotic values. We consider the following general form of Bosonic Lagrangian: L"(!g[!R#G R ?R @gIJ!kKRFK FR gIHgJM!lKRFK 夹FR gIHgJM] , (358) ?@ I J IJ HM IJ HM where FK ,R AK!R AK are Abelian "eld strengths with charges (pK, qK)"((1/4p)FK, IJ I J J I (1/4p) [kKR夹FR#lKRFR]) and kKR, lKR are moduli dependent matrices. We restrict our attention to static ansatz for the metric ds"e3 dt!e\3c dxK dxL , KL where for spherically symmetric con"gurations c c dq # (dh#sin h du) , c dxK dxL" KL sinhcq sinhcq

(359)

(360)

where q runs from !R (horizon) to 0 (spatial in"nity). The function ; satis"es the boundary conditions ;Pcq as qP!R and ;(0)"1. The equations of motion for ;(q) and ?(q) can be derived from the following 1-dimensional action L

d; d ? d @ #G " #e3 and ?"z?/(> with >"1! z?z ?. In ? general, the ADM mass of the extreme solution has the form [524]: "m !n t!Q AG" A GA M " A , .1 2(1!ttM ! AGA M G) G where

(388)

m ,q#iq , n ,iq !q, Q ,iqG !q, t,X/X, AG,XG/X . A A GA G For the following "xed-point values of the moduli "elds, which satisfy (356),

(389)

tM "n /m , A M G"Q /m , A A GA A the ADM mass M takes the minimum value [524]

(390)

g 0 1 M" ("m "!"n "!"Q ")"p(q q ) A GA 0 g 2 A

q

. (391) q For other embedding X"SXS\ of S;(1, n) into Sp(2n#2) related via the symplectic transformation S3Sp(2n#2), the ADM mass is given in terms of new symplectic ;(1) charges (qYq )2"(qq )S\ by [524] g 0 qY S2 . (392) M"p(qYq )S 0 g q

5.5.2.5. General quadratic prepotential. We discuss the case where the lower-component of < is proportional to the upper component [62]:

¸'

¸'

, (393) M R ¸) ( () where R "a !ib with real matrices a and b . Note, it is su$cient to consider the case '( '( '( '( '( where R "!ib , since the most general case with a O0 is achieved by applying the symplectic '( '( '( transformation "

1

0

3Sp(2n#2) a 1 '( on the con"guration with a "0. '( The ADM mass and entropy transform under this symplectic transformation similarly as (392).


111

By solving (356) with (393), one obtains the ADM mass

(RM !R)\ (R!RM )\R i M" (qq ) RM (R!RM )\ RM (RM !R)\R 2 b\ !b\a 1 " (qq ) !ab\ b#ab\a 2

q

q

q

. (394) q The case R"!ig, where g is an S;(1, n) invariant metric, is the CP(n!1, 1) model, i.e. (394) reduces to (391). The most general extreme solution to this model has the form [597] ds"!e\3 dt#e3 dx ) dx , F' "e R HI ', G "e R H , IJ IJM M (IJ IJM M ( >,ZM ¸"i(R!RM )\(RM HI !H) ,

(395)

where

e3"i(H2 HI 2)

(RM !R)\

(R!RM )\R

RM (R!RM )\ RM (RM !R)\R

q' q HI '" hI '# , H " h # ( . ( ( r r

H HI

, (396)

The asymptotically #atness condition leads to the following constraint on hI ' and h in (396): ( (RM !R)\ (R!RM )\R h "!i . (397) (h2 hI 2) RM (R!RM )\ RM (RM !R)\R hI

5.6. Quantum aspects of N"2 black holes Supersymmetric "eld theories respect remarkable perturbative non-renormalization theorems. In N"1 theories, superpotentials are not renormalized in perturbation theory [321,375]. N54 theories are "nite to all orders in perturbative quantum corrections [473,573]. So, the classical BPS solutions in N54 theories are exact to all orders in perturbative corrections. (Cf. Some classical solutions of N"4 theories are also exact solutions [47,48,173,174,369}371,518,607,608,614,615] of conformal p-model of string theory and, therefore, exact to all orders in a-corrections.) For N"2 theories, prepotentials, which determine the Lagrangians, receive perturbative quantum corrections up to one-loop level [9,209,321,548]. Hence, one has to study quantum e!ect on prepotentials for complete understanding of solutions in N"2 theories. In the following, we study the quantum aspects of black holes in the e!ective N"2 theories of compacti"ed superstring theories. It is conjectured that the E ;E heterotic string on K3;¹ and the type-II string on a Calabi}Yau manifold are a N"2 string dual pair [13,14,25,119, 257,396,397,418]. Since the dilaton}axion "eld S belongs to a vector multiplet (hyper multiplet) of

112


the heterotic (type-II) theory, moduli space of hyper multiplet (prepotential for a vector multiplet) is exact at the tree level, due to the absence of neutral perturbative couplings between vector multiplets and hypermultiplets [13,118,167,168,418,435]. Thus, applying the duality between heterotic and type-II strings, one can compute the exact prepotential for vector multiplets (hyper multiplet superpotential) of the heterotic (type-II) theory. The prepotentials of the N"2 e!ective "eld theories of these string theories contain the cubic terms: F(X)"d X?X@XA/X , (398) ?@A plus correction terms that include part of quantum corrections, instanton and topological terms which cannot be included in (398). For the type-IIA string on a Calabi}Yau three-fold, real coe$cients d are the topological intersection numbers d , J J J , where J 3H(>, Z) ?@A ?@A ? @ A ? are the KaK hler cone generators. For the heterotic string on K3;¹, d describe the classical parts ?@A and the non-exponential parts of perturbative corrections to the prepotential. The KaK hler potential associated with (398) is K(z, z )"!log(!id (z!z )?(z!z )@(z!z )A) . (399) ?@A General double-extreme black holes and a special class of extreme black holes with non-constant scalars of the N"2 theory with (398) are discussed in [50]. In the following, we study the e!ect of the quantum correction terms of the prepotential on the classical solutions [50,51,54,123,514]. 5.6.1. E ;E heterotic string on K3;¹ At generic points in moduli space, the E ;E heterotic string on K3;¹ is characterized by 65 gauge-neural hypermultiplets (20 from the K3 surface and 45 from the gauge bundle) and 19 vectors (18 from vector multiplets and 1 from the gravity multiplet). So, the moduli z? (a"1,2, n ) in the vector multiplets consist of the axion}dilaton S, the ¹ moduli ¹ and ;, and Wilson lines "i#>M ,jO0. Note, the symplectic magnetic charge in the perturbative basis de"ned by (408) is expressed in terms of the charges pH and qR in the original basis as q( "(p, q , p,2). By solving the stability equation P!PM "iQ with the following perturbative prepotential: d >?>@>A #ic(>), >K,ZM XK , F(>)" ?@A >

(416)

one obtains the following expression for the entropy (cf. See (323)):

S (p) "!2(q !2cj) j# . p j

(417)

Here, j in (417) is obtained by solving the following equation derived from (356): d p?p@pA 3p q " ?@A #2cj, q "! d p@pA . ? j j ?@A

(418)

For the case cpO0, one can express j in terms of charges as 3pq #p?q ?, j" 6cp

(419)

from which one sees that charges satisfy the following constraint when cp"0: 3pq #p?q "0 . ? For the case c"0 and q O0, j is d p?p@pA q , j"$ ?@A

(420)

(421)

116


with the sign $ determined by the condition q j(0 that the entropy should be positive. As a special case, when c"p"0 the entropy is S "2(q d p?p@pA , ?@A p

(422)

with the solution having only n #1 independent charges, i.e. q "0. In particular, with the cubic ? prepotential (>) >>> #a , F(>)"!b > > entropy (422) becomes S "2(!q (bppp!a(p)) . p

(423)

The explicit solution with non-constant scalar "elds is obtained by just substituting the symplectic charges by the associated harmonic functions in the stabilization equations. For the case where the prepotential is the general cubic prepotential, i.e. (416) with c"0, the solution has the form [50]: ds"!e\3 dt#e3 dx ) dx, e3"(H d H?H@HA , ?@A F? "e R H?, F "R (H )\, z?"iH H?e\3 , KL KLN N K K where the harmonic functions in the solution are

(424)

p? q H?"(2 h?# , H "(2 h # . r r

(425)

Here, the constants h's are constrained to satisfy the asymptotically #at spacetime condition: 4h C h?h@hA"1 , ?@A and, therefore, the ADM mass of the solution is

(426)

q 1 M" # p?h C h@hA . (427) 4h 2 ?@A When h's take the values at the horizon expressed in terms of charges as h "q /c and h?"p?/c, the ADM mass (427) reduces to the entropy S"pM" in (422).

As a special case, we consider the following prepotential corresponding to the S¹; model with the cubic quantum correction: (X) XXX #a . F" X X

(428)

For this case, the metric component has the form

q e3"4 h # r

p h! r

p h! r

p p h# #a h# . r r

(429)


117

Note, the quantum correction term acts as a regulator, smoothing out singularity, for example those of massless black holes [48,49,179,318,406,408,488,578], of classical solutions. 5.6.2. Type-II string on Calabi}Yau manifolds Type-IIA string on a Calabi}Yau three-fold [115,373] gives rise to N"2 theory with h #1 vector "elds and h #1 hypermultiplets (h and h being the Hodge numbers of the three-fold), with the additional hyper multiplet and vector "eld coming from those associated with the dilaton and the gravity multiplet, respectively. The moduli in the vector multiplets consist of the KaK hlerclass moduli t? (a"1,2, n "h ), where h "dim(H(M, Z)). The general form of type-IIA prepotential is [115,373]

sf(3) 1 1 # nP ¸i e i d t? , (430) F''"! C t?t@tA! ?@A ? 2(2p) (2p) 2 B2BF 6 B BF ? where nP 2 F are the rational instanton numbers of genus 0 and s is the Euler number. Here, the B B prepotential is de"ned inside of the KaK hler cone p(K)"+ t?J " t?'0,, where J are the (1, 1)? ? ? forms of the Calabi}Yau 3-fold M. In the large KaK hler class moduli limit (t?PR), only the classical part in the prepotential, which is related to the intersection numbers, survives. To the general form (430) of the prepotential, one can add an extra topological term which is determined by the second Chern class c of the Calabi}Yau 3-fold: F c )J (431) F'' " ?t?; c ) J , c J . ? ? 24 + ? The e!ect of adding such term to the prepotential is the symplectic transformation corresponding to constant shift of the h-angle [630] (cf. the paragraph below (384)). In particular, the type-IIA model dual to the Heterotic string on K3;¹ with n "4 and n"2 (an S¹;< model) [117] corresponds to the compacti"cation on the Calabi}Yau three-fold P (20) [397] with h "4 and Euler number s"!372 [71]. The transformations be tween the moduli in the pair of these type-IIA and heterotic theories are

t";!2?>@>A c ) J # ?>>? , F''(>)"! ?@A 24 6>

(436)

where >?,ZM X?. Note, the prepotential is determined by the classical intersection numbers C "!6d and the expansion coe$cients c ) J "24= of the 2nd Chern class c of the ?@A ?@A ? ? 3-fold. Note, the above form of prepotential can be obtained by imposing the symplectic transformation of the following form on the prepotential without 2nd Chern class terms:

p K q R

"

0

pK

= 1

qR

1

.

(437)

The general expression for the entropy with p"q "0 and = "0 is obtained in (422). By ? ? imposing the symplectic transformation (437), one obtains the following entropy for the type-IIA theory with the prepotential (436) and the charge con"guration p"q "0: ? S "2((q != p ?)d p @p Ap B . ? @AB p

(438)

5.6.3. Higher-dimensional embedding The above D"4 black holes in string theories with the cubic prepotential arise from the compacti"cation of the following intersecting M-brane solution

1 1 du dv#H du# C H?H@HA dx#H?u , ds " ? (C H?H@HA) 6 ?@A ?@A

(439)

due to the duality [635] among the heterotic string on K3;¹, the type-II string on a Calabi}Yau 3-fold (C>) and M-theory on C>;S [103,261,262]. This solution corresponds to 3 M 5-branes (with the corresponding harmonic functions H?, a"1, 2, 3) intersecting over a 3-brane (with the spatial coordinates x), and the momentum (parameterized by the harmonic function H ) #owing along the common string. Here, the intersection of 4-cycles that each M 5-brane wraps around is determined by the parameters C and each pair of such 4-cycles intersect over the 2-dimensional ?@A line element u . ? Compactifying the internal coordinates and the common string direction, one obtains the following D"4 solution with spacetime of the extreme Reissner}Nordstrom black hole:

1 1 dt# ! HC H?H@HA dx . ds"! ?@A 6 (!HC H?H@HA ?@A

(440)


119

6. p-branes The purpose of this section is to review the recent development in p-branes and other higherdimensional con"gurations in string theories. (See also [284] for another review on this subject.) Study of p-branes plays an important part in understanding non-perturbative aspects of string theories in string dualities. The recently conjectured string dualities require the existence of p-branes within string spectrum along with well-understood perturbative string states. Furthermore, microscopic interpretation of entropy, absorption and radiation rates of black holes within string theories involves embedding of black holes in higher dimensions as intersecting p-brane. This section is organized as follows. In Section 6.1, we summarize properties of single-charged p-branes. In Section 6.2, we systematically study multi-charged p-branes, which include dyonic p-branes and intersecting p-branes. In the "nal section, we review the lower-dimensional p-branes and their classi"cation. We also discuss various p-brane embeddings of black holes. 6.1. Single-charged p-branes In this section, we discuss p-branes which carry one type of charge. Such single-charged p-branes are basic constituents from which `bound statea multi-charged p-branes, such as dyonic p-branes and intersecting p-branes, are constructed. p-branes in D dimensions are de"ned as p-dimensional objects which are localized in D!1!p spatial coordinates and independent of the other p spatial coordinates, thereby having p translational spacelike isometries. Note, the allowed values of (D, p) for which supersymmetric p-branes exist are limited and can be determined by the bose-fermi matching condition [2]. (The details are discussed in Section 2.2.3.) The e!ective action for a single-charged p-brane has the form:

1 1 1 d"x (!g R! (R )! e\?N(F , (441) I (p)" N> " 2 2(p#2)! 2i " where i is the D-dimensional gravitational constant, is the D-dimensional dilaton and " F ,dA is the "eld strength of (p#1)-form potential A . Here, the parameter a(p) given N> N> N> below is determined by the requirement that the e!ective action (441) and the p-model action (443) scale in the same way [242] under the rescaling of "elds 2(p#1)(p #1) 2(p#1)(p #1) "4! , a(p)"4! D!2 p#p #2

(442)

where p ,D!p!4 corresponds to spatial dimensions of the dual brane. 6.1.1. Elementary p-branes Electric charge of (p#1)-form potential A in I (p) is carried by the èlementarya p-brane. N> " The èlementarya p-brane has a d-function singularity at the core, requiring existence of singular electric charge source for its support so that equations of motion are satis"ed everywhere. Namely, the electric charge carried by the èlementarya p-brane is a Noether charge with the Noether

120


current associated with the p-brane worldvolume p-model action:

1 (p!1) (!c S "¹ dN>m ! (!ccGHR X+R X,g e?N(N># G H +, N N 2 2

1 ! eG2GN>R X+2R N>X+N>A 2 N> , G G + + (p#1)!

(443)

where X+(mG) (M"0, 1,2, D!1; i"0, 1,2, p) is the spacetime trajectory of p-branes, ¹ is N the elementary p-brane tension, c (m) (g (X)) is the worldvolume (spacetime) metric and GH +, in (443) is related to the canonical metric A "A 2 N> dX2dXN>. The metric g +, N> + + g in the Einstein-frame e!ective action (441) through the Weyl-rescaling g "e?N(N>g . +, +, +, The èlementarya p-brane is a solution to the equations of motion of the combined action I (p)#S . In particular, "eld equation and Bianchi identity of A are " N N> d夹(e\?N(F )"2i (!1)N> 夹J , dF "0 , (444) N> " N> N> where the electric charge source current J "J+2+N>dX 2dX N> is + + N> d"(x!X) J+2+N>(x)"¹ dN>m eG2GN>R X+2R N>X+N> . (445) N G G (!g

Here, 夹 denotes the Hodge-dual operator in D dimensions, i.e. 1 (夹 N> +"\N\ 1 " where SN> surrounds the elementary p-brane. In solving the Euler}Lagrange equations of the combined action I (p)#S to obtain the " N elementary p-brane solution, one assumes the P ;SO(D!p!1) symmetry for the con"guraN> tion. Here, P is the (p#1)-dimensional Poincare group of the p-brane worldvolume and N> SO(D!p!1) is the orthogonal group of the transverse space. Accordingly, the spacetime coordinates are splitted into x+"(xI, yK), where k"0,2, p and m"p#1,2, D!1. Due to the P invariance, "elds are independent of xI, and SO(D!p!1) invariance further requires that N> this dependence is only through y"(d yKyL. In solving the equations, it is convenient to make KL the following static gauge choice for the spacetime bosonic coordinates X+ of the p-brane: Q" N

XI"mI, >K"constant ,

(447)

where k"0, 1,2, p (m"p#1,2, D!1) corresponds to directions internal (transverse) to the p-brane. The general Ansatz for metric with P ;SO(D!p!1) symmetry is N> ds"eWg dxI dxJ#e Wd dyK dyL . (448) IJ KL


121

By solving the Euler}Lagrange equations with these AnsaK tze, one obtains the following solution for the elementary p-brane: ds"f (y)\N >N>N >g dxI dxJ#f (y)N>N>N >d dyK dyL , IJ KL e("e(f (y)\?N, A

I2IN>

e?N( "! e 2 N> f (y)\ , I I Ng

(449)

where Ng is the determinant of the metric g and f (y) is given by IJ

2e?N(i ¹ 1 " N 1# , p '!1 , ( pJ #1)X yNJ > N > f (y)" e?N(i ¹ " N ln y, p "!1 . 1! n

(450)

By solving the Killing spinor equations with the P ;SO(D!p!1) symmetric "eld AnsaK tze, N> one sees that the extreme èlementarya p-brane preserves 1/2 of supersymmetry with the Killing spinors satisfying the constraint: (1!CM )e"0 ,

(451)

where 1 CM , eGG2GNR X+R X+2R NX+NC 2 N , G G G ++ + (p#1)!(!c

(452)

which has properties CM "1 and Tr CM "0 that make (1$CM ) a projection operator. This orig inates from the fermionic i-symmetry of the super-p-brane action. The extreme èlementarya p-brane (449) saturates the following Bogomol'nyi bound for the mass per unit area M " d"\N\y h : N 1 i M 5 "Q "e?N( , " N (2 N

(453)

where h is the total energy-momentum pseudo-tensor of the gravity-matter system. +, 6.1.2. Solitonic p -branes The magnetic charge of A is carried by a solitonic p -brane, which is topological in nature and N> free of spacetime singularity. Since magnetic charges can be supported without source at the core, solitonic p -branes are solutions to the Euler}Lagrange equations of the e!ective action I (p) alone. " is de"ned as The `topologicala magnetic charge P of A N> N

1 P" F , N (2i N> N> 1 "

(454)

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where SN> surrounds the solitonic p -brane. The magnetic charge P is quantized relative to the N electric charge Q via a Dirac quantization condition: N n Q P N N" , n3Z . (455) 2 4n Solitonic p -brane solution has the form: ds"g(y)\N>N>N >g dxI dxJ#g(y)N >N>N >d dyK dyL , IJ KL (456) e("e( g(y)?N, F "(2i P e /X , N> " N N> N> where e is the volume form on SN> and g(y) is given by N> (2e\N>(?NN>N >i P 1 " N . (457) g(y)"1# yN> (p#1)X N> The `solitonica p -brane (456) preserves 1/2 of supersymmetry and saturates the following Bogomol'nyi bound for the ADM mass per unit p -volume: 1 M 5 "P "e\?N( . N (2 N

(458)

Note, the sign di!erence in dependence of the mass densities (453) and (458) on the dilaton asymptotic value . So, in the limit of large , the mass density M (M ) is large (small), and vice N N versa. 6.1.3. Dual theory We consider the theory whose actions II (p ) and SI are given by (441) and(443) with p replaced by " N p "D!p!4. So, in this new theory, p -branes carry electric charge QI and p-branes carry N magnetic charge PI of (p #1)-form potential AI . N N> Since a p-brane from one theory and a p -brane from the other theory are both èlementarya (or `solitonica), it is natural to assume that these branes are dual pair describing the same physics. One assumes that the graviton and the dilaton in the pair actions are the same, but the "eld strengths are related by FI "e\?N( 夹F . (Thereby, the role of "eld equations and F and FI N> N> N> N> Bianchi identities are interchanged.) Then, it follows that since Q "PI and P "QI the Dirac N N N N quantization conditions for electric/magnetic charge pairs (Q , P ) and (QI , PI ) lead to the following N N N N quantization condition for the tensions of the dual pair èlementarya p-brane and p -brane: (459) i ¹ ¹ ""n"p . " N N By performing the Weyl-rescaling of metrics to the string-frame, one sees that the p-brane (p -brane) loop counting parameter g (g ) is N N g "e"\?N(N>, g "e"\?N (N > , (460) N N with a(p )"!a(p). It follows that the brane loop counting parameters of the dual pair are related by (461) (g )N>"1/(g )N > . N N


123

Thus, the strongly (weakly) coupled p-branes are the weakly (strongly) coupled p -branes. In particular, a string (p"1) in D"6 are dual to another string (p "6!1!4"1), thereby strongly (weakly) coupled string theory being equivalent to weakly (strongly) coupled dual string theory. Other interesting examples are membrane/"vebrane dual pair in D"11, string/"vebrane dual pair in D"10 [219,239,240], self-dual 3-branes in D"10 and self-dual 0-branes in D"4. Note, in D"11 supergravity strong and weak coupling limits do not have meaning due to absence of the dilaton. 6.1.4. Blackbranes We discuss non-extreme generalization of BPS `solitonica p -brane in Section 6.1.2. Such solution is obtained [464] by solving the Euler}Lagrange equations following from I (p) with " R;SO(p#3);E(p ) symmetric "eld AnsaK tze. The non-extreme p -brane solution is ds"!D D\N >N>N > dt#D\D?NN>\ dr > \ > \ #rD?NN> dX #DN>N>N > dxG dx , \ N> \ G e\("D?N, F "(p#1)(r r )N>e , (462) \ N> >\ N> where D ,1!(r /r)N> and i"1,2, p . The magnetic charge P and the ADM mass per unit ! ! N p -volume M of (462) are N X X (463) P " N> (p#1)(r r )N>, M " N>[(p#2)rN>!rN>] . >\ N > \ N (2i 2i " " The solution (462) has the event horizon at r"r and the inner horizon at r"r , and therefore is > \ alternatively called `blacka p -brane. The requirement of the regular event horizon, i.e. r 5r , > \ leads to the Bogomol'nyi bound (2M 5"P "e\?N(. N N In the limit r "0, and A become trivial, i.e. P "0, and spacetime reduces to the product \ N> N of (D!p )-dimensional Schwarzschield spacetime and #at RN . In the extreme limit (r "r ), the > \ symmetry is enhanced to that of the BPS p -brane, i.e. P ;SO(p#3), since g "g G G. Such RR VV N> extreme solution is related to the BPS solution (456) through the change of variable yN>"rN>!rN>. In the extreme limit, the event-horizon and the singularity completely disap\ pear, i.e. becomes soliton with geodesically complete spacetime in the region r'r "r . > \ 6.2. Multi-charged p-branes In this section, we discuss p-branes carrying more than one types of charges. These p-branes are `bound statesa of single-charged p-branes. Multi-charged p-branes are classi"ed into two categories, namely `marginala and `non-marginala con"gurations. The `marginala (BPS) bound states have zero binding energy and, therefore, the mass density M is sum of the charge densities Q of the G constituent p-branes, i.e. M" Q . Such bound states with n constituent p-branes preserve at least G G ()L of supersymmetry. The marginal bound states include intersecting and overlapping p-branes. The `non-marginala (BPS) bound states have non-zero binding energy and the mass density of the

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form M" Q. The quantized charges Q of `non-marginala bound states take relatively prime G G G integer values. In general, non-marginal p-brane bound states are obtained from single-charged p-branes or marginal p-brane bound states by applying the S¸(2, Z) electric/magnetic duality transformations and, therefore, preserve the same amount of supersymmetries as the initial p-brane con"gurations (before the S¸(2, Z) transformations). In particular, intersecting p-branes are further categorized into orthogonally intersecting p-branes and p-branes intersecting at angles. 6.2.1. Dyonic p-branes In D"2p#4, i.e. dimensions for which p"p , p-branes can carry both electric and magnetic charges of A . Examples are dyonic black holes (p"0) in D"4; dyonic strings (p"1) in D"6; N> dyonic membranes (p"2) in D"8. Such dyonic p-branes can be constructed by applying the D"2(p#2) S¸(2, Z) electric/magnetic duality transformations on single-charged p-branes. #+ These dyonic p-branes have P ;SO(D!p!1) symmetry and are characterized by one harN> monic function, just like single-charged p-branes, since the S¸(2, Z) transformations leave the Einstein-frame metric intact. In particular, in D"2 mod 4, the (p#2)-form "eld strengths satisfy a real self-duality condition F "!夹F and, thereby, electric and magnetic charges are N> N> identi"ed, i.e. Q "!P . N N The S¸(2, Z) electric/magnetic duality transformations of 2k-form "eld strength F in D"4k #+ I can generally be understood as the ¹ moduli transformations of D"(4k#2) theory compacti"ed on ¹ [311]. We consider the following D"(4k#2) action

I " dI>x (!gR#a [dCH#H 夹H] , I>

(464)

where C is a (4k#2)-form potential with the "eld strength H"dC. We compactify the action (464) on ¹ with the following AnsaK tze for the "elds 1 ("q" dy#2Re q dx dy#dx) , ds(M )"ds(M )# I> I Im q C"B dy#A dx,

H"G dy#F dx ,

(465)

where q is the moduli parameter of ¹, (x, y) are coordinates of ¹ (i.e. x&x#1 and y&y#1), and A, B [F, G] are (2k!1)-forms [2k-forms] in D"4k. By applying the self-duality condition [625] of H, one "nds that 2k-form G is an auxiliary "eld, which can be eliminated by its "eld equation as G"Im q 夹F!Re q F, and one "nds that F" dA. The "nal expression for the D"4k action is

1 dq dq #a FG , I " dIx (!g R! I 2 (Im q)

(466)

where a real constant a is, in the convention of [388], given by a"2[(2k)!]\. The complex scalar q, which is expressed in terms of real scalars o and p as q"2o#ie\N, parameterizes the target space manifold M"S¸(2, Z)!S¸(2, R)/;(1), which is the fundamental domain of S¸(2, Z) in the upper half q complex plane. Here, ;(1) is a subgroup of S¸(2, R) which preserves the vacuum


125

expectation value 1q2. The "eld equations are invariant under the following S¸(2, R) electric/ #+ magnetic duality transformation of F:

a b aq#b ; A" 3S¸(2, R) . (F, G)P(F, G)A\, qP cq#d c d

(467)

Note, this S¸(2, Z) transformation is not S-duality of string theories, since p is not the D"4k #+ dilaton. (In (464), the dilaton is set to zero.) The following electric Q and magnetic P charge densities form an S¸(2, R) doublet: #+ 1 1 G, P" F. (468) Q" X X I I This S¸(2, R) transformation on single-charged (2k!2)-branes yields (2k!2)-branes which #+ carry both electric and magnetic charges of A . Such dyonic p-branes preserve 1/2 of superI\ symmetry. Charges (Q, P) and (Q, P) of two dyonic (2k!2)-branes satisfy the generalized Nepomechie-Teitelboim quantization condition [482,591]:

QP!QP3Z .

(469)

When such dyonic (2k!2)-branes are uplifted to D"(4k#2) through (465), the solutions become self-dual (2k!1)-branes [230,238]. The electric and magnetic charges of the dyonic (2k!2)-branes are interpreted as winding numbers of the self-dual (2k!1)-branes around the x and y directions of ¹. The dyonic (2k!2)-branes (k"1, 2) uplifted to D"11 describe p-brane which interpolates between the M 2-brane and the M 5-brane, i.e. a membrane within a 5-brane (2"2 , 5 ). + + In the following, we speci"cally discuss the k"2 case [311,388]. The associated action ((466) with k"2) is type-IIB e!ective action (155) consistently truncated and compacti"ed to D"8. The N"2, D"8 supergravity has an S¸(3, R);S¸(2, R) on-shell symmetry, whose S¸(3, Z);S¸(2, Z) subset is the conjectured U-duality symmetry of D"8 type-II string. The R-R 4-form "eld strength and its dual "eld strength transform as (1, 2) under S¸(3, R);S¸(2, R). This U-duality group contains as a subset the SO(2, 2, Z),[S¸(2, Z);S¸(2, Z)]/Z T-duality group. The S¸(2, Z) factor (in S¸(3, Z);S¸(2, Z)) is the electric/magnetic duality (467), which is a subset of `perturbativea T-duality group. All the `non-perturbativea transformations are contained in the S¸(3, Z) factor. The following dyonic membrane solution with 1q2"i is obtained by applying the ;(1)L S¸(2, R) transformation with a parameter m to purely magnetic membrane: #+ ds "H\ ds(M)#H ds(E) , F " cos m(夹 dH)# sin m dH\e(M) , sin(2m)(1!H)#2iH , (470) q" 2(sinm#H cosm) where ds(M) [ds(E)] is the metric of D"3 Minkowski space M (the "ve-dimensional Euclidean space E), e(M) is the volume form of M, 夹 is the Hodge-dual operator in E and H is a harmonic function given by (457) with p"2. Here, ;(1) is the subgroup of S¸(2, R) that #+ preserves 1q2"i. In the quantum theory, the ;(1) group breaks down to Z due to Dirac

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quantization condition, resulting in either electric or magnetic solutions when applied to purely magnetic solution. (But the solution in (470) with an arbitrary m satis"es the Euler}Lagrange equations following from (466) and, therefore, can be taken as an initial solution to which the ìnteger-valueda duality transformations are applied.) To generate dyonic solutions with an arbitrary 1q2 and are relevant to the quantum theory, one has to apply the full S¸(2, R) transformation to (470). The pair of electric and magnetic charges of #+ such dyonic solutions take co-prime integer values. The ADM mass density of this extreme dyonic membrane saturates the Bogomol'nyi bound: M5[e6N7(Q#21o2P)#e\6N7P] ,

(471)

and therefore 1/2 of supersymmetry is preserved. When uplifted to D"10, this dyonic membrane becomes the following self-dual 3-brane of type-IIB theory [238]: ds "H\[ds(M)#dv]#H[ds(E)#du] , (472) F "( 夹 dH) du# dH\e(M) dv , where the coordinates (u, v) are related to the coordinates (x, y) in (465) through (y, x)"(v, u)A\ (A3S¸(2, Z)). The electric and magnetic charges of the above D"8 dyonic membrane are respectively interpreted as the winding numbers of this D"10 self-dual 3-brane around x and y directions of ¹. The D"8 dyonic membrane (470) uplifted to D"11 is a special case of orthogonally intersecting M-brane interpreted as a membrane within a 5-brane (2"2 , 5 ): + + ds "H(sinm#H cosm)[H\ ds(M)#(sinm#H cosm)\ ds(E)#ds(E)] , 1 1 3 sin 2m F" cos m 夹 dH# sin m dH\e(M)# dHe(E) . (473) 2 2 2[sin m#H cos m] This M-brane bound state interpolates between the M 5-brane and the M 2-brane as m is varied from 0 to p/2. As long as magnetic charge is non-zero (mO0), (473) is non-singular, thereby singularity of the D"8 dyonic membrane (470) is resolved by its interpretation in D"11. By compactifying an extra spatial isometry direction of (473) on S, one obtains 3 di!erent types of dyonic p-branes in type-IIA theory: (i) a membrane within a 4-brane (2"2, 4), (ii) a membrane within a 5-brane (2"2, 5) and (iii) a string within a 4-brane (1"1, 4). One can construct dyonic p-branes in DO2(p#2) by compactifying purely electric or purely magnetic p-branes down to D"2(p#2) (or D"2(p #2)), applying electric/magnetic duality transformations of p-branes (or p -branes) in D"2(p#2) (or D"2(p #2)), and then uplifting the

The above procedure can be applied to intersecting p-branes to generate p-brane bound states which interpolate between di!erent intersecting p-branes, e.g. the interpolation between the intersecting two (p#2)-branes and the intersecting two p-branes; the interpolation between the intersecting p-brane and (p#2)-brane and the intersecting (p#2)-brane and p-brane [151].


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dyonic solution to the original dimensions. In particular, the Z subset of the D"2(p#2) (or D"2(p #2)) electric/magnetic duality transformations relates electric p-brane and magnetic p-brane. Thus, the elementary p-brane in DO2(p#2) is interpreted as the electrically charged partner of magnetic p-brane, establishing electric/magnetic duality between electric èlementarya p-brane and magnetic `solitonica p-brane in DO2(p#2). To enlarge this Z electric/magnetic duality symmetry in D"2(p#2) (or D"2(p #2)) to the S¸(2, Z) symmetry so that one can generate dyonic p-branes from single-charged p-branes, one has to turn on a pseudo-scalar "eld. For example, the dyonic 5-brane in D"10 type-IIB theory is constructed in [72] by applying the S¸(2, Z) ;S¸(2, Z) transformation of the truncated type-IIB theory in D"6. (This '' #+ SO(2, 2),S¸(2, Z) ;S¸(2, Z) group is a subgroup of SO(5, 5) U-duality group of type-II string '' #+ on ¹.) There, it is found out that non-zero R-R "elds (which are related to the pseudo-scalar "eld) are needed for the solution to have both electric and magnetic charges. The type-IIB S¸(2, Z) S-duality transformation leads to dyonic p-brane whose electric and '' magnetic charges coming from di!erent sectors (NS-NS/R-R) of string theory. General dyonic solutions where form "elds carry both electric and magnetic charges are generated by additionally applying the S¸(2, Z) electric/magnetic transformation in D"6 [72]. In particular, the elec#+ tric/magnetic duality transformation that relates the solitonic 5-brane and elementary 5-brane is the product of (Z ) and (Z ) transformations. Such dyonic 5-brane solutions preserve 1/2 of '' #+ supersymmetry. We comment on generalization of dyonic p-branes discussed in this section. In [230], general dyonic p-branes within consistently truncated heterotic string on ¹, where truncated moduli "elds are parameterized by 3 complex modulus parameters ¹G (i"1, 2, 3) of 3 ¹ in ¹, is constructed. Thereby, the O(6, 22, Z) T-duality symmetry of heterotic string on ¹ is broken down to S¸(2, Z). The general class of multi-charged p-brane solution is then characterized by harmonic functions each associated with ¹G and the dilaton}axion scalar S. Such solutions break more than 1/2 of supersymmetry. With trivial S, ()L of supersymmetry is preserved for n non-trivial ¹G. With non-trivial S, additional 1/2 of supersymmetries is broken unless all of ¹G are non-trivial. With all ¹G non-trivial, () or none of supersymmetry is preserved, depending on the chirality choices. In particular, a special case of general class of solution with S and only one of ¹G non-trivial corresponds to generalization of D"6 self-dual dyonic string, when such solution is uplifted to D"6. This dyonic solution is parameterized by 2 harmonic functions, which are respectively associated with electric and magnetic charges of 2-form potential. In the self-dual limit, i.e. when the electric and magnetic charges are equal, the solution becomes the D"6 self-dual dyonic string. Within the context of non self-dual theory, the solution preserve only 1/4 of supersymmetry, whereas as a solution of self-dual theory it preserve 1/2 of supersymmetry. 6.2.2. Intersecting p-branes 6.2.2.1. Constituent p-branes. Before we discuss intersecting p-branes, we summarize various single-charged p-branes in D"10, 11, which are constituents of intersecting and overlapping p-branes. These p-branes are special cases of èlementarya p-branes and `solitonica p -branes discussed in Section 6.1. They are characterized by a harmonic function H(y) in the transverse space (with coordinates y ,2,y ) and break 1/2 of supersymmetry. D"11 supergravity has 3-form N> "\ potential. So, the basic p-branes (called M p-branes) are an electric èlementarya membrane and

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a magnetic `solitonica "vebrane: ds"HN>[H\(!dt#dx#2#dx)#(dy #2#dy )] , N N N N> c F ?"! R ?H\, a"3,2,10 (for p"2) , RV V W 2 W N c F ?2 ?" e 2 R ?H , a "6,2, 10 (for p"5) , W W G 2 ? ? W N

(474)

where c"1(!1) for (anti-) branes and harmonic function cQ N N H "1# N " y!y "\N is for M p-brane located at the +0, 1,2, p, hyperplane at yG"yG . The Killing spinor e of these M p-branes satis"es the following constraint: (475) C 2 e"ce , N where C 2 ,C C 2C is the product of #at spacetime gamma matrices associated with the N N worldvolume directions. In D"10, there are 3 types of p-branes, depending on types of charges that p-branes carry. The electric charge of NS-NS 2-form potential is carried by NS-NS strings (or fundamental strings): ds"H\(!dt#dx)#dy#2#dy, e("H\ . (476) The magnetic charge of NS-NS 2-form potential is carried by NS-NS 5-branes (or solitons): ds"!dt#dx#2#dx#H(dy#2#dy), e("H . The charges of R-R (p#1)-form potentials are carried by R-R p-branes:

(477)

ds"H\(!dt#dx#2#dx)#H(dy #2#dy) , N N> (478) e("H\N\, F 2 N G"R GH\ , W RV V W where p"0, 2, 4, 6 (p"1, 3, 5, 7) for R-R p-branes in type-IIA (type-IIB) theory. These R-R p-branes of the e!ective "eld theory are long distance limit of D p-branes in type-II superstring theories [193]: the transverse (longitudinal) directions of R-R p-branes correspond to coordinates with Dirichlet (Neumann) boundary conditions. The Killing spinors of D"10 p-branes satisfy one constraint. The left-moving and the rightmoving Majorana}Weyl spinors e and e have the same (opposite) chirality for the type-IIB * 0 (type-IIA) theory, i.e. C e "e [C e "e and C e "!e ]. So, spinor constraints are *0 *0 * * 0 0 di!erent for type-IIA/B theories: E E E E

NS-NS strings: e "C e , e "!C e * * 0 0 IIA NS-NS "vebranes: e "C 2 e , e "C 2 e 0 * * 0 IIB NS-NS "vebranes: e "C 2 e , e "!C 2 e 0 * * 0 R-R p-branes: e "C 2 e . * N 0


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Whereas in type-II superstring theories there are D p-branes with p"!1, 0,2, 9, R-R p-branes in massless e!ective "eld theories cover only range p46. So, there is no place in the massless type-II supergravities for R-R 7- and 8-branes. Although the R-R 7-brane in type-IIB theory can be related to 8-form dual of pseudo-scalar of type-IIB theory [293], it cannot be T-dualized to R-R p-branes in type-IIA theory since it is speci"c to the uncompactixed type IIB theory, only. In [76], it is proposed that D p-branes of type-II superstring theories with p'6 can be realized as R-R p-branes of massive type-II supergravity theories. In the following, we discuss R-R 7- and 8-branes in some detail, since their properties and T-duality transformation rules are di!erent from other R-R p-branes. The R-R 8-brane is coupled to 9-form potential. The introduction of 10-form "eld strength into the theory does not increase the bosonic degrees of freedom (therefore, is not ruled out by supersymmetry consideration), but leads to non-zero cosmological constant. In fact, the massive type-IIA supergravity constructed in [204] contains such cosmological constant term. It is argued [498] that the existence of the massive type-IIA supergravity with the cosmological constant is related to the existence of the 9-form potential of type-IIA theory. In [76], new massive type-IIA supergravity is formulated by introducing 9-form potential A whose 10-form "eld strength F "10dA is interpreted as the cosmological constant once Hodge-dualized. (Note, the cos mological constant m in the massive type-IIA supergravity is independent of the dilaton in the string-frame, which is typical for terms in R-R sector.) In this new formulation, the cosmological constant m is promoted to a "eld M(x) by introducing A as a Lagrange multiplier for the constraint dM"0 that M(x)"m is a constant: the "eld equation for M(x) simply determines the new "eld strength F , while the "eld equation for A implies that M(x)"m. It is conjectured in [154] that the massive type-IIA supergravity is related to hypothetical H-theory [44] in D"13 through the Scherk}Schwarz type dimensional reduction (see below for the detailed discussion), rather than to M-theory. The conjectured type-IIB Sl(2, Z) duality requires the pseudo-scalar s to be periodically identi"ed (s&s#1), which together with T-duality between massive type-IIA and type-IIB supergravities implies that the cosmological constant m is quantized in unit of the radius of type-IIB compacti"cation circle, i.e. m"n/R (n3Z). The massive type-IIA supergravity admits the following 8-brane (or domain wall) as a natural ground state solution: ds"H\g dxI dxJ#H dy, e\("H , (479) IJ and the Killing spinor e satis"es one constraint CM e"$e. The form of harmonic function H(y), W which is linear in y, depends on M(x). When M(x)"m everywhere, H"m"y!y ", with a kink singularity at y"y . When M(x) is locally constant, the corresponding solution is interpreted as a domain wall separating regions with di!erent values of M (or cosmological constant). An example is H"!Q y#b [H"Q y#b] for y(0 [y'0], where 8-brane charges Q are \ > ! de"ned as the values of M as yP$Rand b is related to string coupling constant e( at the 8-brane core. This 8-brane solution is asymptotically left-#at (right-#at) when Q "0 (Q "0). The multi \ > 8-brane generalization can be accomplished by allowing kink singularities of H at ordered points y"y (y (2(y . L R-R 9-brane of type-IIB theory is interpreted as D"10 spacetime.

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The R-R 8-brane of massive type-IIA supergravity can be interpreted as the KK 6-brane of D"11 supergravity [76]. Namely, after the R-R 8-brane is compacti"ed to 6-brane in D"8, the 6-brane can be lifted as the R-R 6-brane of massless type-IIA supergravity, which is interpreted as the KK monopole of M-theory on S [601]. Under the T-duality, the R-R 8-brane is expected to transform to R-R 7-brane or 9-brane in type-IIB theory. First, T-duality transformation of massless type-II supergravity along a transverse direction of the R-R 8-brane leads to the product of S and D"9 Minkowski spacetime, which is 9-brane. (Note, the direct dimensional reduction requires H to be constant.) Second, T-duality transformation leading to type-IIB 7-brane is much involved and we discuss in detail in the following. T-duality transformation involving massive type-IIA supergravity requires construction of D"9 massive type-IIB supergravity. D"9 massive type-IIB supergravity is obtained from massless type-IIB supergravity through the Scherk}Schwarz type dimensional reduction procedure [528], i.e. "elds are allowed to depend on internal coordinates. This is motivated by the observation that the `StuK ckelberga type symmetry, which "xes the m-dependence of "eld strengths in type-IIA supergravity, is realized within type-IIB supergravity as a general coordinate transformation in the internal direction, which requires some of R-R "elds to depend on the internal coordinates. Namely, an axionic "eld s(x, z) (i.e. R-R 0-form "eld) is allowed to have an additional linear dependence on the internal coordinate z, i.e. s(x, z)Pmz#s(x), where x is the lowerdimensional coordinates. Since s appears always through ds in the action, the compacti"ed action has no dependence on the internal coordinate z. The result is the massive supergravity with cosmological term. The massive type-IIA supergravity compacti"ed on S through the standard KK procedure is related via T-duality to the massless type-IIB supergravity compacti"ed on S through the Scherk}Schwarz procedure. This massive T-duality transformation generalizes those of massless type-II supergravity in [81]. The explicit expression for m-dependent correction to the T-duality transformation can be found in [76]. Under the massive T-duality, the type-IIA 8-brane transforms to the type-IIB 7-brane, which is the "eld theory realization of D 7-brane of type-IIB superstring theory. By applying the T-duality of massless type-II supergravities to this 7-brane, one obtains a 6-brane of type-IIA theory, whereas transformation to 8-brane of type-IIA theory requires the application of massive T-duality. This generalized compacti"cation Ansatz for s is a special case of the generalized compacti"cation on a d-dimensional manifold M where an (n!1)-form potential A (n4d) for which nth B L\ cohomology class HL(M , R) of M is non-trivial is allowed to have an additional linear dependence B B on the (n!1)-form u (z), i.e. A (x, z)"mu (z)#standard terms [154,443]. (All the other L\ L\ L\ "elds are reduced by the standard KK procedure.) Here, du represents non-trivial nth cohomolL\ ogy of M . (In the case of compacti"cation of type-IIB theory on S, the cohomology dz is the B volume form on S.) Since A appears in the action always through dA , the lower-dimenL\ L\ sional action depends on A only through its zero mode harmonics on M , only. The constant L\ B m manifests in lower dimensions as a cosmological constant and, thereby, the compacti"ed D-dimensional action admits domain wall, i.e. (D!2)-brane, solutions. The general pattern for mass generation is as follows. First, the KK vector potentials always become massive. Second, a "eld that appears in a bilinear term in the Chern}Simons modi"cation of a higher rank "eld strength acquires mass if it is multiplied by A (with general dimensional reduction Ansatz). L\


131

Third, when axionic "eld AGHI associated with D"11 3-form potential A is used for the +,. Scherk}Schwarz reduction, the lower-dimensional theory contains a topological mass term. In these mechanisms, the "elds associated with the StuK ckelburg symmetry (under which the eaten "elds undergo pure non-derivative shift symmetries) get absorbed by other "elds to become mass terms for the potentials that absorb them. The consistency of the theory requires that the "elds that are eaten should not appear in the Lagrangian. Note, whereas the original Scherk}Schwarz mechanism [528] is designed to give a mass to the gravitino, thus breaking supersymmetry, and do not generate scalar potentials, in our case a cosmological constant is generated and the full supersymmetry is preserved. In addition to single-charged p-branes, there are other supersymmetric con"gurations which are basic building blocks of p-brane bound states. These are the gravitational plane fronted wave (denoted 0 ), called `pp-wavea, and the KK monopole (denoted 0 ). Their existence within p-brane U K bound states are required by duality symmetries. First, the KK monopole in D"11 is introduced [601] in an attempt to give D"11 interpretation of type-IIA D 6-brane [367]. The KK monopole is the magnetic dual of the KK modes of D"11 theory on S, which is identi"ed as R-R 0-brane (electrically charged under the KK ;(1) gauge "eld) [635]. The D"11 KK monopole, which preserves 1/2 of supersymmetry, has the form [601]: ds "!dt#dy ) dy#P N>P N>Q IJ H N>Q ? H Q>P # N>Q (dx?)#(H H )d dyG dyH , N>P N>Q GH H N>P ?Q> e\("HN>P\HN>Q\ , N>P N>Q F 2 Q G"R GH\ , F Q>2 Q>P G"R GH\ . (493) V W RV V W W N>Q RV W N>P We discuss intersecting p-branes which contain NS-NS p-branes. First, (0"1 , p ) with 04p48, ,1 0 is nothing but open strings that end on D-brane. This type of con"gurations can be obtained by "rst compactifying (0"2 , 2 ) on S along a longitudinal direction of one of M 2-brane (resulting in + + (0"1 , 2 )), and then by sequentially applying T-duality transformations along the directions ,1 0 transverse to the NS-NS string. Second, (p!1"5 , p ) with 14p46, are interpreted as D p-brane ,1 0 ending on NS-NS 5-brane. Namely, NS-NS 5-branes act as a D-brane for D-branes. This interpretation is consistent with the observation [46] that M 5-branes are boundaries of M 2-branes. This type of intersecting branes can be constructed by "rst compactifying (1"2 , 5 ) on S along an + + overall transverse direction (resulting in (1"5 , 2 )), and then by sequentially applying T-duality ,1 0 transformations along the longitudinal directions of the NS-NS 5-brane. Third, intersecting NS-NS p-branes can be obtained in the following ways: (i) compacti"cation of (3"5 , 5 ) along an + + overall transverse direction leads to (2"5 , 5 ), (ii) compacti"cation of (1"2 , 5 ) along a relative ,1 ,1 + + transverse direction which is longitudinal to M 2-brane leads to (1"2 , 5 ), (iii) the type-IIB ,1 ,1 S-duality on (!1"1 , 1 ) yields (!1"1 , 1 ). 0 0 ,1 ,1 We comment on the case some or all of harmonic functions depend on the relative transverse coordinates [75]. These types of intersecting p-branes can be constructed by applying the general harmonic superposition rules, taking into account of dependence of harmonic functions on the relative transverse coordinates. In particular, the metric components associated with the relative coordinates (that harmonic functions depend on) have to be the same so that the equations of motion are satis"ed. First, the second type of intersecting p-branes, i.e. one harmonic function depends on the relative transverse coordinates, can be constructed from the "rst type of intersecting p-branes, i.e. all the harmonic functions depend on the overall transverse coordinates, by letting one of harmonic functions depend on the relative transverse coordinates. Thus, the classi"cation of the second type is the same as that of the "rst type. The third type of p-branes, i.e. all the harmonic functions depend on the relative transverse coordinates, have 8 relative transverse coordinates (n"8) for a pair of p-branes. It is impossible to have more than two p-branes with each pair having n"8. In D"11, the only con"guration of the third type is (1"5 , 5 ) [287]. This M-brane + + preserves () of supersymmetry, since the Killing spinor satis"es two constraints of the form (475), each corresponding to a constituent M 5-brane. By compactifying an overall transverse direction of

M 2-brane with its longitudinal coordinates given by the overall longitudinal and overall transverse coordinates of (1"5 , 5 ) can be further added without breaking any more supersymmetry. The added M 2-brane intersects the M 5+ + branes over strings and is interpreted as an M 2-brane stretched between two M 5-branes.


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(1"5 , 5 ) on S, one obtains (1"5 , 5 ), which was "rst constructed in [423]. Further application + + ,1 ,1 of the type-IIB SL(2, Z) transformation leads to (1"5 , 5 ). 0 0 The series of application of T-duality transformations, then, yield a set of overlapping 2 R-R p-branes with n"8. (Complete list can be found in [287].) These overlapping p-branes correspond, at string theory level, to D-brane bound states with 8 ND or DN directions, and therefore should be supersymmetric. When 2 R-R p-branes intersect in a point, one can add a fundamental string without breaking any more supersymmetry. This type of con"gurations is interpreted as a fundamental string stretching between two D-branes. For the case where 2 R-R p-branes intersect in a string, one can add pp-wave along the string intersection without breaking anymore supersymmetry. Another third type of intersecting p-branes in D"10 can be constructed by compactifying (1"5 , 5 ) along a relative transverse direction, resulting in (1"4 , 5 ), followed by series of + + 0 ,1 T-duality transformations along the longitudinal directions of the NS-NS 5-brane, resulting in (p!3"p , 5 ) with 34p48. One can further add R-R (p!2)-branes to these con"gurations; 0 ,1 these con"gurations are interpreted as a D (p!2)-brane stretching between D p-brane and NS-NS 5-brane. 6.2.2.3. Other variations of intersecting p-branes. So far, we discussed intersecting p-branes with p!p"0 mod 4. Existence of such classical intersecting p-branes that preserve fraction of supersymmetry is expected from the perturbative D-brane argument [501,504]. One can construct such solutions by applying the harmonic superposition rules discussed in the above. In this subsection, we discuss another type of p-brane bound states which do not follow the intersection rules discussed in the previous subsection. Such p-brane bound states contain pair of constituent p-branes with p!p"2 and still preserve fraction of supersymmetry. These p-brane con"gurations can be generated by applying dimensional reduction or T-duality along a direction at angle with a transverse and a longitudinal directions of the constituent p-branes [96,519,152]. (Hereafter, we call them as `tilteda reduction and `tilteda T-duality.) These p-brane con"gurations can also be constructed by applying òrdinarya dimensional reduction and sequence of òrdinarya duality transformations on (2"2 , 5 ) (473), which preserves 1/2 of supersymmetry. + + In fact, it con#icts with di!eomorphic invariance of the underlying theory that one has to choose speci"c directions (which are either transverse or longitudinal to the constituent p-branes) for dimensional reduction or T-duality transformations [152]. So, the existence of such new p-brane con"gurations is required by (M-theory/IIA string and IIA/IIB string) duality symmetries and di!eomorphism invariance of the underlying theories. We now discuss the basic rules of `tilteda T-duality and `tilteda dimensional reduction on constituent p-branes. Before one applies `tilteda T-duality and dimensional reduction to p-branes, One can further add a fundamental string along the string intersection without breaking any more supersymmetry. T-duality along the fundamental string direction leads to type-IIB (1"5 , 5 ) with pp-wave propagating along the string ,1 ,1 intersection. Note, the former con"guration preserves only 1/8 of supersymmetry, rather than 1/4, if regarded as a solution of type-IIB theory. It is argued [501,504] that D-brane bound state with p!p"6 is not supersymmetric and is unstable due to repulsive force. Although a solution that may be interpreted as (0"0 , 6 ) is constructed in [152], its interpretation is 00 00 ambiguous due to abnormal singularity structure of harmonic functions, and it cannot be derived from (2"2 , 5 ) though + + a chain of T-duality transformations.

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one rotates a pair of a longitudinal x and a transverse y coordinates of a (constituent) p-brane by an angle a (a di!eomorphism that mixes x and y):

x

"

y

cos a !sin a

x

sin a

y

cos a

.

(494)

(Note, x and y have to be di!eomorphic directions so that one can compactify these directions on S.) Then, one compacti"es or applies T-duality along the x-direction. These procedures preserve supersymmetry. When a"0 [a"p/2], such compacti"cation or T-duality transformation is the compacti"cation or T-duality transformation along a longitudinal direction x (a transverse direction y). Thus, as the angle a is varied from 0 to p/2, the resulting bound state interpolates between the corresponding two limiting con"gurations. First, we discuss the `tilteda reduction R of solutions in D"11. The `tilteda reduction on ? M-branes leads to type-IIA p-brane bound states, interpreted as `brane within branea: 0? (4"4 , 5 ) , 0? (1"1 , 2 ) , 5 P (495) 2 P ,1 0 + 0 ,1 + where the subscript A means type-IIA con"guration. From 0 and 0 in D"11, one obtains the K U following type-IIA bound states: 0? (0 "6 ) , 0 P 0? (0 "0 ) , 0 P (496) K U 0 U K 0 where (0 "0 ) is interpreted as a `boosteda D 0-brane. U 0 Second, we discuss `tilteda T-duality transformations ¹ . The o!-diagonal metric component ? g induced by the coordinate rotation (494) is transformed to the same component B of 2-form VYWY VYWY potential under T-duality. So, the T-transformed solution has diagonal metric and non-zero F "B #2paF , where F is the worldvolume gauge "eld strength [445]. For R-R p-brane IJ IJ IJ IJ bound states, the corresponding perturbative D-brane now, therefore, satis"es the modi"ed boundary condition R XI!iFIR XJ"0. The induced #ux F is related to a as F "!tan a. L J J VYWY VYWY The ADM mass of the transformed con"guration is of the form M&Q#Q, a characteristic of non-threshold bound states. First, the `tilteda T-duality on type-IIA/B p-branes leads to the following type-IIB/A bound states: 2? (0 "5 ), (p#1) P 2? (p"p , (p#2) ) , 2? (0 "1 ), 5 P (497) 1 P U ,1 ,1 K ,1 0 0 0 ,1 where (0 "1 ) is simply a boosted fundamental string. The existence of the D-brane bound states U ,1 (p"p , (p#2) ) preserving 1/2 of supersymmetry is also expected from the perturbative D-brane 0 0 considerations. One can apply `tilteda T-duality transformation more than once to obtain new p-brane bound state con"gurations. For example, by applying `tilteda T-duality transformations to D 2-brane in two di!erent directions, one obtains D (4, 2, 2, 0)-brane bound state [96]. Second, `tilteda T-duality on 0 and 0 in type-IIA/B theory yields the following type-IIB/A bound states: K U 2? (0 "1 ) . 2? (0 "5 ), 0 P (498) 0 P K ,1 U U ,1 K Next, we discuss p-brane bound states obtained by "rst imposing a Lorentz boost along a transverse direction and then applying T-duality transformation or reduction along the direction of the boost: T-duality along a boost and reduction along a boost, respectively denoted as ¹ and R . T T The Lorentz boost yields non-threshold bound state of a p-brane and pp-wave. This bound state


141

interpolates between extreme p-brane and plane wave, as the boost angle a (see below for its de"nition) varies from zero (no boost) to p/2 (in"nite boost). The ADM energy E of such non-threshold bound state of extreme p-brane (with mass M) and pp-wave (with momentum p) has the form E"M#p, a reminiscent of relativistic kinematic relation of a particle of the rest mass M with the linear momentum p, rendering the interpretation of such bound state as `boosteda p-brane. The Lorentz boost with velocity v/c"sin a along a transverse direction y has the form [519]: tPt"(cos a)\(t#y sin a), yPy"(cos a)\(y#t sin a) .

(499)

In the above, the angle a is related to the boost parameter b as cosh b"1/cos a. After the Lorentz boost (499), one compacti"es or applies T-duality transformation along y. First, we discuss the reduction R along a boost. Since the momentum of pp-wave manifests as T the KK electric charge after reduction along the direction of momentum #ow, the resulting bound state always involves R-R 0-brane. The following type-IIA bound states are obtained from the compacti"cation with a boost of con"gurations in D"11: 0T (0"0 , 5 ) , 0T (0"0 , 2 ) , 5 P 2 P 0 0 + 0 ,1 + 0T (0 "0 ) , 0 P 0T (0"0 , 6 ) . 0 P U 0 U K 0 0

(500)

Second, we discuss the T-duality ¹ along a boost. Under the T-duality, the linear momentum of T a pp-wave is transformed to the electric charge of the NS-NS 2-form potential [359]. So, the T-dualized con"gurations always involve a fundamental string with non-zero winding mode. We have the following type-IIB/A bound states from type-IIA/B con"gurations: 2T 1 , 1 P ,1 ,1

2T (0 "1 ), p P 2T (1"1 , (p#1) ) , 5 P ,1 K ,1 0 ,1 0

2T (0 "1 ), 0 P 2T (0 "1 , 5 ) . 0 P U U ,1 K K $ ,1

(501)

The di!eomorphic invariance and duality symmetries require new type of bound states in M-theory that preserve 1/2 of supersymmetry. These new M-theory con"gurations can be constructed by uplifting new type-IIA con"guration discussed in this subsection. For example, by uplifting (0 "5 ) or (4"4 , 6 ) , one obtains the M 5-brane and the KK monopole bound state in K ,1 0 0 D"11. Another example is the M 2-brane and the KK monopole bound state uplifted from (1"1 , 6 ) or (0 "1 ) . ,1 0 K ,1 This boost angle a can be identi"ed with the angle a of coordinate rotation in (494). Namely, the non-threshold typeIIA (q , q ) string bound state obtained from D"11 pp-wave through tilted dimensional reduction at an angle a, followed by T-duality, can also be obtained by reduction along a boost of M 2-brane with the same boost angle a, followed by T-duality transformation. One can straightforwardly apply this procedure to intersecting M-branes, followed by sequence of T-duality transformations, to construct p-brane bound states that interpolate between those that preserve 1/4 of supersymmetry and those that preserve 1/2 of supersymmetry as a is varied from 0 to p/2 [151].

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Non-threshold type-IIB (q , q ) string, obtained from R-R or NS-NS string through S¸(2, Z) duality, can be related to D"11 pp-wave compacti"ed at angle [519]. This is understood as follows. When the compacti"cation torus ¹ (parameterized by isometric coordinates (x, y)) is rectangular, the angle a of coordinate rotation de"nes the direction of (q , q ) cycle in ¹, around which the D"11 pp-wave is wrapped, as cos a"q /(q#q. (So, choice of di!erent angle a corresponds to di!erent choice of a cycle in ¹.) Starting from pp-wave propagating along x, one performs coordinate rotation (494) in the plane (x, y), where y is an isometric transverse direction of the pp-wave, and then compacti"es along y-direction, which is the direction of (q , q )-cycle of ¹ with coordinates (x, y). The resulting type-IIA con"guration is a non-threshold bound state of pp-wave and D 0-brane. Subsequent T-duality transformation along y leads to type-IIB (q , q ) string solution. This is related to the fact that the type-IIB S¸(2, Z) symmetry is the modular symmetry of the D"11 supergravity on ¹ [81,538,539] (see Section 3.7 for detailed discussion). Had we started from the bound state of M 2-brane and pp-wave along a longitudinal direction of the M 2-brane, we would end up with boosted type-IIB (q , q ) string that preserves 1/4 of supersymmetry. (The M 2-brane charge is, therefore, interpreted as a momentum of the type-IIB (q , q ) string.) On the other hand, the non-threshold type-IIB (q , q ) 5-brane can be related to M 5-brane. Namely, one "rst compacti"es M 5-brane at an angle to obtain (4"4 , 5 ) and then applies 0 ,1 T-duality transformation along the relative transverse direction to obtain type-IIB (q , q ) 5-brane bound state. Following the similar procedures, one can obtain the non-threshold type-IIB (q , q ) string from M 2-brane. 6.2.2.4. Branes intersecting at angles. In [89], it is shown that one can construct BPS D-brane bound states where the constituent D-branes intersect at angles other than the right angle. We "rst summarize formalism of [89]. Then, we discuss the corresponding classical solutions [33,55,97,151,285] in the e!ective "eld theory. In the presence of a D p-brane, two spinors e and e (corresponding to N"2 spacetime supersymmetry) of type-II string theory satisfy the constraint: N e "C e " : eIC e , (502) N G I G where e is an orthonormal frame spanning D p-brane worldvolume. For N numbers of constituent G D-branes, it is convenient to de"ne the following raising and lowering operators from gamma matrices: !iC ), aI"(C #iC ), k"1,2, N , aR"(C I I\ I I I\ which satisfy the anticommutation relations:

(503)

+aH, aR,"dH , +aH, aI,"0"+aR, aR, . (504) I I H I The lowering operators aI de"ne the `vacuuma "02, satisfying aI"02"0. Under an S;(N) rotation zGPRG zH of the complex coordinates zI,x?I#ix@I spanned by D p-branes, the raising and the H lowering operators transform as aIPRIaH, aRPRR HaR . H I I H

(505)


143

One can construct intersecting D p-branes at angles in the following way. One starts from two D p-branes oriented, say, along the directions Re zG and applies S;(p) rotation zGPRG zH H to one of D p-branes. For this type of intersecting D p-branes at angles, the spinor e satis"es the constraint: N N (aR#aI)e" (RR HaR#RIaH)e . I I H H I I

(506)

So, the resulting con"guration has unbroken supersymmetries "02 and N aR"02. These two I I spinors have the same (opposite) chirality for p even (odd). One can further compactify this intersecting D p-branes at angles on a torus and then apply T-duality transformations to obtain other types of intersecting D p-branes at angles. Alternatively, one can start from (q"(p#q) , (p#q) ) and rotate one D (p#q)-brane relative to the other by applying the SO(2p) 0 0 transformation. The resulting con"guration is supersymmetric when the S;(p)LSO(2p) transformation is applied. When these intersecting D-branes are further compacti"ed on tori, the consistency of toroidal compacti"cation imposes the quantization condition for the intersecting angles h's in relation to the moduli of tori [32,89]. When intersecting D-branes at angles are compacti"ed on a manifold M, the unbroken supersymmetry should commute with the `generalizeda holonomy group (de"ned by a modi"ed connection } with torsion } due to non-zero antisymmetric tensor backgrounds) of M. Here, the spinor constraint g" eIC g de"nes an action of discrete generalized holonomy. Generally, G G I starting from an intersecting D p-brane at angles with FK "F#B"0, where F [B] is the worldvolume 2-form "eld strength [the NS-NS 2-form], one obtains a con"guration with FK O0 when T-duality is applied. For intersecting D 2-branes at angles, the necessary and su$cient condition for preserving supersymmetry is that FK is anti-self-dual [89]. Classical solution realization of intersecting D-branes at angles is "rst constructed in [97]. Starting from n parallel D 2-branes (with each constituent D 2-brane located at x"x , a"1,2, n, ? and its charge related to l '0) lying in the (y, y) plane, one rotates each constituent D 2-brane by ? an S;(2) angle a in the (y, y) and (y, y) planes, i.e. (z, z)P(e ??z, e\ ??z) where z"y#iy ? and z"y#iy. The solution in the string frame has the form: ds"(1#X

L 1 !dt# (dyH)# X +[(R )dyG]#[(R )dyH], ? ?G ?H 1#X H ?

# (dxG) , G

dt L A" X (R )dyG(R )dyH ? ?G ?H 1#X ? L ! X X sin(a !a )(dydy!dydy) , ? @ ? @ ?@ e("(1#X;

L L X, X # X X sin(a !a ) , ? ? @ ? @ ? ?@

(507)

144


where R? (a"1,2, n) are (block-diagonal) SO(4) matrices that correspond to the above mentioned rotation of constituent D 2-branes and harmonic functions

l 1 ? X (x)" ? 3 "x!x " ? are associated with constituent D 2-branes located at x"x . This con"guration preserves 1/4 of ? supersymmetry and interpolates between previously known con"gurations: (i) a "0, p/2 case is ? orthogonally oriented D 2-branes, (ii) a "a , ∀a, case is parallel n D 2-branes oriented in di!erent ? direction through S;(2) rotations, etc. The ADM mass density of (507) is the sum of those of constituent D 2-branes, i.e. A L m" l , ? 2i ? and is independent of the S;(2) rotation angles a . The physical charge density is also simply the ? sum of those of constituent D 2-branes, although the charge densities in di!erent planes (yG, yH) of the intersecting D 2-branes depend on a . ? T-duality on (507) yields other types of D-brane bound states. The T-duality along transverse directions leads to angled D p-branes with p'2. The T-duality along worldvolume directions leads to more exotic bound states of D-branes. Namely, since the constituent D 2-branes intersect with one another at angles, the worldvolume direction that one chooses for T-duality transformation is necessarily at angle with some of constituent D 2-branes. Consequently, the resulting con"guration is exotic bound state of D p-brane (pO2) and bound states of the type studied in [96] (e.g. bound state of D (p#1)-brane and D (p!1)-brane, and D (4, 2, 2, 0)-brane bound state) obtained by applying the `tilteda T-duality transformation(s) on a D p-brane. The above intersecting D-branes at angles and related con"gurations are alternatively derived by (i) applying the `tilteda boost transformation on the orthogonally intersecting two D-branes, followed by the sequence of ¹-S-¹ transformations of type-II strings [55], or (ii) applying `reduction along a boosta followed by T-duality transformations [151]. For the former case [151], the resulting con"gurations are mixed bound states of R-R branes that necessarily involves fundamental string. It is essential that one has to turn on both D-brane charges of original orthogonally intersecting D-branes and apply the S-duality between two T-duality transformations to have con"gurations where the D-branes intersect at angles. Intersecting p-branes at angles in more general setting, starting from M-branes with the #at Euclidean transverse space EL replaced by the toric hyper-KaK hler manifold M , are studied in L [285]. We "rst discuss the general formalism and then specialize to the case of intersecting p-branes in D"10, 11. 4n-dimensional toric hyper-KaK hler manifold M with a tri-holomorphic ¹L isometry has the L following general form of metric: ds "; dxG ) dxH#;GH(du #A )(du #A ) (i, j"1,2, n) , &) GH G G H H

(508)

The manifold M is tri-holomorphic i! the triplet KaK hler 2-forms X"(du #A ) dxG!; dxG;dxH are indepenL G G GH dent of u , i.e. L GX"0. G ..P


145

where u (which are periodically identi"ed } to remove a coordinate singularity } u &u #2p, G G G thereby parameterizing ¹L) correspond to the ;(1) isometry directions of M and L xG"+xG " r"1, 2, 3, parameterize n copies of Euclidean spaces E. The n"1 case is Taub-NUT P space. The hyper-KaK hler condition relates n 1-forms A "dxH ) x with "eld strengths F "dA to G GH G G ; through ePQJRJ ; "FPQ "RPuQ !RQ uP . (So, a toric hyper-KaK hler metric is speci"ed by GH H IG HIG H IG I HG ; alone.) This implies that ; are harmonic functions on M , i.e. ;GHR ) R ;"0. Generally, GH GH L G H a positive-de"nite symmetric n;n matrix ;(xG) is linear combination ,+N, (509) ; ";# ; [+p,, a (+p,)] GH K GH GH + , N K of the following harmonic functions speci"ed by a set of n real numbers +p "i"1,2, n,, called G a `p-vectora, and an arbitrary 3-vector a: pp G H ; [+p,, a]" . (510) GH 2" p xI!a" I I The vacuum hyper-KaK hler manifold EL;¹L with moduli space Sl(n, Z)!Gl(n, R)/SO(n) has the metric (508) with ; "; (so, A "0). Regular non-vacuum hyper-KaK hler manifold is representGH GH G ed by harmonic functions ; [+p,, a] associated with a 3(n!1)-plane in EL de"ned by 3-vector GH equations L p xI"a. The hyper-KaK hler metric (508) is non-singular, provided +p, are coprime I I integers. The Sl(n, Z) transformation ;PS2;S (S3Sl(n, Z)) on the hyper-KaK hler metric (508) leads to another hyper-KaK hler metric with the Sl(n, Z) transformed p-vector S+p,. The angle h between two 3(n!1)-planes de"ned by two p-vectors +p, and +p, is given by cos h"p ) p/(pp. Here, the inner product is de"ned as p ) q"(;)GHp q , which is invariant under Sl(n, Z). The solution G H (508) is, therefore, speci"ed by angles and distances between mutually intersecting 3(n!1)-planes associated with harmonic functions ; [+p,, a (+p,)]. GH K The special case where *;,;!; is diagonal (i.e. the p-vectors have the form (0,2, 1,2, 0) and *; "d (1/2"xG"): there are only n intersecting 3(n!1)-planes) describes n GH GH fundamental BPS monopoles in maximally broken rank (n#1) gauge theories found in [444], thereby called LWY metric. When additionally ; is diagonal (so that ; is diagonal), n 3(n!1)planes intersect orthogonally (cos h"0) and M "M ;2;M . In this case, one can L always choose u such that F "dA and ; are related as F "夹d; , where 夹 is the Hodge-dual GH G G GG G GG on E. The hyper-KaK hler manifold M preserves fraction of supersymmetry. It admits (n#1) L covariantly constant SO(4n) spinors (in the decomposition of D-dimensional Lorentz spinor representation under the subgroup Sl(n, R);SO(4n)) if the holonomy of M is strictly Sp(n), which L corresponds to the case where 3(n!1)-planes intersect non-orthogonally, i.e. ; is non-diagonal. These covariantly constant SO(4n) spinors arise as singlets in the decomposition of the spinor representation of SO(4n) into representations of holonomy group of M , i.e. Sp(n) for this case. L The only toric hyper-KaK hler manifolds whose holonomy is a proper subgroup of Sp(n) are those corresponding to the òrthogonallya intersecting or `parallela 3(n!1)-planes. For this case, M "M ;2M (i.e. product of n Taub-NUT space) with Sp(1)L holonomy and diagonal L ; (thereby, 3(n!1)-planes intersecting orthogonally), and more supersymmetry is preserved GH since the non-singlet spinor representations of Sp(n) are further decomposed under the proper

146


subgroup Sp(1)L. A trivial case corresponds to the case ; ";, i.e. the vacuum hyper-KaK hler GH GH manifold: since the holonomy group is trivial, all the supersymmetries are preserved. The starting point of general class of intersecting p-branes is the following D"11 solution, which is the product of the D"3 Minkowski space and M : ds "ds(E)#; dxG ) dxH#;GH(du #A )(du #A ) , (511) GH G G H H where i"1, 2. For a general solution with non-diagonal ; , thereby with the Sp(2) holonomy for GH M , (511) admits (n#1)"3 covariantly constant spinors. Namely, 32-component real spinor in D"11 is decomposed under Sl(2, R);SO(8) as 32P(2, 8 )(2, 8 ). Two SO(8) spinor representaQ A tions 8 and 8 are further, respectively, decomposed under Sp(2)LSO(8) as 8 P5111 and Q A Q 8 P44. So, 3/16 of supersymmetry is preserved. When ; is diagonal (so, M "M ;M and A GH 3-planes orthogonally intersect), the holonomy group is Sp(1);Sp(1). Under the Sp(1);Sp(1) subgroup, non-singlet Sp(2) spinor representations 5 and 4 are, respectively, decomposed as 5P(2, 2)(1, 1) and 4P(2, 1)(1, 2). So, 8/32"1/4 of supersymmetry is preserved. One can generalize the solution (511) to include M-branes without breaking any more supersymmetry, resulting in `generalized M-branesa, where the transverse Euclidean space is replaced by M . The harmonic functions (associated with M-branes) are independent of the ;(1) isometry L coordinates u , thereby M p-branes are delocalized in the u -directions. G G First, one can naturally include an M 2-brane to the solution (511), since the transverse space of M 2-brane has dimensions 8: ds "H\ds(E)#H[; dxG ) dxH#;GH(du #A )(du #A )] , GH G G H H F"$u(E)dH\ ,

(512)

where u(E) is the volume form on E, the signs $ are those of M 2-brane charge and H"H(xG) is a harmonic function (associated with M 2-brane) on M , i.e. ;GHR ) R H"0. The G H SO(1, 10) Killing spinor of this solution is decomposed into the SO(8) spinors of de"nite chiralities 8 and 8 , which are related to the signs $. So, depending on the sign of M 2-brane charge, either A Q all supersymmetries are broken or 3/16 of supersymmetry is preserved. Second, one can add M 5-branes to the solution (511) if M "M ;M , i.e. ;"diag(; (x), ; (x)). For this purpose, it is convenient to introduce 2 1-form potentials AI G (i"1, 2) with "eld strengths FI which can be related to the harmonic functions H (x) and H (x) G (associated with 2 M 5-branes) as dH "夹FI . (This is analogous to the relations d; "夹F G G G G satis"ed by the diagonal components of ; and the "eld strengths F "dA of the solution (511) G G when both *; and ; are diagonal.) Here, 夹 is the Hodge-dual on E. The explicit solution has the form: ds "(H H )[(H H )\ ds(E)#H\[; dx ) dx#;\(du #A )] # H\[; dx ) dx#;\(du #A )]#dz] , F"[FI (du #A )#FI (du #A )]dz . The subscripts s and c denote two possible SO(8) chiralities.

(513)


147

Generally with non-constant ; and H , (513) preserves 1/8 of supersymmetry, provided the proper G G relative sign of M 5-brane charges is chosen. In the following, we discuss the intersecting (overlapping) p-brane interpretation of solutions obtained via dimensional reduction and duality transformations of the D"11 solutions (511)}(513). Due to the triholomorphicity of the Killing vector "elds R/Ru , the Killing spinors G survive in these procedures. As for the p-branes associated with the harmonic functions ; , there is GH a one-to-one correspondence between 3(n!1)-planes and p-branes, and the intersection angle of p-branes is given by the angle between the corresponding p-vectors, which de"ne 3(n!1)-planes. First, we discuss intersecting (overlapping) p-branes related to (511) and (512). Since (511) is a special case of (512) with H"1, we consider p-branes related to (512), and then comment on the H"1 case. First, one compacti"es one of the ;(1) isometry directions of M , say u without loss of generality, on S, resulting in a type-IIA solution, and then applies the T-duality transformation along the other ;(1) isometry direction, i.e. the u -direction, to obtain the following type-IIB solution: ds"(det ;)H[H$det ;)\ds(E)#(det ;)\; dxG ) dxH#H\ dz] , # GH

(514)

(det ; ; , i D"u(E)dH\ , B "A dz, q"! #i I G G ; ; where is the dilaton, q,l#ie\( (l"R-R 0-form "eld), B (i"1, 2) are 2-form potentials in G the NS-NS and R-R sectors, D is the 4-form potential, and z,u . This solution is interpreted as D 3-brane (with harmonic function H) stretching between 5-branes along the z-direction. The 5-branes in this type-IIB con"guration are speci"ed by a set of intersecting 3-planes L p xI"a I I in E. From the expression for q in (514), one sees that the Sl(2, R) transformation ;P(S$2;S\ (S3Sl(2, R)) in M is realized in this type-IIB con"guration as the type-IIB Sl(2, R) symmetry aq#b qP cq#d of equations of motion, where

S"

a b c

d

.

The condition that Sl(2, R) is broken down to Sl(2, Z) so that M with the coprime integers +p , p , remains non-singular after the transformation is translated into the type-IIB language that the Sl(2, R) symmetry of the equations of motion is broken down to the Sl(2, Z) S-duality symmetry of type-IIB string theory. In the following we discuss particular cases of (514). We "rst consider the solution (514) with ;"diag(H (x), H (x)) and H"1. In this case, M "M ;M with holonomy Sp(1);Sp(1), thereby preserving 1/4 of supersymmetry. The corresponding solution is òrthogonallya intersecting (2"5 , 5 ): ,1 0 ds"(H H )[(H H )\ds(E)#H\dx ) dx#H\dx ) dx#dz] , (515) # where harmonic functions H "1#(2"xG")\ (i"1, 2) are respectively associated with NS-NS G 5-brane and R-R 5-brane [285].

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A particular case of (514) with ;"1"H and a single 3-plane in E (de"ned by +p , p ,) is the bound state of NS-NS 5-brane and R-R 5-brane with charge vector (p , p ). So, the restriction that M is non-singular, i.e. +p , p , are coprime integer, manifests in the type-IIB theory that the corresponding p-brane con"guration is a non-marginal bound state of NS-NS 5-brane and R-R 5-brane. There is a correlation between the D"11 Sl(2, Z) transformation, which rotates a 3-plane in E, and the type-IIB Sl(2, Z) transformation, which rotates the charge vectors of the 2-form "eld doublet B . G The general type-IIB solution (514) with non-diagonal ; is interpreted as an arbitrary number of 5-branes intersecting or overlapping at angles. p-vectors and ; specify orientations and charges of 5-branes, and a determines distance of 5-branes from the origin. Since the corresponding M has the Sp(2) holonomy, 3/16 of supersymmetry is preserved. Next, we discuss the intersecting p-branes in type-IIA string and M-theory related to (512). The intersecting p-branes in type-IIA theory are constructed in the following way. First, one T-dualizes the type-IIB solution (514) along a direction in E to obtain the following type-IIA `generalized fundamental stringa solution, which can also be obtained from (511) by compactifying on a spatial direction in E: ds"H\ds(E)#; dxG ) dxH#;GH(du #A )(du #A ) , GH G G H H (516) B"u(E)H\, "! ln H . Subsequent T-dualities along u and u lead to the type-IIA solution: ds"H\ds(E)#; (dxG ) dxH#duGduH) , GH (517) B"A duG#u(E)H\, " ln det ;! ln H , G where B is the NS-NS 2-form potential and is the dilaton. This solution is interpreted as an arbitrary number of NS-NS 5-branes intersecting on a fundamental string (with a harmonic function H), generalizing the solutions in [423]. The case with diagonal ; represents orthogonally intersecting NS-NS 5-branes. More general case with non-diagonal ; represents intersecting NS-NS 5-branes at angles and preserves 3/16 of supersymmetry. The following solution in D"11 is obtained by uplifting the type-IIA solution (517): ds "H(det ;)[H\(det ;)\ds(E) # (det ;)\; (dxG ) dxH#duG duH)#H\dy] , (518) GH F"[F duG#u(E)dH\]dy . G When ; is of LWY type, this solution represents parallel M 2-branes which intersect intersecting 2 M 5-branes (orthogonally when ; is diagonal as well) over a string. For more general form of ;, the solution represents the M 2-branes intersecting arbitrary numbers of M 5-branes at angles and preserves 3/16 of supersymmetry. T-duality on the type-IIA solution (517) along the fundamental string direction leads to intersecting type-IIB NS-NS 5-branes with a pp-wave along the common intersection direction. Further application of Z LS¸(2, Z) S-duality transformation leads to the following solution involving R-R 5-branes, which preserves 3/16 of supersymmetry when 5-branes intersect at


149

angles: ds"(det ;)[dt dp#Hdp#; (dxG ) dxH#duG duH)] , # GH (519) B"A duG, q"i(det ; , G where B is the R-R 2-form potential. The H"1 case is classical solution realization of intersecting D-branes at angles of [89]. In [89], the condition for unbroken supersymmetry is given by the holonomy condition arising in the KK compacti"cations. This corresponds to the holonomy condition on the hyper-KaK hler manifold of [285]. Namely, intersecting R-R p-branes preserve fraction of supersymmetry if orientations of the constituent R-R p-branes are related by rotations in the Sp(2) subgroup of SO(8). This is seen by considering spinor constraints of intersecting two D 5-branes, where one D 5-brane is oriented in the (12345) 5-plane and the other D 5-brane rotated into the (16289) 5-plane by an angle h. For this con"guration, type-IIB chiral spinors e (A"1, 2) satisfy the constraints C e"e and R\(h)C R(h)e"e, where R(h)" exp+!h(C #C #C #C ), is the SO(1, 9) spinor representation of the above mentioned SO(8) rotation. (The rotational matrix R(h) is associated with an element of ;(2, H) Sp(2) that commutes with quaternionic conjugation, which is the rotation mentioned above.) It can be shown [285] that (i) for h"0, p, 1/2 of supersymmetry is preserved since the second spinor constraint is trivially satis"ed, (ii) for h"$p/2, 1/4 is preserved, and (iii) for all other values of h, 3/16 is preserved. Finally, we discuss intersecting (overlapping) p-branes related to (513). The compacti"cation on one of the ;(1) isometry directions followed by the T-duality along the other ;(1) isometry direction yields the following type-IIB solution: ds"(H H ; ; )[(; ; H H )\ds(E)#(; H )\dx ) dx # #(; H )\dx ) dx#(H H )\dz#(; ; )\dy] , (520) H ; . B"A dz#AI dy, B"A dz#AI dy, q"i H ; This solution represents 2 NS-NS 5-branes in the planes (1, 2, 3, 4, 5) and (1, 6, 7, 8, 9) and 2 R-R 5-branes in the planes (1, 5, 6, 7, 8) and (1, 2, 3, 4, 9) intersecting orthogonally. Since the spinor constraint associated with one of the constituent p-branes is expressed as a combination of the rest three independent spinor constraints, the solution preserves () of supersymmetry. Related intersecting p-brane is constructed by applying T-duality, oxidation and dimensional reduction. One can further include additional p-branes without breaking any more supersymmetry, provided the spinor constraints of the added p-branes can be expressed as a combination of spinor constraints of the existing p-branes.

6.3. Dimensional reduction and higher dimensional embeddings The lower-dimensional (D(10) p-branes can be obtained from those in D"10, 11 through dimensional reduction. Reversely, most of lower-dimensional p-branes are related to D"10, 11 p-branes via dimensional reduction and dualities. In particular, many black holes in D(10 originate from D"10,11 p-brane bound states, which makes it possible to "nd microscopic origin

150


of black hole entropy. It is purpose of this section to discuss various p-branes in D(10. We also discuss various p-brane embeddings of black holes. There are two ways of compactifying p-branes to lower dimensions. First, one can compactify along a longitudinal direction. It is called the `double dimensional reductiona (since both worldvolume and spacetime dimensions are reduced, bringing a p-brane in D dimensions to (p!1)brane in D!1 dimensions diagonally in the D versus p brane-scan) or `wrappinga of branes (around cycles of compacti"cation manifold). Since target space "elds are independent of longitudinal coordinates, one only needs to require periodicity of "elds in the compacti"cation directions. Second, one can compactify a transverse direction of a p-brane. It is called the `direct dimensional reductiona (since this takes us vertically on the bran-scan, taking a p-brane in D dimensions to a p-brane in D!1 dimensions) or `constructing periodic arraysa of p-branes (along the compacti"ed direction). Since "elds depend on transverse coordinates, direct dimensional reduction is more involved [286,422,461]. For this purpose, one takes periodic array of parallel p-branes (with the period of the size of compact manifold) along the transverse direction. Then, one takes average over the transverse coordinate, integrating over continuum of charges distributed over the transverse direction. The resulting con"guration is independent of the transverse coordinate, making it possible to apply standard Kaluza}Klein dimensional reduction. In the double [direct] dimensional reduction, the values of p [p] and D are preserved; in the direct dimensional reduction, the asymptotic behavior of the "elds (which goes as &1/"y"N ) changes. Conventionally, the direct dimensional reduction uses the zero-force property of BPS p-branes, which allows the construction of multicentered p-branes. Note, however that it is also possible to apply the vertical dimensional reduction even for non-BPS extreme p-branes [461] and non-extreme p-branes [463,442], contrary to the conventional lore. Namely, since the equations of motion (of a non-extreme, axially symmetric black (D!4)-brane in D dimensions, for the non-extreme case) can be reduced to Laplace equations in the transverse space with suitable choice of "eld AnsaK tze, one can still construct multi-center p-branes for non-BPS and non-extreme cases as well. For the non-extreme case, when an inxnite number of non-extreme p-branes are periodically arrayed along a line, the net force on each p-brane is zero and the conical singularities along the axis of periodic array act like `strutsa that hold the constituents in place. Furthermore, since the direction of periodic array is compacti"ed on S with each p-brane precisely separated by the circumference of S, the instability problem of such a con"guration can be overcome. One can also lift p-branes as another p-branes in higher-dimensions, so-called &dimensional oxidation'. First, the oxidation of a p-brane in D dimensions to a p-brane in D#1 dimensions (i.e. the reverse of the direct dimensional reduction) is never possible in the standard KK dimensional reduction, since the oxidized p-brane in D#1 dimensions has to depend on the extra transverse

Such staking-up procedure breaks down for (D!3)-branes in D dimensions, due to conical asymptotic spacetime [461]. This is also related to the fact that (D!2)-branes reside in massive supergravity, rather than ordinary massless type-II theory. Or one can promote such isometry directions as the spatial worldvolume of a p-brane, leading to an intersecting p-brane solution [287,424,461,611].


151

coordinate introduced by oxidation. (This is analogous to the `dangerousa T-duality transformation.) Second, the oxidation of a p-brane in D dimensions to a (p#1)-brane in D#1 dimensions (i.e. the reverse of the double dimensional reduction) is classi"ed into two groups. A p-brane in D dimensions is called `rustya if it can be oxidized to a (p#1)-brane in D#1 dimensions. Otherwise, it is called `stainlessa. Thus, p-branes in bran scan are KK descendants of stainless p-branes in some higher dimensions. A p-brane in D dimensions is `stainlessa when (i) there is not the antisymmetric tensor in D#1 dimensions that the corresponding (p#1)-brane couples to, or (ii) the exponential prefactors a and a for (p#2)-form "eld strength kinetic terms in D#1 and "> " D dimensions do not satisfy the relation 2(p #1) . a "a ! "> " (D!2)(D!1) (This relation is satis"ed by the expression for a in (531), provided D remains unchanged in the dimensional reduction procedure.) In this section, we focus on p-branes in D(10 with only "eld strengths of the same rank turned on, comprehensively studied in [243,455}463]. A special case is black holes, which are 0-brane bound states. We also discuss their supersymmetry properties and interpretations as bound states of higher-dimensional p-branes. 6.3.1. General solutions We concentrate on p-branes in D"11 supergravity on tori. Bosonic Lagrangian of D"11 supergravity is (521) L "(!GK [R K ! FK ]#FK FK AK , % where FK "dAK is the "eld strength of the 3-form potential AK . So, such p-branes have interpreta tion in terms of M-theory or type-II string theory con"gurations. Although one can directly reduce the D"11 action down to D(11 by compactifying on ¹\" in one step, it turns out to be more convenient to reduce the action (521) one dimension at a time iteratively until one reaches D dimensions. Namely, one compacti"es 11!D times on S, making use of the following KK Ansatz: ds "e?Pds #e\"\?P(dz#A ) , "> " (522) A (x, z)"A (x)#A (x)dz , L L L\ where u is a dilatonic scalar, A "A dxI is a KK 1-form "eld, A is an n-form "eld arising from I L AK and a,1/(2(D!1)(D!2). The explicit form of resulting D(11 action can be found elsewhere [456]. The advantage of such compacti"cation procedure is that spin-0 "elds are

Such a p-brane in D dimensions should rather be viewed as a p-brane in D#1 dimensions whose charge is uniformly distributed along the extra coordinate. This is interpreted as the limit where one of charges of intersecting two p-branes in D#1 dimensions is zero [461]. Contrary to the conventional lore that all the p-branes in D(10 are obtained from those in D"10, 11 through dimensional reductions, there are stainless p-branes in D(10 which cannot be viewed as dimensional reductions of p-branes in D"10, 11. So, the conventional brane scan is modi"ed [459].

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manifestly divided into two classes in the Lagrangian. Namely, only dilatonic scalars

"( ,2,

) appear in the exponential prefactors of the n-form "eld kinetic terms. The \" dilatonic scalars originate from the diagonal components of the internal metric and are true scalars. The couplings of to n-form potentials A? are characterized by the `dilaton vectorsa a in the L ? following way: 1 (523) e\L "! ea? (F? ) . L> LU 2n! ? The complete expressions for `dilaton vectorsa, which are expressed as linear combinations of basic constant vectors, are found in [456]. On the other hand, the remaining spin-0 "elds coming from the o!-diagonal components of GK and the internal components of AK are axionic, being +, associated with constant shift symmetries, and should rather be called 0-form potentials, which couple to solitonic (D!3)-branes. (Note, there are no elementary p-branes for 1-form "eld strengths.) Up to present time, study of p-branes within the above described theory has been mostly concentrated on the case where only n-form potentials of the same rank are turned on, with the restrictions that terms related to the last term in (521) (denoted as L from now on) and the $$ `Chern}Simonsa terms in (n#1)-form "eld strengths are zero. These restrictions place constraints on possible charge con"gurations for p-branes. These constraints become non-trivial when a pbrane involves both undualized and dualized "eld strengths, i.e. when the p-brane has both electric and magnetic charges coming from di!erent "eld strengths. The former [later] type of constraint is satis"ed as long as the dualized and undualized "eld strengths have [do not have] common internal indices i, j, k. 6.3.1.1. Supersymmetry properties. The supersymmetry preserved by p-branes is determined from the Bogomol'nyi matrix M, which is de"ned by the commutator of supercharges Q " Re C !t dR per unit p-volume: ! C .

N dR &eR Me , (524) .R where N "e C !d t is the Nester's form de"ned from the supersymmetry transformation rule C ! of D"11 gravitino t : (525) N "e C !D e #e C!!e F # e C !2!e F 2 . !! ! ! ! The "rst term in N gives rise to the ADM mass density and the last two terms respectively contribute to the electric and magnetic charge density terms in M. The Bogomol'nyi matrix for the 11-dimensional supergravity on (S)\" is in the form of the ADM mass density m term plus the electric and magnetic Page charge density [490] (de"ned respectively as (1/4u ) "\L夹F and L "\L 1 (1/4u ) LF ) terms. L 1 L [Q , Q ]" C C

In particular, to set the 0-forms AGHI to zero consistently with their equations of motion, the bilinear products of "eld strengths that occur multiplied by AGHI in L should vanish. $$


153

Since the Bogomol'nyi matrix is obtained from the Hermitian supercharges, its eigenvalues are non-negative. The matrix M has zero eigenvalues for each component of unbroken supersymmetry associated with the Killing spinor e satisfying d t "0. Since D"11 spinor has 32 components, C the fraction of preserved supersymmetry is k/32, where k is the number of 0 eigenvalues of M (i.e. the nullity of the matrix M) or equivalently the number of linearly independent Killing spinors. The amount of preserved supersymmetry is determined as follows. First, one calculates the ADM mass density m from the p-brane solutions. Then, one plugs m, together with the Page electric and magnetic charge densities of the p-branes, into the Bogomol'nyi matrix. The multiplicity k of 0 eigenvalues of the resulting matrix M determines the fraction of supersymmetry preserved by the corresponding p-branes. 6.3.1.2. Multi-scalar p-branes. General p-branes with more than one non-zero p-brane charges are called `multi-scalar p-branesa, since such p-branes have more than one non-trivial dilatonic scalars. The Lagrangian density has the following truncated form:

1 1 ea? (F? ) , (526) L"(!g R! (R )! N> 2 2(p#2)! ? where "eld strengths F? "dA? are de"ned without `Chern}Simonsa modi"cations. In this N> N> action, the rank p#2 of "eld strengths is assumed to not exceed D/2, namely those with p#2'D/2 are Hodge-dualized. This is justi"ed by the fact that the dual of "eld strength of an elementary (solitonic) p-brane is identical to the "eld strength of solitonic (elementary) (D!p!2)-brane, with the corresponding dilaton vector di!ering only by sign. We consider extreme p-branes with N non-zero (p#2)-form "eld strengths, each of which is either elementary or solitonic, but not both. For the simplicity of calculations, the SO(1, p!2);SO(D!p#1) symmetric metric Ansatz (448) is assumed to satisfy (p#1)A# (p #1)B"0 [457], so that the "eld equations are linear. The p-brane solutions are then determined completely by the dot products M ,a ) a of the dilaton vectors a associated with non?@ ? @ ? zero (p#1)-form "eld strengths F? (a"1,2, N). In solving the equations, it is assumed that N> M is invertible, which requires the number N of non-trivial F? to be not greater than the num?@ N> ber of the components in , i.e. N411!D. For such p-branes, only N components u ,a ) of ? ?

are non-trivial. If one further takes the Ansatz !eu #2(p#1)AJ (M\) u , then M @ ?@ @ ?@ ? takes the form: 2(p#1)(p #1) M "4d ! . ?@ ?@ D!2

(527)

The conditions on "elds that linearize the "eld equations and lead to M of the form (527) are also ?@ dictated by supersymmetry transformation rules for the BPS con"gurations. Thus, a necessary condition for `multi-scalar p-branesa to be BPS is for the dilaton vectors a associated with ? participating "eld strengths to satisfy (527). The following extreme multi-scalar p-brane solution is

When M is singular, analysis depends on the number of rescaling parameters [456]. The only new solution is the ?@ case a "0, which yields solutions with a"0 and (F?)"F/N, ∀a. ? ?

154


obtained by taking further simplifying Ansatz discussed in [457]: , , eCP?\N>"H , ds" H\N >"\ dxI dxJg # HN>"\ dyK dyK , (528) ? ? IJ ? ? ? where harmonic functions 1 j H "1# ? (y,(yKyK) ? p #1 yN> are associated with p-branes carrying the Page charges P "j /4, and the "eld strengths are given, ? ? respectively, for the electric and magnetic cases by F? "dH\dN>x, F? "夹(dH\dN>x) . (529) N> ? N> ? The elementary and solitonic p-branes are related by P! . The ADM mass density is the sum of the mass densities of the constituent p-branes, i.e. m" , P . Multi-center generalization is ? ? achieved by replacing harmonic functions by [424]. j ?G H "1# ? "y!y "N > ?G G 6.3.1.3. Single-scalar p-branes. We discuss the case where the bosonic Lagrangian for 11-dimensional supergravity on (S)\" is consistently truncated to the following form with one dilatonic scalar and one (p#2)-form "eld strength [456,459]:

1 1 e?((F ) , L"(!g R! (R )! N> 2 (p#2)!

(530)

where we parameterize the exponential prefactor a in the form: 2(p#1)(p #1) a"D! . D!2

(531)

This expression for a is motivated from (422), now with an arbitrary parameter D replacing 4. By consistently truncating (526), one has (530) with (F ), (F? ) and a, given by (for the case ? N> N> M is invertible) ?@ \ a" (M\) , "a (M\) a ) . (532) ?@ ? ?@ ?@ ?@ By taking AnsaK tze which reduce equations of motion for (530) to the "rst order, one obtains [459] the `single-scalar p-branea solution with the Page charge density P"j/4:

e("H?CD,

ds"H\N >D"\ dxI dxJ g #HN>D"\ dyK dyK , IJ

where 1 (Dj H"1! . 2(p#1) rN>

(533)


155

The mass density is m"j/2(D. Although inequivalent charge con"gurations give rise to the same D, i.e. the same solution, supersymmetry properties depend on charge con"gurations. Note, although one can obtain p-brane solutions (533) for any values of p and a, hence for any values of D, only speci"c values of p and a can occur in supergravity theories. The value of D is preserved in the compacti"cation process, provided no "elds are truncated. For p-branes with 1 constituent, D is always 4, as can be seen from the form of a in (442), determined by the requirement of scaling symmetry, and always 1/2 of supersymmetry is preserved. The value D"4 can also be understood from the facts that D"4 in D"11, since there is no dilaton in D"11, and the value of D is preserved in dimensional reduction not involving "eld truncations. When the "eld strength is a linear combination of more than one original "eld strengths, D(4. With all the Page charges P "j /4 of `multi-scalar p-branea (528) equal, one has `single-scalar ? ? p-branea with the Page charge P"j/4 (j"(Nj ). By substituting M in (527) into (532), one has ? ?@ D"4/N. So, `single-scalar p-branesa with D"4/N (N52) are bound states of N single-charged p-branes (with D"4) with zero binding energy, and preserve the same fraction of supersymmetry as their multi-scalar generalizations. Only `single-scalar p-branesa with D"4/N (N5Z>) and `multi-scalar p-branesa can be supersymmetric. (Non-supersymmetric p-branes in this class is related to supersymmetric ones by reversing the signs of certain charges.) And only single-scalar p-branes with D"4/N (N52) have multi-scalar generalizations. 6.3.1.4. Dyonic p-branes. In D"2(p#2), p-branes can carry both electric and magnetic charges of (p#2)-form "eld strengths. There are two types of dyonic p-branes [456]: (i) the "rst type has electric and magnetic charges coming from di!erent "eld strengths, (ii) the second type has dyonic "eld strengths. As in the multi-scalar p-brane case, the requirements that L "0 and the $$ Chern}Simons terms are zero place constraints on possible dyonic solutions in D"2(p#2)"4, 6, 8. Such restrictions rule out dyonic p-branes of the "rst type in D"6, 8. For the second type, dyonic p-brane in D"8 is special since it has non-zero 0-form potential A [388], thereby requiring non-zero source term FK FK e+,./0123, and can be obtained +,./ 0123 from purely electric/magnetic membrane by duality rotation, unlike dyonic D"6 string and D"4 0-brane of the second type. Dyonic p-branes of the second type include self-dual 3-branes in D"10 [238,367], self-dual string [242] and dyonic string [230] in D"6, and dyonic black hole in D"4 [456]. There are two possible dyonic p-branes (associate with (530)) of the second type with the Page charge densities j /4 [456]: (1) a"p#1 case (i.e. *"2p#2) with the solution G 1 1 j j , e?(\N>"1# , e\?(\N>"1# a(2 rN> a(2 rN>

(534)

For p-branes with DO4, the scaling symmetry of combined worldvolume and e!ective supergravity actions is broken. The solutions for dyonic p-branes of the "rst type have the form (528) with Lagrangian (526) containing both Hodgedualized and undualized "eld strengths.

156


and (2) a"0 case (i.e. D"p#1) with the solution

1 j#j 1

"0, e\N>"1# . 2 p#1 rN>

(535)

The ADM mass density for (534) is m"(1/2(D)(j #j ), whereas for (535) m"(1/2(D)(j#j. The solution (535) is invariant under electric/magnetic duality and, there fore, is equivalent to the purely elementary (j "0) or solitonic (j "0) case. For (534) with j "j , the "eld strength is self-dual and "0, thereby (534) and (535) are equivalent, but for (535) j and j are independent. When j "!j , (534) corresponds to anti-self-dual massless string with enhanced supersymmetry. Note, the solutions (534) and (535) are not restricted to those obtained from the D"11 supergravity on tori. For D"8, 6 and 4, which are relevant for the D"11 supergravity on tori, D's for (534) and (535) are respectively +6, 3,, +4, 2, and +2, 1,. So, (534) and (534) with p"2 (i.e. D"8) are excluded. 6.3.1.5. Black p-branes. We discuss non-extreme p-branes. Non-extreme p-branes are additionally parameterized by the non-extremality parameter k'0. There are two ways of constructing non-extreme p-branes. The "rst method involves a universal prescription for `blackeninga extreme p-branes, which deforms extreme solutions with [243]: k (r,"y") eD"1! rN> dtPeD dt, drPe\D dr ,

(536)

while modifying harmonic functions associated with p-branes as H"1#k sin h2d / ? rN >P1#k sinh d /rN >. The resulting non-extreme p-branes, called `type-2 non-extreme p? branesa, have an event horizon at r"r "kN >, which covers the singularity at the core r"0. > The ADM mass density has the generic form m& ((Q )#k, which is always larger than the ? ? extreme counterpart, and all the supersymmetry is broken since the Bogomol'nyi bound is not saturated. For type-2 non-extreme p-branes (with p51), the PoincareH invariance is broken down to R;EN because of the extra factor eD in the (t, t)-component of the metric. For 0-branes, the metric remains isotropic but the quantity (p#1)A#(p #1)B no longer vanishes. In the second method, the metric Ansatz (448) remains intact but instead general solution to the "eld equations is obtained [455,462] without simplifying AnsaK tze, e.g. (p#1)A#(p #1)B"0, that linearize "eld equations. (In solving the "eld equations without simplifying AnsaK tze, one encounters an additional integration constant interpreted as non-extremality parameter.) So, the resulting non-extreme p-branes, called `type-1 non-extreme p-branesa, preserve the full PoincareH invariance (in the worldvolume) of extreme p-branes. So, type-1 non-extreme p'0 solutions do not overlap with the type-2 non-extreme counterparts. But type-1 non-extreme 0-branes contain type-2 non-extreme 0-branes as a subset. The equations of motion for single-scalar p-branes and dyonic p-branes of the second type are, respectively, casted into the forms of the Liouville equation and the Toda-like equations for two


157

variables, which are subject to the "rst integral constraint. The equations of motion for dyonic p-branes are solvable when a"p#1 (i.e. D"2(p#1)) or a"3(p#1) (i.e. D"4(p#1)). When a"p#1, the equations of motion are reduced to two independent Liouville equations. Since D44 in supergravity theories, only dyonic strings in D"6 and dyonic black holes in D"4 are relevant, with only dyonic strings having BPS limit. When a"3(p#1), the equations of motion are reduced to S;(3) Toda equations. Only dyonic black holes are possible in supergravity theory for this case. In the extreme limit, such black holes preserve supersymmetry when either electric or magnetic charge is zero. For multi-scalar p-branes with N "eld strengths, the equations of motion are Toda-like in general, but when the extreme limit is BPS (i.e. dilaton vectors satisfy (527)) the equations of motion become N independent Liouville equations. The requirements that non-extreme p-branes are asymptotically Minkowskian and dilatons are "nite at the event horizon (thereby the event horizon is regular) place restrictions on parameters of the solutions. 6.3.1.6. Massless p-branes. For multi-scalar p-branes and a dyonic p-brane (535) of the second type, the ADM mass density has the form m& j . So, they can be massless when some of the ? ? Page charges are negative. In this case, there are additional 0 eigenvalues of the Bogomol'nyi matrix, enhancing supersymmetry. Generally massless p-branes are ruled out if one requires the Bogomol'nyi matrix to have only non-negative eigenvalues, since the Bogomol'nyi matrix is obtained from the commutator of the Hermitian supercharges. Since some of the Page charges are negative, the massless p-branes have naked singularity. On the other hand, if one allows negative eigenvalues, one can have p-branes preserving more than 1/2 of supersymmetry and some of non-BPS multi-scalar p-branes can become supersymmetric due to the appearance of 0 eigenvalues with suitable sign choice of Page charges (but their single-scalar counterparts are non-BPS, since Page charges have to be equal in the single-scalar limit) [457]. 6.3.2. Classixcation of solutions In this subsection, we classify p-branes discussed in the previous subsection according to their supersymmetry properties. Since single-scalar p-branes and their multi-scalar generalizations preserve the same amount of supersymmetry (except for the special case of massless p-branes), the classi"cation of multi-scalar p-branes is along the same line as that of single-scalar p-branes. Single-scalar p-branes are supersymmetric only when D"4/N (N3Z>) and the dilaton vectors a (associated with the participating "eld strengths) of their multi-scalar p-brane counterparts ? satisfy relations (527). Spin-0 "elds, i.e. dilatonic scalars and 0-form "elds, form target space manifold of p-model. The target space manifold has a coset structure G/H, where GKE (R) (n"11!D) is the (realL>L valued) ;-duality group and H is a linearly realized maximal subgroup of G. Under the G-transformations, the equations of motion are invariant. When the DSZ quantization is taken

In the exceptional case of 4-scalar solution with 2-form "eld strengths in D"4, it is possible to have massless p-branes where the Bogomol'nyi matrices have no negative eigenvalues [457]. For D46, this is the case only when all the "eld strengths are Hodge-dualized to those with rank 4D/2 [458].

158


into account, G and H break down to integer-valued subgroups. The subgroup G(Z) is the conjectured ;-duality group of type-II string on tori. The asymptotic values of spin-0 "elds, called `modulia, de"ne the `scalar vacuuma. The asymptotic values of dilatonic scalars and 0-form "elds are, respectively, interpreted as the `coupling constantsa and `h-anglesa of the theory. One can parameterize spin-0 "elds by a G-valued matrix B> as

p pG 0 " 0 pH p * *

0

p( *

,

(557)

where p "[nR#mR(B!G)]aH, p "[nR#mR(B#G)]aH, m,n3RB> . 0 * Here, aH is the basis vector of the lattice dual to KB>:

(558)

aGH"(eGH, 0), a'H"(!A'eGH, E'H) , (559) G where E'H [eGH] are dual to E [e ], i.e. B eIeHH"dH [ E?E(H"d( ]. ' G I G I G ? ' ? ' The heterotic string with the action (551) has an O(d#16, d, Z) ¹-duality symmetry. This group is a subgroup of the following O(d#16, d#16, Z) transformation that preserves the triangular form of E in (553): EPE"(aE#b)(cE#d)\,

a b c

d

3O(d#16, d#16, Z) ,

(560)

and (p , p ) in CBB> transforms as a vector. T-duality is proven [307] to be exact to all orders in 0 * string coupling. The mass of perturbative states for heterotic string on a torus is [322] 1 1 +(p )#2N !1," +(p )#2N !2, , (561) M" 0 * 8j * 8j 0 where N are left- and right-moving oscillator numbers, j is the vacuum expectation value of * 0 the dilaton (or string coupling). We now identify string states with black holes. The mass of the BPS purely electric black holes in heterotic string on ¹, which preserves of the N"4 supersymmetry, is [337,544,558,560,564]: 1 m" a?(M#¸) a@ , (562) ?@ 16j where a is the charge lattice vector on an even, self-dual, Lorentzian lattice K with the O(6, 22) metric ¸ and the subscript (0) denotes asymptotic values. Here, M is the moduli matrix of ¹ de"ned in (121). Under the T-duality, the moduli matrix and the charge lattice transform

By contracting with eGH and E'H, one obtains the background "elds in (551) from E. For example, I ? G?@"2G (E'H)?(E(H)@ and BIJ"2B (eGH)I(eHH)J, etc. '( GH

170


as [560] MPXMX2, KP¸X¸K, X3O(6, 22)

(563)

and the BPS mass (562) is invariant. With a choice of the asymptotic values j"1 and M"I , the mass takes a simple form: #¸) a@"(a ), a? ,(I $¸) a@ . (564) m" a?(I ?@ 0 0* ?@ The string momentum (winding) zero modes are identi"ed with the quantized electric charges of KK (2-form) ;(1) gauge "elds, i.e. a "p . Then, m"M, provided N " [248]: the BPS black 0* 0* 0 holes are identi"ed with the ground states of the right movers. With a further inspection of (561), one "nds N in terms of a [248]: * (565) N !1"((a )!(a ))"a2¸a , * * 0 leading to a2¸a5!2. So, the various BPS black holes in the heterotic string on a torus are identi"ed [248] as string states with the corresponding value of N [or a2¸a]. * Non-extreme black holes are identi"ed with string states with the right movers excited as well. Identi"cation of black holes in other dimensions and in type-II theories with string states is proceeded similarly as above. For type-II string theories, the mass of perturbative string state is 1 1 M" +(p )#2N !1," +(p )#2N !1, . (566) 0 * 8j 0 8j * Since the type-II strings have supersymmetry in both the right- and left-moving sectors, perturbative string states can (i) preserve supersymmetry in both sectors (N "N "1/2), leading to 0 * short supermultiplet; (ii) preserve supersymmetry in one sector, only (N 'N "1/2), leading 0* *0 to intermediate supermultiplet; (iii) break supersymmetry in both sectors (N , N '1/2), leading to 0 * long supermultiplet. Further study of equivalence of string states and extreme black holes, including spins of string states and dipole moments of rotating black holes, is carried out in [237,249].

7.2. BPS, purely electric black holes and perturbative string states In the previous section, we showed that BPS electric black holes in the string low energy e!ective actions are identi"ed with perturbative string states. Thus, it is natural to infer that the microscopic degeneracy of black holes originates from the degenerate string states in a corresponding level. In general, non-extreme black holes are also identi"ed with perturbative string states. However, non-extreme solutions are plagued with (unknown gravitational) quantum corrections and, therefore, the ADM mass cannot be trusted. In fact, the number of states in non-extreme black hole grows with the ADM mass M like &e+ [594], whereas the string state level density grows with the string state mass M as &e+ . Thus, if one is to identify string states with black hole states, one is forced to identify M with M [517,583]. In [583], Susskind attributes the discrepancy to the mass renormalization due to unknown quantum corrections. (See also Section 8.2, where it is discussed that the Bekenstein}Hawking entropy of non-extreme black holes has to be evaluated


171

at the speci"c string coupling at the black hole and microscopic con"guration (D-branes and fundamental string) transition point.) As "rst pointed out by Vafa [583], the BPS solutions do not receive quantum corrections [485] due to renormalization theorem of supersymmetry. Such class of solutions are, therefore, suitable for testing the hypothesis that the statistical origin of black hole entropy is from the degenerate string states with mass given by the ADM mass of black hole. So, one can calculate the `statisticala entropy by taking logarithm of the string level density. This yields the "nite non-zero entropy &(N . However, the `thermala entropy of the BPS purely * electric black holes in heterotic string is zero. In [564], Sen circumvented with problem by postulating that the `thermala entropy of the BPS black hole is not the event horizon area, but the area of a surface close to the event horizon, a so-called `stretcheda horizon [586,587,596]. Although the BPS electric black hole solutions are free of quantum corrections, they receive (classical) stringy a corrections due to the singularity at the event horizon. This leads to the shift of the event horizon by the amount of an order of a. Originally, the stretched horizon is de"ned [587] as the surface where the local Unruh temperature for an observer, who is stationary in the Schwarzschield coordinate, is of the order of the Hagedorn temperature [331]. Namely, it is a surface where the string interactions become signi"cant. In [419,475,582], it is observed that the transverse size of strings diverges logarithmically and "ll up a region at the stretched horizon, melting to form a single string. Thereby, information in the string states is stored and thermalized with black hole environment in the region near the stretched horizon [452,454,475,582,584], and black hole states are in one-to-one correspondence with single string states. So, the statistical entropy is due to degenerate strings states in equilibrium with the black hole background at the stretched horizon [588]. In this section, we summarize [495,564] to illustrate this idea. The electric black hole considered in [564] is a special case of the general solution [178] discussed in Section 4.2.1. But for the purpose of illustrating the idea of perturbative string state and black hole correspondence, we follow Sen's parameterization of solution in terms of left-moving and right-moving electric charges, rather than in terms of KK and 2-form electric charges. 7.2.1. Black hole solution In Sen's notation, the most general non-rotating, electric black hole solution in the heterotic string on ¹, in the Einstein-frame, is [564] D r(r!2m) dt# dr#D(dh#sinh du) , g dxI dxJ"! IJ r(r!2m) D

(567)

where D,r[r#2mr(cosh a cosh b!1)#m(cosh a!cosh b)].

It is argued in [350] that the Bekenstein}Hawking formula for entropy, i.e. (entropy)Jhorizon area), ceases to hold for extreme case and entropy of an extreme black hole is always zero, displaying discontinuity in going from non-extreme to extreme case. This is attributed [350] as being due to di!erence in Euclidean topologies for the two cases. The cure for this discontinuity is proposed in [291], where it is suggested that one has to extremize after quantization, rather than quantizing after extremization.

172


The ADM mass and the electric charges are 1 m(1#cosh a cosh b) , M " & 2G , m g nG sinh a cosh b for 14i422 , Q (2 QG" m g pG\ sinh b cosh a for 234i428 , (2 Q

(568)

where n [p] is an arbitrary 22 [6] component unit vector. The left- and right-handed charges are de"ned as nG 1 !¸) QH" * g m sinh a cosh b , QG , (I GH * 2 (2 Q nG 1 #¸) QH" 0 g m sinh b cosh a , QG , (I GH 0 2 (2 Q

(569)

and the 28-component left- and right-handed unit vectors n and n are similarly de"ned. The * 0 solution (567) is in the frame where the O(6, 22) invariant metric ¸ (127) is diagonal. This parameterization of black hole solution has a convenient form in which only left (right) handed charges are non-zero when b"0 (a"0) with all the parameters "nite. The solution has 2 horizons at r"r "2m, 0. The event horizon area is >\

A" dh du(g g " >"8pm(cosh a#cosh b) . FF PP PP

(570)

The surface gravity at the event horizon is 1 i" lim (gPPR (!g " " . P RR F 2m(cosh a#cosh b) PP>

(571)

7.2.2. Extreme limit and string states The extreme limit is de"ned as a limit where the inner and outer horizons coincide, i.e. mP0. In taking the `non-extremality parametera m to zero, one has to let one (or both) of the boost parameters a and b go to in"nity so that the electric charges (569) do not vanish. Since we are interested in the BPS solutions, we let the ADM mass depend only on the right-handed electric charge. This is achieved by taking the limit bPR and mP0 such that m( "me@ remains as a "nite non-zero constant, while a remaining "nite. In this limit, the ADM mass and the electric charges are 1 m( cosh a , " .1 2G , 1 QG " g nG m( sinh a, * (2 Q * M

1 QG " g nG m( cosh a , 0 (2 Q 0

(572)


173

thereby the ADM mass depends on the right-handed electric charge, only: 1 Q , M " .1 8g 0 Q where G "2. In this limit, the solution has the form (567) with m"0 and , D"r(r#2m( r cosh a#m( ) .

(573)

(574)

The event horizon area (570) is zero in the BPS limit. However, the string states are degenerate. One can circumvent such problem by calculating entropy at the `stretched horizona right above the event horizon. To "nd a location of the stretched horizon, one considers a region close to the event horizon in the `string framea metric: r dS,g dxI dxJK! g dt#g dr#gr(dh#sin h du) Q Q IJ m( Q "!r dtM #dr #r (dh#sin h du) ,

(575)

where r ,g r and tM ,t/m( . Note, in the frame (tM , r , h, ), all the dependence on the other parameters Q has disappeared. One can show that the other background "elds also become independent of the parameters near the event horizon, if one performs a suitable O(6, 22) transformation. Thus, the location of stretched horizon, i.e. the location where higher-order stringy corrections become important, is unambiguously estimated to be located at r "C, a distance of order 1 (in unit of string scale) from the event horizon. In terms of the original coordinate, the stretched horizon is located at r"C/g ,g. Q The stretched horizon area, calculated from (567) with m"0 and (574), is AK4pgm( "4pm( C/g , Q where only the term leading order in g is kept, and therefore the thermal entropy is

(576)

A p m( C S , " . & 4G 2 g Q , To compare this expression with the statistical entropy, one expresses S in terms of electric & charges by using the relation

Q m( "4 M ! * & 8g Q derived from (572):

Q 2pC M ! * . (577) S " & 8g & g Q Now we compare the thermal entropy (577) with the degeneracy of string states. Since string states identi"ed with BPS black holes have the right movers in ground state (N "), the string 0 state degeneracy is from the left movers with N given in terms of electric charges as (details are *

174


along the same line as in Section 7.1):

Q 4 (578) N K M ! * , .1 8g * g Q Q for large Q. So, the statistical entropy associated with the degeneracy of string states is * 8p Q M ! * . (579) S ,ln d(N )K4p(N K 122 * * g & 8g Q Q This entropy expression has the same dependence on M and Q as the thermal entropy (577) .1 * calculated at the stretched horizon, and the two expressions agree if one chooses C"4 in (577). Note, it is crucial that the constant C does not depend on parameters of black holes; otherwise, the dependence of S (579) on Q and M cannot be trusted because of the unknown dependence of 122 * & C on these parameters.

7.3. Near-extreme black holes as string states In the previous section, we saw that thermal entropy of the BPS, non-rotating, electric black holes agrees (up to numerical factor of order one) with statistical entropy associated with the degeneracy of string states, if it is evaluated at the `stretched horizona. However, the rotating black hole case is problematic for the following reasons. Since the electric, rotating black hole (285) in the BPS limit with all the angular momenta non-zero has naked singularity, thermal quantities cannot be de"ned. The BPS limit with a horizon is possible in D56 with at most 1 non-zero angular momentum [366]. Even for this case, not only the event horizon is singular (i.e. the event horizon and the singularity coincide) and has zero surface area, but also the area of the stretched horizon (which is assumed to be independent of parameters of the black hole) is independent of angular momenta. We surmise that this is due to the unknown dependence of the location of the stretched horizon on physical parameters, unlike the non-rotating black hole case. The determination of the stretched horizon location may require understanding of a corrections with rotating black hole as the target space con"guration, which is di$cult to estimate at this point. We propose [184] an alternative way to circumvent the problems of the BPS electric black holes. Instead of de"ning the thermal entropy of the BPS black holes at the stretched horizon, we propose to calculate the thermal entropy of near extreme black holes at the event horizon. Then, the thermal entropy of near-extreme black holes takes suggestive form which can be interpreted in terms of string state degeneracy. We attempt statistical interpretation of such thermal entropy expression by using the conformal "eld theory of p-model with the near-extreme solution as a target space con"guration and with angular momenta identi"ed with [(D!1)/2] ;(1) left-moving world-sheet currents. 7.3.1. Thermal entropy of near-BPS black holes The proper way of taking the near-BPS limit of rotating black holes is to take the limit in such a way that the angular momentum contribution to the thermal entropy is not negligible compared with the contribution of the other terms, while ensuring the regular horizon so that thermal quantities can be evaluated at the event horizon. This is achieved as follows. First of all, the


175

near-BPS limit is de"ned as the limit in which the non-extremal parameter m'0 is very small and the boost parameters d are very large such that the combinations meBG (i"1, 2) remain as "nite, G non-zero constants. Then, as long as l are non-zero, J (287) do not vanish. Second, the requirement G G of the regular event horizon restricts the range of the parameters of the solution [478], e.g. m5"l " for D"4 and m5("l "#"l ") for D"5. For an arbitrary D, we write such a constraint generally as . QQ<J 2 "\

Third, for the thermal entropy to be macroscopically non-negligible, the electric charges have to be very large, i.e. Q<m"O(1). This is required also for the statistical entropy, since the system has to be very large so that statistical quantities approach the exact values. Fourth, while keeping the angular momenta small so that the regular horizon is always ensured as m+0, one has to make sure that angular momenta contribution to entropy is not macroscopically negligible compared with the other terms. This is achieved by taking the limit > 夹FGH"(1/2(G)eGHIJF . IJ 7.5.1.1. General four-dimensional, static, BPS black hole. We consider the case where 4-dimensional transverse part of the metric has the form G "f (x)g where f"0, , (g) and g is GH GH GH a hyper-KaK hler metric with a translational isometry in the x-direction. The D"6 part (u, v, x,2, x) of (612) then takes the special form: ¸"F(x)Ru[RM v#K(x)RM u#2A(x)(RM x#a (x)RM xQ)]#R ln F(x)#¸ , Q , ¸ "f (x)k(x)(Rx#a (x)RxQ)(RM x#a (x)RM xQ)#f (x)k$x)RxQRM xQ , Q Q #b (x)(RxRM xQ!RM xRxQ)#R (x) , Q

(616)

where xQ"(x, x, x) are non-compact coordinates and compact coordinates are x"y and u"y . Here, we chose A in (612) to take the form A "A and A "Aa so that the D"4 metric G Q Q has no Taub-NUT term. The Lagrangian (616) is invariant under T-duality transformations in the x-direction (P P and qP!q): fPk\, kPf \, a b , AP( f k)\A , Q Q

(617)

and in the u-direction (Q Q ): FPK\, KPF\, U(F)PU(K$ .

(618)

186


When A"0, the Lagrangian has remarkable manifest invariance under D"4 S-duality, under which (618) transforms as u x and FPf \, KPk\, fPF\, kPK\ ,

(619)

and under the D"6 string}string duality (GPe\UG, dBPe\U夹dB, UP!U) between the heterotic string on ¹ and the type-II string on K3: FPf \, KPK,

fPF\, kPk .

(620)

Note, the invariance of (616) under the T-duality is manifest only when all the charges associated with 4 harmonic functions F, K, f and k are non-zero. The self-dual case F"K\"f \"k and a "b corresponds to the D"4 Reissner}NordstroK m solution. As expected, the combined Q Q transformation of T-duality and the string}string duality yields the D"4 S-duality (619). One can obtain D"4 black hole solution which is exact to all orders in a by solving (613) with (616) and all the background "elds depending on non-compact transverse coordinates xQ, only. Solutions for background "elds are expressed in terms of harmonic functions f, k, F and K, which satisfy (linear) Laplace equations. Particularly, A is given in terms of harmonic functions by A"q k\#q f k (q " const). If one further assumes the asymptotic #atness condition (i.e. : q "!q . kP1, fP1, AP0 as r"(xQx PR), coe$cients in A are restricted such that q " Q The solutions for background "elds are: Q P Q F\"1# , K"1# , f"1# , r r r

P k\"1# , r

a dxQ"P (1!cos h)du, b dxQ"P (1!cos h)du , Q Q

(621)

r#P q r#(P #P ) , eU"Fe(" , A" ) r#P r#Q r where q,2q (P !P ). Since the resulting (conformal invariance condition) equations are of Laplace-type, one can superpose harmonic functions to obtain multi-center generalization of the above. The D"4 spherically symmetric solution in (200) is obtained by applying the standard KK procedure with all the background "eld in (612) properly identi"ed with those in (611) and setting u"y , v"2t, x "y . 7.5.1.2. General xve-dimensional, rotating, BPS black hole. We consider the case where M-part is (locally) SO(4)-invariant. The D"6 part of chiral null model Lagrangian is again given by (616) with A "(A , 0) and the transverse part ¸ replaced by G K , ¸ "f (x)RxKRM xK#B (x)RxKRM xK#R (x) , , KL

(622)

where m, n,2"1,2, 4, and the "elds are given in terms of a harmonic function f (x) (Rf"0) R "!e R f. By solving the conformal by " ln f, G "fd and H "!2(GGNJe KL KL KLI KLIN J KLIJ J


187

invariance conditions (613) with above AnsaK tze, one obtains Q P f"1# , F\"1# , r r

Q K"1# , r

(623)

r#P c c e "Ff" , A " sin h, A " cos h , P P r#Q r r U

where r,(xKxK and c is related to the angular momenta as J "J "(p/4G )c. Note, the , conformal invariance condition (e\(FKL)"0 is solved by imposing the #at-space anti-selfK > duality condition F "0 on the "eld strength of the potential A , and as a result the 2 angular >KL K momenta are the same. By superposing harmonic functions, one obtains the multi-center generalization of the above. The dimensional reduction along u leads to D"5, rotating, BPS black hole with charge con"guration (Q "Q, Q "Q, P), where P is a magnetic charge of the NS-NS 3-form "eld strength (or an electric charge of its Hodge-dual). The Einstein-frame metric is

c ds"!j(dt#A dxK)#j\ dxK dx "!j dt# (sinh du #cos h du ) # K K r #j\[dr#r(dh#sin h du#cosh du)] r j"(F\Kf )\" . [(r#Q )(r#Q )(r#P)]

(624)

7.5.2. Level matching condition To calculate statistical entropy of black holes, one has to relate macroscopic quantities of black holes to microscopic quantities of perturbative string states through `level matching conditiona [189]. Strictly speaking, level matching process is possible for electric solutions, only, since perturbative string states do not carry magnetic charges and string momentum [winding] modes are matched onto èlectrica charges of KK [2-form "eld] ;(1) "elds. Furthermore, magnetic solutions can be supported without source at the core (cf. magnetic solutions are regular everywhere including the core), since they are topological in character. However, it turns out [174] that the dyonic solution found in [178], which is a `bound statea of fundamental string and solitonic 5-brane, still needs a source for its support and satis"es the same form of level matching condition as the fundamental string. The crucial point in the level matching of such dyonic solutions onto the perturbative string spectrum is that as in the purely magnetic case "elds are perfectly regular near the horizon (or the throat region), making it possible to describe the solutions at the throat region in terms of WZNW conformal model [289,437,484,632]. Since the solution is regular and the dilaton is "nite near the event horizon (implying that the classical a and the string loop corrections are under control), such e!ective WZNW model near the horizon can be trusted. For large magnetic charges (or large level), the theory e!ectively looks like a free p-model for perturbative string theory with the string tension rescaled by magnetic charges. Namely, for large magnetic charges, the dyonic

188


solutions are matched onto perturbative string states with string tension rescaled by magnetic charges. To match background "eld solutions onto the macroscopic string source at the core, one considers the combined action of string p-model and the e!ective "eld theory. Among the equations of motion of the combined action, the relevant parts are the Einstein equations for target space metric, equations of motion for XI and the Virasoro conditions. Requiring that all the solutions are supported by sources, one obtains the level matching condition:

1 p0 E(u), du E(u)"0, (E(u),[F(x)K(u, x)] "0) . V 2pR

(625)

This condition is satis"ed without modi"cation even when solutions carry magnetic charges. 7.5.3. Throat region conformal model and magnetic renormalization of string tension 7.5.3.1. Four-dimensional dyonic solutions. The p-model (616) of dyonic black hole (193) with charge con"guration (P, Q, P, Q),(P , Q , P , Q ) takes the following form near the hor izon (rP0) [174,609,612]:

1 1 dp ¸ " dp(e\XRuRM v#Q Q\RuRM u) I" P pa pa

P P # dp(RzRM z#Ry RM y #RuRM u#RhRM h!2 cos hRy RM u) . pa

(626)

This is the S¸(2, R);S;(2) WZNW model with the level i"(4/a)P P . (Since the level has to be an integer, one has the quantization condition 4P P /a3Z.) Here, the coordinates are de"ned as z,ln(Q /r)PR, u ,(Q\Q P P )\u, v ,(Q Q\P P )\v, and y ,P\y #u. For large P (or i), the transverse (o, y , u, h) part of (626) looks like a free theory of perturbative string with the string tension ¹"1/(2pa) renormalized by P : 1 P P P P 1 P " " , aR a a a ,

(627)

where R "(a is the radius of the internal coordinate associated with P . 7.5.3.2. Five-dimensional dyonic solution. The p-model with the target space con"guration given by the 3-charged, BPS, non-rotating black hole, i.e. (263) with J"0, takes the following form in the limit rP0 [609,612]:

1 1 dp ¸ " dp(e\XRuRM v#Q Q\RuRM u) I" P pa pa #

P dp(RzRM z#Ru RM u #Ru RM u #RhRM h!2 cos hRu RM u ) , 4pa

(628)


189

where z,ln(Q /r)PR. This is the S¸(2, R);S;(2) WZNW model with the level i,(1/a)P. In the limit of large P or large i, the transverse part (o, u , u , h) of (628) reduces to free perturbative string theory with the renormalized string tension: 1 P 1 P " . 4a a a ,

(629)

7.5.4. Marginal deformation The degeneracy of micro-states responsible for statistical entropy is traced to the degrees of freedom associated with oscillations or marginal deformations around the classical solutions. The marginal deformations lead to a family of all the possible solutions (obeying conformal and BPS conditions) with the same values of electric/magnetic charges but di!erent short distance structures that depend on a choice of oscillation pro"le function. Thus, one has to consider the region near the horizon (at r"0) to determine the microscopic degrees of freedom, since in this region degeneracy of solutions is lifted. General chiral null model action which represents deformation from the classical BPS solutions in Section 7.5.1 and preserves the BPS and conformal invariance properties is given by (612) with K and A "(A , A ) having an additional dependence on u, where u,z!t with the longitudinal G K ? direction z satisfying the periodicity condition z,z#2pR. Here, A &q (u)/r and A "q (u)/r K K ? ? (r,x x ) are respectively `deformationsa in the non-compact xK and the compact y? directions. K K On the other hand, the perturbation K(u, x)"h (u)xK#k(u)/r does not contribute to the K degeneracy, since h (u)xK drops out and k(u) has zero mean value. K The perturbations K, A and A represent various `left-movinga waves propagating along the ? K string and are invisible far away from the core. The mean values q (u) and q(u) are related to the ? K oscillation numbers of the macroscopic string at the core as p pa NK" q (u), N?" q(u) , (630) * * 16G K 16G ? , , thereby contributing to the microscopic degrees of freedom. These marginal deformations do not contribute to the microscopic degeneracy of black holes with the same order of magnitude [612]. This can be inferred from the fact that the classical BPS black holes are solutions of both heterotic and type-II string. Namely, although the thermal entropy is the same whether one embeds the solutions within heterotic or type-II string, one faces the discrepancy in factor of 2 in the statistical entropies within the two theories, if one takes the degeneracy contributions of all the oscillators to be of the same order of magnitude. In fact, as can be seen from the conformal invariance condition (e\(FGH )"0, the perturbations A in the G > ? compact directions y are decoupled from the non-trivial non-compact parts of the solution. On ? the other hand, the perturbations A in the non-compact directions x are non-trivially coupled to K K the magnetic harmonic functions, with the net e!ect being the rescaling of q (u) terms by the K The chiral null model corresponding to rotating black hole (624) also approaches the S¸(2, R);S;(2) WZW model in the throat limit. The requirements that the level i"P/a is an integer and u "u #2cP\Q\u (u ,u #u ) has the period 4p lead to the quantization of P and J: P"ai and J"iwl with k, l3Z (w" string winding number). The regularity of the underlying conformal model requires that J(imw, i.e. the `Regge bounda.

190


magnetic charges. (This is related to the scaling of the string tension in the transverse directions by the magnetic charge(s).) So, the marginal deformation contributions from the compact directions, which are di!erent for the two theories, are suppressed relative to those of non-compact directions by the factor of the inverse of magnetic charge(s), thereby negligible for a large magnetic charge(s) or the large level i. Only the marginal deformations from the non-compact directions and the compact direction associated with non-zero magnetic charge(s), which are common for both theories, have the leading contribution to the degeneracy. Only these 4 string coordinates get their tension e!ectively rescaled by the magnetic charge(s). Furthermore, the marginal deformations on the original black hole solutions have to be only left-moving (i.e. depend only on u, not on v) so that marginally deformed p-models are conformaly invariant. This is related to the `chirala condition on the p-model; only left-moving deformations lead to supersymmetric action. Thus, for large magnetic charge(s), the statistical entropy calculations within heterotic and type-II strings agree. As pointed out, the marginal deformations A (u, x)"q (u)/r in the compact directions ? ? y contribute to the microscopic degeneracy to sub-leading order (suppressed by the inverse of ? magnetic charge(s)), which can be neglected for large value of magnetic charge(s). But the zero modes q of the Fourier expansions of q (u)"q #q (u) (q (u) denoting the oscillating parts) ? ? ? ? ? produce additional left-handed electric charges [174] of D"6 strings. Namely, the internal marginal deformation A (u)"q (u)/r on the p-model associated with D"4 4-charged BPS black ? ? hole (193) leads to 5-charged BPS black hole solution (621) with the zero mode q corresponding to ? an additional charge parameter q. The mean oscillation values q (u) of the marginal deformations A (u, x)&q (u)/r K K K (q (u)"q #q (u) with q and q (u) respectively denoting the zero modes and oscillating parts) K K K K K in the non-compact directions x contribute to N to the leading order. Meanwhile, the zero modes K * q have an interpretation as angular momenta [610] of black holes. Namely, the rotational K marginal deformation corresponding to S;(2) Cartan current deformation A (u, x) dxK" K (c(u)/r)(sin h du #cos h du ) on the p-model action of the D"5 3-charged non-rotating solution leads to the rotating solution (624) [609,610]. To calculate the statistical entropy associated with degeneracy of solutions (i.e. all the possible marginal deformations of the original classical solution), one has to determine N of the macro* scopic string at the core. For this purpose, we write the marginally deformed p-model action (612) in the throat region in the form:

1 dp(marginal deformation terms) I"I# P pa

1 dp [e\XRuRM v#E(u)RuRM u]#other terms , " pa

(631)

where I stands for the throat limit WZNW models (626) and (628) of the (undeformed) classical solutions. First, for the D"4 dyon, the marginal deformation (612) gives rise to the following expression for E(u): E(u)"Q Q\!(P P )\Q\q(u)!Q\q(u) L ? "(P P )\Q\[Q Q P P !q(u)!P P q(u)] . L ?

(632)


191

Note, since the string tension a (627) of the transverse parts is rescaled by the magnetic charges, , the coe$cient in front of the term q(u) is rescaled by (P P )\. Applying the level matching L condition E(u)"0 (625), one "nds that q(u)"P P (Q Q !q ), where q denote the zero modes L ? ? of the oscillations q (u) in the compact directions. Thus, the statistical entropy is [174] ? p (P P (Q Q !q ) , (633) S +2p(N " ? * 2G , in agreement with the thermal entropy. Second, we consider D"5 solutions. For the non-rotating solutions, the marginal deformation (612) leads to E(u)"Q Q\#Q\k(u)!P\Q\q (u)#O(P\) . (634) K Applying the level matching condition E(u)"0, one "nds that q (u)"Q Q P for large P, which K reproduces the thermal entropy [609,612]: p (Q Q P . (635) S +2p(N " * 2G , For the rotating solution, one introduces the marginal deformation A in a non-compact K direction. Then, one has E(u)"Q Q\!P\Q\c(u)"P\Q\[Q Q P!c(u)] , (636) where c(u)"c#c(u) with c and c(u) respectively denoting zero and oscillating modes of c(u). From the level matching condition E(u)"0, one has c(u)"Q Q P!c, reproducing entropy of the rotating black hole [609,610] in the limit of P +P and Q large: p (Q Q P!c . (637) S +2p(N " * 2G , 8. D-branes and entropy of black holes Past year or so has been an active period for investigation on microscopic origin of black hole entropy. The construction of general class of BPS black hole solution in heterotic string on ¹ [178] motivated renewed interest [441] in the study of black hole entropy within perturbative string theory. The explicit calculation of statistical entropy of BPS solution in [178] by the method of WZNW model in the throat region of black hole reproduced the Bekenstein}Hawking entropy. Realization [498] that D-branes in open string theory can carry R-R charges motivated the explicit D-brane calculation of statistical entropy of non-rotating BPS solution in D"5 with 3 charges [580]. This is generalized to rotating black hole [98] in D"5, near extreme black hole [109,368] in D"5 and near extreme rotating black hole [95] in D"5. Meanwhile, the WZNW model approach was generalized to the case of non-rotating BPS D"5 black hole [609] and rotating BPS D"5 black hole [612]. D-brane approach was soon extended to D"4 cases: non-rotating BPS case in [392,471], near extreme case in [361] and extreme rotating case in [361]. Later, it is shown [365] that the microscopic counting argument in string theory can be extended even to

192


non-extreme black holes as well, provided the entropy is evaluated at the proper transition point of black hole and D-brane (or perturbative string) descriptions. In this section, we review the recent works on D-brane interpretation of black hole entropy (for other reviews, see Refs. [468,360]). With realization [498] that R-R charges, which were previously known to be decoupled from string states, can couple to D-branes [193], it became possible to do conformal "eld theory of extended objects (p-branes) within string theories and to perform counting [90,91,566,567,619,620] of string states that carries R-R charges as well as NS-NS charges. To apply D-brane techniques to the calculation of microscopic degeneracy of black holes, one has to map non-perturbative NS-NS charges of the generating black hole solutions of heterotic string on tori to R-R charges by applying subsets of U-duality transformations. In D-brane picture of black holes [109], the microscopic degrees of freedom are carried by oscillating open strings which are attached to D-branes. Whereas the e!ect of magnetic charges in the chiral null model and Rindler geometry approaches is to rescale the string tension, the e!ect of R-R charges on open strings in the D-brane description is to alter the central charge (i.e. the bosonic and fermionic degrees of freedom) of open strings from the free open string theory value. In the D-brane picture of [109] (see also Ref. [196]), the number of degrees of freedom of open strings is increased relative to the free open string value because of an additional factor (proportional to the product of D-brane charges) related to all the possible ways of attaching the ends of open strings to di!erent D-branes. So, the net calculation results of statistical entropy in both descriptions are the same. This chapter is organized as follows. In Section 8.1, we summarize the basic facts on D-branes necessary in understanding D-brane description of black holes. In Section 8.2, we discuss the D-brane embeddings of black holes. The D-brane counting arguments for the statistical entropy of black holes are discussed in Section 8.3. 8.1. Introduction to D-branes We discuss basic facts on D-branes necessary in understanding D-brane description of black holes. Comprehensive account of the subject is found, for example, in [193,391,445,498,501,504], which we follow closely. The basic knowledge on string theories is referred to [316,535]. Each end of open strings can satisfy two types of boundary conditions. Namely, from the boundary term (1/2pa) M dp dXIR X , where RM is the boundary of the worldsheet M swept by L I . an open string and dXI[R X ] is the variation (the derivative) of bosonic coordinates XI parallel to L I (normal to) RM, in the variation (with respect to XI) of the worldsheet action, one sees that the ends of the string either can have zero normal derivatives R XI"0 , L called Neumann boundary condition, or have "xed position in target spacetime XI"constant ,

(638)

(639)

called Dirichlet boundary condition. In order for the T-duality to be an exact symmetry of the string theory, open string has to satisfy both the Neumann and Dirichlet boundary conditions [193]. Under the T-duality of open string theory with the coordinate XG"XG (z)#XG (z ) compacti"ed on S of radius R , R PR"1/R and 0 * G G G G


193

XGP>G"XG (z)!XG (z ). So, the Neumann and the Dirichlet boundary conditions get inter0 * changed under T-duality:

Rz Rz Rz Rz RXG! RM XG" R>G# RM >G"R >G , R XG" O L Rq Rq Rq Rq

(640)

where q is the worldsheet time coordinate, which is tangent to RM. Starting from the D"10 open string with the Neumann boundary conditions and with the coordinates XG (i"p#1,2, 9) compacti"ed on circles of radii RG, one obtains open string theory with the ends of the dual coordinates >G con"ned to the p-dimensional hyperplane in the R P0 limit (or the decompacti"caG tion limit, i.e. 1/R PR, of the dual theory). Such p-dimensional hyperplane is called D-brane G [498]. A further T-duality in the direction tangent (orthogonal) to a Dp-brane results in a D(p!1)-brane (D(p#1)-brane). D-brane is a dynamical surface [498] with the states of open strings (attached to the D-brane) interpreted as excitations of #uctuating D-brane. The massless bosonic excitation mode in the open string spectrum is the photon with the vertex operator < "A R X+, where R is the + R R derivative tangent to RM. So, the bosonic low energy e!ective action is that of N"1, D"10 Yang-Mills theory [637]:

1 dx Tr[F FIJ] . IJ 2g

(641)

In the T-dual theory, the vertex operator < of the photon is decomposed into < " N A (XI)R XI and < " u (XI)R XG, corresponding respectively to ;(1) gauge boson GN G N I I R P and scalars on the p-brane worldvolume. The scalars u are regarded as the collective coordinates G for transverse motions of the p-brane. The bosonic low energy e!ective action of the T-dual theory is, therefore, obtained by compactifying (641) down to p#1 dimensions [637]. In this action, the worldvolume scalars u have potential term (p)!>(0)"

p

dp R >"(2p#h !h )R , N H G

(644)

where R"a/R is the radius of the circle in the dual theory. Note, without the Chan-Paton factors taken into account, both ends of open strings of the dual theory are con"ned to the same hyperplane up to the integer multiple of periodicity 2pR of the dual coordinate >. The similar argument can be made for unoriented open string theories. With SO(N) Chan-Paton symmetry, the Wilson line can be brought to the form: diag(h ,!h ,2, h ,!h ) . , ,

(645)

Note, in the dual coordinate >K, worldsheet parity reversal symmetry (zP!z ) of the original theory is translated into the product of worldsheet and spacetime parity operations. Since unoriented strings are invariant under the worldsheet parity, the T-dual spacetime is a torus modded by spacetime parity symmetry Z . The "xed planes >K"0, pR under spacetime parity symmetry are called orientifolds [193]. Away from the orientifold plane, the physics is that of oriented open strings, with a string away from the orientifold "xed plane being related to the string at the image point. Open strings can be attached to the orientifolds, but the orientifolds do not correspond to the dynamic surface since the projection X"#1 [355,356,509] removes the open string states corresponding to collective motion of D-branes away from the orientifold plane. In this T-dual theory, there are N/2 D-branes on the segment 04>K(pR and the remaining N/2 are at the image points under Z . Open strings can stretch between pairs of Z re#ection planes, as well as between di!erent planes on one side.


195

With a single coordinate X compacti"ed on S of radius R, the mass spectrum of the dual open string is

M"

[2pn#(h !h )]R 1 G H # (N!1) . 2pa a

(646)

Thus, the massless states arise in the ground state (N"1) with no winding mode (n"0) and both ends of the string attached to the same hyperplane (h "h ): G H aI "k, ii2, , where SIJ"! t I tJ are the fermionic part of G P \P P the D"10 Lorentz generators. These 2 representations are physically equivalent for open strings. The GSO projection picks out 8 and, therefore, the ground state of the open string theory is 8 8 , Q T Q forming a vector multiplet of D"10, N"1 theory. Including the Chan-Paton factors, the gauge group G of the N"1, D"10 super-Yang-Mills theory is ;(N) [SO(N) or ;Sp(N)] for an oriented [an unoriented] open string theory. For an open string theory with 9!p coordinates compacti"ed, the massless spectrum of Dp-brane worldvolume theory of dualized open string is described by the D"10, N"1 supersymmetric gauge theory compacti"ed to D"p#1. We brie#y discuss some aspects of type-II closed string relevant for understanding Dp-branes of open string theory. For type-II string, i.e. the closed string theory with supersymmetry on both leftand right-moving modes, the two choices of the GSO projections in the R sector are not equivalent. So, there are two types of type-II theories de"ned according to the possible inequivalent choices of the GSO projections on the left- and the right-moving modes. The massless sectors of these two type-II theories are Type IIA: (8 8 )(8 8 ) , T Q T A (649) Type IIB: (8 8 )(8 8 ) . T Q T Q The massless modes 8 8 in the NS-NS sector of the both theories are the same: dilaton, gravitino T T and the 2-form "eld. In the R-R sector, the massless modes 8 ;8 [8 ;8 ] of type-IIA [type-IIB] Q A Q Q theory are 1- and 3-form potentials (0-, 2- and self-dual 4-form potentials) [275]. The massless modes in NS-R and R-NS sectors contain 2 spinors and 2 gravitinos of the same (opposite) chirality for the type-IIB (type-IIA) theory. When an oriented type-II theory with a coordinate compacti"ed on S is T-dualized [193,216], the chirality of the right-movers gets reversed. So, when odd (even) number of coordinates in type-IIA/B theory are T-dualized, one ends up with type-IIB/A (type-IIA/B) theory. The e!ect of òdda T-duality, which exchanges type-IIA and type-IIB theories, on massless R-R "elds is to add (remove) the indices (of (p#1)-form potential) corresponding to the T-dualized coordinates, if those indices are absent (present) in the (p#1)-form potential. For example, the T-duality on B [B ] along xM (kOoOl) produces B [B ]. IJ IM IJM I A worldsheet parity symmetry X in a closed string, de"ned as pP!p or zPz , interchanges left- and right-moving oscillators. The unoriented closed string is de"ned by projecting only even parity states, i.e. X"t2"#"t2, as in the open string case. When type-II string is coupled to open superstring (type-I string), the orientation projection of type-I string picks up only one linear combination of 2 gravitinos in type-II theory, resulting in an N"1 theory. The only possible consistent coupling of type-I and closed superstring theories is between (unoriented) SO(32) type-I theory [108,503] and unoriented N"1 type-II theory. But in the T-dual theory, type-II theories without D-branes are invariant under N"2 supersymmetry, with orientation projection relating a gravitino state to the state of the image gravitino. The chiralities of these 2 gravitinos are the same

T-duality on the type-II theory leads to 16 D-branes on a ¹ /Z orbifold. (The restriction to 16, i.e. SO(32) gauge \N symmetry, comes from the conservation of R-R charges.) However, in non-compact space, one can have a consistent theory with an arbitrary number of D-branes.


197

(opposite) if even (odd) number of directions are T-dualized. In the presence of D-branes, only one linear combination of supercharges in the T-dual type-II theory is conserved, resulting in a theory with 1/2 of supersymmetry broken (N"1 theory), i.e. the BPS state [498]. For this case, the leftand right-moving supersymmetry parameters (of the T-dual type-II theory) are constrained by the relation [326,501,504] e "C2CNe . (650) 0 * The conserved charges carried by D-branes are charges of the antisymmetric tensors in the R-R sector. The worldvolume of a D p-brane naturally couples to a (p#1)-form potential in the R-R sector, with the relevant space-time and D p-brane actions given by

1 G 夹G #ik N> N 2 N>

C , (651) N> NU where k"2p(4pa)\N is the D p-brane charge [498]. The worldvolume action is given by the N following Dirac}Born}Infeld type action [269,445] describing interaction of the world-brane ;(1) vector "eld and scalar "elds with the background "elds [445]: !¹

N

dN>m e\P det(G #B #2paF ) , ?@ ?@ ?@

(652)

where ¹ is the D p-brane tension [199,311], and G and B are the pull-back of the spacetime N ?@ ?@ "elds to the brane. In the amplitudes of parallel D-brane interactions, terms involving exchange of the closed string NS-NS states and the closed string R-R states cancel [498], a reminiscence of no-force condition of BPS states. Furthermore, the D-brane tension, which measures the coupling of the closed string states to D-branes, has the g\ behavior [445], a property of R-R p-branes [367,381,601]. The Q "eld strengths which couple to D p- and D(6!p)-branes are Hodge-dual to each other, and the corresponding conserved R-R charges are subject to the Dirac quantization condition [482,498,591] k k "2pn. \N N 8.2. D-brane as black holes In Section 8.1, we observed that D p-branes have all the right properties of the R-R p-branes in the e!ective "eld theories. As solutions of the e!ective "eld theories, which are compacti"ed from the D"10 string e!ective actions, black holes can be embedded in D"10 as bound states [224,448,637] of D p-branes. Here, p takes the even (odd) integer values for the type-IIA [type-IIB] theory. Such p-branes of the e!ective "eld theories correspond to the string background "eld con"gurations [104,105] (with the p-brane worldvolume action being the source of (p#1)-form charges) to the leading order in string length scale l "(a and describe the long range "elds away Q from D p-branes. As long as the spacetime curvature of the soliton solutions (at the event horizon in string frame) is small compared to the string scale 1/l, the e!ective "eld theory solutions can be Q trusted, since the higher-order corrections to the spacetime metric is negligible. The e!ective "eld theory metric description of solitons in superstring theories is valid only for length scales larger than a string. The D-brane picture of black holes, or more generally black p-branes, is as follows.

198


The string-frame ADM mass M of p-branes carrying NS-NS electric charge [187,380], NS-NS magnetic charge [577] and R-R charge [381] behaves as &1,&1/g and &1/g (in the unit where Q Q l &1), respectively. Since the gravitational constant G is proportional to g, the gravitational Q , Q "eld strength (JG M) of the NS-NS electric charged and the R-R charged p-branes vanishes as , g P0. Namely, in the limit g P0, strings live in the #at spacetime background. Note, in the limit Q Q g P0, the description of R-R charged con"gurations in terms of black p-branes is not valid, since Q the size of the p-brane horizon (A&g) is smaller than D-brane size; the black hole is surrounded Q by a halo which is large compared to its Schwarzschield radius. In the limit g N potential charge and can be estimated by comparing the ADM mass of (653) to the mass of D-brane state carrying one unit of the (p#1)-form potential charge. The Killing spinors of this solution are constrained by [367]: e "C2CNe , e "!C2CNe , (654) * * 0 0 where e denotes the left/right handed chiral spinor (Ce "e ). One can construct solutions *0 *0 *0 for bound states of p-branes by applying the intersection rules. (See Section 6.2.2 for details on intersection rules.) In particular, the dilaton is the product of individual factors associated with those of the constituent p-branes: e\P"f N \2f NI I\. N N To add a momentum along an isometry direction xG, one oscillates p-brane so that it carries traveling waves along the xG-direction. (See Section 7.4.1 for the detailed discussion on the construction of such solutions.) At a long distance region, the solutions approach the form where all the oscillation pro"le functions are (time or phase) averaged over. So, the long-distance region p-brane solution carrying a momentum along the xG-direction is obtained by just imposing SO(1, 1) boost among the coordinates (x, xG), with the net e!ect on the metric being the following substitution: !dt#dxP!dt#dx#k(dt!dx ) , (655) G G G where k"cN/r\N with N interpreted as a momentum along the xG-direction. The momentum N along the xG-direction adds one more constraint on the Killing spinor: e "CCGe , e "CCGe . (656) 0 0 * * In general, the intersecting n p-branes preserve at least 1/2L of supersymmetry; since a single p-brane breaks 1/2 of supersymmetry with one spinor constraint (654), as one increases the number of constituents more supersymmetry get broken. One obtains BPS con"gurations if spinor constraints of constituent p-branes are compatible with non-zero spinor e . The intersecting D p0* and D p-branes preserve 1/2 of supersymmetry i! p"p mod 4 [224]. All the supersymmetries are broken when the dimension of the relative transverse space is neither 4 nor 8. When there is

It is shown [431] that the stringy BPS black holes with non-zero horizon area are not possible for D56. This can be seen from the explicit solutions discussed in Section 4.5.

200


a momentum in the xG-direction, the additional Killing spinor constraint (656) breaks 1/2 of the remaining supersymmetry. Black holes in lower dimensions are obtained by compactifying (intersecting) p-branes in D"10. In the language of p-branes, this compacti"cation procedure corresponds to wrapping p-branes along the cycles of compact manifold. Since the compacti"ed space is very small, the con"guration looks point-like (0-brane) in lower dimensions. In the following subsections, we discuss various D p-brane embeddings of D"4, 5 black holes having the regular BPS limit with non-zero horizon area. 8.2.1. Five-dimensional black hole We discuss D"5 type-IIB black hole originated from intersecting Q D1-branes (along x) and Q D5-branes (along x,2, x) with a momentum P #owing in the common string direction [109], i.e. the x-direction. The 1- and 5-brane charges are electric and magnetic charges of the R-R 2-form "eld, and the momentum corresponds to the KK electric charge associated with the metric component G. To obtain a black hole in D"5, one wraps Q D1-branes around S (along x) of radius R and wrap Q D5-branes around ¹"¹;S. Here, ¹ has coordinates (x ,2, x ) and volume

. (675) M" S $ / >

This conformal "eld theory has central charge c"6(Q#1), since the real dimension of this $ manifold is 2(Q#2). $ First, we consider the non-rotating black hole (671) with the thermal entropy (672) [580]. The statistical entropy of the BPS solutions is given in terms of degeneracy d(n, c) of conformal states with the left-moving oscillators at level ¸ "n (n (with NS-type quantiz> ation in ND directions of the R sector) which is positive-chiral under the external group SO(1, 1);SO(4) LSO(1, 5) (thereby a spacetime fermion). # Here, the spatial rotation group SO(4) (external to the D-brane con"guration) is isomorphic to # S;(2) ;S;(2) , which is identi"ed with the symmetry of (4, 4) superconformal theories. This is 0 * related to the fact that worldsheet fermions, which carry angular momenta, manifest themselves as spacetime fermions. The ;(1) LS;(2) charges F are related to angular momenta J of the *0 *0 0* spatial rotational group SO(4) as in (678). # The e!ect of angular momenta is to reduce the total oscillation numbers [98]: 3F 3F N "N ! *, N "N ! 0 , 0 0 * * 2c 2c

(689)

where N (N ) are the total oscillation numbers of states that do not carry the ;(1) charges *0 *0 *0 (have the ;(1) charges F ). To obtain the statistical entropy of black holes, one plug this *0 *0 expression for N into (680). *0 For non-extreme or near-extreme black holes, which do not saturate the BPS bound, the right moving as well as the left moving supersymmetries are preserved, and thereby F are both 0* non-zero. The statistical entropy of near-extreme black holes is:

c S &2p ((N #(N ) , * 0 6

(690)

where N are given in (689). *0 However, for D"5 BPS black holes, F "0 (leading to J "J ":J) since only the left-moving 0 supersymmetry survives (i.e. (0, 4) superconformal theory). So, for BPS black holes, N "N "0 0 0 and N "N !(6/c)J, leading to the statistical entropy [361]: * *

c c S &2p (N "2p N !J . * 6 6 *

(691)

212


For the D"5 rotating black hole with charge con"guration described in the above, c"6Q Q and N "N. So, the statistical entropy (691) correctly reproduces the Bekenstein}Hawking * entropy: S "2p(NQ Q !J. To obtain D"4 rotating black hole with regular extreme limit, one "rst applies T-duality along the common string direction of the intersecting D 1- and D 5-branes with open strings wound around the common string direction. The resulting con"guration is bound state of D 2- and D 6-branes with momentum #owing along one of the common intersection directions. To have non-zero horizon area in the BPS limit, one adds solitonic 5-brane with charge Q . Here, D 6-branes, D 2-branes and solitonic 5-brane, respectively, wrap around ¹"¹;S;S, S;S and ¹;S, and momentum #ows along S. Since the right moving supersymmetry breaks in the presence of solitonic 5-branes, the ;(1) charge F in the right moving sector vanishes (i.e. 0 J "J "J). Particularly, D"4 extreme rotating black hole corresponds to the minimum energy con"guration which is regular with non-zero angular momentum, which happens when N "0 * [361]. So, the left movers are constrained to carry angular momentum, only. In this case, N "N 0 0 and N "6J/c. By plugging these into (683), one obtains * a 6 N " J! [p!p] , * 0 c 4 0 leading to the statistical entropy:

c ac S &2p (N "2p J! [p!p] . 0 * 6 24 0

(692)

For the D-brane bound state corresponding to the D"4 black hole described above, c"6Q Q Q (since there are 4Q Q Q species of bosons and fermions) and (a/4)[p!p]"N is the total * 0 momentum mode of open strings around S. So, the statistical entropy S "2p(J#NQ Q Q agrees with the Bekenstein}Hawking entropy. For further reading, see Refs. [15,16,28,31,53,78,153,172,176,256,402,417,433,502,508,616].

Acknowledgements I would like to thank M. Cvetic\ for suggesting me to write this review paper, for discussion during the initial stage of the work and for proof-reading part of the review. Part of the work was done while the author was at University of Pennsylvania. The work is supported by DOE grant DE-FG02-90ER40542.

The extra factor of Q comes from the fact that whenever D 2-branes (which intersect the solitonic 5-brane along S) cross the solitonic 5-brane they can break up and ends separate in di!erent positions in ¹. Thus, Q D 2-branes break up into Q Q D 2-branes.


213

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J. Main/Physics Reports 316 (1999) 233}338

USE OF HARMONIC INVERSION TECHNIQUES IN SEMICLASSICAL QUANTIZATION AND ANALYSIS OF QUANTUM SPECTRA

JoK rg MAIN Institut fu( r Theoretische Physik I, Ruhr-Universita( t Bochum, D-44780 Bochum, Germany


233


Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra JoK rg Main Institut fu( r Theoretische Physik I, Ruhr-Universita( t Bochum, D-44780 Bochum, Germany Received June 1998; editor: J. Eichler

Contents 1. Introduction 1.1. Motivation of semiclassical concepts 1.2. Objective of this work 1.3. Outline 2. High precision analysis of quantum spectra 2.1. Fourier transform recurrence spectra 2.2. Circumventing the uncertainty principle 2.3. Precision check of the periodic orbit theory 2.4. Ghost orbits and uniform semiclassical approximations 2.5. Symmetry breaking 2.6. expansion of the periodic orbit sum 3. Periodic orbit quantization by harmonic inversion 3.1. A mathematical model: Riemann's zeta function

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3.2. Periodic orbit quantization 3.3. The three-disk scattering system 3.4. Systems with mixed regular-chaotic dynamics 3.5. Harmonic inversion of cross-correlated periodic orbit sums 3.6. expansion for the periodic orbit quantization by harmonic inversion 3.7. The circle billiard 3.8. Semiclassical calculation of transition matrix elements for atoms in external "elds 4. Conclusion Acknowledgements Appendices References

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Abstract Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows one to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking e!ects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories } for both the Berry}Tabor formula and Gutzwiller's trace formula } and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a "nite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for various open and bound systems with underlying regular, chaotic, and even mixed classical dynamics. The e$ciency of the method is increased, i.e., the number of orbits required for periodic orbit quantization is reduced, when the harmonic inversion technique is generalized to the analysis of 0370-1573/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 3 1 - 8

J. Main / Physics Reports 316 (1999) 233}338

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cross-correlated periodic orbit sums. The method provides not only the eigenenergies and resonances of systems but also allows the semiclassical calculation of diagonal matrix elements and, e.g., for atoms in external "elds, individual non-diagonal transition strengths. Furthermore, it is possible to include higher-order terms of the expanded periodic orbit sum to obtain semiclassical spectra beyond the Gutzwiller and Berry}Tabor approximation. 1999 Elsevier Science B.V. All rights reserved. PACS: 05.45.#b; 03.65.Sq Keywords: Spectral analysis; Periodic orbit quantization

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1. Introduction 1.1. Motivation of semiclassical concepts Since the development of quantum mechanics in the early decades of this century quantum mechanical methods and computational techniques have become a powerful tool for accurate numerical calculations in atomic and molecular systems. The excellent agreement between experimental measurements and quantum calculations has silenced any serious critics on the fundamental concepts of quantum mechanics, and there are nowadays no doubts that quantum mechanics is the correct theory for microscopic systems. Nevertheless, there has been an increasing interest in semiclassical theories during recent years. The reasons for the resurgence of semiclassics are the following. Firstly, the quantum mechanical methods for solving multidimensional, nonintegrable systems generically imply intense numerical calculations, e.g., the diagonalization of a large, but truncated Hamiltonian in a suitably chosen basis. Such calculations provide little insight into the underlying dynamics of the system. By contrast, semiclassical methods open the way to a deeper understanding of the system, and therefore can serve for the interpretation of experimental or numerically calculated quantum mechanical data in physical terms. Secondly, the relation between the quantum mechanics of microscopic systems and the classical mechanics of the macroscopic world is of fundamental interest and importance for a deeper understanding of nature. This relation was evident in the early days of quantum mechanics, when semiclassical techniques provided the only quantization rules, i.e., the WKB quantization of one-dimensional systems or the generalization to the Einstein}Brillouin}Keller (EBK) quantization [1}3] for systems with n degrees of freedom. However, the EBK torus quantization is limited to integrable or at least near-integrable systems. In non-integrable systems the KAM tori are destroyed [4,5], a complete set of classical constants of motion does not exist any more, and therefore the eigenstates of the quantized system cannot be characterized by a complete set of quantum numbers. The `breakdowna of the semiclassical quantization rules for non-regular, i.e., chaotic systems was already recognized by Einstein in 1917 [1]. The failure of the òlda quantum mechanics to describe more complicated systems such as the helium atom [6], and, at the same time, the development and success of the `moderna wave mechanics are the reasons for little interest in semiclassical theories for several decades. The connection between wave mechanics and classical dynamics especially for chaotic systems remained an open question during that period. 1.1.1. Basic semiclassical theories 1.1.1.1. Chaotic systems: Gutzwiller1s trace formula. The problem was reconsidered by Gutzwiller around 1970 [7,8]. Although a semiclassical quantization of individual eigenstates is, in principle, impossible for chaotic systems, Gutzwiller derived a semiclassical formula for the density of states as a whole. Starting from the exact quantum expression given as the trace of the Green's operator he replaced the exact Green's function with its semiclassical approximation. Applying stationary phase approximations he could "nally write the semiclassical density of states as the sum of a smooth part, i.e., the Weyl term, and an oscillating sum over all periodic orbits of the corresponding classical system. For this reason, Gutzwiller's theory is also commonly known as


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periodic orbit theory. Gutzwiller's semiclassical trace formula is valid for isolated and unstable periodic orbits, i.e., for fully chaotic systems with a complete hyperbolic dynamics. Examples of hyperbolic systems are n-disk repellers, as models for hyperbolic scattering [9}15], the stadium billiard [16}18], the anisotropic Kepler problem [8], and the hydrogen atom in magnetic "elds at very high energies far above the ionization threshold [19}21]. Gutzwiller's trace formula is exact only in exceptional cases, e.g., for the geodesic #ow on a surface with constant negative curvature [8]. In general, the semiclassical periodic orbit sum is just the leading order term of an in"nite series in powers of the Planck constant, . Methods to derive the higher-order contributions of the

expansion are presented in [22}24]. 1.1.1.2. Integrable systems: the Berry}¹abor formula. For integrable systems a semiclassical trace formula was derived by Berry and Tabor [25,26]. The Berry}Tabor formula describes the density of states in terms of the periodic orbits of the system and is therefore the analogue of Gutzwiller's trace formula for integrable systems. The equation is known to be formally equivalent to the EBK torus quantization. A generalization of the Berry}Tabor formula to near-integrable systems is given in [27,28]. 1.1.1.3. Mixed systems: uniform semiclassical approximations. Physical systems are usually neither integrable nor exhibit complete hyperbolic dynamics. Generic systems are of mixed type, characterized by the coexistence of regular torus structures and stochastic regions in the classical PoincareH surface of section. A typical example is the hydrogen atom in a magnetic "eld [29}31], which undergoes a transition from near-integrable at low energies to complete hyperbolic dynamics at high excitation. In mixed systems the classical trajectories, i.e., the periodic orbits, undergo bifurcations. At bifurcations periodic orbits change their structure and stability properties, new orbits are born, or orbits vanish. A systematic classi"cation of the various types of bifurcations is possible by application of normal form theory [32}35], where the phase space structure around the bifurcation point is analyzed with the help of classical perturbation theory in local coordinates de"ned parallel and perpendicular to the periodic orbit. At bifurcation points periodic orbits are neither isolated nor belong to a regular torus in phase space. As a consequence the periodic orbit amplitudes diverge in both semiclassical expressions, Gutzwiller's trace formula and the Berry}Tabor formula, i.e., both formulae are not valid near bifurcations. The correct semiclassical solutions must simultaneously account for all periodic orbits which participate at the bifurcation, including `ghosta orbits [36,37] in the complex generalization of phase space, which can be important near bifurcations. Such solutions are called uniform semiclassical approximations and can be constructed by application of catastrophe theory [38,39] in terms of catastrophe di!raction integrals. Uniform semiclassical approximations have been derived in [40,41] for the simplest and generic types of bifurcations and in [37,42}44] for nongeneric bifurcations with higher codimension and corank. With Gutzwiller's trace formula for isolated periodic orbits, the Berry}Tabor formula for regular tori, and the uniform semiclassical approximations for orbits near bifurcations we have, in principle, the basic equations for the semiclassical investigation of all systems with regular, chaotic, and mixed regular-chaotic dynamics. There are, however, fundamental problems in practical applications of these equations for the calculation of semiclassical spectra, and the development of techniques to overcome these problems has been the objective of intense research during recent years.

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1.1.2. Convergence problems of the semiclassical trace formulae The most serious problem of Gutzwiller's trace formula is that for chaotic systems the periodic orbit sum does not converge in the physically interesting energy region, i.e., on and below the real axis, where the eigenstates of bound systems and the resonances of open systems are located, respectively. For chaotic systems the sum of the absolute values of the periodic orbit terms diverges exponentially because of the exponential proliferation of the number of orbits with increasing periods. The convergence problems are similar for the quantization of regular systems with the Berry}Tabor formula, however, the divergence is algebraic instead of exponential in this case. Because of the technical problems encountered in trying to extract eigenvalues directly from the periodic orbit sum, Gutzwiller's trace formula has been used in many applications mainly for the interpretation of experimental or theoretical spectra of chaotic systems. For example, the periodic orbit theory served as basis for the development of scaled-energy spectroscopy, a method where the classical dynamics of a system is "xed within long-range quantum spectra [45,46]. The Fourier transforms of the scaled spectra exhibit peaks at positions given by the classical action of the periodic orbits. The action spectra can therefore be interpreted directly in terms of the periodic orbits of the system, and the technique of scaled-energy spectroscopy is now well established, e.g., for investigations of atoms in external "elds [47}56]. The resolution of the Fourier transform action spectra is restricted by the uncertainty principle of the Fourier transform, i.e., the method allows the identi"cation of usually short orbits in the low-dense part of the action spectra. A fully resolved action spectrum would require an in"nite length of the original scaled quantum spectrum. Although periodic orbit theory has been very successful in the interpretation of quantum spectra, the extraction of individual eigenstates and resonances directly from periodic orbit quantization remains of fundamental interest and importance. As mentioned above the semiclassical trace formulae diverge at the physical interesting regions. They are convergent, however, at complex energies with positive imaginary part above a certain value, viz. the entropy barrier. Thus the problem of revealing the semiclassical eigenenergies and resonances is closely related to "nding an analytic continuation of the trace formulae to the region at and below the real axis. Several re"nements have been introduced in recent years in order to transform the periodic orbit sum in the physical domain of interest to a conditionally convergent series by, e.g., using symbolic dynamics and the cycle expansion [9,57,58], the Riemann}Siegel look-alike formula and pseudo-orbit expansions [59}61], surface of section techniques [62,63], and heat-kernel regularization [64}66]. These techniques are mostly designed for systems with special properties, e.g., the cycle expansion requires the existence and knowledge of a symbolic code and is most successful for open systems, while the Riemann}Siegel look-alike formula and the heat-kernel regularization are restricted to bound systems. Until now there is no universal method, which allows periodic orbit quantization of a large variety of bound and open systems with an underlying regular, mixed, or chaotic classical dynamics. 1.2. Objective of this work The main objective of this work is the development of novel methods for (a) the analysis of quantum spectra and (b) periodic orbit quantization. The conventional action or recurrence spectra obtained by Fourier transformation of "nite range quantum spectra su!er from the fundamental resolution problem because of the uncertainty principle of the Fourier transformation. The broadening of recurrence peaks usually prevents the detailed analysis of structures near


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bifurcations of classical orbits. We will present a method which allows, e.g., the detailed analysis of bifurcations and symmetry breaking and the study of higher order corrections to the periodic orbit sum. For periodic orbit quantization the aim is the development of a universal method for periodic orbit quantization which does not depend on special properties of the system, such as the existence of a symbolic code or the applicability of a functional equation. As will be shown both problems can be solved by applying methods for high resolution spectral analysis [67]. The computational techniques for high-resolution spectral analysis have recently been signi"cantly improved [68}70], and we will demonstrate that the state of the art methods are a powerful tool for the semiclassical quantization and analysis of dynamical systems. 1.2.1. High precision analysis of quantum spectra Within the semiclassical approximation of Gutzwiller's trace formula or the Berry}Tabor formula the density of states of a scaled quantum spectrum is a superposition of sinusoidal modulations as a function of the energy or an appropriate scaling parameter. The frequencies of the modulations are determined by the periods, i.e., the classical action of orbits. The amplitudes and phases of the oscillations depend on the stability properties of the periodic orbits and the Maslov indices. To extract information about the classical dynamics from the quantum spectrum it is therefore quite natural to Fourier transform the spectrum from ènergya domain to `timea (or, for scaled spectra, àctiona) domain [45}47,71}77]. The periodic orbits appear as peaks at positions given by the periods of orbits, and the peak heights exhibit the amplitudes of periodic orbit contributions in the semiclassical trace formulae. The comparison with classical calculations allows the interpretation of quantum spectra in terms of the periodic orbits of the corresponding classical system. However, the analyzed experimental or theoretical quantum spectra are usually of "nite length, and thus the sinusoidal modulations are truncated when Fourier transformed. The truncation implies a broadening of peaks in the time domain spectra because of the uncertainty principle of the Fourier transform. The widths of the recurrence peaks are determined by the total length of the original spectrum. The uncertainty principle prevents until now precision tests of the semiclassical theories because neither the peak positions (periods of orbits) nor the amplitudes can be obtained from the time domain spectra to high accuracy. Furthermore, the uncertainty principle implies an overlapping of broadened recurrence peaks when the separation of neighboring periods is less than the peak widths. In this case individual periodic orbit contributions cannot be revealed any more. This is especially a disadvantage, e.g., when following, in quantum spectra of systems with mixed regular-chaotic dynamics, the bifurcation tree of periodic orbits. All orbits involved in a bifurcation, including complex `ghosta orbits [36,37,44], have nearly the same period close to the bifurcation point. The details of the bifurcations, especially of catastrophes with higher codimension or corank [38,39], cannot be resolved with the help of the conventional Fourier transform. The same is true for the e!ects of symmetry breaking, e.g., breaking of the cylindrical symmetry of the hydrogen atom in crossed magnetic and electric "elds [54] or the `temporal symmetry breakinga [55,56] of atoms in oscillating "elds. The symmetry breaking should result in a splitting of peaks in the Fourier transform recurrence spectra. Until now this phenomenon could only be observed indirectly via constructive and destructive interference e!ects of orbits with broken symmetry [54,55]. To overcome the resolution problem in the conventional recurrence spectra and to achieve a high-resolution analysis of quantum spectra it is "rst of all necessary to mention that the

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uncertainty principle of the Fourier transformation is not a fundamental restriction to the resolution of the time domain recurrence spectra, and is not comparable to the fundamental Heisenberg uncertainty principle in quantum mechanics. This can be illustrated for the simple example of a sinusoidal function with only one frequency. The knowledge of the signal at only three points is su$cient to recover, although not uniquely, the three unknown parameters, viz. the frequency, amplitude and phase of the modulation. Similarly, a superposition of N sinusoidal functions can be recovered, in principle, from a set of at least 3N data points (t , f ). If the data points G G are exact the spectral parameters +u , A , , can also be exactly determined from a set of 3N I I I nonlinear equations , f " A sin(u t ! ) , (1.1) G I IG I I i.e., there is no uncertainty principle. The main di!erence between this procedure and the Fourier transformation is that we use the linear superposition of sinusoidal functions as an ansatz for the functional form of our data, and the frequencies, amplitudes, and phases are adjusted to obtain the best approximation to the data points. By contrast, the Fourier transform is not based on any constraints. In case of a discrete Fourier transform (DFT) the frequencies are chosen on a (usually equidistant) grid, and the amplitudes are determined from a linear set of equations, which can be solved numerically very e$ciently, e.g., by fast Fourier transform (FFT). As mentioned above the high precision spectral analysis requires the numerical solution of a nonlinear set of equations, and the availability of e$cient algorithms has been the bottleneck for practical applications in the past. Several techniques have been developed [67], however, most of them are restricted } for reasons of storage requirements, computational time, and stability of the algorithm } to signals with quite low number of frequencies. By contrast, the number of frequencies (periodic orbits) in quantum spectra of chaotic systems is in"nite or at least very large, and applied methods for high precision spectral analysis must be able to handle signals with large numbers of frequencies. This requirement is ful"lled by the method of harmonic inversion by ,lter-diagonalization, which was recently developed by Wall and Neuhauser [68] and signi"cantly improved by Mandelshtam and Taylor [69,70]. The decisive step is to recast the nonlinear set of equations as a generalized eigenvalue problem. Using an appropriate basis set the generalized eigenvalue equation is solved with the "lter-diagonalization method, which means that the frequencies of the signal can be determined within small frequency windows and only small matrices must be diagonalized numerically even when the signal is composed of a very large number of sinusoidal oscillations. We will introduce harmonic inversion as a powerful tool for the high precision analysis of quantum spectra, which allows to circumvent the uncertainty principle of the conventional Fourier transform analysis and to extract previously unattainable information from the spectra. In particular, the following items are investigated: E High precision check of semiclassical theories: the analysis of spectra allows a direct quantitative comparison of the quantum and classical periodic orbit quantities to many signi"cant digits. E Uncovering of periodic orbit bifurcations in quantum spectra, and the veri"cation of ghost orbits and uniform semiclassical approximations. E Direct observation of symmetry breaking e!ects.


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E Quantitative interpretation of the di!erences between quantum and semiclassical spectra in terms of the expansion of the periodic orbit sum. Results will be presented for various physical systems, e.g., the hydrogen atom in external "elds, the circle billiard, and the three disk scattering problem. 1.2.2. Periodic orbit quantization The analysis of quantum spectra by harmonic inversion and, e.g., the observation of `ghosta orbits, symmetry breaking e!ects, or higher-order corrections to the periodic orbit contributions provides a deeper understanding of the relation between quantum mechanics and the underlying classical dynamics of the system. However, the inverse procedure, i.e., the calculation of semiclassical eigenenergies and resonances directly from classical input data is of at least the same importance and even more challenging. As mentioned above, the periodic orbit sums su!er from fundamental convergence problems in the physically interesting domain and much e!ort has been undertaken in recent years to overcome these problems [9,57}66]. Although many of the re"nements which have been introduced are very e$cient for a speci"c model system or a class of systems, they all su!er from the disadvantage of non-universality. The cycle expansion technique [9,57,58] requires a completely hyperbolic dynamics and the existence of a symbolic code. The method is most e$cient only for open systems, e.g., for three-disk or n-disk pinball scattering [9}15]. By contrast, the Riemann}Siegel look-alike formula and pseudo orbit expansion of Berry and Keating [59}61] can only be applied for bound systems. The same is true for surface of section techniques [62,63] and heat-kernel regularization [64}66]. We will introduce high resolution spectral analysis, and in particular harmonic inversion by "lter-diagonalization as a novel and universal method for periodic orbit quantization. Universality means that the method can be applied to open and bound systems with regular and chaotic classical dynamics as well. Formally, the semiclassical density of states (more precisely the semiclassical response function) can be written as the Fourier transform of the periodic orbit recurrence signal C(s)" A d(s!s ) , (1.2) with A and s the periodic orbit amplitudes and periods (actions), respectively. If all orbits up to in"nite length are considered the Fourier transform of C(s) is equivalent to the non-convergent periodic orbit sum. For chaotic systems a numerical search for all periodic orbits is impossible and, furthermore, does not make sense because the periodic orbit sum does not converge anyway. On the other hand, the truncation of the Fourier integral at "nite maximum period s yields

a smoothed spectrum only [78]. However, the low resolution property of the spectrum can be interpreted as a consequence of the uncertainty principle of the Fourier transform. We can now argue in an analogous way as in the previous section that the uncertainty principle can be circumvented with the help of high-resolution spectral analysis, and propose the following procedure for periodic orbit quantization by harmonic inversion. Let us assume that the periodic orbits with periods 0(s(s are available and the semiclassi cal recurrence function C(s) has been constructed. This signal can now be harmonically inverted and thus adjusted to the functional form of the quantum recurrence function C (s), which is a superposition of sinusoidal oscillations with frequencies given by the quantum mechanical

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eigenvalues of the system. The frequencies obtained by harmonic inversion of C(s) should therefore be the semiclassical approximations to the exact eigenenergies. The universality and wide applicability of this novel quantization scheme follows from the fact that the periodic orbit recurrence signal C(s) can be obtained for a large variety of systems because the calculation does not depend on special properties of the system, such as boundness, ergodicity, or the existence of a symbolic dynamics. For systems with underlying regular or chaotic classical dynamics the amplitudes A of periodic orbit contributions are directly obtained from the Berry}Tabor formula [25,26] and Gutzwiller's trace formula [8], respectively. As mentioned above, the harmonic inversion technique requires the knowledge of the signal C(s) up to a "nite maximum period s . The e$ciency of the quantization method strongly

depends on the signal length which is required to obtain a certain number of eigenenergies. In chaotic systems periodic orbits proliferate exponentially with increasing period and usually the orbits must be searched numerically. It is therefore highly desirable to use the shortest possible signal for periodic orbit quantization. The e$ciency of the method can be improved if not just the single signal C(s) is harmonically inverted but additional classical information obtained from a set of smooth and linearly independent observables is used to construct a semiclassical crosscorrelated periodic orbit signal. The cross-correlation function can be analyzed with a generalized harmonic inversion technique and allows calculating semiclassical eigenenergies from a signi"cantly reduced set of orbits or alternatively to improve the accuracy of spectra obtained with the same set of orbits. However, the semiclassical eigenenergies deviate } apart from a few exceptions, e.g., the geodesic #ow on a surface with constant negative curvature [8] } from the exact quantum mechanical eigenvalues. The reason is that Gutzwiller's trace formula and the Berry}Tabor formula are only the leading-order terms of the expansion of periodic orbit contributions [22}24]. It will be shown how the higher-order corrections of the periodic orbit sum can be used within the harmonic inversion procedure to improve, order by order, the semiclassical accuracy of eigenenergies, i.e., to obtain eigenvalues beyond the Gutzwiller and Berry}Tabor approximation. 1.3. Outline The paper is organized as follows. In Section 2 the high precision analysis of quantum spectra is discussed. After general remarks on Fourier transform recurrence spectra in Section 2.1, we introduce in Section 2.2 harmonic inversion as a tool to circumvent the uncertainty principle of the conventional Fourier transformation [79]. In Section 2.3 a precision check of the periodic orbit theory is demonstrated by way of example of the hydrogen atom in a magnetic "eld. For the hydrogen atom in external "elds we furthermore illustrate that dynamical information which has been unattainable before can be extracted from the quantum spectra. In Section 2.4 we investigate in detail the quantum manifestations of bifurcations of orbits related to a hyperbolic umbilic catastrophe [44,80] and a butter#y catastrophe [37,81], and in Section 2.5 we directly uncover e!ects of symmetry breaking in the quantum spectra. In Section 2.6 we analyze by way of example of the circle billiard and the three disk system the di!erence between the exact quantum and the semiclassical spectra. It is shown that the deviations between the spectra can be quantitatively interpreted in terms of the next order contributions of the expanded periodic orbit sum.


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In Section 3 we propose harmonic inversion as a novel technique for periodic orbit quantization [82,83]. The method is introduced in Section 3.1 for a mathematical model, viz. the calculation of zeros of Riemann's zeta function. This model shows a formal analogy with the semiclassical trace formulae, and is chosen for the reasons that no extensive numerical periodic orbit search is necessary and the results can be directly compared to the exact zeros of the Riemann zeta function. In Section 3.2 the method is derived for the periodic orbit quantization of physical systems and is demonstrated in Section 3.3 for the example of the three disk scattering system with fully chaotic (hyperbolic) classical dynamics [82,83], and in Section 3.4 for the hydrogen atom in a magnetic "eld as a prototype example of a system with mixed regular-chaotic dynamics [84]. In Section 3.5, the e$ciency of the method is improved by a generalization of the technique to the harmonic inversion of cross-correlated periodic orbit sums, which allows to signi"cantly reduce the number of orbits required for the semiclassical quantization [85], and in Section 3.6 we derive the concept for periodic orbit quantization beyond the Gutzwiller and Berry}Tabor approximation by harmonic inversion of the expansion of the periodic orbit sum [86]. The methods of Sections 3.5 and 3.6 are illustrated in Section 3.7 for the example of the circle billiard [85}88]. Finally, in Section 3.8 we demonstrate the semiclassical calculation of individual transition matrix elements for atoms in external "elds [89]. Section 4 concludes with a summary and outlines possible future applications, e.g., the analysis of experimental spectra and the periodic orbit quantization of systems without scaling properties. Computational details of harmonic inversion by "lter-diagonalization are presented in Appendix A, and the calculation and asymptotic behavior of some catastrophe di!raction integrals is discussed in Appendix C.

2. High precision analysis of quantum spectra Semiclassical periodic orbit theory [7,8] and closed orbit theory [90}93] have become the key for the interpretation of quantum spectra of classically chaotic systems. The semiclassical spectra at least in low resolution are given as the sum of a smooth background and a superposition of modulations whose amplitudes, frequencies, and phases are solely determined by the closed or periodic orbits of the classical system. For the interpretation of quantum spectra in terms of classical orbits it is therefore most natural to obtain the recurrence spectrum by Fourier transforming the energy spectrum to the time domain. Each closed or periodic orbit should show up as a sharp d-peak at the recurrence time (period), provided, "rst, the classical recurrence times do not change within the whole range of the spectrum and, second, the Fourier integral is calculated along an in"nite energy range. Both conditions are usually not ful"lled. However, the problem regarding the energy dependence of recurrence times can be solved in systems possessing a classical scaling property by application of scaling techniques. The second condition is never ful"lled in practice, i.e., the length of quantum spectra is always restricted either by experimental limitations or, in theoretical calculations, by the growing dimension of the Hamiltonian matrix which has to be diagonalized numerically. The given length of a quantum spectrum determines the resolution of the quantum recurrence spectrum due to the uncertainty principle, *E ) *¹& , when the conventional Fourier transform is used. Only those closed or periodic orbits can be clearly identi"ed quantum mechanically which appear as isolated non-overlapping peaks in the quantum

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recurrence spectra. This is especially not the case for orbits which undergo bifurcations at energies close to the bifurcation point. As will be shown the resolution of quantum recurrence spectra can be signi"cantly improved beyond the limitations of the uncertainty principle of the Fourier transform by methods of high resolution spectral analysis. In the following, we "rst review the conventional analysis of quantum spectra by Fourier transformation and discuss on the example of the hydrogen atom in a magnetic "eld the achievements and limitations of the Fourier transform recurrence spectra. In Section 2.2 we review methods of high-resolution spectral analysis which can serve to circumvent the uncertainty principle of the Fourier transform. In Section 2.3 the harmonic inversion technique is applied to calculate high-resolution recurrence spectra beyond the limitations of the uncertainty principle of the Fourier transformation from experimental or theoretical quantum spectra of ,nite length. The method allows to reveal information about the dynamics of the system which is completely hidden in the Fourier transform recurrence spectra. In particular, it allows to identify real orbits with nearly degenerate periods, to detect complex `ghosta orbits which are of importance in the vicinity of bifurcations [36,37,44], and to investigate higher order corrections of the periodic orbit contributions [22}24]. 2.1. Fourier transform recurrence spectra According to periodic orbit theory [7,8] the semiclassical density of states can be written as the sum of a smooth background . (E) and oscillatory modulations induced by the periodic orbits, 1 . (E)". (E)# Re A (E)e P1# , (2.1) p P with A (E) the amplitudes of the modulations and the classical actions S (E) of a primitive periodic orbit (po) determining the frequencies of the oscillations. Linearizing the action around E"E yields (2.2) S(E)+S(E )#dS/dE" (E!E )"S(E )#¹ (E!E ) , # with ¹ the time period of the orbit at energy E . When Eq. (2.2) is inserted in Eq. (2.1) the semiclassical density of states is locally given as a superposition of sinusoidal oscillations, and it might appear that the problem of identifying the amplitudes A and time periods ¹ that contribute to the quantum spectrum . (E) can be solved by Fourier transforming . (E) to the time domain, i.e., each periodic orbit should show up as a d-peak at the recurrence time t"¹ . However, fully resolved recurrence spectra can be obtained only if the two following conditions are ful"lled. First, the Fourier integral is calculated along an in"nite energy range, and, second, the classical recurrence times do not change within the whole range of the spectrum. The "rst condition is usually not ful"lled when quantum spectra are obtained from an experimental measurement or a numerical quantum calculation. In that case the Fourier transformation is restricted to a "nite energy range and the resolution of the time domain spectrum is limited by the uncertainty principle of the Fourier transform. It is the main objective of this section to introduce high-resolution methods for the spectral analysis which allow to circumvent the uncertainty principle of the Fourier transform and to obtain fully resolved recurrence spectra from the analysis of quantum spectra with "nite length. However, the second condition is usually not ful"lled either. Both the


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amplitudes and recurrence times of periodic orbits are in general non-trivial functions of the energy E, and, even worse, the whole phase space structure changes with energy when periodic orbits undergo bifurcations. For the interpretation of quantum spectra in terms of periodic orbit quantities it is therefore in general not appropriate to analyze the frequencies of long ranged spectra . (E). The problems due to the energy dependence of periodic orbit quantities have been solved by the development and application of scaling techniques. 2.1.1. Scaling techniques Many systems possess a classical scaling property in that the classical dynamics does not depend on an external scaling parameter w and the action of trajectories varies linearly with w. Examples are systems with homogeneous potentials, billiard systems, or the hydrogen atom in external "elds. In billiard systems the shapes of periodic orbits are solely determined by the geometry of the borders, and the classical action depends on the length ¸ of the periodic orbit, S " k¸ , with k the wave number. For a particle with mass m moving in a billiard system it is therefore most appropriate to take the wave number as the scaling parameter, i.e., w"k"(2mE/ . For a system with a homogeneous potential \

p e WW\V8W\V!p

((4y!3x)G2y(4y!3x

.

(2.33)

The terms of Eq. (2.33) can now be compared to the standard periodic orbit contributions Eq. (2.26) of Gutzwiller's trace formula. In our example the "rst term is related to the orbit 0!#!! (with a multiplicity factor of 2 for its time reversal 0!!!#), and the other two terms are related to the orbits 00#! and ###!!! for the upper and lower sign, respectively. The phase shift in the numerators describe the di!erences of the action *S and of the Maslov index *k"G1 relative to the reference orbit 0!#!!. The denominators are, up to a factor cw, with c"0.1034, the square root of "det(M!I)", with M the stability matrix. Fig. 9 presents the comparison for the determinants obtained from classical periodic orbit calculations (solid lines) and from Eqs. (2.32) and (2.33) (dashed lines). The agreement is very good for both the real and complex ghost orbits, similar to the agreement found for *s in Fig. 8. The constant c introduced above determines the normalization of the uniform semiclassical approximation for the hyperbolic umbilic bifurcation which is "nally obtained as [44,80] (2.34) (EI , w)"(c/p)s wW(aw(EI !EI ), bw)e QU\pI , with s and k denoting the orbital action and Maslov index of the reference orbit 0!#!!, and the constants a,b, and c as given above. The comparison between the amplitudes Eq. (2.26) of the conventional semiclassical trace formula for isolated periodic orbits and the uniform approximation Eq. (2.34) for the hyperbolic umbilic catastrophe is presented in Fig. 10 at the magnetic "eld strengths c"10\, c"10\, and c"10\. For graphical purposes we suppress the highly oscillatory part resulting from the function exp[i(S / !(p/2)k )] by plotting the absolute value of A(EI ,w) instead of the real part. The dashed line in Fig. 10 is the superposition of the isolated periodic orbit contributions from the A


261

Fig. 9. Same as Fig. 8 but for the determinant det(M!I) of the periodic orbits (from [44]).

Fig. 10. Semiclassical amplitudes (absolute values) for magnetic "eld strength (a) c"10\, (b) c"10\, and (c) c"10\ in units of the time period ¹ . Dashed line: amplitudes of the standard semiclassical trace formula. Dash-dotted line: ghost orbit contribution. Solid line: uniform approximation of the hyperbolic umbilic catastrophe (from [44]).

262


four orbits involved in the bifurcations. The modulations of the amplitude are caused by the constructive and destructive interference of the real orbits at energies EI 'EI and are most pronounced at low magnetic "eld strength (see Fig. 10c). The amplitude diverges at the two bifurcation points. For the calculation of the uniform approximation Eq. (2.34) we numerically evaluated the catastrophe di!raction integral Eq. (2.28) using a more simple and direct technique as described in [103]. Details of our method which is based on Taylor series expansions are given in Appendix C.1. The solid line in Fig. 10 is the uniform approximation Eq. (2.34). It does not diverge at the bifurcation points but decreases exponentially at energies EI (EI . At these energies no real orbits exist and the amplitude in the standard formulation would be zero when only real orbits are considered. However, the exponential tail of the uniform approximation Eq. (2.34) is well reproduced by a ghost orbit [37,36] with positive imaginary part of the complex action. As can be shown, the asymptotic expansion of the di!raction integral Eq. (2.28) has, for x g (w)e\ QUdw"!i d e\ UIQ . (3.17) C (s)" I 2p \ I C (s) is a superposition of sinusoidal functions with frequencies w given by the Riemann zeros and I amplitudes d "1. (It is convenient to use the word `frequenciesa for w referring to the sinusoidal I I form of C(s). We will also use the word `polesa in the context of the response function g(w).) Fitting a signal C(s) to the functional form of Eq. (3.17) with, in general, both complex frequencies w and amplitudes d is known as harmonic inversion, and has already been introduced I I in Section 2.2 for the high-resolution analysis of quantum spectra. The harmonic inversion analysis is especially non-trivial if the number of frequencies in the signal C(s) is large, e.g., more than a thousand. It is additionally complicated by the fact that the conventional way to perform the spectral analysis by studying the Fourier spectrum of C(s) will bring us back to analyzing the non-convergent response function g(w) de"ned in Eq. (3.9). Until recently the known techniques of spectral analysis [67] would not be applicable in the present case, and it is the "lter-diagonalization method [68}70] which has turned the harmonic inversion concept into a general and powerful computational tool. The signal C(s) as de"ned by Eq. (3.16) is not yet suitable for the spectral analysis. The next step is to regularize C(s) by convoluting it with a Gaussian function to obtain the smoothed signal,

i ln(p) > 1 e\Q\K NN (3.18) C(s)e\Q\QYN ds" C (s)" N pK (2pp N K (2pp \ that has to be adjusted to the functional form of the corresponding convolution of C (s). The latter is readily obtained by substituting d in Eq. (3.17) by the damped amplitudes, I (3.19) d PdN"d e\UIN . I I I

288


The regularization Eq. (3.18) can also be interpreted as a cut of an in"nite number of high frequencies in the signal which is of fundamental importance for numerically stable harmonic inversion. Note that the convolution with the Gaussian function is no approximation, and the obtained frequencies w and amplitudes d corrected by Eq. (3.19) are still exact, i.e., do not depend I I on p. The convolution is therefore not related to the Gaussian smoothing devised for Riemann zeros in [121] and for quantum mechanics in [122], which provides low resolution spectra only. The next step is to analyze the signal Eq. (3.18) by harmonic inversion. The concept of harmonic inversion by "lter-diagonalization has already been explained in Section 2.2 and the technical details are given in Appendix A.1. Note that even though the derivation of Eq. (3.18) assumed that the zeros w are on the real axis, the analytic properties of C (s) imply that its representation by I N Eq. (3.18) includes not only the non-trivial real zeros, but also all the trivial ones, w "!i(2k#), k"1,2,2, which are purely imaginary. The general harmonic inversion proceI dure does not require the frequencies to be real. Both the real and imaginary zeros w will be I obtained as the eigenvalues of the non-Hermitian generalized eigenvalue problem, Eq. (A.7) in Appendix A.1. 3.1.3. Numerical results For a numerical demonstration we construct the signal C (s) using Eq. (3.18) in the region N s(ln(10)"13.82 from the "rst 78498 prime numbers and with a Gaussian smoothing width p"0.0003. Parts of the signal are presented in Fig. 22. Up to s+8 the Gaussian approximations to the d-functions do essentially not overlap (see Fig. 22a), whereas for s(w)AK ,"tr+GK >(w)(AK AK ), , ?@ ? @ ? @ and Eq. (3.59) can be applied to the product AK AK . However, we do not want to restrict the ? @ operators to those commuting with HK , which obviously would be a severe restriction especially for chaotic systems, and the problem is now to "nd a semiclassical approximation to Eq. (3.55) for the general case of two arbitrary smooth operators AK and AK . A reasonable assumption is that the ? @ amplitudes A in Eq. (3.56) have to be multiplied by the product of the classical averages, a a , of these two observables, i.e. ? @ C (s)" a a A d(s!s ) . (3.60) ?@ ? @ Although no rigorous mathematical proof of Eq. (3.60) will be given here, we have strong numerical evidence, from the high-resolution analysis of quantum spectra, part of which will be


313

given below, that the conjecture of Eq. (3.60) is correct. Details on semiclassical non-trace type formulae like Eq. (3.60) are given in [139]. Eq. (3.60) is the starting point for the following application of harmonic inversion of cross-correlation functions. Note that all quantities in Eq. (3.60) are obtained from the classical periodic orbits. The idea of periodic orbit quantization by harmonic inversion of cross-correlated periodic orbit sums is to "t the semiclassical functions C (s) given in a "nite range 0(s(s to the functional ?@

form of the quantum expression Eq. (3.55). As for the harmonic inversion of a one-dimensional signal (see Section 3.2) the frequencies of the harmonic inversion analysis are then identi"ed with the semiclassical eigenvalues w . The amplitudes b are identi"ed with the semiclassical approxiL ?L mations to the diagonal matrix elements 1n"AK "n2. Here we only give a brief description how the ? harmonic inversion method is extended to cross-correlation functions. The details of the numerical procedure of solving the generalized harmonic inversion problem Eq. (3.55) are presented in [68,137,138] and in Appendix A.2. As for the harmonic inversion of a single function the idea is to recast the nonlinear "t problem as a linear algebraic problem [68]. This is done by associating the signal C (s) (to be inverted) with a time cross-correlation function between an initial state U and ?@ ? a "nal state U , @ (3.61) C (s)"(U ,e\ XK QU ) , ? ?@ @ where the "ctitious quantum dynamical system is described by an e!ective Hamiltonian XK . The latter is de"ned implicitly by relating its spectrum to the set of unknown spectral parameters w and L b . Diagonalization of XK would yield the desired w and b . This is done by introducing an ?L L ?L appropriate basis set in which the matrix elements of XK are available only in terms of the known signals C (s). The Hamiltonian XK is assumed to be complex symmetric even in the case of a bound ?@ system. This makes the harmonic inversion stable with respect to `noisea due to the imperfections of the semiclassical approximation. The most e$cient numerical and practical implementation of the harmonic inversion method with all relevant formulae can be found in [137,138] and Appendix A.2. The method of harmonic inversion of cross-correlated periodic orbit sums will be applied in Section 3.7 to the circle billiard. As will be shown, for a given number of periodic orbits the accuracy of semiclassical spectra can be signi"cantly improved with the help of the crosscorrelation approach, or, alternatively, spectra with similar accuracy can be obtained from a periodic orbit cross-correlation signal with signi"cantly reduced signal length. 3.6. expansion for the periodic orbit quantization by harmonic inversion Semiclassical spectra can be obtained for both regular and chaotic systems in terms of the periodic orbits of the system. For chaotic dynamics the semiclassical trace formula was derived by Gutzwiller [7,8], and for integrable systems the Berry}Tabor formula [25,26] is well known to be precisely equivalent to the EBK torus quantization [1}3]. However, as already has been discussed in Section 2.6, the semiclassical trace formulae are exact only in exceptional cases, e.g., the geodesic motion on the constant negative curvature surface. In general, they are just the leading-order terms of an in"nite series in powers of the Planck constant and the accuracy of semiclassical quantization is still an object of intense investigation [109,110,140]. Methods for the calculation of the higher-order periodic orbit contributions were developed in [22}24]. In Section 2.6 we have

314


demonstrated how the periodic orbit quantities of the expanded trace formula can be extracted from the quantum and semiclassical spectra. It is an even more fundamental problem to obtain semiclassical eigenenergies beyond the Gutzwiller and Berry}Tabor approximation directly from the expanded periodic orbit sum. Note that the expansion of the periodic orbit sum does not solve the general problem of the construction of the analytic continuation of the trace formula, which is already a fundamental problem when only the leading-order terms of the expansion is considered. Up to now the expansion for periodic orbit quantization is restricted to systems with known symbolic dynamics, like the three-disk scattering problem, where cycle expansion techniques can be applied [23,24], and semiclassical eigenenergies beyond the Gutzwiller and Berry}Tabor approximation cannot be calculated, e.g., for bound systems with the help of Riemann}Siegel-type formulae [60,61] or surface of section techniques [62,63]. In this section we extend the method of periodic orbit quantization by harmonic inversion to the analysis of the

expansion of the periodic orbit sum. The accuracy of semiclassical eigenvalues can be improved by one to several orders of magnitude, as will be shown in Section 3.7 by way of example of the circle billiard. As in Section 2.6 we consider systems with a scaling property, i.e., where the classical action scales as S "ws , and the scaling parameter w, \ plays the role of an inverse e!ective Planck constant. The expansion of the periodic orbit sum is given (see Eq. (2.65)) as a power series in w\, 1 ALe QU . (3.62) g(w)" g (w)" L wL L L The complex amplitudes AL of the nth-order periodic orbit contributions include the phase information from the Maslov indices. For periodic orbit quantization the zeroth-order contributions A are usually considered only. The Fourier transform of the principal periodic orbit sum

1 > C (s)" g (w)e\ QU dw" Ad(s!s ) 2p \

(3.63)

is adjusted by application of the harmonic inversion technique (see Section 3.2) to the functional form of the exact quantum expression

1 > d I C(s)" e\ UQ dw"!i d e\ UIQ , I 2p w!w #i0 \ I I I

(3.64)

with +w , d , the eigenvalues and multiplicities. The frequencies w obtained by harmonic I I I inversion of Eq. (3.63) are the zeroth-order approximation to the semiclassical eigenvalues. We will now demonstrate how the higher-order correction terms to the semiclassical eigenvalues can be extracted from the periodic orbit sum Eq. (3.62). We "rst remark that the asymptotic expansion Eq. (3.62) of the semiclassical response function su!ers, for n51, from the singularities at w"0, and it is therefore not appropriate to harmonically invert the Fourier transform of Eq. (3.62), although the Fourier transform formally exists. This means that the method of periodic orbit quantization by harmonic inversion cannot straightforwardly be extended to the expansion of the periodic orbit sum. Instead we will calculate the correction terms to the semiclassical eigenvalues separately, order by order, as described in the following.


315

Let us assume that the (n!1)th-order approximations w to the semiclassical eigenvalues IL\ are already obtained and the w are to be calculated. The di!erence between the two subsequent IL approximations to the quantum mechanical response function reads [86]

d d *w d I I IL I ! + , g (w)" L #i0 (w!wN #i0) w!w #i0 w!w IL\ IL IL I I

(3.65)

with wN "(w #w )/2 and *w "w !w . Integration of Eq. (3.65) and multiplicaIL IL IL\ IL IL IL\ tion by wL yields

!d wL*w I IL , G (w)"wL g (w) dw" L L w!wN #i0 IL I

(3.66)

which has the functional form of a quantum mechanical response function but with residues proportional to the nth-order corrections *w to the semiclassical eigenvalues. The semiclassical IL approximation to Eq. (3.66) is obtained from the term g (w) in the periodic orbit sum Eq. (3.62) by L integration and multiplication by wL, i.e.

1 1 . G (w)"wL g (w) dw"!i ALe UQ#O L L w s

(3.67)

We can now Fourier transform both Eqs. (3.66) and (3.67), and obtain (n51)

1 > C (s), G (w)e\ UQ dw"i d (w )L*w e\ UIQ L L I I IL 2p \ I 1 "!i ALd(s!s ) . s

(3.68)

(3.69)

Eqs. (3.68) and (3.69) are the main result of this section. They imply that the expansion of the semiclassical eigenvalues can be obtained, order by order, by harmonic inversion (h.i.) of the periodic orbit signal in Eq. (3.69) to the functional form of Eq. (3.68). The frequencies of the periodic orbit signal Eq. (3.69) are the semiclassical eigenvalues w . Note that the accuracy I of the semiclassical eigenvalues does not necessarily increase with increasing order n. We indicate this in Eq. (3.68) by omitting the index n at the eigenvalues w . The corrections *w to the I IL eigenvalues are obtained from the amplitudes, d (w )L*w , of the periodic orbit signal. I I IL The method requires as input the periodic orbits of the classical system up to a maximum period (scaled action), s , determined by the average density of states [82,83]. The amplitudes A are

obtained from Gutzwiller's trace formula [7,8] and the Berry}Tabor formula [25,26] for chaotic and regular systems, respectively. For the next order correction A explicit formulae were derived y Gaspard and Alonso for chaotic systems with smooth potentials [22] and in [23,24] for billiards. With appropriate modi"cations [87,88] the formulae can be used for regular systems as well. As an example we investigate the expansion of the periodic orbit sum for the circle billiard in the next section.

316


3.7. The circle billiard In Section 3.5 (see also [85]) we have introduced harmonic inversion of cross-correlated periodic orbit sums as a method to signi"cantly reduce the required number of periodic orbits for semiclassical quantization, and in Section 3.6 (see also [86]) we have discussed the expansion of the periodic orbit sum and the calculation of semiclassical eigenenergies beyond the Gutzwiller [8] and Berry}Tabor [25,26] approximation. We now demonstrate both methods by way of example of the circle billiard. The circle billiard is a regular system and has been chosen here for the following reasons. 1. The nearest-neighbor level statistics of integrable systems is a Poisson distribution, with a high probability for nearly degenerate states. The conventional method for periodic orbit quantization by harmonic inversion requires very long signals to resolve the nearly degenerate states. We will demonstrate the power of harmonic inversion of cross-correlated periodic orbit sums by fully resolving those nearly degenerate states with a signi"cantly reduced set of orbits. 2. All relevant physical quantities, i.e., the quantum and semiclassical eigenenergies, the matrix elements of operators, the periodic orbits and their zeroth- and "rst-order amplitudes of the

expanded periodic orbit sum, and the periodic orbit averages of classical observables can easily be obtained. 3. The semiclassical quantization of the circle billiard as an example of an integrable system demonstrates the universality and wide applicability of periodic orbit quantization by harmonic inversion, i.e., the method is not restricted to systems with hyperbolic dynamics like, e.g., pinball scattering. The circle billiard has already been introduced in Section 2.6.1. The exact quantum mechanical eigenvalues E" k/2M are given as zeros of Bessel functions J (kR)"0, where m is the angular K momentum quantum number and R, the radius of the circle. In the following we choose R"1. The semiclassical eigenvalues are obtained from the EBK quantization condition, Eq. (2.71), kR(1!(m/kR)!"m" arccos("m"/kR)"p(n#) , and the expansion of the periodic orbit sum reads (see Eq. (2.73)) 1 1 g(k)" ALe lI , g (k)" L kL (k L (k L 1

with (see Eqs. (2.74) and (2.75))

p l e\ pI>p , A" 2 m 1 5!2 sinc e\ pI\p . A" (pm 3sinc 2 The angle c is de"ned as c,pm /m , with m "1,2,2 the number of turns of the periodic orbit ( ( around the origin, and m "2m ,2m #1,2 the number of re#ections at the boundary of the ( (


317

circle. l "2m sin c and k "3m are the geometrical length and Maslov index of the orbits, respectively. 3.7.1. Harmonic inversion of the cross-correlated periodic orbit sum We now calculate the semiclassical eigenenergies of the circle billiard by harmonic inversion of the cross-correlated periodic orbit sum Eq. (3.60) with A "A (Eq. (2.74)) the amplitudes of the Berry}Tabor formula [25,26], i.e., the lowest-order approximation. To construct the periodic orbit cross-correlation signal C (l) we choose three di!erent operators, ?@ AK "I the identity, AK "r the distance from the origin, and AK "(¸/k) the square of the scaled angular momentum. For these operators the classical weights a (Eq. (3.58)) are obtained as ? a "1 , a "(1#(cos c/tan c)arsinh tan c) , a "cos c . (3.70) Once all the ingredients of Eq. (3.60) for the circle billiard are available, the 3;3 periodic orbit cross-correlation signal C (l) can easily be constructed and inverted by the generalized "lter?@ diagonalization method. Results obtained from the periodic orbits with maximum length s "100 are presented in Fig. 35. Fig. 35a is part of the density of states, . (k), Fig. 35b and

Fig. 35c is the density of states weighted with the diagonal matrix elements of the operators AK "r and AK "¸, respectively. The squares are the results from the harmonic inversion of the periodic orbit cross-correlation signals. For comparison the crosses mark the matrix elements obtained by exact quantum calculations at positions k# ) obtained from the EBK quantization condition Eq. (2.71). In this section, we do not compare with the exact zeros of the Bessel functions because Eq. (3.60) is correct only to "rst order in and thus the harmonic inversion of C (s) cannot provide ?@ the exact quantum mechanical eigenvalues. The calculation of eigenenergies beyond the Berry}Tabor approximation will be discussed in Section 3.7.2. However, the perfect agreement between the eigenvalues k&' obtained by harmonic inversion and the EBK eigenvalues k# ) is remarkable, and this is even true for nearly degenerate states marked by arrows in Fig. 35a. The eigenvalues of some nearly degenerate states are presented in Table 11. It is important to emphasize that these states with level splittings of, e.g., *k"6;10\ cannot be resolved by the originally proposed method of periodic orbit quantization by harmonic inversion (see Section 3.2) with a periodic orbit signal length s "100. To resolve the two levels at k+11.049 (see Table 11)

a signal length of at least s +500 is required if a single periodic orbit function C(s) is used

instead of a cross-correlation function. The method presented in Section 3.5 can therefore be used

318


Fig. 35. Density of states weighted with the diagonal matrix elements of the operators (a) AK "I, (b) AK "r, (c) AK "¸ for the circle billiard with radius R"1. Crosses: EBK eigenvalues and quantum matrix elements. Squares: eigenvalues and matrix elements obtained by harmonic inversion of cross-correlated periodic orbit sums. Three nearly degenerate states are marked by arrows (from [85]).

to signi"cantly reduce the required signal length and thus the required number of periodic orbits for periodic orbit quantization by harmonic inversion. As such the part of the spectrum shown in Fig. 35 can even be resolved, apart from the splittings of the nearly degenerate states marked by the arrows, from a short cross-correlation signal with s "30, which is about the Heisenberg period

s "2p.N (k), i.e. half of the signal length required for the harmonic inversion of a 1;1 signal. With & "ve operators and a 5;5 cross-correlation signal highly excited states around k"130 have even been obtained with a signal length s +0.7s [87,88], which is close to the signal length

& s +0.5s required for the Riemann}Siegel-type quantization [59}61]. The reduction of the

& signal length is especially important if the periodic orbit parameters are not given analytically, as in our example of the circle billiard, but must be obtained from a numerical periodic orbit search. How small can s get as one uses more and more operators in the method? It might be that half of

the Heisenberg period is a fundamental barrier for bound systems with chaotic dynamics in analogy to the Riemann}Siegel formula [61] while for regular systems an even further reduction of the signal length should in principle be possible. However, further investigations are necessary to clarify this point.


319

Table 11 Nearly degenerate semiclassical states of the circle billiard. k# ): results from EBK-quantization. k&': eigenvalues obtained by harmonic inversion of cross-correlated periodic orbit sums. States are labeled by the radial and angular momentum quantum numbers (n,m) k# )

k&'

4 7

11.048664 11.049268

11.048569 11.049239

3 0

1 9

13.314197 13.315852

13.314216 13.315869

3 1

2 7

14.787105 14.805435

14.787036 14.805345

1 5

11 1

19.599795 19.609451

19.599863 19.608981

1 6

15 2

24.252501 24.264873

24.252721 24.264887

n

m

1 0

3.7.2. Periodic orbit quantization beyond the Berry}¹abor approximation The semiclassical eigenvalues obtained by harmonic inversion of a cross-correlated or a su$ciently long single signal are in excellent agreement with the results of the EBK torus quantization, Eq. (2.71). However, they deviate from the exact quantum mechanical eigenenergies, i.e., the zeros of the Bessel functions because the Berry}Tabor formula [25,26] is only the lowest-order approximation of the periodic orbit sum. We now demonstrate the expansion of the periodic orbit sum and apply the technique discussed in Section 3.6 to the circle billiard. The "rst-order corrections to the semiclassical eigenvalues, *k"k!k are obtained by harmonic inversion of the periodic orbit signal C (l) (see Eq. (3.69)), 1 i d k[k!k]e\ IHl , Ad(l!l )" C (l)"!i H H H H l H

(3.71)

with d the multiplicities of states. The signal C (l) in Eq. (3.71) can be inverted as a single function H as has been done in [86], where the accuracy of the eigenenergies was improved by one to several orders of magnitude, apart from the nearly degenerate states marked by arrows in Fig. 35. Here we go one step further and use Eq. (3.71) as part of a 3;3 cross-correlation signal. For the two other diagonal components of the cross-correlation matrix we use the identity operator, AK "I, and the distance from the origin, AK "r. By applying the cross-correlation technique of Section 3.5 we obtain both the zeroth- and "rst-order expansion of the eigenenergies and the diagonal matrix elements of the chosen operators simultaneously from one and the same harmonic inversion procedure. Furthermore, we can now even resolve the nearly degenerate states. The spectrum of the integrated di!erences of the density of states *. (k) dk obtained by harmonic inversion of the 3;3 cross-correlation matrix with signal length l "150 is shown in Fig. 36. The squares mark the

320


Fig. 36. Integrated di!erence of the density of states, *. (k) dk, for the circle billiard with radius R"1. Crosses: *. (k)". (k)!. # )(k). Squares: . (k)". (k)!. (k) obtained from the expansion of the periodic orbit signal.

spectrum for *. (k)". (k)!. (k) obtained from the harmonic inversion of the signal C (s). For comparison, the crosses present the same spectrum but for the di!erence *. (k)" . (k)!. # )(k) between the exact quantum mechanical and the EBK-spectrum. The deviations between the peak heights exhibit the contributions of terms of the expansion series beyond the "rst-order approximation. The peak heights of the levels in Fig. 36 (solid lines and crosses) are, up to a multiplicity factor for the degenerate states, the shifts *k between the zeroth- and "rst-order semiclassical approximations to the eigenvalues k. The zeroth- and "rst-order eigenvalues, k and k"k#*k are presented in Table 12 for the 40 lowest eigenstates. The zeroth-order eigenvalues, k, agree within the numerical accuracy with the results of the torus quantization, Eq. (2.71) (see eigenvalues k# ) in Table 12). However, the semiclassical eigenvalues deviate signi"cantly, especially for states with low radial quantum numbers n, from the exact quantum mechanical eigenvalues k in Table 12. By contrast, the semiclassical error of the "rst-order eigenvalues, k, is by orders of magnitude reduced compared to the lowest-order approximation. An appropriate measure for the accuracy of semiclassical eigenvalues is the deviation from the exact quantum eigenvalues in units of the average level spacings, 1*k2 "1/.N (k). Fig. 37 presents the semiclassical error in units of the average level spacings 1*k2 +4/k for the zeroth-order (diamonds) and "rst-order (crosses) approximations to the eigenvalues. In the zeroth-order approximation the semiclassical error for the low lying states is about 3}10% of the mean level spacing. This error is reduced in the "rst-order approximation by at least one order of magnitude for the least semiclassical states with radial quantum number n"0. The accuracy of states with n51 is improved by two or more orders of magnitude. Finally, we want to note that the small splittings between the nearly degenerate states are extremely sensitive to the higher-order corrections. E.g., in the zeroth-order approximation the splitting between the two states around k+11.05 is *k"6;10\. In the "rst-order approximation the splitting between the same states is *k"0.0242, which is very close to the exact splitting *k"0.0217. The accuracy obtained here for the "rst order approximations to the nearly degenerate states goes beyond the results presented in [86], and is achieved by the combined application of the methods introduced in Sections 3.5 and 3.6, i.e., the harmonic inversion of cross-correlation functions and the analysis of the expanded periodic orbit sum, respectively.


321

Table 12 The 40 lowest eigenstates of the circle billiard with radius R"1. n,m: radial and angular momentum quantum numbers; k# ): results from EBK-quantization; k and k: zeroth and "rst order semiclassical eigenvalues obtained by harmonic inversion of the periodic orbit signal; k: exact eigenvalues, i.e., zeros of the Bessel functions J (kR)"0 K n

m

k# )

k

k

k

0 0 0 1 0 1 0 1 2 0 1 0 2 1 0 2 3 0 1 2 3 0 1 2 0 3 1 4 0 2 1 3 4 0 2 1 3 0 4 5

0 1 2 0 3 1 4 2 0 5 3 6 1 4 7 2 0 8 5 3 1 9 6 4 10 2 7 0 11 5 8 3 1 12 6 9 4 13 2 0

2.356194 3.794440 5.100386 5.497787 6.345186 6.997002 7.553060 8.400144 8.639380 8.735670 9.744628 9.899671 10.160928 11.048664 11.049268 11.608251 11.780972 12.187316 12.322723 13.004166 13.314197 13.315852 13.573465 14.361846 14.436391 14.787105 14.805435 14.922565 15.550089 15.689703 16.021889 16.215041 16.462981 16.657857 16.993489 17.225257 17.607830 17.760424 17.952638 18.064158

2.356230 3.794440 5.100382 5.497816 6.345182 6.997006 7.553055 8.400145 8.639394 8.735672 9.744627 9.899660 10.160949 11.048635 11.049228 11.608254 11.780993 12.187302 12.322721 13.004167 13.314192 13.315782 13.573464 14.361846 14.436375 14.787091 14.805457 14.922572 15.550084 15.689701 16.021888 16.215047 16.462982 16.657846 16.993486 17.225252 17.607831 17.760386 17.952662 18.064201

2.409288 3.834267 5.138118 5.520550 6.382709 7.015881 7.590990 8.417503 8.653878 8.774213 9.761274 9.938954 10.173568 11.063791 11.087943 11.619919 11.791599 12.228037 12.338847 13.015272 13.323418 13.356645 13.589544 14.372606 14.478531 14.795970 14.821595 14.930938 15.593060 15.700239 16.038034 16.223499 16.470648 16.701442 17.003884 17.241482 17.615994 17.804708 17.959859 18.071125

2.404826 3.831706 5.135622 5.520078 6.380162 7.015587 7.588342 8.417244 8.653728 8.771484 9.761023 9.936110 10.173468 11.064709 11.086370 11.619841 11.791534 12.225092 12.338604 13.015201 13.323692 13.354300 13.589290 14.372537 14.475501 14.795952 14.821269 14.930918 15.589848 15.700174 16.037774 16.223466 16.470630 16.698250 17.003820 17.241220 17.615966 17.801435 17.959819 18.071064

322


Fig. 37. Semiclassical error "k!k" (diamonds) and "k!k" (crosses) in units of the average level spacing 1*k2 +4/k.

3.8. Semiclassical calculation of transition matrix elements for atoms in external xelds The interpretation of photoabsorption spectra of atoms in external "elds is a fundamental problem of atomic physics. Although the èxacta quantum mechanics accurately describes the energies and transition strengths of individual levels it has completely failed to present a simple physical picture of the long-ranged modulations which have been observed in early low-resolution spectra of barium atoms in a magnetic "eld [141] and later in the Fourier transform recurrence spectra of the magnetized hydrogen atom [142,71]. However, the long-ranged modulations of the quantum photoabsorption spectra can be naturally interpreted in terms of the periods of classical closed orbits starting at and returning back to the nucleus where the initial state is localized. The link between the quantum spectra and classical trajectories is given by closed orbit theory [90}93] which describes the photoabsorption cross section as the sum of a smooth part and the superposition of sinusoidal modulations. The frequencies, amplitudes, and phases of the modulations are directly obtained from the quantities of the closed orbits. When the photoabsorption spectra are Fourier transformed or analyzed with a high resolution method (see Section 2.1) the sinusoidal modulations result in sharp peaks in the Fourier transform recurrence spectra, and closed orbit theory has been most successful to explain quantum mechanical recurrence spectra qualitatively and even quantitatively in terms of the closed orbits of the underlying classical system [46}48,51]. However, up to now practical applications of closed orbit theory have always been restricted to the semiclassical calculation of low-resolution photoabsorption spectra for the following two reasons. First, the closed orbit sum requires, in principle, the knowledge of all orbits up to in"nite length, which are usually not available from a numerical closed orbit search, and second, the in"nite closed orbit sum su!ers from fundamental convergence problems [90}93]. It is therefore commonly accepted that the calculation of individual transition matrix elements 1 "D"t 2 of the dipole operator D, which describe the transition strengths from the initial state " 2 to "nal states "t 2, is a problem beyond the applicability of the semiclassical closed orbit theory, i.e., is the domain of quantum mechanical methods. In this section we disprove this common believe and demonstrate that individual eigenenergies and transition matrix elements can be directly extracted from the quantities of the classical closed


323

orbits. To that end, we slightly generalize closed orbit theory to the semiclassical calculation of cross-correlated recurrence functions. We then adopt the cross-correlation approach introduced in Section 3.5 to harmonically invert the cross-correlated recurrence signal and to extract the semiclassical eigenenergies and transition matrix elements. Results will be presented for the photoexcitation of the hydrogen atom in a magnetic "eld. The oscillator strength f for the photoexcitation of atoms in external "elds can be written as f (E)"!(2/p)(E!E )Im 1 "DG>D" 2 , #

(3.72)

where " 2 is the initial state at energy E , D is the dipole operator, and G> the retarded Green's # function of the atomic system. The basic steps for the derivation of closed orbit theory are to replace the quantum mechanical Green's function in Eq. (3.72) with its semiclassical Van Vleck}Gutzwiller approximation and to carry out the overlap integrals with the initial state " 2. Here we go one step further by introducing a cross-correlation matrix g "1 "DG>D" 2 ??Y ? # ?Y

(3.73)

with " 2, a"1,2,2,¸, a set of independent initial states. As will be shown below the use of ? cross-correlation matrices can considerably improve the convergence properties of the semiclassical procedure. In the following, we will concentrate on the hydrogen atom in a magnetic "eld with c"B/(2.35;10 T) the magnetic "eld strength in atomic units. As discussed in Section 2.1 the system has a scaling property, i.e., the shape of periodic orbits does not depend on the scaling parameter, w"c\" \, and the classical action scales as S"sw with s the scaled action. As, e.g., in [47] we consider scaled photoabsorption spectra at constant scaled energy EI "Ec\ as a function of the scaling parameter w. We choose dipole transitions between states with magnetic quantum number m"0. Note that the following ideas can be applied in an analogous way to atoms in electric "elds. Following the derivation of [91,93] the semiclassical approximation to the #uctuating part of g in Eq. (3.73) reads ??Y !(2p) (sin 0 sin 0 Y (0 )Y (0 )e QU\pI>p , g (w)"w\ ? ?Y ??Y ("m "

(3.74)

with s and k the scaled action and Maslov index of the closed orbit (co), m an element of the monodromy matrix, and 0 and 0 the initial and "nal angle of the trajectory with respect to the magnetic "eld axis. The angular functions Y (0) depend on the states " 2 and the dipole operator ? ? D and are given as a linear superposition of Legendre polynomials, Y (0)" B P (cos 0) with J? J ? usually only few nonzero coe$cients B with low l. Explicit formulae for the Jcalculation of the J? coe$cients can be found in [91,93] and in Appendix B. The problem is now to extract the semiclassical eigenenergies and transition matrix elements from Eq. (3.74) because the closed orbit sum does not converge. The Fourier transformation of wg (w) yields the cross-correlated ??Y recurrence signals C (s)" A d(s!s ) , ??Y ??Y

(3.75)

324


with the amplitudes (3.76) A "(!(2p)/("m ")(sin 0 sin 0 Y (0 )Y (0 )e \pI>p ? ?Y ??Y being determined exclusively by closed orbit quantities. The corresponding quantum mechanical cross-correlated recurrence functions, i.e., the Fourier transforms of wg (w) read ??Y C (s)"!i b b e\ UIQ , (3.77) ??Y ?I ?YI I with w the eigenvalues of the scaling parameter, and I b "w1 "D"t 2 (3.78) ?I I ? I proportional to the transition matrix element for the transition from the initial state " 2 to the ? "nal state "t 2. I The method to adjust Eq. (3.75) to the functional form of Eq. (3.77) for a set of initial states " 2, a"1,2,2,¸, is the harmonic inversion of cross-correlation functions as discussed in Sec? tion 3.5 and Appendix A.2. We now demonstrate the method of harmonic inversion of the cross-correlated closed orbit recurrence functions Eq. (3.75) for the example of the hydrogen atom in a magnetic "eld at constant scaled energy EI "!0.7. This energy was also chosen for detailed experimental investigations on the helium atom [48]. We investigate dipole transitions from the initial state " 2""2p02 with light polarized parallel to the magnetic "eld axis to "nal states with magnetic quantum number m"0. For this transition the angular function in Eq. (3.76) reads (see Appendix B) Y (0)"(2p)\2e\(4 cos 0!1). For the construction of a 2;2 cross-correlated recurrence signal we use for simplicity as a second transition formally an outgoing s-wave, i.e., D" 2J> , and, thus, Y (0)"const. A numerical closed orbit search yields 1395 primitive closed orbits (2397 orbits including repetitions) with scaled action s/2p(100. With the closed orbit quantities at hand it is straightforward to calculate the cross-correlated recurrence functions in Eq. (3.75). The real and imaginary parts of the complex functions C (s), C (s), and C (s) with s/2p(50 are presented in Figs. 38 and 39, respectively. Note that for symmetry reasons C (s)"C (s). We have inverted the 2;2 cross-correlated recurrence functions in the region 0(s/2p(100. The resulting semiclassical photoabsorption spectrum is compared with the exact quantum spectrum in Fig. 40a for the region 16(w(21 and in Fig. 40b for the region 34(w(40. The upper and lower parts in Fig. 40 show the exact quantum spectrum and the semiclassical spectrum, respectively. Note that the region of the spectrum presented in Fig. 40b belongs well to the experimentally accessible regime with laboratory "eld strengths B"6.0 T to B"3.7 T. The overall agreement between the quantum and semiclassical spectrum is impressive, even though a line by line comparison still reveals small di!erences for a few matrix elements. It is important to note that the high quality of the semiclassical spectrum could only be achieved by our application of the cross-correlation approach. For example, the two nearly degenerate states at w"36.969 and w"36.982 cannot be resolved and the very weak transition at w"38.894 with 12p0"D"t 2"0.028 is not detected with a single (1;1) recurrence signal of the same length. However, these hardly visible details are indeed present in the semiclassical spectrum in Fig. 40b obtained from the harmonic inversion of the 2;2 cross-correlated recurrence functions.


325

Fig. 38. Real parts of the cross-correlated recurrence functions for the hydrogen atom in a magnetic "eld at constant scaled energy EI "!0.7 (from [89]).

To summarize, we have demonstrated that closed orbit theory is not restricted to describe long-ranged modulations in quantum mechanical photoabsorption spectra of atoms in external "elds but can well be applied to extract individual eigenenergies and transition matrix elements from the closed orbit quantities. This is achieved by the high-resolution spectral analysis (harmonic inversion) of cross-correlated closed orbit recurrence signals. For the hydrogen atom in a magnetic "eld we have obtained individual transition matrix elements between low lying and highly excited Rydberg states solely from the classical closed orbit data.

4. Conclusion In summary, we have shown that harmonic inversion is a powerful tool for the analysis of quantum spectra, and is the foundation for a novel and universal method for periodic orbit quantization of both regular and chaotic systems. The high-resolution analysis of "nite range quantum spectra allows to circumvent the restrictions imposed by the uncertainty principle of the conventional Fourier transformation. Therefore, physical phenomena can directly be revealed in the quantum spectra which previously were

326


Fig. 39. Same as Fig. 38 but for the imaginary parts of the cross-correlated recurrence functions.

unattainable. Topical examples are the study of quantum manifestations of periodic orbit bifurcations and catastrophe theory, and the uncovering of symmetry breaking e!ects. The investigation of these phenomena provides a deeper understanding of the relation between quantum mechanics and the dynamics of the underlying classical system. The high-resolution technique is demonstrated in this work for numerically calculated quantum spectra of, e.g., the hydrogen atom in external "elds, three-disk pinball scattering, and the circle billiard. Theoretical spectra are especially suited for the high-resolution analysis because of the very high accuracy of most quantum computational methods. In principle, harmonic inversion may be applied to experimental spectra as well, e.g., to study atoms in external "elds measured with the technique of scaled-energy spectroscopy [45}55]. However, the exact requirements on the precision of experimental data to achieve high-resolution recurrence spectra beyond the limitations of the uncertainty principle are not yet known. It will certainly be a challenge for future experimental work to verify the quantum manifestations of bifurcations, which have been extracted here from theoretically computed spectra, using real systems in the laboratory. We have also introduced harmonic inversion as a new and general tool for semiclassical periodic orbit quantization. Here we brie#y recall the highlights of our technique. The method requires the complete set of periodic orbits up to a given maximum period as input but does not depend on special properties of the orbits, as, e.g., the existence of a symbolic code or a functional equation. The universality and wide applicability has been demonstrated by applying it to systems with


327

Fig. 40. Quantum (upper part) and semiclassical (lower part) photoabsorption spectra of the hydrogen atom in a magnetic "eld at scaled energy EI "!0.7. Transition matrix elements 12p0"D"t 2 for dipole transitions with light polarized parallel to the magnetic "eld axis (from [89]).

completely di!erent properties, namely the zeros of the Riemann zeta function, the three disk scattering problem, and the circle billiard. These systems have been treated before by separate e$cient methods, which, however, are restricted to bound ergodic systems, systems with a complete symbolic dynamics, or integrable systems. The harmonic inversion technique allows to solve all these problems with one and the same method. The method has furthermore been successfully applied to the hydrogen atom in a magnetic "eld as a prototype example of a system with mixed regular-chaotic dynamics. The e$ciency of the method can be improved if additional semiclassical information obtained from a set of linearly independent observables is used to construct a crosscorrelated periodic orbit sum, which can then be inverted with a generalized harmonic inversion technique. The cross-correlated periodic orbit sum allows the calculation of semiclassical eigenenergies from a signi"cantly reduced set of orbits. Eigenenergies beyond the Gutzwiller and Berry}Tabor approximation are obtained by the harmonic inversion of the expansion of the periodic orbit sum. When applied, e.g., to the circle billiard the semiclassical accuracy is improved by at least one to several orders of magnitude. The combination of closed orbit theory with the cross-correlation approach and the harmonic inversion technique also allows the semiclassical calculation of individual quantum transition strengths for atoms in external "elds.

328


Periodic orbit quantization by harmonic inversion has been applied in this work to systems with scaling properties, i.e., systems where the classical actions of periodic orbits depend linearly on a scaling parameter, w. However, the method can even be used for the semiclassical quantization of systems with non-homogeneous potentials such as the potential surfaces of molecules. The basic idea is to introduce a generalized scaling technique with the inverse Planck constant w,1/ as the new formal scaling parameter. The generalized scaling technique can be applied, e.g., to the analysis of the rovibrational dynamics of the HO molecule [94]. By varying the energy of the system, the harmonic inversion method yields the semiclassical eigenenergies in the (E,w) plane. For nonscaling systems the semiclassical spectra can then be compared along the line with the true physical Planck constant, w"1/ "1, with experimental measurements in the laboratory. Acknowledgements The author thanks H.S. Taylor and V.A. Mandelshtam for their collaboration and kind hospitality during his stay at the University of Southern California, where part of this work was initiated. Financial support from the Feodor Lynen program of the Alexander von Humboldt foundation is gratefully acknowledged. Stimulating discussions with M.V. Berry, E.B. Bogomolny, D. BooseH , J. Briggs, D. Delande, J.B. Delos, B. Eckhardt, S. Freund, M. Gutzwiller, F. Haake, M. Haggerty, E. Heller, C. Jung, J. Keating, D. Kleppner, M. KusH , C. Neumann, B. Mehlig, K. MuK ller, H. Schomerus, N. Spellmeyer, F. Steiner, G. Tanner, T. Uzer, K.H. Welge, A. Wirzba, and J. Zakrzewski are also gratefully acknowledged. The author thanks V.A. Mandelshtam for his program on "lter-diagonalization and B. Eckhardt and A. Wirzba for supplying him with numerical data of the three-disk system. Part of the present work has been done in collaboration with K. Weibert and K. Wilmesmeyer, whose Ph.D. thesis [88] and diploma thesis [80] are prepared under the author's guidance. The author is indebted to G. Wunner for his permanent support and his interest in the progress of this work. The work was supported by the Deutsche Forschungsgemeinschaft via a Habilitandenstipendium (Grant No. Ma 1639/3) and the Sonderforschungsbereich 237 Ùnordnung und gro{e Fluktuationena. Appendix A. Harmonic inversion by 5lter-diagonalization In the following we give details about the numerical method of harmonic inversion by "lterdiagonalization. We begin with the harmonic inversion of a single function and then extend the method to the harmonic inversion of cross-correlation functions. A.1. Harmonic inversion of a single function The harmonic inversion problem can be formulated as a nonlinear "t (see, e.g., [67]) of the signal C(s) de"ned on an equidistant grid, c ,C(nq)" d e\ LOUI, n"0,1,2,2,N , L I I

(A.1)


329

with the set of generally complex variational parameters +w ,d ,. (In this context the discrete I I Fourier transform scheme would correspond to a linear "t with N amplitudes d and "xed real I frequencies w "2pk/Nq, k"1,2,2,N. The latter implies the so-called ùncertainty principlea, I i.e., the resolution, de"ned by the Fourier grid spacing, *w, is inversely proportional to the length, s "Nq, of the signal C(s).) The `high-resolutiona property associated with Eq. (A.1) is due to the

fact that there is no restriction for the closeness of the frequencies w as they are variational I parameters. In [68] it was shown how this nonlinear "tting problem can be recast as a linear algebraic one using the "lter-diagonalization procedure. The essential idea is to associate the signal c with an autocorrelation function of a suitable dynamical system, L c "(U ,;K LU ) , (A.2) L where ( ) , ) ) de"nes a complex symmetric inner product (i.e., no complex conjugation). The evolution operator can be de"ned implicitly by ) (A.3) ;K ,e\ OXK " e\ OUI"B )(B " , I I I where the set of eigenvectors +B , is associated with an arbitrary orthonormal basis set and the I eigenvalues of ;K are u ,e\ OUI (or equivalently the eigenvalues of XK are w ). Inserting Eq. (A.3) I I into Eq. (A.2) we obtain Eq. (A.1) with d "(B ,U ) , (A.4) I I which also implicitly de"nes the ìnitial statea U . This construction establishes an equivalence between the problem of extracting spectral information from the signal with the one of diagonalizing the evolution operator ;K "e\ OXK (or the Hamiltonian XK ) of the "ctitious underlying dynamical system. The "lter-diagonalization method is then used for extracting the eigenvalues of the Hamiltonian XK in any chosen small energy window. Operationally, this is done by solving a small generalized eigenvalue problem whose eigenvalues yield the frequencies in a chosen window. The knowledge of the operator XK itself is not required, as for a properly chosen basis the matrix elements of XK can be expressed only in terms of c .The advantage of the "lter-diagonalization L procedure is its numerical stability with respect to both the length and complexity (the number and density of the contributing frequencies) of the signal. Here we apply the method of [69,70] which is an improvement of the "lter-diagonalization method of [68] in that it allows to signi"cantly reduce the required length of the signal by implementing a di!erent Fourier-type basis with an e$cient rectangular "lter. Such a basis is de"ned by choosing a small set of values u in the frequency H interval of interest, qw (u (qw , j"1,2,2,J, and the maximum order, M, of the Krylov

H

vectors, U "e\ LOXK U , used in the Fourier series, L + + W ,W(u )" e LPHU , e LPH\OXK U . (A.5) H H L L L It is convenient to introduce the notations, (A.6) ;N,;N(u ,u )"(W(u ),e\ NOXK W(u )) , HY HHY H HY H for the matrix elements of the operator e\ NOXK , and UN, for the corresponding small J;J complex symmetric matrix. As such U denotes the matrix representation of the operator ;K itself and U,

330


the overlap matrix with elements (W(u ),W(u )), which is required as the vectors W(u ) are not H HY H generally orthonormal. Now using these de"nitions we can set up a generalized eigenvalue problem, UNB "e\ NOUIUB , I I

(A.7)

for the eigenvalues e\ NOUI of the operator e\ NOXK . The column vectors B with elements B de"ne the I HI eigenvectors B in terms of the basis functions W as I H ( B" B W , (A.8) I HI H H assuming that the W 's form a locally complete basis. H The matrix elements Eq. (A.6) can be expressed in terms of the signal c , the explicit knowledge of L the auxiliary objects XK , B or U is not needed. Indeed, insertion of Eq. (A.5) into Eq. (A.6), use of I the symmetry property, (W,;K U)"(;K W, U), and the de"nition of c , Eq. (A.2), gives after some L arithmetics

+ + ;N(u, u)"(e\ P!e\ PY)\ e\ P e LPYc !e\ PY e LPc L>N L>N L L + + , uOu , (A.9) !e +P e L\+\PYc #e +PY e L\+\Pc L>N L>N L+> L+> + ;N(u, u)" (M!"M!n"#1)e LPc . L>N L (Note that the evaluation of UN requires the knowledge of c for n"p,p#1,2,N"2M#p.) L The generalized eigenvalue problem Eq. (A.7) can be solved by a singular value decomposition of the matrix U, or more accurately by application of the QZ algorithm [143], which is implemented, e.g., in the NAG library. Each value of p yields a set of frequencies w and, due to I Eqs. (A.4), (A.5) and (A.8), amplitudes,

( + (A.10) d " B c e LPH . HI L I H L Note that Eq. (A.10) is a functional of the half-signal c , n"1,2,2,M. An even better expression L for the coe$cients d (see [70]) reads I

1 ( 1 ( d" B (W(u ),W(w )) , B ;(u ,w ) , (A.11) I HI H I HI H I M#1 M#1 H H with ;(u ,w ) de"ned by Eq. (A.9). Eq. (A.11) is a functional of the whole available signal H I c , n"0,1,2,2M, and therefore sometimes provides more precise results than Eq. (A.10). L The converged w and d should not depend on p. This condition allows us to identify spurious or I I non-converged frequencies by comparing the results with di!erent values of p (e.g., with p"1 and


331

p"2). We can de"ne the simplest error estimate e as the di!erence between the frequencies w obtained from diagonalizations with p"1 and p"2, i.e. I e""wN!wN" . (A.12) I I A.2. Harmonic inversion of cross-correlation functions We now consider a cross-correlation signal, i.e., a D;D matrix of signals de"ned on an equidistant grid (a,a"1,2,2,D): (A.13) c (n),C (nq)" b b e\ LOUI, n"0,1,2,2,N . ??Y ??Y ?I ?YI I (We choose q"1 for simplicity in what follows.) Each component of the signal c (n) contains the ??Y same set of frequencies w , and the amplitudes belonging to each frequency are correlated, i.e., I d "b b with only D (instead of D) independent parameters b . As for the harmonic ??YI ?I ?YI ?I inversion of a single function the nonlinear problem of adjusting the parameters +w , b , can be I ?I recast as a linear algebra one using the "lter-diagonalization procedure [68,137,138] The crosscorrelation signal Eq. (A.13) is associated with the cross-correlation function of a suitable dynamical system, c "(U ,;K LU ) , (A.14) ??YL ? ?Y with the same complex symmetric inner product as in Eq. (A.2), and the evolution operator ;K de"ned implicitly by Eq. (A.3). The extension from Eq. (A.2) to Eq. (A.14) is that the autocorrelation function (U , ;K LU ) built of a single state U is replaced with the cross-correlation function (U , ;K LU ) built of a set of D di!erent states U . Inserting Eq. (A.3) into Eq. (A.14) we obtain ? ?Y ? Eq. (A.13) with b "(B , U ) , (A.15) ?I I ? which now implicitly de"nes the states U . After choosing a basis set in analogy to Eq. (A.5), ? + (A.16) W "W (u )" e LPH\XK U , ? ?H ? H L and introducing the notations (A.17) UN,;N ";N (u , u )"(W (u ) ,e\ NXK W (u )) , ?Y HY ?H?YHY ??Y H HY ? H for the small matrix of the operator e\ NXK in the basis set Eq. (A.16), we can set up a generalized eigenvalue problem, UNB "e\ NUIUB , I I for the eigenvalues e\ NUI of the operator e\ NXK , which is formally identical with Eq. (A.7). The matrix elements Eq. (A.17) can be expressed in terms of the signal c . The following expression ??YL

332


for the matrix elements of UN is derived in complete analogy with [69,70] with the additional indices a,a being the only di!erence,

+ + ;N (u,u)"(e\ P!e\ PY)\ e\ P e LPYc (n#p)!e\ PY e LPc (n#p) ??Y ??Y ??Y L L + !e +P e L\+\PYc (n#p) ??Y L+> + #e +PY e L\+\Pc (n#p) , uOu , (A.18) ??Y L+> + ;N , (u, u)" (M!"M!n"#1)e LPc (n#p) . (A.19) ??Y ??Y L Given the cross-correlation signal c (n), the solution of the generalized eigenvalue problem ??Y Eq. (A.7) yields the eigenfrequencies w and the eigenvectors B . The latter can be used to compute I I the amplitudes,

( " 1!e\A (A.20) B ;(u ,w #ic) , b " ?YHI ??Y H I ?I 1!e\+>A H ?Y where the adjusting parameter c is chosen so that ;(u ,w #ic) is numerically stable [138]. One H I correct choice is c"!Im w for Im w (0 and c"0 for Im w '0. I I I Appendix B. Angular function Y (0 ) K In Section 2.4.2 we have presented Eq. (2.36) as the "nal result of closed orbit theory for the semiclassical photoabsorption spectrum of the hydrogen atom in a magnetic "eld. Here we de"ne explicitly the angular function Y (0) in Eq. (2.36). K The angular function Y (0) solely depends on the initial state t and the dipole operator D and is K a linear superposition of spherical harmonics: Y (0)" (!1)lBl >l (0, 0) . K K K l K The coe$cients Bl are de"ned by the overlap integrals K

Bl " dx(Dt )(x)(2/rJ l ((8r)>H l (0, u) K K >

(B.1)

(B.2)

(with J (x) the Bessel functions) and can be calculated analytically [93]. For excitations of the J ground state t ""1s02 with p-polarized light (i.e. dipole operator D"z) the explicit result is (B.3) Y (0)"!p\2e\ cos 0 ,


333

and for t ""2p02, i.e., the initial state in many spectroscopic measurements on hydrogen [45}47] we obtain Y (0)"(2p)\2e\(4 cos 0!1) .

(B.4)

Appendix C. Catastrophe di4raction integrals Here we give some technical details about the numerical calculation and the asymptotic expansion of catastrophe di!raction integrals for the hyperbolic umbilic and the butter#y catastrophe. C.1. The hyperbolic umbilic catastrophe The catastrophe di!raction integral of the hyperbolic umbilic (Eq. (2.28)) reads

W(x, y)"

>

dp

\

>

\

dq e N>O>WN>O>VN>O .

(C.1)

By substituting p"s !y , q"s !y , we obtain W(x, y)"e WW\V U(x!y, 2y) ,

(C.2)

with

U(m, g)"

>

ds

\

>

\

ds e Q>Q>KQ>Q>EQQ .

(C.3)

The integral U(m, g) can be expanded into a Taylor series around g"0. Using

> > RLU(m,g) " ds ds iL(s s )Le Q>Q>KQ>Q "iL E RgL \ \

>

\

ds sLe Q>KQ

,

(C.4)

and solving the one-dimensional integrals [144]

>

1 (!m)I k#n#1 ds sLe Q>KQ" C [e L>\Ip#(!1)Le\ L>\Ip] , k! 3 3 \ I

(C.5)

334


we "nally obtain

1 (ig)L (!m)I k#n#1 C [e L>\Ip#(!1)Le\ L>\Ip] , (C.6) U(m, g)" n! k! 3 9 L I which is a convergent series for all m and g. C.2. The butteryy catastrophe The uniform phase integral W(x, y) of the butter#y catastrophe is expanded in a two-parametric Taylor series around x"y"0:

W(x, y),

> exp[!i(xt#yt#t)] dt \

1 xLyK > tL>Kexp[!it] dt . " iL>K n! m! \ L K

(C.7)

With the substitution z"tL>K> we obtain [144]

>

2 tL>Kexp[!it] dt" exp[!izL>K>] dz 2n#4m#1 \

2n#4m#1 2n#4m#1 1 p C , " exp !i 12 6 3 and "nally

p 1 W(x, y)" exp !i 12 3

1 2n#4m#1 C n! m! 6 L K

2 ; x exp !i p 3

L

5 y exp !i p 6

K

,

(C.8)

which is a convergent series for all x and y. The asymptotic behavior of W(x,y) for xP$R is obtained in a stationary-phase approximation to Eq. (C.7) with the stationary points t being de"ned by t (6t#4yt#2x)"0 . (a) xP!R: There are three real stationary points given by t "0 and t"!y#(y!x

(C.9)


335

and we obtain

V\ > W(x, y) & exp[!ixt] dt#2 exp[!i(xt#yt#t)] \ > ; exp [!i(x#6yt#15t) t] dt \

p p p exp i # 4 !x !x#y!y(y!x ;exp+i[2(y!x)!y(y!x)!p], . (C.10) (b) xP#R: The only real solution of Eq. (C.9) is t "0 and W(x, y) is approximated as "

p p V> > exp !i . exp[!ixt] dt" W(x, y) & 4 x \

(C.11)

y the complex zeros In the case of y(0 and x& t "$(!y$i(x!y of Eq. (C.9) are situated close to the real axis. When one considers these complex zeros in a stationary phase approximation, Eq. (C.11) is modi"ed by an additional term exponentially damped for large x: V> W(x, y) &

p p p exp !i # 4 x x!y!iy(x!y ;exp+!2(x!y),exp+!i[y(y!x)!p], .

(C.12)

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N. KonjevicH /Physics Reports 316 (1999) 339}401

PLASMA BROADENING AND SHIFTING OF NON-HYDROGENIC SPECTRAL LINES: PRESENT STATUS AND APPLICATIONS

N. KONJEVICD Faculty of Physics, 11001 Belgrade, P.O. Box 368, Yugoslavia Institute of Physics, 11081 Belgrade, P.O. Box 68, Yugoslavia


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Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications N. KonjevicH Faculty of Physics, 11001 Belgrade, P.O. Box 368, Yugoslavia Institute of Physics, 11081 Belgrade, P.O. Box 68, Yugoslavia Received November 1998; editor: T.F. Gallagher Contents 1. Introduction 2. Theory 2.1. Neutral atom lines 2.2. Ionic lines 3. Experimental determination of line shapes and shifts 4. Other broadening mechanisms 4.1. Natural broadening 4.2. Resonance broadening 4.3. Van der Waals broadening 4.4. Doppler broadening 4.5. Instrumental broadening 4.6. Self-absorption 5. Electron temperature diagnostics

341 342 344 345 346 350 350 350 351 351 352 353 356

6. Regularities in experimental Stark widths and shifts 6.1. Widths 6.2. Shifts 7. Typical experimental procedure 8. Analysis of experimental results 8.1. Neutral atom lines 8.2. Singly ionized atom lines 8.3. Multiply ionized atom lines 8.4. Studies along isoelectronic sequences 9. Conclusions Acknowledgements Appendix References

360 360 361 361 363 364 375 384 387 393 396 396 397

Abstract The present status of the experimental studies of plasma broadening and shifting of non-hydrogenic neutral atom and positive ion spectral lines is discussed with an emphasis to the plasma diagnostic applications. A short overview of the available theoretical results is followed by the description of experimental techniques for the line shape and shift measurements. The in#uence of other broadening mechanisms to the Stark width and shift determination is discussed and a typical experimental procedure described. To select higher accuracy experimental data for plasma diagnostic purposes the available experimental results are analyzed and the uncertainty of recommended ones is estimated. The procedure for application of selected lines for plasma electron density diagnostics is described. The in#uence of ion dynamics to the width and shift of visible helium lines in low electron density plasmas is examined. In order to test theories the selected data are compared with results of various Stark width and shift calculations. Neutral atom lines asymmetry and new techniques for ion-broadening parameter measurements are reviewed. The studies of line widths and shifts along isoelectronic sequences of multiply ionized atoms are also discussed. 1999 Elsevier Science B.V. All rights reserved. PACS: 52.70.Kz; 32.70.Jz Keywords: Plasma spectroscopy; Electron density diagnostics; Stark broadening

0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 3 2 - X

N. Konjevic& / Physics Reports 316 (1999) 339}401

341

1. Introduction Plasma-broadened and shifted spectral line pro"les have been used for a number of years as a basis of an important non-interfering plasma diagnostic method. The numerous theoretical and experimental e!orts have been made to "nd solid and reliable basis for this application. This technique became, in some cases, the most sensitive and often the only possible plasma diagnostic tool. In the early 1960s a number of attempts were made to improve and to check experimentally existing theories of spectral line broadening by plasmas. Most of these early works were concerned with the Stark broadening of hydrogen lines. Owing to the large, linear Stark e!ect in hydrogen, these studies were very useful for plasma diagnostic purposes. However, it is not always convenient to seed plasma with hydrogen, and sometimes this is not possible. Furthermore, due to the large Stark e!ect, hydrogen lines or the lines of hydrogen-like ions are sometimes inconvenient for plasma diagnostic purposes, since they become so broad at high electron densities that, due to the interference with other neighboring lines, it is di$cult to determine their shape correctly. Thus, from the beginning of this "eld of research, there was an interest for the plasma broadening of isolated non-hydrogenic lines of neutral atoms and positive ions. Due to relatively small Stark broadening of non-hydrogenic lines, they can be used for plasma diagnostic purposes at high electron densities and, in particular, at high electron temperatures when hydrogen is fully ionized. Hereafter, the word ìsolateda is used for the lines originating from isolated energy levels in the sense that levels are not degenerate and do not overlap each other. First semiclassical calculations of Stark broadening parameters for isolated non-hydrogenic atomic and singly charged ion lines were carried out by Griem and co-workers [1,2]. After some improvements of the theory (see e.g. [3]) the new comprehensive calculations of Stark-broadening parameters of neutral (helium through calcium and cesium) and singly ionized atom lines (lithium through calcium) were published in 1974 [3]. Later on the calculations for neutral atom lines were extended to some other elements heavier than calcium [4]. Using another semiclassical perturbation method by Sahal-Brechot [5], DimitrijevicH and Sahal-Brechot performed numerous calculations of Stark broadening parameters for the lines of neutral, singly and multiply ionized atoms (see list of publications in [6]). Within Òpacity projecta, using close-coupling approximation, Seaton [7] evaluated a number of Stark broadening data for multiply ionized atoms. Large number of data were produced also using simple semiempirical formula [8] and its modi"ed version [9}11]. A simpli"ed semiclassical formula (Eq. (526) in [3]) is also used extensively for data evaluation and a large set of results is given in [12] together with those from the modi"ed semiempirical formula [9]. Here, only calculations which provide data for a large number of elements and their ions are given. Parallel with the development of the theory of Stark broadening, numerous experiments were performed to provide plasma broadening data and to test theoretical predictions. In order to evaluate and select reliable experimental results, which can be used with con"dence for both, plasma diagnostic purposes and for the testing of theory, KonjevicH and Roberts [13], KonjevicH and Wiese [14], KonjevicH et al. [15,16] and KonjevicH and Wiese [17] carried out analysis of all available experiments for neutral and ionized atoms until the end of 1988. The basis for these experimental reviews and in a good part for this report, was the bibliography on atomic line shapes and shifts [18]. In order to select reliable data, the authors of these reviews [13}17] imposed certain criteria on each experimental paper and, therefore, not all experiments reviewed were included in the "nal

342


results [13}17]. The accuracies of experimental data, coded with letters A ($15%), B ($30%), C ($50%) and D (larger than 50%) are estimated in [13}17] irrespective of the agreement with the theory which was used only as a check for experimental consistency within experiments and between di!erent experiments. Although the theory was used only as a consistency check, an important conclusion is derived from the comparison with experiments: the results of semiclassical calculations [3] may be used for plasma diagnostic purposes with an average estimated accuracy of $20% and $30% for a neutral atom and singly charged ion lines, respectively; see also [3,19]. Here, on the basis of comparison of selected data, we shall make an attempt to extend this analysis and to identify lines of various elements which may be used with higher accuracy for plasma diagnostic purposes. Furthermore, this search will be extended to the lines of higher ionization stages as well. When one intends to use Stark broadening data for plasma diagnostic purposes, problems usually not encountered in experiments set to measure Stark broadening parameters have to be solved. In these experiments, the amount of investigated atoms and ions whose lines are studied is usually controlled so that the distortion caused by the radiative transfer is avoided as much as possible. Furthermore, plasma sources for these experiments are selected in such a way that the in#uence of plasma inhomogeneity to the line shapes is negligible or such that it can be taken into account (e.g. axially symmetric plasmas). In the case when one uses line shapes and shifts to diagnose plasma source, changes or control of plasma conditions is usually not feasible and often (e.g. astrophysical plasmas) not possible at all. So the measurements of line parameters and estimation of the in#uence of other broadening mechanisms will be discussed in detail. Since line widths and shifts also depend upon electron temperature, ¹ , for accurate N plasma diagnostics, ¹ measurements are required and they too will be discussed. In addition, a part of this article is devoted to the regularities and similarities of Stark widths and shifts which arise from atomic energy structure of the line emitter. The knowledge of these regularities and similarities may improve the accuracy or, in some cases, when Stark broadening data are not available, make possible the determination of plasma electron density. Finally, at the end of the general part of this paper typical experimental procedure for electron density plasma diagnostics is also described. An important aim of this work is to summarize results of experimental studies of the Stark broadening and shifting of spectral lines and to discuss the comparison of these data with theory. For this purpose critical evaluation of the recent works (published after 1988) has been performed and the results are used together with earlier data [13}17] to test consistency among experiments and between experiments and theories. The estimated accuracies of selected data will be given and some experimental di$culties underlined. The results related to di!erent atoms and their ionization stages are described separately. In this part the recommended data for plasma diagnostic purposes and their estimated accuracies are given. The recent studies of neutral line asymmetries and line width and shift investigations along with isoelectronic sequences of multiply ionized atoms are also described.

2. Theory The general area of line broadening has been reviewed extensively, the most complete treatment being that of Griem [3]. Here we shall con"ne our discussion towards the application of theoretical results for plasma diagnostic purposes.


343

According to Stark broadening theory the shapes and shifts of plasma-broadened isolated lines are mainly determined by electron impacts with the radiating atom or ion and a smaller contribution arises from the electric micro"elds generated by essentially static plasma ions. The quadratic Stark e!ect due to the quasistatic "eld of ions shifts the energy of the upper and lower level [20] by an amount which depends on the instantaneous local "eld strength. The distribution of "elds in the plasma smears out these shifts and the neutral atom line is unsymmetrically broadened. The parameter A (in earlier literature designated as a), tabulated by Griem [3] is a measure of the e!ect this has on the width in relation to the electron impact width. The in#uence of static "eld to the Stark splitting and shifting of ionic lines is even smaller than for neutral atom lines, see e.g. [20], so the ion contribution to the line shapes and shifts of ionic lines is usually considered to be negligible in comparison with electron impacts. By taking into account the linear dependence of the Stark broadening parameters upon N for ionic lines and very close to linear for neutral atom lines (proven in a number of experiments, see e.g. [3,13}17]), unknown electron density may be determined from the comparison of the measured Stark width, w , and/or shift, d , with results from another experiment where these parameters are

measured at known N and at similar electron temperature ¹ . This procedure has a limited range of application and instead, theoretical data w(N ,¹ ) and d(N ,¹ ) in conjunction with w and

d are used for N plasma determination. In practice, to perform N measurements it is necessary to

determine experimentally the line width and/or shift and to compare with theoretical data for the same ¹ . This, well established procedure, was preceded by numerous experimental and theoretical studies which have proven that experiment agrees with theory. The degree of the discrepancy between various theoretical approaches and experiments was the main task of many earlier publications and it will be discussed here in detail. Two large sets of theoretical data are available for plasma diagnostic purposes. One is published in [3] for neutral atom lines (He through Ca and Cs) and for singly charged ion lines (Li II through Ca II). The other set of data covers large number of neutrals, singly, and multiply ionized atoms (see [6] and references therein). Furthermore, numerous calculations of Stark widths and shifts for ionic lines are performed using simple approximate formulas (see [6,9}12] and references therein) in particular for the radiators without su$ciently complete set of atomic data necessary for semiclassical calculations. The Stark broadening data are usually published in the form of tables where widths and shifts are given for a single electron density and for several electron temperatures. The results for atomic lines are usually presented for N "10 cm\ while data for ionic lines are given at N "10 cm\. For neutral atom lines (Appendix IV in [3]) in addition to electron impact half-halfwidth w (in [6] are always given full halfwidths 2w ) and electron impact shift, d , the corresponding ion broadening parameter, A, is given as well. Namely in [3] impact approximation is used for evaluation of electron collisions contribution to the line width while for ionic part quasistatic approximation is used. In [6] impact approximation is used for both, electrons and ions and the results for ion contribution are given in the form of widths and shifts induced by collisions with di!erent ions present in plasma. Therefore, from the data in [6], the total width and shift is a sum of the electron and all ion impact widths. For neutral atom lines the latter approach [6] always gives symmetric Lorentzian pro"le which is not in agreement with experiments at higher electron densities, where asymmetric line pro"les are detected. The ionic spectral line pro"les in both sets of theoretical data, Appendix V in [3] and [6], are symmetrical and well described by the Lorentzian function which is in a good agreement with experimental results.


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2.1. Neutral atom lines The total theoretical full-width at half-maximum (FWHM) w and the total theoretical shift at the line maximum d may be calculated using the approximate formulas [3] obtained from a numerical "t to the widths of the resultant j (x) pro"les where x is the reduced frequency which 0 is given by x"(u!u !d )/w with u being the frequency, u the center frequency of the unperturbed line, d and w the electron impact shift and width, respectively. In this case, w and d may be calculated from the following equations: w (N ,¹ ) 2w (¹ ) [1#gA (¹ )]N 10\ , (1) , d (N ,¹ ) [d (¹ )$2.0g A (¹ )w (¹ )]N 10\ , (2) , where g"1.75(1!0.75R), with R being Debye shielding parameter, A (¹ )"A(¹ )N 10\ and , g "g/1.75. Due to the asymmetry of plasma-broadened atom lines, the shift at the half-width of the line is slightly di!erent from the one measured at the peak of the line pro"le and may be calculated from [3,21] d

(N ,¹ ) [d (¹ )$3.2g A (¹ )w (¹ )]N 10\ . , From Eqs. (1)}(3) it follows that

(3)

w (N ,¹ ) 2w (¹ ) [1#1.75;10\NA(¹ ) (1!0.068N¹\)]10\N , (3a) d (N ,¹ ) [d (¹ )$2.0;10\NA(¹ )w (¹ ) (1!0.068N¹\)]10\N , (3b) d (N ,¹ ) [d (¹ )$3.2;10\NA(¹ )w (¹ ) (1!0.068N¹\)]10\N . (3c) In the above equations, w (¹ ) and d (¹ ) are electron impact half half-width and shift, respectively, A(¹ ) is the ion broadening parameter, all given in [3] at N "10 cm\, and N and ¹ are the electron density (cm\) and temperature (K). The sign of the ion quadratic contribution to the shift in Eqs. (2) and (3) is equal to that of the low-velocity limit for d . There are certain restrictions on the applicability of Eqs. (1)}(3) and they are R"8.99;10\N¹\40.8 , (4) 0.054A(¹ )N10\40.5 , (5) where R (the Debye shielding parameter) is de"ned as a ratio of the mean inter-ion distance o to the Debye radius o . For values of ion broadening parameter larger than 0.5 the forbidden " component begins to signi"cantly overlap, and the linear Stark e!ect becomes important. Other considerations, such as Debye shielding a!ecting line shapes, widths and shifts are covered in detail in [3]. Simple estimation of the conditions when Debye shielding correction may be omitted and when the assumption of the isolated line approximation is ful"lled, may be derived from the following conditions: , Debye shielding: j <j ??Y , Isolated line: j <j ??Y


345

where j is the wave number (cm\) associated with perturbed level a and nearest perturbing level ??Y is the wave number (cm\) associated with a to which dipole transition is allowed, j plasma frequency given by j

"[(e/pmc)N ]"2.995;10\N , (6) and j is the wave number (cm\) associated with the half-width of the spectral line and given in terms of the broadened line's total half-width w and central wavelength, j by j "w /j . (7) Since impact approximation is used for evaluation of both, electron and ion contributions in [6], the resulting pro"le has always Lorentzian shape and the evaluation of theoretical FWHM w and shift d in this case is very simple whenever all necessary data are available: L w (N ,¹ ) 2w (¹ )10\N # 2w (¹ )10\N , (8) G G G G L d (N ,¹ ) d (¹ )10\N $ d (¹ )10\N , (9) G G G G where 2w , 2w and d and d are electron and ion impact full half-widths and shifts respectively at G G N "10 cm\; N and ¹ are the ion concentration and temperature, respectively. The sign of G G G the ion contribution to the shift in Eq. (9) is speci"ed for all ionic species and they are given in data tables. One can "nd the restrictions of the applicability of Eqs. (8) and (9) in [6] and references therein. Apart from extensive set of Stark broadening data for light elements He through Ca and Cs [3] and some lines of other heavy elements Zn, Ge, Br, Rb, Cd, Sn, Pb and Hg [4] calculated using the same semiclassical code [3], large number of theoretical results evaluated from another semiclassical approach [5] are also available for: E

E

some lines of C, N, Mg and Si calculated by Sahal-Brechot without the contribution of Feshbach resonances, see [22]; contribution of resonances are taken into account in an extensive set of calculations for the lines of 79 multiplets of He, 61 Li, 19 Be, 62 Na, 270 Mg, 25 Al 51 K, 24 Rb, 3 Pd, 31 Se; for some lines of F and Br; see [6] and references therein. Recently, extensive sets of data for neutral Sr [23] and Ba [24] lines have become available.

In the case when w , d and A data are not available one can use simple formulas [25,26] for their estimation. 2.2. Ionic lines The theoretical full half-width w and shift d of singly ionized atom lines may be calculated from data in [3]: w (N ,¹ ) 2w (¹ )10\N , d (N ,¹ ) d (¹ )10\N ,

(10) (11)

346


where w and d are electron impact half half-width and shift at N "10 cm\, respectively. Here, ion broadening is assumed to be negligible. For evaluation of the theoretical widths and shifts from [6] one can use Eqs. (4) and (5) so the ion contribution may be included as well. Here it should be noted that in [6] electron impact full half-widths (2w ) and shifts (d ), and ion impact full half-widths (2w ) and shifts (d ) are calculated G G usually for the perturber concentration 10 cm\ and thus in Eqs. (8) and (9), 10\ changes to 10\. If one neglects the ion contribution to the total line width, reduced Eqs. (8) and (9) become identical with Eqs. (10) and (11). The omission of ion contribution which is usually of the order of several percent for ion lines may be easily justi"ed for most of applications. Since detailed composition for majority of plasmas is not known the omission of the ion contribution sums in Eqs. (8) and (9) is the only way to determine N . Semiclassical data for singly charged ions Li through calcium are available in [3]. Data for a number of alkali-like ion lines Be II, Mg II, Ca II, Sr II Ba II and Si II and Ar II are available, see [6] and references therein. Extensive calculations of Stark broadening parameters are also performed by DimitrijevicH and Sahal Brechot for 29 multiplets of Li II, 30 Be II, 52 Mg II, 3 Fe II, 2 Ni II; see [6] and references therein. For Ba II lines see [24]. Furthermore data for 23 multiplets of A III, 10 Sc III, 10 Ti IV, 19 SI IV, 90 C IV, 5 O IV, 19 O V, 30 N V, 25 C V. 51 P V, 30 O VI, 21 S VI, 10 F VII, 20 Ne VIII, 8 Na IX, 7 Al XI, and 9 Si XII; data for certain lines of Ga II, Ga III, Si II, Br II, Hg II, N III, S IV and F V also exist, see [6] and references therein. Recently, data for Be III, B III [27] and P IV [28] have been published. Modi"ed semiempirical approach [9}11] has been used for the radiators where there is not a su$ciently complete atomic data set for reliable semiclassical calculations, see [6] and references therein. In case when theoretical data for electron impact widths and shifts are not available one can use for E E E

singly ionized atom lines: semiempirical formula [8] for estimation of w and d , multiply ionized atom lines: simpli"ed semiclassical formula (Eq. (526) in [3]) for w , singly and multiply ionized atoms: modi"ed semiempirical formulas [9}11] for w and d , and classical-path approximation approach [29] for w .

3. Experimental determination of line shapes and shifts In principle, line shape determinations are straightforward. For the ideal case of homogeneous steady-state plasmas, which are approximately realized in end-on observations of stabilized arcs, and in fast homogeneous pulsed sources, one simply scans over the pro"le with a spectrometer of su$cient resolution. To carry out the very fast spectral scanning of pulsed plasmas, optical multi-channel analyzers, fast rotating mirror systems and fast scanning Fabry}Perot interferometers have been developed. Side-on observations of stabilized arcs and plasma jets are more tedious. In this case one has to obtain the intensity distribution over the cross section of the radially symmetric plasma for many "xed wavelength positions over the line pro"le. Then, by applying an Abel inversion technique (see e.g. [30,31]), one obtains the radial intensity distribution at each wavelength and can thus construct a pro"le for the radial positions of interest. An advantage of this method is that one may obtain pro"les for a range of electron densities corresponding to the


347

Fig. 1. Set of Abel-inverted pro"les for the Ar I 425.9 nm line: 15 di!erent radial positions starting from the axis of a wall stabilized arc [32].

di!erent radial positions, i.e., one may measure the functional dependence of line shape parameters on electron density and temperature from a single experimental run, see example in Fig. 1. In line shape determinations from pulsed sources with standard monochromators, one may advance the instrument stepwise in small wavelength increments from shot-to-shot over the range of the line pro"le. Good reproducibility of the pulsed plasma source and convenient monitoring setup are required for this operation. With modern waveform recorders (digital oscilloscopes) this technique may be considerably improved. Time evolution of the light intensity at di!erent wavelengths may be stored so the analysis of the data may be performed in various times of plasma evolution and decay. This technique is useful in particular for repetitive plasma sources where the signal averaging technique may be employed. One can achieve a similar goal by using a boxcar averager instead of an oscilloscope. In this case signals are averaged only in a preselected time of the plasma existence. Of course, the most advanced technique of spectral line recording is to use optical multi-channel analyzer (OMA) whose detector head is made of a large number of diodes or charged coupled detectors (CCD) which may be gated and used for time-resolved spectra recordings. Unfortunately, the "nite dimensions of diodes or CCD (typically 15 or 25 lm) frequently limit spectral resolution of the detection system, therefore OMA is mainly used for studies of broader lines. Line shift is measured by comparing the investigated line with unshifted reference line generated in low-pressure lamps (see Fig. 2). The main di$culty one may encounter in these experiments is to obtain a comparable signal from the main discharge and from a low-intensity reference source. This is di$cult in particular, if the line shapes from powerful pulsed plasma sources are studied. Comparable intensities to those from the reference source may be achieved by selecting the transmittivity of the partially re#ecting mirror (see Fig. 2A). Another way is to extend the observation time (see e.g. [34]) during reference line recording. If this technique is combined with wavelength positioning of the monochromator (best results achieved with the stepping motor) and automatic radiation chopper control (see Fig. 2B) with all events synchronized and controlled by

348


Fig. 2. Shift measurement techniques: (A) simultaneously recorded spectra from plasma and reference source (RS) and (B) alternatively recording radiation from plasma and reference source. PRM and S are partially re#ecting mirror and monochromator's slit respectively. Examples of spectra recordings for (A) and (B) are taken from [33] and [34], respectively.

PC, one obtains a powerful apparatus for line shift measurements. An alternative to the extension of reference source observation time is an increase of sensitivity of the radiation detection system during reference spectra recordings. This can be achieved with an externally controlled detector power supply which is commercially available nowadays. The line shift measurements are most accurate if one can use the same unshifted line from the reference source for comparison. In this way uncertainty in wavelength of the unshifted line cancels out. The line shift may be determined at the maximum of the line pro"le or at its half-width. In the case of asymmetric line pro"le (e.g. neutral atom lines) these two shifts slightly di!er, see Eqs. (2) and (3). For homogeneous steady-state plasmas, shift measurements may be e$ciently done by the photographic method because of the ease of spectra recording on a photographic plate and measuring small wavelength di!erences. A special method applicable to pulsed sources is to use as a reference line the line emission from this same source at a very late stage of the discharge, when the electron density has decreased to a very small value [35]. Another technique for shift measurements is to use an auxiliary low current discharge within the plasma source to determine an unshifted line position which is later during pulsed plasma evolution used for line shift measurements (see e.g. [36]). Here, one has to mention techniques of width and shift measurements based on linear and nonlinear laser spectroscopy (see e.g. [37]). These techniques are, in particular, advantageous for


349

the study of small widths and shifts typical for low-pressure gas discharges. Here we draw attention to the use of optogalvanic signal from the reference source low-pressure discharge, as a wavelength indicator of unshifted line position [38]. Recently, tunable lasers have been used in absorption and induced #uorescence experiments for electron density diagnostics, see e.g. [39}42]. For this application tunable semiconductor diode lasers are in particular very popular and widely used in numerous laboratories. They have typical bandwidth of several tenths of megahertz while diode lasers with the bandwidth of 0.1 MHz are commercially available. The simplest way of semiconductor laser tuning is to change diode temperature, while for "ne wavelength scanning diode current is being used. In a typical experiment the continuous frequency scanning laser beam is directed through plasma and by monitoring either the intensity of transmitted radiation or by recording the induced #uorescence intensity it is possible to determine plasma line shape and in conjunction with the wavelength reference source, the line shift. If all necessary theoretical data are available, the measured half-width and/or shift can be used to determine N from Eqs. (1)}(3). All precautions related to the in#uence of radiative transfer on the line shapes, see Section 4.6, may be applied to induced #uorescence experiments. In case of the absorption experiment the intensity of transmitted radiation, I , through a homogeneous plasma slab of thickness l is H given by

J (12) I (l)"I exp ! k dx , H H H where I is the intensity of incident laser radiation. In the case of homogeneous plasma, H determination of the absorption coe$cient k is rather simple. For inhomogeneous plasma one has H to determine local absorption coe$cient and this may not be an easy task. In a special case of cylindrical plasmas one has, like in emission spectroscopy, to apply the Abel inversion procedure to determine radial distribution of k . Once k is deduced it may be further used for determination of H H plasma line shape function from the following relationship (see e.g. [43]): H g g N j (13) k " A N 1! , H H 8p g g N where j is the central wavelength for the observed transition from energy level 1 to energy level 2, A is the probability for spontaneous emission from 2 to 1 and g and g are the degeneracies of the levels in question. The line shape function is normalized to one. So H by measuring absorption coe$cient k it is possible to determine the line shape pro"le irrespectH ive of the broadening mechanism. One should take great care in recording k so as to H avoid saturation e!ects which may be induced by laser radiation. Particularly in cases of low current, low-pressure discharges where the number of collisions per unit time interval is relatively small, so due to the narrow laser bandwidth, even a laser beam of a few milliwatts only, may produce considerable saturation. This would result in a broader line pro"le and consequently larger plasma electron densities or gas temperatures would be detected. To avoid saturation e!ects it is necessary to reduce the intensity of the laser probing beam (usually with neutral density "lters) until the recorded line width does not depend upon the intensity of incident laser radiation.


350

4. Other broadening mechanisms The experimentally observed line shapes are usually the result of several broadening factors from which the Stark pro"le component has to be isolated and retrieved. While the experiments reporting plasma broadening data are usually designed so as to have Stark broadening as the dominant cause of the width, the conditions can rarely be made so ideal that all broadening factors become insigni"cant. For the line shifts, the situation is less complex insofar as van der Waals shifts are the only competing factor. When line shapes and shifts are used for plasma diagnostic purposes, in most of the cases, one cannot change plasma conditions so the contribution of other broadening mechanisms has to be carefully estimated and, if necessary, measured widths and shifts corrected. Here we shall list the other broadening mechanisms which are present in plasmas and may in#uence Stark broadening parameter measurements, see also [21]. All widths given below are FWHM (full-widths at half-maximum). 4.1. Natural broadening Natural broadening arises because even an unperturbed level has a "nite lifetime, q, due to spontaneous emission. For a transition between states n and m,

(14) w (cm)"j A # A /2pc , KYK LYL * KY LY where A is the transition probability between the state m and any àlloweda level m. Natural KYK broadening is largest when one of the two levels is dipole-coupled to the ground state. Even in this case, it is usually negligible (of the order of 10\ nm). However, it may be of some importance for low electron density plasmas generated in low-pressure gas discharges. 4.2. Resonance broadening Resonance broadening occurs for transitions involving a level that is dipole-coupled to the ground state. Ali and Griem [44] derived an equation for width evaluation, w (cm)"1.63;10\(g /g )jj f N , 0 G I 0 0

(15)

where j is the wavelength (in cm) of the observed radiation. Here, N is the density of groundstate particles which are the same species as the emitter, and g and g are the statistical weights G I of the upper and lower state; j and f are the wavelength and f-value, respectively, of the 0 0 resonance transition from level `Ra. The level `Ra is the upper or lower level of the observed transition, which happens also to be the upper level of a resonance transition to the ground state. Not all transitions involve such a level. The shifts due to resonance broadening are negligible. For 4s[3/2]!4p[3/2] transition of neutral Ar, j"810.37 nm; j (3p S!4s[3/2])" 0 106.67 nm; g "1; g "3; f "0.061 at N "1;10 cm\, w "0.00212 nm. G I 0 0


351

4.3. Van der Waals broadening Van der Waals broadening results from the dipole interaction of an excited atom with the induced dipole of a neutral ground-state atom of density N . This is a short-range C /r interaction. Griem's estimate [45] for the full-width at half-maximum w can be written as w (cm)"8.18;10\j(aN R)(¹ /k)N where R"R !R . 3 *

(16)

(17)

R is the di!erence of the squares of coordinate vectors (in a units) of the upper and lower level, and k is the atom-perturber reduced mass in a.m.u. (k"19.97 for excited Ar perturbed by Ar atoms). Values of aN , the mean atomic polarizability of the neutral perturber, are tabulated for di!erent elements by Allen [46]: for argon aN "16.54;10\ cm. If the required value of aN is not tabulated, it can be estimated either from the expression given by Allen [46] or by Griem [45]: aN "(9/2)a(3E /4E ) , (18) & #6! where E is the ionization potential of hydrogen (109737.32 cm) and E is the energy of the "rst & #6! excited level of the perturber. In the Coulomb approximation the values of R and R in Eq. (17) * 3 may be calculated from R"nH/2[5nH#1!3l (l #1)] , (19) H H H H H where the square of e!ective quantum number nH is H nH"E /(E !E ) . (20) H & '. H Here E is the ionization potential of the studied element and E is the energy of the upper or lower '. H level of the transition. van der Waals broadening causes a red wavelength shift which is two-third of the size of the width w . For 4s[3/2]}4p[3/2] transition of neutral Ar, and with Ar as perturber; j"810.37 nm; I"127 109 cm\; E "106 087 cm\ and E "93751 cm; k"19.97; aN "16.54;10\. At 3 * ¹ "7000 K and N "1;10 cm\, w "0.00356 nm. 4.4. Doppler broadening For a Doppler-broadened spectral line the intensity distribution is Gaussian whose full halfwidth w is given by (21) w "7.16;10\j((¹ /M) , " where ¹ and M are gas kinetic temperature (in K) and mass of radiating atom (in a.m.u.), respectively.

352


4.5. Instrumental broadening The standard practice to determine the apparatus pro"le of a spectrometer is to scan over a line whose intrinsic width is very small compared to the apparatus width so that the latter determines its shape. Typically, such narrow lines are obtained from low-pressure discharges, like Geissler tubes or hollow cathode lamps. The instrumental line shapes obtained with good spectrometers are usually approximately Gaussian pro"les in particular central core which usually is being used for deconvolution procedure. The far wings of the instrumental pro"les often decay more slowly. Since the Stark pro"les of ionic lines are, to a very good approximation, of dispersion type (Lorentzian shape), the well-known Voigt pro"le analysis (see [47] and references therein) can be applied to derive the Stark width from the observed line pro"le. In the case of neutral atom lines which are asymmetric due to the ion broadening contribution, another deconvolution procedure [48] has to be applied. The use of the deconvolution procedure [47] to the asymmetric line pro"les as applied by some authors introduces systematic error in Stark width (Lorentzian part of the experimental pro"le) measurements. This is well illustrated in Fig. 3 where coe$cients K and K , % * K "w /w , (22) % % K K "w /w , (23) * * K needed for evaluation of Gaussian w and Lorentzian w , half-widths from the measured width % * w are given as a function of the ratio of w /w where w and w are measured line widths K

Fig. 3. An example of K and K dependence upon w /w ratio for a symmetric Voigt pro"le (thick solid line) [47] % * and for asymmetric pro"les for o /o "0.40 (see Eq. (4)) and various ion broadening parameters A ranging from 0.1 " to 0.5 [48].


353

at one-tenth and one-half of the maximum intensity. The solid line in Fig. 3 is used for deconvolution of symmetric Voigt pro"les [47] while for asymmetric line pro"les whose asymmetry depends upon the ion broadening parameter, A, coe$cients K and K have di!erent values for the same % * w w ratio. So the application of deconvolution procedure for the Voigt pro"le to the asymmetric line may introduce an error in determination of Gaussian and Stark widths whose magnitude depends upon ion broadening parameter A and may be as large as 25%; see Fig. 2 in [48]. In some cases, Fabry}Perot interferometer is employed for line pro"le measurements. The determination of the apparatus function is similar to the procedure described above but the analysis and deconvolution procedure is more complex (see [49,50] and references therein). Generally speaking it is best if one can virtually eliminate the instrumental broadening contribution by using a high-resolution spectrometer whose instrumental width is of the order of one-tenth or less than the observed experimental pro"le. 4.6. Self-absorption Self-absorption will have the e!ect of distorting and especially of broadening lines and will therefore produce an apparently large width. If the self-absorption originates mostly from the cooler boundary layer of much lower electron density and if the spectral resolution is su$cient, the line center exhibits readily recognizable self-reversal. But more often, self-absorption (especially when originates within homogeneous plasma layers) only slightly distorts the shape of a dispersion pro"le, even if it is appreciable [13]. Thus, it is very di$cult to judge the amount of self-absorption from the observed shape of a line. In fact, good adherence of Stark broadened lines to dispersion shapes may have sometimes mislead authors to the conclusion that line shapes have not su!ered self-absorption. A variety of well-established techniques exists to determine the presence of self-absorption e!ects. Particularly straightforward is the method of checking line intensity ratios within multiplets which are expected to adhere to LS-coupling. The well-known ratios for LS-coupling line strengths [46,51] are taken as the basis to which the observed line intensity ratios are compared, and reduction in the observed intensity of the strongest line in the multiplet relative to a weaker one indicates that self-absorption is present. In the case where transition probabilities within a multiplet are available, one can use them to evaluate line strengths (see e.g. [52,53]) and to compare those with experimentally determined values. In this way one is not related to the LS-coupling scheme only. The compilations of selected atomic transition probabilities data are available [52}54]. Furthermore, one can "nd vast transition probability data in the TOP base (the complete package of the Opacity Project data with the database management system which is usually referred to as TOP base) [55,56]. For new sources of transition probabilities data see also introduction of [54]. A modi"cation of this technique for self-absorption check, in a Stark parameters measurement, is to vary (if possible) the concentration of the species under study and to observe variations of the intensity ratio (and of the line widths) with decreased concentration. If the ratio of the line intensities (or line widths) remains constant within certain lines in multiplet these are optically thin and may be used for plasma diagnostic purposes without any correction for self-absorption.

354


Fig. 4. Self-absorption test: (A) with external concave mirror; (B) doubling the plasma length by moving an auxiliary electrode within discharge and (C) recorded line pro"les I (j) and I (j) with single and double plasma length, respectively.

Another popular technique to check for self absorption is to double the optical path length by placing a concave mirror at two focal lengths behind the plasma (see Fig. 4A) and to "nd out if the increase in signal intensity, except for re#ection and transmission losses, also doubles. In practice, one scans over the line pro"le with and without an external mirror and observes whether the pro"les are proportional everywhere to the same factor. Self-absorption is present if the proportionality factor is not maintained for the high-intensity region near the line center but instead becomes smaller. If the optical depth k l (k is absorption coe$cient and l the plasma H length) is not large, k l(1, one may correct the measured intensity to the limit of an optically thin H layer [57]. Recently, another technique for self-absorption check in axially homogeneous pulsed sources has been developed [58,59]. Within a linear pulsed discharge one auxiliary movable electrode is introduced between the cathode and anode (see Fig. 4B). By changing the position of the movable electrode one can change the observed plasma length without changing plasma impedance. In this


355

way, plasma parameters remain constant while observation length is being changed. By measuring line shapes with two plasma lengths it is possible to determine k l. H In order to evaluate correction for self-absorption for the case described in Fig. 4A and B we start from the expression for the intensity of radiation I from the homogeneous plasma layer H (length l) in LTE: I "B [1!exp(!k l)] , (24) H H2 H where B is the Planck's function and k the absorption coe$cient which is related to the emission H2 H coe$cient e "k B . Depending upon the value of k l one can distinguish three cases: H H H2 H 1. k l;1, a very low absorption when exp(!k l) 1!k l so Eq. (24) may be written as H H H I B k l e l . (25) H H2 H H 2. k l B> B> B> B> B> B> B> B> B>

[147] [148] [149] [149] [149] [149] [149] [149] [149] [149]

11 800}13 000 13 000 11 000}14 000 12 800 12 400 22 000 12 800 22 000 22 000 12 400 22 000 12 800 13 000 12 400 22 000 22 000 22 000

0.87} 0.90 0.91 0.39}0.51 0.61 0.55 0.94 0.57 0.88 0.94 0.71 1.02 0.60 0.91 0.68 0.96 0.92 1.05

0.30}0.30 0.36 0.16}0.20 0.24 0.22 0.32 0.24 0.32 0.35 0.29 0.34 0.24 0.36 0.27 0.34 0.34 0.40

B B B B> B> B> B> B> B> B> B> B> B B> B> B> B>

[144] [145] [146] [147] [148] [149] [147] [149] [149] [148] [149] [147] [145] [148] [149] [149] [149]

13 000 12 800 22 000 13 000 11 000}14 000 12 800 12 400 22 000 22 000 22 000 13 000 22 000 13 000 12 800 12 400 22 000

1.02 0.60 1.01 1.02 0.62}0.56 0.67 0.79 1.02 0.96 1.15 1.02 1.18 1.02 0.57 0.73 1.07

0.32 0.20 0.29 0.32 0.22}0.19 0.23 0.26 0.30 0.28 0.32 0.32 0.34 0.32 0.19 0.24 0.32

B B B> B B B> B> B> B> B> B B> B B B> B>

[145] [147] [149] [145] [146] [147] [148] [149] [149] [149] [145] [149] [145] [147] [148] [149]

396.84 391.48 394.43 387.53 403.88 399.20 393.12 3p4s}3p(P)4p

P}P (6)

480.60

493.32 497.22 473.59 484.78

500.93 506.20 P}D (7)

434.81

442.60

443.02 426.65 433.12 437.97


382 Table 13 (continued) Transition

Multiplet (No)

P}S (10)

j (nm)

Temperature range (K)

w /w

w /N

Accuracy

Reference

417.84 428.29

22 000 22 000

1.20 0.90

0.32 0.25

B> B>

[149] [149]

372.93

12 400 22 000 13 000 12 400 22 000 13 000 12 800 12 400 22 000

0.69 1.17 1.01 0.80 1.22 1.01 0.62 0.84 1.17

0.18 0.23 0.26 0.21 0.24 0.26 0.16 0.22 0.23

B> B> B B> B> B B> B> B>

[148] [149] [145] [148] [149] [145] [147] [148] [149]

13 000 12 800 12 400 22 000 22 000 13 000 12 800 12 400 22 000 12 800 12 400 22 000

1.00 0.68 0.78 1.24 1.34 1.00 0.66 0.75 1.27 0.79 0.80 1.30

0.38 0.26 0.30 0.36 0.39 0.38 0.25 0.29 0.37 0.30 0.31 0.38

B B> B> B> B> B B> B> B> B> B> B>

[145] [147] [148] [149] [149] [145] [147] [148] [149] [147] [148] [149]

385.06

392.86

P}P (15)

454.50

488.90 465.79

476.49

The average value for 10 900(¹(13 900 K.

Mult.7, w /w "1.089!5.28;10\¹ for 11 000(¹(22 000 K ,

Eu.:$20%, ADT"1.5% , Mult.10, w /w "0.756#1.98;10\¹ for 12 500(¹(22 000 K ,

Eu.:$17%, ADT"9.8% , Mult.15, w /w "0.592#3.16;10\¹ for 12 500(¹(22 000 K ,

Eu.:$18%, ADT"13.7% . Calcium: Ca II. All experimental and most of the theoretical results for the Stark widths of Ca II resonance lines are given together in Fig. 1 [17]. The experimental low temperature data points fall into two groups separated by more than a factor of two, see also selected experiments in Table 14. Comparisons with other experiments (see Fig. 1 [17]) are di$cult to interpret since they were carried out at signi"cantly higher temperatures and the temperature dependence of the widths is not clearly established. Interestingly, the Stark widths of Goldbach et al. [153] for the two lines are


383

Fig. 10. Same as Fig. 6 but for the Ar II lines from multiplets 6 and 7, respectively. The line strength values, S, were taken from [53]. The experimental results: (A) multiplet 6 䢇 [144], 䊏 [145], 夹 [146], £ [147], # [148], 䉭 [149]. (B) multiplet 7 䊏 [145], ; [146], * [147], £ [148], 䉭 [149].

larger for their higher temperature point, while all theories predict decreasing widths with increasing temperature in this range. Similar increasing temperature trend of Mg II experimental Stark widths for the data of Goldbach et al. [156] and Roberts and Barnard [157] may be detected in Fig. 1 [16]. Thus it is very di$cult to recommend best set of experimental data, but it is important to note that the results by Goldbach et al. [153] have the smallest estimated uncertainty. In this situation and with the large scatter of results in Table 14 determination of the best "t through data was not performed.


384 Table 14 Numerical data for Ca II Transition

Multiplet (No)

j (nm)


w /w

w /N

Accuracy

Reference

4s}4p

S}P (1)

393.37

12 000 19 000 13 000 12 240 & 13 350 18 560 13 000 7450 12 240 & 13 350

0.71 0.69 0.78 0.41 & 0.50 0.75 0.78 0.67 0.38 & 0.44

0.20 0.17 0.22 0.11 & 0.14 0.19 0.22 0.21 0.11 & 0.12

B B B B> B B B> B>

[150] [151] [152] [153] [154] [152] [155] [153]

396.85

Table 15 Numerical data for Xe II Transition

j (nm)


w /w

w /N

Accuracy

Reference

(P )6s[2]}(P )6p[1]

460.30

10 000 12 700}15 300 10 000 8000 13 000}15 300

1.11 2.13 0.89 1.07 1.59

0.44 0.68 0.48 0.67 0.69

B B B B B

[147] [158] [147] [159] [158]

10 000 11 000 11 100}15 300 8000 11 000 12 300}15 300 10 000 13 000}15 300

0.84 1.23 1.86 0.99 1.20 1.60 0.88 1.51

0.43 0.59 0.80 0.60 0.59 0.68 0.58 0.79

B B B B B B B B

[147] [160] [158] [159] [160] [158] [147] [158]

10 000 12 500}13 000 11 100}15 300 12 500}15 300 12 500}13 000 11 300}15 300

0.80 1.40 2.22 1.21 1.39 1.88

0.37 0.55 0.83 0.46 0.68 0.90

B B B B B B

[147] [160] [158] [158] [160] [158]

537.24

(P )6s[2]}(P )6p[2]

529.22

533.94

597.65 (P )6s[2]}(P )6p[3]

484.43

489.01 541.92

Xenon: Xe II. The results of four experiments [147,158}160] are compared with results of MSE [9] calculated in [161] using jK coupling scheme. Too large a scatter of data is evident from Table 15. New results are needed to clear the large discrepancy between various experiments. 8.3. Multiply ionized atom lines An intensive activity in this "eld is initiated by both, need of reliable data for theory testing and lack of data for high temperature, high electron density plasma diagnostics (primarily laser


385

produced and astrophysical plasmas). So, after the publication of last critical review [17] a large number of experimental results became available. The shapes and shifts of many spectral lines were remeasured or numerous new transitions were investigated. For example, Stark widths of N III spectral lines multiplet 3sS}3pP were studied intensively for more than 20 years (see Table 17 and [16,17]). As for singly charged ions, the experimental results for multiply ionized atom lines are presented in tabular form. For each set of data the source of semiclassical theoretical results taken for comparison is given. Although ion-impact broadening data are available for a number of investigated lines only electron impact contribution is taken into account. The normalized line widths w /N in data tables are given in [0.1 nm/(10e/cm)] units.

Carbon: C I

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