BW2003 WORKSHOP
Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model Perspectivesof the Balkan Collaborations
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BW2003 WORKSHOP
Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model Perspectives of the Balkan Collaborations Vrja&a Banja, Serbia and Montenegro 29 August - 3 September 2003
Editors
G. Djordjevic L. Nesic University of Nis, Serbia and Montenegro
J. Wess Ludwig-Maximillianss University University & Max Planck Institute, Germany
World Scientific NEW JERSEY * LONDON * SINGAPORE BElJlNG * SHANGHAI * HONG KONG - TAIPEI BANGALORE
Published by
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MATHEMATICAL, THEORETICAL AND PHENOMENOLOGICAL CHALLENGES BEYOND THE STANDARD MODEL Perspectives of the Balkan Collaborations Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd
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V
Preface The Balkan Workshop (BW2003): Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model - PERSPECTIVES OF BALKAN COLLABORATIONS - was held from 29 August to 2 September 2003, in VrnjaEka Banja, Serbia, right after the Fifth General Conference of the Balkan Physical Union (BPU5). The main purpose of the workshop was to foster the communication among the researchers in the Balkan region as well as between their international colleagues. This meeting created the opportunity for the researchers working in the general area of high energy physics (HEP) a t different institutions in the Balkan countries to present their work and results. It stands as a logical and natural extension of “Wissenschaftler in globaler Verantwortung (WIGV)” - Scientists in global responsibility initiative - an initiative for the advancement of the scientific contact between Germany and the countries which have emerged from the former Yugoslavia. As expected, a creative and supportive environment facilitated the establishment of new, closer collaborative ties as well as strengthening of the existing regional and interregional collaborations. Attendance of about 50 participants from 17 countries has encouraged closer contacts and cooperation between their faculties, universities and institutions, initiating an exchange of scientific personnel and enabling joint applications for support and participation in international projects. We expect that this scientific integration will have a positive influence on the society in the Balkan/Southeast Europe countries, in general. Between many topics considered during the workshop, we emphasized on: strings and superstrings; supersymmetry and conformal field theory; noncommutative, gauge and string field theories; D-branes and matrix models; cosmology, quantum gravity, extra dimensions, grand unification; particle physics, neutrino physics and various aspect of noncommutative, q-deformed and nonarchimedean models. A few rather poor mathematical topics, but also related to the main scope of the meeting, e.g. integrability of some mechanical systems, were also considered. The invited lectures, ten of which were presented in this volume, gave an excellent review of the “hot” topics in theoretical HEP, Q F T and cosmology. We believe that many readers of this book, as well as the young PhD students who attended the workshop, will benefit a lot from this assembly of excellent papers.
vi
Shorter and more topical papers of other lecturers cover many of all actual problems in theoretical and mathematical problems in HEP. That highlighted the interest of researchers from this region as well as of from about 15 participants from West Europe (mainly from Germany), USA, Russia and South America. In total, more than 80% of the speakers sent their contribution; we would like to thank them for this huge collection. Regrettably, a written document cannot record the stimulating atmosphere and the fruitful informal discussions that took place during lunches and coffebreaks, as well as during a visit to the Serbian orthodox monasteries from Middle Ages, located near VrnjaEka Banja We would like to thank Dragoljub Dimitrijevic for great help in preparing this proceedings. Kind help of Frank Mayer is also warmly acknowledged. We would like to thank also World Scientific Publishing Company (especially to Lance Sucharov, Katie Lydon and Rhaimie Wahap) for their goodwill and interest in publishing this proceedings. Goran Djordjevib LjubiSa NeSi6 Julius Wess NiS, Munich, September 2004
vii
Advisory Committee A. Albrecht (Davis) G. Altarelli (CERN) I. Antoniadis (CERN) E. Arik (Istanbul) TBC C. Bachas (Paris) J . Bagger (Baltimore) S. Bellucci (Frascati) S. Dimopoulos (Stanford) J . Ellis (CERN) B. Guberina (Zagreb) R. Jackiw (MIT) J . Louis (Halle) D. Luest (Berlin) S. Meljanac (Zagreb) L. Mezincescu (Miami) H. Nicolai (Golm) S. Randjbar-Daemi (ICTP) A. Sen (Allahabad) G. Senjanovic (ICTP) E. Sezgin (Texas AM Univ.) I. Todorov (Sofia) J. Wess (Munich) G. Zoupanos (Athens)
Organizing Committee B. Aneva (Sofia) A. Balaz, (Belgrade) G. L. Cardoso (Berlin) G. Djordjevic (Nis, Munich), chairman V. Dragovic (Belgrade, Trieste) L. Moeller (Munich) Lj. Nesic (Nis) A. Nicolaidis (Thessaloniki) C. Sochichiu (Chisinau, F’rascati)
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List of Participants INRNE, Bulgarian Academy of Sciences, 1784 Sofia Bulgaria
[email protected] Gleb Arutyunov Max-Plank-Institut fuer Gravitationphysik Am Muhlenberg 1, D-14476 Golm, Germany
[email protected] Paolo Aschieri Sektion Physik der Ludwig-Maximilians-Universitat Theresienstr. 37, D-80333 Munchen, Germany
[email protected] Borut Bajc Jozef Stefan Institute, Jamova 39, 1001 Ljubljana Slovenia,
[email protected] Antun Balaz Institute of Physics, P.O.Box 57, 11 001 Belgrade Serbia and Montenegro ant
[email protected] .ac.yu Klaus Behrndt Max-Plank-Institut fuer Gravitationphysik Am Muhlenberg 1, D-14476 Golm, Germany
[email protected] Faculty of Physics, University of Belgrade, P.O. Box 368 Maja Buric 11 001 Belgrade, Serbia and Montenegro majabQphy. bg.ac.yu Institut fur Physik, Humboldt University Gabriel Lopes Cardoso Newtonstrasse 15 D-12489 Berlin, Germany
[email protected] Masud Chaichian Helsinki Institut of Physics, P.O. Box 64, FIN-00014 Helsinki, Finland
[email protected] Dragoljub Department of Physics, Faculty of Sciences Dimitrijevic P.O. Box 224, 18000 Nis, Serbia and Montenegro
[email protected] Dragan Djordjevic Department of Mathematics, Faculty of Sciences P.O. Box 224, 18000 Nis, Serbia and Montenegro
[email protected] Goran Djordjevic Department of Physics, Faculty of Sciences P.O. Box 224, 18000 Nis, Serbia and Montenegro gorandj @junis .ni.ac. yu
Boyka Aneva
ix
X
Berkol Dogan
Vladimir Dragovic
Dan Radu Grigore
Sinisa Ignjatovic
Nemanja Kaloper
Ivan Kostov
Dusko Latas
George Lazarides
Katarina Matic
Alejandra Melfo Aleksandar Mikovic
Alexei Morozov
Bogazici University, Department of Physics, Bebek, 80815 Istanbul, Turkey
[email protected] Mathematical Institute Serbian Academy of Sciences and Arts P.O. Box 367, 11001 Beograd, Serbia and Montenegro
[email protected] Department of Theoretical Physics Inst. Atomic Physics, Bucharest-Magurele MG 6, Romania grigoreQtheor1.theory.nipne.ro Faculty of Natural Sciences and Mathematics Mladena Stojanovica 2, Banja Luka Republic of Srpska, Bosnia and Herzegovina
[email protected] Department of Physics, University of California Davis, CA 95616 USA kaloper @physics.ucdavis .edu Service de Physique Theorique, CNRS-URA 2306 C.E.A-Saclay, F-91 191 Gif-Sur-Yvette, F'rance
[email protected] Faculty of Physics, University of Belgrade P.O. Box 368, 11 001 Belgrade, Serbia and Montenegro lat
[email protected] .ac .yu Physics Division, School of Tehnology Aristotle University of Thessaloniki Thessaloniki GR 54124, Greece
[email protected] Faculty of Physics, University of Belgrade P.O. Box 368, 11 001 Belgrade, Serbia and Montenegro
[email protected] CAT, Universidad de Los Andes, Merida, Venezuela
[email protected] Departamento de Matematica e Ciencias de Computacao Universidade Lusofona de Humanidades e Tecnologias Av. do Campo Grande, 376, 749-024 Lisboa, Portugal
[email protected] State Science Center of Russian FederationInstitute of Theoretical and Experimental Physics
xi
B. Cheremushkinskaja, 25 MOSCOW, 117218, Russia
[email protected] Department of Physics, Faculty of Sciences Ljubisa Nesic P.O. Box 224, 18000 Nis, Serbia and Montenegro
[email protected] Argyris Nikolaidis Theoretical Physics Department University of Thessaloniki 54124 Thessaloniki, Greece
[email protected] Bojan Nikolic Faculty of Physics, University of Belgrade P.O. Box 368, 11 001 Belgrade, Serbia and Montenegro
[email protected] .ac.yu Institute for Nuclear Research and Nuclear Energy Todor Popov Bulgarian Academy of Sciences Tsarigradsko Chaussee 72, BG-1784, Sofia, Bulgaria
[email protected]. bg Voja Radovanovic Faculty of Physics, University of Belgrade P.O. Box 368, 11 001 Belgrade, Serbia and Montenegro
[email protected] .ac.yu Riccardo Rattazzi Theoretical Physics Division CERN CH-1211 Geneva 23, Switzerland
[email protected] Branislav Sazdovic Institute of Physics P.O.Box 57, 11 001 Belgrade, Serbia and Montenegro
[email protected] .ac.yu Volker Schomerus Service de Physique Theorique, CEA Saclay F-91191 Gif-sur-Yvette CEDEX, France vschomer @aei-potsdam.mpg .de Goran Senjanovic International Centre for Theoretical Physics Trieste, 34 014, Italy
[email protected] Corneliu Sochichiu INFN - Laboratori Nazionali di Frascati Via E. Fermi 40, 00044 Frascati, Italy Didina Serban
[email protected] Service de Physique Thorique CEA/Saclay - Orme des Merisiers F-91191 Gif-sur-Yvette Cedex, France
[email protected] xii
Sergei Solodukhin International University Bremen School of Engineering and Science P.O.Box 750561, 28759, Bremen, Germany
[email protected] Jelena Stankovic Department of Physics, Faculty of Sciences P.O. Box 224, 18000 Nis, Serbia and Montenegro
[email protected] Gordan Stanojevic Department of Physics, Faculty of Sciences P.O. Box 224, 18000 Nis, Serbia and Montenegro
[email protected] Harold Steinacket Sektion Physik der Ludwig-Maximilians-Universitat Theresienstr. 37, D-80333 Munchen, Germany
[email protected] Svjetlana Terzic Faculty of Sciences and Mathematics Cetinjski put BB, 81000 Podgorica Serbia and Montenegro
[email protected] Rudjer Boskovic Institute Tomislav Terzic Theoretical Physics Division P.O. Box 180, HR-10002 Zagreb, Croatia
[email protected] Helsinki Institut of Physics, P.O. Box 64, FIN-00014 Anca Tureanu Helsinki, Finland
[email protected] Mihai Visinescu Department of Theoretical Physics National Institute of Physics and Nuclear Engineering Magurele, P.O.Box MG-6 76900 Bucharest, Romania
[email protected],ro Marko Vojinovic Faculty of Physics, University of Belgrade P.O. Box 368, 11 001 Belgrade, Serbia and Montenegro
[email protected]. yu Sektion Physik der Ludwig-Maximilians-Universitat Julius Wess Theresienstr. 37, D-80333 Munchen, Germany
[email protected] Marija Zamaklar International Centre for Theoretical Physics Trieste, 34 014, Italy
[email protected] ...
Xlll
Alexei Zamolodchikov
Laboratoire de Physique Mathematique Universite Montpellier 11, P1. E. Bataillon 34095 Montpellier, France zamolodQLPM .univ-montp2 .fr
Guests Martin Huber
Zvonko Maric
Vladimir Kouzminov
Ilija Savic
Metin Arik
European Physical Society 4 rue des Freres Lumieres F-68200 Mulhouse, France
[email protected] Serbian Academy of Arts and Sciences Knez Mihailova 35, 11001, Belgrade Serbia and Montenegro
[email protected] ROSTE Regional Bureau for Science in Europe Palazzo Zorzi, Castello 4930 30122 Venice, Italy
[email protected] Serbian Physical Society Pregrevica 118, 11080 Zemun Serbia and Montenegro
[email protected] Bogazici University, Department of Physics, Bebek, 80815 Istanbul, Turkey
[email protected] CONTENTS
Preface Organizers and Committees List of Participants Conference Photo
V
vii ix xiv
I. Invited Lectures Integrable Structures in the Gauge/String Corespondence G. Arutyunov
3
Fluxes in M-theory on 7-manifolds: Gz-, SW(3)- and SU(2)-structures K. Behrndt, C. Jeschek
16
Noncommutative Quantum Field Theory: Review and its Latest Achievements M. Chaichian
32
Shadows of Quantum Black Holes N . Kaloper
47
Yukawa Quasi-Unification and Inflation G. Lazarides, C. Pallis
56
Supersymmetric Grand Unification: The Quest for the Theory A . Melfo, G. SenjanoviC
71
Spin Foam Models of Quantum Gravity A . Mikovic'
88
Riemann-Cartan Space-time in Stringy Geometry B. Sazdovic'
94
xv
xvi
Can Black Hole Relax Unitarily? S. N . Solodukhin
109
Deformed Coordinate Spaces Derivatives J. Wess
122
11. Short Lectures Deformed Coherent State Solution to Multiparticle Stochastic Processes B. Aneva
131
Non-Commutative GUTS, Standard Model and C, P, T Properties from Seiberg-Witten Map P. Aschieri
142
152
Seesaw, Susy and SO(10) B. Bajc On the Dynamics of BMN Operators of Finite Size and the Model of String Bits S. Bellucci, C. Sochichiu Divergencies in &expanded Noncommutative Yang-Mills Theory M. BuriC and V. Radovanovic'
162
SU(2)
Heterotic String Compactifications with Fluxes G. L. Cardoso, G. Curio, G. Dall'Agata and D. Lust Symmetries and Supersymmetries of the Dirac-type operators on Euclidean Taub-NUT space I. I. Cotciescu, M. Visinescu Real and pAdic Aspects of Quantization of Tachyons G. S. Djordjevic', Lj. NeSiC Skew-Symmetric Lax Polynomial Matrices and Integrable Rigid Body Systems V. DragoviC, B. GajiC
171
181
186
197
208
xvii
Supersymmetric Quantum Field Theories
220
D. R. Grigore Parastatistics Algebras and Combinatorics
231
T. Popov Noncommutative D-branes on Group Manifolds
241
J . Pawelczyk, H. Steinacker High-Energy Bounds on the Scattering Amplitude in Noncommutative Quantum Field Theory
247
A . Tureanu Many Faces of D-branes: from Flat Space, via AdS to pp-waves
258
M.Zamaklar Abstracts and Titles of Reports not Included in the Volume
269
Acknowledgements Sponsors Statement of intention Epilogue
273 275 277 279
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I. INVITED LECTURES
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INTEGRABLE STRUCTURES IN THE GAUGE/STRING CORRESPONDENCE
G. ARUTYUNOV Max-Planck-Institut fur Gmvitationsphysik Albert-Einstein-Institut Am Muhlenberg 1, 0-14476Potsdam, Germany E-mail:
[email protected] We discuss the integrable structures both of the classical closed string sigma model with the Ads5 x S5 taget space and of the planar maximally supersymmetricYangMills theory. By using the Biicklund transformations we show that in the sector of highly energetic and spinning strings the integrable structures of string and gauge theories match precisely up to two loops.
1. Gauge/String Duality
One of the fundamental questions of modern theoretical physics is the connection between gauge and string theories. In 1997 J. Maldacena conjectured a new surprising relation between gauge theories and strings.l According to the AdS/CFT duality conjecture, certain quantum supersymmetric conformal field theories have a dual formulation in terms of a closed superstring theory on the Anti-de-Sitter (Ads) background. To fully appreciate the non-triviality of this statement we recall that closed string theory contains gravity, and now it appears to have an alternative description in terms of a non-gravitational theory! In this lecture we report on a recent exciting progress towards understanding the fundamental example of the gauge/string dual pair, which involves four-dimensional maximally supersymmetric Yang-Mills theory and type IIB superstring propagating in the Ad& x S5 space-time (the product of a five-dimensional Ads space and a five-sphere). Already a first inspection shows that both theories mentioned above possess the same amount of symmetry, which can be taken as an initial evidence that they indeed might relate to each other in a non-trivial way. However, this reasoning is kinematical, and the real question is whether 3
4
these theories also share the same dynamical features. This is much harder to answer. By duality strongly coupled Yang-Mills theory is equivalent to weakly coupled string and vice versa. Because of a lack of adequate theoretical methods we neither have much insight into the strongly coupled regime of the gauge theory, nor on the structure of the spectrum of strings propagating in a curved space-time. Fortunately, due to so far poorly understood reasons, parts of the gauge and the string spectra, both accessible by existing mathematical tools, seem to allow for a direct comparison. The modern development of the gauge/string duality was initiated by Berenstein, Maldacena and Nastase2 (BMN) who noticed that in a certain (Penrose) limit the corresponding string theory becomes solvable and its excitations can be identified with gauge theory operators of a certain type. Later on it became clear that there exists even a larger sector of string state^^^^^^ which is accessible by semiclassical6 methods. In parallel, the recent advances in gauge theory are due to an important observation that the planar JV = 4 super Yang-Mills theory is integrable in the one-loop approximation7t8 and, very likely, at higher loops as well.g 2. Integrable Structure of Gauge Theory
The maximally supersymmetric SYM theory has the following field content: six scalar fields 42, i = 1,.. . ,6, a vector field A,, and four Majorana fermions $,: where T = 1,.. . ,4. All fields are in the adjoint representation of SU(N). The action is
ri)are ten-dimensional Dirac matrices in the Majorana-Weyl Here (P, representation. In what follows it will be convenient to use the ’t Hooft coupling X = g 2 N as a natural loop counting parameter. The theory we consider is known to be finite, i.e. the beta-function vanishes at any loop order. The absence of scale implies that SYM remains conformal even at the quantum level. Supersymmetry together with conformal invariance are combined in a larger superconformal group known as PSU(2,214), which contains the bosonic subgroup S0(4,2) xSO(6): PSU(2,214) 3 Su(2,2)xSU(4)~z S0(4,2)xSO(6)
5
The basic physical quantities of this theory are the local, gauge-invariant composite operators O ( x ) ,i.e. the operators constructed as products of elementary fields. They transform in unitary irreducible representations of the superconformal group. Therefore, to each operator one can associate scaling (conformal) dimension A, the Lorentz spins s1,sa and three additional Dynkin labels ai related to the internal SU(4) symmetry:
[A,s i , ~ 2a ;i, a2, a31 . These are the quantum numbers associated to any highest weight state (also called a superconformal primary state) of a supersymmetry multiplet. An important class of operators we will be concerned in here is
O ( x )= Tr (@pa$@$) + ...,
@1 =
4’
+ i$2,
etc.
The dots indicate arbitrary orderings of the scalars inside the trace. All such operators realize irrep of SU(4) with labels [J2 - J3, J1 - J2, J2 J3]. Note that these operators are holomorphic and their dimension in free theory is
+
A = J1 + J2 + J3.
In general, due to quantum fluctuations, scaling dimensions get shifted from their classical values and acquire an “anomalous” piece. Conformal primary operator 0 is an eigenstate of the dilatation operator D , which is one of the generators of the conformal algebra, and conformal dimension is its eigenvalue
DO = A(& N ) O , k=O
It should be stressed that the existence of anomalous dimensions is one of the most important concepts of conformal field theory as they constitute the spectrum of the theory. At the same time calculation of anomalous dimensions proved to be one of the hardest problems of quantum field theory. Until recently little was known about the general behaviour of anomalous dimensions in super Yang-Mills theory, see however Ref. 10, 11. The major problem arises, even at one loop, due to the complicated mixing between operators differing by the ordering of the scalars inside the trace. However, for the planar case N = 00, a new very important feature arises:’ The problem of diagonalizing the one-loop dilatation operator becomes equivalent to the problem of finding the spectrum of the so-called XXX Heisenberg magnet, the famous integrable short-range spin chain. Moreover, as was conjectured in Ref. 9, the integrability extends to higher loop orders!
6
To convey the basic idea about the spin chain description of gauge theory we restrict ourselves to the simplest case of operators made of two complex scalar fields
... .
n(a:1!@2Jz+
On the space of renormalized fields {Oi} (here index i labels different orderings of scalars inside the trace) the one-loop dilatation operator acts in the non-diagonal fashion
DOi = AijOj. The mixing matrix with elements Aij encodes the Feynman diagrammatics and its eigenvalues are the scaling dimensions of primary operators. The mapping to the XXX spin chain is constructed by identifying the (a1 and $2 fields with ‘hp” and “down” spins (see Fig.1).
Figure 1. Periodic spin chain. The Hamiltonian H acts as 2 J x 2’ matrix, where J = J1 J2 is the length of the chain. We look for the eigenstates of H in a sector with the total spin fixed.
+
Upon this identification the dilatation operator becomes identical to the Hamiltonian of the spin chain. The importance of this observation is difficult to overestimate. For operators of sufficiently small dimensions the mixing matrix can be diagonalized by hand or by a computer. As its rank grows (especially in the thermodynamic limit we are interested in) this becomes rapidly unfeasible. Fortunately integrability saves the day - there is an efficient method, based on existence of local commuting charges, which allows one to determine the spectrum of the Hamiltonian (and simultaneously the one of all the commuting charges). This is so-called algebraic Bethe Ansatz (see Ref. 12 for a comprehensive review). Let us associate to each lattice cite of the periodic spin chain the following Lax operator
7
Here Sf,S3 are the standard spin operators represented by Pauli matrices and cp is a spectral parameter. Matrix L,(cp) is a 2 x 2 matrix in the auxiliary two-dimensional space. The monodromy around the chain is
Using the fundamental commutation relations between the elements of the monodromy operator one can show that the trace of T(cp) is in fact a generating function of commuting charges 5-2
,
[Qk Q m ]
=0.
k=O
Simultaneous eigenstates of Q k can be found by first defining the vacuum state vacuum state 10) (with all spins “up”)
An eigenstate of
Q k
with M = Jz spins down is given by @M = B(cpl)...B(cpM)R
provided the Bethe roots cpj tions
7
= cp(pj) obey the set of algebraic Bethe equa-
where cp(p) = ;cot(ip). The Bethe equations have a beautiful physical interpretation in terms of quasi-particles, called magnons. The state @ M describes an excitation with M magnons, each of them carrying momentum p j . Thus, the Bethe equations can be viewed as quantization condition for the magnon momenta. The total phase gained by the magnon with momentum p j traveling around the chain of length J is equal to the sum of pairwise phase shifts which arise due to its elastic scattering with other M 1magnons. The factorized scattering is one of the remarkable consequences of complete integrability of the model. Further simplification of the Bethe equations occurs in the thermodynamic limit, which amounts to sending the length of the chain J and the number of magnons M = Jz to infinity while keeping the filling fraction
8
9
a= constant. To perform this limit we present the first equation in (1) in the logarithmic form
2nnj + p j =
where the mode (winding) numbers nj define the branch of the log function. Now one can easily recognize on the r.h.s. of this equation the difference analog of derivative. Therefore, in the large J limit the Bethe equations reduce to
Further we assume that the roots c p j condense onto certain smooth contours -+ cp and that
C in the complex plane of the continuous variable cp:
y
y)
the root distribution is described by a density p(cp) = ~~~1 6 (cp normalized as J " d p p(cp) = a. Taking into account that pj -+ 1 in the %. limit we consider we see that Eqs.(2) transform into a singular integral equation
where nC are mode (winding) numbers which are constant on each smooth component of the density support C. From the point of view of soliton theory the density p(cp) describes the continuous spectrum of the model. Thus, in the thermodynamic limit the discrete eigenvalues of the monodromy matrix (the Bethe roots respectively) condense and form the continuous part of the spectrum. Having found the density one can define the generating function of all local commuting charges Qk (resolvent):
In particular, the eigenvalues of the dilatation operator coincide with the value of the second charge Q2. For the simplest ground state solution13 the one-loop gauge theory resolvent was determined in Ref. 14 and it reads as
9
Here and below K(q), E(q) and II(m2,q ) are the standard elliptic integrals of the first, the second and the third kind respectively; the parameters a and b are a = 1/4K(q), b = a/-. Expanding in cp we find explicit form of the first few charges for the ground state solution:
Q6
26 = T K ( ~ )~ 2(8 - 8q
[
+ 3q2)E(q) + (16 - 24q + 18q2 - 5 q 3 ) K ( q ) ],
The gauge theory modulus q is expressed via the filling fraction a through the following transcendental equation
To conclude, the problem of diagonalizing the one-loop dilatation operator is integrable and can be solved by using the algebraic Bethe Ansatz. The spectrum of the theory is encoded in the resolvent whose explicit form for the ground state is Eq.(3). Now we turn our attention to string theory. 3. Integrable Structure of String Theory
String theory we consider can be described by a non-linear two-dimensional sigma model whose target space is a supersymmetric extension of the bosonic Ad& x S5 space-time. Since the corresponding action is highly non-linear and contains fermions, the quantization problem appears to be extremely complicated; at present the full spectrum of the quantum string is beyond our reach. However certain regions of the quantum spectrum can be well approximated by semiclassical string configurations. Typically these configurations are solutions of the classical string equations of motion (supplemented by the Virasoro constraint), which carry “large” energy and spins. Picking one such solution one can approximately determine the string spectrum by performing a semiclassical quantization around it. Thus, we are led to the problem of studying “spinning” strings,i.e. classical strings rotating in the background space-time with large angular momenta. The bosonic sigma model describing propagation of our classical string is a two-dimensional integrable model which can be thought of as a non-trivial
10
matrix generalization of the famous sine-Gordon equation. The action is a sum of S0(2,4) and SO(6) sigma models
\,.
where
Here
X M , M = l , ..., 6 ,
Y M , M = O ,..., 5
are the the embedding coordinates of R6 with the Euclidean metric in L s and with ~ M = N (-1, +1, +I, +1, +1, -1) in LAds respectively; A and i i are the Lagrange multipliers. What are the relevant spinning string configurations? As in the flat space-time the simplest configurations are those corresponding to rigid strings, i.e. to strings whose shape is independent of time. These configurations carry finite energy and can be viewed as solitons (the so-called finite-zone solutions) of the sigma model. A remarkable fact about these solitonic solutions discovered in Ref. 15 is that they are naturally classified in terms of periodic solutions of the Neumann integrable system. This is a finite-dimensional integrable system describing a three-dimensional harmonic oscillator constrained to move on a two-sphere (or a hyperboloid in the non-compact case). Historically this model, discovered by C. Neumann in 1859, is one of the first examples of a completely integrable Hamiltonian system. To see the appearance of the Neumann system we consider the simplest case: string rotates in S5 and is trivially embedded in Ads5 as Y5 iY0 = einT with Yl,...,Y4 = 0. Ansatz for periodic motion with three non-zero angular momentum components Ji reads as:
+
x1+ ix2 = q ( ~ eiwlT, ) ~3 + i xq = z ~ ( GeiwZT ) , ~5 + i& = ~ ( 0eiwzT ) , where
11
Upon substituting this ansatz into the sigma model evolution equations they reduce to the 1-d ( ''mechanical") model (0is time now !)
which is the Neumann integrable system. If the coordinates xi(.) are complex then one gets16 another integrable system known as the NeumannRosochatius model. It is worth noting that the Neumann system inherits its integrable structure from that of the two-dimensional sigma model. Rigid strings appear to be of two types - folded, with the topology of a rod, and circular, with the topology of a circle (see Fig.2).
Figure 2. Folded and circular rigid strings (generically hyperelliptic three-spin solutions). Elliptic two-spin solutions arise when string stretches along the equator in the ("1,121 plane.
The angular momentum components (spins) are
From here one can determine the frequencies wi in terms of spins Ji: wi = Wi(J1,J 2 , J 3 ) .
The general three spin solutions are described in terms of the hyperelliptic functions. Elliptic two spin solutions arise when x3(0) = 0. Explicitly the two-spin solution corresponding to the folded string can be written in
12
terms of Jacobi elliptic functions q ( c r ) =d n ( o 6 , t)
,
22(cr) =
d?sn(cr&,
t) .
(4)
Since we are dealing with closed strings the coordinates xi(.) must obey the periodicity condition: q ( c r 27r) = zi(a). This leads t o the following equation
+
Solving for w’s in terms of spins
the modulus t is then found from
Ji,
2
4
(&) =gX.
J2
(K(t)-E(tJ2-
As was already mentioned in Sect. 1, the rigid strings provide a simple and useful tool to probe matching of the spectra of gauge and string theories. The rigid two-spin solutions are completely described by two integrals of motion which are the spins J1 and J2. However, when we embed a rigid soliton into the two-dimensional sigma model it inherits from its integrable structure an infinite set of local commuting integrals of motion, all of them appear as the non-trivial functions of J1 and 5 2 . How to compute these integrals and compare with the gauge theory charges discussed in the previous section? This problem was completely solved in Ref. 14, 17. An important technical tool which allows to determine the spectrum of commuting integrals is the so-called Backlund transformations. The Backlund transformations transform one solution, X , of the evolution equations into another one, X ( p ) , and they depend continuously on a (spectral parameter) p:
The generating function of the local commuting charges of the sigma model cx)
k=2
can be obtained from the Backlund solution as follows
+
o and = r - (T are the light-cone coordinates. It is a where [ = r very non-trivial problem to solve the Backlund equations (we refrain from presenting them here) and a general solution is unknown. However, the
13
corresponding Backlund solutions X(y) were found14 starting from either two-spin folded (4) or circular string solitons. We will not go into the further details here and refer the reader to the original The final result for an exact generating function of string commuting charges (for the folded two-spin solution) is:14
and z satisfies
Here w1,2 and t are certain functions of 51, J2. Now we are ready to compare the integrable structures of gauge and string theories. 4. Matching the Spectra of Gauge and String Theories
Resolvent (5) provides the “all-loop” result and to compare with our findings in gauge theory one needs to detect from Eq. (5) the “one-loop” contribution. To this end we consider the slightly modified string theory resolvent
where & ( p ) is determined by Eq. (5). Here the string resolvent is viewed as a function of the rescaled spectral parameter ‘p and the so-called BMN coupling constant
We will refer to p and ‘p as to the string and gauge spectral parameters respectively. Clearly, the Backlund resolvent can be expanded in a double series cn k=O
= Q(l’(‘p)
n,k=O
+
+ X 1 2 Q ( 3 ) ( ‘ p ) + O(A’3).
In particular, the leading term Q(l)(‘p) represent the “one-loop” contribution and should be directly compared to our one-loop gauge theory result W3).
14
A remarkable fact14 is that Q(l)(cp) appears precisely the same (after a certain Gauss-Landen modular transformation which relates the string to the gauge moduli) as the one-loop gauge theory resolvent H(cp) generating the Heisenberg charges: Q q P ) = H(cp). (6) Although derived by using the particular gauge/string theory solutions, this formula is universal and does not depend on particular solutions. It exhibits the matching of the spectra of gauge and string theories at the lowest order of perturbation theory. 5. Discussion In this lecture we confined ourselves to demonstrate the matching of the gauge/string spectra at the one-loop approximation. Recently this relationship between gauge and string theories received further spectacular confirmation. Of course, a question of primary importance is to understand what happens at the higher loop level. Very recently the matching of the spectra of gauge and string theories has been extended to two This became possible due to the important observation18 that the two-loop dilatation operator (and the three-loop as well!) can be emulated by the Hamiltonian of another long-range integrable spin chain known as the Inozemtsev spin chain.22 The situation with the three-loop matching is less clear at p r e s e n t l ’ ~and ~ ~ further work is required to clarify it. It is also interesting to analyze occurrence of finite-dimensional integrable systems inside the two-dimensional string sigma model. The theory of finite zone integration allows one to construct solutions of the original equations of motion starting from the stationary periodic solutions of the hierarchy of the evolution equations generated by the higher Hamiltonians. The stationary solutions themselves should be described in terms of certain finite-dimensional integrable systems (presumably generalizing the Neumann model). One could try to identify them explicitly and relate to the theory of Backlund transformations. Perhaps this would lead t o new insights into the gauge/string duality.
Acknowledgments I would like to thank Goran Djordjevic and all other organizers of the B W2003 Workshop in Vrnjacka Banja (Serbia) in August-September 2003 for an inspiring conference, and their warm hospitality.
15
References 1. J. M. Maldacena, Adw. Theor. Math. Phys. 2, 231 (1998), hep-th/9711200. 2. D. Berenstein, J. M. Maldacena and H. Nastase, JHEP 0204, 013 (2002), hepth/0202021. 3. S. Frolov and A. A. Tseytlin, Nucl. Phys. B668, 77 (2003), hep-th/0304255. 4. S. Frolov and A. A. Tseytlin, JHEP 0206, 007 (2002), hepth/0204226. 5. S. F'rolov and A. A. Tseytlin, JHEP 0307, 016 (2003), hepth/0306130. 6. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Nucl. Phys. B636, 99 (2002), hepth/0204051. 7. J. A. Minahan and K. Zarembo, JHEP 0303, 013 (2003), hep-th/0212208. 8. N. Beisert and M. Staudacher, Nucl. Phys. B670, 439 (2003), hepth/0307042. 9. N. Beisert, C. Kristjansen and M. Staudacher, Nucl. Phys. B664, 131 (2003), hepth/0303060. 10. G. Arutyunov, S. Penati, A. C. Petkou, A. Santambrogio and E. Sokatchev, Nucl. Phys. B643, 49 (2002), hepth/0206020. 11. G. Arutyunov, B. Eden, A. C. Petkou and E. Sokatchev, Nucl. Phys. B620, 380 (2002), hepth/0103230. 12. L. D. Faddeev, hep-th/9605187. 13. N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, JHEP 0309, 010 (2003), h e p t h/0306 139. 14. G. Arutyunov and M. Staudacher, JHEP 0403,004 (2004), hep-th/0310182. 15. G. Arutyunov, S. Frolov, J. Russo and A. A. Tseytlin, Nucl. Phys. B671, 3 (2003), hepth/0307191. 16. G. Arutyunov, J. Russo and A. A. Tseytlin, hep-th/0311004. 17. G. Arutyunov and M. Staudacher, Proceedings of the 5th International Workshop on Lie Theory and Its Applications in Physics, Varna, Bulgaria, (2003), hepth/0403077. 18. D. Serban and M. Staudacher, hep-th/0401057. 19. V. A. Kazakov, A. Marshakov, J. A. Minahan and K. Zarembo, h e p th/0402207. 20. M. Kruczenski, hep-th/0311203. 21. M. Kruczenski, A. V. Ryzhov and A. A. Tseytlin, hepth/0403120. 22. V. I. Inozemtsev, Phys. Part. Nucl. 34, 166 (2003); Fiz. Elem. Chast. Atom. Yadra 34, 332 (2003), hepth/0201001.
FLUXES IN M-THEORY ON 7-MANIFOLDS: G2-, SU(3)- AND SU(2)-STRUCTURES
K. BEHRNDT Max-Planck-Institut fur Gravitationsphysik, Albert Einstein Institut A m Muhlenberg 1, 14476 Golm, Germany E-mail: behrndtQaeimpg. de
C. JESCHEK Humboldt Universitat zu Berlin, Institut fur Physik, Newtonstrasse 15, 12489 Berlin, Germany E-mail: jeschekQphysik. hu-berlin. de
We consider compactifications of M-theory on 7-manifolds in the presence of 4form fluxes, which leave at least four supercharges unbroken. Supersymmetric vacua admit G-structures and we discuss the cases of G2-, SU(3)- as well as SU(2)structures. We derive the constraints on the fluxes imposed by supersymmetry and determine the flux components that fix the resulting 4-dimensional cosmological constant (i.e. superpotential).
1. Introduction An essential input in lifting the continuous moduli space might be non-zero fluxes on the internal space. By now one can find a long list of literature about this subject. A starting point was the work by Candelas and Rainel for an un-warped metric which was generalized later in Ref. 2 (for an earlier work on warp compactification see Ref. 3) and the first examples, which preserveN = 1supersymmetry appeared in Ref. 4. The subject was revived around 10 years later by the work of Polchinski and S t r ~ m i n g e rwhere ,~ flux compactifications in type I1 string theory was considered. In the M-theory setting, different aspects are discussed in Refs. 6-14. Fluxes induces a non-trivial back reaction onto the geometry, because for the Killing they appear as specific con-torsion spinor. The resulting spaces are in general non-Kahlerian, which reflects 16
17
the fact that the moduli space is (partly) lifted. In order to see which moduli are fixed, one can deriving the corresponding superpotential as function of the fluxes in a way discussed in Ref. 23, but this approach becomes subtle if the fluxes are not related to closed forms (due to Chern-Simons terms). In this talk we discuss M-theory compactifications in the presence of 4form fluxes, which keep the external 4-d space time maximal symmetric, i.e. either flat or anti deSitter (Ads), where in the latter case the superpotential remains non-zero in the vacuum giving rise to a negative cosmological constant. We start by making the Ansatz for the metric and the 4-form field strength and separate the gravitino variation into an internal and external part. In addition, we have to make an Ansatz for the 11-d Killing spinor, which decomposes into internal 7-d spinors and the external 4-d spinors. In the most general case, the solution will be rather involved and we use G-structure to classify possible vacua (Section 3). These structures are defined by a set of invariant differential forms and are in one to one correspondence to the number of internal spinors, which will enter the 11-d Killing spinor. Using these differential forms, one can formely solve the BPS equations (Section 4), but explicit solutions require the construction of these forms. Note, the case of the G2- and SU(3)-structures have been discussed already before1°-12 and we will be rather short. 2. Warp Compactification in the Presence of Fluxes
In the (flux) vacuum, all Kaluza-Klein scalars and vectors are trivial and hence we consider as Ansatz for the metric and the 4-form field strength
+ habdy"dyb] , P = 5 epLypAdxpA dx" A d x P A d x A + $Fabcd
ds2 = e Z A[ g g d x p d z "
dy" A dyb A dy" A dyd , (1) where A = A ( y ) is a function of the coordinates of the 7-manifold with the metric h a b , m is the Freud-Rubin parameter and the 4-d metric g,$ is either flat or anti deSitter. Unbroken supersymmetry requires the existence of (at least) one Killing spinor 77 yielding a vanishing gravitino variation of 11-dimensional supergravity 0 = dQM = [ a M + 'GRS 4 M rRS where:
+&(rMF-
~ ~ F M ) ] V ,
P E F M N p Q r M N P Q , FM E F M N p Q r N P Q , etC.
Since,
(2)
18
one can bring the variation also in the more common form. Using the convention { F A , r B }= 2vAB with 77 = diag(-, +, + . . . +), we decompose the r-matrices as usual p
L
=y
with p = 0..3, a = 1..7, and .AS&,
- -E,upx^ 1
ra+3 +5
=
= i.$0.$1.$2.$3, 1 = iy1y2y3y4y5y6y7yields €abcdrnnp
ZYY -
(4)
.$58Ya,
abcd
= [a b c d ]
YUPX 7 'YrnnP=Y -7 Y Y Y * (5) 3! 3! The spinors in 11-d supergravity are Majorana and we take all 4-d .$pmatrices are real and T5 as well as the 7-d ya-matrices are purely imaginary and ant isymmet ric. With this notation, we can now split the gravitino variation into an internal and external part. In order to deal with the warp factor, we use 1 ds2 = e2A&2 -+ DM = DM -r i$%vA (6) 2 and find for the external components of the gravitino variation 1 im 1 0 = [V, 8 l F .$,.$5 8 (5 d A -) -e-3A $ ., 8 F ] v , (7) 36 144
+
+
+
7
+
v,
where F = F a b c d Y a b c d , Fa = F a b c d Y b c d , etc. and is the 4-d covariant derivative. In the same way, we get for the internal variation
1 im 1 -ya) - '"j.$'Vp 8 y a - - e-3A .$5 8 Fa]77 , (8) 2 48 4 12 where we eliminated the term N -yaFr] by using Eq. (7). In order to solve these equations, we have to decompose the spinor and introduce the superpotential yielding the negative cosmological constant. The 11-d Majorana spinor can be expanded in all independent spinors as 0 = [W @ (VP' - -aaA
+
N i=l
where and Oi denote the 4- and 7-d spinors, resp. If there are no fluxes, all of these spinors are covariantly constant and N 5 8 gives the resulting extended supersymmetries in 4 dimensions. With non-trivial fluxes one can however impose a relation between the spinors and N does not refer to the number of unbroken supersymmetries (see last Section) , but gives nevertheless a classification of supersymmetric vacua. In fact, with these spinors one can build differential forms that are singlets under a subgroup G c spin(7) and hence define a G-structure, where the number of spinors is directly related to the group G (see next Section). By definition, the spinors
19
are singlets under G and therefore obey certain projector conditions, which annihilate all non-singlet components and, at the same time, can be used to derive simple differential equations for the spinors and constraints on the fluxes (see last Section). If the 4-d spinors are covariantly constant, the resulting vacuum will be a 4-d flat space, but for an anti deSitter vacuum the spinors satisfy
V,€i
N
-i;(W,ij + i T 5 W?) €j .
(9)
Note, the resulting 4-d cosmological constant will be: -IWI2 and we did not take into account a Kahler potential, i.e., our superpotential will not be holomorphic. If there is only a single spinor this equation simplifies to
V,E
N
9, (Wl + i=y5W2)€ ,
and if E is a Weyl spinor it becomes V,E = +,eKI2 WE*with the complex superpotential W = W1 + i W2. If ~i are a set of Weyl spinors, we introduce the superpotential by a 11-d spinor satisfying the equation
[ V , @ F ] q = (T,@W)ij
with:
fj=Wij~i@f?ej*+cc.
(10)
This way of introducing the superpotential might be confusing. Recall, we set constant all 4-d scalars as well as vector potentials and hence the superpotential should just be a number fixing the cosmological constant for the given vacuum. Since we introduced the superpotential in the 11-d Killing spinor equation it will, on the other hand, depend on the fluxes and the warp factor and thus it is in general not constant over the internal space. The correct 4-dimensional superpotential is of course obtained only after a Kaluza-Klein reduction, i.e. after an integration over the internal space and to make this clear we will denote this constant superpotential by W(O).We do not want to discuss issues related to a concrete Kaluza-Klein reduction (over a not Ricci-flat internal space) and want instead determine the flux components that are responsible for a non-zero value of W(O) 3. G-Structures Supersymmetric compactifications on 7-manifolds imply the existence of differential forms, which are singlets under a group G c spin(7) and which define G-structures.a These globally defined differential forms can be constructed as bi-linears of the internal Killing spinors as f?i-Yq...a,f?j,
follow here basically the procedure initiated in the recent discussion by Ref. 17
20
and the group G is fixed by the number of independent spinors 8i which are all singlets under G. E.g. if there is only a single spinor on the 7manifold, it can be chosen as a real G2 singlet; if there are two spinors, one can combine them into a complex SU(3) singlet; three spinors can be written as Sp(2) N SO(5) singlets and four spinors as SU(2) singlets. Of course, all eight spinors cannot be a singlet of a non-trivial subgroup of SO(7) and G is trivial. The 7-dimensional y-matrices are in the Majorana representation and satisfy the relation: ( Y ~ ~ . . . ~=, (-) ) ~ 2 T ~ ~ . . . ~which ,, implies that the differential form is antisymmetric in [i,j ] if n = 1,2,5,6 and otherwise symmetric [we assumed here of course that Bi are commuting spinors and the external spinors are anti-commuting]. This gives the wellknown statement that having only a single spinor, one cannot build a vector or a 2-form, but only a 3-form and its dual 4-form [the 0- and 7-form exist trivially on any spin manifold]. If we have two spinors 6'11p), we can build one vector and one 2-form (and of course its dual 5- and 6-form). Since the spinors are globally well-defined, also the vector field is well defined on X, and it can be used to obtain a foliation of the 7-d space by a 6manifold X 6 . Similarly, having three 7-spinors we can build three vector fields as well as three 2-forms and having four spinors the counting yields six vectors combined with six 2-forms. In addition to these vector fields and 2-forms, one obtains further 3-forms the symmetrized combination of the fermionic bi-linears. We have however to keep in mind, that all these forms are not independent, since Fierz re-arrangements yield relations between the different forms, see Refs. 9, 17 for more details. Using complex notation, we can introduce the following two sets of bilinears [et = (e*)T]:
Ra ,... a k = Oty,,...,,8
and
= 8T
yal...ak8,
where dropped the index i , j which counts the spinors. The associated k-forms becomes now
ak = k!1
and
- ~ ~ ~ . . . ~ ~ e ~ l . " ~ k
1 fik = -f22,1...akea1"'ak. k!
(11)
If the spinors are covariantly constant the group G coincides with the holonomy of the manifold. If the spinors are not covariantly constant, then neither can be these differential forms and the deviation of G from the holonomy group is measured by the intrinsic torsion. In the following we will discuss the different cases in more detail.
21
3.1. G z Strtdures In the simplest case, the Killing spinor is a G2 singlet and reads O = e z6 0 ,
Or
(12)
where is a normalized real spinor. Due to the properties of the 7-d y-matrices (yielding especially OFyaOo = 0), only the following differential forms are non-zero
1 = O,TOo
,
T i f a b c d m n p = 60 TabcdmnpOO .
They are G2-invariant since 00 is a G2 singlet, i.e. it obeys the appropriate projector constraints. Note, the Lie algebra 50(7) is isomorphic to A' and a reduction of the structure group on a general X7 from SO(7) to the subgroup G2 implies the following splitting: SO(7) =
82
@ 8;
.
(14)
This induces a decomposition of the space of 2-forms in the following irreducible G2-modules, R2 = A;
(15)
@
where
A; = {
T X T }= { a E A'I
U J ~ ~E U
Af4 = {Q E A2 I * ('p A-Q) + a = 0)
* ( ~ A - Q-)2 a = 0 } , 82,
with the abbreviation u 1'p = umpmnpand 'p denotes the G2-invariant 3index tensor, which is expressed as fermionic bi-linear in (13). The operator *(pA a ) splits the 2-forms correspondingly to the eigenvalues 2 and -1. These relations serve us to define the orthogonal projections P , onto the k-dimensional spaces:
where .II, = *p. To be concrete, the Gz-singlet spinor satisfies the condition
22
which is equivalent to
,
Tab80 = @abcyc80 Tabcf% = ('@abc TabcdeO =
+ $abcdT d)
(18)
60
( - $abed - 4ip[abcTd])eO 7
where the second and third conditions follow from the first one. These relations can now be used to re-cast the Killing spinor equations into constraints for the fluxes and differential equations for the warp factor as well as the spinor 8. In the generic situation this spinor is not covariantly constant, which reflects the fact that fluxes deform the geometry by the gravitational back reaction. This can be made explicit by rewriting the flux terms as con-torsion termsb v,e = (v, - -41T , ~ " T ~ ~=) oo . Fkom the symmetry it follows that T has 7 x 21 = 7 x (7+ 14) components, but if €J is a G2-singlet the 14 drops out and hence T E A1 @ gi. These components decompose under G2 as
49 = 1 -t- 7
+ 14 + 27 = + + + 7 2 7 , 77
71
714
where ~i are called G2-structures. Since the Killing spinors define p and $, these torsion classes can be obtained from d p and d$ as follows
d p E A4 = A; @ A$ @
d$ E R5 = A$
@
Ai7 , (19)
A:, ,
where the 7 in A$ is the same as in A: up to a multiple. For a general 4-form 0, the different projections are
R(P) =
P
7
W P ) = -&-l P ,
(20)
p27(P)ab = $ ( P c d e { a $ b } C d e ) O
7
where in (-)O we removed the trace. Thus, the different components in the differentials d p can be obtained from dl)
--
where 714 and
1d p
*d$ 727
-
--
,
$(*d$) J ?I,
d7) ,
have to satisfy:
7(27)
93 A
p-1dp,
(dpcde{a$b}cde)O
(21) 7
Az7 = 9 3 A 714 = 0.
bThere is also an ongoing discussion in the mathematical literature, see Ref. 24.
23
3.2. SU(3) Structures Having a G = SU(3), one can find two singlet spinors on X 7 , which are equivalent to the existence of a vector field v. This in turn can be used to combine both spinors into one complex spinor defined as
where the constant spinor 00 is again the Gz singlet and Z is now a complex function. The vector w is globally well-defined and gives a foliation of X7 by a 6-manifold X S and both spinors, 6 and its complex conjugate 8*, are chiral spinors on x6. In this case, we have to distinguish between the forms R and fi as defined in (11) and findlo-12
and all other forms are zero or dual to these ones. The associated 2-form to the almost complex structure on x6 is w and with the projectors (W f i w ) we can introduce (anti) holomorphic indices so that R(370)can be identified as the holomorphic (3,0)-form on XG. There exists a topological reduction from a Gz-structure to a SU(3)-structure (even to a SU(2)-structure). The difficulties arise by formulating the geometrical reduction. Using the vector v the explicit embedding of the given SU(3)-structure in the Gz-structure is:
i
with the compatibility relations e-ZiIm(Z) o ( 3 , O ) A w =
(x+ + i X - ) A w
=0,
24
Now, the projectors (18) for
80
imply for the complex 7-d in ( 2 2 )
+ v a + i'fabcvb'YC)eO , T a b 6 = 5 ( i ( p a b c y c + i(pabcVc + '$abcdvcYd 2 v [ a Y b ] ) e O, 'Yabce = $ ( i ( p a b c + '$abcdYd + 3 i v [ a ( p b c ] d Yd - '$abcdvd 4'&'[abcYd]Vd)eO , 'Yabcde = $(-'$abed - 4'@[abcYd] - 5'$[abcd'Ye]ve 'Yae
=
ez
z ( Y a
-
-
'Yabcdee Yabcdefe
=
5
-4iv[a(pbcd] (-5'$,[abcd'Ye]
= L(-.
ZEabcdefgY'
- 4v[a'$bcd]eYe)e0 - i&abcdefgYgvf
7
- 5v[a'$bcde]
+ EabcdefgvhYj(pghi
- 2 o i v [ a ( p b c d ' Y e ] ) ~ ,O
- i & a b c d e f g v g )00
*
Again, these relations can be used to rewrite the Killing spinor equations in terms of constraint equations- for the fluxes and a differential equation for the warp factor as well as the spinor. The corresponding torsion componentsz5 are now related to the differential equation obeyed by the forms: v,w , R and their dual. As next case one would consider S P ( 2 ) structures implying three (real) singlet spinors. An example is a 7-d 3-Sasaki-space (i.e. the cone yields an 8-d Hyperkahler space with Sp(2) holonomy), with the Aloff-Walach space N1il as the only regular examplesz6 (apart from S7);non-regular examples are in Ref. 27. We leave a detailed discussion of this case for the future and investigate instead the S U ( 2 ) case in more detail.
3.3. SU(2) Structures On any 7-d spin manifold exist three no-where vanishing vector fieldsz8, which implies that one can always define SU(2) structures. The corresponding four (real) spinors can be combined in two complex S U ( 2 ) singlet spinors O 1 p . The three vector fields v,, a = 1 , 2 , 3 can be chosen as Vl = e
1 1
212
= e
2
,
'u3
= (p(Ul,V2),
and they parameterize a fibration over a 4-d base space X,.The embedding of the S U ( 2 ) into the Ga structures is then given by
Since the vector fields are no-where vanishing, we can choose them of unit norm and perpendicular to each other, i.e. ( v , , v ~ ) = hap, and using the 3-form (p, one obtains a cross product of these vectors. One can pick one
25
of these vectors, say v3, to define a foliation by a 6-manifold and on this 6manifold one can introduce an almost complex structure by J = 213 -I ‘p E T * M 6 8 T M 6 . The remaining two vectors, which can be combined into a holomorphic vector‘ v1 iv2 imply that this 6-manifold is a fibration over the base &. On this 4-manifold we can define a basis of anti-selfdual 2-forms whose pullback correspond to the w,. Note, on any general 4-d manifold we have the splitting
+
A~=A:wP., where we can take { w l ,w2, w3) as a basis of A: and this splitting appears in group theory as: 50(4) 2 su(2) @ su(2), which is equivalent to reducing the structure group from SO(4) to SU(2). The 2-forms satisfy the algebraic relations wi2 =2vo1q
wiAwj=O
for,i#j,
and the associating complex structures fulfill the quaternionic algebra (note: the orientation on the 4-fold is negative). We can further split the 2-form bundle into a symplectic 2-form, say w = w3, and the remaining can be combined into complex (2,O)-form. Thus, the subbundle AT decomposes as
A: r X 2 > 0 ~ R ~ . So, besides the symplectic form w , let us introduce the complexified 2-form: x=w1+iw2
which is, with respect to w , a holomorphic (2,O)-form. The SU(2) singlet spinors can again be constructed from the G2 singlet spinor 00 by (29) where v, = vF3;n. Using the expressions from before, it is straightforward to verify the following relations
(v1v2 - iv3)e0= o
,
(vl - iv2)e2= (vl
+ iv2)e1= 0 ,
which imply v,(gff)klel
=ek
,
wek = i e k ,
v,vpek = i € , p x ( a x ) k l e l ,
xek = - i ( a 2 ) k 1 8 ; ,
“Meaning, that it annihilated by the projector: (W - J )
26
where w
= wmnymn, X G Xmnymn and with the Pauli matrices u 3 = ( l0 -1O ) .
(31)
4. BPS Constraints Now we can come back to the BPS equations from Section 2. With the superpotential as introduced before, equation (7) becomes 1 o = f j + [9518( 2-1d A + -)i36m -e-3A ( F IF~) ] q , (33) 144
+
and if: ij = e-+q, equation (8) yields im Ya)ij - i 9 5 y ae - 4 6 o = w 18(vih) +48
-
- e-3A y- 5 B ~ ~ i j (34) . 12
It is useful to decompose the 35 components of the 4-form field strength under Gz as 35 + 1 7 27 with
+ +
where 3(l), F(7)and 3(27) are the projection introduced in Eq. (20). The cases of G2 and SU(3) structures have been discussed already in the literature and we will summarize only the main results.
4.1. G2 Structure In this case, the 11-d spinor is a direct product, i.e. q=E@e,
(36)
and since the 11- and 7-d spinor are Majorana also the 4-d spinor E has to be Majorana (a more detailed discussion is given in Ref. 29). One finds that all internal 4-form components have to vanish
Fabcd = 0 ,
w1 = 0
, m = -36W2.
(37)
27
The Eq. (34) gives a differential equations for the spinor e Z & , which implies
aaz= 0 . The differential equations for 00 fixes the 7-manifold to have a weak G2 holonomy and hence is a Einstein space with the cosmological constant given by the Freud-Rubin ~ a r a m e t e r . ~ ~This > l Oin turn implies, that the 8d space built as a cone over this 7-manifold has Spin(7)holonomy. In fact, after taking into account the vielbeine, this gives the known set of first order 7 differential equations for the spin connection 1-form wab: Wab$9abc = 36 m e b , where m was the Freud-Rubin parameter [note w is here the spin connection and should not be confused with the associated 2-form introduced before]. Using the differential equation for the 7-spinor, it is straightforward to verify that d$=O,
and therefore only dl)is non-zero. The 4-d superpotential is only given by the Freud-Rubin parameter, ie.
which fixes the overall size of the 7-manifold. In the limit of flat 4-d Minkowski vacuum, the Freud-Rubin parameter has to vanish and we get back to the Ricci-flat G2-holonomy manifold. In order to allow for nontrivial fluxes one has to consider SU(3) instead of G2 structures. 4.2. SU(3) Structure
In this case, there is one (complex) 7-d spinor and the 11-d Majorana spinor reads = E 86
+ E*
80*.
(39)
where the 4-d spinors E and E* have opposite chirality (y5e = E ) . More details about this case can be found in Refs. 1 0 , l l . The solution of Eq. (33) read now
w = w1+ i w2= l6e - ( K / 2 + 3 A ) +
[$ 4
(1)
V a a a e 3 ~= + ~ ( 1 ) v a ~ ab ( 2 7 ) v 1b
m=O,
- vaF:;7)vb
+ i v " F p ], (40)
28
and
[the flux components were introduced in (35)]. In addition, one obtains a differential equation for the spinor with the non-trivial torsion components as introduced in Eq. (19)
--
,
dl)
w 2
7A7)
48 w1
-
9 3(l) + $ (PabcvbFi7)+ 27.F;i7)vb . 21,
(43)
To make the set of equations complete, we have to give the differential equations obeyed by the vector field u,which is straightforward if we use the differential equation for the spinor
recall w& = 'p&Uc. Note, unVmun = 0, which is consistent with 1uI2 = 1. Using the decomposition (35) one finds
v [ m u n ]= (drZ64 2 7 1(dd;:
V{,Vn) = --(dmn
+Z
1 + -21' $ " m n a b ) ~ ~ ~ 7 ) ' U C +' U b-'pmn"(d," 4
- V,Vb);Fb(7) ,(45)
1 9) - Zu{m'pn}abu,F~7)
- u(mun})
+ w;w:).F;;7)
- 2dmnF:~7)u"vb. 1
(46)
The first term in the anti-symmetric part is the projector onto the 7,see Eq. (16), and by contracting with 'p and employing Eqs. (41) and (42), one can verify that:" d(e3Av)= 0. One can project the flux components onto X s and using the symplectic 2-form w we can introduce (anti) holomorphic indices. As result, we can define a 3-form H and 4-form G on X S and find for the superpotential
whereas the 4-form has to fulfill the constraint: R 1G = 0 and de3A1w = ~ w ~ H a s w e l l a s u ~ d e ~ ~ = ~ ~ ~ ~ G .
29
4.3. SU(2) Structure Finally, in the SU(2) case we write the 11-d spinor as
7=
Be1 + e 2 B e 2 + c C ,
and we choose chiral4-d spinors with
y~5 E i
'
=Ea.
Eq. (33) gives
1 im 1 W&j* + ( - d A + - + - F ) O j ] . 2 36 144
(49)
If one dos not impose any constraints on the spinors 8 , one finds14
wij with the 4-form
-
eiFej = F
-I
6(4)as derived in (32). Defining the 2-forms:
Gap = V ~ V ~ F m n a b X a b ,
F,p = V z V ; F m n a b W a b ,
we can write Wij as matrix: W ( ~ ~ P Y G , p o ~ ) owith 2 the 0, as Pauli matrices. It would be identical zero if G = 0, but instead we can also impose: ~ ' W i j= 0 so that Wij projects out one of the 4-d spinor as we would need for an N = 1 vacuum. This implies that: det W = 0 which gives one constraint on the complex 2-form G. As next step, the contraction with 6: yields N
m =0,
-
~ A - S2(') I
N
F 1d4),
which implies that: d,A e a p r F p T (with 8, = v,"d,). Finally, one has to contract with Bya as well as with 8tya (with the index a projected onto the base) and if we assume that the dbA = 0 (ie. the warp factor is constant over the 4-d base), we get as further contraints on the fluxes =0,
eyaFe = 0 .
-
These constraints are solved, e.g., if the only non-zero components of the are: v, Avp A w ; ie. are contained in Fap and Gap = 0 (as defined above). These are all constraints on the fluxes, but from the internal variation (34) we get differential equations. Setting, m = 0 and ij = 0, we find
V,&
N
FmnpqynPqei.
If only the components in Fap are non-zero, it is straightforward to further simplify this equation by using the relations in (30). On the other hand,
30
this equation fkes also the corresponding differential equations obeyed by the differential forms.
For the 2-forms eg., our constraints on the fluxes imply that w and X are closed, when projected onto the 4-d base, which is therefore a hyper Kahler space. Unfortunately, we have to leave a detailed analysis of these equaions for the future.
Acknowledgments K. B. is partially supported by a Heisenberg grant of the DFG. C . J. is supported by a Graduiertenkolleg grant of the DFG (The Standard Model of Particle Physics - structure, precision tests and extensions).
References P. Candela and D. J. Raine, Nucl. Phys. B248,415 (1984). B. de Wit, D. J. Smit, and N. D. Hari Dass, Nucl. Phys. B283,165 (1987). B. de Wit and H. Nicolai, Phys. Lett. B148,60 (1984). M. A. Awada, M. J. Duff, and C. N. Pope, Phys. Rev. Lett. 50,294 (1983); B. de Wit, H. Nicolai, and N. P. Warner, Nucl. Phys. B255,29 (1985). 5. J. Polchinski and A. Strominger, Phys. Lett. B388, 736 (1996), hep-
1. 2. 3. 4.
th/9510227. 6. B. Brinne, A. Fayyazuddin, T. Z. Husain, and D. J. Smith, JHEP 03, 052 (2001), hepth/0012194. 7. K. Dasgupta, G. Rajesh and S. Sethi, JHEP 9908, 023 (1999), hepth/9908088; B. S. Acharya and B. Spence, hep-th/0007213. K. Becker and M. Becker, JHEP 11, 029 (2000), hep-th/0010282. 8. P. Kaste, R. Minasian, M. Petrini, and A. Tomasiello, JHEP 09,033 (2002), hep-t h/0206213. 9. J. P. Gauntlett and S. Pakis, JHEP 04, 039 (2003), hep-th/0212008; D. Martelli and J. Sparks, Phys. Rev. D68,085014 (2003), hep-th/0306225; J. P. Gauntlett, D. Martelli, and D. Waldram, Phys. Rev. D69, 086002 (2004), hep-t h/0302158. 10. K. Behrndt and C. Jeschek, JHEP 04, 002 (2003), hep-th/0302047; K. Behrndt and C. Jeschek, hep-th/03llll9. 11. P. Kaste, R. Minasian, and A. Tomasiello, JHEP 07, 004 (2003), hepth/0303127. 12. G. Dall’Agata and N. Prezas, Phys. Rev. D69, 066004 (2004), hepth/0311146. 13. K. Behrndt and M. Cvetic, Nucl. Phys. B676, 149 (2004), hep-th/0308045. K. Behrndt and M. Cvetic, hep-th/0403049.
31
14. K. Behrndt and C. Jeschek, Class. Quant. Gruv. 21, S1533 (2004), hepth/0401019. 15. A. Strominger, Nucl. Phys. B274,253 (1986). 16. K. Becker, M. Becker, K. Dasgupta and P. S. Green, JHEPO304, 007 (2003), hep-th/0301161; K. Becker, M. Becker, K. Dasgupta and S. Prokushkin, Nucl. Phys. B666,144 (2003), hep-th/0304001; K. Becker, M. Becker, P. S. Green, K. Dasgupta, and E. Sharpe, hep-th/0310058. 17. 3. P. Gauntlett, D. Martelli, S. Pakis, and D. Waldram, Commun. Math. Phys. 247,421 (2004), hep-th/0205050. 18. T. Friedrich and S. Ivanov, math-dg/0112201. 19. J. Louis and A. Micu, Nucl. Phys. B626,26 (2002), hep-th/0110187; S. Gurrieri, J. Louis, A. Micu, and D. Waldram, Nucl. Phys. B654, 61 (2003), hep-th/O211102. 20. G. L. Cardoso et al., Nucl. Phys. B652, 5 (2003), hep-th/0211118; G. L. Cardoso, G. Curio, G. Dall’Agata, and D. Lust, JHEP 10,004 (2003), hepth/0306088. 21. G. Dall’Agata, hep-th/0403220. 22. A. R. Frey, hep-th/0404107. 23. S. Gukov, Nucl. Phys. B574,169 (2000), hep-th/9911011; C. Beasley and E. Witten, JHEP 07,046 (2002), hep-th/0203061; G. Curio, JHEP 03,024 (2003), hepth/0212211. 24. T. Friedrich, I. Kath, A. Moroianu, and U. Semmelmann, Journal of Geometry and Physics 2 3 , 4 1 (1997); T. Friedrich and S. Ivanov, [math-dg/0102142]; I. Agricola and T. Friedrich, Class. Quant. Graw. 20,4707 (2003), [mathdg/0307360]; S. Karigiannis, math-dg/0301218. 25. S. Chiossi and S. Salamon, math-dq/0202282. 26. T. Friedrich and I. Kath, Commun. Math. Phys. 133,543 (1990). 27. K. G. C.P. Boyer and B. Mann, Journ. Reine u. Angew. Math. 455, 183 (1994). 28. E. Thomas, Bull. Americ. Math SOC.75,643 (1969). 29. A. Bilal, J.-P. Derendinger, and K. Sfetsos, Nucl. Phys. B628, 112 (2002), hep-th/Oll1274.
NONCOMMUTATIVE QUANTUM FIELD THEORY: REVIEW AND ITS LATEST ACHIEVEMENTS
M. CHAICHIAN
P.
Department of Physical Sciences, and Helsinki Institute of Physics, 0. Box 64, 00014 University of Helsinki, Helsinki, Finland E-mail:
[email protected] Some properties of quantum field theories on noncommutative space-time are reviewed. Studying the general structure of the noncommutative (NC) local groups, we present a no-go theorem for NC gauge theories. This no-go theorem imposes strong restrictions on the NC version of the Standard Model (SM) and in resolving the standing problem of charge quantization in noncommutative QED. We also consider the phenomenological implications of noncommutative y on the spectrum of the H-atom and derive a bound on the noncommutativity parameter 8. Finally, in the framework of noncommutative quantum field theories (NC QFT), we show the general validity of the CPT and spin-statistics theorems, with the exception of some peculiar situations in the latter case.
1. Introduction It is generally believed that the notion of space-time as a continuous manifold should break down at very short distances of the order of the Planck length Xp M 1.6 x 10-33cm. This would arise, e.g. from the process of measurement of space-time points based on quantum mechanics and gravity arguments. Arguments for noncommutativity arise also from string theory with a constant antisymmetric background field, whose low-energy limit, in some cases, turns up to be a noncommutative quantum field theory (NC QFT).2 This in turn implies that our classical geometrical concepts may not be well suited for the description of physical phenomena at very small distances. One such direction is to try to formulate physics on some noncommutative space-time. 1-3 If the concepts of noncommutative geometry are used, the notion of point as elementary geometrical entity is lost and one first expectation is that an ultraviolet cutt-off appears. In Ref. 4 this expectation was shown not to be fulfilled in general. Instead, a peculiar 32
33
UV/IR mixing appear^.^ The usual way of constructing a noncommutative theory is through the Weyl-Moyal correspondence: in a NC space-time the coordinate operators satisfy the commutation relation:
where 8’”” is a constant antisymmetric matrix of dimension (length)2. In QFT the operator character of the space-time coordinates (1) requires that the product of any two field operators be replaced by their *-product, or Weyl-Moyal product. The *-product compatible with the associativity of field products is given by:
An important step in constructing a physical noncommutative model is t o develop the concept of local gauge symmetry. Intuitively, because of the inherent nonlocality of noncommutative field theories, the notion of local symmetry in the noncommutative case should be handled with special care. As a result, the pure noncommutative U(1) theory behaves similarly to the usual non-Abelian gauge theories, but now the structure constants depend on the momenta of the fields.6 This feature induces a charge quantization problem,’ in the sense that the electric charges in the noncommutative quantum electrodynamics (NC QED) based on NC U(l) group are quantized only to f 1 , O . The solution of this problem was sought in the construction of a noncommutative version of the Standard Model (NC SM),8 based on a no-go t h e ~ r e m and , ~ is discussed in Section 11. In Section 111 phenomenological implications of the noncommutativity are also addressed on a concrete model of the H-atom, for which we present the noncommutative corrections to the spectrum and, using the data for the Lamb shift, we find a bound on the noncommutativity parameter 0.” In Section IV, we show that a breaking of the spin-statistics relation in NC QFT could occur only in the case of theories with NC time. We also present in Section V a general proof that the CPT theorem remains valid in NC field theories, for general form of noncommutativity, although the individual symmetries C,T and P are broken.”
34
2. Noncommutative Gauge Groups. A No-go Theorem 2.1. Charge Quantization Problem in NC QED In Ref. 7 it was shown that in NC QED based on the NC U(l) group, one can encounter only fields with charge +1:
$74= V ( X ) * $(). 9 Dp$ = a,$ - iA, * 4,
$(XI
+
(3)
fields with charge -1: $(XI
+
$'(.I D,$
= $(XI = a,$
*u - w
+ i$
1
* A,,
(4)
and fields with charge 0:
Xb)
-+
X'(X)
=W
X:)
*X(X) *U-W ,
D,x = a,x
+ i[x,A,]* .
(5)
This immediately raises the question about other known charges, i.e. the fractional charges of quarks. The simple extension
D p ~ ( n=)
- inA,
* $(n) ,
(6)
with $(4--+ $44 = U*"* $'"'
(7)
for the field $ with integral multiple n of a (conventional) unit charge fails to transform covariantly. In conclusion in NC QED, charge is quantized only to 0,fl. A possible way out from this situation is to construct a NC version of the Standard Model, to which end we have to choose the gauge groups and their representations and also define the direct product of group factors. 2.2. A No-go Theorem The following result was partially obtained in Ref. 12 in the framework of noncommutative gauge groups and extended to a no-go theorem in Ref. 9. In general, as discussed in Ref. 13, it is not trivial to define the noncommutative version of usual simple local groups, as the *-product will destroy the closure condition. Consequently, the only group which admits a minimal noncommutative extension is U ( n ) (we will denote its extention by U*(n)). However, the NCSO and U S p algebras have been constructed in a more involved way. l3
35
To define the pure NC U * ( n ) gauge theory we take as generators of the u*(n) algebra: T”, a = ,n2 - 1 ( n x n su(n) generators) and T o = -&lnxn. The u*(n) Lie-algebra is defined with the star-matrix bracket : 1 , e . a
[f,9l* = f
*9
-
9 *f
,
f , 9 E u*(n>.
(8)
The U,(n) gauge theory is described by the u,(n) valued gauge fields: n2-1
G, =
C
G$(x)TA,
(9)
A=O
with the infinitesimal gauge transformation
G,
4
GG = G,
+ i8,X + g[X,G,],
, X E u*(n).
(10)
Under the above tranformation, the field strength G,u = d[,Gv]
+ i9[G,, GI*,
(11)
transforms covariantly:
leaving invariant the action of the pure U*(n) gauge theory:
4n
dDa: n(G,, *G’””)
One peculiar feature to be noticed in the case of the pure U*(n)gauge theory is that, fixing the number of gauge field degrees of freedom (which is n2) the dimension of the matrix representation is automatically fixed, i.e. the gauge fields must be in the n x n matrix form. The main physical implication is that the matter fields coupled to the U * ( n ) gauge theory can only be in fundamental, antifundamental, adjoint and singlet states. Another nontrivial point in the noncommutative gauge theories is to define the direct product of NC gauge groups. In the commutative case, if GI and G2 are gauge groups, then G = G1 x GZ is defined through:
9 = 91 x 92 19) = 9‘1 x 9: 1 g i ,9: E Gi , 9 , g t E G , 9 . 9 ) = (91 x 92) * (9; x 9;) = (91 .9;) x (92 * 9:) *
(14)
In the noncommutative case, let G1 = U*(n) and G2 = U*(m). But now, the group products involve the *-product so that the group elements can not be re-arranged. As a result, the definition of direct product cannot be straightforwardly generalized to the NC case and consequently the matter fields cannot be in fundamental representations of both U,(n) and U*(m).
36
The only possibility left is for a matter field to be in the fundamental representation of a gauge group and the antifundamental representation of another:
9 -+ 9' = u 9 * V - l ,
U E U,(n), V E ~,(rn).
(15)
In the general case of n gauge groups N
the matterfields can be charged under at most two of the U,(ni) factors. 2.3. N C Standard Model. A Solution to the Charge Quantization Problem Based on the above no-go theorem, we have built a noncommutative version of the Standard Model.8 The model is based on the gauge group U,(3)x U,(2) x U,(1)(the general elements of the respective group factors will be denoted by U E U,(3), V E U,(2), II E U,(l)) and comprises: one gauge field, B, , valued in u,(1), four gauge fields of u,(2) :
where d ,i = 1,2,3 are the Pauli matrices and fields of u,(3):
go = 1 z X 2
and nine gauge
R
A=O
where T", a = 1,2,. . , 8 are the Gell-Mann matrices and To = 1 3 x 3 . This choice of the gauge group is due to the fact that there i s no straightforward noncommutative extension of the S U ( n ) groups. However, compared t o the commutative Standard Model, two additional gauge fields have appeared, corresponding to the extra U(1) factors. The reduction of the extra U(1) factors is achieved through a Higgs-type of mechanism, in two stages. First the mechanism is run with the symmetry-reducing scalar field with U1 E U(1)C U,(3)and V1 E U(1)c U,(2).In the second stage, the symmetry is reduced eventually to that of hyper-charge, throught the scalar particle @z(.)
+
s(.)@2.-l()
1
(20)
37
with sEU(l)residual and v ~ U * ( l ) After . the symmetry reduction, two of the gauge fields become massive (Go and W o )and the gauge field corresponding to the residual U ( 1 ) symmetry will be the (masless) hyper-photon Y . When coupling matter fields to the U*(3) x U*(2) x U,(1) theory, we have to keep in mind that, according to Ref. 9, the fields can be only in the fundamental and/or anti-fundamental representation of the group factors. It is interesting to note that the no-go theorem allows six different types of charged particles in the case of three simple group factors and the matter content of the original Standard Model (including the Higgs particle) exhausts those possible types of charges. By properly taking the representations of the matter fields and performing the U(1) symmetry reduction introduced earlier, it is straightforward to show that the couplings of all matter fields to the hyper-photon Y, are realized through the usual hypercharges.' Moreover, after performing the spontaneous symmetry breaking of the original Standard Model, all particles will couple to the photon A, through the usual electric charges, i.e. 1, -1, 0, -I/?, 2/3, so this model provides a solution to the NC charge quantization problem. Another proposal for a noncommutative version of the Standard Model is based on the Seiberg-Witten (SW) map,14 which assigns to commutative gauge configurations the noncommutative equivalent configurations, linked by field-dependent noncommutative gauge transformations. This version of the NC SM is constructed from NC fields realized by SW map as a tower of commutative fields, transforming under G = U(1) x S U ( 2 ) x SU(3). There are no additional U(1) gauge fields, so there is no need for the U(1) factor reduction. The gauge symmetry is considered on Lie algebra level and not Lie group level. Consequently, arbitrary (fractional) U (1) charges are admissible. However, this last point can be considered as a disadvantage: in the NC SM based on the no-go theorem, the U(1) factor reduction fixes the correct (hyper) charges for all SM particles.
3. Lamb Shift in NC QED. Bounds on 8 In this section we focus on the hydrogen atom and, using the non-relativistic limit of NC QED results, we propose the Hamiltonian describing the NC H-atom. Given the Hamiltonian and assuming that the noncommutativity parameter ( & j ) is small, we study the spectrum of H-atom. We show that because of noncommutativity, even at field theory tree level, we have some corrections to the Lamb shift (2+, -+ 2&/2 transition)." Hereafter, we shall consider the electron of the H-atom moving in the
38
external field of the proton. However, similar results (up to a numerical factor) would be obtained by treating the proton as a composite particle, e.g., in the naive quark m0de1.l~The latter analysis infirms the treatment of Ref. 16, where the proton is taken as an elementary particle, thereby obtaining no noncommutative corrections for the H-atom spectrum at tree level. To start with, we propose the following Hamiltonian for the noncommutative H-atom:
with
p i ,3y = ieij , pi,fijl = itisij , bi16j] =0.
(22)
The NC Coulomb potential
Ze2 e Zer V ( r )= --r - -(e 4ti x P) . (-$
+ o(e2),
with e i j = i f i j k e k can be obtained either as the nonrelativistic limit from the NC photon exchange diagram or from the change of variables:
xi = xi+ -eijfij, 2ti
(24)
Pi = Pi 7
where the new variables satisfy the usual canonical commutation relations: [Zi,Zj]
=0
,
[Pi,Pj] =0
,
(25)
[Zi,Pj] =m i j .
Using the usual perturbation theory, the leading corrections t o the energy levels due to noncommutativity, i.e. first order perturbation and in field theory tree level, are:
1 f o r j = l f i a n d f n , l = n34+3)(1+1). The case of our interest, the 2P112--f 2S1p transition (Lamb shift), for the noncommutative H-atom, besides the usual loop effects, depends on the j , quantum number (only for the 2P112 level, as the levels with 1 = 0 are not affected) and is there, even in the field theory tree level. Hence we call it polarized Lamb shift. New transition channels are opened (notation nlp), i.e. 2PG;l2 4 2P://2"and a split of
the usual Lamb shift occurs: 2Pt;:
+ 2S1p
and 2Pz;l2 --$
25'112.
39
One can use the data on the Lamb shift to impose some bounds on the value of the noncommutativity parameter 8. Of course, to do it, we only need to consider the classical (tree level) results, Eq. (26). Comparing these results, the contribution of (26) should be of the order of smaller than the usual one loop result and hence,
The same bound is obtained also from the violation of Lorentz invariance, based on the clock-comparison experiments, which monitor the difference between two atomic hyperfine or Zeeman transition frequencies, searching for variations as the Earth rotates.17
4. Noncommutative Quantum Field Theory and Spin-statistics Theorem Pauli’s spin-statistics relationls is responsible for the entire structure of the matter and for its stability. Experimentally, the relation has been verified to high accuracy. Theoretically up to now there has been no compelling argument or logical motivation for its breaking. However, the violation of Lorentz invariance, as well as the intrinsic nonlocality of noncommutative field theories, may suggest that the (presumably very small, of the order of lPVlm2)breaking of this fundamental theorem, as well as of the CPT theorem, might be possible. Pauli demonstrated18 the spin-statistics relation based on the following requirements: (i) The vacuum is the state of lowest energy; (ii) Physical quantities (observables) commute with each other in two space-time points with a space-like distance; (iii) The metric in the physical Hilbert space is positive definite. In the noncommutative case the physical quantities (observables) which are in general products of several field operators, are no more local quantities and could therefore fail to fulfil the above requirement ii). For instance, taking the normally ordered product : qb2(x) : for a real scalar field with mass rn, its noncommutative version : 4(x) 4(z) : could give a nonvanishing equal-time commutation relation (ETCR). In particular, the matrix element between vacuum and a two-particle state, on a d-dimensional space, when Bose statistics is used, is:”
*
40
-
d
2i
1 (e--ip'x--ipy
+
e--ipx--ip'y
)
G
where wk = ko = and = (kl, ..., kd). The r.h.s. of (28) is nonzero only when Ooi # 0. This holds for observables expressed as any power of both bosonic fields and their derivatives, with *-product analogous to (28), and spinor fields and their derivatives, with anti-commutation relation used in the latter case. The study of NC QFT also showed a violation of both causalitylg and unitarity" conditions, for theories with noncommutative time (Ooi # 0). Indeed, while the low-energy limit of string theory in a constant antisymmetric background field B"", which exhibits noncommutativity, reduces to field theory with the *-product when Ooi = 0, for the case Ooi # 0 there is no corresponding low-energy field theory limit. The field theories with light-like noncommutativity, OpVOp, = 0, i.e. Ooi = --eli, become very interesting from this point of view as they preserve unitarity.21 In this case, however, the microcausality in the sense of ETCR (28) is still violated.ll If the field theory with light-like noncommutativity is indeed the lowenergy limit of string theory, as stated in Ref. 21, it is then intriguing that the theory is unitary but acausal (as it is known that a low-energy effective theory should not necessarily be unitary, as is the case, e.g., for the Fermi four-spinor interaction). 5. CPT Theorem in NC Field Theories
The CPT t h e ~ r e m (see ~ ~ also t ~ ~Ref. 24 for a review) is of a universal nature in that it is valid in all the known field theories. Here we shall recapitulate essential features of the CPT transformation and then extend the CPT theorem to noncommutative field theories. First, we shall summarize the common properties of anti-unitary transformations, including time reversal and CPT transformation. An antiunitary transformation denoted hereafter by 4 is a generalization of complex conjugation and satisfies
(a' ,a')
= (a
,a).
(29)
The transformation of state vectors corresponds to the Schrodinger picture and we can also attribute the same transformation to operators correspond-
ing t o the Heisenberg picture by
(9' ,Q@')
= (a , Q ' 9 ) .
In what follows we shall mainly discuss the latter approach. a) The transformation of operators obeys the following rules: (CIA
+ cgb)'
= clA' (AB)' = B'A',
+ czB'
(linearity) ,
where c1 and c g are c-number coefficients. b) Let us assume that
Q' = EQ,( E = f l ) and that 9 is an eigenstate of Q with the eigenvalue q,
&@==a; then ' 9 is also an eigenstate of Q and QQ' = Eq9'.
5.1. The C P T h n s f o r m a t i o n of Local Elementary Fields
+
In what follows we shall use the symbol exclusively for the C P T transformation and we shall first define it for local elementary fields. Let $Ia, Ga and $ A ~ . . . A ~be local elementary fields representing spinors and tensors, respectively; then the C P T transformation is specified by:25 $I%)
= (iY5)cu&d--2)
7
= $p(--S)(iYdpa
7
' $34
4 A 1 , . . A n (-2)
= (-')n$A1...An(-z)
.
(35)
This set of rules completely specifies the transformation of any local elementary field carrying definite spinor and/or tensor indices. Then the C P T theorem for local field theories can be formulated in the following form: 5 . 2 . C P T Theorem for Local Fields Let $Ia, $a and $A~.,.A,, be local but composite fields representing spinors and tensors, respectively; then they are transformed exactly in the same form as Eq. (35) for local elementary fields. In what follows we shall clarify the significance of this theorem.
42
1) Let us consider local composite scalar fields of which free and interaction Lagrangian densities, as well as interaction Hamiltonian densities, are typical members and we have:
LfI(.
, L,?,,>.(
= Lf(-).
= Lint (-.)
,
HL&) = H i n t ( - . ) .
(36) (37)
In Ref. 25, Eq. (37) has been referred to as the CPT theorem and its proof has been given there so that we skip it. When asymptotic conditions are valid, the CPT invariance of the S matrix follows from it:
s' =s.
(38) 2) Next, let @ A be a local composite vector field and q 5 a~ local elementary vector field, respectively; then a composite scalar field @ = q 5 ~ @is ~ transformed as (36) or (37) and q 5 ~as (35). From the above information we deduce:
'
= -@A(-)
(39) and similarly we can prove Eq. (35) for spinors and tensors. As an example of local composite vector fields we choose the electric current density j ~ ( z ) ; then the conserved electric charge Q transforms as: @A(.)
Q' =
/
d 3 2 j t ( 2 )= -
/
7
d3zjo(-2) = -Q.
3) The energy-momentum vector PA can be expressed as the space integral of the energy-momentum tensor of the second rank. Therefore, we immediately conclude
P!
= PA.
(41) 4) The generators of the Lorentz transformation Mpu can be expressed as the space integral of a tensor of the third rank, so that we have:
M:u
-Adpu. (42) This indicates that the spin of a particle defined in terms of the PauliLubanski operator should reverse its direction under CPT. In general, the CPT transformation of an operator is determined by the tensorial rank of its density. 5 ) We assume the validity of the LSZ asymptotic conditions;26then on the basis of their definition of the asymptotic fields, it is shown straightforward that the CPT transformation turns incoming fields into outgoing fields and vice versa. =
43
5.3. C P T Theorem for Noncommutative Fields
The validity of CPT theorem for noncommutative QED has been discussed in Ref. 27, where it was concluded that CPT is accidentally preserved, although the charge conjugation and time reversal symmetries are broken due to noncommutativity. However, in Ref. 27 the specific version of NC QED of Ref. 7 was studied, where the photon couples only to particles with the electric charges +1,-1 and 0. The latter is usually referred to as the "charge quantization problem". In Ref. 28 the CPT invariance of the noncommutative Yang-Mills theories has been shown using the SeibergWitten map. In the following, we shall show the general validity of the CPT theorem for any noncommutative quantum field theory of the type described in Sect. 11, without reference to any specific model or to the Seiberg-Witten map. Let H ( z ) be the Weyl-Moyal product (2) of field operators representing the interaction Hamiltonian in a noncommutative field theory. It is understood that H ( z ) stands for a normal product in the interaction representation. The CPT theorem is given by
In order to prove it we shall choose as an illustration a n-linear form for H ( z ) ,namely,
H(z)=
c c
fal...an+il,(z>*...*+~n(z)
il.. .in
= eD
. ...2,.
fi ,...in+:,
(21)...~~n(2,)111= ...=In~a: 1
(44)
2,
where ij with j = 1,...,n stand for spinorial or tensorial indices and the coefficients fil...in are so chosen as to make H ( z ) a scalar under proper Lorentz transformations, in the local limit. D stands for the differential operator of the form
with general W. Then the CPT transform of H ( z ) is given by:
44
where f' is given by f!2 1 ...2,. = (-1)W".
21
...2,
(47)
7
and F stands for the number of the Fermi fields involved in H ( z ) . When we reverse the order of multiplication back to the original one in (44), we obtain:
H'(.)
c c ...in4:,
=e
D
fz l... z,4~l(-z~)..,4~,(-z~)121 =...=2,=2
il
=
. ...2.,
...in fz,
(-z)
* ... * 4;,
(-z) = H ().-
.
(48)
21
Thus the CPT theorem is valid not only in local field theories but also in noncommutative field theories. This can be also seen from the fact that, when we expand the interaction Hamiltonian density in powers of 8, the first term is the local limit of the Hamiltonian expressed in terms of the Weyl-Moyal product. It is a local but composite scalar density. The coefficients of other terms are local but composite tensor fields of even ranks obtained by differentiating the fields involved in the first term, an even number of times. Therefore, they transform in the same way as the first term under CPT. From this point of view it is intuitively clear that the Hamiltonian density expressed in terms of the Weyl-Moyal product transforms in the same way as the local ones under CPT. As seen from the proof presented above, the CPT theorem is valid for any form of noncommutativity, including the case Oo2 # 0. Individual discrete transformations P, C and T The individual transformations P, C and T are violated in many cases and we shall comment on them only by comparison with the local (commutative) limit of the noncommutative field theory in question. In the case of only space-space noncommutativity (8'2 = 0), the parity of a noncommutative field theory is the same as for its commutative limit, while charge conjugation and time reversal are broken, even if they hold for the commutative limit. This is due to the fact that C and T imply a complex conjugation, that would change the sign of the phase in (45). In the case of a space-time noncommutative theory (8" # 0) - whose commutative limit is P, C and T invariant - all these discrete transformations are violated, as in the NC QED case.27
6. Conclusions In the framework of noncommutative gauge theories, we present a no-go theorem according to which the closure condition of the gauge algebra im-
45
plies that: 1) the local NC u ( n ) algebra only admits the irreducible n x n matrix-representation. Hence the gauge fields are in nxn matrix form, while the matter fields can only be in fundamental, adjoint or singlet states; 2) for any gauge group consisting of several simple-group factors, the matter fields can transform nontrivially under at m o s t two NC group factors. In other words, the matter fields cannot carry more than two NC gauge group charges. This no-go theorem imposes strong restrictions on the NC version of the Standard Model and in resolving the standing problem of charge quantization in noncommutative QED. Elaborating on the phenomenological implications of noncommutativity we have calculated the noncommutative corrections to the spectrum of the H-atom and obtained a bound on 0 from the data on the Lamb shift. We have found that the CPT theorem is generally valid in NC FT, irrespective of the form of the noncommutativity parameter O p v involved, although Lorentz invariance is violated. The spin-statistics theorem holds in the case of field theories with space-space noncommutativity, which can be obtained as a low-energy limit from the string theory. A violation of the spin-statistics relation in the case of NC time can not be justified, given the pathological character of such theories. The case of light-like noncommutativity (compatible with unitarity) deserves, however, more attention. In conclusion, it is of importance to study further the light-like case, as to determine whether it can indeed be obtained as a low-energy limit of string theory. Questions concerning a possible breaking of the spin-statistics relation are of utmost importance, since such a violation, no matter how small, would have a crucial impact on the structure and the stability of matter in the Universe. The issue, on the other hand, is of fundamental interest by itself, since up to now no theoretical argument or motivation for such a breaking has been presented.
Acknowledgments The financial support of the Academy of Finland under the Project no. 54023 is acknowledged.
References 1. S. Doplicher, K. Fredenhagen and J. E. Roberts, Phys. Lett B331,39 (1994); Comm. Math. Phys. 172, 187 (1995). 2. N. Seiberg and E. Witten, JHEP 9909, 31 (1999), hepth/9908142. 3. A. Connes, Noncommutative Geometry, Academic Press, New York (1994).
46
4. T. Filk, Phys. Lett. B376, 53 (1996); M. Chaichian, A. Demichev and P. Prehajder, Nucl. Phys. B567, 360 (ZOOO), hep-th/9812180. 5. S. Minwalla, M. Van Raamsdonk and N. Seiberg, JHEP 9906, 020 (2000), hepth/9912072. 6. M. M. Sheikh-Jabbari, JHEP 9906, 015 (1999), hep-th/9903107. 7. M. Hayakawa, Phys. Lett. B478, 394 (ZOOO), hep-th/9912094. 8. M. Chaichian, P. Presnajder, M. M. Sheikh-Jabbari and A. Tureanu, Eur. Phys. J. C29, 413 (2003), hepth/0107055. 9. M. Chaichian, P. PreSnajder, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Lett. B526, 132 (2002), hep-th/0107037. 10. M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001), hep-th/0010175. 11. M. Chaichian, K. Nishijima and A. Tureanu, Phys. Lett. B568, 146 (2003), h e p t h/0209008. 12. S. Terashima, Phys. Lett. B482, 276 (2000), hep-th/0002119. 13. L. Bonora, M. Schnabl, M. M. Sheikh-Jabbari and A. Tomasiello, Nucl. Phys. B589, 461 (2000), hep-th/0006091. 14. X. Calmet, B. JurEo, P. Schupp, J. Wess and M. Wohlgennant, Eur. Phys. J.C23, 363 (2002), hepph/Olllll5. 15. M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, hep-th/0212259. 16. P.-M. Ho and H.-C. Kao, Phys. Rev. Lett. 88,151602 (2002), hep-th/0110191. 17. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001), hep-th/0105082. 18. W. Pauli, Phys. Rev. 58, 716 (1940); Progr. Theor. Phys. (Kyoto) 5, 516 (1950). 19. N. Seiberg, L. Susskind and N. Toumbas, JHEP 0006, 044 (2000), h e p th/0005015; L. Alvarez-GaumB and J. L. F. Barbon, Int. J. Mod. Phys A16, 1123 (2001), hep-th/0006209. 20. J. Gomis and T. Mehen, Nucl. Phys. B591, 265 (ZOOO), hep-th/0005129. 21. 0. Aharony, J. Gomis and T. Mehen, JHEP 0009, 023 (2000), hepth/0006236. 22. G. Luders, Dansk. Mat. Fys. Medd. 28, 5 (1954). 23. W . Pauli, Niels Bohr and the Development of Physics, W. Pauli (ed.), Pergamon Press, New York (1955). 24. R. F. Streater and A. S. Wightman, CPT, Spin, Statistics and All That, W. A. Benjamin, Inc., New York (1964), and references therein. 25. K. Nishijima, Fundamental Particles, W. A. Benjamin, Inc., New York (1963). 26. H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cimento 1, 1425 (1955); 6, 319 (1957). 27. M. M. Sheikh-Jabbari, Phys. Rev. Lett. 84, 5265 (2000), hep-th/0001167. 28. P. Aschieri, B. JurEo, P. Schupp and J. Wess, Nucl. Phys. B651, 45 (2003), h e p t h/02052 14.
SHADOWS OF QUANTUM BLACK HOLES
N.KALOPER Department of Physics, University of California Davis, C A 95616, USA E-mail:
[email protected] We discuss our recent conjecture that black holes localized on a brane in AdSD+I should be interpreted as quantum-corrected D-dimensional black holes, rather than classical ones, in the dual CFT coupled to gravity. Thus in 4D they include the corrections from Hawking radiation.
1. Black Holes in AdS/CFT with a Cutoff: the Role of Tunneling The semi-infinite Randall Sundrum modell is based on a bulk geometry of AdSD+I space ending on a D - 1-dimensional domain wall, or brane. A prototype is the RS2 model where Ads5 ends on a 3-brane, which should model our 3+1 dimensional world. It is natural to ask what is a suitable description of a black hole in this scenario. The attempts to find exact, static, asymptotically flat black hole solutions localized on the brane in AdSD+l>4, with regular horizons both on and off the brane, have been marred with very serious It has even been suggested that static, asymptotically flat, spherical black holes on the brane might not altogether exist in the RS2 model. In contrast, exact static solutions localized on a 2-brane in Ad& have been found in Refs. 7, 8. This is surprising in two ways. First of all, the RS2 model in 5 0 is believed to have a dual description in terms of purely 4 0 physics, as a strongly coupled CFT with a UV cutoff, coupled to weak gravity and perhaps Standard Model-like excitations. One would expect that such a theory should admit the conventional black holes, which are no more difficult t o construct than is usual. On the other hand, the same argument applied to a 3 0 CFT coupled to gravity would suggest that there should not be any black holes to begin with, since as is well known,g there are no black hole solutions of 2 1 Einstein gravity in asymptotically flat spaces. So why do
+
47
48
such objects appear in the variant of RS2 in the 4 0 bulk? In order to resolve this, in the recent work with Emparan and Fabbri," we have proposed a connection between the bulk and dual CFT+gravity interpretation of black holes in RS2, based on a modification of AdS/CFT correspondence" for the RS2 m ~ d e l . ~ ~Our - ~main ' result was the following conjecture:
The black hole solutions localized o n the brane in the AdSD+1 braneworld which are found by solving the classical bulk equations in AdSD+1 with the brane boundary conditions, correspond to quantum-corrected black holes in D dimensions. rather than classical ones. This comes about as follows: according to AdS/CFT, the classical dynamics in the AdSD+1 bulk encodes the quantum dynamics of the dual D-dimensional conformal field theory (CFT), in the planar limit of a large N expansion. Cutting the bulk with a brane introduces a normalizable D-dimensional graviton while on the dual side this same Ddimensional gravity mode is merely added to the CFT, which is also cutoff in the ultraviolet. Then, solving the classical D 1-dimensional equations in the bulk is equivalent to solving the D-dimensional Einstein equations G,, = 87rG~(T,v)CFT,where the CFT stress-energy tensor incorporates the quantum effects of all planar diagrams. These include particle production in the presence of a black hole, and possibly other vacuum polarization effects. We have shown that the conjecture is fully consistent with the existence of black holes in 2+1 CFT+gravity, which emerge solely due to the quantum corrections in the CFT sector of the the0ry.l' In fact, this result provides for an interesting spinoff, showing that in 2 1 dimensions, the quantum dynamics of a CFT naturally serves as a cosmic censor, regulating the (6function) naked singularities, and dressing them in a horizon. Thus the black holes in 2 1 dimensions are generic, once quantum corrections are included. This is true independently of whether the CFT is strongly or weakly coupled, and is more efficient when there is more CFT degrees of freedom. We have also found a consistent reinterpretation of black holes in the physically more relevant case of a 3-brane in AdS5.l' The main point is that the CFT+gravity dual allows us to reinterpret the alleged obstruction for finding a static black hole3 as a manifestation of the backreaction from Hawking effects. As long as the bulk is asymptotically Ads, the conformal
+
+
+
49
symmetry of the dual CFT is valid in the infrared, and so there is no mass gap. Thus any black hole at a finite temperature will emit CFT modes as a thermal spectrum of Hawking radiation, which on the bulk side is captured by a deformation of the bulk geometry close to the brane. Since in the asymptotically flat space Hawking radiation escapes t o infinity, carrying away the black hole energy, the black hole mass must be time-dependent, and hence the geometry is not stationary. The mechanism of the tunneling suppression plays an important role in the determination of the Hawking effects from the bulk side, and we will review this below. We begin with several key aspects of the AdS/CFT d i c t i ~ n a r y . ~ ~ ~ ~ ~ Since we want to discriminate between classical and quantum effects, we must keep ti in our formulas, while setting c = 1. Then, the 4-dimensional Newton’s constant G4, Planck length l 4 , and Planck mass M4 are related to each other as
ti M4 = -
-e4
G4 = -
M4 l
e4
In Ads braneworlds the 5 0 bulk Newton’s constant and the bulk cosmological constant A5 = -10/L2 together determine the Newton’s constant induced on the D-dimensional brane as 1
G4=-G5.
L
(2)
In light of our discussion from previous sections, this means that we are taking the volume of the Calabi-Yau space to be negligible compared to the volume of the brane throat, which is the opposite limit t o that we worked in previously. The precise details of the dual CFT depend on the specifics of the string/M-theory construction that yield the Ads background. For our purposes here it is enough to determine the effective number of degrees of freedom of the CFT, g*. For D = 4, the dual pair are IIB string theory on Ads5 x S5 of radius L e1,(g,N)1/4 and N = 4 S U ( N ) super Yang-Mills theory, leading to N
where we have used Eq. (2) to get the final expressions, 9%is taken to be a large number, in order to keep small quantum corrections to the supergravity approximation to string/M-theory. For the CFT, this is a large N limit where planar diagrams give the leading contribution. The Planck brane that cuts off the Ads bulk denotes that very high energy states of the dual CFT are integrated out, breaking the conformal invariance of the
50
theory in the UV. However, the breaking washes into the low energy theory only through irrelevant operators, generated by integrating out the heavy CFT states at the scale p u v h/L. In the IR, at energies E < p u v , the effects of the conformal symmetry breaking are suppressed by powers of E / p u v , meaning that the bulk geometry far from the brane is Ads. Cutting off the bulk yields also a normalizable graviton zero mode localized on the brane; this same D-dimensional gravity mode is added to the dual theory. However, note that the CFT cutoff p u v is not equal to the'induced D-dimensional Planck mass. Instead,
-
which is much smaller than the Planck mass on the brane, and is what we would expect for the single throat limit of the "octopus", as discussed previously.
1.1. Resolving the Mystery of the Missing 3 -/- 1 Black Hole As we have said above, as long as the bulk is Ads5 far from the brane, the dual CFT is conformal in the IR, without a mass gap separating the CFT modes from the vacuum. Hence any black hole at a finite temperature will have unrestricted access t o a large number of light CFT modes, and will them with a thermal spectrum, which is precisely the Hawking radiation.a On the bulk side, this must be described by a deformation of the bulk geometry near the brane, which arises because the black hole appears as a source in the classical bulk gravity equations. Computing these effects entails the usual complications involving the choice of the vacuum for a quantum field theory in a black hole background, with the possibilities being (1) the Hartle-Hawking state, (2) the Unruh state, and (3) the Boulware state. Then by our conjecture, the black hole on the RS2 brane must correspond to one of these choices, with the corrections from the backreaction included. This immediately shows why the search for a static, "We should note that a step in this direction for the case of RS2 in Ads5 was made by T. Tanaka,20 and, simultaneously, by R. Maartens and the author, in order t o explain the results of Ref. 3. A naive argument that the bulk dynamics encodes the backreaction from Hawking radiation would lead one to expect that all asymptotically flat branelocalized black holes are time-dependent. This would be in conflict with the exact static 2 1 solutions of Refs. 7, 8. Our conjecture that the classical bulk dynamics encodes all quantum corrections at the level of planar diagrams completely resolves this conflict. These exact solutions in fact strongly support the conjecture as presented in here."
+
51
asymptotically flat black hole solution on the brane has failed so far: the state (1) is not asymptotically flat, ( 2 ) is not static, and (3) does not have a regular horizon. This leads us to considering a radiative solution as the leading-order description of the exterior of a black hole localized on the brane. The detailed description of this geometry on the bulk side would require either the exact bulk solution, which has been missing so far, or a much better approximation than the existing ones. On the side of the 3 f l CFT+gravity, a description at the same level of rigor would require a careful backreaction analysis, where we should start with a classical Schwarzschild black hole and perturb it by means of the (Tpy) in the Unruh state evaluated in the classical background geometry. The far-field outgoing metric encodes the flux of Hawking radiation pouring out of the black hole, which is described by the stress-energy tensor
where u is the retarded null coordinate and L(u) is the flux luminosity. The perturbed geometry is
ds2=-
(1 - 2G4y(u)) du2 2drdu + r2dS22, -
where = -L(u). To check our conjecture, we should recover the relation between L and M from leading-order corrections to the black hole geometry induced from the bulk. The precise calculation would require the detailed matching of the far-field solution (6) to a near horizon one, which should then be matched onto the interior. In order to circumvent these details, we have considered the radiative collapse of a very massive dust cloud, forming a black hole of large mass. This collapsing cloud of dust, whose interior is described by the bulk dynamics encompassing leading order quantum CFT corrections, determined in Ref. 3 can be matched to an outgoing Vaidya metric (6), following the work of Ref. 21. The quantum corrections propagate through the matching regions, and this relates the outgoing flux of radiation to the subleading correction in the interior star geometry, which is c( ( G ~ M J ~ ) as ~ calcu/R~, lated in Ref. 3, r.h.s. of their Eq. (6) (we only consider the limit Q = A = 0 of this equation, which is sufficient for our purposes). Comparing to Eq. (5) we find L G 4 ( M L ) 2 / R t hg,(G4M)2/R:, where Ro is the radius of the matching surface. For a large collapsing mass, this will be near 2 G 4 M , so L hg*/(G4M)2. This corresponds to a flux of Hawking radiation of N
N
-
52
-
g* degrees of freedom of the CFT, at a temperature TH ti/(G4M), as required. Replacing M ( u ) by M is consistent since L c( ti and we are working in an expansion in ti. While this does not reproduce a detailed formula with accurate numerical coefficients, it does give the correct scalings with the black hole and CFT parameters, in complete accord with our conjecture. We should still check the consistency of the matching of geometries across the horizon. A simple way to check this is to compare the quantum trace anomalies of the backreacted states in the exterior and interior. The trace anomaly of the quantum stress tensor is a local geometric quantity independent of which vacuum the field is in.22>23 It has been studied in detail in the AdS/CFT context,24and in particular in the case of Ads braneworlds in Refs. 25-27. It gives us further insight into our problem, in that it provides a simple leading-order consistency check, which a configuration must pass in order to be described by the leading-order effects in the duality pair. In the case of D = 4 N = 4 SU(N) SYM at large N N
Note the absence of the term RPVapRFYap.Ref. 24 showed how this anomaly is precisely reproduced from a computation in the AdSs bulk. This result is perturbatively identical to the familiar quadratic stress-energy correction terms that appear in the effective long distance 3 1 gravity equations in Ads braneworlds,28 which can be checked explicitly recalling g* N2.“J6 The matching to the far-field Vaidya metric (6), is consistent with this form of the anomaly, because the tracelessness of the radiation stress-energy implies R P , , = 0, and so the anomaly vanishes, with no contributions from the R P y a p R P V a p terms. Although this argument by itself does not fully guarantee that the bulk will be free from singularities, it passes the anomaly check with only minimal assumptions which are physically well-motivated. It is now straightforward to correctly interpret the ”no-go theorem” of Ref. 3. The anomaly matching requires that the exterior is radiative, leading to a time-dependent evaporating black hole (6). Therefore, the classical bulk dynamics does require braneworld black holes to be time-dependent. However, this is simply a natural consequence of black hole quantum mechanics, and is generated as a leading order quantum correction. Understanding this picture from the point of view of the full bulk Ads5 spacetime, and in particular the details of the dual description of the Hawking radiation as a classical bulk process represents a very
+
-
53
interesting challenge. In Ref. 10 we have touched upon some aspects of this picture. The mechanism of tunneling ~ u p p r e s s i o nplays ~ ~ a very important role for understanding the difference between the rapidly evaporating large black holes and slowly evaporating small ones. For the small black holes, the description in terms of a 3 1 theory of gravity+CFT breaks down. A black hole of size r H 0. In order to allow for a more sizable violation of YU, we further extend the model by including q5 with the coupling q5h'h. To give superheavy masses t o the color non-singlets in 4, we introduce one more superfield 4 with the coupling 64, whose coefficient is of order MGUT.
60
The terms $4 and $H"B" imply that, after the breaking of Gps t o GSM, 4 acquires a superheavy VEV of order MGUT.The coupling &'h then generates SU(2)R violating unsuppressed bilinear terms between the doublets in h' and h. These terms can certainly overshadow the corresponding ones from the non-renormalizable term HcHcK'h. The resulting SU(2), violating mixing of the doublets in h and h' is then unsuppressed and we can obtain stronger violation of W.
3. The Yukawa Quasi-Unification Condition
To further analyze the mixing of the doublets in h and h', observe that the part of the superpotential corresponding to the symbolic couplings i'h', &'h is properly written as mtr (h't7L'c)
+ ptr (~ ' ~ 4 7 L,t )
(2)
where EE is the antisymmetric 2 x 2 matrix with €12 = +1, t r denotes trace taken with respect t o the SU(4), and SU(2), indices and tilde denotes the transpose of a matrix. After the breaking of Gps t o GSM,4 acquires a VEV (4) MGUT. Substituting it by this VEV in the above couplings, we obtain N
tr(h'&'E)
= ?lchi
+ hl,eh; + . .. ,
- = -tr(h'EashE) (4) - tr(h'e4he) d
(3)
(4) = = -(h:ehz
Jz
- k1&!J
,
(4)
where the ellipsis in Eq. (3) contains the colored components of h', h' and a3 = diag(1, -1). Inserting Eqs. (3) and (4) into Eq. (2), we obtain
m%E(hh - alhz) + m($
+ a1i1)eG
with
a1
= -p($)/z/2m.
(5)
So, we get two pairs of superheavy doublets with mass m. They are predominantly given by
The orthogonal combinations of h l , h', and hz, hh constitute the electroweak doublets
The superheavy doublets in Eq. (6) must have vanishing VEVs, which readily implies that ( h i ) = - a l ( h l ) , (ha) = al(h2). Equation (7) then
61
+
gives (h;") = (1 la112)1/2(h1), (h;") = (1+ l a 1 1 ~ ) l / ~ ( hFrom 2 ) . the third generation Yukawa couplings y33F3hFi1 2yi3F3h'Fi, we obtain
where p = yi3/y33. From Eqs. ( 8 ) and (9), we see that YU is now replaced by the YQUC,
h t : h b : h, = ( l + c ) : (1 - c ) : (1+3c), with 0 < c = p a l / &
0, A0 = 0, m b ( M ~=) 2.888 GeV and cu,(Mz) = 0.1185.
3000
2500
0
'LOO
200
300
m,B
400
600
600
700
(GeV)
Figure 3. The mass parameters m A and Msusy versus m ~ for~ p p> 0, A0 = 0, A, = 1, a,(Mz) = 0.1185 and with m b ( M z ) = 2.684 GeV (dashed lines), 3.092 GeV (dotted lines) or 2.888 GeV (solid lines).
65
0
The lower bounds on mLSp are not so sensitive to the variations of mb(MZ)* The lower bound on mLSp from Eq. (15) overshadows all others. The upper bound on mLSp from Eq. (11) is very sensitive to the variations of mb(MZ). In particular, one notices the extreme sensitivity of the almost vertical part of the corresponding line, where the LSP annihilation via an A-boson exchange in the s-channel is36 by far the dominant process, since mA, which is smaller than 2 m ~ s pis , always very close to it as seen from Fig. 2. This sensitivity can be understood from Fig. 3, where V Z A is depicted versus mLSp for various mb(MZ)’s. We see that, as mb(MZ) decreases, mA increases and approaches 2 m ~ s p .The A-pole annihilation is then enhanced and RLsph2 is drastically reduced causing an increase of the upper bound on mLsp. For A?2 < 0.25, bino-stau c~annihilations~~ take over leading to a very pronounced reduction of R ~ s h2, p thereby enhancing the upper limit on mLSp.
For p > 0, a,(Mz) = 0.1185 and mb(MZ) = 2.888 GeV, we find the following allowed ranges of parameters: 176 GeV 5 mLSp
5 615 GeV,
0 5 A, 585tanP559, 0.145~50.17.
5 1.8,
7. The Inflationary Scenario One of the most promising inflationary scenarios is hybrid inflation37, which uses two real scalars: one which provides the vacuum energy for inflation and a second which is the slowly varying field during inflation. This scheme is naturally i n ~ o r p o r a t e din~ ~ SUSY GUTS, but in its standard realization has the following problem39: if the GUT gauge symmetry breaking predicts monopoles (and this is the case of Gps), they are copiously produced at the end of inflation leading to a cosmological catastrophe4’. One way to remedy this is to generate a shifted inflationary trajectory, so that GPS is already broken during inflation. This could be achievedl8 in our SUSY GUT model even before the introduction of the extra Higgs superfields, but only by utilizing non-renormalizable terms. However, the introduction of 4 and 4 very naturally gives rise17 to a shifted inflationary path with the use of renormalizable interactions only.
66
7.1. The Shifted Inflationary Path The superpotential terms which are relevant for inflation are given by
W = r;S(H"I? - M 2 ) - pSq52 + m&
+ X$H"H",
(17)
where M , m MGUT N 2.86 x 10l6 GeV, and K , p and X are dimensionless coupling constants with M , m, K , X > 0 by field redefinitions. For simplicity, we take p > 0. (The parameters are normalized so that they correspond to the couplings between the SM singlet components of the superfields.) The scalar potential obtained from W is given by N
+ 12pS4 - mr$I2 + Im4 + X H c 8 c \ 2 + x6l2 ( 1 ~ ~ +1 21 1 ~ 1 ~ ) + D - terms. (18)
V = I K ( H ' P - M 2 ) - ,@b2l2
+
(Ks
Vanishing of the D-terms yields H" * = ei'HC (H", H c lie in their right handed neutrino directions). We restrict ourselves to the direction with 2f) = 0 which contains the shifted inflationary path and the SUSY vacua (see below). Performing appropriate R and gauge transformations, we bring S , H" and H c to the positive real axis. From the potential in Eq. (18), we find that the SUSY vacuum lies at
HCHC
-2M2
II
(z)2
7
1 = - (I2
0 and a constant potential energy density VOgiven by
which can be used as inflationary path. VO# 0 breaks SUSY on this path, while the constant non-zero values of H", H" break the GUT gauge symmetry too. The SUSY breaking implies the existence of one-loop radiative correction^^^ which lift the classical flatness of this path, yielding the necessary inclination for driving the inflaton towards the SUSY vacuum. The one-loop radiative corrections to V along the shifted inflationary trajectory are calculated by using the Coleman-Weinberg formula42:
67
where the sum extends over all helicity states i, Fi and Mi2 are the fermion number and mass squared of the ith state, and A is a renormalization mass scale. In order to use this formula for creating a logarithmic slope which drives the canonically normalized real inflaton field (T = d m S / X towards the minimum, one has first to derive the mass spectrum of the model on the shifted inflationary path. This is a quite complicated task and we will skip it here.
7.2. Inflationary Observables The slow roll parameters are given by (see e.g. Ref. 43)
where the primes denote derivation with respect to the real normalized inflaton field (T and mp 21 2.44 x 10l8 GeV is the reduced Planck scale. The conditions for inflation to take place are E 5 1 and lql 5 1. Calculating the number of e-foldings NQ that our present horizon scale suffered during inflation, we obtain the following relation (see e.g. Ref. 43):
where uf [(TQ] is the value of o at the end of inflation [when our present horizon scale crossed outside the inflationary horizon] and T, N lo9 GeV is the reheat temperature taken to saturate the gravitino c o n ~ t r a i n t ~ ~ . The quadrupole anisotropy of the cosmic microwave background radiation can be calculated as follows (see e.g. Ref. 43):
which is its central value from the cosmic Fixing ( ~ T / T )E Q6.6 x background explorer (COBE)45 (assuming that the spectral index n = l ) , we can determine one of the free parameters (say ,6) in terms of the others (m, K and A). For instance, we find ,6 = 0.1, for m = 4.35 x 1015 GeV and IC = X = 3 x In this case, the instability point of the shifted path lies at (T, 2 3.55 x 10l6 GeV, af 21 1.7 x 1017 GeV and OQ N 1.6 x 10l8 GeV43. Also, M 21 2.66 x 10l6 GeV, NQ 21 57.7 and n N 0.98. Note that the slow roll conditions are violated and, thus, inflation ends well before reaching the instability point at ( T ~ We . see that the COBE constraint can be easily satisfied with natural values of the parameters. Moreover, superheavy SM
68
non-singlets with masses 0. We found that there exists a wide and natural range of parameters consistent with the data on the CDM abundance in the universe, b + sy, the muon anomalous magnetic moment and the Higgs boson masses. Moreover, the model gives rise to a new version of the shifted hybrid inflationary scenario, which avoids overproduction of monopoles at the end of inflation by using only renormalizable interactions.
Acknowledgments We would like to thank M.E. Gbmez, R. Jeannerot and S. Khalil for fruitful and pleasant collaborations from which this work is culled. This work was supported by European Union under the RTN contracts HPRN-CT-200000148 and HPRN-CT-2000-00152.
69
References 1. G. L. Kane, C. Kolda, L. Roszkowski and J. D. Wells, Phys. Rev. D49, 6173 (1994), hepph/9312272. 2. G. Lazarides and C. Panagiotakopoulos, Phys. Lett. B337, 90 (1994), hepph/9403316; S. Khalil, G. Lazarides and C. Pallis, ibid. 508, 327 (2001), hep-ph/0005021. 3. L. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D 50, 7048 (1994), hepph/9306309; M. Carena, M. Olechowski, S. Pokorski and C. E. M. Wagner, Nucl. Phys. B426, 269 (1994), hepph/9402253. 4. D. Pierce, J . Bagger, K. Matchev and R. Zhang, Nucl. Phys. B491,3 (1997), hepph/9606211. 5. M. Carena, D. Garcia, U. Nierste and C. E. M. Wagner, Nucl. Phys. B577, 88 (2000), hep-ph/9912516. 6. S. F. King and M. Oliveira, Phys. Rev. D63,015010 (200l), hepph/0008183. 7. S. Abel et al. (SUGRA Working Group Collaboration), hep-ph/0003154. 8. M. E. G6mez, G. Lazarides and C. Pallis, Nucl. Phys. B638, 165 (2002), hepph/0203131. 9. H. Baer, J. Ferrandis, K. Melnikov and X. Tata, Phys. Rev. D66, 074007 (2002), hepph/0207126. 10. T. Blaiek, R. DermEek and S. Raby, Phys. Rev. Lett. 88, 111804 (2002), hepph/0107097; Phys. Rev. D65, 115004 (2002), hepph/0201081. 11. D. Auto et. al., J . High Energy Phys. 06, 023 (2003), hep-ph/0302155. 12. U. Chattopadhyay, A. Corsetti and P. Nath, Phys. Rev. D66, 035003 (2002), hepph/0201001; C. Pallis, Nucl. Phys. B678, 398 (2004), hep-ph/0304047. 13. G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B181, 287 (1981); G. Lazarides and Q. Shafi, ibid. B350, 179 (1991). 14. M. E. G6mez, G. Lazarides and C. Pallis, Phys. Rev. D67, 097701 (2003), hepph/0301064; C. Pallis and M.E. G6mez, hep-ph/0303098. 15. D. Spergel et al., Astrophys. J . Suppl. 148, 175 (2003), astro-ph/0302209. 16. R. Barate et al. (ALEPH Collaboration), Phys. Lett. B429, 169 (1998); K. Abe et al. (BELLE Collaboration), ibid. 511, 151 (2001), hep-ex/0103042; S. Chen et al. (CLEO Collaboration), Phys. Rev. Lett. 87, 251807 (2001), hepex/0108032. 17. R. Jeannerot, S. Khalil and G. Lazarides, J . High Energy Phys. 07, 069 (2002), hep-ph/0207244. 18. R. Jeannerot, S. Khalil, G. Lazarides and Q. Shafi, J. High Energy Phys. 10, 012 (2000), hepph/0002151. 19. G. Lazarides, hepph/0011130; R. Jeannerot, S. Khalil and G. Lazarides, hepph/0106035. 20. I. Antoniadis and G. K. Leontaris, Phys. Lett. B216, 333 (1989). 21. G. Lazarides and Q. Shafi, Phys. Rev. D58,071702 (1998), hepph/9803397. 22. M. E. G6mez, G. Lazarides and C. Pallis, Phys. Rev. D61, 123512 (2000), hepph/9907261; Phys. Lett. B487, 313 (2000), hepph/0004028. 23. M. E. G6mez and C. Pallis, hepph/0303094 (in the SUSYO2 Proceedings). 24. J . Ellis, K. A. Olive, Y. Santoso and V.C. Spanos, Phys. Lett. B565, 176
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25.
26. 27. 28. 29. 30.
31. 32. 33. 34.
35. 36. 37. 38. 39.
40. 41. 42. 43. 44. 45. 46. 47.
(2003), hepph/0303043; A. B. Lahanas and D. V. Nanopoulos, ibid. 568,55 (2003), hep-ph/0303130; H. Baer and C. Balbs, JCAP 05, 006 (2003), hepph/0303114; U. Chattopadhyay, A. Corsetti and P. Nath, Phys. Rev. D68, 035005 (2003), hepph/0303201. H. Goldberg, Phys. Rev. Lett. 50, 1419 (1983); J. R. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. A. Olive and M. Srednicki, Nucl. Phys. B238, 453 (1984). G. BBlanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 149, 103 (2002), hep-ph/0112278. J. Ellis, T. Falk, K. A. Olive and M. Srednicki, Astropart. Phys. E13, 181 (2000); ibid. 15, 413 (2001), hepph/9905481. A. Djouadi, J. Kalinowski and M. Spira, Comput. Phys. Commun. 108, 56 (1998), hepph/9704448. A. L. Kagan and M. Neubert, Eur. Phys. J. C7, 5 (1999), hepph/9805303; P. Gambino and M. Misiak, Nucl. Phys. B611,338 (2001), hepph/0104034. M. Ciuchini, G. Degrassi, P. Gambino and G. Giudice, Nucl. Phys. B527, 21 (1998), hepph/9710335; G. Degrassi, P. Gambino and G. F. Giudice, J . High Energy Phys. 12, 009 (2000), hepph/0009337. S. Martin and J. Wells, Phys. Rev. D64, 035003 (2001), hep-ph/0103067. M. Davier, hepex/0312065 (to appear in the SIGHAD03 Proceedings). G. W. Bennett et al. (Muon 9-2 Collaboration), Phys. Rev. Lett. E89, 101804 (2002); ibid. 89, 129903 (2002), hepex/0208001. ALEPH, DELPHI, L3 and OPAL Collaborations, The LEP Higgs working group for Higgs boson searches, hepex/0107029; LHWG-NOTE/2002-01, [http://lephiggs.web.cern.ch/LEPHIGGS/papers/July2002SM/index.html] . S. Heinemeyer, W. Hollik and G. Weiglein, hepph/0002213. A. B. Lahanas, D. V. Nanopoulos and V. C. Spanos, Phys. Rev. D62,023515 (ZOOO), hepph/9909497. A. D. Linde, Phys. Rev. D49, 748 (1994), astro-ph/9307002. E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart and D. Wands, Phys. Rev. D49, 6410 (1994), astro-ph/9401011. G. Lazarides and C. Panagiotakopoulos, Phys. Rev. D52, 559 (1995), hepph/9506325; R. Jeannerot, S. Khalil and G. Lazarides, Phys. Lett. B506, 344 (2001), hepph/0103229. T. W. B. Kibble, J. Phys. A9, 1387 (1976). G. Dvali, R. Schaefer and Q. Shafi, Phys. Rev. D73, 1886 (1994), hepph/9406319. S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973). G. Lazarides, Lect. Notes Phys. 592, 351 (2002), hep-ph/0111328; hepph/0204294. M. Yu. Khlopov and A. D. Linde, Phys. Lett. B138, 265 (1984); J. Ellis, J. E. Kim and D. V. Nanopoulos, ibid. 145, 181 (1984). C. L. Bennett et al., Astrophys. J . L1, 464 (1996), astro-ph/9601067. G. Lazarides, R. K. Schaefer and Q. Shafi, Phys. Rev. D56, 1324 (1997), hepph/9608256. C. Panagiotakopoulos, Phys. Lett. B459, 473 (1999), hep-ph/9904284.
SUPERSYMMETRIC GRAND UNIFICATION: THE QUEST FOR THE THEORY
A. M E L F O ( ~ )G. , SENJANOVIC(~) ('1 Centro d e Fisica Fundamental, Universidad de Los Andes, Mkrida, Venezuela (2) International
Centre for Theoretical Physics, 34100 Trieste, Italy
With the advent of neutrino masses, it has become more and more acknowledged that SO(10) is a more suitable theory than SU(5): it leads naturally t o small neutrino masses via the see-saw mechanism, it has a simpler and more predictive Yukawa sector. There is however a rather strong disagreement on what the minimal consistent SO(10) theory is, i.e. what the Higgs sector is. The issue is particularly sensitive in the context of low-energy supersymmetry.
1. Introduction Supersymmetric Grand Unification has been one of the main extensions of the Standard Model (SM) for now more than two decades. Today, however, it is in search of a universally accepted minimal, consistent model. With the growing evidence for neutrino masses,' it is becoming more and more clear that the SU(5) theory is not good enough: it contains too many parameters in the Yukawa sector. The situation is much more appealing in the SO( 10) scenario, which is custom fit to explain small neutrino masses in a simple and fairly predictive manner. The main dispute lies in the breaking of SO(10) down to the Minimal Supersymmetric Standard Model (MSSM), in the delicate question of the choice of the Higgs superfields. Roughly speaking, there are two schools of thought: one that sticks to the small representations, which guarantees asymptotic freedom above MGUT,but must make use of higher dimensional operators, suppressed by M p l ; one that argues in favor of the renormalizable theory only, even at the price of becoming strong between MGUT and the Planck scale. Each program has its pros and cons. The first one in a sense goes beyond grand unification by appealing to the string picture in order to provide additional horizontal symmetries needed to simplify the theory plagued by many cou71
72
plings. The second one is based on pure grand unification, with the hope that the Planck scale physics plays a negligible role. It is the second one that we discuss at length in this talk.
2. Why grand unification and why supersymmetry ?
No excuse needs to be offered for the natural wish to unify the strong and electro-weak interactions. This appealing idea has two important generic features: proton decay and the existence of magnetic monopoles. They are by themselves sufficient reason to pursue the unification scenario. There are three important reasons to incorporate low-energy supersymmetry in this program: i) the hierarchy problem of the Higgs mass, ii) the gauge coupling unification, and iii) the Higgs mechanism in the form of radiative symmetry breaking. Let us briefly discuss them. 0
0
0
Supersymmetry per se says nothing about the smallness of the Higgs mass (the hierarchy problem), it just keeps the perturbative effect small, the way that chiral symmetries protect small Yukawa couplings. The old feelings that this might not be such a big deal, since the cosmological constant does not get protected in a similar way, are becoming more widespread today. Gauge coupling unification of the MSSM is a rather remarkable phenomenon, but its meaning is not completely clear. Namely, if one believes in a desert between MWand M G ~ T then , this becomes a crucial ingredient. The desert is a property of the minimal gauge group, SU(5), which is not a good theory of neutrino masses. In SO(lO), on the other hand, supersymmetry is not essential at all; the theory works even better without supersymmetry since then it predicts intermediate scales in the range lo1' - 1014GeV, ideal for neutrino masses via the see-saw mechanism13and for lept~genesis.~ It is worth stressing though that supersymmetric grand unification was anticipated already in 1981, and it gave a rationale for a heavy top quark with a mass around 200GeV (needed to increase the p parameter and help change sin2Bw from its then accepted value of 0.20 to the current 0.23). Radiative symmetry breaking and the Higgs mechanism. The tachyonic property of the Higgs mass term has bothered people for a long time. It is of course purely a question of taste, for either sign of M$ is equally probable. Since the charged sfermion mass terms are definitely not tachyonic, in supersymmetry one could
73
ask why is the Higgs scalar so special. The answer is rather simple: if the nature if the scalar mass terms is determined by some large scale, and if all m2 > 0, it turns out that the Higgs doublet coupled to the top quark rather naturally becomes tachyonic at low scales due to the large top Yukawa coupling. This was kind of prophetic more than twenty years ago, and it could be a rationale for such a heavy top quark. Admittedly a little fine-tuning is still needed between the so-called p term and the stop mass, but wether this is a small or a large problem is still disputable. Suppose we accept low energy supersymmetry as natural in grand unification. We then face the task of identifying the minimal consistent supersymmetric grand unified theory and then hope that it will be confirmed by experiment. Due to the miraculous gauge coupling unification in the MSSM, we are then tempted to accept the idea of the desert. Since the desert is a natural property of SU(5), it is not surprising that SU(5) was considered for a long time the main candidate for a supersymmetric GUT. Why not stick to this idea? The point is that SU(5), at least in its simplest form, favors massless neutrinos. Higher dimensional operators can only provide small neutrino masses, not large enough to explain Amzdot I I 10-'eV2 and especially Am; I I10-3eV2. While we can try to remedy this in one way or another, the fact is that the realistic theory needs too many parameters to be predictive. Let us try to justify this claim by analyzing SU(5) in some detail.
3. Supersymmetric SU(5) The minimal Higgs sector needed to break the symmetry completely down U ( l ) e mx s U ( 3 ) , consists of the adjoint 2 4 and ~ two fundamentals 5 8 and 5 ~ The . Higgs superpotential is quite simple
td
+ X ( 2 4 ~ +) @~ H
W H =m(24p)
5H
+a 5 H 2
4 S ~H
(1)
and so is the Yukawa one WY = Y d 1oF 5 F 5 H
+ y.u 1oF 1oF 5 H
(2)
since the charged fermions belong to 5~ and 1 0 ~ . The above theory is usually called the minimal supersymmetric SU(5) theory. It apparently has a small number of parameters: 3 real
Yd
(after diagonalization)
74
0
0 0
2 x 6 = 12 real yu (yu is a symmetric matrix in generation space) 2 real p , m (after rotations) 2 x 2 = 4 real A, a
In total, 21 real parameters. The trouble is that the theory fails badly.5 Neutrinos are massless, and thus Ve = 1; furthermore, the relations me = md at MCUT fail, except for the third generation. The most conservative approach would be to blame the failure in the absence of higher dimensional operators, this way no change in the structure of the theory is needed. Neutrino masses are then given by the Weinberg-type operator
W," = y v
5 5 5 5
H
MPl
(3)
giving 0
6 x 2 = 12 new real parameters in yv.
Similarly, we must add higher dimensional operators to (2),
(we have omitted SU(5) indices, and represented contraction with indices in 2 4 with ~ a bracket). This means 0
9x 2x2
+ 6 x 2 x 2 = 60 real parameters
and then most predictability is gone. What remains? First, mb = mT is still there, still a success. Gauge couplings unify as we know, but the GUT scale is not predicted precisely as often claimed in the literature. The point is that, for the sake of consistency, one should add higher dimensional operators to (l),so that one expects AWH = CI-
Tr24; MP1
(33-24:)' +cz
Mp1
(5)
and if the coupling X in (1) were to be small, these terms would become important. But X is a Yukawa type coupling, i.e. it is self-renormalizable, so it can be naturally small. This point is worth discussing further. At the renormalizable tree level, one gets the same masses for the color octet and the s U ( 2 ) ~triplet in 2 4 ~ :m8 = m3. This is almost always assumed when the running from MGUT to MW is studied. Now, if X is small, the ci terms in ( 5 ) can dominate; if so, one gets m3 = 4mg. This
75
fact alone suffices to increase MGUT by an order of magnitude above the usually quoted value MSUT N 10l'GeV (calculated with ci = 0). Similarly, the masses of the colored triplets T and T in 5H and S H would get increased by a factor of about 30, and the d = 5-induced proton life-time by about lo3. More precisely, one obtains
where the superscript
denotes the tree-level value m3 = ma. In this case
ma
N
M&JT MP1
so that
With MSUT N 101'GeV, this means
MGUT N lOM&,, , mT
N
32mT. 0
(10)
It should be stressed that X small is natural technically, as much as a small electron Yukawa coupling. Taking X N 0(1)and ruling out the theory would be equivalent to finding that the SM does not work with all the Yukawa couplings being of order one, and insisting on this as if you didn't know fermion masses. Taking into account non-renormalizable interactions can thus save the theory. It is important to recall that without them, the minimal SU(5) does not make sense anyway, predicting as it does m, = 0 and md = me;once this is corrected the theory is still valid. Of course, if one prefers the renormalizable theory, one needs new states such as 45H (in order to correct md = me),or 1 5 to~ give neutrino masses, or three (at least two) singlet right-handed neutrinos. This introduces even more uncertainties in the computations of MGUT and ~ p In . short, the minimal realistic supersymmetric SU(5) theory is not yet ruled out. It is indispensable to improve the experimental limit on ~p by two-three orders of magnitude. Grand unification needs desperately a new generation of proton decay experiments.
76
In the supersymmetric version of SU(5), there is yet another drawback.
As much as the MSSM, it allows for the d = 4 proton decay through terms like
AW = mX' 1 0 S ~F S F
(11)
which contains both
+
A' (vcDcDc QLD") (12) This is a disaster (unless A' 5 10-l'). A way out is assumed through the imposition of R-parity, or equivalently matter parity M : F -+ -F, H -+ H , where F stands for the fermionic (matter) superfields and H for the Higgs ones. Grand unification ought to do better than this, and SO(10) does it as we shall see. In any case, SU(5) does a poor job in the neutrino sector and in the charged fermion sector it is either incomplete or it has too many parameters. One would have to include extra horizontal symmetries, and this route is in some sense beyond grand unification and often needs strings attached. If we stick to the pure grand unification, we better move on to SO(10). 4. Towards unification: Pati-Salam symmetry Quark-Lepton unification can be considered a first step towards the complete SO( 10) unification of a family of fermions in a single representation. Many interesting features of SO(10) GUTS, such as a renormalizable seesaw and R-parity conservation, are already present in partial unification based on the Pati-Salam group GPS = SU(4), x S u ( 2 ) ~ x S u ( 2 ) ~so , it is instructive to review the situation there. Later, when we turn to SO(lO), decomposition of representation under the Pati-Salam subgroup will prove to be the most useful. To simplify the discussion, imagine a two-step breaking of the PS symmetry down to the MSSM
- -
s u ( 2 ) L x s u ( 2 ) R x su(4)c
su(2)L x s u ( 2 ) R x U(1)B-L x S u ( 3 ) ~ SU(2)L x W(1)y x SU(3),.
(13)
The first steps breaks GPS down to its maximal subgroup, the LR (LeftRight) group,8 and it is simply achieved through the vev of and adjoint representation (the numbers in parenthesis indicate the GPS representations)
A = (15,1,1).
(14)
77
In turn, the breaking of the L R group can be achieved by having s U ( 2 ) ~ triplets fields, with B - L = 2, acquiring a vev. Triplets will couple to ferniions and give a mass to right-handed neutrino, providing the see-saw mechanism at the renormalizable level. Right-handed doublets could also do the job, but then non-renormalizable operators have to be invoked, which means effective operators resulting from a new theory at a higher scale, but this theory we will discuss explicitly in the next section. There is a more profound reason for preferring the triplets. They have an even B - L number, and thus preserve matter parity as we defined above. This in turn means R-parity is not broken at a high scale. But then it can be easily shown that it cannot be broken afterwards, at the low energy supersymmetry breaking or electroweak scale. More precisely, a spontaneous breakdown of R-parity through the sneutrino VEV (the only candidate) would result in the existence of a pseudo-Majoron with its mass inversely proportional to the right-handed neutrino mass. This is ruled out by the Z decay ~ i d t h . ~ ? This l O fact is completely analogous to the impossibility of breaking R-parity spontaneously in the MSSM, where the Majoron is strictly massless. In terms of PS representations, the LR triplets are contained in the fields
C(3,1, lo), Z(3,1, fO), Cc(1,3, CO), zc(l,3,lO).
(15)
The matter supermultiplets are
$J(2,1,4), &(I, 2,a)
(16)
and the minimal light Higgs multiplet is
@,
231).
(17)
The most general superpotential for the fields (15) is
+
W = mTrA2 + M T r ( C Z + CcCc) TT(CAC- C,AC,)
(18)
where we assume the following transformation properties under Parity C4C,,
Z-C,,
A + -A.
(19)
We choose A to be a parity-odd field in order to avoid flat directions connecting left- and right-breaking minima. It is straightforward to show that the SM singlets in A , C, and C, take vevs in the required directions to achieve the (in principle two-step) symmetry breaking
=Mc
=Mr,
=Mr,
(20)
78
with
As discussed in detail in,l1?l2the SU(2)~-breakingvev lies in a flat direction that connect them with charge-breaking vacua. It can be eliminated if the soft breaking terms break also su(2)R. If not, one would have to appeal to operators coming from a more complete theory as studied in the next section. The interesting point here is that the breaking in the minimal model leaves a number of fields potentially light.13 There is a larger, accidental S U ( 3 ) symmetry broken down to SU(2) by the right-handed triplet fields, hence five Nambu-Goldstone bosons. But the gauge symmetry s u ( 2 ) R x U ( ~ ) B - Lis broken down to U(l)y, so that three of them are eaten, leaving us with states S$+, 8$+ that acquire a mass only at the scale of supersymmetry breaking. These states are common in supersymmetric theories that include the Left-Right group, and have been subject of experimental search.14 In a similar way, a color octet in A has a mass of order M:/M,, and could in principle be light. The unification constraints give the interesting possibility 103GeV 5 M R
< 107GeV
10"GeV
5 M, 5 10I4GeV
opening up the possibility of the LHC discovering them at the TeV scale. For larger M R , which would be necessary if one wants to fit neutrino masses without additional fine-tuning, these particles become less accessible to experiment. However, the large number of fields in this theory implies the loss of perturbativity at a scale around lOM,, and non-renormalizable effects suppressed by this new fundamental scale can be shown to guarantee that they have comparable masses.12 Namely, if these effects are included, the only consistent possibility is the single-step breaking
MR N M,
I IlOl0GeV
(22)
Surely the most interesting feature of a low scale of PS symmetry breaking is the possibility of having U ( ~ ) B - Lmonopoles, with mass r n =~ lOM,. If produced in a phase transition via the Kibble mechanism, the requirement that their density be less than the critical density then implies M, 1012GeV. We see that the single-step breaking at M , M R lOl0GeV (in a theory including non-renormalizable terms) offers the interesting possibility of potentially detectable intermediate mass monopoles, as long as one manages to get rid of the false vacuum problem of supersymmetric theories.
- -
> 7 1 . Now, when the time t is of the order of 7 1 but much less than 1-2 the asymptotic expansion (7) takes place. The corrections to the leading term are multiplied by the factor e-TLT and are small. The 2-point function thus is exponentially decaying in this regime. It seems that the system has almost lost information about the initial perturbation (at t = 0). But it is not true: as time goes on and approaches the second time scale t 72 the corrections to the leading term in (7) become important and the system starts to collect its memory about the initial perturbation. The information is completely recovered as t = 72 and the time-periodicity is restored. This example is instructive. In particular, it illustrates our point that there can be thermalization in the finite volume for relatively small intervals of time, i.e. when t > 1 is the central charge. The parameter k plays the role of N in the usual terminology of large N CFT. According to the prescription (see Ref. 4),each AdS space which asymptotically approaches the given two-dimensional manifold should contribute to the calculation, and one thus has to sum over all such spaces. In the case of interest, the two-manifold is a torus ( ~ , 4 )where , 1/T and L are
115
the respective periods. There exist two obvious Ads spaces which approach the torus asymptotically. The first is the BTZ black hole in Ads3 and the second is the so-called thermal Ads space, corresponding to anti-de Sitter space filled with thermal radiation. Both spaces can be represented (see Ref. 14) as a quotient of three dimensional hyperbolic space H 3 , with line element 12
ds2 = -(dzdZ
Y2
+dy2),
y
> 0.
In both cases, the boundary of the three-dimensional space is a rectangular torus with periods L and 1/T. We see that the two configurations (thermal AdS and the BTZ black hole) are T-dual to each other, and are obtained by the interchange of the coordinates T H q5 and L t) 1/T on the torus. In fact there is a whole SL(2, Z) family of spaces which are quotients of the hyperbolic space. In order to find correlation function of the dual conformal operators, one has to solve the respective bulk field equations subject to Dirichlet boundary condition, substitute the solution into the action and differentiate the action twice with respect to the boundary value of the field. The boundary field thus plays the role of the source for the dual operator O(,,h). This way one can obtain the boundary CFT correlation function for each member of the family of asymptotically Ads spaces. The total correlation function is then given by the sum over all SL(2,Z) family with appropriate weight. However, for our purposes it is sufficient to consider the contribution of only two contribution^'^
( O ( t ,~ ) o (0)) o ,= e-"TZ(O
o')BTZ
+ e-'Aa
(O O ' ) A d S
,
(9)
where SET,= - h L T / 2 and S A d S = -kn/2LT are Euclidean actions of the BTZ black hole and thermal AdS3, respectively.16 On the Euclidean torus ( )BTZ and ( ) A d s are T-dual to each other. Their exact form can be computed ex~licit1y.l~ For our purposes it is sufficient to note that the (realtime) 2-point function coming from the BTZ part is exponentially decaying, ( )BTZ e-2?rhTt even though it is a correlation function in a system of finite size L. On the other hand, the part coming from the thermal Ads is oscillating with period L, as it should be for a system at finite size. Thus, the total 2-point function (9) has two contributions: one is exponentially decaying and another is oscillating. So that (9) is not a quasi-periodic function of time t. This conclusion does not seem to change if we include sum over SL(2, Z) in Eq. (9). There will always be contribution of the BTZ black hole that is exponentially decaying. This can be formulated also in N
116
terms of the poles in the momentum representation of 2-point function (see Refs. 11, 18). The poles of ( )BTZ are exactly the complex quasi-normal modes (2) while that of ( )Ads are the real normalizable modes (3). Depending on the value of LT, one of the two terms in Eq. (9) dominates.16 For high temperature (LT is large) the BTZ is dominating, while at low temperature (LT is small) the thermal AdS is dominant. The transition between the two regimes occurs at 1/T = L. In terms of the gravitational physics, this corresponds t o the Hawking-Page phase t r a n ~ i t i 0 n . l ~ This is a sharp transition for large k, which is the case when the supergravity description is valid. The Hawking-Page transition is thus a transition between oscillatory relaxation at low temperature and exponential decay at high temperature.
3.4. The Puzzle and Resolution Thus, the AdS/CFT correspondence predicts that the CFT dual to gravity on Ad& is rather peculiar. Even though, it is in finite volume, the relaxation in this theory is combination of oscillating and exponentially decaying functions. This immediately raises a puzzle: how this behavior is consistent with the general requirement for a unitary theory in finite volume to have only quasi-periodic relaxation? A resolution of this puzzle was suggested in Ref. 6. It was suggested that additionally to the size L there exists another scale in the game. This scale appears due to the fact that in the dual CFT at high temperature the typical configuration consists of multiply wound strings which effectively propagate in a much bigger volume, Leff kL. The gravity/CFT duality however is valid in the limit of infinite k in which this second scale becomes infinite. So that the exponential relaxation corresponds to infinite effective size Leff that is in complete agreement with the general arguments. At finite k the scale Leff would be finite and the correlation function is expected to be quasi-periodic with two periods: 1/L and l/Leff. The transition of this quasi-periodic function to combination of exponentially decaying and oscillating functions when Leff is infinite then should be similar to what we have observed in the case of free fermions when L was taken to infinity. N
4. Black Hole Unitarity: Finite k
That relaxation of black hole is characterized by a set of complex frequencies (quasi-normal modes) is a mathematically precise formulation of the lack of unitarity in the semiclassical description of black holes. The unitarity prob-
117
lem was suggested to be resolved within the AdS/CFT correspondence. l5 Indeed, the theory on the boundary is unitary and there should be a way of reformulating the processes happening in the bulk of black hole space-time on the intrinsically unitary language of the boundary CFT. The analysis of the relaxation is helpful in understanding how this reformulation should work. Before making comments on that let us note that the loss of information in semiclassical black hole is indeed visible on the CFT side. It is encoded in that exponentially decaying contribution to the 2-point correlation function. For the CFT itself this however is not a problem. As we discussed above, the finite size unitarity is restored at finite value of k. This however goes beyond the limits where the gravity/CFT duality is formulated. Assuming that the duality can be extended to finite k an important question arises: What would be the gravity counter-part of the duality at finite k ? Obviously, it can not be a semiclassical black hole. The black hole horizon should be somehow removed so that the complex quasinormal modes (at infinite k) would be replaced by real (normal) modes when k is finite. Below we consider two possibilities of how it may happen.
4.1. Fractal Brick Wall
It was suggested in Ref. 7 that the quantum modification of the black hole geometry, needed for the restoring the PoincarC recurrences, can be modeled by the brick wall. Here we elaborate on this interesting idea. The brick wall is introduced by placing a boundary at small distance E from the horizon and cutting off a part of the space-time lying inside the boundary. The effect of the boundary on the quantum fields is implemented by imposing there the Dirichlet boundary condition. Originally, the brick wall was introduced by 't Hooft2' for regularizing the entropy of the thermal atmosphere out-side black hole horizon. With this regularization the quantum entropy S, correctly reproduces the proportionality of the black hole entropy to the horizon area A rf-'. Assuming that E is taken to be of the order of the Planck length, so that Newton's constant is G c d P 2 , one can argue that the black hole entropy is correctly reproduced in this approach. Later on it was, however, realized that the brick wall divergence is actually a UV divergence. One can introduce a set of the Pauli-Villars fields with masses set by parameter p, which plays the role of the UV regulator. Taking into account the contribution of the regulator fields in the entropy of the quantum atmosphere the brick wall can be removed.21 The entropy then is proportional to certain power of the UV regulator, S,
-
-
-
-
118
In our story of black hole relaxation the brick wall indeed gives the wanted effect: once the brick wall has been introduced the quasi-normal modes disappear completely and are replaced by a set of the real (normal) modes. This happens because the effective infinite size region near horizon is now removed and the whole space is the finite size region between the brick wall and the boundary at spatial infinity. In such a system we expect periodicity with the period set by the brick wall parameter E as tbw l/Tln(l/c). This periodicity shows up in the boundary CFT correlation functions rather naturally. Indeed, these correlation functions are constructed from the bulk Green’s function which describes propagation of the perturbation between two points on the boundary through the bulk. In the present case the perturbation from a point 4 on the boundary goes along null-geodesic through the bulk, reflects at the brick wall and returns to the same point 4 on the boundary. The time which the perturbation travels gives the periodicity for the boundary theory and it equals tbw. Matching tbw and l/Leff gives the relation between brick wall regulator E and parameter k of the large N boundary CFT. This probably should be enough for the explaining and reproducing the second time scale of the boundary CFT from the gravity side. The time tbw is however much smaller than the Poincar6 recurrence time which is A expected t o be of the order, tp e: . So how t o get this time scale in the model with the brick wall? We notice that the brick wall should not be ideally spherical. The possible complexity of the shape is not restricted. It may even be fractal. In order to serve as a regulator for the quantum entropy calculation brick wall should just stay at mean distance E from the horizon, but its shape can be arbitrary. For the recurrence time the shape is however crucial. In the absence of the spherical symmetry the perturbation emitted from the point 4 on the boundary (which is still a circle) at spatial infinity goes along null-geodesic through the bulk, reflects from the brick wall, goes back and arrives at completely different point 4’ on the boundary at spatial infinity. Only after a number of back and forth goings between two boundaries the perturbation can manage to arrive on the boundary at the same point where it was initially emitted. This number can be very large and it sets the periodicity for the boundary theory. The emerging geometric picture is standard set up for the system having classical chaos. Indeed, generic deviations from the spherical symmetry of one of the boundaries leads to chaotic behavior of the geodesics. This means that the 2-point functions on the boundary would generically have chaotic time evolution. The optical volume V between two boundaries N
N
119
seems to be the right quantity to measure the size of the phase space of the chaotic geodesics. Since S, V the recurrence time t p ev gives the right estimate for the Poincark time. In this picture the information sent to black hole eventually comes back. The characteristic time during which it should happen is set by the Poincark recurrence time t p . The classical chaos of the geodesics manifests in the (normal) frequencies. The latter are the eigenvalues of the Laplace-type operator considered on the classical geometry. As we know from the relation between classical and quantum chaos, the chaos of the geodesics in the classical system manifests in that the eigen values of the quantum problem are randomly distributed. Thus, the normal frequencies will be random numbers. This again means that the 2-point function on the boundary (we expect that the normal modes are still poles in the momentum representation of the correlation function) is chaotic function of time. The irregularity of the shape of the brick wall may actually be physically meaningful. It can model the fluctuating quantum horizon. It may also be a way of representing the so-called stretched horizon (see Ref. 2 2 ) . N
N
4.2. Worm-hole Modification: BTZk
The horizon can be removed in a smooth way by modifying the black hole geometry and making it looks like a worm-hole. As an example we present here a modification of the BTZ metric ( l ) , 1 ds2 = -(sinh2 y -) dt2 d y 2 cosh2 y d 4 2 , k2 which we call BTZk. The horizon which used t o stay at y < 0 disappears in metric (10) if k is finite. The whole geometry now is that of worm-hole with the second asymptotic region a t y = -m. The two asymptotic regions separated by horizon in classical BTZ metric can now talk to each other leaking the information through the narrow throat. The metric (10) is still asymptotically AdS although it is no more a constant curvature space-time. The Ricci scalar 2 R=[(k2 1) 3k4 sinh4 y 5k2 sinh2 y] (11) (k2 sinh2 y 1)2
+
+
+
+
+ +
+
+
approaches value -6 at infinite y and -2(k2 1) at y = 0 where the horizon used to stay. The normal frequencies in the space-time with metric (10) are real and are determined by the normalizability and the Dirichlet boundary condition at both spatial infinities. Since the space-time (10) is asymptotically AdS one can use the rules of the AdS/CFT duality and
120
calculate the boundary correlation function. Technically it is more difficult than in the standard BTZ case since (10) is not maximally symmetric space. But the result should be a periodic in time function with the period set by parameter k. It would be interesting to do this calculation and see if this correlation function makes sense from the point of view of the expected behavior of the boundary CFT a t finite k. One can calculate the entropy of the thermal atmosphere in the metric (10). It is now finite with no need for introducing the brick wall. The entropy then behaves as S, kA that is the right answer for the Bekenstein-Hawking entropy of BTZ black hole. Thus, the modification (10) gives us the right entropy and solves the unitarity problem. N
Acknowledgments
I would like t o thank D. Birmingham and I. Sachs for enjoyable collaboration and many useful discussions. I also thank G. Arutyunov, J. Barbon, A. Morozov and N. Kaloper for important discussions. References 1. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill Book Company (1971). 2. L. Susskind, hep-th/0204027. 3. S. W. Hawking, Phys. Rev. D14, 2460 (1976). 4. 0. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rev. 323,183 (2000). 5. G. T. Horowitz and V. E. Hubeny, Phys. Rev. D62, 024027 (2000). 6. D. Birmingham, I. Sachs and S . N. Solodukhin, Phys. Rev. D67, 104026 (2003). 7. J. L. F. Barbon and E. Rabinovici, JHEP 0311,047 (2003). 8. I. Sachs, Fortsch. Phys. 52,667 (2004). 9. J. L. F. Barbon and E. Rabinovici, Fortsch. Phys. 52,642 (2004). 10. V. Cardoso and J. P. S. Lemos, Phys. Rev. D63, 124015 (2001); D.Birmingham, Phys. Rev. D64,064024 (2001). 11. D. Birmingham, I. Sachs and S. N. Solodukhin, Phys. Rev. Lett. 8 8 , 151301 (2002). 12. V. Balasubramanian, P. Kraus and A. E. Lawrence, Phys. Rev. D59, 046003 (1999). 13. P.Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, New York, USA: Springer (1997). 14. S. Carlip and C. Teitelboim, Phys. Rev. D51,622 (1995).
15. J. M. Maldacena, hepth/0106112. 16. J. M. Maldacena and A. Strominger, JHEP 9812,005 (1998).
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17. L. Chekhov, hep-th/9811146; E. Keski-Vakkuri, Phys. Rev. D59, 104001 (1999). 18. U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Nucl. Phys. B563, 279 (1999). 19. S. W. Hawking and D. N. Page, Commun. Math. Phys. 87,577 (1983). 20. G. 't Hooft, Nucl. Phys. B256,727 (1985). 21. J. G. Demers, R. Lafrance and R. C. Myers, Phys. Rev. D52, 2245 (1995). 22. N. Iizuka, D. Kabat, G. Lifschytz and D. A. Lowe, Phys. Rev. D68, 084021 (2003).
DEFORMED COORDINATE SPACES DERIVATIVES
J. WESS Universitat Munchen, Fakultat fur Physik Theresienstr. 37, 0-80333 Munchen, Germany and Max- Planck- Institut f u r Physik Fohringer Ring 6, 0-80805 Munchen, Germany E-mail:
[email protected] The concept of derivatives on quantum spaces is worked out in detail. Special example of the @-deformedcoordinate space is analysed. It is shown that it is possible t o construct a deformed Lorentz symmetry for this space. Fields are defined in such a way that they transform with respect to the deformed symmetry.
1. Introduction
This lecture is based on joint work with Marija Dimitrijevi6, Laxisa Jonke, Frank Meyer, Lutz Moller, Efrossini Tsouchnika and Michael Wohlgenannt.' The aim of this lecture is to clarify the concept of derivatives on quantum spaces.2 These derivatives are an essential input for the construction of deformed field equations such as the deformed Klein-Gordon or Dirac equation^.^ These deformed field equations are in turn the starting point for field theories on quantum spaces. For a given coordinate space there are in general many ways t o define derivatives.* We shall try to develop a general concept of such derivatives into which all the different sets of derivatives fit and that allows us by adding additional requirements - usually based on symmetries - to reduce the number of possible derivatives. 2. Deformed Coordinate Spaces Let me first remind you of the concept of deformed coordinate spaces (DCS) which we will use as quantum spaces. DCS are defined in terms of coordi122
123
nates F , p = 1. . . n and relations. Examples of such relations are 1. Canonical relations5
[P, 27 = i P V ,
(1)
for constant 0 it leads to the so called &deformed coordinate space (0-DCS). 2. Lie-type relations6 where the coordinates form a Lie algebra = iCf”P,
[F, 27
(2)
Cr” are the structure constants. Among these is the r;-deformed quantum space (K-DCS).~ 3. Quantum group relations: 2Pg” = -RP”,$PP“ 1
(3)
9
where the R-matrix defines a quantum group.8 These are the q-deformed spaces (q-DCS). The DCS is the algebra d n , this is the factor space of the algebra freely generated by the elements ri-p divided by the ideal generated by the relation^.^ We not only consider polynomials in d, but formal power series as well. In short all polynomials of the coordinates 2” that can be transformed into each other by using the relations are linearly dependent. For the examples listed it can be shown that the dimensions of the vector spaces of polynomials with given degree are the same as for commuting coordinates. This is the socalled Poincark-Birkhoff-Witt property.1°
3. Derivatives Derivatives are maps of the DCS
5:dn+d*.
(4)
They are usually defined by maps on the coordinates, and therefore on the free algebra defined by them. To define a map on the factor space DCS, derivatives have to be consistent with the relations defining the DCS. They also should lead to a Leibniz rule. A very general ansatz for the action of a derivative on the coordinates is:
[&, 2 7
= 5(;
+
c
A;p’-.pJ
j
5 .. . P1
*
(5)
124
The coefficients AY’””3 are complex numbers. They have to be chosen such that Eq. ( 5 ) is consistent with the relations. Having found such coefficients a Leibniz rule can be derived because ( f i j )can be computed from Eq. ( 5 ) , f and ij are elements of A,. Maps can also be defined on the set of derivatives:
E : 6-+Sl,
6;
= E/(6)&.
(6)
The matrix E depends on the derivatives 8 only, not on the coordinates. Because the derivatives 8 are maps on DCS the new derivatives will be as well. If E is invertible and if the matrix E starts with the Kronecker symbol as derivative-independent term we obtain from Eq. (6) again derivatives in the sense of Eq. (5). All derivatives satisfying the consistency condition that have been found up to now are related by such transformations. We shall discuss the 8-DCS here, this is the simplest case. The relations (1) are consistent with
[a,, ?”I
= 6;
.
(7)
A short calculation shows: ( p y- y p - ip”)= ( p p- p
af
p
-~ 2. 9P”
)af .
(8)
This is sufficient t o prove consistency. Eq. (7) leads to the Leibniz rule by applying Eq. (7) to the product of the two functions f i j
4 ( f i j ) = (&,f)ij + f ( 8 P i j )
*
(9)
A short calculation shows that
[6&]
=0
(10)
is compatible with the relations (7). We can assume that the derivatives commute and define an algebra that way. The Leibniz rule (9) can be algebraically formulated as a comultiplication:
A i P = 6,, @ 1 + 1 8
aP.
It is compatible with the Lie algebra (10): [A6P,A&] = 0 , (12) and it is coassociative. Thus, Eqs. (10) and (11) define a bialgebra, the q-deformed bialgebra of translations in the 8-DCS.
125
Other sets of derivatives can be obtained from bp by a transformation (6). In general, such derivatives will not have defining relations that are linear in 8 such as Eq. (7). They will also have more complicated comultiplication rules. Thus the definition (7) singles out a specific type of derivatives. Moreover they will transform linearly under a &deformed orthogonal or Lorentz group. We shall now show that.
4. Deformed Symmetry Algebra
A deformed orthogonal group or a deformed Lorentz group will be a deformation of the transformations
where wpu are the parameters of the infinitesimal orthogonal or Lorentz transformations. The corresponding Lie algebra satisfies:
[L, 1s: = L x , t , (w x w’),” = - (w,Owb - w; “w):
.
(14)
The map (13) can be obtained from a differential operator (angular moment um)
6,
= -xuwupap,
This concept can be lifted to the 8-DCS.
This result was first obtained in Ref. 11. For 8 = 0 Eq. (16) agrees with the undeformed equation (13). In Eq. (16) coordinates transform into derivatives. The additional terms are needed to make the deformed Lorentz transformation compatible with the relation (1). The map &, is really a map on A?. This can be shown in a short calculation, applying Eq. (16) to the relations (1). We find
,j(i~y -y p -i p u )
=
( p j y - p p - iew)$,
+ ( p p - p p - ipw) wpv + ( y p - p p v
- i8w) wp” .
(17)
Analagous to Eq. (15), the transformation (16) can be generated by a differential operator
126
This allows us to calculate the transformations of the derivatives:
and the algebraic relations of
iW:
[iw,
&]
=iwxw,
That iwis a map on d, follows from the fact that 6 and 2 are. The comultiplication can be calculated by applying Eq. (18) to the product of two functions f i j . We find: i A&, = iw 8 1 1 8 iw - - ( o U p w , p - e’pwup)6p 8 (21) 2 This result has recently been obtained by M. Chaichian et al. in Ref. 12. This coproduct is coassociative because iufij& is associative: iwfijwiL = (iwfij)& + fij(iwwi-L)-2i ( ~ p ~ -, pe’pwup)(8pfij)8p&
ap.
+
=
(iuf)ij& + f(&wijk)- ~ ( B y ’ ” W u p- O u p w u p ) ( ~ p f ) 8 p ( i j &(22) ).
The Lorentz algebra by itself does not form a bialgebra. Derivatives appear in the comultiplication rule (21). We can, however, interpret Eq. (21) as a comultiplication rule for the Poincar6 algebra (translation included). Then Eqs. (lo), (19) and (20) define an algebra, the &deformed Poincar6 algebra with the comultiplication (11) and (21). We have obtained the 8deformed Poincark bialgebra. The algebra relations are the same as for the undeformed Poincar6 algebra, the comultiplication is deformed. &deformed Poincar6 bialgebra:
[&, *
$4 = 0 , [iw, Bp] = wptL2p, A
[6,,6:]
u
= iwwxw‘, (w x w ) ; v = - ( u p :
‘
(23)
-wpuw:),
A& = b p 8 1 + 1 8 G p , i
~i~= iw8 1 + 1 8 iw+ -2 (ep’w,p
-v
~
~
& .8
p 8)
~
(24)
That the algebraic relations and the comultiplication rules are compatible can be verified directly. 5 . Fields
On our way to a field theory we have to define fields. They are elements of with certain transformation properties. For a scalar field we define:
4
A
=
n
-Cpapq5
A
,
.
A
n
and 6 ~ q 5= -JWq5.
(25)
127
The translation is parametrized by the constant vector
' of n oscillators. Alternatively a solution of the non-homogeneous algebra with 2-terms on the right hand side of Eq. ( 1 ) can be found trough a relation to a lower-dimensional quantum space realized equivalently as a lower-dimensional deformed Heisenberg algebra. Proposition 1.1. The boundary vectors with respect to which one determines the stationary probability distribution of the n-species diffusion process are generalized, coherent or squeezed states of the deformed Heisenberg algebra underlying the algebraic solution of the corresponding quadratic algebra. We first review the known basic properties of the deformed oscillator coherent states and then define a deformed squeezed state of a pair of deformed oscillators by analogy with the conventional squeezed states as the eigenstate of the deformed boson operators linear combination and study their squeezing properties. Such a q-generalization of the conventional undeformed squeezed states is not known. As a physical application of the deformed coherent and the considered squeezed states we obtain the boundary problem solution of the general n-species stochastic diffusion process.
2. Coherent States of a q-Deformed Heisenberg Algebra We consider an associative algebra with defining relations
aa+ - qa+a = 1,
qNa+ = qa+qN,
N
- -1
4 a-9
(5)
aqN,
where 0 < q < 1 is a real parameter and a+a = = [ N ] . A Fock representation is obtained in a Hilbert space spanned by the orthonormal basis %lo) = In), n = 0, 1,2, ... and (nln') = dnn,: N
alO) = 0 ,
+
aln) = [n]'/21n - 1) ,
+
a+ln) = [n 1]1/21n 1). ( 6 )
The Hilbert space consists of all elements = Cr=o fnln) with complex fn and finite norm with respect to the scalar product = C,"==, The q-deformed oscillator algebra has a Bargmann-Fock representation on the Hilbert space of entire analytic functions. Generalized or q-deformed coherent s t a t e ~ are ~ ~ defined '~ as the eigenstates of the deformed annihilation operator a and are labelled by a con-
If)
(flf)
lfnI2.
134
tinuous (in general complex) variable z:
&.
These vectors belong to the Hilbert space for 1 . ~ 1’),in terms of the internal geometry. The other Hodge components, i.e. H(3t0)and H(073), must vanish in order to have unbroken supersymmetry. In the following, let us review the rewriting leading to Eq. (2).
2. Fluxes and Torsion
The bosonic part of the Lagrangean up to second order in a’ is given by7
This action is written in the string frame and its fermionic completion makes it supersymmetric using the three-form Bianchi identity given by
dH = a’ (tr R+ A R+ - t r F A F ) ,
(4)
where the curvature RS is the generalized Riemann curvature built from the generalized connection V+ (i.e. from wf = w H ) . In the search for a BPS rewriting of Eq. ( 3 ) , 6 we will assume that the ten-dimensional space is given by the warped product of four-dimensional Minkowski spacetime with a six-dimensional internal space admitting an SU(3) structure. In order to consistently obtain that setting to zero the BPS-like squares implies a solution to the equations of motion, we also impose that the only degrees of freedom for the various fields are given by expectation values on the internal space and are functions only of the internal coordinates. To simplify the discussion we limit ourselves to the case with dilaton and warp factor identified, i.e. q5 = A, but the generalization of the following results is straightforward. After various manipulations, the action (3) can
183
be written as6
-
14 J d6y &eg+
Nmnpgmqgnrgps N~~~
In this expression the traces are taken with respect to the fiber indices a, b, . . ., whereas the Hodge type refers to the base indices m, n,. . . of the curvatures. The other geometrical objects appearing in the above expression are the Lee-form 3 0 JAdJ = - J"" qrnJnpldxP , (6) 2 the Nijenhuis tensor
NmnP= JmqdiqJnlP- Jnq6jq J,,',
(7)
and the generalized curvature R, which is constructed using the Bismut connection built from the standard Levi-Civita connection and a totally antisymmetric torsion T B proportional to the complex structure,
The action ( 5 ) will now be used to find the conditions determining the background geometry by demanding the vanishing of ( 5 ) . Setting the squares to zero yields 0
0
the vanishing of the Nijenhuis tensor the vanishing of some components of the generalized Riemann curvature constructed from the V+ connection, the vanishing of
1 d4+-0=O, 8
(9)
184
the vanishing of
1 H+-*e-'+d(e'+J) 2 0
=0,
(10)
the vanishing of
The vanishing of the Nijenhuis tensor states that the internal manifold is complex. The conditions on the R+ curvature can be translated into the requirement of SU(3) holonomy for the V- connection. The proof requires the identity b R :
cd
= R z ab - (dH)abcd
(11)
7
which relates the R+ and R- curvatures with the base and fiber indices swapped. Using this identity and the fact that d H gives higher order terms in a', the conditions on the base indices of R+ become conditions on the R- fiber indices, to lowest order in a', R-
(2,o) = R- (0,2)
= JabR-
ab
=0
'
(12)
These conditions precisely state that the generalized curvature R- is in the adjoint representation of SU(3) c SO(6) and therefore its holonomy group is contained in SU(3). The conditions in the gauge sector are that the gauge field strength is of type (1,l)and J traceless. On a complex manifold, the condition (10) yields
=0 , H(3,0)+(0>3)
(13)
where we also used (9). On the solution, R+ A R+ and F A F are of type (2,2). Therefore, the Bianchi identity (4) implies that (dH)(311)f(193)= 0, which is indeed satisfied. We conclude our discussion with remarks about the possible superpotential describing such vacua in the effective four-dimensional theory. It has been argued that a candidate superpotential describing the N = 1 vacua of the heterotic theory in the presence of fluxes is given by6,'
where R denotes a holomorphic (3,O)-form on Ms.This conjecture is motivated by the hope of getting the conditions (13) out of W = DW = 0.
185
For instance, assuming that H is constant under variation of the almost complex structure moduli and that &s2 = kiR xi,where xi are a set of ( 2 , l ) forms and ki are constants over the internal manifold, one obtains t h a t t h e (1,2) part of H $ d J is vanishing. On a complex manifold, this yields the locking of the t h r e e f o r m flux onto the geometry, namely
+
+
H(ly2)+ 8 J = 0 , and since H is a real form, it follows that
References 1. A. Strominger, Nucl. Phys. B274, 253 (1986). 2. G. L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, P. Manousselis and
G. Zoupanos, Nucl. Phys. B652, 5 (2003). 3. K. Becker, M. Becker, K. Dasgupta and P. S. Green, JHEP 04, 007 (2003). 4. J. P. Gauntlett, D. Martelli and D. Waldram, Phys. Rev. D69,086002 (2004). 5. J. P. Gauntlett, N. w. Kim, D. Martelli and D. Waldram, JHEP 0111, 018 (2001). 6. G. L. Cardoso, G. Curio, G. Dall’Agata and D. Lust, JHEP 0310,004 (2003). 7. E.A. Bergshoeff and M. de Roo, Nucl. Phys. B328, 439 (1989). 8. K. Becker, M. Becker, K. Dasgupta and S. Prokushkin, Nucl. Phys. B666, 144 (2003).
SYMMETRIES AND SUPERSYMMETRIES OF THE DIRAC-TYPE OPERATORS ON EUCLIDEAN TAUB-NUT SPACE
I. I. COTAESCU West University of Tamigoara, V. P6rvan Ave. 4, RO-1900 Timigoara, Romania E-mail: [email protected]
M. VISINESCU Department of Theoretical Physics, National Institute for Physics and Nuclear Engineering, P.O.Box M. G.-6, Magurele, Bucharest, Romania E-mail: [email protected]
The role of the Killing-Yano tensors in the construction of the Dirac-type operators is pointed out. The general results are applied to the case of the four-dimensional Euclidean Taub-Newman-Unti-Tamburino space. Three new Dirac-type operators, equivalent to the standard Dirac operator, are constructed from the covariantly constant Killing-Yano tensors of this space. The Dirac operators are related among themselves through continuous or discrete transformations, In this space there is also a non covariantly constant Killing-Yano tensor connected with hidden symmetries. The Runge-Lenz operator for the Dirac equation in this background is written down pointing out its algebraic properties.
1. Introduction The (skew-symmetric) Killing-Yano (K-Y) tensors, that were first intrcduced by Yanol from purely mathematical reasons, are profoundly connected to the supersymmetric classical and quantum mechanics on curved manifolds where such tensors do exist.2 The K-Y tensors play an important role in theories with spin and especially in the Dirac theory on curved spacetimes where they produce first order differential operators, called Dirac-type operators, which anticommute with the standard Dirac one, D,.3 Another virtue of the K-Y tensors is that they enter as square roots in the struc186
187
ture of several second rank Stackel-Killing tensors that generate conserved quantities in classical mechanics or conserved operators which commute with D,. The construction of Carter and McLenaghan depended upon the remarkable fact that the (symmetric) Stackel-Killing tensor K,, involved in the constant of motion quadratic in the four-momentum p p
has a certain square root in terms of K-Y tensors f,,: Kpv =
The K-Y tensor here is a 2-form
fpu
fpxf.,A . .
(2)
= -f,,, which satisfies the equation
These attributes of the K-Y tensors lead to an efficient mechanism of supersymmetry, especially when the Stackel-Killing tensor K,, in Eq. (1) is proportional with the metric tensor gPv and the corresponding K-Y tensors in Eq. (2) are covariantly constant. Then each tensor of this type, fi, gives rise to a Dirac-type operator, Di, representing a supercharge of the superalgebra {Di, Dj} 0: D2&. The general results are applied to the case of the four-dimensional Euclidean Taub-Newman-Unti-Tamburino (Taub-NUT) space. The Euclidean Taub-NUT metric is involved in many modern studies in physics. This metric might give rise to the gravitational analog of the Yang-Mills i n ~ t a n t o nThe . ~ Kaluza-Klein monopole of Gross and Perry5 and of Sorkin‘ was obtained by embedding the Taub-NUT gravitational instanton into fivedimensional Kaluza-Klein theory. On the other hand, in the long-distant limit, the relative motion of two monopoles is approximately described by the geodesics of this space.? The Euclidean Taub-NUT space which is a hyper-Kahler manifold possessing three covariantly constant K-Y tensors. Using these covariantly constant Killing-Yano tensors it is possible to construct new Dirac-type operators3 which anticommute with the standard Dirac ~ p e r a t o r . * ~ ~ The Taub-NUT space also possesses a Killing-Yano tensor which is not covariantly constant. The corresponding non-standard operator, constructed with the general rule3 anticommutes with the standard Dirac operator but is not equivalent to it.lo This non-standard Dirac operator is connected with the hidden symmetries of the space allowing the construction of a conserved vector operator analogous to the Runge-Lenz vector of the Kepler problem. We shall discuss the behavior of this operator under
188
discrete transformations pointing out that the hidden symmetries are in some sense decoupled from the discrete symmetries studied here. 2. Dirac Equation on a Curved Background In what follows we shall consider the Dirac operator on a curved background which has the form (4)
D, = y p V p .
In this expression the Dirac matrices y,,are defined in local coordinates by the anticommutation relations {y,,y’} = 2gp’I and V, denotes the canonical covariant derivative for spinors. Carter and McLenaghan showed that in the theory of Dirac fermions for any isometry with Killing vector R, there is an appropriate ~ p e r a t o r : ~
which commutes with the standard Dirac operator (4). Moreover, each Killing-Yano tensor f p u produces a non-standard Dirac operator of the form A
1
D, = - i Y Y f ( f , ” V u - p ” r V p u ; p )
(6)
1
which anticommutes with the standard Dirac operator D,.
3. Dirac Operators of the Taub-NUT Space Let us consider the Taub-NUT space and the chart with Cartesian coordinates z@(p, v = 1,2,3,4) having the line element
+
+
where 2 denotes the three-vector 2 = ( T , 0, cp), (dZ)2 = ( d ~ l ) ( ~ d ~ ~ ) ~ ( d ~ ’ and ) ~ A’ is the gauge field of a monopole divA’=O,
-
2 B =rotA=p-.
r3
(8)
The real number p is the parameter of the theory which enters in the form of the functions P+r f ( r ) = g-l(T) = V-’(r) = -. (9) r
189
In the Taub-NUT geometry there are four Killing vector^.'^^^^ On the other hand, in the Taub-NUT geometry there are known to exist four KillingYano tensors of valence 2. The first three are covariantly constant
D,
fr' = 0 ,
i, j , k = 1 , 2 , 3 .
( 10)
These first three Killing-Yano tensors of the Taub-NUT space16 are rather special since they are covariantly constant. The f i define three anticommuting complex structures of the Taub-NUT manifold, their components realizing the quaternion algebra fifj+ fjfi
=-2&,
fifj
- f j f i = - 2 & . .k f k .
(11)
The existence of these Killing-Yano tensors is linked to the hyper-Kahler geometry of the manifold and shows directly the relation between the geometry and the N = 4 supersymmetric extension of the theory.2*18 For the theory of the Dirac operators in Cartesian charts of the TaubNUT space, it is convenient to consider the local frames given by tetrad fields .a(.) and 2"(x) while the four Dirac matrices y" satisfy {y", yP} = 2S@, where all of them are self-adjoint. In addition, we consider the matrix y5 = yiyiyiyd which is denoted by yo in Kaluza-Klein theory explicitly involving the time." The standard Dirac operator of the theory without explicit mass term is defined D, = y"V&, where the spin covariant derivatives with local indices, V", depend on the momentum operators, Pi = -i(ai - A&) and P4 = -4,and spin connection,12such that the Hamiltonian 0 p e r a t o r l ~ 7 ~ ~
can be expressed in terms of Pauli operators,
involving the Pauli matrices, ~ i .These operators give the (scalar) KleinGordon operator of the Taub-NUT space: A = -V,gp""V, = We specify that here the star superscript is a mere notation that does not represent the Hermitian conjugation because we are using a non-unitary representation of the algebra of Dirac operators. Of course, this is equivalent to the unitary representation where all of these operators are self-adjoint.12
190
Moreover we can give a physical interpretation of these Killing-Yano tensors defining the spin-like operators,
that have similar properties to those of the Pauli matrices. In the pseudoclassical description of a Dirac particle27l8the covariantly constant KillingYano tensors correspond to components of the spin which are separately conserved. Here, since the Pauli matrices commute with the Klein-Gordon operator, the spin-like operators (14) commute with H 2 . Remarkable the existence of the Killing-Yano tensors allows one to construct Dirac-type operators3
which anticommute with D , and y5 and commute with H.l0 Another Dirac operator can be defined using the fourth Killing-Yano tensor, but this will be discussed separately in Sec. 6. 4. Equivalent Representations
In Ref. 12 we have shown that in the massless case the operators Qi (i = 1,2,3) and the new supercharge QO = iD, = iy5H form the basis of an N = 4 superalgebra obeying the anticommutation relations {QA, Q B ) = ~
~ A B H A,B ~ ,... , =0,L2,3,
(16)
linked t o the hyper-Kahler geometric structure of the Taub-NUT space. In addition, we associate to each Dirac operator Q A its own Hamiltonian operator QA = - i y ' Q ~ , obtaining thus another set of supercharges
Qo =H ,
Qi = i [ H , C i ] ,
(17)
which obey the same anticommutation relations as Eq. (16). Thus we find that there are two similar superalgebras of operators with precise physical meaning. Obviously, since all of these operators must be self-adjoint we have to work only with unitary representations of these superalgebras, up to an equivalence. The concrete form of these supercharges depends on the representation of the Dirac matrices which can be changed at any time with the help of a non singular operator T , such that all of the 4 x 4 matrix operators of the Dirac theory transform as X -+ X' = T X T - l . In this way one obtains
191
an equivalent representation which preserves the commutation and the anticommutation relations. We used such transformations for pointing out that the convenient representations are equivalent to a unitary one.12 We note that some properties of the transformations changing representations in theories with two Dirac operators and their possible new applications are discussed in a paper by K1i~hevich.l~ For example, simple and convenient transformations can be chosen of the form:
U(d
E SU(2). where O(p,f) = e - i P 6 ( f i E U(2) = U(1) 8 SU(2) with This is because among these transformations one could find those linking equivalent Dirac operators. It is interesting to observe that the SU(2) transformations are generated just by the above defined spin-like operators as
~~ As in the case of the Dirac operators (15), one can use f Y for defining the fifth Dirac operator
called here the non-standard or hidden Dirac operator t o emphasize the connection with the hidden symmetry of the Taub-NUT problem. It is denoted by Q,' to point out its relation to the standard Dirac operator since it can be put in the form
Q,'
= iT [Qo
P
(:
g,v-l)] 0
9
where B, = 3 .Z / r . We showed that QF commutes with QO = H and anticommutes with QO and y5.lo This operator is important because it allowed us to derive the explicit form of the Runge-Lenz operator, 2, of the Dirac field in Taub-NUT background establishing its properties." We recall that the components of the conserved total angular momentum, f, and the operators Ri = F-lKi with F2= P42 - H 2 are just the generators of the dynamical algebra of the Dirac theory in Taub-NUT b a ~ k g r o u n d . ' ~ ? ~ ~ Starting with Q,' we can construct a new orbit, Ry, of GQ defining
(for k = 1 , 2 , 3 ) and observing that P Q Z P = -QZ , A = 0 , 1 , 2 , 3 . From the explicit form (32) we deduce that, in contrast with the operators of the orbits RQ and those of the orbit RY have more involved algebraic properties. We can convince that calculating
a~, +
H2(Q,')2 = H4
(33)
and it is worth comparing it with Eq. (16). The Dirac-type operators Q A are characterized by the fact that their quanta1 anticommutator close on the square of the Hamiltonian of the theory. No such expectation applies to the non-standard, hidden Dirac operators QZ which close on a combination of different conserved operators. Also from Eq. (33) it results that ( Q f ; ) 2#
194
(QZ)2 if A # B (because ? does not commute with u k ) . Moreover, one can show that the commutators [QX, Q';] have complicated forms which can not be expressed in terms of operators Q,'. Therefore, neither the commutator nor the anticommutator of the pairs of operators of this orbit do not lead t o significant algebraic results as the anticommutation relations (16) of the operators Q A , ( A = 0 , 1 , 2 , 3 ) . The hidden symmetries of the Taub-NUT geometry are encapsulated in the non-trivial Stackel-Killing tensors Kip",(i = 1,2,3). For the Dirac theory the construction of the Runge-Lenz operator can be done using products among the Dirac-type operators Q y and Qi. Let us define the operator:8
Ni = m { Q Y , The components of the operator lowing commutation relations
Q i } - JiP4.
fl commutes with H
[Ni, p4] = 0 , [Ni, Jj] = i & i j k N k , [Ni, Qo] = 0 [Ni, Q j ] = i E i j k Q k p 4 9
[Ni, N j ] = i S i j k J k F 2 +
(34) and satisfy the fol-
>
(35)
2 -&ijkQiH,
2
where F 2 = P42 - H 2 . In order to put the last commutator in a form close to that from the scalar case,16917 we can redefine the components of the Runge-Lenz operator, g , as follows:
Ki = Ni
+ -21H - l ( F
- P4)Qi,
having the desired commutation relation:8
7. Concluding Remarks In the study of the Dirac equation in curved spaces, it has been proved that the Killing-Yano tensors play an essential role in the construction of new Dirac-type operators. The Dirac-type operators constructed with the aid of covariantly constant Killing-Yano tensors are equivalent with the standard Dirac operator. The non-covariantly constant Killing-Yano tensors generates non-standard Dirac operators which are not equivalent to the standard Dirac operator and they are associated with the hidden symmetries of the space.
195
The Taub-NUT space has a special geometry where the covariantly constant Killing-Yano tensors exist by virtue of the metric being self-dual and the Dirac-type operators generated by them are equivalent with the standard one. The fourth Killing-Yano tensor fY which is not covariantly constant exists by virtue of the metric being of type D.The corresponding non-standard or hidden Dirac operator does not close on H as it can be seen from Eq. (33) and is not equivalent to the Dirac-type operators. As it was mentioned, it is associated with the hidden symmetries of the space allowing the construction of the conserved vector-operator analogous to the Runge-Lenz vector of the Kepler problem. Acknowledgments
M.V. thanks Professor Goran Djordjevic for his kind invitation and pleasant hospitality. This work is partially supported by a grant of the Romanian Academy. References 1. K. Yano, Ann. Math. 55,328 (1952). 2. G. W. Gibbons, R. H. Rietdijk and J. W. van Holten, Nucl. Phys. B404,42 (1993). 3. B. Carter and R. G. McLenaghan, Phys. Rev. D19,1093 (1979). 4. S. W . Hawking, Phys. Lett. A60,81 (1977). 5. D. J. Gross and M. J. Perry, Nucl. Phys. B226,29 (1983). 6. R. Sorkin, Phys. Rev. Lett. 51, 87 (1983). 7. M. F. Atiyah and N. J. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton, Princeton University Press (1987). 8. I. I. Cotbscu and M. Visinescu, Gen. Rel. Grav. 35,389 (2003). 9. I. I. Cotbscu and M. Visinescu, Class. Quant. Grav. 21, 11 (2004). 10. I. I. Cotbscu and M. Visinescu, Phys. Lett. B502,229 (2001). 11. I. I. Cot&scu and M. Visinescu, Mod. Phys. Lett., A15, 145 (2000). 12. I. I. Cotbscu and M. Visinescu, Int. J. Mod. Phys. A16,1743 (2001). 13. I. I. Cotbscu and M. Visinescu, Class. Quant. Grav. 18,3383 (2001). 14. I. I. Cotbscu and M. Visinescu, J. Math. Phys. 43, 2978 (2002). 15. M. Visinescu, Int. J. Mod. Phys. A17,1049 (2002). 16. G. R. Gibbons and P. J. Ruback, Comrnun. Math. Phys. 115,267 (1988). 17. G. W. Gibbons and N. S. Manton, Nucl. Phys. B274,183 (1986). 18. J. W. van Holten, Phys. Lett. B342,47 (1995). 19. V. V. Klishevich, Class, Quant. Grav. 17,305 (2000). 20. A. 0. Barut and R. b c z k a , Theory of Group Representations and Applications, Warszawa, PWN (1977). 21. H. S. M. Coxeter and W. 0. J. Moser, Generators and Relations for Discrete Groups, Berlin, Springer-Verlag (1965).
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22. D. Vaman and M. Visinescu, Phys. Rev. D54,1398 (1996). 23. D. Vaman and M. Visinescu, Fortschr. Phys. 47, 493 (1999).
REAL AND P-ADIC ASPECTS OF QUANTIZATION OF TACHYONS
G.S. DJORDJEVIk AND LJ. NESIC Department of Physics, Faculty of Sciences P.O.box 22.4, 18001 NiS, Serbia and Montenegro E-mail: gorandj, [email protected]. yu A simplified model of tachyon matter in classical and quantum mechanics is constructed. p-Adic path integral quantization of the model is considered. Recent results in using padic analysis, as well as perspectives of an adelic generalization, in the investigation of tachyons are briefly discussed. In particular, the perturbative approach in path integral quantization is proposed.
1. Introduction An increasing number of researchers are testing again an (almost twenty year) old idea that padics and padic string theory' can be useful in attempts to understand ordinary string theory, D-brane solutions and various aspects of tachyon^.^?^ Possible cosmological implications4 are also an interesting matter. pAdic strings have many properties similar to ordinary strings, but p adic ones are much simpler. For example, one can find an exact action for this theory, as well as for practically all variations of padic string theory. In addition, it turns out that padic string could be a very useful model for testing Sen% conjecture on tachyon condensation in open string field theory (see e.g. Ref. 5). Corresponding padic classical solution in string field theory can be explicitly found. An important fact should be also considered, that is, padic string field theory in the p 4 1 limit reduces to the tachyon effective action.6 It might be the case that results for padic strings are applicable t o boundary string field theory. The next reason for further investigation is similarity between padic D-branes (without problems with strong coupling problems) and the results found in vacuum cubic string filed theory. It is interesting that the effective energy-momentum tensor is equivalent to that of nonrotating dust.' 197
198
Generally speaking, after almost twenty years of “discoveringL‘ padic strings, our understanding of physics on padic spaces is still very poor. From many point of view, including pedagogical one, it is very useful to understand (quantum) mechanical analogies of new, unfamiliar objects, including tachyons. Considering “padic tachyons L L we stress results in foundation of padic quantum mechanics, padic quantum cosmology and their connection with standard theory on real numbers (and corresponding spaces: Minnkowsky, Riemman ...) and adelic quantum theory. It has been noted that path integrals are extremely useful in this a p proach. Following S. Kar’s’ idea on the possibility of the examination of zero dimensional theory of the field theory of (real) tachyon matter, we extend this idea to the padic case. In addition, we note possibilities for adelic quantum treating of tachyon matter. After some mathematical background in Section 2, we present padic path integrals in Section 3. Section 4 is devoted to padic strings and tachyons. Simple quantum mechanical analog of padic tachyons is considered in Section 5. The paper is ended by a short conclusion and suggestion for further research.
2. pAdic Numbers and Related Analysis Let us recall that all numerical experimental results belong to the field of rational numbers Q. The completion of this field with respect to the standard norm I (absolute value) leads to the field of real numbers R = Qoo. Completion of Q with respect to the padic norms yields the fields of padic numbers Q p ( p is a prime number). Each non-trivial norm (valuation) on Q, due to the Ostrowski theorem, is equivalent either to a padic norm I I p or to the absolute value function. Any padic number x E Q p can be presented as an expansiong
lo
x = x”(x0 + x1p
+ x2p2 + . * . ),
v E 2,
(1) where xi = 0,1, . . . , p - 1. pAdic norm of any term in (1) is P-(”+~). The padic norm is the nonarchimedean (ultrametric) one. There are a lot of exotic features of padic spaces. For example, any point of a disc &( a ) = {x E Qp : Ix - alp 5 p”} can be treated as its center. It also leads to the total disconnectedness of padic spaces. For the foundation of the path integral approach on padic spaces it is important to stress that no natural ordering on Q p exists. However, one can define a linear order as follows: x < y if lxlp < Iylp, or when 1xIp = Iylp, when there is an index m 2 0 such that the following is satisfied:
199
z o = yo,z1 = yl, ..., x ,-1
= ym-l, z , < ym.l0 Generally speaking, there are two analysis over Q p . One of them is connected with map 4 : Q p -+ Q p , and the second one is related to the map ?c, : Q p + C. In the case of padic valued function, derivatives of 4(z) are defined as in the real case, using padic norm instead of the absolute value. pAdic valued definite integrals are defined for analytic functions M
n=O as follows:
Jd
M
b
4(t)dt =
4n x(bn+l
-ant1). (3) n=O In the case of mapping Q p -+ C , standard derivatives are not possible, and several different types of pseudodifferential operators have been introduced.’~~~ Contrary, there is a well defined integral with the Haar measure. Of special importance is Gauss integral
~ where xp(u) = exp(27ri{u},) is a padic additive character, and { u } denotes the fractional part of u E Q p . X,(CY) is an arithmetic, complex-valued function.’ To explore the existence of a vacuum state in padic quantum theory we need
where 2, is the ring of padic integers (a set of all padics z, lzlp 5 1) and R is the characteristic function of 2,. It should be noted that R is the simplest vacuum state in padic Quantum Theory. There is quite enough similarity between real numbers and padics and the corresponding analysis for the so called adelic approach in mathematics” and physics (see e.g. Ref. 4), that in a sense unifies real and all padic number fields. 3. Path Integral in Ordinary and p-Adic Quantum Mechanics According to Feynman’s idea,13quantum transition from a space-time point (z’,t’) to another (d’”’’) is a superposition of motions along all possible
200
paths connecting these two points. Let us remind the corresponding probaz*i s bility amplitude is (x”,t”(x’,t’) = C , e T [,I. Dynamical evolution of any quantum-mechanical system, described by a wave function $ ( x , t ) ,is given by $(XI’,
t”) = J
K(x”,t”;X I , t’)$(x‘, t’)dx‘,
Q-
(6)
where K(x”,t”;X I , t’) is a kernel of the unitary evolution operator U(t”,t‘). In Feynman’s formulation of quantum mechanics, K(x”,t”;x’, t’) was postulated to be the path integral
where x” = q ( t ” ) and x’ = q(t’). As we know, for a classical action S(X”,t”;x’, t’),which is a polynomial quadratic in XI‘ and X I , the corresponding kernel K (for one-dimensional quantum system) reads
( i 823 > + exp (2ai -S(X”, t“;x‘, t’) K(X”,t”;2 1 , t’) = -hax~~ax/ h It can be rewritten in the form more suitable for generalization (at least from the number theory point of view)
,/-
where & = = lalz2A,(-a). In (9), x,(a) = exp(-27ria) is an additive character of the field of real numbers R. D-dimensional generalization of the transition amplitude is:
where ,A
is defined as
(,) and x = (x,), a = 1 , 2 , . . . , D. By defining AO this A,-function satisfies the same properties as A.,
= 1, one can see that
201
In padic quantum mechanics dynamical differential equation of the Schrodinger type does not exist and padic quantum dynamics is defined by the kernel Kp(x”, t”;x’,t’) of the evolution operator:
$,(x”, t”) = Up(t”,t’)$,(x’, t’) =
s,,
Kp(x”,t”;x’,t’)$,(z’, t’)dx’. (12)
All general properties which hold for the kernel K(x”,t”; t’) in standard quantum mechanics also hold in padic case, where integration is now over Q,. pAdic generalization of (7) for a harmonic oscillator was done in” starting from
( h E Q and q , t E Q,). In (13) d q ( t ) is the Haar measure and padic path integral is the limit of a multiple Haar integral. This approach was extended in Ref. 14. A rather general path integral approach, valid for analytical classical solutions, was developed for quadratic padic quantum systems in Ref. 15 (in one-dimension)
Kp(x”,t”;
t’) = A,
1
s2s
(-2hm)
1 4 xp(--S(x”, 1t”;x‘, t’)) l%mlp h a2S
(14) and Ref. 16 (two-dimensional case). The obtained padic result (14) has the same form as (11) in the real case. The higher-dimensional padic kernel was also c0nsidered.l’ Considering real-ordinary and all padic quantum mechanics on the same foot, adelic formulation is also possib1e.l’ It could be a good starting point to consider quantum mechanical analog of “real“, “padic“ and, possibly, ‘adelic“ tachyons . 4. p-Adic Strings and Tachyons
The padic open string theory can be deduced from ordinary bosonic open string theory on a D-brane by replacing the integral over the real worldsheet by padic integra1.l’ A tachyon was defined as a particle that travels faster than light, and consequently has negative mass2. Surely, it is not a convincing case for the tachyon. Quantum field theory offers a much better framework for considering such a pretty exotic physical model. If we would carry out
202
perturbative quantization of the scalar field by expanding the potential around $J = 0, and ignore higher (cubic, ...) terms in the action, we would find a particle-like state with mass2 = V”(0).In the case of V”(0)< 0 we have again a particle with negative mass2, i.e., a tachyon. The physical interpretation is that the potential V ( 4 ) has a maximum at the origin and hence a small displacement of 4 will make it grows exponentially. It is associated with the instability of the system and a breakdown of the theory. Conventional formulation of string theory uses a first quantized formalism. In this formulation one can get a state-particle with negative muss2, i.e. tachyons. The simplest case appears in 26 dimensional bosonic string theory. This approach is, unfortunately, not suitable for testing tachyon‘s solutions,2 but there are supersting theories defined in (9+1) dimensions that have tachyon free closed string spectrum. In addition, some string theories contain open string excitations with appropriate boundary conditions at the two ends of the string. So, one can ask: is there a stable minimum of the tachyon potential around which it is possible to quantize the theory. In the last few years there many papers devoted to this problem have made some progress, but we will not consider them in this paper. In padic string theory all tree level amplitudes involving tachyons in the external states can be computed. The padic (open) string theory is obtained from ordinary bosonic (open) string theory on a D-brane by replacing the integrallg over the real world-sheet coordinates by padic integral associated with a prime number p. There have been somewhat different approaches, but we will not consider the constructions of all these theories. Let us see the exact effective action for the ptachyon field. It is described by the lagrangian21
This form, obtained by computing Koba-Nielsen amplitudes for a prime p, makes sense for all (integer) values of p . The classical equation of motion derived from (15) is
Besides the trivial constant solutions $J = 0,1, a soliton solution is admitted. The equations separate in the arguments and for any spatial
203
direction x we get
a gaussian lump whose amplitude and spread are ~ o r r e l a t e d . ~
5. Quantum Mechanical Analogue of Tachyon Matter Now, we will concentrate on a relatively new field theory - the field theory of tachyon matter was proposed by Sen a few years ago.2o The derivation of its action is based on a rather involved argument. The obtained form is pretty strange and different from the actions we used to be familiar ones
Let us consider padic analogue of the above action, originally considered as real one, i.e. 700 = -1, ,,v = ,S where p , v = 1,2, ..., n. T ( x ) is a padic scalar tachyon field and V ( T )is the tachyon potential: V ( T )= exp(-crT/2). In the bosonic case, n = 25, cr = 1, and for superstring n = 9, Q = fi.The square root appearing in action (18) (and its multiplication to tachyon potential) makes this theory so unusual. Here we examine a lower (zero-dimensional) mechanical analogue of the field theory of padic tachyon matter (whatever it would physically mean). As usually, the correspondence can be obtained by the correspondence xi + t , T + x , V ( T )+ V ( x ) .The corresponding zero-dimensional action reads
so = -
I
dtV(X)drn,
(19)
where integration has to be performed over padic time. From the above action it is not difficult to get the classical equation of motion
where function f ( x ) denotes
) . ( f
=
---.1 d V
v dx
Partial differentiation of padic valued function is well defined, although in this case it can be replaced by the ordinary one, because V = V ( x ) .
204
Keeping in mind that exponential padic function (the tachyon potential V ( z )= exp(-ax) should be understood as an analytic function with corresponding radius of convergenceg r l/p, we obtain as in the real case N
?+a* 2 = a .
By the replacements motion
j. = ~ y a, y =
mji
6 and
(21) = g we get the equation of
+ pg2 = mg,
(22) which describes motion of a particle of mass m moving in a constant (say gravitational, Newtonian) field with quadratic friction. It is interesting that this equation can be derived from the (padic) action
Surprisingly or not, the zero dimensional analog of the (Sen's) field theory of tachyon matter offers an action integral formulation for the system under gravity in the presence of (quadratic) damping. The solution of the equation of motion (22) reads Y = y o + - lm n(
2P
:),
g-Pv2 g-,v
with initial t = 0 conditions for position y(0) = yo and velocity w(0) = wo. This solution has the same form in the real and padic case, but the radius of convergence is rather different. Faced with the increasing interest in various aspects of tachyon field theory, including its padic aspect, this connection with the field theory of tachyons thorough action integral formulation seems worth mentioning and examining in general. Also, quantization of the theory in path integral language might be very useful and, as we know, very general (for real, p adic and adelic path integrals see, e.g. Ref. 17). However, a kernel of the operator of evolution that corresponds to the action (23) is still unknown, even in the real case. Because of that the square root and exponential for small P should be expanded. If we treat p padicaly small, in respect to padic norm, we obtain
205
We have already calculated the path integral for the particle in constant external field.14 Here we have a slightly changed form (y” = Y(T), = Y(O), h = 1)
It makes it possible to check the existence of the simplest tachyonic vacuum state (invariant in respect t o the evolution operator), of the corresponding quantum mechanical model, i. e.
1
Kp(Yll,7; d ,O)fl(lY’lp)dY’. (27) IY’ll Using (5) and (26) we find that for the existence of the “ground“ (52) state of (quantum) padic tachyons (here some technical details and case p=2 are omitted) the following is necessary 5 1 for 5
52(lY”lp) =
1, or 12y” -
$1
5 1 for P
I &l p >
15
(s) I I5I P
1. Possible physical implication on
constraints for quantities related to the starting tachyon action (18) will be discussed elsewhere. The existence of R state opens the “door“ for further adelic generalization and investigation of higher-excited states. As in the real case8 a quadratic damping effect could enter explicitly into the play treating it as a perturbation over classical solution of the equation for a particle in constant external field without friction. Damping effect would be ascertained and understood through its dependence of ,B term.22 6. Conclusion
In this paper we show that quantum mechanical simplification of the tachyon field theory, besides the real case, is possible in a padic context. Also, an adelic generalization looks possible, i. e. without some obvious principal obstacles. Path integral formulation of zero-dimensional padic tachyons has been done and some ”minimal” conditions for their existence have been found. Of course, how much this approach could be useful for deeper understanding of the whole string theory and of its tachyon sector requires time and further, in-depth research. The fact, that the exact effective tachyon action in the usual string theory is not known, while in
206
padic string theory it is, is quite enough motivation for this and similar and investigation. We would propose a few promising lines for further investigation. The exact formula for quadratic quantum padic systems in two and more dimensions17 could be useful for multidimensional generalization of padic tachyons. It is tempting to extend our approach t o 1 1 dimensional field theories, even nonlinear field theories would be here quite nontrivial problem. Finally, padic string theory could be a very useful guide to difficult question in the usual string theory. It requires deeper understanding of padic string theory itself, especially of closed padic strings (strings on p adic valued worldsheet and target space as well). It is a worthwhile task to explore padic strings in nontrivial backgrounds. It will naturally lead to noncommutative formulation on padic quantum theory and examination of the corresponding Moyal-product, introduced in a context of the Noncommutative Adelic Quantum Mechanics.23 Recently, the Moyal-product has been applied in the calculation of the Noncommutative Solitons in padic string theory.24
+
Acknowledgments The research of both authors was partially supported by the Serbian Ministry of Science and Technology Project No 1643. A part of this work was completed during a stay of G. Djordjevic at LMU-Munich supported by DFG Project "Noncommutative space-time structure-Cooperation with Balkan Countries". Warm hospitality of Prof. J. Wess is kindly acknowledged.
References 1. I.V. Volovich, Theor. Math. Phys. 71 (1987) 337. 2. A. Sen, Tachyon Dynamics in Open String Theory, hepth/0410103. 3. D. Ghoshal and T. Kawano, Towards p-Adic String in Constant B-Field, hepth/0409311. 4. G. S. Djordjevic, B. Dragovich, Lj. D. Nesic and I. V. Volovich, Int. J.Mod.Phys. A17 (2002) 1413. 5. D. Ghoshal and A. Sen, N d P h y s . B584 (2000) 300. 6. A.A. Gerasimov and S.L. Shatashvili, JHEP 0010 (2000) 034. 7. A. Sen, Znt. J . Mod.Phys. A18 (2003) 4869. 8. S. Kar, A simple mechanical analog of the field theory of tachyon matter, [hep-th/0210108],
207
9. V. S. Vladimirov, I. V, Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore 1994. 10. E. I. Zelenov, J. Math. Phys. 3 2 (1991) 147. 11. D. Dimitrijevic, G. S. Djordjevic and LJ. Nesic, Fourier Transformation and Pseudodifferential Operator with Rational Part, Proceedings of the Fifth General Conference of the Balkan Physical Union BPU-5, Vrnjacka Banja, Serbia and Montenegro, August 25-29, (2003) 1231. 12. I. M. Gel’fand, M. I. Graev and I. I. Piatetskii-Shapiro, Representation Theory and Automorphic Functions (Saunders, London, 1966). 13. R. P. Feynman, Rev. Mod. Phys. 20 (1948) 367; R. P. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York, 1965. 14. G. S. Djordjevik and B. Dragovich, O n p-Adic Functional Integration, in Proc. of the I1 Math. Conf. in PriStina, PriStina (Yugoslavia) (1996) 221. 15. G. Djordjevik, B. Dragovich, Mod. Phys. Lett A12 (1997) 1455. 16. G.S. Djordjevic, B. Dragovich and Lj. Nesic, p-Adic Feynmads Path Itegrals, Int. Conference: FILOMAT 2001,Nis, 26-29 August 2001. FILOMAT 15 (2001) 323. 17. G. S. DJordjevic and LJ. Nesic, Path Integrals f o r Quadratic Lagrangians in Two and More Dimensions, Proceedings of the Fifth General Conference of the Balkan Physical Union BPU-5, Vrnjacka Banja, Serbia and Montenegro, August 25-29, (2003) 1207. 18. G. Djordjevic, B. Dragovich and Lj. Nesic, Infinite Dimensional Analysis, Quantum Probability and Related Topics 6 (2003) 179. 19. P. G. 0. Freund and E. Witten, Phys. Lett. B199 (198’7) 191. 20. A. Sen, JHEP 0204 (2002) 048. 21. L. Brekke, P.G.O. Freund, M. Olson and E. Witten, Nucl. Phys. B302 (1988) 365. 22. G.S. Djordjevic and Lj. NeSic, Quantum p-tachyons, work in progress. 23. G.S. Djordjevic, B. Dragovich and Lj. Nesic, Adelic Quantum Mechanics: Nonarchimedean and Noncommutative Aspects, Proceedings of the NATO ARW ”NONCOMMUTATIVE STRUCTURES IN MATHEMATICS AND PHYSICS”, Kiev, Ukraine, September 2000, Eds. S. Duplij and J. Wess, Kluwer. Publ. (2001) 401-415. 24. D. Ghoshal, JHEP 0409 (2004) 041.
SKEW-SYMMETRIC LAX POLYNOMIAL MATRICES AND INTEGRABLE RIGID BODY SYSTEMS
V. DRAGOVIC AND B. GAJIC Mathematical Institute SANU Kneza Mihaila 35, 11 000 Belgrade, Serbia and Montenegro E-mail: [email protected], [email protected] Skew-symmetric matrix Lax polynomials are considered. Few rigid body systems with such representations are presented. Lagrange bitop is completely integrable four-dimensional rigid body system. Algebro-geometric integration procedure, using Lax representation, for that system is performed. This integration is based on deep facts from the theory of Prym varieties such as the Mumford relation and Mumford-Dalalyan theory. Class of isoholomorphic integrable systems is established and importation class of its perturbations is observed, generalizing classical Hess-Appel’rot system.
1. Introduction
Lax representation is one of the most powerful tools in the modern theory of integrable systems. The Lax equation
,with L(X),A(X) being matrix polynomials in so called spectral parameter X were studied by different authors in the middle of seventies. Approach, based on the Baker-Akhiezer functions, is developed by Dubrovin (see Ref. 1). Different version of that theory, based on Ref. 2 is presented by Adler and van Moerbeke in Ref. 3. Both of the theories were applied to the rigid body motion in Refs. 4,5, 3, 6 and 7. Recently in Refs. 8 and 9 we have found Lax representation for the classical Hess-Appel’rot system of rigid body, and for a new completely integrable four-dimensional rigid body system that we called Lagrange bitop. It appears that in both cases matrices L and A are skew-symmetric. This was not the case in Manakov top.4 In the case of Lagrange bitop matrix 208
209
L(X) satisfies (in proper basis) Ll2 = L2l = L34 = L43 = 0,
(1)
which is excluded in both theories (Dubrovin’s and Adler-van Moerbeke’s). Analysis of the spectral curve and the Baker-Akhiezer function shows that dynamics of the system is related to certain Prym variety ll (which splits according to the Mumford-Dalalyan theory and evolution of divisors of some meromorphic differentials 0;. Then the condition (1) requires that some of these differentials have to be holomorphic during the whole evolution. Compatibility of this requirement with dynamics put a strong constraint on the spectral curve: its theta divisor should contain some torus. The conditions (1)and holomorphicity conditions create a new situation from the point of view of the existing integration techniques. Such systems we call isoholomorphic systems (see Ref. 9). Recently, in Ref. 13 we have constructed families of partially integrable rigid body systems with skew-symmetric Lax matrices. These families represent higher-dimensional generalizations of the Hess-Appel’rot system in any dimension. They are certain perturbations of n-dimensional Lagrange top and Lagrange bitop. 2. Lagrange bitop
We will consider motion of a heavy n-dimensional rigid body fixed at a point. Equations on semidirect product so(n) x sa(n) in moving frame are introduced in Ref. 6
Ail = [ M R I + [F’X],
= [F’RI 7
(2)
+
where the matrix I is diagonal, diag(l1,. . . ,In).Here Mij = (Ii Ij)Rij E so(n) is the kinetic momentum, R E so(n) is the angular velocity, x E so(n) is a given constant matrix (describing a generalized center of the mass), r E so(n). Then Ii Ij are the principal inertia momenta. The Lagrange bitop is system defined by (see Ref. 9):
+
0 13 = 1 4 = b and ’‘=I2=’
x12
.=(-:2
0
with the conditions a
# b,
0
0
0
x3 O4)
-x34
0
# 0 , l x l ~# l 1x341.
x12,x34
(3)
210
Proposition 2.1. T h e equations of m o t i o n ( 2 ) under the conditions (3) have a n L - A pair representation L(X) = [L(X),A(X)] ,(see Refs. 8,9) where
L(X) = X2C+ AM and C = ( u
+ r,
A(X)
= XX
+ R,
(4)
+b ) ~ .
One can observe that both leading terms in the operators L and A (matrices C and x) are skewsymmetric, while in Refs. 1 and 14 C is symmetric and M is skew-symmetric. Before analyzing spectral properties of the matrices L(X),we will change the coordinates in order to diagonalize the matrix C. In this new basis the matrices L(X) have the form L(X) = U-lL(X)V,
The spectral polynomial p(X, p ) = det
(E(X) - p . 1) has the form
where
P(X) = AX4 + BX3 + DX2+EX
+ F,
&(A)
= GX4
+ HA3 + I X 2 + J X + K .
21 1
Their coefficients
A = C t 2 + C i 4 = (C+,C+)+(C-.,C-), B=2C34M34+2C12M12 = 2 ( ( C + , M + ) + ( C - , M - ) ) ,
D
=~ =
3 + 22c12r12 ~ + 2c34r34
, 2+, M ? ~+ M ; ~+ M ? ~+ ~
(M+,M+)+(M-,M-)+2((c+,r+)+(c-,r-)),
+ + 2r14w4 + 2 r 2 3 ~+223r 2 4 ~+224r 3 4 ~ 3 4 = 2 (F+, M+) + F-, M-)), F= + ri3+ r:4 + r;3 + r;4+ ri4= (r+,r+)+ (r-,r-), E
= 2r12~1 2r13w3 2
G = c 1 2 c 3 4 = (c+, c-),
+
H = c34M12 + c12M34 = (c+, M - ) (c-,M + ) , I = c34r12r34c12M12M34 M23M14 - M13M24 = (C+,F-) + (C-,F+) + ( M + , M - ) , J = ~~~r~~ + ~~~r~~ + ~~~r~~~~~r~~ - r 1 3 ~-2r42 4 ~ 1 3
+
+
+
+
W+W + w-,r+), r34r12 + r23r14 - r13r24 = (r+,r-). integrals of motion of the system (a), (3). =
K are
=
M+, M- E R3 which correspond to
Mij E
We used two vectors so(4) according to
(7)
Here M$ are the j-th coordinates of the vector M+. The system (2), (3) is Hamiltonian with the Hamiltonian function
1 2
= -(M13013+M14fl14+M23023+M12012 +M34034) Sx12r12 +x34r34*
The algebra so(4) x so(4) is 12-dimensional. The general orbits of the coadjoint action are 8-dimensional. According to Ref. 6, the Casimir functions are coefficients of A', A, A4 in the polynomials [ d e t z ( A ) ] 1 / 2and - I2T T ( L ( A ) )and ~ they are J , K , E , F . Nontrivial integrals of motion are B , D, H, I , and they are in involution. When 1x121 = 1x341, then 2H = B or 2H = -B and there are only 3 independent integrals in involution. Thus, for 1x121 # 1x341, the system (2), (3) is completely integrable in the Liouville
212
Ratiu introduced in Ref. 6 two families of integrable Euler-Poisson equations: the generalized symmetric case, defined by the conditions
x arbitrary;
I1 = . . . = I,,,
and the generalized Lagrange case is defined by I1 =
IZ = a, I3 = ... = I,,
= b, xij = 0 if ( i 7 j )
6 ((17 2)7 (271)).
The system (2), (3) doesn’t fall in any of those families and together with them it makes the complete list of systems with the L operator of the form
+
L ( A ) = A2C AM
+ r.
More precisely, if x 1 2 # 0 then the Euler-Poisson equations (2) could be written in the form ( 4 ) (with arbitrary C ) if and only if the equations (2) describe the generalized symmetric case, the generalized Lagrange case or the Lagrange bitop, including the case X ~ = Z f ~ 3 4 . ~ ~ ’ Starting from the well-known decomposition so(4) = so(3) CB so(3), introducing
1 1 M2 = -(M+ - M - ) M I = -(M+ + M - ) 2 2 (and similar for R, r,x),where M+, M- are defined with (7), equations (2) become
a1+ rl x xl), = 2(hf2 x a2+r2x x 2 ) ,
A& = 2 p 1 x
r1= 2(r1 x nl), r2= 2(r2 x 02),
(8)
and
As it is shown in Ref. 9 after changing of variables equations (8) can be written in the form: 2i; = Pi(?&),
i = 1,2,
P ~ ( u=) -4u3 - 4uZBi+ 4uCi + Di,
i = 1,2;
where the constants Bi, Ci, Di, i = 1,2 depends of the first integrals and the parameters of the system. For more details see Ref. 9. From the previous relations, we have
213
So, the integration of the system (8) leads to the functions associated with the elliptic curves El, E2, given with:
Ei : y2 = P ~ ( u )i, = 1 , 2 .
(9)
3. Properties of spectral curve The L(X) matrix (4) for the Lagrange bitop (2), (3) is a quadratic polynomial in the spectral parameter X with matrix coefficients. There are two general theories describing the isospectral deformations with matrix coefficients. The first one, based on the Baker-Akhiezer function is developed by Dubr0vin.l The other one, given by Adler and van Moerbeke in Ref. 3 was based on Ref. 2. Both of these theories were applied to the rigid body motion in Refs. 4, 5 and 3, 6, 7, respectively. As it is shown in Ref. 9, non of these two theories can be directly applied in our case. So, we are going to make certain modifications, and then we will integrate the system (2), (3). As usual in the algebro-geometric integration, we consider the spectral curve
I? : det i ( A ) - p . 1) = 0.
(
By using ( 5 ) , ( 6 ) , we have
r : p4+p2(A;, +
+ 4p3p3*+ 4p4p4*)+[A12A34+ 2i(p3*p4- p3~4*>i2 = 0.
There is an involution (T : (A, p ) -+ (A, - p ) on the curve r, which corresponds to the skew symmetry of the matrix L(X). Denote the factor-curve by rl = r/o.
Lemma 3.1. (a) The curve rl is a smooth hyperelliptic curve of the genus g(r1) = 3. The spectral curve is a double covering of r l . The arithmetic genus of is ga(r)= 9. (b) The spectral curve r has f o u r ordinary double points Si, i = 1,. . . , 4 . The genus of its normalization is five. (c) The singular points Si of the curve are fixed points of the involution (T. The involution o exchanges the two branches of I' at Si . In general, whenever matrix L(X)is antisymmetric, the spectral curve is reducible in odd-dimensional case and singular in even-dimensional case.
214
Before starting the study of the analytic properties of the BakerAkhiezer function, let us give the formulae for (nonnormalized) eigenvectors of the matrix L(A). An eigen-vector of L , i.e. such that L(A)f = pf is given by
f1 =
+ p2)(iA34
- p ) - 2p(P3P3*
f 2 = W P 3 - iP4)(iP$
+ P4P4*) + 2 A 1 2 ( @ 3 f l i - P 4 P ; ) ,
+Pi),
+ '$4) [(iAi2- P ) ( i A 3 4 - P ) + 2i(P3Pi P;P4)] , f4 = (iP: + P i ) [(iAi2+ P ) ( i A 3 4 - P ) -t2i(P3Pi - P;P4)] . f 3 = (-P3
-
Corollary 3.1. The eigenvectors f' normalized by the condition
f: + f; + f; + fi = 1, f',
have different values in the points Sl,S,!l E points SiE I?.
which cover the singular
Following ideas of Dubrovin and Krichever (see Refs. 15, 16, 1, and bibliography therein) , we consider the next eigen-problem
where $k are the eigenvectors with the eigenvalue p k . Then $ k ( t , A) form 4 x 4 matrix with components $ i ( t ,A). Denote by cp: the corresponding inverse matrix. Let us introduce gj(t1 (A, P k ) )
=
$i(t,A)
'
$(t, A)
or, in other words g ( t ) = $ k ( t ) 18 p(t)k. Matrix g is of rank 1,and we have d$/at = --A$, dcp/dt = cpA, dgldt = [ g ,A].w e can consider vector-functions $ k ( t , A) = ($i(t,A), ...,$:( t , A ) ) ~ T
as one vector- function $(t,(A, p ) ) = ($'(t, (A, p ) ) , ...,7,b4(t,(A, p ) ) ) on the curve r defined with lCli(t,(A, p k ) ) = $ i ( t , A). The same we have for the matrix 9:. The relations for the divisors of zeroes and poles of the functions $2 i cpi in the afine part of the curve I? are:
(g;), = d j ( t )
+ dZ(t) - D,
-
D$,
(10)
where D, is the ramification divisor over X plane (see Ref. 1) and D , is the divisor of singular points, D: 5 D,. One can easily calculate deg D , = 16, degD, = 8.
215
Lemma 3.2.
(a) The matrix g has a representation
+
+
where a1 = L , a2 = P E L 2 ,a3 = P L L3. (b) For the Lax matrix L and for X i such that &(Xi) = 0 , it holds a3 = 0. (c) The matrix g has no poles at the singular points of the curve F. From now on we will consider all the functions in this section as functions on the normalization ? of the curve F. The matrix elements gj(t,(A, P k ) ) are meromorphic functions on the curve I?. We need their asymptotics in the neighborhoods of the points Pk, which cover the point X = 00. Let be the eigenvector of the matrix z ( X ) normalized in Pk by the condition = 1, and let be the inverse matrix for We will also use another decomposition of matrix elements of g : 9;.= dJ:p$ = $i+$.It is an immediate consequence of the proportionality of the vectors dJk and (and p k and g k ) . Using asymptotics of the functions i g$ in neighborhoods of the points Pk we getg that the divisors of matrix elements of g are
'& 4;
4:.
+:
I .
&
(9;) = 2
4;
+ J j - D, + 2 (PI + P2 + P3 + P4)
-
where the divisors c&, & are of the same degree deg & thet are given with:
+ P2, 2' = d1 + P2,
21 = d1
+ Pi, 2' = d2 + P i ,
2 2 = d2
Pa - Pj,
=
deg
&
=
5 and
+ P4, 2 4 = d4 + P3, J3 = d3 + P4, J4 = d4 + P3. 23
= d3
Let us denote with @ ( tA) , the fundamental solution of
(g+
A ( h ) ) q t , A)
= 0,
normalized with Q ( T ) = 1. Then, if we introduce functions
'$(t,T,(A,
Pk))
@ t ( hA)h"(T,(A, P k ) ) ,
= S
where h" are the eigenvectors of L(A) normalized by the condition h s ( t ,(A, p k ) ) = 1, it follows that
c,
216
Proposition 3.1. Ref. 9)
The functions
Gi satisfy the following properties (see Gi
(a) I n the a f i n e part o f f ' the function has 4 time-dependent zeroes which belong to the divisor d i ( t ) defined by formula (lo), and 8 time-independent poles, e.q.
(Gi(t,T , (A, p k ) ) ) a = d i ( t ) - 9, (b) At the points follows:
4 , the functions
di@,
7, (A,
4 2
degV = 8.
have essential sangularities as
P ) ) = exp [-(t -
G i ( t ,7, (A, P ) )
where Rk are given with
and di are holomorphic in a neighborhood of Pk,
Gi(~,7,(X,p)) = h i ( 7 , ( A , p ) ) , di(t,7,Pk) = S t + v : ( t ) z + 0 ( z 2 ) , with
Observeg that the following relation takes place on the Jacobian Jx(f'):
+
A(dj(t) odi(t))= A(dj(7)
+odj(7))
where A is the Abel map from the curve f' to Jac(f'). Thus, the vectors A ( d i ( t ) ) belong to some translation of the Prym variety Il = Pryrn(f'lr1). More details concerning the Prym varieties one can find in Refs. 10, 17 and 12. The natural question arises to compare two two-dimensional tori Il and El x Ez, where the elliptic curves Ei are defined in (9). 4. Geometry of the Prym variety II and Mumford's relation
Lemma 4.1.
The curves Ei are Jacobians of the curves
C1 :
v2 = P(A) + Q(A), 2
C2 : v2 =
Ci
-- &(A). 2
given by:'
217
Theorem 4.1. (For details see Ref. 9) (a) The Prymian II is isomorphic to the product of the curves Ei:
n = Jac(C1) x Jac(C2). The curve f' is the desingularization of rl x p CZ and C1 x p r l .
(b) (c) The canonical polarization divisor E of II satisfies
E = El x
0 2
+ 0 1 x Ez,
where 0i is the theta - divisor of Ei. Theorem 4.1 is based on Mumford-Dalalyan theory and it gives the connection between the curves E l , E2 and the Prym variety II. The Baker - Akhiezer function Q satisfies usual conditions of normalized 4-point function on the curve of genus g = 5 with the divisor 2, of degree d e g B = 8, see Refs. 16, 14. It was proven by Dubrovin in the case of general position, that 0; = g i j d X , i = 1, ..., 4 are a meromorphic differentials having poles at Pi and Pj, with residues vj and -v! respectively. But here we have that the four differentials O i l Oq, fli, 0: are holomorphic during the whole e v o l ~ t i o n . ~ We can say that the condition (1) implies isoholomorphicity. Let us recall the general formulae for v from14
where U = C X ( ~ ) U is( ~ certain ) linear combination of b periods U ( i )of the differentials of the second kind Og) , which have pole of order two at Pi; X i are nonzero scalars, and
(Here v is an arbitrary odd non-degenerate characteristics.) Thus, from
vz1 = 2112 = v43 = 2134 = 0 ,
(12)
using (11)we get that holomorphicity of some of the differentials 0; implies that the theta divisor of the spectral curve contains some tori (see ref. 9). In a case of spectral curve which is a double unramified covering
T : Fj r l ;
218
with g(r1) = g, g(p) = 2g - 1, as we have here, it is really satisfied that the theta divisor contains a torus, see Ref. 10. More precisely, following," let us denote by II- the set
IT-
= L E Pic2g-2?INmL = Krl, ho(L)is odd},
{
where Krl is the canonical class of the curve l?l and N m : Picf' -+ P i c r l is the norm map, see Refs. 10, 12 for details. For us, it is crucial that ITis a translate of the Prym variety II and that Mumford's relationlo holds
II- c QF. Complete formulae for Baker-Akhiezer function are given in Ref. 9. 5 . Conclusion
New family of rigid body systems with skew-symmetric Lax matrices have been constructed recently in Ref. 13. These systems are higher-dimensional generalizations of Hess-Appel'rot system constructed in any dimension. They are certain perturbations of higher-dimensional Lagrange tops. The four-dimensional Hess-Appel'rot system is perturbation of Lagrange bitop and belongs to the class of isoholomorphic systems. Thus, for the integration of the system we need to apply approach given it this paper.
Acknowledgement The research of both authors was partially supported by the Serbian Ministry of Science and Technology Project No 1643.
References B. A. Dubrovin, Funk. Analiz i ego prilozheniya 11,28 (1977 [in Russian]). P. van Moerbeke and D. Mumford, Acta Math. 143,93 (1979). M. Adler and P. van Moerbeke, Advances in Math. 38,318 (1980). S. V. Manakov, Funkc. Anal. Appl. 10, 93 (1976) [in Russian]. 0. I. Bogoyavlensky, Soviet Acad Izvestya 48,883 (1984) [in Russian]. T. Ratiu, American Journal of Math 104,409 (1982). T. Ratiu and P. van Moerbeke, 32,211 (1982). V. DragoviC, B. GajiC: An L-A pair for the Hess-Apel'rot system and a new integrable case for the Euler-Poisson equations on so(4) x so(4). Roy. SOC.of Edinburgh: Proc A 131,845 (2001). 9. V. DragoviC, B. GajiC, American Journal of Math., (2004), (to appear) 10. D. Mumford, A collection ofpapers dedicated to Lipman Bers (Acad. Press.) New York, 325 (1974). 1. 2. 3. 4. 5. 6. 7. 8.
219
11. S. G. Dalalyan, Uspekhi Math. Naukh 29, 165 (1974) [in Russian]. 12. V. V. Shokurov, Algebraic Geometry 111, 219 (Berlin: Springer-Verlag, 1998). 13. V. DragoviC, B. GajiC, ”Systems of Hess-Appel’rot type”, Preprint SISSA, (2004). 14. B. A. Dubrovin, Uspekhi Math. Nauk. 36,11 (1981) [in Russian]. 15. I. M. Krichever, Uspekhi Math. Naukh 32, 183 (1977). 16. B. A. Dubrovin, I. M. Krichever and S. P. Novikov, Dynamical systems l V , Berlin: Springer-Verlag, 173. 17. J. D. Fay, Lecture Notes in Mathematics vol. 352, Springer-Verlag, (1973).
SUPERSYMMETRIC QUANTUM FIELD THEORIES
D. R. GRIGORE Department of Theoretical Physics, Institute of Atomic Physics, Bucharest-Miigurele, Romcinia E-mail: [email protected] We consider some supersymmetric multiplets in a purely quantum framework. A crucial point is to ensure the positivity of the scalar product in the Hilbert space of the quantum system. For the vector multiplet we obtain some discrepancies with respect to the literature in the expression of the super-propagator and we prove that the model is consistent only for positive mass. The gauge structure is constructed purely deductive and leads to the necessity of introducing scalar ghost superfields, in analogy t o the usual gauge theories. Then we consider a supersymmetric extension of quantum gauge theory based on a vector multiplet containing supersymmetric partners of spin 3/2 for the vector fields. As an application we consider the supersymmetric electroweak theory. The resulting self-couplings of the gauge bosons agree with the standard model up to a divergence.
1. Introduction
The supersymmetric gauge theories are constructed using the so-called vector supersymmetric multiplet.’ The justification for this choice comes from the analysis of the unitary irreducible representations of the N = 1 supersymmetric extension of the Poincark group; there are two irreducible massive representations R l p N [m,01 @ [m,1/21 @ [m,1/21 @ [m,I], and 01 [m,1/2]@[m,11@[m, 11 @ [m,3/21, containing a spin 1 system; here [m,s] is the irreducible representation of mass m and spin s of the Poincark group.) The standard vector multiplet is constructed such that the oneparticle subspace of the Fock space carries the “simplest” representation
-
%/2.
We use entirely the quantum f r a m e ~ o r k ~avoiding -~ the usual approach based on quantizing a classical supersymmetric theory. In this way we do not have the complications associated to the proper mathematical definition of a super-manifold and we do not need a quantization procedure. A rigorous treatment of the perturbative aspects of these models can be pro220
221
vided using the Epstein-Glaser framework. We start this program analyzing the layout of the model, that is the construction of the quantum multiplet, its gauge structure and the expression of the interaction Lagrangian (or,in the language of perturbation theory, the first order chronological product). The main results are the following. The vector model is consistent only for positive mass. The origin of this result comes from the condition of positivity of the scalar product in the Hilbert space. In the standard literature one constructs the Hilbert space by applying polynomials in the free fields of the model on the vacuum. (In the language of the reconstruction theorem from axiomatic field theory this amounts to the construction of the Borchers algebra). However, this is not enough: one needs to provide the expression of the scalar product and prove that it is positively defined. The comparison with the literature dedicated to this subject shows that this point has not been analyzed. The structure of the Hilbert space of the model is never described explicitly. One can infer it only from the Feynman rules, more precisely the expressions of the (super)propagators. The point is that the assumption that the BRST quantization procedures gives automatically a positively defined Hilbert space structure is wrong. An explicit check should be made and this leads to some conditions on the mass of the supersymmetric multiplet. (For another point of view concerning supersymmetric positivity see Ref. 5). We can determine the gauge structure of this model in a deductive way: it coincides essentially with the expression from the literature but, because the mass of the multiplet is positive, we need to introduce some scalar ghost superfields. We are also able to determine the general expressions for the Feynman super-propagators in a purely deductive way; some discrepancy with the standard literature appears. Similar results are available for extended supersymmetries.6 Next we construct a new vector multiplet based on the representation R1. This multiplet exists for all values of the mass, its gauge structure is very similar to the gauge structure of the usual gauge theories and the interaction Lagrangian can be obtained by simply replacing fields by superfields in the Lagrangian of the standard model. In this way we obtain a supersymmetric extension of the standard model. One can conclude that the new vector multiplet proposed for the first time in Ref. 4 and based only on chiral superfields is a more natural object and it remains as a serious candidate for a possible supersymmetric extension of the standard model.
222
2. Quantum Supersymmetric Theory We remind here the definition of a N = 1 supersymmetric theory in a pure quantum context. Suppose that we have a quantum theory of free fields; this means that we have the following construction: (a) H is a Hilbert space of Fock type (associated to some one-particle Hilbert space describing some choice of elementary particles) with the scalar product (., .); (b) R E H is a special vector called the vacuum; (c) U a ,is~a unitary irreducible representation of inSL(2,C) the universal covering group of the proper orthochronous Poincark group such that a E R4 is translation in the Minkowski space and A E SL(2,C); (d) b j , j = 1,.. . ,NB (resp. f ~ ,A = 1,.. . , N F ) are the quantum free fields of integer (resp. halfinteger) spin. We assume that the fields are linearly independent up to equations of motion and that the equations of motion do not connect distinct fields. In practice, one considers only particles of spin s 5 2. For the standard vector model we consider only the case s 5 1. For the new vector model we consider the more unusual case 1 5 s 5 3/2. If one considers higher-spin fields (more precisely s 2 l), as we have done in Refs. 4, 7, it is necessary to extend somewhat this framework: gauge fields must be considered and we will describe them in the indefinite metric approach (Gupta-Bleuler). That is, we assume the existence of the following objects: (a) A gauge charge operator Q verifying Q 2 = 0 ; (b) A non-degenerate sesqui-linear form < . > which becomes positively defined when restricted to a factor Hilbert space K e r ( Q ) / l m ( Q ) Q ; is called gauge charge and we denote by At the adjoint of the operator A with respect to < ., . >; (c) Hphys = K e r ( Q ) / l m ( Q )will be the physical space of our problem; (d) The interaction Lagrangian t ( x ) is some Wick polynomial acting in the total Hilbert space H and verifying the conditions a,
[Q,t(x)l=ia#’(x)
(1)
for some Wick polynomials t”(x). This condition guarantees that the interaction Lagrangian t ( x ) factorises to the physical Hilbert space K e r ( Q ) / l m ( Q )in the adiabatic limit, i.e. after integration over x; the condition (1) is equivalent to the usual condition of (free) current conservation. The condition (1) has far reaching physical consequences: under some reasonable additional assumptions one can prove that the usual expression of the interaction Lagrangian for a Yang-Mills model is unique, up to trivial terms. It is desirable to generalize this scheme to supersymmetric theories. We note here that our framework is different from the usual treatment of quantum gauge invariance based on the construction of a classical
223
field theory with BRST invariance, some quantization procedure and the quantum action principle imposed on the Green functions. It seems that for ordinary gauge theories the two formalisms give identical results. But this is not the case for the supersymmetric gauge theories. This makes our alternative approach far from being superfluous. Now we define the notion of supersymmetry invariance of the system of Bosonic and Fermionic fields considered above. Suppose that in the Hilbert space H we also have the operators Q a , a = 1 , 2 such that: (i) the following relations are verified:
[&a, P p ]
UiIQaUA = AabQb ,
= 0,
(4)
here Pp are the infinitesimal generators of the translation group and Q b (&b)t; (ii) The following commutation relations are true: i [ Q a , b ] =p(a)f >
{&a,
f}= q(a)b,
=
(5)
where b (resp. f ) is the collection of all integer (resp. half-integer) spin fields and p , q are matrix-valued polynomials in the partial derivatives 8, (with constant coefficients). These relations express the tensor properties of the fields with respect to (infinitesimal) supersymmetry transformations. If these conditions are true we say that Qa are super-charges and b, f are forming a supersymmetric multiplet. As emphasized in Ref. 4, the matrix-valued operators p and q are subject to various constraints: ( 1 ) From the compatibility of (5) with Lorentz transformations it follows that these polynomials are Lorentz covariant. (2) Next, we start from the fact that the Hilbert space of the model is generated by vectors of the type 9 = b(zp) f(x,) R E H . It follows that the relations (3) are true if the left hand sides commute with all the fields of the model. Using the (graded) Jacobi identities we obtain:
n
n
[&a, [ Q b , W ] ]
[&a,
[QG,
w ] l + [&a,
[&a,
= -(a
~ 1 =1 -2i
++
b) ,
apw;
7
(6)
here w = b , f and [.,.I is the graded commutator. (3) The equation of motion are supersymmetric invariant. (4) The (anti)commutation relations have the implication that one and the same vector from the Hilbert space H can be expressed in distinct ways. This means that the supercharges are
224
well defined via (3) iflsome new consistency relations are valid following again from graded Jacobi identities; the non-trivial ones are of the form:
[b(z),{ f ( ~ 7 &all ) = -{f(y)
[&a, b(z)l}
.
(7)
(5) If a gauge supercharge Q is present in the model, then it is usually determined by relations of the type (5) involving ghost fields also, so it means that we must impose consistency relations of the same type as above. Moreover, it is desirable to have
{Q,Qa) = 0 , {Q,Q E } =
(8)
0;
this implies that the supersymmetric charges Qa and Qa factorizes to the physical Hilbert space Hphys = K e r ( Q ) / l m ( Q ) . These new consistency relations are of the type ( 6 ) with one of the supercharges replaced by the gauge charge: {&a,
[Q,bl}
= - {Q,[&a, bl>
>
[&a,
{Q,f}] = - [Q,{ & a , f}]
*
(9)
( 6 ) A relation of the type (7) must be also valid for the gauge charge:
[b(z),{ f ( ~Q)1 ) , = -{f(~),[Q,b ( z ) l ) .
(10)
(7) To have Q2 = 0 we must also impose
{ Q , [Q,bl)
=0
[Q,{ Q ,f)] = 0 .
(11)
(8) In the presence of a gauge structure one can relax the condition (3). Indeed, every operator A which (anti)commutes with the gauge charge Q factorizes t o an operator 7r(A) : K e r ( Q ) / l m ( Q ) -+ K e r ( Q ) / l m ( Q ) . In particular, the supercharges anticommute with Q by construction and Pp also commute with Q. So, we can impose Eq. (3) only on the physical Hilbert space7 i.e. we replace Qa -+ 7r(Q,), QE -+ 7 r ( & ) , Pp -+ r(P,>. All these conditions are of pure quantum nature i.e. they can be understood only for a pure quantum model. Some of them do not have a classical analogue so we can interpret the obstacles in constructing supersymmetric quantum models (associated to some classical supersymmetric theories) as some quantum anomalies. It seems to be an essential point to describe supersymmetric theories in superspace. We do this in the following way. We consider the space HG = G 8 H where G is a Grassmann algebra generated by Weyl anticommuting spinors 8, and their complex conjugates & = (8,). and perform a Klein transform such that the Grassmann parameters 0, are anticommuting with all Fermionic fields, the supercharges and the gauge charge.
225
The field operators acting in HG are called superfields. Quite generally one associates to every Wick polynomial w (x) its supersymmetric extension W = S(W) according t o the formula:
W ( x ,8,e) E exp ( i P Q , - i6'Q8)
W(Z)
exp (-ida&,
+ ie"Qa) ,
(12)
and we interpret the exponential as a (finite) Taylor series. Of special interest are the superfields constructed as in Refs. 2, 8 according to the preceding formulz taking w = b, f . It is a remarkable fact that only such type of superfields are really necessary, so in the following, when referring to superfields we mean expressions given by Eq. (12). We will call them super-Bose and respectively super-Fermi fields. For convenience we will denote frequently the ensemble of Minkowski and Grassmann variables by = @,8,s). Moreover, one postulates that the interaction Lagrangian t should be of the form
x
(13) for some supersymmetric Wick polynomial T . This hypothesis makes possible the generalization of the Epstein-Glaser approach to the supersymmetric case as it is showed in Ref. 4. Concerning the gauge invariance of the model there are two possible attitudes. One is to impose only Eq. (1); this "minimal" possibility is certainly consistent from the physical point of view but in general one loses the unicity results concerning the interaction Lagrangian. One can hope to keep this unicity result if one finds out a supersymmetric generalization of Eq. (1). A natural candidate would be the relation:
[Q,T ( X ) ]= ia,T.(X)
+ .. . ,
(14)
where by . . . we mean total divergence expressions in the Grassmann variables. It is clear that Eq. (14) implies Eq. (1) but not conversely. We call Eq. (14) the condition of supersymmetric gauge invariance because it involves only superfields in contrast to Eq. (1). In Ref. 4 we have showed that the stronger condition (14) can be imposed for the 01-vector model and indeed the unicity argument concerning the interaction Lagrangian holds. However for the Ol,z vector model the situation is not so good. If one uses only the LLminimal" gauge invariance condition (1) then one loses the unicity of the interaction Lagrangian. (However some other limitations might come from the condition of gauge invariance in higher orders of perturbation theory.) If one tries to impose the supersymmetric version
226
(14) one finds out that the usual expressions for the interaction Lagrangian suggested by the literature do not fulfill it. Of course, it is in principle possible to find alternative expression for the interaction Lagrangian such that Eq. (14) is true, but this possibility seems to be rather unprobable. So our results concerning the construction of the interaction Lagrangian for the R 1 p vector model must be considered as a criticism of the traditional approaches. The construction of the ghost and anti-ghost multiplets (which are needed in the construction of supersymmetric gauge theories) is done by inverting the spin-statistics assignment: the integer (resp. half-integer) spin fields have Fermi-Dirac (resp. Bose-Einstein) statistics.
3. The Vector Multiplet By definition, the vector multiplet has the following content: the Bosonic fields are some (real) scalars b ( j ) , j = 1,.. . ,s and a real vector field b,; the Fermionic fields are some Majorana fields of spin one-half f(A), A = 1 , . . . ,f . Let us consider that C is one of the scalar fields b(j) (or a linear combination of them). Now we define the following superfield: V E s(C) using formula (12); it is clear that one has the reality condition V t = V. Moreover the generic expression of V must be V(X,
e,B) = C ( X ) + ex(.) + ~z(x)+ e2 4 ( ~+)e-2 4t (x) +(e&7) .,(x) + e2 E ~ ( x+) e2 ex(^) + e2g2 d ( ~ ) ,
(15)
where the fields appearing in the expansion must linear combinations of the basic fields taking into account Lorentz properties and statistics. The action of the supercharges on the components of the multiplet is given in a compact way if we introduce the covariant derivative operators
acting on superfields. Then one imposes
i [ ~ V ( X~, 8,,
e)]= ~ , v 8,( e)~, ,
i[Q,,v
( e,8)] ~ , = D , V ( ~ ,e , e )(17)
and determines the action of the supercharges on the component fields by expanding both sided in the Grassmann variables. The explicit formulae are given in Ref. 7 and are compatible with the Jacoby identities (6); they also coincide with the formulae from the l i t e r a t ~ r e ~ after ~ ' ~some field redefinitions.
227
If C verifies the Klein-Gordon equation (for mass m ) , then the superfield V verifies the Klein-Gordon equation so all the components of the multiplet are verifying Klein-Gordon equation of mass rn. These equations are compatible with the supersymmetry action, i.e., they are left invariant by the supercharges Qa and Q f i . The multiplet (C,4l ZI,, d, A, x a )is irreducible; in particular it follows that the indices j and A take four values, C and d are real scalar fields of mass m, b, is a real vector field of mass m, 4 is a complex scalar field of mass m and xa and A, are Dirac fields of mass m. We now determine the supercommutator of the vector field. Let us consider the causal commutator
[ V ( X l ) V(X2)I , = -iD(X1; X2) 1 .
(18)
The expression D ( X 1 ; X 2 ) is a distribution in the variables xj and a polynomial in the Grassmann variables B j , j = 1 , 2 and verifies the following properties: (a) it is Poincarh covariant; in particular it depends only on the difference 2 1 - 2 2 ; (b) it has causal support; (c) verifies Klein-Gordon equation; (d) verifies the Hermiticity condition: D(X1;X2) = -D(X2; X I ) ; (e) verifies the antisymmetry condition: D(X2;X I ) = -D(X1; X2); and (f) verifies the consistency condition:
+
(0: D2)D(X1;X2) = 0 .
(19)
All properties except (f) are immediate. The property (f) follows from the Jacobi identity [&a,
[V(X1)1V(X2)11+ XI), [V(X2)7&all
+ [V(X2),
[&a,
V(X1)ll = 0
(20) and formula (17). It is easy to prove that the generic solution of the conditions (a)-(f) of the preceding proposition is a linear combination of four expressions. The condition of positivity gives an important result, namely the vector multiplet exists only for m > 0 and also some constraints on the coefficients of the development. If we compare our results with the usual quantization procedure by means of path integrals we can find out the corresponding values of these coefficients and it turns out that the constraints are not verified. One can argue that some of the fields are gauged away in the Wess-Zumino gauge; however, Wess-Zumino gauge can be proved to be in conflict with the supersymmetric transformation rules. Now we mention the (quantum) gauge structure of the vector model. In ordinary quantum gauge theory, one gauges away the unphysical degrees of freedom of a vector field v, using ghost fields. Suppose that the vector field is of positive mass
228
m; then one enlarges the Hilbert space with three ghost fields u,ii, 4 such that: (a) all three are scalar fields; (b) the fields u,ii are Fermionic and 4 is Bosonic. (c) all them have the same mass m as the vector field. (d) the ut = u, ut - = --u;and (e) the Hermiticity properties are; q5t = 4, commutation relations are:
[dW,d(Y)l = -2
{u(x),qY)} = -i Om(a:- Y) 7 (21)
- Y) 7
and the rest of the (anti)commutators are zero. Then one introduces the gauge charge Q according to:
QR=O,
[Q,up] = i a p u ,
Qt=Q,
[Q,41 = i m u ,
{Q,u}= 0 , {Q,ii} = -2 (dpup + m 4). (22) It can be proved that this gauge charge is well defined by these relations i.e. it is compatible with the (anti)commutation relations. Moreover, one has Q2 = 0 so the factor space K e r ( Q ) / l m ( Q )makes sense; it can be proved that this is the physical space of an ensemble of identical particles of spin 1. For details see Refs. 11, 12. One can generalize this structure in the supersymmetric context if one introduces ghost superfields. One obtains essentially the expressions from the literature;" the main difference is due to the fact that we have m > 0. One can compare this gauge structure to the classical gauge structure;' there is no straightforward correspondence, but at least for ordinary Yang-Mills theories, one finds out that the analogy is rather miraculous. It is natural to try the same idea in a supersymmetric context. However, we have found a negative result (the details are in Ref. 7). There are other multiplets associated to the vector multiplet as the linear multiplet and the rotor multiplet. The algebraic structure of these multiplets is also clarified in Ref. 7, but there are obstacles in the construction of a consistent supersymmetric extension of the standard model also in these cases.
4. The New Vector Multiplet We will consider here a new vector multiplet which has the nice property that the corresponding gauge structure is similar to the usual gauge theories. If this model is consistent with the phenomenology it brings new physics. The new vector multiplet is obtained from the Wess-Zumino multiplet if we transform the scalar (resp. the spinor) field in a vector (resp. a Rarita-Schwinger) field wp,Gpa by simply adding the index p ; this multiplet is subject to the following consistency conditions: (a) Hermiticity
229
= Gpii; (b) all fields verify Klein-Gordon (resp. Dirac) equation of mass M . (c) the action of the supercharges can be obtained from the corresponding formulz for the Wess-Zumino multiplet:
[Qalupl = 0 ,
i [Qa,ut1 = 2
+pa
= CJl6avup ;
(23) and (d) (anti)commutation relations can be obtained starting from the commutation relation for the vector field u, and using the consistency relation (10). We call this new multiplet the RS vector multiplet. The associated superfield can be easily constructed using the supersymmetric extension formula. The gauge structure of this multiplet can be obtained if we replace fields by superfields v, + v,, q5 + @, u U, G + 0 where Ul 0 are ghost and anti-ghost multiplets; all these multiplets are of the same positive mass M . One can obtain the action of the supercharge on the field components if one substitutes in the preceding relations the expressions of the superfields in terms of component the fields and the Grassmann variables. It also can be showed that, as for the usual gauge case, the factor space K e r ( Q ) / l m ( Q )describes a system of identical I;21 super-symmetric systems. So, the analogy with the usual gauge case is really remarkable. Moreover, it is quite easy to obtain consistent gauge invariant couplings if we substitute fields by superfields in the known formulz.12 One can prove4 that in this way a consistent supersymmetric extension of the standard model can be obtained. {&a, $pb}
=
EabMup
7
{Qal
$p6}
-
References 1. 2. 3. 4. 5. 6.
7. 8. 9. 10.
11.
S. Ferrara, 0. Piguet, Nucl. Phys. B93,261 (1975). F. Constantinescu, M. Gut, G. Scharf, Ann. Phys. 11,335 (2002). D. R. Grigore, European Phys. Journ. C21,732 (2001). D. R. Grigore, G. Scharf, Annalen der Physik 12,5 (2003). F. Constantinescu, Lett. Math. Phys. 62,111 (2002). D. R. Grigore, G. Scharf, Quantum Extended Supersymmetries, h e p th/0303176, to appear in Annalen der Physik (2003). D. R. Grigore, G. Scharf, Annalen der Physik 12,643 (2003). F. Constantinescu, G. Scharf, Causal Approach to Supersymmetry: Chiral Superfields, hep-th/0106090. P. P. Srivastava, Supersymmetry, Superfields and Supergmuity: an Introduction, IOP Publ. (1986). S. J. Gates Jr., M. T. Grisaru, M. RoEek, W. Siegel, Superspace or One Thousand and One Lessons in Supersymmetry, Cummings (1983), hepth/0108200. D. R. Grigore, Journ. Phys. A33, 8443 (2000).
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12. G. Scharf, Quantum Gauge Theories: A Tme Ghost Story, Wiley (2001).
PARASTATISTICS ALGEBRAS AND COMBINATORICS
T. POPOV Institute f o r Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, bld. Tsarigradsko chauske 72, 1784 Sofia, Bulgaria E-mail: [email protected]. bg We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise t o generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.
1. Introduction Wigner' was the first to remark, back in 1950, that the quantum mechanical equations of motion allow for a commutation relations more general than the canonical commutation relations
A few years latter (in 1953) Green wrote the relations' now known under the name parastatistics commutation relations
Let us define the parafermi algebra p$ and the parabose algebra p B to be the associative algebra (over C ) generated by the creation at and annihilation ai operators with relations (l),(2) and (3) with the upper and lower sign, respectively. These are not all parastatistic relations, one obtains new ones by hermitean conjugation. We shall only consider quantum systems with finite number D degrees of freedom, i, j, k = 1 , . . . ,D. 23 I
232
The parastatistics commutation relations are generalization of the cannonical commutation relations
Indeed, the quadratic relations (4) imply the trilinear parastatistics relations ( l ) , (2) and (3). The representations of the parafermionic and p% parabosonic algebras relevant in physics are labelled by the non-negative integer number p , called the order of parastatistics. A creation operator u! acting on the vacuum state creates one particle state, and the annihilation operator ai annihilates it. One has
ps
uiu; 10) = p6ij 10)
.
(5)
The representation p = 0 is trivial (ui = ul = 0) and corresponds to the vacuum representation. The case p = 1 yields particular representation of (p%) in which the relations (4) hold and one retrieves the ordinary fermionic (bosonic) Fock space. For p 2 2 fermionic generalized Fock spaces are characterized by the property that one can accomodate up to p identical particles in a state. This property is a manifestation of a generalized exclusion principle and the Greens parastatistics2 is the first consistent example of generalized statistics. We denote by (p%+) the associative algebra, subalgebra of ps(p%) generated by the creation operators ui subject of the relations (3). Every generalized parafermionic (parabosonic) Fock space labelled by integer p arises as a quotient of the the creation algebra ( p S + ) , i e . , by imposing further pdependent conditions. Remark The parafermionic parabosonic p% algebra with D degrees of freedom has been identified with the universal enveloping algebra U(so(2D 1)),3 U ( o ~ p ( 1 1 2 D ) )respectively. ~ However, we shall not make use of these identifications in what follows.
ps
ps+
ps+
ps,
+
2. Homogeneous Algebras
A homogeneous algebra of degree N or N-homogeneous algebra is an algebra of the form5
A = A ( E ,R ) = T ( E ) / ( R ) ,
(6)
where E is a finite-dimensional vector space over C, T ( E ) is the tensor algebra of E and ( R )is the two-sided ideal of T ( E )generated by a vector
233
subspace R of E @ N .The homogeneity of ( R ) implies that A is a graded algebra A = e n E ~ A with n A, = Em'" for n < N and
A,=E@'"/
E @ r @ R @ E @for s n>N,
(7)
r+s=n-N
where we have set E@'O= C. Thus A is a graded algebra A = anE~.Rn which is connected ( A= C), generated in degree 1 and such that the A, are finite-dimensional vector spaces. When N = 2 or N = 3 we shall speak of quadratic or cubic algebras respectively. We shall refer to the subspace R as relation space. We set E = @:,Cul N C D . When the parastatistical order p = 1 then the symmetrical S ( E ) = en>oSnE and the exterior A ( E )= @,>o - An E algebras provide the basises of the bosonic and fermionic Fock space respectively. The grading is the number of particles in a state. The creation algebras pS+ and p%+ are cubic algebras. The creation parafermionic algebra pS+ is the cubic algebra6
PS+ = A ( E ,R ) = T ( E ) / ( " x Y, l @ l .I@) Yl t E E , (8) where [x,y]@= x @ y - y @ x and the subspace R is associated to the 7
2 1
relations (3). The creation parabosonic algebra p%+ is the cubic algebra6
P%+
.I@)
= A(E1 R) = T ( E ) / ( [ { x , Y ) @ ,
7
x1 Y l z E E
1
(9)
where the R means that instead of commutators one has t o take supercommutators, ie., the relation space R is the super-counterpart of R. The parabosons are super-parafermions. Note that in this approach the parafermions are even and the parabosons are odd. Choosing a basis in E l which means fixing the isomorphism E N C D ,we identify the linear group G L ( E )with GL(D,C). One has a natural action of the linear group GL(D,C) (denoted GL(D) from now on) on the linear space E N C D given by left matrix multiplication of the column vector ( G z ) ~= Gixj,
2 = X ~ U ;E
E,
G E GL(D) .
The natural action of GL(D) can be continued t o every tensor power
E@'"as G(x~@ ~2 @ . . . x,) = G z @ ~ G x 8.. ~ . Gx,
.
The natural action of the linear group G L ( D ) on E@' preserves the subspaces R and R, ie., they are invariant submodules. It follows that GL(D)-action passes to the quotients pS+ and p%+ respecting the grading.
234
ps+
This implies that G L ( D )acts on and p%+ by automorphisms, i.e., the algebras p$+ and p%+ are modules of the linear group GL(D). 3. The Symmetric and the Linear Group We briefly recall some general facts about the intimately related representation theories of the symmetric and the linear group. Let X = (X1,Xz ,... Xk) with A1 2 Xz 2 .. . 2 X k > 0 be partition of k
n,
C X i = n.
A partition X of n will be identified with a Young diagram
i=l
X with n boxes and k rows, where X i is the length of the i-th row. The number of boxes in X is denoted by 1x1. The filling of a Young diagram with numbers non-decreasing on rows (from left t o right) and strictly increasing on columns (from top t o bottom) is called a Young tableau. A standard Young tableau is a Young tableau strictly increasing in rows, i.e., without repetition in the entries. The group ring C[Sn]is endowed with structure of a right (left) S -, module in a natural way. The regular representation is the representation of S, into C[S,]. The dimension fx of an irreducible representation Sx of the symmetric group S, (with X partition of n) is equal t o the number of the standard tableaux with shape A. It is well known fact from the theory of finite groups that the regular representation contains all irreducible representations and each representation enters with multiplicity equal t o its dimension. Therefore the left S,-module C[Sn]has the decomposition into a direct sum of modules
C[Sn]N
@ (SX)@fX,
fx
= dim Sx ,
(10)
IXI=n
where Sx are the irreducible S,-modules. The Young projectors Yx(T) are idempotent elements of the group algebra C[Sn]of the symmetric group S,. They are labelled by a partition X and a standard tableau T with shape A. The projectors Yx(T)form an orthogonal system
Yx(T)Yx,(T’) = SAA/STT’YX(T).
(11)
When acting from the right on C[Sn]the Young projectors give rise t o the left irreducible representations of the symmetric group S,
S’ = c[s~IY~(T), 1x1 = n .
(12)
235
The Sn-modules SX arising through different Young projectors Yx(T)with one and the same X axe isomorphic. One can endow the tensor degree EBn with a right action of Sn which permutes the slots in E B n . The right &-action commute with the natural left GL(D)-action on E@l"(Schur-Weylduality). Hence the projection E B n Y ~ ( Tis) stable under the natural left GL(D)-action. The physical meaning of the right &-action is a place permutation of the creation operators a+ while the left So-action (which is a subgroup of GL(D)) operates as particle permutations. Schur module is the irreducible representation E X given by
E X21 E@1"Yx(T),
1x1 = 72.
(13)
The Schur modules arising through Young projectors with one and the same X are isomorphic. So the finite-dimensional irreducible representations E X of the linear group GL(D) are parametrized by Young diagrams too. The character & ( E x ) of the representation E X is the Schur polynomial associated to a partition X
and therefore the decomposition of the GL(D)-module EBn
EBn = @ (EX)@fA IXI=n
follows from the well known formula
4. The Algebras p B + and p 5 + as GL(D)-modules
The standard tableaux for X = ( 2 , l ) are two, thus for n = 3 one has @ E(3) = r\3E @ E(2,1)@ E(2,1)@ S 3 E . E@3 E(l,l,l)@ E(2y1)@ E(2~1)
The Jacobi and super-Jacobi identities
b, [Y,.IF1 + [Y,[z, 471 + b, b,YlFl = 0 imply that the GL(D)-modules R and R (corresponding to p$+ and p%+ ) are neither symmetric nor antisymmetric. It follows that R and R are two orthogonal GL(D)-modules associated to the Young diagram ( 2 , l )
RN
N
I?.
(16)
236
Let T be the maximal element in S3, 7 = slszsl = S Z S ~ S Z . One can choose the Young projectors qz,l)(T1)= Y + and Y&)(T2) = Y - to be the eigenvalues of the place permutation action of T , the so called flip (flip(u@bbc) = ( a @ b @ c ) T = c @ b @ u )
f l i p ( y * ) = y * T = &Y* ,
(17)
which determines Y + and Y - uniquely as S('tl)-modules. The corresponding Schur modules isomorphic to E('?l) are
R = EB3Y+,
R = EB3Y-.
(18)
This simple observation will be very useful later.
Theorem 4.1. Each irreducible representations E x of GL(D) appears exactly once in the decomposition of creation parafermionic pg+ and parabosonic p?B+ algebras
pg+ N @EX N p?B+.
x In the classical textbook on paras tat is tic^^ this result has a tedious proof. Chaturvedi' seems to be the first who used combinatorial identities in parastatistics (in this context). Here we give a short proof to the theorem which boils down to an exercise in the Fulton's Proof: Let us endow the space E @ A'E with the bracket defined by [z, y] = z A y if z and y are both in E and [z,y] = 0 otherwise. The so defined bracket is a Lie bracket and E @ A'E is a graded Lie algebra for this bracket if one ascribes the degree 1to the elements of E and the degree 2 to the elements of A'E. By definition pg+ is the universal enveloping algebra of the graded Lie algebra E @ A'E
pg+ = U ( E CB 2 ~ ) . In view of the PoincarbBirkoff-Witt theorem U ( E@ A'E) is isomorphic as graded vector space and as a graded coalgebra to the symmetric algebra
S ( E @ A'E) = S ( E ) @ S(A'E).
(19)
The character of the symmetric algebra S ( E )over the space E (of degree 1
n (l--zi)-l. It follows from (19) that the character D
elements) is ch S ( E ) =
i=l
of pg+ is given by the left hand side of the Schur combinatorial identity
237
while the right hand side is a sum (over all Young diagrams A) of the characters of Schur modules which implies the decomposition of In view of isomorphisms (16) and p%+ must be isomorphic as GL(D)modules which ends the proof. Remark. The direct proof" of the decomposition of p%+ uses the PoincarB-Birkhoff-Witt theorem for super-Lie algebras. One of the definitions of Schur polynomial is
ps+.
ps+
sx(z) =
czT,
where
zT = z1
... zg
,
(21)
T
where ai(T)is the number of times the entry i appears in the Young tableau T and the sum is over all Young tableaux T which are fillings of the diagram X with numbers from the set (1,. . . , D}. Therefore the Young tableaux T with shape X are in 1-1 correspondence with the monomials in the Schur polynomial s~(z),hence with the basis of the representation EX.Due to the homogeneity of the Schur polynomials sx(t,.
. . ,t ) = tlX1sX(l,. . . , 1) = tl'l dim E X ,
one obtains the PoincarQ series of the algebra p$+ (and also of p%+)
5 . The Plactic Algebra
We now introduce another cubic algebra coming from the combinatorics. The set of Young tableaux can be endowed with a structure of associative monoid, the so called plactic monoid" (see also Ref. 9). Every Young tableau T can be encoded with a word written with the entries of T and one can define algebraic operations on these words, i.e., on the set of Young tableaux. The plactic monoid is generated by ordered set of elements { e l , . . . ,e g } (for tableaux with entries in (1,. . . ,D})subjects of the Knuth relations
eizei3ei, = ei,eil ei3 if
il
< i 2 5 i3
= ei3eilei2 if
il
5 i2 < i 3
eilei3ei,
I
(23)
Let us choose the basis ( e i ) of the linear space C D 21 E to be the canonical basis of C D . The set { e l , . . . , e g } is ordered with the natural ordering
238 el < e:! < ..- < e D . To the Knuth relations we associate in an obvious way the subspace Rv c EB3 which generates the ideal (Rv).The plactic algebra p is the cubic algebra6
'$ = A ( C D ,Rv).
(24)
In contrast to R and R, the relation space Rv depends on the basis ( e k ) and on the ordering of ( e k ) . There is no natural action of GL(D) on $ I because GL(D)-action spoils the ordering. We have seen that the Schur module E X has a linear basis labelled by Young tableaux which are fillings of the diagram X with entries out of the set { 1,2,. . . ,dim E } . On the other hand the homogeneous independent elements of the plactic monoid (and so the plactic algebra) are identified with Young tableaux. We conclude that the Poincarh series of the plactic algebra and the creation parafermionic (parabosonic) p5+ ( ~ 2 3 ' ) algebra coincide
P&)
= Ppg+( t )= PPB+( t ).
(25)
This is not just a coincidence, it turns out that the plactic algebra can be obtained by means of the deformation of the algebras pS+ and pB+. 6. Hecke Algebra and the Quantum Linear Group The Hecke algebra '&(q) is the algebra generated from bi, i = 1,.. . ,n - 1 and the unit element with relations
bibi+lbi = bi+lbibi+l bibj = bjbi b: = 1 ( 4 - q-l)bi
+
Z=l, ..., n - 2 , Ji - j l 2 2, i = l , ...,n - 1 .
These are the same relations as for the symmetric group Sn except for the last one which is relaxation of .s: = 1. When q is not root of unity the Hecke algebra NHn(q) is isomorphic to the group algebra C[Sn]
which allows to index the idempotents in 7-lFln(q) called generalized Young projectors Y:(T)I2 in the same way as Yx(T),ie., again by standard tableaux. One has the q-analog of Eq. (11)
Yx"(T)Y,4(T') = bXA'bTT/YXQ(T).
(27)
239
An irreducible ',Jf,(q)-module stems from the counterpart of (12)
'FIX((?) = ',Jfn(q)YXQ(T) 7
1x1 = n .
The right action of 'H,(q) commutes with the left action of the quantum linear group GL,(D) (quantum Schur-Weyl duality). An irreducible GL,(D)module or q-Schur module is defined in the manner of (14)
1x1 = n .
E A ( q )= E@"YZ(T), 7. Deformation of pS+ and p B + and $!3
Let T, be the maximal element in ' , J f 3 ( q ) . The quantum flip, which is the place permutation (ie., the right) action of T,, has two idempotent eigenvalues Yq* q-flip(Y") = Y P f T , = fY,*.
Yqf are generalized Young projectors Y&)(T).
(28)
We now define two
GL,(D)-modules isomorphic to the q-Schur module E(271)(q) R, = E@'3Yq+,
R -E@3yQ-.
(29)
4-
Deformed creation parafermionic p$,f and deformed creation parabosonic p23: algebra are the following cubic algebras"
Ps:
= A(& Rq)
,
p%;
ii,) .
= A(E,
An outcome of lengthy calculations (using a representation of the Hecke algebra given by a R-matrix of Hecke type) are the explicit expressions for the relations of ps: and p23: (see Ref. 10 and also Ref. 6 for D = 2)
t t t t bf2,[ais,ai1lrlq2 + daf,, [ai2,aillrl =0 r.13, .lll,,al2I42-1- q"al,, "Z',lr,az',] = 0
< i2 I i3 il I i2 < i3 il
}
*
(30)
At the classical point q = 1 with the help of the (super)Jacobi identity we recover the relations (3). In the application of the quantum groups in two-dimensional statistical mechanics, the parameter q has merely the sense of temperature, q = e - h . The point q = 0 corresponding to the absolute zero temperature T = 0 is a singular point for the GL,(D)-symmetry. Figuratively speaking at q = 0 the symmetry is frozen. Nevertheless, the relations (30) are regular at q = 0 and the evaluation in this point called the crystal limit yields
240
Comparing the crystal limit of the pTi-relations (the upper sign in (30)) with the Knuth relations (23) we come to the conclusion that
is a crystal limit of the deformed creation that is, the plactic algebra parafermionic algebra p T i . By analogy the crystal limit of pBl is an algebra that might be called the super-plactic algebra
One can speculate that the ordinary parastatistics is the high temperature limit T 4 00, i e . , q = 1 of the more general deformed (or quantum) parastatistics. Then the algebra of the (super)plactic monoid is the zero temperature limit q = 0.
Acknowledgments
I wish to thank to Michel Dubois-Violette for many inspiring discussions. References 1. E. Wigner, Phys. Rev. 77,711 (1950). 2. H.S. Green, Phys. Rev. 90, 270 (1953). 3. C. Ryan, E.C.G. Sudarshan, NucZ. Phys. 47, 207 (1963). 4. A. Ganchev, T. Palev, J . Math. Phys. 21,797 (1980). 5. R. Berger, M. Dubois-Violette, M. Wambst, J . Algebra 261,172 (2003). 6. M. Dubois-Violette, T. Popov, Lett. Math. Phys. 61,159 (2002). 7. Y. Ohnuki, S. Kamefuchi, Quantum field theory and parastatistics, SpringerVerlag (1982). 8. S. Chaturvedi, hepth/9509150. 9. W. filton, Young tableaux, Cambridge University Press (1997). 10. T. Popov, Ph.D. thesis, http://qcd.th.u-psud.fr/preprints_labo/physiquemath/art2OO3/. 11. A. Lascoux, M.P. Schutzenberger, Quaderni de “La ricerca scientzfica” 109, Roma, CNR 129 (1981). 12. D. Gurevich, Algebra i Analiz 2, 119 (1990).
NONCOMMUTATIVE D-BRANES ON GROUP MANIFOLDS
J. PAWELCZYK Institute of Theoretical Physics, Warsaw University, H o i a 69, PL-00-681 Warsaw, Poland E-mail: [email protected]
H. STEINACKER Institut f u r theoretische Physik, LMU Munchen, Theresienstr. 37, 0-80333Munchen, Germany E-mail: Harold.Steinacker@physik. uni-muenchen. de
We propose an algebraic description of (untwisted) D-branes on compact group manifolds G using quantum algebras related to U,(g). It reproduces the known characteristics of D-branes in the WZW models, in particular, their configurations in G, energies, and the set of harmonics.
1. Introduction This report is a brief review of the quantum algebraic description of Dbranes on group manifolds as proposed in Ref. 1. The structure of D-branes in a B field background has attracted much attention recently. The case of flat branes in a constant B background has been studied extensively and leads to quantum spaces with a MoyalWeyl star product. A rather different situation is given by D-branes on compact Lie groups G, which carry a B field which is not closed. It is known from their conformal field theory (CFT) descriptions2 that stable branes are given by certain conjugacy classes in the group manifold. On the other hand, it is expected that these branes are formed as bound states of DO-branes. Attempting to unify these various approaches, we proposed in Ref. 3 a matrix description of D-branes on S U ( 2 ) . This was generalized in Ref. 1,giving a simple and compact description of all (untwisted) D-branes on group manifolds G in terms of quantum algebras related to the quantized universal enveloping algebra U,(g).The main result is that a model based 241
242
on the reflection equation (RE) algebra leads to precisely the same branes as the WZW model. It not only reproduces their configurations in G, i.e. the positions of the corresponding conjugacy classes, but also gives the same (noncommutative) algebra of functions on the branes.
2. CFT and the Classical Description of Untwisted D-branes The CFT description is given in terms of a WZW model, which is specified by a compact group group G and an integer level k.4 We concentrate on the case G = S U ( N ) ,but all constructions work for S O ( N ) and U S p ( N ) as well. The WZW branes can be described by certain boundary states of the Hilbert space of closed strings. We consider here only "symmetrypreserving branes" (untwisted branes), given by the Cardy (boundary) states. They are labeled2v5by a finite set of integral weights
X E I'z = {A
E
P+; X.8 5 k } ,
(1)
(here 8 is the highest root of g), corresponding to integrable irreps of G. Hence untwisted branes are in one-to-one correspondence with X E P l . The energy of the brane X is given by
The CFT also contains the description of branes as quantum manifolds, in terms of boundary primary fields. Their number is finite for any compact WZW model. In the k -+ oa limit, these boundary primaries generate the (noncommutative) algebra of functions on the b r a n e ~ ,see ~ )also ~ Section 4. For finite k , the corresponding algebra, as given in Ref. 6, is not associative. It becomes associative after "twisting", so that it can be considered as algebra of functions of a quantum manifold. Then the primaries become modules of the quantum group U,(g). On a semi-classical level, the D-branes are simply conjugacy classes of the group manifold, of the form
C ( t ) = {gtg-';
g E G} .
(3)
One can assume that t belongs to a maximal torus T of G. These conjugacy classes are invariant under the adjoint action of the vector subgroup Gv
243
GLx GR of the group of motions on G. This reflects the breaking 5~X
~ -+ R
iV.
A lot of information about the spaces C ( t ) can be obtained from the harmonic analysis, i.e. by decomposing scalar fields on C ( t ) into harmonics under the action of the (vector) symmetry G v . One finds
F(C(t))2 @ muzt,,( K t ) v,.
(4)
XEP+
Here X runs over all dominant integral weights P+, V, is the corresponding highest-weight G-module, and r n u l t ~is~the ) dimension of the subspace of V,+ = V,, which is invariant under the stabilizer Kt o f t . As discussed above, there is only a finite set of stable D-branes on G (up to global motions) in the CFT description, one for each integral weight X E P:. They correspond to C ( t x ) for
t,
tn
= q2(HA+HP)
(5)
q=ek+sv.
Here gv is the dual Coxeter number. The location of these branes in G is encoded in s,
= t r(gn) = tr(t") ,
g E C(t), n E
N,
(6)
which are invariant under the adjoint action. For the classes C ( t x ) , they can be easily calculated: s, = trVN ( 42 7 G P + H A ) ) =
c
q274P+A)+
(7)
UCvN
where VN is the defining representation. The s, completely characterize C ( t x ) , and their quantum analogs (11) can be calculated exactly. An equivalent characterization of these conjugacy classes is provided by a characteristic equation of the form P x ( M ) = 0.l 3. Quantum Algebras and Symmetries for Branes We now define the quantum space describing G and its branes in terms of a non-commutative algebra M , which transforms under a quantum symmetry. The quantized algebra M of functions on G is generated by elements Mj with indices i,j in the defining representation of G, subject to some commutation relations and constraints. The relations are given by the socalled reflection equation (RE),' which in a short notation reads R2iMiRi2M2 = M2R21MiRi2.
(8)
244
Here R is the R matrix of U,(g)in the defining representation. For q = 1, this reduces to [M:, Mf]= 0. Because M should be a quantization of G, there must be further constraints. In the case G = S U ( N ) , these are det,(M) = 1, where det, is the so-called quantum determinant (12), and suitable reality conditions imposed on the generators M j . The RE appeared more than 10 years ago in the context of the boundary integrable model^.^ M is covariant under the the transformation
M; -+ ( s - w t ) ; ,
(9)
where si and ti generate algebras BL and GR respectively, which both coincide with the well-known quantum groups Fun,(G), as defined in Ref. 8; i.e. s2slR = RslsZ, t2tlR = Rtlt2 etc. This (co)action is consistent with RE if we impose that (the matrix elements of) s and t commute with M , and in addition satisfy s2tlR = Rtls2. Formally, M is then a right 8' GR - comodule algebra; seeRef. 1 for further details. Notice that Eq. (9) is a quantum analog of the action of the classical isometry group GL x GR on classical group element g. Furthermore, GL B R GR can be mapped to a vector Hopf algebra GV with generators T , by si @I 1 -+ rj and 18 tj -+ T ; . The (co)action of Gv on M is then
M;
-+ (r-lMr);.
.
(10)
The (generic) central elements of the algebra (8) are given by c, = t r q ( M n )E trv,,,
(M"
W)
E
M ,
(11)
where v = T ( q - 2 N p ) is a numerical matrix which satisfies S'(T) = W - ~ T W for the generator T of Gv. These elements c, are independent for n = 1 , 2 , ...,rank(G). One can also show that the c, are invariant under Gv. As we shall see, the c, for n = 1,...rank(G) fix the position of the brane configuration on the group manifold, i.e. they are quantum analogs of the s n , Eq. (7). There is another central term det,(M), the quantum determinant, which is invariant under the full chiral quantum algebra GL BR GR. Hence we can impose the constraint det,(M) = 1.
(12)
Furthermore, there is a realization (algebra homomorphism) of the RE algebra (8) in terms of the algebra U,(g), given by M = (T 8 i d ) ( R z i R i z ) where T is the defining representation. Equation (8) is then a consequence of the Yang-Baxter equation for R.8
245
4. Representations of M and Quantum D-branes We claim that the quantized orbits corresponding to the D-branes of interest here are described by irreps 7rx of M . On any irrep, the Casimirs c, (11) take distinct values, i.e. they become constraints. In view of their explicit form, an irrep of M should be considered as quantization of a conjugacy class C ( t ) , whose position depends on c,. To confirm this interpretation, one can calculate the position of the branes on the group manifold, and study their geometry by performing the harmonic analysis on the branes. The “good” irreps of the algebra M coincide with the highest weight representations Vx of U,(g)for X E P z : they are ~ n i t a r y have , ~ positive quantum-dimension, and are in one-to-one correspondence with the integrable modules of the affine Lie algebra g. As shown in Ref. 1,these irreps 7 r ~ of M for X E P z describe precisely the stable D-branes C ( t x ) , denoted by Dx. It is an algebra of maps from Vx to Vx which transforms under the quantum adjoint action of U,(g).For “small” weights, this algebra coincides with Mat(Vx). There is clearly a one-to-one correspondence between the (untwisted) branes in string theory and these Dx,since both are labeled by X E P z . 4.1. Position of DX
The values of the Casimirs c, on Dx are as follows:1
Here XN is the highest weight of the defining representation VN, and the sum in Eq. (14)goes over all v E VN such that X v lies in P z . The value of q ( X ) agrees precisely with the corresponding value (7) of s1 on C ( t x ) . For n 2 2, the agreement of cn(X) with sn on C ( t x ) is only approximate, becoming exact for large A. The discrepancy can be blamed to operator-ordering ambiguities. M also satisfies a characteristic equation1 similar to the classical one. Therefore the position and “size” of the branes essentially agrees with the results from CFT. Furthermore, the energy of the D-brane is given by the quantum dimension of the representation space
+
vx
246
4.2. The Space of Harmonics on DA
For simplicity, assume that X is not too large. Then
Dx
2
Mut(Vx) = Vx @ V i % CB,N;~+V, ,
(16)
since the tensor product is completely reducible. Here N f x + are the usual fusion rules of 8. This has a simple geometrical meaning if p is small enough: comparing with Eq. (4),one can show' that
D x 2 F(C(ti))1
(17)
. differs slightly from up t o some cutoff in p, where t i = e x p ( 2 ~ i h ) This Eq. (5), by a shift X + X p. The structure of harmonics on Dx is however in complete agreement with the CFT results. Moreover, it is known5 that the structure constants of the corresponding boundary operators are essentially given by the 6 j symbols of V,(g), which in turn are precisely the structure constants of the algebra of functions on Dx. Therefore, our quantum algebraic description not only reproduces the correct set of boundary fields, but also essentially captures their algebra in (B)CFT. Particularly interesting examples of degenerate conjugacy classes are the complex projective spaces @PN-l, which in the simplest case of G = SU(2) become (q-deformed) fuzzy spheres" Si,N.
+
References 1. J. Pawelczyk, H. Steinacker, H. Nucl. Phys. B638, 433 (2002). 2. A. Yu. Alekseev, V. Schomerus, Phys. Rev. D60, 061901 (1999). 3. J. Pawelczyk and H.Steinacker, JHEP 0112,018 (2001). 4. J. Fuchs, AfJine Lie Algebras and Quantum Groups, Cambridge University Press (1992). 5. G. Felder, J. Frohlich, J. Fuchs, C. Schweigert, J. Geom. Phys. 34, 162 (2000).
6. A. Yu. Alekseev, A. Recknagel and V. Schomerus, JHEP 9909, 023 (1999). 7. E. Sklyanin, J. Phys. A21,2375 (1988). 8. L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, Algebra Anal. 1, 178 (1989). 9. H. Steinacker, Rev. Math. Phys. 13, No. 8, 1035 (2001). 10. H. Grosse, J. Madore, H. Steinacker, J. Geom. Phys. 38, 308 (2001).
HIGH-ENERGY BOUNDS ON THE SCATTERING AMPLITUDE IN NONCOMMUTATIVE QUANTUM FIELD THEORY
A. TUREANU
P.
Department of Physical Sciences, and Helsinki Institute of Physics, 0. Box 64, 00014 University of Helsinki, Helsinki, Finland E-mail: [email protected]
In the framework of quantum field theory (QFT) on noncommutative (NC) space time with SO(1,l) x SO(2) symmetry, which is the feature arising when one has only space-space noncommutativity (&i = 0), we prove that the Jost-LehmannDyson representation, based on the causality condition usually taken in connection with this symmetry, leads to the mere impossibility of drawing any conclusion on the analyticity of the 2 + 2-scattering amplitude in cos 8,0 being the scattering angle. A physical choice of the causality condition rescues the situation and, as a result, an analog of Lehmann’s ellipse as domain of analyticity in cos 8 is obtained. However, the enlargement of this analyticity domain to Martin’s ellipse and the derivation of the Froissart bound for the total cross-section in NC QFT is possible only in the special case when the incoming momentum is orthogonal to the NC plane. This is the first example of a nonlocal theory in which the cross-sections are subject t o a high-energy bound. For the general configuration of the direction of the incoming particle, although the scattering amplitude is still analytic in the Lehmann ellipse, no bound on the total cross-section has been derived. This is due to the lack of a simple unitarity constraint on the partial-wave amplitudes.
1. Introduction The development of QFT on NC space-time, especially after the seminal work of Seiberg and Witten,l which showed that the NC QFT arises from string theory, has triggered lately the interest also towards the formulation of an axiomatic approach to the subject. Consequently, the analytical properties of scattering amplitude in energy E and forward dispersion relations have been ~ o n s i d e r e d ,Wightman ~>~ functions have been introduced and the CPT theorem has been p r o ~ e n , and ~ ’ ~as well attempts towards a proof of 247
248
the spin-statistics theorem have been made.5a In the axiomatic approach to commutative QFT, one of the fundamental results consisted of the rigorous proof of the Froissart bound on the high-energy behaviour of the scattering amplitude, based on its analyticity properties.lOyll Here we aim at obtaining the analog of this bound when the space-time is noncommutative. Such an achievement, besides being topical in itself, will also prove fruitful in the conceptual understanding of subtle issues, such as causality, in nonlocal theories to which the NC QFTs belong.12 In the following we shall consider NC QFT on a space-time with the commutation relation [ Z p , ZCYI
= iepv
7
(1)
where 8,, is an antisymmetric constant matrix (for a review, see, e.g., Refs. 13, 14). Such NC theories violate Lorentz invariance, while translational invariance still holds. We can always choose the system of coordinates, such that 813 = 823 = 0 and OI2 = -OZ1 = 8. Then, for the particular case of space-space noncommutativity, i.e. 8oi = 0 , the theory is invariant under the subgroup SO(1,l) x SO(2) of the Lorentz group. The requirement that time be commutative (8oi = 0 ) discards the well-known problems with the unitarity15 of the NC theories and with ~ a u s a l i t y . ~ ~ ? ~ ~ As well, the Boi = 0 case allows a proper definition of the S - m a t r i ~ . ~ In the conventional (commutative) QFT, the Froissart bound was first obtained1' using the conjectured Mandelstam representation (double dispersion relation)," which assumes analyticity in the entire E and cos@ complex planes. The Froissart bound,
expresses the upper limit of the total cross-section ctot as a function of the CMS energy E , when E -+ cy). However, such an analyticity or equivalently the double dispersion relation has not been proven, while smaller domains of analyticity in cos 0 were already known.lg One of the main ingredients in rigorously obtaining the Froissart bound is the Jost-Lehmann-Dyson representation20'21 of the Fourier transform of the matrix element of the commutator of currents, which is based on aIn the context of the Lagrangian approach to NC QFT,the CPT and spin-statistics theorems have been proven in general in Ref. 6; for CPT invariance in NC QED,see Refs. 7, 8, and in NC Standard Model Ref. 9.
249
the causality as well as the spectral conditions (for an overall review, see Ref. 22). Based on this integral representation, one obtains the domain of analyticity of the scattering amplitude in cos0. This domain proves However, the domain to be an ellipse - the so-called Lehmann’s e1lip~e.l~ of analyticity in c o s 0 can be enlarged to the so-called Martin’s ellipse by using the dispersion relations satisfied by the scattering amplitude and the unitarity constraint on the partial-wave amplitudes. Using this larger domain of analyticity, the Froissart bound (2) was rigorously provenll (for a review, see Ref. 23). In NC QFT with 8oi = 0 we shall follow the same path for the derivation of the high-energy bound on the scattering amplitude, starting from the Jost-Lehmann-Dyson representation and adapting the derivation to the new symmetry SO(1,l) x SO(2) and to the nonlocality of the NC theory. In Section 2 we derive the Jost-Lehmann-Dyson representation satisfying the light-wedge (instead of light-cone) causality condition, which has been used so far, being inspired by the above symmetry. In Section 3 we show that no analyticity of the scattering amplitude in cos 0 can be obtained in such a case. However, with a newly introduced causality condition, based on physical arguments, we obtain from the Jost-Lehmann-Dyson representation a domain of analyticity in cos 0 , which coincides with the Lehmann ellipse. In Section 4 we show that the extension of this analyticity domain to Martin’s ellipse is possible in the case of the incoming particle momentum orthogonal t o the NC plane (x1,x2), which eventually enables us to derive rigorously the analog of the Froissart-Martin bound (2) for the total cross-section. The general configuration of incoming particle momentum is also discussed, together with the problems which arise. Section 5 is devoted to conclusion and discussions. 2. Jost-Lehmann-Dyson Representation The Jost-Lehmann-Dyson representation2’t21 is the integral representation for the Fourier transform of the matrix element of the commutator of currents:
where X
X
f ( x ) = ( p ’ l [ j l ( ~ ) , j 2 ( - ~ ) I l p7 )
(2)
satisfying the causality and spectral conditions. The process considered is the 2 -+ 2 scalar particles scattering, k + p -+ k’ + p ’ , and j , and j 2 are the
250
scalar currents corresponding to the incoming and outgoing particles with momenta k and k' (see also Refs. 22, 24). For NC Q F T with SO(1 , l ) x SO(2) symmetry, in Ref. 25 a new causality condition was proposed, involving (instead of the light-cone) the light-wedge corresponding to the coordinates xo and x3, which form a two-dimensional space with the S O ( 1 , l ) symmetry. Accordingly we shall require the vanishing of the commutator of two currents (in general, observables) a t space-like separations in the sense of S O ( 1 , l ) as: X
X
for ~ 2 = x ~ -< ox . ~
[ j l ( 2- ) , j 2 ( - - 2) ] = 0 ,
(3)
The spectral condition compatible with (3) would require now that the physical momenta be in the forward light-wedge:
p2 =pE
-pi
> 0 and
po
>0.
(4)
The spectral condition (4) will impose restrictions on f ( q ) (see Ref. 12 for details), in the sense that f ( q ) = 0 in the region outside the hyperbola Po -
Jm