Introduction to superstrings and M-theory

Graduate Texts in Contemporary Physics R N. Mohapatra: Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics R E. Prange and S. M. Girvin (eds.): The Quantum Hajj Effect, 2nd ed.

M. Kak:u: Introduction to Superstrings and M-Theory, 2nd ed. J. W. Lynn (ed.): High Temperature Superconductivity H. V. Klapdor (ed.): Neutrinos

J. H. Hinken: Superconductor Electronics: Fundamentals and Microwave Application

Michio Kaku

Introduction to Superstrings and . M-Theory •. Second Edition :: .

:. Wich 45 llIuslrolions •

Springer

Michio Kaku Department of Physics City College of the City University of New York New York, NY 10031, USA

Series Editors Professor Joseph Birman Department of Physics City College of CUNY New York, NY 10031 USA

Professor R. Stephen Berry Department of Chemistry Uni versity of Chicago 5735 South Ellis Avenue USA

Professor Mark Si1vennan Department of Physics Trinity College Hartford, CT 06106 USA

Professor H.E. Stanley Center For Polymer Studies Physics Department Boston University Boston, MA 02215 USA

Professor Jeffrey Lynn Department of Physics University of Maryland College Park, MD 20742 USA Professor Mikhail Vo10shin Theoretical Physics Institute Tate Laboratory of Physics University of Minnesota Minneapolis, MN 55455 USA

Library of Congress Cataloging-in-Publication Data Kaku, Michio. Introduction to superstrings and M-theory I Michio Kaku. -2nded. p. crn.--{Graduate texts in contemporary physics) Includes bibliographical references and index. ISBN 0-387-98589-1 (alk. paper} I. Superstring theories. I. Title. H. Series. QC794.6.s85K35 1998 539.7'258--; a . '.. that acted instanlly over a distance. EiMtein proposed Ihat gravitation was . .. . by the CUf\I'dturc of space-time. The naive merger of general relativity and mechanics produces a divergent theory, quantum gravity. which assumes is cuuscd by the exchange of particle-like gravitons. Superstring proposes that gravitation is caused by the exchange of closed !>tring~.

other

~~~::~:~r~!~:~~h~ow~~Cvtr, is qui te unlike its predcccssors in its historical Unlike physical theories, superstring theory has perhaps

!

in science, with more twislS and turns than a roller

young physicists Veueziauo and Suzuki [21,221. independently ;ov,,,d its qunntum theory when they were thumbing through a mathbook and accidentally noted that the Euler Bela function satisfied postulatcs of the S-matrix for hadronic intcldctions (except un.itarity). Schwarz, and Ramond [23-25] quickly generalized the theory to inspinning particles. To solve the problem ofunitarity, Kikkawa, Saki Is, {26] proposed that the Euler Beta function be treated as the Born to a perturbation scrit.'S. Finally. Kaleu, YlI, Lovelace, and Alessandrini the quantum theory by calculating boson ic multiloop dia· hOlVevcr. was still formulated entirel y in renns of on-shell twO

'v.i.,oco

and Ooto []4, 35] realized that lurki ng behind these scanering was a classical relativistic string. In one sweep, they revolutionthe entire theory by revenling the unifying, classical piCf\lre behind the The relationship between the clossical theory and the quanrum theory y ;~:~;;:G~O~ldstone, Goddard, Rebbi, and Thorn 136] and further b: [3 7]. The Ihcory, however, ""'as still fonnulated as a

16

1. Path Integrals and Point Particles

first quantized theory, so that the measure, the vertices, the counting of graphs, etc., all had to be postulated ad hoc and not deduced from first principles. The action (in particular gauge) was finally written down by Kaku and Kikkawa [38]. At last, the model could be derived from one action strictly in terms of physical variables, although the action did not have any symmetries left. However, when it was discovered that the theory was defined only in 10 and 26 dimensions, the model quickly died. Furthermore, the rapid development of QCD as a theory of hadronic interactions seemingly put the last nail in the coffin of the superstring. For 10 years, the model languished because no one could believe that a 10- or 26-dimensional theory had any relevance to four-dimensional physics. When Scherk and Schwarz [39] made the outrageous (for its time) suggestion that the dual model was acruaUy a theory of all known interactions, no one . took the idea very seriously. The idea fell like a lead balloon. The discovery in 1984 by Green and Schwarz [40] that the superstring theory is anomaly-free and probably finite to all orders in perturbation theory has·· revived the theory. The £8 ® £8 "heterotic string" of Gross, Harvey, Martinec,.· and Rohm [41] was considered to be the best candidate for unifying gravity·· with physically reasonable models of particle interactions. . One of the active areas of research now is to complete the evolution of the < theory, to discover why all the "miracles" occur in the model. There has been . a flurry of activity in the direction of writing down the covariant action using methods discovered in the intervening 10 years, such as BRST. Finally, intensive work on duality in the early 1990s, and especially the work of Witten and Townsend in1995, established M-theory as the latest, most ad~ . vanced formulation of string theory. The five superstring theories could now be unified into a single theory in 11 dimensions, perhaps in terms of membranes. Furthermore, the vast web of dual relationships in lower dimensions has given. us a "road map" by which to probe nonperrurbative solutions to string theory. What is still missing, however, is the geometry on which the theory ultimately lies. Understanding this geometry may allow us to formulate the entire theory into a simple, coherent equation. This could complete the evolution of the •. theory, which has been evolving backward for the past 30 years. .

Quantum theory -+ Classical theory -+ Action -+ Geometry. Let us summarize some of the promising positive features of the superstring • model: (1) The gauge group includes E 8 ® E 8 which is much larger than the minimal group SU(3) ® SU(2) 0 U(l). There is plenty of room for phenomenology in this theory. (2) The theory has no anomalies. These small but important defects in a quantum field theory place enormous restrictions on what kinds of theories are

1.2 Historical Review of Gauge Theory

17

The symmetries of the superstring llieory. by a series of cancel all of its potential anomalies. from the theory ofRiemano surfaces indicate that the is to all orders in perturbation theory (although a rigorous still lacking). is very linle freedom to play with. Superstring models arc nodifficult to tinker wilh without destroyi ng their miraculous Thus, we do not have the problem of 19 arbitrary coupling thcory includes GUTs, super-Yang-Mills, supergravity, and Kaluza-::: Klein thcories as subsets. Thus, many ofthe features of the phenomenology :::: for these theo ries carry over info tbe string theory. ,pe;~.ing

thoary, crudely speaking, unites the various forces and particles same way thai 8 violin string provides a unifying description of the tones. By themselves. the noteS A, B. C, ele., are not fundamcntal. the violin string is fundamental; one physical object can explain ;";i,tios of musical notes and eveD the harmonics we can construct from much the same way, the superstring provides a unifying description of ;""twyparticles and forces. In fact, the "music" created by the superstring forces and particles of Nature.

~~'~~~~~:::~~~:~!

theory, because of its fabu lously large set of symme-

' cancellations of anomalies and divergences, we must

C!

present a balanced picture and point out its shortcomings. To be fair we "~::: list the potential problems of the theory that have been pointed out Cl

of the model:

is impossible experimentally to reach the tremendous cnergies found the Planck sCllle. Therefore. the theory is in !>nme sense unlestable. A th"o,y "hat is untcslable is not an acceptoble physical theory. one shred of experimental evidence has been found fO confirm. the ofsuIP,,,,ym,,,,,'ry, let lone superstrings. . presumptuous to assume that there will be no surprises in the "desert" ::: betwecn 100 and 10 19 GeV. New, totally uncxpected phenomena have alup when we have pushed the energy scalc ofouraccelerators. theory, however, wakes predictions over the next 17 orders of which is unhcan.l of in the history of science. does not ~ Iain why the cosmological constant is zero. Any thaI claims to bc II ''theory of everything" must surely explain puzzle of a vanishing cosmological conSlant, but it is not clear how

""""no
D;ui2d / ",

(1.3.15)

K(a, b)K{b. c)Dx/t

(1.3.16)

22

1. Path Integrals and Point Particles

and

L ~ /DX =

lim /rIIldxi,n,

N ..... oo

paths

(1.3.17)

;=1 n=1

where the index Il labels N intermediate points that divide the interval between the initial and the final coordinate. We will take the limit when N approaches ··. infinity. . •.. It is absolutely essential to understand that the integration Dx is not the ... ordinary integration over x. In fact, it is the product of aU possible integrations • over all intermediate points Xi,nbetween points a and h. This crucial difference . between ordinary integration and functional integration goes to the heart of the path integral formalism. This infinite series of integrations, in turn, is equivalent to summing over all . possible paths between a and h. Thus, we will have to be careful to include normalization factors when performing an integration over an infinite number .• of intermediate points. Ifwe take the simple case where L = all functional integrations can actually be performed exactly. The integral in question is a Gaussian, which is fortunately one of the small number of functional integrals that can actually be · performed. One of the great embarrassments of the method of path integrals • . is that one of the few integrals that can actually be performed is

!mx;,

"f xx d

2n

_, 2.(x )

~

= (xli}llq'>

+ m 2q'>2].

One of the most powerful techniques we used to explore the first theory was symmetry. We would now like to study the question ofsyrnrnletri~W within the second quantized formalism. First, let us calculate the equations of motion by making a small in the field and requiring that the action be invariant under this variation:

8S

f

= 0=

dDx (8L 8q'> +

8q'>

~8aflq'» 8i}fl¢

.

Let us now integrate by parts, using 8I1(1.q'> = afl8¢:

8S =

!

dDx

G~

-

a(1.

o~~¢) o¢ +

f

dDxi}1l

(8:~¢ o¢) .

lfwe temporarily ignore the surface term, the action is stationary if we hli;;i@E:l the following equation of motion:

8L oL aJ.L -oaJ.Lq' > oq'>

= 0

(l ....

"'0'1

L . ...

1.9 Current:; lind Second Quantization

41

im"rt the Lagrangian into Ibis equation. we obtain the equations of which reproduce the constraint found earlier in the first quantized noW make n small change in the fields, pammetriz.ed by a small, as ,,,Ui,J, Ilwnber 0":

8¢ = 8¢ as".

'c·

( 1.9.6)

this into the previous equation for the variation of the action, keep term intact, Illld assume that the equations of motion are satisfied,

have the following:

!s~fdDxa" (~'¢)"-' JJ ¢ at"

(1.9.7)

Il

~efin,

the tensor in the parentheses as the current: !ii, 8¢

" J" =

aa ll ¢, oE!"

(1.9.8)

, .",ve the important equation

oS = / dDx0I'J:Ssa.

(1.9.9)

action S is stationary under this variation. we have a conserved (1.9.10)

I

w"' ,hi" ",""ri,," over and over again in the discussion ofslrings when

extrnct the currcnt for supersymmetry and conformal invariancc. note that the integrated eharge Q" associmed with the current is

in time:

f

d D xal' JI"J =

f

d D - 1x8rJJo..

+ surface term,

( 1.9.11 )

(1.9. 12) we wish to construct yet another conserved current associated with Let us make a small variation in the spacHime variable: (1.9.13) charge, the volume clement of the integral changes as

odDx

=dDxallax".

(1.9. 14)

42

I. Path Integra1s and Point Partic1es

Therefore, the variation of the action under this change is

f

dDx[LafLDX fL

+ 8L].

DS

=

DL

= DXfLollL + -8¢' + --8all ¢,.

8L

8L

8¢'

8all¢'

Now, if we assume that the equations of motion are satisfied, we have

SS =

!

D d x all { (

+L8~ - 8!~¢' au¢,) ox

u } •

If we now define the energy-momentum tensor as 8L TJ.l.u = sail¢' au¢,

then we have the equation

SS =

!

- 1)J.l.u L ,

dDxoll(T!'-vS xv ).

So if the action is invariant under this change, then the tensor is conserved: aJ.l.p v = O.

emlrg:y-{nOlnellltu~a~

(1.9.1 (

For example, for the scalar particle action, the energy-momentum becomes TJ.l.v = a/1¢,a,,¢, -

1]/1V

L

which is conserved if the equation of motion holds. Lastly, it is instructive to investigate how the various quantization dures treat the Yang-Mills field (see Appendix). Let us begin with the' invariant action:

where

F;v

= a/1A~ -

avA~ - roc AtA~.

The action is invariant under SA"/1!i = a N l - f abe AbJ.I. A C ' where A a is a gauge parameter. The path integral method begins with functional

Z=

tenlsdj~m

fn

dA/1(x)i fh -(I/4)F; ••

J.I. .x

Now we consider the three methods of quantization.

prclcei~~

Currenl~

1.9

and Second Quantization

43

invariancc permits us to take thc gauge

"ViAr = O.

,"gro,t, 0"" 'h' Ao component because it has no time derivatives, so formulation is explicity ghost-free. (The price we pay for this, of loss of manifest Lorentz invariance, which must be checked by :- :. this gauge, the action becomes

~(F;jl

L = +HaoAn2 -

+ " ',

(1.9.25)

,field, are D'ansverse. This is the canonical form for the Lagrangian.

of the Gupta-BJculcr fonnulation is that we cao keep manifest 'Y'"m,,"y without viol:~,~;~I; '~·;It1~~~~:. overcenain topologies, which must be rnenns thlll tulilatily is not obvious in the first quantized

"1:

Later, we wiU sec that this problem in the first quantized point . theory carries over directly into the first quanlized string theory. In the

r~~:~~;:~:ri:l:' :~;hl:~::: all topologies can be derived explicitly action.

~,n

from a first [0 a second quantized description is straightin the path integral formulation. For example, the propagator can be

;,";lh"6"1. o,sec.

Then let us factor out X a from the action (2.1.7): L = __ 1_IX~211/2(l_

2Jta'

where

v~)1/2,

r

r

vis the velocity component perpendicular to the string: _ Vi

=

VkX~

Vi -

I

-a-X;. Xj

The boundary conditions that we derive from this gauge-fixed action UJ\JJUULV.

v~ = 1. This means that the e.nds of the classical string travel at the speed of light. We can also calculate the energy of the classical string. Let us assume the string is in a configuration that maximizes its angular momentum, Le., it , a rigid rod that rotates with angular velocity w around an axis labeled by unit vector r. The string can be parametrized as

X = err,

2.1 Bosonic SlTi llg~

r :::: w x

55

r,

r · w :::: 0,

(2.1.19)

o~1.

>~~~.~;. ::.:the energy and the angular mornenrum of the system. we must ~

the Lorentz generators associated with the ~tring: M il":::: /

da(P~ X/I

- P"' X ") .

(2. 1.20)

that this generates the algcbrd. of the Lorentz group if we impose the brackets: [XII(q), P..(q')l "" 11... ~ 8«(J - a ' ).

(2.1.21)

now calculate the energy and the angulor momeorum from th(: )O,ots of the UlrenlZ generator [5 J:

J = -1- / ' d. ;.,----'...','~-;,c;),cor> [r x (w 2Ha '

_I

(I

w-.....

,•

X

T)]

... w(4a'w·1)-1

""' wE 2a'.

(2.1.22)

tbe angular momentum of me rolating string is proportionai lo tlte square of the system: (2. 1.23) p lot the energy squared on the x-axis and the angular momentum on we obtain a curve called the Regge trajectury. The slope of the is: given by a' and the curve is linear. Thus, we have obtained Regge trajectory for the classical motion of a rigid rotator. We this book. lake the oonnalization a' :::: This is an arbitrary However, we will see laler that the intercept Qo of the leading must be equal to one, which is fixed by confonnal invariaoce once the theory. Thus, we set

i.

,

,

a :::: 2' ao = 1.

(2. 1.24)

we quantize the system, we will find that there is an infin ite number cb IP",IlI,,I Regge trajectories, but with increasingly negative },-interceplS. we bave stressed, there is a crucial difference between the point particle and the string, which is tbat the string system has a larger set of conthat generate the gauge group of teparametrizations. For example, if

56

2. Nambu-Goto Strings

we introduce the canonical Poisson brackets, then we can explicitly show tll~~{~1 the constraints generate an algebra. To calculate this algebra, we decompose the X and P in terms of nOlrni!~~]1 modes for the open string:

+2 L 00

X"(o-) = xl"

pl"(o-) =

n=1

!I

pI"

1 -X~ cos no-,

..jii

+ [; In p~L cos no-

I·

Notice that the X decomposition is given strictly in terms of cosines. This because, when we calculate the equations of motion of the string, we mti$~:j:m integrate by parts and hence obtain unwanted surface terms at 0- = n: aIi~~:~~ 0- = O. In order to eliminate these surface terms, we must impose

X'I" = 0 at the boundary. This boundary condition eliminates all sine modes of tll~::@ string. (These are called Neumann boundary conditions. It was once th~mllnt::W that Dirichlet boundary conditions, where the derivative of the string V 0,

+ 2a' p"r + j ../20"

."

L ~e-I'"

cos nO" .

(2.2. 11 )

",.0 II

in this basis, tbe Hamiltonian takes on an especially simple f('lnn (see

~

..,

= '"' + O"p'/'-' L natIt/'- aJl. n

(2.2.12)

have made an infinite shift in "the zcro point energy. At this point. of the lowcst-ordcr particle is not well defined because we made ;0;1. 'mill, but we will later show thai this lowest particle is actually n We will show that the intercept of tbe model is fixed at L

'~:~:~:,:~:~,:~~I~;::;~:~l~ is basically uncoupled from the other oscil-

n'

. In fact, the Hamiltonian is diagonal in the Fock space ofbatmonic

excitations. Taking this specific representation oflhe string function

62

2. Nambu-Goto Strings

s_

10> kl1 a?10>lkl1 a;1110>

FIGURE 2.3. Regge trajectories for the open string. The x-axis corresponds to t~~~~11 energy squared and the y-axis to the spin. The particle farthest to the left is t'~~:~~ tachyon, which corresponds to the vacuum of the Fock space. The massless ~ni .. ~,,·~·»!w. particle is the Maxwell or Yang-Mills field, which corresponds to a single creaQ(~~~m operator acting on the vacuum. There is an infinite number of Regge trajectlori(~~J~m.: corresponding to the infinite excitations of a relativistic string or the infinite nUlnbl~~W. of states in the Fock space. from the infinite number of possibilities is a great advantage because lowed eigenstates of our Hamiltonian are now simply the products of the Fo(~~::::=i spaces of all possible harmonic oscillators: ( eigenstates:

rr

[a!.IL} 10} ,

n.1L

where the vacuum is defined as

n > O. The spectrum of the lower lying states can be categorized as (see Fig. tachyon -+ 10) , massless vector -+ a[1L 10) , massless scalar -+ klLai lL 10} ,

(2.2.15:)iW

massive spin-2 -+ ai lL ai 10) , massive vector -+ lL 10} . v

ai

(The fact that the string theory is so simple to quantize can be traced t~~m the fundamental fact that the Hamiltonian is quadratic in the string this means that it decomposes into an infinite series of free particles. A va:i~~:~m collection of vacuum solutions to string theory can be constructed because, i:t.@;§!§ essence, it is a free theory. However, this simplicity breaks down c~:~;~~7ij1~~ when we analyze membranes in M-theory. We will find that the I~ is now quartic, making the quantization intractable. In contrast to the sinllp].ic~:m~ ity of string theory, there is still no satisfactory method for quantizing fte,~@m membranes.) As expected, we recover the leading Regge trajectory that we obtained earli~~:m*l from classical arguments, and also an infinite number of daughter tra,jectori~~:mi

2.2 Gupta-meuler Quantization

63

increasingly negative y-intcrccpl. In this gauge, the propagator or function takes on a simple form: K(a, b):;:: (Xul e- m • IX b )

=

r" Dxex p -jdad,!-I-(X! - X )1 , (2.2.16) lx. 41!'a' '2 /I

DX=

TITI dX'(u) = TI dX:. ,

0

(2.2. 17)

,."

careful to note that the functional integral over DX is an infinite over each point along the string,. or each Fourier mode of ;;'",I.y" this integration can be carried out cxplicitly. Let the particle , :;:: 0 to r = 00. Because the Hamiltonian is diagonal on the harmonic oscillators, we can perfonn the integration over 't: exactly.

(2.2.18) is the zeroth Fourier component of the Virasoro generators (2.1.28). ,"'or,ds, I,ro.pal,ator for the fn."C theory is

0'"

(2.2.1 9)

id"d brlwoo. stales of the string. However, becausc wc have the identity IX)

j

DX (XI = I

(2.2.20)

~:"'m.vm string:

(Lo - Lo) I¢} = O. (The interpretation of this constraint is that the closed string must be mdlep1ew. dent of the origin of the 0' -coordinate. For example, the operator f dO't>ia:(:Lo...;;tiiXED can be interpreted in two ways. First, it generates rotations in O'-space, so w.~ED average over a rotation of 2IT in a -space. Second, if we perform the i'Iltel¥~!~WI we have B(Lo - Lo), which is constraint (2.2.23) when applied to the H'.1lbj~m:~ space. We will return to this constraint later.) The Fock space consists of (see Fig. 2.4): tachyon

~

massless spin-2 ~

10>

IO} ,

aitlai" IO} ,

v, ,

k /I k a+/I i+PIO>

FIGURE 2.4. Regge trajectories for the closed string. The Fork space is of two commuting sets of hannonic oscillators. The massless spin-2 particle graviton, which corresponds to the product of both types of operators acting ontMM vacuum.

2.2 Gupta--BJeuler Quantization

6S

massless scalar ....... k"kllar"ar~ 10).

a massless spin-2 particle occurs in the speclruOl of che closod "n th:' "'fig model was first being interpreted as a model ofhadroos. of this gravilonlikc spin·2 particle was

II.

great embarr.lssment.

were made to associate this particle with the Pomeron trajectory . S-matrix theory. It can be shov.'t1 that, when we generalize to trees this OllIssless spin-2 particle has n gauge invariance equivalent to of Einstein's theory. By dropping the earlier interpretation of the th."y as a model ofhadrons, we find a natural place for £his gravilonlikc

graviton itself. ~:~~~~,t~h~:e, Gupla--Blculer formalism in the confonnal gauge appears ir mainJy because we arc allowing ghosts to appear throughtheory reduces to the simplest possible string theory, a free string. we must pay for this simplicity, however, is the imposition of the the Fock space. We ean writo the Virasoro generators as [til:

(2.2.25) 00

1.0 = .

..L,a_~/J.a~ + !a~

.

states of the theory must therefore satisfy Ln 14I} = 0. (L. - I) I ~ ) = O.

II >

0, (2.2.26)

generated by these operators is

[Ln. L .. J = (11

~

D , m)L,,-t .. + 12(n - n)1in._m ,

(2.2.27)

is the dimension of space-time. The fact that there is a c-number appearing in this equation at first sounds surprising, but this can ,1"IOd explicitly by taking the vacuwn expectation product of a simple

(OI(L,.

D

Ld 10) = '2'

(2.2.28)

of this e 0,

{RIS} = 0, for some integer n and some state Ix}. (If we take the matrix elemeqt onfifi~ state with a physical state, the scalar product always vanishes bel::aULse< destroys a physical state.) Now let us construct the spurious state:

It} = [L-2 + aL~tll L _h Lo, which generates

(2.7.7)

easily calculate the transformation of a vertex function induced by ea L,

V(y)e- 0,

Let us define a spurious state as one that does not couple to real states:

(SIR)

= O.

We can conveniently represent such a state as IS) = L_n Ix)

for some state X. We now wish to show that spurious states do not couple ... trees, i.e., (S] Tree) = O.

To show this, we need two more identities:

[Ln - Lo - n + lJVo = Vo[Ln - Lo + 11, 1 1 [L n -Lo+11 1= [Ln- L o-n+l]. L0 Lo +n - 1 (This can easily be proved from (2.7.6), which in tum crucially depends on vertex function having conformal weight 1 with external tachyons ,,"'t-id='lifli~W&

2.9 Ghost E.limination

91

1. Thus, thc restriction on the ronfonnal weight of the vertex is crucial ;mination of all ghosts in the theory.) these two identities, we can now easily show

[t" - Lo - n + I )VoDVQD··· Vo 10) =

o.

(2.9.6)

· is the direct result; it shows tlllll the L 's can be pushed to lhe right until · finally annihilate on the vacuum. • """,mary, we have shown (hat Spurious slates do nOI rouplc to trees:

L" ITree) = 0

~

-+

(SI Tree) = O.

(2.9.7)

:~~'~::i we do 1I0t have to make any speciaJ change in the functional for because of the presence of ghosts. The ghosts stales propagat-

the tree amplitude automatically cancel by themselves. However. we . that the loop functions do indeed have problems with gh~ts propagating · I . This is because

(SI Tree IS)

f

O.

(2.9.8)

we naively trace over the trees to obtain loops, we thus inadvertently ud,,,h,, presence of ghosts, which must be facLOred out explicitly:

th",p =

L (n I Tree In) .

(2.9.9)

•

explicitly contoins ghost slales.

;~;~~~:~~.E~:l~ ~d~":':IU;;:plehappens from trees andooly contribute loop diagrams in the Yang--Mills case. to Faddecv-Popov

~

to the Yang-Mills theory do not contribute to the IrCCS of theory, but they do contribute to the loops. The Faddcev-Popov ghost . . is (see (1 .9.30): (2.9. 10)

that this produces the foUowing interaction of the gauge field A with ghost field c:

L, '" cAe.

(2.9.11)

,i, """piing means that a single ghost cannot couple to a tree diagram congauge fields. They can only contribute to loops, where the e fields circulate internally. Thus, particular attention was required to guarantee lhc Faddeev--Popov ghosts canceled against lhe negative metrie Slates in loops. m(p b" + p aP 8" X/,../i/](cx.' secolld-order form :

- I L = _../ig"bJ"X",abX". 4:rra'

L = ~(Xl X fl

nonlinear form:

21f0"

".

_

(2.10. 1)

(X X'I')2 )1/2 I'

•

us summarize the similarities fOund in the point particle case and the

:th,co,y v,hen expressed in pi:lth integral language: xI'(r)

Dx =

n

--t-

X,, (u, r ),

length -+ area,

d.tp( t )

--t-

DX =

P.T

n

dXI'(a, r),

I',r.a

(fI ""') - (fI ""X') ' I_I

I_I

graph _ manifold,

"

P +m =0-+

(P,,+-X; )' =0. 2Jra'

'~:::;:, for quantizing the Siring action is

\0 write down the symmetry ~I extract the currents from this symmetry. calculate the a lgebra currents, and apply these currents onto the Hilbert space in o ,Itim;,"" ghosts. The Slmtegy we follow throughout the book is _ Symmetry -) Current _ Algebra ~n'~"t; ot"

--t

Constraints

--t

Unitarity.

possess repar~trization symmetry, which generates the

(2. 10.2)

96

2. Nambu-Goto Strings

The algebra generated by these operators is

[L", Lml

= (n -

m)Ln+m

D 3 + 12 (n -

(2.1·~·:_

n)8n,_m'

As in the point particle case, there are three ways in which to qUirntiz¢.~W theory.

Gupta-Bleuler Quantization In the Gupta-Bleuler fonnalism, we fix the gauge: gab

= 8ab

and the resulting action breaks reparametrization invariance but mElml:alrlsJl subgroup of conformal transformations:

L = _1 (X 2 _ X'2). 2][ I' I' (,

This Lagrangian, of course, propagates negative metric ghosts associated . the timelike mode of X. To eliminate them, the Gupta-Bleuler qUirnt'izalti~~ method states that the state vectors of the theory must satisfy

L" I¢} = 0, (Lo - 1) I¢} = o.

n > 0,

This means that the constraints are constructed to vanish on the Hilbert Sp~(~~ Although the action in this formalism is quite elegant, the price we "","""tc" as an invnriancc of an action, was first discovered in the Gervais and Sakita [1] showed that an extension of the usuaJ possessed a symmetry that converted bosons into fermion:>. it languished for many years because the supcrsymmetry of model was a two-dimensional supcrsymmetry on the world until relatively recently that it was fml:l.!Iy proved that the ~~!:;;~"'''''d borh two-dimensional and IO-dimensional spaoo-tlmc our di.scussion by t:oWiitlcring the siwph::st possible spinning ,e spi~'~i"g point particle. In addition to the \f'.uiable x)J.' which locates of the pOin! particle, let us introduce Dirac spinors 0", where A is In general, a Dimc spinar the Dirac matrix in D dimensions ;n"ioo;b", 2''''''"compiex components. We can writc p. 3J

r" .

'f -'( '

S=i

e

..t)J. -

'£j"r)J. B")'d r.

lv

(HI)

particle action is invariant under

80 A = t: A ,

B8 A = Ox)J.

e,

=ieA r"8 A ,

(3.1.2)

Be = O.

(3.1.3) ~:~~:I, all by itself under this transformation. Thus, any expression

combination will be invariant under this symmetry. The strange action, however. is that half of the components of the fennioruc from the action by itself. "Ying,th'"'' .f.)J.. and e fields. we can derive several equations of motion: IB

n 2 = 0, Ii" =: 0,

r·no =o.

(3. 1.4)

(3.1.5)

aJ!~:~~i,.~~::'J'l~;: :~!"~~:~~~~;~ vanish. Butsincc8 always appears p'e". half they. ncomponents of the Dirac spinor are

:

104

3. Superstrings

actually missing from the action. Thus (J is not an independent splnolf::.t)j satisfies a constraint that reduces its components by half. The reason for this decoupling is that the action is invariant under yet an(lftijijjij local symmetry [2]: '

{

8(JA

= i r • OKA.

8 X IL

= i~ArILS(JA.

Se = 4ee A KA.

Furthelmore, there is also another bosonic invariance of the action: S(JA

= ),J)A.

8X IL = {

ieArIL(j(JA,

lie = O.

When we try to commute two supersymmetry operations given in (3 .1'~I':'"' find that the algebra does not close unless we use the equations of mCIt1o~::::;:: [8 1• 82]eA = (2ir IL Kte Br.

orILK~ +4ir· oKteBK~)-(l ~ 2).

(This is actually typical of supersymmetric actions. Notice that the nU1J1lj:~ of components of x lL and (J A do not necessarily match off-shell. which 1l1~~~ that auxiliary fields are in general necessary to close the algebra.) Lastly, if we calculate the canonical conjugate to the coordinates, we A _

1T:(J -

8L _.

-.- -

(j(JA

I

r IL nile A .

This poses vast complications for the covariant quantization. Because the~t-e:: an explicit dependence on x within the canonical conjugate, the qUJmtiiz.a1.ft relations become nonlinear and hence exceedingly difficult to solve. Wij~1 it turns out that the term r il OIL becomes a projection operator in the th~~~ meaning that we cannot invert the transformation and solve for (JA. In q:~g words, a naive covariant quantization of the spinning point particle dotlS;:~ijffi seem to exist at all! This is a warning that the supersymmetric string is:)~m§ going to be as simple as super-Yang-Mills or supergravity theory. As sequence, we will discuss the simpler two-dimensional world sUI)en;yn:Up:i¢~~ of the N eveu-Schwarz-Ramond (NS--R) model first, and then the more .Cj9~~ plicated lO-dimensional space-time supersymmetry of the Grleenr-SlcMv:im~ model.

3.2

Two-Dimensional Supersymmetry

Given the problems associated with superparticles (such as lack of a cO'vM'iJPll quantization and lack of an algebra that closes off-shell), let us tenlpoltW;!m:: abandon space-time supersymmetry and discuss the two-dimensional

3.2 Two-Dimensional Supersynunl:try

lOS

ym",e~

of the simplest possihle action involving free strings and free . This action will already be gauge fixed in tile conformal gauge, but all the essential features oftwo-dimcnsional supert.ymmctry. In dixod fonnulation, we will impose the gauge COllstrnims on the Fock hand. Lalcr, we will present the complete action, where we will be lhese cOllstraints starring from a locally symmetric action.

gauge, we have [J] 1 L = --2 (a~ X",a~ X" - iy,JJ p "a.. Yt14 ), n

(3.2.1)

I, 2 and labe ls two-dimensional vectors, and J.L is a space-lime is a bizarre object, an anticommuting Majorana spinor in two

,im" ,md a vector in real space-time. Let us define

!/1M =

(~~ ) ,

pO =

(~ ~j ),

, (0 i)

p...

Ip", p'} =

j

0

~jJ ~ iftupo,

'

(3 .2.2)

-2,"".

this out explicitly in components, v.'C have (3.2.3) action is gauge fIXed, it is still inv:lriant under the global

jrmatio",

SX" = e",". Sl/f" = - ;p"a. xl't.

(3.2.4)

abandoning OUf attempt to produce a theory of strings spucHme sp inors, we have achieved a two-dimensional theory that is quite s.imple, involving free bosonic and fie lds. the NS- R theory [4-6J was the first successful attempt at spin inlo the dual model. It was also the first example of a lin. action and was soon followed by fOUI·dimensional actions [7. 8}. wri tten down our two-dimensional supcrsymmetric acn'tn,,, the "lOP"'" '"ed in the previous chapter to solve the system. !.1ep is to extrnct the currents associated with these symmetries, then the algebra that these currents generate, and fmally apply these conon the Hilbert space. TIle sequence we will follow in this section is

"U'

106

3. Superstrings

a straightforward generalization of the steps we developed in the chapter: Action -* Symmetries -)- Current -)- Algebra -* Constraints -)-

nrp',';"

UUHi1I(i~(

Following this strategy, let us now calculate the supersymmetric curretll:m~ sociated with this symmetry. As we saw in the last chapter, the eXllste:n¢I~:; conformal symmetry is sufficient to generate a conserved current. h1 ( we saw that this conserved current could be written as

r=~ 8¢. 8 ap,¢ 8s a

p,

By inserting (3.2.4) into (3.2.6), we find Ja =

! pb p" 1fJ/'/' abx

Jl"

To check that this is conserved, we must first write down·the motion for the system, which is especially easy since it is a~ee action:

[ar + Orr ]1fJt; = 0, [or - arr ]1fJi = o. It is now easy to show

Oa 1"=0. Written out in components, this equals (h

= !(Jo ± J 1»)

L = 11fJt;(o, - orr)X/./" J+ =

!1fJi(ar + aCT)X IL .

U sing the equations of motion, we can show that

(a r

+ arr)L =

(ar

-

arr)J+

0,

= O.

In addition to the supersymmetric current, we also have the momentum tensor, which as we saw earlier in (1.9.17) can be as

8L P' = - 0 A. v SOIL¢ V'Y

-

wr.l~M

LSI'

v'

This can easily be adapted for our purposes. Inserting (3.2.1) into (3.2.1"fj:;::~~ find

Tab = OaX/'/'ObX IL

i -

i -

+ 41fJ IL Pa Ob1fJp, + 41fJ/'/'PbO,,1fJ1L o"T"b = 0,

(Trace),

3.2 Two-Dimensional Supersynunctry

\07

is traceless because we have explicitly subtractcd out the explicitly, this is

1'2

== l(X

+ X ,:z ) -

t rac~.

i" i .. '41Jro (a, - iJ,,)1Jrou - '41Jr1 (a r -+ B")1JrI I,,

== Till

(32.14)

. Ii.. iu = X· X + 40/0 (Of - 8,,)Vto.. - 41P1 {a f

+ (1,,)"11"

i strategy (3.2.5) is to use these currents to restrict the Fock space so ghosts are eliminated:

(T,,/>l =

o.

~

o.

(J.)

(3.2.15)

strcs..~

again, however, that we are simply imposing Ihis constraint the Pock space. Because me action is not locally supcrsymmetric, ,Ier;;vethis constraint from first principles. Later, when we present symmetric action, we will see that these constraints emerge because

""

" .. -DO

(3.2.29) ~ ...d ,,!f1~, H,'mil"mi,,,, ,ow cOlodilion [Lo 10;..1:) = 0 with a'k2 = is quite remarkable. The state fO; k}, which used to represent the old "m "," at (1' k 2 = I , has now suddenly transfotmed into the tachyon state

V

t.

114

3. Superstrings

= !.

Thus, in the F2 formalism, the tachyon state and the new va(~u~~m at a' k 2 are the same particle. (y./e should be careful in noting that the same synl11jQru§§ 10; k) can represent either the old vacuum in the FJ formalism or the tachy(j~:§§ in the new F2 formalism.) This shifting of the Hilbert space is a novel feature not found in point l?;~i tide field theories. In fact, when we discuss the conformal field theory l1 next chapter, we will find that this strange "picture changing" phlenc)lru:~9im~ arises whenever we construct irreducible representations of the sU]:lerc:onJtO'J mal group. Hence, it is not a trick, but an essential feature of the group thp;,,,, (We should also mention that when we make the model space-time SU~ler~rvtJj~ metric, we will eliminate the tachyon state as well. Thus, the tachyon de(~Ou~t~~~ from the true superstring Hilbert space, leaving a unitary theory.) . Let us summarize the differences between these two formalisms: vertex = V, propagator = (Lo - 1)-1, tachyon:

kfJ.· b~I/2 10; k) (afe =

vacuum:

IO;k) (a f k2 = 1),

k),

(

vertex = V,

F2 :

I

propagator = (Lo tachyon = Vacuum:

krl, IO;k) (afe = ~).

The advantages and disadvantages of the two formalisms are as follows: (I) In the FI formalism, manifest cyclic symmetry is much easier to However, we must carry along the excess baggage of the vacuum which decouples completely. (2) In the F2 formalism, cyclic symmetry is obscure, but ghost and gauge transformations are easy to perform. The advantage is are now working in a smaller Fock space. Using the F2 formalism, it is not hard to evaluate the four-point furLci%9mm exactly: (0, -kd V(k2)

I I V(k3 ) 10; 1.4) Lo - '2

=

r(1 - a(s))r(l - a(t)) , r(l - a(s) - aCt)) .

(3.3:~91§1

wherea{s) = 1 +ais. There is also a straightforward generalization to the N -point function. ~W N -point function is represented as 1:

3.3 Trees ~.~I,

115

this expression contains many factors that are tedious to work out

method of obtaining the entire resuh is to usc the relation

-V(y) ~

.;;

f

dO exp{ik· X

+ 9k· "".;;).

(3.3.22)

that i f we power expand the exp onential. only the linear teno survives 'j"",,&",u,on, and we wind up with the previous cllpression for tbe vertex. at this point, done nothing. Now, leI us use two identities:

(01 "'"(,,) ,,"(y,) 10) ~

../Yi...!Yi I

V(y,) V(n)

../Yi ../Yi

10) =

f

."" y, - 12

dO, dfh. exp[k 1 . k1ln(y.

-)'2 -

91th)] (3.3.23)

bard to generalize the above relation to lhc N -point tachyon amplitude:

;~:;:~.~ '~:'~~:'~:'~~;~: is that We can now read off the various terms

[:1

the

by successively power expanding and integrating

Grassmann variables. to tachyon-tachyon scattering amplitudes, we can also calculate of massless vector particle!; (which correspond 10 Maxwell and

. The choice of the vertcx function for the massless vector by thl: mclthat it must hnve conformal weight 1 and thc spin. A natural choice for this vertex is

vcclorofthe vector particle amI kl = {·k = O. The of the vertex is equal 10 the sum of the conformal weights of factors. Since '" has conformal weight! and flU has conformal (J'k 2 , this means that the conformal weight of this vcrtex is given by

~+~+O= I , is the desired weight. This vcrtcx is guaranteed to satisfy the G and L conditions. It is not difficult to calculate the N -point scattering trus massless gauge particle us ing thc same fonnalism developed tachyon. For example, the scattcring amplitude for four massless gauge is givcn by

116

3. Superstrings

where the kinematic factor K is given by K =

+ SU/;23;14 + tU/;12;34) . + I s(k I4 k 32l;z4 + k23 k41 /;13 + k 13k 4Z /;Z3 + k24 k 31/;14)

-i(stl;I3;Z4 1

+ !t(k2I k 43 l;31 + k 34k 12 /;24 + k 24k 13 /;34 + k 3I k 42 l;lZ) + ~U(k12k43l;32 + k 34k 21 l;14 + k 14 k23 /;34 + k32 k 41 /;li), where

(The scattering involving fermions can also be calculated; they will also Mj!~ the basic form, except that the kinematic factor K will depend on the ext:eJ·m.«m spinors.) We should add that another advantage of working with the Pi t~~~~(~~~11 that we can explicitly show the invariance of the amplitude under the (

8Yi = eje, 8ej = e. This generates the group Osp(!, 2) (see Appendix), which is the ric generalization of the proj ective group SL(2, R). This group is geller:ate.ct:~~..:3. the algebra formed by the set G±I/2,

L±h

La.

Let Q be an element of the group OspO, 2). Then the proof of invariance can be shown by noting: Q 10; 0)

= 10; 0) .

(Notice that, as in the bosonic case, the vacuum state 10; 0) is not a nn'VSUJa state. Thus n, which does not in general kill physical states, can annihilate vacuum state.) The vertex rotates as nv(y, e)Q-l = V(yl, ( 1 ),

where V(y, e) is the vertex function before we integrate out over 9. Now that we have established the properties of the three-boson vertex tion in the Neveu---Schwarz formalism, the fermion-fermion-boson (with an external boson line) for the Ramond model can also be We choose V(k) =

where

r :e ik .X :,

3.4 Locnl Two-Dimensiona! Supersyrnmecry

117

Yll is the product of the Dirac matrices. In addition to this vertex.

we also need the propagator: D

F. = -FoI = -'-0 ,

(3 .3.3 1)

Fo is given in (3.2.20), We have chosen tbe intercept to be zero (which see is the proper choice in order to guarantee confonnal invariance; lit stage, however, we cannot yet motivate the choice afzero intercept.) the vacuum

Slate

is DOW a spin-~ fermion,. given by

LIO;q), u, .

(3 ,3,32)

o

we sum over the spinar indices a of the spinar 14",. The scattering for a fennien interacting with several bawns is now given by (3,3,33) we have only exhibited amplitudes with two external fermion lines. Out first guesses, based on simple intuition, have all been suethe action in Ule confonnal gauge was thm of a free thcory, model has been exceedingly simple. There is a price to be paid for principle. because lIle suing model can be fdetorized in any channel, hn"ld be possible to factorize the R model in thc meson channel Wld the NS model or to extract out the multifermion vertex functioll. this is quite difficult. In particular, the fennion vertex. function (with :X~::::~~fermion leg coupled 10 internal meson and fermion lines) is an ~ difficult object to work wi th. This, in tum, makes it difficult to multifermion amplitudes in the NS-R fonnulism. Ith'nuJ!b;', ""n be shown that the R model can be factorized in (he various to derive the NS model , the NS-R. formalism is ac ruaU y quite clumsy cak.-ulating multifermion amplilUdes. In the next chapter, we will see ... the techniques of conformal fi cld theory make the covariant calculation of .. amplirudes possible.

Local Two-Dimensional Supersymmetry

no< th'" "C imposed the foHowing conditions by band: (7;"")

(Jo )

= 0, = o.

(3.4.1)

was given. We merely appealed to the fact thallhcrc must elliSI i from which these constraints can be derived ab initio. will now describe the local gencraii7.3lion of the previous theory. The the construction of this locally sypcrsymmetric action is to add more

118

3. Superstrings

fields to the theory. In addition to the supersymmetric pair (X il ,1/ril)

we introduce a two-dimensional vierbein into the theory and supersymmetric partner: (e~,

X,,),

where Greek letters such as Cl and f3 label two-dimensional vectors in c·llii··Y~~ space, Greek letters like f.1, and v continue to label IO-dimensional soatce-4iii:.l1 vectors, Roman letters such as a, b, c label two-dimensional vectors space, where X" is a two-dimensional spinor as well as a tw.)-diffilens:i9~m vector, and where we have suppressed two-dimensional spinor ind_i.~~ce~s~~'::a~~mm both the vierbein and the spinor Xa have four components, as n supersymmetry. Let us now construct the complete action, fir!,t written d9V~ by Brink, Di Vecchia, Howe, Deser, and Zumino [11-13]: L = - - 14I Jg[g"P fJ"XilfJpXil - itil p"V" 1/rJ.l,

net

+ 2X"p{3 i1/rJ.l,ap

+ 4tJ.l,1/rJ.l,x"pP p"Xp]. Notice that the p matrices used above are actually defined in tw()-diffil~ns.iQ~~~ curved space because they are multiplied by the vierbein:

This action is invariant under 8XJ.I, = s1/rJ.l" 81/rJ.l, = -ip"s(a"Xil - tJ.l,X,,).

2DSUSY:

oe p = -2isp"Xp,

8Xa = V"s. It is also invariant under Weyl (scale) transformations: oX il = O.

Weyl:

o1/rJ.l, = -tu1/rl'. ueap = ue ap • ~

ox" = 4a x". It is also invariant, due to certain identities in two dimensions, under 0Xa

= ip"TJ.

oep = 01/r1' = 8X il

= O.

Finally, by construction, it is also manifestly invariant under local .tw.!ffi:: dimensional Lorentz transfonnations and reparametrizations because presence of the vierbein. (The presence of ordinary derivatives like aa utlt.fJ~::::

3.5 Quantization

n;::~,:O::f,:C,~U.r",:e~d~,,:~p;:a::c;~e,~covariant derivatives D" i

:-f.

. .-

119

is duc to the fact that

field vanishes in the action.) arc in a position to derive the constrain[S thaI we imposed on the hu hand. Notice that the variation ofthe action with respect to the

(3.4.8) .nat;''" of the vierbein's supersymmctric partner yields

p"pil8"X1,l/!IJ. - ~(..p.1/f)P"p/lXa = O.

(3.4.9)

the constraints that we promised earl ier, which are now derived as a of the variation of fields.

e"" = 8"If' (3.4.10) that there are enough symmetries on the X" field to eliminate it entirely, ,!o"~~'" of two-dimensional supc rsymmett}' (3.4.5) and the symmetry the constraints reduce to Tap = i:J,.XIlJtJX" .fa

+ ~iJiJp("afJl'Ir -

(trace) = 0,

= ~p~p"atJx!J,l/FjI =0,

(3.4.11) (3.4.12)

the parentheses symbolize taking one-half of the symmetrized sum). or course, are the constraints we originally introduced al the begiruting discussion in (3.2.7) and (3.2.13). Thus, we now have derived these from an invariant action, rathcr than simply imposing them OIl the . of the sySLCm.

.8 ±t"~/.,fi).

projection operalor has the follow ing properties: ·, p" p.' P±.8p~*=*,

P±· , 8.8., p" =F = 0 , Ill!>

=

"

p!>.8/{1

-

K'· -- P·'K' + .8'

(3.7.5)

128

3. Superstrings

where the supersymmetric parameter is given by K Aaa , where A = 1 or:~:::;i;i: represents a two-dimensional vector index, and a is a spinorial mdlex<m::m: dimensions. We will usuaUy suppress the spinorial index. Notice that two equations state that K is self-dual (anti-self-dual) for A = 2 (1). Then the action can be shown to be invariant under '

8e A = 2ir . naKAcynam~m~ transformation that equates the two supersymmetry parameters . This that the open Type I superstring has only N = I supersymmetry. the signs in the previous equation, yielding opposite chiralities, would supersymmetry totally impossible.) The normal mode expansion thus

L 00

SIII(a,

T)

=

2- 1/ 2

S~e-ill(T-a\

n=-OO

L 00

S2a(u,

r)

=

2- 1/ 2

S;e- ill (T+al,

It=-oo

where

IS:, S!} =

(jlIb(jn._m.

For the closed string (Type II), however, we actually have two options; the fields can either be chiral or not. Closed strings are, by periodic in sigma, which yields the following normal mode expansion: 00

L L

S11lI e-2in(r-a) '

n=-co 00

S2a(a, .) =

S~e-2i,,( 64 states. ~lli) -+ 64 states.

(3.8. 16) (3.8. 17)

N = 2 level, we have 11 52 bosons lind nn equal number offemllons:

a~I Q:~1 Ik)

-+ 288 states,

~I~~I Ii} -+ 224 states,

1152 bosons:

a~llj) -+ 64 stales. a ~lS~ 1 Ib)

-+

~2Ib) -+

64 slales,

Q: ~1~!

-+

U)

~2Ij) -+-

1152 fermions:

a~IO:~J la)

512 stales. 512slales.

64 states, -+-

288 stales,

a~2Ia) -+ 64 states,

s.'. I S~ 1 Ie) -+- 224 Slates,

(3.8. 18)

132

3. Superstrings

This procedure can be repeated at the next level, where we have 1 states. Not surprisingly, we can also regroup these massive multiplets ac(;¢.t1tm ing to 0(9). This is because massless D-dimensional supergravity, w~iW:::: compactified, yields D - 1 massive states. Thus, the N = 1 sector, .,,'dm~ 128 bosons and 128 fennions, can be regrouped as 44 + 84 bosons reducible 0(9) representations, and a single spin-~ 128 multiplet fennions. At the N = 2 level, we have the bosons being regrouped 9 + 36 + 126 + 156 + 23 I + 594 representations of 0(9), while the ferrnl~imi can be reassembled in 16 + 128 + 432 + 576. For the Type II closed string, the spectrum becomes even more ~:!~~:l~t1!IiOO because we obtain supergravity at the massless level, with 128 b9sons and"';;;;.'-1~ fennions: ~

SUSY multlplet:

128 Bosons Ii} Ij} ; la} Ib} , { 128 Fennions Ii) la) ; la} Ii} ,

where the first state refers to the S oscillators and the second to the S If the fermions have the opposite handedness (Type IIA), then this the N = 2, D = IO-dimensional reduction of ordinary N = 1, D supergravity. However, if the fennions have the same handedness, then there are c9'!)~::::: plications. This theory contains a fourth-rank antisymmetric tensor dimensions, with 35 independent components in eight dimensions. It has shown that no covariant action for such a particle exists! Thus, we unusual situation where a supersymmetric action does not exist for this tor. The theory, of course, is still well defined. The light cone theory S-matrix elements are explicitly calcUlable. However, the S-matrix, speaking, does not seem to arise from a covariant action [33]. Finally, if we take the restriction to symmetrized states of the Type I mel9fYi we obtain N = I, D = 10 supergravity. Let us summarize the zero slope of these theories:

= 10 super-Yang-Mills,

Type I ---+ N = I,

D

Type 1---+ N = I,

D = 10 supergravity,

Type IIA ---+ N = 2,

D = 10 supergravity,

Type lIB ---+ nonexistent. At higher and higher levels, it becomes prohibitive to continue this of the spectrum to show supersymmetry. We can prove, however, that the spectrum, at arbitrarily high orders, is supersymmetric simply by that supersymmetry generators exist with commutation relations that with the Hamiltonian. We will now show this by explicitly supersyrnmetry generators to all orders.

18 Light Cone Quantizatioll of the GS Action

133

light cone gauge, the two supersymmetry transformations (3.7.6) and become quite simple: 8S' = (2p+/,2'1 a ,

18X

i

=

0,

&5" =

- j

1

Jjj+pt:V"."'rt;'" and propagators to obt..'lin trees. The great of this approach is tbat we can calculate multifennion amplitudes the same ease with which we calculate multiboson amplitudes. """""e, the propagator and the amplitude for the scattering of fermions are given by

D

~

I--,f: a~"a~

+ nS'!.."s,; + ~p2 ) -'

AN = (0; kd V(k 2)D· . . V(kN _I) 10; kH) ,

(3.9.8) (3.9.9)

128

3. Superstrings

where the supersymmetric parameter is given by K Aaa , where A = 1 or:~:::;i;i: represents a two-dimensional vector index, and a is a spinorial mdlex<m::m: dimensions. We will usuaUy suppress the spinorial index. Notice that two equations state that K is self-dual (anti-self-dual) for A = 2 (1). Then the action can be shown to be invariant under '

8e A = 2ir . naKAcynam~m~ transformation that equates the two supersymmetry parameters . This that the open Type I superstring has only N = I supersymmetry. the signs in the previous equation, yielding opposite chiralities, would supersymmetry totally impossible.) The normal mode expansion thus

L 00

SIII(a,

T)

=

2- 1/ 2

S~e-ill(T-a\

n=-OO

L 00

S2a(u,

r)

=

2- 1/ 2

S;e- ill (T+al,

It=-oo

where

IS:, S!} =

(jlIb(jn._m.

For the closed string (Type II), however, we actually have two options; the fields can either be chiral or not. Closed strings are, by periodic in sigma, which yields the following normal mode expansion: 00

L L

S11lI e-2in(r-a) '

n=-co 00

S2a(a, .) =

S~e-2i,,( 64 states. ~lli) -+ 64 states.

(3.8. 16) (3.8. 17)

N = 2 level, we have 11 52 bosons lind nn equal number offemllons:

a~I Q:~1 Ik)

-+ 288 states,

~I~~I Ii} -+ 224 states,

1152 bosons:

a~llj) -+ 64 stales. a ~lS~ 1 Ib)

-+

~2Ib) -+

64 slales,

Q: ~1~!

-+

U)

~2Ij) -+-

1152 fermions:

a~IO:~J la)

512 stales. 512slales.

64 states, -+-

288 stales,

a~2Ia) -+ 64 states,

s.'. I S~ 1 Ie) -+- 224 Slates,

(3.8. 18)

132

3. Superstrings

This procedure can be repeated at the next level, where we have 1 states. Not surprisingly, we can also regroup these massive multiplets ac(;¢.t1tm ing to 0(9). This is because massless D-dimensional supergravity, w~iW:::: compactified, yields D - 1 massive states. Thus, the N = 1 sector, .,,'dm~ 128 bosons and 128 fennions, can be regrouped as 44 + 84 bosons reducible 0(9) representations, and a single spin-~ 128 multiplet fennions. At the N = 2 level, we have the bosons being regrouped 9 + 36 + 126 + 156 + 23 I + 594 representations of 0(9), while the ferrnl~imi can be reassembled in 16 + 128 + 432 + 576. For the Type II closed string, the spectrum becomes even more ~:!~~:l~t1!IiOO because we obtain supergravity at the massless level, with 128 b9sons and"';;;;.'-1~ fennions: ~

SUSY multlplet:

128 Bosons Ii} Ij} ; la} Ib} , { 128 Fennions Ii) la) ; la} Ii} ,

where the first state refers to the S oscillators and the second to the S If the fermions have the opposite handedness (Type IIA), then this the N = 2, D = IO-dimensional reduction of ordinary N = 1, D supergravity. However, if the fennions have the same handedness, then there are c9'!)~::::: plications. This theory contains a fourth-rank antisymmetric tensor dimensions, with 35 independent components in eight dimensions. It has shown that no covariant action for such a particle exists! Thus, we unusual situation where a supersymmetric action does not exist for this tor. The theory, of course, is still well defined. The light cone theory S-matrix elements are explicitly calcUlable. However, the S-matrix, speaking, does not seem to arise from a covariant action [33]. Finally, if we take the restriction to symmetrized states of the Type I mel9fYi we obtain N = I, D = 10 supergravity. Let us summarize the zero slope of these theories:

= 10 super-Yang-Mills,

Type I ---+ N = I,

D

Type 1---+ N = I,

D = 10 supergravity,

Type IIA ---+ N = 2,

D = 10 supergravity,

Type lIB ---+ nonexistent. At higher and higher levels, it becomes prohibitive to continue this of the spectrum to show supersymmetry. We can prove, however, that the spectrum, at arbitrarily high orders, is supersymmetric simply by that supersymmetry generators exist with commutation relations that with the Hamiltonian. We will now show this by explicitly supersyrnmetry generators to all orders.

18 Light Cone Quantizatioll of the GS Action

133

light cone gauge, the two supersymmetry transformations (3.7.6) and become quite simple: 8S' = (2p+/,2'1 a ,

18X

i

=

0,

&5" =

- j

1

Jjj+pt:,~'~erbein and the X field, we get the NS-R tbeory in the (3 . 10.3)

of the invariancc of the original action, which is locally super, we can construct the following currents and set them equal to

10 = ~/Pd1/l1'&bXI" Toh :::: 8aX I' &hX"

(3. 10.4)

i _

i -

+ 41/1" p"8h1/l,, + 41/11'Ph8a1/l1'

- (Trace). (3.10.5)

calculate the momenls of these consttainls, they fonn the algebra: [L",. L it ]:::: (m - n)L", ..."

NS:

D

+ S(m

,

- m)o",._n,

[L m • G, J:::: (-~m - r)G",+"

{G" C s }

::::

2LrH

+ ~ D(r 2 -

[L .. , Ln] = (m - n)L",+n

D

~)O,._S,

+ 8m

,

[L"" Fn]:::: {~m - n)F",+n.

0",._,., (3.10.6)

{F",. f~l = 2L,.+n + ~Dm2o,"._",

the bosonic case, it is straightforward to make the transition from path to the hannonic oscillator formalism. The Hamiltonian is diagonal

138

3. Superstrings

in the Fock space of harmonic oscillators, so we can remove all i'Iltelrtnc functional integrations. We find that the tachyon vertex function with weight I can be \XT1'itt"n kJl1/fJ.i Vo.

However, we find that there are two distinct ways or "pictures" in can write down the N -point amplitude, called the FJ and Fz tbrmall1slm rules for forming the N -point functions are ' vertex = V, I propagator = - - Lo - I tachyon = kJl . b- 1/2 10; k) ,a'k 2 =

4,

2

a'k = 1,

vacuum = 10;k), vertex = V, propagator =

I J '

Lo - 2 tachyon = vacuum = 10; k) ,

2 "/k ... -

The advantages and disadvantages of the two formalisms are as

!2' lUlJlUW

(I) In the Fl formalism, manifest cyclic symmetry is much easier to However, we must carry along the excess baggage of the which decouples completely in the theory. (2) In the F2 formalism, cyclic symmetry is obscure, but ghost is easy to perform. The advantage of the F2 picture is that we working in a smaller Fock space.

,",UJ""""J

The N -point function is represented as

An = jd fL N (0;01 V(k1,Yl), ... , V(kN,YN) Yl

YN

=1 0 ;0).

Although the NS-R model with GSa projection can be shown to same number of fermion and boson states, the proof that the theory is space-time supersymmetric is prohibitively difficult. As a result, we space-time supersymmetry by postulating the new GS action, where genuine space-time spinors. The GS action is S=

4a'~

j

daar{Jgg"fJ IT" . ITfJ

- 2s"fJ81rJ.io"eI82rJ.iofJe2},

where

+ 2is"fJ o"xJ.i(8 1r J.iOfJe 1 -

2 8 r J.i .

(3. '

References

139

'action is invarianl under

S9 A = e A ,

sx" = 8(J'" = 2; r

ii A r"9'\

(3.10.13)

. n...K"" •

ax" = iO"r

ll

8l/ .",

8(Ji8 ..~) = - 16Jg(p:ri'fJ oy 8'

+ p:rI(2fJ(jy02),

(3.10. 14)

~~,:~;~n~~,:~~~~:S~~S, formulation is that it is spare-time supersymmetric

.u levels.

the price we pay for this is that covariant quantization

. theory is nOl0riously difficult, because it is a coupled system even al the Therefore, as we will go to the light cone gauge, where the theory a linear theory:

S=

2411"0"

J

du dr(J"XiaO Xi - i Sp"abS).

(3. 10. 15)

",,,ion is also spaco-timc supersymmetric, with generators given by Q~

= (2p+),n SO. 00

i Q'" = (p+)-I f1".,. r , " ""... L ... _~a:".

(3 . 10.16)

Intiico,,,"wt,"i,," relations between these generators are given by

I

IQ"Q'I~2P+""

IQ", QIi ) =

../2y:iJp i,

(3. 10. 17)

IQ', Q'I ~ 2H"'.

(3. 10.18) ,we will investigate the conformal field theory, which combi nes the best of both the NS-R and formalisms.

as

J, L. Gervai s and B. Sakila, Nucl. Phys. 834, 632 (1971).

W. Siegel, PhYi. Lett. 1288,397 (1983); Nue!. Phys. 8263, 93 (\985); Classical ua.,;m Oral'if)' 2, L95 (1985). Brink and J. H. Schwarz, Phys. Leu. 1008, 3 10 ( 1981). P. Ramond, Phys. Rev. D3, 2415 (197 1).

140 [5] [6] [7] [8] [9] [10] [II] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

3. Superstrings

A. Neveu and 1. H. Schwarz, Nucl. Phys. B31, 86 ( 1 9 7 1 ) . . > / A. Neveu, J. H. Schwarz, and C. B. Thorn, Nucl. Phys. B31, 529 (1971······· 1. Wess and B. Zumino, Nucl. Phys. B70, 39 (1974). See also Y. A. Gol'fand and E. P. Likhtman, JETP Lett. 13,323 (197 Volkov and V. P. Akulov, JETP Lett. 16, 621 (1972) (English, p. 438). M. A. Virasoro, unpublished. Y. Aharonov, A. Casher, and L. Susskind, Phys. Rev. D5, 988 (1972). L. Brink, P. Di Vecchia, and P. Howe, Phys. Lett. 65B, 471 (1976). S. Deser and B. Zumino, Phys. Lett. 65B, 369 (1976). For an earlier attempt, see Y. Iwasaki and K. Kikkawa, Phys. Rev. (1973). N. Ohta, Phys. Rev. D33, 1681 (1986). M. Ito, T. Morozumi, S. Nojiri, and S. Uehara, Progr. Theoret. Phys. (1986). 1. Schwarz, Suppl. Progr. Theoret. Phys. 86,70 (1986). F. Gliozzi, J. Scherk, and D. Olive, Nucl. Phys. B122, 253 (1977). C. B. Thorn, Phys. Rev. D4, 1112 (1971). L. Brink, D. Olive, C. Rebbi, and 1. Scherk, Phys. Lett. 45B, 379 (1 S. Mandelstam, Phys. Lett. 46B, 447 (1973). ,. J. H. Schwarz and C. C. Wu, Phys. Lett. 47B, 453 (1973). D. Bruce, E. Corrigan, and D. Olive, Nuc!. Phys. B95, 427 (1975). M. Green and J. H. Schwarz, Phys. Lett. 136B, 367 (1984). M. Green and J. H. Schwarz, Nucl. Phys. B198, 252, 441 (1982). T. Hori and K. Kamimura, Progr. Theoret. Phys. 73,476 (1985). M. Kaku and J. Lykken, in Symposium on Anomalies, Geometry, and (edited by W. A. Bardeen and A. R. White), World Scientific, Singap10re See also 1. Bengtsson and M. Cederwall, Goteborg preprint 84-21. T. 1. Allen, Cal Tech preprint CALT-68--I373. T. Hori, K. Kamimura, and M. Tatewaki, Phys. Lett. 185B, 367 (I C. Crnkovic, Phys. Lett. 173B, 429 (1986). R. E. Kallosh, Pis 'ma JETPh 45, 365 (1987); Phys. Lett. 195B, 369 (1 1. A. Batalin, R. E. Kallosh, and A. Van Proeyen, KUL-TF-87117. N. Marcus and J. H. Schwarz, Phys. Lett. 115B, 111 (1982).

4

nformal Field Theory Kac-Moody Algebras

Confonnal Field Theory the mysteries of the superstring is the existence of cwo ways of forthe theory: the first is the NS-R model (after the Gsa projection)

~

:~;~~~I:~(~~'V~'::'~'O;":~', the second is town he GSdistinct modeladvantages with genuine . Eacb ~"'~~d formulation has its and discuss theconfonnal field theory, which to sec the dynamic link between the two formalisms. The confonnal

. theory of Friedan and Shenker [1] combines the best feature s of both . . Conformal field theory allows us to:

""xl""

covariant anticommuting spinor fields based entirely on free

The GS formalism, on the other hand, is based on complicated

in",,,,,;','g fields that make covariant quantization exceedingly difficult.

Construct explicit covariant tree gmphs for multi fermion scattering. 10 NS-R formalism. however. this is prohibitively difficult because it is ":~~~:~,to introduce complicated projection operators that extract out tI . The confonna! field theory replaces these awkward projection operators with free Faddccv- Popov ghosts, which are easy to manipulate. and NS-R formulations and see their relation, ~::~'7.I~~: between the ~,~ . . It provides the bridge by which we can interpret the resulls of one formulati on in tenns of the other:

as

conformal field theory

++- {

as model. NS-Rmodel.

142

4. Conformal Field Theory and Kac-Moody Algebras

(4) Construct the covariant supersymmetry generators. This is ImpOi,Slljli the NS-R formalism, and is possible in the GS formalism only in cone gauge. (5) Describe both the NS and R sectors of the theory with the same . . . . .. rather than using the clumsy formalism of two distinct Hilbert on the vacua IO)NS and IO)R U a . This is accomplished by a process bosonization, i.e., constructing fermions out ofbosons in two' ..

There is a small price we must pay, however, for confolmal field Ghosts and antighosts proliferate quite rapidly in this formalism, p"r"'ii" for superstrings, and a strange phenomenon called "picture changing" introduced. Fortunately, these ghosts and antighosts are free fields, easy to manipulate. Furthermore, conformal field theory does not Setlm derived from any action, as in the GS and NS-R formalisms. Coni()mll:ll theory stresses the group-theoretic behavior of the fields, rather than nrC"'''t from an action. Therefore, we suspect that there is a higher, yet uU~'1i'''V' first quantized action that exists beyond the GS and NS-R actions. (We shoul4 stress that conformal field theory is not a field theory in of second quantization, i.e., where we start with the formalism of :SC1IWl Tomonaga, and Feynman. The second quantized field theory of strings discussed in Part II of this book.) The essence of conformal field theory is that it stresses the use of invariance alone to calculate correlation functions between different 6]. It is quite remarkable that conformal invariance by itself is determine almost completely the structure of N -point scattering amlPU For example, in (2.6.9) we have encountered operators that have the matrix elements:

where the function f can be a power or a logarithm. For example, the element of tree amplitudes between two bosonic strings is (X(w)X(z))

~

log(w - z),

while the matrix element of two normal ordered vertices is (elkX(w)e-lkX(z)) ~

(w _

zrk2.

So far, we have used explicit representations of the fields in order to the matrix elements. However, the process can also be reversed. If all about a field's group properties, we should be able to calculate elements and even reconstruct the field. The essential idea behind conformal field theory is to use the properties of the field ¢ to completely determine all of its matrix elelnel1 even reconstruct the field. In conformal field theory, this is acc:oITlpllst

4.1 Confonna1 Field Theory

14]

the short-distance behavior ofleft-moving or right-moving fields (4.1.2)

all possible matrix elemenLS can be calculated from the oflhefields themselves. field theory, we also construct a spinfield Sa that transfonns as space-time Lorentz transfonnations and has eonfannal

. However, coming from the Faddeev-Popov ghost sector, we find yet j,,,-fie.ld 'with ,on 'o=a""ight ~ . It is the product of these two fields, one field and the other being a ghost field, that allows us to construct ~~r!~e~~ fermion vertex with the proper weight one. This vertex., although ~\

exponentials of fields, is defined totall y in tenns of free fields and solves the problem of constructing a simple fermion vertex function.

which tbis spin field is actually introduced is through the process i " [7,81. i.e., creating a femti OD out of a boson. II is easy, to create a boson out ofrwo fennions. However (in two dimensions) the option of being able to create fermions out ofbosons, which was thought to be impossible. Ac(utllly. the hint of how to "bosonize" was given in the previous chapter, when we introduced the venex function (4.1.3) (2.,6.22) we saw that (4.1.4) ti~'':;;~:~bY choosing various values of the momenta in the exponential, we

create operators that satisfY the relation

Ii

(4.1.5) words, we can create a field with fennionic commutation relations ,fl,,,oniici,,,me.ni, oscillators. The key to this construction is the nonnal of oscillator modes in two dimensions. This feature does not carry .• -,~ . fie ld theories. (In hindsight. it is possible to see wby and bosons are so closely linked in two dimensions but nOI in higher For the Lorentz group O( I, I), which has only one generator, tile called "spin" in one spatial dimension does not have much meaning.) method of bosonization via nonnal ordering of fields will also be the ~ ,onst""cting the confonnal field theory. We will construct the fermion Sa via normal ordering of exponentials of bosonic fields. From this, ;11 ,w"truct the covariant $upersymmeuy operator. The advanlage of this we can now discuss spaco--time spinors using one common vacuum bosons and fennions and that the entire construction is based on free

' ' ' m".

we discuss the superconformal case, let us begin our discussion by conformal field theory. Let us make the most general conformal

144

4. Confonnal Field Theory and Kac-Moody Algebras

transfonnation on the world sheet variable z:

z --+ z(z). Under this conformal transfonnation, we say that a primary analytic transfonns with conformal weight h if it transfonns as .

¢(z) =

¢(Z) (dZ)h dz

(Secondary fields transfonn as derivatives of ¢(z).) This conformal the same concept we introduced back in (2.7.6) when we were dlsCU~fSj conformal properties of vertices. We see now that mathematically mal weight is an index to label irreducible representations of the group generated by the Virasoro algebra. We can now construct objects that are invariant under a transfonnation: ¢(z) dz h = ¢(Z) dz h. Once we have defined how fields transfonn under a conformal trilJ1St()lti:l we mustthen check that we have closure of the group under two such operations. Let us say that we make two successive confonnal traJlstl:)11l:

z --+ ZI(Z)

--+ Z2[ZI(Z)].

Then the field transforms as V1¢(z)V]- 1 = ¢(z]) (dZI)h dZ ]

2

V2V]¢(z)Vi Vi = ¢[Z2(Z](Z))]

'

(dZ 2 dZ])h dz] dz

Thus, the confonnal transfonnations fonn a group with the CO]TIPoslti, V3 = V2V], Z3

= Z2(Z] (z)).

The closure of the algebra can most easily be seen if we take an imal conformal transfonnation. Consider an infinitesimal vru'i/'lticm coordinate: 8z = s(z). Under the transformation (4.1.7), we have the infinitesimal traJlstlJrII 8¢e = [sa

(We will abbreviate az by arrive at

a.)

+ h(as)]¢.

If we commute two such variations,

4.1 Confonnal Field Theory

[El. e2l = £13£2 -

£2a£l.

145

(4.1.1 6)

this relation with (2.1.30).) shown once again that the group closes with arbitrary weight h. understanding of the meaning afthe conformal weight h, let liS conformal weight a f the string. Under the infinitesimal variation we use the chain rule 10 show that the string transforms as

(4.1.17) the string field was weight O. Likewise, il is easy to show thaI the of the string field has weight 1:

S3X .. (z) = rea

+ (3e)J3X,,(z).

(4.1.18)

swnmarize the weights of some common string fie lds: Field

Weight

x ax ax· ax

o

,

(4.1.19)

2

this means thai the energy-momcntum tensor or the Virasoro have 2. . . The product of two fields of weights h) and h2 produces + h2 at the same point: ¢(hd(Z)¢(/!l\z:) = ¢(h l +~lJ (z:).

(4.1.20)

the important fac t, which w ill be used throughout thi s book, that the of an objcct of weight I is an invariant:

8

f

l,b(ll(z)dz = fr vi.l¢ + (i.lv)4>Jdz

=

f al'¢ld,

=0.

(4.1.2 1)

use this fac t in constructing vertex operators and also tbe action of the q"',"';;'od theory. us investigate in detail how the energy-momentum tensor acts on

,Ic,~:~~~: of the theory.

e

is written in tenns of X,,(z. z),

s=

_, 2rr

f

d 2zaX 1'i.lX" ,

(4.1.22)

'''''~!)'-ID''m''nn"," tensor in (1.9 .17) associated with this action is

Ts = - ~ax"axl"

(4.1.23)

146

4. Confonnal Field Theory and Kao--Moody Algebras

The transformation of fields under conformal transformations is by the integrated energy-momentum tensor parametrized by a small

s:

1,

Te = _1_.

ko

21T I

dzs(z)T(z),

where we take the line integral that encircles the origin in z-space. portance of the energy-momentum tensor T is that it is the(Z. e) under a transformation:

[L m• :1/I$«z)1/r"'{ w) + other tenns.

(4 .3.1 0)

calculate the operator product of '" and S. we consider the three-point

(4.3.11) the vacuum is the NS vacuum. In the limit of z\ -4 00 and Z3 -4 0, field changes the NS vacuum into a vacuum with spinor quantum

mb'''', i.e., the R vacuum jOk 5.(0) 10)Ns = 101. u. ;

(0 1",5.(00) =

U.

(4.3.1 2)

(0 1..

that the spin operator allows us to go from the NS vacuum to the R ;,""m" which was not possible in tbe earlier NS-R theory.) This means that J J) can be written as

(4.3.13)

'" c~ly the zero modes of the '" field survive this vacuumexpcctalion product, we are left with the matrix element of a Dirac matrix: I 1 1J.r1' (z )S.,(w) ""'" -Ji (z _ w)1/2(yJol)eS.6(z) + ....

(4.3.14)

in tum, means that 1/1 occurs in the operator product of tv.'o S's:

S.(z)S'(w) -

I .(y )',,"(z)+ .. ·. -Ji(z _ w)l/4 JoI tI

(4.3.15)

, we need to know the short.distance behavior between two spin fields. saw that the confonnal weight of the spin fi eld is ~. Thus,

S,,(z)S.6(w) "" - ~:(z

-

W) - 5/4 +"',

(4.3.16)

~ is twice the dimension of the spin field. Finally, we can bring (4.3. 10), (4.3.14), and (4.3.16) together, which yields short-distance behavior of two spin fields. In summary, invariance arguhave established the short-distance behavior of the spin field, which can represented as p

-1

.6

S,,(z:)S (w) ...... (z: _ w)5/4~"

I

+ -Ji(z _

.6

W)l/4 (y" ),,, '"

I'

(z:)

158

4. Conformal Field Theory and Kac-Moody Algebras

+ .J2(z ~ W)1/4 (y !-,V)"p Vr!-'(z) Vrv(z) + ....

i.

We saw that the spin field has dimension What we want is a vertex of dimension 1, suitable for use in a multifermion amplitude. the missing factor of ~, let us now tum to the ghost sector of the theeq - ).., n ::: - eq + )...

(4 .4.27)

zeroth component of the ghost number current and the Lo have the action on these SUItes:

i, lq) ~ qlq ). L~ Iq ) = !eq(Q

+ q) Iq) .

(4.4.28)

'rt;,,,h,,.the last identities sbow that the only nonvanishing matrix elements (-q - Q lq )

~

1.

(4.4.29)

"',;,,' way'o show this is take the matrix elements ofthe currents between

162

4. Conformal Field Theory and Kac-Moody Algebras

different vacua, where each vacuum is labeled by the number q:

(q'ILo Iq) •

(q'ljo Iq) .

All the matrix elements are zero, except when q' = -q - Q. This is highly unusual. In the usual Veneziano model, there was unique vacuum. Now, there appears to be an infinite number of vacua by q for the ghost sector of the NS and R models! The existence ofan

number of Fermi and Bose vacua is one of the unusual features of . . u,Hu, field theory. This means that there is an anomaly in the ghost current. The prO'Olelll in (4.4.22) and (4.4.23), i.e., the fact that the U(l) ghost current-has anol~ commutation relations with the energy-momentum tensor. The an()Dli term corresponds to the violation of the conservation of the ghost In fact, the divergence of the current is given by 8zj z = ~ Q ..fiR(2) , two-dimensional curvature density. The ultimate origin of all these .. actually lies back in (4.1.36), when we calculated the Faddeev-Popov . minant over \/z and \/z. A careful analysis of the eigenvalues of these ooe:ratf shows that we have to remove the zero modes, or else the determinants sense. When we exponentiate these determinants by expressing them in ofFaddeev-Popov ghosts, these zero modes, in turn, correspond to solutions to the equations 8zcz = 8zbzz = O. We cannot give. a full UU)\.",,; of these zero modes, unfortunately, because it requires new inionna1tioll will not be discussed until Chapter S, when we analyze anomalies in (Briefly, this anomaly can be shown to be related to the topology of mann surface swept out by the string. By integrating the divergence for the U( 1) ghost current, we can use the Gauss-Bonnet theorem to d 2z,JgR(2) = -Sn(g - 1), where g is the number of holes or 11]

for arbitrary cI> has a vanishing anti commutation relation with the because Q is nilpotent. However, all these vertices are spurious. are null and do not couple to real states IR), which satisfy the QBRST

IR) = 0

so they cannot be used as vertex functions. They simply produce elements with the physical sector of the theory. However,there is a function for which this reasoning does not apply: VI / 2 =

2[QBRST,

;-V- I / 2 ].

Normally, we would expect that such a vertex is also spurIous and couple to real states of the theory. However, ;- VI/2, as we said v~. . .v. part of the irreducible Fock space of the theory, and thus we cannot that this commutator vanishes on contraction with real states. This vertex not necessarily vanish because

After working out the details, we find that this vertex is equal to VI/2

= ua(k )e ik . X[e O/2) has weight ~. and e(lf214> has weight Thus, our vertex 1 weigl'tl is given by

i.

i·

v _ I/l.a --

- i.

Sa ,-( l{2I4>.,/~' x. . ..

(4.6.14)

':'e~::,:~~bave an explicit representation of the spin field, we can calculate

tri

for fermion- fermion scattering. us calculate the four·fennion scattering amplitude, represented by the ofthrce independent factors involving X. Sa, and e- f1/ 2\f1:

(4.6.15) ZI

=

00,Z2

=

I ,Z)

=Z , l. = 0.

us calculate each factor separately. Using the formula derived earlier in we find tbal the X -dependent ractors are equal to

"p'ee :',

( V(k l .Z I ) V (k 2• Z2)'" V(k ...,. ZN ») =

D (li -

Zj)u

J

•

(4.6.16)

llej

V (k. z)

the sum or all k, equals zero.

= ;eiU{lI:

(4.6. 17)

170

4. Confonnal Field Theory and Kac-Moody Algebras

Let us now specialize to the case of the ghost contribution, which (e-(l/2)rf>(OO)e-(l/2)rf>(l)e-(l/2)rf>(z)e-(l/2)rf>(O))

= [z(1-

z)r l / 4 .

Lastly, the spin field contribution is

(S,,( 00 )S{3( 1)Sy(Z)S8(0)) =

[zO - z)r3/ 4 {0 - z)(y/i),,{3(YI'J,,8 - Z(y/i),,8(Y/i){3y} ,

Putting all three pieces together, we arrive at

A4 =

g211

x

dzl

J •k2 -

0-

1

Z)kJ .... -1

{(1- z)(y/i)"{3(Y/i)y8 -

Z(y/i),,8(Y/i){3y},

Rewriting this amplitude, we finally arrive at

A4 = g2

{BO -

'-'

~t, -~s)(y/i)"{3(Y/i)Y8

- B(-~t, 1 - ~s)(y/i)"8(Y/i){3y}. This calculation and others involving N -point fermion scattering amp}f are not that difficult with conformal field theory [9, 10], but would extremely difficult in the older covariant NS-R and GS formalisms.

4.7

Kac-Moody Algebras

Although the results of conformal field theory are powerful, to the the various rules and conventions may seem a bit arbitrary and appears, at first glance, that conformal field theory depends on and accidents rather than anything fundamental. In reality, the underlying consistency and elegance of conformal are due to new infinite-dimensional Lie algebras, called Kac-Moody [11-22], which are powerful extensions ofthe usual . gebras. They were discovered by mathematicians V. G. Kac and R. in 1967, although one form of them was already known to nh'VSl,dst rnid-1960s as current algebras. Together with the superconformal two dimensions, the SO(IO) Kac-Moody algebra provides the U"'.•u.,. framework for conformal field theory. Many of the matrix elements formal field theory, in fact, can be viewed as Clebsch-Gordan cO()mlCl Kac-Moody algebras. We define a Kac-Moody algebra to be a generalization of an algebra such that its generators obey I i Tj] - 'fijlTm+n [Tm' n - I

+ k rno~ih0m,-n·

The algebras looks very much like an ordinary Lie algebra, exc:epI infinite integer index rn on each generator and the COl1stant k,

4.7 Kao--Moody Algebras

171

i The zeroth component of the T 's is nothing but the algebra of a finite . We will often find it convenient to rewrite the generators of the ~~~;:~; algebra as the Fourier components of a single functi on defined (4.7.2)

• also break down the generators of the Kac-Moody algebra into the subaigebra and ils eigenvectors:

£.(9).

(4.7.3)

a Kao-Moody algebra looks like an ordinary Lie algebra smeared over us now construct what is called the "basic representation" of the K.acalgebra using vertex operators. This representation holds only for laced groups (i.e., groups with roots of equal length, which are A, E Lie groups) y..ith level one (k = 1). Let us first define the string

(4.7.4) introduce the basis vectors for the lattice of our Lie algebra such that (4.7.5)

,,110'" us 10 write the string variable and the vertex function as vectors lallice: j and the - sign fork < j. the ilh and jth eiemenlS commute, the H 's are generalizations of the subalgebra, i.e., the set of mutually conunuting elements of the Lie

= Ct -

e)

172

4. Conformal Field Theory and Kac-Moody Algebras

Let us now calculate the commutator between these generators. In same way as before, we find that the product of two vertex functions

V(a, 8) = :eimp(O):, V(a, 8)V(,8, 8') = !':!..(8 - 8,)-a·{3:e ia q,(O)+i{3q,(O'):, where !':!..(8-8') = e- i(l/2)(O-O')(l_(l_s)e- i(O-O'))-1 ~ 8 -~,

+ is +....

We now have all the identities necessary to compute the commutator all the generators. We find, by direct computation,

[Hi(8), Hj (8')] = 0, [Hi(8), Ea(e')] = -2n8(8 - 8')aiEa(8). Notice that the H's are mutually commuting, as they are in the ornin&~.

(4.9.6)

y = eT/.

in turn. finally allows us to write the vertex function with conformal l on-shell: V_ 1/2 = u"e--{I /2}1; S«tI~·x

(4.9.7)

has conforma l weight (4.9.8)

t

order to write amplitudes that can contract w ith the - vertex operator, a corresponding vertex function with +~, which is given by

= u"(k)e r.X [e(I ,l2lt; (aX"

+ iik ·l/il/I")(y"kpSP + ie3,,/2T/bS«].(4.9.9)

these two vertices, we can now construct multifennion scattering litud".

176

4. Conformal Field Theory and Kac-Moody Algeb!as

One drawback of this formalism, however, is that we have an infinite of vacua and hence an infinite number of vertex functions. These vacua· constructed as eqo1"S;"" equivalent to a sphere. There are two types of loop diagrams create out of the closed string. Let us cut 2N holes in the sphere and mark the orientation of points along the circular edge of each hole. i;';~':::~rs oftlOles to obtain a sphere with N handles. The two types

PI,n., d,i,s",ns, where the orientation of the circular edge of each pair of is preserved when we resew the diagram. A doughnut, for example, II planar single-loop diagram. ~o"ori.' nl"bl, diagrams, where the pairs of points along the circular edges the holes are joined by reversing tbe orientation of the holes. A Klein , for example, is a nonorientable diagram (in Fig. 5.3, we see how bottles can be assembled from fiat two-dimensional surfaces that "vt"h,'~ boundaries identified). (We note that only Type I strings, whieh no orientation or direction, have Klein bottles in their perturbation Type n strings carry an intrinsic orientation and cannot p roduce bottles.)

",i".

r.

~th~'~fu?~nfc~ti~o~na~I~"~o~n ~:aliJ.~sm[~,~w~c~sa~W~"'~H~Cf'~th~'~t :"'~I~"~C~'~1~'~~

spbere[9-II] . . method of calculating this Newmann function was to conformaJly were conor sphere to the upper half-plane or the entire complex plane. We a method developed in electrostatics, the method of images, to the Neumann funclion: G(z, z) = In Iz - z'l

+ In Iz - tl ·

(5.1.8)

we will generaJjze this discussion 10 Riemann surfaces with holes. Formathematicians long ago wrole down the Neumann function for the sphere with N holes. In fact, Burnside [12] solved this problem in ! The solutions 10 this classical problem are given in terms of automorphic wruch we will now analyze.

Single-Loop Amplitude let us set up the functional integral for the single-loop diagram:

A=

1

DX dtte lS

s

fI

elk, x, ,

(5.2.1)

1=1

we functionally integrate over a horizontal strip in the complex plane

182

5. Multiloops and Teichmiiller Spaces A

B

C

0

C

·0

A

0-

C

•

B

,

A

B

0

C

E

0

o

FIGURE 5.3. The Klein bottle. By identifying the opposite sides ofa rp("hn can obtain either a torus or (ifthe orientation ofthe sides is reversed) a shown here. The Klein bottle is a two-dimensional closed surface with only .,

(see Fig. 5.4) that has finite length and then identifY the left and right this way, we construct a surface topologically equivalent to a disk The functional integral, of course, can be calculated explicitly, I P,,'VIn the factor exp

[L.:kiN(Zi, zj)kjJ,

where N is the Neumann function. Now, let us topologically deform the horizontal strip into the surface. Consider an annulus, defined as the region in the upper that has an outer radius of rb and an inner radius ofra and a ratio w Now impose the fact that the outer perimeter is to be identified with perimeter. This means that a point z on the surface of the outer pe.ri·m~)t¢J

,0'''' '.' . Confonnal surface of the single-loop open string diagram. In Ihe p Ihe surface corresponds to a rectangle of width :r and of arbitrary length, we identify the ends. Rectangles of constant width with varying lengths confonnally inequivalent, so we must integrate over all lenglhs in the path The wavy lines correspond [0 "zero width" strings or tachyons, which may to the surface on cilher the upper or lower boundary. In lhe ~.plane, the

corresponds 10 a narrow tube that is bent into a half-circle. with external

!' ,m,,";';. from the ends. id,,.,;fied with the point on the inner perimeter with the same polar angle:

z -+

WZ.

(5.2.3)

identification creates a semicircular tube in the upper half-plane. The

,fo,,,,,,1 '''''J'pi'g from the horizontal strip 10 this tube isjust Ihe exponential. tube, in can be mapped inlO a disk with a ho le by stretching one of the tube until it becomes a large circle and then shrinking the other When we construct Neumann functi ons in this annulus, we obviously are 'rest,d in functions that have the property ~(z) ~ ~(wz).

(5.2.4)

take arbitrary powers of w, this idemificalion acrua!ly divides the upper . in fi nite number of concentric circles. Each concentric circle radius w~. By identitying the outer perimeter of one annulus with its inner

184

5. Multiloops and Teichmilller Spaces

perimeter, we can create an infinite succession of tubes. Thus, the entfi't!tl\ilii

half-plane can be decomposed into an infinite sequence of these £uu'eS{ QUl are interested in only one of them. Thus, we have divided up the upper half-plane by making an 1·(lentlti,c~ concentric circles generated by multiplication by the number w. This thus parametrized the disk with a hole. This is called the Teichmiiller for the single-loop diagram. These parameters have a: natural geller:ali:l~~' surfaces with N holes. . In general, we can also divide up the upper half-plane by using an projective transformation, which, as we saw earlier,maps the real the real axis. (In general, circles are mapped into cirfles under a transformation.) We define anautomorphicjUnction as one that has the 1/I(Z) = 1/I(z'),

.0

where we make a projective or SL(2, R) transformation:

,

Z =

SL(2, R):

az + b , cz +d

_

0

{ ad - bc = 1.,

for real a, b, C, and d. (Phases can sometimes enter into the erties of these functions.) Fortunately, mathematicians have function. The Neumann function is N(z, z')

"al"Ula.,

= In 11/I(z' Iz, w)1 + In 11/1(2' Iz, w)l,

where we demand, up to a phase, that the function be periodic: 1/I(z, w) = 1/I(wz, w).

Explicitly, this periodic function is In 1/1 (x, w)

= ln(l

- x) -

! In x +

ln2 x 2 In w

00

+ ~)ln(l -

wnx)

+ In(l -

w nIx) - 21n(l - w n)

n=1

Notice that this function explicitly has the required periodicity mentioned earlier. Thus, when we exponentiate the Neumann turlctlq momentum factor in the integrand contains factors like

nn(Zi N

WnZj)kiok j •

n i<j

This function can also be rewritten in terms of Jacobi theta turlCtJ01l!~;: .

.

2

1/I(x, w) = -21fl exp(lJ!'~ Ir)

8,(~lr)

, (I ' 8, 0 r)

5.3 Harmonic Oscillators

185

Inx

~:::::2Jri' In w r=2Irj'

(5.2. 12)

are four Jacobi theta functions, given by

~

__ oo

..n, ~

= 2f(ql)q 1/ 4 sin Jr LI

(1 - 2q 'bt cos 2lT V + q"'), (5.2.13)

L ~

0 J (v I r ) =

q~ ' e21"~" ~

.., +

= f(q2) n (l

2q 2,, - 1cos 2:rr v + q411 - 1) •

~

f(q') = TI(I - q"'),

q =~If I,

(5.2.14)

'~::':'~~;:d~function. is thus

;:

confonnal im'atiance, we can also determine the factors a and dlL . ~~~:~ we will find it convenient simply to swnmanze the results of the ; i oscillator approach, which also arrives at the correct integmtion

Harmonic Oscillators convenient language in which to discuss the planar, nooplanar, and

~[~:i;;Single-l00P diagrams is the operator (anguage, which. ofcoursc. representation of lhe functional integral. Lei us begin with a strip

~

I

plane r + ia t the sets orall points from a = 0 to If. Now place with exlemallines along the x-axis. As before, if we take this

186

5. Multiloops and Teichmilller Spaces

to be our surface for the functional integral, then we can insert a COlmp of intermediate states everywhere: IX)

f

DX{.t (XI

=

1,

and we obtain for the functional integral

... f

DX I (XII VO IX 2)

f

DX2 (X21

1

Lo - 2

....

As before, we note that the above expression for the functional possible because the Hamiltonian is diagonal in the harmonic v~,,,ucnVJ By eliminating the functional integration at all intermediate points ....... strip, we obtain the first planar loop amplitude, in operator language:

Ap =

f

,)

d D p Tr[VoDVoD .. : VoD].

Carefully choosing the gauge for the multiloop calculation is v. u.v.u •••• tree amplitudes, we saw in (2.9.5) that ghost states do not couple to they can essentially be ignored. However, ghost stp.tes do couple to trees because of (2.9.9), so they can propagate internally in a loop are carefully eliminated. An identical problem occurs when qmmtizing Mills theory (which is not surprising, since both Yang-Mills and gauge theories.) There are three standard ways of eliminating the ghosts in loop First, we can insert projection operators which explicitly remove from the Hilbert space. However, this technique is rather cumbersome hibitively difficult for fermion loops. Second, we can allow ghost states to propagate and cancel the ghost state,s: Using BRST ~...~'" mal field theory, even higher fermion loop amplitliQes can be vu.'v ....,,,.v'" third, we can use the light cone gauge, which we will use in this ch~LPt~ The advantage of choosing the light cone condition is that it vastly the calculation. Ghost states, for the most part, can simply be H111I{"\r,'11 In addition to the light cone gauge, we will also choose a specific frame for the external legs of a multiparticle amplitude:

k+ =

o.

(We should caution the reader that certain complications exist WlLHU to the kinematics of enforcing this for an arbitrary number of external with arbitrary spin. For example, we may have to perform analytic tions of the external components of momenta in order to preserve A careful analysis shows that we can always choose this frame for 26 external boson legs and less than 10 supersymmetric legs. inherent problems, however, with the light cone approach because ways perform a Lorentz rotation on the vertex functions to a frame wbl~t~ij + components are nonvanishing. The light cone gauge is compatible

5.3 Harmonic Oscillators

187

...,/ ""Iw,,,,f uh, + components of the momenta. As a result, we will omit discussion arlhis delicate point in our discussiOlL) i~I1W""'ly. Ihis trace is easily evaluated using the coherent state formalism we used for the tree diagram;

I Ii,., ! dx}

dO P T rrVo(k .. x, )Vo{k1 , XIXl )' .. VolkN. X,XI •.. .IN }w"'-l ).

(S.3.S) (5.3.6)

trace can be explicitl y calculated using cohercIlI Slate methods. We usc

Tr(M)

=

~

f

d 2At-llf (AI M I'>").

(5.3.7)

the different fac tors, we have AN =

I Ii,., d DP

T - f(w)

I 1 dx/x: /2Ir.-

r,

(5.3 .8)

'-DO O~ 1(1- w"'-Ic}/)( l w-)2

(I

I<J ,. ... 1

ul"Cftl» ) i, '~J

(5.3.9)

00

few) =

0(1 - w"), ".,

(SJ.l0)

ell = Pi/Pi.

also explicitly perform the p integmtion:

Id 0• aP

I_ I

( 1m,; -

XI

-

-2Jt) n,. (" -

In w

0

1~I<Js.N

[e

-1 (1

JI

['

exp In Cjl 2lnw

JJ'" (5.3. 11 )

""')1 = I/I(Cj/, w), VI

= tnA / lnw,

(5.3. 13)

188

5. Multiloops and Teichmuller Spaces

and the f) function here simply orders the various Vi factors along This is the final result for the planar single-loop amplitude. l~011ce features about the result: (1) As predicted by the path integral method, the integrand is an au.~,~\, function. (2) The measure is easily evaluated in the harmonic oscillator aPl:)roz calculation is a bit harder in the path integral approach.) (3) The integral diverges at q = 0, which corresponds to the inner".v •.,,,:'~ ing to zero radius. The divergence is a mild one and can be eli·rnin:~~ we add superpartners to the strings. .

For future reference, we will define the planar (P), nonplanar nonorientable (NO) integrands at the same time: 2

_ 1- x ( In x ) Vrp(x, w) - .jX exp 2ln w (In2 x) _ I +x VrNP ( x, w) .jX exp 2In w

VrNO

(x,

W

)

2

( In x ) .jX exp 2ln w

_ 1- x -

CXl n n=1

CXl n n=1

CXl n n=!

{(1-

n n w x)(l - w /X)} (1 -;- ivn)2

{o{(1 -

n w x)(l +wn/X)} < (1 _ wn)2 ..... (_w)nx)(1 -

0 _ (_

These functions, in turn, can be reexpressed in a form in which Jacobi theta functions is more apparent: - 2rr. Vrp(x, w) = lnq smrrv

IJ {(I - (1(1-

2q2n cos 2rrv q2n)2

CXl

n_1

+ q4n) }

.

'

-4rr. 1 rrCXl {(1-2(-Jij)nCosrrv+qn) VrNO(X, w) = lnq sm2: rrv n=1 (_Jij)n)2 VrNP(X,

_ -rr -1/4 CXl w) - In q q n=!

IT

{(I - 2q)2n-1(1 _cos 2n)2 2ftv + q4n-2) } . q

Written explicitly in terms of theta functions, we have

Vrp(x,

vi) =

-2rri exp X

VrNP(X,

(;~::)

In x Ell ( - . 2m

In w ) I 1 ( - . El2rrz !

In2 X ) w) = 2rr exp ( 2ln w

° lnw) 2rr i '

5.3 Hannonic Oscillators

x E)' (_In_X _In_W_) 9,-1 (0 Inw) -

2;rri

"",(x" w) = -2rri exp

x9 1

2;rri

2:rr i

I

(In'

Inx

( 2n,

-.

189 (5.3.21)

'

x )

21n w

lnw

-2rr-j

[)"'_1(

+2

\,"JI

0

In WI) -. + -2 . (5.3.22) 2Jrl

that we have calculated the planar single-loop graph, let us calculate . single-loop function. The trace we want to evaluate is

ANO =

JdDpTr(~VoDVo'"

VoD),

(5.3.23)

n is the twist operator in (2.7.13). Notice that the twist can he placed along the chain, and nothing changes. The trace can be evaluated same coherent state techniques, and the major change is that w turns The final result is

(5.3.24)

that the imegralion region is twice the usual one from 0 to I. This is for the external lines to go completely around the Mobius strip, the go around the edge twice (see Fig. 5.5). This fact will eventually crucial role in the cancellation of anomalies in Chapter 8. A

8 A

-

L-,.-----'

8

8 A

L -_ _- '

A

-

-

A

8 A

8

L-----r---'

A

8

A

L -_ _--'

8

5.5. Integrating around a Mobius strip. For the Mobius strip, an external travel twice the length of the strip in order to make a complete path around Thus, the nonorientable strip has an integration region that has hvice

190

5. Mu1tiloops and Teichmill1er Spaces

The nonplanar diagram can also be evaluated. Notice that placing ...... .. number of twists on the loop separates the external lines into two ....la"".," that revolve around the outer edge and those that revolve around the final answer depends on which lines are on the inner and outer ""'E,"" ..• .-:; place the twists so that the first to the Lth lines are split apart from the ANP

f

= .

D

.

d pTr(QDVJD··· VLQDVL+ J ··· DVN ).

The integral can again be perfonned, with the answer [15] ANP =

1 1 -t N-J IT dVi

R i=J

J

0

d

q

(

2

-In 7r q

2)N (f(q2)r24 IT(VrNP,ij ,~ or i<j

where we use VrNP if the ij lines are on opposite sides of the disk they lie on the same side. The integration region for the lines reflects that there are two disjoint regions of the disk. The external lines are mt,egt sequentially, except that there are now two disjoint regions. For convenience, let us now put all three amplitudes in the same expressions: ,

where J = P, NO, NP,

Rp = {O < VI < ... :::: VN = I}, RNO = {O :::: VI < ... < VN = 2}, fp(q2) = fNP(q2), fNO(q2) = f( _q2), q

4 = qp = qNP = qNO'

and where the nonplanar region of integration reflects the "fact that two disjoint regions of integration. Now that we have written explicit representations for the various ( string amplitudes, let us draw some rather remarkable conclusions . amplitudes.

(1) Closed Strings from Open Strings One of the strange features of the nonplanar single-loop diagram is that more poles than those found by factorizing on the usual open string [ 16]. By examining the factor

5.3 Hannonic Oscillators 2

19 1

3

• 4

1 2

3

1

4

5.6. Emergence or the closed string theory rrom open strings. A curious the open string theory is that, at the first-loop level. it already contains string theory as a "bound state." The nooplanar diagram ror open strings stretched unlil it becomes a cylinder, which in tum can be ractorized into two cylinders. Thus, the intermediate state must be a closed string. the ~s factor comes from the momentum-dependent integrand) we find there are extra poles at s/4:::: -2, 0,2,4,6, ... , which are precisely Ihe of the poles of the closed string sector. Tbus, the open SIring sector I the closed string sector. This can most easily be shown dual diagrams by thinking of the nonplanar diagram being a cylinder, with sets of extcrnal lincs that can rotate around the top and bottom edges of However, by factorization, we can slice the cylinder horizontally, the. intermediate state is a closed loop. Thus. the closed SIring emerges "bound state " o/the open string sector (see Fig. 5.6). The closed string by itself, is an entirely unitary theory. However, the open string sector, is not. We see that the presence of open strings demands the existence strings as intennediate states. or else the theory is not unitary. In fact, by Lovelace [171. this unwanted singularity in the complex plane oonplanar diagram is 8ctu81ly a cut (which would be disastrous), but )"". 'pole only in 26 dimensions. In fac t. this was the first indication that model was consistent only in 26 dimensions.

:h'h"

Renormalization

"'" th," the divergence of the planar diagram arises from

l

o

divergence at

Q --Jo

ldq , QJ

l idQ. 0

(5.3.30)

Q

0 corresponds to the hole in the disk shrinking to

"~i);~'~f~~;~;~;~:,:"~iSCS from the fact that we arc summing over an infinite ~

stales propagating in the interior or the loop.

192

5. Multiloops and Teichmiiller Spaces

This is not, however, the ultraviolet divergence that we usually with Feynman diagrams. For point particles, the divergences of amplitudes arise when we deform the local topology of a particular .. such that a propagator shrinks to a point. Thus. divergences are associated with deformations of the local topology of the graphs. } In string theory, however, because of conformal invariance, we '.' a propagator to a point. Thus. conformal invariance on the world out ultraviolet divergences. However, we still have the infrared diverli:et the interior points shrinking to zero. Again, conformal invariance tells us that we can always map the ing hold into a dilaton or tachyon vanishing into the vacuum. We . "pinch" this shrinking hole and extract a closed string resonance with quantum numbers vanishing into the vacuum. Thus, conformalm'lVaJ~Ul1Jl()1 us an entirely new interpretation of the divergences of string theory. interpretation associates with each divergence a closed string state "~,i"" off from the hole, with zero momentum, corresponding to tachyons divergence 'or dilatons for the q - I divergence vanishing into the va(:uUti! Fig. 5.7). When we pinch the shrinking hole and extract a closed string vanishing momentum, we notice that the remaining diagram looks tree without the hole. Thus, by extracting out the divergent pole C0I1ID what we have left is a finite tree. This, in tum, allows us to treat the as a redefinition of the free parameter, the Regge slope a'. This prc)ce!,s "slope renormalization," works fine for the dilaton q-I pole, but it is how to handle the tachyon pole contribution q -3 . Thus, the bOSOnIC might not be totally free of divergences.

(3) Finite Superstrings Experience with supersymrnetric loop calculations for point particle has shown that the internal bosonic line cancels against the internal line, yielding amplitudes that are much less divergent than expected. thing occurs for superstrings. Let us now tum to the superstring graph:s.? we will be able to perform this "slope renormalization" [18, 19] ~hIJU',H~j will find that the q-3 divergence in the open string Type I theory "''''TIN''. itself (this is also expected because the theory has no tachyons). only the q-I pole, so slope renormalization is possible. The truly renml1' thing, however, is that the Type II theory is actually finite by itself, slope renormalization!

5.4

Single-Loop Superstring Amplitudes

The single-loop open superstring can be calculated in either the GS NS-R formalism. In the GS formalism, we might use the light cone forl[jj'g

5.4 Single-Loop Superstring Amplitudes

193

o

5.7. Emission of the dilaton. By confonnal invariance, we can defonn the string diagram and pinch the interior loop. The resulting diagram 10 the emission of a tachyon or dilu ton into the vacuum, which results divergence. Super.;tring theory is constmcted so that these poles do so the theory is fonnally finite.

,,~~::~,:,~; satisfactory covariant onc does not exist. The advantage arthis ~

however, is that the graphs are manifestly space-time supersym. In the NS-R fonnalism, we must either use a projection operator to

bu,,, out the ghosts or use the SRST techniques that allow the ghost to prop-

and cancel the negative metric ghosts. Unfonunately, supersymmetry is I",mifes, until we add bosonic and fermionic loop contributions separately insen the GSQ projection operator into each loop. us use OS fonna lism developed in Section 3.9, where supersymmetry As before, the massless vector boson vertex is

(5.4.1)

Bi(r) = p i(r)

+ k J R1j(r),

Rij (r) = ~S(r)yij- S(r), 00

S'(r) =

L "", - co

S:e-'''·.

(5.4.2)

194

5. Multiloops and Teichmiiller Spaces

where we have also set t + = 0 for the polarization vector of a ma.ssl'es.$ The fermion vertex is

where Fa(r)

= i(p+)-1/2(S(r)y.

P(r) - ~:Rij(r)k;S(r)yj:t.

Let us consider only the trace over external bosons. There are a greati simplifications. For example, the trace over the So operators eight of these operators. Thus, amplitudes with two and three f>yt,>rri, vanish all by themselves. As a result, there are no self-energy corrections at all in the theory. The first nonzero amplitude occurs for four external legs. Even amplitude actually vanishes, except for the contribution from the

t; k j R~ :e ik .X :. In fact, the only diverging contribution to the trace is '. Tr(wDI/2)n.LuY-Su) = f(W)8. ··0:::.>:.

As expected, this cancels against the other contribution coming from .•••> loop. Thus, the final single-loop open superstring amplitude is .

Aloop

=

K

t Jo

n

8(vl+I - vI)dvI

1=1

t Jo

dq TI(1/I/J)k l .kJ , q 1P" h,lfCpllane for the open string case, now becomes the entire complex

;'~";n "n;n',g...;on 0'''" "othat it is independent :ti~:'::~~~:':;:~:~~, of space. 0'

must sum over different orderings of the vertex functions. extemallines, which were once attached to the boundary oftbe strip, not attached to the interior of the complex surface .

= 0 to a = 21l' , such top and bottom edges are identified with each other. For the single we must now also identify the left and right edges. By the exponential we then map the finite horizontal strip to the entire complex plane. the usual coherent state methods, we find [20J (taking the slope ex' == ~)

. 5.8, we see the horizontal strip defined from

Cf

A= jd Dp Tr (V! DV2 ... VND)

(5.5.1)

Vj

= {21l'ir! InZ I l. 2 '··l.j.

VJi

==

v} -

T

=

V,w

w= eji

==

V;.

== (2ni)-lln w,

ZlZ2 "'ZN,

Zi+1Z/+2" 'Zj,

(5.5.2)

196

5. Multiloops and Teichmiiller Spaces

t a

A ,...--...........-----/---+---, A D 'D

C

C

8

8

a = 27r

A

A

8

z=e

T +

ia

FIGURE 5.8. Conformal surface for the single-loop closed string dmgra.n:i.' plane, the surface is a rectangle of width 2n and arbitrary length, with edges identified. In the z-plane, the surface corresponds to a doughnut. can attach themselves to any point within the surface. and 2

(

)TI OO

m

)_ (ln lzl) -1/2(1{(l,-W Z)(l-W X z, w - exp 21n Iwl Z Z m=\ .,. (1 _ wm)2 Xij

= X (Cj; , w) = 2n exp

(

-n(lm V.;)2) 8\(v·; I r) J J.. 1m r 8;(0Ir)

Let us rewrite this as

where

Notice that the integrand is doubly periodic:

xCv + 1, r) = xCv + r, r) = xCv,

r).

m ;

5.5 Closed Loops

197

easy to see because the theta function has the following properties: 8 1(1.1+ II f)=8 1(v I f), 9 1(1.1 + fir) = _e- l lI'(2w+f)8 1(v

I f).

(5.5.6)

' is important, because it shows that a naive integration over the v varidrastically overcounl the proper region of integration. Consider the ~1~log~~formed by the origin and the points 0, I, f, and I Y. When are identified, this becomes topologically equivalent to a torus )Ughm... Notice that the double periodicity divides up the complex p lane infinite number of these parallelograms. Thus, we want to integrate only one parallelogram. or else we will have infinite overcounting. Thus. we must restrict the integration over the v variables or else we will over an infinite number of copies of the same thing. We w ill I

+

0 :5 1m

Vj ::::

1m Y,

-! (5.5.7)

';"'gly. in addition to this truncation in v space, we must perform yet truncation in r space. The integrand of the closed-loop diagram is invariant under yet another ttansformation. given by

,

ay

+b

, -- c::--,--; cr +d'

(5.5.8)

C, d are al l integers and ad - be = I. This generates what is cal led ::::~~:;';: .SL(2. Z). Let us show that the integral is invariant unde r d1y _

le r

+ d l- 4 d 2y,

Imr ~·.IC't+dl -2 1mY.

(5.5.9)

1",,10""",. following are actualty invarianl under this modular transformad'f d 2r Tm2 r = 1m2 T .

(5.5. 10)

lei us calculate how the other lenns transform under a modular (5.5. 11 )

(5.5.12)

X(

b) cr+d

"a, + .

cr+d

=Ier+d l-'x(v.y}.

(5.5.13)

198

5. Multiloops and Teichmiiller Spaces

Thus, F(i) = F(i).

The integrand is therefore invariant under a modular transionnaltiof was first pointed out by Shapiro [20]. But what is the intuitive me:anll~: symmetry? The Polyakov action tells us that we must integrate over all cOll,tbti inequivalent surfaces. We first observe that two parallelograms values of i are confonnally inequivalent when we identify .... Thus, naively we expect that the integration over i automatically·· over all confonnally inequivalent surfaces. However, this is not the . .• There are actually two kinds of reparametrizations of a surface carefully distinguished. The first are the reparametrizations that be defonned back to the identity, i.e., the set of smooth rep'aram~1 that contains the identity map. The second is the set oflreparame:tri,~al cannot be smoothly defonned back to the identity. Global dltJteolmolrp of this category. For example, take the parallelogram and identify of opposite sides. This creates a tube. Nonnally, we would bring of the tube together to make a torus. However, now twist one of the of the tube by 2rr and then resew the ends together. By examing surface, we find that this has caused a genuine reparametrization but that the identity map cannot be represented in this way. This a "Dehn twist" and it generates a discrete group. For the torus, it that the group generated by Dehn twists is the modular group Sp(2 Concretely, the integral is invariant under, i -+ ;-1 Ii and i -+ successive applications of these two transfonnations, we can generate the entire modular group. But carefully examining the two transfonnations shows that they simply interchange the b01Ullcia parallelogram, generating Dehn twists. In summary, this second symmetry, called modular invariance, the fact that we must gauge-fix not only reparametrizations that reach the identity map but also global diffeomorphisms that are not to the identity map. Thus, we must divide up the complex i-plane only integrate over one surface invariant under the.transfonnations and i -+ -liT. Thus, the complex i-plane is divided up into an of redundant copies. To eliminate this infinite overcounting, we following fundamental region of integration:

fundamental region

(see Fig. 5.9).

~

I

I -2I < Re i < 2' 1m i > 0,

Iii 2: 1,

5.5 Closed Loops

199

I -j

j

5.9. Fundamental region for the s ingle-loop closed string amplitude. Modol"h~ amplitude divides (he complex plane into an infinite number of regions. Thus, ..... e must choose only one such regioll, oreise the amplitude and with ITI . The most convenient region lies between ReT = gr

(5 .7.4)

' : ~:,';,~~:e~:~. the delenrunant will always mean deleting the zero mode. ill

function (5.2.14), which contained the divergence of the single comes out of this detenninant.

ow,ov"., the functional integration over the metric is considerably more because of the presence of gauge parameters and also the presence . As we saw in (2.4.1), the measure is invariant under 2D general

iari.m,..md rescaling: (5.7.5)

·we""",•• h" C \ri,,,oifel symbols, we can rewrite this expression covariantly: (5.7.6)

parametrizes a Weyl rescaling, and OVa parametrizes a reparametrizaoor",,, two-dimensional surface. In general, this means that the measure of over gab is actually infinite. Notice that gab has three independent and that 5va and du also have three components, so that naively we can set all the components oftbe metric to the delta function:

(5.7.7) choosing the conformal gauge is equivalent to factoring out the inteinfinite volume due to Weyl rescalings and to two-dimensional over the surface. Gauge fixing thus means replacing the integration over metrics with

Dg"h -

Dgab(QOitr) -l(flweyt)-l,

(5.7.8)

the infinite volume of the space can be represented as (sec (1.6.7» QD iff

=

f

OVa,

212

5. Multiloops and Teichmiiller Spaces QWeyl

= / DO'.

For spheres and disks without holes, this is actually· true. surfaces with larger numbers ofloops, or handles, this isno longer true . .•••.. • of complications due to the parametrization of the loops. . In general, a sphere with N holes or handles requires a number eters to describe the location and size of each hole. Basically, for a holes we need one parameter to label the radius of each hole and parameters to give us the coordinates of the center of each hole. 3N parameters to describe N holes. (For a sphere witli!. handles, we .. . complex parameters to describe N pairs of holes.) As we saw parameters can be fixed overall (and set equal, for example, to 0, 1, . . so a disk with N holes can be described by 3N -3

real parameters, or double that number if the surface is a sphere with N •.. •• Thus, gah cannot be set equal to flab for surfaces with higher . •••... However, any two-dimensional metric is conformally equivalent to /> of constant curvature. By a conformal transformation, we can ... . curvature to a constant. Thus, the space of metrics over which integrate is the space of constant curvature metrics divided out by morphisms on the surface M. Moduli space is the space of constant · metrics when we eliminate the overcounting arising from rp"' the

discussion has been general. To actually extract out these extra that describe the holes of a Riemann surface, we need to use the spaces.

of the manipulations are a bit involved, so we must always clearly goal in mind, which is to rewrite the functional measure DEal! in of '~'P"''"''''''' OVa D(1 and the 3N - 3 Teichmiiller parameters Dc/_

214

5. Multiloops and Teichmiiller Spaces

Thus, our goal is to establish

Dg ab = P,DvaDa Dti.

objective:

Then, by simply dividing out by DVa and Da, we will have eliminated the infinite redundancy introduced by reparametrization mvarlance. Let us now rewrite the variation of the metric tensor (5.7.6) in a venient form, revealing the fact that gab is also a function of ti, the parameters. Using the chain rule, we can formally represent the OelDeti of the metric on the Teichrnillierparameters via 8tilJ/8ti:

8g ab = [Va8Vb

+ Vb8va -

(V c8VC)gabJ

+ C'Vc8V'C)gab + 28agab +

where

. agab T' = - . - (trace), at' where we have explicitly taken out the variation ofthe metric as a the 3N - 3 parameters (Teichrniiller parameters) that we label t i with the N loops. Notice that we have subtracted out the trace in and that, at this point, we do not have to specify precisely how depends on the various ti. This variation can be rewritten simply as 8gab = PI (8V)ab + 28agab + 8t iai gab, where

PI (8v)ab = Va8Vb

+ Vb8va -

gab V c8v c.

The operator PI plays a crucial role in the theory ofTeichrnuller SP3LCe$ that it is an elliptic operator that maps vectors into .traceless syInrrLetri¢ Let ker PI represent the kernel of the operator, i.e., the set of vp."tn·r~ mapped to zero by the operator. Ker PI is called the set of "ccmf()mLIi vectors." For surfaces with genus N, the dimension of the kernel nt1th" is given by dim ker PI = 6 dim ker PI = 2 dim ker PI = 0

for genus 0, for genus 1, for higher genus.

We also wish to define the adjoint of PI, which we will call PIt. of course, we first have to define how to take the inner product.

118ga bl1 2 = 118va bll 2

=

f f

d2zyggacgbd8gab8gcd, d2zyggab8va8vb.

5.7 Riemann Surfaces and Teichmiillcr Spaces

2! 5

have defined a scalar product, this allows us to define the adjoint definition (a IPb) = (pta lb}:

Pt

(5.7.23)

pt,

i.e., the space of vectors that are annihi· now analyze the space ker ]fwe rew rite everything in terms of l and ~. then the elements of satisfies

p/.

(5 .7.24)

I of Pt is spanned by what are called quadratic diffe rentials. Forthe dime nsions of the spaces of quadratic differentials for Riemann are known:

dim ker

pt = 0

for genus 0,

= 2 for genus I, dim ker Pit = 6N - 6 for genus N.

dimkcr

Pit

(5.7.25)

nwnberofparamcters within the kernel of Pit isequallo the numberof ,tin,Olle"",,,.,".,,,, needed to describe N loops or handles. Symbolically,

summarize how 10 divide me measure of integration into its constiruent (5.7.26) has a rather simple meaning. It says that tbe components within up into three parts, a dilation part, a traceless part, and also Teichmuller parameten;. 11 also means that we are very close to (5.7.16), but there are some subtle complications. make a change of variables and calculate the Jacobian ofthe . Lei us first assume that there are no Teichmiiller parameters Then we make a change of variables from hub (which is the pan of the metric) and r (which is the trace of 8gb) to oVa and 0' ;

r

h)]

ae"~ Dgub = del a(O', v) Do-DI)G'

IX] PI

= det [ 0

= detP I = [detPt PIt 11/2 ,

(5 .7.27)

(5 .7.28)

th,,,,,lue of X is arbitrary, since it drops out of the determinant. that the square root of the delerminant of PI Pit is just the Faddee\'detenninant for the confonnal gauge, which in turn can be written in BRST ghosts. Thus, we can write the Faddeev--Popov determinant in (2.4.3) as (5.7.29)

216

5. Multiloops and Teichmiiller Spaces

Next, we wish to calculate the Jacobian factor due to the fact thaltijf~ sure actually depends on the Teichrniiller parameters t,. The the Teichmiiller parameters are not orthogonal to Pl 8va • so the Jac:ohili contain cross terms that have to be factored out. Let us begin our discussion by introducing a set of 3N - 3 cornpI!i.ti~ 1fta that will be an orthogonal basis ofker PIt. Let us now del;oIlo.pC)Se:tb;e: [22,23J:

into the following identity: Ti =

(1-

PI

+ PIt) PI PI

Ti

+ PI-;-plT i • PI PI

It is important to note that we have done nothing. We have only acld.~li subtracted the same term. However, the term that we have mtrocluced:¢qj~. the operator PI acting on another state. Thus, the above identity 1~:::~ because it allows us to extract the piece of Ti that lies along the rurl~~~ PI:

where

. 1 t· v' - - - P T'

- Pt PI

(1ft", Ti) =

I

'

f d2z..;glcgdeT~dl/r~e·

As expected, there is a piece of Ti that lies along the direction PI accounted for when we write the Jacobian. Let us now insert this back into the measure for 8gab :

118gab l1 2 = 118all + (T i , l/r°)(l/ra, l/rb)-I(l/rb, Tj)8ti 8tj + IIP I DVa Il 2 • This, finally, gives us the Jacobian that includes the contribution .. Teichrniiller parameters. We have also explicitly taken into account .. : •• : that Ti originally contained a piece that lay in the direction of PI : Dgob = Du DVaDti det

1{2

t

(PI

Pd

det{l/ra, Tb) 1{2

•

det (l/ra, 1ftb) This, in turn, allows us to put the entire functional integration I" the following factor:

f

DXe-

s

DgabQDilffnw~YI =

f DVanDilffn~eyl 1{2

t

du

x Dt, det (PI PI)

dete1ft°, Ti) 1{2 . det (l/r0, l/rb) .

5J:i Confonnal Anomaly ,

X (

J cf2:..;g der'( -

217

) - (1/2)0

(5.7.37)

"11)

finally attained our goal , (5.7.16), up to the question of anomalies ," mo·il scale invariancc. this discussion migllt appear long and difficult, the end result is

~

i~~;~;;Th~ie;:fi~n::a;!I~:an~~"~v~e~r ;Sh~O~:ws~~that thebemeasure integration, ,can written of totally in lennsincluding of dcterPI and . There are three parts to the measure: is the FaddL'Cv-POpov tenn, which is written as the square root of the

run", of P1 pt; (2) the second is the term involving T

i,

which arises the moduli parameters ti are not pe:fCndicular to the basis vectors fonn an orthogonal basis for kcr PI ; and (3) the third factor is the of the Laplacian, which we will also show can be VoTitten in terms

P,. , it is not yet possible to remove the integration over the scalar Cf in (5.7.37). Unfortunately, the various measure terms contain thc ~,~~:~~,;~. We will find, in fact, that in general the scale factor CUIUlot IT from the mcasure terms at all, breaking conformal invariance. section, we will show that there is an obstruction, an anomaly, to ;~:'..:;~;' called the confonnal anomaly, which can be removed only if !iI of space-time is 26. This fixes the dimension of space-time. process, we will show how to write al l detenninants totally in tenns P1 operator.

ConfonnaI Anomaly that we can cancel the integration over the volume of the ,mclt;;"" 'ion group, because

nDi~

!

DVa

= 1.

(5.8.1)

we wish to eliminate the Weyl rescaling term:

nW~l

f

DCf

= 1.

(5.8.2)

"~;:';:~:,,:in~:,:egration over the Cf rescaling term is considerably more comCf terms exist in the other factors of the measure as well. we must very carefully extract all a-iXIl'II (The 1{1 variable remains unchanged under a Weyl rescaling.) Thus, the only term we have to worry about is

)}-(1/Z

det'ptPl lI/Z{detl (-VZ

)D

f d 2z,,[K

[ det( 1{Ia, 1{Ih)

(The prime still means that we have extracted out the zero determinant. ) Fortunately, the extraction of the Weyl rescaling parameter can be:~1i simultaneously on both terms if we use a few facts about Ri~:Iru:riUi· The trick is to rewrite PI, PIt, and V Z in a way such that all three be expressed in terms of the same differential operator.. Let Tijlkmn ... abcdef···

represent an arbitrary tensor in the two-dimensional Riemann ·sw.;W~~ course, we can always rewrite this tensor in terms of complex';';'",K;.-A z and Z. In these coordinates, we denote as K n the set of all teIllsQj~l transforms as

(e.g., see (2.7.6) and (4.1.7)). Clearly, the operator ViT = (g1"~",,

'!>in

m.tncs

.

mo.bcJJ'O);i

OX'

f f D.p'

Dx.,-s.

SlnlOlllre.

(5.9.11) addition here is the intcgration over the two anticommuting vector and also the sum over all spin structures. sh"w,'
(5.9.12) ~OW.

in analogy with the bosonic string. coostruct two operators Pi n such that

(5.9.13)

10 the integration over moduli. we now have to integrate over mo,dul;·, which are defined as the space of traceless Xa that cannot be away by a local supersymmctry LTansformation. Thus, they satisfy

(5.9.14)

224

5. Multiloops and Teichmiiller Spaces

For a surface of genus N, the dimension of the supennoduli space '·:. t the dimension ofker P1/ 2 : ...•, ·•.·c·.c.·c.:.,

o o

if N = 0, if N = 1 (periodic-periodic), if N = 1 (otherwise),

4N -4

if N :::: 2,

2

(where periodic--periodic stands for boundary conditions on the l".::lC':",m;,::n We wish to to construct the Jacobian for

DgabDXa' The Jacobian for the fermion field is calculated as before: 4N-4

DXa = (detPi/2Pl/2)-1/2e-(11/2)S(0')D~DAn dai, ;=1

where ai are the supermoduli, the counterpart of the Teichmiiller p~1~i and where the integration over the fermionic A and ( replresents'11~~1ii1 gration over the redundancy introduced by supersymmetry and SU~I~!I transformations. We define S=

4~Jt

f d2z~(gabaaaaba + /L2 (e"

- tiAya baA

- 1) +

r 3/ 2 /LAAe(2 j 2)a + ! Rga ),

where R is the curvature tensor. Putting everything together, we x(M)

(2p)!

Z~2p)

(i),

(5.10.5)

is the Euler munber (p is the number of zero mode.rol""~ exploring the propenies of the ·\miversal moduli space" of all surfaces, including i.nfinite genus.

li:,~~.~~.t1ri:::~~~:;:::' was too ambitious and complex to extract mean-

("

. Ilowever, two dcvelopments have given some impetus to this

Bclavin and Knizhnik l34] have shown that the measure for the I bosonic amplirude is simply the absolute value of a certain 38 - 3 fonn. This is cnlled "holomorphie fa ctorization ." it may D1lIke possible wriring the multiloop meas ure by inIn practice, however, there arc eertain problems in spectfYing of the period matrix beyond three loops. This is called " HOlomorphic factorizat ion has made the two- and measure almost nivinl, but the Schottky p roblem prevents the measure from being easily written. 1984 mathematicians finally solved the Schottky problcm. Thus, a key " "",iion LO utilizing holomorphic factori7.ation seems to be removed. the Sehottky problem, moreover, provides us with an even more w,,;fullooi. It allows us to describe an infinite-dimensional space, called ··Grassmannian" (Gr), in which all Ricmunn surfaces of genus g are as single points [35-37). Thus, by studying the properties of the in the Grassmaonian, it may be possible to manipulate the set of all irt,,,b,a'ion diagrams at once.

~~~i"~'~ Grassmannian finally provides us with the conceprual frameto treal the entire penurbation Geries as a whole. we sliU do how to sum the entire series. For example, phase space reduces the . all po~sihle positions and momenta of a panicle to a series of points. . space, in principle, conlains all possible motions of all possible in the Universe. However, this says everything and it says noth.i.ng. have to impose equations of mol ion IUId boundary conditions to exmeaningful informa tion from phase space. The same applies to the

in

ambitious program requires fu lly exploiting the modular transfonna,[BU"n"," surfuces of gellus g. Of particular importance is the mapping l'O"P :MC;G •. wl,ii,h reduces to the modular group SL(2, Z) for the tolUS. want is a way of studying the mapping class group for arbitrary will be the constructiOD. of theta funct ions defined on Riemann

'''''Y

228

5. Multiloops and Teichmilller Spaces

surfaces of arbitrary genus. This is in contrast to the Schottky metl10;fi: studied in Section 5.6, where modular invariance was not so In Fig. 5.12, we have written the a- and b-cycles for an artlltriitM~ Riemann surface, which we call the "canonical homology basis.'· :"",,,;I,l" ~::IHI tisymmetric symbol (a, b) represent whether or not two cycles int,'(i' 'fii~~ equal to 0 if they do not intersect and equal to ± I if they do. ror:ex--Jl consider the torus, where we have only the a- and h~cycles. NotlCc~tI1tlWl is equal to I, because the a- and b-cyc1es intersect, but (a, a) Then the four possible combinations for the torus create a ma,trDe: (a, a) = (b, b) = 0, { (a, b) = -(b , a) = 1. The elements of the mapping class group do not change this j'I1ltersectim1ii However, we know, by definition, that the group that leaves this is Sp(2, Z). (See the Appendix.) To describe the elements a Dehn twist Da created by slicing the surface along the a-cycle; cut by 2n, and then resplicing the surface back again. Under ll~'crl~l= the b-cycle converts into the sum ofthe a-cycle and the b-cycle (s Thus,

Da(a) = a, Now represent this in terms of matrix language. Let the Dehn the column vector [a, b]. Then the Dehn twist can be represented as

Similarly, we can describe the mapping class group for the (W(H9(QRJH by taking Dehn twists along ai, a2, hi, ~ as well as the cycle aj"I/.f':"l'~ ::)ly.J;MJ! a circular line that encircles the two holes. Using same reasOIl~'~~~.~~ , show that the Dehn twists, operating on the column vector [ai, be represented as [38]

the

1 0 0 0

o .'

100

101 010

1 0 I 000

0 1

001 000

I

0 0

0

o o o

100

1 0 0 010

0 1 0 101

001 000

",

5. J I Moduli Srace and Gmssmannians I D II ,· ' • •

=

0

0 -I

0

0

0 0 0 -I

0

let us Ireut Ihe closed Rieman surface of arbilrary genus. The matrix can be represented as (aj,aj ) =(b/.bJ)=O,

!

(ai, b) j =-(b1 ,Qj)=8/j

(5.11.5) ,

in,>th,ing bUi a block-diagonal matrix. with equation (5.1 1.1) in each group that preserves this matrix is Sp(2g, Z). We might suspect, , that the mapping class group is j ust Sp(2g. Z). This is not quite right. 'ie~:::;~D~e.hu twists D~ around cycles that arc homologically trivial and ::(I the a- or b-cycJes. Thus. they are represented as the unit lhe basis that we have been using. Ncvenhelcss, these Dehn twists tim.to g lobal diffeomorphisms that must be included in the mapping This subgroup is called the Torelli group T, and hence we finally , d",;",d result

MeG

----,,--

~

Sp(2g. Z).

(5.11.6)

the effect of the Torelli group on spin structures is trivi81, so we funho< discussion of it) we wish to describe the period matrix for the Riemann surface and "::~~):~ under Sp(2g, Z). In addition to the 28 cycles we caD write on ~ surface, we can also write 2g independent harmonic one·forms on the surface (see Appendix). Because we have equal numbers of one· form s (due to the Hndgo-de Rahm thoorem), we can always ze "h. integration over a- imo N pairs, labeled ill and a/ . These are called a-cycles. If line between any pair of a-cycles, we obtain b-!~::~tl>ll how to formulate M-theory in a satisfactory first quantized forlllili.lisltll/f~~~iiI a second quantized one. Thus, our remarks about second qu:mtiza"tio;~j!itIl confined strictly to superstrings.) At first, the second quantized approach seems to be totally redlundlii)ij~~~ usual first quantized approach. Perturbatively, we simply reproduce diagrams as the first quantized approach. However, there are several it',~"P".i advantages to the second quantized field theory: " " ::~:::;:~

>

(1) Interactions are introduced through a new gauge group whose close on interacting strings. There is now a group-theoretical for introducing the interactions of string theory. (2) The theory is manifestly unitary because the Hamiltonian is H¢~~ All weights of perturbative diagrams are fixed at the very l)el~l1llll.in: (3) Most important, we have, in principle, a method for calculating dYiiJiil l effects in the theory.

Historically, it was once thought that a field theory of strings was u"n"]":w:&.i because it would violate a host of fundamental and cherished n';'rl;;~ quantum mechanics. Specifically: (1) A field theory of strings would be a nonlocal theory riddled problems, such as the violation of causality. ::: (2) A field theory of strings would necessarily be off-shell, yet ""tJ"i:~:;jifj properties of the Veneziano model, such as cyclic symmetry;." on-shell. Thus, a field theory of strings would not replro(luclm::~~~1 model. (3) A field theory of strings would not be both Lorentz invariant a.n;~~~ at the same time. This is because a quantization program for not exist that is both Lorentz invariant and unitary off-shell. quantized theorem. this is not important because the theory is the field theory of strings is necessarily off-shell and hence either Lorentz invariance or unitarity. (4) Most, important, a field theory of strings would be plagued of unitarity because of severe problems with overcounting of I)~~I diagrams. Field theories add the sum of s- and t-channel poles'~¢i~WJ!tl which violates duality.

Fortunately, a field theory of strings that answers each oeQ.(~~ objections is, 4,ldeed, possible. First, string field theory does not violate causality because the "iJltl~ such as the breaking of a string, take place instantaneously. information concerning the change in the topology of the string tra~r#~ the string at or less than the speed of light. In other words, smng::w:~ multilocal. Thus, string field theory is the only known nonlocalfi:":"~"~~~UI consistent with the principles of quantum mechanics.

6.1 Why String Field Theory?

249

IOnd,smin, fi"ld theory generates Green's functions that necessarily viimportant properties of the Veneziano model. But this is m; s~;;:.": these Green's functions correctly reproduce the Veneziano 01 and only onoShell matrix. clements can be measured. BRST method explicitly breaks unitarity off-shell with Faddccv, . This makes DO difference. however, because the Faddeev-Popov -: cancel with the unitary ghosts on-shell. Similarly, the light cone method , unitarity but breaks Lorentz invariance off-sh.ell. The point is tbat the theory is both Lorentz invariant and unitary, so the theory is still

"'.th'

string field theory actually breaks duality off-sheU, but this makes on-shell . For example, in light cone string field theory we sum s- and t-channel poles in the Fcynman series:

"A,

A = L.,. f

"A,

l+L

s - M,

J

(6.1.1)

2+ · .. ·

f-M J

theory of Slrings solves (he problem of double counting by breaking amplitude into its separate t- and s-chatlllel parts, consistent with a in~::,::;::~:~ (see Fig. 6.1). Thus, sln'ngfield theory breaks manifest I, it at lhe level of the S-matr1x. Although string field theory solves the problem of un itarity, it does so at the price of breaking duality, which is recovered only at the end when we sum over all cio,sed strings, we also find that the individual Feyrunan diagrams break modular invariaucc. Onlythc sum is modular invarianl. Thus, the light

, 2 4

,

,

.

2

.

,

4

by the light cone field theory. As in any field theory, .I i duality. Only the sum is dua1. Thus, light eooc theory of strings solves the duality is explicitly broken for each diagram. Only the . duol.

250

6. Light Cone Field Theory

cone field theory is manifestly unitary (because 'the Hamiltonian is . Hermitian) but the price we pay is that we break manifest modular ~V~~4 First quantization (modular invariance) ---* second quantization (u.~~~~ We will begin a discussion ofthe second quantized theory string staJ'®it~ the light cone formalism [I], because this retraces the historical CI·4~vellt1fH; of the theory and because it is the most fully developed tbr:malllSD1; because the light cone gauge is'a gauge-fixed formalism, in which aH~q'@~ gauge degrees of freedom have been explicitly removed, there is no::trfI:il~ the elegant group-theoretical formalism from which string field the()l1~1iaiii derived. Thus, for pedagogical reasons we begin our discussion .un-",:·" cone field theory. We begin by once again discussing the field theory ofpointpartlCI.es;::;Vl trace how Feynman derived the Schr6dinger equation from the '~U'.H"\ A-lei",~If2f _00

"a''' )dll·

!

x l/I(X,t) + '1 aw ax + '211 axl

(6.2.7)

can be cxplicilly performed. First, the integration constant can be to be A~

(-21riE)'" m

(6.2.8)

the terms in the right-hand side, we notice thai the only terms that the Gaussian integration arc those for which 11 appears with an even the integrand. Finally, we are left w ith .81{1 j

at :::: -

I a1 1{1 2m &x2 .

(6.2.9)

have now derived the SchrOdinger equation, slarting only from the that L :::: and the basic assumptions of quantwn mechanics. the effects of a potential term and extend the expression to all directions, the derivation is basically unchanged, and we arrive at

!mx7, i

a1/l = __ , I V 2 1/1 + V(x)y,. "

(6.2.10)

m

'1:::;,:';;;' with e;oo;ternnl potentials entirely and introduce 1/13_ or1{l4directly into the action. This is the second quanti7M analog

:r;

over y ~shaped or X~shapcd topologies in the first quantized point

:th,ory. Feynman's original derivation of the Schrodinger equa~

'" ''" ,,,lou',,,;n, the time evolution of the wave. However, there is yet

252

6. Light Cone Field Theory

another way in which we cart make the transition from the first to mef:iijj;1ffil quantized formalism in which the derivation is more direct. Let us nQ'i\'-~ to the path integral action and show that" we can make the tral[lsili6bd& second quantized formalism starting from the Green's functions theittiS~ffi rather than appealing to the equations of motion. In Chapter I, when we made the transition from the Hami.ltOiliiti(~~ Lagrangian formalism in the first quantized point particle theory, ...."··:u·,, an infinite set of intermediate states, the eigenvectors of the x-coordii:iiU l=lx],t;)

f

DXj(xj,t;!

into the expression AU = (Xj, tilxj,tj)

at each intermediate point between Xi and x.j. This allowed us to' transition between the Hamiltonian and the Lagrangian approaches .. Now we wish to replace this with the integration over a cOlnpl¢i(!=:$Jj second quantized fields: 1 = IVi)

f

D 2 te-(tlt) (Vii·

Following (1.8.21), we define t(x) = (xl1fr), Vi*(x) = (tlx), 2

D t =

TI dVi(x)dt*(x). x

Thus, at each intermediate point between the initial and final states H~::!"::1!'!1'I'l particle, we are now going to introduce an infinite set of intermediate fi"~. states IVi) rather than x-eigenstates Ix). The easiest way to check ttt~'lJJj~ of (6.2.13) is to take the following matrix element and insert a cOInpJ'~m intermediate string states: 8(x - y)

= (xly) = (xilly)

• . (xly)

=

f

f

2

2

D Viexp {-

f f f (1frlz)

D Vit*(x)t(y) exp {-

DZ(Zlt)} (.,.,:t.\l:t.::,,7.'!

DzVi*(z)t(z)

1

(where as usual we drop ovemll normalization factors in the path ';t"¢Igm treat 1jF(x) as an element of a column vector labeled by discrete· .' in'

6.2 Deriving Point Particle Field Theory

253

Jizingthe expressioo.. we find

;" - Jr;Id";dV,"';"',ex ~ ";,,, 1 p[-

(6.2. 16)

relation is just ( 1.7.11). Thus, we have now shown thai we can move

'"~::~:~i from firs t quantized basis elements Ix) to second quantized i~ I"'). such thut 1j!(x ) = (xlw). ' " u, beboln our derivation of the Green's fun ction for the SchrOdinger entirely in lenns of sccond quantized field functionals, without rethe equations of motion or Huygens' principle. We will joserr the identity at every intermediate point along the path :

l1Jrl}

f D2"'I D21/tlexp{-1jI~("'2

-1/1)) -

1Jr;l/rd ("'21 .

(6.2.17)

"'t,

,p,ove this identity by functionally integrating the expression over reduces to the completeness expression written in lenns of the tfl

the above expression between two infinitesimally close position in (6.2. 12):

= (xlle- iH"!x2) = (XdX2) - i (xd Hot IX2) + ... = ! LY1fr12{Xd wl}{¥rzlx2 ) e;o;p[ -l/rj(1/I2 - 1/Id - i

f

2

D ""m4 (x d 1/l1)

, which is simultaneously a composite of all pOl~sib!l;~J~ configurations. To make things concrete, let us now introduce a specific reI're1;en~~~'ij1~ IX) eigenstates in terms of harmonic oscillators. We want the X to act on the string eigenvector IX) such that we reproduce tL. ••'.)!J';:::::::~

6.3 Light Cone Field Theory

255

w oment, lei us make the assumption that the eigenvector IX) can be

IX} = (l IO) - k

ncxp(aX~~ + bXI,"a,~ + ca:,,,a,~,,) ..

(0) .

(6.3.6)

L.

C, and k are arbitnuy constants.

X,

these constants a, b, and

c by acting on this slate vector

XI ,,. IX ) = 00- 1 ~i ("i... - ai~n)n 10)

= n} = L¢(n)l{n}) {nl

to the following

(XI ct» =

L ¢(n1 H(n1(X)e- L ;JI Xl. •• {nl

The inner product between two such fields functionals is easily "al"'H~~ (\III ct» =

L(nl t/!{:)¢(n)'

We can also show B[nl.(m)

=

f

¢[nl¢(ml exp (-

L ¢f

n ))

(n)

d¢,

6.3 Light Cone Field Theory

257

,,,n","u« of integration for this space is given by

D2¢ =

n,.,

(6.3.22)

d¢l"i dq,(nj-

tdentititls, we can now show that lhe number I can bl: wrillcn in a oJO!,OU, 10 (6.2.13),

1

~

f

11 n' ("I""pl-("'I"'I).

(6.3.23)

we

equation can he proved by power expanding each of variouS oCliooo11 , via (6.3. 19) anu explicitly performing the inlcgrdtion overthc

. Thus,

1~

[I: f.,., 110 n] nd'.'m' II: .i"IIPII ) ,-I.'.' 1m)

(n)

L ,prnll{n}) ({plitJIjr) / L Ifn)) ({p}16

Ipl

d2¢lmle-l:!f,~·,I:

(,,).(p).)nrl

=

(6.3.24)

1"j.lpl·

IIIJ.(pl

are justified in taking this expansion of !.he number I given by no" i, ci;';~:{' number of these first quantized intermediate states between every ill.~·~t~~ interval between the initial and final points:

where L = P X - H, Thus, we could go back and forth between Hamiltonian and ~~jR formalisms in the first quantized string theory, N ow let us repeat all our steps, inserting a complete set of seconcttU:imm field functionals into our action. Let us start with an II'mn1te1;unatJ~ amplitude 1l.12

= (XII e- iHdr IX2)

= (X,IX2 ) =

f

i (XII H dT IX2)

+,.,

1;fct>d X tlct>I)(ct>1IX2 } exp{-(ct>dct>2 - ct>1) - (

- i

f

x exp

1

D ct>1234(X 1 lct>l) {ct>21 H dr 1ct>3} (ctJ 4IX 2 )

1- i~

(;JCPj+, - ctJ i )

-

(ctJHdctJ;)

I+, ".

6.3 Light Cone Field Theory we now take the limit as

L I;I ,+, - ,)

,

~

f

259

d' I"'I.

We are now in a position to derive the canonical commutation the second quantized string field theory. Because we have power field in terms of orthonormal polynomials, it is easy to show that [1 [ct>p+(X, T), ct>t+(Y, T)] = 8(p+ - Q+)

T1 T18(X;(a) -

Yi(a)).

rr

i

Let us denote the vacuum ofthe A oscillators by 10)). Notice that ml:SV~ll'>l state is the product of the vacua of all the higher spin fields cOIltaine(J'W, . This state is the vacuum of the infinite-component field theory ",,-,',1;·»:1 has nothing to do with 10). Using the previous identities, find an eX]~(¢j~ for the Green's function in terms of the field functionals: ll.12

= ((01 ct>p+(X[, T[)ct>;+(X 2 • T2) 10)) = 8(p+ - q+) / DXeifLdrrdr

IJ

x T18(X(cr', T2) - X 2 (cr)). rr'

..

8 (X(cr, T[) - X[(cr))·

6.4 Interactions

26 1

initial and final states are labeled by X(a , rd and X(a', s oflhe second quantized field theory. Again, we will insert the identity (6.4.1) integral in order to extract the vertex function. "o,;, ~lly, many oflhe early pioneers in qUlIntum physics, sueh as HeisenYukawa, looked into the question of non local field theories and found theorics violated causality, i.e., interactions could propagate faster !..peed of light. The nonlocal interactions, which can in,'olve two di.sXI and X2 , cOllld trnnsmit informAtion fasler thRo the speed oflighl, is fo rbidden.

.~:g~Jfield theory solves this problem. The solution is simple field theory does not violate causality and the laws of quan!!~:;:'~:~i~'~trm~ it is not a nonlocal theory, it is actually a multilocal I

The interactions of the string, in which·strings can break or refonn, are

1JI.

We will write explicitly what some of these interactions are, cOI]t;ii!~ the conditions of locality. Once we have written specific rep'resent~~~

6.4 Interactions

263

interactions, we must eheck that they reproduce thc known results first quantized theory. i;~~;,:::,i~:ntcrdction involves the breaking ofa string into two smaJler ti' the condition o flocality, the string can break only at a single along the string. The disturbances from this rupture shou ld laler ,k,ng the string at velocities less than or cqual to the speed of light. arc led to pos tulate that the points along the ~tri ng are continuous Ie l)o,..d'>«:1 o,ftb""""",tic''''. The uniquc fonn of the vertcx function, with momentum conservation and locality, is a series of Dirac della ,,,hal continuity of the three s(("ings. Our vertex is [lJ

",u"e

dp~8

(t J p+F)

DX m ¢J t (X ) ¢I'( X])¢I(X 2)om

+ ac.,

(6.4.4)

0, = DX 1 DX 2DX 3•

(6.4.5)

. 6.3). We will usc the notation

o S :s: JT(al + 0:1). (1

al =

a

for

a2 = a - ;ra] O'J

0 .:: a ::: JTO:]. ;rO: l::: a ::: ;r(a ] + a 2).

for

= ;rr(Q'] + Q'2) -

(J

for

0.:::

(f :::

;r(al

(6.4.6)

+ az),

i length of ellch string is given by ;ra; and whcre the onl y thc transverse modes of the s tring . . clillllcrually perform the D X integration in the above vertcx it is a simple Gaussian. Lct us defi ne

" ;(X) = (XI";).

(6.4.7)

" " 0-0

"

PaTlimelrization of thc Ihree-string vertex. The parametrization length ill given by Ira,. The lIum of all three lTtX( is equal 10 zero.

264

6. Light Cone Field Theory

Then we can rewrite (6.4.4)

S3

f dp+r (t p+r)

=

8

{I1Sl: demonstration of their explicit cancellation was performed in [6] t{\Y::fh;~' point function. The cancellation for the arbitrary case, including l,c"ip'IP'~~ performed in [12].) Thus, we have reduced the Jacobian to a trivial fa~~~1 12], the product of the various Koba-Nielsen variables. Putting ~ together, we have AN =

f fl 'lI(P?;) f dJL I.n

x

n

exp

r<J

(~L fda' da" pt)(a')N(a', i,;a", is) r.S

If we take external tachyons instead of arbitrary external find

as before. Thus, the N -point amplitude is recovered.

6.7

Four~Slri ng

Interaction

275

Four-Siring Interaction i led us to postulate tne eltistencc ot'a four-string interacwnere two strings can combine at their interiors and instantaneously ~eth,,;'r local topology. AI first. it doesn 'I appear as if this diagram should the Neumann function method. since the upper half-planc was al'.:::~.~ into a configuration thaI was planar. HowevCT. the fact that the ~i (and all the other) interaction tenns are presenl in the Veneziano can be secn if we rigorously examinc the region of integnllion of variables. begin with a four-string mapping and sel XI = t, .f2 = 00, X3 = 0, a nd . Then the four-string mapping becomes

:oba-N;"ls,'o

p=a: 1 In(z-I)+a3 In z+a:.In(z-x).

(6.7. 1)

where the mapping is singular, let us set

dp =

dz

o.

(6.7.2)

og ,';s equation yields Ihe turning points of the transformation:

(6.7.3)

(6.7.4)

":::~~;cthC two solutions of the turning point equation, given by the

:::

±

rool, show thaI there are (wo turning poinls on the Riemann have the freedom of moving past each other. In the s- and t-channel the world sheets of thc strings smoothly transform into each other. a n interesting thing happens when the imaginary parts of the two point'> in the p-pJanc coincide.

~;~~~2;~lei:t~u~s~ls~:t:UdY the Feynman graphs for the f - and II -channel . (sec Fig. 6.5). We sec Ihat these two graphs ealUlot

~

into cach other. The sirings meet either near the top or near the : of the diag:ram.. so there is 00 way of continuously defonniog these 'g"'ms iOlo each other. But this is impossible. The conformal map, by was a smooth one Ihat aUowed us 10 go continuously from the t to ita,,,,,,,1and viee versa. In oth~r 1II0nl.t. a piece of the integration region To see this, let us calculate precisely when the 1- and u-channcl

276

6. Light Cone Field Theory

----------------------~~~ 1 _________ ~ 2

1

4

---------- ---::;-, L_____________ __________ ~

~

2

FIGURE 6.5. The four-string interaction. It is impossible to the t- and u-channel four-string scattering diagrams into each other three-string vertices. Since the confonnal mapping is continuous, there is a missing piece of the integration region, which can be SUI)pli:e:~:~ postulating a new four-string interaction.

diagrams meet We set /). equal to zero and then solve for x:

Notice that there are two solutions that give us the t- and u-(:ha~~~t~HI points. The region (x+ < x < (0) gives us one diagram, aW",,:LI1:"" (x_ > X > -(0) gives us the other diagram, but what about between? This is the missing piece. It represents a continuous de:torm2Iticlp::!~19i channel graph into the other graph, such that the local 'VfJ'UllJ'''J,,'V'''::J~ graphs is only instantaneously deformed into the other (see Fig. ~,,~,~~::«::::;

FIGURE 6.6. Topology of string interactions. The equipotential lines on a disk with external charges are isomorphic to the interactions of we have like charges located on opposite sides of the disk, then the lines collide in the very center and rearrange their topology. This is interaction. Likewise, all five allowed interactions can easily be fushion.

.".:S:tl

ft'

6.7 FQur.String lnlcrac.cion

277

278

6. Light Cone Field TheoI)'

In this intermediate region, the local topology of the four such that the strings reconnect in a different sequence. graphical proof that the four-string interaction that we lier is, indeed, part of the Veneziano formula (for the t- and ,"E',,,·...= graphs). Without this missing piece, the string field theory is actu:af.~ complete and violates conformal invariance and other n1"rln"rt;":~:: S-matrix. The four-string interaction, therefore, requ4"es an extra int~~gr:ati(I.lt: This is like a "zipper" that allows us to locally change the topolo&i;=:6) four strings. At first, we might suspect that this interaction takes pl!i~~~1 than the speed of light. After all, the four-string interaction instantaneously in time because the da integration happens lstalntlyt:)Va to be violating our postulate of locality, which we originally ImlPOsedJ,i: a causal theory. The resolution ofthls puzzle is that the four-string interaction to the Coulomb interaction term that arises in Yang-Mills theories ··f.ii*-fi~ tizing in the Coulomb or light cone gauge. In the Coulomb gauge, appears quadratically in Ao'112 Ao (without any time derivatives) . in the coupling to ferrnions Ao y °1/t. By functionally integrating otdi:~i! we find . m'

'i.\I"

t

which is the four-fermion Coulomb term. Notice that the o~:~~~lil an instantaneous operator (Le., has no time dependence). This ently violates relativity, but the S-matrix is actually strictly in the presence of this term. It is an artifact of the gauge any apparent violation of causality disappears when we _..';";4';'."" S-matrix. The apparent violation of special relativity is only an lllUISl0rt; integration dIY over the zipper in the four-string interaction is COll~f~~~.1 special relativity. Let us write down the four-string interaction term for the tion. We stress that the four-string interaction term can be postulating locality but is also consistent, as we have seen, with amplitude. The interaction term is [1]

where J.L is a measure term and

6.7 Four-String Intcl1\ction

279

,., > 0.

a)

Q"L
Let us discuss the OS action in light cone language, because explicitly supersymmetric. Our first quantized light cone ""L.'V" (3.8.3) Sic

=

4n~'

J

2

d z(8u X

i

arzxi -

(2i ct')p+OUyCl o",oa),

where e1,2 are spinors with eight components in SO(8) space aiia!::SJJiIIl neously spinors with two components in two dimensions. RecalH~[ml dimensions a Dirac spinor has 32 complex components, a MajoraLIll1,~P.l.:. 32 real components, a Majorana--Weyl spinor has 16 real cOlnp,ori~:*~~ the light cone gauge the spinor has eight real components traJlsfi)tii:i#lJ@I SO(8). The problem with quantizing this action is that the fermion fiel~~i conjugate. We have the equation

Thus: {o'la(a), elb(a')} = {e 2u (a), e 2h (a')}

~ 8ab 8(a -

{Ola(a), 02b(a')} = O.

Notice that this is not in canonical form. These fields are self-clon:iu~ fermionic fields form a Clifford algebra, while we would prefer to ·ml.;'!fK~ mann states without the delta function on the right-hand side of

6.8 Superstring Field Theory

281

way out of th is problem is to divide up the eight states in the spinor 4 staleS, with one sct of four states being the conjugate of the other One way to do this is to use the SU(4) subgroup of SO(8}: SO(8) :0 SO(6) ® 0(2) = SU(4) ® U(I).

(6.8.4)

decomposition, the 8 of 50(8) decomposes intO

8 = 4 ED 4. ,w d~""op are open string fields and \II are closed string fields . . OUf basic field functional is now given by (where we have plalce:l[t~ labels on the field):

ab[X(O'), 0 1(0'), e2 (0')] = -ba[X{rra - 0'), e2 (rra - a), el where the 0' variable is purely symbolic but was added to show hm~: tl1'D transforms under a twist Following (6.3.44) fOf the bosonic case, we can construct quantization relations:

1

.

P

.

[ct>ab(l), q,cd(2)] = 2 +o(al +(2){8aCobd~16[Zl(0') -Z2(0'

- oadC"c~16[ZI(0') - z2(rr!a21- a)]}, where

z = (Xl, elA , e2A ). Now that we have established the free theory of superstrings, let:Us,:t'I! the difficult task of constructing the interacting vertices for sUf,e;'~f~J will find several complications:

(1) There will be two sets of oscillators, not one, and sef,antte: R:9.I~~~ conditions arising from the overlap delta function. sets of oscillators commute with each other and do not mix. (2) There will be extra terms defined at the joining point of the tnr~(~:$J In general, extra fields cannot be introduced along the string b~·.~~~j will violate Lorentz invariance and conformal invariance. HCIWIl~\i~i!:~ can be placed at the precise point where the string breaks. We~:m1i:ili careful, however, because of singularities that exist at that po;trit~::}~~mil (3) The greatest restriction is supersymmetry, which will cOlnpll¢(~:ij::~ mine the nature of these insertions at the breaking point. We begin our discussion of the vertex function by postulating ~#>:t~:iH it will take. Based on an analogy with the bosonic case, we poi5tur~t~~~lml superstring vertex must look like

IV} = Zi

expJ~o + ~s] IO} o( ~ar};( LP:}S( LaAA

where the Zi fields are insertions at the breaking point and ~o bosonic term found in (6.4.23): 1300

~ = _ '\' ~ air) Frs a(s) o 2 ~ ~ -m mn -n r.•=1 m.n=1

300

+~ ~ Fl' air ) p ~~ m -m r=1 m=1

_ ~ p2 2ot'

6.8 Superstring Field Theory

283

the bosonic theory, n~;:n~' ;:~:'~:: ~~~::';,f~~!~:~:~~l:~~o:with operators. Let us adoptthat the decomposition:

e1';: =

_ ,_

8 2';: =

~

AlA

L

RA "~I,,alk..l ,

h. "

L RA e-j""/~1 I L

.J2a.

=

n

n

RAeina/lrrl

Jml.l" ,1.211

,

=

"

'

L RAe-i-/I,,1

1

~

./2:rrlal"

.

{R~. RB "I = as,,,+,,.08 I1 B• ",,3

lis =

_003

L L

R~'~A U~~R~~A

+L

m.n .. I,...=1

__

L

V; R~!~eA.

(6.S. iR)

",_ 1 r=1

U and V matrices are totally unkno ....'Yl and I· . A e A- = _(£110 1 _ R l ).

.,

is no justification for this fonn (6.S. 18) other than that it satisfies boundary conditions thaI we will now apply. We will enforce the

(6.8.19)

is + 1(-1) if the string state is incoming (outgoing) and the tilde ',"uh,,''',md oscillator. Tllesc conditions, when written oul in Fourier resemble the conditions found for the conservation of momentum.. I we gcnerdlize (6.4.17)

tf../ii A~!(R~);' ,=1 ,,=1

t f: ~A~!(R~)'

J 1,. ".1

/1;1

"II"

a,

+ R~;') -

-

R~!A)IV} =0.

I

h.BmEl' IV}

~ 0,

(6.S.20)

enforce the continuity conditions

LA:(O) ~ 0,

" -,

~J..,(n)=O .

(6.8.21)

284

6. Light Cone Field Theory

These continuity conditions, in tum, require the following CO]rrdltlOWfiil Fourier modes: 3

00

• - : : :.

LL -In A~~(R~)A - R~~A) IV) = 0.• •••• a, :.. r=l n=]

3001

" " _A(r) (R(')A (

L L -In

mn

r=1 11=1

11

a) IV\ - 0 ..•.•

1 + R(r)A) + -8 -_ -n ..fi mae A

1-

.•... , ..

We now have enough conditions with which to solve for the matrices. The calculation is arduous, but we finally find

Next, we wish to construct the supersymmetry operators in the theci@~iOOl ing on our experience with the generators of the free sUJ)er:syrnmletritilrliill~ Let the first quantized supersymmetric generators in (3.8.22) be then the second quantized supersymmetric generators Q are related free level by

1 f 00

Q2

=

ada

16

D Z[Tr

~-",q¢a + *-",q*a]'

Notice that the canonical quantization conditions (6.8.14) gmrrarlte~ q's form a supersymmetric algebra, then the Q's must also. In n'n-N;;~fil~m first quantized q's obey {q-A, q-B} = 2h8AB, {q-A, q-B}

= {q-A, q-B} =

O.

oftheise.t#l~11r

Now we wish to construct the second quantized version Specifically, the interacting part of the second quantized generator .... :~;.I:}'~ Q-A

_

= ... + A

x

~16

f Dda,D

I6

Zr8

(t

(ts;Z;) (~tI (~21

a)

(,., ,. , 11'111.

(6.9.4)

11> satisfies the string SchrOdingcr equations, we can power expand

functional in eigenfunctions of solutions to the string SchrOdingcr

1-

• (r,Xi o/p+Jnl

J

d PiC.J(p.• ··EjolflA p*.",.ln)·

(6.9.5)

288

6. Light Cone Field Thea!),

Notice that A is the creation lannihilation operator for creating or q~1~~ possible excitations of a string. Thus, it cOll'esponds to an u· UinJte.'11O;ii1 field theory. We impose the standard canonical commutation rell'ltio~ force us to choose

[Ap~ ' Pi.lnl'

A!.... q"tmd =

8(p+ - q+)8(Pi - Qi)8{nl.(ml·

Now we can power expand the field function c.I> in terms of these elgl~lD

(XI".

D.12 = ((01 c.I>;.,.(X t , + ·[L-(f(A:J"')l/2]p~I"'.

197

(7.1.27)

afthis was to show thai the action can be written in a form thai is ,10- (- l)¢'¢>*¢>o.

,

(7 .7.49)

~l:;(~~D exists, then we can show that it is nilpotent. Now insert

~

back inlo the Lagrangian, which now becomes

L = <J:I

* D¢+ ~g(J)3.

(7.7.50)

326

7. BRST Field Theory

If we can make the identification that this D is equal to the "~,,..a·; then we have shown that the action is precisely the BRST action W,itb)if, term! Thus, the usual BRST action might emerge when we eXIj~i~:j1 new classical solution of the theory. The novel feature of this approach is that nowhere have we maIM:~Ei tion ofthe background space-time metric. The background OC(::tlT!::C:i>li;jliP. kinetic term, not the interacting term. In fact, the choice of the M~ metric emerges when we expand about a classical solution to · ~t~1il of motion. This is why this approach is called the "pl·egeOIne1tr.l~3.f:~: In principle, the geometric of space-time should emerge as one aml~ji: possible vacua. In practice, however, we must be careful to check for the CO'~$~Wi1 our approach. The key equation was (7.7.49). Under certain asS~~ appears possible to find solutions for this equation where D satiisfile~~!iii for the BRST string field theory. In this case, it appears that the usU:aU BRST string field theory is nothing but one solution of the cI>3 a~~11 solutions of this action presumably will yield BRST theories d different classical metrics. For example, let us choose

~

cI>o = Q~l,

len;-mrnQ:'~'~~~i

where Q is the BRST operator defined only on the midpoint (for open strings) and I is the identity operator (which e~l~ left half and right half of the string coincide and zero .) BRST Q operator is equal to QL + QR. Ifwe make the Ibs1titi.iI~~~ original action cI> = QLl +gcI>,

we find that we recover the original BRST action (7.5.7). (The of Q emerges in the manipulations because QRI = -QLI .) Surprisingly, these definitions can be shown to be consis u,;i·h: Hbli.::h~ five axioms in Section 7.5, so that the usual BRST string actioDi :S~~~ one among many possible solutions to (7.7.45). So far, it can be shown that flat space is a consistent solution . remains to be seen, however, what other kinds of classical bac;kgro,",ij;1 found as solutions to the equations of motion. Lastly, let us now discuss some of the new directions taken U'::'l;!:!:.l:l theory. The first approach is to correct a hidden defect in the theory describ~d earlier. The superstring field theory is _~ ... l". [57]. For example, when calculating the four-point scattering aiD:p~~jt1;(~ three-string vertices meet on the world sheet, such that two pic:ttu;·~~~ operators X collide, yielding a divergence. We can show, U"'ll!$;::J;!~ffll! field theory, that two picture changing operators, defined at the.~tij~1! yields an infinity. Not only are the amplitudes divergent when j'tlte~~~ij the moduli space of the disk, but the gauge symmetry itself

7.7 Closed Strings and SuptfStrings

327

""uco changing operators collide at the midpoint when multiplying two ,. lIm, . the problem is hidden within the algebra itself. have been some anempts to solve this problem by playing with the changing operators, by altering tho ghost number of the ~trings. un~v,~:~~"~~ disappeari. Since all pictures must necessarily yield the q superstring amplitudes, one solution is to change the picwhich the theory is formulated. In other words, the action, which is . may be defined in various pictures, but the on-shell S-matrix is

,

development concerns the closed string, perhaps the most urgent facing string field theory. Naively, we might start with a closed string such thlit all strings have the same lengths and then construct a '~";~~ Howevcr, Kaku and Lykken [58] have shown that the resulting ;d not fully reproduce the Shapiro-Virasoro amplitude, i.e.• there region of the integration region. They posrulated the existence "~:~~,~:~~iJ\teraction necessary to restore modular invariance and the il region. By unitarity, there arc therefore an infmite number diagrams necessary to restore modular invariance, i.e., the action must Kaku and gmups at MIT and Kyoto l59, 60, 61J established the 'p',Il",o,,'al closed string field theory. Not only is Ihe gauge invari~:~:;,;,:~o;:,::p.~olynomlal interactions of closed ~trings, the action is also 'fi that the complcte modular region is recovered, piece by integrating over the moduli space of N -point functions. A generalof ' hi·,' bosonic string field theory to the superstring case, however. has successful. Because tlle closed string field thcory ill nonpolynomial, of colliding pictun;: changing operators. Now, there which can collide sheet. development is the construction of a nonpolynomial closed string for the D = 2 clnsed string ease [62]. In fWO dimensions, string be ,cJ~nmul..ed in terms of a Liouville lhcory. In fWO dimensions, retains its hasic nonpolynomial structure, except that the states are i'"!,ler and the ghost insertions at the midpoints arc different. in order counting eoru;tmints. development concerns a geometric string field theory, one based posrulates which may unify all these various string field theories · geometric theory gauges the string length. so we can "interpolate" · Witten's formulation (based on fixed string lengths) and the light· formulation (based on variable string lengths). This unifies the two nU>aI'i,nu,. ,which are now seen to be specific gauges of the same theory. first BRST theory is defined in the " midpoint" gauge, interact at their midpoints, while the second BRST theory is "endpoint" gauge, wbere strings interact at their endpoints, as

328

7. ERST Field Theory

Lastly, we should conclude by saying that string field tneqry:::£@. evolving subject. As a result, it has not yet lived up to its proliDjl~¢::~am nonperturbative information about string theory. Although is defined independent of perturbation theory, it is still too form nonperturbative calculations. By contrast, M-theory has alt(~~ a wealth of information about the nonperturbative nature although in a nonrigorous fashion. It remains to be seen if M-theory can be formulated in a sec,()l:i!~ijji fashion, as in string theory.

7.8

Summary

The origin of the covariantized gauge approach was to cor[s~~im invariant under 00

lJ 1tm us begin a discussion of the mathematics behind anomalies. Spl~cijti.qjlii will show that the anomaly can be written in terms of generalized chliFn~ classes that have been studied by mathematicians. Then we will sh()W)!fj elegant part of the theory, namely that the integrals over these cl;l~@.!!j classes yield various index theorems. Thus, our strategy is to "he".;;' Anomalies --+ Characteristics classes --+ Index th"n"'~"""ii:

8.4

Anomalies and Characteristic Classes

To study characteristic classes [10, 11], which will give us a S~E~t~=m ysis of all possible topological terms, we will use the language Appendix.) The theory of forms, in some sense, only produces re~i~l1~ili be derived using the usual analytical methods of calculus. dimensions, the number of indices rapidly proliferates beyond the powerful shorthand notation of the theory of forms allows us nipulate tensors of arbitrary rank in any dimension, which is standard tensor calculus. In Chapter lOwe will see that the theoryi'i;if.1mi the most convenient language for the theory of cohomology and .. Armed with the theory of forms, we now construct a series of classes that will allow us to write by inspection the set of aIllOp1olO;gJ,:qj for practically any dimension and any group. Let us now define an invariant polynomial that has the nTe>np1rru:~ilili~~e~ P(a)

=

Peg-lag),

where D! and g are group matrices. Examples of invariant po.lynortll~] Det(l

+ a).

Tr ea.

Now let n be a curvature two-form:

n =dw+w!\w that satisfies the Bianchi identities

dn +w!\ n - (-ly"n!\ w

= O.

In the Appendix we prove two important statements for .' mials:

[1] [2]

dP(n) = 0, p(n)=dQ.

for some Q. That is, an invariant polynomial based on cUJ'Valtun~f~ltJ closed and exact. The theory of forms allows an explicit COllS1J1X~I~~!II

8.4 Anomalies and Characteristic Classes exampla. in

349

me Appendix we show

Trn" = nd =

11 dlt~-'

Tr IA(dA +IAl)"-I } (R.4.6)

d{j)n _l.

,~:~~~:,~~~I'i~iP';:~~: of curvature two-fonns is exact and closed. This ~~ in the language of forms o f previous identities that were hand. f orcxampJc. in follI' dimensions we havc (&.2.7) and (8.2.17). dimension we have (8.2.16). This new form tU~_1 is called the mU,M Jun,,,, which we encountered in (82.R). is important, bccause earlier we showed that in two and fourdimensions polynomials in the curvature tensor that are rotal derivatives. But unable 10 construct these higher topological tcnn.t coupling limns: LI

iltl'· 1 ="ih ¥r"YCI(JII"2(l

L2 =

S

- y )01,." ,

/6(h~a"hTa)~ .. rCl"·~( 1 i

L)

_ YS)l/r",

(8.6.20)

_ (J"h"")¥ray ~ ,,, ...

= 2"(Ja h"v -

.. 'g'''", we can also write the general one-loop amplitude as 1m =

221:+1 i

Ml R(lj). pU')Z(£li). p(J).

(R.6.2I)

Z represents the scattering of an internal charged vector meson intcra constant electric field. leI us write an effective theory of charged 'me• .,,, that reproduces Z to onc loop: ~d'

lo

Z => Trln H = _

_

,

Tr~ -·H,

+ ~iAa)2q,,, + ~jF"v¢".

H¢ .. = -(aa

(8.6.22)

(8.6.23)

'en",;,,,,,,,,,,,," be performed, and we arrive at

n.

24"+1

::: _i(21fr2k-1 R(.s{i1.

pi})

"'~' =

al' l

A../"_...... , + cyclic perm.

,

(8.6.25)

""""" the constraint that it is self-dual: F

-

I'I"'U,,,, - (2k

+ I)!

t;.Ul.O.!·- .......,

"' - ""~I

F

",u,···",,·,·

(8.6.26)

fe~~:;~ :~~:~'~:. By the Bianchi identities, we can show tbat this relation ;j,

to the equlItions of motion. In other words. tbe naive action L __ p1

","',,,''',

(8.6.27)

both self-dual and antidual states, not jllst the self-dual ones. Thus,

': ':~'~~,~~';ln fact, we can prove that a covariant action that propagates bl stlltes does not ex.ist (14]. V~;:~, ~:'o:~ use a trick to calculate the scattering amplitude. Although

d(

seem to be a covariant action. the Feynman rules for such a

362

8. Anomalies and the Atiyah-Singer Theorem

particle can be written covariantly. The trick is to use a spinor nel[9':tij: the antisymmetric tensor field. The energy-momentum tensor for this field is

1 "I "'1l2k 1 2 TILv = (2k)! FP.al·'·"lk Fv - 2(2k + I)! gp.u Fot \'''''2k+1' Fortunately, all the Feynman rules for this antisymmetric F tenso~:~~m. even if its action is not known. Now we will rewrite this antisYIIUlq¢tt}iii in terms offermion fields by a trick. Let us define N

N " ' Tr (r P.1/l.2"·P.. )otp FP.tlll"·P.. • = 2- /4 '~

A.. ~,,,{J

11=0

We can also invert this A.."P • FILIP.l " ' /L. -- 2- N / 4(r" . .. ·Ill ){Ja~

The advantage of using this embedding is that fermion fields are ~~~!~ to use than antisymmetric tensors. The two-point function is .. IJ:'

(¢"p(q )¢y~( -q)) = (2 q 2) - I«y5 y /LqlL)"y (ySyp.ql' )(J~

+ q2 S"Y

"",d - Fbcl'J ad - Fadl'J hc + FbdO ac )' We can insert explicit expressions into the trace calculation of F6,,'li~mB Tr F6 = (I - 32)tr F6

+ 15 tr F2 tr F4.

In order to satisfy the factorization condition, we must be able tOI~Atl.Wi number of terms, including the sixth power of the curvature this is possible if I = 32, which has precisely 496 generators. back again to SO(32). The proof that E8 ® Es also satisfies (8.6.37) is a bit more lllV'Ul~.t:U" to know whether or not the Clebsch-Gordon coefficients of the us to write independent invariants that allow us to contract Mathematically, we need to know if there are independent CasinW!~~ of orders four and six. Fortunately, there is a theorem which homotopy group X 2n-1 (E8) contains the integers, then there is Casimir operator of order n. (Homotopy is a way of geller.atlllg~q~ classes whose members are continuous maps, rather than spaces It can be shown that the first homotopy groups satisfying this coli{ij~~i and XIS:

x/(Es) = 0

for 3 < i
- M is oricnlable,

(LI;!

= 0

-7

M is a spin manifold.

(8.8.8)

370

8. Anomalies and the Atiyah-Singer Theorem

The index theorems, in turn, are usually written in terms of invariant polynomials: Todd class = td(M) =

x· . I _ ~-Xi '

TI

n [

Hirzebruch class

= L(M) =

~Xi '

I

I

a roof polynomial = A(M)

= IT .. zx; smh

i

.

i:Xi

theore~~1

The most useful of the index theorems is the Dirac index concerns the number ofpositive chirality zero eigenvalue solutu:ms:tt);tlJ; equation minus the negative chirality solutions. Using the sigma model, we can show the following: IndexCP) = TrC _l)F e-

rD2

=

~! d DxfJjJ.JjJ.5

=

(2n

. )(1/2)D

_1-

.

:. ··!i~i.

!TreFdeCI/2[(!R) - lSinh! ·.· •.

..

,2

2 :. __ '"

Armed with this theoretical apparatus, we can calculate the galJig~. ~ia itational anomalies found in supergravity and in string theory. In,¢'1*-.~B.g gravitational and gauge anomaly contributions arise because the • can be either: (1) a chiral spin-! fermion; (2) a chiral spin-~ fermion; or (3) an antisymmetric tensor that has no covariant action.

The calculation of the anomaly contribution is carried out Feynman diagrams. The calculation, however, simplifies we can make certain assumptions about the polarization tensor lines, thus reducing its spin. Thus, the difficult problem of COI1I@~~ the various indices is reduced to a much simpler problem of COill®:l~ a lower-spin particle. The final results are

8.8 Summary

371

everything together, we find the total anomaly contribution from and gravitationa l sectors: ~I 6 -TrF

720

I

,

+ 24·48 TrF'TrW

",la,u,ly,we can set this 10 zerO if we make a few assumptions. first. we 10 be 496. Then we assume th~t we can facto ri ze the anomaly into ,duo' of"vo tenns:

III - (Tr R2 XK =

+ k Tr p2)x 8,

2~ Tr p4 ~ 72~O(Tr F2)2 1

2:0 Tl' F2/\ Tr R2

(8.8.13)

1

+ 8 Tr R4 + 32(Tr R2)2. 10 supergravity is immediately ruled out because the id"nrity canna! be satisfied. Hoy.'ever, string theory has one big adover :mpergravity. The presence of higher spin fields in superstring thai the zero slope limit of the theory does not have to reto supergravity. In particular, the couplings o f the B field in arc such thatlhey can. in principle, cancel the tenns shown

,atu.1 proof, however, that the anomaly term disappears must be carried the full single- loop hexagon graph in the string formalism, We use a 'V~;';,:d;~: regularization on the intennediate lines and then sum over nonorienlable loops. The anomaly is proportional to

iii

.,

I1n€(C,k)

/ d ,"(1 pTr Lo+m

2Vo(l) ' "

16)) rJ , Lo+m 2VO(

('.'. 14)

(8.8.16)

372

8. Anomalies and the Atiyah-Singer Theorem

Finally, adding the planar and nonorientable diagrams together J 5

yi~r~M~® . . ~~jmm

'0 ~~(n-,+~:.a)lt(..tJA2· .. ::

11 uVi i= l

x 9(IJi+J -

Vi)

(01 Vo(k J, ZI)'"

For this term to be zero, we must have II = 32

l= {

~

-1

Usp(n), U(n), SO(n).

Thus, the gauge group must be 0(32).

References (1969).

1. E. Paton and H. M. Chan, Nucl. Phys. B10, 51 S. L. Adler, Phys. Rev. 177,2426 (1969). J. S. Bell and R. Jackiw, Nuovo Omento 60A, W. A. Bardeen, Phys. Rev. 184, 1848 (1969) •. [5] E. Witten, in Symposium on Anomalies, Ge(nnet~ A. Bardeen and A. R. White) World Scientific, [6] P. H. Frampton and T. W. Kephart, Phys. Rev. 028, 1010 (1983). [7] P. K. Townsend alld G. Sierra, Nucl. Phys. [8] B. Zumino, Y. S. Wu, and A. Zee, Nucl. Phys. [9) J. Wess and B. Zumino, Phys. Lett. 37B, 95 (I [10] T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. [11] C. Nash and S. Sen, Topology and Geometry New York, 1983. (12) L. Alvarez·Gaume and E. Witten, Nucl. Phys. [13] 1. S. Schwinger, Phys. Rev. 82,664 (1951). [14] N. Marcus and J. H. Schwarz, Phys. Lett. [15] M. B. Green andJ. H. Schwarz,Phys. Lett. [16] M. F. Atiyah and 1. M. Singer, Ann. Math. 87, (1971). [17] L. Alvarez-Gaume, Commun. Math . Pllys. 90, [18] D. Friedan and P. Windey, Nucl. Phys. B235 [1] [2] [3] [4]

9

Strings and "fication

Compactification problem facing string

is the question

reduction can be made, the theory lacks any real contact quantities. breaking canbe dODe within the framcworkoffie ld theory, the best we can do is look at dassical solutions where spontaneous moos"""h", alrcadytaken place. In this chapter thc heterotic :.tring, which incorporates the groups Eg ® ER that arise when we compactify a 26-dimcnsional space down ~w

in the previous chapter, the cancellation of anomalies allows for of either 0(32) or E8 ® £8 . We saw, however, thai thc Chan,,"md does not allow for the possibility of cx~ptional groups. Thus, use yel anotber method, arising from compactification on a self-dual achieve an £8 ® £8 model. Although the heterotic string is a closed , it contains within il the super-Yang---Mills field. whieh usually s :,,~;~:::t~~ string sector for Type I strings. ~E

string takes advantage of the deceptively simple identity

26 = 10 +16.

(9.1.1)

that a 26-dirnensional string, when compactified down to a JOstring, leaves 16 extra dimensions that may be placed on [be root E8 13 £8, which is known to yield an anomaly-free lheory. TIlis was made by Freund Pl.

374

9. Heterotic Strings and Compactification

The heterotic string takes advantage of the fact that the CIO:S~tJi::S1 two independently moving sectors, the right-moving and lett-Illlovifilf: depending on whether they propagate as functions a +r or a - r. 1·; f1.f.l::lrreti string makes fundamental use of this splitting. The word "~'~~~i!~lll "hybrid vigor." This means that the asymmetric treatment 0 movers creates a hybrid theory considerably more sophisticated.fuil··ifj.:i6.~~ Type I and IT superstrings. It has been shown that it is has .... . anomalies, and is finite to one loop. Before we begin a discussion of the heterotic string, hmNe"et., descnoe the process of compactification by describing the sin}pli~~1:1OH case, that of a scalar particle in a periodic one-dimensional that we make the identification

x=x+2nR, where R is the radius of this space, which is just the real axis one-dimensional lattice r oflength 2n R:

divli:l~j;1;:

Rl

Sl =

r'

A field defined in this periodic space must therefore satisfy

¢(x) = t~~~I~~\~~';~~~~~:~;::~::: ::I~~ 16 to yield an irreducible representation ~I This means that, in general. the Fock space of the helerotic string manifestly recombine into E, ® E, multiplets. At the next level, for , the states form irreducible representations of the algebra only in the

of the calculation. . the next level, where m 1 = 8, for the right-moving sector we have 128 '. and 128 fcrmions : 128 bosons

I a~dj)/C· I

. I 28 'icnmons

~I Ib)II'

a~ l la)II' N .J_I I i)~ .

(9.3. 10)

that we have fonned bosons out of the tensor product of two fermions

above equation. ,lcft-.no·" " 8 "''',''~. considerably more involved, with a total of73.764 ' . We bave a tolal, therefore, of256 x 73,764 = 18,883,584 states! The

"" scalars -+

Ip' , (pl)2 = 4)L -+ 61,920 slales, al. 1 Ipi , (pi)2 = 2)L -+ 7680 states, a~la:1 10h -+ 136 states,

(9.3.1 1)

a~lIO)L -+ 16states, of 69,752 scalar states. (To count the states in Ipl. (pl)2 = 4), we "Iact thaI the number of points oflengtb squared equaI to 2m for integer lattice r l6 is 480 times the sum ofllie seventh powers of the divisors for this states, there are 480(1 + 21) = 61.920 states r6J.) it is not obvious ataIl tbatthese 69,752 scalarstate.s can be rearranged of E, ® £8 multiplels. However, a careful anal)'sis shows that the)' •. b,ok,," down into the following £8 ® EB multiplets:

1)+ (1. 3875) + (248. 248) + (248. I ) +(1.248)+ (I. 1) + ( I , I). (9.3. 12) vectors in tbc lcft-moving sector arc arranged as

I

ci'_1 p I , (p')' = vectors -+

[

2)/. -+

3/W0 statCS,

a~2 10) I. -+ 8 stales,

a~IQ~[ jO}L -+ 128 states.

(9.3.13)

386

9. Heterotic Strings and Compactification

and we have the tensor state

a~la~l 10) L ~ . 36states. The total number of states in the left-moving sector is thus 73, At higher levels, the degeneracy climbs exponentially as d(M) ~ e:' chwart.: (antiperiodic) boundary conditions on oscillators. we can represent the same [cft.moving isotopic (9.4.5) = J. 16. This, of course, yields the light cone representation we used when we broke Lorena covariance. (It may seem strange at first to left and' sectors labeled by ±. which seems to single out a oon",.,tiO? io cwo dimensions. However, the above fonnulation reparametrization invariant because the ± directions are placed in thc space. Thus. two-dimensional repammetrizalion mwriunce remains let us write the entire space-{ime action (minus the isospin part) that both left- and right-moving sectors in a covariant fashion:

/ d 2 (:e

H(e; iJg x"i -1; ljrl-' p - e': ilg W" + ~ i(Xgp' pit w,ll)ajl X",}. (9.4.6)

<etion has ;'N = ~" supersymmetry: !e~

=

il:p_~o,

6X .. = -2Vg E. 8X" = ;E1/II', l!l/J/I. =

(9.4.1)

{a.. x" +; x.. \11'" p"]t.

us, instead of choosing the light cone gauge, choose the conformal the heterotic action reduces to

(9.4.8)

of all this was to show that a covariant version of the heterotic exists and thltt we can use either fcnnion or bosonic fields to rcpresen! port of the field". We are not bound to a light cone formulation in on"",," fields.

388

9. Heterotic Strings and Compactification

It should be noted, however, that even in covariant •...,LU"'U2 · P. +P1·kJP, · k,P , 'Pl

+ Pr - k.}P4 · k 1Pl ' Pl + Pl' ~Pl' k,p, . p.l - ~1(p, ·k,~ · k 2 PJ · p, +PJ ·x..PI ·K.2P2 ·P.

+ Pl·I4PI • k,P) ' P4 + P3' k , P4' k 2P2' p.] + PJ' k.P2· kiP, . P. + PI ·k.Pl ·k)p)· P. + P3 ·k2P. ·k,P, -Pl] .

- !U[PI . klP,, ' k,P) ' P2

(9.5. 14)

is a phase factor and

s = - (k , +kd. r = - (11.2 + k))2, II

= -(k\ + *))1, s+t+u=O,

S = (KI

+ Kd ,

(9.5 . 15)

+ K)2, U = (K, + K )}1, T = (Kl

S+T+U=8.

Single-Loop Ampli tude ill ',",of "heth"o,),. however. is the calculation ofthe single-loop diagr.un : where we demand Ihat the theory be fini le. It is now a straightfor'"'''''' to wrile the single-loop diagram in terms of these vertices and

A""

= T,(6 V(N)6 .. · 6V( I».

(9.6. 1)

consider scattering by charged gauge mcsons. • the trace calculation is loog but slnlightforward. After the trnce is we have

A.ioop "" i K

f Ii d2zilwl-l[_4n]S w-I

f(W)-24

1.1

)( n

1 9<j~'

In Iwl [x(eJI. wW'f2Jt··tj (';(Cj/, w}]"'r K, I. .

(9.6.2)

392

9. Heterotic Strings and Compactification

where

[ln2IZI]

w =ex

p 2lnlwl

X(z,) .1,(- -)

=

'I' Z, W

L(w, Zi. Kd

"" ..;z n

l- z

(l-wmz)(l-w", , ' : (1-

m=1

. ::.', '

[ln2z] l-z n"" (0- wmz)(l-w ,,;z w

x e p 2 In w

=L PeA

m=I

(1 _

·'

m

m

.

exp [41n W (p - t lln~ Qi)2] , ;= 1

nw

and where i- I

Qj ==

LK

j ,

j=1

loz)

= L - 2., j=1 rr I j

Vi

lnw

i=--

2rri'

ejl

= ZIZ;+I ..• Zj'

where the sum over PEA represents the sum over all points in'::'tl}im K is a kinematic factor identical to the one found for the cause the final result bears so much resemblance to the amplitude, it is not hard to show that the amplitude is mvari,ant.:W'i:4] ::::::~:m

J Vi ~ + I, l Vi ~ Vj + i. Vj

Slightly more difficult is the proof that the integrand is j'IMl.ljWi;@~ + 1 and i ~ - i -\ • which is necessary to prove modular ttwwa, Fortunately, most of the terms in the integrand are j'(lerrtic~IH~ in a single-loop amplitude in (5.5.1). However, we must vm"'I1'.: ~ of the terms that are different, i.e., the factors dependent on As before, the theta function, if we make a modular trrurlstiq:mt~ arguments, transforms as follows:

i

8

1

.(

V ...

ci+d

ai+b)= (ccr

+d

S

i

+ d)l/2 exp (-irrcv2): - d a + . ... .';~:.:~...,u.

where SB = 1. Since the X function can be written in terms .. : we have

ar+b) = cr+d

1

la+dl

:'.. ' x(vli). :

. ::

9.6 Singl e-Loop AmplilUde

J9J

"rt;,;o,n nlDction few) also transforms as f(w ):::: (i / T)'fl w -

' [2 W, I/ 24 f(w'),

lnw T :::: -

- 1

- 2lTi

r

In w'

-:::: -

-f-

2lri

(9.6.9)

important transformAtion is the one for L;

(-2>1' ) ' ~ exp [2>1"w( t;:"Q.lntn '-nri )'

'"

;: L(w. ZI . K,·)::::

W

+

tn

P-

l.i

2 ~W (t,Q'ln"Y]·

(9.6 .10)

identity depended crucially on the lattice being self-dual, which is of the strongest arguments for this restriction on the Janice. Let these factor.; togeth.er:

n

1Jr(i'i j ;l r l K,.K' L(t ,vj)

:~. (w'r' f(ii/)- 24 e2[ f1 ,,( -~Ij ":9~!>4

::

::·

.::

·· · ··

- I

L (T

'

T) (iT s exp -VI

T

( (. )') ilT

T {;; Qli'ii

.

(9.6. 11 )

factors in (9.6. 111 iliat apparently violate modular invariance, metors of T and the e~poneutia l s involving r. Fortunately, both sets ",n;sh. The factors of r cancel because £\\'0

us with r '2-4-1

:::: I.

The exponential faclOrs also vanish because

eliminate the factors of f and thc exponentials of t in (9.6.111, we this combination is modulnr invariant. "''",,. of the alDplirudc under v, -4 VI + 1 and Vi -4 V; + T means truncate its region of inlcgration: 0< 1m VI < 1m" -! ...-:: Re v,....-::~.

(9.6.12)

9. Heterotic Strings and Compactification

394

Also, as in the case of the closed single-loop diagram under r ~ r + I and r -+ -1/ r, we are free to choose a hlI:ldalll'¢:iiUti {

-!2 -< Rer < ! - 2'

Irll.

,

Once again, the choice of this fundamental region allows potential singularity at r :::: 0, and hence we have a finite On(Hol()lj:.~ In this discussion, modular invariance was crucial in showing u.~~=I tude, like the usual superstring, is finite. We can simply choose ~ region where no singularities are present. However, the calcula~i~~;ijfi precisely why the theory is modular invariant. Let us amIlYi~e,j~.~ case and isolate the point at which modular invariance comes matters, let us consider the vacuum one-loop amplitude with Let us first define a function F (which appears in the trace q~~i,ijjij

(9.6.1)):

L

F(r, X) =

e-irrr(L-XY,

LEA

where we sum over lattice sites, X is an arbitrary 16-(lirrlensi(JtUalf:v(i~i root lattice space, and each site is labeled by integers n,:

L=

Lllief.

Notice that F is periodic: Fer, X)

=

F(r, X

+ ed

because this shift can be simply absorbed into the integers n;. this function in terms of its Fourier transform F:

L

Fer, X) =

e- 2iM .X FCr, M).

MeA"

Notice that, because F is periodic, the M's must necessarily

lattice: 16

MI =

L m e7 i

1

•

;=1

Now take the reverse Fourier transform to find F in terms of dl6X FCr M) _e2ircM·X F(r X) .>.

M

f

'-.JT8T

"

where is the volume of the torus. Now insert the eXT)re~)si(jnJ-ot:f.~ previous equation. Perform the integration, and we arrive at for F:

- M) = -1r - 8 em · M'/ r. F(r,

M

9.7 £1 and Kac-Moody Algebras

395

would likc 10 insert this identity into the expression thai actually the one-loop vacuum c.1leulation. We must calculate the traee over l.:.~!:,~'~~;; which contains the piece (pI? Thus, in the lrnt:e calculation M f appears: f(r) = ,"

L

e - lttrL.L.

(9.6.21 )

'" 14). we have F(r, 0) = 1(,), ......

;. ..... substitutc the expression for

fer)

= -

=

F. and we have an eJtPression for f:

I ( I)' L

./fiT

- -

'

(9.6.22)

e'tf.,.M/ r

MEr. '

~r(-D·

(9.6.23)

is ooLhing but the usual f defined in (9.6.21) ovcr the dUll/lauiee, key result. that the modular transformation, _ - l iT has il t/redua/lanice. Thus. for modular invariam:ewedemand Janice be sclf-dual. In fact.., this is the origin of the condition that the self-dual. Here, we see the tight link between self-duality (which us cithcr E, ® £8 Spin(32)/Z2) and modular inNlh.ough the original choice oflhese groups came from the anomaly "~;:~;we scc that we need precisely these groups for modular invariance lit' . of the nmplirudc.

";:;';;:;;;;,;we::,:s;ee

restricts to

re-

or

; £8 aDd Kac-Moody Algebras i; wo ""V Ih" thespcctrum ofstates was nol manifestly £s®E~ invariant.

that

the lowesT-lying stales could difficulty were we able to show in irreducible representations of the group. Because the number rapidly proljfcrates into tens of millions, it becomes prohibitive (""'Hnl!' the SIllies into E. ® £8 multiplets. section we will use the techniques of Kao-Moody algebras (8--10] in Cllapter 4 to show, to all orders, thai the heterotic string docs :.have a spectrum that is E. ® E, symmetric. We will use the vertex .. defined in that chapter to geoerate the representation of a .K.1C'all!eb" (which wo rks only if the lattice is even. self-dual, and the level I).

to

;:~~~:~~:;~ that we desire is the Chcvalley basis. where the 496 up into 16 mUlually commuting generators (which form

i)J

396

9. Heterotic Strings and Compactification

the Cartan subgroup) and the 480 root vectors. Notice that obey the condition

fnf1:,:-I:/;'i·'....t-::,

[pI, pI] = O.

Trivially, we can always represent the Cartan subalgebra as sinlply~f~j~11 commuting vectors pl. More difficult, however, 1s the COllsttllC1ti(;jifl!ij~ root vectors. The simplest operator with 480 states is the vertex 01p()rator,J.~l~111 write in terms of a line integral surrounding ilie origin: E(K) =

J 2d~

V(K, z)C(K).

r

nlZ where the vertex function V is defined over the lattice (rather time):

tlL~~%iI~~

and where

(Kif = 2, Z=

e2i (r+ Bj j • and Af fields are by constant background fields. Notice thai the total number . in this approach is easily found by counting the number of modes within these fields:

;q(q+l) 16q

I

(9.8.9)

iq(q - I) number of pq parameters. Now quantize this system, assuming that approximated by

Xi = 2all '

+ q'(r).

(9.8.10)

9. Heterotic Strings and Compactification

400

Now substitute back this expression into the action, quantizing " . .c:J holomorphic z = X + Jy

analytic z = x + iy

(Intuitively, the process of patching together neighborhoods ~~!:1! agonal resembles the "elevator problem" in general rel:ativritv. :.... ~";"'" coordinate transformation, we can always locally transform 1O,e::'Q symbols to zero at any specific point, which corresponds to ....!'~!~~ an elevator without gravity. In general, we cannot globally tra.i~$ts!OOi Christoffel symbols away entirely, or else the space is fiat, but",,'.'~r¢l:iiJ point on the manifold enter the elevator frame.) . At first, this definition might seem verbose fora rather However, certain 2N -dimensional real manifolds can be criterion. For example, the spheres S2N are not complex IIl3lni1uI~~:'~ S2, because S2 = C PI). Thus, it is not at all obvious that a real manifold can be rewritten as an N-dimensional comple~XX!;[;fjiJ Now let us discuss differential forms on these complex n complexify our basis differentials as

= dx j + idyj.

dzi

Now the concept of a p-form can be generalized to a (p, W

=

W,r]'k ···U.l "j-k- ···u-

di

j

u

1\ dz ..• dz 1\

dz i

1\

dz1 ... dz u ,

with p unbaqed and q barred indices. We can now licitcaJCl ~a

20d =

= ~il'

This is a powerful identity because it states that the VlIl:10l1S:::]:;i1 that we can fOlm on a Kahler manifold are all the same. ~~II! confusion when using different Laplacians for the Kahler ' (3) If a manifold is to admit a Kahler metric, then the even co~~!ijj the Betti numbers must be greater than or equal to one, an(Vl:tsk~J:ti numbers must be even: @~rs

V"1.1 ~er

b2 p > 1, { b2n + 1 = even,

--+

for integers p and n. (This simple criterion rules out a large "l' ''1':~l;i:~~J folds. For example, this rules out S3 x S3 as a manifold that metric, because b2 = O. It also rules out P N but allows C PN '.:: (4) If the J::' tensor is covariantly constant, then the metric . ... converse is also true for complex manifolds.

va J% =

0

B-

Hermitian metric is Kahler.

(5) If the torsion fonn that we construct on a complex manifold v~l~~t~BfI the metric is Kahler. (6) The only nonvanishing Christoffel symbol of a Kahler roanm;)m:~

r abc =

cj;

g gat.h·

The only nonvanishing component of the curvature tensor

R:

bc

= -

r:b,c

and other curvature components related by symmetry conjugation. The contracted Ricci tensor becomes

R -- -

ab -

a

2(lndet g) ---=,'-

ozaaib

10.4 Kahler Manifolds

423

(10.4.26)

a direct consequence, we can show that Kahler manifolds have U(N)

In facl. this caD be used as an alternative definition of Kiihler C;~~:~~.. Notice that lhe holonamy group is the rotation group generated closed paUlS around the manifold. which is a function of ( 10.4.27) M ;j

is Lbe rotation matrix for some 2N-dimensiona! tangent space

The coefficients of this rotation, we sec, are

funCl io n~

ofwc 3nti-

Riemann tensor R"bed. which CW1 1l0W be viewed as an element a 2N x 2N antisymmcnic rotation matrix, Th\ls, in general, we have holOGamy for a general manifold. However, if the manifold is

we can show that che restriction on the curvature tensor we found reduces the SO(2N) rotation matrix down to U(N), which is a ofS0(2N). The restriction that (10.4.24) is the only oon7.ero aC the curvature tensor breaks SO(2N) symmetry.

manifolds. which have U(N) holonomy, can be further restricted if demand that they have vanishing first Chern c1a~s. In this casc, U(N) '~;:~:~:~r:educes 10 SU(N) hnlonomy. In fact, we have the theorem of D which works for all N, thaI a Kahler manifold oj vanishing Chern class always admits a Kahler metric ofSU(N) holonomy. In , it can be shown tbat the vanh;hing of the first Chern class is equiv-

!:;~C:~':'~~:~\I:/a

Ricci-flat mC!:tric. Thus, wc will use these two eoncepts

.1.;" now be instructive to consider explicit examples ofKiihler manifolds. ,em""" Sumce . oriented Riemann surface admits a Kahler metric. Because the line of any Riemann surface can be put in the form (10.4.28)

is Kabler because we ean always find a Klihler pOtential ¢ such thaI

a' azaz¢ =

e".

(10.4.29)

surfaces are trivially Kahler because any two-form on a Riemann including the Kahler (onn, is clO.'led.

424

10. Calabi-Yau Spaces and Orbifolds

(2) Complex N Space Complex N space eN is trivially Kahler because its standard

Imlfe1i~

ds 2 = Lldlf. i

can always be put into Kabler form. (3) Sphere We note that SN, in general, does not admit a Kahler metric. T'~f$i:~~~i the even Betti numbers h2p for p equal to a nonzero integer .~'t~~~!m.1 equal to zero. Hence, they cannot be greater than or equal to vu,,,,,.·.,,. one of the conditions on a Kahler manifold. The exception i$ d'iit:1~ S2, which has no even (nonzero) Betti numbers. To show n.···.',K;';':"'; admits a Kahler metric, notice that a two-sphere has the line e'llettJ&1J dx 2 + dy2

2

ds =

(1

+x2 + y2)2

.

From this, we can write the Kabler form Q

= !i dz 1\ dz 2

(1

+ zz)2

if we put it into complex form. This Kahler form is exact, and li¢j~~ admits a Kahler metric. Although S2 is Kahler, we can sh()V;('itij~ therefore it is not Ricci flat. We notice that Sp x Sq is a complex manifold if p and q However, this does not mean that they are Kabler. In fact, S3 are both complex manifolds, but they are not Kahler. is Kiihler (but is not Ricci flat). (4) Complex Projective N -Space To prove that C PN is Kabler, we note that it is possible to a Fubini-Study metric on C PN : Q =

!iaaln (1 +

W"~f'ilill

tziZi). r=]

(5) Complex Sqbmanifolds of C P N It is trivial to see that complex submanifolds of C PN are also K.aJ~~t~ use precisely the metric defined for C PN for the sul)m;am·told,)~x! be careful to take only the components of the metric tensor to the submanifold. Since the original metric was Kabler, th ..;·itM'i> submanifold (which is the same metric) must also be L,."'...." •.~;

lOA Klihler Manifolds

415

,-ton" T2 acrualJy has vanishing first Chern class c) == O. We can also the four-dimensional Lorus T4 is Kabler. Finally, we can show that ,·t'oru" T6 is both Kahler and Ricci flat. Thus, compactification on the . appears to have the desirable fcature that N = 1 supersymmclIy is . However, che drnwback oflhe six-torus is that too much symmetry ~"'ed . Tn fact, N = 4 supcrsymmctry is preserved on T,» making it

,ptable from the point of view of phenomenology. now summarize some of these results in a table:

R-Kiililer mean~ Ricci flat aod Kahler, Y, (N) means yes (no), X equals andn == 1,2, J ... the condition of being Ricci flat places even more restrictions for example, it can be shown that a Kahler manifold complex dimensions has c] = 0 if and only ifthere exists a co\'nriantly nOIlV"dnishing hoiomorphic three-form w. This shows, for example, S2 x S2 does oot admit a Ricci-flat Kahler metric. (We know thai this hilS b3 =: O. Thus, by definition,mere are no harmonic three-fonns on mil'o"li.. But this also means that there are no holomorphic three-forms, , it cannot admit a Ricci-flat Kiihler metric.) ,i,"p'" cOI1$equenlZ of This is Ttl::l.l ::I.oy IUlnnonir. (1'.0) in thr~r. dimensions can be multiplied by w, thus creating a hannonie (10.4.34)

'ntity is true because w is covariantly constant and hence acts like under the laplacian. Thus, a harmonic form remains a hannonic multiplication by w. Also, a (P . O)-form becomes a (0, 3 ~ p)-form . we are contracting with the Hermitian metric tensor gat; and not gu/). in tun" allow~' us to eliminateafmost all the various Hodge coefficients. various reflection symmetries and the previous symmetry, we can It ooly·" •.. •. /12.1 • II 1.11 survive as independent components of a manifold holonomy. Of these, we c"n also eliminatc h 1•Ofor the foll(lwing

426

10. Calabi-Yau Spaces and Orbifolds

reason. We know that the Laplacian can always be expanded 04i@~:ffill £!"d = _'12

+ curvature terms.

Acting on a one-form, the various curvature tl;:nsors reduce to tl,lMjij~i But the Ricci tensor is zero on a Ricci-flat manifold. Thus, form must also be covariantly constant. This means that the ~"'¢t!it:1J:\J a harmonic one-form must be zero: b l = O. But it also mea~Oij~~j by (10.4.15). Putting everything together, we can now sholvi : fil¥:~: manifold,

x=

L (-l)P+qhp,q = 2(hl,l - h

2 , 1).

p.q=n

This last identity for a Ricci-flat metric will become extremely ~~~II we discuss the generation problem. It turns out that h 1.1 and the number of positive- and negative-chirality fermions we "ciitt:dil~i manifold, so that the above equation states that the gellerlltiOlillj[ijffi the Euler number generation number =

! IX I.

Thus, we have a topological derivation of the generation

10.5

Embedding the Spin Connection

Armed with some of these elementary results from algebraic 19tJIQ~~~ now return to string phenomenology and apply these results. >,. , Earlier, we saw that the condition of N = I ' ', existence of a covariantly constant spinor, which in tum ' dimensional manifold K was Kahler, Ricci flat, and had N = 1 supersymmetry -+ covariantly constant -+ Kahler, Ricci flat, SU(3) holonomy.

Now we wish to exploit the remaining condition, which is the 13i~~Mi (10.1.9): ' '

Tr R A R =

1 30

Tr F A F.

This is actually a strange identity, because we have the R.H!m~uw;,t~ left and the Yang-Mills tensor on the right. The equation u::~:ill identity, dev~~d of any content, but in the presence of our a! d¢ = H =0,

we find that the equations are actually quite difficult to s~~;=r.il preservation of this identity is nontrivial because the system is 4 One attractive way to satisfy this odd identity is to set pi:tJr:t:1~~~ E8 gauge fields equal to the Riemann spin connections, whjlch::

10.5 Embedding the Spin Connection

427

. This produces a nontrivial link between the spin connection and ",~-M' gauge field. We can perform this embedding as follows on the

(10.5.3) is the spin connection, which occ upies pari afme gauge field matrix. embed the spin connection fie ld into the Yang- Mills gauge field, we .find a .subgroup of the gauge group that contains SU(3). This means, of , that we are breaking the original gauge symmetry of the Yang-Mills simplest decomposition is

(10.5.4) check, however, that the C lebs.:h·.(Jordon coefficient:; afC such that . idenlilY is satisfied exactly. In particular, we must show that we the factor of in the Bianchi identity (10. 1.9). kn"wthaJt' ,e Yang-Mills gauge fie lds arc in the adjoin t representation has 248 elemenlS. We must now find a breakdown of these 248 representations of SU(3) ® E 6 • which is always possible to do.

io

w ,,,' """,'0

248 = (3. 27) Ell (3. 27) Ell (8, 1) Ell (1, 78).

(10.5.5)

(iO.5. 1) is satisfied, let us convert from the adjoint rcpresenlation 10 the 3 CilJ,1f A is a generator of SUP), then we wish to fin d the between the trace of this matri x squared in the 8 representation '3 represe ntation. The answer is ( 10.5.6) focus on the SU(3) content oC(10.S.5). Notice that we have 27 sets transforming in the 3 e'3 representation of SU(3), and also one sel . But the sum of the rrace of the squares of the OCtcts, as we saw in be multiplied by 3 when we convert to the 3 @'3 representation matrices. Thus, the total redundancy in the 3 63 '3 representation

27+3xl=30 fac tor of 30. Because of spin embedding, where thc curvature tensor in the same spaee as the Yang-Mills tensor, we enn now satisfy missing factor aOO emerges when we count the specific in (10.5.5). E, reasons,

,

group, by contrast, docs not have complex representations, to describe crural fermions. but £6 does. The 27 multiplet, in multip let for fermions for model building with £,.

428

10. Calabi-Yau Spaces and Orbifolds

The group E6 is also good from the point ofview onow-energy because the 27 of the fermions can form a supersymmetric mtlltij~J~~~ 27 of the Higgs. :::m::1m In summary, we have the phenomenologically acceptable cOlil¢l~Mii Tr R

A

R=

1 30

Tr F

A

F ~ spin embedding -+ 27 tenmo'ti$~:

Because the fennions are now in the 27 representation, question: How many generations are there? The GUT, as we with the problem of generations. There was no reason for as!mntiP:~:ij~ one generation. In the superstring picture, the situation is precisely the now show that we get too many generations!

10.6

Fermion Generations

One of the most powerful applications of algebraic nomenology of string theory is the calculation of the from topological considerations. To calculate the number of generations predicted by the the:o~M¢o."¢::u.1iij calculate the number of massless particles. The 1O-dimensional ~.',l,~~~~m operator for the particles in question becomes, after c~;~~~~.~lific~~l~~1

6t tw(n>Jem~crolrllpmmUl.bJ..Lhuur'ad.s;.y~\mm~j.$l\!'2:~. 010lfr = (04 + 06)lfr =

o.

In general, 0 6 will have eigenvalues denoted by m 2 , that is, so that our wave equation becomes

(04

+ m 2)lfrm =

O.

We are interested in the massless sector in four dimensions, so only the zero eigenvalues of the 0 6 operator. Thus,

04¥r

=

06¥r =

O.

This is an important equation because it has two mt1erpretati()ns. of course, that the four-dimensional fermions are massless. means that ¥r is a harmonic form in six dimensions. massless modes in four dimensions will be related to the forms that we can , write for the six dimensions manifold. Wei$a~Je (10.3.18) that the number of harmonic forms of degree p is ecm~n~ Betti number. Thus, topological arguments alone should give.~..H~m ofgeneratio1lS! In summary:

m 2 =0-+

{

¥r is harmonic in six dimensions,.....•• 1jr is massless in four

' .:

10.6 Fennion Generations

429

expect that the number of generatioos is a topological number because Dirac index theorem.. We know that the solutions of the D irac equation in ,i,,,,,,,,I, have zero modes:

i(P}'"

~

O.

(10.6.4)

index of this operator is equal 10 the difference between the positive ""tiyo chiralilies of the zero modes: ( 10.6.5)

index is a topological quantity defined on II spin manifold, so we that it can be related to the chara~teristic classes we found earlier. In discussing SU(3) holonomy, we will find thai

,in"

i"d,,

Ind..(P} ~ Ilx(M)I.

(10.6.6)

is also equal to the generation number, because we will be

only fcnnions of one specific chirality. Thus, the precise relation generation number and the Dirac index, or the Euler number, generation nwnber = 4Ix(M)I.

(10.6.7)

this, C()nsidcr a manifold of SU(3) holonomy. with Betti (Hodge) that have two indices p , q. The Euler number can be written as

,

X(M) ~ L(-I)''''h".

(10.6.8)

bc

i down, which i:ifi;~~~~O:;fh:Ih;:'~:~~~~:~~""~d~~1~;~~~~:, : ~~can r related Betti numbers. We find. if compare the hclicityi1ctcr-of WE

~nme"i'

pairs with their multiplicity, (2.~) (~. I) (l,~) (!,O)

ho.c, hO.h

(2hl.c+hc.I),

(ho.o

(10.6.9)

+ h l . l + h2.l).

oaI:y" the spin-~ fennion sector, given by C!. 0), wc find that their number equals ferotion multiplicity = ho.c + h l . l

+ h2.1.

(10.6.10)

il:j:i~;~i~; x:'~~v:~. is too large. We want only the subset of this figure ()f

the 27 and 27 of the spin- ! fcnnions. in (10.5.5). we found that the Z48 of EB ® Es can be decomposed 3) and (27.3). It can be shown that the multiplicity associated with

430

10. Calabi-Yau Spaces and Orbifolds

each of these two representations is equal to

(27,3)

h 2•1 ,

(27,3)

hl.l.

Thus, the generation number is equal to

#(27) - #(27) = h 2 •1 - h l •l , where # represents the multiplicity of the 27 representation. But in turn, is precisely half the absolute value of the Euler number, (10.4.36). The relation between the generation number and the Euler mun~~tiii surprising because there is no reason to believe that any """"'l\Jlli~I).~'~ between the two. The generation number is a function of the Yarlg-MIU group E8 ® E8, while the Euler number is a function OOf~!th~:e~n~~~111 Naively, we do not expect the two to be correlated. But the n the two is established because we performed spin embedding, gauge group and produces an intimate relationship between the grQuV:·@jj K6 and the gauge group. Thus, the essence of this important result of compactification and spin embedding. Next, we would like to calculate the Euler number, and hence tt·1~:~~ of generations, for a series of Calabi--Vau metrics. We showed earlier that a submanifold ofthe Kahler metric on it has the same Hennitian metric gil, is also Kahler. Thus, we wanflhi.~l6'ii! the set of submanifolds of C PN that have vanishing first Chern This can be accomplished by placing constraints on the z's by polynomials to zero. Consider the submanifold of C P4 obtained by the constraint 5

Lzi = o. 11=1

It can be shown that this manifold has vanishing first Chern

clal~fC8ro:it:

:tr~~~111

X(M) is equal to -200. Let y(N;dl.dl •...• dtl

represent the submanifold obtained by setting k homogeneous P~l~I;1 degrees d; to zero within C PN • Fortunately, the fonnula for tl: class, not just the first Chern class, of these sub manifolds is knj~w)Q:;::~!~ holonomy, the total Chern class equals (1 C

=

k

+

J)k+4

fT=1 (1 + d;J)

'

where J is a certain two-fonn obtained by nonnalizing the ~Q[i~iill C Pk+3 • By expanding this fonnula and then setting the first,

10.6 Fennian Generations

431

, 2: d; =k+ 4.

(10.6. 16)

h., 1

we find only five possibilities with vanishing first Chern class: X(Y(4:.5) = -200, X (Y(3;l.4)) = - 176, X(Y(5;3.])

X(Y(6:l .2.2) X(Y(l;2.l.2 .2)

= - 144, = -144,

( 10.6. 17)

= - 121t

the generation numbers are unacceptably large! This is phenomenologundesirable because we know ftom arguments from llucleosynthcsis, and asymptotic freedom in QeD that we want very few such as three or four.

, we can still reduce the generation number and make it much . Let us divide the origmanifold Mo by a discrete symmetry group G that acts freely au the '. . (i.e., no fixed points), yielding a manifold M . If the number of disgenerators of G is n( G), then the Euler number of the original manifold by the discrete group G is o

X (M)

= X(M,)

,,(G) ,

M,

M =G'

(

10618) ..

( 10.6.19)

consider the previous example of the submnnifold of C P4 with

,

2: zi=o

(10.6.20)

thai this polynomial i:;; invariant under the following synunetry mhO." --+ (ZS. Zlo Z2 •.•• , Z4). 4 (Z I, Z2, . .. ,Z;) --+ (ZI, aZ2 . 0:2 z), ... ,0 '::5), (ZI, . then SU(4) is and each o rlhe four supersymmetry generators in four dimensions r%:::,~:;'~:""":·;ves. However, if we start with an orbifold by divjding by ~c ZJ as beiongingEo Ihc SU(3) s ubgroup ofSU(4). Thus, by Zl necessarily breaks SU(4) symmetry, since there are no ilirce~ion.' representatio ns of SU(4). But only one of the four components 4 rurvives the division by ZJ, RIld hence only N = 1 supcrsymmetry , whic h is fortunate . •• umn",cy, orbifolds can be used to break the overa ll symmctry of the because only symmetries that commute with the discrete group survive. be used to break the ga uge group as weU . :: g be a spt."eific element oftbc JO·dimensional gauge group such that

'!",,""'oa.

(l0.8. 17) integer m. Then we demand tbat the s tates of the theory Bre those that with the combined action of

g x Z ...

(10.8. 18)

we have a mechanism for breaking both gauge and supersymmeuy

= 1, which is an element of SU(3). the s tates of the theory he invariant under the combined

we might take g3

438

10. Calabi-Yau Spaces and Orbifolds

."

operation yields the breaking of E8: ~

E8 ® E8

Es ® SU(3) ® E 6 •

Now let us calculate the Euler number on this surface and of generations. Once again, we must cut off the 27 fixed pOilllts divide by Z3, and then sew back on 27 patches. The Euler number of the surface T is zero, so the manifold T . disks d located at the fixed points has Euler number

X(T - d) = -27. Ifwe divide by E 3 , the Euler number becomes -9. Now we mUlst.~~~. disks d. This time, however, each disk has SU(3) holonomy and'~:'~~j~:wi 3, so the total Euler number is X

(T-d) + - = Z3

27XCd)

72.

Thus, we have 36 generations offermions. The previous example was only a toy model in six dirneIlSi()ns,.:lt;~lii however, to construct orbifold models that have much fewer ge~i~~iHii low as two or four.

10.9 Four-Dimensional Superstrings In model building using orbifold compactification to produce fOl~·~!im~ stringent constraints must be met, such as modular invariance an(Ufii~1fJii:i1 conditions. These conditions are nontrivial, because modular up the boundary conditions. For example, when studying modular invariance, we can six-torus T6 divided by a discrete point group P = Zn such boundary conditions for the orbifold T6/ P are as follows: X (0'\

+ 2](, 0'2) =

hX(O'l, (2).

+ 2Jl') =

gX(O'l, (2),

X(ul.O'2

where h and g are elements of P of order n, that is, gn the usual bosonic or fermionic boundary conditioIlS, we have ." to ±. However, when compactifying on orbifolds, this generalized. The presence of g and h, of course, can break the overall theory, both for space-time and for the internal group. The ~c;~~~al the compactification is the subgroup that is unaffected by this subgroup that commutes with g and h. To study how the one-loop trace is affected by the co~n~~ct!~~~~1 diagonalize g and h in terms of their eigenValues, g. h ~

10.9 Four-Dimensional Supel"l'itrings

439

so that they are of order fl. For simplicity, we take the following odid,.y condition: X(a

+ 2:rr) = e2>riv X(a}.

(10.9.2)

that the Fourier decomposition of the string modes is now shifted. we must now expand the string in terms of a new set of X(o-) = p"'''n~' of the

Li(m+.. p X m. '"

(10.9.3)

t.' w ithin the Fourier modes changes the trace ca1culation one-loop amplitude in a subtle but important fashion, which wiJl put

rl~:'~:~~:;::~I~' on the Vi · calculation., the presence ofll enters in the zero point energy. For

;I

bosonic string, for example, the unregularized Hamiltonian contains ~ Ln a_nan + anl"L n . This, of course, has infinite matrix elements be nonnul ordered. T~ehn i cally speaking, normal orderi.ng the ere' ..d ."ni,h· operaEOrs creates an infinite zero-point energy given by This infinite zero-point energy can be handled in several wa)use n(a» I'"~ m>ultipl>, !ht, wehavc lheeonstraints basis elements bi of 8 :

air."1

'oy,man"" .

II(b l )

= omod 8.

n(b/n bj )=O mod4. n(b l n h} nb t il b,) = Omod2,

( 10.9.31)

..,llas the cODStraints emerging from modular invariance: ifa. {3 E 2 . th .

I P) ~ [

C(a

I fJ) = e!I1~C({J I a). I cr) = GFG"C(cr I F),

C(a C(a

±I

C(a

I p)C(a I y)

~

0

5.C(a

o

eI'WlSC,

(10.9.32)

I py) .

• {to y wi thin the collection 8. we have the constraints coming from the confonnal anomaly cancelwhich must always be strictly observed. Let us usc the fenn ionic (not

o~,:; ;:~~>:;';~ti~·~~)~Of the lattice compnctification. This was used for the

:t1

. We musl carefu lly include the fcnnionic cOlltncution

conformal anomaly in the supcr-VirdSOro algebra. It is rather remarkthat a representation of the fermionic partner T,,· to the energy-momcntum 1>, g;'ven strictly in teons of the adjoint representation of fermions:

",T.cru,

(10.9.33) we calculate the anomaly arising from this term, we find thai the ru~~~:contribution to the anomaly is ~ N, w here N is the number of !,Il in the group. left-moving seclor, the number of compaetificd dimensions is 26 - D , D is the number of spaec-iime dimensions. We represent the eompactby fermions in the adjoint representation. In this seclor, the three

"',to,

446

10. Calabi-Yau Spaces and Orbifolds

contributions to the anomaly are XI-'

left movers:

{

b, c ghosts 1{ru

Since the sum of the three contributions to the anomaly must.•• ~·\t~~=WI obviously have N = D) fennions in the set 1/fu. Now let us analyze the right movers, where the anomalies mllst::M!:l:l:zero:

2(26 -

right movers:

XI',1{r1'

{

b'Ct~'Y

I {-2:;11 I. 3D/2

--+

The condition for zero anomaly is therefore N = 3(10 - D). tn:i;iJ@jiil (D - 2)NS-R fennions (in the light cone gauge) contained in D - 2 + 3(10 - D) fermions for the right movers (in the light COl1ll;1:g~ summary, we must have the following number of fennions in O[{tet::lQ:~ml the anomalous term in the Vrrasoro algebra: left movers --+ 2(26 - D), { right movers --+ D - 2 + 3(10 - D). The anomaly cancellation is automatic if we have this many left- and right-moving sectors. Now that we have explicitly expressed all our constraints, our follows: (a) We first calculate the total number of space-time fermions 1/f.~.'~II fermions 1{rG that satisfy (10.9.36). which removes COlltOrm:~~,: from the theory. Then we calculate the total number of spin . arise from this set, which we call F. (b) We next randomly choose a collection of spin structures as within B that includes F, the complete set. (c) We then check for closure under multiplication (10.9 .27), whlCti;~ the full group B compatible with this initial choice. (By chclosj;IjgJi,p initial sets for 8, we can reproduce different cOlnpactifi(:ation::~M Cd) We then calculate the square matrix C(a I /3) defined ov¢nll~ that satisfies our constraints. Some arbitrary phases are it·ltrOlq\tl~~:4i~ fashion, whichallow us to generate more than one solution fqr:M(#.f. We will now discuss some of the properties of the soluti~.tiltI!~ equations. Remarkably, we find that all consistent solutions nelc~~@tnl gravitons, dilatons, and antisymmetric tensors. We also find thaLttl~¢:l. ofthe massless spin-~ field is sufficient to prove the absence ofta¢ft}1~ the vanishing of the cosmological constant at the single-loop l
(1 0 .10. 14)

ulee oo""lb" ofRicci-ftat Kllhler manifolds is usually quite we can always reduce thil! numbe r drascically by taking a subman iexample, we can divide by 8 discrete group thaI preserves a certain 'O"ual in the coordinates. A good examp le is C p~ divided by Zs x 2 5• not connected. This mllOifold has four generations. the model even further to the Standard Model SU(3}® N = I supersymmctry. Normally, Ihis is quite schemes break the group down to the standard model supcn;ymmetry. One solution to this problem is to use the o f Wilson lines. We saw earlier that the manifolds K6 that we are

452

10. Calabi-Yau Spaces and Orbifolds

considering are not simply connected in order to produce a number. For nonsimply connected manifolds, the Wilson loop

U = P exp

(l

A/LdX/L)

does not necessarily equal 1 even if the curvature tensor because we cannot always shrink closed looped into points. break E6 symmetry down to a subgroup G by choosing an elelll~.D.:~:j~ that G commutes with all elements of this element. For eX~IIDJ'le;;it:.wt:ui element of U that is Z5 x Z5, we find

E6 = SU(3)c ® SU(2)L ® SU(2)R ® U(l) ® U(l). Z5 x Z5 Unfortunately, we see that, in addition to arriving at the stand:arili)~~ also have unwanted V(l) groups. Thus, Wilson lines -+ SU(3) ® SU(2) ® U(lt . Orbifolds are another way of compactifying the ably the limiting case ofa Calabi-Yau space. Orbifolds are a torus T6 and dividing by a discrete group Zn, which allows

T6 Zn (These fixed points apparently do not spoil the properties of the Nontrivial constraints are placed on orbifolds by modular j"lllV~j®m-{ mixes the boundary condition nontrivially. The boundary orbifold:

+ 2IT, (12) = hX«(1J, (12), X«(1], 0"2 + 2IT) = g X«(11, (12), X(O"J

'~~::~e~i~;llllli

(where hand g are elements of Zn) can break the model. The subgroup of the space-time group and th~ internal vives the process is the sub group that commutes with g and h. Ih\'~(I~}lO)1j g and h to equal the elements e 2Jriv, , where VI = ri / n for some 41tl~~flj the zero-point energy of the Hamiltonian is shifted by an am.OUl[H to Vi(Vj - 1). Because the trace over the single loop is serls"l~~l~:~~~fi~~1 point energy, this means that modular invariance places a n eigenvalues:

where ri are the eigenvalues from the right-moving sector, ai1<MI1~~M of eigenvalues '1.2;; are from the left-moving E8 ® E8 sectot.:J~':~~ Wilson lines, then the group that survives the compactification that commutes with g. h, and the Wilson lines.

References

453

p~:;:;:::,;:~~~:.\::,~ there are many solutions to this compactification, some

and the group SU(3)®SU(2)® U(l)n. which yields too I) factors Unfortunately. the method of Wilson lines docs not change rank of the we will have too many U(l) factors. over Calabi- Yau spaces is that they arc sim~ of them can be cx-plicilly constructed. Unfortunately, I there is no systematic way to catalogue these """,II, of solutions. step in this direction is to usc modular invanance and the absence of '~Y''"s and anomalies to systematically derive all possible solutions. We with the amplitude wrilten as a sum over spin strucrures:

til

(10.10.20)

we dem&nd that the C's faCtori7.e properly, have modular invariance. and models that have no tachyons or anomalies. Remarkably, it is possible the constraints on C and obtain solutions. The simple ones reproduce of the known compaclifications, but thousands of other compactifications . . This yields the hope iliat we may be able to exhaust . a realistic model.

'Y ;~~~~~:~;~';::7;~~;:~large classes of fo ur-dimensional soluh . The heterotic string is an example of an compl3.ctification, i.e., treating and right-moving sectors dif. Asynunelric orbifolds are the largest class of orbifold s found SO far

left-

using

. ,:::;;;.~~::;:::~S~ii~gn~ificanf[Y with the work of others different types of ~f addition to asymmetric orbifo[ds, we should also menpossibility of non-abelian orbifolds. i.e .• orbifolds based on dividing finite groups such as the crystal groups. The advantage of constructions is that they help us eliminate some ofthc unwanted U(I) and also yield three and four generations. See [40].)

G. Horowitz, A. Strominger, and E. Witten, Nl./c/. Phys. 8258,46 in Unified String Theories (edited by M. Green and D. Gross), , Singapore, 1986. S. Manton, Phys. Lett. 120B, IDS (1983). Geometry und Topology: A Symposium in Honor Qf S. University Press, Princeton, NJ, 1957 . . Yau, Proc . Natl. Acad. Sci. U.S.A. 74, 1798 ( 1977). . YlIU, in Symposi!lfn orr Anomalies, Geometry, and Topology (edited by W.

Bardeen and A.lt While), World Scientific, Sinilapore, 1985. Hosomni, Pl!y~·. Lett. 126B, 303 (! 983).

454

J O. Calabi-Yau Spaces and Orbifolds

[8] L. Dixon, J. Harvey, C. Vafa, and E. Witten, Nuc!. Phys. B261, B274, 286 (1986). [9] C. Vafa, Nucl. Phys. B273, 592 (1986). [10] L. E. Ibanez, H. P. Nilles, and F. Quevedo, Phys. Lett. 187B, 25 [I I] L. E. Ibanez, J. E. Kim, H. P. Nilles, and F. Quevedo, Phys. Lett. ... (1987). . [12] D. X. Li, Phys. Rev. D34 (1986). ... [13] D. X. Li, in Super Field Theories (edited by H. C. Lee), Plenum, .. 1987. •••. [14] V. P. Nair, A. Sharpere, A. Strominger, and F. Wilczek, Nue!. Phys.> . (1986). .. :~:~mim [15] B. R. Greene, K. H. Kirklin, P. J. Miron, and G. G. Ross, Phys. (1986); Nucl. Phys. B278, 667 (1986). [16] A. Strominger, Phys. Rev. Lett. 55,2547 (1985). [17] P. Ginsparg, Harvard preprint HUTP·86/A053, 1986. [18] S. Karlara and R. N. Mohapatra, LA·UR·86·3954, 1986. [19] M. Dine, V. Kaplunovsky, M. Mangano, C. Nappi, and N. Sieberg, B259, 46 (1985). .. [20] S. Cecotti, J. P. Deredinger, S. Ferrara, L. Girardello, and M. KOlIlCft!qeIl~ Lett. 156B, 318 (1985). ·::::7":;::;:;:; [21] J. P. Deredinger, L. Ibanez, and H. P. Nilles, Nue!. Phys. B267, [22] 1. Breit, B. Ovmt, and G. Segre, Phys. Lett. 158B, 33 (1985). [23] A. Strominger and E. Witten, Comm. Math. Phys. 101,341 (l [24] G. Segre, Schladming Lecture Notes (1986). [25] H. Kawai, D. C. Lewellen, and S. H. H. Tye, Phys. Rev. Lett. 57, 12·•2.(.-2R when we interchange n

+'>-

m.

11.4 T-Duality

46J

'unusual symmetry, one which links the large-scale behavior structure. Unlike point particle field theory, the differentiate bern:een these two regions. Notice that this dual ity which intCIchanges winding modcs with K.a1uza- Klein modes, is a result of the geometry of string theory and does not appear in point thcories. rewrite tills duality transformation in the familior language o f ~~'::;;:~nfield theory, this transformation is equivalent to making the

ax ~ ax, &x --+ -ax.

( 11.4.4)

sign change for onc set of movers, When we apply thi::; same du. theory in theNeveu-Scbwat"'.l-Ramond !,N8--R) fonnalism, fiod

( 11.4.5)

-, this transfonnatioll of the ninth lcft-moving oscillator also reverses the ,olfth, IO-dimcnsional lcfi-moving chirality operator which is constructed the fermionic zero modes: (11.4.6)

the chirality of the left-movers. The Type IIA spinors, defined in tcmlS ,,;"tive and negative chiralities, are transformed by T -duality into a theory the same type of chirality. i.c., the Type IlB theory. In nine dimensions, the followi ng duniity emerging [10): T:

IlA~rm.

(11.4.7)

use the symbol ~ to represent symbolically the duality relationship.) therefore view Typc IlA and Type UB as merely two extreme points tbe same continuum of vacua labeled by R. As R -+ 00 or R _ 0, we the usual Type II string theories in 10 dimensions. . the five supe rstring theories have now been reduced down to fOUf. I ,oolh" T -duaJity can be found by analyzing the 1::8 ® £8 and the SO(32) string. COllsider first the Narain lattice. Let us compactifY the bctdown to d dimensions. This rne3ns we must compactify 26 - d dimensious and 10 - d right-moving modes. Lei us eornpaclify a lattice r I. and r /I. We recall that by constructing the heterotic string amplitude and demanding that il satisfY modular we have the additional constraint:

ft.·rt.-rll· r R=Z, Z refers lo lhe integers.

( 11.4.8)

464

11. M-Theory and Duality

The fact that there is a minus sign in the metric means that this . lattice. Modular invariance also demands that these lattices be vVsnl;!~~ NS-NS sector Ns-NS:

{¢, gllv, Bllv } •

But the R-R sector for the Type IIA theory contains the ad(li~'g~~11

R-R:

{GjL, AjLvp} .

11.5 S-Dua1iry

467

let US write the bosonic part of the effective action of Type ITA theory lowest order} in tcnns of massless fields. which is just IO·dimensional nchirnlsupergravicy with N = 2 supersymmetry. If we let K = dC, H = and G = d A, then the action for the massless fields (10 lowest order) is

s=

f

d'"x{J

g'-"[R+4Id~I'-1IHI']

- J dlKI' + ,iIGI']l + I~

f

GAG

A

8.

( 11.5.8)

ago, it was noticed that if we compactify the: I I-dimensionsl su· theory on 1:r

Ti

M-theory on

1

M-theory on

s,

Ti

+f"

llA .

B

TlB.

(\ 1.5.27)

we may puzzle over several strange mcts. First, it appears odd

1~;.~~~:~':nb~' supergravity cannol reproduce a chiral theory after but M-theory can. In fact, the lack of a chiral interprel8-

la

472

II. M-Theory and Duality

tion for ll-dimensional supergravity was an important reason Ior:OM:! eliminating it on physical grounds. For example, M-theory compactified on T2 reduces to D = .-,:..:-~~,;.= supergravity coupled to a tower of KK states plus higher terms. ~1't~il that we let the area of the torus go to zero for fixed shape, we n.,,,nv.e'.j,, supergravity theory. So starting from a nonchiral theory, we chiral one. Where do the chiral states come from? The answer is that M -theory has higher corrections to II-dimeW~~)iI pergravity. In particular, M-theory also contains massive "LaLt;:;. wrapping up a membrane on T2 • In the limit that the area of thetQi~ii to zero for fixed shape, we find that these massive modes ~~~:l~IJI such that the effective theory reproduces D = 10 chiral Type lIB •... theory. The lesson here is that M -theory contains, in additioo:n~i~t~0~aI~~le1~i:111 supergravity, higher p-brane states which cure the many "c mer theory. These higher p-brane states, in particular, are im])ortanUtl:W1 all five superstring theories into one. A second mystery is related to the fact that the Type UB two background fields given by the tensor fields Band B'. It that the Type lIB string acts as a source for the B field, via .... sij 8; Xli 8j XV Bliv , and hence has a charge under this field. Type.: . cannot act as sources for the B' fields as well. But SL(2, Z) . and B', so there must be another object, besides the Type lIB g~ir~~I~ a charge under Bf. But what is it? Like the Kaluza-Klein modes ~ compactifying M-theory down to Type IIA theory, we will ~;tjrq;11 are new BPS states which carry charge under Bf and yield the symmetry. Before ending our discussion of Type lIB theory, let us factors of cp which appear in the various low-energy effective theory. We recall that the Euler number for a Riemann X = 2 - 2g. For a sphere with no handles, g=O and the Euler path integral over e- s picks up a factor of g;X, which gives us This means that the effective string action defined on the . accompanied by a factor of g;2 = e-zq,. Indeed, we find this . string actions for fields coming from the NS-NS sector. For the ..... lIB actions, these tenns include the R, cp, and H fields. However, the massless background fields coming from the not have any cp dependence at all, since these fields do not the string world sheet. They cannot couple to the string world usual way, yia products of ai X Ii. Instead, they couple via spinor as ST1t11t2 .4 n S. In general, this means that there will be world sheet. This also means that the tenns in the Type II actions R-R fields do not contain any cp dependence. Thoughout this discussion, we will find that the R-R sec:to~:)~~:~ theory contains highly nontrivial information concerning the nOll~¢iW.Wj!lm structure of string theory.

11.5 S-DuaJity

473

E8 ® E8 fIeterotic String .. far, by considering N = 2 superstring theories, we have reduced five ~r.;tri,"gs down to three. But what about dural superstring theories with = J? In particular, what about the £8 ® E, heterotic string and the '"ypc I Ooce again, wc find many surprises [20). begin by oompac(irying the sri II-mysterious I I-dimensional theory, low-energy sector is II-dimensional supergravity, On a finite line scgi.e., we first compactify aD SI, giving us Type IIA theory, and Lhen thc action of 'L}.. This discrete symmet ry acts on SI by the following:

·__ X".

dividing by ~, we also reduce the N = 2 symmetry of the T)pc HA by half: down to N :.= I. We also wish 10 keep the bosonic Siaies which under 'Lt, which include: the 10-dimcnsionai metric gil"' the scalar and the antisymmetric tensor A,...II. Notice that these bosonic fields = I supersymmetry make up precisely the spectrum of the EI ® EB string (minus the gauge multiplet). by tbe line segment introduces anomalies into the theory, which be eaneeled because M-theory is anomaly-free. To solve this problem, naturally introduce two additional pieces to the action, each one associated the endpoints of the line segment, which form (WO hYJ>Crplanes. We recall must have 496 vectors in an N = 1 theory to kilt the anomalies. This be solved by having 248 vectors defined on each hyperplane. We know E. can give us precisely 248 slales, and we also know that ~ symmetry thaI we have equal gauge groups on the ends of the line segment, so tbeory has gauge symmelry E, ® £3 or SO(32). In other words, we anomaly by placing a super-E. Yang-Mills theory at each of the located at the endpoints of the line segment. so that the resuiling E B® E. symmetry. (At present, p~isely how the super~ Yang-Mills emerges from the compactification of M-theory 00 a line segment is understood.) we find that the larger the string couplillg constant, the l:Jrger the between these two hyperplanes, i.e., R = gl/J. Thus, to any finite perturbation theory, we will never see tbe direct cffr..'Cts ofthese This is the reason why siring theory previously missed these and the nonpcrturbalivc J>tructure of the heterotic string. ! this symbolicnl ly by slating;

S: M-theory on SI/'L}.

++

EI ® £8.

(11.5.28)

Type 1 Slrings ••'. .ave reduced Ihe five superstrings down to lWO. We are still left with I strings, which, unlike the others, have both open and elosed strings. I . obtained by comparing Type I and S0(32) heterotic both have SO(32} symmetry. Althougb they realize the 80(32)

474

11. M-Theory and Duality

symmetry in entirely different ways (Type I strings via ChaJl-':Pl,\jt~i~ii and SO(32) strings via the Frenkel-Kac construction}, they ~~~~:::!~g low-energy sectors. We suspect that they may be S-dual [2 As before, let us now make a prediction about this theory. Let us first compactify M-theory on Sl/Zrz with a We arrive at the Es0Es heterotic string. And now let us with radius given by R, which breaks the symmetry down to We have then compactified M-theory on a cylinder, with raO'lUS: L. Now let us compare this resulting theory with Type 1 and the compactified on a circle of radius R. The effective low-energy action is given by

Notice that the cfJ terms appearing in the Type I effective lOVlr~tiififfi~ have the correct character. The R and dcfJ terms appear mJ;lltil[1H!~~:a~jj~ as expected, since they are defined on the sphere. (However, field is missing.) The F term is accompanied by a factor of e~jfW~[. Yang-Mills terms are associated with the open string sector, w.l)ltcJi:ls::M on the disk (with Euler number I) rather than the sphere. comes from the R-R sector, and hence has no rp dependence. the Type I action have the correct world sheet structure and rp del?¢~{(~@jl The SO(32) heterotic action is given by

S=

f

d 10x,J=ge-2 [ R+4Id.p12]_e~ TrIF12_~I H '12 1 ·

( 11.6.15)

SO(32) heterotic action is given by

s=

f

dIOx..;'-ge-2·[R+4Idtf;lz-iIHI2-a'Tr lFll ].

,"",,,;,," !h,,~ two theories compaetified on Sl

(11.6. 16)

with M-theory compaetificd

Sl/'L) and then by SI, we find R 81 = 1/8so(31) = L·

( 11.6. 17)

summary, when compactified on a circle :>'1, we found the following \lu" !;.;,, in nine dimensions: T:

!

Type IlA Es® Es

+->-

Type nB,

+->-

S0(32).

(11.6. 18)

; '>S. ,m turn, reduces the various supcrstring theories dov.n 10 two cmegories,

with N = 2 supe rsymm~try (the 1'ype II tl1eories) and those with N = 1 heterotic theories).

480

II. M-Theory and Duality

We also found the following S-duality relations in 10 diIllen.sions:!: S:

{TYPe lIB Type I

>j~ naturally lead us to dualities in lower dimensions. In PaIl1C·U:l~f,,: the nonperturbative region of one compactified string rnp,nrv

12.9 Summary

string theory.

507

string theory

,o~:::~~:~:~'lk~'::~~:,~ For example, Type IIA rn on K J is dual to the heterotic smng compactified on T... .: There are man}' ways to see how this duality works. The simplest way is : compare two seemingly unrelated string theories which have the same lowSpecifically. we shall analyze the supersymmetry generators survive the compactificalion process. lfwc begin with a supcrsymmctry Q" in 10 or 11 dimensiODS and then begin 10 compactify it down . , we find !hat it decomposes into Q .. .I. where a tabels the 'in:.~:;:~e~ in a lower dimension and i labels the number of supersynunetly m In this way, once we know the holonomy group of the manifold o

•

which we arc compactifying a theory, we can calculate the number of ,.",,.,="t'Yg~,,,",,,!

N tlla! survives the comp:lctific:ltiol1 process.

particular, case D = 6 has been analyzcd cxtcnsively. We begin with fact that M-theory in II-dimensions contains both a membrane and its dual, '·b'~ne (wh,;ch is given to us by analyzing the BPS supersymmetry algebra). can compaclify both the membrane and 5-branc on a five-dimensional 'c.~;,f.:~:~ by product of a one-dimensional space M, and a four~. space M 4 • The resulting theories in six dimensions should be to each other, sinee they were dual to eaeh other in !I-dimensions. can let Mf == S, or Ml = S';~. Also, we can let M4 = Xl or M~ = 74. way, we now have four ways in which La compactify the membrane and In this way, we now obtain four possible dualities. interesting case is D = 6and N = (I, I), which establishes II duality. Evidence for this duality is given by analyzing the stnlcrure of both theories, which are not obviously the same. . space of the heterotic string compactified on T4 is given by the

the

SO(4.20) 80(4) 0 SO(20) 0 T'

(12.9.13)

is 4 x 20 = RO dimensional. This, in tum, must match the numher of

",::c~:.~r:h::'~'P~1:;:~I!'~:~TO see Ihis, we note that the metric gil-v

= 16 afit;rCOUlpactificatioll, Similarly, the Yang--Mills field A~ gives us 4 x 16 = 64 scalar field.,. The sum 80 scalar fields, as expected:

B"v yidd 4; for these two eascs will shed much light 011 the physically relevant case D = 4, N = I. Not surprisingly, we find tbat these oornpactifica tions are

=

=

:j,il~~~::~!~. bccausc oftbe in[rOduction ofCalabi-Yau manifolds. the case of D = 4, N = 2. By checki ng the supersymmetry chan,

::;

can see that the heterntic ~tring can yield this symmetry if we rompnclify a six-dimcnsional manifold with SU(2) holonomy, of which there is only choice: K 10 T2. Using the Bianchi identity, we can show thai

dH=TrPJ\F+TrR J\ R.

(12.9. 18)

:WI,,, the left-hand side is zero, and we integrate over the right-hand side, we that the second·Chern class ~ defined over the gauge space £ . 0 E, is ~qua l

to the second-Chem class defined over the manifold K J ® T2• which is 24. satisfy this relationship, we must have 24 inslantons for the· gauge group. must howe 12 - It instantons for one £8. and another 12 + /I insumtoos for

other Ea. We suspect that this theory is dUlil to a Type n theory compactified on n c:,~~~~:~manifold with SU(3) bolonomy. labeled by some inde:o: n. Fortu~: is n simple way in which to construct such Calabi-Yau manifolds. make the oh.~ervation that a certain class of K3 manifolds can be writas elliptic fibrations, or, crudely speaking, manifolds with lorii T2 deftned ."""h point of a base manifold. In thi:; case, the base manifold is given by P I. Then wc can write tbe following sequence o f relationships between fibe r

K} over P I = [T2 over P.] over P I

= T2 over [P lover PI In = T2 over F ft •

(12.9.l9)

last step involves the fiber bundle defined by P I fibers defined over a base . given by PI , which is given by the Hirzcbruch space Fn. Thus, the defined by compactifying the heterotic string on KJ ® Tl the index labeling Hirzebrueh spaces Fn when we compaclify the If string on n particular Calabt.-Yau IlUInifold. we have £8 ® £8 on K) 0 T2 we have 12 -

-H-

IlA on [T2 over Fnl.

n instantons in one £8 sector and

(12.9.20)

12 + 11 instanlons in the

510

12. Compactifications and BPS States

Many of these results can, in turn, be derived by postulating the . ' a 12-dimensional theory which, when compactified, becomes Type $~$I··· theory. We recall that we can define an SL(2. Z) symmetry at eacliiip.~r· the manifold, which d~fines a fiber ~un.dle. We. there~ore define F-th~9~r.~~. theory when compactdied on an elhptlc fibratlOn, YIelds the Type II:$.:tfr We can summarize many of these relationships via .... .. . F-theory on T2

*+

lIB,

F-theory on A

*+

F-theory on TdZ2

*+

lIB on B, SO(32),

F -theory on K 3

+*

E8 ® E8 on T2 •

for A being an elliptic fibration of B. Whether F -theory is fundamental theory remains to be seen.

References [1] M.1. Duff, M-theory (the Theory Formerly Known as Strings), hep,-JIV9:1;i 1996 '-:-:-:':':-:.:1 [2] 1. H. Schwarz, Lectures on Superstring and M- Theory Dualities. Tl~i:/ School, June 1996. [3] C. M. Hull and P. K. Townsend, Nucl. Phys. 8438, 109 (1995). [4] P. S. Aspinwall, K3 Suifaces and String Duality, hep-th/9611] [5] M. Duff, R. Minasian, and E. Witten, Nucl. Phys. 8465,413 (I [6] E. G. Gimon and 1. Polchinski, hep-th/9601038. [7] G. Aldazabal, A. Font, L. E. Ibanez, and F. Quevedo, hep-thl9602o.St [8] D. R. Morrison and C. Vafa, Nuc!. Phys. 8473,74 (1996). [9] C. Vafa, Nucl. Phys. 8469,403 (1996). [iO] M. P. Blencowe and M. J. Duff, Nue!. Phys. 8310,387 (1988). [11] T. Kugo and P. K. Townsend, Nue!. Phys. 8266,440 (1983).

13

olitons, D-Branes, d Black Holes

Solitons seen thai these BPS saturated states are essential to prove the duality . we have conjecrured. The important point is that it is possible aU of these p-brnne stales as solitons, solitonlike solutions (called

or as supcrsymmetric actions. Solitons are II powerful way in which the noopcrturbattve ~tructurc of II theory, so it is not surprising that and soliton like objccrs will play an im portant part in filling oul the 'S ,,,.,osof lh, theory. Even if we start with a (hoory which consists purely of we are forced to admit the presence of these membrane sLlItcs because are solutions 10 the classical equations ormation. , start with D = 11 ~upergre.v ity, we can construct classical solutions . i correspond to membranes as follows. We break-up the vector X'" = , y"'). with JL = O. 1, 2 representing the membrane coordinates, and m =

... 10.

rne D =

I I supergravity equations of motion are satisfied by the

'owing mctric corresponding to a membrane 11 J:

=

I + y~ )'" (dy'l + idOD , (I + ky~ )-'" dx" dXIl- + (k

(13.1.1)

is the volume form for the S, sphere, and the four·formfield strength to me dual of me volume form on S7. (This solution. it can be I diverge nt at the origin, meaning that it probably corresponds fundllmental solution.)

mil"l,y. the 5-brane soliton of the D = I I supcrgravity theory is given the vector X M = (x", y''') where 11. = 0, 1,2,3,4,5 and m =

512

13. Solitons, D-Branes, and Black Holes

6, ... , 10. Then the metric tensor is given by [2]: ds 2 =

(1 + k~)-1/3 Y

dxi-Ldx ll

+

(1 + k~)2/3

(d/

+ y2dQU

Y

••

.....

and the four-form field strength is proportional to the volume 'V,~:'~1:I;:~~ (Unlike the membrane solution, the 5-brane solution is finite at thfi~J!tgjM~~ it is probably not fundamental.) Thus, the p-brane states form an integral part of the web of BPS st~jt¢'§~11 ing with a D = 11 supergravity theory, we necessarily introduce the ... and 5-brane solutions (as demanded by the central charges of the alg¢.lit solitons or solitonlike solutions. This is an essential point. The ne,;.>::jjJjifj phy of M -theory is that the various strings are just different vacua()t~~:~OO theory. Perturbation theory just probes the vicinity of each vacua, t:i.!W~~ take us from one vacua to another. However, when nonperturbative introduced, then we necessarily find p-branes emerging. Thus, sense have lost their central role in this web of dualities (other mBIq/(p+l) +

p;

.., 2 •••••.p+1 (l X' M ) (J. X Mp+1 A '" . "p+1 M, .... ,Mp+1

+ I)! 6· •

where gU is the metric on the world volume of the p-brane. represents the generalization of the Nambu-Goto action for a last term represents the coupling of the massless p + I rank field to the p-brane variables. To solve these coupled equations, we start with the ansatz ¢ = A i-L1.JL2 .. ·JLp+1 -- -deCI(g11" )6JL,JLl .. ·i-LpH'

ds 2 = e2A d x JL dX Il

+ t?B dym dym,

1

13.2 SupermembrancActions

513

11 = 0, I, 2, ... , p represent the p world volume parameters, and = p + 1, P + 2 .... , D - I. Similarly, we split the p-branc coordinates as (XM = XI', Y"'),and XI' =~I' . and ym = constant. TIlen a solution can be given as D-p-3 A = 2(0 2) (C - a¢o/2).

p+l B = -2(0

,¢ a

3)(C .-a¢ol2).

p

a'

= .(C - a",f2)

+ a¢ol2.

a' = 4 - 2{p + I)(D - p - 3)/(D - 2),

,-c =

I

e-(J4Jr,/2

+ klyp+l,

e-a~!2 _ (K1T Irr)

In y ,

D - P - 3 > 0, D - P - 3 = 0,

(13.1.6)

(13.1.7)

k = belT I(D - p - 3)QO-p-2. Q is the volume ofa hyperspbere. and is the vtlcuum expectation value of ¢. :: From this, for example, we can calculate the electric charge of the p-brane QE

=

;'1

">12K

e-~~~ F

= .../2/{T(_llO- I'-I )(!'+21.

( 13.1.8)

SD_.,-,

The point is ilial even if we originally started with a theory of strings. we ,:::~a~~~,;are forced to introduce solitonlike p-brnnes intu the theory. For :ir below 10, we find a large number of p-branes, so it is important we systematice.lly calculate their action and theirpropcrtics. ln thc process, will find many surprises.

Supermernbrane Actions to constructing the solitons as classical solutions oflhe equations of we can construct the locally supersymmetric actions for these various '. lct us first count the physical degrees offreedom for the p-brane. Becausc p-brane action is defined in a p + I world volume with reparametrization . the coordinate X I' has JJ - P - I physical degrees offi"eedom. To a supersymmetric theory, this in tum must equal the number of dcgrees ,,,do,n within the spinor

D - P - I = ~MN,

(13.2 .1)

the dimension ofthe spinor and N is the number of supcrsymtheory. We have to divide by 4, since, by local kappa invariancc.

514

13. Solitons, D-Branes, and Black Holes

we halve the number of fermion fields, which is halved again wnen::l~ffi1 shell. This relation is easily satisfied for the string (in 10 membrane (in II dimensions) where the right-hand side is equal tice that it fails for the D-branes and the 5-brane in 11 au'nellSlIJnSi'j\It).W such as vectors and tensors on the world volume, will have to be correct this defect.) In particular, we find the following possible solutions [4]:

p = 0: p = 1: p=2: p = 3:

D = 2,3,5,9,

p = 4:

D = 9, D = 10.

p = 5:

D = 3,4,6, 10, D=4,5,7,1l, D = 6,8,

To write the supermembrane action [5], we start with a ization of the original Green.-Schwarz action. We begin with .. XIl(aj, a2, ... , ap+d defined on the (p + I)-dimensional VYv..... a p-dimensiona1 membrane moving in space. As before, we generalized derivative as . IT; = OiXIl - iOrlLOie,

e

illl

where is a spinor defined in D-dimensional space. combination is invariant under the global supersymmetry transtiotjija~~ oX Il = ierlle,

oe =

8.

Then the first part of the action is given by a simple Nambu-Goto action S[ = -T

!

dP+1aJ-det ITi . IT j

.

Although S[ is trivially invariant under a global supersymmetry traJli.sf!Q~~1l it fails to transform correctly under a local supersymmetry ~:~~I~I hence the number of fermionic and bosonic degrees of freedom .... :. To correct this problem, we introduce a second contribution ~:(t;I~~:~gg which is a Wess-Zumino tenn. We begin by introducing an h defined by c.

h=

2'ip.

-

ITllp ... ITIl! der Il! ...Ilp de.

Notice that this p-fonn h is invariant under the previous glolb~l(~~til metry transformation. The point of introducing h is that we can no,&::1l'!p' a (p - I)-fonn b, where h = db,

13.2 Supemlcmbr.me Actions ,.". ." demand that dh = O. Because dn" = i dBf"

(dBr'"

d(J)(dOr "'_.", dO) =

515

d(J, we have (13.2.8)

0.

: This, in ttlrn, forces us (0 have

{r'" P)I"" (r" '''·'''P)y~) =

(13.2.9)

0,

,h"co 1D is the chirality projection operator, if 0 is a chual spinor. For the case = I, this yields the well-known constraint that D = 3.4, 6, 10, which gives the Green-Schwarz string. This new identity, however, forces us to obey 8 constraint., which is given by contracting the identity with ( r ~)"fI. After 3 • J we lind once again thaI D - P - 1 = M N /4, as before. : Thcn the action is given by (13.2.10)

!t we write this out in detail for the supennembmne, we find for {he WcssS2 = -

i; /

dlu

{(s/}tBrl'ui1;O) [n j n;

+ in j'O r a1o U

- (lxe r'a)o)(iir"a,Olll.

(13.2. 11)

. let us cbeck that this action is locally supersymmemc. Let J(J be· at thill point Then we find I _ ll 6b = - nil, ... n \ i do r 1'1 ..1,~ 8(J. ( 13.2. 12) pi the case p = 2. we find: oS = 2iT / d l uo8

"" iT

!

(.J

ggil r i - Is;jkriJ) [hO

d l u 60(l - cr)J l:gll r ,ajlJ.

,~

= 6 I1-g eIJtn''n"n' i I r ,,~p' v

w,,,,,, therefore that lJS = 0 jfwc choose c = ±l,

! ~ (I

( 13.2. 13)

± r)'(u).

( 13.2. 14) ( 13.2.15)

± r) is a projection operator.

~

~~~~t~~'~~i~,~th~,~t,~thi'~inijtro~!d::"~,~ti~on~O~h~WCSS-ZwninO

locaUy supersymmetric, termsuch intothat the of the 32~componen t spinor. (We can by going on-shell. leaving

516

13. Solitons, D-Branes, and Black Holes

us with the desired eight components to match the eight cOlnpI:ment$, bosonic theory.)

13.3

Five-Brane Action

By counting states, we find that the 5-brane does not satisfy the usual '. of p-branes. To construct the 5-brane action, we must introduce a "'4'''',,,,",= rank tensor which is described by a self-dual field tensor Fmnl . Let us count the number of physical modes

-2)

(D-p-l)+ ( P 2

=~MN,

where the first two terms represent the string and two-form degrees respectively. The solution for this is p = 5, giving us a 5-brane. This poses new problems, however, since in general a covariant " a propagating self-dual field is extremely difficult to construct. ' theorems exist, in fact, which actually show that it is impossible covariant action for a propagating self-dual field under certain ' Solutions to this long-standing problem have been proposed, but an infinite number of auxiliary fields. Recently, however, a COllVllncj~$.:~$!j1 for the 5-brane has been written involving just a single auxiliary U~,: need a new nonperturbative object, called D-branes, which can for the R-R background fields. We recall that the tensor field B /LV background field arising .... sector couples directly to Type II strings, so that Type II strings .• ::. under this tensor field. However, the R-R fields couple to the ... their field tensors F/-tl .... /-t ' . So the NS-NS and R-R bac~kgJroundfi#lt~:~i to the string via

where Sa are the standard spin fields defined on the string . the string has no charge under the R- R fields. This poses duality will in general mix the two sets of charges. For example, ••... that the Type IIB string has two massless tensor fields B and B'.::.·~ '(!'tf: duality group SL(2, Z) turns one into another. But since the TYiPi.tUB carries no charge under B', it means that we must introduce : : ~·H·..ij.j~~:~ the D-brane, to carry this charge. For example, if (I, 0) reJ)re~;tm't,$JW Type lIB string which has charge one under the NS-NS field but: Z¢j(~:~ under the R-R field, then an SL(2, Z) rotation will in general .tr:@~!f{iJOO (1, 0) string into a (m. n) string, which can never be seen to anyfj·:j~1~jw~m~ perturbation theory. Second, there is a surprisingly large family of such U-I)ralles, ~ij.J~j~ Ill .

>

mensions. If we analyze the super translation algebra in I 1 ;~IIIII recall that we only have p = 2 and p = 5 M-branes. But if we •. : •• super translation algebra, then the p = 2 and p = 5 sources in .. :. decompose into a very large family of p-branes in lowe~~~·:. :·~. '~ . ~illl of which correspond to D-branes. In general, we will have D-branes for the Type IIA string, and odd dimensional IIB string. We noticed earlier that there was a simpJe relationship the dimension of a super p-brane. However, the D-branes pactifying M-branes do not, in general, obey this relationship . . to introduce a new vector field in addition to the p-brane generalize the brane-scan. A vector field on the (p + 1)-dimensional world volume p - I degrees of freedom to the physical states. So we must formula D - P - I = M N to the following

!

D-brane:

D - 2 = !MN.

Notice that the p has dropped out of the calculation. Compaffil~H~ the R-R states generated by Type IIA, lIB, and I strings, we hn,1··rnllN'

13.4 [)..Hranes

5 19

match them perfectly. This allows us to complete the coullting of BPS

which is a powerful check 0 0 the self-consistency ofour nonpemtrbative of duality. D-branes were discovered by analyzing T -duality in Type I when we made the standard T -duality transfonnation. closed

were mapped into closed strings. We found that (13.4.3) aT-dual ity transformation. So T -duality simply linked d ifferent closed , However, jfwc try to perform a T -duality lransfomlation 00 Type we find that the boundary conditions interfere with the analysis. the Neumann boundary condition at Ihe end of an open string, duality transfonnation, turns into a Dirichlet bowtdary cond ition. us compactify lhe JLth dire Q. in suitable units. For an extrema! black hole. At = Q. For our wc have an extremal black bole when r + = r _ = ro.) For this metric, we find the following Charge relationship QIQ5 = r:r:V/gl.

(13 .7.1 2)

The momentum is given hy

NV

P = 2g2 Sinh{2a)(r~ -r:).

( 13.7.13)

will shortly take the limit a -+ 00 as r _ -+ r +, such that P remains Then the energy E and area A become

RV

[2(,! + ,i) + cosh2a(,i .- ,:,)] ,

E = 2g2

,!Rcosh

A = 4rr 1

(1

Jr~ r:. -

(13.7.14) (13.7.15)

'0ckgrrmnd equations_ task is now to find 11 statistical derivation of this re:ationship using wm.nun counting arguments. The key observation is that we know how to the BPS D·brane Sl in string theory. Even if we originaUy start with a theory purely ,eventually we must solve for the equations of motion!';, which allow and solitonlikc objects. Specifically, we can write tbe membrane )Iut;o" of Il-dimensional supergravity by introducing the following ansatz:

:! ds

2

=

(I

;~

+

r

21J

dx" dx" +

(I

+

;~

yo

(dj

+ jdS1;) ,

( 13.8.1)

I.L = O. 1. 2. m = 3..... 10, dQ 7 is the volume form for thc 57 sphere,

the four-form field strength is proportional to the dual of the volume form

5,. Similarly, the 5-'oranc solution of II-dimensional supergravitycan be written

2 (I + ~,~rlJ3 dx"dx,,+ (I + ~~yjl (di +jdQn.

:: ds =

( 13.K2)

the four-form field strength is proporuonaJ to the volwne form on S4. ~~ Not surprisingly, we can now present the general case of the p-branesolution '. D dimensions. Wc start with the Ilction '.

5= _1 !d{;X,J R(R~t{a¢l2K

1

2(d+l)!

e- a4> F'f.....2 )

,

(13.8.3)

r '

Fp+z is the usual amisymmetric field strength. We couple thi!'; to the p-brane action

5p = T _

f

d P+ 1/;

(p

(_~JYyiJ&jXMaiXN8MNea(d)4IJ(p+ 11 +

fl- XM,.. I A ) +I I)! Si;il.--.i.....;a.,XMI , _ .. ',-1 MI •.... M,..."

p;

1.JY (13.8.4)

yli is the metric on the world volume of the p-branc.

solve these coupled equations, we start with Inc ansatz

ds 2 ::::'eV·tLxJ1.dx,. +e2R dy m dym,

(13.8.5)

538

13. Solitons, D-Branes, and Black Holes

where /.l. = 0, 1,2, ... , p and m = p + 1, p + 2, ... , D - 1. ~m111~tl~ split the p-brane coordinates as follows: (XM = XI', YTJI ), and XI' ym = constant Then a solution can be given as D- p-3 A = 2(D _ 2) (C - ar/Jo/2),

p+l B = - 2(D _ P _ 3) (C - a¢o/2),

a

'2¢ =

a2

'4(C - a¢o/2) + a¢o/2,

a 2 = 4 - 2(p + I)(D - P - 3)/{D - 2), and k yP+I' K2T e- ao/2 - In y,

e-ao/2

e-c

=

+ --

D - P - 3 > 0, D - p - 3 = 0,

J!

where k = 2K2T I(D - p - 3)QD- p-2 and Q is the volume ofa So far, we have only constructed classical solutions :~~~:~~~l~d!~N;M~:~ITi membrane and 5-brane. To construct the explicit supermembrane (1\i~,I;!;I1! introduce the generalized Nambu-Goto term SI

= -T

f

dp+1aJ-det

TIl' TI j ,

where nf' = &/XIL-iePai 8. Thistermbyitselfisnot so we must introduce a Wess-Zumino term. We first introduce the i h = -2,TII'P ... ni+ lb·

.

" ·.. ' .... 1

.

Written out explicitly, the Wess-Zumino term for the membrane " ,. S2 = -

i; f

d 3a

leePer I'vo/8) [TIiTIk + iTIier ok8 V

- (i)(erl'oj8)(er ak 8)]) , V

where

13.8 Summary

539

We see therefore that 8S = 0 if we choose

,=

c= ± l , where

(I

±

(13.8. 14)

r),(a),

!(l ± r) is a projection operator.

T he I I-dimensional mtmlbrane and the 5-brane arc sometimes called thc M-branes ofM-theory. However, we are also interested in the lower p-branes fliz,atit'" aiM-theory. Unless M-theory is quantized. we will never underits spectra and rigorously understand its properties. Unfortunately, even membrane actions cannot be qllantized with kilO"",," techniques. if3 interactions. At present, v(..T)' little is known about memExci tation~ of O-branes may be given in tenns of strings, there are large gaps in our understanding of how membranes interact with other. ~~;:::;,~,~~g.~ th;~e of dualitie!J' down to = 4. Although a considerable In ha... b(..'Cn done on dualities in D = 8 and D = 8, very little still known uoout duali ties in D = 4 with N = I supersymmetry, which is

web

D

~:~;;:;;r!\~:~~~::m:iw~re~~st~.~!~ give us

M· theory still cannot explain why superinto the cosmological constant.

544

13. Solitons, D-Branes, and Black Holes

However, given the astonishingly rapid rate at which the theory ' . oping, there is considerable optimism that the mystery behind . soon be revealed. Perhaps some of the readers of this textbook will...,,,,-:~,,,,,,w those who accomplish this feat.

III

References [I] M. J. Duffand K. Stelle, Phys. Leu. 8253, 113 (1991). [2] R. Gueven, Phys. Lett. B276, 49 (1992). [3] For a review, see: M. J. Duff, R. R. Khuri, and 1. X. Lu, Phys. (1995). [4] A. Achucarro, J. Evans, P. Townsend, and D. Wiltshir, Phys. Leu. ( 1987). [5] B. E. Bergshoeff, E. Sezgin, and P. K. Townsend, Phys. Lett. U~t~7,. · ri);·I,::t~ Ann. Phys. 185,300 (1988). [6] M. Aganaic, 1. Park, C. Popescu, andlH.Schwarz, hep-th/97011 [7] P. Pasti, D. Sorokin, and M. Tonin, Phys. Rev. D52, 4277 (I: th/970 1149. . [8] 1. Dai, R. G. Leigh, and 1. Polchinski, Mod. Phys. A4, 2073 (I [9] R. G. Leigh, Mod. Phys. Lett. A4, 2767 (1989). [10] P. K. Townsend, Phys. Lett. B373, 68 (1996). [II] C. Schmidhuber, Nucl. Phys. B467, 146 (1996). [12] E. Witten, Nuc!. Phys. B460, 335 (1995). [13] T. Banks, W. Fischler, S. H. Senker and L. Susskind, Phys. Rev..: (1997). •• [14] For a review of matrix models, see: A. Bilal, hep-thl97 101 36. [15] U. H. Danielsson, G. Ferretti, and B. Sundborg, hep-thl9603081 [16] D. Kabat and P. Pouliot, hep-th/9603127 (B.9) [17] M. R.Douglas, D. Kabat, P.Pouliot, and S. Shenker, hepl-thI96ID80IA'K [18] K. Becker and M. Becker, hep-thl9705091. [19] B. de Wit, M. Luscher, and H. Nicolai, Nud. Phys. B30S[FS23], [20] I. 1. Bekenstein, Nuov. Cimento Lett. 4,737 (1972); Phys. Rev. D7, Phys. Rev. D9, 3292 (1974); Phys. Rev. D12 3077 (1975). [21] S. Hawking, Nature, 248, 30 (1974); Comm. Math. Phys. 43, 199 . Rev. D13, 191 (1976). [22] C. Vafa and A. Strominger, Phys. Leu. B383, 44 (1996). [23] G. Horowitz, gr-qc/960405I. [24] J. M. Maldacena and A. Strominger, hep-thl9603060.

Appendix

the mathematics of superstring theory has soared to such dizzying we bave included this snort appendix to provide the reader with 8 understanding of some of the concepts introduced in this

thai we must necessarily sacrifice a certain degree of i rigor if we are to cover a wide range of topics in this appendix. the interested reader is advised to consult some of the references later for more mathematical details.

A Brief Introduction to Group Theory . group G is a collection of eiemenJS gi such tbat: There is an identity ciement 1. There is closure under multiplication:

Every clement has an inverse: gl X g,-, = I .

Multiplication is associative: (8, x gJ) x 8t = Ri x (gi x 8t). Groups come in a variety offarms. SpecmcaUy, we have the discrete groups, a finite number ofelemcnts, and the continuous groups, such as the which have an infinite number of clements. Examples of discrete

Appendix

546

(1) The Alternqting groups, Zn, based on the set ofpennutations (2) The 26 sporadic groups, which have no regularity. The interesting of the sporadic groups is the group F], cornmont "Monster group," which has 2 46 .3 20 . 59 .7 6 • 112 . 13 3 • 17· 19 . 23 . 29 . 31 ·41 .

elements. In this book, however, we mainly encounter the continuous have an infinite number of elements. The most important of groups are the Lie groups, which come in the following four "la";:il,\~\W:ijJ series A, B, e, D when we specialize to the case of compact, An = SU(n + 1), Bn = SO(2n + 1), en = Sp(2n), J)~1r...sQ(2{'J.. J,

as well as the exceptional groups: G2; F4; £6; £7; £8;

of which E6 and E8 are the most interesting from the sta:nClI'olllti phenomenology. Let us give concrete examples of some of these groups b~~11111 set of all real or complex n x n matrices. Clearly, the set of invertible matrices satisfies the definitions of a group and hence group GL(n, R) or GL(n, e). The notation stands for a generalline:l¢:~ n x n matrices with real or complex elements. If we take the subset· or GL(n, C) with unit determinant, we arrive at SL(n, R) and group of special linear n x n matrices with real or complex eleme:p:~k

O(n) Now let us take a subgroup of GL(n, R), the orthogonal group consists of all possible n x n real invertible matrices that are Ox OT

= 1.

This obviously satisfies all four of the conditions for a group. matrix can be written as the exponential of an anti symmetric m"tri,i';' 0= eA.

It is easy to see that OT

=e AT =e -A = 0-1 .

A. I A Brief Introduction to Group Theory

547

general, an onhogonaJ matrix has ~n(1/ - I)

!

~~::~~:l:;~~::.;:~~:. Thus, we can always choose a set of ~fI(n - 1) linearly matrices, called the generators Ai, such that we can write any

"

nf2". r~ - 1)

O=exp

[ L

]

/J..;

.

(A. 1.6)

1'-1

real numbers pi are called Ihcparametcrs oflhc group, and there are (hus I) parameters in O(n). The number of parameters of a !..ie group is !:~::!,~:::dimension. The coounutator of two ofthe..~c gcncldtors yields anolher (A. 1.7)

the

f's

arc callemlations preserve the scalar product XiX/:

= invarianl

(A.L23)

= x 0'0 x = x,x/_

(A.!.24)

XiX;

XjX; "

invariant can be wrineo (A. US)

the metric is Oij' In principle, we could also have a metric with alter"tingpositi've and negative signs along the diagonal, T!ij. which would create parameter space that is noncompnci. If we have N positive Rod M negaelements in "fIij. then the set of matrices that preserve this foim is called VV·,M), (OT)ijJ')jkO~1 = 1711. 7/1/

=

e(j)~I)'

(A.1.26)

~:(i)=±1.

the elements of E arc positive, this gives the group O(n). If the signs arc It"n"u"',thcn the group is noncompact. Special cases include projective group:

0(2, I),

Lorentz group:

0(3, I),

de Sitter group:

0(4, I),

an ti-dc Sitter group:

0(3,2),

confonnal group:

0(4,2).

(A 1.27)

For example, the de Sitter group can be eonstnletoo by taking the generators 'U'14. I) and then wriring the fifth romponenl as

(A.1.28) the algebra becomes [P",P' J ~M·',

0(4, I ):

I

[pO. , Mb~] = pbf/cr _ per/'b, [M o.b , Mcd] = f/~( MtlJ - ....

(A,1.29)

that this is almost the algcbra ofthc Poincare group. In facI, ifwc make substirurioo

po.

---.)0

±r po. ,

(A 1.30)

the omy commutator that changes is

[pO. pb ] = --'-- MOb ,

r2

'

(AI.3I)

550

Appendix

r is called the de Sitter radius. This means that if we go around a cm.~~::ffl de Sitter space and return to the same spot, we will be rotated by a f;"~f~~ transformation from our original orientation. Notice that if r goes to mtl4H¥f;~i!~ have the Poincare group. Thus, r corresponds to the radius universe such that, ifr goes to infinity, it becomes indistinguishable frori1~~l~~ four-dimensional space of Poincare. Letting the radius go to infinity IS:j:ia1jfio! the Wigner-Inonu contraction and will be used extensively in theories. After the contraction, the de Sitter group becomes the

SU(n) The group SU (n) consists of all possible n x II complex matrices determinant and are unitary:

vv t = 1. The notation stands for special unitary n x II matrices with complex Any unitary matrix can be written as the exponential of a Ht = H:

We can show that

V t =e _iHt =e -iH = V-I . Let n elements in a complex vector Uj transform linearly under sl)r@~~~~iliW

The n complex vectors Ui generate thefundamental representation nH'h"·",;;.liill~~ Then an invariant can be constructed: U7U; = invariant.

If u; = Viju j, it is easy to check that

'*i

I

i

u u = u

t) V *(U i

*i

re:r~:t~~il~i

= u Ui· The metric tensor for the scalar product is again 8ij. If we \\ some of the signs in this diagonal matrix, the groups that would pre:s,~t.i~ metric are called SU(N, M). An example of this would be the SU(2, 2). .. Any n x n complex traceless Hermitian matrix has n 2 - 1.: elements and hence can be written in terms of n2 - 1 linearly' matrices Ai. Thus, any element ofSU(n) can be written as ij

jkUk

A.I A Brieflntrodllclion to Group Theory

55l

that the group closes underthis ;~r:~!:~~::·~~~:S:;~;~~:l'::~~~::::i~~;:l~;~ and thllt we can the algebra of the group as (A.1.39) knowledge ofthc s tructure constants determines the algebra completely. can also construct representations of SU(n) out of SpinOIS. If we have group O{2n). then SU(n) is a subgroup. ff we construct the clements

(A. 1.40)

: wh". rlJ 3re Grassmann variables. then the generators ofSU(IJ) can be written

}... =

LATJ (AQ)JiA~.

(A.1.41 )

J"

" "0'. we have an explicit representation fo r the inclusion SU(n) C 0(2n).

(A. 1.42)

symplectic groups are define:-:.;.:.:.:.z....:.z.:-:

dai == (dOli)

+ Cljd( ~ l)d•.

N ow assume that -+

0:

g -1 o:g .

For g close to unity, we can always write

== 1 + W, 80: == -(J) A a + ex /\ w. g

_..... __ .. .. .... .

::::::::::::::::::::&.~~::::::~

and let u,s cal~late the variation of a homogeneous polynomial P of9~~t~fn~]~~t~ under thiS shlit: .~::::::::::::»~::::::::: dP(Q)

as we claimed.

== 0

AJ A Brief mtroduction to the Theory ofFonns

565

The second part of the proof is a bit more involved. Let us define

n =dW+WAW,

+ W' 1\ w'.

(A.3.35)

pen') - pen) ~ dQ

(A.3.36)

0.' = dul Our plan is now to show that

for some form Q. First we wall! 10 wrile a curvature fonn that Allows us to interpolate continuously between n and n'. Let

w,=w+t1]. , 1] =W - ( 0 .

(A.3.37)

Notice iliat the variable t allows us 10 interpolate between the two fonns:

t=O-+w.=w. t = 1 -+ WI = w'.

(A.3.3S)

It is easy now to find the curvature form that interpolates between the Iwo, as 8

function of f:

=dw, +w, I\W"

(A.3.39)

r = 0 -+ Q, = Q.

(A.3.40)

Q,

where Notice that the fonn varies between n and Q ' as t varies from zero to one. Now let us write the form q such that (A.3.41)

q(f3.a) =rP(/3,a.a,,·((r - I)-times)···a).

(J is repeated (r I)-times in the invariant polynomiaL This fOnD q will be the key to showing that P is an eXile! form. By the reasoning given earlier, if we differentiate q, we find

where the fonn

d'l{f/, Q,) = r cJ p(rl, 0, ... «r - 1) -lil1le~) . .. n,)

= q(D1], Qr> - r(r - I)t P(1], 0, A '1 -

1]

A 0" Q" ...• f.l,).

(A.3.42) But we also have the identity, due to the fact that q is an invariant polynomial,

2q(TJ 1\ 'I,

n,) + r(r

- t )P('I,

a, 1\ 1) -

1) 1\

n" ... , a,) = O.

(A.3.43)

In the previous equation, we used

8= I

+ 'I.

(A.3.44)

Putting both equations together, we find

dq(l'J, f.!,) = q{D1), Q,) + 2,q(TJ

1\

11,

n,) =

d pen,) dt

.

(A.3.45)

566

Appendix

Thus, we arrive at

1 1

o

dP(nr) = P(Q') _ P(Q) dt

= d

11

q(w' - w, Qr)dt = dQ.

We thus obtain our final result:

Q=

11

P(Q') - pen) = dQ,

q(w' - w, Q,)dt + closed forms.

Thus, an invariant polynomial based on curvature two-forms is both b).

(A.5.4)

The fields transform under a local gauge transfoftlliltioll as

6h~ = i1l'e

A

+ h!eC f£s'

(A.S.')

Now form the COmml!tator of rwo covariant derivatives [VI" Vv] =

R:.M. .

(A.S.o)

574

Appendix

where

R: LJ

= afi.h~

- avh~

+ h~ h~ f2B'

In component fonn, we have

R~LJ(P)

== alLe~ + w~bee ~ (11- B- v), ch - (II ~ V) Rab(M) = ap.. Wah + wacw J.L \J /.L l.! II

f"V

R:v(Q) = OM 1ir~

?

+ if vU)~)rrab ~ (J.l 8- V).

The variation of the curvature can easily be shovro. to be 8R:v == R!v sC f~B' The action for supergravity is now given by

S=

f

4

d xs

I.I.vprr {R

(ab

~v M) Rpa(M

.. ':::::: {~~~~~~~~~~~~~~~~~~~ ()fJ( ) .......;...."""'.,.........• + R~v Q RfJa Q YSC}f~~tJ~~~~~~~t;

()Q'

)cd Cabcd

(A)$~ ~ Q:j~~~~~~~~~~~~~;

If we make a variation of this action, we find that the action is no(:t~~li~~~~~~~~~~~~~~~ invariant unless we set ".TIUDC1ric generators of the Osp(N /4) group have spin-to r[we take the maximum helieity state of the graviton and then operate on it with all possible Q's, we find that the series eventually mus t terminate or else we create particles of spin-~ and 3: QuQ~ · ·· Q~

Igraviton).

(A.S.2S)

Theories ofspin-! and 3, although they can be constructed as free theories, are thought to be inconsistent when coupled to other panicles. Thus, since thcre are eight half-steps between 2 and - 2, we find that N can only equal eight in the above series. Thus, 0(8) is the largest supergravity theory thllt does not have spins greater thlJl 2.

A.6

Notation

For the purpose of clarity, we have deliberately dropped lhe normalization factors appea ring in thc functional path integrals and the N -point amplitude A 1/ . This should not be D problem because they can be easily restored by the reader. In our units, the Planck length and Planck mass are equal to

L = rfiGc-1J = 1.6 x 1033 em, M = lFicG - t ] = 2.2 x to-I gm = 1.2 We set c and h = I.

X

10 19 GeV/c.

I

· i,

I

·\. · '.

578

Appendix

We use the space---time Lorentz metric (-, +, +, ... +).OntheworldsnE:er~::::::::: we use the two-dimensional metric ( -, +), where the first (second) index retem:~:}~:~:~ to r (a). OUf two-dimensional matrices are I

pO

=(

-i)

0 i

0

'

where -

0

1/f = ljJp . In general, we will use kJL to represent the physical momentum and Pl1 " "~(;)::::::::::::~ represent the operator whose eigenvalue is the momentum. However, as in .."1:,,:.0:.:.-:.'.:-:.: literature, we will often drop this distinction and use both interchangeabl}io:::: We use the following choices for the Regge slope )~ ::::::::::::::::::

c/ == ~ for open strings, (X'

==

i for closed strings.

Whenever possible, we use Greek letters to denote curved space indi'c"~¢"!~~::~:::~:::~:: We use f.i.. and lJ to denote 26- and 10-dimensional Lorentz indices in space. We use Greek letters, a, fJ to denote two-dimensiona1 curved sheet indices. Flat space indices, either in Lorentz or in two-space, are in Roman letters, a, b, c. When the context is clear, we \vill sometimes use metric ( +, +, ... , +) for the fiat tangent space. We choose our gamma matrices such that

{r A, r B}

== -2g A B,

r D +1 = for] ... ID-l. In an even number of dimensions, they are nonnalized so that

(rD+1i

== +1 if D == 4k + 2,

/~I!~~.~.~l

(r D+] i == -1 if D = 4k. When the ganuna matrices appear with more than one index, we take the _~~ of all antisyrrunetric combinations of indices. For example, we nVlljWll'j~lf~~~:j~ the gamma matrices so that

these

rAB

= ~[rArB _

rErA].

We define the light cone gamma matrices as :.,-'

r+

==

['- =

_l_(rO

-J2

+ r D- 1),

JzC['O -

['D-I).

When specializing to the case of 10 or 4 dimensions, we will often just us~:" symbol y f-L • " "",", :}}~::~=~{~~:

References

579

When we make the reduction down to SO(8) for the light cone fonnalism. we will use the direct products of2 x 2 Pauli matriccs: y' = (

~y" y,:,o ) .

yJ = -iT] ® 1"2 0 1"2 . y2 = il@fl ®f2. yl=il® Tl®rZ' y4 = i r l0rz01. yS = iTl®TZ® I, y6= ir z®l®rl' y7 = irz® I ®T~.

YS=l®l® l , , Ij _!( I j _ l'wl' -

2

>'w,;>'''b

j i) >'d,lYilb'

where 1"1 BTe the Pauli matrices, and each gamma matrix is a direct product of

three 2 x 2 blocks.

References For an introduction to group !hoary, see: [I ) R. Gilmore. /J# Groups. Lie Algebra, and Some oj Their Represelllations, Wilcy-Inlerscience. New York, 1974. (2J N. Jacobson, Lie Algebras, Dover. New York, 1962. (3 ) H. Georgi. Lie Algebros in Particle Physics. BenjaminICummings. Reading, MA, 1982. (4) R. N. Calm, Semi·Simple Lie ,Algebras llIId Their Represen/atiotU. BI".'f1jamin/Cummings, Reading, MA, 1984.

For an introduction to gcncl1I.l relativity, see: [5] C. W. Misner, K. S. Thome, and 1. A. Wheeler. Gravitation, Freeman, San Froncisco, 1973. [6] S. Weinberg. Gravitation and Cosmology, Wiley, New York, 1972. [7] S. W. Hawking lind G. f . R. Ellis, The Large Scale Structure o/Space-Time,

Cambridge University Press, Cambridge, 1973. (8] R. Adler, M. Bazin. and M. Schiffer, IlIlrodllction l(J Ge'lf~ral Re/(l/ivity, McGr.tw· Hitl, New York. 1965. [9] M. Canneli. Group ThcOIY and General Relativity, McGraw-Hill, New York. 19 77 .

For an introduction to tile theory offorms. ~ee: {IO] T. Eguchi, P. B. Gilkey, nnd A. J. /lansen, Phys. Rep. 66, 213 (1980). 111 ] C. Nash and S. Sen, Tbpology and GeamerT}' Jor PhysicLw', Al:adelllil;

New York, 1983 .

», ~~.

580

Appendix

{l2] C. von WestenholtZ Difforential Forms in Mathematical Physics, t

Holland, Amsterdam~ 1978, [13] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Ac~ya/~·fnR:ef.::::::: Index Theorem, Publish or Perish~ Wilmington, DE. E14] S. J. Goldcerg, Curvature and Homology, Dover~ New York, 1962. For an introduction to supersymmetry and supergravity, see: [15J P. van Nieuwenhuizen, Phys. Rep. 68C, 189 (1981). [16] S. J. Gates, M. Gtisaru, M. Rocek, and W. Siegel, Superspace or One n:lf:~'~~:m and One Lessons in Supersymmetry, Benjamin/Cummings, Reading, MA, i: [17] M. Jacob, ed., Supersymmetry and Supergravity, North-Holland~ .

]986. [18] P. West, Introduction to Supersymmeny and Supergravity, World Scie~ti:~·]~:~:~::~~~: Singapore, 1986. :::::::: [19] 1. Wess and J. Bagger, Introduction to Supersymmetry, Princeton Unive#j Press, Princeton, NJ, 1983. ::::::::: [20] R. N. Mohapatra, Unification and Supersymmetry, Springer- Verlag, New rq~~~::::::: 1986.

,,'

.

. .. . .....

Index

(\clion, firt! qU(lntize(\ bo~onic

poinl particle, 25

bosonic string. 52 confonnal field theory. 152, 158, 175 first order fonn, 28, 58 ghosta:lion, 152, 158--1 59. 175 Hamiltonian, 28. 58 heterotic string, 378, 400

second order (onn, 28, 58 superpanicle, J 03 superstring (conformal gauge), 105 supcrsl:ing (OS), 126, 138 superst:ing(N~R), 11 9,1 36.152 Action. second quantized

BRST string field theory, 300, ]03 light cone fie ld theory, 259, 263 , 278 point panicle, 40, 252 pregeometry, 326 superstring, 28 1, 324 Almost complex structure, 419 A~. 546 Anomaly

cancellarion in strings. 366-368 characeristic classes, 352 chiral, 347- 348 confonnal anomaly, 211

graviutional,357_JS$:

hexagon diagrams, 344, 37 1 A-roof genus, 352, 369 Atiyah--Singer Theorem. 338, 357 Automorphic fux.ti on

multiloop amplitude, 204-207 single loop ampli tude, 184

Betti number, 413, 4 15 Bianchi ide nti[)" 36 1,406, 427,562 black holes" 458., 5 11, 532

Bw. 546 Bosouizatl0n, 143, 163, 169-1 70 BPS re larions., 459 BPS stales, 458, 472,482, 505, 5 11,

m

BRST, first quattized bosonic siring, 70, 97 conronnal field theory, 160 current, 160-161 fermionic string, 121- 122 point particle, 30-33 BRST, second quantized

ax iomatic formulation, 309, ]30 point particle, 32-33 prcgoometry, 326

superstrings, ) 18 Calabi-Yau manifold, 405 Cartan subalgcbra, 168, 389, 554

· 582

Index

Cartan-Weyl representation, 168-169, 554 Chan-Paton factor, 339--341, 368 Characteristic classes, 348 Chern class, 350, 369 Chem-Simons form, 349, 576 Chiral invariance, 342-345 Closed form, 410 Closed strings field theory of closed strings, 317-319 modular invariance, I 97 multi loop amplitude, 209--210 single loop amplitude, 195 tree amplitude, 87-90 Cn ,546 Cocycle, 168, 171 Coherent state method, 82, 195 Cohomology de Rahm, 409-412 Dolbeault, 421 Hodge theorem, 229, 421 Compactification,374 Conformal anomaly, 217 Conformal field theory fermion verte~ 165, 175-176 multi loop, 234-236 operator product expansion, 147 spin fields, 143, 155-158, 169--170 superconforrnal ghosts, 158 Conformal gauge mu Itiloops, 211- 214 single loop, 60, 98 Conformal ghost, 70, 121,158-165 Conformal group, 549 Conformal Killing vector, 214 Conformal weight, 144, 158-165, 175 C Pn , 418, 424 Currents, 41, 105, 119, 160, 342-344,347,355 Curvature tensor, 6, 13, 405~ 559 D-branes, 458,504,51 1,517, 518, 521 , 539 DDF state, 94 Dehn twist, 198, 213, 228--229 de Rahm cohomology, 409-410

de Sitter group, 549 Differential forms, 409--411, 414-416 Dilaton, 192 Dirac index, 353-356, 370 Divergences bosonic single loop, 190-191 .. • closed bosonic loop, 198-200·.· fermionic single loop, 200 .••• • multiloop, 204, 210; 226 Selberg zeta function, 226, 24 ••

-P(N j M), I I, 12 Partition function, 187, 190,205 Path integral anomaly, 34 6 muitiloops, 181,210 point particle, 18-25 sIring field theory, 224 trees, 72-78 Period matrix

cohomology, 413 multiloop, 229 Schottky problcm, 229 single loop, 209 Pictures confonmd field theory, 162 F J an d Fl pictures, 1\3---11 4,

137-138

picture changing operators, 32J-324 Planck length, J 0 Poincare duality, 41 6 Polyakov action, 57, 2 10 Pontryllgin class, 35 1 Pregeometry, 32~ Prime fOITTI, 235, 242

5135

Ricci flat, 406, 409, 425 Ri emann-Roch theorem confonnal field theory, 162 index theorem, 357 moduli spacc, 212, 227 supcrmoduli space, 224 Riemann surface, 200, 210, 423 Riemann tensor, 558 S-dua1ity, 465, 4 76, 50 1, 503

S-T-U dual ities, 458 Schottky group, 200, 205, 240 Schottky problem, 227, 2H- 237, 242 Second quantization advantage over first qua ntization, 35,44 BRST field thcory, 291 light cone field theory, 254-290 point particle, 34--44 Selberg zcta fWiction, 225-226, 241 Shapiro-Virnsoro model amplitude, 87 modular invariance, 197 multiloop. 2 10 problems with BRST string field Iheory, 317 si ngle loop. 195 Simplex, 411--412 Singularities hexagon graph, 344, 371 multiioop, 205-207, 2 J 0, 226 single loop, 190-191, 198-200 SL(2, R), 85, ISO, 184 SL(2, Z), 197 SO(N),546 SO(8), 2RO, 579 SO(IO), 161l, 170,175, 176 50(32),364

SO(44),399 Q uadratic diffe ren tial, 215 Quantum gravity, II , 15 Quatemion, jj2-jj4 Ramond sector, 107,--111, 136-137, 31' Regge behav ior, g I Regge slope, 55 Regge trajectory, 55, 62

Sp(N),546,55 1,555 Sp(2g, Z), 229 Spectrum bosonic string, 63 closed string, 65 OS su perstring, 13 1 heterotic string, 384-386 NS-R superstring, 110, 136-137 Spi n(N ),553

586

Index

Spin(32), 373 Spin embedding, 426-427 Spin fielcL 143, 155-158, 169-170 Spin manifold, 352 Spin structure multi loop, 222 single-loop, 222 Spinor covariantly constant, 407, 450 Dirac, 358 Majorana, 127,358 Majorana-Weyl,358 Weyl,358 Spurious states, 66 Stiefel-Whitney class, 350, 352, 369 String field theory BRST,291-331 closed string, 317 duality, 249 equivalence of, 311 four-string interaction, 275-280 light cone, 254-290 objections to string field theory, 248,286 superstrings, 280-286, 324 Witten's, 308-309, 324 Strings, types four-dimensional, 443 heterotic, 383-388 Type r, HAB, 130-132

SU(N), 550-551 SU(N 1M), 566 SU(6, 6), 566 Sugawara fonn, 172 Superconfonnal ghosts, 121, 158, 164,322-323 Superconfonnal group, 109, 122, 137,320 Supercurrent, 106--107 Superfield chiral,570 vector, 569 Supergravity ":i N == 1,10,388,574 N =8,11,574-575,577 problems with finiteness, 12 problems with phenomenology, 12 Yang-MiIls, 576 Supennoduli space, 162, 223

Superspace, 567 Supersymmetry two-dimensional world sne~el.llJ.