Modular invariance of twisted string theories on group manifolds

Volume 20 l, number l

PHYSICS LETTERS B

28 January 1988

MODULAR INVARIANCE OF TWISTED STRING THEORIES ON GROUP MANIFOLDS Wei CHEN a, Chao-shang HUANG

b,a,~

and Wei-dong Z H A O

~,c

a Institute of Theoretical Physics, Academia Sinica, Beijing, P.R. China b Centre of Theoretical Physics, CCAST (World Laboratory), Beijing, P.R. China c Physics Department of Tongji University, ShanghaL P.R. China Received 23 October 1987

We use twisted affine algebra to investigate the modular transformation properties of the partition functions of string theories on group manifolds with twisted boundary conditions and construct the modular invariant twisted string theories on group manifolds. We analyze the possibilities that there exist massless states in the spectra.

String theories in a curved space that is M d X G (where M d is a d-dimensional Minkowski space-time and G is a compact group manifold) have attracted much recent interest and critical dimensions and mass spectra for both the bosonic and fermionic string in these theories have been given [ 1,2 ]. In particular, Gepner and Witten [ 2 ] have constructed the modular invariant theories in such a space. In all these theories the normal boundary conditions (periodic for the bosonic string and periodic or antiperiodic for the fermionic string) are assumed. But it is also possible to assume twisted boundary conditions for constructing a string model [ 3,4 ]. String theories with twisted boundary conditions o n Md)< G have been examined in ref. [4]. But their considerations have been only at the level of the free string theory. We have investigated the modular invariance of interacting twisted string theories o n M d X G. In the present letter we present the results for bosonic string theories. The results for spinning string models and details will be presented elsewhere ~ Let g be the Lie algebra associated with G and/z be the diagram automorphism ofg. We have the decomposition ~=go+gl

ifO=2;

g=go+g~+gz

ifO=3

where gi is the eigenspace of/t and O is the order of # (i.e.,/t~= 1). When we assume the twisted boundary conditions

J(a, ~)=u{J(a+~, T)}

(1)

(where J is the right-moving or left-moving current on G), we obtain the twisted affine algebra g(A) satisfied by the complete set of Fourier components of J which contains the untwisted affine algebra associated with go as a subalgebra [4]. The ground state (the tachyonic state) in the theory may be characterized by the highest weight vector of the irreducible integrable highest weight representation L (A) of g (A) and the entirety of string states which are generated from the tachyonic ground state IA) by operating with the creation operators ja_ n (the Fourier components of J) constitutes the representation L(A), i.e., each string state corresponds to a weight 2eP(A), where P(A) is the set of weights of L(A). This work was supported in part by the National Natural Science Foundation of China. ,i See ref. [ 5] for details.

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Volume 201, number 1

PHYSICS LETTERSB

28 January 1988

Now we consider the modular transformation properties of the partition functions. The full partition function at one loop on M d x G is Z(z) = Tr exp[2niz(Lo - % ) ] exp[ - 2rtiz*(Lo - O~o)]= z M Z ~

(2)

where Z u and Z ~ are the partition functions for the Minkowski space Md and the group manifold G, respectively. It is well known that in the light cone gauge Z ~ is given by Z ~ = I (Im z) -(d-Z)/2~/(Z) -2(d-2) exp{ - ( 4 n i ~ / 2 4 ) [ 2 4 - ( d - 2 ) ] } l,

(3)

and I (Im Z)-(d-2)/2~/(Z)-z(d-2)[ is modular invariant n2. The contribution to Z ~ associated with the twisted boundary condition (1) is

(4a)

Z G =FZI, F = l e x p [ - ( 4 n i T / 2 4 ) ( K d i m G)/(CA + K ) ] l,

(4b, c)

Zr =A~e'~mod Ca IAA(P)('r) I ~

where K and CA are the level of L(A) and the second Casimir of the adjoint representation of G, respectively. p~k = {2ep_~ I (2, ~) = k} where p~_ = {2eh* [ (2, or) eZ+ for all ore1-1' v} is the dominant weight lattice associated to the adjacent root system 3' of g(A) (the set of simple coroots in 3' is 1-1 and h* is dual to the Cartan subalgebra h of g (A), and ~ is the basic imaginary root [6 ]. A,~(P')(t) is defined by TM

A) (p') =Af~+p./A'p. ,

(5)

with

A~=

~

~(w)O.,~m,

(6)

w~W' mod T

where W' = W ~
O~.k(z, T, t ) = e x p ( - 2 n i k t )

~M+~k_,~_exp[rtikz I V l / - 2nik(?, z)],

(7)

~(w) = ( - 1 )t(~) and l(w) is the length ofw. p ' e h * such that (p', a ) = 1 for all ore//'~ [6]. In eq. (4) we have taken all left-right symmetric integrable representations, each appearing once. Substituting eqs. (3) and (4) into (2), one has Z ("t') --~z M Z~l

= [(Im T) --(d--2)/27](T)--2(d--2) e x p { - (4niz/24) [ 2 4 - ( d - 2 ) - ( K d i m G)/(CA +K)]} [ ZI (z).

(8)

Thus, even in the critical dimension (i.e., 2 4 - ( d - 2 ) - (K dim g ) / ( C A + K ) =0), Z ( z ) is not modular invariant because Z~ (z) is not so. Under S (S= (° ff~) and T= (~ ]) are the generators of the modular SL (2, Z)), S Zl(~') ~

Z I ( - - I/T) =

E

A~P~ mod Cg~

IA~)(T) [2---Z2,

(9)

where

A ~ ") =.4A+p/A.

(10)

with ~2See,for example, ref. [2 ]. 55

Volume 201, number I

AA=

~

w e W rood T

PHYSICS LETTERSB

e(W)Ow(A),

28 January 1988 (11)

and P~ = {2ep+ I (2, J) = k} with p+ = {2eh*l (2, a ) eZ+ for all c ~ / F } , the dominant weight lattice of g(A).

peh* such that (p, a ) = 1 for all c ~ F F . Modular transformations are global diffeomorphisms o f a torus which may interchange the loops in the tr and z directions and also interchange the boundary conditions on the currents. Therefore, it is clear that ~(z) has no modular invariance because it contains only the contribution associated with the twisted boundary conditions (1) in the tr direction. Furthermore, when two strings join to form a single string we obtain an untwisted bundle from the two twisted ones. In order to ensure modular invariance it is necessary to include the untwisted sector and sum over a modular invariant combination of boundary conditions on the currents J and f Because the relevant quantities associated with the adjacent root system A' transform to the corresponding quantities associated with the conventional root system A under S and there are three different kinds of A' for twisted affine algebras [ 6 ], it is suitable to discuss how to construct the modular invariant twisted string models according to the following three cases. (I) A' ---A. The adjacent root system A' of g(A) is isomorphic to the root system of g(A) when g(A) =E~ 2>, D~2) , and D~3) . Consider a string theory o n MaX G for G = E6, D3 and D4. Its untwisted sector is exactly the WZW model plus the conventional Minkowski one. The one-loop partition function Z~ for the group manifold G with the periodic boundary condition is given by [ 2 ]

ZCu(z)=F(z) Z~(z), Z , ( r ) =

~

Ih~P)(z)l 2,

(12a, b)

~.eP~ mod Ct~

where p÷ is the dominant weight lattice of the untwisted affine algebra ~(~= E~~), D~~) and D~)). Define the projection operator ~ by

~jA~-~ =exp(i2~ts/e) jg

(s=O, 1..... e-- 1)

for JAEp,~ where g= Y.~-d g~. Then we have the contribution to the partition function from the twisted boundary condition in the z direction Tr exp[2rtiz(f~oC - 1) - 2niz*(L0~ - 1)]

~=FZ2,

(13)

where the quantities labeled with a h a t " ^ " are obtained with the untwisted boundary condition in the tr direction. Similarly, for the twisted sector, we have Tr exp[2xiz(Lo~ - a o ) - 2niz*(L0G - c % ) ]

JfG

=FZ,

(14)

and Tr {exp[2rtiz(LoG - O t o ) - 2 n i z * ( f _ ~ - a o ) ] ~i~}=FZ3,

(15)

.g/G

where Z3 =

IC~')(z) 12,

(16)

A ~ P ' ~ rood C~

where (17)

C ~ I) =CA+,,,/C,,, ,

with

exp{xi[2( w(,4), v) + kvZ]} O~C .(,+ )~kM ,.

CA= E ~(w) we~V

56

y a M ' rood M

(18)

Volume 201,

number l

PHYSICSLETTERSB

28 January 1988

Collecting all pieces together we find the full partition function to be

Z(z) = [ (Im r) -td-2)12~l(r)-2ta-2) exp{-- (4niz/24) [24-- ( d - 2 ) - ( K d i m G)/(CA +K)]} I Z(z) - I ( z ) 2(z),

(19)

where Z ( T ) = Z u -~-Z 1 -.[-Z2 -~-Z 3.

(20)

It can be proved from transformation properties of theta functions under SL(2, Z) and some standard properties of the Weyl group [ 5 ] that Zt and Z2 exchange each other and Z3 is invariant under S, Zi and Z3 exchange each other and Z2 is invariant under T. It is well known [2] that Zu is invariant under S and T. Thus, in the critical dimension, the full partition function Z(z) is modular invariant. (II) A' ofg(A) _A ofg(A'). The adjacent root system A' of the algebra g(A) is isomorphic to the root system A of another algebra g(A'). Ifg(A) =A~t_ z) l or D~_2)1,then g(A') =D~+2)1 or A~l~, respectively [6]. In this case we take G=A21_ l × Dr+ ~. The adjacent root system A' of Dt+2)l (respectively A~2_)t) can be realized from the string model on A2z_j (respectively Dt+~) with the twisted boundary condition in the tr direction. The full partition function is Z(z) = l ( z ) zAD(z),

(21)

with zAD(z) =zA Z~ + zDI Z~ + Z~ Z o + Z~3Z~,

(22)

where the superscript A or D implies whether the labelled quantities are associated with the algebra A~ 21 or D/+2)~, respectively. Modular invariance ofeq. (19) in the critical dimension can be proved by using the transformation properties of Zi (i = u, 1, 2, 3) under S and T. (III) A' =A. The adjacent root system d' ofg(A) is equal to the root system A ofg(A). There is only a kind of algebras, A~2) (/= 1, 2, ...), which have this property in all twisted affine algebras. In this case Z1 =Zz and the modular invariant full partition function can be easily obtained. Now we turn to study the mass spectra of the twisted models. It is noted [4] that the mass spectra of the twisted models are different from those of the corresponding untwisted models. In the twisted sector the quantum mass operators in M d of the twisted closed bosonic string is given by m 2 =2ao "at'2CR/(CA +K) +NM +]VM+NG +NG, ao = ~6 [(10--D) + (dim go/dim g) ( D - 2 6 ) ]

(23)

in the critical dimension D = 2 6 - (K dim g)/(CA + K), where Ca is the second Casimir of the highest weight representation L(A) of go. (Note that only the singlet of ~o has been considered in ref. [4].) Na, Nd are the number operators for Ma and Nc, 37c are those for G. From eq. (21) it is obtained that the massless states arise for the twisted model with g---D~3) , D = 6 and the tachyonic states forming the 27-dimensional representation of G2. For most other cases, no massless states have been found. It is a pleasure to thank Yuan-ben Dai, C.S. Lam and Shi-kuen Wang for helpful discussions.

References [ 1 ] V. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83; D. Nemeschansky and S. Yankielowicz, Phys. Rcv. Lett. 54 (I 985 ) 620; S. Jain, R. Shankar and S. Wadia, Phys. Rev. D 32 (1985) 2713;

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P. Goddard and D. Olive, Nucl. Phys. B 257 (1985) 226; A.N. Redlich and H.J. Schnitzer, Phys. Lett. B 167 (1986) 315; E. Bergshoeffet al., Nucl. Phys. B 269 (1986) 77. [2] D. Gepner and E. Witten, Nucl. Phys. B 278 (1986) 493. [3] C. Vafa and E. Witten, Phys. Lett. B 159 (1985) 265; Y. Watabiki, Tokyo Institute of Technology preprint TIT/HEP-93 (1986). [ 4 ] B.I. Nepomechie, University of Washington preprint 40048-02 P6 ( 1986); C.S. Huang, W.D. Zhao and Z.Y. Zhao, preprint IC/87/208. [ 5 ] C.S. Huang et al., AS-ITP-preprint, in preparation. [ 6 ] V. Kac, Infinite dimensional Lie algebra (Birkh~iuser, Basel, 1983); V. Kac and D. Peterson, Adv. Math. 53 (1984) 125.

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