a physical interpretation of cocycle factors in vertex operator representations

Volume 231, number 3 PHYSICS LETTERS B 9 November 1989

A PHYSICAL I N T E R P R E T A T I O N OF COCYCLE FACTORS IN VERTEX O P E R A T O R R E P R E S E N T A T I O N S

Makoto SAKAMOTO Department of Physics, Kobe University, Nada, Kobe 657, Japan

Received 3 May 1989

It is shown that cocycle factors, which appear in vertex operator representations of affine KaY-Moody algebras or in compacti- fled closed string theories on tori and/or orbifolds, have a simple geometrical meaning and that multiplying vertex operators by cocycle factors is equivalent to modifying commutation relations for zero modes. The origin of cocycle factors is revealed in the second quantized picture.

String theories provide us a number o f interesting aspects and give new concepts to "particle" physics. One of them is the appearance of huge gauge symmetries in compactified closed string theories on tori besides the ones expected from the Kaluza-Klein picture [ 1 ]. This is a "stringy" phenomenon which cannot be explained from a point particle point of view since the winding of closed strings around non- contractive loops of tor i is responsible for the appearance of gauge degrees of freedom. A key ingredient to understand the enhancement of symmetries is the construction due to Frenkel and Ka6 [2 ] of representations of (untwisted) affine KaY-Moody algebras by use of string vertex operators. In the construction, to remove annoying factors vertex operators are multi- plied by certain suitably chosen functions of mo- menta, which are called cocycle operators or gener- alized Klein factors. Frenkel and Ka6 have shown that cocycle factors can be constructed without introducing any more degrees of freedom. Cocycle factors for twisted affine KaY-Moody algebras have also been investigated by several authors [3 -5 ] . It has been shown that the construction of cocycle factors re- quires new degrees o f freedom, which are called fixed points by physicists. Our main purpose o f this paper is to reveal the origin o f the cocycle factors and to clarify their physical meaning from the point o f view of string theories.

In the case of string theories compactified on tori and /o r orbifolds [6] , the same situation arises. A

three string interaction will be represented in the form Cp :exp [2ip.X(z)]: in the language of the operator formalism, where p is the compactified momentum of the emitted state and Cp is the cocycle operator. The extra factor Cp will be necessary to ensure the correct commutat ion relations and the duality o f amplitudes [ 1 ]. To see this, let us consider the four-particle scattering amplitude for two different " t ime"- orderings depicted in fig. 1. We naively guess a vertex operator which describes the emission of a particle with the compactified left- and right-moving mo- menta PL, PR as being of the form ~

V(pL,PR;Z) = : exp[2iPL'XL(Z) - -2 ipR 'XR(g) ] : .

(1)

Here we have omitted contributions from uncom- pactified coordinates, which will not be concerned with the following argument. The product of two ver-

#1 Our conventions are the same as in ref. [ 5 ].

! !

PL,PR PL~Pn

I Z Z !

(a)

! !

PL'PR PL,PR

I Z ! Z

(b)

Fig. 1. The 4-particle amplitude for two different "time"-orderings. (a) Izl > Iz' I, (b) Iz ' l> Izl.

258 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Hol land Physics Publishing Division)


tex operators can be written in the normal ordered form

t t . ! V(pL, pR;Z) V(pL, p,~,Z )

= ( z - z ' ) P L & ( g - - r )P~',i~

×: exp [2iPL'XL (Z) -- 2ipR "XR (2)

+ 2 i p ' c ' X L ( Z ' ) -- 2ip~'XR(~' ) 1:

- -V (pL , PR,PL; PR, ' 'Z ,Z ' ) f o r l z l > l z ' l . (2)

Reversing the order of the vertex operators, we find

V ( p [ , p~; z' ) V(pL, PR; Z) = ( -- 1 )m_'P~.--PR'P~

× V ( p L , PR, PL; PR, ' 'Z ,Z ' ) f o r l z ' l > l z l • (3)

The different orderings of the two vertex operators correspond to the different "time"-orderings. To obtain the scattering amplitude, we must sum over all possible "time"-orderings for the emission of parti- cles. We must then establish that each contribution is independent of the order of the vertex operators to enlarge the regions of integrations over z variables [ 7 ]. In the above example, the two diagrams in fig. 1 must be summed over. The expressions (2) and (3) are identical except for the phase factor

( - 1 )PL'PL--PR'Pk

which appears in the right hand side of (3). This annoying factor is the reason for the necessity of a cocycle operator which will remove it.

As explained above, cocycle operators are attached to vertex operators in an ad hoc manner to obtain correct scattering amplitudes. Their physical mean- ings are however obscure. In this paper, we shall show that usually a s s u m e d commutation relations for zero modes should be modified and that the modified commutation relations have a simple geometrical meaning. It turns out that the effects of cocycle factors are automatically included in the modified commutation relations, that is, correct amplitudes can be obtained without introducing any ad hoc factors by hand. We shall also discuss the origin of cocycle factors in the second quantized picture.

Consider closed bosonic strings on a D-dimen- sional torus T °. Zero modes of interacting closed strings on the torus consist of x I, pi, QI and ( I = 1,...,D). The x I is the center o f mass coordinate of a string and p, is the center of mass momentum.

The L I denotes the degrees of freedom of the winding number which must be included in the spectrum of interacting closed strings on the torus [ 1 ] and QI is the canonical "coordinate" conjugate to L I. We pro- pose the following commutation relations ~2:

[ x l, pS] = i~iJ , ( 4a )

[a I , L J] =ialJ , (4b)

[x*, Q J] = -i½na IJ , (4c)

otherwise zero. (4d)

Note that the commutation relation (4c) is usually a s s u m e d to be zero. In the language of the left- and right-moving coordinates, the above commutation relations are rewritten as follows:

[x(_, p~_] =i½alJ= - [x~ , p~] , (5a)

1 J [XL, XL] =i¼nB u = [x~, x~] , (55)

I J [XL, XR] = i ~ n ( a ' - - B U) , (5c)

otherwise zero, (5d)

where

xI= x~ + x~ ,

I I I IJ J J Q = X L - - X R + B ( X L + X R ) ,

p I = p l __pl _F O*J(pJ + p ~ ) ,

L t = p [ + p ~ . (6)

Here, B *J is an antisymmetric constant background field which will appear in the first quantized action [9 ]. (To realize affine Kae-Moody algebras in vertex operator representations B *J must be appropri- ately chosen [ 5 ]. ) It should be emphasized that the

I I left- and right-moving coordinates XL, XR are usually a s s u m e d to mutually commute [ 1 ]. We now show that the above commutation relations have a simple physical meaning. Let Ix'*, LI=O) be a string state whose center of mass coordinate and winding number are given by x 'I and L I = 0, respectively. The operator exp(iL.Q) is a shift operator of the winding number by L t. It follows from (4c) that

[x I, exp(iL.Q) ] = ½nLI exp(iL.Q) , (7)

which implies that the state exp( iL.Q)Ix 'I, L I = 0 )

a2 Similar commutat ion relations have been proposed in ref. [ 8 ], with no physical explanation.

259


has the e igenvalues of x ' t+ ½ ztL t and LI for the center of mass coordinate and the winding number, respectively, i.e.

exp ( iL .Q) l x ' t , L l=O)=lx ' I+½rtLZ , L I) , (8)

up to a phase factor. At first sight it seems strange that the center of mass coordinate of a string changes by ½1tL I when the winding number changes by L I.

However, this is geometrically acceptable as explained below. We first note that the "center of mass coordinate" of a string on a torus is not a well-defined notion in the presence of non-zero winding numbers because the string winds around the toms. However, it will still be a useful notion on the covering space o f the toms. It turns out that on the covering space of the toms the "center of mass coordinate" of the string state (8) is just located at x'Z+ ½nL t as shown in fig. 2.

The above observation will be restated as follows: The string coordinate XI(z , a) at z=0 is expanded as

X I( z = O, a) = X'I'~ - L la+ (oscillators). (9)

It should be noticed that the "center of mass coordinate" of the string is x a in the absence of the winding number but

x1-= i d--a X1(z=0' a) =xtl-~ - ½7~L 1 , ( 1 0 ) 7~

0

in the presence of LI, that is, x I is the "center of mass coordinate" on the covering space of the torus but not x '1. Assuming that x '1 commutes with Q I and that the relation (4b) holds, we find

X ~ Z t I _J_ 7 r L I

P M ~1 l ; ~ t

X 1

Fig. 2. Two strings are depicted on the covering space ofa torus. The "center of mass coordinates" and the winding numbers of the two strings are (a) (x'~,0), (b) (x ' t+ ½nL ~, L~), respectively.

[x I, Q J] = - i½~8 IJ . ( 11 )

This is nothing but the commutatioh relation (4c) we proposed. In other words, the operator x J which appears in vertex operators is not x 'z itself in (9) but the combination of (10). This is quite natural because x I defined in (10) has a simple geometrical meaning in the presence of the winding number rather than x'*. Indeed, this is the case as we shall show in the second quantized picture.

We next examine the effects of modifying the commutation relations and show that vertex operators are equipped with cocycle factors without adding ad hoc factors by hand. The string coordinate XZ(z, tr) obeys the boundary condition

XI(r , a+ n) =X~( r, a) + n L I , (12)

where L % A = {E~=1 ni~, nieZ}. The ei are the basis vectors of the lattice A. Since the wave function 7'(x ~) must be periodic, i.e., ~ ( x l + n L I ) , the allowed ei- genvalues of the momentum pl are given by pIe2A* where A* = { E n I . , _ i= lm,~ , ei e ) - 8 o, mieZ}. Consider the following operator which is part of the vertex operator ( 1 ):

Vo (PL, PR) = exp (2ipL "XL -- 2ipR 'XR)

= e x p ( i p . x + i L . Q ) . (13)

It follows from the commutation relations (4) or ( 5 ) that

Vo(PL, PR) Vo(p~., P'R )

= ( - 1 )pL.pl. -PR'Vk Vo (p [ , p'R ) Vo (PL, PR ), ( 14 )

where we have used the fact that p.L~ 2Z. The factor

( -- 1 )PL'Vl.--VR'V~

in (14) is just what we wanted. This phase factor compensates the one that arises in the right hand side of (3). Therefore, we conclude that modifying the usually assumed commutation relations to (4) or (5) is equivalent to multiplying vertex operators by cocycle operators.

So far, we have considered strings on tori. The gen- eralization to orbifolds is straightforward. An orbifold is obtained by modding out of a toms by discrete rotations which are the symmetry of the lattice defin- ing the toms. In other words, the identification of the

260


point {X I} to the one { U U x J + n L ~} gives an orbifold, where U u is a rotation matrix which transforms the lattice points into themselves. Then strings on such an orbifold obey the following boundary conditions:

X I ( z , a + ~) = UIJXJ(z , a ) + n L ~ . ( 15 )

The case of UIJ=61J is called the untwisted sector whose boundary condition is the same as the toms. The non-trivial case ( U U # 6tJ) is called the twisted sector. In the mode expansions o f the twisted string there only appears the center o f mass coordinate x ~ as zero mode:

X I ( z, a) = x I + (o sc i l l a to r s ) , ( 16 )

where x I satisfies the fixed point equation

XI= UIJxJ'] - ItL 1 . ( 17 )

I n the twisted sector, x I is just the "center o f mass coordinate" because

Xl= i dtTxl('c' o') . ( 1 8 ) 7[

o

It follows from ( 17 ) that the quantization conditions in the twisted sector are

Ix I, X J ] = O, [ QI, QJ] = 0 ,

[M, Q J¼ = - i n ( 1 - U ) -~u , (19)

or equivalently,

[x / , x J ] = i¼ rr [ - (61K--B l/c) ( 1 -- U ) - 1/cJ

+ (~JK--BJ/C) ( 1 -- U ) - I/C/¼,

1 J [Xg, XR] = i l n [ + (8I/C+BI/C) ( 1 -- U) - l r J

- (6J /c+B J/c) ( 1 - U ) - l/cI] ,

[x / , x J ] =i¼7r [ ((~I/C--BI/c) ( 1 -- g ) - I K J

+ (6s/C+B s/c) ( 1 - U) -IKI] , (20)

where we have used the relations (6) . The above commutat ion relations have been derived in ref. [ 5 ]. The authors have shown that cocycle factors in vertex operator representations of twisted affine KaY- Moody algebras given by Ka6-Peterson [ 3 ] are represented in the commutat ion relations (19) or (20).

We finally discuss the origin o f the cocycle factors in the second quantized picture. The three string ver-

tex in the coordinate representation will be given by [10,11]

H(~(XI(a3)-XI(tT1)OI-XI(a2)02). (21) t~

The X~(ar) ( r = 1,2,3) is the coordinate at z = 0 of the rth string depicted in fig. 3.

i ar X~(ar) = x ~ + L r ] - ~ + (oscil lators) , (22)

where

a l = a for 0~<a~<oq 7r,

0"2 = 0"-- a l 7~ f o r Otl/r ~< o'~< (OQ + a 2 ) 7~ ,

0"3 = ( O/1 "Jr- Og2 ) 7~-- t7 f o r 0~< tT~< ( a l + O~2)/ t ,

O I = 0 ( O t l / ~ - - 6 ) ,

O2 = 0 ( 0 " - - a l 7~) ,

cq + a 2 +or3 = 0 . (23)

When referring to fig. 3, we imagine that oq, or2> 0 and a 3 < 0 . The delta-functional (21) are defined through their Fourier modes. The zero mode part o f the vertex will yield the following three conditions:

(t~t +t~2)n

da [ X ~ ( a 3 ) - X t l ( a l ) O t - X ~ ( ~ 2 ) O 2 1 = 0 , 0

(24a)

X { ( a I ~ ) = X ~ ( O ) , (24b)

3 [x~( I a r I n ) - X ~ ( 0 ) ] = 0 . (24c)

r = l

The first condition means that the "center of mass coordinate" of two strings 1 and 2 together coincides with that of string 3. The second condition means that at z = 0 there is no gap between strings 1 and 2. The third condition turns out to be the winding number conservation. (Precisely speaking, the equalities in the first two conditions of (24) should be under-

2 1 o~2~ 0 ~ I "a" 0

0 (c~1 + c~2)~r 3

Fig. 3. Strings 1 and 2join at r=0 to form string 3.

261


stood up to the torus identification.) The zero mode part o f the three string vertex is now written as

Vo(Xtr, Lr)

= ~ ( x ~ l + J I~L 3 -fll(X'll +½~Ll)-fl2(x'21+ t~L2)' )

× ~(x'[ + rtL~ -x'2t)OL, + L 2 + L 3 , 0 , (25)

where fir= ar/(al + a2) and we have omitted the os- cillator modes whose inclusion will not change our conclusions. Fourier transforming (25) into the momentum representation, we have

f dx~ dx~ dx~ exp(ip~ .x~ +ip2 .x~ +iP3 .x~)

× Vo(x'r, Lr)

= ~LI + L2+L3,0 ~pl +p2+p3,0 exp [ i•P2 "LI

+iztp3"[(fl2+½fl,)L~ +½fl2L2-½L3]}, (26)

up to a normalization factor. To make contact with on-shell vertex operators in the operator formalism, we take the limit o f ot l--'0 [ 11 ]:

~LI +L2+L3,0 ¢~pl+p2+p3,0 exp [i½nPt "L3

- i½n(p l 'L~ +p2"L2 +p3"L3) ] , (27)

where we used the momentum and the winding number conservations. The second term in the exponent o f (27) will, be absorbed in the field redefinition. Fi- nally, making the correspondences

fl--,p I, LI3~L ,, (28)

where pl is the momentum of the emitted on-shell particle, we have the factor

exp( i½nP'L) . (29)

This factor implies that the following combinat ion o f the operators

exp [ip. ( x ' + ½nL) ] , (30)

appears in the vertex operators rather than exp ( ip .x ' ) , as announced before.

Here we should make two remarks: The first is concerning the limit of al--,0. It will be problematic in the presence o f non-zero winding numbers since strings which wind around tori cannot reduce to a point. Thus much attention should be paid in taking

the limit of a~- ,0 [ 12,13 ]. Nevertheless, the appearance o f the factor (29) strongly suggests the correct- ness of our discussion even in the presence of non- zero winding numbers. The second is a comment on other works of string field theories on tori [ 14 ] or orbifolds [ 15 ]. In their formulations, cocycle factors are required to ensure the Jacobi identity for a three string vertex. The left- and fight-moving string coordinates are assumed to be separated from the begin- ning so that the geometrical interpretation is less clear and in particular it will be hard to derive the commutat ion relation obtained here.

I am greatly indebted to M. Maeno and H. Takano for many observations about string field theories.

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a physical interpretation of cocycle factors in vertex operator representations

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