Conformal and current algebras on a general Riemann surface

Nuclear Physics B282 (1987) 308-328 North-Holland, Amsterdam

CONFORMAL AND CURRENT ALGEBRAS ON A GENERAL RIEMANN SURFACE Tohru EGUCHI and Hirosi OOGURI

Department of Physics, Universityof Tokyo, Hongo, Bunkyo-ku, Tokyo 113,Japan Received 19 August 1986

Starting from a path integral formulation, Ward identities are derived for conformal and current algebras on a general Riemann surface. An n + 1-point amplitude with energy-momentum tensor insertion is related by the Ward identity to an n-point amplitude and its derivative with respect to the modular parameters. Similarly, an n + 1-point function with current insertion is related to an n-point function and its derivative with respect to an harmonic background gauge field. We discuss how conformal and current algebras, defined in each coordinate patch, are glued together to form a global algebraic structure on the Riemann surface.

1. Introduction

Recently superstring theory has been receiving much attention as a possible candidate for the unification of gauge and gravitational interactions. While the fundamental principle underlying string theories has yet to be discovered, recent investigations suggest that one of their essential ingredients is the conformal invariance on the two-dimensional world sheet. String theories are constructed in such a way that the invariance under diffeomorphism and Weyl scaling of the metric remains after quantization and the theory depends only on the complex structure of the world sheet [1, 2]. Thus, string theory can be regarded as a quantum field theory defined on Riemann surfaces, i.e. conformally equivalent classes of two-dimensional manifolds. The space of Riemann surfaces, the moduli space, on the other hand, has also a complex structure when suitable coordinates are used. This fact may possibly be utilized to determine multiloop vacuum amplitudes of the closed string theory [3]. Thus the complex analytic geometry of Riemann surfaces and the moduli space appear to play a powerful role in the analysis of string theories. Conformal field theory is also interesting as a model of critical phenomena in statistical physics. Field operators appearing in such a model form representations of the Virasoro algebra and their correlation functions obey linear differential equations derived from algebraic relations among degenerate conformal fields. In order to enumerate the possible content of field operators, Cardy, Itzykson and Zuber [4] have proposed considering a model on the torus and imposing modular 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

T. Eguchi, H. Ooguri / Conformal and current algebras

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invariance on its partition function expressed in terms of characters of the Virasoro algebra. Here, in contrast to the case of string theories, mainly algebraic structures of the conformal symmetry have been exploited so far. In this paper we try to combine both the analytic and algebraic approach in order to gain more information on the structure of conformal field theory. In particular, we would like to discuss in some detail the conformal and current algebras on the general Riemann surface. In sect. 2 we derive the Ward identity for insertion of the energy-momentum tensor into the correlation functions of conformal fields by taking infinitesimal variations of the two-dimensional metric. Because of conformal invariance, metrical variation preserving the complex structure of the surface simply generates Lie derivatives of conformal fields, while the effect of metrical variation deforming the complex structure is expressed by the derivative of the correlation function with respect to the moduli of Riemann surfaces. Thus the Ward identity relates an n + 1-point amplitude with an energy-momentum tensor insertion to an n-point amplitude and its derivative with respect to the modular parameters. The operator product expansion among the energy-momentum tensor and conformal field is readily derived from this result. In sect. 3 it is shown that the Virasoro algebra, originally defined in each coordinate patch, may be consistently glued together to form a global structure on the Riemann surface. In particular, the central charge c, scaling dimension h and h and the Kac determinant are invariant quantities of the algebra independent of the choice of local coordinates. The current algebra on a Riemann surface may be discussed in the same spirit. In this case the vacuum amplitude with non-zero harmonic background fields is the basic ingredient, which reduces to the character of an affine Kac-Moody algebra in the case of a torus. In sect. 4 we derive the Ward identity for insertion of currents into correlation functions and show that correlation functions of products of currents are obtained inductively from the vacuum amplitude. During the course of this work we have received an interesting paper by Sonoda [5], where a somewhat different form of Ward identity is obtained for conformal field theory. Toward the completion of this paper we also received a paper by Friedan and Shenker [6], where the idea of the energy-momentum tensor being a connection on the moduli space is discussed. It seems that our work provides a concrete base for their rather abstract arguments.

2. Ward identity for energy-momentum tensor insertion Let us consider a conformal field theory coupled to a zweibein field e~ (/~: world index, a: local Lorentz index) on a surface. We assume that the theory possesses three types of local symmetry; reparametrization, local Lorentz rotation and Weyl scaling. In terms of the expectation values of a product of conformal fields they are

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310

expressed as (1) reparametrization

= 1-

E $p.(Xk) 0

(1)

OI(X1)...ON(XN))e:,

k=l

(2) local Lorentz rotation

(,I(Xl)...ON(xN))e:+~,°~e~= 1+i E s~O(xk) ol(x,)...o,,,(xN))e:, (2) (3) Weyl scaling

(OI(x1)...ON(XN))(I_8o)e~,a=

(N 1-

Z dk3°(xk) k~l

C

)

(OI(XI)'''ON(XN))e"a,

2

Here (0~(Xl)...0N(xN))~: and Z~, mean an expectation value and partition function in the presence of the zweibein e~ on the surface. It is to be noted that in eqs. (1)-(3), the field 0 ( x ) is treated as a world-sheet scalar and a tensor under local Lorentz transformation, s and d are the spin and the anomalous dimension of the field and R is the scalar curvature on the surface, c is a numerical factor which becomes the central charge of the Virasoro algebra. Next, by using (1)-(3) and also the relation

le..(P~W

- ~ a~e°+a.~ e o -

"

"'vA o

(4)

where P is the standard differential operator taking a vector into a traceless symmetric tensor [2] ( P ~ ) "p. = G"uooV"~ ° .

--

u

p.

pp.

O""oo - ~;~o + 6o6o - g""g,o.

(5)

and % is the spin connection on the surface, we obtain

~f

d~=~ e:( Pe)'p.(T;(x)O,... ON) ~" -- k-lE

+~

C

+iSk[.8OolTo.,.o+(gq,.~,~ 2

(01'''0N)

fd x¢~ R~vAox <01.--0N>-

(6)

T. Eguchi, H. Ooguri / Conformal and current algebras

311

If we switch to complex coordinates, ( P ~ ) ~ = 2Vz~ ~ and the above equation is rewritten as

f d:zv~{V'U+ v~(r~(~),,(w,, (T=(z)q'x(w,,~,)"' q'u(wu,~'N))} ~,)...,,,(w~,, ~)) N

k-1

{

"'

hkWw,~W'+ ~ ( O~, + iskww,)

+hkV~,~ +~'(a~,+iSk°~,)

) (~'(W"W')"'dPu(wU'wN))

c fdZzv/~($'a,R+~'a~R)x(ep,(w,,~,t)...epN(wN,

48~r

Wu)>.

(7)

Here h = ½(d + s) and h = ½ ( d - s) are the scaling dimensions of the field. We use the notation V~~) = 0N + i s ~ hereafter. Let us next introduce a Green kernel for the operator P, 3g-3

1- ~'~>(z- w)- E g~,7"~,j(z,e)h~/(w). Cg j=l

V'G'~(z,w)=

(8)

Here h ~ / are holomorphic quadratic differential on the surface, v~h.z/= 0 and rise,, are Beltrami differential dual to h~/, f d2zv~ gZr~.Th./= 8/. Eq. (8) applies to the case of surfaces with genus g >/2. In the case of a sphere (g = 0) and torus ( g = 1) there exist zero and one quadratic differential, respectively. The Green function has the singularity 1

1

G~**(z, w) = 2rr z - w '

z = w.

(9)

We note that G~J(z, w) has a non-analytic piece in it due to zero modes. We next substitute ~ = Gw~~ into (7) and obtain

- Eh.~. f d vqg ~ ,,, •

.

j

2z

z3

z

t N

= -- E {hkl~TwGW%~(Wk, W) + G k-1 C

48~"

~,

ww(Wk,

w)v'¢'}(~,...~N> (lo)

and a corresponding equation where z, w are replaced by E, and ~. The second term

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312

in the l.h.s, of the above equation corresponds to a metrical variation deforming the complex structure of the manifold, 8g~ = 2E¢,-n~8v~ where yJ's are the jozz'l z,J .," Teichmiiller (modular) parameters. By defining the TeichmiJller deformation of the correlation function as 1

(11) J

( Z is the partition function) the l.h.s, of (10) is rewritten as 1

8

(Tw~q'l... q'u) - Ehw~./-~ ayJ ((¢'1"" ~'u)Z)"

(12)

J

By taking N = 0 in (10) we find • 0

c

(T~) = ~_,h~J-ffTyjlog Z - -4-~ f d2wC~ Gw~( w, z ) 8,~R

(13)

J

and thus we obtain the Ward identity for a single insertion of the energy-momentum tensor (r,,¢l...

- (7",,)(,1

...

N k-I

0

+ Y~.hz~.J-~yj(Ol...ON).

(14)

J

Eq. (14) shows that the effect of the energy-momentum insertion induces diffeomorphism and Weyl scaling (lst term on the r.h.s.) and also deformation of the conformal structure (2nd term). The curvature term in (13) generates the center of Virasoro algebra in the Ward identity for a double insertion of the energy-momentum tensor, as we see in the following. This term, however, may be dropped in the end since we usually consider a constant curvature metric on the Riemann surface. Later we will discuss the Teichmiiller term, ~.jhz.,J(a/SyJ)(ep 1"''q~N), in more detail. A double insertion formula may be obtained by taking the variation of (6) with respect to the metric. Equations necessary for the calculation are given by 1 6

1

~~

po

vx'~

qr~ 8(¢gRgrJiP) = - Vx~X½Gp,,+'~gr~,Vflgp° + ½(grx~xA - R~XVx)gpodg p°. (15)

T. Eguchi, H. Oogun / Conformaland current algebras

313

We can then derive

f d2vt/g 2~v~°(TooT~.dp,... ~bN) = - 2(2V.~ ~ + ~*V'w)(Twwq't -.- q'u) N (sD - 2 Y'~ (hkw, k~w*+ ~w~ g:':~ )(T,,,~b 1 ... dpN) k~l C

2

+ 2-~7 f d ~vqR(z)v~z × ( r ~ l . . . , ~ > Again by substituting/~v =

¢

-

24¢r -

C

3

w

a-G(v~) ~ ( < . . . ~ > .

06)

GO we obtain,

( v ~ ) 3 o w= ( w , z ) < < . . . ~ , , )

(2V, C~=(w,

z) + GW~(w, z)v~}(rw~¢,... ~'N)

N

- E {h,v~,c = ( w , , z ) + a

~(w,,

k-1 h

j

0 (17)

J

It is an instructive exercise to check these identities (14), (17) in the case of free field theory on the torus. The solution to (8) is given by the elliptic theta-function

G~w(z, w)

=

1 # ~ ( z - wit) I m ( z - w) 2¢r 01(z - wiT) + i Im¢

(18)

Periods corresponding to two non-contractible cycles are taken to be 1 and r. Note that GZww has a non-analytic piece in (18). In order to make our formulas transparent let us use ~'-function and Weierstrass .~-function which are related to the theta-function as ~'(z) = - -

O,(zlT)

+ 2'/1" z,

n e 2~rinr )

,7,=~(~1=(2~) ~ ~(z) = -~"(z).

~- F.

~--e-~.,.. , (19)

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T. Eguchi, 1t. Ooguri / Conformal and current algebras

We also use standard notations of classical elliptic function theory, ~(~)=el'

~

(l+r) T

=e2'

g2=2(e~+e2+e~),

(2)

~@

=e3'

g3=nele2e3,

.~,(z) = ( ~ ( z ) - ev) 1/2 val O~+l(zlr) - v%+-----~ v~i(zlr)

'

J'= 1,2,3,

(20)

where theta constants are defined by val = (d/dz)Ol(zlz)fz=o, v~ = v%(0lr). ~ and ~ functions have Laurent expansion 1 = -g +

~,(z) =

1 _

_

og2z 2 + . . . ,

~~e v z +

. . . .

(21)

Z

We also recall formulas for ,~ and v% and relationship between rh, g2 and 0 i, ( ~ ' ( z ) ) 2 = 4 ( ~ ( z ) - e , ) ( ~ ( z ) - e2)(~@(z ) - e3), a~Ov(zD- ) = 47ri O,v%(zl~-) ,

gz=48(~ri a,~ll + (~1,)2), ~t=

1 O(" 6 va---]- = -

O,01 z~ri v~

(22)

On the other hand, by varying ~- in the standard euclidean metric on the torus ds z = l d z l 2, z = o l+Toz, 0 < o 1,o 2 < 1 we find 3gzz=3~/2iIm~', 3gee= 3 T / - 2i Im z. Beltrami and quadratic differentials are given by i

~f~=

2Im~

-,

hz~ = - i .

(23)

Note that rff~= 3r(Im z / I m r ) and Im z/Im z is just the angular variable in the y-direction. We can then explicitly evaluate the Teichmiiller term in (10)

=~dz+2if d2z Im im--~z O~(T~fi~l .. q~u).

(24)

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315

The first term comes from the discontinuity of Im z/Im r and the integral is along a cycle in the o~ direction. Using the correspondence with the operator formalism we find d z (Tzzdpl...

*N)

=

-- ( L o * I

" ' " dPN) =

iO - 2-,, o, ( ¢ ' * N )

--

-~q-~qtr(qt'°eOz "'" *u)

+~

i ( O ) ~ l o g z (,,... ¢,,), q = e 2'~''.

(25)

(The relative minus sign between T~ and the Virasoro operator L 0 is due to the change of variable z + e 2"~ mapping the complex plane onto the toms.) On the other hand, the second term of (24) picks up a contribution from z = w k (k = 1 . . . . . N) where analyticity of T., breaks down. Using (10) it can be calculated as Im z

- 2if

d2z - -

Imr

N{

N

a, E {h,O~p

w k

k-a

_,

= i k-1 ~_, hk -2--i--~m - ~+

=(wk,~)+a z~(w, Z)Ow~}{<...,~,> w k

wk I lm--~ aw* ( ~ z . - . q ~ u ) ,

Im

(26)

where we have used

20~GWzz(w,z)=- 8 ( 2 ) ( z - w ) -

1 ) Imr

(27)

and translational invariance. Collecting formulas (10), (18), (19), (25), (26) we find the Ward identity in the case of a toms ( T ~ z . . - ~N) - (T~z)(q~, ... ~N) 1

N

= 2--7 ~-" {hk(~(Z--Wk)+ 2~I)+(~(Z--Wk)+

271'Wk) Owk}(~l"''~U)

k-1

0 + i -0-Tr( ~ .-. ~N ),

(28)

(T::)

0 = i-b-~¢log Z.

(29)

Similarly, the Ward identity for a double insertion of the energy-momentum tensor

316

T. Eguchi, H. Ooguri / Conformal and current algebras

is given by

cr.,:r,~w~,,... ,,,,) - (r,+)(Lw,¢,,...

,,,,)

C

1

- 487rz-~"(z, - w)(qh...#'N) + 2Ir X { 2 ( ~ ( z - w) + 2r/t) + (~'(z - w) + 2r/,w) Ow } (T~,.4,t... *N) 1

N

+~--~ E {hk(d~(Z--Wk)+ 2~t)+(~(Z--Wk)+2"O,Wk)Ow,} k-1

x (LJ#,

0 -.- *,v) + i-ff~ ( L w ¢ , . - .

¢N).

(30)

Note that the non-analytic piece of the Green function has been cancelled in (28) and (30). In the case of a free scalor field q~(z), its partition function and the 2-point function are given by

Z(r) =

--, (Im ~') t/2rt ( ~')r/('r)

1

( ¢ ( z , e ) ¢ ( w , ~,)) = - z - l o g

01(z

wl'r )

,7(+')

(31) 2

1 (Im(z - w))2 + , 2 Imr

(32)

where ~(~-)= ((1/2¢r)0~) 1/3= ql/2q-l~_l(1- qn) is the Dedekind ~-function. On the other hand, the expectation value of the energy-momentum tensor is given by ( (T,,) = tim z

-

afl~(z, Y.) a,+~(w, ~))

[1( ~ ( z - w )

= 1jmz ~

1 2~r 7t

,),

1

1

4~" (z- w) 2

(z-w) 2 +~

rh

]

,

4Imr

1 4ImT "

(33)

We can check the identity (29) by using the relation (22). We may also check the

T. Eguchi, H. Ooguri / Conformal and current algebras

317

Ward identity (30) for N = 0. The 1.h.s. of (30) is given by (T,,Tw,,,) - (T,,)(Tw,,,)

= 2(( O,q,(z,,~)aw~(w, ~)))z

1(

= 8~r2 ~ ( z - w ) + 2 * / 1

Im,

(34)

and this agrees with the r.h.s., with c = 1 due to

(~,(z)) 2 = g ., ~, ~", , , I z ) + ~ g 2

(35)

and the relation (22). In the case of a free spinor field ~(z), there are three types of two-point functions corresponding to different spin structures on the torus

(4,(z)~/(w))=-~i~v(z-w), 1

v = 1,2,3.

(36)

Here v = 1, 2, 3 corresponds to (P, AP), (AP, AP), (AP, P) boundary conditions in the o 1 and e 2 direction, respectively (P and AP means periodic and anti-periodic). Partition function and expectation values of the energy-momentum tensor are given by

z( • ) = ( '~'+ 1(, ) ~/~: ' ,1(,) /

(T~z)=lim[(~N/(z)Ow6(w))

1

w--,z

1

]

4~r z - w

= ~((e,:)~ - e,:,~,).

1 8rr ev '

(37)

Since the spin structure is left unchanged under infinitesimal diffeomorphism our Ward identities should hold for each of them separately. We can check (29) and (30) with N = 0 for each v, for instance, if we use identities

0

e,=-4~ri-~Tl°g 2

0,+1(~-) ~-(~-~)

,

v = 1 , 2 , 3.

(38)

61g2 27he , - ui O.e, = O.

(The second set identities in (38) may be proved by converting them into identities involving g2 and g3 and checking the property under modular transformation.) In the case of a torus the non-analytic piece in the Green function has been cancelled by factors coming from Teichmiiller deformation. Now we point out that

T. Eguchi, H. Ooguri / Conformal and current algebras

318

this is also the case for a general surface with g >/2. Let us write the Beltrami differential as ~*~.~= 0~( 7. If ~ is a single-valued function, ~zZ . I . becomes a total derivative and does not generate a deformation of complex structure. So we consider ~ which is multiple-valued on the Riemann surface. By introducing a Green function which obeys 1 (39) v zG-~ww(z,w)= - ~ 8(2)(z - w) we have

U = f d 2 w ~ G"~ww(Z, w)g ~

*lw~,,i,

3g-3

( G - G ) zw~(z,w)= -

~_, ~z,(z)hww ,i ( W ).

(40)

i--1

The Teichmiiller term is now written as

Z h , / f d2wvq g~%W~,,(r,.Ox... O,,) i

= fd~w¢7 (G-a)~=(w,z)g~a~(T~.,Ox...,u) 3g-3 +

d w 8~ w (T~w0,...0N). i'-'-I ' / c -

(41)

ycles

In the second term integration is over those cycles across which ~ has a discontinuity 8~ ~. On the other hand, the first term picks up a contribution from z = w k (k = 1 . . . . . N) where analyticity of T= breaks down. It can be evaluated using 1 (42) v a"*~ ( z , w ) = ¢ ~ 8(2)(z - w) and the Ward identity itself. Its net effect is to replace G by G everywhere in the Ward identity. Thus eqs. (14), (17) are rewritten as N

< ~ o , . . . o ~ > = - E { h~v~F-w,**(wk,z) + a-~, z,(wk, z),,,w, '-('*)

}(0,. .. O~)

k-1

3g-3

+ ~ i- 1 g

(T,~TwwO~... ON)

dw 8~w( T ~ 0 a . . . 0N),

h~.~

(43)

ycles

24---~(Vw)3G~(w, z) ~W

~W

- ( 2 v ~ a = + a =v~ } ( r w ~ o , . . . o~> N '~" (

hkw~G~w,~ + ~rw~ --

Iu,(s,)

gZ " w k

} (T~wO1 On) •

• •

k-1 3g-3

+ E h=,'~c i= 1

dvr$~(TooLwO,..-ON). ycles

(44)

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T. Eguchi, H. Ooguri / Conformaland current algebras

In these formulas the analyticity of T , is clear; however, its single-valuedness is somewhat obscured.

3. Virasoroalgebra on theRiemann surface Let us now go on to discuss Virasoro algebra on the general Riemann surface. The short-distance expansion in the energy-momentum tensor and conformal field can be read off from the Ward identity

1

(

(21r)2-~c

1_ (z

1

w) 4 + 3 - - + ( z - w2 )

1(2

+~

( z - w) 2 + z 1 ( ~W

t~w-- OwF~w~- ~ 1

lOwt~}

tw~

6 z---w

1}

(45)

w o,~ T,~w(w),

x2

(46)

w~) ,

r . ( z ) , ( w , ~) = - ( hypos. + O',zV~" ),(w, ~) --

-

(z

-

w) 2

+

~ + ~-w z-w

(Ow+isw~

ep(w,~).

(47)

Here / ' ~ is the usual Christoffel connection FZz~ = O~log g~. We note that in the term linear in Tww (eq. (45)), connections have cancelled out. Eq. (45) may be brought into the standard form of Belavin-Polyakov-Zamolodchikov [7] if we redefine the energy momentum tensor as ¢

L~ = T~ + ~

t~.

(48)

1 Since O~t~ = - ~g,~a,R, t~ is analytic for a constant-curvature metric and the redefinition of T~ does not disturb its analyticity property. Under a holomorphic change of coordinate, z ~ z' = f ( z ) , it changes as

Q ' r d z ' 2 = Q~ d z 2 - ( f, z} dz 2,

(49)

where ( f , z } is the schwarzian derivative

(f,z}=

dz3/dz

2 dz~ldz /

(5o)

320

T. Eguchi, H. Ooguri / Conforrnal and current algebras

Thus 7~ is not a tensor but generates a schwarzian under a holomorphic coordinate change. In terms of 1~, we have

lc/2

1 ( 2

L * ( z ) L * ( w ) = (2~r) 2 ( z - w ) '

+

~

1 + - (z-w) 2 z-w

) a~ 7~ww(w) (51)

which is the same as in the case of planar geometry [7]. (In the case of natural metrics on the Riemann surface d s 2 = I d z l 2 / ( l + [z12) 2 ( g - - 0 ) , Idsl 2 = Idzl 2 ( g = 1), ds 2= Idzj2/(Imz) 2 ( g > 2), (Poincar6 metric) tzz vanishes. An atlas of local coordinates on a Riemann surface can be chosen in such a way that coordinate transformation between neighbouring patches is an isometry of the natural metric. With these metrics, one may ignore the distinction between T~z and 7~z~since the schwarzian vanishes for coordinate transformations which are isometrics of these metrics ((sub)-group of SL(2, R)).) Up to now we have regarded the conformal field ~(x) as a world scalar and a tensor under local Lorentz transformation. We now switch to a description where ¢ ( z ) is considered as a world tensor and a scalar under local Lorentz rotation. By introducing complex notation for the local Lorentz frame, g,~ = e~e~ where A = 1 + i2 (~,T= 1 - i2) (g~, = g~ = 0 implies e~ = eft = 0), we define -

A

h

(e , ) . ~-

7,

(52)

q~(z, ~) is a world tensor ~'(z, z.')dz'hdz, d' = ~(z, 2 ) d z h d ~ ~ and is a scalar under local Lorentz transformation. If we rewrite (47) in terms of • we find

1}

(z-w) 2 +-°'z-w

(53)

Again the connections disappear and we have the same form of short-distance expansion as in the planar geometry [7]. It is well-known that when we define Virasoro operators by 1

"r(z) = ~ E L,(w)(z - w ) - , - 2

(54)

short-distance expansions (51) and (53) are equivalent to the Virasoro algebra [ L , ( w ) , Lm(w)] = ( n - m)L,+m(w ) + ~c(n 3 - n)3,+,,.o,

(55)

and the heightest-weight state condition

L,(w)~(w)=O,

(n ~ 1 )

L_~(w)~(w) = a,~(w).

(56)

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321

Since our original short-distance expansion was manifestly covariant under a holomorphic coordinate transformation, the above result implies that the Virasoro algebra is in fact well-defined globally on the Riemann surface. In order to see this more explicitly let us next look at the transformation property of the Virasoro operators. Under a holomorphic coordinate change, z ~ z' = f ( z ) , (54) transforms as

E L . ( w ) ( z - w) -"-z = E L . ( f ( w ) ) ( f ( z ) - f ( w ) ) - " - z × dz ] n

+ he(f,

z}.

(57)

Hence

L.(w) =

.1_ -- d z

w).+t(

/,z}

+ ( f ' ( w ) ) - " L , ( f ( w ) ) + ½(1 - n ) f " ( w ) ( f ' ( w ) ) - " - 2 L , + l ( f ( w ) ) + (61(2 ×(f

t

n ) f ' ( w ) f " ( w ) + ~(n 2 + 9n + 2 8 ) / " ( w ) 2)

(w))

-.-4

L,+2(f(w))+....

(58)

These two sets of operators satisfy the same Virasoro algebra. On the other hand, we note that L,(w) transforms into a sum of L,,(f(w)) with m > n and the contribution from the schwarzian in (58) is non-zero only for L,, n < - 2 . Hence the highest-weight condition (56) is also unchanged by the coordinate transformation. Therefore the Virasoro algebra, defined in each coordinate patch, is glued together consistently to form a global algebraic structure on the Riemann surface. Invariants of the algebra are c, h, and h and consequently the Kac determinant. Then the standard representation theory of the degenerate representation of Virasoro algebra [8] can be readily applied to our case. Let us now derive a linear differential equation for the correlation functions which is a generalization of the one given by Belavin-Polyakov-Zamolodchikov [7] to the case of the general Riemann surfaces. When c < 1 or > 25, there appear null conformal fields X(Z) among the descendants {L_,~(z)L_,2(z)... L_~k(z)~b(z); n, > 1 ) of a primary conformal field ~ ( z ) . Null conformal fields create states with zero norm and are constrained to vanish in a theory with positive-definite Hilbert space. Inserting X(Z) into a correlation function and by seting it to zero we obtain a differential equation. In the case of the Ising model ( c = ~) for example, we get X(Z)=(L_2(z) 3L_l(Z)2)fl)h_I/2(Z ) as a null field. (fI)h=l/2(Z) is a primary field with h = ½ and

T. Eguchi, H. Ooguri / Conformal and current algebrcL~

322

is actually a free spinor field.) On the other hand, if we consider the case of a torus, we have

(L_~(wD,,(~,),~(w~)... a,,,(w,,)) dz

= ~ T (~-

w,)-'(r.(~)*,(w,)*~(w~)...*~(w~))

= (2h,~, + 2~,w, &,)(*,(w,)*~(w~)...*~(w~)) N

+ ~ {hk(~(wl--wk)+2"l)+(;(wl--wk)+2,tWk)Ow.} k=2

x (,,( w,)*2( w 2)... 2~ri 8 + T ~[(*'(w')*2(

*~( w N ) )

w2) ,,,(w,,))z],

(59)

by making use of (28). Then we obtain

(I aw~, -,,-

2,7,w, &,)(,,(w,),2(w~)... ,,~(w,~))

N

-

E {h,(d~(w,-w,)+2"ll+(~(w,-w,l+2"lW,)Ow,} k-2

x <,,( w,),~( w2)... *,,,(wN)) -

2~i 8 Z 8'r

[(*,(w,)dP2(w2)...~,(w~))Z ]

(60)

Let us check this identity when N = 2 and (/}2 is also a free spinor field h = I. The l.h.s, of (60) is

{1 8 ~ - . , - ~ ( ~ ( . ) + 2.,) + (~(~)- 2.,2) ~.}~(~) = (2~ri 8. -

¼e.)..@.(z),

(61)

where z = w t - w 2 and relations (19). (20). (22) have been used. v = 1,2,3 again stands for different spin structures. Eq. (61) agrees with the r.h.s, of eq. (60) 2~ri 8 Z 8'r [.@,,(z)Z], where Z is given by (37).

(62)

T. Eguchi, H. Ooguri / Conformal and current algebras

323

4. Ward identity for current insertion It is straightforward to extend our analysis to the case of current algebra. We consider a conformal field theory coupled to a background gauge field A~ (a is a group index). We postulate that the theory possesses the following local symmetries:

(4) vector gauge invariance

N

- i Z (t~t.j + tg,j)Xa(xj)(qbt(Xt)''' ~N(XN))A~ ' (63) j=l

(5) axial-vector gauge invariance

N

- i ~_, (t~,]--tg,j)Xa(X])(qbt(Xl)...~N(XN))A, j-1

k ,fdxv

4~r

•

2

, F, X(x)× Itv

a

(II~I...(1)N)A

.

(64)

Here D is the covariant derivative (Dh)~a= ~7~ka+fabcAab~ c and fabc is the structure constant of the gauge group G. t~ and t~_ are the representation matrices for right and left component of the conformal field. F~ is the field strength of the background field and k is an integer which becomes the center of current algebra. Since A~ is coupled to the current J~, from (63), (64) we obtain

(DX)2(J

I ..

N

= -- E t ~ t . j ~ a ( w j ) ( ~ l ( W l ) j--1

"'" ¢~N(WN)~

k i f d2zv/g g~Ira~rh"(z) X (~1..-~N) 4~r

(65)

and a corresponding equation for the left-handed counterpart. By varying (65) with

324

T. Eguchi, H. Ooguri / Conformal and current algebras

respect to A ~, we obtain

f d:zv~ (Dx)~ - ~/~o~XA w ) N

E

tb, jXb(wk)( J~ati~l(Wl)''" dlIN(WN)>

j-1

k 4,r ( D X ) ~ ( * I

4~r j v ~ 5

...

*u>

g,~r~ X (z)(J~*,.

..

ON>.

(66)

By putting A~ = 0 (65) and (66) become

f d2zfg g~ 3 ~ < J ~ 1 . . .

N

~>

R,

""

(67)

i-I

N

= if~b~Xb(W)-- ~ t~,,)lb(w,) i=1

k

4 awX~(w)<*l ...

*N>

(68)

Next we introduce a Green function G~(w, z) which satisfies 1

g

O~-G=(w, z) = gw,-~8(Z)(w - z) + i E ~,.gl°,.j •

(69)

j=l

Here % ( j - - 1 , 2 .... , g) are a set of abelian differentials (holomorphic 1-forms) normalized as ~a~j = 8~j, ~b~j = ~ij. a, and bi (i = 1, 2 . . . . . g) are canonical basis of HI(M ) and 9~j is the period matrix. ~'y ( j = 1,2 . . . . . g) are a different set of abelian differentials normalized as Re ~a~'j = 0, Re ~b,~) = 8ij. Using

fMo^#= Y'. •

j-I

o #-

g o

(70)

j

for any closed 1-form 0, 0, we find fdEzfggZ~oZ..y~.k = iSjk. Hence, the r.h.s, of

325

T. Eguchi, H. Ooguri / Conformal and current algebras

(69) is orthogonal to abelian differentials. We note that G~(w,z) contains a non-analytic piece due to zero modes. Substituting ha(W ) = 8gG~(w, z) into (67), (68) we find g

w~,-

2W j--1

N

= - ~ G~(wi, z)t~t.,(~l...~N),

(71)

i--1

( S~aJ~dP1 ... dPN) + i E ~,g f dZvvf~ g-°v~S~,gk J-''~'h~t" Jo -. ~N) j~l = --i/a~cC,(w,~)(J~*l...*N) N

--

, Z )IR, i( J~I

. . . dpN)

i-1

k 4~r

8"t'0 G (w, z ) ( ~ , . . . ~N)"

(72)

-w-g,

Eqs. (71) and (72) are the Ward identity for current algebra on a general Riemann surface. Let us first look at the short-distance expansion implied by the identity

8 '~b k/2 i f,,t,c b - - J ~ ( ) ,c J; ( z ) J ~ ( w ) = (2~r)2 (z - w) 2 + 2~r z - w 1 J:(z)O(w)

t~ - -

- . 2,~ z - w

~(w).

W

(73)

(74)

These formulas have the same form as the ones in the flat geometry case [9]. When written in terms of the moments of the currents

1 g;(~) = ~ E J a ( w ) ( z -- w)-"-~,

(75)

n

eqs. (73), (74) are equivalent to the current algebra and highest-weight state

T. Eguchi, H. Ooguri / Conformal and current algebras

326

condition,

[Jna(w),Jb(w)]

ifa°cjC+m(W)+2kn~ 8n+m,O,

=

l

J~(w)dP(w)=O,

ab

.

(76)

( n > 1),

J~(w)+(w)=t~q~(w).

(77)

As in the case of conformal algebra, it is easy to see that the algebraic structure (76), (77) is preserved under holomorphic coordinate transformations. Therefore the current algebra is also globally well-defined on the Riemann surface. Let us next analyze the structure of the second term in the l.h.s, of (71), (72). We introduce a real harmonic function ~i(z, ~) = RefZ~'j. ~i is continuous around the a, cycles, however, jumps discontinuously by 6ij when it completes a rotation along b,. We also use a Green function satisfying 1

O+G,(w,z)=g~=-~8(2)(w - z).

(78)

We then have g

(G - G)z(w, z) = i E ~j(w, ~,)%,j(z).

(79)

j=l

We can now rewrite the zero mode term as

E,+,,fd ¢~g 2w

w~-

¢~./J:°+,...+~5

J g

: i E %,+@ dw j= 1

(80)

aj

+ifd2w V~(O - d)z(w, z)v~
(J~el-..

es)

= -

E ~(w,, z)q,.,(+a(W,)... +,~(wN)) i--1 N

+ E +:.+~=dw (S~=+,... +~,5. j:l

+

(81)

327

T. Eguchi, H. Ooguri / Conformal and current algebras

Similarly, a b

=

-

ifo

fiAw,

N

- E G~(wi, z)tR,a i ( J b~ l . " ffJN) i--1

g

k/2~abOwGz(w,z)(O1...~N) + E ~z.j~ do 2~r

j- 1

a

b

cN). (82)

a.t

In the case of torus Gz(w, z) = (1/21r)[~'(w - z) - 2~1(w - z)] + i Im(w - z ) / I m r, (~z(w, z) = (1/2~r)[~'(w - z) - 2Th(w - z)] - i Im z/Im'r and % = 1. The nonanalytic piece, -ilmz/Imr, in G~(w,z) cancels out in eqs. (81) and (82) since correlation functions are singlet under gauge group G. If we consider free spinor theory, for instance, the above identities can be checked using addition formulas of the ~v function. In the case of a toms, we note that when the index a belongs to the Cartan subalgebra of gauge group G, 1

~ d w (J~q~l"" ~N) = ~

d

o o

doatr{ qL°e'° J ° ~ x . . . ~ N } ,

(83)

by using the correspondence with the operator formalism. When N = 0, the r.h.s. becomes the derivative of the character function of the affine Lie algebra G X ( O ) = t r ( qLoexp ( i~OaJ~ a ) } '

(84)

where a runs over the generators of the Cartan subalgebra H of G. Eqs. (81), (82) suggest that in the case of a general surface g >/2 one may define generalized characters X( 01, 02..... Os) with the property 0 ^alOg dU i

x(O) = ~.. dw (J~).

(85)

a,

Generalized characters keep track of the dependence of the vacuum energy on the strength of the background field 07 and the correlation functions of products of currents are completely determined in terms of these functions. We would like to thank our mathematician colleagues for discussions.

328

T. Eguchi, H. Ooguri / Conformal and current algebras

References [1] A.M. Polyakov, Phys. Lett. 103B (1981) 207, 211 [2] D. Friedan, in Recent advances in field theory and statistical mechanics (Les Houches, Summer 1982), eds. J.-B. Zuber and R. Stora (North-Holland, Amsterdam, 1984); O. Alvarez, Nucl. Phys. B216 (1983) 125 [3] A.A. Belavin and V.G. gniT.hnik, Phys. Lett. 168B (1986) 201; Landau Institute preprint (April 1986); Y. Martin, Phys. Lett. 172B (1986) 184; R. Catenacci, M. Cornalba, M. Martellini and C. Reina, Phys. Lett. 172B (1986) 328; J.B. Bost and T. Jolicoeur, Phys. Lett. 174B (1986) 273; C. Gomez, Phys. Lett. 175B (1986) 32; A. Kato, Y. Matsuo and S. Odake, Univ. of Tokyo preprint UT-489 (July 1986) [4] J.L. Cardy, Nucl. Phys. B270 (1986) 186; C. Itzykson and J.-B. Zuber, Saclay preprint PhT 85-019 (January 1986) [5] H. Sonoda, Berkeley preprint LBL-21363, (April, 1986) [6] D. Friedan and S. Shenker, Chicago preprint EFI 86-18A (May 1986) [7] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333 [8] V.G. Kac, Lecture Notes in Phys. 94 (1979) p. 441; B.L. Feigin and D.B. Fuchs, Functs. Anal. Pril. 16 (1982) 47 [Funct. Anal. App. 16 (1982) 114]; D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett 52 (1984) 1575 [9] V.G. Knizhrtik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83