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Super-Riemann surfaces, loop measure, etc…
Nuclear Physics B310 (1988) 79-100 North-Holland, Amsterdam
SUPER-RIEMANN SURFACES, LOOP MEASURE, ETC... l M. BERSHADSKY2 Centerfor Theoretical Physics, Laboratoryfor Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 25 May 1988 An explicit expression for the loop measure over supermoduli space is derived for a specific choice of local coordinates. Geometrical interpretation of these local coordinates over supermoduli space allows one to construct a matrix of superperiods and suggests the generalization of theta functions. A non-analytic factor over supermoduli space turns out to be det(3--R -J-L) 5 where JR (:Y-L)is a matrix of superperiods corresponding to the right (left) geometry.
1. Introduction T h e correct formulation of the superstring theories at higher loop level is a p r o b l e m of great interest. There are several approaches to evaluating the measure over supermoduli space [1-4]. The approach that we are going to discuss here is based on conformal field theory and some conformal operators that determine the shifts over the supermoduli space (see ref. [4]). This approach is close to that based on picture-changing operators developed in ref. [1] but is not identical. W e use the path-integral formulation suggested by Polyakov [5] which has an explicit geometric meaning: the path integral runs over the super-Riemann surfaces. This path integral m a y be reduced to the path integral over matter fields, appropriately defined systems of ghost fields and a finite-dimensional integral over superm o d u l i space. We are not going to elaborate this procedure and fix our attention on the integral over supermoduli space. The super-Riemann surfaces were originally i n t r o d u c e d in refs. [6-8] as 212-dimensional real manifolds equipped with an additional structure. The supermoduli space is the space of all superconformal structures on a super-Riemann surface (SRS). The supermoduli space of SRS was discussed in a set of papers [9-11]. For the scattering amplitudes, the path integral runs over the SRS with punctures. There are also supermoduli related to the punctures. This provides the complicated structure of emission vertices (see, for 1 This work is supported in part by funds provided by the US Department of Energy (D.O.E.) under contract no. DE-AC02-76ER03069. 2 Address since September 1, 1988: School of Natural Science, Institute for Advanced Study, Princeton, NJ 08540, USA 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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M. Bershadsky / Super-Riemannsurfaces
example, papers on the fermionic vertex [12,13]. The formalism developed below allows one to consider the supercoordinates over the supermoduli space itself and the supercoordinates related to the emitted particles on the same footing. Below we will evaluate an explicit expression for the measure over supermoduli space written for a specific choice of coordinates. Originally, the supermeasure was evaluated in the pioneering work by the Verlindes [1]. Our choice of supercoordinates over the supermoduli space also corresponds to point-like Beltrami differentials but is different from Verlindes' choice. In the resulting expression, the measure over the moduli space, after integrating over the odd variables, is not a correlation function with some picture-changing insertions but is more complicated. On the other hand, one has a very natural geometric interpretation of the odd parameters. There is a (2g - 2)-dimensional family of mappings which map the split SRS into a family of SRS with non-trivial superconformal structures (see sect. 2). The problem of holomorphic factorization has been extensively explored during the last years [11-16]. The measure over the supermoduli space is not a product of homomorphic and anti-holomorphic contributions coming from the right and the left movers. For the bosonic string theory this anomaly factor equals (det Im T)-13. The case of supertheory is more subtle due to the presence of the correlation function of supercurrents, which is not holomorphic over the supermoduli space. It will be shown that for the heterotic string theory or for the superstring theory all the anomaly contributions may be rearranged in the anomaly factor det(.Y-R - - g L ) -s, where J R ('Y-L) is the matrix of (super) periods corresponding to the right (left) geometry. As long as one considers even and odd coordinates over the supermoduli space on the same footing, there is no ambiguity in the definition of the measure. Problems immediately arise when one tries to integrate over the supermoduli space. Recently, it was discovered by Verlinde and Verlinde [1] that under the change of superBeltrami differentials the integrand over the moduli space shifts by a derivative term. There is another problem which is deeply connected with the one mentioned above. The partition function for superstring theories does not vanish after integrating over odd variables but is equal to a total derivative [1]. As it was first noticed by Atick et al. [17] (see also refs. [18,19]) the solution of this problem lies in the proper definition of the integral over the supermoduli space. They also pointed out that there are two possibilities. The first is to change the boundaries of the moduli space by nilpotents, the s e c o n d - to shift the integrand over the moduli space by a derivative term. The reason for this is that integration over supermanifolds (and also over the supermoduli space which is more complicated than the supermanifold because of the cusps) is a very delicate procedure. Only the measures that are equal to zero on the boundaries may be easily integrated over supermanifolds [20, 21]. Otherwise, the naive integral over odd parameters is not well defined. In the case of superstring theories the measure for a definite spin structure tends to infinity at the boundaries of the supermoduli space because of tachyon's and dilaton's poles. So
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M. Bershadsky / Super-Riemann surfaces
one may hope that after summation over different spin structures all the poles decouple and the measure could be integrated over the supermoduli space. Note that the body of the supermoduli space is not the moduli space but the spin-moduli space. This means that there is only summation over different classes of spin structures which are not related to each other by modular transformations. If one nevertheless insists on summing over all the spin structures, there will be another problem: whether it is possible to make this summation in a modular invariant way (see ref. [22]). It seems unlikely that the condition of modular invariance could be compatible with holomorphicity of the gauge slice. Many things are unclear and the problem is far from solved. So it seems very important to understand the relations between the existing approaches.
2. Super-Riemann surfaces In the covariant approach of Polyakov [5] the path integral runs over the internal geometries and the matter fields that determine the embedding into space time. For the backgrounds satisfying the equation of motion, the conformal anomaly cancels out and the path integral over the internal geometries reduces to a finite-dimensional integral over the space of superconformal structures of super-Riemann surfaces (the supermoduli space) [9-11]. A mathematically correct definition of super-Riemann surfaces (SRS) may be found in refs. [23, 24] (see also refs. [25, 26]) but the only thing which is important for us now is that any SRS may be represented as the collection of coordinate patches L)~ ~ £~ = (z~, 0~) pieced together with superconformal transformations zo 0a =
+ ,
1/2
1 ( "t xl/2~
,
(2.1)
One may think of z~ (0~) as an even (odd) element of some exterior algebra A = A 0 • Ap The transition functions f~B (%~) have to be even (odd) holomorphic functions mapping A 0 -~ A 0 (A 0 ~ A D. The superconformal transformations (2.1) rescale the form co = dz + 0d0 by the scale factor M which is equal to
M = f ' + {'{ + 20{'(f') 1/2.
(2.2)
It is possible to define the forms of weight (a, ~) with respect to superconformal transformations by • (2) ~ t~(k) = M - a ~ c - a ~ ( 2 ( ~ , ) ) .
(2.3)
Two collections of superconformal coordinate patches {U~) and (1~) which are
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M. Bershadsky / Super-Riemann surfaces
related to each other by superconformal coordinate transformations correspond to the same superconformal structure. Otherwise, these coverings correspond to different points in the supermoduli space. So one may move over the supermoduli space by making transformations which are not analytic. Before we start to investigate the deformations of superconformal structure let us first introduce the notation. The SRS is called a split SRS if it can be represented by eq. (2.1) with transition functions
zo =LB(zB), t" ¢, ~1/2t~v[~.
OOt = \JotS]
(2.4)
Let us come back to the supermoduli space. The tangent space to the supermoduli space coincides with the first cohomology group Hi(U, TX)= ~2(l'1/2)/D~2(-a+O) which we represent using Cech cohomology (see ref. [27]). Let 01 . . . . . UN, /)0 = ( 2 - - (U ~ ) ) be a covering of X ()? is an SRS). One has to assume that U, n U/~= 0 for a =~ft. Consider an infinitesimal change of coordinate patches/)+ (a >/1) given by meromorphic vector fields E+ having poles only in P, ~ /)~. This change of coordinate patches corresponds to an element in Hi(U, TX) given by /~0 n 0, ---, E+,
(2.5)
which is not zero if and only if there is no global meromorphic vector field over 2 having the same pole structure in/.~, as E~. In this case, the change of coordinate patches U~ (a>~ 1) induced by vector fields E~ determines the shift over the supermoduli space. Now we are ready to discuss the representation of SRS which will be used in this paper. Consider a family X(t) of RS represented as a collection of holomorphic coordinate patches. We assume that transition functions f~¢(* It) depends holomorphically on the moduli (i.e. on t). Now after specifying the spin structure one may construct a family of split SRS 2o( 0 as a collection of coordinate patches/2~ ~ ~ = (z~, 0~) pieced together by coordinate transformations (2.4). For a split SRS the even and odd coordinates do not mix and thus the components of superforms transform separately as the forms of appropriate weights on X(t). The superconforreal structure on X0(t ) is uniquely induced by the conformal structure on X(t). In order to describe a SRS with arbitrary superconformal structure let us mark the set of points P1. . . . . P2g-2 on X(t) (or on 20(0). Let g~ = (z~, 0~) be the local parameters in the neighborhood of P+ such that z,(P,) = 0. In the vicinity of each marked point, consider an odd-parametric family of coordinate transformations given by meromorphic functions = zo + OoxoCzo)
~ = 0~ + X ~ ( z ~ )
1
(2.6)
M. Bershadsky / Super-Riemann surfaces
83
A
X(t,X)
=
= O + X Z -I
Z,O
A
Xo(t)
=
1 Z
X(t)=
Fig. 1. For a general (P~) the odd coefficients Xl---X2g-2 play a role of local parameters over a supermoduli space. Note that coefficients X~ are not invariantly defined. U n d e r the change of local parameters z, ~ )t~z~ one also has to rescale the odd coefficients X~ ~ ()ka)3/2Xa. In this case the superconformal structure does not change. The construction of SRS discussed above is schematically represented in fig. 1. We are going to say that if X, ¢ 0 then in the neighborhood of P, the coordinate ~ ceases to be proper and the proper coordinate system is defined by eq. (2.6). In spite of the last circumstance we are going to express different objects, such as sections of line bundles, in terms of 2~ variables, keeping in mind that one may always rewrite them in terms of proper variables. So it is possible to define the objects on SRS in terms of the objects given on RS. Consider t A = (t ~, X~) as local coordinates on the supermoduli space. The parameters X ~ correspond to points P~, and t ~ are some even parameters over supermoduli space. We represent small variations of superconformal structure using the m e r o m o r p h i c vector fields E A where A ~ (a, a). The vector fields E~ are uniquely determined by the diagram ^
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N a m e l y ~x+sx = ~x - X S X ~ x 2 + ~^x~X(x ~, ~x+sx = - " , and thus Ex(~ ) = - X ~ 2~/x~x 1. Changing the variables ~ --* 2 one obtains E, =X~(z~)
2+ 2 0 ( z ~ ) - 1
2+
(2.7)
W h a t about the vector fields E 2 The only thing it is necessary to care about is that the vector fields E A have to correspond to some choice of coordinates over a supermoduli space. That implies the vector fields E A have to commute with each other in the sense that
eB(... I/'+ ~/.A) o E~(... I/') = EA(... I/'+ ~/'8) o es(... I/')-
(2.8)
N o t e that eq. (2.8) is not an ordinary commutation relation because the vector fields E A, E B correspond to two different points in the supermoduli space. The vector fields E A ( . . . ) determine a set of vectors in the tangent space to the supermoduli space labeled by index A. If one has the family of vector fields EA(... 1/') depending on /', one can construct the set of vector fields over supermoduli space. The condition (2.8) insures that these vector fields are tangent to some coordinate lines. It turns out to be very convenient to deal with vector fields corresponding to some choice of coordinates instead of the coordinates itself. Below we represent two choices of vector fields defined on a curve that correspond to some choice of local coordinates over supermoduli space E ~ ( 2 ) = (z~)-1¢0 -x ,
(2.9a)
E ~ ( ~ ) = [ X ~ ( Z ~ ) - 2 + 20~(z,)
1]~o
1,
(2.9b)
where we have introduced the additional set of ( 3 g - 3) points Q~ and the local coordinates z a in the neighborhood of each Q,. The points Q~ correspond to the choice of local even coordinates over supermoduli space. It is obvious that vector fields defined in neighborhoods of different points c o m m u t e with each other. Considering the naive limit of Qa ~ P~ (for a ~< 2g - 2) one finds that vector fields cease to commute. For the coinciding points one has to modify this set of vector fields Ea(2 ) = ( z a ) - 2 w 1,
a~<2g-2,
Ea(2 )=(za)-l,0
a>2g-2,
1,
Co(2) = [Xo(zo) -2 + 20o(zo)-1],o
Qa=P,,
(2.10a)
(2.lOb) '
(2.10c)
M. Bershadsky / Super-Riemann surfaces
85
3. Geometryof Super-Riemannsurface (SRS) In sect. 2 we described a family of mappings that map the split SRS into a family of SRS X(t, X ) '
x
)~o(t) "
(3.1)
This construction allows us to express different objects on SRS in terms of objects defined on split SRS. The notion of superholomorphic line bundles was introduced in refs. [28,29]. Here, we consider only the bundles of forms ~2~°. Projecting the bundle g2~° on split SRS one finds it is given by two bundles 12~° and I2~ ° ® K ~ 1/2, where Kn ~1/2 is the spin bundle associated to SRS (we also assume that n is an even non-singular spin structure). The question of the existence of superholomorphic (or meromorphic) sections of I2~ ° may be reduced to the existence of some meromorphic sections qo(z)~ ~2~°, ~b(z)~ g2~?°® Kn~1/2. Let us assume first that section ~ I2~ ° is superanalytic in the neighborhood of P and thus it can be expanded in Taylor series. Let us express the section • in the neighborhood of a marked point P in terms of a £-variable
• (-~) = [M(2)I J~(~(_~)),
(3.2)
where M(£) is the scale factor (2.2). One may easily see that the components q0(z) and ~b(z) ( q ) ( £ ) = q~(z)+ Off(z)) have to be meromorphic with poles of the first and second order, respectively. The coefficients in the polar parts are not arbitrary but satisfy the equations ~-1
= X~o.
(3.3a)
1]./_1 = (1 - 2j)Xq~a,
(3.3b)
~b_ 2 = - 2 jXCPo,
(3.3c)
where cpj (+j) are the coefficients of Laurent series of cp(z) (@(z)). One can check that the equations (3.3) are invariant under the change of local parameter z ~ f ( z ) , in spite of the fact that Laurent coefficients are not invariantly defined (note that one also has to rescale X). Consider the space of superholomorphic half-differentials. The dimension of this space is equal to 0]g (for an even non-singular spin structure!). The superholomorphic half-differentials play exactly the same role in the theory of SRS as the abelian differentials in the theory of ordinary RS. They may be integrated over cycles via fv(Ber) tb.
(3.4)
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M. Bershadsky / Super-Riemannsurfaces
We denote by (Ber) a covariant volume of integration. This integral depends only on the homotopy class of the contour ,1 and not on the contour itself in spite of the presence of the poles. This happens because of the vanishing of the residue of +(z) when j = ½ (see eq. (3.3)). Thus, it is possible to define the (g × g) matrix of super periods [4,16, 28]. Let us find an explicit expression for superholomorphic half-differentials. In order to do this we introduce the bases of meromorphic differentials {p~(dz) 1/2} and { I~ dz }, having a definite asymptotic behavior
.o(zo)-(zo) zo- e°,
(3.5a) (3.5b)
We also assume that all I~(z)dz have zero a-periods. Then, the bases (3.5a) and (3.5b) are uniquely defined. Finally, let ~%(z)dz be the normalized basis of holomorphic one-differentials (9~ ¢0kdz = 8~k). The basis in the space of superholomorphic half-differentials may be written in the form (3.6) The coefficients A~ and C~ are determined by the condition that the components of half-differentials have to satisfy eqs. (3.3a)-(3.3c). Solving these equations one finds A=coX(1-pxHx)
1,
C=cox(1-uxHx)-lpX,
(3.7)
where the matrices co, X, P, H are defined as follows co= (co,(P~)),
X = (X~6~t~},
(3.8a)
v={v~(P~)}~,l ~,
H={I~(P¢)}~¢.
(3.8b)
Note that only the off-diagonal elements of p and H contribute to the expressions (3.7). The matrix elements u,¢ and H~¢ may be expressed through the correlation functions of fermion fields and currents (3.9a)
H~B = ((OzXIz=pa O~Xl~=p,)) - w(w+(Im T ) - x w ) ~ B. We introduce the notation ( ( A B . . . ) ) = ( A B . . . ) / ( I ) . a crucial role in our discussion.
(3.9b)
The last identities will play
M. Bershadsky / Super-Riemann surfaces
87
Now we are ready to define the matrix of superperiods. The half-differentials • k(2) are normalized in such a way that
f~ (Ber) ~)k= 3,k.
(3.10)
So one define the matrix of superperiods in the usuat way YT,k = £(Ber)q~k = T,k + 2~ri(~x(1 - v x H x )
1
t
(3.11)
vxco )ik'
where T~k is the ordinary period matrix of the Riemann surface X ( t ) (see fig. 1). The matrix elements J/k are even elements of exterior algebra on 2 g - 2-elements A = d~k(X1. . . . X 2g- 2). Note that the matrix of superperiods defined above explicitly depends on spin structure of SRS through the matrix v. It would be very interesting to understand the general properties of the matrix of superperiods. First of all, J - is obviously symmetric. Under an infinitesimal shift over the supermoduli space given by E ~ Hi(U, TX), the variation of Y is given by
8 k- f(Ber)e(a,,(Da,,) +
(3.12)
If E is cohomologous to zero, then 3 Y = 0. One may also try to define a modular transformation of the matrix of superperiods. The modular group changes the basis F
F
of cycles { ai, b i } ~ { a', b/' }. But it also acts on the spin structure n ~ m. So one may naively expect that under modular transformation the matrix of superperiods transforms by F
5~ ~ ( A J m + B ) ( C J m + D)
x.
(3.13)
But it is not so simple. The main problem is not to define a modular transformation of the matrix of superperiods but to understand whether or not the choice of coordinates (X~} could be expanded globally in a way compatible with modular invariance (or covariance). Note that it is also possible to formally define the Jacobi map from the ordinary Riemann surface ( X ( t ) - ( 0 P~)) into the coset space J = A~)/(I, ,Y')
z ~ Z k= f ( a.
zB
er)~k.
(3.14)
The coset space J is an analogue of the jacobian variety. It is also not a problem to
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M. Bershadsky / Super-Riemann surfaces
define an analogue of the theta function (generalized theta function) via O[oI(ZI'Y-) =
exP(½m'J'm~+m'Zt),
E
(3.15)
mEZg
for Z ~ A~. The generalized theta function with non-zero characteristic may be defined as usual. One may prove that the infinite sum (3.15) converges and that the generalized theta function is a well-defined object. (For an ordinary theta function see ref. [30]). Note that O(Z(z)I.Y- ) is a meromorphic section of a line bundle on X(t) with poles of first order at P,. We hope that the generalized theta function (see eq. (3.15)) will turn out to be a useful object in the theory of SRS.
4. Measure
In this section we derive an expression for the measure over the supermoduli space. We are going to consider the scattering amplitudes and the partition function on the same footing. The only difference is that one has to consider SRS with punctures. So the amplitudes are given by measures over supermoduli spaces of SRS with punctures. Note that for SRS there are punctures of two k i n d s - N e v e u Schwarz punctures and Ramond punctures. In other words, it is necessary to insert the emission vertices into correlation function. These insertions simulate the proper asymptotic behavior in the vicinity of punctures. Asymptotics are predicted by operator expansion rules. For example, the insertion of exp(ik~X(R,)) means that the correlation function containing the current I(z) = OzX has to be a meromorphic one-differential over z with poles in R , and residues equal to k,. So one may forget about vertices and evaluate the path integral with these boundary conditions (asymptotic conditions) in the vicinity of punctures. Let us recall that emission vertices, for example, for massless particles, may be written in the form
VF1/2 = u,S~Sgc e i?x,
(4.1a)
VB1 = E,~"vc e i?x,
(4.1b)
where u,, % are polarizations. (We present here only the chiral part of the vertices.) At the first glance, something is missing. The vertex (4.1a) is not a full vertex for the spin particle constructed in refs. [12,13] but only a part of it. But, as was shown in F ref. [31], the second part of the vertex V +Fl / 2 -_ [Q, ( V -1/2] m a y be understood as the result of integration over the supermoduli. In order to follow the dependence on the supermoduli explicitly one has to consider vertices written in the form (4.1). Let us pay attention to how the vertices (4.1) depend on ghost fields. Sg(w) is the ghost field of dimension (3) corresponding to Ramond puncturing. It forces the
M. Bershad~ky / Super-Riemann surfaces
89
ghost fields fl(z) and X(z) to have branches ~k( Z )Sg( W ) = ( Z -- w )l/2( ~koSg) At- . . . . ~(Z)gg(W)
= (t
-w)
-1/2
(8_1 S 8)+
(4.2a) ....
(4.2b)
The field p(w) has dimension ~ and makes the supervector field ~(z) go to zero when " z " tends to the puncture
X ( z ) p ( w ) = ( z - w) +l(Xl/2U ) + . . . ,
(4.3a)
¢(z).(w) = ( z - w)-l(B_ j )
(4.3b)
+ ....
Thus, in evaluating the path integral one may relax the class of fields and say that the path integral runs over fields that may have branches, poles or zeroes as predicted by eqs. (4.2) or (4.3). That implies that one has more zero modes than for a smooth SRS without punctures. Zero modes are now meromorphic differentials having poles in the Neveu-Schwarz punctures and having branches in the Ramond punctures. In spite of the fact that zero modes have poles or branches we will call them holomorphic, keeping in mind that they are holomorphic everywhere except the punctures. The dimension of the supermoduli space is the number of zero modes and is equal to ( 2 g - 2 + N + R [ 3 g - 3 + N + ½R) where N (R) is the number of the Neveu-Schwarz (Ramond) punctures. Note that in order to derive the standard expressions for the vertices it is necessary to consider the singular limit where ( N + Q) or ( N + ½R) points associated with odd coordinates tend to the positions of the vertices. Let us come back to the measure. Let t A be the local coordinates over the supermoduli space. Under the local change of variables, t A ~ YcAtc the measure Z has to transform via Z ~ (Ber Y)-1Z. Consider an ansatz for the measure given by the ratio (correlation function) Ber(zero modes)
(4.4)
The detailed structure of the denominator will be explained at the end of this section. It is constructed from zero modes - superholomorphic ~-differentials dual to the choice of coordinates over the supermoduli space. This insures the desirable transformation properties for the measure. Consider an infinitesimal change of local parameters X ~ X + 8X. The variation of the correlation function is given by 8×(corr. funct.) =
f(Ber) axE×(W...),
(4.5)
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M. Bershadsky / Super-Riemann surfaces"
where W= ½J + OT is the super stress-energy tensor. The vector field E× corresponds to the infinitesimal shift X --* X + 8X and is given by eq. (2.9b). Rewriting eq. (4.5) in terms of t-variables living on a split SRS one can show that the correlation function satisfies the system of differential equations 0 OX---£( . . . ) = ( L 3(P~) .--) + ( J - 3 / 2 ( P ~ ) . - - ) ,
(4.6)
where L _ 3 ( P a ) = f ~ p T ( z ) ( z - P a ) - 2 d x , J _ 3 / 2 ( P a ) = fDpJ(z)(z - Pa) 1 dz. Integrating these equations one finds that the correlation function is given by the expression (4.7)
(H(1-x,,J(P,,))...).
Points denote operators that do not depend on X~. All the dependence on odd parameters X, comes through the insertion operators ( 1 - X~J(P~)) which determine the shifts in the odd directions. Note that the operator (1 - x J ( P ) ) simulates the proper asymptotic behavior. Using the operator product expansion rules, one can show that the function F(~) = <(¢p(z) + 0~b(z))(1 - x J ( P ) ) . . . ) ,
(4.8)
is superanalytic in the sense of eq. (3.3). So we have defined the correlation function in terms of the fields living on ordinary RS. When all the X~ become equal to zero, the correlation function has to reproduce the product of different determinants. If we are interested in amplitudes one also has to insert vertices (4.1) into correlation function (4.7). In order to define the correlation function (4.7) properly one has to insert (3g - 3) operators 3(b(w))= b(w) and ( 2 g - 2 ) operators 6(fl(x)) that absorb the zero modes contributions. Thus the correlation function depends on the position of the screening operators 8(b(w)) and 8(fl(x)). The screening operator 8(fl(x)) is determined by its operator product expansion rules /~(X)~(]~(X))
= (Z-- X)(/~_5/2~(~))
= (z-
x)
-4- . . . .
(x3/28(B)) + . . . .
(4.9a) (4.9b)
The ghosts fields b(z) and c(z) do not interact with the field 8(fl). We also demand the field 8(fl) to be fermionic. One may easily derive that the dimension of 8(fl) is equal to - 3. Note that it is possible to bosonize the field 8(fl)= e ¢ in accordance with the rules suggested in ref. [12].
M. Bershadsky / Super-Riemannsurfaces
91
The correlation function (4.7) does not depend on the choice of coordinates over supermoduli space. This dependence comes through the denominator - Berezinian of zero modes. Consider the set of vector fields EA(2 ) corresponding to the choice of coordinates over the supermoduli space. We define the basis of superholomorphic 32-differentials BC(2) dual to the vector fields E A via
(F_,,[Bc)
= ~ ( B e r ) Ea (2)
BC(~.) =
3if( - 1) y
(4.10)
Under the change of local coordinates over the supermoduli space differentials BA(£) transforms exactly like tn: BA(2) --, y~BC(£). Keeping in mind that screening fields inseted into correlation functions are not superfields, we construct the matrix of zero modes from the components of BA(£). The components of superfields are defined with respect to coordinates on a split SRS. Let b ( z ) = ((--1)AbA(z)), ~(X)= (flA(Z)) be the vector rows where bA(z) and flA(Z) are the components of superfields B A ( 2 ) = flA(z)+ ObA(z). We construct the matrix of zero modes from the rows b(w~) and ~(x,). The resulting expression for the measure for the heterotic string is written as follows
(I-I 2g- 2(1 - X,,J(P,~))[I 2g- 23( fl(x~))l-13 g- 3b(w~) vertices} Z(/',t)=
Ber[b(%)"'b(w3g
3)~(X1)'''~(X2g 2 ) ] ~
(4.11)
For the heterotic string there is no left supermoduli and so we suppress the contribution of the left movers. It is easy to see that expression (4.11) transforms as a measure under the change of coordinates. Note that Berezinian of zero modes is a meromorphic 2-differential (3-differential) over each variable % (x~) just like the correlation function. Thus, the ratio (4.11) has to be at least the meromorphic function over each variable %, x~. We will demonstrate below the measure (4.11) does not depend on the position of the points %, x , at all. This is a superanalog of Quillen's theorem [32]. The measure for the superstring theory may be schematically written as Z(F, t) = ( [ I . ( 1 - x J ( P . ) ) ( 1
- ~ . a ; ( R . ) ) (screening fields) }
Ber(right)Ber(left)
(4.12)
We assume that right and left movers correspond in general to different spin structures. The dependence on the position of screening fields is eliminated by the Berezinian of zero modes. These ansatz for the measures (4.11) and (4.12) provides a very natural way to investigate the holomorphic properties of the measure. This will be done in sect. 5.
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M. Bershadsky / Super-Riemann surfaces 5. Calculations
In this section we calculate the expression for the measure in terms of meromorphic differentials on RS and determinants of ordinary 0-operators. For simplicity, we consider the partition function. There are three different contributions for the measure (1) Berezinian of zero modes; (2) Correlation function of matter fields; (3) Correlation function of ghost fields. Let us briefly sketch the evaluation of a determinant of zero modes. We choose the superholomorphic -32-differentials to be dual to the vector fields given by eq. (2.12). As was shown in sect. 3 , the superholomorphic j-differentials may be written in terms of meromorphic j and ( j + ±)2 differentials (here one has 3 and 2 differentials). To specify the basis of 2-differentials we mark the points P2g 1..... P3g 3 except the points P1 ..... P2g 2. Assuming that the Latin (Greek) indices run over the set 1, 2 . . . . . 3 g - 3 (1,2,..., 2 g - 2) we define the following bases in the spaces of meromorphic 3 and 2 differentials fl~(z)(dz)3/2:
z ~ P~,
"/~(z)(dz)3/2:
z ~ P~,
/3,~(PB) = 8,~/~,
(5.1a)
y,,(z) - 8,#~(zB) 1 + O(zB),
(5.1b)
b~(z)(dz)2;
z~P~,
ba(Pc) = Sac ,
(5.1C)
d~(z)(dz)2:
z ~Pb,
d ~ ( z ) - 8~e(zb) -x + O ( z b ) ,
(5.1d)
f~(z)(dz)2:
z ~ P b,
f~(z) - 8,~B(zl,) 2 + O(zb).
(5.1e)
The components of superholomorphic K-differentials are meromorphic v3 and 2-differentials satisfying the eq. (3.3). Omitting rather tedious calculations one obtains that Berezinian of zero modes is equal to Ber = d e t ( b ' + 28X'~'xd')-1(det/~) ×det[1 + 2 8 X ( V ' -
l(det b)
"y(~)-lflt)X6~(I))-l--36X'~(/~)
1xf(b)-l],
(5.2)
where we have introduced the following matrices (b')a c = b~(Pc),
(d'),~ = d:(P,),
(fl')~,, = flZ(P~,),
(-~') ~, = y~' ( P , ) ,
(b),,c = b,,(wc),
(c[),,a = d,~(w,,),
(fl)~ = b~(x,),
('~),~, = V,~(x~,).
(8)a~ = 8 ~ ,
(5.3a) (5.3b)
(f)~
= f,(w,),
(5.3c) (5.3d)
M. Bershadsky / Super-Riemann surfaces
93
The correlation functions of (b, c) or (/3, X) ghost systems may be evaluated using the Wick theorem. One only has to remember the statistics of (b, c) and (fl, X) systems. Let us introduce the notations ( ( X ) ) = ( X(screening fields) )/(screening fields).
(5.4)
By "screening fields" we mean the products of operators that absorb the contribution of zero modes [l-Ib(%),l-IS(/3(x~)) or lqb(wo)Fl,~(/3(x~))]. Then the Green functions are given by
ac, (z, w) = ((¢(z)b(w))),
(5.5a)
Gv,,(z, w) = ( @ ( z ) / 3 ( w ) ) ) .
(5.5b)
Note that these Green functions may be written in terms of differentials of type (5.1) _ dz(w),
(5.6a)
Gx,,(z, w) : ~/z(x,)(fl-1)~" fl,(w) - yz(w),
(5.6b)
Gc, b(Z, W) : dz(wa)([)
1)akbk(W )
where dz(w ) (yz(w)) is meromorphic 2 (~) differential over "w" with the only pole in " z " and having zeroes in P1..... P3g-3 (Px ..... P2g-2). It turns out that the ghost correlation function
may be evaluated in terms of meromorphic differentials defined by eq. (5.1). We remind the reader that supercurrent Jgh(Z) is given by the expression Jgh = b X 3fl 0~c - 2( Ozfl)c. Let us integrate in correlation function (5.7) over superghost fields /3 and X. Then correlation function (5.7) reduces to the average ( ( exp [3b (P~)X,((X (P,)/3(P.))),.xX~e'(P~)
+
(5.8)
From the path-integral point of view one may easily derive that the average (5.8) is nothing else than the determinant of operator [1 - / ( ( 0 1)-~] where the kernel o f / (
94
M. Bershadsky / Super-Riemannsurfaces"
is given by the expression
8(e- ,o)a(,-
h)= 2 z
(5.9) Matrices /3',-/',/q, "~, are defined by eqs. (5.3b) and (5.3c), (/3)~ = &(P~)= 8,~, ('Y)~/~='/~(Pp)l~,p. Note that the kernel of (0-1) ~ coincides with the Green function ((c(z)b(w))). Due to the singular structure o f / ( the only matrix elements that contribute to det[1 -/~(}_~)-1] are ((c(P~)b(P~))) = (el(b) lb - d),~. Evaluating this determinant one gets the result ( ( I ~ e x p [ - X~Jgh( P.)] ) ) = det[1 + 28X(~/-
~(~)-l~')Xd(h) 1
_38X-~(~)-lXf (~9) 1]. (5.10) This is exactly the last term in expression (5.2). Substituting the formulas (5.2) and (5.10) into the expression for heterotic measure one finds
Z(t,i) = (I~Ia (1-XaJmat(ea)))det(b'+28Xg'X
d')
× (II~b(w,,)H,~8(~(x.))) det/~(left ghosts) det b
(5.11)
det(left)
Now it is obvious that measure (5.11) does not depend on the position of screening operators. For the measure in superstring theory one can get the expression
z ( L t ) = (I-Iexp-[XaJmat(Pa)
-
X~&at(R~)] )det(b' + 28XY'xd')
(Hb(w.)H6(B(x,~))) ×
det/~
detdet
/~ (left ghosts) Ber(left)
(5.12)
Let us discuss the analytic properties of the measures (5.11) and (5.12). For the bosonic case the nonanalytic factor over moduli space coincides with (det Im T) -13. This factor comes from the determinant of scalar laplacian. The case of supertheory is more subtle due to the presence of the correlation function of supercurrents Jmat(Z) =~b 3 z X which is not holomorphic over moduli space. The correlation
M. Bershadsky / Super-Riemannsurfaces
95
function 1--/exp( - X a Jmat (Pa))
) x, ~,
(5.13)
may be easily computed using the same trick considered for ghosts. Integrating over scalar fields one gets
Idet0012(detlmT)-5(I-Iexp-[q~(P~)x.G(P~,Pa)xa+(Pa)]), a
where
G(P~, PB) is current-current Green function given by G( P~, Pa) = H~¢ - ~r( ~0t(Im T )-1~0 ) ~ .
(5.15)
The matrices H, o~ were defined in eq. (3.8). Averaging over fermionic fields one obtains
(1--Iexp__XaJmat(P,~)) = [det ~01 lO(det ~,/2)S(det 01/2 )m 5 × det(1 -
vxH x)s(det Im T) -5
xdet[1 + ~ r ( 1 - p x H x ) - l ~ , X o £ ( I m T ) - l o ~ X ]
5, (5.16)
where n ( ~ ) is the spin structure related to the right (left) movers. Using the identity det(1 + AB)= det(1 + BA) -1 that is valid for odd matrices A and B, one may rearrange the two last determinants from eq. (5.16) into the determinant of the matrix of superperiods (see eq. (3.12)) (det Im T) -5 det[1 + ~r(1 -
vXHX)- lvxcot(Im T)-I~oX] 5
=(detlmT)_Sdet[1 +~(ImT)_Xox(I_pxHx) -
det( J R - TL) 5
lpxoot] 5 (5.17)
Now substituting expression (5.16) into (5.12) one finds that the measure for the heterotic string theory may be written in the form
Z,,~,(?, i) = det(YR _ ~L)5 ,
(5.18)
M. Bershadsky / Super-Riemann surfaces
96
where the chiral measure Zn(f ) is given by Zn(~) = (det0o) 5(d e t -01/2) n s det(1 -
vxHx) 5
28X'l'xd')(~b(Pa) ~8(fl(P,,))).
× det(b' +
(5.19)
The dependence on the odd parameters is now manifest. Contributions from the left movers G~(i) depend on the model and are merely a product of different determinants. Note that all ingredients of formula (5.19) may be expressed through theta functions. Now let us evaluate the anomaly factor in the superstring theory. All nonanalytic contributions come through the correlation function of matter fields. Following the arguments for the heterotic case one can evaluate the correlation function of matter fields ( ~ exp
- [XaJmat(P,~) + ~,]mat (Ra)]>
= Idet 0o1-1°( det -3 1"/ 2 ) 5(det
Oq~/2)5det(1
-
x det(1 - ~ H 2 ) 5 ( d e t Im T) 5 det(1 +
vxHx) s AB) 5,
(5.20)
where the matrices A, B are given by the expressions (1
-
pXHx)-lvxoo t
0
A =
0 (Imr) B=
1~ox
_ (Im T)-l~o x
(1 - ~ 2 H 2 ) - ' ~ ' - (Im T ) - I ~ ] . (Im T)
(5.22)
1~ ]
The matrices ~o, X, v, H are defined exactly in the same way as the matrices w, X, v, H (see eq. (3.8)) except for the complex conjugation, change of spin structure n-~ m and the replacement P,---, R~. Using the same trick as in the heterotic case one can rearrange the last two determinants in eq. (5.20) into determinants of the matrix of superperiods (det Im T ) - 5 det(1 +
BA)-5
=(detlmT)-SdetIl+~r(ImT) -Tr(ImT) - det( J R -- J-L )-5
l~ox(1-uxHx)-lvx ~°t 1~(1-~2H--~)
1~
t
1] 5 1 (5.23)
M. Bershadsky / Super-Riemannsurfaces
97
As the result the superstring measure (5.12) may be written in the form
=
det(,y-R - J-L ) 5'
(5.24)
where J R ( g L ) is the matrix of superperiods (see eq. (3.11)) corresponding to the right (left) supergeometry. Note that °JR and g e are not conjugate to each other. Both the anomalous factors in the heterotic and superstring theories may be written in the form (JR-
TR)ij = (q~l~°j),
(JR--gL),j
= (~]q)~),
(5.25) (5.26)
where ~ff (qbL) are superholomorphic }-differentials corresponding to the right (left) supergeometry and normalized by having zero a-periods.
6. Discussion
In the previous sections we constructed a measure over supermoduli space using some specific choice of local coordinates. Let us first establish the relations with other approaches. The approach based on picture changing operator (see the work by the Verlindes [1]) corresponds to point-like super-Beltrami differentials ~ ' , = O08(z - P~) that do not depend on odd variables X~- Our choice of coordinates is quite different. The super-Beltrami differentials used in this paper explicitly depend on odd variables via ~ = O06(x- P~)+ Ox~'(x- P~). To be precise, we work not with Beltrami differentials ~ , but with meromorphic vector fields E~. The relations between them is quite clear . / g ~ - DE,. The Verlindes' choice of superBeltrami differentials ./g~ and ours .//4, differ from one another by some gauge transformation. As was mentioned in sect. 1, on the other hand our choice of coordinates has a very explicit geometric interpretation. The odd parameters coincide with the residues of the transformation functions that map from split SRS into general SRS. Recently, an operator formalism for the conformal field theories on a Riemann surface was developed in refs. [33-35]. This formalism allows one to represent the loop measure via ( AIAI8((Ex ] B ) ) [ @ ,
(6.1)
where [q~) is the state corresponding to RS or SRS; 8((E~ ]B)) operators eliminating the contributions of zero modes. The set of vector fields E A correspond to some
M. Bershadsky / Super-Riemannsurfaces
98
choice of coordinates over moduli or supermoduli space. (For the details see the original papers [34, 35].) It is easy to understand the relations between formalism considered in the present paper (see also ref. [4]) and the operator formalism. One may notice that the measure (4.11) or (4.12) may be written in the form
{FLs(< f, i~>)rlo (1 - xoJ(e~) ) split SRS Ber(
(6.2)
where the vector fields/~A = (/~a, /~) are given b y / ~ = (z - w~) 1, ]~a = (Z -- Xa) -1. Keeping in mind that superholomorphic ~-differentials B°(£) are dual to the vector fields E A corresponding to the choice of coordinates over supermoduli space, one may recognize that the denominator in eq. (6.2) is nothing but the jacobian factor Ber((/~AIBD)(--1) D'A+I)) = Ber(EA --,/~A)-
(6.3)
As a result, the expression for the measure may be rewritten as follows ( 1-I~ (( E A]B)) 1-I (1 - Xfl(P~))) A
(6.4) split SRS
This expression coincides with eq. (6.1) after the identification [ ~ ) = F [ ~ ( 1 Xfl(P~))Icp> where the state [~) corresponds to split SRS and the operator FI~(1 - X f l ( P ~ ) ) shifts among odd directions over supermoduli space. All approaches suffer exactly from the same problems. It is not known whether or not it is possible to define unambiguously the integral over supermoduli space, whether or not their exists a holomorphic and modular invariant choice of gauge slice, whether or not the supermoduli space split, etc . . . . The main obstacle is that it is impossible to globally choose points P~(t) to be holomorphically dependent on moduli space. In the intersection of coordinate patches (t, X) and (f, ~) one has two different sets of points P~(t) and /~(t'). The transition functions are, in general, of the form X~ = X~(f, )~),
(6.5a)
t a=ta(t ") h-ha(t ",X),
(6.5b)
where we extract nilpotent contribution h~(f, ~) from transition function ta(?, ~). Therefore, t or f are even coordinates over supermoduli space and they cannot be considered as coordinates over moduli space. It might be possible only if there exists such covering of supermoduli space that in the overlapping regions all nilpotent parts of transition functions (6.5b) are equal to zero. That exactly means that supermoduli space splits. It is not difficult to demonstrate that, for arbitrary two
M. Bershadsky / Super-Riemannsurfaces
99
sets o f points, the c o o r d i n a t e systems (t, X){po} a n d (/~, 2)(~o) are related to each o t h e r b y t r a n s f o r m a t i o n with n o n - z e r o n i l p o t e n t p a r t in eq. (6.6b) except for two cases: (a) surfaces of genus two - it is the reason why the s u p e r m o d u l i space of SRS of g e n u s two splits; (b) the p o i n t s { / 5 } m a k e p e r m u t a t i o n of { P~ }. U n f o r t u n a t e l y , the last c o n d i t i o n is too restrictive. P r o b a b l y we d e m a n d too m u c h a n d it is p o s s i b l e to consider n o t h o l o m o r p h i c b u t m e r o m o r p h i c slices (see ref. [17]) o r a n o t h e r c o o r d i n a t e system. F o r e x a m p l e one m a y m a r k N p o i n t s P 1 , - - - , PN on R S a n d a s s o c i a t e with each p o i n t Pj the set of o d d p a r a m e t e r s X~j) . . . . . Y(J) such z~. n2 t h a t all o d d p a r a m e t e r s satisfy ( E n j - 2 g + 2) relations. But it is again n o t the g e n e r a l c o o r d i n a t e system. Actually, there are a lot of possibilities b u t p r o b a b l y it is i m p o s s i b l e to choose a g o o d h o l o m o r p h i c slice at all. W e are n o t going to go too d e e p into d e t a i l s a n d repeat a r g u m e n t s a n d refer the r e a d e r to the original p a p e r s [18,191. I w o u l d like to t h a n k A. Eskin, J. Lattore, P. Nelson, A. R a d u l a n d C. Vafa for m a n y fruitful discussions. I a m especially very grateful to P. N e l s o n for r e a d i n g the manuscript.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
E. Verlinde and H. Verlinde, Phys. Lett. B192 (1987) 92 M. Baranov and A. Schwarz, Int. J. Mod. Phys. A2 (1987) 1173; A3 (1987) 28 A. Voronov, Funk. Anal. Pill. (USSR) 21 (1987) 312 M. Bershadsky, Phys. Lett. B201 (1988) 67 A.M. Polyakov, Phys. Lett. B103 (1981) 211 D. Friedan, in Workshop on unified string theories (ITP, Santa Barbara, 1985) E. Martinec, Phys. Rev. D28 (1983) 2606 M. Baranov, A. Schwarz, Pisma ZhETF. 42 (1985) 340; M. Baranov, I. Frolov and A. Schwarz, Theor. Math. Phys. 70 (1987) 64 D. Friedan, E. Martinec and S. Schenker, Nucl. Phys. B271 (1986) 93 G. Moore, P. Nelson and J. Polchinsky, Phys. Lett. B169 (1986) 47 E. D'Hoker and D.H. Phong, Nucl. Phys. B278 (1986) 255 D. Friedan, S. Shenker and E. Martinec, Phys. Lett. B160 (1985) 55 V. Khizhnik, Phys. Lett. B160 (1985) 403 G. Moore, J. Harris, P. Nelson and I. Singer, Phys. Lett. B178 (1987) 167 E. D'Hoker and D.H. Phong, Nucl. Phys. B292 (1987) 317 H. Sonoda, The holomorphic anomaly in superstring theory, preprint 1987, LBL 24409 J.J. Atick, J.M. Rabin and A. Sen, Nucl. Phys. B299 (1988) 279 J.J. Atick, G. Moore and A. Sen, Nucl. Phys. B308 (1988) 1 J.J. Atick, G. Moore and A. Sen, Nucl. Phys. B307 (1988) 221 J. Berhshtein and D. Leites, Funk. an. 11 (1977) 55; 70; B. de Witt, Supermanifolds (Cambridge University Press, Cambridge, 1984); A. Rogers, J. Math. Phys. 16 (1985) 384; 27 (1986) 710 M.J. Rothstein, Trans. Am. Math. Soc. 299 (1987) 387 E. Witten, in Proc. Syrup. on Anomalies, geometry and topology (World Scientific, Singapore, 1985) D. Leites, Russian Math. Surv. 35 (1981) 1 M.H. Rathstein and C. Le Brun, Moduli of super-Riemann surfaces, IAS preprint (1987)
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M. Bershadsky / Super-Riemann surfaces"
[25] P. Nelson, Introduction to supermanifolds, Harvard preprint HUTP-86/A024 [26] S.B. Giddings and P. Nelson, The geometry of super-Riemann surfaces, Harvard preprint HUTP-87A070 [27] P. Griffiths and J. Harris, Principles of algebraic geometry (Wiley/Interscience, New York, 1978) [28] A.A. Voronov, A.A. Rosly and A.S. Schwarz, Geometry of superconformal manifolds (I), (II) ITEP preprints no. 107, no. 115 (1987) [29] S.B. Giddings and P. Nelson, Line bundles on super-Riemann surfaces, Harvard preprint, HUTP87/A080 [30] D. Fay, Theta-functions on Riemann surfaces (Springer, Berlin, 1973) [31] V. Knizhnik, Phys. Lett. B178 (1986) 21 [32] D. Quilen, Funk. Anal. Pill. 19 (1987) 48 [33] C. Vafa, Phys. Lett. B190 (1987) 48 [34] L. Alvarez Gaumd, C. Gomez and C. Vafa, Nucl. Phys. B303 (1988) 455 [35] L. Alvarez Gaum~, C. Gomez, P. Nelson, G. Sierra and C. Vara, Fermionic strings in the operator formalism, Harvard preprint HUTP-88/A027
SUPER-RIEMANN SURFACES, LOOP MEASURE, ETC... l M. BERSHADSKY2 Centerfor Theoretical Physics, Laboratoryfor Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 25 May 1988 An explicit expression for the loop measure over supermoduli space is derived for a specific choice of local coordinates. Geometrical interpretation of these local coordinates over supermoduli space allows one to construct a matrix of superperiods and suggests the generalization of theta functions. A non-analytic factor over supermoduli space turns out to be det(3--R -J-L) 5 where JR (:Y-L)is a matrix of superperiods corresponding to the right (left) geometry.
1. Introduction T h e correct formulation of the superstring theories at higher loop level is a p r o b l e m of great interest. There are several approaches to evaluating the measure over supermoduli space [1-4]. The approach that we are going to discuss here is based on conformal field theory and some conformal operators that determine the shifts over the supermoduli space (see ref. [4]). This approach is close to that based on picture-changing operators developed in ref. [1] but is not identical. W e use the path-integral formulation suggested by Polyakov [5] which has an explicit geometric meaning: the path integral runs over the super-Riemann surfaces. This path integral m a y be reduced to the path integral over matter fields, appropriately defined systems of ghost fields and a finite-dimensional integral over superm o d u l i space. We are not going to elaborate this procedure and fix our attention on the integral over supermoduli space. The super-Riemann surfaces were originally i n t r o d u c e d in refs. [6-8] as 212-dimensional real manifolds equipped with an additional structure. The supermoduli space is the space of all superconformal structures on a super-Riemann surface (SRS). The supermoduli space of SRS was discussed in a set of papers [9-11]. For the scattering amplitudes, the path integral runs over the SRS with punctures. There are also supermoduli related to the punctures. This provides the complicated structure of emission vertices (see, for 1 This work is supported in part by funds provided by the US Department of Energy (D.O.E.) under contract no. DE-AC02-76ER03069. 2 Address since September 1, 1988: School of Natural Science, Institute for Advanced Study, Princeton, NJ 08540, USA 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
80
M. Bershadsky / Super-Riemannsurfaces
example, papers on the fermionic vertex [12,13]. The formalism developed below allows one to consider the supercoordinates over the supermoduli space itself and the supercoordinates related to the emitted particles on the same footing. Below we will evaluate an explicit expression for the measure over supermoduli space written for a specific choice of coordinates. Originally, the supermeasure was evaluated in the pioneering work by the Verlindes [1]. Our choice of supercoordinates over the supermoduli space also corresponds to point-like Beltrami differentials but is different from Verlindes' choice. In the resulting expression, the measure over the moduli space, after integrating over the odd variables, is not a correlation function with some picture-changing insertions but is more complicated. On the other hand, one has a very natural geometric interpretation of the odd parameters. There is a (2g - 2)-dimensional family of mappings which map the split SRS into a family of SRS with non-trivial superconformal structures (see sect. 2). The problem of holomorphic factorization has been extensively explored during the last years [11-16]. The measure over the supermoduli space is not a product of homomorphic and anti-holomorphic contributions coming from the right and the left movers. For the bosonic string theory this anomaly factor equals (det Im T)-13. The case of supertheory is more subtle due to the presence of the correlation function of supercurrents, which is not holomorphic over the supermoduli space. It will be shown that for the heterotic string theory or for the superstring theory all the anomaly contributions may be rearranged in the anomaly factor det(.Y-R - - g L ) -s, where J R ('Y-L) is the matrix of (super) periods corresponding to the right (left) geometry. As long as one considers even and odd coordinates over the supermoduli space on the same footing, there is no ambiguity in the definition of the measure. Problems immediately arise when one tries to integrate over the supermoduli space. Recently, it was discovered by Verlinde and Verlinde [1] that under the change of superBeltrami differentials the integrand over the moduli space shifts by a derivative term. There is another problem which is deeply connected with the one mentioned above. The partition function for superstring theories does not vanish after integrating over odd variables but is equal to a total derivative [1]. As it was first noticed by Atick et al. [17] (see also refs. [18,19]) the solution of this problem lies in the proper definition of the integral over the supermoduli space. They also pointed out that there are two possibilities. The first is to change the boundaries of the moduli space by nilpotents, the s e c o n d - to shift the integrand over the moduli space by a derivative term. The reason for this is that integration over supermanifolds (and also over the supermoduli space which is more complicated than the supermanifold because of the cusps) is a very delicate procedure. Only the measures that are equal to zero on the boundaries may be easily integrated over supermanifolds [20, 21]. Otherwise, the naive integral over odd parameters is not well defined. In the case of superstring theories the measure for a definite spin structure tends to infinity at the boundaries of the supermoduli space because of tachyon's and dilaton's poles. So
81
M. Bershadsky / Super-Riemann surfaces
one may hope that after summation over different spin structures all the poles decouple and the measure could be integrated over the supermoduli space. Note that the body of the supermoduli space is not the moduli space but the spin-moduli space. This means that there is only summation over different classes of spin structures which are not related to each other by modular transformations. If one nevertheless insists on summing over all the spin structures, there will be another problem: whether it is possible to make this summation in a modular invariant way (see ref. [22]). It seems unlikely that the condition of modular invariance could be compatible with holomorphicity of the gauge slice. Many things are unclear and the problem is far from solved. So it seems very important to understand the relations between the existing approaches.
2. Super-Riemann surfaces In the covariant approach of Polyakov [5] the path integral runs over the internal geometries and the matter fields that determine the embedding into space time. For the backgrounds satisfying the equation of motion, the conformal anomaly cancels out and the path integral over the internal geometries reduces to a finite-dimensional integral over the space of superconformal structures of super-Riemann surfaces (the supermoduli space) [9-11]. A mathematically correct definition of super-Riemann surfaces (SRS) may be found in refs. [23, 24] (see also refs. [25, 26]) but the only thing which is important for us now is that any SRS may be represented as the collection of coordinate patches L)~ ~ £~ = (z~, 0~) pieced together with superconformal transformations zo 0a =
+ ,
1/2
1 ( "t xl/2~
,
(2.1)
One may think of z~ (0~) as an even (odd) element of some exterior algebra A = A 0 • Ap The transition functions f~B (%~) have to be even (odd) holomorphic functions mapping A 0 -~ A 0 (A 0 ~ A D. The superconformal transformations (2.1) rescale the form co = dz + 0d0 by the scale factor M which is equal to
M = f ' + {'{ + 20{'(f') 1/2.
(2.2)
It is possible to define the forms of weight (a, ~) with respect to superconformal transformations by • (2) ~ t~(k) = M - a ~ c - a ~ ( 2 ( ~ , ) ) .
(2.3)
Two collections of superconformal coordinate patches {U~) and (1~) which are
82
M. Bershadsky / Super-Riemann surfaces
related to each other by superconformal coordinate transformations correspond to the same superconformal structure. Otherwise, these coverings correspond to different points in the supermoduli space. So one may move over the supermoduli space by making transformations which are not analytic. Before we start to investigate the deformations of superconformal structure let us first introduce the notation. The SRS is called a split SRS if it can be represented by eq. (2.1) with transition functions
zo =LB(zB), t" ¢, ~1/2t~v[~.
OOt = \JotS]
(2.4)
Let us come back to the supermoduli space. The tangent space to the supermoduli space coincides with the first cohomology group Hi(U, TX)= ~2(l'1/2)/D~2(-a+O) which we represent using Cech cohomology (see ref. [27]). Let 01 . . . . . UN, /)0 = ( 2 - - (U ~ ) ) be a covering of X ()? is an SRS). One has to assume that U, n U/~= 0 for a =~ft. Consider an infinitesimal change of coordinate patches/)+ (a >/1) given by meromorphic vector fields E+ having poles only in P, ~ /)~. This change of coordinate patches corresponds to an element in Hi(U, TX) given by /~0 n 0, ---, E+,
(2.5)
which is not zero if and only if there is no global meromorphic vector field over 2 having the same pole structure in/.~, as E~. In this case, the change of coordinate patches U~ (a>~ 1) induced by vector fields E~ determines the shift over the supermoduli space. Now we are ready to discuss the representation of SRS which will be used in this paper. Consider a family X(t) of RS represented as a collection of holomorphic coordinate patches. We assume that transition functions f~¢(* It) depends holomorphically on the moduli (i.e. on t). Now after specifying the spin structure one may construct a family of split SRS 2o( 0 as a collection of coordinate patches/2~ ~ ~ = (z~, 0~) pieced together by coordinate transformations (2.4). For a split SRS the even and odd coordinates do not mix and thus the components of superforms transform separately as the forms of appropriate weights on X(t). The superconforreal structure on X0(t ) is uniquely induced by the conformal structure on X(t). In order to describe a SRS with arbitrary superconformal structure let us mark the set of points P1. . . . . P2g-2 on X(t) (or on 20(0). Let g~ = (z~, 0~) be the local parameters in the neighborhood of P+ such that z,(P,) = 0. In the vicinity of each marked point, consider an odd-parametric family of coordinate transformations given by meromorphic functions = zo + OoxoCzo)
~ = 0~ + X ~ ( z ~ )
1
(2.6)
M. Bershadsky / Super-Riemann surfaces
83
A
X(t,X)
=
= O + X Z -I
Z,O
A
Xo(t)
=
1 Z
X(t)=
Fig. 1. For a general (P~) the odd coefficients Xl---X2g-2 play a role of local parameters over a supermoduli space. Note that coefficients X~ are not invariantly defined. U n d e r the change of local parameters z, ~ )t~z~ one also has to rescale the odd coefficients X~ ~ ()ka)3/2Xa. In this case the superconformal structure does not change. The construction of SRS discussed above is schematically represented in fig. 1. We are going to say that if X, ¢ 0 then in the neighborhood of P, the coordinate ~ ceases to be proper and the proper coordinate system is defined by eq. (2.6). In spite of the last circumstance we are going to express different objects, such as sections of line bundles, in terms of 2~ variables, keeping in mind that one may always rewrite them in terms of proper variables. So it is possible to define the objects on SRS in terms of the objects given on RS. Consider t A = (t ~, X~) as local coordinates on the supermoduli space. The parameters X ~ correspond to points P~, and t ~ are some even parameters over supermoduli space. We represent small variations of superconformal structure using the m e r o m o r p h i c vector fields E A where A ~ (a, a). The vector fields E~ are uniquely determined by the diagram ^
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M. Bershadsky / Super-Riemannsurfaces
N a m e l y ~x+sx = ~x - X S X ~ x 2 + ~^x~X(x ~, ~x+sx = - " , and thus Ex(~ ) = - X ~ 2~/x~x 1. Changing the variables ~ --* 2 one obtains E, =X~(z~)
2+ 2 0 ( z ~ ) - 1
2+
(2.7)
W h a t about the vector fields E 2 The only thing it is necessary to care about is that the vector fields E A have to correspond to some choice of coordinates over a supermoduli space. That implies the vector fields E A have to commute with each other in the sense that
eB(... I/'+ ~/.A) o E~(... I/') = EA(... I/'+ ~/'8) o es(... I/')-
(2.8)
N o t e that eq. (2.8) is not an ordinary commutation relation because the vector fields E A, E B correspond to two different points in the supermoduli space. The vector fields E A ( . . . ) determine a set of vectors in the tangent space to the supermoduli space labeled by index A. If one has the family of vector fields EA(... 1/') depending on /', one can construct the set of vector fields over supermoduli space. The condition (2.8) insures that these vector fields are tangent to some coordinate lines. It turns out to be very convenient to deal with vector fields corresponding to some choice of coordinates instead of the coordinates itself. Below we represent two choices of vector fields defined on a curve that correspond to some choice of local coordinates over supermoduli space E ~ ( 2 ) = (z~)-1¢0 -x ,
(2.9a)
E ~ ( ~ ) = [ X ~ ( Z ~ ) - 2 + 20~(z,)
1]~o
1,
(2.9b)
where we have introduced the additional set of ( 3 g - 3) points Q~ and the local coordinates z a in the neighborhood of each Q,. The points Q~ correspond to the choice of local even coordinates over supermoduli space. It is obvious that vector fields defined in neighborhoods of different points c o m m u t e with each other. Considering the naive limit of Qa ~ P~ (for a ~< 2g - 2) one finds that vector fields cease to commute. For the coinciding points one has to modify this set of vector fields Ea(2 ) = ( z a ) - 2 w 1,
a~<2g-2,
Ea(2 )=(za)-l,0
a>2g-2,
1,
Co(2) = [Xo(zo) -2 + 20o(zo)-1],o
Qa=P,,
(2.10a)
(2.lOb) '
(2.10c)
M. Bershadsky / Super-Riemann surfaces
85
3. Geometryof Super-Riemannsurface (SRS) In sect. 2 we described a family of mappings that map the split SRS into a family of SRS X(t, X ) '
x
)~o(t) "
(3.1)
This construction allows us to express different objects on SRS in terms of objects defined on split SRS. The notion of superholomorphic line bundles was introduced in refs. [28,29]. Here, we consider only the bundles of forms ~2~°. Projecting the bundle g2~° on split SRS one finds it is given by two bundles 12~° and I2~ ° ® K ~ 1/2, where Kn ~1/2 is the spin bundle associated to SRS (we also assume that n is an even non-singular spin structure). The question of the existence of superholomorphic (or meromorphic) sections of I2~ ° may be reduced to the existence of some meromorphic sections qo(z)~ ~2~°, ~b(z)~ g2~?°® Kn~1/2. Let us assume first that section ~ I2~ ° is superanalytic in the neighborhood of P and thus it can be expanded in Taylor series. Let us express the section • in the neighborhood of a marked point P in terms of a £-variable
• (-~) = [M(2)I J~(~(_~)),
(3.2)
where M(£) is the scale factor (2.2). One may easily see that the components q0(z) and ~b(z) ( q ) ( £ ) = q~(z)+ Off(z)) have to be meromorphic with poles of the first and second order, respectively. The coefficients in the polar parts are not arbitrary but satisfy the equations ~-1
= X~o.
(3.3a)
1]./_1 = (1 - 2j)Xq~a,
(3.3b)
~b_ 2 = - 2 jXCPo,
(3.3c)
where cpj (+j) are the coefficients of Laurent series of cp(z) (@(z)). One can check that the equations (3.3) are invariant under the change of local parameter z ~ f ( z ) , in spite of the fact that Laurent coefficients are not invariantly defined (note that one also has to rescale X). Consider the space of superholomorphic half-differentials. The dimension of this space is equal to 0]g (for an even non-singular spin structure!). The superholomorphic half-differentials play exactly the same role in the theory of SRS as the abelian differentials in the theory of ordinary RS. They may be integrated over cycles via fv(Ber) tb.
(3.4)
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M. Bershadsky / Super-Riemannsurfaces
We denote by (Ber) a covariant volume of integration. This integral depends only on the homotopy class of the contour ,1 and not on the contour itself in spite of the presence of the poles. This happens because of the vanishing of the residue of +(z) when j = ½ (see eq. (3.3)). Thus, it is possible to define the (g × g) matrix of super periods [4,16, 28]. Let us find an explicit expression for superholomorphic half-differentials. In order to do this we introduce the bases of meromorphic differentials {p~(dz) 1/2} and { I~ dz }, having a definite asymptotic behavior
.o(zo)-(zo) zo- e°,
(3.5a) (3.5b)
We also assume that all I~(z)dz have zero a-periods. Then, the bases (3.5a) and (3.5b) are uniquely defined. Finally, let ~%(z)dz be the normalized basis of holomorphic one-differentials (9~ ¢0kdz = 8~k). The basis in the space of superholomorphic half-differentials may be written in the form (3.6) The coefficients A~ and C~ are determined by the condition that the components of half-differentials have to satisfy eqs. (3.3a)-(3.3c). Solving these equations one finds A=coX(1-pxHx)
1,
C=cox(1-uxHx)-lpX,
(3.7)
where the matrices co, X, P, H are defined as follows co= (co,(P~)),
X = (X~6~t~},
(3.8a)
v={v~(P~)}~,l ~,
H={I~(P¢)}~¢.
(3.8b)
Note that only the off-diagonal elements of p and H contribute to the expressions (3.7). The matrix elements u,¢ and H~¢ may be expressed through the correlation functions of fermion fields and currents (3.9a)
H~B = ((OzXIz=pa O~Xl~=p,)) - w(w+(Im T ) - x w ) ~ B. We introduce the notation ( ( A B . . . ) ) = ( A B . . . ) / ( I ) . a crucial role in our discussion.
(3.9b)
The last identities will play
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87
Now we are ready to define the matrix of superperiods. The half-differentials • k(2) are normalized in such a way that
f~ (Ber) ~)k= 3,k.
(3.10)
So one define the matrix of superperiods in the usuat way YT,k = £(Ber)q~k = T,k + 2~ri(~x(1 - v x H x )
1
t
(3.11)
vxco )ik'
where T~k is the ordinary period matrix of the Riemann surface X ( t ) (see fig. 1). The matrix elements J/k are even elements of exterior algebra on 2 g - 2-elements A = d~k(X1. . . . X 2g- 2). Note that the matrix of superperiods defined above explicitly depends on spin structure of SRS through the matrix v. It would be very interesting to understand the general properties of the matrix of superperiods. First of all, J - is obviously symmetric. Under an infinitesimal shift over the supermoduli space given by E ~ Hi(U, TX), the variation of Y is given by
8 k- f(Ber)e(a,,(Da,,) +
(3.12)
If E is cohomologous to zero, then 3 Y = 0. One may also try to define a modular transformation of the matrix of superperiods. The modular group changes the basis F
F
of cycles { ai, b i } ~ { a', b/' }. But it also acts on the spin structure n ~ m. So one may naively expect that under modular transformation the matrix of superperiods transforms by F
5~ ~ ( A J m + B ) ( C J m + D)
x.
(3.13)
But it is not so simple. The main problem is not to define a modular transformation of the matrix of superperiods but to understand whether or not the choice of coordinates (X~} could be expanded globally in a way compatible with modular invariance (or covariance). Note that it is also possible to formally define the Jacobi map from the ordinary Riemann surface ( X ( t ) - ( 0 P~)) into the coset space J = A~)/(I, ,Y')
z ~ Z k= f ( a.
zB
er)~k.
(3.14)
The coset space J is an analogue of the jacobian variety. It is also not a problem to
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M. Bershadsky / Super-Riemann surfaces
define an analogue of the theta function (generalized theta function) via O[oI(ZI'Y-) =
exP(½m'J'm~+m'Zt),
E
(3.15)
mEZg
for Z ~ A~. The generalized theta function with non-zero characteristic may be defined as usual. One may prove that the infinite sum (3.15) converges and that the generalized theta function is a well-defined object. (For an ordinary theta function see ref. [30]). Note that O(Z(z)I.Y- ) is a meromorphic section of a line bundle on X(t) with poles of first order at P,. We hope that the generalized theta function (see eq. (3.15)) will turn out to be a useful object in the theory of SRS.
4. Measure
In this section we derive an expression for the measure over the supermoduli space. We are going to consider the scattering amplitudes and the partition function on the same footing. The only difference is that one has to consider SRS with punctures. So the amplitudes are given by measures over supermoduli spaces of SRS with punctures. Note that for SRS there are punctures of two k i n d s - N e v e u Schwarz punctures and Ramond punctures. In other words, it is necessary to insert the emission vertices into correlation function. These insertions simulate the proper asymptotic behavior in the vicinity of punctures. Asymptotics are predicted by operator expansion rules. For example, the insertion of exp(ik~X(R,)) means that the correlation function containing the current I(z) = OzX has to be a meromorphic one-differential over z with poles in R , and residues equal to k,. So one may forget about vertices and evaluate the path integral with these boundary conditions (asymptotic conditions) in the vicinity of punctures. Let us recall that emission vertices, for example, for massless particles, may be written in the form
VF1/2 = u,S~Sgc e i?x,
(4.1a)
VB1 = E,~"vc e i?x,
(4.1b)
where u,, % are polarizations. (We present here only the chiral part of the vertices.) At the first glance, something is missing. The vertex (4.1a) is not a full vertex for the spin particle constructed in refs. [12,13] but only a part of it. But, as was shown in F ref. [31], the second part of the vertex V +Fl / 2 -_ [Q, ( V -1/2] m a y be understood as the result of integration over the supermoduli. In order to follow the dependence on the supermoduli explicitly one has to consider vertices written in the form (4.1). Let us pay attention to how the vertices (4.1) depend on ghost fields. Sg(w) is the ghost field of dimension (3) corresponding to Ramond puncturing. It forces the
M. Bershad~ky / Super-Riemann surfaces
89
ghost fields fl(z) and X(z) to have branches ~k( Z )Sg( W ) = ( Z -- w )l/2( ~koSg) At- . . . . ~(Z)gg(W)
= (t
-w)
-1/2
(8_1 S 8)+
(4.2a) ....
(4.2b)
The field p(w) has dimension ~ and makes the supervector field ~(z) go to zero when " z " tends to the puncture
X ( z ) p ( w ) = ( z - w) +l(Xl/2U ) + . . . ,
(4.3a)
¢(z).(w) = ( z - w)-l(B_ j )
(4.3b)
+ ....
Thus, in evaluating the path integral one may relax the class of fields and say that the path integral runs over fields that may have branches, poles or zeroes as predicted by eqs. (4.2) or (4.3). That implies that one has more zero modes than for a smooth SRS without punctures. Zero modes are now meromorphic differentials having poles in the Neveu-Schwarz punctures and having branches in the Ramond punctures. In spite of the fact that zero modes have poles or branches we will call them holomorphic, keeping in mind that they are holomorphic everywhere except the punctures. The dimension of the supermoduli space is the number of zero modes and is equal to ( 2 g - 2 + N + R [ 3 g - 3 + N + ½R) where N (R) is the number of the Neveu-Schwarz (Ramond) punctures. Note that in order to derive the standard expressions for the vertices it is necessary to consider the singular limit where ( N + Q) or ( N + ½R) points associated with odd coordinates tend to the positions of the vertices. Let us come back to the measure. Let t A be the local coordinates over the supermoduli space. Under the local change of variables, t A ~ YcAtc the measure Z has to transform via Z ~ (Ber Y)-1Z. Consider an ansatz for the measure given by the ratio (correlation function) Ber(zero modes)
(4.4)
The detailed structure of the denominator will be explained at the end of this section. It is constructed from zero modes - superholomorphic ~-differentials dual to the choice of coordinates over the supermoduli space. This insures the desirable transformation properties for the measure. Consider an infinitesimal change of local parameters X ~ X + 8X. The variation of the correlation function is given by 8×(corr. funct.) =
f(Ber) axE×(W...),
(4.5)
90
M. Bershadsky / Super-Riemann surfaces"
where W= ½J + OT is the super stress-energy tensor. The vector field E× corresponds to the infinitesimal shift X --* X + 8X and is given by eq. (2.9b). Rewriting eq. (4.5) in terms of t-variables living on a split SRS one can show that the correlation function satisfies the system of differential equations 0 OX---£( . . . ) = ( L 3(P~) .--) + ( J - 3 / 2 ( P ~ ) . - - ) ,
(4.6)
where L _ 3 ( P a ) = f ~ p T ( z ) ( z - P a ) - 2 d x , J _ 3 / 2 ( P a ) = fDpJ(z)(z - Pa) 1 dz. Integrating these equations one finds that the correlation function is given by the expression (4.7)
(H(1-x,,J(P,,))...).
Points denote operators that do not depend on X~. All the dependence on odd parameters X, comes through the insertion operators ( 1 - X~J(P~)) which determine the shifts in the odd directions. Note that the operator (1 - x J ( P ) ) simulates the proper asymptotic behavior. Using the operator product expansion rules, one can show that the function F(~) = <(¢p(z) + 0~b(z))(1 - x J ( P ) ) . . . ) ,
(4.8)
is superanalytic in the sense of eq. (3.3). So we have defined the correlation function in terms of the fields living on ordinary RS. When all the X~ become equal to zero, the correlation function has to reproduce the product of different determinants. If we are interested in amplitudes one also has to insert vertices (4.1) into correlation function (4.7). In order to define the correlation function (4.7) properly one has to insert (3g - 3) operators 3(b(w))= b(w) and ( 2 g - 2 ) operators 6(fl(x)) that absorb the zero modes contributions. Thus the correlation function depends on the position of the screening operators 8(b(w)) and 8(fl(x)). The screening operator 8(fl(x)) is determined by its operator product expansion rules /~(X)~(]~(X))
= (Z-- X)(/~_5/2~(~))
= (z-
x)
-4- . . . .
(x3/28(B)) + . . . .
(4.9a) (4.9b)
The ghosts fields b(z) and c(z) do not interact with the field 8(fl). We also demand the field 8(fl) to be fermionic. One may easily derive that the dimension of 8(fl) is equal to - 3. Note that it is possible to bosonize the field 8(fl)= e ¢ in accordance with the rules suggested in ref. [12].
M. Bershadsky / Super-Riemannsurfaces
91
The correlation function (4.7) does not depend on the choice of coordinates over supermoduli space. This dependence comes through the denominator - Berezinian of zero modes. Consider the set of vector fields EA(2 ) corresponding to the choice of coordinates over the supermoduli space. We define the basis of superholomorphic 32-differentials BC(2) dual to the vector fields E A via
(F_,,[Bc)
= ~ ( B e r ) Ea (2)
BC(~.) =
3if( - 1) y
(4.10)
Under the change of local coordinates over the supermoduli space differentials BA(£) transforms exactly like tn: BA(2) --, y~BC(£). Keeping in mind that screening fields inseted into correlation functions are not superfields, we construct the matrix of zero modes from the components of BA(£). The components of superfields are defined with respect to coordinates on a split SRS. Let b ( z ) = ((--1)AbA(z)), ~(X)= (flA(Z)) be the vector rows where bA(z) and flA(Z) are the components of superfields B A ( 2 ) = flA(z)+ ObA(z). We construct the matrix of zero modes from the rows b(w~) and ~(x,). The resulting expression for the measure for the heterotic string is written as follows
(I-I 2g- 2(1 - X,,J(P,~))[I 2g- 23( fl(x~))l-13 g- 3b(w~) vertices} Z(/',t)=
Ber[b(%)"'b(w3g
3)~(X1)'''~(X2g 2 ) ] ~
(4.11)
For the heterotic string there is no left supermoduli and so we suppress the contribution of the left movers. It is easy to see that expression (4.11) transforms as a measure under the change of coordinates. Note that Berezinian of zero modes is a meromorphic 2-differential (3-differential) over each variable % (x~) just like the correlation function. Thus, the ratio (4.11) has to be at least the meromorphic function over each variable %, x~. We will demonstrate below the measure (4.11) does not depend on the position of the points %, x , at all. This is a superanalog of Quillen's theorem [32]. The measure for the superstring theory may be schematically written as Z(F, t) = ( [ I . ( 1 - x J ( P . ) ) ( 1
- ~ . a ; ( R . ) ) (screening fields) }
Ber(right)Ber(left)
(4.12)
We assume that right and left movers correspond in general to different spin structures. The dependence on the position of screening fields is eliminated by the Berezinian of zero modes. These ansatz for the measures (4.11) and (4.12) provides a very natural way to investigate the holomorphic properties of the measure. This will be done in sect. 5.
92
M. Bershadsky / Super-Riemann surfaces 5. Calculations
In this section we calculate the expression for the measure in terms of meromorphic differentials on RS and determinants of ordinary 0-operators. For simplicity, we consider the partition function. There are three different contributions for the measure (1) Berezinian of zero modes; (2) Correlation function of matter fields; (3) Correlation function of ghost fields. Let us briefly sketch the evaluation of a determinant of zero modes. We choose the superholomorphic -32-differentials to be dual to the vector fields given by eq. (2.12). As was shown in sect. 3 , the superholomorphic j-differentials may be written in terms of meromorphic j and ( j + ±)2 differentials (here one has 3 and 2 differentials). To specify the basis of 2-differentials we mark the points P2g 1..... P3g 3 except the points P1 ..... P2g 2. Assuming that the Latin (Greek) indices run over the set 1, 2 . . . . . 3 g - 3 (1,2,..., 2 g - 2) we define the following bases in the spaces of meromorphic 3 and 2 differentials fl~(z)(dz)3/2:
z ~ P~,
"/~(z)(dz)3/2:
z ~ P~,
/3,~(PB) = 8,~/~,
(5.1a)
y,,(z) - 8,#~(zB) 1 + O(zB),
(5.1b)
b~(z)(dz)2;
z~P~,
ba(Pc) = Sac ,
(5.1C)
d~(z)(dz)2:
z ~Pb,
d ~ ( z ) - 8~e(zb) -x + O ( z b ) ,
(5.1d)
f~(z)(dz)2:
z ~ P b,
f~(z) - 8,~B(zl,) 2 + O(zb).
(5.1e)
The components of superholomorphic K-differentials are meromorphic v3 and 2-differentials satisfying the eq. (3.3). Omitting rather tedious calculations one obtains that Berezinian of zero modes is equal to Ber = d e t ( b ' + 28X'~'xd')-1(det/~) ×det[1 + 2 8 X ( V ' -
l(det b)
"y(~)-lflt)X6~(I))-l--36X'~(/~)
1xf(b)-l],
(5.2)
where we have introduced the following matrices (b')a c = b~(Pc),
(d'),~ = d:(P,),
(fl')~,, = flZ(P~,),
(-~') ~, = y~' ( P , ) ,
(b),,c = b,,(wc),
(c[),,a = d,~(w,,),
(fl)~ = b~(x,),
('~),~, = V,~(x~,).
(8)a~ = 8 ~ ,
(5.3a) (5.3b)
(f)~
= f,(w,),
(5.3c) (5.3d)
M. Bershadsky / Super-Riemann surfaces
93
The correlation functions of (b, c) or (/3, X) ghost systems may be evaluated using the Wick theorem. One only has to remember the statistics of (b, c) and (fl, X) systems. Let us introduce the notations ( ( X ) ) = ( X(screening fields) )/(screening fields).
(5.4)
By "screening fields" we mean the products of operators that absorb the contribution of zero modes [l-Ib(%),l-IS(/3(x~)) or lqb(wo)Fl,~(/3(x~))]. Then the Green functions are given by
ac, (z, w) = ((¢(z)b(w))),
(5.5a)
Gv,,(z, w) = ( @ ( z ) / 3 ( w ) ) ) .
(5.5b)
Note that these Green functions may be written in terms of differentials of type (5.1) _ dz(w),
(5.6a)
Gx,,(z, w) : ~/z(x,)(fl-1)~" fl,(w) - yz(w),
(5.6b)
Gc, b(Z, W) : dz(wa)([)
1)akbk(W )
where dz(w ) (yz(w)) is meromorphic 2 (~) differential over "w" with the only pole in " z " and having zeroes in P1..... P3g-3 (Px ..... P2g-2). It turns out that the ghost correlation function
may be evaluated in terms of meromorphic differentials defined by eq. (5.1). We remind the reader that supercurrent Jgh(Z) is given by the expression Jgh = b X 3fl 0~c - 2( Ozfl)c. Let us integrate in correlation function (5.7) over superghost fields /3 and X. Then correlation function (5.7) reduces to the average ( ( exp [3b (P~)X,((X (P,)/3(P.))),.xX~e'(P~)
+
(5.8)
From the path-integral point of view one may easily derive that the average (5.8) is nothing else than the determinant of operator [1 - / ( ( 0 1)-~] where the kernel o f / (
94
M. Bershadsky / Super-Riemannsurfaces"
is given by the expression
8(e- ,o)a(,-
h)= 2 z
(5.9) Matrices /3',-/',/q, "~, are defined by eqs. (5.3b) and (5.3c), (/3)~ = &(P~)= 8,~, ('Y)~/~='/~(Pp)l~,p. Note that the kernel of (0-1) ~ coincides with the Green function ((c(z)b(w))). Due to the singular structure o f / ( the only matrix elements that contribute to det[1 -/~(}_~)-1] are ((c(P~)b(P~))) = (el(b) lb - d),~. Evaluating this determinant one gets the result ( ( I ~ e x p [ - X~Jgh( P.)] ) ) = det[1 + 28X(~/-
~(~)-l~')Xd(h) 1
_38X-~(~)-lXf (~9) 1]. (5.10) This is exactly the last term in expression (5.2). Substituting the formulas (5.2) and (5.10) into the expression for heterotic measure one finds
Z(t,i) = (I~Ia (1-XaJmat(ea)))det(b'+28Xg'X
d')
× (II~b(w,,)H,~8(~(x.))) det/~(left ghosts) det b
(5.11)
det(left)
Now it is obvious that measure (5.11) does not depend on the position of screening operators. For the measure in superstring theory one can get the expression
z ( L t ) = (I-Iexp-[XaJmat(Pa)
-
X~&at(R~)] )det(b' + 28XY'xd')
(Hb(w.)H6(B(x,~))) ×
det/~
detdet
/~ (left ghosts) Ber(left)
(5.12)
Let us discuss the analytic properties of the measures (5.11) and (5.12). For the bosonic case the nonanalytic factor over moduli space coincides with (det Im T) -13. This factor comes from the determinant of scalar laplacian. The case of supertheory is more subtle due to the presence of the correlation function of supercurrents Jmat(Z) =~b 3 z X which is not holomorphic over moduli space. The correlation
M. Bershadsky / Super-Riemannsurfaces
95
function 1--/exp( - X a Jmat (Pa))
) x, ~,
(5.13)
may be easily computed using the same trick considered for ghosts. Integrating over scalar fields one gets
Idet0012(detlmT)-5(I-Iexp-[q~(P~)x.G(P~,Pa)xa+(Pa)]), a
where
G(P~, PB) is current-current Green function given by G( P~, Pa) = H~¢ - ~r( ~0t(Im T )-1~0 ) ~ .
(5.15)
The matrices H, o~ were defined in eq. (3.8). Averaging over fermionic fields one obtains
(1--Iexp__XaJmat(P,~)) = [det ~01 lO(det ~,/2)S(det 01/2 )m 5 × det(1 -
vxH x)s(det Im T) -5
xdet[1 + ~ r ( 1 - p x H x ) - l ~ , X o £ ( I m T ) - l o ~ X ]
5, (5.16)
where n ( ~ ) is the spin structure related to the right (left) movers. Using the identity det(1 + AB)= det(1 + BA) -1 that is valid for odd matrices A and B, one may rearrange the two last determinants from eq. (5.16) into the determinant of the matrix of superperiods (see eq. (3.12)) (det Im T) -5 det[1 + ~r(1 -
vXHX)- lvxcot(Im T)-I~oX] 5
=(detlmT)_Sdet[1 +~(ImT)_Xox(I_pxHx) -
det( J R - TL) 5
lpxoot] 5 (5.17)
Now substituting expression (5.16) into (5.12) one finds that the measure for the heterotic string theory may be written in the form
Z,,~,(?, i) = det(YR _ ~L)5 ,
(5.18)
M. Bershadsky / Super-Riemann surfaces
96
where the chiral measure Zn(f ) is given by Zn(~) = (det0o) 5(d e t -01/2) n s det(1 -
vxHx) 5
28X'l'xd')(~b(Pa) ~8(fl(P,,))).
× det(b' +
(5.19)
The dependence on the odd parameters is now manifest. Contributions from the left movers G~(i) depend on the model and are merely a product of different determinants. Note that all ingredients of formula (5.19) may be expressed through theta functions. Now let us evaluate the anomaly factor in the superstring theory. All nonanalytic contributions come through the correlation function of matter fields. Following the arguments for the heterotic case one can evaluate the correlation function of matter fields ( ~ exp
- [XaJmat(P,~) + ~,]mat (Ra)]>
= Idet 0o1-1°( det -3 1"/ 2 ) 5(det
Oq~/2)5det(1
-
x det(1 - ~ H 2 ) 5 ( d e t Im T) 5 det(1 +
vxHx) s AB) 5,
(5.20)
where the matrices A, B are given by the expressions (1
-
pXHx)-lvxoo t
0
A =
0 (Imr) B=
1~ox
_ (Im T)-l~o x
(1 - ~ 2 H 2 ) - ' ~ ' - (Im T ) - I ~ ] . (Im T)
(5.22)
1~ ]
The matrices ~o, X, v, H are defined exactly in the same way as the matrices w, X, v, H (see eq. (3.8)) except for the complex conjugation, change of spin structure n-~ m and the replacement P,---, R~. Using the same trick as in the heterotic case one can rearrange the last two determinants in eq. (5.20) into determinants of the matrix of superperiods (det Im T ) - 5 det(1 +
BA)-5
=(detlmT)-SdetIl+~r(ImT) -Tr(ImT) - det( J R -- J-L )-5
l~ox(1-uxHx)-lvx ~°t 1~(1-~2H--~)
1~
t
1] 5 1 (5.23)
M. Bershadsky / Super-Riemannsurfaces
97
As the result the superstring measure (5.12) may be written in the form
=
det(,y-R - J-L ) 5'
(5.24)
where J R ( g L ) is the matrix of superperiods (see eq. (3.11)) corresponding to the right (left) supergeometry. Note that °JR and g e are not conjugate to each other. Both the anomalous factors in the heterotic and superstring theories may be written in the form (JR-
TR)ij = (q~l~°j),
(JR--gL),j
= (~]q)~),
(5.25) (5.26)
where ~ff (qbL) are superholomorphic }-differentials corresponding to the right (left) supergeometry and normalized by having zero a-periods.
6. Discussion
In the previous sections we constructed a measure over supermoduli space using some specific choice of local coordinates. Let us first establish the relations with other approaches. The approach based on picture changing operator (see the work by the Verlindes [1]) corresponds to point-like super-Beltrami differentials ~ ' , = O08(z - P~) that do not depend on odd variables X~- Our choice of coordinates is quite different. The super-Beltrami differentials used in this paper explicitly depend on odd variables via ~ = O06(x- P~)+ Ox~'(x- P~). To be precise, we work not with Beltrami differentials ~ , but with meromorphic vector fields E~. The relations between them is quite clear . / g ~ - DE,. The Verlindes' choice of superBeltrami differentials ./g~ and ours .//4, differ from one another by some gauge transformation. As was mentioned in sect. 1, on the other hand our choice of coordinates has a very explicit geometric interpretation. The odd parameters coincide with the residues of the transformation functions that map from split SRS into general SRS. Recently, an operator formalism for the conformal field theories on a Riemann surface was developed in refs. [33-35]. This formalism allows one to represent the loop measure via ( AIAI8((Ex ] B ) ) [ @ ,
(6.1)
where [q~) is the state corresponding to RS or SRS; 8((E~ ]B)) operators eliminating the contributions of zero modes. The set of vector fields E A correspond to some
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98
choice of coordinates over moduli or supermoduli space. (For the details see the original papers [34, 35].) It is easy to understand the relations between formalism considered in the present paper (see also ref. [4]) and the operator formalism. One may notice that the measure (4.11) or (4.12) may be written in the form
{FLs(< f, i~>)rlo (1 - xoJ(e~) ) split SRS Ber(
(6.2)
where the vector fields/~A = (/~a, /~) are given b y / ~ = (z - w~) 1, ]~a = (Z -- Xa) -1. Keeping in mind that superholomorphic ~-differentials B°(£) are dual to the vector fields E A corresponding to the choice of coordinates over supermoduli space, one may recognize that the denominator in eq. (6.2) is nothing but the jacobian factor Ber((/~AIBD)(--1) D'A+I)) = Ber(EA --,/~A)-
(6.3)
As a result, the expression for the measure may be rewritten as follows ( 1-I~ (( E A]B)) 1-I (1 - Xfl(P~))) A
(6.4) split SRS
This expression coincides with eq. (6.1) after the identification [ ~ ) = F [ ~ ( 1 Xfl(P~))Icp> where the state [~) corresponds to split SRS and the operator FI~(1 - X f l ( P ~ ) ) shifts among odd directions over supermoduli space. All approaches suffer exactly from the same problems. It is not known whether or not it is possible to define unambiguously the integral over supermoduli space, whether or not their exists a holomorphic and modular invariant choice of gauge slice, whether or not the supermoduli space split, etc . . . . The main obstacle is that it is impossible to globally choose points P~(t) to be holomorphically dependent on moduli space. In the intersection of coordinate patches (t, X) and (f, ~) one has two different sets of points P~(t) and /~(t'). The transition functions are, in general, of the form X~ = X~(f, )~),
(6.5a)
t a=ta(t ") h-ha(t ",X),
(6.5b)
where we extract nilpotent contribution h~(f, ~) from transition function ta(?, ~). Therefore, t or f are even coordinates over supermoduli space and they cannot be considered as coordinates over moduli space. It might be possible only if there exists such covering of supermoduli space that in the overlapping regions all nilpotent parts of transition functions (6.5b) are equal to zero. That exactly means that supermoduli space splits. It is not difficult to demonstrate that, for arbitrary two
M. Bershadsky / Super-Riemannsurfaces
99
sets o f points, the c o o r d i n a t e systems (t, X){po} a n d (/~, 2)(~o) are related to each o t h e r b y t r a n s f o r m a t i o n with n o n - z e r o n i l p o t e n t p a r t in eq. (6.6b) except for two cases: (a) surfaces of genus two - it is the reason why the s u p e r m o d u l i space of SRS of g e n u s two splits; (b) the p o i n t s { / 5 } m a k e p e r m u t a t i o n of { P~ }. U n f o r t u n a t e l y , the last c o n d i t i o n is too restrictive. P r o b a b l y we d e m a n d too m u c h a n d it is p o s s i b l e to consider n o t h o l o m o r p h i c b u t m e r o m o r p h i c slices (see ref. [17]) o r a n o t h e r c o o r d i n a t e system. F o r e x a m p l e one m a y m a r k N p o i n t s P 1 , - - - , PN on R S a n d a s s o c i a t e with each p o i n t Pj the set of o d d p a r a m e t e r s X~j) . . . . . Y(J) such z~. n2 t h a t all o d d p a r a m e t e r s satisfy ( E n j - 2 g + 2) relations. But it is again n o t the g e n e r a l c o o r d i n a t e system. Actually, there are a lot of possibilities b u t p r o b a b l y it is i m p o s s i b l e to choose a g o o d h o l o m o r p h i c slice at all. W e are n o t going to go too d e e p into d e t a i l s a n d repeat a r g u m e n t s a n d refer the r e a d e r to the original p a p e r s [18,191. I w o u l d like to t h a n k A. Eskin, J. Lattore, P. Nelson, A. R a d u l a n d C. Vafa for m a n y fruitful discussions. I a m especially very grateful to P. N e l s o n for r e a d i n g the manuscript.
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