Fermionic string and universal moduli space

Nuclear Physics B317 (1989) 323-343 North-Holland, Amsterdam

F E R M I O N I C STRING A N D U N I V E R S A L M O D U L I SPACE A.S. SCHWARZ Moscow Physical Engineering Institute, Kashirskoe Shosse 31, 115409 Moscow, USSR

Received 6 October 1988

A space is defined containing in some sense supermoduli spaces for all genera-universal moduli space (UMS). It is proved that a super Mumford form (holomorphic square root from the string measure) can be extended on UMS. A simple expression for this extension is given. 1. Introduction String theory in the Polyakov approach leads to an expression of the partition function and the string amplitudes through the integrals over finite-dimensional spaces J/gp (moduli spaces). In the case of the fermionic string JZp is the superconformal moduli space (moduli space of superconformal manifolds of genus p ) [1]. The contribution of J/lp corresponds to the p-loop contribution in perturbation theory. If we do not want to restrict ourselves to perturbation theory we must replace the spaces ./kip by some kind of universal moduli space. Such an approach for the bosonic string was sketched in refs. [2-7]. This approach is far from complete; however it leads to a very promising operator formulation of string theory. We have shown that for the fermionic string one can go much further [8]. The present paper contains a detailed exposition of ref. [8] and some new results in the same direction. A recent preprint [9] contains some interesting results about the operator formulation of fermionic string theory; however, it has no essential intersections with ref. [8] (which is not noted by the authors of ref. [9]) and the present paper. Refs. [3-7] are based on the consideration of the infinite-dimensional grassmannian manifold Gr. To define the manifold Gr one can use as the starting point the space H of functions on the circle [z t = 1; then the points of Gr are linear subspaces of H that are, in some sense, near to the subspace H spanned by z", n < 0. (There exist different modifications of the rigorous definition of Gr; one of these modifications will be given below.) The moduli spaces ~ ' p can be imbedded in G r by means of the so-called Krichever construction. (For every complex curve N and local coordinate z in N one can construct W(N) ~ Gr as a subspace of H, consisting of functions on the circle Iz] = 1 that can be extended holomorphically on the c o m p l e m e n t of the disk Iz[ < 1 in N.) The definition of the grassmannian manifold 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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G r and of the Krichever construction can be easily generalized to the supercase [8-10]. In the bosonic case the space W(N) obtained by means of the Krichever construction satisfies the condition W J-= F W, where F ~ / " and W 1 denotes the space of functions orthogonal to W with respect to the bilinear inner product 9~f(z)g(z) dz. (We use the symbol /" for the group of non-vanishing functions; the functions f e W multiplied by F ~ / " run over the subspace FW.) A similar assertion holds for the supercase. Therefore in both cases we introduce the space U M S ("universal moduli space") as the submanifold of Gr singled out by the condition W ± = F W . The main result of ref. [8] is an explicit formula giving a continuation of the so-called super Mumford form on UMS. (Remember that the super Mumford form [11] is a holomorphic square root from the string measure on supermoduli space [12].) Namely, the Mumford form is expressed through the super-analog of Sato's ~--function. To define the super T-function we introduce for every W ~ G r a finite-dimensional space d ( W ) as a space of functions f ~ W that can be extended holomorphically to the disk Iz[ ~< 1. Given a basis w of ~(W) = d ( W ) + / / ~ ¢ ( W ±) we can construct a basis ~ of W determined up to unimodular transformation. (Here /7 denotes the parity reversion.) The super ~--function ~(w, w',W, F ) , where W ~ Gr, F E F, w is a basis of W and w' is a basis of F W , is by definition the determinant of the matrix connecting the bases F~ and ~' in F W . (Note that this determinant is ill-defined in the bosonic case; therefore in the supercase the definition of the T-function is simpler then in the bosonic one.) The function

"r(/]rw ', w, W, F 3)

e(w, w', W) =

"r(IIw', w', W, F ) 3

can be considered as a holomorphic extension of the Mumford form on the UMS. (Here W ~ UMS, W ± = F W , F ~ F, w, w' and Hw' are bases in N(F3W), Z ( F W ) and N ( W ) = H 2 ; ( F W ) respectively.) More precisely, let W be obtained from the superconformal manifold N by means of the Krichever construction: W = W(N). The space ~ ¢ ( F k N ) can be identified with the space F(~ok) of holomorphic fields of type k on N and by this identification ~g/(w, w',W(N)) transforms into super Mumford form described in refs. [11,12, 17]. In the present paper we shall prove this main result of [8] and complete it by the expression of the super T-function through finite-dimensional determinants. It is known that the Mumford form can be expressed through holomorphic fields on super-conformal manifolds and their zeroes [11]; our expression for the super T-function is closely related with this assertion. The paper is organized as follows. For simplicity we begin with the bosonic case (sect. 2). We repeat the main assertions for the supercase (sect. 3) and then proceed to the results specific for this case (sect. 4). In appendix A we give some proofs and show that our results and proofs become more transparent if some concepts of modern mathematics are used (in particular, we use essentially the notion of Reidemeister torsion).

A.S. Schwar: / Fermiomc string

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2. The bosonie case Let H d e n o t e the linear space of sequences { a,,}, n ~ 7/, where a,, 4:0 only for finite n u m b e r of negative n and la,I < c n for some c. The generating function f ( z ) = Ea,,z n of a sequence { a n ) ~ H is holomorphic for ]z I < c 1, z 4: 0; at z = 0 this function can have a pole. Therefore H can be characterized as a space of germs of m e r o m o r p h i c functions at z = 0. The bilinear inner product in H can be defined by the formula

-+ 1

(f'f)

= 2rri

(z)f(z)dz=

Z

amfi,,.

(1)

(Here 7 is a circle ]z[ = e and e is sufficiently small.) Sometimes it is convenient to m o d i f y the definition of H. For example, one can define H as the space of square-integrable functions on the circle [z[ = 1. Then side by side with the bilinear inner p r o d u c t (1) one can introduce in H the hermitian inner product

(f,f)

=

1•/

l

,-v~v

_

~

(z)f(z)z ldz=~jo'f(~lf(m)d,~=Ea,,a,,. n

N o w 7 is the circle ]z] -- 1, z = e i~, f(cp) = Z a , , e inC. The standard basis z n in H is o r t h o n o r m a l with respect to this hermitian inner product. The modification of the definition of H is irrelevant for most of our considerations, but sometimes it is necessary (see appendix B). For every linear subspace W c H we define linear subspace W J- as a space of vectors g ~ H satisfying {g, f } = 0 for every f ~ W. The subspace s p a n n e d by the functions z", n < 0 will be denoted by H ; it is easy to check that H ± = H . The natural projection of H onto H will be denoted by 7 r . (If f = ~ a , , z n then ~" f=Y~,,
(2)

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A.S. Schwarz / Fermionic string

The n u m b e r ind(W) = dim d ( W )

(3)

- dim sff(W ±)

is by definition the index of Fredholm subspace W. (It coincides with index of Fredholm operator ~rw.) Let us define infinite-dimensional grassmannian Gr as a space consisting of such linear subspaces W c H that both W and W ± are Fredholm and (W 1)± = W. The non-zero elements of H form a group with respect to multiplication of functions; this group will be denoted by F. The operator f ( z ) ~ F(z)f(z), where f ~ H, F ~ F, acts in H; it generates a transformation of Gr which will be denoted by the same symbol F. (One has to check that F W ~ G r if W ~ Gr. The proof is based on obvious relation ( F W ) " = F - 1 W a where W c H, F e F.) Let N be a complex curve (compact one-dimensional complex manifold), z a local coordinate in a neighborhood U of n ~ N, (n corresponds to z = 0). Let us define a space W(N) as a subspace of H, consisting of functions which can be extended holomorphically on N. If L is a holomorphic line bundle over N and the trivialization of L over U is fixed, the function f ~ H can be considered as a section of L over a subset of U and we say that f ~ W(N, L) if this section can be extended to a holomorphic section over N \ n . It is easy to check that

(4)

W(N,L) ± = W(N,L* ® ~),

where ~0 denotes the canonical line bundle on N (i.e. holomorphic line bundle with transition functions O~/Oz, where ~ and z are complex local coordinates in N). R e m e m b e r that holomorphic sections of ~0 are holomorphic differentials. Therefore for f ~ W(N, L), g ~ W(N, L* @ ~0) the extension of fg on N \ n is a holomorphic differential and ( f , g ) = ~vfgdz = 0 because the circle 3' entering in the definition of the inner product is homologous to zero in N \ n . We see that W(N,L* @ ~o) c W ( N , L ) ± ; the inverse inclusion can be proved by means of R i e m a n n - R o c h theorem. It is obvious that ~¢(W(N, L)) coincides with the space f'(L) of holomorphic sections of L. Using this assertion and eq. (4) we obtain W(N, L) ~ Gr and i n d W ( N , L ) = d i m F ( L ) - d i m F ( L * ® w) -- m - p

+ 1,

(s)

where p denotes the genus of N and m denotes the degree of L. (Of course (5) follows from R i e m a n n - R o c h theorem.) It is easy to verify that for every F ~ F the subspace F W ( N , L ) coincides with W(N, L') for another line bundle L' and for arbitrary line bundles L, L' over N one can find such an element A ~ F and W(N, L') = AW(N, L). For example, to prove the last assertion one can use that W(N, L) = AW(N) where L is arbitrary holomorphic line bundle, A is a function which can be extended to a non-vanishing holomorphic section of L over N \ n . (Remember that we have fixed a trivialization

A.S. Schwarz / Fermionic string

327

of L over U and therefore A ~ H can be considered as a section of L over a subset of N.) C o m b i n i n g this assertion with eq. (4) we obtain the following: for every space W = W(N, L) one can choose F ~ F in such a way that (6)

W i = FW.

(In other words W and W ± belong to the same orbit of F in Gr.) The set of points W ~ G r satisfying (6) will be denoted UMS. ( U M S is an abbreviation for "universal m o d u l i space". We see that moduli spaces of complex curves of all genera are in s o m e sense i m b e d d e d in UMS.) Let us denote by F~(W) the direct sum s~'(W) + zac(W ±). If W = W(N, ~0k) then 2 ( W ( N , ~ok)) = F(~o k) + F(~o l - k ) = Z k

(7)

where/~(~o k) is the space of holomorphic k-differentials. The M u m f o r d form can be considered as a function J g ( e , f ) where e is a basis in Z ( W ( N , ¢o2)) = ~2 and f is a basis in ~ ( W ( N ) ) = ~ ( W ( N , ~ o ) ) = S 1. (If p > l we have ~2 =F(~0 2) and e consists of 3p - 3 quadratic differentials.) The weight of M u m f o r d form ~ ( e , f ) with respect to e is equal to - 1 and with respect to f is equal to 13. (In other words by the change of bases e i ~ a~ej, f k ~ blkfl the M u m f o r d form is multiplied by (det a ) - X ( d e t b)13.) In the case p > 1 the arguments of the M u m f o r d form ~¢'(e, f ) can be considered as bases in F ( w 2) and F(co) respectively. If c~, f i e F(¢o) (i.e. c~, fl are abelian differentials) then we can define hermitian inner product (c~, fl) = f ~ A ft. T h e M u m f o r d form determines a real measure on the moduli space J/gp by the formula /~(e)=[Jd(Y,f)]

2,

(8)

where f is an orthonormal basis in F(~0), e is a basis in /'(w2) * and Y is a basis in f'(~0 2) dual to e. (The function # ( e ) has weight 1; hence this function determines a m e a s u r e on the tangent spaces to ~ ' p and therefore on JC/p.) The measure (8) coincides with the string measure. (This assertion is equivalent to the Belavin K n i z h n i k t h e o r e m [13].) Let w = ( % . . . . . wr) denote a basis in S(W), W ~ Gr. At first we will assume that s~e(W ") = 0 and therefore S ( W ) = d ( W ) , the operator ~r m a p s W o n t o H . We construct a basis ~b in W adding to the basis % .... , % the vectors v 1. . . . . v , , . . , satisfying 7 r _ ( v , , ) = z - ' . Of course, the vectors v i are not d e t e r m i n e d uniquely - one can replace vi by vi + Zajwj with arbitrary aj. However this a m b i g u i t y is not essential for our aims, because corresponding change of the basis "3 is described by a unimodular matrix. U n i m o d u l a r transformation of the basis w generates unimodular change of ,3 too. If s d ( W J - ) g 0 there exists a construction of the basis "3 having the same properties. Without loss of generality one can assume that the basis w = (w I . . . . . Wr) of Z ( W ) consists of the basis w I . . . . . % of ~ ( W ) and the basis %+1 . . . . . wr of s / ( W J - ) . Let us fix the elements

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A.S. Schwarz / Fermionic string

Ua Ur s E H satisfying
3. Superspaee T h e consideration of the preceding section can be generalized to the case in superspace without serious modifications. We will consider the superspace having b o t h even and odd part isomorphic to the space H defined above. One can interpret the even points of this superspace as meromorphic functions

O(z,O)=f(z) +ep(z)O where f(z) is an even function and ~ ( z ) is an odd one. We introduce an odd inner p r o d u c t in ~ as follows

OtdzdO- Jlzt I.)d.++tzl;(z)d. (As earlier ~, is a circle [z I = e , e--+0.) The subspace of JY spanned by the functions z", z"O, n < 0 will be denoted by ~ . The symbol ~)f+ denotes the subspace of holomorphic functions in J~. If W is a linear subspace in ~ then the o r t h o g o n a l subspace W " will be defined by means of the inner p r o d u c t (9). The kernel on the natural projection ~ rw of W into ~ will be denoted by sff(W) (i.e. s~'(W) = W A Jr'+). In full analogy with the bosonic case we define Fredholm subspaces of ~ " and the infinite-dimensional supermanifold Gr. However the inner p r o d u c t (9) is odd and therefore (2) must be replaced by (J~/Im~W)

* = H ( W ± A Jr+) = H a g ( W l ) ,

(10)

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A.S. Schwarz / Fermionic string

where /7 denotes the parity reversion. (If E is a linear superspace, the H E consists of the same points with reversed parity. For example, H ~ p' q = l~ q' P.) Let N be a (1, 1)-dimensional compact complex supermanifold, n is a point of N, (z, 0) are local coordinates in a neighborhood U of n. As above we define the linear subspaces W(N) c ~ and W(N, L) c . . ~ where L is a holomorphic line bundle over N and the trivialization of L over U is fixed. It is evident that zC'(W(N, L)) = Y(L) consists of holomorphic sections of L. It is possible that /'(L) cannot be considered as superspace. (The supermanifold N is considered here as A-manifold, i.e. as a manifold with coordinates from Grassmann algebra A. Therefore Y(L) is a A-module. If this A-module is free we say that f'(L) is a superspace; see ref. [14] for details.) It is easy to check that I~(L) is finite-dimensional. We will assume that Y(L) is a superspace. Then one can prove that W(N, L) ~ Gr. The proof is based on the existence of line bundle ~0, satisfying W ( N , L)* = W ( N , L * ®co)

(11)

for every holomorphic line bundle L. (The transition functions of holomorphic line bundle ~0 are (super) Jacobians ~(£,O)/~(z,O), where (Z, 0) and (z,O) are complex local coordinates on N. The sections of o~ can be identified with complex measures on N. Eq. (11) can be deduced from the results of ref. [15], where ~ is proved to be a dualizing sheaf on N.) The manifold N is called normal if all holomorphic functions on N are constant [14]. In this case W(N) ~ Gr and W(N) l = W(N, ~o) ~ Gr. Let us denote by F the subspace of S consisting of even invertible functions. One can consider F as a supergroup, acting in aU by means of multiplication. As in bosonic case ( F W ) ' = F - 1 W ±, where W c ~ g ', F~F. Using this relation we obtain F W ~ G r if W ~ G r and therefore Y acts in Gr. As earlier we single out UMS from Gr by means of requirement W ± = F W for certain F ~ Y and prove that W(N) ~ UMS if N is a normal manifold. For W ~ G r we define 27(W) by the formula X(W) =d(W)

+ ( Y d / I m ~w) * = d ( W )

+

IId(W±).

(12)

The same construction as in bosonic case gives a basis ~b in W for every basis w in Z(W). (The basis ~ is defined up to unimodular transformation.) If W = W(N), W ± = F W then z J ( F k W ) = F(~o k) is the space of holomorphic sections of the line bundle ~k and ~¢((FkW) ±) = d ( F - k W I ) = ~ ( F I - k W ) = Y(o~1 k). We see that 2~(FkW(N)) =

F(wk) +//p(o~l-k)

= 2:k(N) "

The super Mumford form / g ( e , f ) is a function depending on the basis e in •3(N) and of the basis f in •I(N) and having weights - 1 and 5 with respect to e and f

A.S. Schwarz / Ferrnionic string

330

[11, 12,16]. Our aim is to construct the continuation of J g ( e , f ) on UMS. Namely we have to construct a function JCZ(e,f,W), where W ~ U M S , e is a basis in X ( F 3 W ) , f is a basis in X ( F W ) , coinciding with ~ ( e , f ) for W = W ( N ) . This problem will be solved in the following section. 4. R e s u l t s for s u p e r s p a c e

Till now our consideration in superspace was very similar to the bosonic case. An essential difference arises in the definition of a superanalog of Sato's r-function. (For definition a of the r-function in the bosonic case, see for example ref. [16]). We will show that the super r-function is much simpler than the usual one. Let us consider W ~ G r , F ~ F and the bases w of X(W) and w' of X ( F W ) . The r-function will be defined by the formula

r(w, w',W, F ) = det[Ffflff' 1.

(13)

(Here ff and if' are the bases in W and F W , corresponding to w and w', Fff is the basis of F W obtained from the basis ~ by means of the transformation F. We use the notation det[blc] for determinant of matrix connecting the bases b,c.) It is important to stress that r.h.s, of eq. (13) is well-defined in the superspace but the corresponding expression in the usual case is meaningless. To analyze eq. (13) we note that in usual case infinite-dimensional determinant det T is well-defined if the matrix T - 1 belongs to the trace class. (Note that in this case the determinant is well-defined without regularization. The determinant of elliptic operator on compact manifold does not exist in the sense above, however it can be defined by means of regularization, for example by means of proper time cutoff). The condition of existence of the superdeterminant (berezinian): det T = det(T0orll 1 - rolrlxlr, orll 1)

(14)

is weaker. (Here Too denotes the even-even part of T, T0x denotes the even odd part of T, etc.) For example, it is sufficient to require that T~ t and Tx0 are bounded and the matrices 7ol and TooTll I - 1 belong to the trace class. (Then TooT111 - TolTlllTloTll 1 - 1 belongs to the trace class too.) We will prove under certain conditions that the determinant (13) is well-defined in the supercase (see appendix B). It is important to note that the calculation of r-function (13) can be reduced to the calculation of finite-dimensional determinants. Let us consider the case when the function F is holomorphic at the point z = 0, 0 = 0. Then F..~'(W) c d ( F W ) . If F - a is holomorphic at z---0, 0 = 0 too (i.e. if F(0, 0) is an invertible element of Grassmann algebra) we have ~(FW)

= Fsg(W), 5g((F~ff/) ±) ==~'(F xYU ±) = F - l d ( W ± ) .

A.S. Schwarz / Fermionicstring

331

These relations establish one-to-one correspondence between 2;(W) and ! ; ( F W ) ; we will denote this correspondence by a. One can check that (15)

T(W, W', W, F ) = det[c~wlw' ] .

Let us consider now the case F = z k, k > 0. We will choose arbitrary bases Ul,..., u t in z~c(W) and tt . . . . . t, in H s ¢ ( ( z k W ) J - ) = H d ( z k W ' ) . The bases in d ( z k W ) and H d ( W ±) will be chosen in the form (zkua,...,ZkUl, Vl . . . . . Vr) and (zktl . . . . . zkt~.,ql ..... qp). (In other words these bases are obtained by adding arbitrary elements vi, qj to the elements zku,, zkt~.) Rectangular matrix A = (a,x, a[x ), i = 1 . . . . . r, )t = 1 . . . . , k will be defined as the matrix of the coefficients in the decomposition

k

E as

?~=1

k

E a s'

(16)

h=l

The entries of the matrix B = (b/x, bj'x) will be defined by the formula

(qj, z k+x) = bjx,

(qj, z-k+~'O) = bj'x,

j = 1 ..... p,

~ = 1 ..... k.

(17)

One can prove that

(A)

r ( w , w ' , Y I / ' , z k ) =_+det ( B T ) - I

,

(as)

where ( B T ) - 1 denotes the left inverse to B T. In other words the matrix (BY) -1 can be found from the equation: (B T) 1BT= 1.

(19)

(The freedom in the choice of (B T) 1 has no influence on the value of the determinant in eq. (18)). Note that r + p = 2k and therefore the matrix in eq. (18) is quadratic. The proof of eqs. (15) and (18) will be given in appendix A. It is important to stress that eqs. (15) and (18) permit us to calculate the T-function for arbitrary F. Really it is evident that

T(W, w " , W , FG) = T ( W , w ' , W , G ) T ( w ' , w", GW, F ) , ~'(w,w',W,G)=T (here w, w' and Using eq. (20) we holomorphic at z holomorphic at z

I ( w ' , w , G W , G-1)

(20)

w" denote arbitrary bases in d ( W ) , ~¢(GW) and d ( F G W ) . ) can reduce the calculation of T(W, w',W, G) to the case when G is = 0, 0 = 0. In the last case G = zkGo where G o and Go I both are = 0, 0 = 0.

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A.S. Schwarz / Fermionicstring

Let us introduce the function J//(e, f , W ) by the formula

~(e,

"c(Hf, e , W , F 3) f , W ) : "c(Hf, f , W , F ) 3 "

(21)

Here W ± = F W (i.e. W ~ UMS), e is a basis in Z ( F 3 W ) , f is a basis in 2J(FW), and H f is a basis in Z(W). (We have used Z ( F W ) = ~;(W ±) = ~ ' ( W l ) + H a l ( W ) = H E ( W ) . ) It is easy to check that JY(e, f , W ) has weight 1 with respect to e and the weight 5 with respect to f. It is easy to check that in the case when there exists a freedom in the choice of F the expression (21) does not change by the variation of F. If W = W(N) (i.e. W is obtained by means of the Krichever construction from superconformal manifold N) then eq. (21) depends on the same arguments as the M u m f o r d form and has the same weights. In the bosonic case these facts would be sufficient to assert that J g ( e , f , W ) coincides (up to constant multiplier) with the M u m f o r d form. The situation in the supercase is more complicated. Remember however that we have expressed the ~--function through the finite-dimensional determinants. Using this fact we obtain an expression for Jk'(e, f , W ) through the m e r o m o r p h i c sections of line bundles over N. One can check that the expression of this type obtained in refs. [11,17] coincides with the one we have obtained and therefore (22) gives analytic continuation of super Mumford form on UMS.

Appendix A The formulations and the proofs of our results can be simplified by means of the notion of Reidemeister torsion. Let us consider a chain complex (a differential graded vector space) CO J0)C1

J,~ . . .

a" ~C".~

(A.1)

Here C i are vector spaces, the linear operator d i maps C i in C i+1, di+ld i = 0. (One can construct the space C as the direct sum of C i. Then C is a graded vector space, and the operators di, differentials, generate a map d of C into itself satisfying d 2 = 0.) We will denote the chain complex simply by C. The homology groups Hi(C) are defined as Z i / B i where Z / = K e r d i, W = Im di 1- In such a way the element of Hi(C) is a class of elements of Z i (class of cycles); if z 1, z 2 ~ Z i, z 1 - z 2 ~ B i (i.e. z l - z 2 is a boundary) the cycles z I and z 2 belong to the same in C i and bases h i = (h iL. . . . . i class. Let us fix bases c i = ( c~,. .., c,,(i)) i in Hi(C). Then the torsion can be defined by the formula tt

T o r c ( c ' , hi) = I-I d e t [ d i _ l ( b i - 1 ) h i b ' l c i ] i=O

(

w.

(A.2)

A.S. Schwarz / Fermionic string

333

Here h~ denotes a set of vectors in Z i ¢ C ~ satisfying %(/~i) = h ~, where ¢r~ is the natural projection Z ~~ Hi(C), b i denotes such a set of vectors in C i that d y is a basis in B ~+ 1 = Im di and det[a]b] denotes the determinant of the matrix connecting the bases a and b (i.e. the determinant of the matrix ak/ satisfying a k = Y~ldklbl). If all spaces are finite-dimensional it is easy to verify that T o r c ( c ~, h') does not depend on the choice of b' and h~. If there are infinite-dimensional spaces among C ~ the determinant det[di_l(b' 1)hibilci ] is ill-defined by unlucky choice of b i,/~i, of course we have to make this determinant well-defined by means of proper choice of b i,/~. The torsion does not change by variation of b ~,/~ only in the case when the determinants of matrices connecting corresponding bases are well-defined. The definition of Torc can also be applied in the case when C' are linear superspaces. Then det denotes the super determinant (berezinian) as always in this paper. It is evident that Torc is a function of the weight ( - l y +1 with respect to the basis c ~ and a function of the weight ( - l y with respect to the basis h ~. If the function of the basis of linear superspace E has weight 1 it can be considered as a measure on E, if such a function has weight - 1 it can be considered as a measure on E* where E* denotes the dual space. We will introduce the spaces (~ and I21 by the formulae:

d= Z c~ + Z (ck) *, k even

I:I=

k odd

T_, Hk( c)+ Z (H~(C)) *- (A.3) k even

k odd

The remarks above show that Torc can be considered as a function of measures in (~ and in H, having the weight - 1 with respect to the first measure and the weight + 1 with respect to the second one. One can say that the torsion generates one-to-one correspondence between the measures in (~ and I2I. (The bases c ~ in C' determine a measure in (~, the bases h ~ determine the measure in I2I. We say that the measure in (~ corresponds to the measure in lq if T o r c ( c ~, h ~) = 1. The correspondence between measures in (~ and IZI can be constructed in the following simple way. Let us denote by re(E) the one-dimensional linear space of measures in linear space E. It is easy to check that m ( E * ) -- m ( E ) * ,

m ( E l / E 2 ) = m(E1) ® m (E2)* ,

rn (Ea) = m ( E 2 ) ® m ( E 1 / E 2 )

(A.4)

(the sign = denotes canonical isomorphism). It follows from (A.4) that m(C) =

1~ m ( C k ) ® 1-I m ( C k ) *, k even

m(I~I) =

17 m ( H k ( C ) ) ® l - I m ( H k ( C ) ) *k even

(A.5)

k odd

(A.6)

k odd

F r o m the other side B k+l = C k / Z k (the differential d k maps C k onto B k+l and the

A.S. Schwarz / Ferrnionicstring

334

kernel is Zk), Hk(C) = Zk,/B k. We see that rn(C k) = m(B k+l) ® m ( Z k ) ,

(A.7)

m ( Z k) = m ( H k ( C ) ) ® rn(Bk).

(A.8)

Using eqs. (A.7), (A.8) we obtain immediately the canonical isomorphism between rn((~) and m(I2I). This isomorphism can be considered as an invariant version of the torsion. It is shown in refs. [18,19] that the torsion arises very naturally in physical considerations. (Note that in ref. [18] we have considered the torsion of elliptic complexes and used the determinants of elliptic operators by the definition of torsion. In the present paper we use the determinants of infinite matrices in our definitions). The connection between torsion and partition function established in ref. [18] permits one to give a physical explanation of the correspondence between measures in C and I2I, see ref. [19]. (We have used in ref. [19] a modified definition of (~ and I2I. Namely we have replaced (Ck) * and (Hk) * for k odd in the definition of (~ and I2I by H C k and H H k where H denotes the parity reversion. This is possible because m ( H E ) = m ( E * ) = re(E)* for every linear space E.) Let us consider the chain complex C = C o + C 1 where C o = W ~ Gr, C ~ = H (or C 1 - - a ~ in the supercase), d coincides with ~ w r on C o = W. It is obvious that H~{C) = C1/Im d = H / I m T r W = s f f ( W ± ) *

(A.9)

in the bosonic case, and H~(C) = C a / I m d = 3f°_/Im vrw = H ~ ( W ±) *

(A.10)

H°(C) = Ker T w r = s t (W)

(A.11)

in the supercase,

in both cases ((A.9) follows from (2) and (A.10) from (10)). The above consideration shows that the measure in = H°(C) + Hi(C) * = Z(W)

(A.12)

generates the measure in (~ = C ° + (C1) *. Noting that the standard basis in C ~ determines a measure in (C1) * we see that the measure in 2J(W) generates a measure in W = CO; if the measure in IJ(W) corresponds to a basis w it is easy to check that the measure in W corresponds to the basis ~. Let us consider the chain complex cO ,to,C1

d~, - . . --*C ~

(A.13)

A.S. Schwarz / Fermionicstring

335

and suppose that C' is a subspace of C i and d,(C ~) c C i+1. Then C i form a subcomplex and the spaces (~i= C , / ~ form a factor complex of C g0

ao,~l

a,,

...

~,n

Go

do ) E 1

dl :. • - • --~ d n .

(A.14) (A.15)

(The differential d, induces a map (~i ~ d,+l; we will denote this map by the same letter.) The homology groups of complexes C , C , C form an acyclic chain complex 0 ~ H°(C) ~ H°(C) ~ H ° ( d ) ~ Hi(C) . . . ~ Hk(C- ) ~ Hk(C) ~ H k ( d ) --, Hk+~(~) -~ . . .

(A.16)

(This chain complex is known as an exact sequence of the pair (C-, C).) The map Hk(C) ~ H k ( c ) is induced by the imbedding C k c C k and the map Hk(C) --* Hk(d) by the natural map ~r: C k ~ (~k. The map Hk(C) ~ H~+I(C-) is so-called boundary homomorphism. (If the cycle z in d k determines the homology class [z] in Hk(d) and u ~ C k obeys ~r(u) = z then the boundary homomorphism transforms [z] into the homology class of dkuECk.) Let us fix the bases in Ck,(~k,d k and Hk(C), Hk(C--), Hk(C). Then we can calculate the torsions $(C), T(C), T(C) and ~-(H) of complexes (A.13-A.16). Moreover for every k, 0 ~< k ~< n we can construct an acyclic chain complex c k -o C k ¢ r (~k;

(A.17)

the torsion of this complex will be denoted by ~'k- One can prove [20], that

T(c) = _+~(C)T(~)T(H) l~I T,(-1)*.

(A.18)

k=0

This equation is correct if all chain complexes are finite-dimensional. However under some conditions one can use (A.18) for infinite-dimensional chain complexes. We will apply (A.18) to the calculation of the super T-function. Let F(z, O)EJ¢' denote a function which can be extended holomorphically at z = 0, 0 - - 0 . We fix the bases w in 2J(W) and w' in Z ( F W ) , where W ~ Gr. The v-function T(w, w',W, F ) coincides with torsion of (infinite-dimensional) chain complex W --* F W

(A.19)

with respect to the bases • in W and ~' in F W . The proof follows immediately from definitions. We are able now to express the T-function through finite-dimensional determinants. Namely we will prove that the T-function coincides (up to sign)

A.S. Schwarz / Fermionic string

336

with the torsion ~ ( H ) of the finite-dimensional chain complex. 0 ~ . ~ ( W ) - * . ~ ( F W ) ~ ~ ~ l l . ~ (W l ) * ~ / 7 . ~ ¢ ' ( ( F W ) ±) *

(A.20)

where ~ is the kernel of the map ~r F o r ..st' onto 0f". (The homomorphisms in eq. (A.20) and the bases in the spaces of this complex will be described below). The proof is based on eq. (A.18) applied to the chain complexes C °=W Co=0

~-,~=C

1,

,~=~1.

(A.21) (A.22)

It is easy to see that the factor-complex W -~.~/o@

(A.23)

is isomorphic to the complex 120 = F W

" , ~=

121.

(A.24)

The complex (A.16) constructed for the triple (A.21, A.22, A.24) coincides with (A.20) because H°(C) =.if(W),

H°(C) = 0,

H i ( C ) = / ] - d ( W ±) * ,

H°((~) = . N ' ( F W ) ,

Hi(C) = 9 ,

HI((~) = / - I d ( ( F W ) ± ) * .

We assume that the bases in H°(C), Hi(C), H°(12), H1(12) generate the bases w and w' in ,~(W) and ~;(FW). In other words, if for example the bases u ° in H°(C) = d ( W ) and u 1 in Hi(C) = - / T d ( W ' ) * are fixed then the basis w in ,~(W) = sc'(W) + /-/zac(W l ) consists of the basis u ° and of the dual basis to u 1. We fix the bases ~, ~ ' in C o = W, 12o = F W and the standard basis z', z'O, n < 0, in C 1 = 121 = J d ' . The basis in 9 = •1 = HI(•) together with vectors ~r (Fz"), 7r(Fz'O) must give a basis in .X~_, connected with the standard one by means of a unimodular matrix. By this choice of bases "ro (the torsion of the complex 0--, W--, F W ) coincides with "r-l(w, w',W, F), ~1 (the torsion of the complex ~ ~ Of'_~ J{'_/~ = $fL) is equal to 1 (the last equation can be considered as a definition of the basis in 9 ) . Further, the torsions of the complexes C, C and 12 are equal to I too. (For C and 12 this assertion can be considered as a definition of the connection between bases w and ~, w' and ~'). We see that 1 =.(C) = +~(C)~(C)T(H)T,

.1 ~ = + 1 " 1. ~'(H)~-~(w, w',W, F ) - 1

A.S. Schwarz / Fermionic string

337

and therefore •

r) = -+*(m.

To give an explicit expression for ~-(H) we have to calculate the homomorphisms in eq. (A.20). The multiplication operator F generates the homomorphisms aag(W) - - * d ( F W ) and z J ( ( F W ) J - ) = d ( F - I W ±) - - * d ( W ' ) . The first of these homomorphisms and the dual to the second one enter in eq. (A.20). The map d ( F W ) --* transforms u e d ( F W ) into 7r ( F - l u ) . At last the map ~ H d ( W X ) * transforms the element v e @ in an odd functional g on a~c(W x) given by the formula ~(g) = @, F g ) = (Fv, g). In other words the dual map zac(W ~) ~ H ~ * transforms g ~za/(W *) into the odd functional ~ ( v ) = @, Fg) on 9 . All these assertions follow immediately from the definitions. If F and F - 1 a r e holomorphic at z = 0, 0 = 0 then ~ = O. In this case we can take the bases u 1. . . . . u / in z~c(W), FUl,..., Fu I in sC'(FW), t 1. . . . . t, in H~'((FW)±)=H,N'(F lW~), Ft t ..... Ft, in H ~ ( W ~) and the dual bases in H ~ C ( ( F W ) ")*, H s c / ( W ' ) * . It is easy to check that by this special choice of bases we have ~-(H) = 1 and therefore the ~--function is also equal to 1 (up to sign) by the corresponding choice of bases in ,Y(W), 2~(FW). This proves eq. (15). (It is sufficient to prove eq. (15) for special choice of the bases, because both sides of (15) transform identically by the change of bases). Let us consider now the case F = z ~. In this case ~ is spanned by z x z-XO, 2t = 1 ..... k, and hence ~ is isomorphic to (;k,k. To calculate the map ~ ¢ ( F W ) ~ = C k'x we must therefore expand u e s¢(FW):

u = Laizi+ i=0

La'iziO i=0

and take first k terms in both sums: k-1

k

= Z ,=o

1

k

k

Z ,=o

X=l

Instead of the map ~--* Hzag(W ±)* we will describe the dual map aag(W ±) ~ H.@*. If g ~ d ( W -L) is equal to g=

L g , z i + Lg,'ziO i=0

i=0

then the corresponding element of H ~ * is an odd functional on ~ taking the value g;( k - t at z -x and the value gx-k 1 at z x0. Now we calculate the ~--function .r(w, w',W, z k) for the case when the bases w, w' in

z(w) =d(w) +

+ Ud(

-kW

338

A.S. Schwarz / Fermionicstring

are constructed by means of bases (u a. . . . . ut),(ZkUl . . . . . zkul, vl . . . . . Vr), (tl . . . . . t , ) , ( z k t l . . . . . Zkts, ql . . . . . qp) in ~'(W), d ( z k W ) , H ~ C ( z - k W ±), H ~ c ( W +). As we have shown this r-function coincides (up to sign) with torsion r ( H ) of the acyclic complex (A.20) with respect to the same bases in s¢(W), ~aC(z~W) and dual bases in H ~ c ( W ± ) * , H ~ C ( z ~ W ± ) *. It is easy to choose the vectors b ~ in the expression (A.2) for the torsion r ( H ) in such a way that only one determinant in (A.2) differs from 1. It remains to check that this determinant coincides with eq. (18).

Appendix B Let us denote by ~ the group of invertible bounded operators in Hilbert space (i.e. T ~ ~ if T and T -1 are bounded). The set of operators belonging to the trace class will be denoted by ~ (i.e. T~.Y- if SpLT [ = Sp~@-VT < ~c). It is known that the determinant of infinite-dimensional matrix A is well-defined if A - - 1 + T, T ~ 3-; the set of matrices satisfying this condition will be denoted by 5". (We identify the matrices with operators in Hilbert space.) One can prove that det(AB) = det A • det B and det V a A V = det A (here A, B ~5O, V ~ ~ ) . Using these properties one can extend the definition of the determinant on larger classes of operators. Namely let Y/" denote such a subgroup of ~ that det A = 1 for every A ~ Y/'r~5O. Let us consider the set Y/'5O of operators having the form VS where V ~ Y/', S ~ ,5". One can say also that A ~ Y/'5O if A can be represented in the form V + T where V ~ ;v', T ~ 3-. If the operator K ~ ~/'5O is represented in the form K = VS, V ~ ;g', S ~ 5O we define the determinant of K as det S. This definition is unambiguous. Really, let us suppose that K = VS = V'S', (V, V' ~ Y/-, S , S ' ~5O). T b e n S ' = VV' ~. V ' S V ' a. I f S ' = l + T ' , V'SV'--~= I + T, V V ' 1 = 1 + Q then T ' = Q + T + Q T . Using the fact that T a n d T ' b e l o n g to the trace class and Q is bounded we obtain that Q belongs to the trace class too. We see that V V ' a = 1 + Q ~ 3 ¢ ~ c 3 5 ° and det VV' 1 = 1 , d e t S ' = d e t V V ' - ~ det V ' S V ' - t = det S. It is easy to verify that det K 1 K 2 = det K x • det K 2 for every Ka, K 2 ~ ;v'SO. (If K a = VIS a, K 2 = VzS 2, V 1, V 2 ~ ¢/', S a, S 2 ~ 50, then K 1 K 2 --- V1V2 • V f l S y z S 2 and det K 1 K 2 = det Vz aS1V2S2 = det V21S1V2 • det S 2 = det S~ • det S 2 = det K1 • det K2). The above consideration can be applied to the supercase too. However in supercase we have to modify the definition of 5°. Namely, we will say the A = 1 + T~SO if T~3Y- and o d d - o d d block of A is invertible (i.e. it belongs to N). Let us consider the subgroups ~/'+ and ~ of ~ consisting of matrices V satisfying vii = 1, v~j = 0 for i > j (respectively vii = 1, vii = 0 for i < j ) . It is evident

A.S. Schwarz / Fermionic string

339

that det A = 1 if A ~ Y/'+5" or A ~ Y/'5p and therefore we can define the determinants det+A and det_A on y/'+ow and "//'5p. In the supercase we can consider a group Y/~0c ~ consisting of operators with coinciding even-even and o d d - o d d blocks and vanishing odd-even block. An operator V E YFo can be represented in the form

(i.e. we denote coinciding even-even and o d d - o d d blocks of V by A and even-odd blocks by B). If V E "/#0n 5 p then the determinant (berezinian) of V is equal to det A • (det A) 1 = 1 and therefore one can define the determinant of every operator K ~ Y/~0,5". If the operator K ~ . ~ is equal to V + T where T ~ J - and V is a Fredholm operator of the form (B.1), it is easy to prove that K e ~05p. (We have assumed that K is invertible. Therefore V has index zero and it can be written in the form V = V0 + To where To ~ 3"- and Vo is an invertible operator having the form (B.1); i.e. Vo ~ Yf0.) Under certain conditions the operators arising by calculation of the super ~'-function belong to Y/~03° and therefore have a well-defined determinant. To formulate exact results we will modify the definition of the spaces H, .~, F, Gr and UMS. Let us assume that the even and the odd part of ..~ coincide with the space H of square-integrable functions on the circle Izl = 1. The even elements of o~ can be represented as even (super)functions Here (respectively ¢p(z)) is an even (odd) function of z, [z I = 1. We define F as a set of smooth even invertible elements of ..~. (The function is invertible if the number part of does not vanish on the circle Izl = 1.) The set f" can be considered as a supergroup with respect to the multiplication of functions. Side by side with bilinear inner products

F(z, O)=f(z)+ cp(z)O.

f(z)

F(z, O)=f(z)+ cp(z)O

f(z)

(f'f)=

__1 2~ri fr-~1=lf(z)flz)dz

fL-~l=lf(z)~(z)dz+2~ifL~,-i f(z)q)(z)dz~_ 1

£F'b~)=--2~-il we introduce in H and ~

1 (s' s-) = 2,%-7 (F,k~) =

the hermitian inner products

'd., 1

1-~f(z)~(z)z Idz + ~ ( z ) f i ( z ) z - l d z . 2~ri

Considering the subspaces of H and ..~ we will assume that they are closed with respect to topology determined by means of these hermitian inner products.

340

A.S. Schwarz / Fermionic string

As earlier we will denote by 9(Y (•+) the subspace spanned by z ' , ZnO, n < 0 (respectively, by z ' , z"O, n >/0). The space J f can be considered as a direct sum of ~f~ and ,,v{+. Projections of ,g~ onto ~'~ and Jt~+ will be denoted by v and v+; the same projections considered only on the subspace W c ~ f will be denoted by vrw and vw. We will say that the subspace W c ~ f ° belongs to Gr if 7rW is a Fredholm map. This definition is similar to the definition in ref. [16]. (The subspace W of H belongs to Gr in the sense of ref. [16] if vrw is a Fredholm operator map and w is a compact operator). The same arguments as in ref. [16] permit us to prove that the supergroup F acts in Gr. The proof is based on the assertion that the map ~r FTr+ of ~ + into aug and the map ~+FTr of ~ _ into S + belong to the trace class if F is smooth. As earlier we define Jac(W)= W N ~ + , ,Y(W)--z~'(W)+ H d ( W ±) and construct for every basis w in ,Y(W) a basis ~ in W • Gr. Let us consider at first the super ~--function in the case when the maps Trw and v gw of W and F W onto Y are isomorphic (i.e. ,Y(W)= 2 / ( F W ) = 0). Then the T-function can be interpreted as the determinant of the operator v F ( v w) 1 in ~ (i,e. as the determinant of the corresponding matrix in the standard basis). The operator under consideration can be represented in the form ~r F(vrw) -1 = ~r F(~r +~r+)(~rw)

I =7

F + ~r F~r+(~rW) 1 .

The operator v _ F ~ + belongs to the trace class, therefore ~ F ~ + ( v w ) 1 belongs to the trace class too. Operator ~r F has the form (B.I). It follows from the theory of singular integral equations that v F is a Fredholm operator. We see that invertible operator 7r F(~W) -1 belongs to zv'05P. In such a way it has a determinant and the ~'-function is well defined. To analyze the general case we will introduce the notion of admissible basis in W ~ Gr. Let u be a basis in W • Gr. Let us denote the index of W by (a, a). (We define the index of W as the index of ~rw. In other words i n d W = ind ~rw = dim ~¢(W) - dim . J ( W ±). As always the dimension of superspace is considered as a couple of integers). The map of ~ onto W transforming the standard basis of Yf into the basis u will be denoted by qu- We will say that the basis u is admissible if ~ r _ q u - R ~ , ~ • , Y - . (Here R~,~ denotes the linear operator in defined by formulae Ru,~(z" ) = ~r (z'+~), R ~ , o ( z " O ) = vr (z'+~O).) It is easy to check that R ~._~ .R~,~ - 1 • J - . Using this fact we obtain that for an admissible basis q ~ - R ~, ~r • J - . Let us consider the matrix [Fu[u'] connecting two bases Fu and u' in F W . (Here u is an admissible basis in W and u' is an admissible basis in F W , F • F.) We will prove that this matrix belongs to Y/'0 5p and therefore it has a determinant. The matrix [Fu]u'] coincides with the matrix of the transformation qF~Fq~ in the standard basis of ~ . Let us represent v q~ and qF~ in the form v _ q u = R , , ~ + T, = R

,,.

(B.2) +

(B.3)

A.S. Schwarz

where T, T' E J ,

/

341

F e r m i o n i c string

i n d W = (a, a), ind F W = (a', d). Then

qF)Fqu = qF2F~r q~ + qF,~Frr +q, =R

~,

,~r F R ~ , ~ + T ' F q u + R

~,

~,Tr F T + R

_~,

~,Tr F~r+q,.

(B.4)

Using the fact that T, T' and ~r FTr+ belong to the trace class we obtain that all summands in r.h.s, of (B.4) except the first one belong to the trace class too. The first s u m m a n d is a Fredholm operator having the form (B.1). (This follows from the corresponding assertion for 7r_F and from the equation ind F W - ind W = ( a ' , c~') - (a, c0 = (k, k). We obtain that the invertible operator qF,tFq, belongs to ;v0Y; therefore the same assertion is true for the matrix [Fulu']. One can check that for every basis w in X(W), W E Gr, the corresponding basis ~b in W is admissible. Using this fact we conclude that the super ~--function is well-defined if we use the modified definition of Gr. Taking as a starting point this definition of G r we define UMS as earlier. Then the above consideration gives an extension of the Mumford form onto UMS. (To define W(N) and W(N, L) we must suppose that local coordinate in the neighborhood U of n ~ N runs over a domain in C, containing the disk [z[ ~ 1). Two admissible bases in W ~ Gr are connected by a matrix belonging to 5p. Therefore we can define an admissible measure in W ~ Gr as a function of admissible bases in W having weight 1. The admissible measures in W form a one-dimensional linear space; this space will be denoted by m(W). The correspondence between bases in 2 ( W ) and W determines an isomorphism between re(W) and m(X(W)). The line bundle having Gr as a base and re(W) as a fibre over W ~ G r is called a determinant bundle [16]. The action of F on Gr can be extended to the action of F on the determinant bundle. (This follows immediately from the results above.) In other words an element F ~ F determines an isomorphism aFw between m(W) and m ( F W ) ; it is easy to check that Ogw FG

_-ecgw

•

Olw F ,

W~Gr

,

F,G~F.

(B.5)

If FIW = F 2 W then the isomorphisms aft: m ( W ) ~ m(FlW ) and a ~ : m ( W ) - ~ m(F2W ) do not coincide in general (i.e. the isomorphism r e ( W ) = m ( F W ) is not canonical). Let us fix now a subspace W0 ~ Gr and subspaces Wl, W 2 e F W 0 (i.e. we suppose that Wo, Wl, W 2 belong to the same orbit of the group F in Gr). If W 1 = F1Wo, W 2 = F2W0, F I ~ F, ~2 ~ F, we define W ~ Gr as F1F2Wo. Using the isomorphisms c~o between m(W0) and re(W1) and C~FW 2 between m(W2) and m(W) we obtain isomorphism m ( W ) ® re(W0) = m ( W l ) ® m(W2).

(B.6)

342

A,S. Schwarz / Fermionic string

It is important that the space W and the isomorphism (B.6) do not depend on the choice of F x and F 2. (If W I = F { W 0, W 2 = F 2 ' W 0 then F{=F1G1, F2'=F2G > G1Wo = GzW0 = W o. We see that F{F2'W 0 = F1F2Wo. To prove that the isomorphism (B.6) does not change if we replace F a by F{, F 2 by F 2' one must use (B.5).) If W 0 = W(N), W 1 = W(N, L1), W 2 = W(N, L2), where L1, L 2 are holomorphic line bundles over (1,1)-dimensional compact manifold N, then W = W(N, L 1 ® L2). The isomorphism (B.6) gives m ( W ( N , L 1 ® L2) ) ® m ( W ( N ) ) = m ( W ( N , L1) ) ® m ( W ( N , L2) ) .

(B.7)

The isomorphism (B.7) was used in refs. [11,17] to obtain the construction of the super M u m f o r d form. Our construction of the extension of this form on UMS is closely related with the constructions given in refs. [11, 17]. Let us discuss briefly the bosonic case. In this case the determinant entering in our definition of r-function is ill-defined in general. However we can define Sato's r-function by the formula

r(w, w ' , W , F ) = det [F~l~b' ] where w is a basis of X ( W ) , W ~ G r , w' is a basis of X ( F W ) , F E F . = F c ~ H .(It is easy to check that the matrix [/:~l ~'] connecting the bases F ~ and ~ ' in F W belongs to ; ~ ' ~ in our case.) The r-function is closely related with the action of F = F A H in the determinant bundle over Gr. The action of the group F on Gr cannot be extended to an action in the determinant bundle, however a central extension of F acts in the determinant bundle over Gr. It follows from this fact that the canonical isomorphisms (B.6) and (B.7) do not exist in the bosonic case. (It is well known that in the bosonic case m ( W ( N , L 1 ® L2) ) ® m ( W ( N ) ) = m ( W ( N , L1) ) ® m ( W ( N , L 2 ) ) ® [ L , , L 2 ] , where ILl, L2] denotes so called Weil-Deligne pairing of the line bundles L 1, L2. ) We have seen that from our viewpoint the supercase is simpler than the bosonic case. In the forthcoming paper by S. Dolgikh and A.S. Schwarz it will be shown that the case of N = 2 supersymmetry is simpler than the case of N = 1 supersymmetry. In particular it will be proved that the action of F in G r can be extended to the action of F in the determinant bundle over Gr and that m(W1) and m(W2) are canonically isomorphic if W 1 and W 2 belong to the same orbit of F in UMS (in other words the isomorphism between m(W 1) and m(W 2) where W 2 = F W 1 generated by the action of F ~ F in the determinant bundle does not depend on the choice of F).

References [1] M. Baranov and A. Schwarz, JETP Lett. 42 (1986) 419 [Pis'ma Zh. Eksp. Teor. Fiz. 42 (1985) 340] [2] D, Friedan and S. Shenker, Phys. Len. B175 (1986) 287

A.S. Schwarz / Fermionic string [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Yu. Manin, Funk. Anal. 20 (1986) 88 C. Vafa, Phys. Lett. B190 (1987) 47 L. Alvarez-Gaum6, C. Gomez and C. Reina, Phys. Lett. B190 (1987) 55 A. Morozov, JETP Lett. 45 (1987) 585 [Pis'ma Zh. Eksp. Teor. Fiz. 45 (1987) 457 L. Alvarez-Gaum6, C. Gomez, G. Moore and C. Vafa, CERN-TH 4883/87 A. Schwarz, Pis'ma Zh. Eksp. Teor. Fiz. 46 (1987) 340 [JETP Lett. 46 (1987) 438] L. Alvarez-Gaum6, C. Gomez, P. Nelson, G. Sierra and C. Vafa, CERN-TH 5018/88 Yu. Manin, and A. Radul, Commun. Math. Phys. 98 (1986) 65 A. Voronov, Funk. Anal. 22 (1988) 67 M. Baranov and A. Schwarz, Int. J. Mod. Phys. A2 (1987) 1773 A. Belavin, and V. Knizhnik, Phys. Lett. B168 (1986) 201 A. Rosly, A. Schwarz and A. Voronov, Preprints ITEP N107 115 (1987) I. Penkov, Invent. Math. 71 (1983) 501 G. Segal, and G. Wilson, Publ. Math IHES 61 (1985) 1 A. Rosly, A. Schwarz and A. Voronov, Preprint ITEP N81 (1988) A. Schwarz, Commun. Math. Phys. 67 (1979) 1-16 A. Schwarz and Yu. Tyupkin, Nucl. Phys. B242 (1984) 436 G. de Rham, S. Maumary and M.A. Kervaire, Lecture Notes in Math. 48 (1967) 1-101

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