Algebraic—geometric quantization of interacting strings and superstrings

.

8, 9

i0 ii 12 13 14

O. A. Ladyzhenskaya, Mathematical Problems of the Dynamics of Viscous Incompressible Fluids [in Russian], Moscow (1970). O. A. Ladyzhenskaya and V. A. Solonnikov, Zap. Nauchn. Sem. LOMI, 38, 46-89 (1973). D. Joseph, Stability of Fluid Motions [Russian translation], Moscow (1981). J. G. Heywood and R. Rannacher, J. fur Reine Angew. Math., 375 , 1-34 (1986). D. V. Emel'yanov, Sb. Nauchn. Tr. Leningr. Korablestr. In-ta, 34-38 (1990). A. P. Oskolkov, Zap. Nauchn. Sem. LOMI, 171, 73-83 (1989). A. P. Oskolkov and R. D. Shadiev, ibid., 181, 146-185 (1990). J. L. Lyons, Some Methods of Solving Nonlinear Boundary-Value Problems [Russian translation], Moscow (1967).

ALGEBRAIC-GEOMETRIC QUANTIZATION OF INTERACTING STRINGS AND SUPERSTRINGS

G. A. Pel'ts UDC 517.9

The method of algebraic-geometric quantization is used to find an explicit expression for the measure on the (super)moduli space for all possible interaction diagrams of open and closed (super)strings.

INTRODUCTION

The use of the functional integral method within the framework of the Polyakov approach to the theory of interacting (super)strings leads to representation of multiloop (super)string amplitudes in the form of finite-dimensional integrals over (super)moduli spaces, the integration measure for which is expressed in terms of the Fredholm determinants of the differential operators on (super-)Riemann surfaces. However, the functional integral method does not provide explicit information on the structure of (super)string state space or the algebra of the operators acting on this space. Therefore, methods that combine the Polyakov approach with operator formalism have been proposed in [i-ii, 12, 13]. The method formulated in [12, 13] differs from the others in that it does not require the use of specific uniformization, and hence offers the possibility of describing the process of (super)string interaction in parametrically covariant terms. This method is based on the algebraic-geometric interpreta- tion of the space states of scattered strings and the BRST operators acting on them as co- homologous complexes of local cochains of the algebra of quasiconformal transformations. Hereinafter, we therefore will call it the method of algebraic-geometric quantization. We have used this method to derive explicit equations for the measure in the (super)moduli space for diagrams of the most general kind, including open and closed with a world-sheet of arbitrary orientability. The author would like to thank V. N. Popov for his help in the work and

for discussing the results.

i. The World-Sheet of a System of Interacting Strings

In the theory of interacting strings, the world-sheets, which represent two-dimensional surfaces with an edge, act as the Feynman diagrams. Such a surface WP always can be repre-

sented as:

W~ (1.1) where W is a compact surface and KN 0 is a finite set in W whose elements are called Koba- Nielsen points. Here the open-string vertices of the diagram correspond to the boundary Koba-Nielsen points, and closed-string vertices to the interior points. It was shown in [14] that an arbitrary surface W(P) can be obtained, regardless of its orientability and the nature of the boundary, by factorization of the closed oriented surface E(P), which is called its double, with respect to the odd involution I: Z(P) § E(P). The double of a connected surface is a two-component double if it is closed and orientable, and is connected in all

other cases.

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 122-145, 1991.

3016 0090-4104/92/6205-3016512.50 �9 1992 Plenum Publishing Corporation

The relation between the surfaces E and zP is given by the equation

ZP:E\ Kx

where p: E § W is a natural surjection. The number of points in the set KN is equal to N = N I + 2N2, where N I and N 2 are the numbers of boundary and interior points from KN 0, respectively.

The first homology group H4([c~ Z) is a discrete Abelian group on which the integral skew-symmetric bilinear form of the intersection coefficient is defined:

The involution I induces the antisimplectic automorphism ~([(P, ~) defined by the equation

<I~, I~> =-<~,~>. (1.3)

As a result, the subgroup of H,([(P)~) defined by the equation

H: (~..(p~ Z)= {CE ~ ([r ZC:CJ ' ( 1 . 4 )

is isotropic, i.e.,

A,s>=0 Z)). (1.5) <

Note t h a t H ~ ( Z ( g ~ ) i s t h e maximum i s o t r o p i c subgroup o f H~(Z cP), ~ ) , i . e . , i t c a n n o t be ex - panded t o any o t h e r i s o t r o p i c s u b g r o u p . We w i l l c a l l a c l o s e d o r i e n t e d s u r f a c e w i t h a f i x e d maximum i s o t r o p i c subgroup o f a f i r s t homology group a s u r f a c e w i t h a s t r u c t u r e .

Now let us turn to a topological classification of the objects considered here. We will characterize the topology of a compact orientable surface with an edge by the vector genus

g = (gz, g2), where gl is the number of components of the boundary of the surface and g2 is the number of handles. We will place the same vector genus as that of the orientable surface with an edge in the form of whose double the surface can be represented, in correspondence with the compact surface with a structure. Note that all compact surfaces with a structure can be realized in this way, and hence the vector genus is their general topological char- acteristic. When gl > 0, the scalar genus of a surface with a structure is expressed in terms of its vector genus according to the equation

q =~ + ~-4. (1.6)

We will call the genus of the double of a closed compact nonorientable surface its pseudogenus. We will characterize the topology of a compact nonorientable surface with an edge

by the vector pseudogenus • = (X~, X2), where Xz is the number of components of the boundary of the surface, and X~ is the pseudogenus of the closed nonorientable surface obtained from the original surface by contraction. We will call the vector genus of the double of a nonorientable surface its vector genus. The relation between the vector genus and the vector pseudogenus of a nonorientable surface is expressed by the equation g = g(x~,), where

&

Inserting this equation in Eq. (1.6), we obtain the relation

:X,+#: (1.8) As an example, let us present the topological characteristics of the simplest nonorient-

able surfaces.

(I) The projective sphere ~:~ :(0,0), ~=(4,0), ~:0o (2) The Klein bottle: 9=(O,4), ~=(~,0),9=4" (3) The MSbius strip: ~=(4,O), ~=(~,O),~ =4.

The conformal structure of the world-sheet W p induces an analogous structure on the surfaces Z and E(P). Specifying a positively defined conformal structure on an oriented

3017

surface is equivalent to specifying a complex structure on it. Therefore, for a Euclidized string the surfaces Z(P) are Riemann surfaces. Denote by mJ a sheaf of j-differentials on ~. The use of symbols of the type S +, where S is a nonopen set, to designate the domain of the space of sections of the sheaf will indicate that we are examining the colimit of spaces of the sections over the region obtained by replacing S + by an open set that contains S. For example,

If S is a closed set, we also may consider the space

(exS).

By A 2 we will denote the diagonal injection ~'-~ ~-x~-. On ~-~>- we will consider sheaves

of the two-argument forms ~0JS~ J . We will denote the symmetric and skew-symmetric parts of

sheaf ~0J~ j by ~0JS5~0 j and 60J~(0 j, respectively. We will denote by ~ and ~P the presheaves on 7 and l p defined by the equation

(1.9)

where u is a closed contour, and [~]cp) is the corresponding element of H4(5-~F),Z).

Hereafter, the section G ~ I]~04(-A~O-)) such that

will play an important part. By using the Riemann-Roch theorem, it is not hard to show

that such a section indeed exists uniquely. For any section from ~04~ (04 (-A~([)) that satisfies condition (i.i0), the following equation is satisfied:

Hence, with consideration for Eq. (1.5) it follows that Ge/]m/](-A=(W)). Further noting that condition (i.i0) is equivalent to the condition

and taking into account the uniqueness of G, we see that 6E/2Ss/2(-A~(Z)) The Laplace equation with the Neumann boundary conditions is the classical equation of

motion for the string coordinates Qv. The solutions of these equations have the form

Or(p;,) = ge XV( ),

where X v are holomorphic functions on I p, the imaginary part of which is constant on every

component of the set ~I: =P-4~P " The imaginary part of the function X v may be nonunique.

However, its differential dX v is unique and is the integral section of the presheaf ~P. i Denote by ~0 the subspace m~ which consists of locally constant functions that

satisfy the condition

Let ~ (K~4[c 8 c ~l) be an open set of trivial topology. Consider the space

o 6 -4

where ~:~0~ ~ is the morphism of differentiation and ~6 '~)~176 ~) is the contraction

operator. This space turns out to be independent of the choice of E and also can be deter-

mined as follows:

3018

(1.12)

2. The Fubini-Veneziano Operators

In this section we will describe a finite-dimensional line bundle on Z, of which each

fiber C~&(~e~)is a space of coordinate string states associated with the puncture at point

Q. This bundle should have the following structure.

(i) A line bundle Cd A of the space CdQ such that

is placed in correspondence with every topologically trivial closed set A that contains Q (we will use upper case Latin letters to denote such sets).

(2) The topology of the spaces Cd A is defined so that the injection operators

are continuous and their image is everywhere dense in Cd A. _ __ -- ~r

(3) The relativistic momentum operators ~EF~G~A are defined [v = (i ..... D)] such that

[ O A , A �9

""1~ ^V (4) An operator, defined modulo eZCP~A~ , of the complexified coordinate XBA(~):C~A-->C~B

is placed in correspondence with every point Z~8\A(~EAc~) so that

Xcs(Z)C =XfA(Z) (zec\B) & ^~ All

Cdc XSA (Z)=X~ (Z) (Z~ B\A).

Here the operator XBA is continuous on B \ A and holomorphic on (B \ aB) \ A and, on passing around point Q counterclockwise, aquires a displacement 2~iP~A.

It is easy to see that the operators r , -4 ^V

(2.1)

act endomorphically on the space Cd A. Thus, the linear mapping

A

t:::t : ~@ co~ (2,2)

is defined.

(5) The following commutation relations hold:

8'\ B A~\A , ( 2 . 3 )

Hence in particular follow the relations

3019

~A XAB " (2.4)

Denote by OCZA8 and OCd+A8 (8 G A k~A) the subset ~oD(Cg~8,C6~A) formed by the operators X;8(~)

and X~6(~) -- X~(~') ( ~ , ~ A k S ) , respectively.

It is easy to see that

The space

forms a subspace in E~(C~A).

the spaces Og~ and ~D@m~

the subspace OC~ of the form

where

ocG o 6 G , oc4 =s176 - .

-~~ OCi<0)

Mapping (2.2) defines a one-to-one correspondence between

We call the space of the annihilation operators Og~ A

Denote by ~a~ the space of vacuum states on which the annihilation operators act trivially.

Denote by (Os • the subspace OC~ ~ which consists of operators that commute with all oper-

ators from Og~;. Since the operators from OC~A commute with each other, the following exact

sequence is obtained:

This enables us to identify the factor space

with the subspace of the space E~(C~).

Let us consider the mapping

definedby the equation

The elements ~ and ~ (V =~ , . . . ,D ) fo rm a basis in the space OC&~ ~ . We place in correspon-

dence with every gE~\{O} the representation ~ of operators from OC~ =c in the space ~@

C~(~)~ defined by the equation

where Sv are the coordinates in ~D.

The conditions

are fulfilled for the operators

3020

" / 4

This enables us to determine C,~, v~ as the space obtained by factorization of the se.t ~] @C ~

(~D)x ~)~\{@] with respect to the equivalence

($,r =(U(r We also assume that for an arbitrary ~ (~-~A) we have ~ = C~. We will define the space

C~A as the space produced by the action on G~ a~ of operators from OC~: with commutation

relations (2.4). According to the definition of C~ ~ given here, the following natural surjection holds:

~ere, according to relations (2.5) the mapping

is homogeneous in the power Ph �9 This enables us to define the integral section of {~]

of the sheaf c0~@C~ a~ by the equation

~ ( ~p<-~ ~ ), ~ ) ={ P } F ~ . (2.7)

The space ~/J~ decomposes into the direct sum

where

o t' ~-~ , a - { e l m+({Q] x {~]): = a , cocr ].

where

This decomposition induces the decomposition into a direct sum of the space

ocs = oc& oc&,

= aug ~m+({a]-~ {~])). (2. s)

Fixing g~\{~], we expand this decomposition to the decomposition of the space OOd e as follows :

oca, : oc4 �9 (2.9)

ocs lOCd-, t

By using the symmetry of G i t is not hard to show that the operators from 0C~$, l ike

the operators from OCd,~ , commute with each other. By virtue of this we may introduce normal

ordering: ...:r for the product of the operators from OC~ e , in which the elements O~{[ are

arranged to the left of the elements of OCt{re. This procedure runs continuously until there is normal ordering of products of the type

A A ~. f ,\ c~.....o,~.c~ ,a,.eocd,, ~+,,~, A~. =A O.

The coordinate Green function <Xv(Z),X/*(~)>& is defined by the equation

By using Eqs. (2.1), (2.3), and (2.5) it is not hard to show that

3021

We will give the name by the equations

(i)/

(ii)/

-

(~)>~ =-a(z,~,)a, <X(a),~X (z)>~=o, ~ . , . (,<X"(z) X (~)>~-m,~(z).se)=o (d~(a)=r IJ'e,

(Z.lO)

"prime form" to the sectionP of the sheaf ~F~ -~ over ~X%, defined

Pli~=-O , -2

~,=~o (2.11)

where the subscript by the symbol d indicates the order of the variable with respect to which differentiation is performed. When gl = 0, the prime form is well defined only on the set

~§ U ~., where ~-• are components of the connection of ~ ~

The vertex operator: e/J~p(-~:8(~):~E ~0~(~T~ ~ 5~) (~ ~ , ~ ~(A\~A)\ ~) depends, through

normal ordering, on the choice of the linear differential ~ at the point of puncture and on

the global Riemannian structure of the double ~. However, the operator ~ -differential

^ ~ ~ P^V (2.12)

turns out to be independent of these two factors. This operator differential is not unique

on (A\~A)\8 �9 However, when

the section

is unique. ~Z

If here ~ =~, (A\~A)\8 ~0,

correctly defines the state

3. Scattering Amplitudes

Hereafter we will use the notation

where f is some finite set in

at the point ~0EKNo has the form

then equation

eAcl o

(2.13)

= % , q Ci~, The space of coordinate states associated with the puncture

Cd) (~ = f~Q,). The scattering amplitude of such states is a linear N 1 + N 2 functional

X. ~J p-~~

The linear functional

naturally corresponds to this functional.

Let A~E~(g~a)(~e~). Denote by AQ the element E~(~ ~) obtained by the action of the

operator A on the component of the tensor product that corresponds to point Q.

3022

Let

be the states of the scattering strings�9 If $c~ @~ ~0 (~o: =@e,~ ~ )' the average of

the physical quantity corresponding to the operator for the scattering process is determined from the equation

<A>= 5~, '

Because the complexified coordinate is a common parameter to all scattered states, we

A A

<X~(~)> =<X a, (~)> -- <X(g)> have

Here, according to the information presented in Sec. here, we obtain

In particular,

(~,~' ~KN).

1, <o~X'> ~.-~P(zP) . Using Eq. (1.13)

). (3.1)

The equation for closed-string states follows from Eq. (3.2a) b): A

= (a O ao). %%~ -'

Now let the scattered states be eigenstates for the relativistic momentum operators:

~ p g . I t follows from Eq. (3.3) that

where

(3.2)

(3.3)

~, ~p~=

is the total relativistic momentum of the state ~o and S is a function on W that vanishes

at the boundary and is equal to unity at interior points. Denote by P~No the element (~m)~%

formed by the set of vertex momenta ~mo (~o~KNo), As a result of (3.2a), in order for

the scattering amplitude to be nonzero, the following equation must be fulfilled:

Denote by (~D)~ N~ the subspace of (~D) K% identified by this condition.

If we use Eqs. (3.1), the calculation of the scattering amplitudes of arbitrary states reduces to calculation of the scattering amplitudes of the vacuum states. Let us therefore

consider the process of scattering of vacuum states with momenta ~mo(~OE KNo], which form the

�9 ^9 element ~kHo~(~); N~ The operators ~ (QcKN)act correctly on the vacuum states. Accord-

ing to Eq. (2.5), their averages can be calculated from the equation

V

3023

where ga (~ ~-;<N) are the differentials to the points KN that are included in the definition V

of the operators x . We will assume that they were chosen to be involution symmetric as

follows: (~)~z(~) =tm. It follows from the existence of limits in the right side of Eq. (3.4)

that the vacuum averages <~Xv>~=z have simple poles with the following residues at points

6~,='I<N : - I )~

('I +5(p(O.))) Ppa. By using the Riemann-Roch theorem it is not hard to show that these conditions uniquely de-

fine the section <~X~>v~/~P(~-e). The mapping ~, +<~XV>v=z is linear. The operators of the

real coordinates of the center of a string

Cao =(4 +SCao )) ,T-a~i'a~ ;a (ao~ KNo ) satisfy the condition

and hence, in contrast to the operators of complexified coordinates, act endomorphically on the spaces of closed-string states. The averages of these operators for the scattering process of the vacuum states are determined from the equation

V ^V V

(~)v~ " "" "^'h2/o§ ( 3 . 5 )

g,,I ~o(~)1 cO(a) -- Co).

Denote ~'~No the element of (~D) Kn~ formed by the vertex coordinates ~Oo" The functions ger <XvC%)>v= z are determined up to an additive constant by integration of the linear differentials

<~XV(~)>v~. The vector ~no thus is defined in Eq. (3.5) modulo the space ~KNo(~O), where

AKNo: ~-e (~)~N0 is a diagonal injection. Thus, the vertex coordinates are related to the

vertex momenta by the linear mapping (~D)~ .o. KNo

where

Since the spaces (~D)~ N~ and (~D)KN~ are mutually dual, the following bilinear form naturally

corresponds to this linear mapping I) KNo

As calculations show, this form is symmetric. The established relation among the vertices, momenta, and coordinates turns out to be fulfilled if the scattering amplitudes of the vacuum states in the momentum representation are given by the equation

where

If the double is connected (go ~ 0) and if all open-string punctures are concentrated on one of the components ~W, then Eq. (3.6) assumes the form

3024

a~W

Here we use the branch of the prime form that is uniquely defined on E • E, where a is some open, topologically trivial set that is involution symmetric and that contains KN. Let KN~ be the set obtained from KN 0 by moving one of the boundary points Q0 to point Q~, which lies in the interior region of W. Converting to the limit Q~ + Q0 in Eq. (3.7), we obtain

=/4). (3.8)

This equation also is valid when the open-string punctures lie on different components of aw - a fact that makes it possible to compute the amplitude in this case as well. For closed orientable W Eq. (3.6) assumes the form

p~a 9 pca~) . (3 .9 )

We now will introduce the numbering~N={~ .... ~NJ (N=N~+~N~)on the set KN, and we

hereafter will use the notations ~=P~, ~j =~, and so forth. Amplitudes (3.7)-(3.9) are

smooth functions of the position in Z of the set KN, which is involution symmetric. The analytic extension of this function to a multivalent function of the position in Z of an arbitrary set of N points has the form

D N ~

The action of vertex operators (2.11) on the states of the scattering strings is equivalent to the inclusion of additional vacuum vertices in the diagram, i.e.,

Here the choice of the branch for Sod in the right side of Eq. (3.11) is based on the states of which of the scattering strings the vertex operators act on and in what order.

As we will see below (see Sec. 7), the physical states of a string can be obtained from the vacuum state by the sequence of transformations (2.13). By using Eq. (3.11), the scattering amplitudes of such states are expressed in terms of the sequence of residues of expressions of type (3.10) at the points where the arguments match.

4. Atiyah Algebras

This section introduces the basic concepts necessary for further understanding of the work.

Let X be some analytic manifold. We will denote by O x the sheaf of analytic ftmctions

on X, and by ~ the sheaf of analytic vector fields on X.

Let ~ and ~i be bundles of O x moduli on X. Then the sheaves ~@~i and Am(~),S~(~),

the dual sheaf~ ~, and the sheaves @~k(~,~i) of the k-th-order differential morphisms from

and ~tare defined, and are also ~ moduli.

The sheaf ~ , which is endowed with the structures of Lie algebra and the O x modulus

such that the adjoint representation ~ has the form of the first-order differential morphism

-->~ (~) , whose composition with the canonical morphism ~ ) - ~ gives ~the Ox-linear

3025

linear surjective morphism ~:~-~ ~ , is called an Atiyah algebra. The morphism v is the homomorphismof Lie algebras. Its congruence kernel ~a:=c~V is aLie algebra withanOx-linear

commutator of the type represented by the algebra of Yang-Mills gauge transformations. We will call . 4 the~-linear homomorphismof Lie algebras V:~ --~(~) the local representation of theAtiyah

algebra ~ in the 0 x modulus of ~ . We will call the analogous homomorphism that is a first-

order differential operator the differential representation. In particular, the adjoint representation is differential. We will call the natural representation of the Atiyah algebra in the sheaf 0 x the trivial representation.

Let Y be a bundle on the base X with surjection ~:~-->X , and let ~ be some sheaf on

X. Denote by ~[6] (g~X) the sheaf on Y defined by the equation

~[6](~):=~(6n('~) (~ ~y). Let us consider a pair of bundles on the common base ~j:Y~ -->X (~=4,~) . Denote by

(~x~):X4• a bundle whose fibers have the form

= Xtx X~. Let ~ be the O~ moduli on the bases of the manifolds Xj. Denote by ~ ~ ~ the

modulus on the basis X~xX 2 , such that

where 6~cW~ , U 6 X

Denote by ~ the sheaf of infinitesimal morphisms of the bundle ~:Y-~X. The sheaf

~[6]~ (6 cY) has the natural structure of an Atiyah algebra and has the natural local

representation in the sheaf Z~y~J.

In the OX modulus of ~, let the local representation of the Atiyah algebra ~ be speci-

fied. We will call the graded sheaf

C"(A, t h e s h e a f o f l o c a l c o c h a i n s o f t h e A t i y a h a l g e b r a ~ w i t h c o e f f i c i e n t s in t h i s r e p r e s e n t a - t i o n . As I have showed e l s e w h e r e [ 1 2 ] , t h i s s h e a f has t h e n a t u r a l s t r u c t u r e o f a cohomol- ogous complex w i t h t h e b o u n d a r y o p e r a t o r 6, which a c t s as a f i r s t - o r d e r d i f f e r e n t i a l morphism.

5. The S t r u c t u r e o f Modul i Space

Le t us c o n s i d e r t h e f o l l o w i n g m a n i f o l d s : +

~ - t h e s p a c e o f c o n f o r m a l c l a s s e s o f compact o r i e n t a b l e s u r f a c e s o f t h e v e c t o r genus

g. ~ - t h e o f modul i o f Riemann s u r f a c e s w i t h t h e s t r u c t u r e o f t h e v e c t o r space compact

pseudogenus X.

~ - the space of moduli of compact Riemann surfaces with the structure of the vector

genus g.

As a result of the material presented in Sec. i, the natural injections ~ ~ ,

- $t~ J,l,~ ~ (~), where ~ ( ~ ) i s f u n c t i o n ( 1 . 7 ) , i s o b t a i n e d . Th i s e n a b l e s us t o c o n s i d e r t h e

manifold ~ as a complexification of the real-analytic manifolds complex-analytic

Let ~ be a topologically trivial open set in J~ . Consider the smooth bundle ~:~-~

whose fibers ~-~:= ~ (n+r are Riemann surfaces with the structure of the conformal classes

3026

corresponding to points ~ For canonical sheaves on J~ we will use short notations:

O=O~, ~=f~ . Let us consider the following sheaves on the basis of E:

- mJ is the sheaf of j-differentials on the fibers of Z, and ~ is the presheaf isolated

from the sheaf ~z by the condition that the periods be trivial with respect to the ele- I

ments H4~,~[) [see Eq. (1.9)].

- o~: =(~)-~(~'~T) is the sheaf of infinitesimal morphisms of the bundle of E.

The sheaf m-l may be viewed as a sheaf of tangents to the fibers of the vector fields on E. Therefore, the exact sequence of canonical morphisms of Lie algebras gives

O-+w-~-~o%-~f-+O. (5.1)

We will denote the Lie derivatives with respect to the sections of these sheaves by ~v

(VE~,~,g0-~) . The conformal Green functions ~(~Z) on the surfaces ~ defined by Eq. (i.i0)

form the section 6~/~sJ](-A(ExE)). The singularity ~6 (~Em-~(5], 6c~) on A~(6) is remov-

able. As a result, the equation d~ =D~ ~ correctly defines the third-order differential

morphism ~oO-~-+~ ~ .

Let C be some closed set of E. For the O modulus X~[Cf\s the following decomposition results:

~J~ [Ok C] = ~ [C ~] +~ ~ I-C]. (5.2)

Hereafter we shall confine ourselves to the multiloop case where ~+3~5 . Here.,~Ja+~OY

[W]=0, and hence decomposition (5.2) is direct. In this case the arbitrary section HEg0"~

~(6• is naturally representable in the form H=M++H_ , where

Now let the section H be chosen so that

(~ -,(%), %~o=~). (5.3)

Then the following identity also holds for the section H+:

This condition uniquely defines H+ as a section over -f(~). If condition (5.3) is observed,

singularity of the section ~H(~0-[65 on A~(6) is removable. This fact allows us to treat it

as a section of ~-~(6z6). Denote by ~p:~-~-~(6z6) --~(6) the linear operator that acts

in such a way that

~p ( ~ ) = v~ (5.4)

We will define the operator ~H:~ (6) --+ ~=(6) by the equation

~. fl = Sp (v~H). (5.5)

Simple calculations show that there always exists a section ~=(6) for which the following condition is fulfilled:

d~ = (5.6)

Consider the linear morphism Kc:~[~t\~]--> 0 defined by the equation

(5.7)

3027

Hereafter we will use the symbol V to denote, in addition to ordinary Lie derivatives,

the morphism ~:~J-+~(~(~J~), such that Vvm~ =~V~" it is not hard to show that

there exists a unique first-order differential morphism~7:~--~O for which the following diagram is commutative:

(5.8)

Here (I) and (2) are the canonical morphisms of (5.1), and (3) is the projector for decomposition (5.2). The morphism div defined in this way satisfies the conditions

v+ vv ' (5.9)

where V, V 2, V 2, and A are arbitrary sections of the sheaves ~ and O , respectively. Conse-

quently, at least locally there exists a volume form~ on J~$ such that

VvA= ,,d rv. (5.1o) Equation (5.10) defines~b uniquely to the accuracy of a constant factor. This volume form will be used later as an integration measure to calculate the string scattering amplitudes.

In describing diagrams with N l open-string vertices and N 2 closed-string vertices, according to Eq. (1.1) one should consider compact surfaces with N z boundary points and N 2

-

marked interior Koba-Nielsen points. Denote by ,ff , J~.~ (~ = (N~, N~)) the spaces of con-

formal classes of such surfaces. The following natural injections result:J~,~,~

~,~ --~J~(~).N, where ~,N is the space of moduli of Riemann surfaces with a structure with

N = N 4 + ~ N 2 marked points. Let ~* be a topologically trivial open set in J~,N �9 We will

denote by ~*: ~-*~* the bundle on the base ~ whose fibers are Riemann surfaces with marked

points from the corresponding points ~$ of the conformal classes. The marked points ~(~)

( ~ ~=~, .... N) form integral sections of this bundle. We will denote the closed sets in ~

corresponding to these sections by

N

bases of the manifolds ']~,N we will consider the following sheaves: On the

_ ~j,s is the sheaf of holomorphic j-differentials on fibers of [~ that have exponential asymptotic behavior with an exponent of at least S.

- ~ (k=~,..., N) is the sheaf of i-differentials at the k-th marked point.

- ~* is the sheaf of infinitesimal morphisms of the bundle of ~" that preserve the

position of the marked points.

The canonical points are: ~:=~ ~=~ ,N ~ ~,N'

6. Representations of Algebras of Quasiconformal Transformations i_nn

the State Space of Scattered Strings

The spaces of coordinate states of scattered strings C~ (k =~, .... N) described in Sec. 2

form infinite-dimensional line bundles on the base~, N . Mappings (2.2) define the morphisms

3028

A D o aj. E cG, (6.1) - - ~

and states (2.7) with fixed relativistic moment P LPe~ )form integral sections ~PJK of the

bundles ~'P~@C~ K. The Atiyah algebra of the quasiconformal transformations has the form

~KN:=~*~*[-~<N] . Therefore, according to information presented in [12, 13], the ghost

bundles 6~k must be defined so that the following morphisms hold:

for which the following anticommutation relations are fulfilled: A A A A

[6(V), ~(f~)]+ =.[C(~), c(~)J+ =0, [C(g),~,(v)]§ < ~,v>. (6.3)

Here <*, *> is a pairing of sections of dual bundles. Let us write J~: * ~" ='~,..#. [{Q,J d. The

vacuum ghos t s t a t e s t h a t van i sh under the a c t i o n of the a n n i h i l a t i o n o p e r a t o r s from ~(A~)d) GK vac

$((J~-K) • form in ~ a one-dimensional subbundle _..~ . We will determine the topology in

~ by assuming that there exists an integral, nowhere-vanishing section {~K of the sheaf

~/-~ 50 C,g vac ~ r v K

There e x i s t s a unique r e p r e s e n t a t i o n of the At iyah a l g e b r a J~ in sheaves of s e c t i o n s of

the bundles C ~ and G~, fo r which morphisms ( 6 . 1 ) - ( 6 . 2 ) and the s e c t i o n s {.~K , { ~ t ~ a r e , ~ -

i n v a r i a n t . For ~)= 26, t he co r r e spond ing r e p r e s e n t a t i o n of ~ in the shea f of s e c t i o n s of

the bundle ~K: = C ~ @ G~, ex tends u n i q u e l y to an anoma ly - f r ee r e p r e s e n t a t i o n of a l g e b r a ~ .

The s c a t t e r i n g ampl i tudes d e f i n e d in Sec. 3 f o r the c o o r d i n a t e s t a t e s form the ~ -

invariant linear morphism ~c~: ~ ~ @~<=~cc~ ~ To determine the scattering amplitudes of the

ghost states we must specify the ~KN-invariant morphism $~: @~=~G~j--)~(T*)a). We will define

the scattering amplitude of the vacuum states by the equation

N N

where ~ is the volume form on ~ defined in See. 5. Then the scattering amplitude of arbi-

trary ghost states is determined by using the equations

~_.~(_,[)1~ ̂ " l (6.5)

Here ,= $=4 tV~b$, where #q~f is the ghost number of the state located at the i - t h vertex:

A~KN is the subspace ~ , . , ~ , identified by the condition V Vr : Z j=4<~,V> =0 . We fix the

scale with respect to ghost numbers by setting the ghost number of the vacuum states to zero.

The ~ -invariance of the amplitude of ~k defined in this way is ensured by the method used N

to specify the measure ~. This amplitude turns out to be nonzero only when ]~=~ ~i = 0.

Therefore, the ghost number of the physical states should be set equal to zero. The states

of the scattering strings ~k (k=4,..0,N) are the integral sections of the bundles ~ o Their

global scattering amplitude S~ is expressed in terms of the integral of the local ones:

N Sz ~ (6.6)

~,N

3029

. p c

7. The BIj_ST Operator and Wave Equations

In the formalism of algebraic-geometric quantization, the BRST operator ~ (k=4, .... N)

acting on a string state located at the k-th vertex is a coboundary-generalized operator of

the Atiyah algebra ~K �9 According to the information presented in [12], this operator must

have the following properties:

A

" ~ l =~ 1;~, ~(v).l ~ [ ~'l<' " l< . , - -l< ' --- l T v ,

r ^ - A a . . ^ D o

Here [e~(~EI)J and {e=Cgg~)] are mutually dual bases in the bundles of a~ and J~:<6~,e#>=8;;

A (~k) --'>A (~K) is the coherent operator for local cochains of the Atiyah algebra ~K A

with trivial coefficients. The action of the operator ~ on arbitrary sections of the sheaf

is defined uniquely by its action on the vacuum states

{n L . L (7.2)

and by Eqs. (7.1). ~" t '~' R,, the operator $K In the absence of an anomaly in the representation %7 '~ -~

defined in this way is nilpotent (6~--0) and ~-invariant. The ~N -invariance of the global

amplitude 5fu~ ensures its BRST invariance:

li~ S~ ~(-'f) =0 (7.3)

The spaces of the physical states of scattered strings Pk K are the cohomology spaces of the complex of generalized local cochains, i.e.,

A

Pk - c~ ~ I oo~o~, ~'~. (7.4)

By virtue of Eq. (7.3), the global amplitude S~ correctly defines the scattering amplitude

g~ ,@N ~--+~ We fix the arbitrariness with respect to the addi- of the physical states ~, K=~

tion of the BRST-exact states by gauge conditions ~(~R)~ =G((~-R)~)~)=0. The state of ~ that

satisfies these conditions is representable in the form ~7 = ~U~@{~}K, where ~' is the section

~ @ ~ . The condition of the BRST invariance of ~ is equivalent to the condition of ~ -

invariance of ~' The simplest ~K -invariant integral sections ~@~K are the tachyonic

vacuum states {~}~ (P~=-~) �9 The vertex operators V(P) form the ~K -invariant sections

~-~ @ $~ ~&K . An arbitrary ~'K -invariant section C~K@~ can be represented as a linear

combination of states obtained from [~(~---'~) by the sequence of transformations (2.13).

The expression for the local scattering amplitude of these states was derived in Sec. 3. In

order to calculate global amplitude (6.6), this local amplitude must be integrated first over

the Koba-Nielsen variables and then, with the measure ~, over the moduli space of the compact

Riemann surface J~g. The integrand ,)~=~~=~ ~ for the scattering amplitude of the BRST-closed

3030

vacuum states also can be defined in a way analogous to that used for the measure #. Here

the sheaves ~, ~-~'~, ~,'~, and ~'-~ act as the sheaves ~, ~ ; ~ , ~, and ~, respectively,

and the section ~<~X~, ~X~/~ should be used instead of G, where <..~ is the average

for the scattering process. The condition of existence of the operator div* for which there

is a commutative diagram analogous to (5.8) defines the asymptotic behavior of <~X ~, ~Xg>~l$

near the vertices of the world-sheet. This in turn determines the vacuum spectrum of string masses. The advantage of this method is that its use does not require compactification of the world-sheet. Consequently, this method is applicable for computing the scattering amplitudes and spectrum of interacting string states, to scattering processes of which the world- sheets of an infinite genus correspond. Here the compactification procedure is impossible. An analogous method also is needed in a treatment of the scattering process for superstrings with Ramond vertices when compactification also is absent.

8. Scattering of Neveu-Schwarz-Ramond Superstrings

To denote the superalgebras of previously introduced objects, we will use the same symbols

as before but with hats: ~ , O , , D ~ ^ , , 5p , and so forth. In addition to the

superanalog of operator (5.4), we will need the operator 5p:~J ~uOV~(gx6)-U>~/~(g), which acts

^ ' ~ ~ ^ ^ , ^ ~_>~I~ so that 5p(~"~ ~)=(D~q~)'~@~. Here f is an odd element ~@d6) , ~W~176 ~6) 4):~O is

the morphism of superdifferentiation, which is a superanalog of the morphism ~:~~ . The

superconformal Green function G and the super prime form are sections of the sheaves c0~ ~0 ~

and u0-~/~5cO -f2 . Here, instead of (2.11) the following conditions are met:

^ 2 ^-4

= o , p ( z , z o ) j Z ( z ) = D 3 ( z o)

Let us consider the morphism /)& :w -~w , and the operator ~)~ : 60"%6)--~0~'(6), which are de-

fined by the equations D~ ^ ^ ~ ^ ^ " ~ " " =~p'Vg~ andD~=$pV~H. Here H is some section from ~-'m~ =(g•

~(6)) , which satisfies conditions analogous to (5.3). It is not hard to show that there

always exists a section ~(6) such that ~'~(6): /)~-oD~=~7~,. The functional~=:

~'~[C~\~] -~ ~ is defined by analogy with (5.7), and then the morphism ~:~F--~O is found

from a diagram analogous to (5.8). The Berezin form ~ corresponding to this morphism de-

fines a regular measure on ~$ for the case of the critical dimension D = i0.

The generalization proposed above is applicable in the scrutiny of a diagram with Neveu- Schwarz vertices. The compactified double of E may not be arbitrary. The condition dim

~e(~)=(410) must be fulfilled for it. Otherwise there does not exist a section ~ ~ •

(-A~(~)) that satisfies a condition analogous to (i.i0). This fact indicates the impossibility

of the corresponding scattering processes.

To calculate amplitudes with Ramond vertices, the method of determination of the measure on the moduli space of noncompact surfaces presented at the end of Sec. 7 must be generalized analogously.

i~ 2. 3. 4. 5. 6. 7.

LITERATURE CITED

A. Neveu and P. West, Phys. Lett., 193B, 187-194 (1987). A. Neveu and P. West, Phys. Lett., 194B, 200-214 (1987). A. Neveu and P. West, Phys. Lett., 200B, 275-279 (1988). A. Neveu and P. West, Nucl. Phys., B311, 79-139 (1988/89). A. Neveu and P. West, Commun. Math. Phys., 114, 613-643 (1988). A. Neveu and P. West, Commun. Math. Phys., 119, 585-607 (1988). P. West, Phys. Lett., 205, 38-48 (1988).

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8. P. West, Nucl. Phys., 3320, No. i, 103-134 (1989). 9. M. D. Freeman and P. West, Phys. Lett., 2.053, No. i, 30-37 (1988).

i0. L. Alvarez-Gaum~, C. Gomez, G. Moore, and C. Vafa, Nucl. Phys., B303, No. 3, 455-521 (1988).

ii. L. Alvarez-Gaum6, P. Nelson, C. Gomez, G. Sierra, and C. Vafa, Nucl. Phys., B311, 333-400 (1988/89).

12. G. A. Pel'ts, Zap, Nauchn. Sem. LOMI, 169, 107-121 (1988). 13. G. A. Pel'ts, Zap. Nauchn. Sem. LOMI, 180, 142-160 (1990). 14. V. A. Alessandrini, Nuovo Cimento, 2A, 321-352 (1971).

EXACTLY QUANTIZABLE MODELS WITH INTERACTIONS

G. A. Pel'ts UDC 517.9

A method is proposed for obtaining nonlinear field-theoretic models for which exact computation of all Green's functions is possible. The connection between the action and the generating function is described for each system.

INTRODUCTION

The use of methods developed for the inverse-scattering problem has lead to explicit solutions for the equations of motion of a number of classical (non-quantum) models, e.g., the Korteweg-de Vries model, the nonlinear Schroedinger equation model, the Heisenberg magnet, the sine-Gordon model, etc. (see, e.g., [i]). A common property of these models is the existence of so-called "dressing" transformations, which, although they are not sym- metries of the system, act on the space of solutions endomorphically. Within the framework of the Hamiltonian approach, it has been shown [2, 4] that the group of dressing transformations possesses a natural Poisson structure. The canonical quantization of such models reduces to the procedure of the quantization group [3], which consists of comparing the noncommuting Poisson group with a noncommuting Hopf algebra,

In this paper, we propose an alternate method for obtaining exactly quantizable models, in which the Hamiltonian formalism is replaced by the functional integral formalism. In this approach, the possibility of exact quantization of these models is contingent on the possibility of exactly carrying out the functional integration in a nonperturbative fashion. It is possible that the method proposed here will lead to an alternative way of formulating the theory of quantum groups.

The author is grateful to V. N. Popov, E. K. Sklyanin, and A. Yu. Alekseev for stimula- ting discussions.

i. Formulation of the Model

Let us consider a dynamic system whose space of fields X is the locally uniform space

of a certain infinite-dimensional Lie algebra ~ . Let us denote the vector fields on X that

correspond to elements of ~ by V~ (where ~ ). The dynamics of this system is deter-

mined by the action S which is a functional on the infinite-dimensional manifold X. Let us

denote the space of functionals on X obtained by varying the action S with respect to the

elements of ~ by~;=~. Let the action be such that the algebra ~ acts endomorphically

on the space ~ extended by the space of constant functionals. This implies that the action

of elements of ~ on the functionals from ~ can be written in the form

(1.1)

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 146-152, 1991.

3032 0090-4104/92/6205-3032512.50 �9 1992 Plenum Publishing Corporation

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Algebraic—geometric quantization of interacting strings and superstrings