JGP - Vol. 5, ii. 4, 1988
Virasoro-Gelfand-Fuks type algebras, Riemanfi surfaces,
operator’s theory of closed strings
I.M. KRICHEVER
S.P. NOVIKOV
Landau Institute ofTheoretical Physics
Kosygma 2, 117334, Moscow, USSR
Dedicated to LM. Gelfand
on his 75th birthday
Abstract. In thispaper we construct the operatorfields of the Ricinarin surfaces of
arbitrary genus. The coiresponding operator theory ofmteracting strings can be considered as the direct development of Virasoro-Mandeistam theory for g 0 and its
unification with Polyakov-Belavin-Knizhnik theory
This review sums up the results of three authors’ papers [1-3]. The theory of the
bosonic string in critical dimension ‘1) = 26, which was proposed by Polyakov-BelavinKnizhnik and others [4-6] is based on thepath integral approach. The g-loops contribution to the partition functions, to the string amplitude can be represented in this approach
in terms of finite-dimensional integrals over the modular space of Riemann surfaces of
the genus g.
On the other hand the ‘sold>> bosonic string theory by Virasoro-Mandelstam et al.
[7-8] had existed only for the case of <<genus g = Os’. This theory was based on the
usual operator’s approach to the quantum theory. The creation and annihilation operators for non-interacting bosonic string were introduced using the Fourier expansion for
the co-ordinates X’~(a), 0 < a < 271. Then in the corresponding Fock space the
Key-Words: Vfrasoro-Gelfand-Fuks algebras, Riemann surfaces, operatorfields, strings.
1980 MSC: 30F30.
632
KRICIIEVER AND NOV1KOV
<<physical>> states can be defined using the action of the Lie algebra of the symmetry
group of this theory the group of reparametrization of the circle S’ (more precisely
the Virasoro-Gelfand-Fuks algebra which is its central extention). Each of these states
generates the so-called Verma-module the infinite-dimensional highest weight representation playing important role in the whole theory. All those algebraic objects are not
clearly presented in the Polyakov et al. theory.
The main purpose of our papers is the construction of the operator fields on the Riemann surfaces of arbitrary genus.
The corresponding operator theory of interacting strings can be considered as the
direct development of Virasoro-Mandelstam theory for g > 0 and its unification with
Polyakov-Belavin-Knizhnik theory.
—
—
1. TENSORS ON THE RIEMANN SURFACES. ANALOGUES OF
THE FOURIER-LAURENT BASES. ALGEBRAIC FUNCTIONS AND
VECTOR FIELDS. SPINORS.
Let’s considerthe simpliest one-string <<diagrams>> (Riemann surfaces) corresponding
to the world sheet ofthe string which can in the intermediate moments break up into a few
components, than glue some of them and etc. The admissible processes can be described
in the following way. The triple (F , P.. P_) would be called a <<one-string diagram>>,
where F-compact Riemann surface ofthe genus g 0, P~,P~-two points on it. The
holomorphic co-ordinates in the neighbourhoods of these points would be denoted by
z~(Q),where z~(P~)= 0.
There exists the unique differential dk with the following properties:
a)
it has simple poles at P~ with the residues ±1 and is holomorphic on F
outside P~
b) the function Re k( z) is single-valued on F (i.e. all the periods of the differential dk on ‘y\( P~U P_) are purely imaginary).
The function Re k( z) would be denoted by r( z) and would be called ‘stimes’. We
shall denote the curves ‘r(z) = const = 7~ by C1., the domains r~< T
r2 by
,
<~
C~.2 (Riemann annulus). The curves C1. for r
±00 tends to small circles around
the points P±,respectively. In the neighbourhoods of points P~ the canonical coordinates z~can be introduced so that the differential dk would have the form:
—‘
(11
.
dk
dz~/z~
dk=—dz_/z_
(near
P~),
(nearP_).
2, z~= z, z_ = w z~1,P~ 0, P_
For
0 domains
wehave:ofFdefinition
CP’ ofScanonical co-ordinates do not intersect.
g > g0 =the
=
oo. For
633
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
REMARK. The general <<multistring diagrams>> can be represented by the set (F, P.~,
P.~,c~,
c~),where P.1.~,P_1 are the points of the surface F and c~,—c~are positive
real numbers, such that
(1.2)
There exists the unique meromorphic differential dk on F with simple poles at
P~1,P_~and the residues c~,
c~’at these points and such that the function Re k(z) is
single-valued on F. Again we shall denote this function by r( z) and call it <<times’.
The theory of multistring diagrams would be considered in detail in our papers to follow.
Let’s consider the tensors of the weight A on the Riemann surfaces. In local coordinates holomorphic tensor ofthe weight is defined as the value of the form:
f
f(z)(dz)~’
with the following transformation law under the changing of local co-ordinate
f(z)~f(z(w))(~A
\ dw
The definition of the tensors of the complex weight A require the introduction of
some additional structures on the Riemann surfaces. We shall consider them in some
special cases below. The definition and investigation of tensors for the integer A can
be obtained without any difficulties. The important case A =
(spinors) requires the
introduction of a spinor structure.
The most important cases which are necessary for the construction of the operator
theory are as follows:
f
A
=
—1
(vector fields)
A= 0
(scalars)
A
=
~-
(spinors)
A
=
1
(differentials)
A
=
2
(quadratic differentials)
Let’s denote the value
S
~-—A(g— 1)
by S = S( A, g). From the Riemann-Roch theorem it follows that:
634
KRICI-IEVER AND NOVIKOV
1. For any ~-.~one-string.~’
diagram (F, P~,P) in the general position, any
integer A, integer n+ (except for thecases which would be listed below) there exists
the unique, up to the constant factor, tensor f~with the following analytical properties:
a) tensor ~ is holomorphic on F except for the points P~, where it possibly
has the poles offinite orders:
b) near thepoints P~it has the fonn
LEMMA
~-
(1.4)
f~= const ~
+
O(z~))(dz~)A.
We shall denote by MA the space of tensors meromorphic on F with the poles only
at the points ~d:~
The exceptional cases are as follows:
g=l,
1
n=~-,
9>1,
f~using the following asymptotics near
In these cases we can define
P~.
~
(1.5)
A=l:f,=0(l)zdz,f~=0(l)zI’dz,l,~<~
The nature of these exceptions is very simple. For g = 1 there exists holomorphic
non-zero differential dz on F corresponding to euclidian co-ordinate z on F. That’s
why any tensor can be globally presented in the form fA = f( z) ( dz) A where f( z)
is a scalar function. In this case there exists no actual difference between the tensors of
different weights.
For A
I and g ~ I there exist g holomorphic differentials. These differentials together with the differential dk which was introduced above, generate the (g +
1)-dimensional space, corresponding to the indices In~<
The choice of the basis
in this space is non-canonical; below we shall use conditions (1.5). The case A = 0 is
dual to A 1 and we shall use the conjugate basis.
There exists the natural scalar product of the tensors of the weights A and (1 A)
~-.
—
(1.6)
(fA,
91_A)
=
.L,
fAgl_A~
viRAsoRo-GELFAND-FuKs TYPE ALGEBRAS
635
From the definitions (1.4, 1.5) it follows that, after the appropriate choice of constant
factors, the bases f,~’and f,~ are dual
(f~,fl_A)
(1.7)
=
Let’s now consider the so-called Baker-Akhiezer function which is widely used in
the theory of periodic difference operators with scalar coefficients (in particular, in the
theory ofperiodic Todalattice, discrete KdV equation [9, 10] and in the theory of general
commutative difference operators [11, 12]). This function F~(n~
z) is defined for any
triple (F,P~) and any set of g points V = (‘y~+
19) in general position. It has
the asymptotics near points
. . .+
F~(n,z)= const .z~(l + O(z±)), nEZ,
and outside them
points P~then
for g
=
—
simple poles at the points ‘yr,... , 19• if all these poles tend to the
2q: F~(n,z)
—+
f~(z) = F~
0(n,z),Vo= qP4
for g=2q+ 1: F~(n,z)~
+
qP_,
=F~0(n,z),V0 =qP~+(q+ 1)P_.
Hence for A = 0 our basis ~ is the particular case of the Baker-Akhiezer function
corresponding to the special choice of the set of poles at points P~. For A ~ 0 the
consideration of the values
F~(n,z)=
f~/f~
reduces the general case to scalar Baker-Akhiezer functions. The divisor VA of the
poles of this function for A ~ 0, 1 in general do not contain the points P~ and are not
special. We shall call tensors f~(
z) = FA( n, z) the ‘stensor Baker-Akhieser functions’.
It is the common eigenfunction ofthe commutative difference (in respect to the variable
n) operators and the curve F is the curve of <<spectral parameter>>.
The general theta-functional formulae for F~(n,z) were obtained in [11]. Hence
the exact formulae for our basic tensors
can be easily obtained from the soliton
theory. From these formulae it follows that coefficients of the difference operators (the
eigenfunctions of which are f~
( a)) are quasi-periodic functions of the variable n.
Hence, one can use the avaraging procedure and obtain the following analogue of the
Fourie-Laurent expansion.
THEOREM I. Let C7 be non-singulari For any smooth tensor fA of the weight A on
C7 expansion
(1.8)
fA(a)
=
~f,~(cr)
x (~-.fA1)fa1daF)
636
KRICHEVER AND NOVIKOV
is valid. The same expansion is valid for the tensors f)~(
z) which are holomorphic in
theRiemann annulus Cr,r. The convergence ofthis series is thesame as in the ordinary
Laurent-Fourier series.
(The theorem is valid for singular contours C7 but smoothness conditions in these
cases are slightly more rigorous. The theorem is valid independently of whether the C7
contour is connected or noL)
The importantproperties of our bases f~(which immediately follow from the definition) are their almost-graduated structure in respect to the multiplication
(1.9)
‘~s Ins
—
‘~~‘
()Aj~,k rA+p
—
~_,
‘~n,ffi
Jni~rn—k’
IkII
(1.10)
[e~,f~]
=
~
R~f~rn_k,9O =
kI<g>
Here and below we use the notations
(1.11)
~
For exceptional cases A,~i= 0, 1; I~I< ~ or
include the additional terms with lkl = + ~ Iki
in [1, 2]).
ml
=
g0
the sum in (1.9, 1.10) must
e where e = 1,2 (see exactly
~-
+
DEFINITION. An almost-graduated (N -graduated) algebra L (or module M over L)
is an algebra (or module) which can be expanded into direct sum of the subspaces
L=~L~,
M=>~M~
so that
L2L, E
>
IkI<N
(1.12)
L.M1 E ~
IkI<N
According to (1.9), we have the commutative almost-graduated algebra A’ of scalar
functions on F with the basis A0 and the N = g0-graduated Lie algebra Lr with its
basis e~.
637
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
All the spaces MA of the tensors are N-graduated modules over algebras Ar ( N =
and IT(N=90,g>1).
The other examples of the almost graduated modules in context of the soliton theory
are discussed by the authors in their work [I]. Here we shall briefly consider only the
simplest generalizations of the modules MA which are particularly important in the case
A=
The corresponding modules M~’~’
depend on the set of data (F, P~,p, ~ v)
where: a is the line which connects the points ~±; p: ~1 (F)
C~is the character
of the fundamental group
711(F);
an (multi-valued)
arbitrary complex
number.
1’is the
spacepofisthe
tensors
of the weight A which
The
module
M~’~’
are holomorphic on F except for the points P~and the line a. Along this line a the
~.
—~
boundary values of such tensors must satisfy the following relation
f(z)
For each closed cycle
e2~”’f(z), a E a.
=
‘~
e
71~(F)the changing of
f”
E MP,~,Pwhen moving along
1~is the multiplication fA by the complex number p(’y).
The cases p( 1) = ±1 correspond to spinor structures on
r.
LEMMA 2. If A =
(e’ p) is a half-integer and p is therepresentation in the ‘sgeneral
position~,there exists the unique, up to the constant factor, spinor CI),,( z, p)
whichin the neighbourhoods ofthepoints P~has the form of
~-,
(1.13)
—
i~(z,p) = const .z~”~(l+
For an integer p the tensor ~
does not dependon a.
The representation p such that p( 1) = ±1 is in general position iff (for general F)
the corresponding spinor structure is even.
The analogues of the almost-graduated properties (1.9, 1.10) are valid for spinors
(1.14)
cI~~(z,p)cD~(z,p)
= ~
IkIf
(1.15)
[e~,11~~J=
~
~
In case p(’y) = ±1 (spinor structures, p = p’) spinors CI~,are square roots from the
meromorphic differentials.
We can unite the data p,
considering them as the character
~
p:ir
1(F—(P~UP_))-->C’.
638
KRICHEVER AND NOVIKOV
2. RIEMANN ANALOGUES OF HEIZENBERG AND VIRASORO ALGEBRAS
The reparametrization group i.e. the group of the diffeomorphisms of the circle
is a natural group of symmetries ofthe theory of closed bosonic string. Its Lie algebra is
the algebra of the vector fields on the circle. For each parametrization p of the circle,
the subalgebras Z÷,Z_, Z0 can be introduced
—
—
(2.1)
where Z0 is one dimensional and generated by the vector field e0
subalgebras Z~are generated by the vector-fields
e0EZ÷,e~0~Z,n>0;
=
z
~,
z
=
e’W and
e0=z0+1~_.
The algebra Z, as it has been shown by Gelfand-Fuks [13], has a single cohomology
class central extension, which is defined by the cocycle
-
(2.2)
x(f,g)
=
—~-—J
48iu s’
(f”g
—
g”’f)dz,
where f = f( a)
g = g(z)
are vector-fields on the circle. In this extention the
commutators of the elements have the form
~-,
~-
[f,g]
-
=
(f’g —g’f)
~+
x(f,g)
.t, [f,t]
=
0.
This extented algebra would be called <<Gelfand-Fuks algebras>> and denoted by Z,~.
At the beginning ofthe 70-ies this algebra was independently discovered by physicists
(Virasoro, Mandelstam and others) in the context of constructing the operator quantization of the string. To be more precise, they have found the Z-graduated subalgebra
L~of Z, which consists of the central element t and all trigonometrical polynomial
vector-fields (i.e. all finite combinations of vector-fields e0 and t)
L~C Z~,L~= ~
(2.4)
L~= (e0), n~0,L~= (e0,t).
nEZ
[e0,
em]
=
(m
—
n) en+m + ~
—
12
This Z-graduated algebra L~is called <<Virasoro algebras’. It has the obvious decomposition, corresponding to (2.1)
=
+
+
L_.
vIRAsoRo-GELFAND-FuKs TYPE ALGEBRAS
639
The most important representations of the Virasoro algebra are the so-called right <<Verma moduless’ ~
The module W,~ has the <<highest vector>> ~pR which satisfies to
the following relations
(2.5)
eotP~~=h~VR,
t~V~?=ckYI~.
L~pRO
The space of the representation W~ is the space of the finite sums of the basis vectors
of the form:
~
fl1~~2....flk>O.
For the general pairs (h, c) the module W,~is irreducible. The reducible modules
W,~correspond to the pairs (h, c) such that Pn,m (e, h) = 0 where ~n,m is one of
the <<Kac-polynomialss’.
The left Verma module W/~ has the generating vector ~ L
pL~~
qIL~
0~qJL
q~~L~=~pL
The details of the representation theory of the Virasoro algebra can be found in [14].
The simple and important example ofthe reducible Verma module corresponds to the
<<vacuum sectors’, where h = 0 and c is arbitrary. If h = 0 then the non-trivial vector
i.jiR = ~~pR
satisfies the relations:
5’~~=—’P~~,
t+’~=c41’~.
L~~1”~0,e0
If W,~is reducible then the irreducible representation can be obtained as the factormodule of ~
overthe ideals which are generated by all the <<singular-vectorss’ ~~JJ?
E
W,~ such that L+’PJ~= 0.
The vacuum sector in the string theory is irreducible. Hence, we must have in this
sector the relation e_
1
= 0.
Let’s consider now the Heizenberg algebra which is more simple then the Virasoro
algebra. This algebra has the basis a0, a t, n> 0, with the following commutators
,
(2.8)
[an,am]
=
[a~,a~]
[a0,a~]
= ~n,m
=
[t,a0]
•~•
This algebra A also has the decomposition
A =A÷+ A0
+
A_
(2.9)
(as) (t) (a).
=
[t,a]
=
0,
640
KRICHEVER AND NOVIKOV
The analogues of right Verma modules for this algebra are well-known in the elementary
quantum theory. The generating vector ‘l’~ in this case is called <<in-vacuums’
(2.10)
A~P~ 0,
t’P~
~~vac
The space of the representation of algebra A (Verma modules) is called the right (or in)
bosonic Fock space of the scalar theory. Its basic vectors have the form
(2.11)
a~
. . .
The left (or out-) Fock space can be defined in similar way. As it was shown at the
beginning of 70-ies the physicists, the operators
(2.12)
=
~-
~
: a0a_,~÷t
: ; a~=
generate Virasoro algebra with central change c
orderings’ has been used:
anam,:
=
1, L~~ —ek. Here the <<normal
aa~:= aa~,
:a0a~:=: a~a0:= a~,a0.
The geometrical realization of Verma modules over Virasoro algebra has been proposed by Feigin-Fuks [14]. For any complex p the tensors of the weight A onthe circle
with the <<multiplicator>> p
2~f(~)
f = f(~)(dw)A,f(~+ 2~)= e
can be defined. In the space M~ of such tensors the tensors
=
are basic.
The finite linear combinations of the basic vectors generate the module M~.Let’s
consider the right semi-infinite forms exterior products of the form
—
(2.16)
~
such that the consequence
(n
1, n2,...
,
This means that for some k and k0, n3
form are
(2.17)
=
n8,...) becomes stable from some number on.
= s + k — 1 > k0. The simplest vectors of such
f~~A f~’4~
A fk~P2A
641
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
The action of the vector fields e0 on the forms (2.16) can be correctly defined with the
help of Leibnit.z rule for n~0. The attempt to define the action of e0 with the help of
the commutator’s relations leads to the representation of the central extention of L, i.e.
to the representation of the Virasoro algebra (2.4) with the central charge
2 + 12A —2.
c= —l2A
The space of all right semi-infinite forms is the direct sum
nuR
“Apt,
—
VVA~p
keZ
where WA~~,k
is generated by Virasoro algebra from vector ‘1’~1~(2.17). The conespondinghighestweightequals h~=~(p+ k—A) x(l —p—k—A).
In the same way the space of the left semi-infinite forms can be defined. The space
~
is the space of finite linear combinations of the left semi-infinite form:
(2.20)
...
A f~’A
...
A
f~’A f~”
where m
8
=
k
—
a + I for a> k0 for some k0.
W~= ~ W,,”T?,~, ‘i’~=
...
A
f~ A f~’1A fAP
kEZ
This construction ofthe representation of the Virasoro algebra is based on special Fourier
basis in the space of tensors on the circle. Our construction of the operator fields on the
Riemann surfaces, the definitions and investigations oftheir vacuum expectation values
(Green functions) widely used the analogues of the semi-infinite form. At this moment,
especially, the analogues of the Fourier-Laurent bases, which were introduced above,
are necessary.
To begin with we shall introduce the analogues of the Heizenberg and Virasoro algebras in the pure geometrical way.
Let (F, ~±) be an arbitrary one-string diagram. The commutative algebra Ar of
the meromorphic functions on F with the poles atthe points P~has the basis A0 =
which was introduced in § 1. The analogue of the Heizenberg algebra is the Lie algebra
with the basis a0, t the commutators of which have the form of
(2.22)
[a0, am]
= Inns
t, [a0,t]
=
0;
1mm =
f
~
~
From (1.4) it follows that
ml
Inns = 0, ni-
ni, mi>
~-;
N=g+2,Inl,lm~<
N
~-.
>
const
g+ 1,
A~dA0.
c~
= N,
~>
~-,
m~<
642
KRICHEVER AND NOVIKOV
For g = 0, P.F = 0, P_ = oc we obtain the ordinary Heizenberg algebra.
The definition of the analogue of the Virasoro algebra requires the introduction of
the <<projective structures’ on the surface F \( P÷U P_), i.e. the introduction of the
systems oflocal coordinates, suchthat they connect each other with the help of projective
transformations from the group SL( 2 C) / ±1. Let’s define the commutators
,
[en,em]
~
=
+
Xy,n
~
(2.23)
[e0,t]
=
0,
where the coefficients c~ in (2.23) are the same as in (1.10). The cocycle x( f, g) for
the pair of vector-fields
ff(z)-~-,
c9z
g=g(z)-~--
in the system of the projective co-ordinates has the form
(2.24)
x(f,g)
g”f)dz,
—~—
f(f”~
—
48irz
,
where ‘1 is the arbitrary element of the homology group
-
-
[1]EH1(F\(P+UP_);Z).
The formula (2.24) is correctly defined because the value f”g —g” f is trasformed as the
1-form under the projective transformation of the co-ordinate a. The cocycle x can be
defined in any (non-projective) system of the co-ordinates with the help of the so-called
projective connection R on F, which is holomorphic on F outside the points P~.
Such connection is defined in any system of co-ordinates as the function R( a), which
is transformed in the following way under the transformation of the co-ordinate:
(~)
2
(2.25)
R(z)
—*
R(z(w))
+
,,,
[~-7
~(~)]
,,
—
2
,z’ =
The difference between the two projective connections is the quadratic differential.
In the arbitrary system of the holomorphic co-ordinates, where R~0, the cocycle
(2.24) can be presented in the form of:
(2.26)
x( f, g)
=
~
j( f”g
—
g”fx
—
2 R( f’g
—
g’f) ) dz.
The cohomology class of these cocycles does not depend on the choice of the projective
connection (i.e. on the choice of projective structure on F) but for our purpose the
choice of the cocycle is important.
VIRASORO-GELFAND-FUKS TYPE
643
ALGEBRAS
THEOREM 2. All almost-graduated central extensions of the algebra Lr (i.e. such
extensions of the algebra Lr that x(e
0, em) = Inns = 0 ~f In + ml > const) are
defined by the formulae (2.24) or (2.26), where the class ofhomology [I] is the class
ofthe cycle C7, which separates the points P~and P_ on F.
.
CONJECTURE.
2(LrR) =H
H
1(F\(P~UP_),R).
It must be mentioned that the Riemann analogue of the Heizenberg algebra which
was defined above with the help of the formulae (2.22), is also almost-graduated if the
homology class [1] is the class of the cycle C7.
Below we shall consider only almost-graduated central extensions of Lr and Ar
because this structures are very important for our futher constructions.
The complex conjugate anti-holomorphic theory can be constructed completely in
the same way. The string theory involves (as for the case g = 0) both the holomorphic
and anti-holomorphic algebras.-But we can confine ourselves to the consideration of
holomorphic part only because holomorphic and anti-holomorphic algebras commute
with each other.
The Riemann analogue of the Virasoro algebra has two filtrations which are generated
by the Taylor expansions near the points P±,respectively
a)
L~H...jL1jL~j
b)
L~:...JL~JL;1D...
The space L is generated by the vector-fields e1 where ~
9o + n (i.e. the
expansion of e1 near P~begins from z~,k n+ 1).
The space L; is generated by the vector-fields e1, 5 < m g~.
We have
1L~L~’ L~
—
1
n’
n4-rn~
mJ
The adjoint algebras
= ~L/L~
1
,L~
both isomorphic to the ordinary Virasoro algebra.
The similar filtrations have the Riemann analogues of Heizenberg algebra.
The algebra L~,Ar and spaces of tensor and spinor fields have the decompositions:
Ar=A++A0+A_
(A=0),
MA=M+,A+MoA+M_A
(A=~-,—l,O,l,2),
644
KRICIIEVER AND NOVIKOV
where A~,L~,M+,A are generated by the basic tensors (or spinors) ~A, n, which at the
point P.,. have the zero of the order s( A)
s( A
s(A
=
=
=
0) = I
—1) = 2
= 0
~-)
for
for
for
A.,.
L.,.
A=
(scalars)
(vector-fields)
(scalars)
~
The subspaces A_, L_, M_,A are defined in the same way.
In the dual spaces A
I
A the similar decompositions are dual by definition
according to the scalar product (1.6). For A = I 2 we have
—~
—
,
M1=M~1+M01÷M~1 (A=l,l-form)
M2
=
M~,2+ M02
+
M_2
(A
= 2,
quadratic differentials)
The space M01 consists of the holomorphic differentials and the differential dk which
has simple poles at the points P~.The subspace M02 is the space of the quadratic
differentials holomoiphic except for the points P±where they have the poles of the
orders not greater than 1. The dimensions of the spaces M0 1,A0 equal to g + 1. The
dimensions of the spaces L0 and M02 equal to 3g + 1.
The subalgebras A.,. A_ are commutative.
Let’s define the subspaces
,
M±AC M~
-
-
as the space of all the tensors of the weight A which are holomorphic on F except for
the point P.,. or P_, respectively. The intersection M~,A fl M_ ,A consists of the tensors
which are holomorphic on F everywhere. The peculiarity of the case A =
(for an
even spinor structure) is the properties that
~-
+
M_~= M~,M÷~
fl ~
=
M~.
=
0.
In this case
The spaces M±~
are dual to each other with respect to the scalar product (1.6).
The analogues of Verma modules for the algebras Ar, L~are defined with the help
of the generating vectors and the following conditions
a)
A.,.
= 0, (right (in) Fock space)
= 0, e90
= h’1’c~(right Verma module)
= c’P~~.
b)
‘I’,~~A_
= 0, (left (out) Fock space)
~
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
=
645
0,’I’~e_90= h’P~,(left Verma module)
=
The operator belonging to A~(A_) are annihilation operators of <<in-s’ (<<out-s’)
states.
From the existence of the filtrations of A’, L~which are generated by the Taylor
expansions at the points P~it follows that the closure of these spaces are isomorphic
to the ordinary free Fock spaces and Verma modules but with different basic states and
different Z-graduated structures.
The <<frees’ basic states are defined with the help of the canonical local co-ordinates
(1.1). The transformation matrices U~from the bases 4 to Am =
are triangular.
The powers 4 correspond to the creation and annihilation operators of the free (in) and
(out) states for r
~oc. Hence, formally the matrix
—~
s= u~’u+
defines the analogue of the Bogolubov transformation from the basis of free in-states
to the basis of free out-states. But for g > 0 the matrix S is ill-defined because the
elements of the product of the infinite matrices U~,U,. are given by the series which
seem to diverge.
The basic elements of the Riemann analogues of the Virasoro algebra can be represented in terms of the generators of the quadratic expressions as in free case g = 0
(Sugawara-type construction). Therefore, the Riemann analogues of the Virasoro algebra are acting in the Fock spaces (in- and out-). The different physical states correspond
to the different Verma modules with the different highest weights h. In particular, the
vacuum sectors in the right and left Fock spaces correspond to h = 0 and are generated
by the vectors ‘f’~ and ~
respectively. The central charge c would be equal to
the dimension of physical space c = V.
It must be mentioned once again that in vacuum sectors the following relations are
valid:
11vac = 0,
e90
I’~
= 0,
e~0
-1 ‘
~1’v’~ce_go
= 0,
~P~~e_
90+1= 0.
(We shall call them <<the regularity conditions of the vacuums’.)
3. THE RLEMANN ANALOGOUS OF THE HEIZENBERG AND
VIRASORO ALGEBRAS IN THE STRING THEORY
The phase space of the classical V-dimensional string in the Euclidean or Minkovsky
spaces is the space of 2 ir-periodic functions X’~(a) and 2 it-periodic 1-form PM( a)
with Poisson brackets
(3.1)
{p”(a’),X~(a)}=
646
KRJCHEVER AND NOVJKOV
where A (a, a’) is the 6-function on the circle (which is a scalar function of the variable
a and 1-form of the variable a’) i.e.
f( a)
(3.2)
=
f
f( a’) A (a, a’) da’
-
This definition is adeguate only in the case of free string because in this case there are
no topological bifurcations ofthe string.
Fora one-string diagram (F, ~±) the contour C.,. plays the role of the string position
at the fixed moment ‘r.
Let X~(Q) PP( Q) be an operator-valued scalars and I-forms for Q C F which
commute with each other at different moments of <<times’ ‘r. Naive quantization of (3.1)
gives us
,
(3.3)
[x~(Q), Pv(QF)]
=
—i~”A,.(Q, Q’),
where A,. is the 6-function on the contour C,.. As it follows from the results of
this 6-function can be represented in the form
(3.4)
~-~--~A0(Q)dw0(Q’),Q,Q’
A,.(Q,Q’)
§
1,
CC,..
Let’s expand X~and P~in our analogue of the Fourier series
X’~(Q)=
(3.5)
P~(Q)= >~Pdw0(Q).
The direct consequence of (3.3), (3.4) is the commutator relations
(3.7)
[P:,x,~j
7l’~’6m,n~
The coefficients of the expansion of the 1-form dArn ~1n 1rnnthL~n are given by the
formulae (2.22). Then the operators a~which are defined from the expansion
M)da— >c~dw
(3.8)
(irP~+ c9aX
are equal to
(3.9)
=
itP
+
0
647
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
LEMMA
(3.10)
3. The operators c~satisfy the commutatorrelations of the Heizenberg algebra
[~,~]1nm7f”’
The complex conjugate differentials, scalars, tensors lead to the definition of the operators o~which commute with a~
X~’(Q)=
.~A0(Q),P~(Q)
=
(5aXP_irP1S)da=~~~n(Q),
P(Q)=
~
{a~,ew~j
= 0.
= fl~1nm,
The Fock spaces ofin- and out-states can be defined using the vacuum vectors W~,
and the conditions
(.
311 )
qiL~ c~v~c~
_~pL~
vac n
vac n
=0
‘
I’~C
n>~,n=—~.
n~—~2
—
In the classical case the densities of the Hamiltonian and momentum are the linear combinations of the values
T= ~-(Xa+7iP)2
=
I~2~
~(x0._irp)2
=
The definitions of the corresponding quantum operators require, as usual, the definition
ofthe<<normal orderings’. Let’s dissect the integer (or half-integer) plane of pairs (n, m)
into two parts ~
such that ~ differs from the integer half-plane m < n only in
the finite number of points. The definition of the normal ordering depend on the choice
(3.12)
(n,m)E~,
(n,m) E ~
In [2] a bit more general definitions of normal ordering were considered.
We can define the quantum operators
~
~m~n’
T(Q)
=
D(Q)
= ~-:
~-
: 12 :=
:
~
:
dthn(Q)dWm(Q),
(3.13)
j2
:=
~-
~
:
~n~m
:
They are quadratic differentials on C,.. Hence, they can be expanded in the series
2fl~(Q), T=~L~d2c?.~(Q).
(3.14)
T=~L~d
The quadratic expressions of L~through a~follow from (3.13).
648
KRICHEVER AND NOVIKOV
DEFINITION. The nonnal ordering is admissible if the conditions of the regularity of
vacuum
r
us R
vac
—
—
r
LIgo_1
us R
vac
—
n.
—
‘•~~
UI
L 1~—g
r
‘vac
—
0—
u, L
~
—
‘vac~’~g0—
are fulfilled.
THEOREM 3. The operators et = —Li, where L~are given by the formulae (3.13),
(3.14) satisfy the commutatorrelations (2.23) ofthe Riemann analogues ofthe Virasoro
algebra with the central charge t = V. The cocycle Xmm depends on the choice of
the nomiai ordering but his cohomology class does not depend on this choice. For the
admissible normal ordering the coffespondingprojective connection is holomorhphic on
F (for othernormal ordering it has thepoles at points P~).
The proof of the theorem is given in [2, 3], where the examples of the admissible
normal ordering are discussed.
Let’sconsider the geometrical realization of the modules over the Riemann analogues
of the Heizenberg and Vitasoro algebras which are similar to (2.16-2.21). The bases of
spaces ofthe right and left semi-infinite forms can be defined by thesame formulae (2.162;21) using the bases f~.We shall denote these spaces by W~= ~ W~, W~ =
~ W,~omitting in this denotation the dependence of these spaces on the diagram
(F,P~) and the number p.
The almost-graduated structure of all modules under consideration provides the simple proof ofthe following lemma.
-
LEMMA 4. The action ofthe vector-fields e0 on the space ofsemi-infinite forms can be
correctly defined with the help ofthe Leibnitz nile (and also the action e0 on f~)for
nI
g~.This action can be extended to the representation ofour Virasoro-type algebra
with the central charge depending on the tensor weight A
2+ 12A—2.
t=c= —l2A
>.
The highest weight ofthe restriction ofthis representation on the subspace W~~(
W~~)
generated by vector tP~R(‘Pt) is given by the formulae (4.9) in [1].
For our purposes the most important cases are as follows:
A
=
~-,
c= I
(for the physical scalar bosonic fields),
A
=
—1,2, c= —26
(fortheghostfields).
649
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
Now we shall consider in detail the first case A =
normalization of the basic Fourier-Laurent-type tensors
(3
.
15)
A
f~(z)
1J ~AZ(I
2~—S~
1
—
—
~-.
Let’s consider the following
+ O(z_))(dzDA,
+
S= S(A,g)
We shall call it- <<in-normalizations’. It is unique up to the transformation a.,.
qz~,f~
qn_Sf,,A. It is natural to call the normalization (3.16)— <<out-normalization>>.
—~
...,
1f~(z)
(3.16)
= {+zn~S
(1
+
J~(z)= (~AY
Inparticular,for A = ~ wedefine f~= (I),,,f~ = ~n’ where n ishalfinteger.
The generating vectors ~P~(‘l’~’~)
for any A in the geometrical realization have the
form
(3.17)
I.fI
—
=(...Af,~_
1 AfkA,)
=
Below we shall use only <<in-normalizations’.
Regularity conditions of the vacuum.
The generating vectors of the form (3.17) can play the role of the in- (out-) vacuum
vector only if the numbers k, k’ is such that the corresponding semi-infinite form coincides (up to the constant factor) with the exterior product of all the positive powers of
the local parameter
(3.18)
<OAI =
t’(H~<~,
2 IOA
Az_Al)
= (IA z~A z~A
~A)
>.
..
(...Az
(in the latest equality at this moment we consider the product
From the asymptotics (1.4), it follows that
(3.19)
LEMMA
k= k(A)
=
S(A,g)
=
~-
fln<k’
—A(g— l),k’= k’(A)
~A
as formal).
—S(A,g)
5. The vacuum vectors (3.18) satisfy the regularity conditions (2.29).
650
KRICI-IEVER AND
NOVIKOV
Later we shall discuss the physical motivation of the regularity conditions (which
were given by A. Polyakov and others in the different language). Later we shall consider also the problem of the regularization of the product U ~
which formally is
n<k
diverging.
There exists the natural scalar product between the spaces of the right and left semiinfinite forms. For the basic forms f E W~,ge W~let’s consider the product f A g.
If this infinite (in bothdirections) form coincides after the permutation with the standard
form (the exterior product of all basic tensors f~)then we define
(3.20)
(f,g)(—l)~,
fAg=(—l)~i~z~,
where e is the sign ofthe corresponding permutation. In other cases the product (f, g) =
0 would be equal to zero. The scalar products of any elements f E W~,g E W~can
be defined by the linearity. The basic forms are the exterior products of the tensors f~
in the <in-normalizations’.
LEMMA 6. If k and k’ are given by the fonnulae (3.19) (i.e. the corresponding generating vectors (3.17) can play the roles of the vacuum vectors), their scaiar product
(‘f’t,’Yk1~0 iffA=~,k—_f,k’=_~.
A few -words should be said about the normalization of the vacuum vectors. It is
natural to define the vector 0 >=
using <<in-normalizations’, and vector < 0 I =
‘f’~-using<<out-normalizations’. The normalized out-vacuum differs from the vector
~
(which is taken in the <.in-normalizations’) up to factor
‘~‘~
(3.21)
<01=
(H~~)
~
n<o
By the definition which was introduced above, we have
(3.22)
(0~I0+)=
(qIL,
Pft)
= I
Therefore,
(u~
\n<O
(We still consider the product in right-hand side of this equality formally. It was shown
by lengo (private communication) that after the regularization it coincides with the determinant of Dirac operator).
LEMMA 7. The operators Ln acting in the spaces ofright and left semi-infinite forms
are self-adjoint in respect to the scalarproduct (,) which was defined above, i.e.
(3.23)
(f,Ln9)
=
(fL~,g),
f E W~,gE W~
651
vmAsoRo.GELFAND-puKs TYPE ALGEBRAS
Below the normalized in- and out- vacuums would be denoted by 0) (01.
Let’s consider the spaces 1-1k of the Dirac fermions. They are generated by the
fermion operators iJ,,, ~1’~with the half-integer indices which satisfy the commutator
relations
(3.24)
~
~
=
=
0,
~
=
The <<vacuums>> OF), (0~I are defined by the relations
(3.25)
=
O,u >0,
0FI~ = (OFI~ =
O,~LL<0.
~vI°F)
~IOF)
=
(
If we suppose that the operators iJ.’,,, ~
(3.26)
7i~= ~
have the <<charges’ 1 or —1, respectively, then
p-charge.
The natural isomorphism between 1i~and W{~,Wi~can be obtained, if we consider
the correspondences
—b
(3.27)
—÷
(OFI
(multiplication on
(differentiation)
4~A
4-
-.
(u~<
0~‘~)
<01,
IO~) — 0).
Let (I),~ = ~~(z,p’) be, by definition, the dual half-differential (spinor). According
to (1.13)
(3.28)
~-~---
f ~
=
Let’s introduce the <<fermionic operator-fieldss’
~(z,p)
=
(3.29)
~(z,p)
=
~~w,p),~(z,p)
=
THEOREM 4. The chronological ordering product of the operators ~( a, p) ~
where i-( a) > r( tu) is defined correctly. For a
w the operator expansion
—*
(3.30)
~(z,p)~(w,p)
+
I(z,p)
+
O(z
—
w)
( w, p),
652
KRICHEVER AND NOVIKOY
is fullfilled. The coefficients ofthe expansions
(3.31)
I(a,p)
=
~a~(p)dw~(a)
satisfy the commutator relations of the generalized Heizenberg algebra (3.32)
(3.32)
[°~n(P)’’~rn(P)]
=
Let’s define the analogue S~(a, w, p) of the Soger kernel. They are uniquely determined by the following analytical properties. With respect to the variables z, w they
are the multi-valued tensors of the weighL
(half-differentials) which are transformed
along the contours on F according to the representations p and p’, respectively. For
the fixed w the kernel S~is
a)
holomorphic on F except for the points P±and a w and the contour a
connecting the points ~
b) for the boundary values of S~,of contour a the relation
~-
e2’~S,~
=
is valid.
c)
in the neighbourhoods of the points P~it has the form
-
S,
a~O(l)(dz~)’1~.
The analytical properties of S~,in respect to w (for the fixed a) are the same as for
a after the interchanging
p—,—p,
Near the diagonal z
(3.33)
S
=
z—’w.
w the kernel S, has the form
(z,w,p) =
p
+
z—w
LEMMA 8. a) the expansion
(334)
S~(z,w,p) =
~
ds~(z,p)+ O(z
—
w).
653
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
is validfor r(z) > r(w); b) the expansion
(3.35)
S~(z,w,p) =
—
cI~(z,p)cI~~(w,p)
~
vp+
is valid for r( a) <r( w). (Here
ii
~-
—
—
p ~ Z in both cases.)
The main term ds~(a, p) of the regular part of the expansion S~near the diagonal
w is 1-differential, which is holomorphic on F except for the points ~ where
it has simple poles with the residues ±p.For p = 0 this differential is holomorphic on
F.
Using the result of lemma for p = 0 we obtain the equality
z
=
(3.36)
I(z,p)
:
=
: ~(z,p)~D.(w,p)
+ ds
0(a,p).
The comparison of (3.36) and (3.31) gives
(3.37)
cr~(p)= ~a~(p)
:
:
where
~-L
J
(3.38)
a~ =
(3.39)
a=.~~t’
~
a~=
f
~2—
A~ds0,
I~I>~
1In_v_~I>~+l, InI~The differential ds0 is holomorphic on F. That’s why
(3.40)
a~=a~(p)=0, InI>~, n=—~~.
(recall, that A_~=
1
according to our choice of the basis 1,3).
DEFINITION. The vacuum expectation value ofthe operator H is given by the formula
‘H’
‘‘p
=
(O~IHI0~)
/00
\p
p
-
654
KRICHEVER AND NOVIKOV
The vacuum expectation value of the operators equals to ([3])
(3.41)
(~(z,p)~(w,p))~=S0(z,w,p).
In the modern physical literature the formula (3.41) is playing the role of the definition
of the propagator of the fermionic fields without any construction of the fields proper.
The following equality comes from (3.37)
(3.42)
(a~(p))~
= a~(p).
For even spinor structure we have from p
S0(a,w,p)
Therefore, ds0(a,p)
=
=
O,p
p~’that
=
—S0(w,z,p)
p~’.
LEMMA 9. Foran even spinorstructures
(a~(p))~
= 0
for all integer (or half-integer) n.
The usual Vick Theorem is valid for the chronological product of the fermionic fields
on any Riemann surface
(3.43)
‘Y(z1) ...W(z~) =
where ‘P (a) are the fields
~(a, p)
fI(’V(z~)P(z1)): flkP(zk)
~±
I
or
~i~(
:,
k~I
s,JEI
z, p). Here in addition to (341) we have
(~(a,p)~(w,p))=O,
(~(z,p)i4’~(w,p))—O.
4. THE ENERGY-MOMENTUM PSEUDO-TENSOR AND OPERATORS EXPANSIONS
As it was mentioned in § 3, the energy-momentum tensor depends on the choice of
the normal ordering. Now we shall introducethe invariant value the energy momentum
pseudo-tensor.
Let’s consider the chronological product of the current operators.
—
655
vIRAS0RO-GELFAND-FUKS TYPE ALGEBRAS
THEOREM 5. The chronological product V~(z)I~(w), where r( z) > r( w), is correctly defined. For z
w the followingexpansion is valid
—‘
(4.1)
~Iz)I~(w)=V(~’~)2
+2D(z)+Q(z—w).
For anyprojective connection R, whichis holomorphic on F outside thepunctures
P±~
the coefficients ofthe expansion ofthe operator-valuedquadratic differential
2~lk(z)
T(a)
=
D(z)
—
~R(z)
=
~Lkd
have the commutator relations of the Virasoro-type algebra (2.23,2.26), corresponding
to theprojective connection R.
The proof is given in [3].
DEFINITION. The value T( a) is called an energy-momentum pseudo-tensor.
THEOREM 6. The chronological product T( z)T( w), T( a) > r( w), is correctly defined.Forz—’v., wehave
(4.2)
D(a)D(w)
V ~
2(a—w)
+
2D(a)
(z—w)
+
T~(z) + 0(1).
z—w
•
The vacuum expectation values of the products I( z~)T( z,),..., can be easily obtained from these definitions and Vick theorem. For example,
(2~(a)I~(w)) lim
z
lim
1-..z w1—.tv
x
\
(
~(z)~(a~)
\\
—
1
~ x
z—z1J
~(w)~(w~)—
W—W1
Using the Vick theorem we obtain that
(4.3)
(D~(z)I~(w))~=
—S0(z,w,p)S0(w,z,p).
The expansion
(4.4)
—S0(z,w,p)S0(w,z,p)
=
+
R~(z)+ 0(a
—
w)
656
KRICHEVER AND NOVIKOV
defines the so-called Segue-type projective connection R~(a). Itmust be specially mentioned that this projective connection does not depend on the punctures P~.The comparison of the formulae (4.3), (4.4) and the definition of T( a) gives
(4.5)
(D(z))~=~R~(z).
EXAMPLE. Let g = 1. We have 3 even hall-periods w~,a
to 3 spinor structures. The Segue kernel have the form
S0(a,w,p)
=
=
1,2,3, which correspond
a(a —w+w,~) e_h1~(~tL~)
w)a(t~)
a(z
—
Finally, we obtain that
=
where a, P-elliptic Weiershtrass functions. This result coincides with the result of [16]
where it was obtained from the different approach, using the Ward identities.
5. A FEW REMARKS ABOUT ~<GHOSTSECTOR>>
The Polyakov-Faddeev-Popov ghost-fields in the string theory have the tensor
weights —1 and 2 and are fermionic. We shall define them by
2Ll~(z),
c(a) = ~c~e~(a).
b(a) = ~b~d
The coefficients C~,Cm have the ordinary commutators
(5.1)
[bn,bml+
=
[cn,cm]+ =
0, [bn,Cm]+ = 5n,m
As it was shown in [17],the definitions of the energy-momentum of the ghost-field and
the operator of the BRST-charge can be easily generalized for the case of the Riemann
surfaces of the genus g > 0 with the help of the bases which were introduced in § 1.
We do not consider in detail these definitions but shall make a few remarks.
The full Fock space includes the tensor product of the <<physical and <<ghosts’ sectors. In particular, the vacuum vector has to be the tensor product of the <<physicals’ and
<<ghosts’ vacuum vectors. The regularity conditions of the ghost vacuum vector have
(according to the results of the § 3 for A = 2) the form
,
I0
2)=’Pd~, k=k(2)=—g0-i-2,
(021 = ‘P,~, k’ = —k(2).
657
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
(5.2)
b~IO2)=
(5.3)
(02Ib~= 0,
0,
c~I02)= 0, n<
n ge—i,
n _9o
+
1,
(02lc~=
0,
90—1,
n> —g~+ 1.
The operators b~can be represented in the spaces of the semi-infinite forms W~,W~
as the operators of the exterior multiplications on f~and the operators c~can be represented as the derivation with respect to f,~.
As it was emphasized in § 3 the scalar product of the ghost-vacuum must be equal
to zero
(5.4)
(02102)=0.
The non-zero expressions can be obtained only in the presence of the insertions. For
example
(O2Ic_icociIO2)=1,
g0,
(02Ib_~c~I02), g=1,
(5.5)
—1
—90+1
(O2Ib90+2...b90_2I02)= (Hw~2)
~0,
p>l.
The operators b~for I~I ~
2,9 > 1, correspond to the holomorphic quadratic
differentials which are the basis of the cotangent bundle over the modular space of the
surfaces of the genus 9. That’s why the square of the modulus of the value (5.6) defines the measure on the modular space. The connection of this measure with PolyakovBelavin-Knizhnik measure will be considered in our paper to follow.
One of the main questions is the definitions of the ghost vacuum expectation value
for arbitrary operators. The same arguments as for (5.4) show that
—
~(02IT’~i02)
= 0.
The value
2
II
/
go—
fl
+
b~I0
2)0.
2
is non-zero, but there exists ambiguity in such definition oftheghost-vacuum expectation
value of TE~,c, beoause we can arrange the operators T~and b~in a different way.
658
KRICHEVER AND NOVIKOV
6. REMARKS ABOUT THE STATES WITH NON-ZERO MOMENTUM
Here we shall consider only the characters p, which correspond to the even spinor
structures, and the integer values of momentum.
Let’s define the states with the momentum p for a one-component bosonic field.
Consider the space ofthe spinors with the character p and the multiplicator exp( 2irip)
along the line connecting the points P~.This space was denoted by MPaP and the
spaces of the semi-infinite forms, which had been built using their Fourier-type basis,
were denoted by Wt’~
W~
In the case of integer p the dependence of these
yP~~P
rP~P
spaces on the line a is absent.
Below it will be shown that the states with integer momentum p which we denote
by I~),(pIareequalto
-
(6.1)
I~)=
(6.2)
(p1
‘V~J =
A
u>p+
iVy,
(~~
‘P~I
\~~-~ /
Here the generating vectors ~PkR, tpt belonging to the spaces Wf~,Wi~were defined
in (2.17).
The motivation of the factor in (6.2) is the same as in case of the definition (0 I It
follows from the difference between the <<in-s’ and <<out-s’ normalization. The correspondance
(6.3)
IO)—’lp),
(OI-~(pI
and the structure of the modules over Lr allow us to plot the so-called <<vertex operators’
~
~
-
In [3] the following properties
(64)
a~Ip)O, n>~,
afIv)pIp),
(pIa~0,
(pla_
m<—~,
1.p(pI
have been proved. These equalities are the consequence of the representation of the
current operator I( z, p) in the form
(6.5)
I(z,p)
=
: ~
:p
~
+
ds~(z,p),
659
vIRASORO-GELFAND-FUKS TYPE ALGEBRAS
where : :~-the normal ordering in respect to the vector p). The vectors I~)and (~I
are annihilated by the operator for n > 9o and n < —9o, respectively, and are the
eigenvectors for the operators L9 and L
(6.6)
L901p)
=
~-Ip), (pIL_90 =
.
The equalities (6.6) mean the following
LEMMA
10. The equality
‘
A(±p,P~,P_)
=
~
=
p
~
(~,v)~’,p>0
p
takes
2/2.place. The value A( ±p,P~P_) depends on P.~.,P.. as the tensor of the weight
p
REMARK. If we redefine the vertex operator as the operator corresponding to the shift
of the indices in out-normalization and redefine respectively the normalization of the
,
vectors we obtain that
A(p,P~,P_)~A’(p,P÷,P_)
and the tensor weight of the amplitued in respect to the variables p.~. will be equal to
P2.
7t~romthe results of the soliton theory the formula
O[p]
(6.7)
~7,,=
O[p]
((L#_
~-) (A(P+)
E2v(P+,P_)
((~+
~-) (A(P~)_A(PJ))
follow, where 8[ p1 -theta-function with even characteristic; E( P~P_) -Prym-form
,
92[m1(A(P~)—A(P_))
(~wi(P+)9i[m]) (~wiP_)9im1)
Let’s choose the local co-ordinates a±so that E2 ( P
1., P_)
regularization gives
(6.8)
fJ~)~(0~I0~) O[pj(O).
=
=
=
I. Then the natural
660
KRICI-IEVER AND N0VIKOV
For the integer p we obtain
(6.9)
ul(p P P
)=
O[p](p(A(P÷)
9[p](0)—A(Pj))
Here and above the value A( P) is the Abelian map from the devisor to the Jacobian
torus of the curve.
This is the situation for a one component bosonic field and the integer p. In the
D-dimensional space we have
—
v-I
(~1v
‘
—
—
‘
M=o
/
(in the Euclidean metric
=
= 1).
The question how to define the amplitude in the case of the Minkovsky space is yet
not absolutely clear for the authors. There are two possibilities which were briefly considered in [3].
In the first version the definition of A( p, P~,P_) goes through a few steps. The amplitude must be defined first for rational and then for all the real values of momentum.
And finally one can extend the result on the imaginary values after the transformation
p
ip0. For the realizadon of this program it is necessary to introduce into the consideration the contour a between the points P~and P_ There are different possibilities
at this point as well. One can consider all possible contours a or non-intersecting <<timelikes’ contours. In any case it is necessary to take the avarage value with respect to the
choice of a. In the case of all possible contours a we obtain
~
—*
-
(6.11)
A(p P P
)=
(9[p](A(P+)
—
A(P_))
\~ E(P~,Pj9[9](0)
-
,‘
This formula can be extended over all complex (for integer p we shall choose the formula (6.11) instead of (6.9)). Therefore, in the Minkovsky case the amplitude has the
form
(6.12)
where A is given by the formula (6.11).
The second version of the definition of A(5~P~,P_), which was briefly discussed
in [3] is based on other definition of the vertex operator (or normalized state (pJ). If
such operator corresponds to the shift of indices of basic form 4,, in out-normalization
for zero component then we obtain even in the case of integer rj’~the formula
___
‘0
0’/
‘
—
~ 1A(p’~P P
Ii.
‘.
‘
+‘
)~
—
I
We hope that our future studies will show which of the two versions is correct.
VIRASORO-GELFAND-FUKS TYPE ALGEBRAS
661
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Manuscriot received: November 9, 1988