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Operator formalism and tau function for supersymmetric ghosts in higher genus
Volume 231, number 1,2
PHYSICS LETTERS B
2 November 1989
O P E R A T O R F O R M A L I S M A N D TAU F U N C T I O N FOR SUPERSYMMETRIC GHOSTS IN HIGHER GENUS O.S. K R A V C H E N K O Physics Department, State University of Moscow, 117 234 Moscow, USSR and A.M. S E M I K H A T O V P.N. Lebedev PhysicsInstitute of the USSR Academy of Sciences, Leninsky prospect 53, 117 924 Moscow, USSR Received 18 July 1989
We derive the relation and establish the consistency between two different approaches to the operator bosonization of (super)ghosts on Riemann surfaces, the global operator formalism and the tau function technique. We solve an apparent puzzle between thefree field representation provided by vertex operators acting on the tau function, and the non-free superghost operator insertions (as dictated by global effects on the Riemann surface). A version of the global operator formalism is proposed which renders all insertions within one fixed coordinate patch free. The bridge between the two bosonization approaches is then provided by an "operator valued algebro-geometric super tau function". This can be explicitly derived from the modified operator formalism and in turn yields the ordinary tau function, showing at the same time an "equivariance" under the action of the vertex operators and fusions with operator insertions.
I. Introduction Ghosts o f the fermionic string constitute the m a i n ingredient o f higher-loop calculations [ 1-7 ]. As long as extracting physically relevant i n f o r m a t i o n from the f u n d a m e n t a l results o f refs. [3,4 ] remains a hard j o b [ 5 7 ], any insight into the structure o f the ( super- )ghost theory is desirable ~1. Recent a t t e m p t s to elucidate different aspects o f the ghost systems in higher genera e m b r a c e d various versions o f the o p e r a t o r f o r m a l i s m [ 9-13 ], including those based on the z-function a p p r o a c h [ 14-18 ]. The a i m o f this p a p e r is to derive the explicit form o f the " s u p e r " r-function from what we believe is a m o r e f u n d a m e n t a l notion, i.e. the global o p e r a t o r f o r m a l i s m on R i e m a n n surfaces [ 10,11 ]. The super z-function will follow naturally by c o m b i n i n g the global o p e r a t o r f o r m a l i s m with the general prescription for building up a zfunction as [ 19 ] z ( t ) = (01 exp [ H ( t ) ] IS )
( 1)
where H is a " h a m i l t o n i a n " which d e p e n d s on a collection o f " t i m e s " and IX) is a g-vacuum [ 14-16] state which in the global o p e r a t o r f o r m a l i s m is explicitly generated from the v a c u u m as IX) = B I 0 ) where B is a " b a c k g r o u n d " o p e r a t o r [ 10 ] associated with the ( b o s o n i z e d ) g h o s t / s u p e r g h o s t theory on the R i e m a n n surface Y. The h a m i l t o n i a n H should be linear in the time variables (which is tied up with the integrability o f non-linear #~ A new source of interest in superconformal-ghost-like theories comes from the recent "bosonization" of the current algebra due to Wakimoto, Feigin, Frenkel and others, see ref. [ 8 ] and references therein. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
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equations satisfied by the r-function). Accordingly, the vertex operators whose action on the r-function represents field insertions, are the usual free ones. All this works quite well for the fermionic bc-(reparametrization ghost ) theories. For the fy- (superconformal ghost) theories, however, this seems to be in contradiction with the fact that in the global operator formalism the "bosonized" fly-operator insertions are non free due to global effects on the Riemann surface [ 11,10 ] We will show that just in the context suitable for constructing the r-function, i.e., when all field insertions are taken to lie within a single coordinate patch, there exists a modification of the global operator formalism which renders these insertions free and attributes all global effects in any field operator insertion to a universal operator S acting on the state 12[). The essential part of our construction thus is the "operator valued r-function" T = ( e x p ( H ) ) S B . We will be able to show that the vertex operators produce the same effect on T as does the fusion with operator insertions required in the global operator formalism. All results concerning the ordinary rfunction (including the explicit form thereof) follow then by merely sandwiching the operator expressions between the (bosonic) vacua ( 0 I... I0 ) ~2. Throughout the paper, we use that "supersymmetric", or phase II bosonization [21,22 ]. This bosonization scheme, although reproducing the same local operator products as the original ("phase I " ) bosonization proposal of ref. [23 ], leads to higher genus correlation functions [ 11 ] which do not coincide with those pertaining to phase I [4 ]. This essentially accounts for the difference between the super r-function which we derive below and the one which can be obtained as a combination of the r-functions postulated in ref. [ 18 ]. The phase II rfunction, rather than that of phase I [ 18 ], should be preferred as a candidate for the algebro-geometric solution to the super-KP (Kadomtsev-Petviashvili) hierarchy [24 ]. Anyway, we check that under the action of vertex operators our r-function does generate the (phase II) correlation functions of (super)ghosts on the Riemann surface. (We actually prove a stronger result, namely, the "equivariance" property of the operator r-function T). Below, in section 2, we explain how the background operator B is constructed in the bosonized theory. Further, in section 3, we construct the hamiltonian H as an operator acting in the bosonized Hilbert space built upon the bosonic vacuum [0 ). The r-function then follows in terms of bosonic quantities as r = (01 exp (H)SBI 0 ). Introducing the vertex operators in section 4, we compute the action of these on T, thereby establishing the consistency with the global operator formalism.
2. The global operator formalism Operators of the bcfly theory and the auxiliary ~t/system were bosonized in ref. [21 ] as b=~,exp(~),
c=~exp(-~),
fl=exp [ ( ~ - O ) r / ] , = exp (~),
r/=~exp(~),
y=exp(O-~)d~ =exp(-(0) (.f- H),
~=~exp(-~),
(2a,b,c,d) (2e,f) (2g,h)
where ~ and ~ are scalars, and ~ and ~ the fermions. The latter may be bosonized in their own turn, yielding the total of three currents, J = O~, .f= t~ and H = ~,~. When promoted to higher genus Riemann surfaces, the scalars become multi-valued, whereas the currents keep single-valued. One should thus use only those combinations of the above field operators, referred to as neutral, which can be expressed through the currents. All these are arbitrary functions of the elementary neutral insertions ~2 This is an advantageof the bosonized formalism:already for the bc theory, with 12~)given as a state in the fermionic Fock space [20 ], it is by no means straightforwardto reproducethe algebro-geometrictau function [27 ] by explicitlycomputingthe action of exp[H(t) ] on I-r}, whereas in the bosonized formalism the z-functionfollowsimmediately, see ref. [ 17 ]. 86
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PHYSICS LETTERSB
'
~u(x)~(y) = E(x, y-----~exp
(i)H
2 November 1989
,
(3a)
Y
exp[~o(x)]exp[-~o(y) ]=exp(f J),
exp[O(x) ]exp[-O(y) ]=exp
Y
(3b,c) Y
(E (x, y) is the prime-form [ 2 5 ] ). The ghost-number anomaly, which determines the predominance of ~'s over the ~'s, etc., in non-zero correlation functions, is saturated by the background operator which in our case is, heuristically [ 11 ], B~
1-I ~,(M) exp[~0(M)] l-I M e ~t"
exp[O(P)]~'(Xo)
exp[-0(Xo)],
(4)
P c t~'
where de and ~ are positive divisors of degree, respectively, g - 1 and 2 ( g - 1 ), g being the genus of the Riemann surface Z, and Xo is a generic point on Z (location of an extra ~ insertion). To give the above formula a precise meaning, we first of all normalize the currents to zero a-periods and assign the following two-point correlators:
( OIH(x)H(y)
[0> =
(5)
and otherwise zero, where co(x, y) = dxdyln E(x, y) is the symmetric meromorphic bi-differential [ 25 ] and I0) is normalized by ( 0 1 0 ) = 1. The three currents are free, which implies in particular that correlation functions of higher order monomials follow by applying Wick's rule. Accordingly, normal ordering of composite operators is also defined as a result of subtracting all possible contractions, the elementary contractions being read off from eqs. (5). After these preparations, we define B (denoting it now by B*) as
B*(dg, ~ ) =S(J)A(H, J, J) O(.//g+~--3Lt+fb(H+J) ) O(~-2d+~bJ) ' A(H,J,J)=exp
(H+J)+ (g
Xexp ~ . )
)z0
exp ~ - - ~ y 2 ( g - - 1 )z0
(H+J+2J)
toj(z) aj
(H+J+2J) ,
dzlnE(xo, z) xo
(6a)
zo
(6b)
zo
where for the current K we have defined S(K) =exp I ~ i ~ x0
[J(z)-H(z)]lnO(z-xo+~-2A+~K)] b
× exp .'co
(overall normal orderings are understood). Apart from the factor (6c), B* is essentially the same as the background operator B of ref. [ 11 ]. In particular, (i)~I is the Riemann class divisor. (ii) The Abel mapping is understood in the arguments of theta functions. (iii) fbJ is the vector of b-periods of the current. (iv) The theta's have characteristics [~], which we suppress, determined by the spin structure. (v) zo in (6b) is a generic point of which the RHS is actually independent. The factor S(J) arises in the z-functional context. When constructing an algebro-geometric z-function, one
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needs to fix a point on the Riemann surface and choose a coordinate system in a neighborhood of the point. The vertex operators with which one is to act on the r-function do depend on this coordinate system and therefore produce correlation functions only of field insertions confined to the coordinate neighborhood. This allows a simplification over the general case considered in ref. [ 11 ]. It was shown there that in higher genus the operator insertions ( 2 c ) - (2h) acquire "extra" operator factors responsible for spurious poles, i.e. exp [ + ~0(x) ] gets replaced with exp[ +~0*(x) ] = e x p [ +~0(x) ]O(X--X+fbj) +-l and similarly, gt and q7 with q/*(x) =~u(x)O(xXo+ ~bJ) - 1 and ~7"(x) = ~(x)O(x-Xo + ~bJ), where j is essentially the same as J, j(z) = J ( z ) - 2dr In E(xo, z). These theta factors, besides rendering the operator insertions non-free, alter drastically the global behaviour of the fields, and are in particular responsible for breaking global supersymmetry at the operator level. However, when all the insertions are to lie within a single coordinate patch, one can do without the extra theta factors carried by individual insertions (and work thus with free field insertions ( 2 ) ). Instead, the background operator can be modified so as to produce all necessary theta factors when fused with free operator insertions provided the latter lie within the coordinate patch. This is the job done by the factor S(J). The integration contour in ~xo encircles the coordinate neighborhood. Note also that once we have chosen a coordinate system in the patch, we need not care about the presence on the RHS's of eqs. ( 3 ) of c-number factors making these expressions into a ( ~, - 3 )_differential (eq. (3b) ) and a ( ½, - ½)-differential (eq. (3c) ) in (x, y) (cf. refs. [ 10,11 ] ). These factors are effectively accounted for by the corresponding term in S(J). Thus, for example, fusing (3b) with S(J) into a normal-ordered expression yields
x
x
y
Y
3,
O(y--xo + ~-2A+~bJ)" \E(xo, x) ]
and similarly for other neutral insertions.
3. The hamiltonian and the "operator super z=function"
Corresponding to the three currents H, J and Jthere are in the r-functional set-up three infinite collections of "times", t = (tn), s = (sn), g= (gn), n>~ 1. The hamiltonian and hence the r-function will depend on these. Presumably, the times are just the evolution parameters of a super-KP hierarchy of the type discussed in ref. [ 24 ] but with "rebosonized" (!?) odd times. Similarly to the case of the fermionic bc system [ 17 ], the hamiltonian is linear in the times [ 19,26 ]
H(t,s, glII, J,i)=~-~ i
[~(t,z-~)H(z)+~(s,z-')J(z)+~(~,z-~)J(z)l, )CO
~(U,Z--I)= ~ Un Z - n , n>~l
U=t,
sor£
(7)
Here and in the sequel, z(N) with Ne X is a local coordinate around Xo with Z(Xo) = O. We need to normal order exp (H). The result is exp [H(t, s, ~IH, J, av) ] =exp [ ½Q(t, t) +Q(s, g) ] :exp [H(t, s, gIH, J, .D ]:,
(8)
Q(u,v)=
(9)
~
m,n >11
UmVnQmn, Q,~.=Q.m,
where amn c o m e from the local expansion of the prime form, dcm.p n
lnE(x,y)=ln(x-y)+
88
~. - - a m n .
m,n>~ l
reel
(lO)
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Now exp (H) can be fused with the background into a normal-ordered expression: introducing the "operator super r-function" T(t, s, g) = exp [H(t, s, sl H, J, J) ]:B*:
( 11 )
(where exp(H) is initially not normal-ordered), we find, performing all necessary fusions,
T(t, s, Y) =exp[ ½Q(t, t) + Q(s, y) ] exp [H(t, s, YlH*, J*, J*) ]A (J¢, J ,
~.~)S(J) 0 ( ~ + ~ - 3A+ ~ b ( ~ + J ) ) 0(~-2d+f~) (12)
Several important things have happened here: first, the (t, s, Y)-extended currents ~,u{,j a n d J of the form
J{=H+i(t), J=J+i(g), J=J+i(s), i(u)= ~
1 0n no~(zo)Un, ~o(~")(z)=~dz~a, lnE(a,z).
(13a)
n>~l
have emerged. These "reciprocal" (i.e., both operator and (t, s, y)-dependent) currents are responsible for the equivalence of the vertex operator/z-function formalism with the global operator formalism. We will see in section 4 that the script currents enter in the expression for T in such a way that it reacts in essentially the same way to the action of the vertex operators and the fusions with the neutral insertions. Second, in the hamiltonian the non-free currents
H*(z)=H(z)-dzlnO(z-xo+~-2A+~J), b
J*(z)=J(z)+dzlnO(z-xo+~-2A+~J)-3dzlnE(xo, z), b
J*(x) = J ( z ) - d z l n
E(xo, z),
(13b)
have appeared, which account for the effects due to interaction of H, J and J with the global holomorphic structure of the Riemann surface, and are in particular responsible for the spurious poles. The d In 0-terms persist in the expression for the c-number super z-function. The latter follows from T by sandwiching T between the bosonic vacua, which amounts to deleting all operatorcurrents from ( 12 ):
z(t,s,~)=A(i(t),i(g),i(s))exp ~
~(s-t,z-1)dzlnO z-xo+~-2A+I(Y)
×exp(~i~(g+3s, z-l)d~lnE(xo, z)) O(~+N-3A+I(t+g)) O(~-2A+I(g))
'
xo
with
I(u) = (fbfi(U) )+=,,....g.
4. Vertex operators and the T-function
Vertex operators should furnish a representation of the field operators involved in the bosonization formulae (2) on the space of formal power series of the time variables tn, sn and gn, n >/1. Denoting by [. ] v the vertex representation, we set 89
Volume 231, n u m b e r 1,2
[~(z)lv=V+(z),
PHYSICS LETTERS B
[exp[+~(z)l]v=V~(z),
2 November 1989
[~(z)lv=Vt(z),
[exp[+g(z)]]v=V~(z),
(14)
where we introduce --~ V + (z) = exp [ + aoln z:~ ~(a, z - I ) ] exp ( + n~, nZnO0) O+ -u -OUo
(15)
(with the conventions that/-= t and (g) =s). We have also introduced auxiliary variables to, So, and ~o of which the r-function is independent [ 14-18 ]. Now combining eqs. (14) and eqs. (2) we get neutral insertions, such as [b(x)c(y) ]v, [~(x)q(y) ]v etc., in the vertex representation. For example,
[fl(x)y(y) ]v=(X/y) s° ~exp[~(g,y-1)--~(s,x-l ×
l
y"-
Osn
+ n~,E n y - ~-
0 ) ] exp [~ ~ Xn__yn_ Os. ] \n~ 1
~(s.-t.)+y -1 (So -
n
to) +
1
~,
(16)
where the open dots indicate a "vertex" normal-ordering, which amounts to placing all derivatives with respect to the times to the right of the times t, s and L Our aim now is to work out how the operator r-function ( 11 ) behaves under the action of the vertex operators. The derivation amounts to finding the behaviour of T ( t, s, g) under the shifts of u~ = in, Sn or "qnby (x ~ - y ~) / n. The characteristic cases are as follows: (i) under the action of V + (x) V7 (y), exp [ ½Q(t, t) ] acquires the factor of x
E(x,y) eXp\.~nt.
O)~o)+~(t,x-~)-~(t,y-~)
);
Y
(ii) similarly, exp (H) yields the structures of the type
( 1 ~n~>~ xn--yn . 1- - zn- ~ K ( z )
exp ~ni
)
xO
=exp
(i)K y
where K is one of the three currents, perhaps shifted by a d In 0-term (see (14) ) or by dzln E(xo, z), and summation over n has been performed by noting that for x and y within the coordinate patch encircled by the contour, Lxl, lYl < Izl; (iii) acting on each i(u) entering in A (J¢, J , J ) (see eq. (13a) ) with the corresponding vertex operator produces extra factors:
NN E(N,x_~) (E(xo, x) a(x) ) Q, E(N, y) \E(xo, y) a(y) where Y equals Jg for u=t, ~ and ~ for u=s, and Q(g-1 ) is the degree of Y; o(. ) is Fay's g/2-differential [251; (iv) in the argument of the theta functions, we get extra terms
~ n~> l l (x~--Y~)m~xo)=im~ • bt
y
In this way we arrive at 90
Volume 231, number 1,2
[b(x)c(y) _
]v T(t,
PHYSICS LETTERSB
2 November 1989
s, g)
1 (a(x)~ 3 1-[ E(x,P) I-1 E(x'M) exp[½Q(t,t)+Q(s,g)]exp[H(t,s, glH*,J*,f*)] E(x, y) k,a(y) ] p~ E(y, P) ~d~ E(y, M)
×exp
(i
(o~(+J)
) (x/y)t°+e°A(~,J,J)S(J) O(x--y+ 0J/( N+-~--2A+~bJ) 3A+ ~b(~ +J) ) '
(17)
Y
[~(x)r/(y) ]v T(t,
1
s, g)
E(xo, x)
--(x/y)tO-~Oexp( E(x, y) E(xo, y)
i ( ~ - ~ ) ) A ( ~ , J , j ) { e x p [ H ( t , s , glH*,J*,J*)]S(J)}~+y_x Y
×exp[½Q(t,t)+Q(s,g)]
O(2y-xo - x + ~ - 23+5~bJ) O(y--xo+~--2A+fbJ)
0(~'+ ~-- 3A+5~b(~+j ) )
O(y-x+~-2A+~bJ)
'
(18)
where [... ] .~,+y_x means that the divisor ~ whenever inside this bracket, should be replaced with ~ + y - x . Similarly, acting on T with the RHS of eq. (16) yields
s
f-I E ( y , p ) exp
( i j)
(x/y)e°exp[1Q(t,t)+Q(s,g)]A(J{,J,3)
Y
O(x-xo + ~ - 2A + ~oJ) Xexp[H(t,s, glH*,J*,Y*)]S(j) O(y--xo+~--2A+fbJ) ×
dyln
E(y, x______+~(y)_~(y)+w,(y) )
E(y, xo)
+(So--to)Y-~+~niO~i(y)
0(de+ ~ - 3 A + ~ o ( ~ + j ) ) 0 ( ~ - - 2A +5~6j)
a~ln O(X--Xo + ~ -2J + ~bJ) O(P- 2 J + ~bJ)
[](z)-H(z)+~(s-t,z-n)dz]OilnO
Z-Xo+~-2A+
)co
y
.
(19)
b
Again, sandwiching these expressions between the bosonic vacua, i.e. deleting the operator currents, one reproduces the correct (t, s, g)-extended correlation functions of the corresponding fields on Riemann surfaces (the ordinary correlation functions are reproduced at zero times). It is clear that the construction works for arbitrary number and arbitrary combinations of neutral insertions. This round-about way of computing the correlation functions has been chosen in order to confirm the validity of the r-function derived in section 3. We can strengthen our consistency result by producing an explicit relation between the global operator formalism and the r-function set-up: We claim that the action on T(t, s, ~) of the vertex operators representing the neutral insertions (such as [b(x)c(y) ]v, [~(x)r/(y) ]v, [fl(x)~,(y) ]v, [~/(x)~(y) ]v etc.) is equivalent to the fusion with T(t, s, s) of the respective insertions represented in the global operator formalism. The latter are given by eqs. (3) for the elementary neutral insertions, and combinations thereof, such as
b(x)c(y)-
1
E ( x , y ) exp
( i )(H+J)
,
~(x)q(y)=exp
Y
fl(x)~,(y)=exp ( f J) [J(y)-H(y) +dyln E(xo, y) ].
(H-J)
,
(20a,b)
Y
(20c)
Y
The point is that, as can be verified by a direct calculation, all extra terms generated from T by the vertex 91
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operators, are equally generated by the fusion with T of the respective insertions (3), (20). Consider, for instance, how the current H + J, which is present in (20a), extends to ~ + J (as on the RHS of eq. ( 17 ) ) under the fusion with T. When fusing (20a) with exp (H) from T, the Campbell-Hausdorffformula yields the normalordering factor
xo
y
Using here the expansion og(u, z ) = Z,>~nz ,-1 Ogxo (n) (u), one gets the desired i(t) and i(y)-currents (see eqs. ( 13 ) ) which combine with H + J into o~(+J. A similar mechanism works in all other instances.
5. Conclusion We have been able to derive explicitly the algebro-geometric super r-function from the underlying global operator formalism on Riemann surfaces. We have introduced an operator version of the r-function and have proved its "equivariance". We have been working in the phase II formalism. The above formulae for the vertex representation of field insertions can therefore be rewritten in an explicitly supersymmetric form; this would be a supersymmetry inside the coordinate neighborhood and as such insensitive to global effects (such as carrying fields around the homologies, i.e. the "boundary conditions", see ref. [ 11 ] ). The super tau function may be further studied in relation with completely integrable hierarchies; one can look for a "bosonized" version of the super KP hierarchy with three independent collections of times, and the corresponding Segal-Wilson construction. We hope also that the super r-function will be helpful in constructing a formalism for gluing handles and sewing Riemann surfaces.
Acknowledgement One of us (O.K.) is grateful to V.V. Fock and A.O. Radul for useful discussions.
References [ 1 ] L. Alvarez-Gaum6, G. Moore and C. Vafa, Commun. Math. Phys. 106 (1986) 1; L. Alvarez-Gaum6, J.B. Bost, G. Moore, P. Nelson and C. Vafa, Commun. Math. Phys. 112 (1987) 503. [2l V.G. Knizhnik, Phys. Lett. B 180 (1986) 247. [3] E. Verlinde and H. Verlinde, Nucl. Phys. B 288 (1987) 357. [4 ] E. Verlinde and H. Verlinde, Phys. Lett. B 192 ( 1987 ) 95. [ 5 ] J.J. Atick and A. Sen, Nucl. Phys. B 296 ( 1988 ) 157. [ 6 ] J.J. Atick, G. Moore and A. Sen, Nucl. Phys. B 308 ( 1988 ) 1; B 307 ( 1988 ) 221. [7] O. Lechtenfeld and A. Parkes, Phys. Lett. B 202 (1988) 75; preprint CCNY-HEP-89/04, UCD-89-8 (April 1989); O. Lechtenfeld, Nucl. Phys. B 309 (1988) 361; A. Parkes, Phys. Lett. B 217 (1989) 458. [ 8 ] A. Gerasimov, A. Marshakov, A. Morozov, M. Olshanetsky and S. Shatashvili, ITEP preprints 64-89, 72-89, 74-89 ( 1989 ). 19l C. Vafa, Phys. Lett. B 190 (1987) 47; L. Alvarez-Gaum6, C. Gomez, G. Moore and C. Vafa, Nucl. Phys. B 303 ( 1988 ) 455; L. Alvarez-Gaum6, P. Nelson, C. Gomez, G. Sierra and C. Vafa, Nucl. Phys. B 311 ( 1988 / 89) 333. [ 10l A.M. Semikhatov, Nucl. Phys. B 315 (1989) 222; Phys. Lett. B 217 (1989) 77; B 220 (1989) 406. [ 11 ] A.M. Semikhatov, The "supersymmetric" bosonization and phase II of the superghost system on Riemann surfaces, to be published. [ 12 ] U. Carow-Watamura, Z.F. Ezawa, A. Tezuka and S. Watamura, Phys. Lett. B 221 (1989) 299; U. Carow-Watamura, Z.F. Ezawa, K. Harada, A. Tezuka and S. Watamura, preprints TU-341, TU-342 (May 1989).
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[ 13 ] P. Di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 211 ( 1988 ) 301; P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, preprint NORDITA-88/47 P; G. Cristofano, R. Musto, F. Nicodemi and R. Pettorino, Phys. Lett, B 211 (1988) 417; B 217 (1989) 59; J.L. Petersen, J.R. Sidenius and A.K. Tollst6n, Phys. Lett. B 214 ( 1988 ) 533. [ 14] M. Ishibashi, Y. Matsuo and H. Ooguri, Mod. Phys. Lett. A 2 ( 1987 ) 119. [ 15 ] L. Alvarez-Gaum6, C. Gomez and C. Reina, New methods in string theory, CERN preprint CERN-TH.4775/87 (1987). [ 16] N.Kawamoto, Y. Namikawa, A. Tsuchiya and Y. Yamada, Commun. Math. Phys. 114 (1988) 247. [ 17] A.M. Semikhatov, Mod. Phys. Lett. A 3 (1988) 1689. [ 18] N. Nohara, preprint UT-Komaba 88-2 ( 1988); N. Nohara, Phys. Lett. B 216 (1989) 103. [ 19 ] E. Date, M. Kashiwara, M. Jimbo and T. Miwa, Transformation groups for soliton equations, in: Proc. RIMS Symp. (Kyoto), eds. M. Jimbo and T. Miwa (World Scientific, Singapore, 1983) p. 39; J.L. Verdier, S6minaire Bourbaki, 34-e ann6e (1981/1982) No. 596, p. 1. [20] G.B. Segal and G. Wilson, Loop groups and equations of the KdV type, Publ. Math. IHES 63 (1985) 5. [ 21 ] M. Takama, Phys. Lett. B 210 ( 1988 ) 153; E. Martinec and G. Sotkov, Phys. Lett. B 208 (1988) 249. [22] G.T. Horowitz, S.P. Martin and R.S. Myers, Phys. Lett. B 215 (1988) 292. [23] D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 271 (1986) 93. [24] A. LeClair, Nucl. Phys. B 314 (1989) 425. [25 ] J.D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352 (Springer, Berlin, 1973). [26] H. Ooguri and N. Sakai, Nucl. Phys. B 312 (1989) 435. [27] I.M. Krichever, Usp. Mat. Nauk. 32, No. 6 (1977) 183; T. Shinta, Invent. Math. 83 (1986) 333.
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O P E R A T O R F O R M A L I S M A N D TAU F U N C T I O N FOR SUPERSYMMETRIC GHOSTS IN HIGHER GENUS O.S. K R A V C H E N K O Physics Department, State University of Moscow, 117 234 Moscow, USSR and A.M. S E M I K H A T O V P.N. Lebedev PhysicsInstitute of the USSR Academy of Sciences, Leninsky prospect 53, 117 924 Moscow, USSR Received 18 July 1989
We derive the relation and establish the consistency between two different approaches to the operator bosonization of (super)ghosts on Riemann surfaces, the global operator formalism and the tau function technique. We solve an apparent puzzle between thefree field representation provided by vertex operators acting on the tau function, and the non-free superghost operator insertions (as dictated by global effects on the Riemann surface). A version of the global operator formalism is proposed which renders all insertions within one fixed coordinate patch free. The bridge between the two bosonization approaches is then provided by an "operator valued algebro-geometric super tau function". This can be explicitly derived from the modified operator formalism and in turn yields the ordinary tau function, showing at the same time an "equivariance" under the action of the vertex operators and fusions with operator insertions.
I. Introduction Ghosts o f the fermionic string constitute the m a i n ingredient o f higher-loop calculations [ 1-7 ]. As long as extracting physically relevant i n f o r m a t i o n from the f u n d a m e n t a l results o f refs. [3,4 ] remains a hard j o b [ 5 7 ], any insight into the structure o f the ( super- )ghost theory is desirable ~1. Recent a t t e m p t s to elucidate different aspects o f the ghost systems in higher genera e m b r a c e d various versions o f the o p e r a t o r f o r m a l i s m [ 9-13 ], including those based on the z-function a p p r o a c h [ 14-18 ]. The a i m o f this p a p e r is to derive the explicit form o f the " s u p e r " r-function from what we believe is a m o r e f u n d a m e n t a l notion, i.e. the global o p e r a t o r f o r m a l i s m on R i e m a n n surfaces [ 10,11 ]. The super z-function will follow naturally by c o m b i n i n g the global o p e r a t o r f o r m a l i s m with the general prescription for building up a zfunction as [ 19 ] z ( t ) = (01 exp [ H ( t ) ] IS )
( 1)
where H is a " h a m i l t o n i a n " which d e p e n d s on a collection o f " t i m e s " and IX) is a g-vacuum [ 14-16] state which in the global o p e r a t o r f o r m a l i s m is explicitly generated from the v a c u u m as IX) = B I 0 ) where B is a " b a c k g r o u n d " o p e r a t o r [ 10 ] associated with the ( b o s o n i z e d ) g h o s t / s u p e r g h o s t theory on the R i e m a n n surface Y. The h a m i l t o n i a n H should be linear in the time variables (which is tied up with the integrability o f non-linear #~ A new source of interest in superconformal-ghost-like theories comes from the recent "bosonization" of the current algebra due to Wakimoto, Feigin, Frenkel and others, see ref. [ 8 ] and references therein. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
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equations satisfied by the r-function). Accordingly, the vertex operators whose action on the r-function represents field insertions, are the usual free ones. All this works quite well for the fermionic bc-(reparametrization ghost ) theories. For the fy- (superconformal ghost) theories, however, this seems to be in contradiction with the fact that in the global operator formalism the "bosonized" fly-operator insertions are non free due to global effects on the Riemann surface [ 11,10 ] We will show that just in the context suitable for constructing the r-function, i.e., when all field insertions are taken to lie within a single coordinate patch, there exists a modification of the global operator formalism which renders these insertions free and attributes all global effects in any field operator insertion to a universal operator S acting on the state 12[). The essential part of our construction thus is the "operator valued r-function" T = ( e x p ( H ) ) S B . We will be able to show that the vertex operators produce the same effect on T as does the fusion with operator insertions required in the global operator formalism. All results concerning the ordinary rfunction (including the explicit form thereof) follow then by merely sandwiching the operator expressions between the (bosonic) vacua ( 0 I... I0 ) ~2. Throughout the paper, we use that "supersymmetric", or phase II bosonization [21,22 ]. This bosonization scheme, although reproducing the same local operator products as the original ("phase I " ) bosonization proposal of ref. [23 ], leads to higher genus correlation functions [ 11 ] which do not coincide with those pertaining to phase I [4 ]. This essentially accounts for the difference between the super r-function which we derive below and the one which can be obtained as a combination of the r-functions postulated in ref. [ 18 ]. The phase II rfunction, rather than that of phase I [ 18 ], should be preferred as a candidate for the algebro-geometric solution to the super-KP (Kadomtsev-Petviashvili) hierarchy [24 ]. Anyway, we check that under the action of vertex operators our r-function does generate the (phase II) correlation functions of (super)ghosts on the Riemann surface. (We actually prove a stronger result, namely, the "equivariance" property of the operator r-function T). Below, in section 2, we explain how the background operator B is constructed in the bosonized theory. Further, in section 3, we construct the hamiltonian H as an operator acting in the bosonized Hilbert space built upon the bosonic vacuum [0 ). The r-function then follows in terms of bosonic quantities as r = (01 exp (H)SBI 0 ). Introducing the vertex operators in section 4, we compute the action of these on T, thereby establishing the consistency with the global operator formalism.
2. The global operator formalism Operators of the bcfly theory and the auxiliary ~t/system were bosonized in ref. [21 ] as b=~,exp(~),
c=~exp(-~),
fl=exp [ ( ~ - O ) r / ] , = exp (~),
r/=~exp(~),
y=exp(O-~)d~ =exp(-(0) (.f- H),
~=~exp(-~),
(2a,b,c,d) (2e,f) (2g,h)
where ~ and ~ are scalars, and ~ and ~ the fermions. The latter may be bosonized in their own turn, yielding the total of three currents, J = O~, .f= t~ and H = ~,~. When promoted to higher genus Riemann surfaces, the scalars become multi-valued, whereas the currents keep single-valued. One should thus use only those combinations of the above field operators, referred to as neutral, which can be expressed through the currents. All these are arbitrary functions of the elementary neutral insertions ~2 This is an advantageof the bosonized formalism:already for the bc theory, with 12~)given as a state in the fermionic Fock space [20 ], it is by no means straightforwardto reproducethe algebro-geometrictau function [27 ] by explicitlycomputingthe action of exp[H(t) ] on I-r}, whereas in the bosonized formalism the z-functionfollowsimmediately, see ref. [ 17 ]. 86
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'
~u(x)~(y) = E(x, y-----~exp
(i)H
2 November 1989
,
(3a)
Y
exp[~o(x)]exp[-~o(y) ]=exp(f J),
exp[O(x) ]exp[-O(y) ]=exp
Y
(3b,c) Y
(E (x, y) is the prime-form [ 2 5 ] ). The ghost-number anomaly, which determines the predominance of ~'s over the ~'s, etc., in non-zero correlation functions, is saturated by the background operator which in our case is, heuristically [ 11 ], B~
1-I ~,(M) exp[~0(M)] l-I M e ~t"
exp[O(P)]~'(Xo)
exp[-0(Xo)],
(4)
P c t~'
where de and ~ are positive divisors of degree, respectively, g - 1 and 2 ( g - 1 ), g being the genus of the Riemann surface Z, and Xo is a generic point on Z (location of an extra ~ insertion). To give the above formula a precise meaning, we first of all normalize the currents to zero a-periods and assign the following two-point correlators:
( OIH(x)H(y)
[0> =
(5)
and otherwise zero, where co(x, y) = dxdyln E(x, y) is the symmetric meromorphic bi-differential [ 25 ] and I0) is normalized by ( 0 1 0 ) = 1. The three currents are free, which implies in particular that correlation functions of higher order monomials follow by applying Wick's rule. Accordingly, normal ordering of composite operators is also defined as a result of subtracting all possible contractions, the elementary contractions being read off from eqs. (5). After these preparations, we define B (denoting it now by B*) as
B*(dg, ~ ) =S(J)A(H, J, J) O(.//g+~--3Lt+fb(H+J) ) O(~-2d+~bJ) ' A(H,J,J)=exp
(H+J)+ (g
Xexp ~ . )
)z0
exp ~ - - ~ y 2 ( g - - 1 )z0
(H+J+2J)
toj(z) aj
(H+J+2J) ,
dzlnE(xo, z) xo
(6a)
zo
(6b)
zo
where for the current K we have defined S(K) =exp I ~ i ~ x0
[J(z)-H(z)]lnO(z-xo+~-2A+~K)] b
× exp .'co
(overall normal orderings are understood). Apart from the factor (6c), B* is essentially the same as the background operator B of ref. [ 11 ]. In particular, (i)~I is the Riemann class divisor. (ii) The Abel mapping is understood in the arguments of theta functions. (iii) fbJ is the vector of b-periods of the current. (iv) The theta's have characteristics [~], which we suppress, determined by the spin structure. (v) zo in (6b) is a generic point of which the RHS is actually independent. The factor S(J) arises in the z-functional context. When constructing an algebro-geometric z-function, one
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needs to fix a point on the Riemann surface and choose a coordinate system in a neighborhood of the point. The vertex operators with which one is to act on the r-function do depend on this coordinate system and therefore produce correlation functions only of field insertions confined to the coordinate neighborhood. This allows a simplification over the general case considered in ref. [ 11 ]. It was shown there that in higher genus the operator insertions ( 2 c ) - (2h) acquire "extra" operator factors responsible for spurious poles, i.e. exp [ + ~0(x) ] gets replaced with exp[ +~0*(x) ] = e x p [ +~0(x) ]O(X--X+fbj) +-l and similarly, gt and q7 with q/*(x) =~u(x)O(xXo+ ~bJ) - 1 and ~7"(x) = ~(x)O(x-Xo + ~bJ), where j is essentially the same as J, j(z) = J ( z ) - 2dr In E(xo, z). These theta factors, besides rendering the operator insertions non-free, alter drastically the global behaviour of the fields, and are in particular responsible for breaking global supersymmetry at the operator level. However, when all the insertions are to lie within a single coordinate patch, one can do without the extra theta factors carried by individual insertions (and work thus with free field insertions ( 2 ) ). Instead, the background operator can be modified so as to produce all necessary theta factors when fused with free operator insertions provided the latter lie within the coordinate patch. This is the job done by the factor S(J). The integration contour in ~xo encircles the coordinate neighborhood. Note also that once we have chosen a coordinate system in the patch, we need not care about the presence on the RHS's of eqs. ( 3 ) of c-number factors making these expressions into a ( ~, - 3 )_differential (eq. (3b) ) and a ( ½, - ½)-differential (eq. (3c) ) in (x, y) (cf. refs. [ 10,11 ] ). These factors are effectively accounted for by the corresponding term in S(J). Thus, for example, fusing (3b) with S(J) into a normal-ordered expression yields
x
x
y
Y
3,
O(y--xo + ~-2A+~bJ)" \E(xo, x) ]
and similarly for other neutral insertions.
3. The hamiltonian and the "operator super z=function"
Corresponding to the three currents H, J and Jthere are in the r-functional set-up three infinite collections of "times", t = (tn), s = (sn), g= (gn), n>~ 1. The hamiltonian and hence the r-function will depend on these. Presumably, the times are just the evolution parameters of a super-KP hierarchy of the type discussed in ref. [ 24 ] but with "rebosonized" (!?) odd times. Similarly to the case of the fermionic bc system [ 17 ], the hamiltonian is linear in the times [ 19,26 ]
H(t,s, glII, J,i)=~-~ i
[~(t,z-~)H(z)+~(s,z-')J(z)+~(~,z-~)J(z)l, )CO
~(U,Z--I)= ~ Un Z - n , n>~l
U=t,
sor£
(7)
Here and in the sequel, z(N) with Ne X is a local coordinate around Xo with Z(Xo) = O. We need to normal order exp (H). The result is exp [H(t, s, ~IH, J, av) ] =exp [ ½Q(t, t) +Q(s, g) ] :exp [H(t, s, gIH, J, .D ]:,
(8)
Q(u,v)=
(9)
~
m,n >11
UmVnQmn, Q,~.=Q.m,
where amn c o m e from the local expansion of the prime form, dcm.p n
lnE(x,y)=ln(x-y)+
88
~. - - a m n .
m,n>~ l
reel
(lO)
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Now exp (H) can be fused with the background into a normal-ordered expression: introducing the "operator super r-function" T(t, s, g) = exp [H(t, s, sl H, J, J) ]:B*:
( 11 )
(where exp(H) is initially not normal-ordered), we find, performing all necessary fusions,
T(t, s, Y) =exp[ ½Q(t, t) + Q(s, y) ] exp [H(t, s, YlH*, J*, J*) ]A (J¢, J ,
~.~)S(J) 0 ( ~ + ~ - 3A+ ~ b ( ~ + J ) ) 0(~-2d+f~) (12)
Several important things have happened here: first, the (t, s, Y)-extended currents ~,u{,j a n d J of the form
J{=H+i(t), J=J+i(g), J=J+i(s), i(u)= ~
1 0n no~(zo)Un, ~o(~")(z)=~dz~a, lnE(a,z).
(13a)
n>~l
have emerged. These "reciprocal" (i.e., both operator and (t, s, y)-dependent) currents are responsible for the equivalence of the vertex operator/z-function formalism with the global operator formalism. We will see in section 4 that the script currents enter in the expression for T in such a way that it reacts in essentially the same way to the action of the vertex operators and the fusions with the neutral insertions. Second, in the hamiltonian the non-free currents
H*(z)=H(z)-dzlnO(z-xo+~-2A+~J), b
J*(z)=J(z)+dzlnO(z-xo+~-2A+~J)-3dzlnE(xo, z), b
J*(x) = J ( z ) - d z l n
E(xo, z),
(13b)
have appeared, which account for the effects due to interaction of H, J and J with the global holomorphic structure of the Riemann surface, and are in particular responsible for the spurious poles. The d In 0-terms persist in the expression for the c-number super z-function. The latter follows from T by sandwiching T between the bosonic vacua, which amounts to deleting all operatorcurrents from ( 12 ):
z(t,s,~)=A(i(t),i(g),i(s))exp ~
~(s-t,z-1)dzlnO z-xo+~-2A+I(Y)
×exp(~i~(g+3s, z-l)d~lnE(xo, z)) O(~+N-3A+I(t+g)) O(~-2A+I(g))
'
xo
with
I(u) = (fbfi(U) )+=,,....g.
4. Vertex operators and the T-function
Vertex operators should furnish a representation of the field operators involved in the bosonization formulae (2) on the space of formal power series of the time variables tn, sn and gn, n >/1. Denoting by [. ] v the vertex representation, we set 89
Volume 231, n u m b e r 1,2
[~(z)lv=V+(z),
PHYSICS LETTERS B
[exp[+~(z)l]v=V~(z),
2 November 1989
[~(z)lv=Vt(z),
[exp[+g(z)]]v=V~(z),
(14)
where we introduce --~ V + (z) = exp [ + aoln z:~ ~(a, z - I ) ] exp ( + n~, nZnO0) O+ -u -OUo
(15)
(with the conventions that/-= t and (g) =s). We have also introduced auxiliary variables to, So, and ~o of which the r-function is independent [ 14-18 ]. Now combining eqs. (14) and eqs. (2) we get neutral insertions, such as [b(x)c(y) ]v, [~(x)q(y) ]v etc., in the vertex representation. For example,
[fl(x)y(y) ]v=(X/y) s° ~exp[~(g,y-1)--~(s,x-l ×
l
y"-
Osn
+ n~,E n y - ~-
0 ) ] exp [~ ~ Xn__yn_ Os. ] \n~ 1
~(s.-t.)+y -1 (So -
n
to) +
1
~,
(16)
where the open dots indicate a "vertex" normal-ordering, which amounts to placing all derivatives with respect to the times to the right of the times t, s and L Our aim now is to work out how the operator r-function ( 11 ) behaves under the action of the vertex operators. The derivation amounts to finding the behaviour of T ( t, s, g) under the shifts of u~ = in, Sn or "qnby (x ~ - y ~) / n. The characteristic cases are as follows: (i) under the action of V + (x) V7 (y), exp [ ½Q(t, t) ] acquires the factor of x
E(x,y) eXp\.~nt.
O)~o)+~(t,x-~)-~(t,y-~)
);
Y
(ii) similarly, exp (H) yields the structures of the type
( 1 ~n~>~ xn--yn . 1- - zn- ~ K ( z )
exp ~ni
)
xO
=exp
(i)K y
where K is one of the three currents, perhaps shifted by a d In 0-term (see (14) ) or by dzln E(xo, z), and summation over n has been performed by noting that for x and y within the coordinate patch encircled by the contour, Lxl, lYl < Izl; (iii) acting on each i(u) entering in A (J¢, J , J ) (see eq. (13a) ) with the corresponding vertex operator produces extra factors:
NN E(N,x_~) (E(xo, x) a(x) ) Q, E(N, y) \E(xo, y) a(y) where Y equals Jg for u=t, ~ and ~ for u=s, and Q(g-1 ) is the degree of Y; o(. ) is Fay's g/2-differential [251; (iv) in the argument of the theta functions, we get extra terms
~ n~> l l (x~--Y~)m~xo)=im~ • bt
y
In this way we arrive at 90
Volume 231, number 1,2
[b(x)c(y) _
]v T(t,
PHYSICS LETTERSB
2 November 1989
s, g)
1 (a(x)~ 3 1-[ E(x,P) I-1 E(x'M) exp[½Q(t,t)+Q(s,g)]exp[H(t,s, glH*,J*,f*)] E(x, y) k,a(y) ] p~ E(y, P) ~d~ E(y, M)
×exp
(i
(o~(+J)
) (x/y)t°+e°A(~,J,J)S(J) O(x--y+ 0J/( N+-~--2A+~bJ) 3A+ ~b(~ +J) ) '
(17)
Y
[~(x)r/(y) ]v T(t,
1
s, g)
E(xo, x)
--(x/y)tO-~Oexp( E(x, y) E(xo, y)
i ( ~ - ~ ) ) A ( ~ , J , j ) { e x p [ H ( t , s , glH*,J*,J*)]S(J)}~+y_x Y
×exp[½Q(t,t)+Q(s,g)]
O(2y-xo - x + ~ - 23+5~bJ) O(y--xo+~--2A+fbJ)
0(~'+ ~-- 3A+5~b(~+j ) )
O(y-x+~-2A+~bJ)
'
(18)
where [... ] .~,+y_x means that the divisor ~ whenever inside this bracket, should be replaced with ~ + y - x . Similarly, acting on T with the RHS of eq. (16) yields
s
f-I E ( y , p ) exp
( i j)
(x/y)e°exp[1Q(t,t)+Q(s,g)]A(J{,J,3)
Y
O(x-xo + ~ - 2A + ~oJ) Xexp[H(t,s, glH*,J*,Y*)]S(j) O(y--xo+~--2A+fbJ) ×
dyln
E(y, x______+~(y)_~(y)+w,(y) )
E(y, xo)
+(So--to)Y-~+~niO~i(y)
0(de+ ~ - 3 A + ~ o ( ~ + j ) ) 0 ( ~ - - 2A +5~6j)
a~ln O(X--Xo + ~ -2J + ~bJ) O(P- 2 J + ~bJ)
[](z)-H(z)+~(s-t,z-n)dz]OilnO
Z-Xo+~-2A+
)co
y
.
(19)
b
Again, sandwiching these expressions between the bosonic vacua, i.e. deleting the operator currents, one reproduces the correct (t, s, g)-extended correlation functions of the corresponding fields on Riemann surfaces (the ordinary correlation functions are reproduced at zero times). It is clear that the construction works for arbitrary number and arbitrary combinations of neutral insertions. This round-about way of computing the correlation functions has been chosen in order to confirm the validity of the r-function derived in section 3. We can strengthen our consistency result by producing an explicit relation between the global operator formalism and the r-function set-up: We claim that the action on T(t, s, ~) of the vertex operators representing the neutral insertions (such as [b(x)c(y) ]v, [~(x)r/(y) ]v, [fl(x)~,(y) ]v, [~/(x)~(y) ]v etc.) is equivalent to the fusion with T(t, s, s) of the respective insertions represented in the global operator formalism. The latter are given by eqs. (3) for the elementary neutral insertions, and combinations thereof, such as
b(x)c(y)-
1
E ( x , y ) exp
( i )(H+J)
,
~(x)q(y)=exp
Y
fl(x)~,(y)=exp ( f J) [J(y)-H(y) +dyln E(xo, y) ].
(H-J)
,
(20a,b)
Y
(20c)
Y
The point is that, as can be verified by a direct calculation, all extra terms generated from T by the vertex 91
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operators, are equally generated by the fusion with T of the respective insertions (3), (20). Consider, for instance, how the current H + J, which is present in (20a), extends to ~ + J (as on the RHS of eq. ( 17 ) ) under the fusion with T. When fusing (20a) with exp (H) from T, the Campbell-Hausdorffformula yields the normalordering factor
xo
y
Using here the expansion og(u, z ) = Z,>~nz ,-1 Ogxo (n) (u), one gets the desired i(t) and i(y)-currents (see eqs. ( 13 ) ) which combine with H + J into o~(+J. A similar mechanism works in all other instances.
5. Conclusion We have been able to derive explicitly the algebro-geometric super r-function from the underlying global operator formalism on Riemann surfaces. We have introduced an operator version of the r-function and have proved its "equivariance". We have been working in the phase II formalism. The above formulae for the vertex representation of field insertions can therefore be rewritten in an explicitly supersymmetric form; this would be a supersymmetry inside the coordinate neighborhood and as such insensitive to global effects (such as carrying fields around the homologies, i.e. the "boundary conditions", see ref. [ 11 ] ). The super tau function may be further studied in relation with completely integrable hierarchies; one can look for a "bosonized" version of the super KP hierarchy with three independent collections of times, and the corresponding Segal-Wilson construction. We hope also that the super r-function will be helpful in constructing a formalism for gluing handles and sewing Riemann surfaces.
Acknowledgement One of us (O.K.) is grateful to V.V. Fock and A.O. Radul for useful discussions.
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