- Email: [email protected]
On the operator formalism in higher genus
Volume 198, number 2
PHYSICS LETTERS B
19 November 1987
O N THE OPERATOR F O R M A L I S M IN H I G H E R G E N U S D. E S P R I U 1,2
The Institutefor AdvancedStudy, Princeton, NJ 08540, USA and Lyman Laboratoryof Physics, HarvardUniversity, Cambridge, MA 02138, USA Received 24 July 1987
We discuss several aspects of the recently proposed operator formalism on Riemann surfaces of genus g~>2, particularly the transformation under the reparametrization group of the states generated by the non-trivial topology.
One o f the most interesting new d e v e l o p m e n t s in string theory is the recent f o r m u l a t i o n by AlvarezG a u m r , G 6 m e z and Reina [ 1 ] and, m o r e explicitly, by Vafa [ 2 ] o f an 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 o f genus greater than one ~. The ability to use an o p e r a t o r f o r m a l i s m on the sphere and the torus has led in the past to considerable progress in the understanding o f t w o - d i m e n s i o n a l conformal field theory (see e.g. refs. [3,4]). A truly r e m a r k a b l e achievement has been the classification o f all " m i n i m a l " ( c < 1 ) conformal and superconformal theories [ 5 ] on the sphere. C o n f o r m a l field theory on the torus has also been a b u n d a n t l y investigated in the context o f string theories with the help o f o p e r a t o r methods. It is therefore potentially i m p o r t a n t to extend the o p e r a t o r f o r m a l i s m to R i e m a n n surfaces o f arbitrary genus in order to cast what we already know in a new, perhaps m o r e powerful, language a n d then see whether something new can be learnt from it. As we have just mentioned, the physical interest o f the o p e r a t o r formulation goes well b e y o n d the confines o f string theory. On the other hand, it turns out to be related to the theory o f representations o f loop groups [ 6 - 8 ] as well as to other m a t h e m a t i c a l p r o b l e m s (see references in refs. [ 1,2]). H o w e v e r we shall mostly ignore these interesting ramifications in this letter. This work in fact c a m e about in an at-
t e m p t to u n d e r s t a n d i n g some o f the relevant results o f refs. [ 1,2] in terms as simple as possible. Let us first s u m m a r i z e the construction o f the relevant Hilbert space. By choosing a generic point on a R i e m a n n surface a n d analytic coordinates at that point, we can d i v i d e the surface o f our preference into the patch Iz l < 1 a n d the rest. The resulting surfaces share the c o m m o n b o u n d a r y Izl = 1. (It is ass u m e d that the original one had no b o u n d a r y . ) O f course all the c o m p l i c a t e d topology lives in the comp l e m e n t a r y o f Izl < 1. The fields defined on the R i e m a n n surface have to satisfy the suitable continuity conditions across the boundary. The path integral on the whole surface then splits into two parts each defining a wave function, in this case a functional o f the fields on the b o u n d a r y ~ [ O ( z ) ] . Integrating over all values o f O(z) is equivalent to "sewing" the two halves o f the surface together. Here we want to focus on two different aspects o f the o p e r a t o r formalism. First we will present the construction o f the relevant operators and Fock space in a slightly different way o f that o f refs. [ 1,2]. This will help us u n d e r s t a n d the second part o f our work, which is the realization o f r e p a r a m e t r i z a t i o n invariance on such a Hilbert space. The wave function that is associated to the I z l < 1 patch is, o f course, very simple. I f ~ is a bosonic free field it is given by a p r o d u c t o f gaussians
Fulbright Fellow. 2 Address after 1 October 1987: Departament de Fisica Terrica, Universitat de Val+ncia,E-46100 Burjassot (Valencia), Spain. "~ For references to previous or related work see refs. [ 1,2]. 0370-2693/87/$ 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 )
~[x(z)]= (x(z) fO)~exp(-~
x~ ) ,
(1)
171
Volume 198, number 2
PHYSICS LETTERS B
where the x, are the coefficients o f the mode expansion o f x ( z ) , x ( z ) = E , x , + ~z". The wave function for free fermions, which is the system that we will be interested in ~2, is likewise trivial; it is given by the semiinfinite product ~'t[~//(Z)] = ( ~ ¢ ( Z ) ~ ( Z ) 1 0 ) ~ ~/,71/2~b¢1/2~/,/3/2~//3/2...
(2)
provided that the Dirac v a c u u m is defined by ~u~10) =qJ~10) = 0 for r > 0 , with ~g(Z)~- 2 ~,ln+l/2 Zn, n
~(Z)=
2 [~n+l/2Ztl " n
(3)
The antiholomorphic part qP(g) leads to a similar wave function ~u[~t(g)] that depends on the conjugate coefficients and the complete wave function is the product o f the two. If we perform a change of local coordinates z--, z' = e (z) the expansion coefficients in ( 3 ) change. The new coefficients are related to the old ones by a Bogolubov transformation ~3 n-I/2=
20lnm~ffm--l/2"q-
m
(4) m
a,,,~u-,,+L/z + Y, B,,,q/m--llZ,
f dzdf ½(~t~+~,)
+b++~c + .
(5)
T++ = - ~ ( V 0 ~ + q/0q~)
8~t = 2 c + O~,+ ½20c + ~,
S=Cexp(-~_,,+,/z( ~)nm~--m+1/2)
that lead to the conserved charge
O/-1
Not all reparametrizations change the wave function (2). U n d e r a change o f coordinates o f the form z - , z + ~ z k+l, the operators get modified by ~/r--) Vr+ ~ [ Lk, ~ ] , and likewise for q/. F r o m the definition o f the Virasoro generator Lk it is obvious that the wave function (2) is invariant under one such ~2 The bosonic theory can be recovered by taking into account the appropriately twisted sectors [ 2 ]. #3 See, for instance, ref. [9] for a description of the properties of Bogolubov transformations.
172
8 ~ = 2 c + OqT+ ½2Oc+ ~ , 8 c + = 2 c + 0 c +,
8b++=22T++,
QBRS = ½C+ ( ~ O~ + ~O~t) --C + OC+ b+ + .
(6)
(8)
(7) is invariant under the BRS transformations
and similar relations for q/'. The matrices a and fl form in fact a representation of D i f f S ~/S ~ [ 10]. The new vacuum is the state annihilated by the new operators V'r, q~'r, r > 0 and is related to the old vacuum by 1 0 ) ' = S I 0 ) with -
(7)
+½(c +Ob++ + 2 0 c + b + + ) .
m
n,m>~l ,
transformation if k > ~ - 1, i.e. Lk[ 0> = 0 for k>_-- 1, as in this case flnm = 0. Generically, however, the vacuum is not invariant and one obtains in this way a bundle over Diff S ~/S ~, with the independent entries o f a-~fl as holomorphic coordinates [ 10]. The curvature o f this bundle with the metric induced on it by the inner product of the Hilbert space at the origin, i.e. for fl, fl = 0, is the usual conformal anomaly [10,11]. So far we have dealt with the easy part, namely the patch Iz I < 1 which is conformally equivalent to a semi-infinite cylinder. We would like to characterize in a similar way the transformation properties under Diff S t/S l of the state created by the non-trivial half o f the original surface. On the surface we have a gauge fixed action defined by
The conjugate fields have been omitted. The ene r g y - m o m e n t u m tensor associated to it is
~ J~nm~'l-m+l/2, m
n,m>_.l,
19 November 1987
(9)
(10)
Following the standard procedure, in order to quantize the above action we have to solve the wave equation 0~u=0 (and similar ones for q/, c + and b++). For the spin-l/2 field such an equation has between 0 and g globally defined solutions on a Riemann surface. We will assume that there is none. Should one exist we would proceed in a form similar to the ghost case which will be discussed shortly. However, the wave equation does admit an infinite n u m b e r o f solutions when a patch is removed. All the information that we need to determine these solutions is contained in the propagator. On general grounds
(g/(z)C,(w) ) = ~
1
19 November 1987
PHYSICSLETTERSB
Volume 198, number 2
((c +(z)b+ +(w))) = H ( c +(z)b+ +( w ) )
(11)
+ ~(z, w).
Z--W
- ~ Hohi(w) (c+(z)b++(wj)) id
~(z, w) is analytic both in z and w, so taking I wl < 1, Iz[ > 1 we can expand the propagator in powers of w, each of the terms in the expansion being a holomorphic function of z, since ~<~/(z)~(w))=0,
Izl>lwl.
(12)
m O r,e - - I ~ ( 0 , 0 ) , Defining B n("2)=0~-~ morphic solutions read
t~l/2)(z) = z - " + E B~I,/~)z'n-I
the
holo(13)
m
Since t~ ~2) (z) is holomorphic on the surface minus the disk, we have 0=
ri.r t O / 2 ) g ~ , ~ t O / 2 ) t ~ = ~ . R ( l / 2 )
-- --R(1/2)
(14)
=H
1 z-w
1 - ~ H,jhi(w) + ~(z, w, w ) , o z-wj
(16) where [ 12] hi is the set of zero modes, H = d e t hi(wj), H o is the/j-minor of H, and (¢(z, w, wj) is an hop omorphic function of the indicated variables. Since their location is arbitrary we can place all the b+ + insertions inside the Izl < 1 disk. We take w to be also in the disk so that we can expand (16) in powers of z. Taking into account that the modified propagator solves the wave equation, we immediately obtain the holomorphic sections for the antighost b+ + on the surface minus the disk
t(,2)(z) = z - n + ~ B(,~z m-' ,
(17)
m
Explicit expressions for these coefficients have been given by Vafa [ 2 ] in terms of 0 functions and the prime form, but we will not need them. It is more interesting to repeat the above construction for the ghost and antighost. Also in this case we can determine all the holomorphic sections from the propagator, but in addition we will learn which is the appropriate way of computing expectation values in the operator formalism. Since dim H ° ( K 2) = 3 g - 3, for g>~ 2, there exist precisely 3 g - 3 zero modes of the antighost field b + + and none of c +. The "naive" ghost propagator ( c + ( z ) b + + ( w ) ) has to be modified to include 3 g - 3 insertions, at arbitrary points, of the b+ + field to soak the zero modes
~(z) = ~ Z_,+lat~"2)(z) .
((c+ (z)b++ (w) ))
(18)
n
~__. ( 3 g - 3
. H=l b++(wi)c+tz)b++(w))
where the B ~ are given by the derivatives of f9 with respect to z and w. Of course, in addition to (17) we still have the sections hi(z)=Eka~z k. Finally, the sections of c + can be obtained from those of b++ using relations of the type (14). We expand the fields ~u, ~t using the wave functions t~"2) that are holomorphic on the surface minus the disk (but not on the disk itself) for the positive frequencies. To form a complete set, for the negative frequencies we take wave functions that lead to a set of canonical commutation relations. These are the z n, n > 0 , so using the convention that t(m)(z~ - - n ~ J =z n if n > 0
(15)
While the "naive" propagator is not meromorphic in z due to the presence of the zero modes, (15) is. There are a number of extra singularities that lie at isolated points that obviously coincide with the location of the zero mode insertions [12]. Applying Wick's theorem one obtains
In the overlap area we can determine the connection between the new coefficients and those defined on the disk n-l/2"~l/'ln-l/2Ji-
E B n( 1m/ 2 ) ,,, W --m+l/2 m
(19)
(and analogous relations for 2). Recalling that BO n ma ) -- -0 if n,m~O we see that the creation operators are the same, but the annihilation operators are modified. This corresponds to a Bogolubov transformation in the Fock space of the disk. The "vacuum" will now be the state annihilated by all the Xr, Zr, r > 0, which of course are the same operators that 173
Volume 198, number 2
PHYSICS LETTERS B
were obtained by Vafa [ 2 ] using the method o f conserved charges
Z, lff2)=~rl~Q)=O,
r>0.
(20)
So finally we have the wave function which is associated to the non-trivial half of the surface. It is simply t/2)~ t/2)~3/2~3/2 . . . .
(21
)
In terms o f the SL2 invariant v a c u u m ll2) = C e x p ( - - ~ ql-n+t/2Bn,,, (l/E) gt-m+,/2)lO). nm
19 November 1987
However it is important to understand the dependece of this constant on the moduli [ 13 ]. Next we ask which is the action o f the Virasoro generators Lk on the state Is'2) defined by (22). We know that Is'2) cannot be invariant under all reparametrizations of the form z--, z' = z + ez ~+t, even with k>~ - 1 , because some combinations o f them correspond to vector fields that cannot be extended over the whole surface and, therefore, correspond to changes of the moduli. After some algebra one finds (for k>~ - 1 )
L k I O ) = -- ½ Z M,s~_~+l/2~_~+l/zlg2) k+½ckl.Q) , r.s>0
(22) The norm of this state is
with
(~21S'2) = [C[ 2 det(1 + B ' B ) ,
(23)
and hence is manifestly positive. This process can be repeated for the ghosts. The only difference is the presence of the 3 g - 3 global zero modes that provide additional operators that annihilate the vacuum, as it can be easily seen by the method o f conserved charges [2], These operators can be expressed solely in terms of creation operators on the disk. One obtains in this way the state
]t2)g, = C' Qt Q2...Q3g_3S' 1 0 ) g h
(24)
,
where S' = e x p
--
t-'-n+2
nm C--m--
1
,
Qk= E c _ , . a ,k. . m
The vacuum I0 ) g, is defined by G I0 ) m = 0 for r>~ - 1 and b A 0 ) - - 0 for r~>2. Note that the Q's c o m m u t e with the operator S' that implements the Bogolubov transformation on the vacuum. We see that there are exactly as many charges as insertions on the disk. If we had not placed these insertions the partition function which is given by 3g- 3
Cg,(01 l--[ b++(w~)QtQz...Q3g_3S'lO)gh
(25)
i=1
would vanish, as expected. In all these expressions there is an arbitrary constant that is not determined by any of the above arguments and is, in fact, completely irrelevant for any one Riemann surface. 174
(26)
M k, = ( k + 2 r - 1 ~uo/2) -~ ( k + 2 s - 1 ~o (1/2) JUg+r,s ]Vr, k+s
- ~. (k-2n+l~l:C(V2) u(l/2) ~l~r.k--n+ll~n.s
,
(27)
n
and
c~
~ ( k - 2 n + '~°"/2) I )On.k_n+
1 •
(28)
n
If k < - 1 there is an additional term on the R H S o f (26) obtained from the action of Lk o n the SL2 vacu u m 10). Under a reparametrization generated by Lk the normalization o f the state changes C--, C+ ec k, and so does the partition function since ( 0 [ I 2 ) = C. Note that c o = c t = c - ~= 0 and, consequently, the partition function is SL2 invariant. Eq. (26) implicitly provides us with the vector fields that are globally defined on the surface minus the disk. If we define N~s = ( k + 2 r - 1 ~uo/2) 1 ~o,/2) lU k+r,s ~ ~ (k+2s]~t~ r , k + s
+ ~ (k_2n+l~oo/2) u~1/2) , l]~r.k-n+l~'n.s
(29)
n
and
kTk=Lk +½ ~ N ~s~g_r+l/z~U_~+t/2,
(30)
vs
the T~ annihilate the state created by the Riemann surface T~ Ig2) = 0 ,
n~>-l.
(31)
In a way they can be viewed as the generalization of the Virasoro generators in higher genus in the sense
Volume 198, number 2
PHYSICS LETTERS B
that they generate reparametrizations that do not change the moduli and leave us with the same surface, i.e. bonafide reparametrizations. One could find which these reparametrizations are in terms of the local coordinates in the disk by determining the coefficients rpq in
Tp= E FpqLq,
(32)
19 November 1987
alternative way to (31 ) and obtains a check of the above formulae. We have not written down them explicitly before, but it should be clear that the conserved charges for the ghost field c + which are associated to the holomorphic sections b+ + (I 7) are S m .~-C m "~- ~ B ( m 2 ) + 2 , k _ l C _ k . k
(37)
q
since the Tp are defined in the Fock space of the disk, where the Lq form a complete set. We have to worry now about how the conformal anomaly reflects itself on these operators. From (26) we note that T~ can be written as
Tp=SLpS-' ,
(33)
where the Lp satisfy
[L,,L,,,]=(n-rn)L,,+,,, +A(m)&,+m,o ,
(34)
and S is the operator that implements the Bogolubov transformation on the vacuum as given (22) (note that S is invertible). It is therefore obvious that
[T,, T,n]=S[L,,,L,,]S -~ = ( n - m ) T n + , , , +A(m)gn+m,o •
(35)
So they have the same central extension. However this is not true if the theory is for instance defined on a non-trivial background and the central extension is not a c-number. In a non-linear sigma model the central extension contains the dilaton beta function (no explicit calculations exist for g~> 2, though). In such a case, the central extension may or may not commute with S, so this may depend on the topology of the world-sheet. Finally we would like to discuss the construction of a BRS-like charge that implements the condition Tq=O for n~> - 1 on the "vacuum" It2>. We can then use this charge to define physical states, exactly as one does on the sphere. First we note that the wave function (21 ) is annihilated by operators of the form -½ ~ ( m + 2 n + 1)~,,+n-,/2Z-n+,/2 ,
(36)
n
which are but the usual Virasoro generators with the creation and annihilation operators on the disk replaced by the ones on the other half of the surface, that we have discussed previously. By using the definition of Z, ~ in terms of ~v and ¢, one arrives in an
Therefore the charge Q= Z m s_mT,, annihilates the state IK2) ® I$2) gh. We would have to supplement Q with the part corresponding to the ghost sector in order to obtain the complete charge, but we have not constructed the equivalent of the T, for the ghosts here. The usual BRS charge, defined by (10), contains the components of the energy-momentum tensor that generate changes in moduli space, in addition to those that correspond to pure reparametrizations. So, in fact, the above construction amounts to dividing up QBgs into two parts: variations of the metric that correspond to pure reparametrizations and moduli deformations. Something that is perhaps worth pointing out is that the operator formalism allows us to obtain a remarkable degree of explicitness in the construction of the BRS charge. There are many other interesting questions that can be studied. For instance, one could characterize the variation of the vacuum state under the reparametrization group by means of a Bogolubov transformation on the operators Zr, ~r (similar to what has been done for the states on the disk) and construct in this way a bundle using the "vacuum" 112> instead of 10>. The Bogolubov coefficients would surely provide a representation of Diff S~/S ~, whose relation to the one defined by the conformal field theory on the sphere would have to be elucidated. The curvature of the bundle so constructed would have to reproduce the conformal anomaly. Yet it would be nice to see how things work out explicitly. The authors would like to thank J.M.F. Labastida, J. Mafies and M. Pernici for discussions. This work has been partially supported by NSF contract PHY82-15249 and US DOE contract DE-AC0276ER02220.
175
Volume 198, number 2
PHYSICS LETTERS B
References [ 1 ] L. Alvarez-Gaum6, C. G6mez and C. Reina, Phys. Lett. B 190 (1987) 55. [2] C. Vafa, Phys. Lett. B 190 (1987) 47. [ 3 ] A. Belavin, A. Polyakov and A. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [4] D. Friedan, E. Martinec and S. Shenker, NucL Phys. B 271 (1986) 93. |5J D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575; Phys. Lett. B 151 (1985) 37. [6] A. Pressley and G. Segal, Loop groups (Oxford U.P., Oxford, 1986);
176
19 November 1987
G. Segal and G. Wilson, Publ. IHES (1985) 61; G. Segal, Commun. Math. Phys. 80 (1981) 301. [ 7 ] I. Frenkel, H. Garland and G. Zuckerman, Proc. Nat. Acad. Sci. USA 83 (1986) 8442. [8] E. Witten, Princeton preprint PUPT-1061 (1987). [9] B. DeWitt, Phys. Rep. 19 (1975) 295. [ 10] K. Pilch and N. Warner, CTP preprint 1457 (1987). [ 11 ] M. Bowick and S. Rajeev, Phys. Rev. Lett. 58 (1987) 535; CTP preprint 1450 (1987), [12] H. Sonoda, Phys. Lett. B 178 (1986) 390; M. Dugan and H. Sonoda, LBL preprint LBL-22776 (1987). [ 13 ] L. Alvarez-Gaum6, C. G6mez, G. Moore, C. Reina and C, Vafa, in preparation.
PHYSICS LETTERS B
19 November 1987
O N THE OPERATOR F O R M A L I S M IN H I G H E R G E N U S D. E S P R I U 1,2
The Institutefor AdvancedStudy, Princeton, NJ 08540, USA and Lyman Laboratoryof Physics, HarvardUniversity, Cambridge, MA 02138, USA Received 24 July 1987
We discuss several aspects of the recently proposed operator formalism on Riemann surfaces of genus g~>2, particularly the transformation under the reparametrization group of the states generated by the non-trivial topology.
One o f the most interesting new d e v e l o p m e n t s in string theory is the recent f o r m u l a t i o n by AlvarezG a u m r , G 6 m e z and Reina [ 1 ] and, m o r e explicitly, by Vafa [ 2 ] o f an 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 o f genus greater than one ~. The ability to use an o p e r a t o r f o r m a l i s m on the sphere and the torus has led in the past to considerable progress in the understanding o f t w o - d i m e n s i o n a l conformal field theory (see e.g. refs. [3,4]). A truly r e m a r k a b l e achievement has been the classification o f all " m i n i m a l " ( c < 1 ) conformal and superconformal theories [ 5 ] on the sphere. C o n f o r m a l field theory on the torus has also been a b u n d a n t l y investigated in the context o f string theories with the help o f o p e r a t o r methods. It is therefore potentially i m p o r t a n t to extend the o p e r a t o r f o r m a l i s m to R i e m a n n surfaces o f arbitrary genus in order to cast what we already know in a new, perhaps m o r e powerful, language a n d then see whether something new can be learnt from it. As we have just mentioned, the physical interest o f the o p e r a t o r formulation goes well b e y o n d the confines o f string theory. On the other hand, it turns out to be related to the theory o f representations o f loop groups [ 6 - 8 ] as well as to other m a t h e m a t i c a l p r o b l e m s (see references in refs. [ 1,2]). H o w e v e r we shall mostly ignore these interesting ramifications in this letter. This work in fact c a m e about in an at-
t e m p t to u n d e r s t a n d i n g some o f the relevant results o f refs. [ 1,2] in terms as simple as possible. Let us first s u m m a r i z e the construction o f the relevant Hilbert space. By choosing a generic point on a R i e m a n n surface a n d analytic coordinates at that point, we can d i v i d e the surface o f our preference into the patch Iz l < 1 a n d the rest. The resulting surfaces share the c o m m o n b o u n d a r y Izl = 1. (It is ass u m e d that the original one had no b o u n d a r y . ) O f course all the c o m p l i c a t e d topology lives in the comp l e m e n t a r y o f Izl < 1. The fields defined on the R i e m a n n surface have to satisfy the suitable continuity conditions across the boundary. The path integral on the whole surface then splits into two parts each defining a wave function, in this case a functional o f the fields on the b o u n d a r y ~ [ O ( z ) ] . Integrating over all values o f O(z) is equivalent to "sewing" the two halves o f the surface together. Here we want to focus on two different aspects o f the o p e r a t o r formalism. First we will present the construction o f the relevant operators and Fock space in a slightly different way o f that o f refs. [ 1,2]. This will help us u n d e r s t a n d the second part o f our work, which is the realization o f r e p a r a m e t r i z a t i o n invariance on such a Hilbert space. The wave function that is associated to the I z l < 1 patch is, o f course, very simple. I f ~ is a bosonic free field it is given by a p r o d u c t o f gaussians
Fulbright Fellow. 2 Address after 1 October 1987: Departament de Fisica Terrica, Universitat de Val+ncia,E-46100 Burjassot (Valencia), Spain. "~ For references to previous or related work see refs. [ 1,2]. 0370-2693/87/$ 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 )
~[x(z)]= (x(z) fO)~exp(-~
x~ ) ,
(1)
171
Volume 198, number 2
PHYSICS LETTERS B
where the x, are the coefficients o f the mode expansion o f x ( z ) , x ( z ) = E , x , + ~z". The wave function for free fermions, which is the system that we will be interested in ~2, is likewise trivial; it is given by the semiinfinite product ~'t[~//(Z)] = ( ~ ¢ ( Z ) ~ ( Z ) 1 0 ) ~ ~/,71/2~b¢1/2~/,/3/2~//3/2...
(2)
provided that the Dirac v a c u u m is defined by ~u~10) =qJ~10) = 0 for r > 0 , with ~g(Z)~- 2 ~,ln+l/2 Zn, n
~(Z)=
2 [~n+l/2Ztl " n
(3)
The antiholomorphic part qP(g) leads to a similar wave function ~u[~t(g)] that depends on the conjugate coefficients and the complete wave function is the product o f the two. If we perform a change of local coordinates z--, z' = e (z) the expansion coefficients in ( 3 ) change. The new coefficients are related to the old ones by a Bogolubov transformation ~3 n-I/2=
20lnm~ffm--l/2"q-
m
(4) m
a,,,~u-,,+L/z + Y, B,,,q/m--llZ,
f dzdf ½(~t~+~,)
+b++~c + .
(5)
T++ = - ~ ( V 0 ~ + q/0q~)
8~t = 2 c + O~,+ ½20c + ~,
S=Cexp(-~_,,+,/z( ~)nm~--m+1/2)
that lead to the conserved charge
O/-1
Not all reparametrizations change the wave function (2). U n d e r a change o f coordinates o f the form z - , z + ~ z k+l, the operators get modified by ~/r--) Vr+ ~ [ Lk, ~ ] , and likewise for q/. F r o m the definition o f the Virasoro generator Lk it is obvious that the wave function (2) is invariant under one such ~2 The bosonic theory can be recovered by taking into account the appropriately twisted sectors [ 2 ]. #3 See, for instance, ref. [9] for a description of the properties of Bogolubov transformations.
172
8 ~ = 2 c + OqT+ ½2Oc+ ~ , 8 c + = 2 c + 0 c +,
8b++=22T++,
QBRS = ½C+ ( ~ O~ + ~O~t) --C + OC+ b+ + .
(6)
(8)
(7) is invariant under the BRS transformations
and similar relations for q/'. The matrices a and fl form in fact a representation of D i f f S ~/S ~ [ 10]. The new vacuum is the state annihilated by the new operators V'r, q~'r, r > 0 and is related to the old vacuum by 1 0 ) ' = S I 0 ) with -
(7)
+½(c +Ob++ + 2 0 c + b + + ) .
m
n,m>~l ,
transformation if k > ~ - 1, i.e. Lk[ 0> = 0 for k>_-- 1, as in this case flnm = 0. Generically, however, the vacuum is not invariant and one obtains in this way a bundle over Diff S ~/S ~, with the independent entries o f a-~fl as holomorphic coordinates [ 10]. The curvature o f this bundle with the metric induced on it by the inner product of the Hilbert space at the origin, i.e. for fl, fl = 0, is the usual conformal anomaly [10,11]. So far we have dealt with the easy part, namely the patch Iz I < 1 which is conformally equivalent to a semi-infinite cylinder. We would like to characterize in a similar way the transformation properties under Diff S t/S l of the state created by the non-trivial half o f the original surface. On the surface we have a gauge fixed action defined by
The conjugate fields have been omitted. The ene r g y - m o m e n t u m tensor associated to it is
~ J~nm~'l-m+l/2, m
n,m>_.l,
19 November 1987
(9)
(10)
Following the standard procedure, in order to quantize the above action we have to solve the wave equation 0~u=0 (and similar ones for q/, c + and b++). For the spin-l/2 field such an equation has between 0 and g globally defined solutions on a Riemann surface. We will assume that there is none. Should one exist we would proceed in a form similar to the ghost case which will be discussed shortly. However, the wave equation does admit an infinite n u m b e r o f solutions when a patch is removed. All the information that we need to determine these solutions is contained in the propagator. On general grounds
(g/(z)C,(w) ) = ~
1
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Volume 198, number 2
((c +(z)b+ +(w))) = H ( c +(z)b+ +( w ) )
(11)
+ ~(z, w).
Z--W
- ~ Hohi(w) (c+(z)b++(wj)) id
~(z, w) is analytic both in z and w, so taking I wl < 1, Iz[ > 1 we can expand the propagator in powers of w, each of the terms in the expansion being a holomorphic function of z, since ~<~/(z)~(w))=0,
Izl>lwl.
(12)
m O r,e - - I ~ ( 0 , 0 ) , Defining B n("2)=0~-~ morphic solutions read
t~l/2)(z) = z - " + E B~I,/~)z'n-I
the
holo(13)
m
Since t~ ~2) (z) is holomorphic on the surface minus the disk, we have 0=
ri.r t O / 2 ) g ~ , ~ t O / 2 ) t ~ = ~ . R ( l / 2 )
-- --R(1/2)
(14)
=H
1 z-w
1 - ~ H,jhi(w) + ~(z, w, w ) , o z-wj
(16) where [ 12] hi is the set of zero modes, H = d e t hi(wj), H o is the/j-minor of H, and (¢(z, w, wj) is an hop omorphic function of the indicated variables. Since their location is arbitrary we can place all the b+ + insertions inside the Izl < 1 disk. We take w to be also in the disk so that we can expand (16) in powers of z. Taking into account that the modified propagator solves the wave equation, we immediately obtain the holomorphic sections for the antighost b+ + on the surface minus the disk
t(,2)(z) = z - n + ~ B(,~z m-' ,
(17)
m
Explicit expressions for these coefficients have been given by Vafa [ 2 ] in terms of 0 functions and the prime form, but we will not need them. It is more interesting to repeat the above construction for the ghost and antighost. Also in this case we can determine all the holomorphic sections from the propagator, but in addition we will learn which is the appropriate way of computing expectation values in the operator formalism. Since dim H ° ( K 2) = 3 g - 3, for g>~ 2, there exist precisely 3 g - 3 zero modes of the antighost field b + + and none of c +. The "naive" ghost propagator ( c + ( z ) b + + ( w ) ) has to be modified to include 3 g - 3 insertions, at arbitrary points, of the b+ + field to soak the zero modes
~(z) = ~ Z_,+lat~"2)(z) .
((c+ (z)b++ (w) ))
(18)
n
~__. ( 3 g - 3
. H=l b++(wi)c+tz)b++(w))
where the B ~ are given by the derivatives of f9 with respect to z and w. Of course, in addition to (17) we still have the sections hi(z)=Eka~z k. Finally, the sections of c + can be obtained from those of b++ using relations of the type (14). We expand the fields ~u, ~t using the wave functions t~"2) that are holomorphic on the surface minus the disk (but not on the disk itself) for the positive frequencies. To form a complete set, for the negative frequencies we take wave functions that lead to a set of canonical commutation relations. These are the z n, n > 0 , so using the convention that t(m)(z~ - - n ~ J =z n if n > 0
(15)
While the "naive" propagator is not meromorphic in z due to the presence of the zero modes, (15) is. There are a number of extra singularities that lie at isolated points that obviously coincide with the location of the zero mode insertions [12]. Applying Wick's theorem one obtains
In the overlap area we can determine the connection between the new coefficients and those defined on the disk n-l/2"~l/'ln-l/2Ji-
E B n( 1m/ 2 ) ,,, W --m+l/2 m
(19)
(and analogous relations for 2). Recalling that BO n ma ) -- -0 if n,m~O we see that the creation operators are the same, but the annihilation operators are modified. This corresponds to a Bogolubov transformation in the Fock space of the disk. The "vacuum" will now be the state annihilated by all the Xr, Zr, r > 0, which of course are the same operators that 173
Volume 198, number 2
PHYSICS LETTERS B
were obtained by Vafa [ 2 ] using the method o f conserved charges
Z, lff2)=~rl~Q)=O,
r>0.
(20)
So finally we have the wave function which is associated to the non-trivial half of the surface. It is simply t/2)~ t/2)~3/2~3/2 . . . .
(21
)
In terms o f the SL2 invariant v a c u u m ll2) = C e x p ( - - ~ ql-n+t/2Bn,,, (l/E) gt-m+,/2)lO). nm
19 November 1987
However it is important to understand the dependece of this constant on the moduli [ 13 ]. Next we ask which is the action o f the Virasoro generators Lk on the state Is'2) defined by (22). We know that Is'2) cannot be invariant under all reparametrizations of the form z--, z' = z + ez ~+t, even with k>~ - 1 , because some combinations o f them correspond to vector fields that cannot be extended over the whole surface and, therefore, correspond to changes of the moduli. After some algebra one finds (for k>~ - 1 )
L k I O ) = -- ½ Z M,s~_~+l/2~_~+l/zlg2) k+½ckl.Q) , r.s>0
(22) The norm of this state is
with
(~21S'2) = [C[ 2 det(1 + B ' B ) ,
(23)
and hence is manifestly positive. This process can be repeated for the ghosts. The only difference is the presence of the 3 g - 3 global zero modes that provide additional operators that annihilate the vacuum, as it can be easily seen by the method o f conserved charges [2], These operators can be expressed solely in terms of creation operators on the disk. One obtains in this way the state
]t2)g, = C' Qt Q2...Q3g_3S' 1 0 ) g h
(24)
,
where S' = e x p
--
t-'-n+2
nm C--m--
1
,
Qk= E c _ , . a ,k. . m
The vacuum I0 ) g, is defined by G I0 ) m = 0 for r>~ - 1 and b A 0 ) - - 0 for r~>2. Note that the Q's c o m m u t e with the operator S' that implements the Bogolubov transformation on the vacuum. We see that there are exactly as many charges as insertions on the disk. If we had not placed these insertions the partition function which is given by 3g- 3
Cg,(01 l--[ b++(w~)QtQz...Q3g_3S'lO)gh
(25)
i=1
would vanish, as expected. In all these expressions there is an arbitrary constant that is not determined by any of the above arguments and is, in fact, completely irrelevant for any one Riemann surface. 174
(26)
M k, = ( k + 2 r - 1 ~uo/2) -~ ( k + 2 s - 1 ~o (1/2) JUg+r,s ]Vr, k+s
- ~. (k-2n+l~l:C(V2) u(l/2) ~l~r.k--n+ll~n.s
,
(27)
n
and
c~
~ ( k - 2 n + '~°"/2) I )On.k_n+
1 •
(28)
n
If k < - 1 there is an additional term on the R H S o f (26) obtained from the action of Lk o n the SL2 vacu u m 10). Under a reparametrization generated by Lk the normalization o f the state changes C--, C+ ec k, and so does the partition function since ( 0 [ I 2 ) = C. Note that c o = c t = c - ~= 0 and, consequently, the partition function is SL2 invariant. Eq. (26) implicitly provides us with the vector fields that are globally defined on the surface minus the disk. If we define N~s = ( k + 2 r - 1 ~uo/2) 1 ~o,/2) lU k+r,s ~ ~ (k+2s]~t~ r , k + s
+ ~ (k_2n+l~oo/2) u~1/2) , l]~r.k-n+l~'n.s
(29)
n
and
kTk=Lk +½ ~ N ~s~g_r+l/z~U_~+t/2,
(30)
vs
the T~ annihilate the state created by the Riemann surface T~ Ig2) = 0 ,
n~>-l.
(31)
In a way they can be viewed as the generalization of the Virasoro generators in higher genus in the sense
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PHYSICS LETTERS B
that they generate reparametrizations that do not change the moduli and leave us with the same surface, i.e. bonafide reparametrizations. One could find which these reparametrizations are in terms of the local coordinates in the disk by determining the coefficients rpq in
Tp= E FpqLq,
(32)
19 November 1987
alternative way to (31 ) and obtains a check of the above formulae. We have not written down them explicitly before, but it should be clear that the conserved charges for the ghost field c + which are associated to the holomorphic sections b+ + (I 7) are S m .~-C m "~- ~ B ( m 2 ) + 2 , k _ l C _ k . k
(37)
q
since the Tp are defined in the Fock space of the disk, where the Lq form a complete set. We have to worry now about how the conformal anomaly reflects itself on these operators. From (26) we note that T~ can be written as
Tp=SLpS-' ,
(33)
where the Lp satisfy
[L,,L,,,]=(n-rn)L,,+,,, +A(m)&,+m,o ,
(34)
and S is the operator that implements the Bogolubov transformation on the vacuum as given (22) (note that S is invertible). It is therefore obvious that
[T,, T,n]=S[L,,,L,,]S -~ = ( n - m ) T n + , , , +A(m)gn+m,o •
(35)
So they have the same central extension. However this is not true if the theory is for instance defined on a non-trivial background and the central extension is not a c-number. In a non-linear sigma model the central extension contains the dilaton beta function (no explicit calculations exist for g~> 2, though). In such a case, the central extension may or may not commute with S, so this may depend on the topology of the world-sheet. Finally we would like to discuss the construction of a BRS-like charge that implements the condition Tq=O for n~> - 1 on the "vacuum" It2>. We can then use this charge to define physical states, exactly as one does on the sphere. First we note that the wave function (21 ) is annihilated by operators of the form -½ ~ ( m + 2 n + 1)~,,+n-,/2Z-n+,/2 ,
(36)
n
which are but the usual Virasoro generators with the creation and annihilation operators on the disk replaced by the ones on the other half of the surface, that we have discussed previously. By using the definition of Z, ~ in terms of ~v and ¢, one arrives in an
Therefore the charge Q= Z m s_mT,, annihilates the state IK2) ® I$2) gh. We would have to supplement Q with the part corresponding to the ghost sector in order to obtain the complete charge, but we have not constructed the equivalent of the T, for the ghosts here. The usual BRS charge, defined by (10), contains the components of the energy-momentum tensor that generate changes in moduli space, in addition to those that correspond to pure reparametrizations. So, in fact, the above construction amounts to dividing up QBgs into two parts: variations of the metric that correspond to pure reparametrizations and moduli deformations. Something that is perhaps worth pointing out is that the operator formalism allows us to obtain a remarkable degree of explicitness in the construction of the BRS charge. There are many other interesting questions that can be studied. For instance, one could characterize the variation of the vacuum state under the reparametrization group by means of a Bogolubov transformation on the operators Zr, ~r (similar to what has been done for the states on the disk) and construct in this way a bundle using the "vacuum" 112> instead of 10>. The Bogolubov coefficients would surely provide a representation of Diff S~/S ~, whose relation to the one defined by the conformal field theory on the sphere would have to be elucidated. The curvature of the bundle so constructed would have to reproduce the conformal anomaly. Yet it would be nice to see how things work out explicitly. The authors would like to thank J.M.F. Labastida, J. Mafies and M. Pernici for discussions. This work has been partially supported by NSF contract PHY82-15249 and US DOE contract DE-AC0276ER02220.
175
Volume 198, number 2
PHYSICS LETTERS B
References [ 1 ] L. Alvarez-Gaum6, C. G6mez and C. Reina, Phys. Lett. B 190 (1987) 55. [2] C. Vafa, Phys. Lett. B 190 (1987) 47. [ 3 ] A. Belavin, A. Polyakov and A. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [4] D. Friedan, E. Martinec and S. Shenker, NucL Phys. B 271 (1986) 93. |5J D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575; Phys. Lett. B 151 (1985) 37. [6] A. Pressley and G. Segal, Loop groups (Oxford U.P., Oxford, 1986);
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G. Segal and G. Wilson, Publ. IHES (1985) 61; G. Segal, Commun. Math. Phys. 80 (1981) 301. [ 7 ] I. Frenkel, H. Garland and G. Zuckerman, Proc. Nat. Acad. Sci. USA 83 (1986) 8442. [8] E. Witten, Princeton preprint PUPT-1061 (1987). [9] B. DeWitt, Phys. Rep. 19 (1975) 295. [ 10] K. Pilch and N. Warner, CTP preprint 1457 (1987). [ 11 ] M. Bowick and S. Rajeev, Phys. Rev. Lett. 58 (1987) 535; CTP preprint 1450 (1987), [12] H. Sonoda, Phys. Lett. B 178 (1986) 390; M. Dugan and H. Sonoda, LBL preprint LBL-22776 (1987). [ 13 ] L. Alvarez-Gaum6, C. G6mez, G. Moore, C. Reina and C, Vafa, in preparation.