- Email: [email protected]
Potentials for topological sigma models
Physics Letters B 271 (1991) 101-108 North-Holland
P I-4YSI C S L ETT ER S B
Potentials for topological sigma models J.M.F. Labastida Departamento de Fisica de Particulas, Universidadede Santiago, E-15706 Santiago de Compostela, Spain
and P.M. Llatas Departamento de Fisica Te6rica, UniversidadAut6norna de Madrid, E-28049 Madrid, Spain Received 9 August 1991
Topological sigma models with potential terms are obtained by twisting N=2 supersymmetric sigma models. These models exist for manifolds with isometries and potential terms in their actions are built out of the corresponding Killing vector fields. It turns out that, as ordinary topological sigma models, the models resulting after twisting N=2 supersymmetry can be generalized to the case of almost hermitian manifolds.
Topological m a t t e r in two d i m e n s i o n s have been shown to play an i m p o r t a n t role in relating topological q u a n t u m field theories to m u l t i - m a t r i x models. The connection discovered by Witten [ 1,2 ] between topological gravity in two d i m e n s i o n s [3,4 ] and matrix models has been generalized to the multi-matrix case [ 5 ] by coupling topological gravity to some specific kind of topological m a t t e r [6 ]. F u r t h e r study o f this connection has been carried out recently [ 7 - 1 0 ]. On the other hand, the study o f topological m a t t e r may provide some insight on the structure o f a class of theories which presumably posses a phase where gravitons are composite particles. Topological m a t t e r in two d i m e n s i o n s provides a rich variety o f two-dimensional topological theories which seem to be closely related to their non-topological counterparts, at least for some specific cases [6,5 ]. We have started a systematic study of the most general form of topological m a t t e r in two d i m e n sions. Our strategy is the following. First we consider an N = 2 supersymmetric model. We then twist it [ 1 1 ] and extend it to the most general possible situation. We have started our analysis with N = 2 topological sigma models of the type considered in ref. [ 12], i.e., E-mail address: [email protected].
N = 2 supersymmetric sigma models with potential terms whose target space is a K~ihler manifold. More general situations [ 13] should also be considered in this framework. F o r example, one would expect that in those situations one might construct a topological version o f W Z W models. In this letter we report on part of our results concerning this analysis on topological matter. A detailed account of the procedure involved in the construction o f these theories will be reported elsewhere [ 14 ]. In N = 2 superspace in two d i m e n s i o n s there exist two basic multiplets, the chiral multiplet, and the twisted chiral multiplet [ 13 ]. Once a precise form of the twist o f N = 2 supersymmetry is chosen, each multiplet leads to a different type o f topological matter [ 14]. Let us consider the s t a n d a r d N = 2 superspace action,
S = f d2zd4OK(X/, X r) Z
z
where I; denotes a Riemann surface, and X ~, X 1 ( I = 1, .... d) are local coordinates of a 2d-dimensional corn-
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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pact K~ihler manifold M whose K~ihler potential is K. The superpotential W ( X ~) is a holomorphic scalar function on M. The local coordinates X t ( X r) may be regarded as chiral or twisted chiral (antichiral or twisted antichiral) N = 2 multiplets. O f course, the 0measure of the F-term in ( 1 ) depends on the kind of multiplet one is considering. In ref. [14] a detailed analysis of the resulting models after twisting ( 1 ) is carried out when only a single type of multiplet appears in the action. It would be very interesting to perform a similar analysis for the case in which one has a mixture of these types of multiplets. One could ask similar questions for the recently discovered N = 2 semichiral multiplet [ 15 ]. Twisting N = 2 supersymmetry in ( 1 ) may lead to twisted actions which are not Lorentz invariant. As is shown in ref. [ 14], for chiral multiplets this is indeed the case unless one does not introduce the Fterm in (1). The resulting theory is a topological sigma model [ 16 ] for the specific case in which the target manifold M is a K~ihler manifold. Of course, no potential terms for topological sigma models seem to appear in this way. While the original N = 2 supersymmetric action cannot be extended to a more general type of manifold, Witten showed [16] that in the topological case the theory can be generalized to the case of almost hermitian manifolds. Before describing how potential terms can be introduced for these models, let us briefly comment on the models resulting after twisting the other type o f multiplet. Twisting a model containing twisted chiral multiplets allows F-terms and therefore one obtains potential terms automatically. The resulting models are a topological version of N = 2 L a n d a u - G i n z b u r g models. For the case of a flat target manifold these models have been obtained in ref. [ 17 ]. Conformal cases of these models correspond to the ones discussed in ref. [ 6 ]. As shown in ref. [12], there exist some N = 2 supersymmetric sigma models which are not contained in the N = 2 superspace formulation (1). These models involve new types of potential terms and their corresponding supersymmetric algebra possesses central charges. Twisting N = 2 supersymmetry for these models leads to the opposite situation to the one observed in the twist o f ( 1 ). Namely, twisted chiral multiplets lead to actions which are not Lorentz invariant while chiral ones lead to Lorentz invariant 102
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actions. Let us therefore consider chiral multiplets but first let us introduce some notation. For future convenience we will depart from the holomorphic notation used in ( 1 ). However, one has to keep in mind that for the m o m e n t we are considering K~ihler geometry, i.e., the metric on M, Gij has only non-vanishing components GIj, and the complex structure J5 ( J * f l S = - a } ) satisfies D a J ) = 0 . Notice that lower case Latin indices run from 1 to 2d, the real dimension of M. The action resulting after carrying out the twist of N = 2 supersymmetry takes the form S-- f d Z z ~
( ½Gog '~p O~x i O~xJ+ ½~al~JO-Oax" O,~Xj
Z l ~O~[3D ~i nJ ~k,.m - ig'~G,jp'. DaZ j - g6 *.o~,,,v,~t'~z x
+ G o V g V J - z ' z J D , 15-lg'~Pp~papD, Vj) ,
(2)
where g ~ and a~p (e~yeY~= - ~ ) denote the metric and complex structure chosen on the Riemann surface Z. In (2), x i ( i = 1..... 2d) denote local coordinates on M, while p~ and Z' ( i = 1, ..., 2d) are anticommuting fields which are sections of the pullback 0 * ( T ) of the tangent bundle T to M. The anticommuting fields p~, have acquired after the twist a vector index a = 1, 2, which is a tangent index to E. p " ( i = 1, ..., 2d) are one forms on Y. with values in ~*(T) which satisfy the self-duality condition p~ = ~,~J'jp~. In (2) D~ denotes the pullback covariant derivative, D M ( = 0 , Z ~ + c~,~xJF'jkZk. Finally V' is a holomorphic Killing vector on M, D, Vj + D j V, = 0 ,
J " J " j D , , , V , , = D , Vj.
(3)
The action (2) is obtained after performing the twist o f N = 2 supersymmetry in the action corresponding to eq. (50) of ref. [ 12] and setting the quantities h and G ~to zero. As we have discussed, h ¢ 0 would lead to an action which is not Lorentz invariant. However, we could have kept the vector field G' in ref. [ 12 ] all along the twisting. Let us not do that for the m o m e n t and let us postpone for later the discussion of additional vector field structures. Twisting N = 2 supersymmetry also provides the Qtransformations of the fields [ 14 ]. They turn out to be
[Q, x q = i z ' ,
{Q,x'}=-iv',
(4)
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{Q, pl}=a,~xi+e,~J(/a~xJ--iE~kZJpk,.
(4 cont'd)
Since the N = 2 supersymmetric model is an on-shell model in the sense that the supersymmetric algebra is realized after using the field equations of motion, we expect (4) to possess the same feature. Indeed, this is the case. Furthermore, since the N = 2 supersymmetric algebra in ref. [ 12 ] has central charges, we do not expect Q to be nilpotent. A simple calculation shows that, after making use of the field equations,
[Q2, xi]=Vi,
[Q2,z']=O:VizJ ,
[QZ, pi] =O:V' p~ ,
(5)
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Furthermore, one can verify that by simply adding the term
- ~ ~ x/-gg"aGijHiHJ/~
(9)
y
to the action (2), it becomes invariant under the first two transformations displayed in (4), and the transformations (6) and (7). Notice that the term (9) manifests the auxiliary nature of the field H i . Now that we possess an off-shell formulation we are in the position to analyze easily if we have an action which is Q-exact. It turns out that indeed, the action (2) plus the term (9) can be written as
i.e., Q2 acts as a Lie derivative with respect to the holomorphic Killing vector field V ~. The fact that Q2 ¢ 0 does not mean that the theory is not topological. In order for the theory under consideration to be topological with respect to the metric chosen in Z the only requirement is that the energy-momentum tensor be Q-exact. For the theory being topological with respect to target space quantities, the action itself needs to be Q-exact. It is simple to find out that, indeed, the energy-momentum tensor corresponding to (2) is Q-exact. To analyze if the action is Q-exact let us first reformulate the model off-shell. The natural way to carry out this reformulation is to introduce auxiliary fields for fields such that Q2 closes only after using the field equations. This is the case for the field p~. Let us redefine the Q-transformation o f p l as
{Q, pl}=Hi +O~xi+e~J~O~xJ--iF~kZ:p~.
(6)
Consistency implies that the auxiliary field H i must be also self-dual, H i = eJJ(/H~. The Q-transformation of H i is easily worked out by demanding that the new transformation o f p l (6) satisfies (5) without using the field equations. One finds [ Q, H i ] ~ _ _ --
i D.Z' - ie.aJ/j D~Zj
~o 21"~mj ~ k Z..... . Z. j~k ,t"c~ + D k V i p ~
1F/k)(. Ha "
-
t
j
k
(7)
To check that we are on the right track one may compute Q2 acting on H i to verify that indeed now the algebra closes off-shell,
[QZ, HI]=OjVi H~.
(8)
{Q' t xfg
[½g~/JG,jpi(O/~x:-½H~)+iGoVizQ}.
Z
(lO) This means that the theory is topological with respect to the target space. Any change in the metric Gij or the complex structure J~j can be written as a Q-exact quantity and therefore, following the general arguments described in ref. [ 11 ], the vacuum expectation values of the observables of the theory remain invariant under those changes. Notice also that the same arguments imply that the Killing vector field entering (10) can be deformed without having any effect on the observables. We will exploit this property later on when discussing observables. After finding a property like (10) we could ask about other possible terms to add. Indeed, one could introduce Q acting on any quantity as long as its Lie derivative with respect to the holomorphic Killing vector field vanishes. However, the three terms entering ( 10 ) are such that they lead to standard potential and kinetic terms in quantum field theory and we will restrict ourselves to them. Having an argument to keep only three types of terms in (10) we may ask about their relative numerical coefficients. The coefficient between the first and the second is the correct one for keeping H i as auxiliary fields. The coefficient of the third one is, however, undetermined. Actually, in ref. [ 12 ] there is a parameter m which in the twisting we have set to 1. This term would have entered in front of the third term of (10). At first sight, one could think that a coefficient 22 in front of the third term could be reabsorbed by a redefinition of the poten103
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tial, V ' - , 2 - J V ' plus the change of variables X;--" ~i "11/2~i )~ I/2Z',• p,--,.~ p , , and therefore it would just account for the scale of the potential. However, this is not entirely true because the path integral measure is in general not invariant under such a change in Z' and p',. Typically the amounts of X; zero modes and p~ zero modes are different and one has therefore a contribution from the measure. Thus, we have to consider the general situation in which there is a constant parameter multiplying the third term in (10). We will introduce it in our discussion of the generalized model. Before carrying out the analysis of observables and some of the consequences of the theory let us try to generalize it to the case in which the target manifold M is almost hermitian. This means that now the complex structure J~ is not covariantly constant but still the metric is such that u~i j - - _rk~,,, Our strata ia j tc:_ Jkm. egy will be first to construct the Q-algebra for this case and then to consider an action similar to the one in (10). For the m o m e n t we will not assume that the vector field V; satisfies conditions (3). Let us keep it for the m o m e n t general. Guided by the construction in ref. [ 16 ] we leave the transformations of x; and Z; as they stand. The transformation ofp~, in (6) is consistent with the self-duality condition satisfied by p~ when the complex structure f ; is covariantly constant. If one checks what fails when J'j is not covariantly constant one finds that simply by adding a term of the form ~ J f f p ~ DkJ~ to the transformation of p~ the consistency condition is satisfied. The transformation o f H i is then defined similarly as we did before, by requiring consistency of the Q-algebra. One finds
D; Vj + D j V, = 0 , V k OkJin + Jik On V k -- Jkn Ok V i = 0 .
( 12 )
These conditions are rather natural. They are just the requirement that the metric and complex structure remain invariant under a variation along the vector field V;. The first one constrains V ~to be Killing while the second one just tells us that the Lie derivative of the complex structure with respect to V i vanishes. Notice that in the case in which the manifold becomes K~ihler the second condition just reduces to the second equation in (3), i.e., the Killing vector becomes holomorphic. Notice that in this construction the non-trivial check is to verify if the Q-algebra closes for H i when there are potential terms. When the potential terms are set to zero that is known to be the case [ 16 ]. Now that we have constructed the Q-algebra let us build up the action. We will take (10) with an arbitrary parameter 2 z multiplying the third term,
S={Q, f x/g [ ½g~PG,jp'.(O/~xJ-½H~)+ i22GoV;zJ]} Z
----
( 2 ~Uo
~o¢ "~
Opx-'
Z
- ig"/JG,jp', [ D ~ Z ; + ½( DkJ/)zkE/7 0yx;] --
l c ~ O Z f l l - r . L / I L / J ..I[ ' d ij . . . . . . fl --
4d~
½R,jk,,~p',p~ff Z'"
-22Z;Z j D; V; - ¼gO~p~,pj/~ D; Vj}.
{Q,z'}=-iv;, {Q, p~ } = H', + 0 , x ; + e,fJ'j O/jxj • , ~p~k + ~l~. i. ~Z kP~i DkJ'j, -1F'j~Z
- ½R,,v;~.'"XJp~ + ½e,fZ'")ffp~l~ D,,, Dx.J'j - ½i e J ( D~,jij )Z k ( Oax j - eaYJi,,, Oyx '" ) 1 k tn n + ~Z Z p ~ ( D k J ji) (D,~J/~) + ½iej~ffH~ DkJ'i
104
Imposing that the action o f Q 2 on H i be the right one to close the algebra, i.e. (8), one finds after a somewhat tedious calculation that V; must satisfy the following conditions:
/ kZ m ] +22Gu V i V j -- ~(DkJip) (D,,,J pj)Po,, P~Z
[ Q , x ; ] = i z i,
+ D , V;pf~ -- ½eofV*pJ~ D , J ' j .
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( 11 )
(13
The action we have obtained coincides with the one given in ref. [ 16 ] when V' is set to zero. Our next task is to analyze the observables of this theory, i.e., the operators which are annihilated by Q. One immediately realizes that these operators are the same as in ref. [ 16 ] with the additional condition that the closed forms entering in them must be orthogonal to the Killing vector field V;. This is in fact rather natural since the non-trivial topology of the manifold M is encoded on the directions orthogonal to V;. No-
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rice also that we can a d d any observable Q o f any quantity which is invariant u n d e r the Lie derivative with respect to V( Let us display here, for example, the observables corresponding to zero forms (on Z ) . These take the form [16] ~A =A,i2...i,,Zi'Z ~2 ... Z i" ,
( 14 )
where Aaa...,, is an a n t i s y m m e t r i c tensor corresponding to a closed form which is ortbogonal to the Killing vector field V e, d A -~ 0
,
V/A/i2...i, = 0 .
( 15 )
The forms corresponding to the Q-descendants of (14) are like the ones in ref. [ 16]. Let us analyze the v a c u u m expectation value of some observables to make manifest the consequences o f the existence o f a theory like the one constructed. In dealing with functional integral computation it is i m p o r t a n t to consider the presence o f zero modes. In topological q u a n t u m field theories where the ghost n u m b e r o p e r a t o r c o m m u t e s with Q, one finds typically a n o m a l o u s contributions from the functional integral measure due to an unbalance between p~ zero modes and Z i zero modes. In the present theory ghost n u m b e r is not a good s y m m e t r y due to the presence of potential terms. However, the difference b e t w e e n p ~ zero modes and Z i zero m o d e s is also i m p o r t a n t because it is sensitive to rescaling o f fields. This difference is given by [ 18 ] ~ Z i zero m o d e s - : ~ p ~ zero modes
=2d(1-g)
+ J X*Cl(M),
(16)
Z
where x represents a m a p o f the h o m o t o p y type under consideration and c~ ( M ) is the first Chern class of the manifold M. By x * c ~ ( M ) we mean the pullback o f the two form on M associated to the first Chern class. Recall that 2d is the real d i m e n s i o n o f M. Since the action (13) is Q-exact we m a y deform the Killing vector field V ~ or change the value o f 2 without making any effect on the observables. In particular, one could set the Killing vector field to zero. In that case the theory just becomes an o r d i n a r y topological sigma model. But one could also do the opposite. Starting with a topological sigma model whose
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target manifold possesses some isometries we could introduce one o f the corresponding Killing vector fields and perform the calculation o f the observables o f the original model. To be able to deform V ~ (as long as it is Killing) and to change the value o f 2 with no effect on the observables m a y have some advantages regarding computations. For example, one could take the large 2 limit reducing the calculation to a sum over the critical points of the Killing field. Analogously, one can find interesting relations regarding the Killing vector field by considering the v a c u u m expectation value o f an observable which is manifestly zero when there is no Killing field in the action. Let us consider for example the case in which Z = S 2 and the maps x i are h o m o t o p i c a l l y trivial. In this situation the n u m b e r o f z ~zero modes is 2d (the constants) while there are no p~ zero modes. In ordinary topological sigma models this implies that one has to consider operators whose ghost n u m b e r is 2d to be able to have a possibly non-vanishing vacuum expectation value. In particular, the partition function is zero. Let us assume that the m a n i f o l d M under consideration admits at least a Killing vector with isolated singular points and let us compute such a partition function. Equating the result to zero will provide some relations which must be satisfied by any Killing field o f the kind under consideration. We will compute the partition function in the large 2 limit. F r o m the bosonic piece of the action (13) one observes that the path-integral gets d o m i n a t e d by configurations where V i is singular ( V ' = 0 ) and x ~ satisfies the instanton equation 0~x'+ e~PJ} 0z~xJ= 0. For the kind o f maps under consideration these are the constant configurations. Near each o f these configurations the path-integral for the non-zero modes becomes simply the ratio of fermionic and bosonic determinants which cancel. Therefore, we are left with an integration over zero modes and a finite sum over singular points of V£ Since there are only Z ~ zero m o d e s for the case under consideration, we have just to consider the term o f the action involving 22Z~Z j D~I~. The integration over anticommuting zero modes is easily carried out, 1-I d z ' exp(22Z'Z j Ds ~ ) 1
-
-
d! ~ 2d(t ....... 2d O, 1 ri2 ... O. . . . g,2, ,
(17) 105
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where the e-symbol is normalized such that e 12.,.2d= 1. On the other hand, the integration over the constant modes o f the bosonic fields leads to
f ~_~_dx'exp(-Z2G, V'VQ,
(18)
which, in the large 2 limit, after replacing V s by its linear term, reduces to the evaluation o f a gaussian,
; 2.=I~ 1 d.x t e x p ( - 2 2 G , j ,_
V~Vj) = ~ - 2 d
1
Id e t ( H ) I '
(19)
where H is a 2 d × 2d matrix defined as
(20)
H,=a,V,.
In obtaining (19) one has to keep in mind that the argument o f the exponent is being e x p a n d e d a r o u n d a singular point o f V i. Notice also that for almost hermitian manifolds as the ones under consideration ( 18 ) is well defined. Since the matrix H is antisymmetric its d e t e r m i n a n t is positive and therefore one m a y disregard the absolute value in (19). Equating to zero the result that we have o b t a i n e d for the partition function one finds the following relation: E
OilVi2...Ol2d_lV,2 d
a
a
=0,
(21)
where the sum extends over all the critical points o f V i. Notice that all d e p e n d e n c e on 2 has nicely cancelled. This can be written in terms o f the pfaffian associated to the a n t i s y m m e t r i c matrix H. Let us define Pf(H)=
(-I)"
2ad!
E
t l i2...i2d
Hi~,2H,3t4...H,2a_,i2a.
(22)
Let us now consider the case in which Z = T 2 and the target manifold M has isometries whose Killing vectors possess isolated singular points. As before we will consider maps which are h o m o t o p i c a l l y trivial. In this situation one has 2dz' zero modes and 2dp~ zero modes. Let us compute the partition function. If one sets the Killing field to zero, standard arguments show that this partition function is just the Euler n u m b e r o f the manifold M. F o r the torus, the computation o f the partition function is equivalent to a trace over the o p e r a t o r ( - 1 )F where F denotes the ghost number. On the other hand, since states and observables are in one-to-one correspondence and the observables are just in one-to-one correspondence with the cohomology classes o f M, one has to compute the trace o f ( - 1 )F just taking care of the ghost n u m b e r and the dimensions o f the cohomology groups o f M. But this is easily taken into account since according to (14) the ghost n u m b e r is just the degree o f the corresponding form. Therefore, ( - 1 )F just reduces to a sum over Betti numbers with alternate signs which indeed is the Euler n u m b e r o f manifold M. According to our ongoing discussion we can now turn on the Killing field into the acticn with no effect on the value o f the partition function. C o m p u t i n g the resulting functional integral will provide an expression for the Euler n u m b e r o f M which involves the Killing vector field. As before, we will carry out the c o m p u t a t i o n in the large 2 limit. The discussion is analogous to the previous one except for the fact that now one has to take care o f the p~ zero modes. The calculation o f the contribution to the partition function is similar to the one carried in (17) after one expands p~ into a basis o f c o h o m o l o g y one forms on the torus. One finds f
Then (21 ) can be written as
i J ~2dI d p ~ e x p ( ~ g1 ¢~flp~p~D, Vs)
i=1 --
since one has det ( H ) = P f ( H ) 2 .
(24)
It is easy to confirm the result (23) for the case in which M is just S 2. Eq. (21 ) is telling us that the curls o f the Killing fields on the north and the south poles have opposite signs. 106
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1
d!4 dei'i2"'i2d c~i, Vi2
"'" al2d
I V/2d"
(25)
One could think that there might be other contributions coming from terms in the action like the one involving the curvature on M. A simple analysis shows that those contributions are suppressed in the large 2 limit. Notice that the leading contribution in 2 is the one in which all powers o f 2 cancel as before. This contribution is just the one resulting after taking
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[ Q, H~] = - i D . X i -ie~zJ~j DnxS - iF~kxJH~
into account (17), (19) and (25): F(dI]-22 [ \ ~'" ]
-2d ~ l" 12...12d~llJ2..,J2d
L
- lie,/S(DkJ~)zk(OisxJ-~/[JJm O~,x'")
X0,,V,:...0i~, , V,~,0,, ~=...0s=, , ~ ]
,
(26)
_1a
where, as before, the sum extends over all the critical points of V i. Using (22) and (24) this expression can be rewritten in a more familiar form which is in fact a standard result in P o i n c a r d - H o p f theory [ 19,20] when the vector field is Killing: z ( M ) = ~ singular points of V i .
(27)
We hope these explicit computations have given a flavour of the kind of results which can be obtained using topological sigma models with potential terms. It would be very interesting to analyze which kinds of invariants are generated when computing other observables. One could try to obtain these results as the Bott theorem [21,22] for the particular case in which the manifold is almost hermitian. Let us now discuss possible extensions of the formulation we have constructed when the manifold M has more than one isometry. Consider a new Killing vector field U i satisfying (12) whose Lie bracket with the previous one vanishes, V j 0sU i - U s 8 i V ' = 0 .
(28)
There seems to exist only one possible extension of the theory we have constructed which does not reduce to a redefinition of the Killing field. The idea is to change the Q-transformations ( 11 ) such that V'--+ V i - U i while the Killing field in the first line of ( 13 ) is changed such that Vi--+ V ' + U i. It is straightforward to realize that since U i also satisfies (12), the resulting Q-algebra, [Q, x i] = i z * ,
{Q, z i } = - i ( V i - U
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') ,
{ O, p', } = H g + Oc,x' + EjsJ!~ Olsx ' --iF~.ZJpk. + ½i e J x ~ p ~ D~J!) ,
(29)
, ,.k Z........ tr~ r, j ), 4Z Po~kL"k"
(D,,,jJ ) + ½ie PxkH% OkJ'j
+ D k ( V ' - Ui)p• - ½E,n( V k - Uk)pSn DkJ's, (29 cont'd) closes in such a way that
[Q2, x ' ] = V i - U ' , [Q2, xi] = OJ( V ' - Ui)x j ,
[Q2, p~] = Oj ( V ' - U i ) p ~ , [Q2, H'.] = cq,( V ' - U ' ) H ~ .
(30)
On the other hand, after using (28) one finds that indeed iGv( V~+ Ui)x J has a vanishing Lie bracket with respect to V ~- U ~ and therefore one obtains a suitable topological action,
Z
+ i22Gv ( Vi+ U')Z-'] }
= f x / g {½Gog °
- ig"~G#p'. [ D/sZJ+ ½( D k J / ) z k e n ~'O~,xi] _ ~4g.a[GaH~HJn+ 2, 1oX u k m P.o,t .l J, f l A ~kA.... - 14(DkJ,,,)( D,,,J p)p ~,p~,~j(" ] -I- X2G,j( V i V a - U i U `) --)L2xixiDi ( Vj + L,!,.) - ~4g"apZpJ/~Di( V j - U:)}.
(31)
Notice that in this construction one is forced to have opposite signs for the quadratic terms in the Killing vector fields. One would have to take into account this fact when considering large 2 limits since one may get divergent contributions. The topological sigma models presented in this paper for the case in which almost hermitian manifolds possess isometries can be applied to analyze the quantum cohomology associated to these manifolds. We hope that this formulation may provide a useful tool in this respect. Another interesting line o f research regarding topological sigma models concerns 107
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the case in w h i c h the m a n i f o l d M has discrete symmetries. F o r e x a m p l e , one c o u l d t h i n k o f a p p l y i n g it to study p r o p e r t i e s o f C a l a b i - Y a u m a n i f o l d s like duality ( o r m i r r o r p a i r s ) [ 2 3 ] , w h i c h h a v e b e e n recently c o n j e c t u r e d [24] to be related to the q u a n t u m c o h o m o l o g y o f t o p o l o g i c a l sigma models. O n e w o u l d like to exploit the p r e s e n c e o f discrete s y m m e t r i e s in c o m p u t i n g the o b s e r v a b l e s o f the t h e o r y in a s i m i l a r spirit to the o n e p r e s e n t e d h e r e for the case o f c o n t i n uous symmetries. Discussions with L. A l v a r e z - G a u m 6 , M. Atiyah and A.V. R a m a l l o are gratefully a c k n o w l e d g e d . P.M.L. w o u l d like to t h a n k the B a n c o E x t e r i o r de Esp~na for financial s u p p o r t a n d the D e p a r t a m e n t o de Fisica de Particulas de la U n i v e r s i d a d e de Santiago, w h e r e this w o r k was carried out, for its hospitality.
References [ 1] E. Witten, Nucl. Phys. B 340 (1990) 281. [2] J. Distler, Nucl. Phys. B 342 (1990) 523. [3] J.M.F. Labastida, M. Pernici and E. Witten, Nucl. Phys. B 310 (1988) 611. [ 4 ] D. Montano and J. Sonnenschein, Nucl. Phys. B 313 ( 1989 ) 258. [5] K. Li, Topological gravity with minimal matter, Caltech preprint CALT-68-1662; Recursion relations in topological gravity with minimal matter, Caltech preprint CALT-681670. [6] T. Eguchi and S.-K. Yang, N= 2 superconformal models as topological field theories, Tokyo preprint UT-564. [7] R. Dijkgraafand E. Witten, Nucl. Phys. B 342 (1990) 486.
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[8] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 348 (1991) 435. [ 9 ] E. Verlinde and H. Verlinde, Nucl. Phys. B 348 ( 1991 ) 457. [10] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 352 (1991) 59. [ 11 ] E. Witten, Commun. Math. Phys. 117 (1988) 353. [12] L. Alvarez-Gaum6 and D.Z. Freedman, Commun. Math. Phys. 91 (1983) 87. [13] S.J. Gates, C.M. Hull and M. Ro~ek, Nucl. Phys. B 248 (1984) 157. [ 14 ] J.M.F. Labastida and P.M. Llatas, Topological matter in two dimensions, Santiago preprint. [ 15 ] T. Buscher, U. Lindstrom and M. Ro~zek,Phys. Lett. B 202 (1988) 94. [16] E. Witten, Commun. Math. Phys. 118 (1988) 411. [ 17] C. Vafa, Mod. Phys. Lett. A 6 ( 1991 ) 337. [ 18 ] E. Winen, Two dimensional gravity and intersection theory on moduli space, Harvard Lecture Notes, IAS preprint IASSNS-HEP-90/45 (1990). [19] Y. Choquet-Bruhat, C. DeWitt-Morette and M. DillardBleick, Analysis, manifolds and physics (North-Holland, Amsterdam, 1982 ). [20] S. Chern, Complex manifolds without potential theory (Springer, Berlin, 1979). [21 ] P.F. Baum and J. Cheeger, Topology 8 (1969) 173. [22] S. Chern and J. Simons, Proc. Natl. Acad. Sci. (USA) 68 (1971) 791. [23] W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B 324 (1989) 427; P. Candelas, M. Lynker and R. Schimmrigk, Nucl. Phys. B 341 (1990) 383; B.R. Greene and M.R. Plesser, Nucl. Phys. B 338 (1990) 15; P. Aspinwall, A. Lutken and G.G. Ross, Phys. Lett. B 241 (1990) 373. [24] M. Atiyah, talk London Mathematical Society Durham Research Symp. on Conformal field theory ( 1991 ).
P I-4YSI C S L ETT ER S B
Potentials for topological sigma models J.M.F. Labastida Departamento de Fisica de Particulas, Universidadede Santiago, E-15706 Santiago de Compostela, Spain
and P.M. Llatas Departamento de Fisica Te6rica, UniversidadAut6norna de Madrid, E-28049 Madrid, Spain Received 9 August 1991
Topological sigma models with potential terms are obtained by twisting N=2 supersymmetric sigma models. These models exist for manifolds with isometries and potential terms in their actions are built out of the corresponding Killing vector fields. It turns out that, as ordinary topological sigma models, the models resulting after twisting N=2 supersymmetry can be generalized to the case of almost hermitian manifolds.
Topological m a t t e r in two d i m e n s i o n s have been shown to play an i m p o r t a n t role in relating topological q u a n t u m field theories to m u l t i - m a t r i x models. The connection discovered by Witten [ 1,2 ] between topological gravity in two d i m e n s i o n s [3,4 ] and matrix models has been generalized to the multi-matrix case [ 5 ] by coupling topological gravity to some specific kind of topological m a t t e r [6 ]. F u r t h e r study o f this connection has been carried out recently [ 7 - 1 0 ]. On the other hand, the study o f topological m a t t e r may provide some insight on the structure o f a class of theories which presumably posses a phase where gravitons are composite particles. Topological m a t t e r in two d i m e n s i o n s provides a rich variety o f two-dimensional topological theories which seem to be closely related to their non-topological counterparts, at least for some specific cases [6,5 ]. We have started a systematic study of the most general form of topological m a t t e r in two d i m e n sions. Our strategy is the following. First we consider an N = 2 supersymmetric model. We then twist it [ 1 1 ] and extend it to the most general possible situation. We have started our analysis with N = 2 topological sigma models of the type considered in ref. [ 12], i.e., E-mail address: [email protected].
N = 2 supersymmetric sigma models with potential terms whose target space is a K~ihler manifold. More general situations [ 13] should also be considered in this framework. F o r example, one would expect that in those situations one might construct a topological version o f W Z W models. In this letter we report on part of our results concerning this analysis on topological matter. A detailed account of the procedure involved in the construction o f these theories will be reported elsewhere [ 14 ]. In N = 2 superspace in two d i m e n s i o n s there exist two basic multiplets, the chiral multiplet, and the twisted chiral multiplet [ 13 ]. Once a precise form of the twist o f N = 2 supersymmetry is chosen, each multiplet leads to a different type o f topological matter [ 14]. Let us consider the s t a n d a r d N = 2 superspace action,
S = f d2zd4OK(X/, X r) Z
z
where I; denotes a Riemann surface, and X ~, X 1 ( I = 1, .... d) are local coordinates of a 2d-dimensional corn-
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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pact K~ihler manifold M whose K~ihler potential is K. The superpotential W ( X ~) is a holomorphic scalar function on M. The local coordinates X t ( X r) may be regarded as chiral or twisted chiral (antichiral or twisted antichiral) N = 2 multiplets. O f course, the 0measure of the F-term in ( 1 ) depends on the kind of multiplet one is considering. In ref. [14] a detailed analysis of the resulting models after twisting ( 1 ) is carried out when only a single type of multiplet appears in the action. It would be very interesting to perform a similar analysis for the case in which one has a mixture of these types of multiplets. One could ask similar questions for the recently discovered N = 2 semichiral multiplet [ 15 ]. Twisting N = 2 supersymmetry in ( 1 ) may lead to twisted actions which are not Lorentz invariant. As is shown in ref. [ 14], for chiral multiplets this is indeed the case unless one does not introduce the Fterm in (1). The resulting theory is a topological sigma model [ 16 ] for the specific case in which the target manifold M is a K~ihler manifold. Of course, no potential terms for topological sigma models seem to appear in this way. While the original N = 2 supersymmetric action cannot be extended to a more general type of manifold, Witten showed [16] that in the topological case the theory can be generalized to the case of almost hermitian manifolds. Before describing how potential terms can be introduced for these models, let us briefly comment on the models resulting after twisting the other type o f multiplet. Twisting a model containing twisted chiral multiplets allows F-terms and therefore one obtains potential terms automatically. The resulting models are a topological version of N = 2 L a n d a u - G i n z b u r g models. For the case of a flat target manifold these models have been obtained in ref. [ 17 ]. Conformal cases of these models correspond to the ones discussed in ref. [ 6 ]. As shown in ref. [12], there exist some N = 2 supersymmetric sigma models which are not contained in the N = 2 superspace formulation (1). These models involve new types of potential terms and their corresponding supersymmetric algebra possesses central charges. Twisting N = 2 supersymmetry for these models leads to the opposite situation to the one observed in the twist o f ( 1 ). Namely, twisted chiral multiplets lead to actions which are not Lorentz invariant while chiral ones lead to Lorentz invariant 102
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actions. Let us therefore consider chiral multiplets but first let us introduce some notation. For future convenience we will depart from the holomorphic notation used in ( 1 ). However, one has to keep in mind that for the m o m e n t we are considering K~ihler geometry, i.e., the metric on M, Gij has only non-vanishing components GIj, and the complex structure J5 ( J * f l S = - a } ) satisfies D a J ) = 0 . Notice that lower case Latin indices run from 1 to 2d, the real dimension of M. The action resulting after carrying out the twist of N = 2 supersymmetry takes the form S-- f d Z z ~
( ½Gog '~p O~x i O~xJ+ ½~al~JO-Oax" O,~Xj
Z l ~O~[3D ~i nJ ~k,.m - ig'~G,jp'. DaZ j - g6 *.o~,,,v,~t'~z x
+ G o V g V J - z ' z J D , 15-lg'~Pp~papD, Vj) ,
(2)
where g ~ and a~p (e~yeY~= - ~ ) denote the metric and complex structure chosen on the Riemann surface Z. In (2), x i ( i = 1..... 2d) denote local coordinates on M, while p~ and Z' ( i = 1, ..., 2d) are anticommuting fields which are sections of the pullback 0 * ( T ) of the tangent bundle T to M. The anticommuting fields p~, have acquired after the twist a vector index a = 1, 2, which is a tangent index to E. p " ( i = 1, ..., 2d) are one forms on Y. with values in ~*(T) which satisfy the self-duality condition p~ = ~,~J'jp~. In (2) D~ denotes the pullback covariant derivative, D M ( = 0 , Z ~ + c~,~xJF'jkZk. Finally V' is a holomorphic Killing vector on M, D, Vj + D j V, = 0 ,
J " J " j D , , , V , , = D , Vj.
(3)
The action (2) is obtained after performing the twist o f N = 2 supersymmetry in the action corresponding to eq. (50) of ref. [ 12] and setting the quantities h and G ~to zero. As we have discussed, h ¢ 0 would lead to an action which is not Lorentz invariant. However, we could have kept the vector field G' in ref. [ 12 ] all along the twisting. Let us not do that for the m o m e n t and let us postpone for later the discussion of additional vector field structures. Twisting N = 2 supersymmetry also provides the Qtransformations of the fields [ 14 ]. They turn out to be
[Q, x q = i z ' ,
{Q,x'}=-iv',
(4)
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{Q, pl}=a,~xi+e,~J(/a~xJ--iE~kZJpk,.
(4 cont'd)
Since the N = 2 supersymmetric model is an on-shell model in the sense that the supersymmetric algebra is realized after using the field equations of motion, we expect (4) to possess the same feature. Indeed, this is the case. Furthermore, since the N = 2 supersymmetric algebra in ref. [ 12 ] has central charges, we do not expect Q to be nilpotent. A simple calculation shows that, after making use of the field equations,
[Q2, xi]=Vi,
[Q2,z']=O:VizJ ,
[QZ, pi] =O:V' p~ ,
(5)
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Furthermore, one can verify that by simply adding the term
- ~ ~ x/-gg"aGijHiHJ/~
(9)
y
to the action (2), it becomes invariant under the first two transformations displayed in (4), and the transformations (6) and (7). Notice that the term (9) manifests the auxiliary nature of the field H i . Now that we possess an off-shell formulation we are in the position to analyze easily if we have an action which is Q-exact. It turns out that indeed, the action (2) plus the term (9) can be written as
i.e., Q2 acts as a Lie derivative with respect to the holomorphic Killing vector field V ~. The fact that Q2 ¢ 0 does not mean that the theory is not topological. In order for the theory under consideration to be topological with respect to the metric chosen in Z the only requirement is that the energy-momentum tensor be Q-exact. For the theory being topological with respect to target space quantities, the action itself needs to be Q-exact. It is simple to find out that, indeed, the energy-momentum tensor corresponding to (2) is Q-exact. To analyze if the action is Q-exact let us first reformulate the model off-shell. The natural way to carry out this reformulation is to introduce auxiliary fields for fields such that Q2 closes only after using the field equations. This is the case for the field p~. Let us redefine the Q-transformation o f p l as
{Q, pl}=Hi +O~xi+e~J~O~xJ--iF~kZ:p~.
(6)
Consistency implies that the auxiliary field H i must be also self-dual, H i = eJJ(/H~. The Q-transformation of H i is easily worked out by demanding that the new transformation o f p l (6) satisfies (5) without using the field equations. One finds [ Q, H i ] ~ _ _ --
i D.Z' - ie.aJ/j D~Zj
~o 21"~mj ~ k Z..... . Z. j~k ,t"c~ + D k V i p ~
1F/k)(. Ha "
-
t
j
k
(7)
To check that we are on the right track one may compute Q2 acting on H i to verify that indeed now the algebra closes off-shell,
[QZ, HI]=OjVi H~.
(8)
{Q' t xfg
[½g~/JG,jpi(O/~x:-½H~)+iGoVizQ}.
Z
(lO) This means that the theory is topological with respect to the target space. Any change in the metric Gij or the complex structure J~j can be written as a Q-exact quantity and therefore, following the general arguments described in ref. [ 11 ], the vacuum expectation values of the observables of the theory remain invariant under those changes. Notice also that the same arguments imply that the Killing vector field entering (10) can be deformed without having any effect on the observables. We will exploit this property later on when discussing observables. After finding a property like (10) we could ask about other possible terms to add. Indeed, one could introduce Q acting on any quantity as long as its Lie derivative with respect to the holomorphic Killing vector field vanishes. However, the three terms entering ( 10 ) are such that they lead to standard potential and kinetic terms in quantum field theory and we will restrict ourselves to them. Having an argument to keep only three types of terms in (10) we may ask about their relative numerical coefficients. The coefficient between the first and the second is the correct one for keeping H i as auxiliary fields. The coefficient of the third one is, however, undetermined. Actually, in ref. [ 12 ] there is a parameter m which in the twisting we have set to 1. This term would have entered in front of the third term of (10). At first sight, one could think that a coefficient 22 in front of the third term could be reabsorbed by a redefinition of the poten103
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tial, V ' - , 2 - J V ' plus the change of variables X;--" ~i "11/2~i )~ I/2Z',• p,--,.~ p , , and therefore it would just account for the scale of the potential. However, this is not entirely true because the path integral measure is in general not invariant under such a change in Z' and p',. Typically the amounts of X; zero modes and p~ zero modes are different and one has therefore a contribution from the measure. Thus, we have to consider the general situation in which there is a constant parameter multiplying the third term in (10). We will introduce it in our discussion of the generalized model. Before carrying out the analysis of observables and some of the consequences of the theory let us try to generalize it to the case in which the target manifold M is almost hermitian. This means that now the complex structure J~ is not covariantly constant but still the metric is such that u~i j - - _rk~,,, Our strata ia j tc:_ Jkm. egy will be first to construct the Q-algebra for this case and then to consider an action similar to the one in (10). For the m o m e n t we will not assume that the vector field V; satisfies conditions (3). Let us keep it for the m o m e n t general. Guided by the construction in ref. [ 16 ] we leave the transformations of x; and Z; as they stand. The transformation ofp~, in (6) is consistent with the self-duality condition satisfied by p~ when the complex structure f ; is covariantly constant. If one checks what fails when J'j is not covariantly constant one finds that simply by adding a term of the form ~ J f f p ~ DkJ~ to the transformation of p~ the consistency condition is satisfied. The transformation o f H i is then defined similarly as we did before, by requiring consistency of the Q-algebra. One finds
D; Vj + D j V, = 0 , V k OkJin + Jik On V k -- Jkn Ok V i = 0 .
( 12 )
These conditions are rather natural. They are just the requirement that the metric and complex structure remain invariant under a variation along the vector field V;. The first one constrains V ~to be Killing while the second one just tells us that the Lie derivative of the complex structure with respect to V i vanishes. Notice that in the case in which the manifold becomes K~ihler the second condition just reduces to the second equation in (3), i.e., the Killing vector becomes holomorphic. Notice that in this construction the non-trivial check is to verify if the Q-algebra closes for H i when there are potential terms. When the potential terms are set to zero that is known to be the case [ 16 ]. Now that we have constructed the Q-algebra let us build up the action. We will take (10) with an arbitrary parameter 2 z multiplying the third term,
S={Q, f x/g [ ½g~PG,jp'.(O/~xJ-½H~)+ i22GoV;zJ]} Z
----
( 2 ~Uo
~o¢ "~
Opx-'
Z
- ig"/JG,jp', [ D ~ Z ; + ½( DkJ/)zkE/7 0yx;] --
l c ~ O Z f l l - r . L / I L / J ..I[ ' d ij . . . . . . fl --
4d~
½R,jk,,~p',p~ff Z'"
-22Z;Z j D; V; - ¼gO~p~,pj/~ D; Vj}.
{Q,z'}=-iv;, {Q, p~ } = H', + 0 , x ; + e,fJ'j O/jxj • , ~p~k + ~l~. i. ~Z kP~i DkJ'j, -1F'j~Z
- ½R,,v;~.'"XJp~ + ½e,fZ'")ffp~l~ D,,, Dx.J'j - ½i e J ( D~,jij )Z k ( Oax j - eaYJi,,, Oyx '" ) 1 k tn n + ~Z Z p ~ ( D k J ji) (D,~J/~) + ½iej~ffH~ DkJ'i
104
Imposing that the action o f Q 2 on H i be the right one to close the algebra, i.e. (8), one finds after a somewhat tedious calculation that V; must satisfy the following conditions:
/ kZ m ] +22Gu V i V j -- ~(DkJip) (D,,,J pj)Po,, P~Z
[ Q , x ; ] = i z i,
+ D , V;pf~ -- ½eofV*pJ~ D , J ' j .
14 November 1991
( 11 )
(13
The action we have obtained coincides with the one given in ref. [ 16 ] when V' is set to zero. Our next task is to analyze the observables of this theory, i.e., the operators which are annihilated by Q. One immediately realizes that these operators are the same as in ref. [ 16 ] with the additional condition that the closed forms entering in them must be orthogonal to the Killing vector field V;. This is in fact rather natural since the non-trivial topology of the manifold M is encoded on the directions orthogonal to V;. No-
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rice also that we can a d d any observable Q o f any quantity which is invariant u n d e r the Lie derivative with respect to V( Let us display here, for example, the observables corresponding to zero forms (on Z ) . These take the form [16] ~A =A,i2...i,,Zi'Z ~2 ... Z i" ,
( 14 )
where Aaa...,, is an a n t i s y m m e t r i c tensor corresponding to a closed form which is ortbogonal to the Killing vector field V e, d A -~ 0
,
V/A/i2...i, = 0 .
( 15 )
The forms corresponding to the Q-descendants of (14) are like the ones in ref. [ 16]. Let us analyze the v a c u u m expectation value of some observables to make manifest the consequences o f the existence o f a theory like the one constructed. In dealing with functional integral computation it is i m p o r t a n t to consider the presence o f zero modes. In topological q u a n t u m field theories where the ghost n u m b e r o p e r a t o r c o m m u t e s with Q, one finds typically a n o m a l o u s contributions from the functional integral measure due to an unbalance between p~ zero modes and Z i zero modes. In the present theory ghost n u m b e r is not a good s y m m e t r y due to the presence of potential terms. However, the difference b e t w e e n p ~ zero modes and Z i zero m o d e s is also i m p o r t a n t because it is sensitive to rescaling o f fields. This difference is given by [ 18 ] ~ Z i zero m o d e s - : ~ p ~ zero modes
=2d(1-g)
+ J X*Cl(M),
(16)
Z
where x represents a m a p o f the h o m o t o p y type under consideration and c~ ( M ) is the first Chern class of the manifold M. By x * c ~ ( M ) we mean the pullback o f the two form on M associated to the first Chern class. Recall that 2d is the real d i m e n s i o n o f M. Since the action (13) is Q-exact we m a y deform the Killing vector field V ~ or change the value o f 2 without making any effect on the observables. In particular, one could set the Killing vector field to zero. In that case the theory just becomes an o r d i n a r y topological sigma model. But one could also do the opposite. Starting with a topological sigma model whose
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target manifold possesses some isometries we could introduce one o f the corresponding Killing vector fields and perform the calculation o f the observables o f the original model. To be able to deform V ~ (as long as it is Killing) and to change the value o f 2 with no effect on the observables m a y have some advantages regarding computations. For example, one could take the large 2 limit reducing the calculation to a sum over the critical points of the Killing field. Analogously, one can find interesting relations regarding the Killing vector field by considering the v a c u u m expectation value o f an observable which is manifestly zero when there is no Killing field in the action. Let us consider for example the case in which Z = S 2 and the maps x i are h o m o t o p i c a l l y trivial. In this situation the n u m b e r o f z ~zero modes is 2d (the constants) while there are no p~ zero modes. In ordinary topological sigma models this implies that one has to consider operators whose ghost n u m b e r is 2d to be able to have a possibly non-vanishing vacuum expectation value. In particular, the partition function is zero. Let us assume that the m a n i f o l d M under consideration admits at least a Killing vector with isolated singular points and let us compute such a partition function. Equating the result to zero will provide some relations which must be satisfied by any Killing field o f the kind under consideration. We will compute the partition function in the large 2 limit. F r o m the bosonic piece of the action (13) one observes that the path-integral gets d o m i n a t e d by configurations where V i is singular ( V ' = 0 ) and x ~ satisfies the instanton equation 0~x'+ e~PJ} 0z~xJ= 0. For the kind o f maps under consideration these are the constant configurations. Near each o f these configurations the path-integral for the non-zero modes becomes simply the ratio of fermionic and bosonic determinants which cancel. Therefore, we are left with an integration over zero modes and a finite sum over singular points of V£ Since there are only Z ~ zero m o d e s for the case under consideration, we have just to consider the term o f the action involving 22Z~Z j D~I~. The integration over anticommuting zero modes is easily carried out, 1-I d z ' exp(22Z'Z j Ds ~ ) 1
-
-
d! ~ 2d(t ....... 2d O, 1 ri2 ... O. . . . g,2, ,
(17) 105
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where the e-symbol is normalized such that e 12.,.2d= 1. On the other hand, the integration over the constant modes o f the bosonic fields leads to
f ~_~_dx'exp(-Z2G, V'VQ,
(18)
which, in the large 2 limit, after replacing V s by its linear term, reduces to the evaluation o f a gaussian,
; 2.=I~ 1 d.x t e x p ( - 2 2 G , j ,_
V~Vj) = ~ - 2 d
1
Id e t ( H ) I '
(19)
where H is a 2 d × 2d matrix defined as
(20)
H,=a,V,.
In obtaining (19) one has to keep in mind that the argument o f the exponent is being e x p a n d e d a r o u n d a singular point o f V i. Notice also that for almost hermitian manifolds as the ones under consideration ( 18 ) is well defined. Since the matrix H is antisymmetric its d e t e r m i n a n t is positive and therefore one m a y disregard the absolute value in (19). Equating to zero the result that we have o b t a i n e d for the partition function one finds the following relation: E
OilVi2...Ol2d_lV,2 d
a
a
=0,
(21)
where the sum extends over all the critical points o f V i. Notice that all d e p e n d e n c e on 2 has nicely cancelled. This can be written in terms o f the pfaffian associated to the a n t i s y m m e t r i c matrix H. Let us define Pf(H)=
(-I)"
2ad!
E
t l i2...i2d
Hi~,2H,3t4...H,2a_,i2a.
(22)
Let us now consider the case in which Z = T 2 and the target manifold M has isometries whose Killing vectors possess isolated singular points. As before we will consider maps which are h o m o t o p i c a l l y trivial. In this situation one has 2dz' zero modes and 2dp~ zero modes. Let us compute the partition function. If one sets the Killing field to zero, standard arguments show that this partition function is just the Euler n u m b e r o f the manifold M. F o r the torus, the computation o f the partition function is equivalent to a trace over the o p e r a t o r ( - 1 )F where F denotes the ghost number. On the other hand, since states and observables are in one-to-one correspondence and the observables are just in one-to-one correspondence with the cohomology classes o f M, one has to compute the trace o f ( - 1 )F just taking care of the ghost n u m b e r and the dimensions o f the cohomology groups o f M. But this is easily taken into account since according to (14) the ghost n u m b e r is just the degree o f the corresponding form. Therefore, ( - 1 )F just reduces to a sum over Betti numbers with alternate signs which indeed is the Euler n u m b e r o f manifold M. According to our ongoing discussion we can now turn on the Killing field into the acticn with no effect on the value o f the partition function. C o m p u t i n g the resulting functional integral will provide an expression for the Euler n u m b e r o f M which involves the Killing vector field. As before, we will carry out the c o m p u t a t i o n in the large 2 limit. The discussion is analogous to the previous one except for the fact that now one has to take care o f the p~ zero modes. The calculation o f the contribution to the partition function is similar to the one carried in (17) after one expands p~ into a basis o f c o h o m o l o g y one forms on the torus. One finds f
Then (21 ) can be written as
i J ~2dI d p ~ e x p ( ~ g1 ¢~flp~p~D, Vs)
i=1 --
since one has det ( H ) = P f ( H ) 2 .
(24)
It is easy to confirm the result (23) for the case in which M is just S 2. Eq. (21 ) is telling us that the curls o f the Killing fields on the north and the south poles have opposite signs. 106
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1
d!4 dei'i2"'i2d c~i, Vi2
"'" al2d
I V/2d"
(25)
One could think that there might be other contributions coming from terms in the action like the one involving the curvature on M. A simple analysis shows that those contributions are suppressed in the large 2 limit. Notice that the leading contribution in 2 is the one in which all powers o f 2 cancel as before. This contribution is just the one resulting after taking
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[ Q, H~] = - i D . X i -ie~zJ~j DnxS - iF~kxJH~
into account (17), (19) and (25): F(dI]-22 [ \ ~'" ]
-2d ~ l" 12...12d~llJ2..,J2d
L
- lie,/S(DkJ~)zk(OisxJ-~/[JJm O~,x'")
X0,,V,:...0i~, , V,~,0,, ~=...0s=, , ~ ]
,
(26)
_1a
where, as before, the sum extends over all the critical points of V i. Using (22) and (24) this expression can be rewritten in a more familiar form which is in fact a standard result in P o i n c a r d - H o p f theory [ 19,20] when the vector field is Killing: z ( M ) = ~ singular points of V i .
(27)
We hope these explicit computations have given a flavour of the kind of results which can be obtained using topological sigma models with potential terms. It would be very interesting to analyze which kinds of invariants are generated when computing other observables. One could try to obtain these results as the Bott theorem [21,22] for the particular case in which the manifold is almost hermitian. Let us now discuss possible extensions of the formulation we have constructed when the manifold M has more than one isometry. Consider a new Killing vector field U i satisfying (12) whose Lie bracket with the previous one vanishes, V j 0sU i - U s 8 i V ' = 0 .
(28)
There seems to exist only one possible extension of the theory we have constructed which does not reduce to a redefinition of the Killing field. The idea is to change the Q-transformations ( 11 ) such that V'--+ V i - U i while the Killing field in the first line of ( 13 ) is changed such that Vi--+ V ' + U i. It is straightforward to realize that since U i also satisfies (12), the resulting Q-algebra, [Q, x i] = i z * ,
{Q, z i } = - i ( V i - U
14 November 1991
') ,
{ O, p', } = H g + Oc,x' + EjsJ!~ Olsx ' --iF~.ZJpk. + ½i e J x ~ p ~ D~J!) ,
(29)
, ,.k Z........ tr~ r, j ), 4Z Po~kL"k"
(D,,,jJ ) + ½ie PxkH% OkJ'j
+ D k ( V ' - Ui)p• - ½E,n( V k - Uk)pSn DkJ's, (29 cont'd) closes in such a way that
[Q2, x ' ] = V i - U ' , [Q2, xi] = OJ( V ' - Ui)x j ,
[Q2, p~] = Oj ( V ' - U i ) p ~ , [Q2, H'.] = cq,( V ' - U ' ) H ~ .
(30)
On the other hand, after using (28) one finds that indeed iGv( V~+ Ui)x J has a vanishing Lie bracket with respect to V ~- U ~ and therefore one obtains a suitable topological action,
Z
+ i22Gv ( Vi+ U')Z-'] }
= f x / g {½Gog °
- ig"~G#p'. [ D/sZJ+ ½( D k J / ) z k e n ~'O~,xi] _ ~4g.a[GaH~HJn+ 2, 1oX u k m P.o,t .l J, f l A ~kA.... - 14(DkJ,,,)( D,,,J p)p ~,p~,~j(" ] -I- X2G,j( V i V a - U i U `) --)L2xixiDi ( Vj + L,!,.) - ~4g"apZpJ/~Di( V j - U:)}.
(31)
Notice that in this construction one is forced to have opposite signs for the quadratic terms in the Killing vector fields. One would have to take into account this fact when considering large 2 limits since one may get divergent contributions. The topological sigma models presented in this paper for the case in which almost hermitian manifolds possess isometries can be applied to analyze the quantum cohomology associated to these manifolds. We hope that this formulation may provide a useful tool in this respect. Another interesting line o f research regarding topological sigma models concerns 107
Volume 27 l, number 1,2
PHYSICS LETTERS B
the case in w h i c h the m a n i f o l d M has discrete symmetries. F o r e x a m p l e , one c o u l d t h i n k o f a p p l y i n g it to study p r o p e r t i e s o f C a l a b i - Y a u m a n i f o l d s like duality ( o r m i r r o r p a i r s ) [ 2 3 ] , w h i c h h a v e b e e n recently c o n j e c t u r e d [24] to be related to the q u a n t u m c o h o m o l o g y o f t o p o l o g i c a l sigma models. O n e w o u l d like to exploit the p r e s e n c e o f discrete s y m m e t r i e s in c o m p u t i n g the o b s e r v a b l e s o f the t h e o r y in a s i m i l a r spirit to the o n e p r e s e n t e d h e r e for the case o f c o n t i n uous symmetries. Discussions with L. A l v a r e z - G a u m 6 , M. Atiyah and A.V. R a m a l l o are gratefully a c k n o w l e d g e d . P.M.L. w o u l d like to t h a n k the B a n c o E x t e r i o r de Esp~na for financial s u p p o r t a n d the D e p a r t a m e n t o de Fisica de Particulas de la U n i v e r s i d a d e de Santiago, w h e r e this w o r k was carried out, for its hospitality.
References [ 1] E. Witten, Nucl. Phys. B 340 (1990) 281. [2] J. Distler, Nucl. Phys. B 342 (1990) 523. [3] J.M.F. Labastida, M. Pernici and E. Witten, Nucl. Phys. B 310 (1988) 611. [ 4 ] D. Montano and J. Sonnenschein, Nucl. Phys. B 313 ( 1989 ) 258. [5] K. Li, Topological gravity with minimal matter, Caltech preprint CALT-68-1662; Recursion relations in topological gravity with minimal matter, Caltech preprint CALT-681670. [6] T. Eguchi and S.-K. Yang, N= 2 superconformal models as topological field theories, Tokyo preprint UT-564. [7] R. Dijkgraafand E. Witten, Nucl. Phys. B 342 (1990) 486.
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[8] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 348 (1991) 435. [ 9 ] E. Verlinde and H. Verlinde, Nucl. Phys. B 348 ( 1991 ) 457. [10] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 352 (1991) 59. [ 11 ] E. Witten, Commun. Math. Phys. 117 (1988) 353. [12] L. Alvarez-Gaum6 and D.Z. Freedman, Commun. Math. Phys. 91 (1983) 87. [13] S.J. Gates, C.M. Hull and M. Ro~ek, Nucl. Phys. B 248 (1984) 157. [ 14 ] J.M.F. Labastida and P.M. Llatas, Topological matter in two dimensions, Santiago preprint. [ 15 ] T. Buscher, U. Lindstrom and M. Ro~zek,Phys. Lett. B 202 (1988) 94. [16] E. Witten, Commun. Math. Phys. 118 (1988) 411. [ 17] C. Vafa, Mod. Phys. Lett. A 6 ( 1991 ) 337. [ 18 ] E. Winen, Two dimensional gravity and intersection theory on moduli space, Harvard Lecture Notes, IAS preprint IASSNS-HEP-90/45 (1990). [19] Y. Choquet-Bruhat, C. DeWitt-Morette and M. DillardBleick, Analysis, manifolds and physics (North-Holland, Amsterdam, 1982 ). [20] S. Chern, Complex manifolds without potential theory (Springer, Berlin, 1979). [21 ] P.F. Baum and J. Cheeger, Topology 8 (1969) 173. [22] S. Chern and J. Simons, Proc. Natl. Acad. Sci. (USA) 68 (1971) 791. [23] W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B 324 (1989) 427; P. Candelas, M. Lynker and R. Schimmrigk, Nucl. Phys. B 341 (1990) 383; B.R. Greene and M.R. Plesser, Nucl. Phys. B 338 (1990) 15; P. Aspinwall, A. Lutken and G.G. Ross, Phys. Lett. B 241 (1990) 373. [24] M. Atiyah, talk London Mathematical Society Durham Research Symp. on Conformal field theory ( 1991 ).