Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity
aa r X i v : . [ h e p - t h ] N ov ITFA-2007-48
Two Dimensional Kodaira-Spencer TheoryandThree Dimensional Chern-Simons Gravity
Robbert Dijkgraaf
Institute for Theoretical Physics & Korteweg-de Vries Institute for MathematicsUniversity of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands andCumrun Vafa
Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Abstract
Motivated by the six-dimensional formulation of Kodaira-Spencer theory for Calabi-Yau threefolds, we formulate a two-dimensional version and argue that this is the relevantfield theory for the target space of local topological B-model with a geometry based on aRiemann surface. We show that the Ward identities of this quantum theory is equivalentto recursion relations recently proposed by Eynard and Orantin to solve the topological Bmodel. Our derivation provides a conceptual explanation of this link and reveals a hiddenaffine SL symmetry. Moreover we argue that our results provide the strongest evidenceyet of the existence of topological M theory in one higher dimension, which in this casecan be closely related to SL Chern-Simons formulation of three dimensional gravity.November 2007 . Introduction
Topological strings have been solved in the context of local toric Calabi-Yau three-folds. In particular the topological vertex can be used to compute all genus amplitudesfor topological A-model on these spaces [1]. On the other hand, using mirror symmetry,this construction can be interpreted as providing a full solution to the B-model topologicalstring with a local Calabi-Yau geometry modelled on a Riemann surface (which we willrefer to as the local B model).There is however a more direct path to obtaining topological strings in the contextof the local B model: Matrix models are conjectured to be equivalent to the topologicalB model on a local geometry [2], where the Riemann surface is identified as the spectralcurve of the matrix model. This gives another solution to the local B-models, namely thelarge N ’t Hooft expansion of the corresponding matrix models. There has been recentspectacular progress in solving these matrix models where it has been shown that the large N description of matrix model can be directly formulated intrinsically on the Riemannsurface in terms of certain recursion relations that essentially follow from the loop equations[3]. This new approach has the advantage that it applies to any local B-model, whether ornot the spectral curve comes from a matrix model. This relation has been recently checkedin the context a number of examples [4]. In the context of the B model the approach of[3] has the remarkable feature that it automatically incorporates the holomorphic anomaly[5]: The partition function depends on the choice of A-cycles on the Riemann surface, andchoosing different basis for A-cycles leads to a generalized Fourier transform of the partitionfunction, as is expected on the basis of the general holomorphic anomaly equation of thetopological string. Usually this fact is formulated as that the topological string partitionfunction transforms as a wave function or a holomorphic block.The aim of this note is to derive the recursion relations of [3] directly in the B modelusing field theory techniques. We will demonstrate that these recursion relations are givenin terms of Ward identities of the B-model field theory, which is the restriction of theKodaira-Spencer theory on the Riemann surface. Quite surprisingly, while proving theserecursion relations, we uncover an SL (2 , R ) current algebra. In this setup, the fact thatthe partition function becomes a wave function is directly related to the fact that one isdealing with a chiral boson on the Riemann surface as the basic field of the gravity, and itis known that this partition function does depend on the choice of a basis for the A-cycles.In fact the best understanding of this phenomenon comes from the interpretation of the1ave function as a state for the three dimensional Chern-Simons theory. Here we alsospeculate about the existence of a topological M-theory, whose restriction to the local casesuggests the existence of a three dimensional gravity theory, which leads to the topologicalstrings as a quantum state. This could potentially explain the appearance of SL (2 , R )current algebra on the Riemann surface.The organization of this paper is as follows: In section 2 we describe the basic setupand formulate the relevant 2d KS theory. In section 3 we show that the Ward identities ofthis theory are the same as those written in [3], and uncover an SL (2 , R ) symmetry. Insection 4 we speculate about embedding this symmetry in one higher dimension.
2. The Basic Setup
Kodaira-Spencer theory is the string field theory of the topological B-model on Calabi-Yau threefolds [6]. This theory can be considered as the quantization of the ∂ operatoron the Calabi-Yau manifold with a fixed complex structure, as captured by a holomorphic(3 ,
0) form Ω. More precisely, it is the quantization of the cohomologically trivial variationsof ∂ , which do not change a fixed background complex structure. Thus the theory is definedin terms of a pair ( ∂, Ω). It is not known, at the present, how to use this formalism tosolve all genus amplitudes for compact Calabi-Yau manifolds. One can compute, however,low genus amplitudes using this approach.As we will see, the situation is much better for the local non-compact threefold modeledon a curve Σ. In the context of these local geometries it is natural to look for a reductionof this structure to the Riemann surface Σ and directly quantize that system. This is theapproach that we will follow in this note.By the local case we shall mean a non-compact Calabi-Yau threefold defined by thehypersurface: vw = H ( x, y ) , where v, w belong to C and x, y belong to C or C ∗ . (To be precise, in the latter casethe appropriate coordinates are e x , e y ∈ C ∗ or, equivalently, x, y ∈ C / πi Z .) In thesecoordinates the holomorphic three-form Ω is given byΩ = dvv ∧ dy ∧ dx. H ( x, y ) = 0 . It is not difficult to see that the periods of the (3 ,
0) form Ω on the local threefold can bereduced, upon integration to the x - y plane, to the integral of the one-form ω = ydx on the Riemann surface. We thus wish to define the quantum Kodaira-Spencer theory ofthe pair ( ∂, ω ), on the Riemann surface Σ given by H ( x, y ) = 0. More precisely, we wishto integrate over all deformation of ∂ which do not affect the cohomology class of ω , justas was the case for the 3-fold case.The variation of the complex structure is captured locally by the deformation ∂ → ∂ − µ∂, where the Beltrami differential µ is a tensor of type µ = µ zz dz ⊗ ∂ z . For the variation to be globally trivial, it means that there is a diffeomorphism by a vectorfield v = v z ∂ z such that µ zz = ∂ z v z . We are interested in quantizing these deformations, while maintaining the cohomologyclass of ω . As such, it is natural to formulate the variation of the ∂ in terms of its actionon ω . The condition of not changing the cohomology class of ω = ω z dz means that δω = dφ, for some function φ . We will use the scalar φ as the basic field of our KS theory. In fact,we can re-express the vector field v z in terms of φ as follows: v z = φω z . To see this, note that the Lie derivative of ω in terms of v can be expressed as δω = L v ω = d ( ι v ω ) − ι v dω = d ( ι v ω ) , ω is closed. Note also that ι v ω = ω · ( φ/ω ) = φ . This leads to the required relation δω = dφ. In terms of the scalar field φ the variation of the ∂ operator takes the form (usingthat ∂v = ∂φ/ω ) ∂ → ∂ − ∂φω ∂. Now before deformation the scalar field φ should satisfy ∂∂φ = 0 . This one can see for example by making more explicit the dictionary from the general KStheory on the threefold with the reduction to the Riemann surface Σ. The KS field A (theBeltrami differential) of [6] is identified as A ∼ ∂φω . The dual form A ′ = Ω · A , which is a (2 ,
1) form in six dimensions, becomes in twodimensions a (0 ,
1) form given by A ′ ∼ ∂φ. Now the closed string field A ′ satisfies the gauge condition b − A ′ = 0. In the KS theorythis becomes ∂A ′ = 0. With the above dictionary, this translates into ∂∂φ = 0. (Closelyrelated to this point of view, we can also think of ω as the classical value of ∂φ . Since thisis a holomorphic (1,0) form, we have (again classically) the equation ∂∂φ = 0.)This means that, at the level of the unperturbed equations, we are dealing with a free(chiral) boson quantum field theory with action (we will not be precise with normalizations) S = Z Σ ∂φ∂φ. We can now capture the effect of the variation of ∂ in terms of an operator. Recallthat for any Beltrami differential µ zz , the operator T ( µ ) which implements this variationon the conformal field theory is given in terms of the holomorphic stress-tensor T zz as T ( µ ) = Z Σ T zz µ zz . Given that we have a free boson system, we can write this very explicitly as T zz = 12 ∂φ∂φ. µ = ∂φ/ω , we therefore have the interaction term Z Σ ∂φ∂φ ∂φω . Note that this operator can be written as total derivative Z d (cid:18) ∂φ∂φ φω (cid:19) , where we use that in perturbation theory ∂φ remains holomorphic. So, if ω has no zeroes,this interaction is trivial. Since for our case ω = ydx , such zeroes can occur if either y = 0or dx = 0. As we will show in the next section, the points where y = 0 do not contribute,but the locus where dx = 0 does. The points where dx = 0 correspond to branch pointsof the Riemann surface H ( x, y ) = 0 on the x -plane. We will thus arrive at the interactionoperator X branch points I P ∂φ∂φ · φω , which will be used in the next section to recover the recursion relations of [3].In fact there is one additional term in the action that we will now explain: Note thatthe action we have thus far can be written as S = Z ∂φ ( ∂ + ∂φω ∂ ) φ However, as we have explained dφ is the variation of ω . In particular ω , being a differentialof type (1 , classical vev of ∂φ . Motivated by this observation weview the first term in the above action as the full ω including the classical piece and arriveat the final form for the action S = Z ( ω + ∂φ )( ∂ + ∂φω ∂ ) φ (2 . λ throughthe usual rescaling ω → ω/λ (a standard relation in KS theory), we obtain the field theory S = Z (cid:20) ∂φ∂φ + 1 λ ω∂φ + λω ∂φ ( ∂φ ) (cid:21) . (2 . A z = ω/λ . Since in perturbation theory the field φ is chiral, this term will only influence the classical free energy, that scales as λ − . Finally,5he third term is capturing the perturbative corrections. (In fact, by shifting ∂φ with thebackground ω the whole action can be put into cubic form, reminiscent of the purely cubicforms encounter in open string field theory [7].)In this action the cubic term is proportional to the string coupling λ . So, up to severalsubtleties related to the chiral nature of this quantum field theory, this model can be solvedusing trivalent Feynman diagrams. In the next section we will show how one can use thisaction to derive the recursion relations of [3].
3. The Recursion Relations
We will now discuss the quantum field theory based on the action (2.1) in terms ofcoordinate space perturbation theory. Following [3] we will consider not just the partitionfunction, but general correlation function of operators ∂φ ( z ) W ( z , . . . , z n ; λ ) = D ∂φ ( z ) · · · ∂φ ( z n ) E con . Here the subscript indicates that we only consider the connected correlators. We computethese correlators in the background of the interaction termexp Z Σ λ ∂φ ( ∂φ ) ω Expanding in the coupling λ brings down these interactions and this defines the perturba-tive correlators. The connected correlators have an expansion of the form W ( z , . . . , z n ; λ ) = X g ≥ λ g − n W g ( z , . . . , z n ) . As noted in the previous section, the interaction can be written as a total derivativeaway from the zeroes of ω . So we are left with contributions of the form X P I P φ ( ∂φ ) ω (3 . P ∈ Σ denote the positions of the zeroes of ω . In the local coordinates the one-form ω is given by ω = ydx , and such zeroes therefore occur either if y ( x ) = 0, or if thedifferential dx vanishes. Let us consider these two cases separately.6f y ( x ) has a zero at x = x , so that y ∼ c ( x − x ), the variable z = x − x is a goodlocal coordinate around this special point and we can expand the quantum field as ∂φ ( z ) = X n ∈ Z α n z − n − . Here α n are the usual creation and annihilation operators. Plugging this relation and ω ∼ zdz into interaction vertex (3.1) we obtain the operator O = I dzz φ ( ∂φ ) ∼ X k + m + n = − k α k α m α n Since z is good local coordinate at this point, the field φ ( z ) has no singularities. In theoperator formalism the operator O should therefore be regarded to act on the vacuum | i .This state satisfies α n | i = 0 for n ≥
0. Because of the condition that k + m + n = −
1, wesee that necessarily the mode expansion of O will have to contain annihilation operators α + n that vanish on this vacuum state. Therefore we obtain the relation O| i = 0 , and the action of the interaction at these zeroes is trivial and can be ignored.As we mentioned, a second source of zeroes of ω = ydx are the points where dx vanishes. If we think of the curve H ( x, y ) = 0 as an orbit in the phase space of Hamiltonianmechanics, these are the turning points. In complex geometry these zeroes are branchpoints of the algebraic curve. Generically, these are simple branch points. At such a pointthe curve is locally described by ( y − y ) = x − x . A good local coordinate is therefore z = y − y = ( x − x ) / , where we clearly see that we are dealing with a branch point. In terms of the variable z we have dx ∼ zdz and therefore also ω ∼ zdz (since y attains the regular value y at thispoint). So ω has indeed a (single) zero at the ramification point z = 0. Now in terms of7he coordinate x the interaction vertex O does not act not on the regular vacuum state | i ,but on a twisted state | σ i , around which the field ∂φ ( x ) has a half-integer mode expansion ∂φ ( x ) ∼ X n ∈ Z α n − ( x − x ) − n − . Equivalently, in terms of “good” coordinate z on the double cover the scalar field satisfiesthe condition φ ( − z ) = − φ ( z ) . (3 . ω + dφ shouldhave the same behaviour as ω , which around the ramification point takes the form zdz .This fixes the boundary condition on the field φ ( z ) and determines in turn the natureof the boundary state | σ i . Working out the decomposition of O in these twisted modeswe easily see that in this case O| σ i 6 = 0. Therefore the branch points give non-vanishingcontributions to the interaction and we are left with X branch points I P φ ( ∂φ ) ω (3 . ∂φ ( z i ) with the operators appearing inthe interaction vertex. If z is the local coordinate around the branch point P this givesterms of the form (cid:28) ∂φ ( w ) I z φ ( z ) ∂φ ( z ) ∂φ ( z ) ω ( z ) · · · (cid:29) To evaluate these kinds of expressions we need to determine the chiral correlator B ( z, w ) = h ∂φ ( z ) ∂φ ( w ) i for a free boson on the surface Σ. This two-point function is well-know to be given by theBergmann kernel. To define it uniquely, we have to fix the loop momenta through a set of A -cycles I A I ∂φ = p I . The standard kernel B ( z, w ) takes these p I = 0. Note that this prescription already breaksthe modular invariance, since Sp (2 g, Z ) transformations, that relate one set of homologycycles to another, will act via generalized Fourier transformation on the correlators of thechiral boson. 8ow first, up to total derivatives, we can consider the contraction of ∂φ ( w ) with φ ( z ).This will be given by the primitive of the Bergmann kernel that we will denote as G ( z, w ) h φ ( z ) ∂φ ( w ) i = G ( z, w ) := Z z B ( v, w ) dv. However we should also take into account the boundary condition (3.2) at the branch point.The field ∂φ should be anti-periodic around the twist field insertion. This we can enforceby inserting an explicit projector to the odd part of the propagator, as is customary in thecomputation of twist field correlation functions in orbifold models. So we get (in the localcoordinate z , close to the branch point) h φ ( z ) ∂φ ( w ) i twist = 12 Z z − z B ( v, w ) dv. Note in particular that at the branch point z = 0 this twisted propagator vanishes, whichis forced by the anti-periodicity. However, in this case there is a matching zero in the de-nominator, because also ω vanishes at the branch point. We can therefore apply l’Hˆopital’srule, and consider instead the limitlim z → h φ ( z ) ∂φ ( w ) i twist ω ( z ) = lim z → R z ˜ z B ( v, w ) dvω ( z ) − ω (˜ z )where z and ˜ z are the two points on the two branches of Σ that project to the same imagein the x -plane (so that ˜ z ∼ − z close to z = 0).Finally we also have to deal with the fact that the interaction consists of a cubic termthat has to be normal ordered. Now recall that this term originated from the stress-tensorinsertion I v z T zz , (3 . v z = φ/ω . The self-interactions in the stress-tensorare usually defined by point-splitting regularization T ( z ) = lim ˜ z → z (cid:20) ∂φ ( z ) ∂φ (˜ z ) − z − ˜ z ) (cid:21) To have a consist perturbation theory we have to insert this definition into equation (3.4)with v z = φ/ω .Summarizing all this, and up to some further subtleties that we discuss in section 3.3,we obtain the recursion relation of [3]. This relation can be considered as the Schwinger-Dyson equations for the interacting boson field theory. Expanding the recursion relation9ives a graphical representation in terms of trivalent Feynman diagrams. There will beboth connected and disconnected contributions. For the connected diagrams with n + 1external legs the recursion relation takes the form W g ( w, z , z , . . . , z n ) = X P Res z = P R z ˜ z B ( v, w ) dvω ( z ) − ω (˜ z ) h W g − ( z, ˜ z, z , . . . , z n )+ g X h =0 X Z = Z ′ ∪ Z ′′ W h ( z, z ′ , . . . , z ′ m ) W g − h (˜ z, z ′′ , . . . , z ′′ n − m ) i (3 . P is summed over all branch points of the spectral curve Σ. The set Z denotes thecollection of “free” marked points Z = { z , . . . , z n } , and in the disconnected piece thereis a sum over all splittings of the set Z into two disjoint (and possibly empty) subsets Z ′ and Z ′′ of order m and n − m . We now turn to the partition function itself, which has an expansion Z = exp F , F = X g ≥ λ g − F g . To derive the final recursion relation of [3] we will employ a rescaling symmetry. Startingpoint will be again the complete action (2.2) that we recall here for convenience S = Z ∂φ∂φ + 1 λ ω∂φ + λω ( ∂φ ) ∂φ. Consider now the action of the vector field λ∂ λ on the free energy F . On the one hand weclearly have λ ∂ F ∂λ = X g ≥ (2 g − F g λ g − . (3 . ∂φ observables inserted anywhere. Onlythe second term in the action will contribute and gives the insertion λ ∂Z∂λ = − λ (cid:10)Z Σ ω∂φ (cid:11) . λ R d ( ωφ ), which gives zero, except when φ has poles. This happens when φ is near thebranch points P . So we can write this term as1 λ X branch points (cid:10)I P ωφ (cid:11) . Near the branch points we write ω = ∂φ cl , where φ cl ( z ) can be interpreted as the classicalvalue of the field φ and so this contour term can be written (using integration by parts) as − λ X P (cid:10)I P φ cl ∂φ (cid:11) = − λ X P I φ cl (cid:10) ∂φ (cid:11) = − λ X P Res z = P φ cl ( z ) W ( z ) (3 . W ( z ) has a perturbative expansion W ( z ) = X g ≥ λ g − W g ( z )Inserting this expansion into (3.7) and comparing this with (3.6) we thus find the recursionrelation of [3] F g = 12 − g X P Res z = P φ cl ( z ) W g ( z ) . SL (2 , R ) symmetry Up to this point the relation of the action (2.2) to the manipulations leading to therecursion relations has not been very precise. Indeed the authors of [3] have remarked thattheir recursion relations and the corresponding graphical solution cannot follow from astraightforward Feynman expansion, since certain contractions and diagrams are missing,and there is a special order in which the vertices are connected. This last point is due tothe fact that the interactions have been rewritten as contour integrals, which we can thinkof as a (local) Hamiltonian formalism. These operators in general do not commute, andthe specific time ordering prescription breaks the general covariance. Exactly the samepoint was met in [8].From our point of view another source of subtleties arise because we are dealing with achiral field theory and up to now we have not consistently implemented the chiral projectionon the scalar field φ . As written in action (2.2) the anti-holomorphic component of φ is apropagating field. This gives unwanted contributions in the contractions of φ with itself.We will now rectify this point. 11t the level of the free theory the projection onto the chiral modes can be done byadding a multiplier (1,0) form γ to the action Z (cid:18) ∂φ∂φ − γ∂φ (cid:19) . Integrating out γ enforces the chiral condition ∂φ = 0. On the other hand, integrating out φ expresses γ as γ = ∂φ. In fact, it is now suggestive to relabel φ = β, to make clear that we are dealing with a bosonic β - γ system with spins 0 and 1. After apartial integration, this βγ system has the action Z Σ (cid:18) ∂β∂β + β∂γ (cid:19) + I ∞ βγ, where by H ∞ here and in the following we mean integration over the boundaries of theRiemann surface if there are any (including the branch points). With this unconventionalaction we get the following Green’s functions for the fields β and γ h β ( z ) β ( w ) i = 0 , h β ( z ) γ ( w ) i = G ( z, w ) , h γ ( z ) γ ( w ) i = B ( z, w ) . (3 . ∂ z G ( z, w ) = B ( z, w ).Using this formalism we can now write the interaction term in an elegant fashion as I ∞ (cid:18) ωλ β + βγ + λω βγ (cid:19) . (3 . (cid:10) γ ( z ) . . . γ ( z n ) (cid:11) . βγ term. Since β only contracts with γ each β in the interaction term βγ gets paired up with a γ ( z i ) leaving two additional γ ’s. At the end when all the β ’s have been contracted, we use the γ correlations and thusrecover the [3] recursion relations as well as the boundary conditions needed to solve them.This reformulation makes our derivation of the recursion relation precise.Interestingly, this way of expressing the action suggests a hidden SL (2 , R ) symmetry.Let us first recall the Wakimoto representation of the SL (2 , R ) k current algebra [9]. Thisconformal field theory consists of another βγ system, that we will write as ( ˆ β, ˆ γ ) and thatnow has spins 1 and 0, together with an extra scalar field χ . In terms of these variablesthe SL (2 , R ) currents, all of spin 1 of course, are expressed as J + ( z ) = ˆ β,J ( z ) = ˆ β ˆ γ + 12 α + ∂χ,J − ( z ) = ˆ β ˆ γ + α + ˆ γ∂χ + k∂ ˆ γ. (3 . α = 2 k − c = kk − . The scalar field χ hasa background charge 1 /α + . Furthermore, in order to compute correlation functions oneneeds to add various insertions of the screening charge S + = I ∞ ˆ β e − χ/α + . (3 . χ does not appear and thus it is natural to viewthis as the special limit of k = 2. In this so-called critical limit, where the central charge c → ∞ and α + →
0, the contribution of the scalar χ decouples and can be ignored. In factthe β - γ system by itself carries a representation of the current algebra SL (2 , R ) at level k = 2. (In the analytic continuation to the case of SU (2) this critical level corresponds tothe value k = − u ( z ) = ( k − T ( z ) = X a J a , (instead of just the identity operator). One way to describe the representations in thiscritical limit, is that the combination v ( z ) = α + ∂χ ( z ) becomes a classical, non-dynamicalscalar field that parametrizes the affine SL (2 , R ) representations. This classical scalar field13an be traded with the expectation values of the rescaled stress-tensor u ( z ). All of this isintimitely connected to the theory of integrable systems, see [10] for more on this.There is a traditional topological twisting of this model related to the KPZ model of2d gravity [11], where the spins of the β - γ system are changed from (1 ,
0) to (0 , SL (2 , R ) currents ( J + , J , J − ) change its spins into (0 , , K Σ on the Riemann surface, i.e. , we haveto pick a meromorphic (1,0) form ω . The twisted fields ( β, γ ) are related to the untwistedfields as ˆ β = ωβ, ˆ γ = 1 ω γ. Ignoring the scalar χ and total derivatives we so find that the triplet of currents can beexpressed as J + ( z ) = ωβ,J ( z ) = βγ,J − ( z ) = 1 ω βγ . (3 . I ∞ (cid:18) ωλ β + βγ + λω βγ (cid:19) , and rewrite it in a more suggestive way as I ∞ J + ( z, λ ) , (3 . J + ( z, λ ) = 1 λ J + ( z ) + J ( z ) + λJ − ( z ) . Here the variable λ can be seen as a spectral parameter that picks a U (1) inside SL (2 , R ),or more geometrically a null plane inside R , . It therefore takes it values on the twistor“sphere”, or perhaps more correctly the twistor upper-half plane (the appropriate realstructure is not quite obvious) λ ∈ H = SL (2 , R ) U (1) . The group SL (2 , R ) acts on this parameter in the usual fashion by fractional linear trans-formations λ → aλ + bcλ + d , ad − bc = 1 . k = 2 limit of the screeningcharge S + given in (3.11). Anyway, there is in the critical case k = 2 a huge algebra thatcommutes with the interaction vertex J + ( λ ), namely the full classical Virasoro algebra.Itis not clear what the precise role of this structure is and toi which extend it is relatedto other models of two-dimensional gravity. In the next section we discuss the potentialmeaning of the SL (2 , R ) structure from a three-dimensional perspective.
4. The 3d Gravity Lift
An important aspect of the formulation of topological B model in the setup of [3] is thefact that it in a natural way explains [5] why the partition function satisfies the holomorphicanomaly equation of [6]. In particular to even formulate the partition function, one has tochoose a set of A-cycles on the Riemann surface, such that the period of the Bergmannkernel vanishes over them. If we chose a different set of A-cycles, the partition functiontransformorms as a wave function as is the interpretation of holomorphic anomaly equation[12], see also [13].In our setup this comes about because we have a chiral boson φ and to define itquantum mechanically we need to specify its periods over only half the cycles, whichin the present case are the A-cycles, where R A i ∂φ = 0. It is known from the study of2d CFT’s [14] that this is indeed consistent with how the chiral blocks are defined, andhow they transform if we choose a different symplectic basis for them. For example, thechiral blocks transform according to Fourier transform, if we switch the A-cycles and B-cycles. This phenomenon for 2d CFT’s is best described by considering the associatedthree dimensinonal Chern-Simons theory [15]: Consider the abelian U (1) Chern-Simonstheory in three dimension, with the action Z A ∧ dA where A denotes the U (1) connection. In particular in the Coulomb gauge we have the term R A x ˙ A y , which means A x and A y are conjugate variables. Viewing the three dimensionalmanifold as Riemann surface times time, and identifying the chiral fields of 2d with therestriction of A = dφ on the Riemann surface, we see the fact that the periods of dφ overthe A -cycles and B -cycles form conjugate variables and do not commute.It is very natural to view this as the natural motivation for us as well. For thegeneral topological string this suggests the existence of a topological M-theory in one15igher dimension, namely 7 for the Calabi-Yau threefold, or 3 for the current case (wherethe CY is based on a Riemann surface). This in fact was one of the motivations for theintroduction of topological M theory [16,17] .However the abelian Chern-Simons theory in 3d cannot be the whole story here fortwo reasons: First of all we have in addition to the free term the interactions in 2d, andthey should lead to some deformations of the 3d theory. Secondly the 2d theory is a gravity theory, and not a current algebra, and so we expect that the 3d theory to also be a gravitytheory. It is natural to ask whether we can relate our theory to the SL (2 , R ) Chern-Simonsformulation of 3d gravity [18] in the geometry Σ × R . Such a 3d theory would give rise tochiral SL (2 , R ) current algebra living on Σ. But this is surprisingly what we have foundin the last section!We are thus led to believe that 3d gravitational Chern-Simons theory underlies whatwe have found in connection with the Kodaira-Spencer theory on the Riemann surface.However we need to better understand the meaning of the insertion terms I ∞ J + ( z, λ ) . First of all, what is the twistorial meaning of λ ? In other words, why should the couplingconstant of topological string parameterize a twistorial sphere and what is the role of the SL transformations acting on the coupling constant? Here it should be stressed thatthe relevant object is the local form ω/λ , which can be viewed as a varying point onthe twistor sphere. This choice of background is what distinguishes the 3d theory fromordinary (chiral) SL (2 , R ) gravity.Secondly, why would one insert this interaction term at the boundaries and branchpoints of the surface? Is it related to the screening operators at the critical level? Oneinterpretation may be that this term shifts the gravitational background so that the con-nection is not flat and would correspond to the background Σ × R . If this is the rightinterpretation one would need to better understand why this particular insertion createsthis gravitational background.Recalling that we have been discussing only the reduction of Kodaira-Spencer theoryto two dimensions, we could lift our findings back to six dimensions. In this context wewould be led to the seven dimensional M-theory formulation, mentioned above. We feelwe have found a first concrete evidence for the existence of topological M-theory. It would16e very important to deepen our understanding of the 3d lift of Kodaira-Spencer theory,and further extend it to the topological M-theory in seven dimensions. Acknowledgements
We would like to thank J. de Boer, V. Bouchard, B. Eynard, A. Klemm, H. Ooguriand E. Verlinde for valuable discussions. We would also like to thank the fifth SimonsWorkshop in mathematics and physics at Stony Brook for inspiring surroundings anddiscussions which led to this project. The research of R.D. was supported by a NWOSpinoza grant and the FOM program
String Theory and Quantum Gravity . The researchof C.V. was supported in part by NSF grants PHY-0244821 and DMS-0244464.17 eferences [1] M. Aganagic, A. Klemm, M. Marino and C. Vafa, “The topological vertex,” Commun.Math. Phys. , 425 (2005) [arXiv:hep-th/0305132].[2] R. Dijkgraaf and C. Vafa, “Matrix models, topological strings, and supersymmetricgauge theories,” Nucl. Phys. B , 3 (2002) [arXiv:hep-th/0206255].[3] B. Eynard and N. Orantin, “Invariants of algebraic curves and topological expansion,”[arXiv:math-ph/0702045].[4] V. Bouchard, A. Klemm, M. Marino and S. Pasquetti, “Remodeling the B-model,”[arXiv:hep-th/07091453].[5] B. Eynard, M. Marino and N. Orantin, “Holomorphic anomaly and matrix models,”JHEP , 058 (2007) [arXiv:hep-th/0702110].[6] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Kodaira-Spencer theory of gravityand exact results for quantum string amplitudes,” Commun. Math. Phys. , 311(1994) [arXiv:hep-th/9309140].[7] G. T. Horowitz, J. D. Lykken, R. Rohm and A. Strominger, “A Purely Cubic ActionFor String Field Theory,” Phys. Rev. Lett. , 283 (1986).[8] R. Dijkgraaf, “Chiral deformations of conformal field theories,” Nucl. Phys. B ,588 (1997) [arXiv:hep-th/9609022].[9] M. Wakimoto, “Fock representations of the affine lie algebra A1(1),” Commun. Math.Phys. , 605 (1986).[10] E. Frenkel, “Lectures on Wakimoto modules, opers and the center at the critical level,”[arXiv:math.QA/0210029].[11] V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov, “Fractal structure of 2d-quantum gravity,” Mod. Phys. Lett. A , 819 (1988).[12] E. Witten, “Quantum background independence in string theory,” [arXiv:hep-th/9306122].[13] R. Dijkgraaf, E. P. Verlinde and M. Vonk, “On the partition sum of the NS five-brane,”[arXiv:hep-th/0205281].[14] R. Dijkgraaf, E. P. Verlinde and H. L. Verlinde, “C = 1 Conformal Field Theories onRiemann Surfaces,” Commun. Math. Phys. , 649 (1988).[15] E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. , 351 (1989).[16] R. Dijkgraaf, S. Gukov, A. Neitzke and C. Vafa, “Topological M-theory as unifica-tion of form theories of gravity,” Adv. Theor. Math. Phys. , 603 (2005) [arXiv:hep-th/0411073].[17] N. Nekrasov, “Z-theory: Chasing M-F-Theory,” Comptes Rendus Physique , 261(2005).[18] E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys.B311