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AND TOPOLOGICAL FIELD THEORY

arXiv:hep-th/9112056v1 19 Dec 1991IASSNS-HEP-91/83December, 1991MIRROR MANIFOLDSAND TOPOLOGICAL FIELD THEORYEdward Witten*School of Natural Sciences,Institute for Advanced Study,Olden Lane,Princeton, N.J. 08540ABSTRACTThese notes are devoted to sketching how some of the standard facts relevant to mirrorsymmetry and its applications can be naturally understood in the context of topologicalfield theory. If X is a Calabi-Yau manifold, the usual nonlinear sigma model governingmaps of a Riemann surface Σ to X can be twisted in two ways to give topological fieldtheories, which I call the A model and the B model.

Mirror symmetry relates the A model(of one Calabi-Yau manifold) to the B model (of its mirror). The correlation functionsof the A and B models can be computed, respectively, by counting rational curves andby calculating periods of differential forms.

This can be proved as a consequence of areduction to weak coupling (as in §3-4 of these notes) or by a sort of fixed point theoremfor the Feynman path integral (see §5). The correlation functions of the twisted modelscoincide, as explained in §6, with certain matrix elements of the physical, untwisted model –namely those that determine the superpotential.

The conventional moduli spaces of sigmamodels can be thickened, in the context of topological field theory, to extended modulispaces, indicated in §7, which are probably the natural framework for understanding thestill mysterious “mirror map” between moduli spaces. * Research supported in part by NSF Grant 91-06210.1

1. IntroductionThe purpose of these notes is to explain aspects of the mirror manifold problem thatcan be naturally understood in the context of topological field theories.

(For other aspectsof the problem, and detailed references, the reader should consult other articles in thisvolume.) Most of what I will say can be found in the existing literature, but to isolate thefacts most relevant to mirror symmetry may be useful.

The points that I want to explainare as follows.First, we will consider the standard supersymmetric nonlinear sigma model in twodimensions, governing maps of a Riemann surface Σ to a target space which for our pur-poses will be a Kahler manifold X of c1 = 0. We will see that there are two differenttopological field theories that can be made by twisting the standard sigma model.

Thereare not standard names for these topological field theories; I will call them the A theoryand the B theory, or A(X) and B(X) when I want to specify the choice of X. The Atheory has been studied in detail in [[1]], where many facts sketched below are explained;the B theory has been studied less intensively.

(The A and B theories were discussedqualitatively, in a general context of N = 2 superconformal field theories, by Vafa in hislecture at this meeting [[2]]. )Unlike the ordinary supersymmetric nonlinear sigma model, the twisted models are“soluble” in the sense that the problem of computing all the physical observables can bereduced to classical questions in geometry.

This is done by a sort of fixed point theoremin field space, which gives the following results: the correlation functions of the A theoryare determined by counting holomorphic maps Σ →X obeying various conditions; thecorrelation functions of the B theory can be computed by calculating periods of classi-cal differential forms. (More generally, “counting” rational curves must be replaced bycomputing the Euler class of the vector bundle of antighost zero modes, as I have ex-plained elsewhere [[3], §3.3]; this has been implemented in the mirror manifold problem byAspinwall and Morrison [[4]].

)For the most direct physical applications one is not interested in the twisted A or Btheories, but in the original, “physical” sigma model. However, there is one very importantcase in which physical observables coincide with observables of the A and B theories.

Thishappens for the Yukawa couplings, which are certain quantities that one wishes to compute2

for Σ of genus zero.1In particular, the 273 and 273 Yukawa couplings of superstringmodels coincide with certain observables of the A and B theories, respectively.TheseYukawa couplings are therefore determined by the fixed point theorem mentioned in thelast paragraph; from this we recover results that were originally found by more detailedarguments, such as the fact that the 273 Yukawa couplings have no quantum corrections,and the instanton sum formula [[5]] for the 273 Yukawa couplings.In §2 we recall the definition of the standard sigma model. The twisted A and Bmodels are described in §3 and §4, along with an explanation of their main properties,including the reduction to instanton moduli spaces and to constant maps in the A and Bmodels respectively.

This latter reduction is reinterpreted as a sort of fixed point theorem in§5. In §6, I explain why certain observables of the standard physical sigma model coincidewith observables of the twisted models.

This occurs whenever the canonical bundle of theRiemann surface becomes trivial after deleting the points at which fermion vertex operatorshave been inserted – in practice, mainly for amplitudes in genus zero with precisely twofermions. In §7 – the only part of these notes containing some novelty – I look a little moreclosely at the A and B models and describe the full families of topological field theories ofwhich they are part.

I strongly suspect that many properties of mirror symmetry that arenot now well understood, including the structure of the mirror map between the parameterspaces, can be better understood in looking at the full topological families.Since “elliptic genera” (see [[6]]) arise in the same supersymmetric nonlinear sigmamodels that are the basis for the study of mirror manifolds, I will also along the way makea few observations about them. In particular we will note the easy fact that if X, Y are amirror pair, they have the same elliptic genus.

This is trivial in complex dimension three(since the elliptic genus of a three dimensional Calabi-Yau manifold is zero), but becomesinteresting in higher dimension.Perhaps I should emphasize that if X and Y are a mirror pair, then mirror symmetryrelates all observables of the sigma model on X to corresponding observables on Y – andnot just the few observables that can be related naturally to the twisted models and thus totopological field theory. The literature on mirror symmetry has focussed on these particular1 The restriction to genus zero appears in two ways: the superpotential that determines theYukawa couplings has no higher genus corrections because of nonrenormalization theorems [[5]];alternatively, in relating the superpotential to an observable of the twisted model, we will requirein §6 that the canonical bundle of Σ with two points deleted be trivial, which is of course trueonly in genus zero.3

observables, which of course are also the ones we will be studying here, because of theirphenomenological importance, and because, since the B model is soluble classically, therelation given by mirror symmetry between observables of the A(X) model and observablesof the B(Y ) model is particularly useful.2. PreliminariesTo begin with, we recall the standard supersymmetric nonlinear sigma model in twodimensions.2It governs maps Φ : Σ →X, with Σ being a Riemann surface and X aRiemannian manifold of metric g. If we pick local coordinates z, z on Σ and φI on X,then Φ can be described locally via functions φI(z, ¯z).

Let K and K be the canonicaland anti-canonical line bundles of Σ (the bundles of one forms of types (1, 0) and (0, 1),respectively), and let K1/2 and K1/2 be square roots of these. Let TX be the complexifiedtangent bundle of X.

The fermi fields of the model are ψI+, a section of K1/2 ⊗Φ∗(TX),and ψI−, a section of K1/2 ⊗Φ∗(TX). The Lagrangian is3L =2tZd2z12gIJ(Φ)∂zφI∂¯zφJ + i2gIJψI−DzψJ−+ i2gIJψI+DzψJ+ + 14RIJKLψI+ψJ+ψK−ψL−.

(2.1)Here t is a coupling constant, and RIJKL is the Riemann tensor of X. D¯z is the ¯∂operatoron K1/2 ⊗Φ∗(TX) constructed using the pullback of the Levi-Civita connection on TX.In formulas (using a local holomorphic trivialization of K1/2),D¯zψI+ = ∂∂z ψI+ + ∂φJ∂z ΓIJKψK+ ,(2.2)with ΓIJK the affine connection of X. Similarly Dz is the ∂operator on K1/2 ⊗Φ∗(TX).The supersymmetries of the model are generated by infinitesimal transformationsδΦI = iǫ−ψI+ + iǫ+ψI−δψI+ = −ǫ−∂zφI −iǫ+ψK−ΓIKMψM+δψI−= −ǫ+∂zφI −iǫ−ψK+ ΓIKMψM−,(2.3)2 This discussion will be at the classical level, and we will not worry about the anomalies thatarise and spoil some assertions if the target space is not a Calabi-Yau manifold.3 Here d2z is the measure −idz ∧d¯z.Thus if a and b are one forms, Ra ∧b=i Rd2z (azb¯z −a¯zbz).

The Hodge ⋆operator is defined by ⋆dz = idz, ⋆d¯z = −id¯z.4

where ǫ−is a holomorphic section of K−1/2, and ǫ+ is an antiholomorphic section of K−1/2.The formulas for the Lagrangian (2.1) and the transformation laws (2.3) have a naturalinterpretation upon formulating the model in superspace, that is in terms of maps of asuper-Riemann surface to X. I will not discuss this here; for references see Rocek’s lecturein this volume.As a small digression, let me note that usually, in discussing superconformal symmetry,one considers ǫ’s that are defined locally (say in a neighborhood of a circle C ⊂Σ if oneis studying the Hilbert spaces obtained by quantization on C). One might wonder whathappens if we can find global ǫ’s.

On a Riemann surface of genus g > 1, this will not occuras there are no holomorphic sections of K−1/2. In genus one K is trivial; if we pick K1/2to be trivial, then there is a one dimensional space of global ǫ−’s.

On the other hand, ifwe pick K1/2 to be a non-trivial line bundle (of order two) then globally ǫ+ must vanish.With the techniques that we will use below, one can readily use the symmetry generatedby ǫ−to prove that the partition function of the sigma model, on such a genus one surface,is independent of the metric of X and so is a topological invariant in the target space; itis in fact the elliptic genus of X. (See [[6]] for an introduction to elliptic genera, and mylecture in that volume for an explanation of the field theoretic approach to them.) As theexistence of a holomorphic section of K−1/2 was essential here, this construction will notgeneralize to genus g > 1.The twisted models that I will explain below and which will be the basis for whateverI have to say about mirror manifolds are the closest analogs of the usual supersymmetricsigma model for which global fermionic symmetries exist regardless of the genus.

In par-ticular the closest analog of the elliptic genus for g > 1 involves the half-twisted modelintroduced at the end of this section.2.1. Kahler Manifolds And The Twisted ModelsWe now wish to describe the additional structure – N = 2 supersymmetry, to beprecise – that arises if X is a Kahler manifold.

In this case, local complex coordinates onX will be denoted as φi; their complex conjugates are φi = φi. (φI will still denote localreal coordinates, say the real and imaginary parts of the φi.) The complexified tangentbundle TX of X has a decomposition as TX = T 1,0X ⊕T 0,1X.

The projections of ψ+in K1/2 ⊗Φ∗(T 1,0X) and K1/2 ⊗Φ∗(T 0,1X), respectively, will be denoted as ψi+ and ψi+.5

Likewise the projections of ψ−in K1/2⊗Φ∗(T 1,0X) and K1/2⊗Φ∗(T 0,1X) will be denotedas ψi−and ψi−, respectively. The Lagrangian can be writtenL = 2tZΣd2z12gIJ∂zφI∂¯zφJ + iψi−Dzψi−gii + iψi+Dzψi+gii + Ri ij jψi+ψi+ψj−ψj−.

(2.4)As for the fermionic symmetries, these are twice as numerous as before because of theKahler structure; this is analogous to the decomposition of the exterior derivative on acomplex manifold as d = ∂+ ∂. I will write the following formulas out in detail becausewe will need various specializations of them in describing the twisted models.

In terms ofinfinitesimal fermionic parameters α−, eα−(which are holomorphic sections of K−1/2) andα+, eα+ (antiholomorphic sections of K−1/2), the transformation laws areδφi = iα−ψi+ + iα+ψi−δφi = ieα−ψi+ + ieα+ψi−δψi+ = −eα−∂zφi −iα+ψj−Γijmψm+δψi+ = −α−∂zφi −ieα+ψj−Γijmψm+δψi−= −eα+∂zφi −iα−ψj+Γijmψm−δψi−= −α+∂zφi −ieα−ψj+Γijmψm+ . (2.5)Now, the twisted models are constructed as follows:(1) Instead of taking ψ+i and ψ+i to be sections of K1/2 ⊗Φ∗(T 1,0X) and K1/2 ⊗Φ∗(T 0,1X), respectively, we take them to be sections of Φ∗(T 1,0X) and K ⊗Φ∗(T 0,1X),respectively.

I will call this a + twist. The terms in the Lagrangian containing ψ+ areunchanged (except that D¯z must now be interpreted as the ¯∂operator of the appropriatebundle).

Alternatively, we can make what I will call a −twist, taking ψi+ and ψi+ to besections of K ⊗Φ∗(T 1,0X) and Φ∗(T 0,1X), respectively. (2) Similarly, we twist ψ−by either a + twist, taking ψi−to be a section of Φ∗(T 1,0X)and ψi−to be a section of ¯K ⊗Φ∗(T 0,1X), or a −twist, taking ψi−to be a section ofK ⊗Φ∗(T 1,0X) and ψi−to be a section of Φ∗(T 0,1X).

Again the Lagrangian is unchanged(Dz is now interpreted as the ∂operator of the appropriate bundle).If one is twisting only ψ+ or only ψ−, it does not much matter if one makes a + twistor a −twist. The two choices differ by a reversal of the complex structure of X, and weare interested in considering all possible complex structures anyway.

However, when wetwist both ψ+ and ψ−, there are two essentially different theories that can be constructed.6

By making a + twist of ψ+ and a −twist of ψ−, we make what I will call the A theory.By making −twists of both ψ+ and ψ−we make what I will call the B theory. When Iwant to make the dependence on X explicit, I will call these theories A(X) and B(X).Locally the twisting does nothing at all, since locally K and K are trivial anyway.In particular, in the twisted models, the transformation laws (2.5) are still valid, butglobally the parameters α, α, etc., must be interpreted as sections of different line bundles.For instance, in the A model, α−and eα+ are functions while α+ and eα−are sections ofK−1 and K−1.

One can therefore canonically pick α−and eα+ to be constants (and theothers to vanish); this gives canonical global fermionic symmetries of the A model that areresponsible for its simplicity. Similarly the B model has two canonical global fermionicsymmetries.

These global fermionic symmetries are nilpotent and behave as BRST-likesymmetries.There is an obvious variant that is possible, and that is to twist only ψ+ or only ψ−,leaving the other untwisted. I will call this the half-twisted model.

The half-twisted modelhas only one canonical fermionic symmetry, just the situation which for g = 1 usually leadsto the elliptic genus.The half-twisted model would appear to be the most reasonableframework for generalizing the elliptic genus to g > 1.It also is of phenomenologicalinterest in the following sense. In superstring compactifications on Calabi-Yau manifoldsof dimC X = 3, the A model is suitable for computing the 273 Yukawa couplings and theB model for computing the 273 Yukawa couplings.

To get a full understanding of thelow energy theory, one also needs the 1 · 27 · 27 and 13 Yukawa couplings; these are notnaturally studied in either the A or B model but involve BRST invariant observables ofthe half-twisted model. The analysis below of the A model can be applied to the half-twisted model to show that correlations of BRST observables reduce to a sum over treelevel computations in instanton fields.Mirror symmetry (since it reverses one of the U(1) quantum numbers in the N = 2superconformal algebra) can be taken to exchange the + and −twists of ψ+ while leavingψ−alone.

As a result, mirror symmetry exchanges A and B models (which differ by thechoice of twist of ψ+, for fixed twist of ψ−) and maps a half-twisted model to another halftwisted model (since in these models one makes no twist of ψ−anyway).Since the elliptic genus of a complex manifold is the same as the partition functionof the half-twisted (or untwisted) model on a Riemann surface of genus one, mirror pairshave the same elliptic genus.We now turn to a detailed description of the A and B models.7

3. The A ModelIn the A model, we regard ψi+ and ψi−as sections of Φ∗(T 1,0X) and Φ∗(T 0,1X),respectively.

It is convenient to combine them into a section χ of Φ∗(TX) (so henceforthχi = ψi+, and χi = ψi−). As for ψ¯i+, it is in the A model a (1, 0) form on Σ with valuesin Φ∗(T 0,1X); we will denote it as ψ¯iz.

On the other hand, ψi−is now a (0, 1) form withvalues in Φ∗(T 1,0X), and will be denoted as ψiz.The topological transformation laws are found from (2.5) by setting α+ = eα−= 0and setting α−and eα+ to constants, which we will call α and eα. The result isδφi = iαχiδφi = ieαχiδχi = δχi = 0δψ¯iz = −α∂zφ¯i −i˜αχ¯jΓ¯i¯j ¯mψ ¯mzδψiz = −˜α∂zφi −iαχjΓijmψmz .

(3.1)The supersymmetry algebra of the original model collapses for these topological trans-formation laws to δ2 = 0, which holds modulo the equations of motion. (By includingauxiliary fields, as in [[1]], one can get δ2 = 0 off-shell.

)Henceforth, we will generally for simplicity set α = eα. (The additional structure thatwe will overlook is related to the Hodge decomposition of the cohomology of the modulispace of holomorphic maps of Σ to X.) In this case, the first two lines of (3.1) combineto δΦI = iαχI.

Also, we will sometimes express the transformation laws in terms of theBRST operator Q, such that δW = −iα{Q, W} for any field W. Of course Q2 = 0.In terms of these variables, the Lagrangian is simplyL = 2tZΣd2z12gIJ∂zφI∂¯zφJ + iψizD¯zχigii + iψizDzχ¯igii −Riijjψi¯zψizχjχj. (3.2)It is now a key fact that this can be written modulo terms that vanish by the ψ equationof motion asL = itZΣd2z {Q, V } + tZΣΦ∗(K)(3.3)whereV = gijψiz∂¯zφj + ∂zφ¯iψjz,(3.4)whileZΣΦ∗(K) =ZΣd2z∂zφi∂zφjgi¯j −∂zφi∂zφjgij(3.5)8

is the integral of the pullback of the Kahler form K = −igi¯jdzidz¯j. ThusRΦ∗(K) dependsonly on the cohomology class of K and the homotopy class of the map Φ.

If, for instance,H2(X, ZZ) ∼= ZZ, and the metric g is normalized so that the periods of K are integermultiples of 2π, thenZΣΦ∗(K) = 2πn,(3.6)where n is an integer, the instanton number or degree. We will adopt this terminology forsimplicity; this involves no essential distortion.Instead of saying that (3.3) is true modulo the ψ equations of motion, we could modifythe BRST transformation law of ψ (by adding terms that vanish on shell) to make (3.3)hold exactly.

We will not spell out the requisite additional terms in the transformationlaw, which do not affect the analysis below, since the operators Oa that we will considerare independent of ψ.3.1. Reduction To Weak CouplingFrom equation (3.3), we can give a quick explanation of one of the key properties ofthe model, which is the reduction to weak coupling.

Suppose that we wish to calculate thepath integral for fields of degree n. With insertions of some BRST invariant operators Oa(the details of which we will discuss presently), one wishes to compute⟨YaOa⟩n = e−2πntZBnDφ Dχ Dψ e−it{Q,RV } ·YaOa. (3.7)Here Bn is the component of the field space for maps of degree n, and ⟨⟩n is the degreen contribution to the expectation value.

We have made use of (3.6) to pull out an explicitfactor e−2πnt which will turn out to contain the entire t dependence of ⟨⟩n.Standard arguments using the Q invariance and the fact that Q2 = 0 show that⟨{Q, W}⟩n = 0 for any W. It is therefore also true that, as long as {Q, Oa} = 0 for all a,(3.7) is invariant under Oa →Oa +{Q, Sa} for any Sa. Thus, the Oa should be consideredas representatives of BRST cohomology classes.Likewise, and this is the key point, (3.7) is independent of t (as long as Re t > 0so that the path integral converges) except for the explicit factor of e−2πnt that has beenpulled out.

In fact, differentiating the other t dependent factor exp(−it{Q,RV }) withrespect to t just brings down irrelevant factors of the form {Q, . .

. }.

Therefore, the pathintegral in (3.7) can be computed by taking the limit of large Re t. This is the conventionalweak coupling limit (for maps of degree n).9

Looking back at the original form of the Lagrangian (2.4), or for that matter at theform of V , one sees that for given n, the bosonic part of L is minimized for holomorphicmaps of Σ to X, that is maps obeying∂¯zφi = ∂zφ¯i = 0. (3.8)The weak coupling limit therefore involves a reduction to the moduli space Mn of holo-morphic maps of degree n. The entire path integral, for maps of degree n, reduces to anintegral over Mn weighted by one loop determinants of the non-zero modes.

(A possiblymore fundamental explanation of this reduction will be given in §5.) In particular, ⟨.

. .⟩nvanishes for n < 0, as there are no holomorphic maps of negative degree.We can also now explain why the model is a topological field theory, in the sensethat correlation functions ⟨Qa Oa⟩are independent of the complex structure of Σ and X,and depend only on the cohomology class of the Kahler form K. This is certainly true ofRΣ Φ∗(K).

For the rest, all dependence of the Lagrangian on the complex structure of Σor X is buried in the definition of V , which appears in the path integral only in the form{Q, V }; varying the path integral with respect to the complex structure of Σ or X willtherefore bring only irrelevant factors of the form {Q, . .

. }.3.2.

The Ghost Number AnomalyThe Lagrangian (3.2) has at the classical level a “ghost number” conservation law,with χ having ghost number 1, ψ having ghost number −1, and φ having ghost number 0.The BRST operator Q has ghost number 1.At the quantum level, the ghost number is not really a symmetry because of theanomaly associated with the index or Riemann-Roch theorem. Let an be the number of χzero modes, that is the dimension of the space of solutions of the equations D¯zχi = Dzχ¯i =0.

Similarly, let bn be the number of ψ zero modes, solutions of D¯zψ¯iz = Dzψi¯z = 0.4 Theindex theorem gives a simple formula for the difference wn = an −bn. In particular, wn isa topological invariant.

For instance, if X is a Calabi-Yau manifold of complex dimensiond, and Σ has genus g, then wn = 2d(1 −g), independent of n. The expression ⟨Q Oa⟩nwill vanish unless the sum of the ghost numbers of the Oa is equal to wn.4 Though not indicated in our notation, an and bn may depend on the particular map Φconsidered; we return to this presently.10

An essential fact is that the equation for a χ zero mode is precisely the linearizationof the instanton equation (3.8), and consequently the space of χ zero modes is preciselyTMn, the tangent space to Mn. In particular, if Mn is a smooth manifold, then an is its(real) dimension and in particular is a constant.The number wn = an −bn is often called the “virtual dimension” of Mn.

The reasonfor this terminology is that in a sufficiently generic situation (which may be unattainablein complex geometry) one would expect that if wn > 0 (the only situation that will be ofinterest) then bn = 0 and hence wn = an.Somewhat more generally, as long as Mn is smooth so that an is a constant, bn willalso be constant. Hence the space V of ψ zero modes will vary as the fibers of a vectorbundle V over Mn.

At singularities of Mn, an and bn may jump.3.3. Observables Of The A ModelTo prepare for actual calculations, we need a preliminary discussion of the observablesof the A model.The BRST cohomology of the A model, in the space of local operators, can be repre-sented by operators that are functions of φ and χ only.5 They have the following simpleconstruction.Let W = WI1I2...In(φ)dφI1dφI2 .

. .

dφIn be an n-form on X. We can define a corre-sponding local operatorOW (P) = WI1I2...InχI1 .

. .

χIn(P). (3.9)The ghost number of OW is n. A simple calculation shows that{Q, OW } = −OdW ,(3.10)with d the exterior derivative on W. Therefore, taking W →OW gives a natural mapfrom the de Rham cohomology of X to the BRST cohomology of the quantum field theoryA(X).

If one restricts oneself to local operators (a more general class is considered in §7),this map is an isomorphism.Particularly convenient are the following representatives of the cohomology. Let H bea submanifold of X (or more generally any homology cycle).

The “Poincar´e dual” of H is acohomology class that counts intersections with H. It can be represented by a differentialform W(H) that has delta function support on H. Hopefully it will cause no confusion ifwe refer to OW (H) as OH. The ghost number of OH is the codimension of H.5 This happy circumstance prevents complications related to the fact that (3.3) only holds onshell or after modifying the ψ transformation laws.11

3.4. Evaluation Of The Path IntegralNow let us carry out the evaluation of the path integral.

We pick some homologycycles Ha, a = 1 . .

. s, of codimensions qa.

We also pick points Pa ∈Σ. We want tocompute the quantity⟨OH1(P1) .

. .OHs(Ps)⟩n = e−2πntZBnDφ Dχ Dψ eitR{Q,V } ·YOHa(Pa).

(3.11)This quantity will vanish unless wn, the virtual dimension of moduli space, is equal toPa qa.The path integral in (3.11) reduces, upon using the independence of t and takingRe t →∞, to an integral over the moduli space Mn of instantons. Moreover, as we havepicked OHa(Pa) to have delta function support for instantons Φ such thatΦ(Pa) ∈Ha,(3.12)the path integral actually reduces to an integral over the moduli space˜Mn of instantonsobeying (3.12).In a “generic” situation, the dimension an of Mn coincides with the virtual dimensionwn.

Moreover, requiring Φ(Pa) ∈Ha involves imposing qa conditions. Hence “generically”the dimension of ˜Mn should be wn −Pa qa = 0.

In such a case, ˜Mn will consist of a finiteset of points. Let # ˜Mn be the number of such points.

In determining the contribution ofany of those points to the path integral, we can take Re t →∞. The computation reducesto evaluation of a ratio of boson and fermion determinants; this ratio is however simplyequal to +1, because of the BRST symmetry which ensures cancellation between bose andfermi modes.6In a generic situation, we therefore have⟨sYa=1OHa(Pa)⟩n = e−2πnt · # ˜Mn.

(3.13)6 In general such a BRST symmetry ensures that the ratio of fermion determinants, in ex-panding around a BRST fixed point, is +1 or −1. In the present case, the ratio is +1 since bosondeterminants are always positive, and the fermion determinant of the A(X) model is also positivesince the χi, ψ¯iz determinant is the complex conjugate of the χ¯i, ψi¯z determinant.12

Summing over n we get “generically”⟨sYaOHa(Pa)⟩=∞Xn=0e−2πnt · # ˜Mn. (3.14)In complex geometry, life is not always “generic” and fMn may well have componentsof positive dimension.

Suppose fMn has real dimension s (or focus on a component of thatdimension). If so, by virtue of the Riemann-Roch theorem, the space V of ψ zero modesis s dimensional and varies as the fibers of a vector bundle V of (real) dimension s overfMn.

7 It can be argued on rather general grounds that the generalization of counting thenumber of points in˜Mn is the evaluation of the Euler class χ(V) of the bundle V:# ˜Mn →Z˜Mnχ(V). (3.15)I refer to §3.3 of [[3]] for an explanation of this key point (and a detailed field theoreticcalculation showing explicitly how χ(V) arises in a representative example); also, see [[8]]for some of the background.The generalization of (3.14) is therefore⟨sYa=1OHa(Pa)⟩=∞Xn=0e−2πnt ·Z˜Mnχ(V).

(3.16)In a real life situation, involving multiple covers of an isolated rational curve in a threedimensional Calabi-Yau manifold, the Euler class of V has been evaluated by Aspinwalland Morrison [[4]], who as a result were able to justify a formula that had been guessedempirically by Candelas et. al.

[[9]].Notice that in deriving (3.14) and (3.16) we have not assumed that Σ has genuszero. The restriction to genus zero will arise only when (in §6) we explain the relation ofthese correlation functions of the twisted model to “physical” correlation functions of theuntwisted model.Although it may be impossible in complex geometry to achieve a “generic” situation(in which the actual dimension of M coincides with its virtual dimension), this can always7 The ψ zero modes were discussed in the subsection on the ghost number anomaly; however,the equation for such zero modes must now be corrected to permit poles at Pa tangent to Ha.

Ageneral extension of index theory to such situations, with the properties essential here, has beengiven by Gromov and Shubin [[7]].13

be done by perturbing the complex structure of X to a generic non-integrable almostcomplex structure. (This is allowed in the A model [[1]].) The importance of (3.16) comesfrom the fact that it is generally impractical to do calculations based on generic non-integrable deformations.

The generic non-integrable deformations are sometimes useful fortheoretical arguments; see the end of §7. See [[7]] for the theory of almost holomorphiccurves in almost complex manifolds.4.

The B ModelNow we will consider the B model in a similar spirit.In the B model, ψ¯i± are sections of Φ∗(T 0,1X), while ψi+ is a section of K ⊗Φ∗(T 1,0X),and ψi−is a section of K ⊗Φ∗(T 1,0X). It is convenient to setη¯i = ψ¯i+ + ψ¯i−θi = gi¯iψ¯i+ −ψ¯i−.

(4.1)Also, we combine ψi± into a one form ρ with values in Φ∗(T 1,0X); thus, the (1, 0) part ofρ is ρiz = ψi+, and the (0, 1) part of ρ is ρi¯z = ψi−.As for the supersymmetry transformations, we now set α± = 0, and set ˜α± to con-stants; in fact, for simplicity we will just set ˜α+ = ˜α−= α. The transformation laws arethenδφi = 0δφi = iαη¯iδη¯i = δθi = 0δρi = −α dφi.

(4.2)The BRST operator is again defined by δ(. .

.) = −iα{Q, .

. .

}, and obeys Q2 = 0 modulothe equations of motion.The Lagrangian isL =tZΣd2zgIJ∂zφI∂¯zφJ + iη¯i(Dzρi¯z + D¯zρiz)gi¯i+iθi(D¯zρzi −Dzρ¯zi) + Ri¯ij¯jρizρj¯zη¯iθkgk¯j. (4.3)This can be rewrittenL = itZ{Q, V } + tW(4.4)14

whereV = gi¯jρiz∂¯zφ¯j + ρi¯z∂zφ¯j(4.5)andW =ZΣ−θiDρi −i2Ri¯ij¯jρi ∧ρjη¯iθkgk¯j. (4.6)Here D is the exterior derivative on Σ (extended to act on forms with values in Φ∗(T 1,0X)by using the pullback of the Levi-Civita connection of X), and ∧is the wedge product offorms.We can now see that the B theory is a topological field theory, in the sense that it isindependent of the complex structure of Σ and the Kahler metric of X.8 Under a change ofcomplex structure of Σ or Kahler metric of X, the Lagrangian only changes by irrelevantterms of the form {Q, .

. .

}. This is obviously true for the {Q, V } term on the right handside of (4.4).

As for W, it is entirely independent of the complex structure of Σ, since itis written in terms of differential forms. It is less obvious, but true, that under change ofKahler metric of X, W changes by {Q, .

. .

}. These observations are “mirror” to our earlierresult that the A theory is independent of the complex structure of Σ and X but dependson the Kahler class of the metric of X.Similarly, the B theory is independent of the coupling constant t (except for a trivialfactor which will appear shortly) as long as Re t > 0 so that the path integral converges.Under a change of t, the t{Q, V } term changes by {Q, .

. .

}. As for the t in tW, this canbe removed by redefining θ →θ/t (since V is independent of θ and W is homogeneous ofdegree one).

Hence, the theory is independent of t except for factors that come from the θdependence of the observables. If Oa are BRST invariant operators that are homogeneousin t of degree ka, then the t dependence of ⟨Qa Oa⟩is a factor of t−Pa ka, which arisesfrom the rescaling of θ to remove the t from tW.

This trivial t dependence of the B theoryshould be constrasted with the complicated t dependence of the A theory, coming fromthe instanton sum.Because the t dependence of the B theory is trivial and known, all calculations can beperformed in the limit of large Re t, that is, in the ordinary weak coupling limit. In thislimit, one expands around minima of the bosonic part of the Lagrangian; these are just theconstant maps Φ : Σ →X.

The space of such constant maps is a copy of X, so the path8 The theory definitely depends on the complex structure of X, which enters in the BRSTtransformation laws.15

integral reduces to an integral over X. We will make this more explicit after identifyingthe observables.This is to be contrasted with the A theory, in which one has to integrate over modulispaces of holomorphic curves.

The difference arises because in the A theory, the t depen-dence becomes standard only after removing a factor of tRΦ∗(K), and after doing this,the rest of the bosonic part of the action is zero for arbitrary holomorphic curves, not justconstant maps. In §5, I will give an alternative and perhaps more fundamental explanationof why calculations in the B theory reduce to integrals over X while in the A theory theyreduce to integrals over instanton moduli space.4.1.

AnomaliesThe fermion determinant of the A model is real and positive (as the χi, ψ¯iz determinantis the complex conjugate of the χ¯i, ψi¯z determinant). In particular, there is no problemin defining this determinant as a function, and the A model, even before taking BRSTcohomology, makes at least some sense as a quantum field theory (perhaps with a cutoff,and not conformally invariant) for any complex manifold X, not necessarily Calabi-Yau.

(In fact, in [[1]], the A model was defined for general almost complex manifolds.) Thefact that the eventual recipe (3.16) for computing correlation functions does not use theCalabi-Yau condition is related to this.The B model is very different.

Because the zero forms η¯i, g¯iiθi are sections of T 0,1Xand the one forms ρi are sections of T 1,0X, the fermion determinant in the B model iscomplex. The B model does not make any sense as a quantum field theory, even withcutoff, without an anomaly cancellation condition that makes it possible to define thefermion determinants as functions (not just sections of some line bundle).

The relevantcondition is c1(X) = 0, that is, X should be a Calabi-Yau manifold.9 Thus, in the Bmodel, the Calabi-Yau condition plays an even more fundamental role than it does in theuntwisted model, where it is merely necessary for conformal invariance.9 In two dimensions, anomalies are quadratic in the coupling of fermions to gravitational andgauge fields. To get an anomaly that depends on the twisting (since the untwisted model is notanomalous) and on X (since the twisted model is a non-anomalous free field theory for X = Cn),we must consider a term linear in the gravitational field, that is the spin connection of Σ, andlinear in the gauge field, that is the pull-back of the Levi-Civita connecton of X.The onlyinvariant linear in the latter is c1(X), and standard considerations show that c1(X) is indeed theobstruction to defining the fermion determinant in the B model.16

Like the A model, the B model has an important ZZ grading by a quantum numberthat we will call the ghost number. The ghost number is 1 for η and θ, −1 for ρ, and zerofor φ. Q is of degree 1.

If X is a Calabi-Yau manifold of complex dimension d, and Oa areBRST invariant operators of ghost number wa, then ⟨Oa⟩vanishes in genus g unlessXawa = 2d(1 −g). (4.7)(There is actually a more refined ZZ × ZZ grading, which we have obscured by setting˜α+ = ˜α−and combining ψi± into ρ.)4.2.

The ObservablesNow we wish to make the simplest observations about the observables of the B model,analogous to our earlier discussion of the A model.Instead of the cohomology of X, as in the A model, we consider (0, p) forms on Xwith values in ∧qT 1,0X, the qth exterior power of the holomorphic tangent bundle of X.10Such an object can be writtenV = d¯zi1d¯zi2 . .

.d¯zipV¯i1 ¯i2...¯ipj1j2...jq∂∂zj1. .

.∂∂zjq(4.8)(V is antisymmetric in the j’s as well as in the ¯i’s. )The sheaf cohomology groupHp(X, ∧qT 1,0X) consists of solutions of ¯∂V = 0 modulo V →V + ¯∂S.For every V as in (4.8), and P ∈Σ, we can form the quantum field theory operatorOV = η¯i1 .

. .

η¯ipV¯i1...¯ipj1...jqψj1 . .

. ψjq.

(4.9)One finds that{Q, OV } = −O¯∂V ,(4.10)and consequently OV is BRST invariant if ¯∂V = 0 and BRST exact if V = ¯∂S for some S.Thus V →OV gives a natural map from ⊕p,qHp(X, ∧qT 1,0X) to the BRST cohomology ofthe B model. This is in fact an isomorphism (as long as one considers only local operators;in §7, we will make a slight generalization).10 In complex geometry, T 1,0X might be called simply TX, but we have used that name forthe complexification of the real tangent bundle of X.17

4.3. Correlation FunctionsNow picking points Pa ∈Σ and classes Va in Hpa(X, ∧qaT 1,0X), we wish to compute⟨YaOVa(Pa)⟩.

(4.11)We will consider only the case of genus zero. It will be clear that (4.11) vanishes unlessXapa =Xaqa = d.(4.12)(This is related to a more precise grading of the theory, by left- and right-moving ghostnumbers, that was alluded to following equation (4.7).

)Taking the large t limit, the calculation reduces as explained earlier to an integralover the space of constant maps Φ : Σ →X.In addition to the bose zero modes –the displacements of the constant map Φ – there are fermi zero modes, which are theconstant modes of η and θ. The nonzero bose and fermi modes enter only via their oneloop determinants; these determinants are independent of the particular constant mapΦ : Σ →X about which one is expanding, and so just go into the definition of the stringcoupling constant.

So one reduces to a computation involving the zero modes only; andcorrelation functions of the B model will reduce to classical expressions.Once we restrict to the space of zero modes, a function of φ, η and θ which is ofpth order in η and qth order in θ can be interpreted as a (0, p) form on X with valuesin ∧qT 1,0X. This of course is where the O’s came from originally.

In multiplying suchfunctions, one automatically antisymmetrizes on the appropriate indices because of fermistatistics.Thus Qa OVa can be interpreted, using (4.12), as a d form with values in∧dT 1,0X. The map⊗aHpa(X, ∧qaT 1,0X) →Hd(X, ∧dT 1,0X)(4.13)is the classical wedge product.What remains is to integrate over X the element of Hd(X, ∧dT 1,0X) obtained this way.The Calabi-Yau condition is here essential; it ensures that Hd(X, ∧dT 1,0X) is non-zero andone dimensional.

The space of linear forms on this space is thus likewise one dimensional;any such non-zero form gives a method of “integration,” unique up to a constant multiple.Of course, the path integral of the B model gives formally a method of evaluating (4.11)and hence of integrating an element of Hd(X, ∧dT 1,0); this procedure formally is unique up18

to a multiplicative constant (a correction to the string coupling constant). We noted in ourdiscussion of anomalies that the B model is anomalous except for Calabi-Yau manifolds.The restriction to Calabi-Yau manifolds amounts to the fact that what can be inte-grated naturally are top forms or elements of Hd(X, ΩdX).

(ΩdX is the sheaf of forms oftype (d, 0).) In general the relation of ΩdX and ∧dT 1,0X is that they are inverses, but inthe Calabi-Yau case they are both trivial, and hence isomorphic.

Indeed, multiplication bythe square of a holomorphic d form gives a map from ∧dT 1,0X to ΩdX. Empirically, thechoice of a holomorphic d form corresponds to the choice of the string coupling constant,though this relation is still somewhat mysterious.5.

The Fixed Point TheoremIn the last section, we explained why calculations in the A model reduce to integralsover moduli spaces of holomorphic curves, while calculations in the B model reduce tointegrals over spaces of constant maps (and ultimately to classical expressions). I will now(as in [[3]], §3.1) explain this in an alternative and perhaps more fundamental way, as asort of fixed point theorem.Consider an arbitrary quantum field theory, with some function space E over whichone wishes to integrate.

Let F be a group of symmetries of the theory. Suppose F actsfreely on E. Then one has a fibration E →E/F, and by integrating first over the fibers ofthis fibration, one can reduce the integral over E to an integral over E/F.

Provided oneconsiders only F invariant observables O, the integration over the fibers is particularlysimple and just gives a factor of vol(F) (the volume of the group F):ZEe−LO = vol(F) ·ZE/Fe−LO. (5.1)We want to apply this to the case in which F is the (0|1) dimensional supergroupgenerated by the BRST operator Q.

This case has some very special features. The volumeof the group F is zero, since for a fermionic variable θ,Zdθ · 1 = 0.

(5.2)Hence (5.1) tells us that if Q acts freely, the expectation value of any Q invariant operatorvanishes.19

To express this in another way, if F acts freely, then one can introduce a collectivecoordinate θ for the BRST symmetry. BRST invariance tells us that L and O are bothindependent of θ, and since the θ integral of a θ-independent function vanishes, the pathintegral would vanish.In general, F does not act freely, but has a fixed point locus E0.

If so, let C be anF-invariant neighborhood of E0 and E′ its complement. Then the path integral restrictedto E′ vanishes, by the above reasoning.

So the entire contribution to the path integralcomes from the integral over C. Here C can be an arbitrarily small neighborhood, so theresult is really a localization formula expressing the path integral as an integral on E0. Thedetails depend on the structure of Q near E0.

If the vanishing of Q near E0 is a generic,simple zero, then the fixed point contribution is simply an integral over E0 weighted bythe one loop determinants of the transverse degrees of freedom. This is analogous to, say,the Atiyah-Bott fixed point theorem in topology.Now let us carry this out in the A and B models.

In the A model, the relevant BRSTtransformation laws readδψ¯iz = −α∂zφ¯i −i˜αχ¯jΓ¯i¯j ¯mψ ¯mzδψi¯z = −˜α∂¯zφi −iαχjΓijmψm¯zδφI = iαχI. (5.3)Requiring δφI = 0, we get that χI = 0 for a BRST fixed point, and setting δψ = 0, we seethat in addition, a fixed point must have∂¯zφi = ∂zφ¯i = 0.

(5.4)This is the equation for a holomorphic curve, and shows the localization of the A modelon the space of such curves.The important part of the BRST transformation law of the B model for our presentpurposes isδρi = −α dφi. (5.5)Setting δρi = 0, we see that the condition for a fixed point is dφi = 0; that is, Φ : Σ →Xmust be a constant map.

Thus we recover the localization of the B model on classical,constant configurations.20

6. Relation To The “Physical” ModelSo far, we have concentrated exclusively on analyzing the twisted A and B theories andtheir correlation functions.

Mirror symmetry is however usually applied to the correlationfunctions of the untwisted, physical nonlinear sigma models. The purpose of the presentsection is to explain why certain correlation functions of the twisted models (either A or B;they can be treated together) are equivalent to certain correlation functions of the physicalmodels.In constructing the twisted models from the physical sigma model, we “twisted” var-ious fields by K1/2 and K1/2 (K being the canonical bundle of a Riemann surface Σ).

Tostate the relation between the physical and twisted models in one sentence, it is simplythat the models coincide (with a suitable identification of the observables) whenever Kis trivial and we choose K1/2 and K1/2 to be trivial, since in that case the twisting didnothing. Although there are other examples of Riemann surfaces with trivial canonicalbundle that might be considered, the important example (for standard applications ofmirror symmetry) is the case that Σ is a Riemann surface of genus zero with two pointsdeleted.Such a surface, of course, can be thought of as a cylinder with a complete, flat metric,say ds2 = dτ 2 + dσ2, −∞< τ < ∞,0 ≤σ ≤2π.

In computing path integrals on sucha surface, we must pick initial and final quantum states, say |w⟩and |w′⟩. Consideringfirst the twisted model, we assume that these are Q invariant, and so are representativesof suitable BRST cohomology classes.

Picking also points Pa ∈Σ, a = 1 . .

.s, and BRSTinvariant operators Oa, we consider the objects⟨w′|sYa=1Oa(Pa)|w⟩(6.1)which can be represented by path integrals if we wish.Of course, (6.1) can be interpreted more symmetrically by compactifying Σ – addingpoints P and P ′ and conformally rescaling the metric to bring them to a finite distance.Then the states |w⟩and |w′⟩will correspond to BRST invariant operators Ow(P) andOw′(P ′). (In the A or B model, the BRST cohomology is spanned by operators OV , whereV is a de Rham cohomology class or an element of some Hp(X, ∧qT 1,0X), respectively;so Ow and Ow′ will automatically be operators of this type for some V ’s.) The matrixelement (6.1) is then equivalent to a correlation function⟨Ow′(P ′)Ow(P)sYa=1Oa(Pa)⟩(6.2)21

on the compactified surface bΣ. This can be evaluated according to the recipes of sectionsthree and four, for the A or B model as the case may be.Now we go back to the open surface Σ, with its trivial canonical bundle and flatmetric, and make the following key observation.

As K is trivial, we can pick K1/2 tobe trivial. If we do so, the twisting by K1/2 does nothing.

Hence, (6.1) is equivalent tosome matrix element in the untwisted model. Of course, the untwisted model has a lot ofmatrix elements (since the physical states have the multiplicity of a Fock space); we willget only a few of them this way.

The operators Oa will correspond to some bosonic vertexoperators of the untwisted model; in fact, as discussed in general terms in [[10]], if theOa are chosen as harmonic representatives of the appropriate BRST cohomology classes,they are standard vertex operators of massless bosons. (In the language of [[10],[2]], theseparticular bosonic vertex operators generate the chiral ring – in fact, the ca or cc chiralring in the case of the A or B model.) Let us call the harmonic representatives Ba.What about the initial and final states in (6.1)?Equivalence of the twisted anduntwisted models depends on choosing K1/2 (and K1/2) to be trivial.This means inthe language of the untwisted model that we are working in the Ramond sector; andthus the initial and final states are fermions (and in fact, harmonic representatives of thecohomology classes in question would be ground state fermions of the untwisted model).Let us call these fermi states |f⟩and |f ′⟩.

From the point of view of the untwisted model,(6.1) might be written⟨f ′|sYa=1Ba|f⟩. (6.3)I stress, though, that in going from the twisted to the untwisted model on the flat cylinderΣ, all that we have changed is the notation.Now, (6.3) is a coupling of two ground state fermions |f⟩and |f ′⟩to an arbitrarynumber of ground state bosons Ba (which are all from the same chiral ring).Of thediversity of possible observables of the untwisted model, these are of particular importanceas they determine the “superpotential.” 11 We have explained how these particular matrixelements can be identified with observables of the twisted model.11 Actually, the cubic terms in the superpotential come from the case s = 1 of the above.

Tocompute higher terms in the superpotential, one must consider the integrated, two form version ofthe O’s, which we will introduce in the next section. The analysis of the relation between twistedand untwisted models on the cylinder is unchanged.22

Of course, if we wish we can compactify Σ in the context of the untwisted model,adding points P and P ′ at infinity and conformally scaling the metric to bring them toa finite distance. At this stage the difference between the twisted and untwisted modelwill come in, as the isomorphism between them depends on a trivialization of K.Inthe untwisted model, when one projects the points at infinity to a finite distance, thestates |f⟩, |f ′⟩will be replaced by fermion vertex operators Vf, Vf ′.

Hence, (6.3) has thealternative interpretation as a correlation function⟨Vf(P)Vf ′(P ′)sYa=1Ba⟩(6.4)on the closed surface bΣ. Note that in contrast to (6.2), which arose from the analogouscompactification in the twisted model, here the “new” vertex operators are of a differenttype from the old ones.

This is possible because the untwisted model has vastly moreobservables than the twisted models. The fact that the Yukawa couplings are derived froma cubic form (with symmetry between the bose and fermi lines), which is usually regardedas a consequence of space-time supersymmetry, is manifest in the representation (6.2) ofthe twisted model, because the “fermions” and “bosons” are represented by the same kindof vertex operators.7.

Closer Look At The ObservablesIn this section we will, finally, take a closer look at the observables of the A and Btheories. We will describe a structure – a hierarchy of q form obervables for q = 0, 1, 2 –which must exist on general grounds.

We will then analyze this hierarchy in some detail inthe A and B models. In the A model we will obtain a simple answer which moreover hasa simple and standard topological description.

The analogous calculation in the B modelturns out to be far more complicated. We will not push it through to the end, but we willgo far enough to identify the relevant structure, which turns out to be somewhat novel.In either the A or B model, we described a family of observables, say OV (P), whereV is a de Rham cohomology class or an element of some Hp(X, ∧qT 1,0X), in the A or Bmodel, and P is a point in a Riemann surface Σ.

Correlation functions⟨sYa=1OVa(Pa)⟩(7.1)23

are independent of the Pa, because of the topological invariance of the theory. We willsystematically exploit the consequences of this fact.

In doing so, we want to think of OVas an operator-valued zero form; to emphasize this we write it as O(0). We fix a particularV and do not always indicate it in the notation.Topological invariance of the theory – the fact that correlation functions of O(0)(P) areindependent of P – means that O(0) must be a closed zero form up to BRST commutators,dO(0) = {Q, O(1)},(7.2)for some O(1).

This formula, read from right to left, means that the operator-valued oneform O(1) is BRST invariant up to an exact form. Hence, we get new observables in thetheory.

If C is a circle in Σ (or more generally a one dimensional homology cycle), thenU(C) =ICO(1)(7.3)is a BRST invariant observable.We can repeat this procedure. Topological invariance means that correlation functionsof U(C) must be invariant under small displacements of C; this means that O(1) must bea closed form up to BRST commutators,dO(1) = {Q, O(2)},(7.4)for some O(2).

Also, (7.4) means that the two form O(2) is BRST invariant up to an exactform, soW =ZΣO(2)(7.5)is a new BRST invariant observable.In this procedure, if O(0) has ghost number q, then O(i) has ghost number q −i, fori = 1, 2.Obviously, if X, Y are a mirror pair of Calabi-Yau manifolds, then the mirror sym-metry A(X) ∼= B(Y ) can be applied to the new observables that we have just described.This is likely to be particularly interesting for applications of mirror symmetry with targetspaces of complex dimension greater than three.Now, (7.4) actually leads to the existence of a more general family of topologicalquantum field theories. If L is the original Lagrangian, and O(0)Va are the operator valued24

zero forms, of ghost number qa, with which the above procedure begins, then we get afamily of topological Lagrangians,L →L +XataZΣO(2)Va . (7.6)Let us call this the topological family.

As the ghost number of O(2)Va is qa −2, Lagrangiansin the topological family do not necessarily conserve ghost number (even at the classicallevel); those that do not are not twistings of standard renormalizable sigma models. In thecase of a mirror pair, the whole topological family A(X) is equivalent to the topologicalfamily B(Y ).

So far mirror symmetry has been applied only to the subfamilies of theoriesthat conserve ghost number classically. It is very likely that aspects of mirror symmetrythat are now not well understood – like the nature of the mirror map between the modulispaces – are more transparent in the context of the full topological family.7.1.

The A ModelNow we will work out the details of the above for the A model. This is easy enough.IfO(0) = VI1I2...InχI1χI2 .

. .

χIn,(7.7)for some n form V , then dO(0) = {Q, O(1)}, whereO(1) = −nVI1I2...IndφI1χI2 . .

. χIn.

(7.8)And dO(1) = {Q, O(2)}, withO(2) = −n(n −1)2VI1I2...IndφI1 ∧dφI2χI3 . .

. χIn.

(7.9)The above field theoretic formulas correspond to the following topological construc-tion. Let M be the moduli space of holomorphic maps of Σ to X of some given homotopytype.

Thus we have a family of maps Φ : Σ →X parameterized by M. Alternatively,one can think of this as a single map Φ : Σ × M →X. Given now an n dimensionalcohomology class V ∈H∗(X), we can pull it back to Φ∗(V ) ∈H∗(Σ × M).

This is an ndimensional cohomology class of Σ × M. To get cohomology classes of M, let γ be an sdimensional submanifold of Σ (for s = 0, 1, or 2) and let i : γ →Σ be the inclusion. Thenby integration over γ, one gets an n−s dimensional class in the cohomology of M, namelyi∗(Φ∗(V )) =ZγΦ∗(V ).

(7.10)25

For instance, if γ is a point P ∈Σ, then integration over P just means restricting to P,and (7.10) corresponds in the quantum field theory description to our old friend O(0)V (P).For γ a one-cycle (say a circle C) or a two-cycle (which must be a multiple of Σ itself), weget the topological counterparts of the objects introduced in (7.3) and (7.5) above.It is not to hard to verify the precise correspondence between the field theoretic andtopological definitions. See [[11]] for further discussion of some of these matters.7.2.

The B ModelTo understand the analogous issues in the B model are more difficult. In fact, becausethe computations involved are rather elaborate, I will first make some qualitative remarksto indicate what must be expected.

Then we will just make a few illustrative computationswhich indicate the form of the general answer.The operator-valued zero forms O(0) of the B model are determined by elements ofHp(X, ∧qT 1,0) for various p and q. The corresponding two forms O(2) are possible pertur-bations of the topological Lagrangian.

The case of p = q = 1 has particular significance,since H1(X, T 1,0) is the tangent space to the space of complex structures on X, and thecorresponding O(2)’s are just the changes in the Lagrangian required by a change of com-plex structure. (The explicit calculation showing this would be just analogous to the onewe will do presently for perturbations determined by elements of H2(X, ∧2T 1,0).) So letus discuss what happens when the complex structure of X is changed.The complex structure of X is determined by the ¯∂operator¯∂=Xiη¯i ∂∂φ¯i .

(7.11)(Mathematically, η¯i would usually be written as the (0, 1) form dφ¯i.) The transformationlaws of the B model (for the fields φ, η, θ from which the basic observables are constructed)are just the commutators with ¯∂.

In (7.11), I have written the ¯∂operator acting on (0, q)forms (that is, functions of φI and η¯i), but one can introduce the analogous ¯∂operatorfor (0, q) forms with values in any holomorphic bundle. In our application, the importantholomorphic bundle is ⊕q ∧q T 1,0X.

(0, q) forms with values in this bundle are simplyfunctions of φI, η¯i, and θj.If one makes a change in complex structure of X, the ¯∂operator changes. To firstorder, we get¯∂→η¯i ∂∂φ¯i + h¯ij ∂∂φj −∂∂φk h¯ij · θj∂∂θk,(7.12)26

where h¯ij is a cocycle representing an element of H1(X, T 1,0). (The perturbed ¯∂operator,acting on functions or (0, q) forms, may be more familiar; it is given by the same expressionwithout the θ · ∂/∂θ term.

This term must be included to give the perturbed ¯∂operatoracting on (0, q) forms valued in ⊕q ∧q T.) When the complex structure of X is changed,the transformation laws of the B(X) model therefore also change; indeed, taking thecommutator with the perturbed ¯∂operator, we findδφi = iαη¯ih¯ij,δθj = −iαη¯i∂jh¯isθs,(7.13)which replace δφi = δθj = 0 in the unperturbed theory. The non-zero transformation lawof θj is not so essential in the following sense: it reflects the change in T 1,0 (of which θis a section) under a change in the complex structure of X, and it can be transformedaway by rotating the θi to a basis appropriate to the new complex structure.

The non-zerotransformation law of φi is unavoidable, as it is a basic expression of the change in complexstructure.A Non-Classical CaseSo even in a “classical” case, where one is just perturbing the complex structure ofX, deformations of the B model require a change in the transformation laws of the basicfields. One must expect this to be true also for other, less classical deformations.As a typical example, let α be a cocycle representing an element of H2(X, ∧2T 1,0).The corresponding BRST invariant operator-valued zero form isO(0) = αi1¯i2j1j2η¯i1η¯i2θj1θj2.

(7.14)We now wish to write dO(0) = {Q, O(1)}, for some O(1). In constrast to the A model,one finds immediately that (i) this is only true modulo terms that vanish by the equationsof motion; (ii) the calculations involved are rather painful.

The second point is almostinevitable (in the absence of a powerful computational framework) given the first; and thefirst point is related, as we will see, to the fact that under perturbation of the Lagrangian,the transformation laws of the fields change.Eventually one finds thatdO(0) = {Q, O(1)} + G,(7.15)27

whereO(1) =iρiDiα¯i1¯i2j1j2η¯i1η¯i2θj1θj2 + 2dφ¯i1α¯i1¯i2j1j2η¯i2θj1θj2−2α¯i1¯i2j1j2η¯i1η¯i2θj1gj2k ⋆dX¯k. (7.16)The ⋆here is the Hodge star operator, andG = 2α¯i1¯i2j1j2η¯i1η¯i2θj1Zj2,(7.17)whereZj = Dθj −iρmη¯jθsRj¯jms −gj¯j ⋆Dη¯j(7.18)vanishes by the ρ equation of motion.The next step is to solve the equationsdO(1) = {Q, O(2)} +XAδLδΦA· ζA.

(7.19)Here ΦA are all the fields of the theory (φ, η, θ, ρ). Moreover, δL/δΦA are the equations ofmotion of the theory, so any expression that vanishes by the equations of motion is of theform PA δL/δΦZ · ζA for some ζA.

(7.19) means that in forming the topological family,the generalized LagrangianeL = L + tZΣO(2)(7.20)is not invariant under the original BRST transformations, but is invariant (to this order;that is, up to terms of order t2) undereδΦA = δΦA + tζA. (7.21)This shows how the modifications of the transformation laws, which we anticipate fromour preliminary discussion of the role of H1(X, T 1,0), depend upon having non-zero ζA.The computation involved in finding O(2) in (7.19) (which is guaranteed to exist bythe general discussion at the beginning of this section) is very complicated, and wouldbe unilluminating if done without powerful computational methods, such as a superspaceformulation.

It is much easier to determine the ζA. The ζA can be determined to eliminateterms in dO(1) that do not appear in any expression of the general form {Q, O(2)}.

I willjust state the results. First of all, one finds that ζη = 0.

This is in fact inevitable, evenwithout computation, to preserve the fact that Q2 = 0 (when acting on φ¯i). The important28

novelty, compared to the derivation of equation (7.15), is that ζθ and ζφ are non-zero. Infact,ζφj = −2iα¯i1¯i2jkη¯i1η¯i2θk(7.22)andζθi = iDiα¯i1¯i2j1j2η¯i1η¯i2θj1θj2.

(7.23)Including the terms necessitated by (7.22) and (7.23), the BRST transformation lawsof the topological family (or rather, the one parameter subfamily determined by the par-ticular O(2) considered here) areδφi = iαη¯iδη¯i = 0δφj = −2itα¯i1¯i2jkη¯i1η¯i2θkδθi = itDiα¯i1¯i2j1j2η¯i1η¯i2θj1θj2. (7.24)What sort of perturbed ¯∂operator will generate such transformations?

Evidently, weneedD = η¯i ∂∂φ¯i −2itη¯i1η¯i2α¯i1¯i2j1j2θj1∂∂φj2 + itη¯i1η¯i2Dkα¯i1¯i2j1j2θj1θj2∂∂θk. (7.25)InterpretationFrom this sample computation, it is possible to guess the general structure, as wewill now indicate.

Like the original ¯∂operator, ¯D is a first order differential operator,which acts on functions of φ, η, θ. As it is the BRST operator of the perturbed model, itobeys ¯D2 = 0 (after adding terms of order t2 and higher, which we have not analyzed).12However, because of the terms of third and higher order in fermions, it is definitely not aclassical ¯∂operator.Let M be the moduli space of complex structures on X, modulo diffeomorphism.

Mparametrizes the B(X) models as we constructed them originally in §4. The “topologicalfamily” of theories (with Lagrangian L →L + Pa taRO(2)a ) is a more general family oftheories parametrized by a thickened moduli space N which contains M as a subspace;the tangent space to N at M ⊂N is TN |M = ⊕dp,q=0Hp(X, ∧qT 1,0X).12¯D is a symmetry of the perturbed model, so ¯D2 is likewise a symmetry, which moreover isbosonic and of ghost number two.

The perturbed or unperturbed model has no such symmetries,so it must be that ¯D2 = 0.29

The moduli space M of complex structures on X is the space of standard ¯∂oper-ators (first order operators whose leading symbol is linear in η¯i and independent of θ),which obey ¯∂2 = 0, modulo diffeomorphisms of the φI. The enriched moduli space N oftopological models is apparently, in view of (7.25), the space of general ¯D operators – firstorder operators on the same space, but with the leading symbol allowed to have a generaldependence on φ, η, θ – obeying ¯D2 = 0, modulo diffeomorphisms of φI, ηi, θi.

(Thus,¯∂operators are classified up to ordinary diffeomorphisms of φ only, but ¯D operators areclassified up to diffeomorphisms of the φ, η, θ supermanifold.) The replacement of classical¯∂operators by the more general ¯D operators has some of the flavor of string theory, whereclassical geometry is generalized in a somewhat analogous, but far more drastic, way.I will call N the extended moduli space.

The analogous extended moduli space in theA model is just ⊕2dn=0Hn(X,C). The study of the extended moduli space is probably theproper framework for understanding the mirror map between the A and B models, and sois potentially rewarding.7.3.

The Mirror MapThe weakest link in existing studies of the consequences of mirror symmetry is theconstruction of the mirror map between the mirror moduli spaces. Given, in other words,a mirror pair X, Y , one would like to identify the mirror map between the A(X) modulispace and the B(Y ) moduli space.

Understanding this map is essential for extracting theconsequences of mirror symmetry. The discussion in [[9]] involved an elegant and verysuccessful but not fully understood ansatz.To construct the mirror map, it suffices to identify some sufficiently rich class ofobservables that can be computed both in the A(X) model and in the B(Y ) model.

I donot know how to do this, but will make a few comments. Since the B(Y ) model reducesto classical algebraic geometry, “everything” is computable in the B(Y ) model.

The A(X)model is another story; observables in the A(X) model depend on complicated instantonsums and so in general are not really calculable (except perhaps via mirror symmetrywhich requires already knowing the mirror map).The association O(0) →O(2) that we have described above gives a natural identifica-tion between the tangent space to the extended moduli space (which is the space of O(2)’s)and the BRST cohomology classes of local operators (the O(0)’s). The two point functionin genus zeroηab = ⟨O(0)a O(0)b ⟩(7.26)30

therefore defines a metric on the extended moduli space. 13 (It is essential here to work withthe extended moduli space N ; the induced metric on the ordinary moduli space M ⊂Nis typically degenerate or even zero.) Moreover, it can be shown that this metric is flat.For topological field theories (like the A and B models for Calabi-Yau manifolds) thatarise by twisting conformal field theories, this was shown in [[13]].

The main ingredientsin the argument are the C∗action on a genus zero surface with two marked points, andthe fact that for the particular BRST invariant local operator O(0) = 1, the correspondingtwo form is O(2) = 0. For the A model, I will give another proof of flatness presently (byshowing that the metric has no instanton corrections); this proof uses the C∗action in adifferent way, and does not require the Calabi-Yau condition.For the A(X) model, the extended moduli space is P2dn=0 H2(X,C), and the flat metricis simply the metric on this vector space given by Poincar´e duality; there are no instantoncorrections, as we will show shortly.

For topological field theories obtained by twistingLandau-Ginzberg models, the metric was computed by Vafa [[14]] up to a conformal factor;that the metric that so arises is flat (after a proper choice of the conformal factor, whichhas not yet been analyzed in the quantum field theory) is part of the rather deep studyof singularities by K. Saito [[15]], as was explained by Blok and Varchenko [[16]]. For theB(Y ) model, the extended moduli space was roughly described above, but the metric isnot yet understood.

It seems to be very hard to find any observables of the A(X) modelexcept the metric (and the related “exponential map” noted below) that are effectivelycomputable, without instanton corrections. So understanding the metric on the extendedmoduli space of the B(Y ) model would appear to be a promising way to understand themirror map.

(If the metric were known for both A(X) and B(Y ), the mirror map wouldbe uniquely determined up to an isometry; isometries depend on finitely many constants,which one might determine by matching a few coefficients at infinity. )Vanishing Of Instanton Corrections To The MetricIt remains to show that instantons do not contribute to the metric of the A(X) model.The proof is essentially dimension counting, taking account of the C∗action on a genus zerosurface with two marked points.

Let X be a complex manifold, not necessarily Calabi-Yau,and Σ a Riemann surface of genus zero. Consider a component U of the moduli space of13 I should stress that this “topological” metric in no way coincides with the Zamolodchikovmetric on the moduli space – which is much more difficult to study and involves what has beencalled “topological anti-topological fusion” [[12]].31

holomorphic maps Φ : Σ →X of virtual dimension w. Let H and H′ be submanifolds (orhomology cycles) of codimension q and q′ with q + q′ = w, and let P and P ′ be two pointsin Σ. Let U′ be the subspace of U parameterizing Φ’s with Φ(P) ∈H and Φ(P ′) ∈H′.

IfU′ is empty, the contribution of this homotopy class to ⟨O(0)H O(0)H′ ⟩is zero. As the virtualdimension of U′ is zero, it appears superficially that U′ need not vanish generically.However, let us take account of the C∗action on Σ fixing P and P ′.

Let eU = U′/C∗.U′ must be empty if eU is. One can think of eU as the moduli space of rational curves C ∈Xthat intersect both H and H′ (without specifying any parameterization of C or map fromΣ to C).

The virtual dimension of eU is −2. Hence, eU is “generically” empty, and willreally be empty at worst after making a generic nonintegrable deformation of the almostcomplex structure of X.

So the instanton corrections to the metric vanish.As a very concrete example of this counting, let X be a particular Calabi-Yau manifold,a quintic hypersurface in P4. The virtual dimension of U is six (regardless of the homotopyclass of the map considered).

Let H, H′ be two homology cycles in X the sum of whosecodimensions is six. So one of them, say H, has codimension at least three.

With thisparticular X, even for generic integrable complex structures, it is believed that the numberof rational curves of given positive degree in X is finite; if so, the union Z of these curveshas codimension four. Since 4 + 3 > 6, H can be perturbed slightly so as not to intersectZ; hence U′ is empty, and the instanton contribution to the metric vanishes.The Exponential MapThe description of the topological family by a family of LagrangianseL = L +XataZΣO(2)a(7.27)shows that once we pick a base Lagrangian L, corresponding to a base point P ∈N ,there is a natural linear structure on N .

This might be described as an exponential mapfrom the tangent space to N at P to N .In the case of the A model, this structurecorresponds to the linear structure on ⊕nHn(X,C). In the case of the B model, it is notpresently understood.

Understanding the exponential map should be roughly similar tounderstanding the metric on the moduli space. It is puzzling, in particular, that in the Amodel, the linear structure on the moduli space does not seem to depend on the choice ofbase point, while in the B model such a dependence seems almost inevitable.32

References[1]E. Witten, “Topological Sigma Models,” Commun. Math.

Phys. 118 (1988) 411.[2]C.

Vafa, “Topological Mirrors And Quantum Rings,” contribution to this volume.[3]E. Witten, “The N Matrix Model and Gauged WZW Models,” IAS preprint, to appearin Nucl.

Phys. B.[4]P.

S. Aspinwall and D. R. Morrison, “Topological Field Theory And Rational Curves,”Oxford and Duke preprints OUTP-91-32P, DUK-M-91-12.[5]M. Dine, N. Seiberg, X.-G. Wen, and E. Witten, “Nonperturbative Effects On TheString World Sheet I,II,” Nucl.

Phys. B278 (1986) 769, 289 (1987) 319.[6]P.

S. Landweber, ed., “Elliptic Curves And Modular Forms in Algebraic Topology,”Lecture Notes In Mathematics 1326 (Springer-Verlag, 1988).[7]M. Gromov and M. A. Shubin, “The Riemann-Roch Theorem For General EllipticOperators,” preprint.[8]M.

F. Atiyah and L. Jeffrey, “Topological Lagrangians And Cohomology,” J. Geom.Phys. 7 (1990) 119.[9]P.

Candelas, P. Green, L. Parke, and X. de la Ossa, “A Pair Of Calabi-Yau ManifoldsAs An Exactly Soluble Superconformal Field Theory,” Nucl. Phys.

B359 (1991) 21.[10]W. Lerche, C. Vafa, and N. P. Warner, “Chiral Rings in N = 2 SuperconformalTheories,” Nucl.

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Baulieu and I. M. Singer, “The Topological Sigma Model,” Commun. Math.

Phys.125 (1989) 227.[12]S. Cecotti and C. Vafa, “Topological Anti-Topological Fusion,” Harvard Universitypreprint (1991).[13]R.

Dijkgraaf, E. Verlinde, and H. Verlinde, “Topological Strings In D < 1,” IASSNS-HEP-90/71.[14]C. Vafa.

“Topological Landau-Ginzberg Models,” Harvard preprint, (1990).[15]K. Saito, “Period Mapping Associated To A Primitive Form,” Publ.

RIMS, KyotoUniv. 19 (1983) 1231.[16]B.

Blok and A. Varchenko, “Topological Conformal Field Theories And The FlatCoordinates,” IASSNS 91-5.33

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