\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

mirror symmetry: Hodge numbers, SYZ, and Yukawa couplings

This article collects a short conceptual route into Calabi--Yau mirror symmetry. The emphasis is on the numerical ``mirror'' in the Hodge diamond, the SYZ picture explaining why dual tori exchange the relevant cohomology groups, and the Yukawa-coupling comparison which historically turned mirror symmetry into an enumerative tool.

1: why is it called mirror symmetry?

Mirror symmetry is a proposed duality between symplectic geometry and complex geometry. In one of its most familiar forms, it starts with a Calabi--Yau manifold \(X\) and predicts another Calabi--Yau manifold \(\check X\), called a mirror, whose complex geometry records symplectic information about \(X\) and whose symplectic geometry records complex information about \(X\). The mirror is not determined by the Hodge numbers alone; the Hodge-number symmetry is the first visible numerical shadow of a stronger correspondence. This slogan is a useful entry point into the more categorical form of mirror symmetry discussed in HMS for Fanos. In the Calabi--Yau setting, the mirror symmetry conjectures predict two related kinds of data:
  • to a suitable symplectic or Kähler Calabi--Yau manifold \(X\), a complex Calabi--Yau manifold \(\check X\) called a mirror of \(X\);
  • a dictionary translating symplectic invariants of \(X\) into complex invariants of \(\check X\), and conversely.
Some influential early papers include [CdlOGP91], [Kon94], [SYZ96], and [GS03]. The purpose of this note is to explain the elementary reason for the word ``mirror'', then to point toward the SYZ and enumerative pictures which motivate later formulations.

definition 0.0.1

In these notes, a Calabi--Yau \(n\)-fold is a compact connected Kähler manifold \(X\) of complex dimension \(n\) whose canonical bundle \[ K_X:=\Lambda^n(T^{1,0}X)^* \] is holomorphically trivial. Equivalently, \(X\) admits a nowhere-vanishing holomorphic volume form \[ \Omega\in H^0(X,K_X). \] Thus a Calabi--Yau manifold carries compatible complex and symplectic data:
  • an integrable complex structure \(J:TX\to TX\);
  • a Kähler form \(\omega\in \Omega^2(X)\), which in particular makes \(X\) a symplectic manifold;
  • a holomorphic volume form \(\Omega\) trivializing \(K_X\).
This convention does not impose the stronger condition \(h^{p,0}(X)=0\) for \(0<p<n\), nor does it require \(X\) to be simply connected. When those extra hypotheses are needed, they will be stated explicitly. Historically, physicists were interested in Calabi--Yau manifolds because of their role in superstring theory: the compact Calabi--Yau directions provide candidates for the hidden internal geometry of space-time. Early expectations suggested that only a small number of Calabi--Yau threefolds might occur up to deformation, but the search for string geometries produced many more examples; see, for example, [Hüb92]. Mathematically, the point is that Calabi--Yau manifolds sit at the meeting place of symplectic, complex, and Riemannian geometry.

theorem 0.0.2

Let \((X,J)\) be a compact Kähler manifold of complex dimension \(n\) with \(c_1(TX)=0\) in \(H^2(X;\RR)\). For every Kähler class \[ \alpha\in H^{1,1}(X;\RR) \] there is a unique Kähler form \(\omega\in\alpha\) whose Ricci form is zero. If \(K_X\) is holomorphically trivial, then with respect to this metric a nowhere-vanishing holomorphic \(n\)-form is parallel, and the holonomy is contained in \(SU(n)\). Conversely, a Kähler metric with holonomy contained in \(SU(n)\) is Ricci-flat and admits a parallel nowhere-vanishing holomorphic \(n\)-form. Yau proved the Calabi conjecture in [Yau78]. Morally, the Calabi--Yau condition is what lets the symplectic, complex, and Riemannian parts of the geometry interact with unusually little friction: the Kähler class determines a Ricci-flat metric, while the holomorphic volume form constrains the complex geometry.

0.1: Hodge diamond and deformation

definition 0.0.3

This definition uses the Hodge decomposition. The Hodge diamond of a compact Kähler manifold \(X\) of complex dimension \(n\) is the array of Hodge numbers \(h^{p,q}(X)=\dim_{\CC}H^{p,q}(X)\) arranged so that entries with fixed \(p+q\) lie on the same row and reflection across the vertical axis interchanges the two indices. Thus the row indexed by \(k\) contains the numbers corresponding to the summands in \[ H^k(X;\CC)=\bigoplus_{p+q=k}H^{p,q}(X). \] For a threefold, the diamond has rows \[ \begin{array}{ccccccc} &&& h^{0,0} &&&\\ && h^{1,0} && h^{0,1} &&\\ & h^{2,0} && h^{1,1} && h^{0,2} &\\ h^{3,0} && h^{2,1} && h^{1,2} && h^{0,3}\\ & h^{3,1} && h^{2,2} && h^{1,3} &\\ && h^{3,2} && h^{2,3} &&\\ &&& h^{3,3} &&& \end{array} \] The diamond is not extra structure beyond the Hodge decomposition; it is a compact way of displaying the dimensions of its summands. Let \(X\) be a compact Kähler \(n\)-fold. Its Hodge numbers satisfy:
  • complex conjugation gives \[ h^{p,q}(X)=h^{q,p}(X); \]
  • Serre duality gives \[ h^{p,q}(X)=h^{n-p,n-q}(X); \]
  • together, these equalities imply that the usual Hodge diamond is symmetric across both the vertical and horizontal axes.
If \(X\) is Calabi--Yau in the sense of definition 0.0.1, then \(K_X\simeq\mathcal O_X\), and Serre duality applied to \(\mathcal O_X\) gives \[ h^{0,q}(X)=h^{0,n-q}(X). \] Thus a connected Calabi--Yau threefold has \[ h^{0,0}=h^{3,0}=h^{0,3}=h^{3,3}=1. \] In complex dimension three, this implies that if \(h^{1,0}(X)=0\), then \(h^{2,0}(X)=0\) as well. For a connected Calabi--Yau threefold satisfying the additional vanishing \(h^{1,0}=0\), the preceding symmetries force the Hodge diamond to have the form Calabi--Yau threefold Hodge diamondAfter the standard symmetries and this vanishing assumption are accounted for, the Hodge diamond is controlled by two numbers: \(h^{1,1}\) and \(h^{2,1}\). These two numbers measure two different deformation directions.
0.0.1: \(h^{1,1}\) and the K\"ahler cone

definition 0.0.4

Let \(X\) be a compact Kähler manifold. Under the Hodge decomposition, the real \((1,1)\)-cohomology is \[ H^{1,1}(X;\RR):=H^{1,1}(X;\CC)\cap H^2(X;\RR). \] The K\"ahler cone of \(X\) is \[ \mathcal K_X :=\{[\omega]_{\mathrm{dR}}\in H^{1,1}(X;\RR)\mid \omega \text{ is a Kähler form on }X\}. \] It is an open convex cone in \(H^{1,1}(X;\RR)\). A point of \(\mathcal K_X\) is the de Rham cohomology class of a compatible symplectic form on the fixed complex manifold \(X\). In mirror symmetry one often uses the complexified K\"ahler parameter \[ [B]+i[\omega]\in H^2(X;\CC), \] where \([B]\in H^2(X;\RR)\) is usually considered modulo integral classes. The number \(h^{1,1}\) controls first-order deformations of the Kähler class while keeping the complex structure fixed. More precisely, the Kähler cone \(\mathcal K_X\) is an open subset of the real vector space \[ H^{1,1}(X)\cap H^2(X,\RR). \] Therefore its real dimension is \(h^{1,1}(X)\). An infinitesimal deformation of the Kähler form is represented by a real closed \((1,1)\)-form \(\eta\). Modding out by exact deformations leaves the cohomology class \([\eta]\). In Dolbeault notation, the corresponding complex vector space is \[ H^1(X,\Omega_X^1)= \frac{\ker(\bar\partial: \Omega^{1,1}(X)\to \Omega^{1,2}(X))} {\operatorname{im}(\bar\partial: \Omega^{1,0}(X)\to \Omega^{1,1}(X))}. \] This is the Hodge-theoretic shadow of the symplectic fact that, on a compact manifold, deformations of symplectic forms in a fixed de Rham cohomology class are equivalent up to isotopy under the hypotheses of Moser's theorem. The takeaway is that \(h^{1,1}\) measures first-order Kähler, hence symplectic, deformations.
0.0.2: \(h^{2,1}\) and complex deformation

definition 0.0.1

Let \(X\) be a compact complex manifold. A first-order deformation of the complex structure on \(X\) is a flat deformation of \(X\) over the dual numbers \(\CC[\epsilon]/(\epsilon^2)\), together with an identification of the special fiber with \(X\). Equivalence classes of first-order deformations are naturally identified with the Kodaira--Spencer group \[ H^1(X,T_X), \] where \(T_X=T^{1,0}X\) is the holomorphic tangent bundle. Analytically, after choosing a splitting, a nearby almost complex structure may be represented by a Beltrami differential \[ s\in \Omega^{0,1}(X,T_X). \] The integrability condition is the Maurer--Cartan equation \[ \bar\partial s+\frac{1}{2}[s,s]=0. \] To first order this reduces to \(\bar\partial s=0\), and infinitesimal changes of coordinates change \(s\) by elements of \(\bar\partial\Omega^0(X,T_X)\). Thus the Zariski tangent space to the deformation functor or Kuranishi space is \(H^1(X,T_X)\), while obstruction classes naturally take values in \(H^2(X,T_X)\). The number \(h^{2,1}\) controls first-order deformations of complex structure on a Calabi--Yau threefold. Let \(J\) and \(J'\) be two nearby almost complex structures. A nearby complex structure can be described by its antiholomorphic tangent bundle \(T^{0,1}_{J'}X\). After choosing the splitting determined by \(J\), this is the graph of a map \[ s:T^{0,1}_JX\to T^{1,0}_JX, \] which is the same as an element \[ s\in \Omega^{0,1}(X,T_X^{1,0}). \]
  • The condition that the new almost complex structure be integrable is the Maurer--Cartan equation \[ \bar\partial s+\frac{1}{2}[s,s]=0. \] To first order, this becomes \(\bar\partial s=0\).
  • Some deformations of complex structure arise from pulling back by a diffeomorphism. Infinitesimally, these lie in the image of \[ \bar\partial: \Omega^0(X,T_X^{1,0})\to \Omega^{0,1}(X,T_X^{1,0}). \]
  • Therefore, the space of first-order deformations of complex structure is \[ \frac{\ker(\bar\partial: \Omega^{0,1}(X,T_X^{1,0})\to \Omega^{0,2}(X,T_X^{1,0}))} {\operatorname{im}(\bar\partial: \Omega^{0}(X,T_X^{1,0})\to \Omega^{0,1}(X,T_X^{1,0}))} =H^{0,1}(X,T_X^{1,0})=H^1(X,T_X). \] On a Calabi--Yau \(n\)-fold, contraction with a holomorphic volume form identifies \(T_X\) with \(\Omega_X^{n-1}\). Thus \[ H^1(X,T_X)\simeq H^1(X,\Omega_X^{n-1})\simeq H^{n-1,1}(X). \] For a Calabi--Yau threefold, this is \(H^{2,1}(X)\).
The Kodaira--Spencer map identifies the tangent space to the moduli space of complex structures with \(H^1(X,T_X)\). On Calabi--Yau manifolds these first-order deformations are unobstructed by the Bogomolov--Tian--Todorov theorem, so locally the deformation space is smooth of dimension \(h^{n-1,1}(X)\). The takeaway is that \(h^{2,1}\) measures first-order complex-structure deformations.

0.2: the mirror flip

This is the elementary origin of the name ``mirror symmetry.'' If \(X\) and \(\check X\) are mirror Calabi--Yau \(n\)-folds, then the expected numerical relation is \[ h^{p,q}(X)=h^{n-p,q}(\check X). \] For Calabi--Yau threefolds satisfying the simplifying vanishing used above, this says in particular that variations of Kähler or symplectic structure on \(X\) should correspond to variations of complex structure on \(\check X\), and conversely. Since the dimensions of these two deformation spaces are governed by Hodge numbers, one expects \[ h^{1,1}(X)=h^{2,1}(\check X),\qquad h^{2,1}(X)=h^{1,1}(\check X). \] Pictorially, for threefolds this mirror relation exchanges the vertical \(h^{1,1}\) entries with the horizontal \(h^{2,1}\) entries: mirror flip of Hodge diamondsMany mirror pairs of Calabi--Yau manifolds were discovered by searching for this kind of numerical symmetry, and those examples provided early evidence for mirror symmetry. The equality of Hodge numbers is only a necessary compatibility check; the stronger mirror statement also identifies deformation theory, enumerative invariants, and eventually categories.

2: constructing mirror manifolds and implications

The Hodge-number flip explains the name, but it does not yet explain where the mirror should come from. The next layer of the story is constructive: one tries to build the mirror family by degenerating \(X\) to a combinatorial, toric, or affine limit, dualizing the relevant data, and then smoothing the result. The constructions below are schematic, but the point is precise: mirror symmetry is usually a statement about matched families near boundary points of moduli, not just about two isolated manifolds.

1.1: constructing mirrors

The first examples of mirror symmetry involved Calabi--Yau hypersurfaces in projective space. Let \(H\subset \CP^n\) be a smooth hypersurface of degree \(d\), so \(H\) has complex dimension \(n-1\). We can compute its first Chern class using the exact sequence \[ 0 \to T^{1,0}H\to T^{1,0}\CP^n|_H\to N_{H/\CP^n}\to 0. \] This tells us that \[ c(T\CP^n)|_H=c(TH)\cdot c(N_{H/\CP^n}). \] Suppose that the hypersurface has degree \(d\), and let \(h=c_1(\mathcal O_{\CP^n}(1))|_H\). Then \[ c(T\CP^n)|_H=(1+h)^{n+1},\qquad c(N_{H/\CP^n})=1+dh. \] Therefore, \[ 1+(n+1)h+\cdots=(1+c_1(TH)+\cdots)(1+dh), \] so \[ c_1(TH)=(n+1-d)h. \] Equivalently, by adjunction, \[ K_H\simeq \mathcal O_H(d-n-1). \] Thus \(d=n+1\) is necessary and sufficient for \(H\) to have trivial canonical bundle. If \(H\) is smooth, a degree \(n+1\) hypersurface in \(\CP^n\) is a compact Kähler manifold with trivial canonical bundle, hence is Calabi--Yau in the sense of definition 0.0.1. The quintic threefold in \(\CP^4\) is the standard example.

definition 1.0.1

Let \[ \pi:\mathcal X^*\to \Delta^* \] be a one-parameter family of Calabi--Yau \(n\)-folds over a punctured disc. Parallel transport around the puncture gives a monodromy operator \[ T:H^n(X_t;\QQ)\to H^n(X_t;\QQ). \] After replacing the punctured disc by a finite cover if necessary, assume that the monodromy is unipotent, and write \(T\) for this unipotent monodromy. Then one can define \[ N:=\log T=(T-I)-\frac{(T-I)^2}{2}+\frac{(T-I)^3}{3}-\cdots . \] The series is finite because \(T-I\) is nilpotent. In this introductory convention, a boundary point \(t=0\) is called a large complex-structure limit, or a maximally unipotent monodromy point, if \[ N^n\neq 0 \] on \(H^n(X_t;\QQ)\). Equivalently, \(N\) has nilpotency index \(n+1\), the maximum possible for a weight-\(n\) variation of Hodge structure. More refined definitions also impose conditions on the limiting mixed Hodge structure; the maximally unipotent monodromy criterion is the part used in this introductory discussion. For the degree \(n+1\) hypersurface family, there is a particularly degenerate limit to the union of the \(n+1\) coordinate hyperplanes. The central fiber is a singular toric normal-crossings variety, not a smooth toric manifold. A general strategy for constructing mirrors is to do mirror symmetry in families. Let \(\mathcal X\to \Delta\) be a degeneration of Calabi--Yau manifolds whose central fiber \(X_0\) is toric or toroidal and possibly singular. Then the strategy is:
  1. find a toric or toroidal degeneration of the family to \(X_0\);
  2. construct a mirror degeneration \(\check X_0\) using the dual toric or affine-combinatorial data;
  3. deform or smooth \(\check X_0\) to obtain the mirror family \(\check{\mathcal X}\), and match the deformation parameters using the mirror map.
There are several motivating reasons for taking this path:
  • The SYZ proposal interprets mirror symmetry as fiberwise \(T\)-duality, which suggests looking at dual tori.
  • Homological mirror symmetry suggests that a Lagrangian torus fiber equipped with a flat unitary local system should correspond to a skyscraper sheaf at a point of the mirror. Thus torus fibrations are natural objects to examine.
  • The topological calculation below shows why dualizing torus fibers exchanges the two Hodge numbers.

definition 1.0.2

Let \((X,\omega,\Omega)\) be a Calabi--Yau \(n\)-fold with its Ricci-flat Kähler metric, and assume that \(\Omega\) is normalized with respect to this metric. For \(\theta\in \RR/2\pi\ZZ\), an oriented real \(n\)-dimensional submanifold \(L\subset X\) is special Lagrangian of phase \(\theta\) if \[ \omega|_L=0,\qquad \operatorname{Im}(e^{-i\theta}\Omega)|_L=0, \] and \[ \operatorname{Re}(e^{-i\theta}\Omega)|_L=\operatorname{vol}_L \] as positive volume forms on \(L\). A special Lagrangian torus fibration of phase \(\theta\) is a continuous map \[ f:X\to B \] to a real \(n\)-dimensional base such that, over a dense open subset \(B_0\subset B\), the restriction \[ f^{-1}(B_0)\to B_0 \] is a smooth fiber bundle whose fibers are \(n\)-tori that are special Lagrangian submanifolds of phase \(\theta\). In the SYZ setting, nontrivial compact examples are expected to have singular fibers over a discriminant locus \(\Delta=B\setminus B_0\).

definition 1.0.3

Let \[ f:X_0\to B_0 \] be a smooth torus fibration with fibers \(F_b\simeq V_b/\Lambda_b\), where \(V_b\) is a real vector space and \(\Lambda_b\subset V_b\) is a lattice. The dual torus fibration is \[ \check f:\check X_0\to B_0, \qquad \check F_b:=\check f^{-1}(b)=\operatorname{Hom}(\Lambda_b,U(1)) \simeq H^1(F_b;\RR)/H^1(F_b;\ZZ). \] Fiberwise, \[ H_1(\check F_b;\ZZ)\simeq H^1(F_b;\ZZ), \qquad H^1(\check F_b;\ZZ)\simeq H_1(F_b;\ZZ). \] If the three-torus fibers are oriented, then Poincar\'e duality, with respect to the chosen orientations, gives the useful identifications \[ H^1(\check F_b;\CC)\simeq H^2(F_b;\CC), \qquad H^2(\check F_b;\CC)\simeq H^1(F_b;\CC). \] These fiberwise identifications induce the corresponding dual local systems over \(B_0\). Here is a motivating calculation from [Gro01]. It deliberately ignores many details about singular fibers, which are essential in the actual theory, but it captures the reason dual tori produce the mirror flip. Suppose that we have a special Lagrangian torus fibration \[ f:X\to B \] of a Calabi--Yau threefold. Assume, for this heuristic calculation, that the fibration is smooth, that the fibers are three-tori \(F=f^{-1}(p)\), that \(X\) has the Hodge diamond shape from 0.1, and that the base \(B\) is a closed oriented three-manifold with \(H^1(B,\CC)=H^2(B,\CC)=0\). A globally smooth torus fibration with simply connected total space is not the actual situation for compact Calabi--Yau threefolds; singular fibers are necessary. The simplified model is still useful because it isolates the cohomological mechanism. Let \[ \mathcal H^q:=R^q f_*\CC \] denote the local system over \(B\) whose fiber at \(p\) is \(H^q(F,\CC)\). The special Lagrangian condition gives an orientation of the fibers: after choosing the phase, \(\operatorname{Re}\Omega\) restricts to a volume form on each fiber. Thus \[ \mathcal H^0\simeq \CC,\qquad \mathcal H^3\simeq \CC. \] The Leray--Serre spectral sequence has \[ E_2^{p,q}=H^p(B,\mathcal H^q)\Longrightarrow H^{p+q}(X,\CC). \] The \(E_2\) page has the form Leray--Serre \(E_2\) pageUsing \(\mathcal H^0\simeq \mathcal H^3\simeq \CC\) and \(H^1(B,\CC)=H^2(B,\CC)=0\), the top and bottom rows become Leray--Serre page with trivial top and bottom rowsAssume further, as part of this simplified model, that the spectral sequence degenerates at \(E_2\) and that the relevant invariant-cycle terms vanish: \[ H^0(B,\mathcal H^1)=H^0(B,\mathcal H^2)=0. \] By duality, this also gives \[ H^3(B,\mathcal H^1)=H^3(B,\mathcal H^2)=0. \] Then the relevant part of the spectral sequence is relevant terms in the Leray--Serre pageConsequently, \[ H^2(X,\CC)\simeq H^1(B,\mathcal H^1), \] and \[ H^3(X,\CC)\simeq \CC\oplus H^1(B,\mathcal H^2)\oplus H^2(B,\mathcal H^1)\oplus \CC. \] Since \(X\) is a Calabi--Yau threefold with \(h^{2,0}=0\), we have \[ \dim H^2(X,\CC)=h^{1,1}(X),\qquad \dim H^3(X,\CC)=2h^{2,1}(X)+2. \] Moreover, Poincar\'e duality on the base with local coefficients, together with fiberwise duality, gives \[ \dim H^1(B,\mathcal H^2)=\dim H^2(B,\mathcal H^1). \] Therefore, \[ h^{1,1}(X)=\dim H^1(B,\mathcal H^1),\qquad h^{2,1}(X)=\dim H^1(B,\mathcal H^2). \] Now look at the dual torus fibration \[ \check f:\check X\to B. \] By definition 1.0.3, fiberwise the dual torus satisfies \[ H^1(\check F,\CC)\simeq H_1(F,\CC)\simeq H^2(F,\CC), \] and similarly \[ H^2(\check F,\CC)\simeq H_2(F,\CC)\simeq H^1(F,\CC). \] Thus the local systems for the dual fibration are swapped: \[ \check{\mathcal H}^1\simeq \mathcal H^2, \qquad \check{\mathcal H}^2\simeq \mathcal H^1. \] Applying the same calculation to \(\check X\) gives \[ h^{1,1}(\check X)=\dim H^1(B,\check{\mathcal H}^1) =\dim H^1(B,\mathcal H^2)=h^{2,1}(X), \] and \[ h^{2,1}(\check X)=\dim H^1(B,\check{\mathcal H}^2) =\dim H^1(B,\mathcal H^1)=h^{1,1}(X). \] This is the topological shadow of the mirror flip of Hodge numbers.

3: where to go from here?

If mirror symmetry relates deformations of symplectic structures to deformations of complex structures, then symplectic and complex invariants should have related deformation theories. One of the original predictions of mirror symmetry was an equality between two different Yukawa couplings.

definition 2.0.1

We use the genus-zero Gromov--Witten invariants from genus-zero Gromov--Witten invariant. Let \(X\) be a smooth projective Calabi--Yau threefold. The \(A\)-model Yukawa coupling is the formal trilinear form on divisor classes \(\alpha_1,\alpha_2,\alpha_3\in H^{1,1}(X;\CC)\) given by \[ \langle \alpha_1,\alpha_2,\alpha_3\rangle_A = \int_X\alpha_1\wedge\alpha_2\wedge\alpha_3 + \sum_{0\neq \beta\in H_2(X;\ZZ)_{\mathrm{eff}}} \langle \alpha_1,\alpha_2,\alpha_3\rangle_{0,\beta}\,q^\beta. \] The first term is the classical triple intersection product. The remaining terms are genus-zero Gromov--Witten quantum corrections. If the complexified Kähler parameter is \([B]+i[\omega]\), then one common convention is \[ q^\beta=\exp\left(2\pi i\int_\beta(B+i\omega)\right). \] Different sign conventions for \(q^\beta\) appear in the literature. In the special case of divisor insertions on a Calabi--Yau threefold, the divisor axiom rewrites this as \[ \langle \alpha_1,\alpha_2,\alpha_3\rangle_{0,\beta} =N_\beta\left(\int_\beta \alpha_1\right) \left(\int_\beta\alpha_2\right) \left(\int_\beta\alpha_3\right), \] where \(N_\beta\) is the relevant genus-zero virtual count of rational curves in the class \(\beta\). If the \(\alpha_i\) are dual to submanifolds \(A_1,A_2,A_3\), this invariant can be heuristically interpreted as counting rational curves in class \(\beta\) which meet \(A_1,A_2,A_3\). Thus the \(A\)-model Yukawa coupling is a power series in Kähler parameters: the classical term is the triple intersection product, and the higher-order terms come from holomorphic curve counts.

definition 2.0.2

Let \(\check X\) be a compact Calabi--Yau threefold, and choose a nowhere-vanishing holomorphic volume form \(\Omega\). The \(B\)-model Yukawa coupling associated to \(\Omega\) is the symmetric trilinear form on first-order complex-structure deformations \[ H^1(\check X,T_{\check X})^{\otimes 3}\to \CC \] defined by \[ \langle a_1,a_2,a_3\rangle_B = \int_{\check X}\Omega\wedge \big((a_1\wedge a_2\wedge a_3)\lrcorner\,\Omega\big). \] Here the \(a_i\) may be represented by \((0,1)\)-forms with values in \(T_{\check X}\), and contraction with \(\Omega\) identifies the \(\wedge^3T_{\check X}\) factor with \(\mathcal O_{\check X}\). In a family, these couplings can be described equivalently in terms of the variation of Hodge structure on \(H^3(\check X)\). Expanding the \(B\)-model Yukawa coupling in flat complex-structure coordinates gives another power series. Unlike the \(A\)-side expression, this series can often be computed from period integrals and the Picard--Fuchs equations for the family \(\check X\).

definition 2.0.3

Let \(X\) and \(\check X\) be a proposed mirror pair of Calabi--Yau manifolds. Given chosen boundary points in the complexified Kähler moduli of \(X\) and the complex-structure moduli of \(\check X\), a mirror map is a local analytic or formal identification of the corresponding parameter spaces: \[ \text{complexified Kähler parameters of }X \quad\leftrightarrow\quad \text{complex-structure parameters of }\check X. \] Depending on convention, either this identification or its inverse is called the mirror map. Near a large-radius limit on the \(A\)-side and a large complex-structure limit on the \(B\)-side, this identification is usually normalized using flat coordinates on the \(B\)-model side obtained from periods of the holomorphic volume form on \(\check X\). It is the coordinate change used to compare \(A\)-model structures on \(X\) with \(B\)-model structures on \(\check X\). Mirror symmetry says that, after the mirror map identifies Kähler parameters of \(X\) with complex-structure parameters of \(\check X\), these two power series agree. The \(A\)-model Yukawa coupling deforms the classical triple product on \(H^{1,1}(X)\) by holomorphic curve corrections, while the \(B\)-model Yukawa coupling is controlled by the variation of Hodge structure on \(H^3(\check X)\). This was the form of mirror symmetry used in [CdlOGP91] to extract enumerative predictions.

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References

[CdlOGP91]Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes. A pair of Calabi--Yau manifolds as an exactly soluble superconformal theory. Nuclear Physics B, 359(1):21--74, 1991.
[Gro01]Mark Gross. Special Lagrangian fibrations i: Topology. AMS/IP Studies in Advanced Mathematics, 23:65--94, 2001.
[GS03]Mark Gross and Bernd Siebert. Affine manifolds, log structures, and mirror symmetry. Turkish Journal of Mathematics, 27(1):33--60, 2003.
[Hüb92]Tristan Hübsch. Calabi--Yau manifolds: A bestiary for physicists. World Scientific, 1992.
[Kon94]Maxim Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, 1994.
[SYZ96]Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Mirror symmetry is T-duality. Nuclear Physics B, 479(1--2):243--259, 1996.
[Yau78]Shing-Tung Yau. On the ricci curvature of a compact Kähler manifold and the complex Monge--Ampere equation, i. Communications on Pure and Applied Mathematics, 31(3):339--411, 1978.