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

Symplectic snippets

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 0.0.1

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 0.0.2

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 0.0.3

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.