\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
\(B\)-model Yukawa coupling
definition 0.0.1 [Mirror symmetry motivation notes]
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 \(H^3(\check X)\).
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