Fukaya category on a symplectic manifold with a B-field

TL;DR

This work extends the Fukaya category to symplectic manifolds equipped with a B-field by using the complexified form $\omega_\mathbb{C}=B+i\omega$ and develops an explicit framework for the $A_\infty$-structure under Lagrangian isotopies. It introduces Z-gradings under $2c_1(TX)=0$ via a global frame $\Theta$ of $\det(T_\mathbb{C}X)^{\otimes 2}$ and defines graded Lagrangian branes with compatible local systems, giving a precise construction of morphisms $CF^*(\hat{L}_0,\hat{L}_1)$ and the higher products $\mu^k$ as counts of $J$-holomorphic discs with weights from relative de Rham data. The disc weights are encoded by a relative pairing $\langle (B,\theta),[u]\rangle$ and allow bulk deformations by the B-field, while the main isotopy result provides an exact formula relating $\mu^k$ under Lagrangian isotopies via $e^{2\pi(f(p_{m+1})-f(p_m))}$. Together, these contributions establish a robust open-string, B-field–aware A$_\infty$-algebra and its deformation behavior, with implications for Homological Mirror Symmetry in the presence of B-fields.

Abstract

We describe the formulation of Fukaya categories of symplectic manifolds with $B$-fields. In addition, we give a formula for how the $A_\infty$ structure maps change as we deform an object by a Lagrangian isotopy.

Figures (3)

• Figure 1: The map $u'$ defined in Equation \ref{["eqn:define u'"]}. The orientation on $\partial \mathbb D\times [0,1]$ is given by the ordered basis $(e_1=\frac{\partial}{\partial \theta}, e_2=\frac{\partial}{\partial r})$.
• Figure 2: Domain (left) and target (right) of a holomorphic disc contributing to $\mu^k$. There is an isotopy between Lagrangians $L_m$ and $L_m'$. This figure illustrates the notations in Section \ref{['sec: isotopy']}, below Equation \ref{['eq: product in isotopy section']}.
• Figure 3: The domain (left) and image (right) of $\phi_2:R_2\to \partial_m\mathbb D\times [0,1]$.

Theorems & Definitions (47)

• Definition 1.1
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Example 4.1
• Example 4.2: associated vector bundles
• Example 4.3
• Lemma 4.4
• proof
• Remark 4.5