Maurer–Cartan equation for the curvature of the family Floer complex

Prove that for a suitable model of Floer theory, the curvature µ^0 of the family Floer complex 𝔠=C^*(B^0;C^*(F_b)⊗O_an) satisfies the Maurer–Cartan equation δµ^0 + (1/2){µ^0,µ^0} = 0 with respect to the classical differential δ and the Chas–Sullivan string bracket on H^*(F_b) ⊗ K[H_1(F_b)].

Background

To address corrections in SYZ mirror symmetry, the author outlines a conjectural framework in which moduli of holomorphic discs determine a curved A∞-algebra structure on a family Floer complex over the smooth locus B⁰ of a Lagrangian torus fibration. The key proposed constraint is that its curvature should satisfy a Maurer–Cartan equation governed by the string bracket, producing a corrected differential.

References

It is conjectured in that, for a suitable model of Floer theory, the curvature $\mu^0$ of $\mathfrak{C}$ satisfies the Maurer-Cartan equation $\delta\mu^{0+\frac12{\mu^{0,\mu^0}=0$}} (or its $L_\infty$ analogue) with respect to the classical differential and the bracket eq:HF-bracket.

eq:HF-bracket:

$\{z^\gamma\, \alpha, z^{\gamma'} \alpha'\}=z^{\gamma+\gamma'}\, \bigl(\alpha\wedge (\iota_\gamma \alpha')+(-1)^{|\alpha|} (\iota_{\gamma'}\alpha)\wedge \alpha'\bigr).\vspace*{-4pt} $

— Lagrangian Floer theory, from geometry to algebra and back again (2510.22476 - Auroux, 26 Oct 2025) in Section 3.1 (Floer theory for families of Lagrangians), final paragraphs

Maurer–Cartan equation for the curvature of the family Floer complex

Background

References

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