Maurer–Cartan property of the curvature in the family Floer complex

Prove that, for a suitable model of Floer theory in the SYZ/family Floer setting, the curvature μ^0 of the curved A_infinity algebra 𝔠 satisfies the Maurer–Cartan equation δμ^0 + 1/2{μ^0, μ^0} = 0 with respect to the classical differential and the Chas–Sullivan string bracket on H^*(F_b) ⊗ K[H_1(F_b)], or its L_infinity analogue.

Background

In the proposed ‘noncommutative’ correction framework for SYZ mirror symmetry, the authors consider an A_infinity algebra 𝔠 built from Floer data with coefficients in a sheaf O_an over the base of a Lagrangian torus fibration. They conjecture that the curvature term μ⁰ of 𝔠 satisfies a Maurer–Cartan equation governed by the string bracket, which would induce a corrected differential δ + {μ^0,·} squaring to zero and encode mirror corrections.

Establishing this property would systematize corrections from Maslov index zero and negative index discs and clarify the algebraic structure underlying the corrected mirror.

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,” subsection discussing negative Maslov index and extended deformations

Maurer–Cartan property of the curvature in the family Floer complex

Background

References

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