Algebra of the Infrared with Curve-Valued Potential

We study an extension of the algebra of the infrared to curve-valued potentials, focusing on the elliptic curve case. Given a finite configuration of points on an elliptic curve, we construct associated $L_{\mathrm{\infty }}$- and $A_{\mathrm{\infty }}$-algebras. In contrast with the classical planar setting, the resulting $A_{\mathrm{\infty }}$-structure depends essentially on the choice of extra data, leading to new phenomena involving the fundamental group of the base curve. We also discuss the expected relation of these constructions to Fukaya-Seidel categories.