Algebra of the infrared and Fukaya–Seidel categories

Abstract

The algebra of the infrared was introduced by Gaiotto--Moore--Witten as an algebraic structure associated with massive two-dimensional (\mathcal N=(2,2)) theories. In the mathematical formulation developed by Kapranov, Kontsevich, and Soibelman, the combinatorics of finite point configurations in the plane, together with the geometry of secondary polytopes, gives rise to natural (L_\infty)- and (A_\infty)-algebras. These algebras encode the higher operations coming from instanton counting and provide an algebraic model for the algebra of the infrared. This dissertation studies an extension of this construction to curve-valued potentials, with particular emphasis on the case where the base curve is an elliptic curve. Starting from a finite configuration of points on an elliptic curve, we construct associated (L_\infty)-algebras by passing to suitable lifts of the configuration and applying the combinatorics of secondary polytopes. After choosing an additional point, or stop, together with a lift to the universal cover, we construct corresponding (A_\infty)-algebras and an (L_\infty)-morphism to the derived derivation space, which is the deformation complex of the (A_\infty)-algebra. A new feature of the curve-valued setting is that the resulting (A_\infty)-structure depends essentially on the choice of lift of the stop. This produces a chamber structure controlled by the fundamental group of the base curve. We formulate the resulting chamberwise algebraic picture and discuss its expected relation to Fukaya--Seidel categories over curves. In this way, the dissertation extends the algebra of the infrared beyond the classical planar setting and reveals new structures arising from the topology of the base curve.