Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions

TL;DR

This paper develops a web-based algebraic formalism for describing boundary conditions and BPS states in 2D massive ${ m N}=(2,2)$ supersymmetric theories, connecting infrared data with $A_ ablafty$ and $L_ ablafty$ structures.

Contribution

It introduces a novel web-based framework to construct the category of boundary conditions from infrared data and relates it to Fukaya-Seidel categories and Morse theory.

Findings

Constructed boundary condition categories from IR data.

Derived algebraic constraints related to $A_ ablafty$ and $L_ ablafty$ algebras.

Established equivalence with Fukaya-Seidel categories in Landau-Ginzburg models.

Abstract

We introduce a "web-based formalism" for describing the category of half-supersymmetric boundary conditions in $1+1$ dimensional massive field theories with ${\cal N}=(2,2)$ supersymmetry and unbroken $U(1)_R$ symmetry. We show that the category can be completely constructed from data available in the far infrared, namely, the vacua, the central charges of soliton sectors, and the spaces of soliton states on $\mathbb{R}$, together with certain "interaction and boundary emission amplitudes". These amplitudes are shown to satisfy a system of algebraic constraints related to the theory of $A_\infty$ and $L_\infty$ algebras. The web-based formalism also gives a method of finding the BPS states for the theory on a half-line and on an interval. We investigate half-supersymmetric interfaces between theories and show that they have, in a certain sense, an associative "operator product." We derive a categorification of wall-crossing formulae. The example of Landau-Ginzburg theories is described in depth drawing on ideas from Morse theory, and its interpretation in terms of supersymmetric quantum mechanics. In this context we show that the web-based category is equivalent to a version of the Fukaya-Seidel $A_\infty$-category associated to a holomorphic Lefschetz fibration, and we describe unusual local operators that appear in massive Landau-Ginzburg theories. We indicate potential applications to the theory of surface defects in theories of class S and to the gauge-theoretic approach to knot homology.