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

TL;DR

The paper constructs a comprehensive web-based framework for 2d massive ${ m N}=(2,2)$ theories, encoding boundary conditions, interfaces, and BPS spectra through interlocking ${L_}$ and ${A_}$-algebraic structures. It derives convolution identities for plane, half-plane, and strip webs, uncovering a unifying ${L_}$/$A_$-homotopical backbone and a categorical wall-crossing mechanism. Representations of webs yield vacuum and brane categories, with MC-equations governing boundary amplitudes and a Brane category ${rak Br}$ underpinning boundary operator algebras, all compatible with interface composition. The LG model serves as a concrete example, linking the web category to Fukaya–Seidel-type categories and illustrating boundary operators, thimbles, and cyclic/rotational isomorphisms; the framework points toward applications to surface defects, knot homology, and wall-crossing phenomena in broader gauge-theoretic settings. The construction provides a principled, categorial account of BPS degeneracies and their wall-crossing in two dimensions, with potential extensions to higher dimensions via interface networks and composite webs.

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.

Figures (179)

• Figure 1: An $ij$ soliton emitted from the boundary. This process is supersymmetric if the soliton is emitted at the proper angle.
• Figure 2: A supersymmetric soliton exchanged between the two branes on the left and the right.
• Figure 3: This figure shows a variety of rigid strip instantons constructed using boundary vertices only.
• Figure 4: A "bulk" vertex that involves a coupling of three BPS solitons. The vacua involved are $i,j,k\in{\mathbb V}$, and the solitons that emanate from the vertex are respectively of types $ij,$$jk$, and $ki$.
• Figure 5: Rigid strip instantons whose construction makes use of bulk vertices. We assume the bulk vertices have no moduli except the ones associated to spacetime translations.