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