An Introduction To The Web-Based Formalism

TL;DR

The work develops a web-based formalism to extract and organize long-distance data from 2D massive N=(2,2) Landau–Ginzburg theories into an A∞-categorical framework of branes, thimbles, and local operators, and relates it to the Fukaya–Seidel category. It then extends this structure to an A∞-2-category of theories and interfaces, enabling flat parallel transport of brane categories across vacuum deformations and providing a categorified perspective on wall-crossing phenomena, including framed BPS states. Central tools include planar and half-plane webs, boosted solitons, ζ-instantons, and the categorified spectrum generator, with explicit interface constructions and composition rules that realize higher-categorical transport. The framework points to significant applications in knot homology and Hitchin-system phenomena, illustrating how categorified wall-crossing and interface calculus can encode rich topological and physical information in a geometrically controlled way.

Abstract

This paper summarizes our rather lengthy paper, "Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions," and is meant to be an informal, yet detailed, introduction and summary of that larger work.

Figures (40)

• Figure 1: Left: An instanton configuration contributing to the differential on the MSW complex. The black regions indicate the locus where the field $\phi(x,\tau)$ varies vary significantly from the vacuum configurations $\phi_i$ or $\phi_j$. The length scale here is $\ell_W$, set by the superpotential $W$. Right: Viewed from a large distance compared to the length scale $\ell_W$ the instanton looks like a straight line $x=x_0$, where the vacuum changes discontinuously from vacuum $\phi_i$ to $\phi_j$. The nontrivial $\tau$-dependence of the instanton configuration, interpolating from a soliton $p_1$ to another soliton $p_2$ has been contracted to a single vertex located at $\tau = \tau_0$. This vertical line with a single vertex on it is the first example of a "web" in the web formalism.
• Figure 2: A pair of Lagrangian submanifolds ${\cal L}_\ell$, $\L_r$ embedded in the $u-v$ plane. ${\cal L}_\ell$ and ${\cal L}_r$ intersect at the one point indicated. $u$ is plotted horizontally and we assume that ${\cal L}_\ell$, ${\cal L}_r$ are embedded in the half-plane $u>0$.
• Figure 3: The rays in the complex $W$-plane that start at critical points and all run in the $\i\kappa$ direction fit into the semi-infinite strip $T_\kappa$, which is shown as a shaded region.
• Figure 4: An instanton in the complex $\mathbb{M}_{{\cal L},j}$. The solitons corresponding to $p_1,p_2 \in {\cal L}\cap R^\zeta_j$, where $R^\zeta_j$ is the right Lefshetz thimble, are exponentially close to the vacuum $\phi_j$ except for a small region, shown in turquoise, of width $\ell_W$. In addition, the instanton transitions from one soliton to another in a time interval of length $\ell_W$, indicated by the green square. At large distances the green square becomes the $0$-valent vertex used in half-plane webs.
• Figure 5: Naive differential on the strip.