Vertex algebras and 4-manifold invariants

TL;DR

This work develops a bridge between four-dimensional gauge theory invariants and two-dimensional half-twisted theories on M5-branes by introducing T[M_4] and its impurity operators. It shows how Seiberg-Witten and multi-monopole invariants arise as chiral correlators of vertex operators in the half-twisted theory, and develops an equivariant localization framework to define and compute equivariant multi-monopole invariants on general 4-manifolds. The authors extend the construction to non-abelian gauge groups, outline non-abelian generalizations, and propose the Coulomb-branch index as a promising diagnostic for new 4-manifold invariants that may surpass classical Donaldson/Seiberg-Witten theories. The results connect M-theory brane setups, Kondo-type impurities, and knot-homology structures into a coherent program for categorified 4-manifold invariants, with concrete predictions for basic classes and fixed-point contributions under torus actions.

Abstract

We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d $\mathcal{N}=(0,2)$ theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations.

Figures (11)

• Figure 1: Various corners of a 4d TQFT with topological "surface operators" or "foams". It assigns vector spaces to knots and 3-manifolds, maps between these homological invariants to the corresponding cobordisms, and numerical invariants to closed 4-manifolds and embedded surfaces.
• Figure 2: An example of a plumbing graph $\Gamma$ (left) and the corresponding link ${\mathcal{L}}(\Gamma)$ of framed unknots in $S^3$ (right). The associated 3-manifold $M_3(\Gamma)$ can be constructed as a Dehn surgery on ${\mathcal{L}}(\Gamma)$.
• Figure 3: An illustration (from Gukov:2017kmk) of a supersymmetric spectrum in 3d ${\mathcal{N}}=2$ theory with an impurity.
• Figure 4: Target space ${\mathbb C}^* \simeq S^1 \times {\mathbb R}$ parametrized by $\sigma + i X$.
• Figure 5: M2-branes ending on M5-branes.