A-models in three and four dimensions

TL;DR

This work introduces a 3d A-model with target a real manifold $X$, detailing its field content $(\phi,\tau)$, BRST structure, and a BRST-exact action augmented by a topological term. It analyzes observables, deformations, and the quantum space of states, showing a de Rham-like BRST cohomology controlled by $X$ and a partition function finite only on certain 3-manifolds, with boundary effects yielding rich instanton-like contributions. The paper then extends to boundary conditions, boundary line operators, a gauged version coupled to a symmetry group $G$, and finally a 4d A-model obtained by twisting a 4d $\mathcal{N}=2$ theory, all while highlighting links to 3d mirror symmetry, geometric Langlands duality, and related dualities. It sketches how these constructions encode 2-categorical boundary data and how their dualities map to known dualities like Montonen–Olive and Rozansky–Witten, suggesting avenues for deeper exploration of boundary phenomena and Langlands-type correspondences.

Abstract

We introduce and study a new 3d Topological Field Theory which can be associated to any compact real manifold X. This TFT is analogous to the 2d A-model and reduces to it upon compactification on an interval with suitable boundary conditions. It plays a role in 3d mirror symmetry as well as in the physical approach to the geometric Langlands duality. A similar TFT can be defined in four dimensions.