2-Morse Theory and the Algebra of the Infrared

TL;DR

This paper develops a formal Morse theory analogue for pairs of commuting gradient-like vector fields, leading to an algebraic structure similar to the infrared algebra in quantum field theory, with concrete Morse-theoretic examples.

Contribution

It introduces a new algebraic formalism for pairs of commuting gradient-like vector fields, connecting Morse theory with the algebra of the infrared in a novel way.

Findings

Construction of an L_infinity-algebra from a manifold with commuting vector fields

Identification of a Maurer-Cartan element within this algebra

Provision of Morse-theoretic examples illustrating the algebra of the infrared

Abstract

We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see [GMW], [KKS]): from a manifold M with the pair of gradient-like commuting vector fields, subject to some general position conditions we construct an L_$\infty$-algebra and Maurer-Cartan element in it. We also provide Morse-theoretic examples for the algebra of the infrared data.