2-Morse Theory and 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.