TQFT, Homological Algebra and elements of K.Saito's Theory of Primitive Form: an attempt of mathematical text written by mathematical physicist

TL;DR

This paper explores the application of Topological Quantum Field Theory (TQFT) to homological algebra and K. Saito's theory of Primitive forms, providing explicit examples and new proofs related to associativity and commutativity equations.

Contribution

It introduces a novel application of TQFT in homological algebra and primitive forms, including explicit constructions and proofs of key algebraic equations.

Findings

Explicit construction of infinity-structure reduction after contraction.

New proof that tree level BCOV theory solves Oriented Associativity equations.

Application of TQFT to derive associativity and commutativity equations.

Abstract

The text is devoted to explanation of the concept of Topological Quantum Field Theory (TQFT), its application to homological algebra and to the relation with the theory of good section from K.Saito's theory of Primitive forms. TQFT is explained in Dirac-Segal framework, one-dimensional examples are explained in detail. As a first application we show how it can be used in explicit construction of reduction of infinity-structure after contraction of a subcomplex. Then we explain Associativity and Commutativity equations using this language. We use these results to construct solutions to Commutativity equations and find a new proof of for the fact that tree level BCOV theory solved Oriented Associativity equations.