Renormalisation and the Batalin-Vilkovisky formalism

TL;DR

This paper develops a renormalisation method for certain quantum field theories on compact manifolds within the Batalin-Vilkovisky formalism, producing new topological invariants that respect symmetries up to homotopy.

Contribution

It introduces a renormalisation approach compatible with the BV formalism for theories like Chern-Simons, leading to manifold invariants with higher algebraic structures.

Findings

Successful renormalisation of Chern-Simons theory respecting symmetries

Construction of new algebraic invariants of smooth manifolds

Extension of BV formalism to quantum field theories on compact manifolds

Abstract

This paper gives a way to renormalise certain quantum field theories on compact manifolds. Examples include Yang-Mills theory (in dimension 4 only), Chern-Simons theory and holomorphic Chern-Simons theory. The method is within the framework of the Batalin-Vilkovisky formalism. Chern-Simons theory is renormalised in a way respecting all symmetries (up to homotopy). This yields an invariant of smooth manifolds: a certain algebraic structure on the cohomology of the manifold tensored with a Lie algebra, which is a "higher loop" enrichment of the natural Lie-infinity structure.