AKSZ-type Topological Quantum Field Theories and Rational Homotopy Theory

TL;DR

This paper reformulates AKSZ-type topological quantum field theories using rational homotopy theory, revealing their structure as gauge theories of higher gauge groups and connecting physical fields to algebraic models.

Contribution

It introduces a new perspective on AKSZ theories through rational homotopy, linking gauge theories to $L_$-algebras and Sullivan models, and explains their topological invariants.

Findings

AKSZ theories are gauge theories of higher gauge groups.

Integration of auxiliary fields corresponds to Sullivan minimal models.

The approach connects topological invariants to physical theories.

Abstract

We reformulate and motivate AKSZ-type topological field theories in pedestrian terms, explaining how they arise as the most general Schwartz-type topological actions subject to a simple constraint, and how they generalize Chern$-$Simons theory and other well known topological field theories, in that they are gauge theories of flat connections of higher gauge groups (infinity-Lie algebras). Their Euler$-$Lagrange equations define quasifree graded-commutative differential algebras, or equivalently $L_\infty$-algebras, the equivalent of the Lie algebra of the gauge group; we explain how integrating out auxiliary fields in physics corresponds to taking the Sullivan minimal model of this algebra, and how the correspondence between fields and gauge transformations realizes Koszul duality. Using this dictionary, we can import topological invariants and notions (e.g.$~$the rational LS-category) to apply to this class of theories.