Involutive categories, colored $\ast$-operads and quantum field theory

TL;DR

This paper develops the theory of involutive colored operads and their algebras, motivated by quantum field theory, introducing involutive structures into operad theory and applying it to algebraic quantum field theory.

Contribution

It introduces colored $ extit{ extbf{}}$-operads and $ extit{ extbf{}}$-algebras, extending operad theory with involutive structures inspired by quantum physics.

Findings

Constructed involutive monoidal categories of symmetric sequences.

Established involutive analogs of change of color and operad adjunctions.

Applied the framework to algebraic quantum field theory, exemplified by the associative $ extit{ extbf{}}$-operad.

Abstract

Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field) theory, we develop the involutive analogs of colored operads and their algebras, named colored $\ast$-operads and $\ast$-algebras. Central to the definition of colored $\ast$-operads is the involutive monoidal category of symmetric sequences, which we obtain from a general product-exponential $2$-adjunction whose right adjoint forms involutive functor categories. For $\ast$-algebras over $\ast$-operads we obtain involutive analogs of the usual change of color and operad adjunctions. As an application, we turn the colored operads for algebraic quantum field theory into colored $\ast$-operads. The simplest instance is the associative $\ast$-operad, whose $\ast$-algebras are unital and associative $\ast$-algebras.