Involutive categories, colored *-operads and quantum field theory

Marco Benini, Alexander Schenkel and Lukas Woike

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 *-operads and *-algebras. Central to the definition of colored *-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 *-algebras over *-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 *-operads. The simplest instance is the associative *-operad, whose *-algebras are unital and associative *-algebras.

Keywords: involutive categories, involutive monoidal categories, *-monoids, colored operads, *-algebras, algebraic quantum field theory

2010 MSC: 18Dxx, 81Txx

Theory and Applications of Categories, Vol. 34, 2019, No. 2, pp 13-57.

Published 2019-02-11.

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