The Cyclic and Modular Microcosm Principle in Quantum Topology

TL;DR

This paper generalizes the microcosm principle to cyclic and modular algebras in quantum topology, providing a unified framework that connects conformal field theories with skein theory.

Contribution

It extends the Baez-Dolan microcosm principle to cyclic and modular algebras, linking local algebraic structures to global topological invariants in quantum topology.

Findings

Unified classification of correlators in conformal field theories.

Extension of genus zero correlators to handlebodies.

Establishes a correspondence between 2D conformal field theory and 3D skein theory.

Abstract

Monoidal categories with additional structure such as a braiding or some form of duality abound in quantum topology. They often appear in tandem with Frobenius algebras inside them. Motivations for this range from the theory of module categories to the construction of correlators in conformal field theory. We generalize the Baez-Dolan microcosm principle to consistently describe all these types of algebras by extending it to cyclic and modular algebras in the sense of Getzler-Kapranov. Our main result links the microcosm principle for cyclic algebras to the one for modular algebras via Costello's modular envelope. The result can be understood as a local-to-global construction or an integration procedure for various flavors of Frobenius algebras that substantially generalizes and unifies the available (and often intrinsically semisimple) methods using for example triangulations or classical skein theory. As the main application of this rather abstract result, we solve the problem of classifying consistent systems of correlators for open conformal field theories and show that the genus zero correlators for logarithmic conformal field theories constructed by Fuchs-Schweigert can be uniquely extended to handlebodies. This establishes a very general correspondence between full genus zero conformal field theory in dimension two and skein theory in dimension three.