Inverse Monoid Topological Quantum Field Theories and Open-Closed Grand Canonical Symmetric Orbifolds

TL;DR

This work generalizes open-closed two-dimensional topological quantum field theories from finite groups to finite inverse monoids by exploiting the semisimplicity of the monoid algebra and its Morita-equivalent matrix-algebra structure. Central to the construction is the Ivanov-Kerov monoid of partial permutations, which provides a grand canonical framework for counting covers with boundaries and connects to symmetric orbifolds. The authors develop parallel viewpoints—groupoid-algebra, representation-theoretic, and Frobenius/state-sum formalisms—and verify Cardy-like consistency conditions in the monoid setting, obtaining explicit expressions for bulk and boundary data, including central idempotents and window elements. A concrete application to grand canonical open-closed symmetric orbifolds is given via the IK monoid, including a permutation-invariant subsector that parallels grand canonical Hurwitz theory and suggests potential holographic interpretations. The framework lays groundwork for further exploration of partial symmetries in topological and conformal field theories, with several open directions such as extending to nontrivial homology and detailed Hurwitz/Gromov-Witten correspondences with boundaries.

Abstract

We present an open-closed topological quantum field theory for inverse monoids which generalizes the theory of principle fiber bundles with finite gauge group over Riemann surfaces with boundary. The theory is constructed using the isomorphism between the semisimple inverse monoid algebra and a matrix algebra which lies at the heart of monoid structure and representation theory. An example that we study in detail is the Ivanov-Kerov monoid of partial permutations. We review motivations from string theory for the resulting grand canonical theory of covers with boundaries.