Automata and one-dimensional TQFTs with defects

Abstract

This paper explains how any nondeterministic automaton for a regular language L gives rise to a one-dimensional oriented topological quantum field theory (TQFT) with inner endpoints and zero-dimensional defects labeled by letters of the alphabet for L. The TQFT is defined over the Boolean semiring \(\mathbb {B}\). Different automata for a fixed language L produce TQFTs that differ by their values on decorated circles, while the values on decorated intervals are described by the language L. The language L and the TQFT associated with an automaton can be given a path integral interpretation. In this TQFT, the state space of a one-point 0-manifold is a free module over \(\mathbb {B}\) with the basis of states of the automaton. Replacing a free module by a finite projective \(\mathbb {B}\)-module P allows to generalize automata and this type of TQFT to a structure where defects act on open subsets of a finite topological space. Intersection of open subsets induces a multiplication on P allowing to extend the TQFT to a TQFT for one-dimensional foams (oriented graphs with defects modulo a suitable equivalence relation). A linear version of these constructions is also explained, with the Boolean semiring replaced by a commutative ring.