Automata and one-dimensional TQFTs with defects

TL;DR

This paper establishes a novel connection between nondeterministic automata for regular languages and one-dimensional TQFTs with defects, providing a new perspective and generalizations involving algebraic structures and topological spaces.

Contribution

It introduces a construction of TQFTs from automata, extends to modules over Boolean semiring, and generalizes to TQFTs for one-dimensional foams with defects.

Findings

Automata induce TQFTs over the Boolean semiring.

Different automata for the same language produce TQFTs differing on decorated circles.

The framework can be extended to modules over rings and to TQFTs for foams.

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 labelled 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 to 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.