Lattice Topological Field Theory on Non-Orientable Surfaces

TL;DR

This paper extends lattice topological quantum field theory to non-orientable surfaces, establishing a correspondence with real associative *-algebras and computing related invariants for various surfaces.

Contribution

It generalizes the lattice TQFT framework to non-orientable surfaces and links it to real associative *-algebras, providing explicit computations for these invariants.

Findings

One-to-one correspondence between real associative *-algebras and topological invariants.

Explicit computation of partition and n-point functions on various non-orientable surfaces.

Application to group rings of discrete groups, such as r[G].

Abstract

The lattice definition of the two-dimensional topological quantum field theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a one-to-one correspondence between real associative $*$-algebras and the topological state sum invariants defined on such surfaces. The partition and $n$-point functions on all two-dimensional surfaces (connected sums of the Klein bottle or projective plane and $g$-tori) are defined and computed for arbitrary $*$-algebras in general, and for the the group ring $A=\R[G]$ of discrete groups $G$, in particular.