One-dimensional topological theories with defects and linear generating functions

TL;DR

This paper investigates the algebraic structure of a one-dimensional topological theory with defects, focusing on the Gram determinant and basis construction for hom spaces, specifically when the generating functions are linear.

Contribution

It provides a detailed analysis of the Gram determinant and constructs bases for hom spaces in the context of decorated unoriented cobordisms with linear generating functions.

Findings

Explicit formulas for the Gram determinant.

Construction of bases for hom spaces.

Insights into the algebraic structure of the theory.

Abstract

We study the Gram determinant and construct bases of hom spaces for the one-dimensional topological theory of decorated unoriented one-dimensional cobordisms, as recently defined by Khovanov, when the pair of generating functions is linear.