Evaluating thin flat surfaces

TL;DR

This paper characterizes recognizable evaluations of thin flat surfaces via algebraic and geometric structures, establishing a correspondence with Frobenius algebras and points on the Hilbert scheme, and constructs associated functors and categories.

Contribution

It introduces a classification of evaluations for thin flat surfaces, linking them to Frobenius algebras and geometric objects, and develops related functorial and categorical frameworks.

Findings

Recognizable evaluations are ratios of polynomials in two variables.

They correspond to isomorphism classes of certain Frobenius algebras.

Associated functors to vector spaces may be non-monoidal.

Abstract

We consider recognizable evaluations for a suitable category of oriented two-dimensional cobordisms with corners between finite unions of intervals. We call such cobordisms thin flat surfaces. An evaluation is given by a power series in two variables. Recognizable evaluations correspond to series that are ratios of a two-variable polynomial by the product of two one-variable polynomials, one for each variable. They are also in a bijection with isomorphism classes of commutative Frobenius algebras on two generators with a nondegenerate trace fixed. The latter algebras of dimension n correspond to points on the dual tautological bundle on the Hilbert scheme of n points on the affine plane, with a certain divisor removed from the bundle. A recognizable evaluation gives rise to a functor from the above cobordism category of thin flat surfaces to the category of finite-dimensional vector spaces. These functors may be non-monoidal in interesting cases. To a recognizable evaluation we also assign an analogue of the Deligne category and of its quotient by the ideal of negligible morphisms.