Classification of Non-Degenerate Symmetric Bilinear and Quadratic Forms in the Verlinde Category $\mathrm{Ver}_4^+$

TL;DR

This paper classifies non-degenerate symmetric bilinear and quadratic forms in the Verlinde category Ver_4^+ over characteristic 2, advancing understanding of symmetric tensor categories in positive characteristic.

Contribution

It provides the first classification of such forms in Ver_4^+ and explores the associated Witt semi-ring structure, extending the theory of symmetric tensor categories.

Findings

Complete classification of non-degenerate symmetric bilinear forms

Complete classification of non-degenerate quadratic forms

Analysis of the Witt semi-ring structure

Abstract

Although Deligne's theorem classifies all symmetric tensor categories (STCs) with moderate growth over algebraically closed fields of characteristic zero, the classification does not extend to positive characteristic. At the forefront of the study of STCs is the search for an analog to Deligne's theorem in positive characteristic, and it has become increasingly apparent that the Verlinde categories are to play a significant role. Moreover, these categories are largely unstudied, but have already shown very interesting phenomena as both a generalization of and a departure from superalgebra and supergeometry. In this paper, we study $\mathrm{Ver}_4^+$, the simplest non-trivial Verlinde category in characteristic $2$. In particular, we classify all isomorphism classes of non-degenerate symmetric bilinear forms and non-degenerate quadratic forms and study the associated Witt semi-ring that arises from the addition and multiplication operations on bilinear forms.