Foams, iterated wreath products, field extensions and Sylvester sums

TL;DR

This paper introduces new foam structures and relations to interpret categorical functors, explores their connections to field extensions and Galois theory, and relates overlapping foams to Sylvester sums, with applications in TQFTs.

Contribution

It presents novel foam relations for categorical interpretations, links foams to field theory, and compares TQFT traces with field extension traces, advancing understanding in these interconnected areas.

Findings

Foams encode functors and natural transformations in wreath product categories.

Patched surfaces with defect circles relate to separable field extensions.

Overlapping foams are connected to Sylvester double sums.

Abstract

Certain foams and relations on them are introduced to interpret functors and natural transformations in categories of representations of iterated wreath products of cyclic groups of order two. We also explain how patched surfaces with defect circles and foams relate to separable field extensions and Galois theory and explore a relation between overlapping foams and Sylvester double sums. In the appendix, joint with Lev Rozansky, we compare traces in two-dimensional TQFTs coming from matrix factorizations with those in field extensions.