Foams, iterated wreath products, field extensions and Sylvester sums

Mee Seong Im and Mikhail Khovanov (with an appendix by Lev Rozansky)

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.

Keywords: Iterated wreath products, categorification, Frobenius algebras, field extensions, separable extensions, matrix factorizations, Sylvester sums, foam evaluation, defect TQFTs

2020 MSC: Primary: 57K16, 20E22, 18N25, 13P15, 57K99, 20C99

Theory and Applications of Categories, Vol. 44, 2025, No. 32, pp 1020-1105.

Published 2025-10-06.

http://www.tac.mta.ca/tac/volumes/44/32/44-32.pdf

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