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arXiv · 2306.08608

Invariants of r-spin TQFTs and non-semisimplicity

Nils Carqueville·Ehud Meir·Lorant Szegedy
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Abstract

For a positive integer r, an r-spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r-fold cover of SO_2 . In particular, such a TQFT assigns a scalar invariant to every closed r-spin surface Sigma. Given a sequence of scalars indexed by the set of diffeomorphism classes of all such Sigma, we construct a symmetric monoidal category C and a C-valued r-spin TQFT which reproduces the given sequence. We also determine when such a sequence arises from a TQFT valued in an abelian category with finite-dimensional Hom spaces. In particular, we construct TQFTs with values in super vector spaces that can distinguish all diffeomorphism classes of r-spin surfaces, and we show that the associated algebras are necessarily non-semisimple.

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Nils Carqueville, Ehud Meir, Lorant Szegedy. 2023-06-14. Invariants of r-spin TQFTs and non-semisimplicity. https://doi.org/10.1016/j.jalgebra.2024.08.039

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