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arXiv · 2605.28650

String nets for twisted pivotal categories

Benjamin Haïoun·William Stewart·Filippos Sytilidis
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Abstract

We develop a graphical calculus for monoidal categories equipped with twisted pivotal structures, which are a generalization of pivotal structures originating from the study of orientation structures in the context of the Cobordism Hypothesis. This graphical calculus depends on a possibly singular foliation, and we use it to construct twisted string net modules for surfaces equipped with a Morse function or a Morse foliation. We prove that, despite the apparent dependence on this Morse function, the twisted string net modules assemble in an oriented categorified 2-TQFT. We study when the twisted string net module of the 2-sphere vanishes, relate it to the distinguished invertible object for finite tensor categories and exhibit examples of non-unimodular finite tensor categories with non-vanishing twisted string net module on the 2-sphere. This vanishing is expected to be the main obstruction for extending our categorified 2-TQFT to a non-compact 3-TQFT.

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BibTeXRIS

Benjamin Haïoun, William Stewart, Filippos Sytilidis. 2026-05-27. String nets for twisted pivotal categories. https://arxiv.org/abs/2605.28650

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