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arXiv · 2009.10099

$SL(2,\mathbb{Z})$ action on QFTs with $\mathbb{Z}_2$ symmetry and the Brown-Kervaire invariants

Lakshya Bhardwaj·Yasunori Lee·Yuji Tachikawa
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Abstract

We consider an analogue of Witten's $SL(2,\mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with $\mathbb{Z}_2$ $(k-1)$-form symmetry. We show that the $SL(2,\mathbb{Z})$ action only closes up to a multiplication by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,\mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional $\mathbb{Z}_2$ gauge theory.

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Lakshya Bhardwaj, Yasunori Lee, Yuji Tachikawa. 2020-09-21. $SL(2,\mathbb{Z})$ action on QFTs with $\mathbb{Z}_2$ symmetry and the Brown-Kervaire invariants. https://doi.org/10.1007/jhep11(2020)141

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