Communications in Number Theory and Physics

Volume 12 (2018)

Number 1

Quantum modularity and complex Chern–Simons theory

Pages: 1 – 52

DOI: https://dx.doi.org/10.4310/CNTP.2018.v12.n1.a1

Authors

Tudor Dimofte (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada; School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.; and Department of Mathematics, University of California at Davis)

Stavros Garoufalidis (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

Abstract

The Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to construct a power series from a Neumann–Zagier datum (i.e., an ideal triangulation of the knot complement and a geometric solution to the gluing equations) and a complex root of unity $\zeta$. We prove that the coefficients of our series lie in the trace field of the knot, adjoined a complex root of unity. We conjecture that our series are those that appear in the Quantum Modularity Conjecture and confirm that they match the numerical asymptotics of the Kashaev invariant (at various roots of unity) computed by Zagier and the first author. Our construction is motivated by the analysis of singular limits in Chern–Simons theory with gauge group $SL(2, \mathbb{C})$ at fixed level k, where $\zeta^k = 1$.

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The work of SG is supported by NSF grant DMS-14-06419. This paper was primarily completed while TD was a long-term member at the Institute for the Advanced Study, supported by the Friends of the Institute for Advanced Study, in part by DOE grant DE-FG02-90ER40542, and in part by ERC Starting Grant no. 335739 Quantum fields and knot homologies, funded by the European Research Council under the European Union’s Seventh Framework Programme. TD is currently supported by the Perimeter Institute for Theoretical Physics; research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.

Received 18 November 2015

Accepted 20 December 2017

Published 27 April 2018