Quantum modularity and complex Chern-Simons theory

Tudor Dimofte, Stavros Garoufalidis

Research output: Contribution to journalArticlepeer-review

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 ζ. 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,C) at fixed level k, where ζk=1.
Original languageEnglish
Pages (from-to)1 – 52
Number of pages53
JournalCommunications in Number Theory and Physics
Volume12
Issue number1
DOIs
Publication statusPublished - 27 Apr 2018

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