Quantum invariants, modular forms, and lattice points II

Kazuhiro Hikami

doi:10.1063/1.2349484

Quantum invariants, modular forms, and lattice points II

Kazuhiro Hikami

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

We study the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant for the Seifert fibered homology spheres with M -exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 12 and 32. By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large- N limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in N→∞, is related to the number of integral lattice points inside the M -dimensional tetrahedron.

Original language	English
Article number	102301
Journal	Journal of Mathematical Physics
Volume	47
Issue number	10
DOIs	https://doi.org/10.1063/1.2349484
Publication status	Published - 2006

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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abstract = "We study the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant for the Seifert fibered homology spheres with M -exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 12 and 32. By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large- N limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in N→∞, is related to the number of integral lattice points inside the M -dimensional tetrahedron.",

author = "Kazuhiro Hikami",

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N2 - We study the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant for the Seifert fibered homology spheres with M -exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 12 and 32. By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large- N limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in N→∞, is related to the number of integral lattice points inside the M -dimensional tetrahedron.

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