Quantum Invariants, Modular Forms, and Lattice Points II

Abstract

We study the SU(2) Witten--Reshetikhin--Turaev 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 1/2 and 3/2. 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 ->\infinity, is related to the number of integral lattice points inside the M-dimensional tetrahedron.

References in corpus (10)

• Chern-Simons Theory and Topological Strings
• q-series and L-functions related to half-derivatives of the Andrews--Gordon identity
• Quantum Knot Invariant for Torus Link and Modular Forms
• Mock (False) Theta Functions as Quantum Invariants
• On the Quantum Invariant for the Brieskorn Homology Spheres
• Transformation Formula of the "2nd" Order Mock Theta Function
• Quantum Invariant, Modular Form, and Lattice Points
• On the quantum sl_2 invariants of knots and integral homology spheres
• Counting Lattice Points by means of the Residue Theorem
• Analytic asymptotic expansions of the Reshetikhin--Turaev invariants of Seifert 3-manifolds for SU(2)

Cited by in corpus (8)

• 3d Modularity
• A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
• On the Quantum Invariant for the Spherical Seifert Manifold
• Quantum modularity of partial theta series with periodic coefficients
• Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds
• $\hat{Z}$ invariants at rational $\tau$
• Witt invariants from q-series $\hat{Z}$
• Witten-Reshetikhin-Turaev Invariants, Homological Blocks, and Quantum Modular Forms for Unimodular Plumbing H-Graphs