BPS invariants for 3-manifolds at rational level K

Hee-Joong Chung

doi:10.1007/jhep02(2021)083

Abstract

We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity q={e}^{frac{2pi i}{K}} with a rational level K = frac{r}{s} where r and s are coprime integers. From the exact expression for the G = SU(2) Witten-Reshetikhin-Turaev invariants of the Seifert manifolds at a rational level obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at a rational level. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the limit in the standard volume conjecture.

Highlights

We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity q
From the exact expression for the G = SU(2) Witten-Reshetikhin-Turaev invariants of the Seifert manifolds at a rational level obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at a rational level
We discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the limit in the standard volume conjecture

Summary

WRT invariant at other roots of unity

PF /QF ) with conditions that Pj’s and Qj’s are coprime for each j, j = 1, . F , and Pj’s are pairwise coprime, the finite sum expression for the WRT invariant in [10] is given by ZK (M3). By considering the Galois action on x, x is replaced with another primitive root of unity, e2πi s 4K. Due to this change, an overall factor appears from the factor B, and we don’t consider it in this paper. An overall factor appears from the factor B, and we don’t consider it in this paper Up to such an overall factor, the integral expression for the WRT invariant at other roots with r s for the Seifert manifolds

WRT invariant at other roots of unity in terms of homological blocks

Properties and general structure

Asymptotic expansion of knot invariants at roots of unity

Asymptotic expansion of q-Pochhammer symbol at the root of unity

Examples: superpolynomial of knots

Discussion