BPS invariants for 3-manifolds at rational level K

Abstract We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity $$ q={e}^{\frac{2\pi i}{K}} $$ q = e 2 πi K with a rational level K = $$ \frac{r}{s} $$ 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.

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