Abstract
We calculate the homological blocks for Seifert manifolds from the exact ex- pression for the G = SU(N ) Witten-Reshetikhin-Turaev invariants of Seifert manifolds obtained by Lawrence, Rozansky, and Mariño. For the G = SU(2) case, it is possible to ex- press them in terms of the false theta functions and their derivatives. For G = SU(N ), we calculate them as a series expansion and also discuss some properties of the contributions from the abelian flat connections to the Witten-Reshetikhin-Turaev invariants for general N . We also provide an expected form of the S-matrix for general cases and the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks.
Highlights
Categorifies the Chern-Simons (CS) partition function or the Witten-Reshetikhin-Turaev (WRT) invariant [1, 2] for closed 3-manifolds is not available yet
We proposed a formula to calculate the S-matrix in general cases and checked that the S-matrix calculated from the formula by using the linking form agrees with the result that is obtained from the calculation of Za’s and Zb’s
We discussed general forms of the WRT invariant in terms of homological blocks and checked that examples that we discussed fit into the expected forms
Summary
We see from several examples that the Weyl orbit of such a t corresponds to one abelian flat connection whose holonomy HolA(h) is a conjugacy class t. We expect that a Weyl orbit of such a t corresponds to an abelian flat connection whose holonomy as in SU(2). If (2, −2) part of (2, −2, 0) led to the non-abelian reducible flat connection, the SU(2) part of the integral on (2, −2) would be given by the residue of the integral, which is not the case. We expect that a Weyl orbit of t ∈ Λ2/HΛ2 in (3.8) which gives HolA(h) ∼= diag (v, v, v−2) corresponds to an abelian flat connection. In sum, we expect that a Weyl orbit of t ∈ Λr/HΛr in (3.8) corresponds to an abelian flat connection of SU(N )
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