In fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N = 2 theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomials. Higgsing the full 3d theories constructed this way recovers theories found previously by Dimofte-GaiottoGukov. We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role played by all flat connections in this operation.
We calculate the homological blocks for Seifert manifolds from the exact expression 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 express 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.In this paper, we consider the G = SU (N ) WRT invariant on general Seifert manifolds X(P 1 /Q 1 , . . . , P F /Q F ) where P j and Q j are coprime for each j = 1, . . . , F and P j 's are pairwise coprime. We provide a formula that calculates Z a 's from which we can calculate the homological blocks exactly for G = SU (2) and as a q-series expansion for G = SU (N ) from the exact expression given by Lawrence and Rozansky [15] and Mariño [16]. We see that examples calculated in this paper fit into the expected structures of the WRT invariant for Seifert manifolds in terms of homological blocks in section 3.2.3 The reason why only abelian flat connections are taken into account, not all flat connections, is not known yet, though the resurgent analysis provide some explanation on it.The q-series in parenthesis takes a form of q 1 120 Z [[q]]. This agrees with the result of [3] on the Poincaré homology sphere Σ(2, 3, 5). As it is an integer homology sphere, there is one homological block from the trivial flat connection, which is (2.20).
The half-BPS boundary conditions preserving N = (2, 2) and N = (0, 4) supersymmetry in 3d N = 4 supersymmetric gauge theories are examined. The BPS equations admit decomposition of the bulk supermultiplets into specific boundary supermultiplets of preserved supersymmetry. Nahm-like equations arise in the vector multiplet BPS boundary condition preserving N = (0, 4) supersymmetry and Robin-type boundary conditions appear for the hypermultiplet coupled to vector multiplet when N = (2, 2) supersymmetry is preserved. The half-BPS boundary conditions are realized in the brane configurations of Type IIB string theory.
In this paper, we apply ideas of Dijkgraaf and Witten [31,6] on 3 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications.
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