Abstract
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-Gaiotto-Gukov. We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role played by all flat connections in this operation.
Highlights
Contour integrals for Poincare polynomialsThese examples indicate how the analysis may be extended to other knots and links (e.g. those whose superpolynomials are found in [23, 26]), and to Poincare polynomials of other homological invariants
Be thought of as the effective three-dimensional theory on the R3 part of the fivebrane world-volume in an M-theory setup: space-time: fivebranes: R5 × CY3 ∪∪ R3 × M3
To much extent we have focused on examples of theories whose partition functions can be identified with homological knot invariants
Summary
These examples indicate how the analysis may be extended to other knots and links (e.g. those whose superpolynomials are found in [23, 26]), and to Poincare polynomials of other homological invariants. In order to interpret (2.1) as a suitable partition function of 3d N = 2 theory in this paper we mainly focus on half-indices and vortex partition functions This gives us enough flexibility to interpret (2.1) and we generically expect that the full set of partition functions for T [M3], labelled by a full set of vacua, corresponds to a complete basis of independent convergent contours for the integrals of section 2. We show that Poincare polynomials can be obtained by directly taking residues of S2 ×q S1 indices and Sb3 partition functions of T [M3]
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