Refined Chern-Simons theory and knot homology

Abstract

The refined Chern-Simons theory is a one-parameter deformation of the ordinary Chern-Simons theory on Seifert manifolds. It is defined via an index of the theory on N M5 branes, where the corresponding one-parameter deformation is a natural deformation of the geometric background. Analogously with the unrefined case, the solution of refined Chern-Simons theory is given in terms of S and T matrices, which are the proper Macdonald deformations of the usual ones. This provides a direct way to compute refined Chern-Simons invariants of a wide class of three-manifolds and knots. The knot invariants of refined Chern-Simons theory are conjectured to coincide with the knot superpolynomials -Poincare polynomials of the triply graded knot homology theory. This conjecture is checked for a large number of torus knots in S 3 , colored by the fundamental representation. This is a short, expository version of arXiv:1105.5117, with some new results included. 1 1 Based on talks presented by M.A. at several conferences and workshops, including String-Math 2011 Conference at U. of Pennsilvania.

Factorisation and holomorphic blocks in 4d

“…Another aspect to investigate is the relation of 4d blocks to integrable systems and to CFT-like correlators. 3d block integrals have been identified with q-deformed Virasoro free field correlators in [57,58]. The possibility to interpret 4d block integrals as free field correlators in an elliptic deformation of the Virasoro algebra will be investigated in [59].…”

Link polynomial calculus and the AENV conjecture

“…According to [1], it is most convenient to begin from the alternative representation for the Hopf link used in ref. [28]:…”

Topological vertex for Higgsed 5d T N theories

“…4 On the other hand the web diagrams corresponding to Higgsed 5d T N theories are not dual to toric geometries and therefore simple application of the refined topological vertex does not work. The formalism to compute their partition function was developed in [21,28] 5 by making use of the technique in [33][34][35][36][37][38]. The prescription to compute the partition function of Higgsed 5d T N theories is a three step process.…”

“…The result of this process is the partition function of the IR theory plus some contributions due to singlet hypermultiplets that are present in the Higgs vacuum and need to be removed to obtain the final result. 2 Recently, it has been noticed that a certain mass deformation of the TN theory induces an RG flow to a linear quiver theory [13][14][15]. 3 As a matter of fact it is possible to see that the enhancement of the flavour symmetry already happens at the level of the Nekrasov partition function after a suitable shift of the Coulomb branch moduli [23].…”

“…It turns out that we can use the same parameterisation as that of the T 3 diagram, which is determined in [21], but the only difference is that we use α 1,2 for the Coulomb branch moduli for the parameterisation 13 In this paper, we do not take into account the perturbative contribution from the Cartan part of the vector multiplet. That contribution is not included in the topological vertex computation.…”