Volume conjecture: refined and categorified

Abstract

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial A(x, y). Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted q or ; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation e-print archive: http://lanl.arXiv.org/abs/1203.2182v1 * Author of an appendix "ATMP-16-6-A3-FUJ" -2013/5/25 -9:47 -page 1670 -#2 ( x, y; q, t). We compute both classical and quantum t-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.

3d-3d correspondence revisited

“…As explained in [40,41] (for t = −1) and [18,22] (for general t), these factors are necessary for quantization but are not associated to classical flat connections. For knots considered in this paper these factors are independent of y, they do not affect the structure of roots of A ref (x, y; t) or A(x, y) at generic fixed x, and therefore they do not modify our discussion here.…”

“…Similarly, if we follow [22] and compactify T [0 1 ] on a circle turning on masses (i.e. complexified scalars in background gauge multiplets) for U(1) x and U(1) t , we find that the theory is governed by an effective twisted superpotential…”

“…Let us start with the Poincaré polynomials P r K (t; q) for colored SU(2) knot homology of a knot K. As found in [18,22,23], the sequence of Poincaré polynomials obeys a recursion relation of the form…”

“…It is further convenient to renormalize the SU(2) polynomial by a factor (−1) r , defining 3 (q 2 (−t) 3 ; q) r (q(−t); q) k (q; q) k (q; q) r−k (−q 2 Specialization a = q 2 leads to Poincaré polynomials of colored SU(2) knot homologies for a certain class of knots, which include unknot, trefoil, and figure-8 knot considered in this paper. In general and for more complicated knots, specialization of superpolynomials to Poincaré polynomials of SU(N ) knot homologies requires taking into account a nontrivial action of differentials [13,15,22]. Note that in [18,22,23] normalized colored superpolynomials were considered, i.e.…”

“…In general and for more complicated knots, specialization of superpolynomials to Poincaré polynomials of SU(N ) knot homologies requires taking into account a nontrivial action of differentials [13,15,22]. Note that in [18,22,23] normalized colored superpolynomials were considered, i.e. divided by the superpolynomial of the unknot colored by the same representation.…”