Ẑ at Large N: From Curve Counts to Quantum Modularity

Abstract

Reducing a 6d fivebrane theory on a 3-manifold Y gives a q-series 3-manifold invariant ẑ(Y). We analyse the large-N behaviour of F_K = Ẑ(M_K), where M_K is the complement of a knot K in the 3-sphere, and explore the relationship between an a-deformed (a = qᴺ) version of F_K and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of F_K in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for a-deformed F_K for (2,2p+1)-torus knots. They suggest a further t-deformation based on superpolynomials, which can be used to obtain a t-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how F_K transforms under natural geometric operations on K indicates relations to quantum modularity in a new setting.

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