Publications

Branches, quivers, and ideals for knot complements

with Tobias Ekholm, Sergei Gukov, Piotr Kucharski, Sunghyuk Park, Marko Stosic, Piotr Sulkowski

We generalize the F_K invariant, i.e. Z for the complement of a knot K in the 3-sphere, the knots-quivers correspondence, and A-polynomials of knots, and find several interconnections between them. We associate an F_K invariant to any branch of the A-polynomial of K and we work out explicit expressions for several simple knots. We show that these F_K invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using R-matrices. We generalize the quantum a-deformed A-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter a, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide t-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d N=2 theory T[M3] and to the data of the associated modular tensor category MTC[M3].

Z at large N: from curve counts to quantum modularity

with Tobias Ekholm, Sergei Gukov, Piotr Kucharski, Sunghyuk Park, Piotr Sulkowski

Reducing a 6d fivebrane theory on a 3-manifold Y gives a q-series 3-manifold invariant Z(Y). We analyse the large-N behaviour of F_K = Z(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 = q^N) version of FK 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.

Computing Modular Data for Pointed Fusion Categoies

with Scott Morrison

A formula for the modular data of Z(Vec_ωG) was given by Coste, Gannon and Ruelle in arXiv:hep-th/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra D_ωG. Further, we have written code to compute this modular data for many pairs of small finite groups and 3-cocycles. This code is optimised using Galois symmetries of the S and T matrices. We have posted a database of modular data for the Drinfeld center of every Morita equivalence class of pointed fusion categories of dimension less than 64.

The database of modular data can be found here.