Multi-cover skeins, quivers, and 3d $$ \mathcal{N} $$ = 2 dualities

Abstract

The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver descriptions of the geometry. Embedding into M-theory, a large class of dualities of 3d N = 2 theories associated to quivers is obtained. The multi-cover skein relation admits a compact formulation in terms of quantum torus algebras associated to the quiver and in this language the relations are similar to wall-crossing identities of Kontsevich and Soibelman. JHEP02(2020)018 6.3 Review of wall-crossing 34 6.4 Wall-crossing as multi-cover skein relations: the pentagon 35 6.5 Operator form of the multi-cover skein relation 37 6.6 Quantum torus algebra and holomorphic curve counting 38 6.7 Beyond the pentagon 40 6.8 Quantum gluing of 3-manifolds along tori 40 6.8.1 Semi-classical limit, disk potentials, and Lagrangian correspondences 41 6.8.2 The quantized Lagrangian correspondence 42A Multi-cover skein relations beyond disks 43 B Nonuniqueness of the quiver for a given knot -4 1 example 45 C Details of generalized multi-cover skein identities 46

Quivers for 3-manifolds: the correspondence, BPS states, and 3d $$ \mathcal{N} $$ = 2 theories

“…The saddle point of the twisted superpotential encodes the moduli space of vacua of T [Q L K ] and defines the quiver A-polynomials [12,13,19,20]:…”

“…The general formula for the quantum quiver A-polynomial corresponding to the quiver with adjacency matrix C is given by [13]…”

“…Geometric and physical interpretation of the knots-quivers correspondence given in [12,13] connects it with Gromov-Witten invariants and counts of holomorphic curves, showing that the quiver encodes the construction of BPS spectrum from the basic states. This spectrum is shared by dual 3d N = 2 theories associated to the knot complement and the quiver, and respecitve counts of BPS states are given by LMOV and DT invariants.…”

“…This spectrum is shared by dual 3d N = 2 theories associated to the knot complement and the quiver, and respecitve counts of BPS states are given by LMOV and DT invariants. The moduli spaces of vacua of these theories are encoded in the graphs of A-polynomials [14][15][16][17][18] and their quiver versions [12,13,19,20]. Another recent results on the knots-quivers correspondence include a relation to combinatorics of counting paths [19], a consideration of more general case of topological strings on various Calabi-Yau manifolds [20], and a connection with the topological recursion [21].…”

“…Moreover, since HOMFLY-PT polynomials admit t-deformation in terms of superpolynomials [48], we can follow [43] and conjecture results for a, t-deformed GM invariants. We also study the physical intepretation of the new correspondence (for clarity we will refer to the knots-quivers correspondence of [1,2] as the standard one), generalising the results of [12,13]. It includes an exploration of the duality between 3d N = 2 theories associated to the knot complement and the quiver, as well as a study of their BPS states, which allows us to introduce knot complement analogs of LMOV invariants via DT invariants.…”