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 mathcal{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.
Highlights
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
There is an interesting relation between quivers and open topological strings that was first observed in applications to knot theory [1, 2]
In [3] we discussed the underlying geometry and physics, in terms of counts of open holomorphic curves ending on a knot conormal LK, and in terms of the 3d N = 2 physics on an M5-brane wrapping LK × S1 × R2
Summary
There is an interesting relation between quivers and open topological strings that was first observed in applications to knot theory [1, 2]. We relate counts of open holomorphic curves, quivers, and certain 3d N = 2 quantum field theories, in a way that takes simple properties of one theory to highly nontrivial statements in the others. This leads to new results both on the mathematical and physical sides, including mechanisms for generating classes of distinct quivers (with different number of nodes) that determine the same physics, multi-cover skein relations, and a large class of 3d N = 2 dualities. The results are not limited to the original knot theory setting of [1, 2] but give connections between quivers and open topological strings in many other situations
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