Physics and geometry of knots-quivers correspondence

Abstract

The recently conjectured knots-quivers correspondence relates gauge theoretic invariants of a knot $K$ in the 3-sphere to representation theory of a quiver $Q_K$ associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of the large $N$ duality of Ooguri and Vafa, that relates knot invariants to counts of holomorphic curves with boundary on $L_K$, the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d $\mathcal{N}=2$ theory $T[Q_K]$ whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of $Q_K$. From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on $L_K$ is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver $Q_K$ and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to $Q_K$ and (doubly) refined version of LMOV invariants.

v2: references updated, v3: updated introduction, matches published version