Physics and Geometry of Knots-Quivers Correspondence

Abstract

The recently conjectured knots-quivers correspondence (Kucharski et al. in Phys Rev D 96(12):121902, 2017. arXiv:1707.02991, Adv Theor Math Phys 23(7):1849–1902, 2019. arXiv:1707.04017) relates gauge theoretic invariants of a knot K in the 3-sphere to the 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 Ooguri-Vafa large N duality (Ooguri and Vafa in Nucl Phys B 577:419–438, 2000), 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 (Ooguri and Vafa 2000; Labastida and Marino in Commun Math Phys 217(2):423–449, 2001. arXiv:hep-th/0004196; Labastida et al. in JHEP 11:007, 2000. arXiv:hep-th/0010102; Aganagic and Vafa in Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots. arXiv:1204.4709; Fuji et al. in Nucl Phys B 867:506–546, 2013. arXiv:1205.1515).

Cross-Check of \(\mathbf {t}\) Refinement

In order to compare refined LMOV invariants for trefoil obtained in Sect. 5.2 with [64] we have to adjust conventions and take care about some subtleties.

Authors of [64] use the convention of $t_{r}$ refinement with $-a^{2}t^{3}$ instead of $-a^{2}t$ in the q-Pochhammer. Moreover, in section 4.7 concerning refined classical LMOV invariants they consider “super-A-polynomials as T-deformation and a-deformation of bottom A-polynomials (that arise for $a=0$ and $T=1$), where $t=-T^{2}$”, which is equivalent to rescaling of $N_{r}^{3_{1}}(a,q,t)$ by the overall a factor in such a way that it starts from 1. Therefore, the refined LMOV generating function for the fundamental representation should be transformed as follows

$$\begin{aligned} \begin{aligned} N_{1}^{3_{1}}(a,q,t)&=\left( 1+a^{2}t\right) \left( a^{2}q^{-2}+a^{2}q^{2}t^{2} +a^{4}t^{3}\right) \\&\rightsquigarrow \left( 1+a^{2}t^{3}\right) \left( 1+q^{4}t^{2}+a^{2}q^{2}t^{3}\right) . \end{aligned} \end{aligned}$$

In order to see 1 clearly, we rescaled also the q power, which however does not matter due to the semiclassical limit. It will also hide the q-shift between definitions of denominators of $N_{r}(a,q,t)$ (we have $1-q^{2}$, they have $q-q^{-1}$), however the sign difference will stay. In consequence

$$\begin{aligned} N_{1}^{3_{1}}(a,q=1,t)\rightsquigarrow -\left( 1+a^{2}t^{3}\right) \left( 1+t^{2}+a^{2}t^{3}\right) . \end{aligned}$$

Finally, authors of [64] put $t=-T^{2}$ and then rescale $a\rightarrow a^{1/2},\ T\rightarrow T^{1/2}$ to reduce the volume of Table 10 in the reference, which contains classical refined LMOV invariants. Unfortunately the last rescaling is not explicitly stated, for which the common author of the two papers apologizes. They consider only terms up to power 5, so summing up

$$\begin{aligned} N_{1}^{3_{1}}(a,q=1,t)\rightsquigarrow -\left( 1-aT^{3}\right) \left( 1+T^{2}-aT^{3}\right) =-1-T^{2}+2aT^{3}+aT^{5}+\ldots \end{aligned}$$

which is exactly equal to $\sum _{i,j}\tilde{b}_{1,i,j}a^{i}T^{j}$ from [64, Table 10] ($\tilde{b}_{r,i,j}$ denotes classical refined LMOV invariants in representation r corresponding to power i of a and power j of T).

We can repeat above steps for $r=2,3,4$ to obtain

$$\begin{aligned} N_{2}^{3_{1}}(a,q=1,t)\rightsquigarrow&T^{2}-aT^{3}+T^{4}-5aT^{5}+\ldots \\ N_{3}^{3_{1}}(a,q=1,t)\rightsquigarrow&-2T^{4}+7aT^{5}+\ldots \\ N_{4}^{3_{1}}(a,q=1,t)\rightsquigarrow&T^{4}-3aT^{5}+\ldots \end{aligned}$$

which again matches results from [64, Table 10] perfectly.