Physics and Geometry of Knots-Quivers Correspondence
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Communications in
Mathematical Physics
Physics and Geometry of Knots-Quivers Correspondence Tobias Ekholm1,2 , Piotr Kucharski1 , Pietro Longhi3,4 1 Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden.
E-mail: [email protected]; [email protected]
2 Institut Mittag-Leffler, Aurav. 17, 182 60 Djursholm, Sweden 3 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland. 4 Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden
E-mail: [email protected] Received: 14 November 2018 / Accepted: 8 June 2020 Published online: 18 September 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 3sphere 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 Mtheory considerations. From the physics side, we show that the quiver encodes a 3d 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). 1. Introduction Over the last 25 years, relations between knot theory and string theory, see e.g. [1,2], have revealed deep interconnections between physics and mathematics. This paper provides
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T. Ekholm, P. Kucharski, P. Longhi
Fig. 1. Physics and geometry of knots and quivers–schematic overview
physical and geometric underpinnings for a recently conjectured correspondence in this area that relates knots to quivers [3,4], see also [5–7]. The basic incarnatio
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