Condensates and instanton - torus knot duality. Hidden Physics at UV scale

TL;DR

This paper reveals a duality linking torus knot superpolynomials with condensates in 5D supersymmetric QED, showing how instanton contributions encode knot invariants and exploring the rich physics of UV degrees of freedom.

Contribution

It establishes a novel duality connecting knot invariants with instanton condensates in 5D supersymmetric gauge theories, expanding the understanding of gauge-knot correspondences.

Findings

Instanton contributions match torus knot superpolynomials.

Summation over knots is necessary for complete gauge theory results.

Connection between superpolynomials and q-deformed DOZZ factors via AGT.

Abstract

We establish the duality between the torus knot superpolynomials or the Poincar\'e polynomials of the Khovanov homology and particular condensates in $\Omega$-deformed 5D supersymmetric QED compactified on a circle with 5d Chern-Simons(CS) term. It is explicitly shown that $n$-instanton contribution to the condensate of the massless flavor in the background of four-observable, exactly coincides with the superpolynomial of the $T(n,nk+1)$ torus knot where $k$ - is the level of CS term. In contrast to the previously known results, the particular torus knot corresponds not to the partition function of the gauge theory but to the particular instanton contribution and summation over the knots has to be performed in order to obtain the complete answer. The instantons are sitting almost at the top of each other and the physics of the "fat point" where the UV degrees of freedom are slaved with point-like instantons turns out to be quite rich. Also also see knot polynomials in the quantum mechanics on the instanton moduli space. We consider the different limits of this correspondence focusing at their physical interpretation and compare the algebraic structures at the both sides of the correspondence. Using the AGT correspondence, we establish a connection between superpolynomials for unknots and q-deformed DOZZ factors.