# Quantum Curves and Quantum Knot Invariants (14w5073)

## Organizers

Vincent Bouchard (University of Alberta)

Mikhail Khovanov (Columbia University)

Motohico Mulase (University of California, Davis)

Alexei Oblomkov (University of Massachusetts)

Marko Stošić (Instituto Superior Técnico, Portugal)

Piotr Sułkowski (University of Warsaw and Caltech)

## Objectives

The goal of the proposed workshop is to explore the mystery surrounding mirror symmetry of quantum knot invariants. More specifically, we aim at building a concrete mathematical understanding of the quantization process, given by the topological recursion of Eynard and Orantin, which is expected to construct quantum knot invariants from the classical knot A-polynomials.

There are two different kinds of knot invariants. One is the class of quantum knot invariants, such as colored Jones and HOMFLY polynomials. The quantum invariants are related to quantum algebras and their representations. Compared to these, the A-polynomials of knots are “classical” invariants, in the sense of classical mechanics versus quantum mechanics. The A-polynomials are derived from the fundamental group of the knot complement as a topological manifold.

Since the time of the celebrated work of Witten, quantum knot invariants are identified as quantum field theoretic expectation values of Chern-Simons gauge theory on a 3-sphere. Large N duality then relates Chern-Simons gauge theory to an A-model topological string theory. In a mathematical language, the A-model string theory can be understood as Gromov-Witten theory of the target variety.

Here comes the mirror symmetry. If quantum knot invariants give rise to an A-model theory, then what is its mirror dual B-model?

The correct answer, noticed by many different groups of physicists, seems to be that the mirror dual is the B-model remodeled on an algebraic curve, and that this algebraic curve is exactly what the knot A-polynomial defines. But we have just said that the A-polynomials are classical invariants. Since we expect that mirror symmetry is a categorical equivalence, each side of the mirror dual should contain the exact same information. This means the following:

1) The A-polynomial contains the information of all quantum knot invariants for any given knot. 2) The procedure of quantization should provide a formula for quantum knot invariants, starting from a given A-polynomial.

Different groups of physicists, including Dijkgraaf, Fuji, Gukov, Sułkowski, and others, discovered last year that the quantization mechanism necessary in the B-model side should be given by the topological recursion of Eynard and Orantin.

In the overview section we mentioned physical procedures known as the loop equation, the Ward identity, and the Schwinger-Dyson equation, appearing in three different contexts. From the mathematical point of view, these equations are exactly the same thing, and its particular manifestation is the Eynard-Orantin formulation. It is a quantization mechanism, and it computes generating functions of various quantum invariants from the classical object, a Riemann surface called the spectral curve of the theory.

It is expected that the solution of the Eynard-Orantin recursion for a given spectral curve determines a tau-function of an integrable system of the KdV/KP type. If the spectral curve satisfies certain conditions (in terms of algebraic K-theory), then this integrable system, in turn, defines a holonomic system, or the quantum curve, that characterizes the partition function of the theory.

Furthermore, already physicists are generalizing all these mechanism, using categorification, to include Khovanov homology and the more recently discovered notion of the super A-polynomials. In this context, Schur functions that appear in quantum knot invariants and integrable systems are deformed into Macdonald polynomials.

The A-polynomial of a knot is defined by considering the moduli space of SL(2)- representations of the fundamental group of the knot complement in the 3-sphere. Thus the procedure explained above is limited to colored Jones polynomials. How can we include the HOMFLY into this scheme?

Here again many different groups of string theorists, including Aganagic and Vafa, and Brini, Eynard and Mariño, speculate that the Kähler parameters of the A-model topological string theory (or the Gromov-Witten theory) are directly related to the parameter N in the group SL(N). Thus by considering complex deformations of the mirror dual B-model, we should be able to obtain HOMFLY polynomials.

This is an amazingly beautiful grand picture. From the mathematical point of view, however, most of it is still a conjecture. A timely workshop is therefore an absolute necessity. Our goal is to form a precise mathematical understanding of the general picture, and to formulate physics speculations in terms of concrete mathematical language. In this way we hope to be able to address, and ultimately solve, the important conjectures of the field.

Already many mathematicians have begun the journey toward the construction of mathematical theories aiming at understanding the grand picture. Search for the mirror symmetric counterpart of the work of Oblomkov and Shende in terms of the remodeled B-model is an example. A mathematical theory of the Eynard-Orantin recursion is also being developed. Although unrelated to knot invariants, many concrete mathematical examples are constructed by Mulase, Shadrin, Sułkowski, and others, with which physics predictions, in particular, the relations to quantum curves and integrable systems, are exhibited and proved.

The appearance of the Bloch group and algebraic K-theory (the second K-group) in connection to knots goes back to the fundamental work of Neumann and Zagier on the value distribution of the volume of hyperbolic knot complements. Zagier noticed the structural similarity between the Bloch groups and the Eynard-Orantin theory. Indeed, the second K-group appears in the work of Gukov and Sułkowski in the context of obstruction to quantizability, in much the same way as the K2-Lagrangian condition of Kontsevich. A recent work of Borot and Eynard indicates the critical importance of modular invariance of the partition function. The combinatorial structure of the Eynard-Orantin formalism under these conditions may be a key to understanding the relation between the hyperbolic volume conjecture of Kashaev, Murakami, and Murakami, and the triangulations of knot complements by ideal tetrahedrons.

What makes the theme of the proposed workshop fascinating is its deep and essential connection to many different areas of mathematics. And being a new subject, to solve the problems in this emerging field requires collaboration of researchers with different backgrounds and expertise. We believe that the proposed workshop is extremely timely, and will attract many young researchers to participate.