American Institute of Mathematics, San Jose, California
Olivia Dumitrescu, Motohico Mulase, Sergey Shadrin, and Piotr Sulkowski
The main topics for the workshop are the following.
Topic 1. The physics origin of quantum curves.
At the beginning of the workshop, we will offer a survey of quantum curves from the physics point of view. As the name suggests, they are quantized algebraic or analytic curves, and take the form of a family of (often infinite-order) ordinary differential equations. Algebraic curves appear in exactly solvable models in high energy physics, such as Seiberg-Witten curves, mirror curves in topological string theory, spectral curves in matrix models, and the A-polynomials in Chern-Simons theory. Each of these curves is conjecturally quantizable, and the resulting differential equation determines the partition function of the theory. Even though the nature of the physics theory is different, a universal formalism of quantization has been proposed, based on the topological recursion formula due to Eynard, Orantin, and others. We plan to identify the mathematical nature of the partition functions that the quantum curve determines.
Topic 2. The mathematical examples of quantum curves.
In recent years numerous rigorous examples of quantum curves have been constructed by mathematicians. Many of the examples are related to certain Hurwitz-type counting problems of coverings, and also Gromov-Witten invariants, of the projective line, and their orbifold generalizations. We will offer a survey of these examples. Other types of differential equations that appear as the result of quantization include classical equations. For example, the quantum curve for the Witten-Kontsevich intersection theory on the moduli of stable curves is the Airy differential equation, and the one for Catalan numbers is the Hermite equation. The Gauss hypergeometric equations also appear in the same context. Most recently, it has been discovered that the spectral curves of Hitchin fibrations are quantizable. The Airy, Hermite, and Gauss equations are all understood in this framework, giving a concrete formula for the non-abelian Hodge correspondence on a curve. A general mathematical theory is sought, based on the abundant rigorous examples.
Topic 3. Filling the gap between physics speculations and mathematical theorems.
The mathematical nature of the topological recursion, when considered locally, has been well understood in terms of Frobenius manifolds and Givental formalism. We hope to study the condition for the existence of a global spectral curve, and seek the role of a quantum curve in terms of Frobenius manifolds. More specifically, we ask when a global spectral curve exists in a Cohomological Field Theory. This direction of research is expected to lead to a discovery of new mathematical relations between different physical theories, and hence different types of quantum invariants.
The appearance of Hitchin fibrations in the context of quantum curves poses a fundamental question: what is the relation between quantization of Hitchin systems and the quantum curves for Hitchin spectral curves? The time seems to be ripe to answer this question. The prototype case for the Verlinde bundles gives a us a guide. A mathematical theory of branes of various types on the moduli of Higgs bundles has been constructed. A generalized topological recursion has been proposed, replacing the affine plane in the original context with the cotangent bundle of an arbitrary algebraic curve. Here, a connection to the Hilbert scheme of points on a surface comes in to provide a mathematical interpretation of the generalized topological recursion, appealing to the idea of geometric quantization. A group of participants will focus their effort on this question.
It has been speculated that behind the scene of all these, there should be a master integrable system and theory of tau-functions. Painleve equations naturally appear in the very context of quantum curves. The question of integrability will also be studied by a group of participants.
The workshop participants include experts from string theory, algebraic geometry, representation theory, integrable systems, topological recursion, complex geometry, and matrix models. We believe they cover the necessary background for the participants to carry out the mission of the workshop.
The workshop schedule.
A report on the workshop activities.
Papers arising from the workshop: