Schedule for: 18w5078 - Geometry and Physics of Quantum Curves

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will give a gentle introduction to the notion of quantum Airy structures proposed by Kontsevich and Soibelman and further studied together with Andersen, Chekhov and Orantin; show how it incorporates the usual topological recursion based on spectral curves of Chekhov, Eynard, and Orantin. In this language, one can identify an action of a group of symplectomorphisms, which allows a transparent dictionary with Givental formalism. Earlier observations of Kazarian on the relevance of affine symplectic geometry in topological recursion, and the relation with cohomological field theories established by Dunin-Barkowski et al. find a natural place in this framework. Therefore, rather than new results, my talk will be about a slight change of point of view, which unifies several aspects known in topological recursion. It also serves as a background for the lecture of Vincent Bouchard on the generalization to higher order Airy structures in which Virasoro constraints are replaced with W(A_r)-algebras and spectral curves can have ramification points of arbitrary order.

We consider the pull-back of a natural sequence of cohomology classes \(\Theta_{g,n}\in H^{2(2g-2+n)}(\overline{\mathcal{M}}_{g,n})\) to the moduli space of stable maps $\overline{\mathcal{M}}^g_n(\mathbb{P}^1,d)$. These classes are related to the Brezin-Gross-Witten tau function of the KdV hierarchy via $$Z^{BGW}(\hbar,t_0,t_1,...)=\exp\sum\frac{1}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}\cdot\prod_{j=1}^n\psi_j^{k_j}\prod t_{k_j}.$$ Insertions of the pull-backs of the classes $\Theta_{g,n}$ into the integrals defining Gromov-Witten invariants define new invariants. In the case of target $\mathbb{P}^1$ we show that these are computable and satisfy the Toda equation.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Under mild assumptions the base of a complex algebraic integrable system carries a natural Kahler metric and a natural affine structure which together constitute what is known as a special Kahler geometry. In this talk we will focus on the case of the Hitchin integrable system. We show that the special Kahler geometry may be computed using the theory of topological recursion. In particular we consider the Donagi-Markman cubic, which measures the difference between the Levi-Civita connection and the affine connection and show that it is given by an Eynard-Orantin invariant. This talk is based on joint work with Zhenxi Huang.

The $GL$-Hitchin system is a Lagrangian fibration whose fibres are Jacobians of spectral curves. Baraglia and Huang showed how the special Kahler geometry of the base of the fibration can be computed from the Eynard-Orantin invariants of the spectral curves. In this talk we describe some other examples of fibrations where these techniques may apply, including compact fibrations by Jacobians and both compact and non-compact fibrations by Prym varieties.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Building on Gaetan Borot's introduction to Airy structures and topological recursion, I will define “higher Airy structures”, whereby quadratic differential operators are replaced by higher order differential operators. I will show how twisted (and untwisted) modules for $W(A_n)$-algebras provide examples of higher Airy structures, and in fact can be used to reconstruct the correlation functions obtained from the topological recursion on spectral curves with higher ramification. If I have time, I may also briefly explore the connection between Airy structures and quantum curves. This is based on joint work with Gaetan Borot, Nitin Chidambaram and Dmitry Noshchenko.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will discuss some general aspects on quantum curves. I will emphasize the need to go beyond WKB-like perturbative schemes, and the usefulness of working with actual Hilbert spaces. I will then present three mathematical mechanisms in our theory of quantum mirror curves which make it possible to go from the discrete world of quantum mechanics to the continuous world of enumerative geometry: Fredholm determinants, large $N$ limits, and Wigner distributions. I will also list some interesting open problems for the future.

I will present work in progress suggesting a conjecturally general correspondence relating quantum (Chern-Simons) invariants of knots in three-space to (Eynard-Orantin) higher genus mirror symmetry, the computation of the former being solved by the latter. The correspondence rests on the identification of a natural group of piecewise linear transformations on both sides of the correspondence.

The Askey–Wilson polynomials are a special case of Macdonald polynomials; their degenerations are organised into a hierarchy called $q$-Askey scheme. In this talk the speaker will give a new approach to study degenerations as well as dualities in the $q$-Askey scheme, based on the fact that all families of polynomials in the $q$-Askey scheme admit a certain algebra of symmetries (the Zhedanov algebra and its degenerations) and in the limit as $q\to 1$ this algebra becomes the coordinate ring of a certain subalgebra of a quantum object that we call quantum cusped character variety.

We reformulate spectral problems in the language of WKB theory and generalise to higher rank. This is based on work with Andrew Neitzke.

Based on the work of Eynard-Orantin and Marino, the Remodeling Conjecture was proposed in the papers of Bouchard-Klemm- Marino-Pasquetti in 2007 and 2008. The Remodeling Conjecture can be viewed as an all genus mirror symmetry for toric Calabi-Yau 3- orbifolds. It relates the higher genus open Gromov-Witten potential of a toric Calabi-Yau 3-orbifold to the higher genus B-model potential which is obtained by applying the topological recursion on the mirror curve.

On any punctured Riemann surface $X$, there are several interesting procedures to deform curves (with additional structure) in the cotangent bundle of $X$ to families of D-modules on $X$ which degenerate to the original curves. The non-abelian Hodge theory provides us one of such constructions. Namely, for each complex number $\lambda$, we have the equivalence between irreducible harmonic bundles and stable $\lambda$-flat bundles of degree $0$, which gives us a deformation of a stable Higgs bundle of degree $0$ to a family of $\lambda$-flat bundles. In this talk, we shall explain an analogue construction from curves (with additional structure) in ${\mathbb C}\times{\mathbb C}^{\ast}$ to families of difference modules which degenerate to the original curves, in the context of non-abelian Hodge theory. Namely, for each complex number $\lambda$, we have an equivalence between irreducible periodic monopoles and stable $2\sqrt{-1}\lambda$-difference modules of degree $0$. It is expected to be a kind of Fourier transform of the equivalences for harmonic bundles on ${\mathbb C}^{\ast}$.

Borcea and Voisin prove that a pair of mirror K3 surfaces with anti-symplectic involutions $(S,i)$ and $(S',i')$ yield two mirror Calabi-Yau three-folds (via products with elliptic curves and quotients). We translate all this into a Landau-Ginzburg model where mirror statements become more transparent, easier to generalize and prove. We provide generalizations in several directions: Calabi-Yau of any dimension with involutions, higher order automorphisms, their fixed loci. In several ways these mirror statements are beyond the Calabi-Yau category; they concern for instance degree-$2d$ hypersurfaces in $\mathbb{P}^{d-1}$. (Work in collaboration wi Kalashnikov and Veniani.)

Moduli spaces of stable parabolic connections on curves are very interesting objects which are related to different area of mathematics like algebraic geometry, integrable systems, mathematical physics and Geometric Langlands conjecture. In this lecture, we will explain about an explicit geometry of the moduli spaces of stable parabolic connections on curves introduced and constructed by Inaba, Iwasaki and Saito and Inaba. Then we will review a work of Arinkin and Lysenko on a rank 2 connections on the projective line with 4 singular points, which is related to Geometric Langlands conjecture in this case. We then explain about the joint work on the moduli space of rank 2 parabolic bundles on the projective line with Simpson and Loray. If time permits, related works of Geometric Langlands conjecture in these cases may be discussed.

The tautological ring of the moduli space $\overline{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points is the subring of the cohomology ring generated by a collection of natural geometric classes. The vanishing results of Looijenga, Ionel, Graber--Vakil and Faber--Pandharipande state that any tautological class of codimension at least $g$ vanishes on the open locus $M_{g,n}$ of smooth curves. This raises the question of finding boundary formulas for such classes. Two large collections of tautological relations were recently proved, both having been earlier conjectured by Pixton: the 3-spin relations, proved by Janda, Pandharipande, Pixton and Zvonkine, and the double ramification relations, proved by Clader and Janda. I will show how these relations can be used to effectively find boundary formulas for tautological classes of codimension at least $g$, and to prove sharper results about these boundary formulas. A corollary is that the tautological ring admits a secondary grading by the number of kappa-markings. This is joint work with Emily Clader, Samuel Grushevsky, Felix Janda and Xin Wang.

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract setup we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies and Zeta-functions based on the simpel closed geodesic length spectrum. We shall see how Geometric Recursion provides us with a kind of categorification of Topological Recursion, namely any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. We will end the talk by applying the machinery to obtain interesting results on expectation values of various statistics of length of simple closed geodesic over moduli spaces of hyperbolic surfaces. The work presented is joint with G. Borot and N. Orantin.

I will describe, by using Eynard-Orantin's topological recursion, the modularity of open-closed Gromov-WItten invariants for certain toric Calabi-Yau 3-folds. When the mirror curve is genus one and is in the hyperelliptic form $y^2=f(x)$, one can analyze the modular behavior in the recursion residue calculus. By combining with the remodeling conjecture (now a theorem) and Eynard-Marino-Orantin's holomorphic anomaly equation (HAE) for the recursion invariants, one obtains the HAE and the Yamaguchi-Yau type equation. This talk is based on the joint works with C.-C. M. Liu, Z. Zong and with Y. Ruan, Y. Zhang, J. Zhou.

I will briefly review the LG/CY correspondence, and what is known so far, and then I will describe new developments in this area, namely gauged linear sigma models (GLSM) which help us understand this phenomenon in greater generality. I will end with an example involving so-called Borcea-Voisin mirror symmetry.

Quantum $A$-polynomials and their generalizations annihilate generating functions of colored knot polynomials, and can be regarded as interesting examples of quantum curves. They also encode open BPS (Labastida-Marino-Ooguri-Vafa) invariants, which can be defined once knots are engineered in string theory in brane systems in the resolved conifold geometry. Recently formulated "knots-quivers correspondence" enables to prove integrality of a large class of such BPS invariants, it provides new viewpoint on homological knot invariants, and reveals new properties of $A$-polynomials – for example it immediately leads to the formulation of their higher dimensional generalizations. In this talks, after a brief summary of the knots-quivers correspondence, I will present its generalization beyond the realm of knot theory, to more general underlying Calabi-Yau geometries, and discuss corresponding quantum curves and their higher dimensional generalizations.

We propose a new link between the moduli spaces of wild Higgs bundles and quantum invariants of 3-manifolds. The construction goes through a class of four-dimensional quantum field theories known as Argyres-Douglas theories. Every such theory realizes a wild Hitchin space as its Coulomb branch and defines a VOA on the Higgs branch. The latter can be used to construct a non-unitary modular tensor category, which leads to $3d$-TQFTs that are generically semisimple but non-unitary. This is based on joint work with Mykola Dedushenko, Sergei Gukov, Hiraku Nakajima and Ke Ye.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.