Irregular Connections, Character Varieties and Physics
Paris VII, March 6-9 2017
Oper structures for connections on curves
There is a classical correspondence between systems of n linear ordinary differential equations (ODEs) of order one and linear ODEs of order n; this may be viewed as a kind of canonical normal form for systems of ODEs. Restated geometrically, the claim is that if E is a vector bundle on a Riemann surface, and $\nabla$ is a connection on E (with arbitrary singularities), then there is a rational basis of E such that $\nabla$ is in the canonical normal form.
The statement makes sense for arbitrary reductive Lie group G (with the case of systems of ODEs corresponding to G=GL(n)): given a connection on a G-bundle on a Riemann surface, we can look for a rational gauge change that brings the connection to the `canonical normal form' (also known as a rational oper structure). This generalized statement turns out to be significantly harder. In my talk, I will prove it for any G and discuss its role in the geometric Langlands program.
Weighted Hurwitz numbers, topological recursion and isomonodromic deformations
The family of 2D-Toda tau functions of hypergeometric type that serve as generating functions for weighted Hurwitz numbers with polynomial weight generating functions have an associated family of spectral curves that are rational. The corresponding quantum spectral curves are given by a family of ODE's with rational coefficients whose monodromy is invariant under the deformations generated by the underlying KP flows. An alternative generating function for the weighted Hurwitz numbers is provided by the multicurrent correlators, which are expressible both as fermionic vacuum expectation values, and directly in terms of the tau function. The WKB series for the Baker function leads to a series of recursion relations between the weighted Hurwitz numbers, fitting within the general framework of the Topological Recursion program. (Based on joint work with A. Alexandrov, G. Chapuy and B. Eynard)
Equivariant Verlinde algebra for Higgs bundles
Motivated by the work of Gukov and Du Pei we discuss a construction of a Frobenius algebra, which computes equivariant indices of line bundles on the moduli space of Higgs bundles.
This is joint work with Andras Szenes.
Stability of the vector bundle along isomonodromic deformations
A result of A. Bolibruch states that some irreducible logarithmic connections on the Riemann sphere can be isomonodromically deformed into a connection on the trivial vector bundle (a fuchsian system). We present a generalisation of this result to the higher genus case, which remains valid for certain irregular connections.
Exact WKB analysis and related topics
Exact WKB analysis, developed by Voros et.al., is an effective method for global study of
(singularly perturbed) ordinary differential equations defined on a complex domain.
After recalling several fundamental facts about exact WKB analysis, I'll talk about
relationships to other research topics, such as cluster algebras, topological recursion,
integrable systems of Painlevé type, etc.
Non-linear Stokes phenomenon in the fifth Painlevé equation
The Hamiltonian system of the sixth Painlevé equation (PVI) has well-defined non-linear monodromy operators, given by meromorphic continuation of solutions along loops around each of its 3 regular singular points, which together carry a great deal of information about the equation. The fifth Painlevé equation (PV) is obtained from PVI as a limit through a confluence of singularities, during which a big part of the monodromy information is lost. This lost information reappears in the form of a non-linear Stokes phenomenon at the irregular singularity at infinity.
The goal of the talk is to explain the relation between the non-linear monodromy of PVI and the non-linear Stokes operators of PV, and how these can be represented via the Riemann-Hilbert correspondence for the associated isomonodromic problem.
Fourier Duality of Quantum Curves
I will discuss the association of meromorphic connections on the formal punctured disc to quantum curves. A special case concerns connections associated to (p,q) minimal model CFT's coupled to gravity. It is known that for these theories there is a duality relating the (p,q) and (q,p) models. I will discuss a description of this duality via the local Fourier transform developed by Bloch-Esnault. This is based on joint work with Albert Schwarz.
Affine Opers with one irregular singularity and the Bethe Ansatz
The ODE/IM correspondence is a conjectural and surprising link between nonlocal observables of integrable quantum field theories and monodromy data of linear analytic ODEs.
In this talk g is a simple Lie algebra over the complex field, (g,1) the corresponding untwisted Kac-Moody algebra and Lan(g,1) the Langlands dual of (g,1).
We prove the following conjecture of Feigin and Frenkel ['11]: If L is an Lan(g,1)-affine oper (of a particular type), its (generalised) monodromy data satisfy the Bethe Ansatz equations of the Quantum g-KdV model.
The talk is based on two papers with A. Raimondo and D. Valeri:
Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case. Comm Math Phys, 344 (2016), no. 3
Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections II: The non simply-laced case . Comm Math Phys 349 (2017), no. 3
BPS states, torus links, and wild character varieties
I will report on recent joint works with Chuang, Diaconescu,
and Donagi, in which we develop a string theoretic framework for
understanding the results and conjectures of Hausel, Letellier, and
Rodriguez-Villegas, and of Hausel, Mereb, and Wong on the topology of
tamely and wildly ramified character varieties. The physics dualities
and constructions lead to a new generalization of the formula of
Hausel, Mereb, and Wong and also provide a colored version of the
conjecture of Shende, Treumann and Zaslow relating the topology of
wild character varieties to knot and link invariants. I will also
explain how the string theoretic approach provides evidence for the
wild variant of the P=W conjecture of de Cataldo, Hausel, and
Migliorini. The main mathematical tool is an exhaustive spectral cover
correspondence which works for all points in the moduli space of
irregular Higgs bundles.
Stokes groupoids and isomonodromic deformations
The Stokes groupoids are the natural domains on which the parallel transport of a meromorphic connection is single-valued and holomorphic---even when the connection has irregular singularities. When applied to the problem of isomonodromic deformation, it leads to a special ``uniformizing solution'', from which all others are obtained by functorial operations of Fourier--Mukai type. I will describe this construction and explain how it interacts with various well-known features of isomonodromy, such as Painlevé transcendents, (shifted) Poisson structures on moduli spaces of connections, and (higher homotopical) mapping class group actions. This talk is based on forthcoming work with Marco Gualtieri.
Gauge theories, wild Hitchin system and Painlevé equations
Abstract: I will show how the BPS spectrum of a class of N=2 supersymmetric gauge theories is described in terms of the Hitchin system
with regular and irregular singularities. I will discuss an algebraic characterisation of the latter in terms of irregular states in the Verma module of the Virasoro algebra.
I will describe the relation between Renormalisation Group Flow of supersymmetric gauge theories, Painlevé confluence diagram and two dimensional Landau-Ginzburg models.
Open de Rham Spaces
By an open de Rham space, we mean a moduli space of meromorphic connections over the trivial bundle over the projective line. We will discuss various aspects of these spaces: their relation to varieties associated to a weighted quiver, the existence of hyperkaehler metrics, the computation of some algebraic invariants, and a generalization of a conjecture of Hausel--Letellier--Rodriguez-Villegas. This is a report on work with Tamás Hausel and Dimitri Wyss.
Stokes phenomenon, Gelfand-Zeitlin systems and quantum groups
We relate Boalch's dual exponential maps to Gelfand-Zeitlin systems.
It motivates a notion of relative Ginzburg-Weinstein linearization.
We then consider a Knizhnik-Zamolodchikov type equation with a pole of order two.
It turns out that the (quantum) Stokes factors at this pole satisfy Yang-Baxter equation.
This implies a relation between irregular connections and Drinfeld's quantum groups. This part is a joint work with Toledano Laredo.
In the end, we explain that the two parts are related via taking semiclassical limit.
Twisted wild character varieties
It is well-known that the meromorphic connections
on a compact Riemann surface with prescribed irregular type
at each singularity are classified by the so-called monodromy/Stokes data.
In this talk I will construct the moduli spaces of monodromy/Stokes data
as algebraic Poisson varieties using the "twisted" quasi-Hamiltonian geometry.
This talk is based on joint work with Philip Boalch.