Home » Lectures
Category Archives: Lectures
January 13, 2021
TITLE: Line defects, UV-IR map and exact WKB
In this talk I’ll give an overview of the relations between class S theories and Hitchin systems, focusing on roles played by line defects in class S theories. Deforming onto the Coulomb branch triggers a UV-IR map for line defects, corresponding to a trace map for certain flat nonabelian connections over a Riemann surface. The UV-IR map admits a q-deformation, which corresponds to a quantum trace map embedding certain skein algebra into a quantum torus algebra. I’ll also briefly describe connections to exact WKB and a potential q-deformation thereof.
January 11, 2021
TITLE: Quadratic differentials as stability conditions
Consider a quasi-projective Calabi-Yau 3-fold which is an affine conic fibration over a two-dimensional surface. I will explain why the space of stability conditions on (a subcategory of) its Fukaya category can be understood in terms of meromorphic quadratic differentials on the surface. This talk reports on old joint work with Tom Bridgeland.
January 11, 2021
TITLE: Introduction to Stokes phenomena and resurgence
January 12, 2021
TITLE: From resurgence to topological strings
The theory of resurgence suggests that the perturbative series that we often calculate
in physics and mathematics are the tip of the iceberg in an extended structure, involving generalized formal power series (also called trans-series), and relations between them, encoded in Stokes constants. In topological field and strings theories, these additional sectors potentially provide new topological invariants for geometric objects. In this talk, after introducing some basic tools of the theory of resurgence, I will discuss the example of complex Chern-Simons theory, where Stokes constants provide an infinite number of integer invariants of hyperbolic knots.
I will also discuss what is known in the case of topological strings and enumerative invariants of Calabi-Yau threefolds, and present some open problems.
January 12, 2021
TITLE: Analyticity and resurgence
I will talk on my recent work with Yan Soibelman on analytic wall-crossing structures, and a hypothetical relation to theory of resurgent series by Jean Ecalle. In particular, our considerations imply the resurgence property of WKB series.
January 15, 2021
TITLE: Geometric description of topological string partition functions from quantum curves and integrability
I will give a progress update on work relating topological string partition functions Ztop for a class of supersymmetric gauge theories to quantum Seiberg-Witten curves through integrability. In particular, I will discuss a geometric characterisation of the Ztop functions in terms of a line bundle over the moduli space of quantum curves, providing evidence for this picture through examples. Part of this discussion will review earlier results which show how the Ztop functions enter certain series expansions of isomonodromic tau functions associated to quantised SW curves. New insight then concerns the existence of certain preferred coordinates on the moduli space of quantum curves, which are defined from the curves via exact WKB analysis and which enter theta-series expansions of appropriately normalised tau functions, in a way that allows to extract the functions Ztop. Understanding these coordinates, how they are related on different patches as a consequence of Stokes phenomena, leads to the proposed geometric characterisation of the tau functions and Ztop.
01/11/2021, 01/13/2021, and 01/14/2021: From Donaldson-Thomas invariants to complex hyperkahler structures (3 lecture series)
January 11, 2021, January 13, 2021, and January 14, 2021
TITLE: From Donaldson-Thomas invariants to complex hyperkahler structures
ABSTRACT: I will report on an ongoing project which aims to use the DT invariants of a CY3 triangulated category to encode a geometric structure on its stability space. The basic idea is to interpret DT invariants as defining non-linear Stokes factors, as in the work of Gaiotto, Moore and Neitzke. Lecture 1 will be mostly background material: I will discuss stability conditions, the wall-crossing formula for DT invariants, and Stokes data. Lecture 2 will be about the particular type of complex hyperkahler structure we expect to find on stability space: I will give a local description involving Plebanski’s second heavenly equation and discuss a (partly conjectural) class of examples relating to moduli spaces of holomorphic connections on rank 2 vector bundles over Riemann surfaces. Lecture 3 will be about attempting to construct the complex hyperkahler structure on stability space from the DT invariants: this involves a class of Riemann-Hilbert problems for maps from the complex plane into a group of symplectic automorphisms; I will discuss their solutions in some simple examples.
Recorded lecture associated with the January 2021 meeting
TITLE: Introduction to Bridgeland stability
The lecture provides a first introduction to triangulated categories and stability conditions on them. The motivation for such notions is discussed from the perspective of (homological) mirror symmetry.
The collaboration will hold a VIRTUAL meeting on 11-15 January, 2021, entitled “Donaldson-Thomas invariants and Resurgence”.
3-Calabi-Yau triangulated categories T are a major area of research in Geometry and String Theory. Examples include derived categories Dbcoh X of coherent sheaves and the derived Fukaya category DbF(X) on a Calabi-Yau 3-fold X, where Kontsevich’s Homological Mirror Symmetry Conjecture explains mirror symmetry between Calabi-Yau 3-folds X, X* as equivalences of categories Dbcoh X = DbF(X*), DbF(X) = Dbcoh X*.
For such a 3-Calabi-Yau category T one can consider the moduli space Stab(T) of Bridgeland stability conditions on T, and for each σ in Stab(T) one can (under good conditions) define Donaldson-Thomas (D-T) invariants DTα(σ), which are rational numbers ‘counting’ σ-semistable objects in class α in Knum(T).
A nice example, which will be covered at the conference, is a class of categories T of derived representations of a quiver with superpotential (Q,W), in which Stab(T) and DTα(σ) can be described by work of Bridgeland and Smith in terms of quadratic differentials on a Riemann surface.
Recent work of Tom Bridgeland and coauthors explains how to encode D-T invariants into interesting geometric structures on Stab(T), involving Stokes phenomena and Riemann-Hilbert problems for singular flat connections, and connected (via Tom’s paper with Ian Strachan) to complex hyperkahler manifolds and twistor theory. All this is related to work of Kontsevich and Soibelman on analytic stability data and resurgence, and to a circle of ideas in String Theory, including work of Gaiotto, Moore and Neitzke, and topics such as resurgence, WKB analysis, and line operators.
The conference will explain these ideas with introductory talks, and aim to promote communication between Geometers and String Theorists, to better understand this new and fast-moving area.
The virtual meeting comes in two parts:
- Recorded lectures which can be viewed by participants at any time, and;
- “Live” lectures streamed via Zoom and YouTube, which can also be watched later. Our live sessions will also include some discussions, which may include discussions of both the live lectures and the recorded lectures. YouTube access for the lectures is available at
Our recorded lectures are as follows. Those who are unfamiliar with Bridgeland stability should endeavor to watch these recorded lectures before the live lectures by Bridgeland.
- Fabian Haiden (Oxford), Introduction to Bridgeland stability
- Dominic Joyce (Oxford), Donaldson-Thomas theory of Calabi-Yau 3-folds
Our live schedule is as follows. – All times are given in British time, in honor of the originally planned venue for the meeting, as well as in Eastern Daylight Time. (Please refer to a time zone converter if you aren’t sure what time it will be in your time zone). There will be a brief conference introduction from Robert Bryant prior to the first talk on Monday.
You can download the programme of this meeting, or consult the version below.
MON 11 JANUARY
TUE 12 JANUARY
WED 13 JANUARY
THU 14 JANUARY
FRI 15 JANUARY
|B. Pym||M. Kontsevich||G. Moore||T. Bridgeland, III||I.-A. Coman|
|T. Bridgeland, I||M. Mariño||F. Yan||S. Donaldson I||S. Donaldson II|
|Meal break||Meal break||Meal break||Meal break||Meal break|
|I. Smith||T. Bridgeland, II||Discussion, led by Maxim Kontsevich and Richard Thomas||Discussion on “complex hyperkähler manifolds”, led by Roger Bielawski||Discussion on “DT invariants and resurgence: Good questions for the future?”, led by Joerg Teschner|
|Questions/Discussion||Questions/Discussion||Discussion, con.||Discussion, con.||Discussion, con.|
Documents associated with Thursday’s discussion:
Documents associated with Friday’s discussion:
The links will take you to abstracts, slides of lectures, and/or video recordings of the lectures (when available).
- Tom Bridgeland (Sheffield), From Donaldson-Thomas invariants to complex hyperkahler structures (3 lectures)
- Ioana-Alexandra Coman (Amsterdam), Geometric description of topological string partition functions from quantum curves and integrability
- Simon Donaldson (SCGP and Imperial College London), Deformations of singular sets and Nash-Moser theory I, II
- Maxim Kontsevich (IHES, Paris), Analyticity and resurgence
- Marcos Mariño (Geneva), From resurgence to topological strings
- Greg Moore (Rutgers), Informal Remarks Complementary To, And Preparatory For,
Fei Yan’s Talk
Andy Neitzke (Yale), Riemann-Hilbert problems, Hitchin systems and the conformal limit
- Brent Pym (McGill), Introduction to Stokes phenomena and resurgence
- Ivan Smith (Cambridge), Quadratic differentials as stability conditions
- Fei Yan (Rutgers), Line defects, UV-IR map and exact WKB