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Higgs bundles in mathematics and physics summer school

10-14 of September 2018
Hamburg

Higgs bundles were introduced by Nigel Hitchin in 1987 while studying the dimensional reduction of the self-duality equations from four to two dimensions. Higgs bundles on Riemann surfaces and their moduli spaces have a rich mathematical structure. They are important for a number of research areas such as gauge theory, Kähler and hyperkähler geometry, integrable systems as well as mirror symmetry and Langlands duality.

This 5-day summer school is addressed to students and researchers in differential geometry, algebraic geometry, theoretical physics and related areas. There will be a number of introductory mini courses covering different aspects of Higgs bundles.

The school poster is available here.

The school programme is available here.

List of speakers, titles and abstracts:

The asymptotic geometry of the Hitchin moduli space
Abstract:
Recently, there have been a number of results about the asymptotic geometry of the Hitchin moduli space.  In these lectures, we'll take a winding path through some classical results before surveying the recent progress.   I'll discuss the centrality of the Hitchin fibration (and resulting semiflat metric) in the asymptotic geometry.  After an introduction to hyperkahler geometry, we'll take a detour into the hyperkahler geometry of ALE and ALG gravitational instantons.  This serves as a warm-up to the conjectures about the asymptotic geometry of the Hitchin moduli space.  We'll survey some of the key ideas in the recent results of Mazzeo-Swoboda-Weiss-Witt, Dumas-Neitzke, Fredrickson about the ends of the moduli spaces.

Higgs bundles and higher Teichmueller spaces
Abstract:
In these series of lectures we consider the moduli space of G-Higgs bundles over a compact Riemann surface X, where G is a semisimple real Lie group, and its correspondence with the moduli space of representations of the fundamental group of X in G. Some connected components of the moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. We will focus on such special components, generally referred as higher Teichmueller components, including split groups (Hitchin components), hermitian groups (maximal Toledo components), and finishing with a general identification of these components for Higgs bundles, and its relation to the notion of positive Anosov representations recently developed by Guichard-Wienhard.

Description and references
Lectures

Topological mirror symmetry for Hitchin systems
1. Definitions: Higgs bundles and their moduli spaces for GL, SL and PGL
2. Hitchin system, and dual SYZ fibrations
3. Cohomology and (stringy) Hodge numbers, topological mirror symmetry
4. Toy example of topological mirror symmetry
5. (stringy) Cohomology of Higgs moduli spaces, computation for SL_2 PGL_2
6. Sketch of proof for SL_n vs PGL_n by p-adic integration
7. Mirror symmetry with branes by equivariant Verlinde formulae
Lectures

Higgs bundles and mirror symmetry Topics:
1. Hodge theory for line bundles on a curve — moduli spaces and geometric properties
2. Nonabelian Hodge theory — Higgs bundles for the general linear group
3. The hyperkaehler metric on the moduli space
4. SYZ mirror symmetry for Calabi-Yau manifolds and the semi-flat metric
5. The integrable system for Higgs bundles
6. Special Kahler geometry and the semi-flat metric
7. SYZ and Langlands duality — the example of odd orthogonal and symplectic groups
8. Mirror symmetry and duality of branes
9. Real forms and Langlands duality
10. Hyperholomorphic bundles
Lecture 1
Lecture 2
Lecture 3
Theories of class S and their associated Hitchin systems
Abstract:  
These lectures aim to give mathematicians some idea of the physical
background and intuition that lies behind the attempts of some physicists to understand
a certain class of four-dimensional quantum field theories with N=2 supersymmetry
using the mathematics of Higgs bundles and Hitchin systems on a punctured Riemann
surface. An extremely ambitious outline is:

Lecture 1: Basic Brane Background
Notes
Motivation From Spectral Networks. Dp-branes And Yang-Mills-Higgs. Geometrization Of The
Higgs Mechanism. D4 branes In T*C And Hitchin Systems. M-theory And M-Branes.
M5 In T*C And Class S. General Remarks: Coulomb Branch Of A d=4 N=2 Field Theory.
Recovering Seiberg-Witten Theory. Relation To Hitchin Moduli Space.

Lecture 2: Defects, Gluing, And BPS States
Notes
An Important Special Case - Linear Superconformal Quivers. General Remarks On Punctures.
Gaiotto Gluing. Other Defect "Operators". General Remarks On BPS States.
A Zoo Of Class S BPS States. Semiclassical Description Of 4d BPS States.

Lecture 3: Spectral Networks And Parallel Transport
Notes
Definition And Construction Of Spectral Networks. Interfaces And Formal Parallel Transport.
Abelianization And Nonabelianization Maps - Cluster-like Coordinates On Hitchin Moduli Space.
Morphisms Of Spectral Networks And BPS States.

Lecture notes:

Talk 84.

Reading Material:

1. talks 46 and 47. (71 is a colloquium-level introduction.)
2. Tachikawa's review

Conformal limits and Morse vs. oper stratifications
Topics:
1. Introduction, nonabelian Hodge correspondance, complex variations of Hodge structure, deformation theory, first variation of harmonic metrics.
2. Morse stratification, opers, lambda connections, the global slice theorem.
3. Conformal limits.

References:
1. C. Simpson, "The Hodge filtration on nonabelian cohomology" arXiv:alg-geom/9604005
2. C. Simpson, "Iterated destabilizing modifications for vector bundles with connection" arXiv:0812.3472
3. D. Gaiotto, "Opers and TBA" arXiv:1403.6137
4. O. Dumitrescu, et.al. "Opers vs. Nonabelian Hodge" arXiv:1607.02172
5. I. Biswas, S. Heller, and M. Roeser, "Real holomorphic sections of the Deligne-Hitchin twistor space" arXiv:1802.06587
6. B. Collier and R. Wentworth, "Conformal limits and the Bialynicki-Birula stratification of the space of lambda-connections", arXiv:1808.01622

Registration and financial support:

Please fill out this form to register. The registration fee is 30 euros, payable on arrival. There will be a social dinner on Wednesday evening. Please indicate on the registration form if you would like to attend the dinner. The dinner will cost an additional 20 euros.

For a limited number of participants we can offer a contribution up to 400 euros to their expenses. You can apply for financial support on the registration form. If you are a student applying for a financial contribution, please arrange for a short letter of support from your advisor to be sent to higgs2018.math [at] lists [dot] uni-hamburg [dot] de.

The deadline for registration is 1 August 2018.
The deadline for applying for financial support is 30 June 2018.

Poster Session:

A poster session will take place on Tuesday afternoon. Participants of the school are welcome to submit their applications for the poster session. Title and abstract of the poster should be sent to higgs2018.math [at] lists [dot] uni-hamburg [dot] de by July 8th, 2018. The applicants will then be notified of the acceptance of their posters.

Venue:

The summer school will take place in the lecture hall 2 (H2) of the Department of Mathematics, Bundesstr. 55, of the University of Hamburg. Here is a map. The closest metro station is "Schlump" on the red line U2 and the yellow line U3. The train station "Sternschanze" (train lines S11, S21 and S31) is also in walking distance. Here is the website for public transport in Hamburg.

Accommodations:

Participants of the summer school are kindly asked to arrange their accommodation on their own. Here is a list of some hotels.

Introductory course in Hamburg:

During the week preceding the summer school there will be an introductory block course of 14 hours on Higgs bundles. This course is aimed for graduate students and researchers in Hamburg as a preparation for the summer school but is open for other participants of the summer school as well. The homepage of the course is here. This course does not require registration, however please send an email to murad.alim_at_uni-hamburg.de if you plan to attend so we can have an estimate of the number of participants.


Organizers:

Murad Alim, Vicente Cortés, Sebastian Heller, Jörg Teschner

Administrative support:

Gerda Mierswa Silva, Heike Wessling

Helpers:

Eren Kovanlikaya, Vadym Kurylenko, Arpan Saha, Danu Thung, Martin Vogrin

If you have any questions or special needs, please feel free to contact the organizers at higgs2018.math [at] lists [dot] uni-hamburg [dot] de.

Sponsored by:
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Verein zur Ausrichtung von Tagungen am Fachbereich Mathematik der Universität Hamburg (VATFBMUHH) e.V.



 
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