Higgs bundles in mathematics and physics summer school
1014 of September 2018
Hamburg
Higgs bundles were introduced by Nigel Hitchin in 1987 while
studying the dimensional reduction of the selfduality 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 5day 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 warmup 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
MazzeoSwobodaWeissWitt, DumasNeitzke, 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
GHiggs
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
GuichardWienhard.
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 padic 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 CalabiYau manifolds and the
semiflat metric
5. The integrable system for Higgs bundles
6. Special Kahler geometry and the semiflat 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 fourdimensional 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. Dpbranes And
YangMillsHiggs. Geometrization Of The
Higgs Mechanism. D4 branes In T*C And Hitchin Systems. Mtheory
And MBranes.
M5 In T*C And Class S. General Remarks: Coulomb Branch Of A d=4
N=2 Field Theory.
Recovering SeibergWitten 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  Clusterlike
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 colloquiumlevel 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:alggeom/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 DeligneHitchin twistor space"
arXiv:1802.06587
6. B. Collier and R. Wentworth, "Conformal limits and the
BialynickiBirula stratification of the space of
lambdaconnections", 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] unihamburg [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] unihamburg [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_unihamburg.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] unihamburg [dot] de.
Sponsored by:
Verein zur Ausrichtung von Tagungen am
Fachbereich Mathematik der Universität Hamburg (VATFBMUHH) e.V.
