Minimal Surfaces in $CH^2$ and their Higgs Bundles by John Loftin
Δημοσιεύτηκε στις 30 Μαΐ 2016
Higgs bundles
URL: http://www.icts.res.in/program/hb2016
DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016
VENUE : Madhava Lecture Hall, ICTS Bangalore
DESCRIPTION:
Higgs
bundles arise as solutions to noncompact analog of the Yang-Mills
equation. Hitchin showed that irreducible solutions of the GL(2,C)
Yang-Mills equation on a compact Riemann surface X are precisely the
polystable Higgs vector bundles on X of rank two and degree zero.
Subsequently Simpson proved that irreducible solutions of the GL(r,C)
Yang-Mills equation on a compact Kaehler manifold X are precisely the
polystable Higgs vector bundles on X of rank r and vanishing Chern
classes.
Hitchin showed that the moduli spaces of stable Higgs
bundles give examples of hyper-Kaehler manifolds and provide examples of
completely integrable systems. Simpson proved basic theorems on
fundamental group of Kaehler manifolds using the identification of Higgs
bundles with the solutions of the Yang-Mills equation.
The
moduli space of Higgs bundles is then full of rich geometric structures,
but the most interesting part of the story is not just this but the
different points of view that one can use for studying this moduli
space, which show this object as the precise mathematical context for
different physical theories. For instance, the moduli space of SL(2,
C)-Higgs bundles was proved by T. Hausel and M. Thaddeus, to be the
first non-trivial example for Strominger-Yau-Zaslow formulation of
Mirror symmetry (symplectic formulation concerning pairs of Calabi-Yau
manifolds) satisfying at the same time the Batyrev-Borisov mirror
symmetry definition (which is comprised of a condition on the Hodge
numbers of those manifolds).
Higgs bundles play a very central
role in the works of G. Laumon, L. Lafforgue and B.C. Ngô in their work
on Langlands program (the last two were awarded Field's medal in 2002
and 2010 respectively). The Geometric Langlands Program, in terms of the
Monotonen-Olive duality conjecture, states that maximally
supersymmetric gauge theory in four dimensions with gauge group G is
isomorphic to a similar gauge theory with gauge group being the
Langlands dual of G. From the physical viewpoint, the Monotonen-Olive
duality can be regarded as a nonabelian generalization of electric
magnetic duality. Gukov and Witten interpreted the ramified (or
punctured) version of the Geometric Langlands Program in physical terms.
There will be two short courses during the program:
1.
Richard Wentworth will lecture on the work of
Hitchin-Donaldson-Corlette-Simpson which prove that solutions of the
Yang-Mills-Higgs equation are precisely the polystable Higgs bundles
with vanishing Chern classes.
2. Laura Schaposnik will lecture on the
completely integrable systems given by the moduli spaces of Higgs
bundles on a Riemann surface.
Some participants will deliver one lecture each and there will be about 30 talks.
ORGANIZERS: V. Balaji, I. Biswas and A. Parameswaran
URL: http://www.icts.res.in/program/hb2016
DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016
VENUE : Madhava Lecture Hall, ICTS Bangalore
DESCRIPTION:
Higgs
bundles arise as solutions to noncompact analog of the Yang-Mills
equation. Hitchin showed that irreducible solutions of the GL(2,C)
Yang-Mills equation on a compact Riemann surface X are precisely the
polystable Higgs vector bundles on X of rank two and degree zero.
Subsequently Simpson proved that irreducible solutions of the GL(r,C)
Yang-Mills equation on a compact Kaehler manifold X are precisely the
polystable Higgs vector bundles on X of rank r and vanishing Chern
classes.
Hitchin showed that the moduli spaces of stable Higgs
bundles give examples of hyper-Kaehler manifolds and provide examples of
completely integrable systems. Simpson proved basic theorems on
fundamental group of Kaehler manifolds using the identification of Higgs
bundles with the solutions of the Yang-Mills equation.
The
moduli space of Higgs bundles is then full of rich geometric structures,
but the most interesting part of the story is not just this but the
different points of view that one can use for studying this moduli
space, which show this object as the precise mathematical context for
different physical theories. For instance, the moduli space of SL(2,
C)-Higgs bundles was proved by T. Hausel and M. Thaddeus, to be the
first non-trivial example for Strominger-Yau-Zaslow formulation of
Mirror symmetry (symplectic formulation concerning pairs of Calabi-Yau
manifolds) satisfying at the same time the Batyrev-Borisov mirror
symmetry definition (which is comprised of a condition on the Hodge
numbers of those manifolds).
Higgs bundles play a very central
role in the works of G. Laumon, L. Lafforgue and B.C. Ngô in their work
on Langlands program (the last two were awarded Field's medal in 2002
and 2010 respectively). The Geometric Langlands Program, in terms of the
Monotonen-Olive duality conjecture, states that maximally
supersymmetric gauge theory in four dimensions with gauge group G is
isomorphic to a similar gauge theory with gauge group being the
Langlands dual of G. From the physical viewpoint, the Monotonen-Olive
duality can be regarded as a nonabelian generalization of electric
magnetic duality. Gukov and Witten interpreted the ramified (or
punctured) version of the Geometric Langlands Program in physical terms.
There will be two short courses during the program:
1.
Richard Wentworth will lecture on the work of
Hitchin-Donaldson-Corlette-Simpson which prove that solutions of the
Yang-Mills-Higgs equation are precisely the polystable Higgs bundles
with vanishing Chern classes.
2. Laura Schaposnik will lecture on the
completely integrable systems given by the moduli spaces of Higgs
bundles on a Riemann surface.
Some participants will deliver one lecture each and there will be about 30 talks.
ORGANIZERS: V. Balaji, I. Biswas and A. Parameswaran
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