Geometry seminar: Rice -- The Hitchin Fibration - an entrée to Langlands duality

Title:The Hitchin Fibration - an entrée to Langlands duality

Speaker: Prof John Rice

The Hitchin fibration exposes the moduli space of ‘Higgs’ fields on a compact Riemann
surface as a completely integrable dynamical system.  Beilinson and Drinfeld established
a strong relationship between quantisation of this system and Langlands duality.  In the
other direction, Donagi and Pantev prove that the fibres of Hitchin fibrations whose
gauge groups are Langlands dual, are dual abelian varieties (with some caveats), and
that this is a classical limit of the conjectured geometric form of Langlands duality.
A larger picture painted by Kapustin and Witten identifies the duality of Hitchin
fibrations as mirror symmetry or T-duality, and argues that geometric Langlands duality
is S-duality for a particular supersymmetric Yang-Mills system, of which the Hitchin
system is a reduction.  

This is huge territory, but the 2007 paper of Hitchin, Langlands
Duality and G2 Spectral Curves, gives a concrete introduction to the phenomena of
Langlands dual groups and the duality of the abelian varieties in his fibrations.  It
also provides the opportunity to consider the basics of lie theory (root systems etc) in
this context and to get to know an exceptional group.  It is a starting point for
insight into the bigger picture of Langlands duality.  

I propose to give a series of
lectures expounding Hitchin’s paper, filling in the background and expanding on the
themes that it touches on.  I would be happy to respond the interests and needs of the
audience in this regard.