Geometric Langlands and non-Abelian Hodge theory

Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve

This is a survey of results and conjectures on mirror symmetry phenomena in the nonAbelian Hodge theory of a curve. We start with the conjecture of Hausel–Thaddeus which claims that certain Hodge numbers of moduli spaces of flat SL(n, C) and PGL(n, C)connections on a smooth projective algebraic curve agree. We then change our point of view in the non-Abelian Hodge theory of the curve, and conce...

Higgs Bundles and Non-abelian Hodge Theory

Gauge Theory and the Geometric Langlands Program

The Langlands program of number theory, or what we might call Langlands duality, was proposed in more or less its present form by Robert Langlands, in the late 1960s. It is a kind of unified scheme for many results in number theory ranging from quadratic reciprocity, which is hundreds of years old, to modern results such as Andrew Wiles’ proof of Fermat’s last theorem, which involved a sort of ...

Representation theory, geometric Langlands duality and categorification

The representation theory of reductive groups, such as the group GLn of invertible complex matrices, is an important topic, with applications to number theory, algebraic geometry, mathematical physics, and quantum topology. One way to study this representation theory is through the geometric Satake correspondence (also known as geometric Langlands duality). This correspondence relates the geome...

Non-Abelian Geometric Phase, Floquet Theory, and Periodic Dynamical Invariants

For a periodic Hamiltonian, periodic dynamical invariants may be used to obtain non-degenerate cyclic states. This observation is generalized to the degenerate cyclic states, and the relation between the periodic dynamical invariants and the Floquet decompositions of the time-evolution operator is elucidated. In particular, a necessary condition for the occurrence of cyclic non-adiabatic non-Ab...