2D Integrable systems, 4D Chern–Simons theory and affine Higgs bundles

Higgs Bundles and Non-abelian Hodge Theory

Matrix String Theory, 2D SYM Instantons and affine Toda systems

Extending a recent result of S.B. Giddings, F. Hacquebord and H. Verlinde, we show that in the U(N) SYM Matrix theory there exist classical BPS instantons which interpolate between different closed string configurations via joining/splitting interactions similar to those of string field theory. We construct them starting from branched coverings of Riemann surfaces. For the class of them which w...

A 2d Inspired 4d Theory of Gravity

Coadjoint orbits of the Virasoro and Kac-Moody algebras provide geometric actions for matter coupled to gravity and gauge fields in two dimensions. However, the Gauss' law constraints that arise from these actions are not necessarily endemic to two-dimensional topologies. Indeed the constraints associated with Yang-Mills naturally arise from the coadjoint orbit construction of the WZW model. On...

Integrable systems and number theory

where is the derivative of with respect to , was derived to describe water waves in shallow channels [KdV95]. It bears the names of Korteweg and de Vries and has become the prototype integrable nonlinear partial differential equation since the work in [Miu68, MGK68, SG69, Gar71, KMGZ70, GGKM74]. The KdV equation is for the field of integrability what the harmonic oscillator is for quantum mecha...

Morse Theory and Hyperkähler Kirwan Surjectivity for Higgs Bundles

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott’s original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperkähler Kirwan map is surjective for the non-fixed determinant case. CONTENTS