Schematic homotopy types and non-abelian Hodge theory

Higgs Bundles and Non-abelian Hodge Theory

One - Form Abelian Gauge Theory As Hodge Theory

We demonstrate that the two (1 + 1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical ingredients that are required in the definition of the de Rham cohomological operators of the differential geometry. The conserved charges, corresponding to the a...

One-form Abelian Gauge Theory as the Hodge Theory

We demonstrate that the two (1 + 1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical ingredients that are required in the definition of the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above...

Homotopy Theory of Simplicial Abelian Hopf Algebras

We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Qu...

The Non-abelian Hodge Correspondence for Non-Compact Curves

In this talk I will describe the non-abelian Hodge theory of a non-compact curve. This was worked out by Simpson in the paper Harmonic Bundles on Non-Compact Curves, and as such almost everything I say here can be found in that paper in more detail. Let X be a smooth non-compact curve, with smooth completion j : X ↪→ X, and S = X \X a finite set of punctures. For simplicity, I’ll assume we have...