We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N = 2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactied on a circle, we get a 3d theory with a su- persymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of classS, that is, for those theories ob- tained by compactifying the six-dimensional (0; 2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A1 theories of classS. Finally, we indi- cate how our results can be used to produce solutions to the A1 Hitchin equations on the Riemann surface C.

Wall-Crossing in Coupled 2d-4d Systems

Suppose $X$ is a smooth quasiprojective variety over $\cc$ and $\rho : \pi _1(X,x) \rightarrow SL(2,\cc )$ is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then $\rho$ factors through a map $X\rightarrow Y$ with $Y$ either a DM-curve or a Shimura modular stack.

/pdf/on-the-classification-of-rank-two-representations-of-3n2zj5xrnj.pdf

On the classification of rank-two representations of quasiprojective fundamental groups

Principal $G$-bundles with parabolic structure over a normal crossing divisor are defined along the line of the interpretation of the usual principal $G$-bundles as functors from the category of representations, of the structure group $G$, into the category of vector bundles, satisfying certain axioms. Various results on principal bundles are extended to the more general context of principal bundles with parabolic structures, and also to parabolic $G$-bundles with Higgs structure. A simple construction of the moduli space of parabolic semistable $G$-bundles over a curve is given, where $G$ is a semisimple linear algebraic group over $C$.

/pdf/principal-bundles-over-projective-manifolds-with-parabolic-2pabpdh5mb.pdf

Principal bundles over projective manifolds with parabolic structure over a divisor

Following ideas of Nori, Biswas, ..., we show that given an integer r>0, a noetherian scheme X, and an effective Cartier divisor D on it, the parabolic vector bundles on (X,D) with weights multiples of 1/r (in the sense of Maruyama-Yokogawa) are equivalent to ordinary vector bundles on an orbifold, the stack of r-th roots associated to (X,D) (a twisted scheme in the sense of Abramovich-Vistoli). We use this fact to get some information on the finite parabolic bundles on the (pointed) projective line.

Fibrés Paraboliques et Champ des Racines

Let χ be an irreducible smooth projective algebraic curve of genus g ≥ 2 over the ground field C and let G be a semisimple simply connected algebraic group. The aim of this paper is to introduce the notion of a semistable and stable parahoric torsor under a certain Bruhat-Tits group scheme G, construct the moduli space of semistable parahoric G-torsors and identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of G. The results give a complete generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles.

/pdf/moduli-of-parahoric-torsors-on-a-compact-riemann-surface-467xz8647m.pdf

Moduli of parahoric -torsors on a compact Riemann surface

https://www.cmi.ac.in/~balaji/Publications/BrauerObstructionForUniversalBundles.pdf

Brauer obstruction for a universal vector bundle

Let k be a field and X a k-form of the projective line. We classify all the isomorphism classes of vector bundles over X.

/pdf/vector-bundles-over-a-nondegenerate-conic-54dxtqp9cn.pdf

Vector bundles over a nondegenerate conic

We give a complete classification of isomorphism classes of real algebraic vector bundles over the scheme defined by a nondegenerate anisotropic conic defined over the field of real numbers.

D. S. Nagaraj

Papers

Principal bundles over projective manifolds with parabolic structure over a divisor

Brauer obstruction for a universal vector bundle

Vector bundles over a nondegenerate conic

Classification of real algebraic vector bundles over the real anisotropic conic

Parabolic ample bundles, II: Connectivity of zero locus of a class of sections