Branes are special submanifolds of hyperkähler manifolds that play an important role in string theory, particularly in the Kapustin–Witten approach to the geometric Langlands program, but which also are of intrinsic geometric interest. More precisely, a brane is a submanifold of a hyperkähler manifold which is either complex or Lagrangian with respect to each of the three complex structures or Kähler forms composing the hyperkähler structure.
Given a totally nonholonomic distribution of rank two ∆ on a three-dimensional manifold M, it is natural to investigate the size of the set of points Xx that can be reached by singular horizontal paths starting from a same point x ∈ M. In this setting, the Sard conjecture states that Xx should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.
The moduli space M_C of Higgs bundles over a complex curve X admits a hyperkaehler metric: a Riemannian metric which is Kaehler with respect to three different complex structures I, J, K, satisfying the quaternionic relations. If X admits an anti-holomorphic involution, then there is an induced involution on M_C which is anti-holomorphic with respect to I and J, and holomorphic with respect to K. The fixed point set of this involution, M_R, is therefore a real Lagrangian submanifold with respect to I and J, and complex symplectic with respect to K, making it a so called AAB-brane.
Let X be a smooth projective curve of genus at least two. Let G be an almost simple affine algebraic group. The universal principal G-bundle is called the Poincaré bundle. It is a principal G-bundle on the product of the curve and the moduli stack of principal G-bundles. We prove that it is stable with respect to any polarization (joint work with I. Biswas and N. Hoffmann).
In this introduction for geometers and topologists, we explain the role that central extensions of L-infinity algebras, a concept we will define, plays in physics. This connection first appeared with the work of physicists D’Auria and Fré in 1982, but is beautifully captured by the “brane bouquet” of Fiorenza, Sati and Schreiber which shows how physical objects such as “strings”, “D-branes” and “M- branes” can be classified by taking successive central extensions of an especially simple L-infinity algebra called the “supertranslation algebra”.
Mumford introduced in the 1960ies an algebraic approach to the construction of (almost) canonical bases of sections of ample line bundles on abelian varieties that permitted him to construct quasi-projective moduli spaces. His construction was later re-interpreted by Welters as a flat projective connection before being generalized by Hitchin to the non-abelian setting.
Let S be a smooth projective surface over C and B a smooth projective curve. A fibration f : S → B is a surjective morphism such that the general fibre is a smooth connected curve.
This talk will focus on some properties of fibrations with general fibre of genus ≥ 2, discussing in particular the existence and number of singular fibres on a fibration.
Let Λ be a D-algebra in the sense of Bernstein and Beilinson. Higgs bundles, vector bundles with flat connections, co-Higgs bundles... are examples of Λ-modules for particular choices of Λ. Simpson studied the moduli problem for the classification of Λ-modules over Kähler varieties, proving the existence of a moduli space of Λ-modules. Using the Polishchuck-Rothstein transform for modules of D-algebras over abelian varieties, we obtain a description of the moduli spaces of Λ-modules of rank 1. We also proof that polystable Λ-modules decompose as a direct sum of rank 1 Λ-modules.
Non-abelian gerbes are a generalization of principal G-bundles, involving the replacement of the Lie group G by a Lie 2-group, or crossed module of groups, not necessarily Abelian. Apart from providing a nice example of categorification in geometry, they have found a number of applications in physics, e.g. in higher gauge theory and topological states of matter.
We generalize the classical MacDonald formula for smooth curves to reduced curves with planar singularities. More precisely, we show that the cohomologies of the Hilbert schemes of points on a such a curve are encoded in the cohomologies of the fine compactified Jacobians of its connected subcurves, via the perverse Leray filtration. A crucial step in the proof is the case of nodal curves, where the weight polynomials of the spaces involved can be computed in terms of the underlying dual graph. This is a joint work with Luca Migliorini and Vivek Schende.