Nahm transform for Higgs bundles

We construct the Nahm transform for Higgs bundles over a Riemann surface of genus at least 2 as hyperholomorphic connections on the total space of the tangent bundle of its dual Jacobian.     

Determinant bundles,manifolds with boundary and surgery

We define determinant bundles associated to the following data: (i) a family of generalized Dirac operators on even dimensional manifolds with boundary, (ii) the choice of a spectral section for the family of Dirac operators induced on the boundary. Under the assumption that the operators of the boundary family have null spaces of constant dimension we define, through the notion ofb-zeta function, a Quillen metric. We also introduce the analogue of the Bismut-Freed connection. We prove that the curvature of a natural perturbation of the Bismut-Freed connection equals the 2-form piece in the right-hand side of the family index formula, thus extending to manifolds with boundary results of Quillen, Bismut and Freed. Given a closed fibration, we investigate the behaviour of the Quillen metric and of the Bismut-Freed connection under the operation of surgery along a fibering hypersurface. We prove, in particular, additivity formulae for the curvature and for the logarithm of the holonomy.     

Twisted quiver bundles over almost complex manifolds

In this paper, we study twisted quiver bundle over general almost complex manifolds. A twisted quiver bundle is a set of J-holomorphic vector bundles over an almost complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of J-holomorphic vector bundles, labelled by the arrows. We prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact almost Hermitian regularized manifold, relating the existence of solutions to certain gauge equations to an appropriate notion of stability for the corresponding quivers. This result can be seen as a generalization of that in [2], [9].     

Second order tangent bundles of infinite dimensional manifolds

The second order tangent bundle T²M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration. It is known [Analele Stiintifice ale Universitatii Al. I. Cuza 28 (1982) 63] that in the case of a finite n-dimensional manifold M, T²M becomes a vector bundle over M if and only if M is endowed with a linear connection. Here we extend this result to M modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite dimensional dynamical systems.     

On algebras,manifolds, and fibre bundles in physics

This article surveys the mathematical or physical aspects of algebraic realizations, fibre bundles, supermanifolds, super Lie groups, and super Lie algebras.An expanded and modified version of an invited talk given at the Special Session on Nonassociative Algebras and Their Connections with Physics, at the 749th meeting of the American Mathematical Society, Purdue University, October 29, 1977.     

Hermitian–Einstein metrics on holomorphic vector bundles over Hermitian manifolds

In this paper, we prove the long-time existence of the Hermitian–Einstein flow on a holomorphic vector bundle over a compact Hermitian (non-Kähler) manifold, and solve the Dirichlet problem for the Hermitian–Einstein equations. We also prove the existence of Hermitian–Einstein metrics for holomorphic vector bundles on a class of complete non-compact Hermitian manifolds.     

Gauge theories on four dimensional Riemannian manifolds

This paper develops the Riemannian geometry of classical gauge theories — Yang-Mills fields coupled with scalar and spinor fields — on compact four-dimensional manifolds. Some important properties of these fields are derived from elliptic theory: regularity, an energy gap theorem, the manifold structure of the configuration space, and a bound for the supremum of the field in terms of the energy. It is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities (this extends a theorem of K. Uhlenbeck).The author holds an A.M.S. Postdoctoral Fellowship     

Graded manifolds of type $$\Delta $$ Δ and n-fold vector bundles

Letters in Mathematical Physics - Vector bundles and double vector bundles, or twofold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids,...     

Construction of twisted products for cotangent bundles of classical groups and Stiefel manifolds

The existence of invariant twisted products (deformations of the associative algebra of C -functions) on the cotangent bundles of classical groups and Stiefel manifolds is proved by explicit constructions. All these products are positive.     

Visible cascade Higgs decays to four photons at hadron colliders

The presence of a new singlet scalar particle a can open up new decay channels for the Higgs boson, through cascades of the form h --> 2a --> X, possibly making discovery through standard model channels impossible. If a is CP odd, its decays are particularly sensitive to new physics. Quantum effects from heavy fields can naturally make h --> 4 g the dominant decay which is difficult to observe at hadron colliders, and is allowed by CERN LEP for m(h) > 82 GeV. However, there are usually associated decays, either h --> 2g2gamma or h --> 4gamma, which are more promising. The decay h-->4gamma is a clean channel that can discover both a and h. At the CERN LHC with 300 fb(-1) of luminosity, a branching ratio of order 10(-4) is sufficient for discovery for a large range of Higgs boson masses. With total luminosity of approximately 8 fb(-1), discovery at the Fermilab Tevatron requires more than 5 x 10(-3) in branching ratio.     

Searching for Higgs decays to four bottom quarks at LHCb

We discuss the feasibility of seeing a Higgs boson which decays to four bottom quarks through a pair of (pseudo-)scalars at the LHCb experiment to argue that the use of b-physics triggers and off-line vertex reconstruction, as opposed to jet triggers with b tagging, may be more effective for this signal. Focusing on inclusive production for the Higgs, we find that for light scalar masses below 20 GeV, signal reconstruction efficiencies of order a few percent may be enough for LHCb to find evidence for a Higgs with a dominant 4b decay channel.