Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact Kähler manifolds and establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we prove some properties of semistable Higgs sheaves.     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case

We review the notions of (weak) Hermitian–Yang–Mills structure and approximate Hermitian–Yang–Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K?hler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be written in terms of the mean curvature of the Hitchin–Simpson connection. We also study some properties of the solutions of the evolution equation associated with that functional. Next, we study the problem of the existence of approximate Hermitian–Yang–Mills structures and its relation with the algebro-geometric notion of semistability and we show that for a compact Riemann surface, the notion of approximate Hermitian–Yang–Mills structure is in fact the differential- geometric counterpart of the notion of semistability. Finally, we review the notion of admissible Hermitian structure on a torsion-free Higgs sheaf and define the Donaldson functional for such an object.     

Regularity and vanishing theorems for Yang–Mills–Higgs pairs

Archiv der Mathematik - We obtain a vanishing theorem for Yang–Mills–Higgs pairs on Euclidean and hyperbolic spaces in dimensions greater than 4, as well as a regularity theorem more...     

On the constraints problem for the Einstein–Yang–Mills–Higgs system

We provide proofs of some key propositions that were used in previous work by Dossa and Tadmon dealing with the characteristic initial value problem for the Einstein–Yang–Mills–Higgs (EYMH) system. The aforesaid proofs were missing, making the considered work difficult to understand. This work is presented with a view to have an almost self-contained paper. With this respect we completely recall the process of constructing initial data for the EYMH system on two intersecting smooth null hypersurfaces as done in the work of Dossa and Tadmon mentioned above. This is achieved by successfully adapting the hierarchical method set up by Rendall to solve the same problem for the Einstein equations in vacuum and with perfect fluid source. Many delicate calculations and expressions are given in details so as to address, in a forthcoming work, the issue of global resolution of the characteristic initial value problem for the EYMH system. The method obviously applies to the Einstein–Maxwell and the Einstein-scalar field models as well.     

Local well-posedness for the (n + 1)-dimensional Yang–Mills and Yang–Mills–Higgs system in temporal gauge

The Yang–Mills and Yang–Mills–Higgs equations in temporal gauge are locally well-posed for small and rough initial data, which can be shown using the null structure of the critical bilinear terms. This carries over a similar result by Tao for the Yang–Mills equations in the (3+1)-dimensional case to the more general Yang–Mills–Higgs system and to general dimensions.     

Morse theory, Higgs fields, and Yang–Mills–Higgs functionals

In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients. The first comes from the Yang?CMills?CHiggs energy, while the second is provided by the Hitchin function. After describing what Higgs bundles are, we explore these functions and how they may be used to extract topological information about the moduli spaces.     

Infrared Variables for the SU(3) Yang–Mills Field

We generalize the self-dual parameterization of the SU(2) Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group SU(3). We demonstrate that the duality property intrinsic to the SU(2) gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products SU(2)³.     

The Goursat problem for the Einstein–Yang–Mills–Higgs system in weighted Sobolev spaces

We establish original Moser estimates to clarify and complete previous works of Christodoulou and Müller zum Hagen concerning local existence and uniqueness results for the Goursat problem associated to second order quasilinear hyperbolic systems. As an application we locally solve, in some weighted Sobolev spaces, the Goursat problem for the Einstein–Yang–Mills–Higgs system using harmonic and Lorentz gauges.     

Morse homology for the Yang–Mills gradient flow

We use the Yang–Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse chain complex. The chain groups are generated by Yang–Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang–Mills gradient flow lines that converge asymptotically to Yang–Mills connections.     

Universal Invariant Renormalization for the Supersymmetric Yang–Mills Theory

We propose a manifestly invariant renormalization scheme for N=1 non-Abelian supersymmetric gauge theories.     

Langevin dynamic for the 2D Yang–Mills measure

Publications mathématiques de l'IHÉS - It is well known that a real analytic symplectic diffeomorphism of the $2d$ -dimensional disk ( $dgeq 1$ ) admitting the origin as a...     

Exact Solutions for Self-dual Yang–Mills Equations

The (constrained) canonical reduction of four dimensional self-dual Yang–Millstheory to 2, (2+1) dimensional sine-Gordon theory and 2 dimensional Liouvilles theory areconsidered. The Bäcklund transformations (BTs) areimplemented to obtain new classes of exact solutions for the reduced 2 dimensional sine-Gordonand Liouville models. Another transformation is developed and used to obtain exact solution forthe 2+1 and the original 3+1 sine-Gordon models.     

The Cauchy Problem for the Massive Yang－Mills－Higgs Equation in Minkowski Space

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Hamiltonian structure of the Yang–Mills functional

Hamilton equations based upon a general Lepagean equivalent of the Yang–Mills Lagrangian are investigated. A regularization of the Yang–Mills Lagrangian which is singular with respect to the standard regularity conditions is derived.