Heat Flow for the Yang–Mills–Higgs Field and the Hermitian Yang–Mills–Higgs Metric

For a parameter > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the -stability of (E, ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.     

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...     

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.     

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.     

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.     

Stable self-similar blowup in energy supercritical Yang–Mills theory

Mathematische Zeitschrift - We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in $$(1+5)$$ -dimensional Yang–Mills theory. A certain self-similar...     

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.     

Large dynamics of Yang–Mills theory: mean dimension formula

We study the Yang–Mills anti-self-dual (ASD) equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover becomes an infinite dimensional and infinite entropy system. We study the mean dimension of this huge dynamical system. Mean dimension is a topological invariant of dynamical systems introduced by Gromov. We prove the exact formula of the mean dimension by developing a new technique based on the metric mean dimension theory of Lindenstrauss–Weiss.     

Perturbed geodesics on the moduli space of flat connections and Yang–Mills theory

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang–Mills connections depending on a parameter ${\varepsilon}$ . In this paper we show that this map is a bijection and maps perturbed geodesics into perturbed Yang–Mills connections with the same Morse index.     

Hermitian Yang–Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kahler Manifolds

Let V be an asymptotically cylindrical Kahler manifold with asymptotic cross-section ■. Let (E■,Φ■)be a st able Higgs bundle over ■, and (E,Φ) a Higgs bundle over V which is asymptotic to (E■,Φ■). In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant projectively Hermitian Yang-Mills metric on (E,Φ).     

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.     

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.     

Renormalization scenario for the quantum Yang–Mills theory in four-dimensional space–time

We consider the renormalization of the Yang–Mills theory in four-dimensional space–time using the background-field formalism.     

Fuzzy knot theory interpretation of Yang–Mills instantons and Witten’s 5-Brane model

A knot theory interpretation of ‘tHooft’s instanton based on hyperbolic volume, crossing numbers and exceptional Lie symmetry groups is given. Subsequently it is shown that although instantons and particle-like states of Heterotic super strings may appear to be different concepts, on a very deep fuzzy level they are not.     

Eliminating gauge anomalies via a “point-less” fractal Yang–Mills theory

A Cantorian fractal extension of Yang–Mills theory can eliminate gauge anomalies caused by the frequent conflict between the classical and the internal symmetries.