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...     

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.     

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.     

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.     

Regularization-Independent Gauge-Invariant Renormalization of the Yang–Mills Theory

A recently proposed renormalization scheme is applied to non-Abelian gauge fields. Explicitly obtained gauge-invariant expressions for the renormalized vertex functions are independent of the choice of the intermediate regularization scheme.     

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.     

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.     

Random Holonomy for Yang–Mills Fields: Long-Time Asymptotics

We study weak and strong convergence of the stochastic parallel transport for time t on Euclidean space. We show that the asymptotic behavior can be controlled by the Yang–Mills action and the Yang–Mills equations. For open paths we show that under appropriate curvature conditions there exits a gauge in which the stochastic parallel transport converges almost surely. For closed paths we show that there exists a gauge invariant notion of a weak limit of the random holonomy and we give conditions that insure the existence of such a limit. Finally, we study the asymptotic behavior of the average of the random holonomy in the case of t'Hooft's 1-instanton.     

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.     

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.     

Chaotic Dynamics of Homogeneous Yang–Mills Fields with Three Degrees of Freedom

Differential Equations - We consider a scenario of transition to chaotic dynamics in a Hamiltonian system of homogeneous Yang–Mills fields with three degrees of freedom in the presence of the...     

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.     

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.     

Lyapunov dimension formula for the attractor of the Glukhovsky–Dolzhansky system

Exact formulas for the Lyapunov dimension of attractors of the generalized Lorenz system and the Glukhovsky–Dolzhansky system are obtained.