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In this paper we consider an $SU(2)$ Yang-Mills field propagating in the $4+1$ dimensional wormhole spacetime. Assuming the spherically symmetric magnetic ansatz the problem reduces to a one dimensional non linear wave equation. This…

We discuss extension of soliton theory and integrable systems to noncommutative spaces, focusing on integrable aspects of noncommutative anti-self-dual Yang-Mills equations. We give wide class of exact solutions by solving a Riemann-Hilbert…

We extend the relation between instanton and monopole solutions of the selfduality equations in SU(2) gauge theory to noncommutative space-times. Using this approach and starting from a noncommutative multi-instanton solution we construct a…

Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mr U(1)$, a $\mr U(1)$-Kepler problem in dimension $(2n-1)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to…

We construct numerically new axially symmetric solutions of SU(2) Yang-Mills-Higgs theory in $(3+1)$ anti-de Sitter spacetime. Two types of finite energy, regular configurations are considered: multimonopole solutions with magnetic charge…

Bertotti-Robinson spacetimes are topologically $AdS_2 \times S^2$ and described by a conformally flat metric. Together with the Coulomb electric potential, they provide a class of static, geodetically complete Einstein-Maxwell solutions. We…

This is a survey of the work of Seiberg and Witten on 4-dimensional N=2 supersymmetric Yang-Mills theory and of some of its recent extensions, written for mathematicians. The point of view is that of algebraic geometry and integrable…

We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a…

We summarize recent results on the resolution of two intimately related problems, one physical, the other mathematical. The first deals with the resolution of the non-perturbative low energy dynamics of certain N=2 supersymmetric Yang-Mills…

We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras,…

In Yang-Mills theory on a Euclidean Cauchy surface, the physical gauge group is often taken to be $\mathcal{G}^I/\mathcal{G}^\infty_0$, where $\mathcal{G}^I$ consists of boundary-preserving gauge transformations asymptoting to a constant,…

We consider a U(2) Yang-Mills theory on M x S_F^2 where M is an arbitrary noncommutative manifold and S_F^2 is a fuzzy sphere spontaneously generated from a noncommutative U(N) Yang-Mills theory on M, coupled to a triplet of scalars in the…

We consider quantum aspects of a class of generalized Gross-Neveu models, which in special cases reduce to sigma models. We show that, in the case of gauged models, an admissible gauge is $A_\mu=0$, which is a direct analogue of the…

In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the…

We propose a Lorentz-covariant Yang-Mills ``spin-gauge'' theory, where the function valued Pauli matrices play the role of a non-scalar Higgs-field. As symmetry group we choose $SU(2) \times U(1)$ of the 2-spinors describing…

We propose a modified version of the Ginzburg-Landau energy functional admitting static solitons and determine all the Painlev\'e-integrable cases of its Bogomolny equations of a given class of models. Explicit solutions are determined in…

Applying Cho-Faddeev-Niemi decomposition for SU(2) Yang-Mills field, we obtain the Abelian-Higgs Lagrangian by some approximation. Abelian-Higgs Lagrangian with a spontaneous symmetry breaking potential has vortex solutions known as…

We consider the partition function for Euclidean $SU(N)$ super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six "fixed" three-spheres on…

We examine integrability of self-dual Yang-Mills system in the Higgs phase, with taking simpler cases of vortices and domain walls. We show that the vortex equations and the domain-wall equations do not have Painleve property. This fact…

We construct a four-parameter class of self-dual instanton solutions of the classical SU(2)-Yang-Mills equations in a closed Euclidean Robertson-Walker space-time.