Non-Abelian Vortices, Super-Yang-Mills Theory and Spin(7)-Instantons

We consider a complex vector bundle E endowed with a connection A over the eight-dimensional manifold R^2 x G/H, where G/H = SU(3)/U(1)xU(1) is a homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A of the Spin(7)-instanton equations on R^2 x G/H and general solutions of non-Abelian coupled vortex equations on R^2. These vortices are BPS solitons in a d=4 gauge theory obtained from N=1 supersymmetric Yang-Mills theory in ten dimensions compactified on the coset space G/H with an SU(3)-structure. The novelty of the obtained vortex equations lies in the fact that Higgs fields, defining morphisms of vector bundles over R^2, are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in N=4 super Yang-Mills theory and show that they have the same feature.Comment: 14 pages; v2: typos fixed, published versio

Orbifold instantons, moment maps, and Yang-Mills theory with sources

We revisit the problem of constructing instantons on ADE orbifolds R4/Γ and point out some subtle relations with the complex structure on the orbifold. We consider generalized instanton equations on R4/Γ which are BPS equations for the Yang-Mills equations with an external current. The relation between level sets of the moment maps in the hyper-Kähler quotient construction of the instanton moduli space and sources in the Yang-Mills equations is discussed. We describe two types of spherically symmetric Γ-equivariant connections on complex V bundles over R4/Γ which are tailored to the way in which the orbifold group acts on the fibers. Some explicit Abelian and non-Abelian SU(2)-invariant solutions to the instanton equations on the orbifold are worked out. © 2013 American Physical Society

Instantons and Yang-Mills Flows on Coset Spaces

We consider the Yang-Mills flow equations on a reductive coset space G/H and the Yang-Mills equations on the manifold R x G/H. On nonsymmetric coset spaces G/H one can introduce geometric fluxes identified with the torsion of the spin connection. The condition of G-equivariance imposed on the gauge fields reduces the Yang-Mills equations to phi^4-kink equations on R. Depending on the boundary conditions and torsion, we obtain solutions to the Yang-Mills equations describing instantons, chains of instanton-anti-instanton pairs or modifications of gauge bundles. For Lorentzian signature on R x G/H, dyon-type configurations are constructed as well. We also present explicit solutions to the Yang-Mills flow equations and compare them with the Yang-Mills solutions on R x G/H.Comment: 1+12 page

Hidden Symmetries and Integrable Hierarchy of the N=4 Supersymmetric Yang-Mills Equations

We describe an infinite-dimensional algebra of hidden symmetries of N=4 supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a generalization of the supertwistor correspondence. Using the latter, we construct an infinite sequence of flows on the solution space of the N=4 SYM equations. The dependence of the SYM fields on the parameters along the flows can be recovered by solving the equations of the hierarchy. We embed the N=4 SYM equations in the infinite system of the hierarchy equations and show that this SYM hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations. Presumably, the existence of such nonlocal symmetries underlies the observed integrable structures in quantum N=4 SYM theory.Comment: 24 page

Rigorous analysis of extremely asymmetrical scattering of electromagnetic waves in slanted periodic gratings

Extremely asymmetrical scattering (EAS) is a new type of Bragg scattering in thick, slanted, periodic gratings. It is realised when the scattered wave propagates parallel to the front boundary of the grating. Its most important feature is the strong resonant increase in the scattered wave amplitude compared to the amplitude of the incident wave: the smaller the grating amplitude, the larger the amplitude of the scattered wave. In this paper, rigorous numerical analysis of EAS is carried out by means of the enhanced T-matrix algorithm. This includes investigation of harmonic generation inside and outside the grating, unusually strong edge effects, fast oscillations of the incident wave amplitude in the grating, etc. Comparison with the previously developed approximate theory is carried out. In particular, it is demonstrated that the applicability conditions for the two-wave approximation in the case of EAS are noticeably more restrictive than those for the conventional Bragg scattering. At the same time, it is shown that the approximate theory is usually highly accurate in terms of description of EAS in the most interesting cases of scattering with strong resonant increase of the scattered wave amplitude. Physical explanation of the predicted effects is presented.Comment: 14 pages, 7 figures; v2: corrections to metadata and bibliographical info in preprin

The Topological B-model on a Mini-Supertwistor Space and Supersymmetric Bogomolny Monopole Equations

In the recent paper hep-th/0502076, it was argued that the open topological B-model whose target space is a complex (2|4)-dimensional mini-supertwistor space with D3- and D1-branes added corresponds to a super Yang-Mills theory in three dimensions. Without the D1-branes, this topological B-model is equivalent to a dimensionally reduced holomorphic Chern-Simons theory. Identifying the latter with a holomorphic BF-type theory, we describe a twistor correspondence between this theory and a supersymmetric Bogomolny model on R^3. The connecting link in this correspondence is a partially holomorphic Chern-Simons theory on a Cauchy-Riemann supermanifold which is a real one-dimensional fibration over the mini-supertwistor space. Along the way of proving this twistor correspondence, we review the necessary basic geometric notions and construct action functionals for the involved theories. Furthermore, we discuss the geometric aspect of a recently proposed deformation of the mini-supertwistor space, which gives rise to mass terms in the supersymmetric Bogomolny equations. Eventually, we present solution generating techniques based on the developed twistorial description together with some examples and comment briefly on a twistor correspondence for super Yang-Mills theory in three dimensions.Comment: 55 pages; v2: typos fixed, published versio

How to find discrete contact symmetries

This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism of the Lie algebra of generators of Lie contact symmetries. Consequently, all contact symmetries satisfy various compatibility conditions. These conditions enable the discrete symmetries to be found systematically, with little effort

One-dimensional structures behind twisted and untwisted superYang-Mills theory

We give a one-dimensional interpretation of the four-dimensional twisted N=1 superYang-Mills theory on a Kaehler manifold by performing an appropriate dimensional reduction. We prove the existence of a 6-generator superalgebra, which does not possess any invariant Lagrangian but contains two different subalgebras that determine the twisted and untwisted formulations of the N=1 superYang-Mills theory.Comment: 12 pages. Final version to appear in Lett. Math. Phys. with improved notation and misprints correcte

Scattering of Noncommutative Waves and Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions

Interactions of noncommutative waves and solitons in 2+1 dimensions can be analyzed exactly for a supersymmetric and integrable U(n) chiral model extending the Ward model. Using the Moyal-deformed dressing method in an antichiral superspace, we construct explicit time-dependent solutions of its noncommutative field equations by iteratively solving linear equations. The approach is illustrated by presenting scattering configurations for two noncommutative U(2) plane waves and for two noncommutative U(2) solitons as well as by producing a noncommutative U(1) two-soliton bound state.Comment: 1+13 pages; v2: reference added, version published in JHE

Reducible connections and non-local symmetries of the self-dual Yang-Mills equations

We construct the most general reducible connection that satisfies the self-dual Yang-Mills equations on a simply connected, open subset of flat $\mathbb{R}^4$. We show how all such connections lie in the orbit of the flat connection on $\mathbb{R}^4$ under the action of non-local symmetries of the self-dual Yang-Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang-Mills equations that are analogous to harmonic maps of finite type.Comment: AMSLatex, 15 pages, no figures. Corrected in line with the referee's comments. In particular, restriction to simply-connected open sets now explicitly stated. Version to appear in Communications in Mathematical Physic