A noncommutative deformation of topological field theory

Cohomological Yang–Mills theory is formulated on a noncommutative differentiable four manifold through the -deformation of its corresponding BRST algebra. The resulting noncommutative field theory is a natural setting to define the -deformation of Donaldson invariants and they are interpreted as a mapping between the Chevalley–Eilenberg homology of noncommutative spacetime and the Chevalley–Eilenberg cohomology of noncommutative moduli of instantons. In the process we find that in the weak coupling limit the quantum theory is localized at the moduli space of noncommutative instantons.     

An Algebraic Proof on the Finiteness of Yang–Mills–Chern–Simons Theory in D=3

A rigorous algebraic proof of the full finiteness in all orders of perturbation theory is given for the Yang–Mills–Chern–Simons theory in a general three-dimensional Riemannian manifold. We show the validity of a trace identity, playing the role of a local form of the Callan–Symanzik equation, in all loop orders, which yields the vanishing of the -functions associated to the topological mass and gauge coupling constant as well as the anomalous dimensions of the fields.     

Noncommutative Supergeometry and Duality

We introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying {Q,Q} =0). We develop the theory of connections on modules over Q-algebras and prove a general duality theorem for gauge theories on such modules. This theorem contains as a simplest case SO(d,d, Z)-duality of gauge theories on noncommutative tori.     

Octonionic Yang–Mills instanton on quaternionic line bundle of Spin(7) holonomy

The total space of the spinor bundle on the four-dimensional sphere S⁴ is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang–Mills instanton on this eight-dimensional gravitational instanton. This is a higher dimensional generalization of (anti-) self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang–Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.     

Topological Contributions in Two-Dimensional Yang–Mills Theory: From Group Averages to Integration over Algebras

We show that keeping only the topologically trivial contribution to the average of a class function on U(N) amounts to integrating over its algebra. The goal is reached first by decompactifying an expansion over the instanton basis and then directly, by means of a geometrical procedure.     

Exact Inhomogeneous Cosmological Models with Yang–Mills Fields

The exact solutions of Einstein–Yang–Mills equations in a class of spherically symmetric cosmological models are found with several coordinate conditions both with the account and without the account cosmological constant.     

Geometrical Lagrangian for a Supersymmetric Yang–Mills Theory on the Group Manifold

Perhaps one of the main features of Einstein's General Theory of Relativity is that spacetime is not flat itself but curved. Nowadays, however, many of the unifying theories like superstrings on even alternative gravity theories such as teleparalell geometric theories assume flat spacetime for their calculations. This article, an extended account of an earlier author's contribution, it is assumed a curved group manifold as a geometrical background from which a Lagrangian for a supersymmetric N=2, d=5 Yang–Mills – SYM, N=2, d=5 – is built up. The spacetime is a hypersurface embedded in this geometrical scenario, and the geometrical action here obtained can be readily coupled to the five-dimensional supergravity action. The essential idea that underlies this work has its roots in the Einstein–Cartan formulation of gravity and in the group manifold approach to gravity and supergravity theories. The group SYM, N=2, d=5, turns out to be the direct product of supergravity and a general gauge group .     

Simple type and the boundary of moduli space

We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the “simple type” condition of Donaldson theory. Using the natural geometric representative of μ(pt) defined in [L. Sadun, Commun. Math. Phys. 178 (1996) 107–113], the boundary region of moduli space contributes of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to the interior of the (k+1)th ASD moduli space, intersected with two representatives of (4 times) the point class, being homologous to 58 copies of the kth moduli space. This is peculiar, since the only known embeddings of the kth moduli space into the (k+1)th involve Taubes gluing, and the images of such embeddings lie entirely in the boundary region.When using the natural de Rham representatives of μ(pt) considered by Witten [Commun. Math. Phys. 117 (1988) 353], the boundary region contributes of what is needed for simple type, again regardless of the topology or geometry of the underlying 4-manifold. The difference between this and the geometric representative answer is not contradictory, as the contribution of a fixed region to the Donaldson invariants is geometric, not topological.     

Quadratic metric‐affine gravity

We consider spacetime to be a connected real 4‐manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi‐Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp‐wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non‐Riemannian solutions. We define the notion of a “Weyl pseudoinstanton” (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non‐Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non‐Riemannian solution as a mathematical model for the neutrino.     

Invariant Solutions of the Yang–Mills Field Equations Coupled to a Scalar Doublet

The Yang–Mills system of field equations which includes coupling to an SU(2) scalar matter doublet is developed. It is shown that an SU(2) current for a scalar matter doublet can be developed. The basic structure which fits the Yang–Mills system is somewhat different from the case of the scalar triplet. Using this form for the scalar current, it is possible to write down the Yang–Mills system which couples to the scalar matter doublet. It is shown that several sets of solutions to this system of equations can be obtained.     

Anti-self-dual Yang–Mills equations on noncommutative space–time

By replacing the ordinary product with the so-called -product, one can construct an analog of the anti-self-dual Yang–Mills (ASDYM) equations on the noncommutative . Many properties of the ordinary ASDYM equations turn out to be inherited by the -product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative , from which one can also derive the fundamental structures for integrability such as a zero-curvature representation, an associated linear system, the Riemann–Hilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin’s Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the -product analogs.     

Super-Matrix KdV and Super-Generalized NS Equations from Self-Dual Yang–Mills Systems with Supergauge Groups

Super-matrix KdV and super-generalized nonlinear Schrödinger equations are shown to arise from a symmetry reduction of ordinary self-dual Yang–Mills equations with supergauge groups.     

N=6 Supergravity on Ad S5 and the SU(2,2/3) Superconformal Correspondence

It is argued that N=6 supergravity on Ad S5, with the gauge group SU(3)× U(1) corresponds, at the classical level, to a subsector of the chiral primary operators of N=4 Yang–Mills theories. This projection involves a duality transformation of N=4 Yang–Mills theory and therefore can be valid if the coupling is at a self-dual point, or for those amplitudes that do not depend on the coupling constant.     

Spinorial Matter in Affine Theory of Gravity and the Space Problem

The purely affine theory of gravity possesses a canonical formulation. For this and other reasons, it could be a promising candidate for quantum gravity. Motivated by these perspectives, we discuss spinorial matter coupled to gravity, where the latter is described by a connection having no a priori relation to a metric. We show that one can establish a truncated spinor formalism which, for special or approximate solutions to the gravitational equations, reduces to the standard formalism. As a consequence, one arrives at "matter-induced" Riemann–Cartan spaces solving the Weyl-Cartan space problem.     

On the infrared behavior of Landau gauge Yang–Mills theory

We discuss the properties of ghost and gluon propagators in the deep infrared momentum region of Landau gauge Yang–Mills theory. Within the framework of Dyson–Schwinger equations and the functional renormalization group we demonstrate that it is only a matter of infrared boundary conditions whether infrared scaling or decoupling occurs. We argue that the second possibility is at odds with global BRST symmetry in the confining phase. For this purpose we improve upon existing truncation schemes in particular with respect to transversality and renormalization.     

A modification of Einstein–Schrödinger theory that contains Einstein–Maxwell–Yang–Mills theory

The Lambda-renormalized Einstein–Schrödinger theory is a modification of the original Einstein–Schrödinger theory in which a cosmological constant term is added to the Lagrangian, and it has been shown to closely approximate Einstein– Maxwell theory. Here we generalize this theory to non-Abelian fields by letting the fields be composed of d × d Hermitian matrices. The resulting theory incorporates the U(1) and SU(d) gauge terms of Einstein–Maxwell–Yang–Mills theory, and is invariant under U(1) and SU(d) gauge transformations. The special case where symmetric fields are multiples of the identity matrix closely approximates Einstein–Maxwell–Yang–Mills theory in that the extra terms in the field equations are < 10^?13 of the usual terms for worst-case fields accessible to measurement. The theory contains a symmetric metric and Hermitian vector potential, and is easily coupled to the additional fields of Weinberg–Salam theory or flipped SU(5) GUT theory. We also consider the case where symmetric fields have small traceless parts, and show how this suggests a possible dark matter candidate.     

Pseudoinstantons in Metric-Affine Field Theory

In abstract Yang–Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible (pseudoinstanton). We suggest a metric-affine action which is a natural generalization of the Yang–Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino.     

Infrared safe observables in super Yang–Mills theory

The infrared structure of MHV gluon amplitudes in planar limit for super Yang–Mills theory is considered in the next-to-leading order of PT. Explicit cancellation of the infrared divergencies in properly defined cross-sections is demonstrated. The remaining finite parts for some inclusive differential cross-sections in planar limit are calculated analytically. In general, contrary to the virtual corrections, they do not reveal any simple structure.     

Exact Solutions for Self-Dual SU(2) and SU(3) Yang–Mills Fields

The (constrained) canonical reduction of four-dimensional self-dual SU(2) and SU(3) Yang–Mills theory to two-dimensional nonlinear Schrödinger (NS) and Korteweg–de Vries (KdV) equations are considered. The Bäcklund transformations (BTs) are implemented to obtain new classes of exact solutions for the reduced two-dimensional NS and KdV models.