A connection dynamic theory of gravity

A gauge theory of gravity with matter sources is investigated. In this theory the fundamental fields are the connections of a principal de Sitter fiber bundle, and the action is a quadratic form of the curvature as in the standard Yang-Mills theory, in which the Einstein-Hilbert action is included. The Hamiltonian formulation and the constraint analysis of it is given. The separation of the self-dual and anti self-dual parts of the connection and curvature is exhibited.     

Multi-instantons localized at the origin

We obtain a family of self-dual Yang-Mills fields in an SU(2) gauge theory. Some of them describe pseudoparticles with arbitrary topological numbers and with action densities concentrated around the origin.     

Beltrami differentials and 2d (super)gravity in the chiral gauge

We consider the Beltrami parametrization of a 2-dimensional metric and work out the basic transformation laws of the constituting fields. These results are then applied to investigate the geometric structure of 2-dimensional gravity in Polyakov's light-cone gauge. Subsequently, we elaborate on the supersymmetric generalization of this theory. The general relation to the superconformai gauge is presented in detail and the corresponding change of coordinates is shown to satisfy simple super Beltrami equations.     

On 3D non-linear vector-vector duality

We consider the duality between topologically massive and self-dual nonabelian vector fields in three dimensions. In Fock-Schwinger gauge we obtain an explicit, albeit nonlocal, self-dual action equivalent to the topologically massive action. The inverse construction is performed for a restricted class of modified self-dual actions and leads to a topologically massive theory with two interacting gauge fields.     

Non-Abelian Tensor Multiplet Equations from Twistor Space

We establish a Penrose–Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly ${\mathcal{N} = (1, 0)}$ and ${\mathcal{N} = (2, 0)}$ supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings.     

Superholomorphic structures,moment maps,and super self-dual Yang-Mills connections over super Riemann surfaces

We consider the space of superconnections with certain curvature constraints over super Riemann surfaces. We define a moment map over that space to the dual of the super Lie algebra of gauge transformations. The zero set of this moment map corresponds to the super self-dual Yang-Mills equations in two dimensions. This result generalizes the recently proposed scheme for the nonsupersymmetric case. The superfield equations also arise from super self-dual Yang-Mills equations in four dimensions by dimensional reduction.     

Reduction of 4-dim self-dual super Yang-Mills equations onto super Riemann surfaces

Recently, a self-dual super Yang-Mills equation over a super Reimann surface was obtained as the zero set of a moment map on the space of superconnections to the dual of the super Lie algebra of gauge transformations. We present a new formulation of the 4-dim Euclidean self-dual super Yang-Mills equations in terms of constraints on the supercurvature. By dimensional reduction, we obtain the same set of superconformal field equations which define self-dual connections on a super Rieman surface.     

On the Gauge Invariant Lagrangian for Seiberg--Witten Topological Monopoles

A topological gauge invariant Lagrangian for Seiberg--Witten monopoleequations is constructed. The action is invariant under a huge classof gauge transformations which after BRST fixing leads to the BRSTinvariant action associated to Seiberg--Witten monopole topologicaltheory. The supersymmetric transformation of the fields involved inthe construction is obtained from the nilpotent BRST algebra.     

Action principle and equations of motion for nonabelian monopoles

Monopoles in gauge theories have an intrinsic interaction with the gauge field. The definition of a monopole as a topological charge implies a certain constraint coupling the gauge field to the coordinates of a particle carrying that charge. Hence, even starting with the free action, the constraint will give equations of motion in which field and particle interact. Applied to electromagnetism, this procedure gives the Maxwell and Lorentz equations. In this paper, we apply the same idea to nonabelian monopoles to deduce their equations of motion which are otherwise unknown. To surmount certain technical difficulties connected with patching, loop space techniques are developed to solve the variational problem. A closed set of equations are obtained, which are analogous to the Maxwell and Lorentz equations, and bear also a formal resemblance to the Wong equations for a “classical” point source of Yang-Mills fields.     

Flat coordinates,topological Landau-Ginzburg models and the Seiberg-Witten period integrals

We study the Picard-Fuchs differential equations for the Seiberg-Witten period integrals in N = 2 supersymmetric Yang-Mills theory. For A-D-E gauge groups we derive the Picard-Fuchs equations by using the flat coordinates in the A-D-E singularity theory. We then find that these are equivalent to the Gauss-Manin system for two-dimensional A-D-E topological Landau-Ginzburg models and the scaling relation for the Seiberg-Witten differential. This suggests an interesting relationship between four-dimensional N = 2 gauge theories in the Coulomb branch and two-dimensional topological field theories.     

Constraints and actions in two and three dimensionalN=1 conformal supergravity

We investigate closure of the gauge algebra and constraints inN=1 conformal supergravity in 2 and 3 dimensions. In the 2 dimensional case, contrary to 3 or higher dimensions, some parts of the gauge fields are algebraically unsolvable in the constraint equations on group curvatures. It will be shown that these unsolvable parts are decoupled from the transformation law as well as from the kinetic multiplets. Hence they are absent in the invariant action for matter multiplets coupled to conformal supergravity which is relevant to the old superstrings. Explicit construction of the invariant actions are illustrated for the case of spinning strings and locally supersymmetric σ-models with the Wess-Zumino term.     

Self-dual solutions to euclidean gravity

Recent work on Euclidean self-dual gravitational fields is reviewed. We discuss various solutions to the Einstein equations and treat asymptotically locally Euclidean self-dual metrics in detail. These latter solutions have vanishing classical action and nontrivial topological invariants, and so may play a role in quantum gravity resembling that of the Yang-Mills instantons.     

A new derivation of the Picard-Fuchs equations for effective N = 2 super Yang-Mills theories

A new method to obtain the Picard-Fuchs equations of effective, N = 2 supersymmetric gauge theories in 4 dimensions is developed. It includes both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. It applies to all classical gauge groups, and directly produces a decoupled set of second-order, partial differential equations satisfied by the period integrals of the Seiberg-Witten differential along the 1-cycles of the algebraic curves describing the vacuum structure of the corresponding N = 2 theory.     

BRST Formalism in Self-Dual Chern-Simons Theory with Matter Fields

We apply BRST method to the self-dual Chern-Simons gauge theory with matter fields and the generators of symmetries of the system from an elegant Lie algebra structure under the operation of Poisson bracket. We discuss four different cases: abelian, nonabelian, relativistic, and nonrelativistic situations and extend the system to the whole phase space including ghost fields. In addition, we obtain the BRST charge of the field system and check its nilpotence of the BRST transformation which plays an important role such as in topological quantum field theory and string theory.     

Properties of the covariant formulation of superstring theories

The covariant two-dimensional action principle that describes the dynamics of free superstrings in a Minkowski background is reviewed. Covariant gauge conditions are formulated, which simplify the equations of motion of the superspace coordinates to free equations. In this gauge there are bosonic and fermionic constraints whose generators give a supersymmetric generalization of the Virasoro algebra. As in certain supersymmetric field theories, closure of the algebra requires using the equations of motion. Covariant constrained bracket relations are obtained for the classical theory, but it is very difficult to extend them to quantum mechanical commutation relations. Interaction vertices satisfying supersymmetry and the necessary gauge conditions are constructed. They reduce in a special frame to ones found in earlier work in the light-cone gauge, and then can be interpreted quantum mechanically.     

Supersymmetry and instantons

We show that the eigenvalue equations for the fluctuation of scalars, fermions and gluon around any classical self-dual solution of the Yang-Mills theory have the same spectrum of non-zero eigenvalues. In the case of a supersymmetric Yang- Mills theory this implies that the one loop correction around any self-dual instanton is just given by a counting of the zero modes of the gluon, fermion and ghost.     

Gauge theories of Josephson junction arrays

We show that the zero-temperature physics of planar Josephson junction arrays in the self-dual approximation is governed by an Abelian gauge theory with a periodic mixed Chern-Simons term describing the charge-vortex coupling. The periodicity requires the existence of (Euclidean) topological excitations which determine the quantum phase structure of the model. The electric-magnetic duality leads to a quantum phase transition between a superconductor and a superinsulator at the self-dual point. We also discuss in this framework the recently proposed quantum Hall phases for charges and vortices in presence of external offset charges and magnetic fluxes: we show how the periodicity of the charge-vortex coupling can lead to transitions to anyon superconductivity phases. We finally generalize our results to three dimensions, where the relevant gauge theory is the so-called BF system with an antisymmetric Kalb-Ramond gauge field.     

The supersingleton I: Free dipole and interactions at infinity

The program of this Letter, and the one following it, is to develop a formulation of supersymmetric electrodynamics in which the massless fields (photons and neutrinos) are bilinear (actually, commutators) of constituent singleton fields. here we lay the foundations, describing the unique gauge invariant interaction that is allowed in the context of conventional quantum field theory. New quantization rules, leading to super QED, are dealt with in the second Letter.     

Gravitational instantons

The path-integral approach to quantum field theory assigns special importance to finite action Euclidean solutions of classical field equations. In Yang-Mills gauge theories, the instanton solutions of classical field equations with self-dual field strength have given rise to a new, nonperturbative treatment of the quantum field theory and its vacuum state. Since gravitation is also a species of gauge theory, one might think that similar phenomena would occur in gravity. The authors recently sought and found a new self-dual solution to Euclidean gravity which plays a role parallel to that of the Yang-Mills instanton. Gravitational instantons now promise to yield new insights into the nature of quantum gravity.This essay received the second award from the Gravity Research Foundation for the year 1979-Ed.