Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Remarks on the Quantization of Self-Dual Fields

We investigate the Srivastava's and the Floreanini-Jackiw's formalisms of self-dual fields. By imposing an additional constraint on phase space we overcome the former's non-unitarity at the quantum level and simultaneously translate the former into the latter. Through introducing a parameter to extend the self-duality condition we show that both formalisms guarantee that it is only the self-dual field that exists in a consistent Hamiltonian quantum theory.     

Gauge-Invariant Formulation of the Chiral Schwinger Model with Faddeevian Regularization

We introduce the Wess-Zumino field to the chiral Schwinger model with Faddeevian regularization. The Wess-Zumino action has an interesting form; it is a self-dual (chiral) boson action. The Schwinger term between the correctly defined Gauss-law constraints G(Χ) and G(y) can be canceled by the WZ action, and the constraint G(Χ) is the first-class one.     

Nonperturbative quantum geometries

Using the self-dual representation of quantum general relativity, based on Ashtekar's new phase space variables, we present an infinite dimensional family of quantum states of the gravitational field which are exactly annihilated by the hamiltonian constraint. These states are constructed from Wilson loops for Ashtekar's connection (which is the spatial part of the left handed spin connection). We propose a new regularization procedure which allows us to evaluate the action of the hamiltonian constraint on these states.Infinite linear combinations of these states which are formally annihilated by the diffeomorphism constraints as well are also described. These are explicit examples of physical states of the gravitational field — and for the compact case are exact zero eigenstates of the hamiltonian of quantum general relativity. Several different approaches to constructing diffeomorphism invariant states in the self dual representation are also described.The physical interpretation of the states described here is discussed. However, as we do not yet know the physical inner product, any interpretation is at this stage speculative. Nevertheless, this work suggests that quantum geometry at Planck scales might be much simpler when explored in terms of the parallel transport of left-handed spinors than when explored in terms of the three metric.     

Linear perturbations of gauge fields

It is shown that under certain weak conditions (the vanishing of the field strength along a family of self-dual or anti-self-dual geodesic two-surfaces), in a curved or flat space-time, the linear perturbations of a given gauge field configuration can be expressed in terms of the solutions of a single second-order linear partial differential equation for a matrix potential. The particular case of the self-dual gauge fields is treated in some detail.     

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.     

Infinite number of local conservation laws for the self-dual SU(2) Yang-Mills system within 't Hooft's ansatz

We derive infinite sets of local continuity equations for the four-dimensional classical self-dual SU(2) Yang-Mills fields subjected to 't Hooft's ansatz. In striking analogy to the two-dimensional CP(n) non-linear sigma model where local conservation laws obtain either from complex Cauchy-Riemann analyticity or from a matrix Riccati equation, our local sets derive from quaternionic Fueter analyticity or a Riccati equation associated with the geometric prolongation structure implied by the Belavin-Zakharov linear spectral problem for the self-dual Yang-Mills system. Our analysis underlines the close connection between local and non-local conservation laws and suggests that infinite sets of local continuity equations should be present in the general self-(antiself-)dual gauge field case.     

Preliminary estimate of branching ratios of weak hadronic decays of bottom baryons emitting charmless scalar mesons

In this paper we study one-dimensional static self-dual non-topological solitons in an Abelian-Higgs Chern–Simons model with a non-canonical kinetic term. We choose a scalar potential with a single minimum and present an analytical soliton solution which corresponds to an electrically charged lump-like soliton without total momentum and located in a region where the scalar field gets maximum. The solitonic boundary conditions impose a discontinuity on the electric field across the soliton line where, then, there is a linear charge density.     

Gauge fields and D-branes

We prove that self-dual gauge fields in type I superstring theory are equivalent to configurations of Dirichlet 5-branes, by showing that the world-sheet theory of a Dirichlet 1-brane moving in a background of 5-branes includes an “ADHM sigma model”. This provides an explicit construction of the equivalent self-dual gauge field. We also discuss type II.     

Quantized self-dual Maxwell field on a null surface

The canonical formalism for a self-dual Maxwell field on a null plane is reviewed. After solution of the second class constraints, the transition to the quantum theory is carried out using a representation in which the self-dual Maxwell field is diagonal. The Gauss law constraint allows us to consider the physical state vectors to be holomorphic functionals of one complex function. Application of reality conditions allows us to define an inner product such that the Hermitean adjoint operators are identified with the classical complex conjugate operators. In going over to the Fourier expansion of the operators, we find that the inner product is formally convergent for positive frequency functionals and formally divergent for the negative frequency functionals. Following similar results of Ashtekar, Rovelli, and Smolin, negative frequency states are functional distributions identified with the helicity opposite to that of the positive frequency states.     

Bogomol'nyi solitons and Hermitian symmetric spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N − 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.     

The Anomaly Line Bundle of the Self-Dual Field Theory

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.     

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.     

Self-duality,helicity and coherent states in non-Abelian gauge theories

Solutions of the non-linear classical Yang-Mills field equations obtained by iteration from self-dual solutions of the linearized equations are shown to be themselves self-dual. In Minkowski space, such solutions are necessarily complex. We give a quantum theory interpretation of these solutions which relates them to the matrix element of the field operators between the vacuum state and a coherent state of spin-one quanta of definite helicity.     

Wave Function of Non-Vacuum Universe in Ashtekar Theory

In this paper, we studied the wave function of the universe in the (anti-) self-dual representation for the cosmological-model in which the closed Robertson- Walker spacetime is minimally coupled to a massless scalar field and the cosmological constant is equal to zero. We found that some wave functions in the (anti-) self-dual representation cannot be mapped into the σ-representation, or the metric representation, and vice versa.     

α Decays in Superstrong Static Electric Fields

Superstrong static electric fields could deform Coulomb barriers between α clusters and daughter nuclei, and bring up the possibility of speeding up α decays. We adopt a simplified model for the spherical α emitter ²¹²Po and study its responses to superstrong static electric fields. We find that superstrong electric fields with field strengths|E|~ 0:1 MV/fm could turn the angular distribution of α emissions from isotropic to strongly anisotropic, and speed up α decays by more than one order of magnitude. We also study the influences of superstrong electric fields along the Poisotope chains, and discuss the implications of our studies on α decays in superstrong monochromatic laser fields. The study here might be helpful for future theoretical studies of α decay in realistic superstrong laser fields.     

Geometric Quantization and the Metric Dependence of the Self-Dual Field Theory

We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. This connection governs metric dependence of the partition function of the self-dual field. We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. As an application, we show that the one-loop determinant of the (2, 0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model.