Space-time topology and a new class of Yang-Mills instanton

Different families of Yang-Mills instantons exist in space-time of different topology. Two (self-dual) examples are provided in Schwarzschild space having SU(2) Pontryagin numbers ±1 and ±2n² (n = integer). The latter solution describes a dyon. The Reissner-Nordstrom geometry also admits of non-self-dual solutions.     

Local stability of deformations of self-dual Yang-Mills fields on S4

We prove that local deformations of irreducible self-dual Yang-Mills fields on S⁴ in the space of smooth soluions not assumed to be self-dual are indeed self-dual.     

Symmetric instantons and the ADHM construction

We construct allSU(2) Yang-Mills instantons onS ⁴ that admit a certain symmetry (“quadrupole symmetry”). This is accomplished by an equivariant version of the “ADHM monad” classification of instantons. This work is part of an attempt to better understand the structure of non-self-dual Yang-Mills connections with the same symmetry. J.S. was supported by NSF Grants DMS-9106807 and DMS-9404468 Part of this work was done at the 1991 Regional Geometry Institute in Park City, Utah     

Semiclassical Yang-Mills theory I: Instantons

The partition functions of quantum Yang-Mills theory have an expansion in powers of the coupling constant; the leading order term in this expansion is called the semiclassical approximation. We study the semiclassical approximation for Yang-Mills theory on a compact Riemannian 4-manifold using geometric techniques, and do explicit calculations for the case when the manifold is the 4-sphere. This involves calculating the Riemannian measure and certain functional determinants on the moduli space of self-dual connections. The main result is that the contribution to the semiclassical partition functions coming from thek=1 connections on the 4-sphere isfinite andcalculable. We also discuss a renormalization procedure in which the radius of the 4-sphere is allowed to tend to infinity.Partially supported by N.S.F. grant DMS-8905211Partially supported by N.S.F. grant DMS-8802885     

Moduli Spaces of Self-Dual Connections over Asymptotically Locally Flat Gravitational Instantons

We investigate Yang–Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel–Hunsicker–Mazzeo. First referring to the codimension 2 singularity removal theorem of Sibner–Sibner and R?de we prove that given a smooth, finite energy, self-dual SU(2) connection over a complete ALF space, its energy is congruent to a Chern–Simons invariant of the boundary three-manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly. Then we introduce framed moduli spaces of self-dual connections over Ricci flat ALF spaces. We prove that the moduli space of smooth, irreducible, rapidly decaying self-dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold. We calculate its dimension by a variant of the Gromov–Lawson relative index theorem. As an application, we study Yang–Mills instantons over the flat , the multi-Taub–NUT family, and the Riemannian Schwarzschild space.     

On the number of parameters of self-dual Yang-Mills configurations

Using pure differential-geometric ideas (Lie groups as R-spaces and related properties) a new method of determining the number of parameters of a self-dual Yang-Mills configuration is proposed. Some connections with the Atiyah-Ward twistor approach are also revealed.     

Strong connections on quantum principal bundles

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS ²→RP ². A certain class of strongU _q (2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialU _q (2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq. This work was in part supported by the NSF grant 1-443964-21858-2. Writing up the revised version was partially supported by the KBN grant 2 P301 020 07 and by a visiting fellowship at the International Centre for Theoretical Physics in Trieste.     

The twistor theory of equations of KdV type: I

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to then-generalized KdV-hierarchy. It also emerges that each flow of then-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group SL _n . It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of then-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of then-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space.A summary of the results of the second part of this work appears in the Introduction.Most of this work was done while Darby Fellow of Mathematics at Lincoln College, Oxford     

Minkowski space Yang-Mills fields from Euclidean self-dual fields

It is examined, if it is possible, to obtain solutions of the SU(2) Yang-Mills field equations in Minkowski space from Euclidean self-dual Yang-Mills fields by method proposed by Bernreuther. It is shown that the conditions, which are imposed on the Euclidean self-dual fields by this method, make every Euclidean self-dual field and the corresponding Minkowski space field obtained from it, equivalent to a pure gauge field, F _ab 0.     

Around Polygons in ℝ3 and S3

We survey certain moduli spaces in low dimensions and some of the geometric structures that they carry, and then construct identifications among all of these spaces. In particular, we identify the moduli spaces of polygons in ℝ³ and S ³, the moduli space of restricted representations of the fundamental group of a punctured 2-sphere, the moduli space of flat connections on a punctured sphere, the moduli space of parabolic bundles on a sphere, the moduli space of weighted points on ℂℙ¹ and the symplectic quotient of SO(3) acting diagonally on (S ²) ⁿ . All of these spaces depend on parameters and some the above identifications require the parameters to be small. One consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given. Received: 20 September 1999 / Accepted: 28 November 2000     

The semiclassical limit for gauge theory onS 2

It is shown that the Yang-Mills measureZ _h ^–1 e^–S()/h[D], whereh>0, describing gauge fields on the two-sphere converges to a probability measure on the moduli space of Yang-Mills connections onS ², ash0.This work was partially supported by NSF Grants DMS-8922941, and PHY-8912067     

The Riemannian geometry of the Yang-Mills moduli space

The moduli space of self-dual connections over a Riemannian 4-manifold has a natural Riemannian metric, inherited from theL ² metric on the space of connections. We give a formula for the curvature of this metric in terms of the relevant Green operators. We then examine in great detail the moduli space ₁ ofk=1 instantons on the 4-sphere, and obtain an explicit formula for the metric in this case. In particular, we prove that ₁ is rotationally symmetric and has finite geometry: it is an incomplete 5-manifold with finite diameter and finite volume.Partially supported by Horace Rackham Faculty Research Grant from the University of MichiganPartially supported by N.S.F. Grant DMS-8603461     

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.     

Lines in space-times

A family of Riemannian metrics on the moduli space of irreducible selfdual connections of instanton numberk=1 overCP ² is considered. We find explicit formulas for these metrics and deduce conclusions concerning the geometry of the instanton space.     

A connection between the Einstein and Yang-Mills equations

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.Andrew Mellon Postdoctoral fellow and Fulbright ScholarSupported in part by NSF grant no. PHY 80023     

Stability and isolation phenomena for Yang-Mills fields

In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods. One of our main results is to prove that any weakly stable Yang-Mills field overS ⁴ with groupG=SU₂, SU₃ orU ₂ is either self-dual or anti-self-dual. The analogous statement for SO₄-bundles is also proved. The other main series of results concerns gap-phenomena for Yang-Mills fields. As a consequence of the non-linearity of the Yang-Mills equations, we can give explicitC ⁰-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields. In this part of the study the nature of the groupG does not matter, neither is the dimension of the base manifold constrained to be four.Laboratoire Associé au C.N.R.S. No. 169Research partially supported by Volkswagen Grant and NSF Grant MCS-77-23579     

Yang-Mills fields which are not self-dual

The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, withSU(2) as a gauge group. The approach is that of equivariant geometry: attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane.     

Non-self-dual Yang-Mills connections with quadrupole symmetry

We prove the existence of non-self-dual Yang-Mills connections onSU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal±1. We study connections equivariant under anSU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dualSU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.Research partially supported by NSF Grant DMS-8806731Most of this research was done while the author was a Bantrell Fellow at the California Institute of Technology, and was partially supported by NSF Grant DMS-8801918     

Self-dual Yang-Mills fields in an Einstein universe

We find exact solutions of the self-consistent Einstein-Yang-Mills system of equations. These solutions are self-dual Yang-Mills fields inI ¹×S³ space-time.     

Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even-dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fieldsF . We derive a topological bound on R⁸, , wherep1 is the first Pontryagin class of the SO(n) Yang-Mills bundle, andk is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.