Stable vector bundles and instantons

Methods of abstract algebraic geometry are used to study rank 2 stable vector bundles on ³. These bundles are then used to give self-dual solutions, called instantons, of the Yang-Mills equation onS ⁴.This article reproduces two lectures I gave in I. M. Singer's seminar on gauge theories, at Berkeley in June 1977. Full details will appear elsewhere [6]Partially supported by NSF grant NSF MCS 76-03423, A02     

Classical solutions of the chiral model,unitons, and holomorphic vector bundles

This paper deals with classical solutions of theSU(2) chiral model on ², and of a generalized chiral model on ²⁺¹. Such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space. With an appropriate boundary condition, the solutions (called 1-unitons in [9]) correspond to bundles over a compact 2-dimensional complex manifold, and the problem becomes one of algebraic geometry.     

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.     

Classification of stationary space-times

A systematic approach to the geometric structure of stationary gravitational fields is presented. The algebraic type of the trace-free Ricci tensor together with the propagation properties of the eigenrays in the background 3-space defined by the Killing trajectories serve as a basis for classifying the solutions of the stationary field equations. The eigenrays are the integral curves belonging to the solutions _A of the eigenvalue problemG _A ^B _B=_A,G _A ^B spinor representing the gravitational field in the background space. Many of the already known stationary metrics can be derived in the present scheme but new solutions of the field equations are also obtained. The possible types of the vacuum and electrovac fields are discussed in their connection with the corresponding exact solutions.Work honoured by a Fifth Gravity Research Foundation Award in 1973.Leverhulme Visiting Fellow.     

Solutions of the Maxwell and Yang-Mills equations associated with hopf fibrings

It is shown that the magnetic pole of lowest strength and the pseudoparticle solution of the Yang-Mills equations correspond to natural connections defined on the principal bundlesU(2)/U(1)=S ₃ S ₂ andSp(2)/Sp(1)=S ₇ S ₄, respectively. This observation leads to a general methods of constructing new, topologically nontrivial solutions of the Maxwell and Yang-Mills equations. Among them is an electromagnetic instanton defined over the two-dimensional complex projective space endowed with the Fubini-Study metric.On leave from the Institute of Theoretical Physics, Warsaw University, Hoza 69, Warsaw, Poland.     

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.     

Noncommutative Sine-Gordon Model

The sine-Gordon model may be obtained by dimensional and algebraic reduction from (2+2)-dimensional self-dual U(2) Yang-Mills through a (2+1)-dimensional integrable U(2) sigma model. It is argued that the noncommutative (Moyal) deformation of this procedure should relax the algebraic reduction from U(2) U(1) to U(2)U(1) × U(1). The result are novel noncommutative sine-Gordon equations for a pair of scalar fields. The dressing method is outlined for constructing its multi-soliton solutions. Finally, the tree-level amplitudes demonstrate that this model possesses a factorizable and causal S-matrix in spite of its time-space noncommutativity.     

Spherically symmetric lightlike radiation and conformally flat space-times

The problem of the coexistence of spherical symmetry of a 3-space, tracelessness of the energy-momentum tensor, conformally flat 4-metrics, and the validity of the Einstein equations is investigated. The assertion is proved that when spherical symmetry is present nonequilibrium lightlike radiation with the energymomentum tensor T_µv = l _µ l _v (l v = 0) cannot serve as the source of the gravitational field corresponding to a conformally flat space-time (type 0 according to the algebraic classification). An exact spherically symmetric solution with a conformally flat metric is obtained which describes dust and equilibrium isotropic radiation without energy exchange between them. This solution is rewritten for a synchronous reference frame in which it is evident that it describes a homogeneous and isotropic universe. In the limit of the absence of radiation the solution changes into the well-known Friedmann solution for an open universe.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 32–35, July, 1984.In conclusion the authors express their gratitude to all the participants in the seminar of the gravitation section of the Scientific and Technical Council of Minvuz of the USSR (the physics faculty of Moscow State University) for a useful discussion of the results of this paper.     

Elliptic-type solutions to the scale invariant Yang-Mills and sigma-model hierarchies

We outline the construction of non-self-dual elliptic solutions by relating the spherically symmetric subsystem of the (scale invariant) Yang-Mills and sigmamodel hierarchies to the hierarchies of ⁴ and Sine-Gordon models in one dimension respectively. The construction is carried out explicitly for the usual Yang-Mills model on ₄, and the first two sigma-models on ₂ and ₄. The solution to the first member of the Yang-Mills hierarchy is related to elliptic solutions found previously.     

Gauge theories,instantons and algebraic geometry

We review the definition of instanton (= pseudoparticle) solutions and their importance in the context of nonabelian gauge (= Yang-Mills) theories, as well as the recent progress, due to Atiyah and Ward, in their construction, using the Penrose twistor transform and methods of algebraic geometry. In particular, we present a proof of the theorem of Atiyah and Ward on the correspondence between SU(2) instanton solutions over the 4-sphere and certain algebraic 2-dimensional complex vector bundles over complex projective 3-space.     

The Krichever map,vector bundles over algebraic curves,and Heisenberg algebras

We study the GrassmannianGr _x ⁿ consisting of equivalence classes of rankn algebraic vector bundles over a Riemann surfaceX with an holomorphic trivialization at a fixed pointp. Commutative subalgebras ofgl(n, H ),H being the ring of functions holomorphic on a punctured disc aboutp, define flows on the Grassmannian, giving rise to classes of solutions to multi-component KP hierarchies. These commutative subalgebras correspond to Heisenberg algebras in the Kac-Moody algebra associated togl(n, H ₎. One can obtain, by the Krichever map, points ofGr _x ⁿ (and solutions of mcKP) from coveringsf: YX and other geometric data. Conversely for every point ofGr _x ⁿ and for every choice of Heisenberg algebra we construct, using the cotangent bundle ofGr _x ⁿ , an algebraic curve coveringX and other data, thus inverting the Krichever map. We show the explicit relation between the choice of Heisenberg algebra and the geometry of the covering space.The research was partially supported by US Army grant DAA L03-87-K-0110 and NSF grant DMS 9106938     

Towards a constructive approach of a gauge invariant,massiveP(φ)2 theory

As part of a possible constructive approach to a gauge invariantP()₂ theory, we consider massive, scalar, polynomially selfcoupled fields in a fixed external Yang-Mills potentialA in two-dimensional euclidean space. For a large class ofA's we show that the corresponding euclidean Green's functions for the fields have a lower mass gap for weak coupling which is uniform inA. The result is obtained by adapting the Glimm-Jaffe-Spencer cluster expansion to the present situation through Kato's inequality, which reflects the diamagnetic effect of the Yang-Mills potential. A discussion of the corresponding gauge covariance is included.     

Topological aspects of Yang-Mills theory

The space of mapsS ³ G has components which give the topological quantum number of Yang-Mills theory for the groupG. Each component of the space has further topological invariants. WhenG=SU(2) we show that these invariants (the homology groups) are captured by the space of instantons. Using these invariants we show that potentials must exist for which the massless Dirac equation (in Euclidean 4-space) has arbitrarily many independent solutions (for fixed instanton number).     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

Calculation of quasi-stationary state parameters within the algebraic version of the resonating-group method

The dynamical equations of the algebraic version of the resonatinggroup method are used to determine the positionsE _res,l and the widths _l of quasi-stationary states or, what is equivalent, to find the polesE _l =E _resl –i _l /2 of theS-matrix in the fourth quadrant of thek-plane. Concrete calculations are performed for the -decay of⁸Be. A comparison with the results of other authors and with experiment is made.     

Cosmological solutions for the coupled Einstein-Yang-Mills-Higgs equations

The coupled system of Einstein-Yang-Mills-Higgs equations is solved numerically for a Robertson-Walker symmetric universe. The open universe must be excluded because, for this situation, the coupled system is unable to produce energy-momentum densityT in agreement with the Robertson-Walker symmetry. For the closed universe, inflation solutions do occur if an early re-collapse is avoided by suitable choice of the initial conditions. The energy exchange between the minimally coupled Higgs and Yang-Mills subsystems is very small so that the inflating power of the Higgs field cannot be disturbed by the presence of the Yang-Mills field. After the inflation phase the influence of the Yang-Mills field is completely negligible (cosmic no-hair theorem).     

The Field of a Vertical Magnetic Dipole in the Presence of a Sphere with a Circular Aperture

We consider diffraction of an arbitrary monochromatic electromagnetic wave by a perfectly conducting, infinitesimally thin sphere with a circular aperture. The initial boundary-value problem is reduced to a system of linear algebraic equations of the second kind in the form (I+H)x=b, where x,bl₂ and H is a compact operator in l₂ (I is the unit operator in l₂). We calculate the spectrum of natural frequencies of a sphere with a circular aperture that correspond to axisymmetric modes of magnetic type. The influence of the resonance regimes (in particular, the mode coupling) of the considered structure on its scattering properties is analyzed.     

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 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     

Momentum and angular momentum in theH-space of asymptotically flat,Einstein-Maxwell space-times

We propose new definitions for the momentum and angular momentum of Einstein-Maxwell fields that overcome the deficiencies of earlier definitions of these terms and are appropriate to the newH-space formulations of space-time. We make our definitions in terms of the Winicour-Tamburino linkages applied to the good cuts of CI⁺. Our transformations between good cuts then correspond to the translations and Lorentz transformations at points inH-space. For the special case of Robinson-Trautman typeII space-times, we show that our definitions of momentum and angular momentum yield the previously published results of Ludvigsen.Part of this work was completed while William Hallidy was supported by a fellowship from the Alexander von Humboldt Foundation.