A symmetric family of Yang-Mills fields

We examine a family of finite energySO(3) Yang-Mills connections overS ⁴, indexed by two real parameters. This family includes both smooth connections (when both parameters are odd integers), and connections with a holonomy singularity around 1 or 2 copies ofRP ². These singular YM connections interpolate between the smooth solutions. Depending on the parameters, the curvature may be self-dual, anti-self-dual, or neither. For the (anti)self-dual connections, we compute the formal dimension of the moduli space. For the non-self-dual connections we examine the second variation of the Yang-Mills functional, and count the negative and zero eigenvalues. Each component of the non-self-dual moduli space appears to consist only of conformal copies of a single solution.This work was partially supported by an NSF Mathematical Sciences Postdoctoral Fellowship     

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     

A Riemann–Hilbert approach to Painlevé IV

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ?¹ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ?¹ are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.     

Nahm's transformation for instantons

We describe in mathematical detail the Nahm transformation which maps anti-self dual connections on the four-torus (S ¹)⁴ onto anti-self-dual connections on the dual torus. This transformation induces a map between the relevant instanton moduli spaces and we show that this map is a (hyperKähler) isometry.     

The prequantum line bundle on the moduli space of flat <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">N</Emphasis>) connections on a Riemann surface and the homotopy of the large <Emphasis Type="Italic">N</Emphasis> limit

We show that the prequantum line bundle on the moduli space of flat SU(2) connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat SU(n) connections, in the limit as n tends to infinity, and \( {\mathbb C}P^\infty \). Applications to the stable moduli space of flat unitary connections are also discussed.     

A largek asymptotics of Witten's invariant of Seifert manifolds

We calculate a largek asymptotic expansion of the exact surgery formula for Witten'sSU(2) invariant of some Seifert manifolds. The contributions of all flat connections are identified. An agreement with the 1-loop formula is checked. A contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series. A 2-loop correction to the contribution of the trivial connection is found to be proportional to Casson's invariant.Work supported by NSF Grant 9009850 and R.A. Welch Foundation.     

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.     

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 semiclassical limit forSU(2) andSO(3) gauge theory on the torus

We prove that forSU(2) andSO(3) quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure converge, asT0, to integrals with respect to a symplectic volume measure µ₀ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly.Research supported in part by LEQSF Grant RD-A-08, and NSF Grant DMS 9400961.     

Electronic-topological transition and shear stability of the β alloys Ni-Al and TiNi

On the basis of a numerical calculation of the band contributions to the shear moduli, this paper discusses a hypothesis expressed earlier that the premartensitic softening of the lattice in Ni-Al and TiNi alloys is caused by the approach of the latter to an electronic-topological transition (ETT). It is found in both alloys that the ETT effects are substantially different in the different shear moduli: They are strong in the modulus C′ (especially in Ni-Al) and weak in C ₄₄. It can be concluded from this result that the observed premartensitic softening of C′ in Ni-Al and TiNi can actually be caused by the approach to the ETT point. At the same time, the anomalous temperature dependence of C ₄₄ observed in TiNi is apparently not associated with ETT effects. Fiz. Tverd. Tela (St. Petersburg) 39, 972–976 (June 1997)     

Real polarization of the moduli space of flat connections on a Riemann surface

We prove that the moduli space of flatSU(2) connections on a Riemann surface has a real polarization, that is, a foliation by lagrangian subvarieties. This polarization may provide an alternative quantization of the Chern-Simons gauge theory in higher genus, in line with the results of [11] for genus one.Supported by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

Dimensional Reduction Over the Quantum Sphere and Non-Abelian q-Vortices

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a K?hler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU _q (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-K?hler quotient construction.     

Enriques Surfaces, Analytic Discriminants, and Borcherds's Φ Function

In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form Φ on the moduli space of marked, polarized Enriques surface of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. In [JT 94] and [JT 96], we defined and studied a meromorphic (multi-valued) modular form of weight 2, which we call the K3 analytic discriminant, on the moduli space of marked, polarized, K3 surfaces of degree 2d; in certain cases, including when , where p _k are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a determinant of the Laplacian on a polarized K3 surface with respect to the Calabi-Yau metric together with the L ² norm of the image of the period map with respect to a properly scaled holomorphic two form. Since the universal cover of any Enriques surface is a K3 surface, we can restrict the K3 analytic discriminant to the moduli space of degree 2 Enriques surfaces. The main result of this paper is the observation that the square of our degree 2 analytic discriminant, viewed as a function on the moduli space of degree 2 Enriques surfaces, is equal to the Borcherd's Φ function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Moody algebras and elliptic curves, and suggests further connections with higher dimensional Calabi-Yau varieties, specifically those which can be realized as complete intersections in some, possibly weighted, projective space. Received: 24 July 1995 / Accepted: 21 March 1997     

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     

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of Yang-Mills theory defined on a projective module for an arbitrary noncommutativity parameter which is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of . The construction of noncommutative instanton contributions to the path integral is described in some detail. In general, there are infinitely many gauge inequivalent contributions of fixed topological charge, along with a finite number of quantum fluctuations about each instanton. The associated moduli spaces are combinations of symmetric products of an ordinary two-torus whose orbifold singularities are not resolved by noncommutativity. In particular, the weak coupling limit of the gauge theory is independent of and computes the symplectic volume of the moduli space of constant curvature connections on the noncommutative torus.     

Complex Anti-Self-Dual Connections on a Product of Calabi–Yau Surfaces and Triholomorphic Curves

We study the adiabatic limit of a sequence of Ω-anti-self-dual connections on unitary bundles over a product of two compact Calabi–Yau surfaces M×N by scaling metrics to shrink N to a point. We show that after fixing gauge transformations, a subsequence of the N-components of these connections converges to a triholomorphic curve from M away from a Cayley cycle in M×N to the moduli space of instantons on M×N modulo gauge equivalence in the Hausdorff topology, and converges on the blow-up locus to a family, which is parameterized by the Cayley cycle, of triholomorphic curves from C ² to . Received: 22 May 1998 / Accepted: 26 August 1998     

Yang-Mills on Surfaces with Boundary: Quantum Theory and Symplectic Limit

The quantum field measure for gauge fields over a compact surface with boundary, with holonomy around the boundary components specified, is constructed. Loop expectation values for general loop configurations are computed. For a compact oriented surface with one boundary component, let be the moduli space of flat connections with boundary holonomy lying in a conjugacy class in the gauge group G. We prove that a certain natural closed 2-form on , introduced in an earlier work by C. King and the author, is a symplectic structure on the generic stratum of for generic . We then prove that the quantum Yang-Mills measure, with the boundary holonomy constrained to lie in , converges in a natural sense to the corresponding symplectic volume measure in the classical limit. We conclude with a detailed treatment of the case , and determine the symplectic volume of this moduli space. Received: 30 June 1996 / Accepted: 22 July 1996     

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     

The temperature dependences of the elastic moduli of solid C60

This paper reports on the determination of the temperature dependences of the complete set of the elastic moduli of solid C₆₀ from sound-velocity measurements made along different crystallographic directions in single-crystal samples within the 100-to 300-K range. Substantial differences in their behavior were revealed, which are accounted for by different relative contributions from relaxation processes to various elastic moduli.     

Moduli Space of BPS Walls in Supersymmetric Gauge Theories

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with N _F charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(N _C) gauge theory with N _F hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(N _C) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.