Richard Wentworth

Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as “Gromov invariants”) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one. Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 E-mail address: bertram@math.utah.edu Department of Mathematics, Brown University, Providence, Rhode Island 02912 E-mail address: daskal@gauss.math.brown.edu Department of Mathematics, University of California, Irvine, California 92717 E-mail address: raw@math.uci.edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Classification of Weil-Petersson isometries

This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmüller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the Thurston classification. The second result concerns the asymptotic behavior of these geodesics. It is shown that geodesics that are equivariant with respect to independent pseudo-Anosovs diverge. It follows that subgroups of the mapping class group which contain independent pseudo-Anosovs act in a reductive manner with respect to the Weil-Petersson geometry. This implies an existence theorem for equivariant harmonic maps to the metric completion.

BIRATIONAL EQUIVALENCES OF VORTEX MODULI

Abstract We construct a finite-dimensional Kahler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the τ-vortex moduli spaces (or τ-stable pairs). The Morse theory of the circle action induces natural birational maps between the reduced spaces for different values of τ which in the case of rank two bundles can be canonically resolved in a sequence of blow-ups and blow-downs.

On the Morgan-Shalen compactification of the SL(2,ℂ) character varieties of surface groups

A gauge theoretic description of the Morgan-Shalen compactification of the

Higgs bundles and local systems on Riemann surfaces

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The integrability criterion in SU(2) Chern-Simons gauge theory

character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the Hitchin map.

Asymptotics of determinants from functional integration

These notes are based on lectures given at the Third International School on Geometry and Physics at the Centre de Recerca Matematica in Barcelona, March 26–30, 2012. The aim of the School’s four lecture series was to give a rapid introduction to Higgs bundles, representation varieties, and mathematical physics. While the scope of these subjects is very broad, that of these notes is far more modest.

Cohomology of U(2,1) representation varieties of surface groups

We prove that the multiplicity spaces appearing in Chern-Simons theory, as defioned by Segal, vanish unless they are associated to integrable representations. This and other links with conformal field theory are examined.

Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles

The expression for the determinant of the Laplace operator is used in terms of functional integration to compute the asymptotic behavior on degenerating Riemann surfaces. In the case of the Arakelov metric, the information is sufficiently precise to give a value for the absolute constants appearing in bosonization formulas.

Families of SU(2) representations for mapping cylinders of periodic monodromy

In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topology of the U(2,1) and SU(2,1) character varieties of surface groups. The main results are a calculation of the equivariant Poincare polynomials, a Kirwan surjectivity theorem in the non-fixed determinant case, and a description of the action of the Torelli group on the equivariant cohomology of the character variety. This builds on earlier work for stable pairs and rank 2 Higgs bundles.