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     

Differential characters and cohomology of the moduli of flat connections

Let \(\pi {:}\, P\rightarrow M\) be a principal bundle and p an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern–Simons differential characters is exploited to define a homology map \(\chi ^{k} {:}\, H_{2r-k-1}(M)\times H_{k}({\mathcal {F}}/{\mathcal {G}})\rightarrow {\mathbb {R}}/{\mathbb {Z}}\), for \(k<r-1\), where \({\mathcal {F}} /{\mathcal {G}}\) is the moduli space of flat connections of \(\pi \) under the action of a subgroup \({\mathcal {G}}\) of the gauge group. The differential characters of first order are related to the Dijkgraaf–Witten action for Chern–Simons theory. The second-order characters are interpreted geometrically as the holonomy of a connection in a line bundle over \({\mathcal {F}}/{\mathcal {G}}\). The relationship with other constructions in the literature is also analyzed.

     

The moduli space of flat SU (2) and SO (3) connections over surfaces

All the connected components of the moduli space of flat connections on SU (2) and SO (3) (trivial and non-trivial) bundles over closed oriented surfaces are determined. The symplectic structure and volumes of the non-maximal strata of the moduli space are also determined.     

Symplectic structure of the moduli space of flat connection on a Riemann surface

We consider the canonical symplectic structure on the moduli space of flatg-connections on a Riemann surface of genusg withn marked points. Forg being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum ofn copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie groupG ^* andg copies of the symplectic structure on the Heisenberg double of the Poisson-Lie groupG (the pair (G, G ^*) corresponds to the Lie algebrag).Supported by Swedish Natural Science Research Council (NFR) under the contract F-FU 06821-304Supported in part by a Soros Foundation Grant awarded by the American Physical Society     

Pre-quantization of the moduli space of flat G-bundles over a surface

For a simply connected, compact, simple Lie group G$G$, the moduli space of flat G$G$-bundles over a closed surface Σ$\Sigma$ is known to be pre-quantizable at integer levels. For non-simply connected G$G$, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction–namely a certain cohomology class in H³(G²;Z)$H^3(G^2;\mathbb{Z})$–that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G$G$.     

Trigonometrical sums connected with the chiral Potts model,Verlinde dimension formula,two-dimensional resistor network,and number theory

We have recently developed methods for obtaining exact two-point resistance of the complete graph minus N$N$ edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving Scott’s conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formulas, the dimension of the space of conformal blocks of SU(2)$SU\left(2\right)$ and SO(3)$SO\left(3\right)$ WZW models may be computed recursively. Our methods are then extended to compute the corner-to-corner resistance, and the Kirchhoff index of the first non-trivial two-dimensional resistor network, 2×N$2\times N$. Finally, we obtain new closed formulas for variant of trigonometrical sums, some of which appear in connection with number theory.     

1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators

Let (P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let D_A be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A(u) of A with tangent vector ω and denote T_ω:= (D_A(u)−f) (u=0. The coefficients of the asymptotic expansion of trace (T_ω · e^-t(D_A−f)2) near t=0 define 1-forms a^(k)_f, K=0, 1, 2, … on (P). In this paper we calculate aa⁽⁰⁾_f, a⁽¹⁾_f, a⁽²⁾_f and study some of their properties. For instance using the 1-form a⁽²⁾_f for suitable functions f we obtain a foliation of codimension 5 of the space of G-instantons of S⁴.     

Entropy bounds and Cardy–Verlinde formula in Yang–Mills theory

Using gauge formulation of gravity the three-dimensional SU(2) YM theory equations of motion are presented in equivalent form as FRW cosmological equations. With the radiation, the particular (periodic, big bang – big crunch) three-dimensional universe is constructed. Cosmological entropy bounds (so-called Cardy–Verlinde formula) have the standard form in such universe. Mapping such universe back to YM formulation we got the thermal solution of YM theory. The corresponding holographic entropy bounds (Cardy–Verlinde formula) in YM theory are constructed. This indicates to universal character of holographic relations.     

Simple type and the boundary of moduli space

We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the “simple type” condition of Donaldson theory. Using the natural geometric representative of μ(pt) defined in [L. Sadun, Commun. Math. Phys. 178 (1996) 107–113], the boundary region of moduli space contributes of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to the interior of the (k+1)th ASD moduli space, intersected with two representatives of (4 times) the point class, being homologous to 58 copies of the kth moduli space. This is peculiar, since the only known embeddings of the kth moduli space into the (k+1)th involve Taubes gluing, and the images of such embeddings lie entirely in the boundary region.When using the natural de Rham representatives of μ(pt) considered by Witten [Commun. Math. Phys. 117 (1988) 353], the boundary region contributes of what is needed for simple type, again regardless of the topology or geometry of the underlying 4-manifold. The difference between this and the geometric representative answer is not contradictory, as the contribution of a fixed region to the Donaldson invariants is geometric, not topological.     

One-vortex moduli space and Ricci flow

The metric on the moduli space of one abelian Higgs vortex on a surface has a natural geometrical evolution as the Bradlow parameter, which determines the vortex size, varies. It is shown by various arguments, and by calculations in special cases, that this geometrical flow has many similarities to Ricci flow.     

Black holes and critical points in moduli space

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. This is a property of a regular special geometry. We also study the critical points in all N 2 supersymmetric theories. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for the study of critical phenomena.     

Fermionic strings on the compactified moduli space

On the compactified moduli space we consider theN=2,N=4 local supersymmetric string theories. It would be proven that theN=2,N=4 fermionic string theories might not develop any tachyon pole, which might imply theg-loop partition functions forN=2,N=4 fermionic string would be finite.     

The dimension formula for the Lorenz attractor

An analytical formula for the Lyapunov dimension of the Lorenz attractor is presented under assumption that all the equilibria are unstable.     

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