Moduli Space Of Holomorphic Bundles Research Articles

On real moduli spaces of holomorphic bundles over M-curves

Let F be a genus g curve and σ : F → F a real structure with the maximal possible number of fixed circles. We study the real moduli space N ′ = Fix ( σ # ) where σ # : N → N is the induced real structure on the moduli space N of stable holomorphic bundles of rank 2 over F with fixed non-trivial determinant. In particular, we calculate H ⁎ ( N ′ , Z ) in the case of g = 2 , generalizing Thaddeus' approach to computing H ⁎ ( N , Z ) .

The Chow ring of the moduli space of bundles on $\mathbb{P}^2$ with charge 1

For an algebraically closed field $K$ with $\mathrm{ch}(K)\neq 2$, let $\mathcal{O}M(1, SO(n,K))$ denote the moduli space of holomorphic bundles on $\mathbb{P}^{2}$ with the structure group $SO(n,K)$ and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line $l_{\infty}$ and has a fixed holomorphic trivialization there. In this paper we determine the Chow ring of $\mathcal{O}M(1, SO(n,K))$.

Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.

Intersection Theory on Moduli Spaces of Holomorphic Bundles of Arbitrary Rank on a Riemann Surface

Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann surface E. This moduli space is a compact Kifhler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri [41]). The subject of this article is the characterization of the intersection pairings in the cohomology ring1 H* (M (n, d)). A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper [2] on the YangMills equations on Riemann surfaces (where in addition inductive formulas for the Betti numbers of M (n, d) obtained earlier using number-theoretic methods [13], [25] were rederived). By Poincare duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson [15] and Thaddeus [47] gave formulas for the intersection pairings between products of these generators in H* (M (2, 1)) (in terms of Bernoulli numbers). Then using physical methods, Witten [50] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in H* (M (n, d)) for general rank n. These generalized his (rigorously proved) formulas [49] for the symplectic volume of M (n, d): For instance, the symplectic volume of M (2, 1) is given by

Chern-Simons and twisted supersymmetry in various dimensions

We introduce special supersymmetric gauge theories in three, five, seven and nine dimensions, whose compactification on two-, four-, six- and eight folds produces a supersymmetric quantum mechanics on moduli spaces of holomorphic bundles and/or solutions to the analogues of instanton equations in higher dimensions. The theories may occur on the world-volumes of D-branes wrapping manifolds of a special holonomy. We also discuss the theories with matter.

Rank Two Bundles on the Blow-up ofC2

In this paper we study holomorphic rank two vector bundles on the blow-up ofC2with vanishing Chern class. The restriction of such a bundle over the exceptional divisor splits as O(j)⊕O(−j) for some integerj. We denote by Mjthe moduli space of holomorphic bundles on the blow-up ofC2whose restriction to the exceptional divisor is O(j)⊕O(−j). We prove that Mjis generically a complex projective space of dimension 2j−3.

ON AN ANALYTIC PROOF OF A RESULT BY DONALDSON

We give an analytic proof of a result by Donaldson which asserts that there is a one to one correspondence between the moduli space of framed instantons on S4 and the moduli space of holomorphic bundles over CP2 trivialized along a line.

Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface

are coprime. This space had long been studied by algebraicgeometers (see for instance Narasimhan and Seshadri 1965 [22]), but a new view-point on it was revealed by the seminal 1982 paper [1] of Atiyah and Bott on theYang-Mills equations on Riemann surfaces. In this paper a set of generators forthe (rational) cohomology ring of

Global aspects of gauged Wess-Zumino-Witten models

A study of the gauged Wess-Zumino-Witten models is given focusing on the effect of topologically non-trivial configurations of gauge fields. A correlation function is expressed as an integral over a moduli space of holomorphic bundles with quasi-parabolic structure. Two actions of the fundamental group of the gauge group is defined: One on the space of gauge invariant local fields and the other on the moduli spaces. Applying these in the integral expression, we obtain a certain identity which relates correlation functions for configurations of different topologies. It gives an important information on the topological sum for the partition and correlation functions.

Two-dimensional topological gravity and intersection theory on the moduli space of holomorphic bundles

We define a two-dimensional topological Yang-Mills theory for an arbitrary compact simple Lie group. This theory is defined in terms of intersection theory on the moduli space of flat connections on a two-dimensional surface and corresponds physically to a two-dimensional reduction and truncation of four-dimensional topological Yang-Mills theory. Two-dimensional topological Yang-Mills theory defines a topological matter system and may be naturally coupled to two-dimensional topological gravity. This topological Yang-Mills theory is also closely related to Chern-Simons gauge theory in 2 + 1 dimensions. We also discuss a relation between SL (2, R ) Chern-Simons theory and two-dimensional topological gravity.