Moduli Space Of Parabolic Bundles Research Articles

Chen–Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface

Let \((X,\,D)\) be an m-pointed compact Riemann surface of genus at least 2. For each \(x \,\in \, D\), fix full flag and concentrated weight system \(\alpha \). Let \(P \mathcal {M}_{\xi }\) denote the moduli space of semi-stable parabolic vector bundles of rank r and determinant \(\xi \) over X with weight system \(\alpha \), where r is a prime number and \(\xi \) is a holomorphic line bundle over X of degree d which is not a multiple of r. We compute the Chen–Ruan cohomology of the orbifold for the action on \(P \mathcal {M}_{\xi }\) of the group of r-torsion points in \(\mathrm{Pic}^0(X)\).

Automorphism group of the moduli space of parabolic bundles over a curve

We find the automorphism group of the moduli space of parabolic bundles on a smooth curve (with fixed determinant and system of weights). This group is generated by: automorphisms of the marked curve, tensoring with a line bundle, taking the dual, and Hecke transforms (using the filtrations given by the parabolic structure). A Torelli theorem for parabolic bundles with arbitrary rank and generic weights is also obtained. These results are extended to the classification of birational equivalences which are defined over “big” open subsets (3-birational maps, i.e. birational maps giving an isomorphism between open subsets with complement of codimension at least 3).Finally, an analysis of the stability chambers for the parabolic weights is performed in order to determine precisely when two moduli spaces of parabolic vector bundles with different parameters (curve, rank, determinant and weights) can be isomorphic.

The symplectic structure for renormalization of circle diffeomorphisms with breaks

The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of [Formula: see text] circle diffeomorphisms with [Formula: see text] breaks, [Formula: see text], with given size of breaks, converge to an invariant family of piecewise Möbius maps, of dimension [Formula: see text]. We prove that this invariant family identifies with a relative character variety [Formula: see text] where [Formula: see text] is a [Formula: see text]-holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group [Formula: see text]. This action allows us to introduce the symplectic form which is preserved by renormalization. The invariant symplectic form is related to the symplectic form described by Guruprasad et al. [Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89(2) (1997) 377–412], and goes back to the earlier work by Goldman [The symplectic nature of fundamental groups of surfaces, Adv. Math. 54(2) (1984) 200–225]. To the best of our knowledge the connection between renormalization in the nonlinear setting and symplectic dynamics had not been brought to light yet.

Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry

Abstract We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

The Witten–Reshetikhin–Turaev invariant for links in finite order mapping tori I

We state Asymptotic Expansion and Growth Rate conjectures for the Witten–Reshetikhin–Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten–Reshetikhin–Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.

Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve

We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead–Ramanan conjecture for stable bundles.

Goldman Systems and Bending Systems

AbstractWe show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and as symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.

Torelli theorems for moduli of logarithmic connections and parabolic bundles

Let Z be a finite subset of a compact connected Riemann Surface X. Let \({\fancyscript{M}_X^{lc}}\) denote the moduli space of pairs (L, D) where L is a line bundle on X and D is a logarithmic connection on L singular along Z. Then \({\fancyscript{M}_X^{lc}}\) has a natural symplectic structure [ωX]. We show that the pair \({(\fancyscript{M}_X^{lc},[\omega_X])}\) determines X and there are no nonconstant algebraic functions on \({\fancyscript{M}_X^{lc}}\). We also prove a Torelli type theorem for the moduli space of parabolic bundles.

Differential forms on moduli spaces of parabolic bundles

Differential forms on moduli spaces of parabolic bundles

The ‘Harder-Narasimhan trace’ and unitarity of the KZ/Hitchin connection: genus 0

Let a reductive group G act on a projective variety X+, and suppose given a lift of the action to an ample line bundle 0. By definition, all G-invariant sections of 6 vanish on the nonsemistable locus X+ss. Taking an appropriate normal derivative defines a map H°(X+,0)G - ► ^(S^V^)0, where VM is a G- vector bundle on a G- variety *SM. We call this the Harder-Narasimhan trace. Applying this to the Geometric Invariant Theory construction of the moduli space of parabolic bundles on a curve, we discover generalisations of Coulomb-gas representations , which map conformal blocks to hypergeometric local systems. In this paper we prove the unitarity of the KZ/Hitchin connection (in the genus zero, rank two, case) by proving that the above map lands in a unitary factor of the hypergeometric system. (An ingredient in the proof is a lower bound on the degree of polynomials vanishing on partial diagonals.) This elucidates the work of K. Gawedzki.

Group-valued implosion and parabolic structures

The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasi-Hamiltonian K -manifolds, where K is a simply connected compact Lie group. The imploded cross-section of the double K × K turns out to be universal in a suitable sense. It is a singular space, but some of its strata have a nonsingular closure. This observation leads to interesting new examples of quasi-Hamiltonian K -manifolds, such as the "spinning 2 n -sphere" for K = SU ( n ). Secondly we construct a universal ("master") moduli space of parabolic bundles with structure group K over a marked Riemann surface. The master moduli space carries a natural action of a maximal torus of K and a torus-invariant stratification into manifolds, each of which has a symplectic structure. An essential ingredient in the construction is the universal implosion. Paradoxically, although the universal implosion has no complex structure (it is the four-sphere for K = SU (2)), the master moduli space turns out to be a complex algebraic variety.

Quotients by Reductive Group, Borel Subgroup, Unipotent Group and Maximal Torus

Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal torus, respectively. Our main result is that there are natural correspondences among quotients of these three actions. In the end, we mention a possible application to the moduli spaces of parabolic bundles over algebraic curves for further research. —Dedicated to Robert MacPherson on the occasion of his 60th birthday

Moduli of Framed Parabolic Sheaves

This paper gives a construction of the moduli space of framed parabolic sheaves on a Riemann surface. This space serves as a universal, master, space for the well known moduli space of parabolic bundles, as well as moduli spaces of vector bundles, which can all be obtained from this space by torus quotients. The construction is given for the structure group SL(N, C), and indeed is adapted to this case. At the end of the paper, an approach is suggested for dealing with the case of arbitrary reductive groups, involving the associated loop group.

Coherent sheaves with parabolic structure and construction of Hecke eigensheaves for some ramified local systems

The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves corresponding to local systems on a smooth, projective curve C to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “ On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11] carry over to our situation. We show that our sheaves descend to the moduli space of parabolic bundles if the rank is ≤3 and that the general case can be deduced form a generalization of the vanishing conjecture of [11]

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.

A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over \(\mathbb{C}\) and let L be a line bundle on X. In this paper we shall construct a moduli space \(\mathcal{P}(L)\) parametrizing \((n-1)\)-cohomology L-twisted Higgs pairs, i.e., pairs \((E,\bar \phi)\) where E is a vector bundle on X and \(\bar \phi \in H^{n-1}(X,\mathcal{E}\kern-0.6em\hbox{\it nd}(E) \otimes L)\). If we take \(L =\omega_X\), the canonical line bundle on X, the variety \(\mathcal{P}(L)\) is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section \(s \in H^0(X,\omega_X^{-1} \otimes L)\), one can define, in a natural way, a Poisson structure \(\theta_s\) on \(\mathcal{P}(L)\). We also analyze the relations between this Poisson structure on \(\mathcal{P}(L)\) and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves.

Rationality of Moduli Spaces of Parabolic Bundles

The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows that if rank and degree are coprime, the moduli space of vector bundles is stably rational, and the bound obtained on the level is strong enough to conclude rationality in many cases.

Group systems, groupoids, and moduli spaces of parabolic bundles

Moduli spaces of homomorphisms or, more generally, twisted homomorphisms from fundamental groups of surfaces to compact connected Lie groups, were connected with geometry through their identification with moduli spaces of holomorphic vector bundles (see [29]). Atiyah and Bott [2] initiated a new approach to the study of these moduli spaces by identifying them with moduli spaces of projectively fiat constant central curvature connections on principal bundles over Riemann surfaces, which they analyzed by methods of gauge theory. In particular, they showed that an invariant inner product on the Lie algebra of the Lie group in question induces a natural symplectic structure on a certain smooth open stratum. Although this moduli space is a finite-dimensional object, generally a stratified space which is locally semialgebraic [19] but sometimes a manifold, its symplectic structure (on the stratum just mentioned) was obtained by applying the method of symplectic reduction to the action of an infinite-dimensional group (the group of gauge transformations) on an infinite-dimensional symplectic manifold (the space of all connections on a principal bundle).

Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections

We construct a family of polarizations of the moduli space of flatSU(n)-connections on a closed 2-manifold of genusg(≧ 2). These are generalizations of various polarizations known until now. That is, our family of polarizations includes Weitsman's real polarizations in the case ofn=2 [17], as well as the Kahler polarizations which are well known since [2] and [18]. Our construction is based on an original formulation of degeneration of Riemann surfaces. The relation between our polarizations and the complex structures of the moduli spaces of parabolic bundles are also studied.

On the mapping class group action on the cohomology of the representation space of a surface

The mapping class group of a d-pointed Riemann surface has a natural C ∞ action on any moduli space of parabolic bundles with the marked points as the parabolic points. We prove that under some numerical conditions on the parabolic data, the induced action of the mapping class group on the cohomology algebra of the moduli space of parabolic bundles factors through the natural epimorphism of the mapping class group onto the symplectic group.