scholarly journals THE VERLINDE FORMULA FOR PARABOLIC BUNDLES

2001 ◽  
Vol 63 (3) ◽  
pp. 754-768 ◽  
Author(s):  
LISA C. JEFFREY
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Conjugacy Class ◽  
Fundamental Group ◽  
Moduli Space ◽  
Boundary Component ◽  
Compact Riemann Surface ◽  
Central Element ◽  
Parabolic Bundles ◽  
Verlinde Formula

Let Σg be a compact Riemann surface of genus g, and G = SU(n). The central element c = diag(e2πid/n, …, e2πid/n) for d coprime to n is introduced. The Verlinde formula is proved for the Riemann–Roch number of a line bundle over the moduli space [Mscr ]g, 1(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component, for which the loop around the boundary is constrained to lie in the conjugacy class of cexp(Λ) (for Λ ∈ t+), and also for the moduli space [Mscr ]g, b(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with s + 1 boundary components for which the loop around the 0th boundary component is sent to the central element c and the loop around the jth boundary component is constrained to lie in the conjugacy class of exp(Λ(j)) for Λ(j) ∈ t+. The proof is valid for Λ(j) in suitable neighbourhoods of 0.

scholarly journals On moduli spaces of Hitchin pairs

2011 ◽  
Vol 151 (3) ◽  
pp. 441-457 ◽  
Author(s):  
INDRANIL BISWAS ◽  
PETER B. GOTHEN ◽  
MARINA LOGARES
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Isomorphism Class ◽  
Compact Riemann Surface ◽  
List Type ◽  
Holomorphic Line Bundle ◽  
Type Number ◽  
Additional Hypothesis

AbstractLetXbe a compact Riemann surfaceXof genus at–least two. Fix a holomorphic line bundleLoverX. Letbe the moduli space of Hitchin pairs (E, φ ∈H0(End0(E) ⊗L)) overXof rankrand fixed determinant of degreed. The following conditions are imposed:(i)deg(L) ≥ 2g−2,r≥ 2, andL⊗rKX⊗r;(ii)(r, d) = 1; and(iii)ifg= 2 thenr≥ 6, and ifg= 3 thenr≥ 4.We prove that that the isomorphism class of the varietyuniquely determines the isomorphism class of the Riemann surfaceX. Moreover, our analysis shows thatis irreducible (this result holds without the additional hypothesis on the rank for low genus).

scholarly journals The dimension of the Hilbert space of geometric quantization of vortices on a Riemann surface

2017 ◽  
Vol 14 (10) ◽  
pp. 1750144
Author(s):  
Rukmini Dey ◽  
Saibal Ganguli
Keyword(s):  
Hilbert Space ◽  
Riemann Surface ◽  
Line Bundle ◽  
Euler Characteristic ◽  
Moduli Space ◽  
Geometric Quantization

In this paper, we calculate the dimension of the Hilbert space of Kähler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.

scholarly journals Stability of the parabolic Poincaré bundle

2018 ◽  
Vol 15 (05) ◽  
pp. 1850081
Author(s):  
Suratno Basu ◽  
Indranil Biswas ◽  
Krishanu Dan
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Compact Riemann Surface ◽  
Stable Bundles ◽  
Parabolic Bundle ◽  
Natural Polarization

Given a compact Riemann surface [Formula: see text] and a moduli space [Formula: see text] of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincaré parabolic bundle on [Formula: see text] is parabolic stable with respect to a natural polarization on [Formula: see text].

scholarly journals CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

2010 ◽  
Vol 21 (04) ◽  
pp. 497-522 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAINAK PODDAR
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Prime Number ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Line Bundles ◽  
Holomorphic Line Bundle ◽  
Stable Vector Bundles

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

PARABOLIC VECTOR BUNDLES AND HERMITIAN-YANG-MILLS CONNECTIONS OVER A RIEMANN SURFACE

1993 ◽  
Vol 04 (03) ◽  
pp. 467-501 ◽  
Author(s):  
JONATHAN A. PORITZ
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Sobolev Space ◽  
Moduli Space ◽  
Vector Bundles ◽  
Weighted Sobolev Space ◽  
Conjugacy Classes ◽  
Stable Bundle ◽  
Yang Mills ◽  
Parabolic Bundles

We study a certain moduli space of irreducible Hermitian-Yang-Mills connections on a unitary vector bundle over a punctured Riemann surface. The connections used have non-trivial holonomy around the punctures lying in fixed conjugacy classes of U (n) and differ from each other by elements of a weighted Sobolev space; these connections give rise to parabolic bundles in the sense of Mehta and Seshadri. We show in fact that the moduli space of stable parabolic bundles can be identified with our moduli space of HYM connections, by proving that every stable bundle admits a unique unitary gauge orbit of Hermitian-Yang-Mills connections.

scholarly journals INTERSECTION COHOMOLOGY OF REPRESENTATION SPACES OF SURFACE GROUPS

2006 ◽  
Vol 17 (02) ◽  
pp. 169-182
Author(s):  
YOUNG-HOON KIEM
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Moduli Space ◽  
Local Structure ◽  
Equivariant Cohomology ◽  
Intersection Cohomology ◽  
Symplectic Reduction ◽  
Surface Groups ◽  
Representation Space ◽  
Splitting Theorem

The representation space X(G) = Hom (π, G)/G of the fundamental group π of a Riemann surface Σ of genus g ≥ 2 is the symplectic reduction of the extended moduli space defined in [6]. Using this description, we study the local structure of X(G) and show that the assumptions of the splitting theorem [11, Theorem 7.7] are satisfied. Hence the middle perversity intersection cohomology is canonically isomorphic to a subspace of the equivariant cohomology [Formula: see text] which can be computed quite explicitly. The case when G = SU(2) is discussed in detail.

scholarly journals A note on the theta characteristics of a compact Riemann surface

2004 ◽  
Vol 76 (3) ◽  
pp. 415-424
Author(s):  
Indranil Biswas
Keyword(s):  
Riemann Surface ◽  
Quadratic Form ◽  
Line Bundle ◽  
Vector Fields ◽  
Compact Riemann Surface ◽  
Line Bundles ◽  
Square Root ◽  
Topological Interpretation ◽  
Theta Characteristics

AbstractLet X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.

scholarly journals Moduli space of branched superminimal immersions of a compact Riemann surface into S4

1999 ◽  
Vol 66 (1) ◽  
pp. 32-50 ◽  
Author(s):  
Bonaventure Loo
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Compact Riemann Surface ◽  
Projective Varieties ◽  
Fibre Products ◽  
Dimension 2 ◽  
Path Connected

AbstractIn this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max {2g, g + 2}, such spaces have the structure of projectivzed fibre products and are path-connected quasi projective varieties of dimension 2d − g + 4. This generalizes known results for spaces of harmonic 2-spheres in S4.

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