scholarly journals Metaplectic quantization of the moduli spaces of flat and parabolic bundles.

1995 ◽  
Vol 1995 (466) ◽  
pp. 145-219
Keyword(s):  
Moduli Spaces ◽  
Parabolic Bundles

scholarly journals Logarithmic Connections, WZNW Action, and Moduli of Parabolic Bundles on the Sphere

2021 ◽  
Author(s):  
Claudio Meneses ◽  
Leon A. Takhtajan
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Riemann Sphere ◽  
Parabolic Bundle ◽  
Parabolic Bundles ◽  
Logarithmic Connection ◽  
Definition Of ◽  
Suitable Notion ◽  
Marked Points ◽  
Explicit Relation

AbstractModuli spaces of stable parabolic bundles of parabolic degree 0 over the Riemann sphere are stratified according to the Harder–Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $$\mathscr {N}_{0}$$ N 0 of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $$\mathscr {S}$$ S is defined as the regularized critical value of the non-compact Wess–Zumino–Novikov–Witten action functional. The definition of $$\mathscr {S}$$ S depends on a suitable notion of parabolic bundle ‘uniformization map’ following from the Mehta–Seshadri and Birkhoff–Grothendieck theorems. It is shown that $$-\mathscr {S}$$ - S is a primitive for a (1,0)-form $$\vartheta $$ ϑ on $$\mathscr {N}_{0}$$ N 0 associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $$-\mathscr {S}$$ - S is a Kähler potential for $$(\Omega -\Omega _{\mathrm {T}})|_{\mathscr {N}_{0}}$$ ( Ω - Ω T ) | N 0 , where $$\Omega $$ Ω is the Narasimhan–Atiyah–Bott Kähler form in $$\mathscr {N}$$ N and $$\Omega _{\mathrm {T}}$$ Ω T is a certain linear combination of tautological (1, 1)-forms associated with the marked points. These results provide an explicit relation between the cohomology class $$[\Omega ]$$ [ Ω ] and tautological classes, which holds globally over certain open chambers of parabolic weights where $$\mathscr {N}_{0} = \mathscr {N}$$ N 0 = N .

scholarly journals Rationality of Moduli Spaces of Parabolic Bundles

1999 ◽  
Vol 59 (2) ◽  
pp. 461-478 ◽  
Author(s):  
Hans U. Boden ◽  
Kôji Yokogawa
Keyword(s):  
Moduli Spaces ◽  
Parabolic Bundles

The symplectic structure for renormalization of circle diffeomorphisms with breaks

2021 ◽  
pp. 2150016
Author(s):  
S. Ghazouani ◽  
K. Khanin
Keyword(s):  
Moduli Spaces ◽  
Symplectic Form ◽  
Symplectic Structure ◽  
Mapping Class ◽  
Character Variety ◽  
Parabolic Bundles ◽  
Dimension Formula ◽  
Renormalization Operator ◽  
Symplectic Dynamics ◽  
Circle Diffeomorphisms

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.

scholarly journals Integrable Systems and Torelli Theorems for the Moduli Spaces of Parabolic Bundles and Parabolic Higgs Bundles

2016 ◽  
Vol 68 (3) ◽  
pp. 504-520
Author(s):  
Indranil Biswas ◽  
Tomás L. Gómez ◽  
Marina Logares
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Integrable Systems ◽  
Algebraic Curve ◽  
Higgs Bundles ◽  
Parabolic Bundles ◽  
Smooth Complex ◽  
Torelli Theorem ◽  
Complex Projective

AbstractWe prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.

scholarly journals GRASSMANNIAN-FRAMED BUNDLES AND GENERALIZED PARABOLIC STRUCTURES

2013 ◽  
Vol 24 (12) ◽  
pp. 1350090 ◽  
Author(s):  
USHA BHOSLE ◽  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE
Keyword(s):  
Riemann Surface ◽  
Moduli Spaces ◽  
Parabolic Bundles ◽  
Universal Spaces ◽  
Equivariant Compactification

We build compact moduli spaces of Grassmannian-framed bundles over a Riemann surface, essentially replacing a group by a bi-equivariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin–Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles (in the sense that all of the moduli can be obtained as quotients), and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.

Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces

2000 ◽  
Vol 43 (2) ◽  
pp. 174-182 ◽  
Author(s):  
Christian Gantz ◽  
Brian Steer
Keyword(s):  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Elliptic Surfaces ◽  
Parabolic Bundles

AbstractWe show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

scholarly journals MODULI SPACES OF PARABOLIC HIGGS BUNDLES AND PARABOLIC K(D) PAIRS OVER SMOOTH CURVES: I

1996 ◽  
Vol 07 (05) ◽  
pp. 573-598 ◽  
Author(s):  
HANS U. BODEN ◽  
KÔJI YOKOGAWA
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Morse Theory ◽  
Smooth Curve ◽  
Higgs Bundles ◽  
Poincaré Polynomial ◽  
Smooth Curves ◽  
Connected Manifold ◽  
Parabolic Bundles ◽  
Simply Connected

This paper concerns the moduli spaces of rank-two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a non-compact, connected, simply connected manifold, and a computation of its Poincaré polynomial is given.

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