scholarly journals Differential forms on moduli spaces of parabolic bundles

2010 ◽  
Vol 40 (6) ◽  
pp. 1779-1795
Author(s):  
Francesco Bottacin
Keyword(s):  
Moduli Spaces ◽  
Differential Forms ◽  
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 Counting curves on surfaces

2017 ◽  
Vol 28 (02) ◽  
pp. 1750012
Author(s):  
Norman Do ◽  
Musashi A. Koyama ◽  
Daniel V. Mathews
Keyword(s):  
Moduli Spaces ◽  
Symplectic Geometry ◽  
Differential Forms ◽  
Combinatorial Problem ◽  
Partition Functions ◽  
Low Dimensional Topology ◽  
Moduli Spaces Of Curves ◽  
Finite Set ◽  
Curves On Surfaces ◽  
Low Dimensional

We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface [Formula: see text], with a finite set of points [Formula: see text] fixed on its boundary, how many configurations of disjoint arcs are there on [Formula: see text] whose boundary is [Formula: see text]? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions and quantum curves.

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

scholarly journals Single-valued integration and double copy

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Francis Brown ◽  
Clément Dupont
Keyword(s):  
Moduli Spaces ◽  
Modular Forms ◽  
Differential Forms ◽  
Smooth Projective Variety ◽  
Genus Zero ◽  
Moduli Spaces Of Curves ◽  
Multiple Zeta ◽  
De Rham ◽  
Zeta Values ◽  
Double Copy

AbstractIn this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary “double copy” formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves {\mathcal{M}_{0,n}} of genus zero with n marked points. We also study a morphism between certain rings of “motivic” periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest.

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 The algebra of cell-zeta values

2010 ◽  
Vol 146 (3) ◽  
pp. 731-771 ◽  
Author(s):  
Francis Brown ◽  
Sarah Carr ◽  
Leila Schneps
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Cohomology Group ◽  
Differential Forms ◽  
Natural Duality ◽  
Connected Component ◽  
Real Numbers ◽  
Zeta Values ◽  
Definition Of ◽  
Formal Version

AbstractIn this paper, we introduce cell-forms on 𝔐0,n, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space 𝔐0,n(ℝ). We show that the cell-forms generate the top-dimensional cohomology group of 𝔐0,n, so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell X. The elements of this basis are called insertion forms; their integrals over X are real numbers, called cell-zeta values, which generate a ℚ-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations.

scholarly journals MODULI SPACES OF TOPOLOGICAL CALIBRATIONS, CALABI–YAU, HYPERKÄHLER, G2 and SPIN(7) STRUCTURES

2004 ◽  
Vol 15 (03) ◽  
pp. 211-257 ◽  
Author(s):  
RYUSHI GOTO
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Finite Dimension ◽  
Type Theorem ◽  
Differential Forms ◽  
Direct Proof ◽  
Stability Theorem ◽  
Point Of View ◽  
Geometric Structures ◽  
New Approach

This paper focuses on a geometric structure defined by a system of closed exterior differential forms and develops a new approach to deformation problems of geometric structures. We obtain a criterion for unobstructed deformations from a cohomological point of view (Theorem 1.7). Further we show that under a cohomological condition, the moduli space of the geometric structures becomes a smooth manifold of finite dimension (Theorem 1.8). We apply our approach to the geometric structures such as Calabi–Yau, HyperKähler, G2 and Spin(7) structures and then obtain a unified construction of smooth moduli spaces of these four geometric structures. We generalize the Moser's stability theorem to provide a direct proof of the local Torelli type theorem in these four geometric structures (Theorem 1.10).

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.

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