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 The mapping class group action on -character varieties

2020 ◽  
pp. 1-15
Author(s):  
WILLIAM M. GOLDMAN ◽  
SEAN LAWTON ◽  
EUGENE Z. XIA
Keyword(s):  
Group Action ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Character Variety ◽  
Character Varieties ◽  
Compact Orientable Surface

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

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 Geometric prequantization on the path space of a prequantized manifold

2015 ◽  
Vol 12 (03) ◽  
pp. 1550030
Author(s):  
Indranil Biswas ◽  
Saikat Chatterjee ◽  
Rukmini Dey
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Form ◽  
Symplectic Structure ◽  
Gordon Equation ◽  
Solution Space ◽  
Path Space ◽  
Klein Gordon ◽  
Class Of Functions ◽  
Compact Symplectic Manifold ◽  
Klein Gordon Equation

Given a compact symplectic manifold M, with integral symplectic form, we prequantize a certain class of functions on the path space for M. The functions in question are induced by functions on M. We apply our construction to study the symplectic structure on the solution space of Klein–Gordon equation.

scholarly journals Picard Groups of Moduli Spaces of Curves with Symmetry

2018 ◽  
Vol 2020 (23) ◽  
pp. 9293-9335
Author(s):  
Kevin Kordek
Keyword(s):  
Moduli Spaces ◽  
Topological Type ◽  
Hyperelliptic Curves ◽  
Mapping Class ◽  
Finitely Generated ◽  
Smooth Curves ◽  
Group Of Automorphisms ◽  
Picard Groups ◽  
Moduli Spaces Of Curves ◽  
Special Cases

Abstract We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the 1st part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the 2nd part of the paper, we show that finite abelian level covers of the hyperelliptic locus in the moduli space of smooth curves have finitely generated Picard groups. We also compute the Picard groups of the moduli spaces of hyperelliptic curves of compact type.

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
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