Benson Farb, Dan Margalit · Princeton University Press eBooks · 2017
This chapter focuses on the moduli space of Riemann surfaces. The moduli space parameterizes many different kinds of structures on <italic>S</italic>ɡ, such as isometry classes of hyperbolic structures on <italic>S</italic>, conformal classes of Riemannian metrics on <italic>S</italic>, biholomorphism classes of complex structures on <italic>S</italic>, and isomorphism classes of smooth algebraic curves homeomorphic to <italic>S</italic>. The chapter first considers the moduli space as the quotient of Teichmüller space before discussing the moduli space of the torus. It then examines the theorem (due to Fricke) that Mod(<italic>S</italic>) acts properly discontinuously on Teich(<italic>S</italic>), with a finite-index subgroup of Mod(<italic>S</italic>) acting freely such that M(<italic>S</italic>) is finitely covered by a smooth aspherical manifold. The chapter also looks at Mumford's compactness criterion, which describes what it means to go to infinity in M(<italic>S</italic>), and concludes by showing that M(<italic>S</italic>ɡ) is very close to being a classifying space for <italic>S</italic>ɡ-bundles.
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Oscar Zariski · University lecture series · 2006
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H. Lawson · Regional conference series in mathematics · 1985
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Clifford Henry · 2012
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Richard Evan Schwartz · Student mathematical library · 2011
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Lecture notes in physics · 2007
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Robert Penner · Contemporary mathematics - American Mathematical Society · 1988
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Luis Anchordoqui, Muldrow Etheredge, Dieter Lust · arXiv (Cornell University) · 2026
In this paper, continuing the discussion about Species Quantum Mechanics, we investigate quantum mechanics in moduli spaces using a mini-superspace approach. From this perspective, moduli-dependent functions can be viewed as operators, and we explore how the taxonomic relations from the Emergent String Conjecture can constrain the non-commutativity between these operators. Next, we study wave functions on moduli spaces, and we find that the geometry of moduli space plays an important role and leads to excited wave functions localised in the bulks of moduli spaces, and with positive energy eigenvalues. For cases when potentials are present, these effects result in moduli localised away from classical minima, and often result in excited, positive energy states.
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Luis Anchordoqui, Muldrow Etheredge, Dieter Lust · ArXiv.org · 2026
In this paper, continuing the discussion about Species Quantum Mechanics, we investigate quantum mechanics in moduli spaces using a mini-superspace approach. From this perspective, moduli-dependent functions can be viewed as operators, and we explore how the taxonomic relations from the Emergent String Conjecture can constrain the non-commutativity between these operators. Next, we study wave functions on moduli spaces, and we find that the geometry of moduli space plays an important role and leads to excited wave functions localised in the bulks of moduli spaces, and with positive energy eigenvalues. For cases when potentials are present, these effects result in moduli localised away from classical minima, and often result in excited, positive energy states.
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Wilderich Tuschmann, David J. Wraith · Oberwolfach Seminars · 2015
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Antón-Sancho, Álvaro · Figshare · 2024
We study the geometry of the moduli space of principal Spin bundles through the study of the subvarieties of fixed points of automorphisms of finite order coming from outer automorphisms of the structure group. We describe the nonstable and the singular locus of the moduli space, prove that nonstable bundles are singular points, and identify the fixed points, in the case of M(Spin(8;C)), in the stable or strictly polystable locus of the moduli space.<b>Published</b> by: Revista de la Unión Matemática Argentina, 57(2), 25-51. ISSN: 1669-9637, 0041-6932https://inmabb.criba.edu.ar/revuma/pdf/v57n2/v57n2a02.pdf<b>Attribution</b>:Antón-Sancho, A. (2016). The moduli space of principal Spin-bundles. Revista de la Unión Matemática Argentina, 57(2), 25-51. https://inmabb.criba.edu.ar/revuma/pdf/v57n2/v57n2a02.pdf<b>License</b>: https://creativecommons.org/licenses/by/4.0/
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Antón-Sancho, Álvaro · Figshare · 2024
We study the geometry of the moduli space of principal Spin bundles through the study of the subvarieties of fixed points of automorphisms of finite order coming from outer automorphisms of the structure group. We describe the nonstable and the singular locus of the moduli space, prove that nonstable bundles are singular points, and identify the fixed points, in the case of M(Spin(8;C)), in the stable or strictly polystable locus of the moduli space.<b>Published</b> by: Revista de la Unión Matemática Argentina, 57(2), 25-51. ISSN: 1669-9637, 0041-6932https://inmabb.criba.edu.ar/revuma/pdf/v57n2/v57n2a02.pdf<b>Attribution</b>:Antón-Sancho, A. (2016). The moduli space of principal Spin-bundles. Revista de la Unión Matemática Argentina, 57(2), 25-51. https://inmabb.criba.edu.ar/revuma/pdf/v57n2/v57n2a02.pdf<b>License</b>: https://creativecommons.org/licenses/by/4.0/
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N. J. Hitchin · 2008
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Stanislav Dubrovskiy · Annals of Global Analysis and Geometry · 2005
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Maloney, Alexander, Witten, Edward · arXiv (Cornell University) · 2020
Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT's to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain's family of two-dimensional CFT's obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like $U(1)^{2D}$ Chern-Simons theory than like Einstein gravity.
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2016
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Byung Chun Kim, Yongjin Song · arXiv (Cornell University) · 2018
A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $ϕ: B_k \rightarrow Γ_{g,b}$ is induced by 3-fold branched covering over a disk with some branch points. In this thesis we give a concrete description of this map and show that it is injective by Birman-Hilden theory. This gives us a new interesting non-geometric embedding of braid group into mapping class group. On the other hand, we show that the map on the level of classifying spaces of groups is compatible with the action of little 2-cube operad so that it induces a trivial homomorphim between stable homology group of braid groups and that of mapping class groups(Harer conjecture). We also show how the lift $\tilde{β_i}$ acts on the fundamental group of the surface and through this we prove that $\tilde{β_i}$ equals the product of two inverse Dehn twists.
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L. Chekhov · 1993
We study the algebraic geometrical background of the Penner–Kontsevich matrix model with the potential Nα tr (−1 2ΛXΛX + log(1 − X) + X). We show that this model describes intersection indices of linear bundles on the discretized moduli space right in the same fashion as the Kontsevich model is related to intersection indices (cohomological classes) on the Riemann surfaces of arbitrary genera. The special role of the logarithmic potential originated from the Penner matrix model is demonstrated. The boundary effects which was unessential in the case of the Kontsevich model are now relevant, and intersection indices on the discretized moduli space of genus g are expressed through Kontsevich’s indices of the genus g and of the lower genera.
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Well-separated Non-abelian Vortices, Toshiaki Fujimori, Giacomo Marmorini et al. · 2010
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Stanislav Dubrovskiy · ArXiv.org · 2001
The action of origin-preserving diffeomorphisms on a space of jets of symmetric connections is considered. Dimensions of moduli spaces of generic connections are calculated. Poincaré series of the geometric structure of symmetric connection is constructed, and shown to be a rational function.
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