scholarly journals Uniformization of p-adic curves via Higgs–de Rham flows

2019 ◽  
Vol 2019 (747) ◽  
pp. 63-108
Author(s):  
Guitang Lan ◽  
Mao Sheng ◽  
Yanhong Yang ◽  
Kang Zuo
Keyword(s):  
Riemann Surfaces ◽  
Algebraic Closure ◽  
Higgs Field ◽  
Close Relation ◽  
Higgs Bundle ◽  
Generic Fiber ◽  
Hyperbolic Curve ◽  
De Rham ◽  
Associated Representation ◽  
Hyperbolic Riemann Surfaces

Abstract Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve {X_{1}} defined over k, there exists a lifting X of the curve to the ring {W(k)} of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over {X/W(k)} . These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group {\pi_{1}(X_{K})} of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin–Simpson’s uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.

scholarly journals A BRIEF SURVEY OF HIGGS BUNDLES

2019 ◽  
Vol 26 (2) ◽  
pp. 197-214
Author(s):  
RONALD A. ZÚÑIGA ROJAS
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Mirror Symmetry ◽  
Higgs Field ◽  
Higgs Bundle ◽  
Higgs Bundles ◽  
Self Duality ◽  
Holomorphic Bundle ◽  
The Subject ◽  
Higgs Fields

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

scholarly journals Hyperbolic three-string vertex

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Atakan Hilmi Fırat
Keyword(s):  
Boundary Value Problem ◽  
Conservation Laws ◽  
Riemann Surfaces ◽  
Higher Order ◽  
Local Coordinates ◽  
Pants Decomposition ◽  
Fuchsian Equation ◽  
String Amplitudes ◽  
Hyperbolic Riemann Surfaces ◽  
Liouville’S Equation

Abstract We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

BMOA(R, m) and capacity density Bloch spaces on hyperbolic Riemann surfaces

1996 ◽  
Vol 29 (3-4) ◽  
pp. 203-226 ◽  
Author(s):  
Rauno Aulaskari ◽  
Peter Lappan ◽  
Jie Xiao ◽  
Ruhan Zhao
Keyword(s):  
Riemann Surfaces ◽  
Bloch Spaces ◽  
Hyperbolic Riemann Surfaces

scholarly journals Banach spaces of bounded solutions of Δu = Pu (P ≥ 0) on hyperbolic riemann surfaces

1974 ◽  
Vol 53 ◽  
pp. 141-155 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Banach Spaces ◽  
Riemann Surfaces ◽  
Space Structure ◽  
Bounded Solutions ◽  
Linear Isometry ◽  
Hölder Continuous ◽  
Hyperbolic Riemann Surfaces ◽  
Hyperbolic Riemann Surface

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

scholarly journals Construction of hyperbolic Riemann surfaces with large systoles

2015 ◽  
Vol 107 (1) ◽  
pp. 187-205
Author(s):  
Hugo Akrout ◽  
Bjoern Muetzel
Keyword(s):  
Riemann Surfaces ◽  
Hyperbolic Riemann Surfaces
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