Determinants of Cauchy-Riemann operators over a Riemann surface

1985 ◽  
Vol 19 (1) ◽  
pp. 31-34 ◽  
Author(s):  
Daniel Quillen
Keyword(s):  
Riemann Surface ◽  
Cauchy Riemann

Universal Series on a Riemann Surface

2011 ◽  
Vol 63 (5) ◽  
pp. 1025-1037
Author(s):  
Raphäel Clouâtre
Keyword(s):  
Riemann Surface ◽  
Holomorphic Function ◽  
Compact Subset ◽  
Fundamental Solutions ◽  
Partial Sums ◽  
Universal Series ◽  
Cauchy Riemann ◽  
Series Of Functions

Abstract Every holomorphic function on a compact subset of a Riemann surface can be uniformly approximated by partial sums of a given series of functions. Those functions behave locally like the classical fundamental solutions of the Cauchy–Riemann operator in the plane.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals AdS2 × S2 × CY2 solutions in Type IIB with 8 supersymmetries

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Yolanda Lozano ◽  
Carlos Nunez ◽  
Anayeli Ramirez
Keyword(s):  
Quantum Mechanics ◽  
Riemann Surface ◽  
Central Charge ◽  
Global Solutions ◽  
Infinite Family ◽  
Dimensional System ◽  
One Dimensional

Abstract We present a new infinite family of Type IIB supergravity solutions preserving eight supercharges. The structure of the space is AdS2 × S2 × CY2 × S1 fibered over an interval. These solutions can be related through double analytical continuations with those recently constructed in [1]. Both types of solutions are however dual to very different superconformal quantum mechanics. We show that our solutions fit locally in the class of AdS2 × S2 × CY2 solutions fibered over a 2d Riemann surface Σ constructed by Chiodaroli, Gutperle and Krym, in the absence of D3 and D7 brane sources. We compare our solutions to the global solutions constructed by Chiodaroli, D’Hoker and Gutperle for Σ an annulus. We also construct a cohomogeneity-two family of solutions using non-Abelian T-duality. Finally, we relate the holographic central charge of our one dimensional system to a combination of electric and magnetic fluxes. We propose an extremisation principle for the central charge from a functional constructed out of the RR fluxes.

scholarly journals Supersymmetric solitons and a degeneracy of solutions in AdS/CFT

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Andrés Anabalón ◽  
Simon F. Ross
Keyword(s):  
Phase Transition ◽  
Riemann Surface ◽  
Magnetic Flux ◽  
Gauge Field ◽  
Wilson Line ◽  
Saddle Points ◽  
Planar Boundary ◽  
Four Dimensions ◽  
Special Value

Abstract We study Lorentzian supersymmetric configurations in D = 4 and D = 5 gauged $$ \mathcal{N} $$ N = 2 supergravity. We show that there are smooth 1/2 BPS solutions which are asymptotically AdS4 and AdS5 with a planar boundary, a compact spacelike direction and with a Wilson line on that circle. There are solitons where the S1 shrinks smoothly to zero in the interior, with a magnetic flux through the circle determined by the Wilson line, which are AdS analogues of the Melvin fluxtube. There is also a solution with a constant gauge field, which is pure AdS. Both solutions preserve half of the supersymmetries at a special value of the Wilson line. There is a phase transition between these two saddle-points as a function of the Wilson line precisely at the supersymmetric point. Thus, the supersymmetric solutions are degenerate, at least at the supergravity level. We extend this discussion to one of the Romans solutions in four dimensions when the Euclidean boundary is S1× Σg where Σg is a Riemann surface with genus g > 0. We speculate that the supersymmetric state of the CFT on the boundary is dual to a superposition of the two degenerate geometries.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

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