The Riemann Surface of Klein with 168 Automorphisms

Author(s):  
Harry E. Rauch ◽  
J. Lewittes
Keyword(s):  
2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang

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.


1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara

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.


2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Yolanda Lozano ◽  
Carlos Nunez ◽  
Anayeli Ramirez

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.


2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Andrés Anabalón ◽  
Simon F. Ross

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.


2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola

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.


2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Davide Cassani ◽  
Grégoire Josse ◽  
Michela Petrini ◽  
Daniel Waldram

Abstract We discuss consistent truncations of eleven-dimensional supergravity on a six-dimensional manifold M, preserving minimal $$ \mathcal{N} $$ N = 2 supersymmetry in five dimensions. These are based on GS ⊆ USp(6) structures for the generalised E6(6) tangent bundle on M, such that the intrinsic torsion is a constant GS singlet. We spell out the algorithm defining the full bosonic truncation ansatz and then apply this formalism to consistent truncations that contain warped AdS5×wM solutions arising from M5-branes wrapped on a Riemann surface. The generalised U(1) structure associated with the $$ \mathcal{N} $$ N = 2 solution of Maldacena-Nuñez leads to five-dimensional supergravity with four vector multiplets, one hypermultiplet and SO(3) × U(1) × ℝ gauge group. The generalised structure associated with “BBBW” solutions yields two vector multiplets, one hypermultiplet and an abelian gauging. We argue that these are the most general consistent truncations on such backgrounds.


2018 ◽  
Vol 234 (5) ◽  
pp. 608-615
Author(s):  
Yu. V. Dymchenko ◽  
V. A. Shlyk

2011 ◽  
Vol 26 (26) ◽  
pp. 4647-4660
Author(s):  
GOR SARKISSIAN

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the N-fold product of the gauged WZW model G/H on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern–Simons theory on a sphere with N holes times the time-line with G and H gauge fields both coupled to two Wilson lines. For the special case of the topological coset G/G we arrive at the conclusion that the phase space of the N-fold product of the topological coset G/G on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern–Simons theory on a Riemann surface of the genus N-1 times the time-line with four Wilson lines.


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