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Register a new userComputational approach to compact Riemann surfaces
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A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. As an application of the code, solutions to the Kadomtsev-Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.
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We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is either given by a list of the branch points, the coefficients of the defining polynomial or a system of cuts for the curve. A canonical basis of the homology is introduced algorithmically for this curve. The periods of the holomorphic differentials and the Abel map are computed with the Clenshaw-Curtis method in order to achieve spectral accuracy. The code can handle almost degenerate Riemann surfaces. This work generalizes previous work on real hyperelliptic surfaces with prescribed cuts to arbitrary hyperelliptic surfaces. As an example, solutions to the sine-Gordon equation in terms of multi-dimensional theta functions are studied, also in the solitonic limit of these solutions.
This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations. 2) Space of meromorphic functions and forms, we classify them with the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The Riemann--Roch theorem that computes the dimension of spaces of functions and forms with given orders of poles and zeros. 5) The moduli space of Riemann surfaces, with its combinatorial representation as Strebel graphs, and also with the uniformization theorem that maps Riemann surfaces to hyperbolic surfaces. 6) An application of Riemann surfaces to integrable systems, more precisely finding sections of an eigenvector bundle over a Riemann surface, which is known as the algebraic reconstruction method in integrable systems, and we mention how it is related to Baker-Akhiezer functions and Tau functions.
We present analytical implementation of conformal field theory on a compact Riemann surface. We consider statistical fields constructed from background charge modifications of the Gaussian free field and derive Ward identities which represent the Lie derivative operators in terms of the Virasoro fields and the puncture operators associated with the background charges. As applications, we derive Eguchi-Ooguris version of Wards equations and certain types of BPZ equations on a torus.
For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras.
We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G is an abelian group over an arbitrary X, and the case of a general G-bundle over an elliptic curve.
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