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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.
Instanton calculations in semiclassical quantum mechanics rely on integration along trajectories which solve classical equations of motion. However in systems with higher dimensionality or complexified phase space these are rarely attainable. A prime
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 obta
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 o
This note is to concern a generalization to the case of twisted coefficients of the classical theory of Abelian differentials on a compact Riemann surface. We apply the Dirichlets principle to a modified energy functional to show the existence of dif
It is proved that the moduli space of static solutions of the CP^1 model on spacetime Sigma x R, where Sigma is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic