1,721,016 research outputs found
CFT partition functions and moduli spaces of canonical curves
No full textConformal field theories (CFT) represent a framework of fruitful interplay between some of the most advanced topics in theoretical physics and algebraic geometry. In particular, the investigation of the CFT partition functions is closely related to the analysis of the correspondence between analytic and algebraic properties of closed Riemann surfaces. In the present thesis, some new aspects of this correspondence, in particular the ones arising in the CFTs associated to string and superstring theories, are considered. More precisely, the algebraic parameters, determining the canonical curve associated to a non-hyperelliptic Riemann surface, are explicitly computed in terms of Riemann theta functions, evaluated at generic points of the curve. Moreover, the techniques here introduced are applied to the analysis of the singular locus of the theta function, also considered with respect to the Andreotty-Mayer approach to the Schottky problem, and to the restriction of the Siegel's measure to the moduli space of canonical curves
Higher genus superstring amplitudes and the measure on the moduli space
No full textRecent derivations of completely gauge-invariant two-loop superstring amplitudes renewed the interest in the possible form of higher genus contributions. In this respect, two main points are discussed: a. At genus greater than 3, the Schottky problem actually represents a formidable obstruction for the explicit expression of the integration measure on the moduli space in terms of Riemann period matrices. As a first step toward such a construction, a modular invariant measure for all genera is derived, corresponding to the restriction to the Schottky locus of the Siegel metric on the upper half-space. b. A class of natural generalizations of the 2 loop D'Hoker and Phong formula is discussed for the higher genus contributions to the 4-gravitons amplitude in type II theories. Such expressions also satisfy the non-trivial requirements of modular invariance and gauge slice independence. © 2007 WILEY-VCH Verlag GmbH & Co. KGaA
Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities
No full textWe characterize genus g canonical curves by the vanishing of combinatorial products of g + 1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g−2)(g−3)/2 theta identities. A remarkable mechanism, based on a basis of H^0(K_C ) expressed in terms of Szegö kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section σ
The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve
No full textA section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian torus defined in terms of Theta_s. It also identifies sections of line bundles on the moduli space of algebraic curves, closely related to the Mumford isomorphism, whose zero loci characterize special varieties in the framework of the Andreotti-Mayer approach to the Schottky problem, a result which also reproduces the only previously known case g=4. This new approach, based on the combinatorics of determinantal relations for two-fold products of holomorphic abelian differentials, sheds light on basic structures, and leads to the explicit expressions, in terms of theta functions, of the canonical basis of the abelian holomorphic differentials and of the constant defining the Mumford form. Furthermore, the metric on the moduli space of canonical curves, induced by the Siegel metric, which is shown to be equivalent to the Kodaira-Spencer map of the square of the Bergman reproducing kernel, is explicitly expressed in terms of the Riemann period matrix only, a result previously known for the trivial cases g=2 and g=3. Finally, the induced Siegel volume form is expressed in terms of the Mumford form
Higher genus superstring amplitudes from the geometry of moduli spaces
Full text linkWe show that the higher genus 4-point superstring amplitude is strongly constrained by the geometry of moduli space of Riemann surfaces. A detailed analysis leads to a natural proposal which satisfies several conditions. The result is based on the recently derived Siegel induced metric on the moduli space of Riemann surfaces and on combinatorial products of determinants of holomorphic abelian differentials
Superstring measure and non-renormalization of the three-point amplitude
Full text linkWe show that a recently conjectured expression for the superstring three-point amplitude, in the framework of the Cacciatori, Dalla Piazza, van Geemen - Grushevsky ansatz for the chiral measure, fails to vanish at three-loop, in contrast with expectations from non-renormalization theorems. Based on analogous two-loop computations, we discuss the possibility of a non-trivial correction to the amplitude and propose a natural candidate for such a contribution. Thanks to a new remarkable identity, it is reasonable to expect that the corrected three-point amplitude vanishes at three-loop, recovering the agreement with non-renormalization theorems
On symmetries of
Full text linkMotivated by an analogous result for K3 models, we classify all groups of symmetries of non-linear sigma models on a torus T^4 that preserve the N=(4,4) superconformal algebra. The resulting symmetry groups are isomorphic to certain subgroups of the Weyl group of E8, that plays a role similar to the Conway group for the case of K3 models. Our analysis heavily relies on the triality automorphism of the T-duality group SO(4,4,Z). As a byproduct of our results, we discover new explicit descriptions of K3 models as asymmetric orbifolds of torus CFTs
Topological defects in K3 sigma models
Full text linkWe consider the topological defect lines commuting with the spectral flow and the N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} = (4, 4) superconformal symmetry in two dimensional non-linear sigma models on K3. By studying their fusion with boundary states, we derive a number of general results for the category of such defects. We argue that while for certain K3 models infinitely many simple defects, and even a continuum, can occur, at generic points in the moduli space the category is actually trivial, i.e. it is generated by the identity defect. Furthermore, we show that if a K3 model is at the attractor point for some BPS configuration of D-branes, then all topological defects have integral quantum dimension. We also conjecture that a continuum of topological defects arises if and only if the K3 model is a (possibly generalized) orbifold of a torus model. Finally, we test our general results in a couple of examples, where we provide a partial classification of the topological defects
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