Liouville and Toda field theories on Riemann surfaces

We study the Liouville theory on a Riemann surface of genus g by means of their associated Drinfeld--Sokolov linear systems. We discuss the cohomological properties of the monodromies of these systems. We identify the space of solutions of the equations of motion which are single--valued and local and explicitly represent them in terms of Krichever--Novikov oscillators. Then we discuss the operator structure of the quantum theory, in particular we determine the quantum exchange algebras and find the quantum conditions for univalence and locality. We show that we can extend the above discussion to $sl_n$ Toda theories.Comment: 41 pages, LaTeX, SISSA-ISAS 27/93/E

Population and Metapopulation Ecology of Childhood Diseases

Researchers have long used mathematical models and empirical data to explore the population ecology of childhood diseases such as measles and whooping cough. These diseases have proven ideal model systems for studying population dynamics over space and time. Here we present a novel dataset of weekly measles and whooping cough case reports in pre-vaccine era U.S. cities and states, along with a previously- studied dataset of measles in England & Wales. We first estimate per-population disease reporting probabilities. We find that disease reporting is highly variable over space and between diseases, and correlated with socioeconomic covariates including ethnic composition and school attendance. Using these reporting estimates, we infer the long-term, marginal distribution of disease incidence for each population. This describes a probabilistic measure of disease persistence that compares favorably with a classic threshold persistence measure, critical community size (CCS). The U.S. and England & Wales exhibit similar patterns of measles incidence distributions: larger populations show higher mean viincidence and lower variance. The per-time probability of local extinction (conditioned on population size) is higher in the U.S. than in England & Wales, likely due to larger distances between U.S. cities. Finally, we use observed persistence and inferred incidence distributions to estimate the per-time probability of true persistence. Estimated persistence of whooping cough is much higher than persistence of measles (conditioned on population size). We find that cryptic persistence (the difference between observed and estimated persistence) of whooping cough is most common in small populations, while for measles cryptic persistence is most common in medium-sized populations that hover at the edge of extinction. Our results show that variation in disease reporting can significantly affect meta- population estimates of disease persistence, such as CCS. The distributional estimates of incidence presented here explicitly account for incomplete reporting, providing summaries of long-term ecological patterns that are comparable between metapopulations. These measures can provide disease control programs with valuable information on where disease incidence is expected to be higher or lower than expected based on population size alone

Toda Fields on Riemann Surfaces: remarks on the Miura transformation

We point out that the Miura transformation is related to a holomorphic foliation in a relative flag manifold over a Riemann Surface. Certain differential operators corresponding to a free field description of $W$--algebras are thus interpreted as partial connections associated to the foliation.Comment: AmsLatex 1.1, 10 page

Large Chiral Diffeomorphisms on Riemann Surfaces and W-algebras

The diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a B.R.S. formulation (for a given order of truncation) leading to a more algebraic set up. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the so called W-algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between KdV flows and W-diffeomorphimsComment: LaTeX file, 31 pages, no figure. Version to appear in J. Math. Phys. Work partly supported by Region PACA and INF

Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

We 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 Szego 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 sigma.Comment: 35 pages. New results, presentation improved, clarifications added. Accepted for publication in Math. An

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The Complex Langevin method: When can it be trusted?

We analyze to what extent the complex Langevin method, which is in principle capable of solving the so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and then point out various mathematical loopholes. The detailed study of some simple examples leads to practical suggestions about the application of the method.Comment: 14 pages, including several eps figures and tables; clarification and minor corrections added, to appear in PR