15 research outputs found
Hyperelliptic integrals modulo and Cartier-Manin matrices
The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field with a prime number of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus . It is known that in this case the total -dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field in this case gives only a -dimensional space of solutions, that is, a "half" of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field can be obtained by reduction modulo of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier-Manin matrix of the hyperelliptic curve. That situation is analogous to the example of the elliptic integral considered in the classical Y.I. Manin's paper in 1961
ON THE LOCAL SUM CONJECTURE IN TWO DIMENSIONS
In this paper we give an elementary proof of the local sum conjecture in two
dimensions. In a remarkable paper [CMN, arXiv:1810.11340], this conjecture has
been established in all dimensions using sophisticated, powerful techniques
from a research area blending algebraic geometry with ideas from logic. The
purpose of this paper is to give an elementary proof of this conjecture which
will be accessbile to a broad readership.Comment: 32 Page
On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version)
We find a relation between the spectrum of solitons of massive quantum
field theories in and the scaling dimensions of chiral fields at the
conformal point. The condition that the scaling dimensions be real imposes
restrictions on the soliton numbers and leads to a classification program for
symmetric conformal theories and their massive deformations in terms of a
suitable generalization of Dynkin diagrams (which coincides with the A--D--E
Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper
subset of this classification. In the particular case of LG theories we relate
the soliton numbers with intersection of vanishing cycles of the corresponding
singularity; the relation between soliton numbers and the scaling dimensions in
this particular case is a well known application of Picard-Lefschetz theory.Comment: 116 pages, HUTP-92/A064 and SISSA-203/92/E