16 research outputs found
The Harmonic Volumes of Hyperelliptic Curves
We determine the harmonic volumes for all the hyperelliptic curves. This
gives a geometric interpretation of a theorem established by A. Tanaka.Comment: 20 pages, 4 figures, to appear in Publ. RIMS, Kyoto Universit
The period matrix of the hyperelliptic curve
A geometric algorithm is introduced for finding a symplectic basis of the
first integral homology group of a compact Riemann surface, which is a
-cyclic covering of branched over 3 points. The algorithm
yields a previously unknown symplectic basis of the hyperelliptic curve defined
by the affine equation for genus . We then explicitly
obtain the period matrix of this curve, its entries being elements of the
-st cyclotomic field. In the proof, the details of our algorithm play
no significant role.Comment: 18 pages, 10 figure
Nonlinear sigma model in discrete complex analysis
We examine a discrete version of the two-dimensional nonlinear sigma
model derived from discrete complex analysis. We adopt two lattices, one
rectangular, the other polar. We define a discrete energy
and a discrete area , where the function is
related to a stereographic projection governed by a unit vector of the model.
The discrete energy and area satisfy the inequality , which is saturated if and only if the function
is discrete (anti-)holomorphic. We show for the rectangular lattice that,
except for a factor 2, the discrete energy and the area tend to the usual
continuous energy and the area as the lattice spacings tend to zero. In the polar lattice, we
section the plane by lines passing through the origin into equal
sectors and place vertices radially in a geometric progression with a common
ratio . For this polar lattice, the Euler--Lagrange equation derived from
the discrete energy yields rotationally symmetric
(anti-)holomorphic solutions in the
zeroth order of . We find that the discrete area evaluated by
these zeroth-order solutions is expressible as a -integral (the Jackson
integral). Moreover, the area tends to in the continuum limit
( and ) with fixed discrete conformal structure
.Comment: v1. 10 pages, 2 figures v2. New title, 19 pages and 3 figures,
Sec.2.3 (EL eq. and its continuum limt) and Sec.3 (polar lattice) adde