7 research outputs found
Rational points in the moduli space of genus two
We build a database of genus 2 curves defined over which contains
all curves with minimal absolute height , all curves with moduli
height , and all curves with extra automorphisms in
standard form defined over with height .
For each isomorphism class in the database, an equation over its minimal field
of definition is provided, the automorphism group of the curve, Clebsch and
Igusa invariants. The distribution of rational points in the moduli space
for which the field of moduli is a field of definition is
discussed and some open problems are presented
Elliptic Loci of SU(3) Vacua
The space of vacua of many four-dimensional, supersymmetric
gauge theories can famously be identified with a family of complex curves. For
gauge group , this gives a fully explicit description of the low-energy
effective theory in terms of an elliptic curve and associated modular
fundamental domain. The two-dimensional space of vacua for gauge group
parametrizes an intricate family of genus two curves. We analyze this family
using the so-called Rosenhain form for these curves. We demonstrate that two
natural one-dimensional subloci of the space of vacua,
and , each parametrize a family of elliptic curves. For these
elliptic loci, we describe the order parameters and fundamental domains
explicitly. The locus contains the points where mutually local
dyons become massless, and is a fundamental domain for a classical congruence
subgroup. Moreover, the locus contains the superconformal
Argyres-Douglas points, and is a fundamental domain for a Fricke group.Comment: 39 pages + Appendices, 5 figures, v2: minor changes and extended
discussion on automorphism