382 research outputs found
A brief survey on singularities of geodesic flows in smooth signature changing metrics on 2-surfaces
We present a survey on generic singularities of geodesic flows in smooth
signature changing metrics (often called pseudo-Riemannian) in dimension 2.
Generically, a pseudo-Riemannian metric on a 2-manifold changes its
signature (degenerates) along a curve , which locally separates into a
Riemannian () and a Lorentzian () domain. The geodesic flow does not have
singularities over and , and for any point and every
tangential direction there exists a unique geodesic passing through the
point with the direction . On the contrary, geodesics cannot pass
through a point in arbitrary tangential directions, but only in
some admissible directions; the number of admissible directions is 1 or 2 or 3.
We study this phenomenon and the local properties of geodesics near .Comment: 23 pages, 14 figure
Uniruledness of orthogonal modular varieties
A strongly reflective modular form with respect to an orthogonal group of
signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new
geometric application of such modular forms: we prove that if the weight is
larger than n then the corresponding modular variety is uniruled. We also
construct new reflective modular forms and thus provide new examples of
uniruled moduli spaces of lattice polarised K3 surfaces. Finally we prove that
the moduli space of Kummer surfaces associated to (1,21)-polarised abelian
surfaces is uniruled.Comment: 14 page
The Modular Form of the Barth-Nieto Quintic
Barth and Nieto have found a remarkable quintic threefold which parametrizes
Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a
(1,3)-polarization and a lecel 2 structure. A double cover of this quintic,
which is also a Calabi-Yau variety, is birationally equivalent to the moduli
space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2
structure. As a consequence the corresponding paramodular group \Gamma_3(2) has
a unique cusp form of weight 3. In this paper we find this cusp form which is
\Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a
character with respect to the paramodular group \Gamma_3. It has several
interesting properties. One is that it admits an infinite product
representation, the other is that it vanishes of order 1 along the diagonal in
Siegel space. In fact \Delta_1 is an element of a short series of modular forms
with this last property. Using the fact that \Delta_1 is a weight 3 cusp form
with respect to the group \Gamma_3(2) we give an independent construction of a
smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).Comment: 20 pages, Latex2e RIMS Preprint 120
Algebraic classification of spacetimes using discriminating scalar curvature invariants
The Weyl and Ricci tensors can be algebraically classified in a Lorentzian
spacetime of arbitrary dimensions using alignment theory. Used in tandem with
the boost weight decomposition and curvature operators, the algebraic
classification of the Weyl tensor and the Ricci tensor in higher dimensions can
then be refined utilizing their eigenbivector and eigenvalue structure,
respectively. In particular, for a tensor of a particular algebraic type, the
associated operator will have a restricted eigenvector structure, and this can
then be used to determine necessary conditions for a particular algebraic type.
We shall present an analysis of the discriminants of the associated
characteristic equation for the eigenvalues of an operator to determine the
conditions on (the associated) curvature tensor for a given algebraic type. We
will describe an algorithm which enables us to completely determine the
eigenvalue structure of the curvature operator, up to degeneracies, in terms of
a set of discriminants. We then express these conditions (discriminants) in
terms of these polynomial curvature invariants. In particular, we can use the
techniques described to study the necessary conditions in arbitrary dimensions
for the Weyl and Ricci curvature operators (and hence the higher dimensional
Weyl and Ricci tensors) to be of algebraic type II or D, and create syzygies
which are necessary for the special algebraic type to be fulfilled. We are
consequently able to determine the necessary conditions in terms of simple
scalar polynomial curvature invariants in order for the higher dimensional Weyl
and Ricci tensors to be of type II or D. We explicitly determine the scalar
polynomial curvature invariants for a Weyl or Ricci tensor to be of type II or
D in 5D. A number of simple examples are presented and, in particular, we
present a detailed analysis of the important example of a 5D rotating black
ring.Comment: 31 pages, 1 figur
The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
We find automorphic corrections for the Lorentzian Kac--Moody algebras with
the simplest generalized Cartan matrices of rank 3:
A_{1,0} =
2 0 -1
0 2 -2
-1 -2 2
and
A_{1,I} =
2 -2 -1
-2 2 -1
-1 -1 2
For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form
\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30
with respect to a 2-congruence subgroup. We find infinite product or sum
expansions for these forms. Our method of construction of \chi_{35} leads to
the direct construction of Siegel modular forms by infinite product expansions,
whose divisors are the Humbert surfaces with fixed discriminants. Existence of
these forms was proved by van der Geer in 1982 using some geometrical
consideration. We announce a list of all hyperbolic symmetric generalized
Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a
lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.Comment: 40 pages, no figures. AMS-Te
A construction of antisymmetric modular forms for Weil representations
We give coefficient formulas for antisymmetric vector-valued cusp forms with
rational Fourier coefficients for the Weil representation associated to a
finite quadratic module. The forms we construct always span all cusp forms in
weight at least three. These formulas are useful for computing explicitly with
theta lifts.Comment: Substantial organizational changes based on reviewer's suggestions.
Previously titled "Computing antisymmetric modular forms and theta lifts". 14
page
K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: the structure of invariant
A holomorphic torsion invariant of K3 surfaces with involution was introduced
by the second-named author. In this paper, we completely determine its
structure as an automorphic function on the moduli space of such K3 surfaces.
On every component of the moduli space, it is expressed as the product of an
explicit Borcherds lift and a classical Siegel modular form. We also introduce
its twisted version. We prove its modularity and a certain uniqueness of the
modular form corresponding to the twisted holomorphic torsion invariant. This
is used to study an equivariant analogue of Borcherds' conjecture.Comment: Section 11 adde
On the Genus of the Moonshine Module
We provide a novel and simple description of Schellekens' seventy-one affine
Kac-Moody structures of self-dual vertex operator algebras of central charge 24
by utilizing cyclic subgroups of the glue codes of the Niemeier lattices with
roots. We also discuss a possible uniform construction procedure of the
self-dual vertex operator algebras of central charge 24 starting from the Leech
lattice. This also allows us to consider the uniqueness question for all
non-trivial affine Kac-Moody structures. We finally discuss our description
from a Lorentzian viewpoint.Comment: 28 pages, 17 table
A generalized Kac-Moody algebra of rank 14
We construct a vertex algebra of central charge 26 from a lattice orbifold
vertex operator algebra of central charge 12. The BRST-cohomology group of this
vertex algebra is a new generalized Kac-Moody algebra of rank 14. We determine
its root space multiplicities and a set of simple roots.Comment: LaTeX, 18 pages, 3 Tables. Published version. Revision contains some
more detail
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