143 research outputs found
Three-dimensional AdS gravity and extremal CFTs at c=8m
We note that Witten's proposed duality between extremal c=24k CFTs and
three-dimensional anti-de Sitter gravity may possibly be extended to central
charges that are multiples of 8, for which extremal self-dual CFTs are known to
exist up to c=40. All CFTs of this type with central charge 24 or higher,
provided that they exist, have the required mass gap and may serve as candidate
duals to three-dimensional gravity at the corresponding values of the
cosmological constant. Here, we compute the genus one partition function of
these theories up to c=88, we give exact and approximate formulas for the
degeneracies of states, and we determine the genus two partition functions of
the theories up to c=40.Comment: 17 pages, harvmac; v2: references added, version accepted in JHE
On the Whitehead spectrum of the circle
The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss and
Williams shows that the homotopy groups in low degrees of the space of
homeomorphisms of a closed Riemannian manifold of negative sectional curvature
can be expressed as a functor of the fundamental group of the manifold. To
determine this functor, however, it remains to determine the homotopy groups of
the topological Whitehead spectrum of the circle. The cyclotomic trace of B
okstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an
expression for these homotopy groups in terms of the equivariant homotopy
groups of the homotopy fiber of the map from the topological Hochschild
T-spectrum of the sphere spectrum to that of the ring of integers induced by
the Hurewicz map. We evaluate the latter homotopy groups, and hence, the
homotopy groups of the topological Whitehead spectrum of the circle in low
degrees. The result extends earlier work by Anderson and Hsiang and by Igusa
and complements recent work by Grunewald, Klein, and Macko.Comment: 52 page
The General Correlation Function in the Schwinger Model on a Torus
In the framework of the Euclidean path integral approach we derive the exact
formula for the general N-point chiral densities correlator in the Schwinger
model on a torusComment: 17 pages, misprints corrected, references adde
The class of the locus of intermediate Jacobians of cubic threefolds
We study the locus of intermediate Jacobians of cubic threefolds within the
moduli space of complex principally polarized abelian fivefolds, and its
generalization to arbitrary genus - the locus of abelian varieties with a
singular odd two-torsion point on the theta divisor. Assuming that this locus
has expected codimension (which we show to be true for genus up to 5), we
compute the class of this locus, and of is closure in the perfect cone toroidal
compactification, in the Chow, homology, and the tautological ring.
We work out the cases of genus up to 5 in detail, obtaining explicit
expressions for the classes of the closures of the locus of products of an
elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally
polarized abelian fourfolds, and of the locus of intermediate Jacobians in
genus 5. In the course of our computation we also deal with various
intersections of boundary divisors of a level toroidal compactification, which
is of independent interest in understanding the cohomology and Chow rings of
the moduli spaces.Comment: v2: new section 9 on the geometry of the boundary of the locus of
intermediate Jacobians of cubic threefolds. Final version to appear in
Invent. Mat
Quantum theta functions and Gabor frames for modulation spaces
Representations of the celebrated Heisenberg commutation relations in quantum
mechanics and their exponentiated versions form the starting point for a number
of basic constructions, both in mathematics and mathematical physics (geometric
quantization, quantum tori, classical and quantum theta functions) and signal
analysis (Gabor analysis).
In this paper we try to bridge the two communities, represented by the two
co--authors: that of noncommutative geometry and that of signal analysis. After
providing a brief comparative dictionary of the two languages, we will show
e.g. that the Janssen representation of Gabor frames with generalized Gaussians
as Gabor atoms yields in a natural way quantum theta functions, and that the
Rieffel scalar product and associativity relations underlie both the functional
equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change
The anomaly line bundle of the self-dual field theory
In this work, we determine explicitly the anomaly line bundle of the abelian
self-dual field theory over the space of metrics modulo diffeomorphisms,
including its torsion part. Inspired by the work of Belov and Moore, we propose
a non-covariant action principle for a pair of Euclidean self-dual fields on a
generic oriented Riemannian manifold. The corresponding path integral allows to
study the global properties of the partition function over the space of metrics
modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual
fields differs from the determinant bundle of the Dirac operator coupled to
chiral spinors by a flat bundle that is not trivial if the underlying manifold
has middle-degree cohomology, and whose holonomies are determined explicitly.
We briefly sketch the relevance of this result for the computation of the
global gravitational anomaly of the self-dual field theory, that will appear in
another paper.Comment: 41 pages. v2: A few typos corrected. Version accepted for publication
in CM
Global analysis by hidden symmetry
Hidden symmetry of a G'-space X is defined by an extension of the G'-action
on X to that of a group G containing G' as a subgroup. In this setting, we
study the relationship between the three objects:
(A) global analysis on X by using representations of G (hidden symmetry);
(B) global analysis on X by using representations of G';
(C) branching laws of representations of G when restricted to the subgroup
G'.
We explain a trick which transfers results for finite-dimensional
representations in the compact setting to those for infinite-dimensional
representations in the noncompact setting when is -spherical.
Applications to branching problems of unitary representations, and to spectral
analysis on pseudo-Riemannian locally symmetric spaces are also discussed.Comment: Special volume in honor of Roger Howe on the occasion of his 70th
birthda
Analytic Representation of Finite Quantum Systems
A transform between functions in R and functions in Zd is used to define the
analogue of number and coherent states in the context of finite d-dimensional
quantum systems. The coherent states are used to define an analytic
representation in terms of theta functions. All states are represented by
entire functions with growth of order 2, which have exactly d zeros in each
cell. The analytic function of a state is constructed from its zeros. Results
about the completeness of finite sets of coherent states within a cell are
derived
Negative discriminant states in N=4 supersymmetric string theories
Single centered BPS black hole solutions exist only when the charge carried
by the black hole has positive discriminant. On the other hand the exact dyon
spectrum in heterotic string theory compactified on T^6 is known to contain
states with negative discriminant. We show that all of these negative
discriminant states can be accounted for as two centered black holes. Thus
after the contribution to the index from the two centered black holes is
subtracted from the total microscopic index, the index for states with negative
discriminant vanishes even for finite values of charges, in agreement with the
results from the black hole side. Bound state metamorphosis -- which requires
us to identify certain apparently different two centered configurations
according to a specific set of rules -- plays a crucial role in this analysis.
We also generalize these results to a class of CHL string theories.Comment: LaTeX file, 32 pages; v2: reference added; v3: added new section 3.
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