505 research outputs found
The Variety of Integrable Killing Tensors on the 3-Sphere
Integrable Killing tensors are used to classify orthogonal coordinates in
which the classical Hamilton-Jacobi equation can be solved by a separation of
variables. We completely solve the Nijenhuis integrability conditions for
Killing tensors on the sphere and give a set of isometry invariants for
the integrability of a Killing tensor. We describe explicitly the space of
solutions as well as its quotient under isometries as projective varieties and
interpret their algebro-geometric properties in terms of Killing tensors.
Furthermore, we identify all St\"ackel systems in these varieties. This allows
us to recover the known list of separation coordinates on in a simple and
purely algebraic way. In particular, we prove that their moduli space is
homeomorphic to the associahedron
Twining characters, orbit Lie algebras, and fixed point resolution
We describe the resolution of field identification fixed points in coset
conformal field theories in terms of representation spaces of the coset chiral
algebra. A necessary ingredient from the representation theory of Kac Moody
algebras is the recently developed theory of twining characters and orbit Lie
algebras, as applied to automorphisms representing identification currents.Comment: Latex, 24 pages. Slightly extended version of lectures by J. Fuchs at
a workshop in Razlog (Bulgaria) in August 199
Polynomial-time Tensor Decompositions with Sum-of-Squares
We give new algorithms based on the sum-of-squares method for tensor
decomposition. Our results improve the best known running times from
quasi-polynomial to polynomial for several problems, including decomposing
random overcomplete 3-tensors and learning overcomplete dictionaries with
constant relative sparsity. We also give the first robust analysis for
decomposing overcomplete 4-tensors in the smoothed analysis model. A key
ingredient of our analysis is to establish small spectral gaps in moment
matrices derived from solutions to sum-of-squares relaxations. To enable this
analysis we augment sum-of-squares relaxations with spectral analogs of maximum
entropy constraints.Comment: to appear in FOCS 201
The Integer Valued SU(3) Casson Invariant for Brieskorn spheres
We develop techniques for computing the integer valued SU(3) Casson
invariant. Our method involves resolving the singularities in the flat moduli
space using a twisting perturbation and analyzing its effect on the topology of
the perturbed flat moduli space. These techniques, together with Bott-Morse
theory and the splitting principle for spectral flow, are applied to calculate
the invariant for all Brieskorn homology spheres.Comment: 50 pages, 3 figure
The eta invariant and equivariant index of transversally elliptic operators
We prove a formula for the multiplicities of the index of an equivariant
transversally elliptic operator on a -manifold. The formula is a sum of
integrals over blowups of the strata of the group action and also involves eta
invariants of associated elliptic operators. Among the applications, we obtain
an index formula for basic Dirac operators on Riemannian foliations, a problem
that was open for many years.Comment: 62 pages, typos correcte
Highest weight Macdonald and Jack Polynomials
Fractional quantum Hall states of particles in the lowest Landau levels are
described by multivariate polynomials. The incompressible liquid states when
described on a sphere are fully invariant under the rotation group. Excited
quasiparticle/quasihole states are member of multiplets under the rotation
group and generically there is a nontrivial highest weight member of the
multiplet from which all states can be constructed. Some of the trial states
proposed in the literature belong to classical families of symmetric
polynomials. In this paper we study Macdonald and Jack polynomials that are
highest weight states. For Macdonald polynomials it is a (q,t)-deformation of
the raising angular momentum operator that defines the highest weight
condition. By specialization of the parameters we obtain a classification of
the highest weight Jack polynomials. Our results are valid in the case of
staircase and rectangular partition indexing the polynomials.Comment: 17 pages, published versio
Quantum tunneling on graphs
We explore the tunneling behavior of a quantum particle on a finite graph, in
the presence of an asymptotically large potential. Surprisingly the behavior is
governed by the local symmetry of the graph around the wells.Comment: 18 page
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