1,208 research outputs found
Elliptic quantum groups and Ruijsenaars models
We construct symmetric and exterior powers of the vector representation of
the elliptic quantum groups . The corresponding transfer
matrices give rise to various integrable difference equations which could be
solved in principle by the nested Bethe ansatz method. In special cases we
recover the Ruijsenaars systems of commuting difference operators.Comment: 15 pages, late
LATTICEEASY: A Program for Lattice Simulations of Scalar Fields in an Expanding Universe
We describe a C++ program that we have written and made available for calculating the evolution of interacting scalar fields in an expanding universe. The program is particularly useful for the study of reheating and thermalization after inflation. The program and its full documentation are available on the Web at http://physics.stanford.edu/gfelder/latticeeasy. In this paper we provide a brief overview of what the program does and what it is useful for
Commuting difference operators with elliptic coefficients from Baxter's vacuum vestors
For quantum integrable models with elliptic R-matrix, we construct the Baxter
Q-operator in infinite-dimensional representations of the algebra of
observables.Comment: 31 pages, LaTeX, references adde
A simple construction of elliptic -matrices
We show that Belavin's solutions of the quantum Yang--Baxter equation can be
obtained by restricting an infinite -matrix to suitable finite dimensional
subspaces. This infinite -matrix is a modified version of the
Shibukawa--Ueno -matrix acting on functions of two variables.Comment: 6 page
Black hole production in tachyonic preheating
We present fully non-linear simulations of a self-interacting scalar field in
the early universe undergoing tachyonic preheating. We find that density
perturbations on sub-horizon scales which are amplified by tachyonic
instability maintain long range correlations even during the succeeding
parametric resonance, in contrast to the standard models of preheating
dominated by parametric resonance. As a result the final spectrum exhibits
memory and is not universal in shape. We find that throughout the subsequent
era of parametric resonance the equation of state of the universe is almost
dust-like, hence the Jeans wavelength is much smaller than the horizon scale.
If our 2D simulations are accurate reflections of the situation in 3D, then
there are wide regions of parameter space ruled out by over-production of black
holes. It is likely however that realistic parameter values, consistent with
COBE/WMAP normalisation, are safetly outside this black hole over-production
region.Comment: 6pages, 7figures, figures correcte
On elliptic CalogeroâMoser systems for complex crystallographic reflection groups
To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G -invariant functions of pairs (T,s), where T is a hypertorus in X (of codimension 1), and sâG is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic CalogeroâMoser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems â an explicit construction as limits of classical CalogeroâMoser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of V. Buchstaber, G. Felder and A. Veselov (1994) [BFV]), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions.National Science Foundation (U.S.) (Grant DMS-0504847)National Science Foundation (U.S.) (Grant DMS-0854764
Dynamical differential equations compatible with rational qKZ equations
For the Lie algebra we introduce a system of differential operators
called the dynamical operators. We prove that the dynamical differential
operators commute with the rational quantized Knizhnik-Zamolodchikov
difference operators. We describe the transformations of the dynamical
operators under the natural action of the Weyl group.Comment: 7 pages, AmsLaTe
Parametrization of semi-dynamical quantum reflection algebra
We construct sets of structure matrices for the semi-dynamical reflection
algebra, solving the Yang-Baxter type consistency equations extended by the
action of an automorphism of the auxiliary space. These solutions are
parametrized by dynamical conjugation matrices, Drinfel'd twist representations
and quantum non-dynamical -matrices. They yield factorized forms for the
monodromy matrices.Comment: LaTeX, 24 pages. Misprints corrected, comments added in Conclusion on
construction of Hamiltonian
Preheating with Trilinear Interactions: Tachyonic Resonance
We investigate the effects of bosonic trilinear interactions in preheating
after chaotic inflation. A trilinear interaction term allows for the complete
decay of the massive inflaton particles, which is necessary for the transition
to radiation domination. We found that typically the trilinear term is
subdominant during early stages of preheating, but it actually amplifies
parametric resonance driven by the four-legs interaction. In cases where the
trilinear term does dominate during preheating, the process occurs through
periodic tachyonic amplifications with resonance effects, which is so effective
that preheating completes within a few inflaton oscillations. We develop an
analytic theory of this process, which we call tachyonic resonance. We also
study numerically the influence of trilinear interactions on the dynamics after
preheating. The trilinear term eventually comes to dominate after preheating,
leading to faster rescattering and thermalization than could occur without it.
Finally, we investigate the role of non-renormalizable interaction terms during
preheating. We find that if they are present they generally dominate (while
still in a controllable regime) in chaotic inflation models. Preheating due to
these terms proceeds through a modified form of tachyonic resonance.Comment: 19 pages, 10 figures, refs added, published versio
Some recursive formulas for Selberg-type integrals
A set of recursive relations satisfied by Selberg-type integrals involving
monomial symmetric polynomials are derived, generalizing previously known
results. These formulas provide a well-defined algorithm for computing
Selberg-Schur integrals whenever the Kostka numbers relating Schur functions
and the corresponding monomial polynomials are explicitly known. We illustrate
the usefulness of our results discussing some interesting examples.Comment: 11 pages. To appear in Jour. Phys.
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