20,280 research outputs found
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
Constant Curvature Algebras and Higher Spin Action Generating Functions
The algebra of differential geometry operations on symmetric tensors over
constant curvature manifolds forms a novel deformation of the sl(2,R)
[semidirect product] R^2 Lie algebra. We present a simple calculus for
calculations in its universal enveloping algebra. As an application, we derive
generating functions for the actions and gauge invariances of massive,
partially massless and massless (for both bose and fermi statistics) higher
spins on constant curvature backgrounds. These are formulated in terms of a
minimal set of covariant, unconstrained, fields rather than towers of auxiliary
fields. Partially massless gauge transformations are shown to arise as
degeneracies of the flat, massless gauge transformation in one dimension
higher. Moreover, our results and calculus offer a considerable simplification
over existing techniques for handling higher spins. In particular, we show how
theories of arbitrary spin in dimension d can be rewritten in terms of a single
scalar field in dimension 2d where the d additional dimensions correspond to
coordinate differentials. We also develop an analogous framework for
spinor-tensor fields in terms of the corresponding superalgebra.Comment: 44 pages, LaTeX, 2 .eps figure
Computer algebra and operators
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions
Wilsonian renormalization, differential equations and Hopf algebras
In this paper, we present an algebraic formalism inspired by Butcher's
B-series in numerical analysis and the Connes-Kreimer approach to perturbative
renormalization. We first define power series of non linear operators and
propose several applications, among which the perturbative solution of a fixed
point equation using the non linear geometric series. Then, following
Polchinski, we show how perturbative renormalization works for a non linear
perturbation of a linear differential equation that governs the flow of
effective actions. Then, we define a general Hopf algebra of Feynman diagrams
adapted to iterations of background field effective action computations. As a
simple combinatorial illustration, we show how these techniques can be used to
recover the universality of the Tutte polynomial and its relation to the
-state Potts model. As a more sophisticated example, we use ordered diagrams
with decorations and external structures to solve the Polchinski's exact
renormalization group equation. Finally, we work out an analogous construction
for the Schwinger-Dyson equations, which yields a bijection between planar
diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given
at the conference "Combinatorics and physics" held at Max Planck Institut
fuer Mathematik in Bonn in march 2007, some misprints correcte
Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds
We consider the spectral behavior and noncommutative geometry of commutators
, where is an operator of order with geometric origin and a
multiplication operator by a function. When is H\"{o}lder continuous, the
spectral asymptotics is governed by singularities. We study precise spectral
asymptotics through the computation of Dixmier traces; such computations have
only been considered in less singular settings. Even though a Weyl law fails
for these operators, and no pseudo-differential calculus is available,
variations of Connes' residue trace theorem and related integral formulas
continue to hold. On the circle, a large class of non-measurable Hankel
operators is obtained from H\"older continuous functions , displaying a wide
range of nonclassical spectral asymptotics beyond the Weyl law. The results
extend from Riemannian manifolds to contact manifolds and noncommutative tori.Comment: 40 page
Open-String Actions and Noncommutativity Beyond the Large-B Limit
In the limit of large, constant B-field (the ``Seiberg-Witten limit''), the
derivative expansion for open-superstring effective actions is naturally
expressed in terms of the symmetric products *n. Here, we investigate
corrections around the large-B limit, for Chern-Simons couplings on the brane
and to quadratic order in gauge fields. We perform a boundary-state computation
in the commutative theory, and compare it with the corresponding computation on
the noncommutative side. These results are then used to examine the possible
role of Wilson lines beyond the Seiberg-Witten limit. To quadratic order in
fields, the entire tree-level amplitude is described by a metric-dependent
deformation of the *2 product, which can be interpreted in terms of a deformed
(non-associative) version of the Moyal * product.Comment: 30 pages, harvma
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