3,803 research outputs found
A Hopf laboratory for symmetric functions
An analysis of symmetric function theory is given from the perspective of the
underlying Hopf and bi-algebraic structures. These are presented explicitly in
terms of standard symmetric function operations. Particular attention is
focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and
twisted products (Rota cliffordizations) induced by branching operators in the
symmetric function context. The latter are shown to include the algebras of
symmetric functions of orthogonal and symplectic type. A commentary on related
issues in the combinatorial approach to quantum field theory is given.Comment: 29 pages, LaTeX, uses amsmat
Lattice worldline representation of correlators in a background field
We use a discrete worldline representation in order to study the continuum
limit of the one-loop expectation value of dimension two and four local
operators in a background field. We illustrate this technique in the case of a
scalar field coupled to a non-Abelian background gauge field. The first two
coefficients of the expansion in powers of the lattice spacing can be expressed
as sums over random walks on a d-dimensional cubic lattice. Using combinatorial
identities for the distribution of the areas of closed random walks on a
lattice, these coefficients can be turned into simple integrals. Our results
are valid for an anisotropic lattice, with arbitrary lattice spacings in each
direction.Comment: 54 pages, 14 figure
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
A Proof of a Generalization of Niven\u27s Theorem Using Algebraic Number Theory
Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary quadratic rings. We then show for which rational multiples of π the squares of the sine, cosine, and tangent functions are rational, providing a generalized form of Niven’s theorem. We end with a discussion of a few related combinatorial identities
String Partition Functions, Hilbert Schemes, and Affine Lie Algebra Representations on Homology Groups
This review paper contains a concise introduction to highest weight
representations of infinite dimensional Lie algebras, vertex operator algebras
and Hilbert schemes of points, together with their physical applications to
elliptic genera of superconformal quantum mechanics and superstring models. The
common link of all these concepts and of the many examples considered in the
paper is to be found in a very important feature of the theory of infinite
dimensional Lie algebras: the modular properties of the characters (generating
functions) of certain representations. The characters of the highest weight
modules represent the holomorphic parts of the partition functions on the torus
for the corresponding conformal field theories. We discuss the role of the
unimodular (and modular) groups and the (Selberg-type) Ruelle spectral
functions of hyperbolic geometry in the calculation of elliptic genera and
associated -series. For mathematicians, elliptic genera are commonly
associated to new mathematical invariants for spaces, while for physicists
elliptic genera are one-loop string partition function (therefore they are
applicable, for instance, to topological Casimir effect calculations). We show
that elliptic genera can be conveniently transformed into product expressions
which can then inherit the homology properties of appropriate polygraded Lie
algebras.Comment: 56 pages, review paper, in honour of J.S.Dowker. arXiv admin note:
text overlap with arXiv:0905.1285, arXiv:math/0006201, arXiv:math/0412089,
arXiv:math/0403547 by other author
The Combinatorics of Iterated Loop Spaces
It is well known since Stasheff's work that 1-fold loop spaces can be
described in terms of the existence of higher homotopies for associativity
(coherence conditions) or equivalently as algebras of contractible
non-symmetric operads. The combinatorics of these higher homotopies is well
understood and is extremely useful.
For the theory of symmetric operads encapsulated the corresponding
higher homotopies, yet hid the combinatorics and it has remain a mystery for
almost 40 years. However, the recent developments in many fields ranging from
algebraic topology and algebraic geometry to mathematical physics and category
theory show that this combinatorics in higher dimensions will be even more
important than the one dimensional case.
In this paper we are going to show that there exists a conceptual way to make
these combinatorics explicit using the so called higher nonsymmetric
-operads.Comment: 23 page
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