4,920 research outputs found
Conversions between barycentric, RKFUN, and Newton representations of rational interpolants
We derive explicit formulas for converting between rational interpolants in
barycentric, rational Krylov (RKFUN), and Newton form. We show applications of
these conversions when working with rational approximants produced by the AAA
algorithm [Y. Nakatsukasa, O. S\`ete, L. N. Trefethen, arXiv preprint
1612.00337, 2016] within the Rational Krylov Toolbox and for the solution of
nonlinear eigenvalue problems
Exclusion Statistics in Conformal Field Theory -- generalized fermions and spinons for level-1 WZW theories
We systematically study the exclusion statistics for quasi-particles for
Conformal Field Theory spectra by employing a method based on recursion
relations for truncated spectra. Our examples include generalized fermions in
c<1 unitary minimal models, Z_k parafermions, and spinons for the su(n)_1,
so(n)_1 and sp(2n)_1 Wess-Zumino-Witten models. For some of the latter examples
we present explicit expressions for finitized affine characters and for the
N-spinon decomposition of affine characters.Comment: LaTeX, 49 pages, some further clarifications added, refs added and
updated. Nucl. Phys. B, to be publishe
Deformed N=2 theories, generalized recursion relations and S-duality
We study the non-perturbative properties of N=2 super conformal field
theories in four dimensions using localization techniques. In particular we
consider SU(2) gauge theories, deformed by a generic epsilon-background, with
four fundamental flavors or with one adjoint hypermultiplet. In both cases we
explicitly compute the first few instanton corrections to the partition
function and the prepotential using Nekrasov's approach. These results allow to
reconstruct exact expressions involving quasi-modular functions of the bare
gauge coupling constant and to show that the prepotential terms satisfy a
modular anomaly equation that takes the form of a recursion relation with an
explicitly epsilon-dependent term. We then investigate the implications of this
recursion relation on the modular properties of the effective theory and find
that with a suitable redefinition of the prepotential and of the effective
coupling it is possible, at least up to the third order in the deformation
parameters, to cast the S-duality relations in the same form as they appear in
the Seiberg-Witten solution of the undeformed theory.Comment: 33 pages, no figures, LaTeX2
On the determinant representations of Gaudin models' scalar products and form factors
We propose alternative determinant representations of certain form factors
and scalar products of states in rational Gaudin models realized in terms of
compact spins. We use alternative pseudo-vacuums to write overlaps in terms of
partition functions with domain wall boundary conditions. Contrarily to
Slavnovs determinant formulas, this construction does not require that any of
the involved states be solutions to the Bethe equations; a fact that could
prove useful in certain non-equilibrium problems. Moreover, by using an
atypical determinant representation of the partition functions, we propose
expressions for the local spin raising and lowering operators form factors
which only depend on the eigenvalues of the conserved charges. These
eigenvalues define eigenstates via solutions of a system of quadratic equations
instead of the usual Bethe equations. Consequently, the current work allows
important simplifications to numerical procedures addressing decoherence in
Gaudin models.Comment: 15 pages, 0 figures, Published versio
O(N) and O(N) and O(N)
Three related analyses of theory with symmetry are presented.
In the first, we review the model over the -adic numbers and the
discrete renormalization group transformations which can be understood as spin
blocking in an ultrametric context. We demonstrate the existence of a
Wilson-Fisher fixed point using an expansion, and we show how to
obtain leading order results for the anomalous dimensions of low dimension
operators near the fixed point. Along the way, we note an important aspect of
ultrametric field theories, which is a non-renormalization theorem for kinetic
terms. In the second analysis, we employ large methods to establish
formulas for anomalous dimensions which are valid equally for field theories
over the -adic numbers and field theories on . Results for
anomalous dimensions agree between the first and second analyses when they can
be meaningfully compared. In the third analysis, we consider higher derivative
versions of the model on , the simplest of which has been
studied in connection with spatially modulated phases. Our general formula for
anomalous dimensions can still be applied. Analogies with two-derivative
theories hint at the existence of some interesting unconventional field
theories in four real Euclidean dimensions.Comment: 44 pages, 8 figure
Renormalization: a quasi-shuffle approach
In recent years, the usual BPHZ algorithm for renormalization in perturbative
quantum field theory has been interpreted, after dimensional regularization, as
a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs,
with values in a Rota-Baxter algebra of amplitudes. We associate in this paper
to any such algebra a universal semi-group (different in nature from the
Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes
associated to Feynman graphs produces the expected operations: Bogoliubov's
preparation map, extraction of divergences, renormalization. In this process a
key role is played by commutative and noncommutative quasi-shuffle bialgebras
whose universal properties are instrumental in encoding the renormalization
process
A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
The Bogoliubov recursion is a particular procedure appearing in the process
of renormalization in perturbative quantum field theory. It provides convergent
expressions for otherwise divergent integrals. We develop here a theory of
functional identities for noncommutative Rota-Baxter algebras which is shown to
encode, among others, this process in the context of Connes-Kreimer's Hopf
algebra of renormalization. Our results generalize the seminal Cartier-Rota
theory of classical Spitzer-type identities for commutative Rota-Baxter
algebras. In the classical, commutative, case, these identities can be
understood as deriving from the theory of symmetric functions. Here, we show
that an analogous property holds for noncommutative Rota-Baxter algebras. That
is, we show that functional identities in the noncommutative setting can be
derived from the theory of noncommutative symmetric functions. Lie idempotents,
and particularly the Dynkin idempotent play a crucial role in the process.
Their action on the pro-unipotent groups such as those of perturbative
renormalization is described in detail along the way.Comment: improved version, accepted for publication in the Journal of
Noncommutative Geometr
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