10,293 research outputs found
An Algebraic Classification of Exceptional EFTs Part II: Supersymmetry
We present a novel approach to classify supersymmetric effective field
theories (EFTs) whose scattering amplitudes exhibit enhanced soft limits. These
enhancements arise due to non-linearly realised symmetries on the Goldstone
modes of such EFTs and we classify the algebras that these symmetries can form.
Our main focus is on so-called exceptional algebras which lead to
field-dependent transformation rules and EFTs with the maximum possible soft
enhancement at a given derivative power counting. We adapt existing techniques
for Poincar\'{e} invariant theories to the supersymmetric case, and introduce
superspace inverse Higgs constraints as a method of reducing the number of
Goldstone modes while maintaining all symmetries.
Restricting to the case of a single Goldstone supermultiplet in four
dimensions, we classify the exceptional algebras and EFTs for a chiral, Maxwell
or real linear supermultiplet. Moreover, we show how our algebraic approach
allows one to read off the soft weights of the different component fields from
superspace inverse Higgs trees, which are the algebraic cousin of the on-shell
soft data one provides to soft bootstrap EFTs using on-shell recursion. Our
Lie-superalgebraic approach extends the results of on-shell methods and
provides a complementary perspective on non-linear realisations
Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
We describe a unification of several apparently unrelated factorizations
arisen from quantum field theory, vertex operator algebras, combinatorics and
numerical methods in differential equations. The unification is given by a
Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff
formula in our study of the Hopf algebra approach of Connes and Kreimer to
renormalization in perturbative quantum field theory. There we showed that the
Birkhoff decomposition of Connes and Kreimer can be obtained from a certain
Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter
operator. We will explain how the same decomposition generalizes the
factorization of formal exponentials and uniformization for Lie algebras that
arose in vertex operator algebra and conformal field theory, and the even-odd
decomposition of combinatorial Hopf algebra characters as well as to the Lie
algebra polar decomposition as used in the context of the approximation of
matrix exponentials in ordinary differential equations.Comment: accepted for publication in Comm. in Math. Phy
Dynkin operators and renormalization group actions in pQFT
Renormalization techniques in perturbative quantum field theory were known,
from their inception, to have a strong combinatorial content emphasized, among
others, by Zimmermann's celebrated forest formula. The present article reports
on recent advances on the subject, featuring the role played by the Dynkin
operators (actually their extension to the Hopf algebraic setting) at two
crucial levels of renormalization, namely the Bogolioubov recursion and the
renormalization group (RG) equations. For that purpose, an iterated integrals
toy model is introduced to emphasize how the operators appear naturally in the
setting of renormalization group analysis. The toy model, in spite of its
simplicity, captures many key features of recent approaches to RG equations in
pQFT, including the construction of a universal Galois group for quantum field
theories
Initial-seed recursions and dualities for d-vectors
We present an initial-seed-mutation formula for d-vectors of cluster
variables in a cluster algebra. We also give two rephrasings of this recursion:
one as a duality formula for d-vectors in the style of the g-vectors/c-vectors
dualities of Nakanishi and Zelevinsky, and one as a formula expressing the
highest powers in the Laurent expansion of a cluster variable in terms of the
d-vectors of any cluster containing it. We prove that the initial-seed-mutation
recursion holds in a varied collection of cluster algebras, but not in general.
We conjecture further that the formula holds for source-sink moves on the
initial seed in an arbitrary cluster algebra, and we prove this conjecture in
the case of surfaces.Comment: 21 Pages, 20 Figures. Version 2: Expanded introduction, other minor
expository changes. Version 3: Very minor corrections. Final version to
appear in the Pacific Journal of Mathematic
Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
In this article we continue to explore the notion of Rota-Baxter algebras in
the context of the Hopf algebraic approach to renormalization theory in
perturbative quantum field theory. We show in very simple algebraic terms that
the solutions of the recursively defined formulae for the Birkhoff
factorization of regularized Hopf algebra characters, i.e. Feynman rules,
naturally give a non-commutative generalization of the well-known Spitzer's
identity. The underlying abstract algebraic structure is analyzed in terms of
complete filtered Rota-Baxter algebras.Comment: 19 pages, 2 figure
Mathematics of CLIFFORD - A Maple package for Clifford and Grassmann algebras
CLIFFORD performs various computations in Grassmann and Clifford algebras. It
can compute with quaternions, octonions, and matrices with entries in Cl(B) -
the Clifford algebra of a vector space V endowed with an arbitrary bilinear
form B. Two user-selectable algorithms for Clifford product are implemented:
'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on
non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used.
Properties of reversion in undotted and dotted wedge bases are discussed.Comment: 24 pages, update contains new material included in published versio
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