Search results for "REPRESENTATION"
showing 10 items of 1710 documents
Differential equations for loop integrals in Baikov representation
2018
We present a proof that differential equations for Feynman loop integrals can always be derived in Baikov representation without involving dimension-shift identities. We moreover show that in a large class of two- and three-loop diagrams it is possible to avoid squared propagators in the intermediate steps of setting up the differential equations.
DEFORMATION QUANTIZATION OF COADJOINT ORBITS
2000
A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored.
Causal representation of multi-loop Feynman integrands within the loop-tree duality
2021
The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops an…
Contractions of Filippov algebras
2010
We introduce in this paper the contractions $\mathfrak{G}_c$ of $n$-Lie (or Filippov) algebras $\mathfrak{G}$ and show that they have a semidirect structure as their $n=2$ Lie algebra counterparts. As an example, we compute the non-trivial contractions of the simple $A_{n+1}$ Filippov algebras. By using the \.In\"on\"u-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the $\mathfrak{G}=A_{n+1}$ simple case) the Lie algebras Lie$\,\mathfrak{G}_c$ (the Lie algebra of inner endomorphisms of $\mathfrak{G}_c$) with certain contractions $(\mathrm{Lie}\,\mathfrak{G})_{IW}$ and $(\mathrm{Lie}\,\mathfrak{G})_{W-W}$ of the Lie algebra Lie$\,\mathfrak{G}$ as…
The Minkowski and conformal superspaces
2006
We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra.
On the maximal superalgebras of supersymmetric backgrounds
2009
17 pages.-- ISI article identifier:000262585300016.-- ArXiv pre-print avaible at:http://arxiv.org/abs/0809.5034
Quantum and Braided Integrals
2001
We give a pedagogical introduction to integration techniques appropriate for non-commutative spaces while presenting some new results as well. A rather detailed discussion outlines the motivation for adopting the Hopf algebra language. We then present some trace formulas for the integral on Hopf algebras and show how to treat the $\int 1=0$ case. We extend the discussion to braided Hopf algebras relying on diagrammatic techniques. The use of the general formulas is illustrated by explicitly worked out examples.
The $q$-calculus for generic $q$ and $q$ a root of unity
1996
The $q$-calculus for generic $q$ is developed and related to the deformed oscillator of parameter $q^{1/2}$. By passing with care to the limit in which $q$ is a root of unity, one uncovers the full algebraic structure of ${{\cal Z}}_n$-graded fractional supersymmetry and its natural representation.
Domain walls in supersymmetric QCD: The taming of the zoo
2000
We provide a unified picture of the domain wall spectrum in supersymmetric QCD with Nc colors and Nf flavors of quarks in the (anti-) fundamental representation. Within the framework of the Veneziano-Yankielowicz-Taylor effective Lagrangian, we consider domain walls connecting chiral symmetry breaking vacua, and we take the quark masses to be degenerate. For Nf/Ncm** there is no domain wall. We numerically determine m* and m** as a function of Nf/Nc, and we find that m** approaches a constant value in the limit that this ratio goes to one.
Instanton Counting, Quantum Geometry and Algebra
2020
The aim of this memoir for "Habilitation \`a Diriger des Recherches" is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how to obtain the instanton partition function and its generalization to quiver gauge theory and supergroup gauge theory in three ways: the equivariant index formula, the contour integral formula, and the combinatorial formula. We then explore the geometric description of $\mathcal{N} = 2$ gauge theory based on Seiberg-Witten geometry together with it…