Left-covariant first order differential calculus on quantum Hopf supersymmetry algebra

2021 ◽  
Vol 62 (3) ◽  
pp. 031702
Author(s):  
H. Fakhri ◽  
S. Laheghi
Keyword(s):  
Differential Calculus ◽  
Supersymmetry Algebra ◽  
First Order

scholarly journals Supersymmetry, T-duality and heterotic α′-corrections

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Eric Lescano ◽  
Carmen A. Núñez ◽  
Jesús A. Rodríguez
Keyword(s):  
Effective Field ◽  
Heterotic String ◽  
Supersymmetry Algebra ◽  
Yang Mills ◽  
First Order ◽  
Duality Symmetry ◽  
Transformation Rules ◽  
Higher Derivative ◽  
String Spectrum ◽  
Fermionic Fields

Abstract Higher-derivative interactions and transformation rules of the fields in the effective field theories of the massless string states are strongly constrained by space-time symmetries and dualities. Here we use an exact formulation of ten dimensional $$ \mathcal{N} $$ N = 1 supergravity coupled to Yang-Mills with manifest T-duality symmetry to construct the first order α′-corrections of the heterotic string effective action. The theory contains a supersymmetric and T-duality covariant generalization of the Green-Schwarz mechanism that determines the modifications to the leading order supersymmetry transformation rules of the fields. We compute the resulting field-dependent deformations of the coefficients in the supersymmetry algebra and construct the invariant action, with up to and including four-derivative terms of all the massless bosonic and fermionic fields of the heterotic string spectrum.

scholarly journals Restrictive metric regularity and generalized differential calculus in Banach spaces

2004 ◽  
Vol 2004 (50) ◽  
pp. 2653-2680 ◽  
Author(s):  
Boris S. Mordukhovich ◽  
Bingwu Wang
Keyword(s):  
Banach Spaces ◽  
Differential Calculus ◽  
Metric Regularity ◽  
Sufficient Conditions ◽  
Normal Cones ◽  
First Order ◽  
Nonconvex Sets ◽  
Generalized Differential ◽  
Necessary And Sufficient ◽  
Generalized Differential Calculus

We consider nonlinear mappingsf:X→Ybetween Banach spaces and study the notion ofrestrictive metric regularityoffaround some pointx¯, that is, metric regularity offfromXinto the metric spaceE=f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case whenfis strictly differentiable atx¯but its strict derivative∇f(x¯)is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.

scholarly journals DIVERGENCES ON PROJECTIVE MODULES AND NON-COMMUTATIVE INTEGRALS

2011 ◽  
Vol 08 (04) ◽  
pp. 885-896 ◽  
Author(s):  
TOMASZ BRZEZIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Hopf Algebras ◽  
Differential Calculus ◽  
Module Structure ◽  
Projective Modules ◽  
Finitely Generated ◽  
Integration By Parts ◽  
Left Module ◽  
First Order

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.

scholarly journals Korevaar–Schoen’s energy on strongly rectifiable spaces

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Nicola Gigli ◽  
Alexander Tyulenev
Keyword(s):  
Energy Density ◽  
Lower Semicontinuity ◽  
Differential Calculus ◽  
Representation Formula ◽  
Target Space ◽  
List Type ◽  
First Order ◽  
Schmidt Norm ◽  
Sobolev Maps ◽  
Limit Energy

AbstractWe extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an $$\mathsf{RCD}$$ RCD space. In this situation it appears that no version of the ‘subpartition lemma’ holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on: the fact that such spaces are ‘strongly rectifiable’ a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim’s metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density, the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential. When the target space is $$\mathsf{CAT}(0)$$ CAT ( 0 ) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.

scholarly journals THE 3D SPIN GEOMETRY OF THE QUANTUM TWO-SPHERE

2010 ◽  
Vol 22 (08) ◽  
pp. 963-993 ◽  
Author(s):  
SIMON BRAIN ◽  
GIOVANNI LANDI
Keyword(s):  
Quantum Group ◽  
Differential Calculus ◽  
Arbitrary Order ◽  
Three Dimensional ◽  
Spectral Triple ◽  
Frame Bundle ◽  
Order Condition ◽  
Spin Geometry ◽  
First Order ◽  
Calculus Formula

We study a three-dimensional differential calculus [Formula: see text] on the standard Podleś quantum two-sphere [Formula: see text], coming from the Woronowicz 4D+ differential calculus on the quantum group SU q(2). We use a frame bundle approach to give an explicit description of [Formula: see text] and its associated spin geometry in terms of a natural spectral triple over [Formula: see text]. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.

THE ASHTEKAR FORMULATION FOR CANONICAL N=2 SUPERGRAVITY

1992 ◽  
Vol 01 (03n04) ◽  
pp. 559-570 ◽  
Author(s):  
HIROSHI KUNITOMO ◽  
TAKASHI SANO
Keyword(s):  
Canonical Formalism ◽  
Poisson Brackets ◽  
Supersymmetry Algebra ◽  
First Order ◽  
Canonical Gravity ◽  
Left And Right ◽  
The Right ◽  
Polynomial Constraints

Ashtekar’s formulation of canonical gravity is extended to N=2 supergravity. The action has a complex first-order chiral form and is invariant under two local supersymmetry transformations. Since we adopt the chiral formulation, left- and right-handed supersymmetries appear in an asymmetric way. The supersymmetry algebra is not closed on auxiliary fields, which is a peculiar result of the chiral formulation, where the right-handed supersymmetries are realized by constrained parameters. By means of the canonical formalism, simple polynomial constraints are derived and we can show that the Poisson brackets of supersymmetry currents are actually closed. This guarantees the consistency of our supersymmetry transformations, which is unclear at the level of the algebra on the fields.

Sign in / Sign up

Export Citation Format

Share Document