scholarly journals The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang–Mills theory on a Minkowskian background

2020 ◽  
Vol 17 (08) ◽  
pp. 2050114
Author(s):  
Luca Accornero ◽  
Marcella Palese
Keyword(s):  
Field Theory ◽  
Geometric Structure ◽  
Arbitrary Order ◽  
Higher Order ◽  
Second Variation ◽  
Jacobi Fields ◽  
Yang Mills ◽  
Conserved Current ◽  
The Second Variation ◽  
Higher Order Lagrangian

We characterize the second variation of an higher order Lagrangian by a Jacobi morphism and by currents strictly related to the geometric structure of the variational problem. We discuss the relation between the Jacobi morphism and the Hessian at an arbitrary order. Furthermore, we prove that a pair of Jacobi fields always generates a (weakly) conserved current. An explicit example is provided for a Yang–Mills theory on a Minkowskian background.

Geometric constrained variational calculus. II: The second variation (Part I)

2016 ◽  
Vol 13 (01) ◽  
pp. 1550132 ◽  
Author(s):  
Enrico Massa ◽  
Danilo Bruno ◽  
Gianvittorio Luria ◽  
Enrico Pagani
Keyword(s):  
Gauge Transformation ◽  
Sufficient Conditions ◽  
Variational Calculus ◽  
Second Variation ◽  
Action Functional ◽  
Covariant Representation ◽  
Jacobi Fields ◽  
Necessary And Sufficient ◽  
The Second Variation ◽  
Variation Part

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

scholarly journals On the notion of Jacobi fields in constrained calculus of variations

2016 ◽  
Vol 24 (2) ◽  
pp. 91-113
Author(s):  
Enrico Massa ◽  
Enrico Pagani
Keyword(s):  
Variational Calculus ◽  
Second Variation ◽  
Action Functional ◽  
Jacobi Fields ◽  
Gauge Transformations ◽  
Groups Of Diffeomorphisms ◽  
Relevant Role ◽  
Fixed End ◽  
The Second Variation

Abstract In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of local gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' strengths [16]. In dis- cussing the positivity of the second variation, a relevant role is played by the Jacobi fields, defined as infinitesimal generators of 1-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this point is presented. An alternative characterization of the Jacobi fields as solutions of a suitable accessory variational problem is established.

On the geometric structure of classical field theory in Lagrangian formulation

1970 ◽  
Vol 68 (2) ◽  
pp. 475-484 ◽  
Author(s):  
Jędrzej Śniatycki
Keyword(s):  
Field Theory ◽  
Conservation Laws ◽  
Geometric Structure ◽  
Classical Field Theory ◽  
Higher Order ◽  
Classical Field ◽  
Lagrangian Formulation ◽  
Symmetry Transformations

AbstractGeometric structure of classical field theory in Lagrangian formulation is investigated. Symmetry transformations with generators depending on higher-order derivatives are considered and the corresponding conservation laws are obtained.

Geometric constrained variational calculus. III: The second variation (Part II)

2016 ◽  
Vol 13 (04) ◽  
pp. 1650038
Author(s):  
Enrico Massa ◽  
Gianvittorio Luria ◽  
Enrico Pagani
Keyword(s):  
Sufficient Conditions ◽  
Variational Calculus ◽  
Second Variation ◽  
Action Functional ◽  
Covariant Representation ◽  
Jacobi Fields ◽  
Gauge Transformations ◽  
Alternative Approach ◽  
The Second Variation ◽  
Variation Part

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.

On a gauge-invariant functional

2010 ◽  
Vol 53 (1) ◽  
pp. 143-151
Author(s):  
Cătălin Gherghe
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Compact Riemannian Manifold ◽  
Second Variation ◽  
Gauge Invariant ◽  
Variation Formula ◽  
First Variation ◽  
Yang Mills ◽  
Invariant Functional ◽  
The Second Variation

AbstractWe define a new functional which is gauge invariant on the space of all smooth connections of a vector bundle over a compact Riemannian manifold. This functional is a generalization of the classical Yang-Mills functional. We derive its first variation formula and prove the existence of critical points. We also obtain the second variation formula.

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

scholarly journals Celestial superamplitude in $$ \mathcal{N} $$ = 4 SYM theory

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Hongliang Jiang
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Scattering Amplitudes ◽  
Momentum Space ◽  
Ward Identity ◽  
Yang Mills ◽  
Conformal Field ◽  
New Perspective ◽  
Celestial Sphere

Abstract Celestial amplitude is a new reformulation of momentum space scattering amplitudes and offers a promising way for flat holography. In this paper, we study the celestial amplitudes in $$ \mathcal{N} $$ N = 4 Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of $$ \mathcal{N} $$ N = 4 SYM theory. We also compute the three-point and four-point celestial super-amplitudes explicitly. They can be identified as the conformal correlation functions of the celestial superfields living at the celestial sphere. We further study the soft and collinear limits which give rise to the super-Ward identity and super-OPE on the celestial sphere, respectively. Our results initiate a new perspective of understanding the well-studied $$ \mathcal{N} $$ N = 4 SYM amplitudes via 2D celestial conformal field theory.

scholarly journals Chiral correlators in $$ \mathcal{N} $$ = 2 superconformal quivers

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Francesco Galvagno ◽  
Michelangelo Preti
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Flat Space ◽  
Field Theories ◽  
Superconformal Field Theories ◽  
Four Dimensions ◽  
Yang Mills ◽  
The Matrix ◽  
Superspace Formalism ◽  
Model Approach

Abstract We consider a family of $$ \mathcal{N} $$ N = 2 superconformal field theories in four dimensions, defined as ℤq orbifolds of $$ \mathcal{N} $$ N = 4 Super Yang-Mills theory. We compute the chiral/anti-chiral correlation functions at a perturbative level, using both the matrix model approach arising from supersymmetric localisation on the four-sphere and explicit field theory calculations on the flat space using the $$ \mathcal{N} $$ N = 1 superspace formalism. We implement a highly efficient algorithm to produce a large number of results for finite values of N , exploiting the symmetries of the quiver to reduce the complexity of the mixing between the operators. Finally the interplay with the field theory calculations allows to isolate special observables which deviate from $$ \mathcal{N} $$ N = 4 only at high orders in perturbation theory.

Sign in / Sign up

Export Citation Format

Share Document