We discuss the (non)-restoration of global and local symmetries at high temperature. First, we analyze a two-scalar model with Z2×Z2 symmetry using the exact renormalization group. We conclude that inverse symmetry breaking is possible in this kind of models within the perturbative regime. Regarding local symmetries, we consider the SU(2)⊗U(1) gauge symmetry and focus on the case of a strongly ...

Chiral perturbation theory is a very general expansion method which can be applied to any dynamical system which has continuous global symmetries and in which the ground state breaks some of these spontaneously. In these lectures we explain at a basic level and in detail how such symmetries are identified in the case of the QCD Lagrangian and describe the steps which are involved in practice in...

Two REDUCE programs are presented which should be of assistance in computing and studying so-called adjoint symmetries of second-order ordinary differential equations. The first program essentially serves to construct the determining equations for adjoint symmetries, whose leading coefficients are allowed to be polynomial functions of the velocities. The second program runs various tests concer...

In this paper we consider the relation between symmetries and first integrals of canonical Hamiltonian equations. Based on a newly established identity (which is an analog of well known Noether’s identity for Lagrangian approach), this approach provides a simple and clear way to construct first integrals with the help of symmetries of a Hamiltonian. The approach is illustrated by a number of ex...

II Symmetries in Classical Particle Mechanics 4 1 Hamiltonian and Lagrangian Formulations of Classical Particle Mechanics 5 1.1 Basic De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Generalized...

In her famous paper [14] Emmy Noether proved that each Noether symmetry associated with a Lagrangian generates a conservation law. (Modern treatment of relationship between components of Noether conserved vector and Lie–Bäcklund operators which are Noether symmetries was adduced in [10,18]. Kara et al [12,13] constructed conservation laws of some classes of PDEs with two independent variables u...

We determine the gauge symmetries of all p–forms in maximal threedimensional gauged supergravity (0 ≤ p ≤ 3) by requiring invariance of the Lagrangian. It is shown that in a particular ungauged limit these symmetries are in precise correspondence to those predicted by the very-extended KacMoody algebra E11. We demonstrate that whereas in the ungauged limit the bosonic gauge algebra closes off-s...

This paper investigates the d = 4, N = 4 Abelian, global Super-Yang Mills system (SUSY-YM). It is shown how the N = 2 Fayet Hypermultiplet (FH) and N = 2 vector multiplet (VM) are embedded within. The central charges and internal symmetries provide a plethora of information as to further symmetries of the Lagrangian. Several of these symmetries are calculated to second order. It is hoped that i...

Duality symmetries are discussed for non-linear gauge theories of (n − 1)-th rank antisymmetric tensor fields in general even dimensions d = 2n. When there are M field strengths and no scalar fields, the duality symmetry groups should be compact. We find conditions on the Lagrangian required by compact duality symmetries and show an example of duality invariant non-linear theories. We also disc...