The geometry of orbit-space and natural minima of Higgs potentials

Abud, M.; Sartori, G.

doi:10.1016/0370-2693(81)90578-5

Cited by 79 publications

(77 citation statements)

References 5 publications

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“…This has been conjectured by Abud & Sartori [1,2]. See [19] for the proof; applications of this theorem to gauge symmetry breaking are given in [20].…”

Section: Theorem 4 the Orbit Space Is Given By ρ(V ) = {Z ∈ Z G(z) Imentioning

confidence: 73%

“…The nontrivial groups (strata) , their restriction d to the diagonal, o as the restriction to off-diagonal elements, the orbit length o of the entire group , the number of pre-images of the primary invariants p (= the orbit length of , acting on the cross product of the space of diagonals and the space of off-diagonal), and the maximal number of leaves of this stratum, l. symmetry group. In the theory of Higgs potentials, the connection between symmetry breaking, energy functions and the orbit space has been known for some time (see the work by Abud & Sartori [1,2]). For phase transitions in crystals, however, the situation is usually more complicated, since the orbit space is high-dimensional.…”

Section: Discussionmentioning

confidence: 99%

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Stored Energy Functions for Phase Transitions in Crystals

Zimmer

2004

Archive for Rational Mechanics and Analysis

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A method is presented to construct nonconvex free energies that are invariant under a symmetry group. Algebraic and geometric methods are used to determine invariant functions with the right location of minimizers. The methods are illustrated for symmetry-breaking martensitic phase transformations. Computer algebra is used to compute a basis of the corresponding class of invariant functions. Several phase transitions, such as cubic-to-orthorhombic, are discussed. An explicit example of an energy for the cubic-to-tetragonal phase transition is given.

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“…This has been conjectured by Abud & Sartori [1,2]. See [19] for the proof; applications of this theorem to gauge symmetry breaking are given in [20].…”

Section: Theorem 4 the Orbit Space Is Given By ρ(V ) = {Z ∈ Z G(z) Imentioning

confidence: 73%

Section: Discussionmentioning

confidence: 99%

Stored Energy Functions for Phase Transitions in Crystals

Zimmer

2004

Archive for Rational Mechanics and Analysis

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“…We will use Abud-Sartori formulation of Michel's theorem [7][8][9]. Stationary points of V are points for which gradient of V is equal to zero.…”

Section: Stationary Points Of V From Abud-sartori-michel Theoremmentioning

confidence: 99%

“…Another advantage is that the problem of finding minima of V becomes another case of the spontaneous symmetry breaking phenomenon. This allows us to use the Abud-Sartori-Michel theorem [7][8][9][10] to discuss some stationary points of V. In the next part of the paper we will use symmetries we found to derive lowest order approximate expressions for the function E and to discuss possible shapes of some molecular types that correspond to minima of V. Already in this simplest approximation one successfully obtains some of molecular shapes that exist in nature. Our approach allows finding approximate relations between bond lengths in a molecule and its vibration frequencies.…”

Section: Introductionmentioning

confidence: 99%

Structure and dynamics of X_n-type clusters (n= 3, 4, 6) from spontaneous symmetry breaking theory

Damljanović¹

2013

Phys. Scr.

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Abstract. In a molecule subjected to no external fields, motion of nuclei is governed by a function V of nuclear coordinates. This function (potential energy) is a sum of two terms: Coulomb repulsion between nuclei and the electronic effective potential E which results from the BornOppenheimer approximation. In this paper we have presented a group of coordinate transformations which are the symmetries of functions E and V. We showed that the formula for dynamical representation, which has fundamental importance in the symmetry analysis of normal modes of a molecule, follows from these symmetries. In addition, we proved that every molecule has at least one normal mode belonging to the totally symmetric (and therefore Raman-active) irreducible representation of the point group of that molecule. Next, we used symmetries of V and E to analyze possible shapes of some molecule types. We applied both Abud-Sartori-Michel theorem and symmetry adapted expansion of the electronic effective potential around united atom. Finally, we derived an approximate relation which predicts order of magnitude of the vibration frequency from the bond length in a diatomic molecule.

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“…M W ⊂ M can be constructed in steps by finding the stationary points of the restriction ofŴ to Σ (H) , one stratum at a time. This useful fact, pointed out in [1], follows from results of Luna [6], Abud and Sartori [7], Procesi and Schwarz [1,8]. In this paper we elaborate further on these results and obtain an algorithm to construct M W which, in some cases, saves us the job of looking for critical points in every stratum, but only on some carefully chosen ones.…”

Section: Introductionmentioning

confidence: 95%

Higgs mechanism and Luna strata in gauge theories

Dotti

2000

Nuclear Physics B

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The classical moduli space M of a supersymmetric gauge theory with trivial superpotential can be stratified according to the unbroken gauge subgroup at different vacua. We apply known results about this stratification to obtain the W = 0 theory classical moduli space M W ⊂ M, working entirely with the composite gauge invariant operatorsφ that span M, assuming we do not know their elementary matter chiral field content. In this construction, the patterns of gauge symmetry breaking of the W = 0 zero theory are determined, Higgs flows in these theories show important differences from the W = 0 case. The methods here introduced provide an alternative way to construct tree level superpotentials that lift all classical flat directions leaving a candidate theory for dynamical supersymmetry breaking, and are also useful to identify heavy composite fields to integrate out from effective superpotentials when the elementary field content of the composites is unknown. We also show how to recognize the massless singlets after Higgs mechanism at a vacuumφ ∈ M W among the moduli δφ using the stratification of M, and establish conditions under which the space of non singlet massless fields after Higgs mechanism (unseen as moduli δφ) is null. A small set of theories with so called "unstable" representations of the complexified gauge group is shown to exhibit unexpected properties regarding the dimension of their moduli space, and the presence of non singlet massless fields after Higgs mechanism at all of their vacua.

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The geometry of orbit-space and natural minima of Higgs potentials

Cited by 79 publications

References 5 publications

Stored Energy Functions for Phase Transitions in Crystals

Stored Energy Functions for Phase Transitions in Crystals

Structure and dynamics of X_n-type clusters (n= 3, 4, 6) from spontaneous symmetry breaking theory

Higgs mechanism and Luna strata in gauge theories

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The geometry of orbit-space and natural minima of Higgs potentials

Cited by 79 publications

References 5 publications

Stored Energy Functions for Phase Transitions in Crystals

Stored Energy Functions for Phase Transitions in Crystals

Structure and dynamics of Xn-type clusters (n= 3, 4, 6) from spontaneous symmetry breaking theory

Higgs mechanism and Luna strata in gauge theories

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Product

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Structure and dynamics of X_n-type clusters (n= 3, 4, 6) from spontaneous symmetry breaking theory