LieART—A Mathematica application for Lie algebras and representation theory

Feger, Robert; Kephart, Thomas W.

doi:10.1016/j.cpc.2014.12.023

Cited by 172 publications

(213 citation statements)

References 17 publications

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“…This leads to non-minimal models with a larger set of fields and bigger cosets. Good sources for the group theoretical material required in some of the calculations are [17][18][19]. In the following, we will denote specific irreps either by their dimensionality or by the symbols F, S n , A n , Ad and Spin for the fundamental, n−symmetric, n−antisymmetric, adjoint and spin.…”

Section: Solution To the Constraintsmentioning

confidence: 99%

Fermionic UV completions of composite Higgs models

Ferretti

Karateev

2014

J. High Energ. Phys.

290

439

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Abstract:We classify the four-dimensional purely fermionic gauge theories that give a UV completion of composite Higgs models. Our analysis is at the group theoretical level, addressing the necessary (but not sufficient) conditions for the viability of these models, such as the existence of top partners and custodial symmetry. The minimal cosets arising are those of type SU(5)/SO(5) and SU(4)/Sp(4). We list all the possible "hyper-color" groups allowed and point out the simplest and most promising ones.

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Section: Solution To the Constraintsmentioning

confidence: 99%

Fermionic UV completions of composite Higgs models

Ferretti

Karateev

2014

J. High Energ. Phys.

290

439

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“…We label the nodes of the E 6 Dynkin diagram by: These are the names used in the Mathematica package LieART [33], which we used to perform computations.…”

Section: A E 6 Representationsmentioning

confidence: 99%

BPS states in the Minahan-Nemeschansky $${E_6}$$ theory

Hollands¹,

Neitzke

2016

Commun. Math. Phys.

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We use the method of spectral networks to compute BPS state degeneracies in the MinahanNemeschansky E 6 theory, on its Coulomb branch, without turning on a mass deformation. The BPS multiplicities come out in representations of the E 6 flavor symmetry. For example, along the simplest ray in electromagnetic charge space, we give the first 14 numerical degeneracies, and the first 7 degeneracies as representations of E 6 . We find a complicated spectrum, exhibiting exponential growth of multiplicities as a function of the electromagnetic charge. There is one unexpected outcome: the spectrum is consistent (in a nontrivial way) with the hypothesis of spin purity, that if a BPS state in this theory has electromagnetic charge equal to n times a primitive charge, then it appears in a spin-n 2 multiplet.

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“…Therefore the representation R must be irreducible and it must be the adjoint representation since all other representations are of larger dimension. For N ≤ 3 one observes from explicit tables of group dimensions [58] that the same reasoning applies and R again equals the adjoint representation. The maximality of PSU(2 N ) in SO(4 N − 1) now directly follows from Dynkin's theorem [59] and the fact that PSU(2 N ) is simple and its adjoint representation is faithful and irreducible (acting on the fundamental representation space of SO (4 N − 1)).…”

Section: B23 Pentagon Conservation Equations For N = 2 Qubitsmentioning

confidence: 98%

Quantum theory from questions

2017

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We reconstruct the explicit formalism of qubit quantum theory from elementary rules on an observer's information acquisition. Our approach is purely operational: we consider an observer O interrogating a system S with binary questions and define S's state as O's 'catalogue of knowledge' about S. From the rules we derive the state spaces for N elementary systems and show that (a) they coincide with the set of density matrices over an N -qubit Hilbert space C 2 N ; (b) states evolve unitarily under the group PSU(2 N ) according to the von Neumann evolution equation; and (c) O's binary questions correspond to projective Pauli operator measurements with outcome probabilities given by the Born rule. As a by-product, this results in a propositional formulation of quantum theory. Aside from offering an informational explanation for the theory's architecture, the reconstruction also unravels new structural insights. We show that, in a derived quadratic information measure, (d) qubits satisfy inequalities which bound the information content in any set of mutually complementary questions to 1 bit; and (e) maximal sets of mutually complementary questions for one and two qubits must carry precisely 1 bit of information in pure states. The latter relations constitute conserved informational charges which define the unitary groups and, together with their conservation conditions, the sets of pure quantum states. These results highlight information as a 'charge of quantum theory' and the benefits of this informational approach. This work emphasizes the sufficiency of restricting to an observer's information to reconstruct the theory and completes the quantum reconstruction initiated in [1].

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LieART—A Mathematica application for Lie algebras and representation theory

Cited by 172 publications

References 17 publications

Fermionic UV completions of composite Higgs models

Fermionic UV completions of composite Higgs models

BPS states in the Minahan-Nemeschansky $${E_6}$$ theory

Quantum theory from questions

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