Anomaly-free supersymmetric $ \frac{{{\text{SO}}\left( {2N + 2} \right)}}{{{\text{U}}\left( {N + 1} \right)}} $ σ-model based on the SO(2N + 1) Lie algebra of the fermion operators

Nishiyama, Seiya; Providência, João da; Providência, Constança; Cordeiro, Flávio

doi:10.1007/jhep02(2011)093

Cited by 2 publications

(10 citation statements)

References 27 publications

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“…As already shown for a fermion system [38,39], it is also easily verified that the result of (7.9) satisfies the gradient of the real function M coset space obtained by van Holten et al [41] and also by the present authors et. al.…”

Section: Scf Condition and Mfh Linear In Jacobi Generatorsupporting

confidence: 89%

“…On the other hand, we also have obtained several years ago the matrixtype Riccati-Hartree-Bogoliubov equation on the compact symmetric coset space SO(2N ) U (N ) for a fermion system accompanying with a general two-body interaction [37]. Along the direction taken by the present authors et al [38,39], here we also use the matrix elements of U and U † as the coordinates on the manifold Sp(2N+2) instead of the manifold SO(2N+2). Even in the case of Sp(2N+2), it turns out that a real line element can be defined by a hermitian metric tensor on the manifold.…”

Section: Scf Condition and Mfh Linear In Jacobi Generatorsupporting

confidence: 60%

“…3) The condition that a certain manifold is a Kähler manifold is that its complex structure is covariantly constant relative to the Riemann connection and that it has vanishing torsions: As in the case of SO(2N +2) [38,39], the hermitian metric tensor G pq rs can be locally given through a real scalar function, the Kähler potential, which takes the well-known form 5) and the explicit expression for the components of the metric tensor is given as…”

Section: Scf Condition and Mfh Linear In Jacobi Generatormentioning

confidence: 99%

“…Notice that the above function does not determine the Kähler potential K(Q B † , Q B ) uniquely since the metric tensor G pqrs is invariant under transformations of the Kähler potential, [38,39], we consider an Sp(2N+2) infinitesimal left transformation of an Sp(2N+2) matrix G to G ′ , G ′ = (1 2N+2 +δG)G, by using the first equation of (B.3):…”

Section: Scf Condition and Mfh Linear In Jacobi Generatormentioning

confidence: 99%

“…Along the direction taken by the present authors et al [38,39], here we also use the matrix elements of U and U † as the coordinates on the manifold Sp(2N+2) instead of the manifold SO(2N+2). Even in the case of Sp(2N+2), it turns out that a real line element can be defined by a hermitian metric tensor on the manifold.…”

Section: Sp(2n+2) U(n+1) Coset Manifoldmentioning

confidence: 99%

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Mean-field theory based on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators

Nishiyama

Providência

2019

Journal of Mathematical Physics

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In this paper, we give an expression for canonical transformation group with Grassmann variables, basing on the Jacobi hsp := semi-direct sum h N ⋊ sp(2N, R) algebra of boson operators. We assume a mean-field Hamiltonian (MFH) linear in the Jacobi generators. We diagonalize the boson MFH. We show a new aspect of eigenvalues of the MFH. An excitation energy arisen from additional SCF parameters has never been seen in the traditional boson MFT. We derive this excitation energy. We extend the Killing potential in the Sp(2N) U (N) coset space to the one in the Sp(2N+2) U (N+1) coset space and make clear the geometrical structure of Kähler manifold, a non-compact symmetric space Sp(2N+2) U (N+1) . The Jacobi hsp transformation group is embedded into an Sp(2N+2) group and an Sp(2N+2) U (N+1) coset variable is introduced. Under such mathematical manipulations, extended bosonization of Sp(2N+2) Lie operators, vacuum function and differential forms for extended boson are presented by using integral representation of boson state on the Sp(2N+2) U (N+1) coset variables.

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Section: Scf Condition and Mfh Linear In Jacobi Generatorsupporting

confidence: 89%

Section: Scf Condition and Mfh Linear In Jacobi Generatorsupporting

confidence: 60%

Section: Scf Condition and Mfh Linear In Jacobi Generatormentioning

confidence: 99%

Section: Scf Condition and Mfh Linear In Jacobi Generatormentioning

confidence: 99%

Section: Sp(2n+2) U(n+1) Coset Manifoldmentioning

confidence: 99%

See 3 more Smart Citations

Mean-field theory based on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators

Nishiyama

Providência

2019

Journal of Mathematical Physics

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A new description of motion of the Fermionic SO(2N+2) top in the classical limit under the quasi-anticommutation relation approximation

Nishiyama,

Da Providencia,

Providencia

2010

Preprint

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The boson images of fermion SO(2N+1) Lie operators have been given together with those of SO(2N + 2) ones. The SO(2N+1) Lie operators are generators of rotation in the (2N+1)-dimensional Euclidian space (N : number of single-particle states of the fermions). The images of fermion annihilation-creation operators must satisfy the canonical anti-commutation relations, when they operate on a spinor subspace. In the regular representation space we use a boson Hamiltonian with Lagrange multipliers to select out the spinor subspace. Based on these facts, a new description of a fermionic SO(2N + 2) top is proposed. From the Heisenberg equations of motions for the boson operators, we get the SO(2N + 1) self-consistent field (SCF) Hartree-Bogoliubov (HB) equation for the classical stationary motion of the fermion top. Decomposing an SO(2N+1) matrix into matrices describing paired and unpaired modes of fermions, we obtain a new form of the SO(2N+1) SCF equation with respect to the pairedmode amplitudes. To demonstrate the effectiveness of the new description based on the bosonization theory, the extended HB eigenvalue equation is applied to a superconducting toy-model which consists of a particle-hole plus BCS type interaction. It is solved to reach an interesting and exciting solution which is not found in the traditional HB eigenvalue equation, due to the unpaired-mode effects. To complete the new description, the Lagrange multipliers must be determined in the classical limit. For this aim a quasi anti-commutation-relation approximation is proposed. Only if a certain relation between an SO(2N+1) parameter z and the N is satisfied, unknown parameters k and l in the Lagrange multipliers can be determined withuout any inconcistency.

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Anomaly-free supersymmetric $ \frac{{{\text{SO}}\left( {2N + 2} \right)}}{{{\text{U}}\left( {N + 1} \right)}} $ σ-model based on the SO(2N + 1) Lie algebra of the fermion operators

Cited by 2 publications

References 27 publications

Mean-field theory based on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators

Mean-field theory based on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators

A new description of motion of the Fermionic SO(2N+2) top in the classical limit under the quasi-anticommutation relation approximation

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