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Sigma Model (sigma + model)

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Physics and Astronomy	100%

Selected Abstracts

Coherent state path integral and super-symmetry for condensates composed of bosonic and fermionic atoms

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 9-10 2007
B. Mieck
Abstract A super-symmetric coherent state path integral on the Keldysh time contour is considered for bosonic and fermionic atoms which interact among each other with a common short-ranged two-body potential. We investigate the symmetries of Bose-Einstein condensation for the equivalent bosonic and fermionic constituents with the same interaction potential so that a super-symmetry results between the bosonic and fermionic components of super-fields. Apart from the super-unitary invariance U(L | S) of the density terms, we specialize on the examination of super-symmetries for pair condensate terms. Effective equations are derived for anomalous terms which are related to the molecular- and BCS- condensate pairs. A Hubbard-Stratonovich transformation from ,Nambu'-doubled super-fields leads to a generating function with super-matrices for the self-energy whose manifold is given by the orthosympletic super-group Osp(S,S | 2L). A nonlinear sigma model follows from the spontaneous breaking of the ortho-symplectic super-group Osp(S,S | 2L) to the coset decomposition Osp(S,S | 2L) \ U(L | S), U(L | S). The invariant subgroup U(L | S) for the vacuum or background fields is represented by the density terms in the self-energy whereas the super-matrices on the coset space Osp(S,S | 2L) \ U(L | S) describe the anomalous molecular and BCS- pair condensate terms. A change of integration measure is performed for the coset decomposition Osp(S,S | 2L) \ U(L | S) , U(L | S), including a separation of density and anomalous parts of the self-energy with a gradient expansion for the Goldstone modes. The independent anomalous fields in the actions can be transformed by the inverse square root of the metric tensor of Osp(S,S | 2L) \ U(L | S) so that the non-Euclidean integration measure with super-Jacobi-determinant can be removed from the coherent state path integral and Gaussian-like integrations remain. The variations of the independent coset fields in the effective actions result in classical field equations for a nonlinear sigma model with the anomalous terms. The dynamics of the eigenvalues of the coset matrices is determined by Sine-Gordon equations which have a similar meaning for the dynamics of the molecular- and BCS-pair condensates as the Gross-Pitaevskii equation for the coherent wave function in BEC phenomena. [source]

The minimal O(N) sigma model at D = 3 as the holographic image of a higher spin field theory on AdS4

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 6-7 2004
T. Leonhardt
We show that to the order included all n -point functions can be expressed by the propagator of the , field and an insertion of the order 4-point function of ,, which is a simple algebraic function. [source]

On (orientifold of) type IIA on a compact Calabi-Yau

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 1 2004
A. Misra
Abstract We study the gauged sigma model and its mirror Landau-Ginsburg model corresponding to type IIA on the Fermat degree-24 hypersurface in WCP4[1,1,2,8,12] (whose blow-up gives the smooth CY3(3,243)) away from the orbifold singularities, and its orientifold by a freely-acting antiholomorphic involution. We derive the Picard-Fuchs equation obeyed by the period integral as defined in [1, 2], of the parent ,, = 2 type IIA theory of [3]. We obtain the Meijer's basis of solutions to the equation in the large and small complex structure limits (on the mirror Landau-Ginsburg side) of the abovementioned Calabi-Yau, and make some remarks about the monodromy properties associated based on [4], at the same and another MATHEMATICAlly interesting point. Based on a recently shown ,, = 1 four-dimensional triality [6] between Heterotic on the self-mirror Calabi-Yau CY3(11,11), M theory on and F -theory on an elliptically fibered CY4 with the base given by CP1 × Enriques surface, we first give a heuristic argument that there can be no superpotential generated in the orientifold of of CY3(3,243), and then explicitly verify the same using mirror symmetry formulation of [2] for the abovementioned hypersurface away from its orbifold singularities. We then discuss briefly the sigma model and the mirror Landau-Ginsburg model corresponding to the resolved Calabi-Yau as well. [source]

(0,2) Gauged linear sigma model on supermanifold

ANNALEN DER PHYSIK, Issue 7-8 2009
Y. Okame
Abstract We construct (0,2), D = 2 gauged linear sigma model on supermanifold with both an Abelian and non-Abelian gauge symmetry. For the purpose of checking the exact supersymmetric (SUSY) invariance of the Lagrangian density, it is convenient to introduce a new operator for the Abelian gauge group. The operator provides consistency conditions for satisfying the SUSY invariance. On the other hand, it is not essential to introduce a similar operator in order to check the exact SUSY invariance of the Lagrangian density of non-Abelian model, contrary to the Abelian one. However, we still need a new operator in order to define the (0,2) chirality conditions for the (0,2) chiral superfields. The operator can be defined from the conditions assuring the (0,2) supersymmetric invariance of the Lagrangian density in superfield formalism for the (0,2) U(N) gauged linear sigma model. We found consistency conditions for the Abelian gauge group which assure (0,2) supersymmetric invariance of Lagrangian density and agree with (0,2) chirality conditions for the superpotential. The supermanifold ,m|n becomes the super weighted complex projective space WCPm-1|n in the U(1) case, which is considered as an example of a Calabi-Yau supermanifold. The superpotential W(,,,) for the non-Abelian gauge group satisfies more complex condition for the SU(N) part, except the U(1) part of U(N), but does not satisfy a quasi-homogeneous condition. This fact implies the need for taking care of constructing the Calabi-Yau supermanifold in the SU(N) part. Because more stringent restrictions are imposed on the form of the superpotential than in the U(1) case, the superpotential seems to define a certain kind of new supermanifolds which we cannot identify exactly with one of the mathematically well defined objects. [source]

(0,2) Gauged linear sigma model on supermanifold

On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

ANNALEN DER PHYSIK, Issue 10 2003
C. Klein
Abstract We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. This leads to equations for three-dimensional gravity where the matter is given by a SU(2,1)/S[U(1,1)× U(1)] nonlinear sigma model. The SU(2,1) invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electro-vacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is shown that the corresponding solutions will always have a non-vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. Harrison transformed hyperelliptic solutions are discussed. As a concrete example we study the transformation of a family of counter-rotating dust disks. To obtain algebro-geometric solutions with vanishing total charge which are of astrophysical relevance, three-sheeted surfaces have to be considered. The matter in the disk is discussed following Bi,ák et al. We review the ,cut and glue' technique where a strip is removed from an explicitly known spacetime and where the remainder is glued together after displacement. The discontinuities of the normal derivatives of the metric at the glueing hypersurface lead to infinite disks. If the energy conditions are satisfied and if the pressure is positive, the disks can be interpreted in the vacuum case as made up of two components of counter-rotating dust moving on geodesics. In electro-vacuum the condition of geodesic movement is replaced by electro-geodesic movement. As an example we discuss a class of Harrison-transformed hyperelliptic solutions. The range of parameters is identified where an interpretation of the matter in the disk in terms of electro-dust can be given. [source]