Superconformal-like transformations and nonlinear realizations (original) (raw)
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Noninvertible N=1 superanalog of complex structure
Journal of Mathematical Physics, 1997
We found another N = 1 odd superanalog of complex structure (the even one is widely used in the theory of super Riemann surfaces). New N = 1 superconformal-like transformations are similar to anti-holomorphic ones of nonsupersymmetric complex function theory. They are dual to the ordinary superconformal transformations subject to the Berezinian addition formula presented, noninvertible, highly degenerated and twist parity of the tangent space in the standard basis. They also lead to the "mixed cocycle condition" which can be used in building noninvertible objects analogous to super Riemann surfaces. A new parametrization for the superconformal group is presented which allows us to extend it to a semigroup and to unify the description of old and new transformations.
Second N=1 Superanalog of Complex Structure
We found another N = 1 odd superanalog of complex structure (the even one is widely used in the theory of super Riemann surfaces). New N = 1 superconformal-like transformations are similar to anti-holomorphic ones of nonsupersymmetric complex function theory. They are dual to the ordinary superconformal transformations subject to the Berezinian addition formula presented, noninvertible, highly degenerated and twist parity of the tangent space in the standard basis. They also lead to the "mixed cocycle condition" which can be used in building noninvertible objects analogous to super Riemann surfaces. A new parametrization for the superconformal group is presented which allows us to extend it to a semigroup and to unify the description of old and new transformations.
On the semigroup nature of superconformal symmetry
Journal of Mathematical Physics, 1991
A semigroup of N = 1 superconformal transformations is introduced and analyzed. Noninvertible ones can describe transitions from body to soul and form a proper ideal containing a set of nilpotent transformations. The projective superspace is also considered. Transformations twisting the parity of a tangent space are brought in. They can be a nonsuperconformal "square root" of noninvertible superconformal transformations and the analogs of the Poincare metric and conformal invariance are suggested for them.
Conformal bridge transformation, 𝒫𝒯- and super- symmetry
2021
Supersymmetric extensions of the 1D and 2D Swanson models are investigated by using the conformal bridge transformation. The latter plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a 𝒫𝒯-invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N=2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the 𝒫𝒯-invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only...
Superconformal mechanics and nonlinear supersymmetry
Journal of High Energy Physics, 2003
We show that a simple change of the classical boson-fermion coupling constant, 2alphato2alphan2\alpha \to 2\alpha n 2alphato2alphan, ninNn\in \NninN, in the superconformal mechanics model gives rise to a radical change of a symmetry: the modified classical and quantum systems are characterized by the nonlinear superconformal symmetry. It is generated by the four bosonic integrals which form the so(1,2) x u(1) subalgebra, and by the 2(n+1) fermionic integrals constituting the two spin-n/2 so(1,2)-representations and anticommuting for the order n polynomials of the even generators. We find that the modified quantum system with an integer value of the parameter alpha\alphaalpha is described simultaneously by the two nonlinear superconformal symmetries of the orders relatively shifted in odd number. For the original quantum model with ∣alpha∣=p|\alpha|=p∣alpha∣=p, pinNp\in \NpinN, this means the presence of the order 2p nonlinear superconformal symmetry in addition to the osp(2|2) supersymmetry.
On conformal supergravity and projective superspace
Journal of High Energy Physics, 2009
The projective superspace formulation for four-dimensional N = 2 mattercoupled supergravity presented in arXiv:0805.4683 makes use of the variant superspace realization for the N = 2 Weyl multiplet in which the structure group is SL(2, C) × SU(2) and the super-Weyl transformations are generated by a covariantly chiral parameter. An extension to Howe's realization of N = 2 conformal supergravity in which the tangent space group is SL(2, C) × U(2) and the super-Weyl transformations are generated by a real unconstrained parameter was briefly sketched. Here we give the explicit details of the extension.
Journal of Physics A: Mathematical and Theoretical, 2012
We survey the salient features and problems of conformal and superconformal mechanics and portray some of its developments over the past decade. Both classical and quantum issues of single-and multiparticle systems are covered. ⋆ Invited review by Journal of Physics A: Mathematical and Theoretical × On leave of absence from V.N. Karazin Kharkov National University, Ukraine N =0 , N =2 [61] and N = 4 [9] superconformal mechanics.
Superconformal geometries and local twistors
Journal of High Energy Physics, 2021
Superconformal geometries in spacetime dimensions D = 3, 4, 5 and 6 are discussed in terms of local supertwistor bundles over standard superspace. These natually admit superconformal connections as matrix-valued one-forms. In order to make contact with the standard superspace formalism it is shown that one can always choose gauges in which the scale parts of the connection and curvature vanish, in which case the conformal and S-supersymmetry transformations become subsumed into super-Weyl transformations. The number of component fields can be reduced to those of the minimal off-shell conformal supergravity multiplets by imposing constraints which in most cases simply consists of taking the even covariant torsion two-form to vanish. This must be supplemented by further dimension-one constraints for the maximal cases in D = 3, 4. The subject is also discussed from a minimal point of view in which only the dimension-zero torsion is introduced. Finally, we introduce a new class of super...
Algebraic structure of Galilean superconformal symmetries
Physical Review D, 2011
The semisimple part of d-dimensional Galilean conformal algebra g ðdÞ is given by h ðdÞ ¼ Oð2; 1Þ È OðdÞ, which after adding via a semidirect sum the 3d-dimensional Abelian algebra t ðdÞ of translations, Galilean boosts, and constant accelerations completes the construction. We obtain Galilean superconformal algebra G ðdÞ by first defining the semisimple superalgebra H ðdÞ which supersymmetrizes h ðdÞ , and further by considering the expansion of H ðdÞ by tensorial and spinorial graded Abelian charges in order to supersymmetrize the Abelian generators of t ðdÞ . For d ¼ 3 the supersymmetrization of h ð3Þ is linked with a specific model of N ¼ 4 extended superconformal mechanics, which is described by the superalgebra Dð2; 1; Þ if ¼ 1. We shall also present the alternative derivations of extended Galilean superconformal algebras for 1 d 5 by employing the Inönü-Wigner contraction method.