Zhanna Kuznetsova

Classification of irreps and invariants of the N-extended Supersymmetric Quantum Mechanics

We present an algorithmic classification of the irreps of the N-extended one- dimensional supersymmetry algebra linearly realized on a finite number of fields. Our work is based on the 1-to-1 (1) correspondence between Weyl-type Clifford algebras (whose irreps are fully classified) and classes of irreps of the N-extended 1D supersymmetry. The complete classification of irreps is presented up to N · 10. The fields of an irrep are accommodated in l different spin states. N = 10 is the minimal value admitting length l > 4 irreps. The classification of length-4 irreps of the N = 12 and real N = 11 extended supersymmetries is also explicitly presented. Tensoring irreps allows us to systematically construct manifestly (N-extended) supersym- metric multi-linear invariants without introducing a superspace formalism. Multi-linear invariants can be constructed both for unconstrained and multi-linearly constrained fields. A whole class of off-shell invariant actions are produced in association with each irreducible representation. The explicit example of the N = 8 off-shell action of the (1,8,7) multiplet is presented. Tensoring zero-energy irreps leads us to the notion of the fusion algebra of the 1D N- extended supersymmetric vacua.

D-module representations of N=2,4,8 superconformal algebras and their superconformal mechanics

The linear (homogeneous and inhomogeneous) (k,N,N−k) supermultiplets of the N-extended one-dimensional supersymmetry algebra induce D-module representations for the N=2,4,8 superconformal algebras. For N=2, the D-module representations of the A(1, 0) superalgebra are obtained. For N=4 and scaling dimension λ = 0, the D-module representations of the A(1, 1) superalgebra are obtained. For λ ≠ 0, the D-module representations of the D(2, 1; α) superalgebras are obtained, with α determined in terms of the scaling dimension λ according to: α = −2λ for k = 4, i.e., the (4, 4) supermultiplet, α = −λ for k = 3, i.e., (3, 4, 1), and α = λ for k = 1, i.e., (1, 4, 3). For λ ≠ 0 the (2, 4, 2) supermultiplet induces a D-module representation for the centrally extended sl(2|2) superalgebra. For N=8, the (8, 8) root supermultiplet induces a D-module representation of the D(4, 1) superalgebra at the fixed value λ=14. A Lagrangian framework to construct one-dimensional, off-shell, superconformal-invariant actions from sing...

Constrained generalized supersymmetries and superparticles with tensorial central charges. A classification.

We classify the admissible types of constraint (hermitian, holomorphic, with reality conditions on the bosonic sectors, etc.) for generalized supersymmetries in the presence of complex spinors. We further point out which constrained generalized supersymmetries admit a dual formulation. For both real and complex spinors generalized supersymmetries are constructed and classified as dimensional reductions of supersymmetries from oxidized space-times (i.e. the maximal space-times associated to n-component Clifford irreps). We apply these results to systematically construct a class of models describing superparticles in presence of bosonic tensorial central charges, deriving the consistency conditions for the existence of the action, as well as the constrained equations of motion. Examples of these models (which, in their twistorial formulation, describe towers of higher-spin particles) were first introduced by Rudychev and Sezgin (for real spinors) and later by Bandos and Lukierski (for complex spinors).

Constrained Generalized Supersymmetries

We present a classification of admissible types of constraint (hermitian, holomorphic, with reality condition on the bosonic sectors, etc.) for generalized supersymmetries in the presence of complex spinors. A generalized supersymmetry algebra involving n‐component real spinors Qa is given by the anticommutators {Qa,Qb} = Zab where the matrix Z appearing in the r.h.s. is the most general symmetric matrix. A complex generalized supersymmetry algebra is expressed in terms of complex spinors Qa and their complex conjugate Q* ȧ. The most general (with a saturated r.h.s.) algebra is in this case given by {Qa,Qb} = Pab{Q*ȧ, Q*ḃ} = P*ȧḃ{Qa,Q*ḃ} = Raḃ where the matrix Pab is symmetric, while Rab is hermitian. The bosonic right hand side can be expressed in terms of the rank‐k totally antisymmetric tensors Pab = Σk(CΓ[μ1 …μ k])abP[μ1 …μ k].The decomposition in terms of anti‐symmetric tensors for any space‐time up to dimension D = 13 is presented. Real type, complex type, and quaternionic type space‐times are class...

The sl(2n|2n)(1) super-Toda lattices and the heavenly equations as continuum limit

The n → ∞ continuum limit of super-Toda models associated with the affine sl(2n|2n)(1) (super)algebra series produces (2 + 1)-dimensional integrable equations in the S1 × R2 spacetimes. The equations of motion of the (super)Toda hierarchies depend not only on the chosen (super)algebras but also on the specific presentation of their Cartan matrices. Four distinct series of integrable hierarchies in relation with symmetric-versus-antisymmetric, null-versus-nonnull presentations of the corresponding Cartan matrices are investigated. In the continuum limit we derive four classes of integrable equations of heavenly type, generalizing the results previously obtained in the literature. The systems are manifestly N = 1 supersymmetric and, for specific choices of the Cartan matrix preserving the complex structure, admit a hidden N = 2 supersymmetry. The coset reduction of the (super)-heavenly equation to the I × R(2) = (S1/Z2) × R2 spaceship (with I a line segment) is illustrated. Finally, integrable N = 2, 4 supersymmetrically extended models in (1 + 1) dimensions are constructed through dimensional reduction of the previous systems.