Francesco Toppan

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...

Critical scaling dimension of D-module representations of N=4,7,8 superconformal algebras and constraints on superconformal mechanics

At critical values of the scaling dimension λ, supermultiplets of the global N-extended one-dimensional supersymmetry algebra induce D-module representations of finite superconformal algebras (the latters being identified in terms of the global supermultiplet and its critical scaling dimension). For N=4,8 and global supermultiplets (k,N,N−k), the exceptional superalgebras D(2, 1; α) are recovered for N=4, with a relation between α and the scaling dimension given by α = (2 − k)λ. For N=8 and k ≠ 4 all four N=8 finite superconformal algebras are recovered, at the critical values λk=1k−4, with the following identifications: D(4, 1) for k = 0, 8, F(4) for k = 1, 7, A(3, 1) for k = 2, 6 and D(2, 2) for k = 3, 5. The N=7 global supermultiplet (1, 7, 7, 1) induces, at λ=−14, a D-module representation of the exceptional superalgebra G(3). D-module representations are applicable to the construction of superconformal mechanics in a Lagrangian setting. The isomorphism of the D(2, 1; α) algebras under an S3 group act...

Twist Deformation of Rotationally Invariant Quantum Mechanics

Noncommutative quantum mechanics in 3D is investigated in the framework of an abelian Drinfeld twist which deforms a given Hopf algebra structure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, anomalous (the anomaly vanishes when the deformation parameter goes to zero). The deformed operators, Taylor-expanded in the deformation parameter, can be selected to minimize the anomaly. We present the deformations (and their anomalies) of undeformed rotationally invariant operators corresponding to the harmonic oscillator (quadratic potential), the anharmonic oscillator (quartic potential), and the Coulomb potential.

Snyder Noncommutativity and Pseudo-Hermitian Hamiltonians from a Jordanian Twist

Nonrelativistic quantum mechanics and conformal quantum mechanics are deformed through a Jordanian twist. The deformed space coordinates satisfy the Snyder noncommutativity. The resulting deformed Hamiltonians are pseudo-Hermitian Hamiltonians of the type discussed by Mostafazadeh. The quantization scheme makes use of the so-called “unfolded formalism” discussed in previous works. A Hopf algebra structure, compatible with the physical interpretation of the coproduct, is introduced for the Universal Enveloping Algebra of a suitably chosen dynamical Lie algebra (the Hamiltonian is contained among its generators). The multi-particle sector, uniquely determined by the deformed 2-particle Hamiltonian, is composed of bosonic particles.

N = 1, 2 SUPER-NLS HIERARCHIES AS SUPER-KP COSET REDUCTIONS

We define consistent finite-superfield reductions of the N = 1, 2 super-KP hierarchies via the coset approach we have already developed for reducing the bosonic KP hierarchy [generating for example the NLS hierarchy from the coset]. We work in a manifestly supersymmetric framework and illustrate our method by treating explicitly the N = 1, 2 super-NLS hierarchies. With respect to the bosonic case the ordinary covariant derivative is now replaced by a spinorial one which contains a spin- superfield. Each coset reduction is associated with a rational super- algebra encoding a nonlinear super--algebra structure. In the N = 2 case two conjugate sets of super-Lax operators, equations of motion and infinite Hamiltonians in involution are derived. Modified hierarchies are obtained from the original ones via free-field mappings [just as an m-NLS equation arises from representing the algebra through the classical Wakimoto free fields].

Symmetries of the Schrödinger equation and algebra/superalgebra duality

Some key features of the symmetries of the Schrodinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving first and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation- dependent notion of on-shell symmetry is introduced. The difference in associating the time-derivative symmetry operator with either a root or a Cartan generator of the sl(2) subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric.

Generalized NLS hierarchies from rational W algebras

Abstract Finite rational W algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. In this letter we address the problem of relating these algebras to integrable hierarchies of equations, by showing how to associate to a rational W algebra its corresponding hierarchy. We work out two examples: the sl(2) U(1) coset, leading to the Non-Linear Schrodinger hierarchy, and the U(1) coset of the Polyakov-Bershadsky W algebra, leading to a 3-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies.

Effects of twisted noncommutativity in multi-particle Hamiltonians

The non-commutativity induced by a Drinfel’d twist produces Bopp-shift-like transformations for deformed operators. In a single-particle setting the Drinfel’d twist allows to recover the non-commutativity obtained from various methods which are not based on Hopf algebras. In multi-particle sector, on the other hand, the Drinfel’d twist implies novel features. In conventional approaches to non-commutativity, deformed primitive operators are postulated to act additively. A Drinfel’d twist implies non-additive effects which are controlled by the coproduct. We stress that in our framework, the central element denoted as ħ is associated to an additive operator whose physical interpretation is that of the Particle Number operator.We illustrate all these features for a class of (abelian twist-deformed) 2D Hamiltonians. Suitable choices of the parameters lead to the Hamiltonian of the non-commutative Quantum Hall Effect, the harmonic oscillator, the quantization of the configuration space. The non-additive effects in the multi-particle sector, leading to results departing from the existing literature, are pointed out.

Twist Deformations of the Supersymmetric Quantum Mechanics

AbstractThe