Generalized quantum statistics (GQS) associated to a Lie algebra or Lie superalgebra extends the notion of para-Bose or para-Fermi statistics. Such GQS have been classified for all classical simple Lie algebras and basic classical Lie superalgebras. In the current paper we finalize this classification for all exceptional Lie algebras and superalgebras. Since the definition of GQS is closely related to a certain Z-grading of the Lie (super)algebra G, our classification reproduces some known Z-gradings of exceptional Lie algebras. For exceptional Lie superalgebras such a classification of Z-gradings has not been given before.
We construct generalized coherent states for the rationally extended Scarf-I potential. Statistical and geometrical properties of these states are investigated. Special emphasis is given to the study of spatio-temporal properties of the coherent states via the quantum carpet structure and the auto-correlation function. Through this study, we aim to find the signature of the `"rationalisation" of the conventional potentials and the classical orthogonal polynomials.
In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagr…
Both the ${\cal N}=7$ superconformal quantum mechanics possessing the exceptional $G(3)$ Lie superalgebra as dynamical symmetry and its associated deformed oscillator with $G(3)$ as spectrum-generating superalgebra are presented. This superconformal quantum mechanics, uniquely defined up to similarity transformations, is obtained via the octonionically-induced "quasi-nonassociative" method employed to derive the exceptional ${\cal N}=8$ $F(4)$ model. To construct the $G(3)$ theories, the covariant embedding of the $7$-dimensional representation of the Lie algebra $g_2$ within the $8\times 8$ m…
The free-field representations of the $D(2,1;\a)$ current superalgebra and the corresponding energy-momentum tensor are constructed. The related screening currents of the first kind are also presented.
We show how the gauge and field structure of the tensor hierarchies in Double and $E_{7(7)}$ Exceptional Field Theory fits into $L_\infty$ algebras. Special attention is paid to redefinitions, the role of covariantly constrained fields and intertwiners. The results are connected to Gauged Supergravities through generalized Scherk-Schwarz reductions. We find that certain gauging-dependent parameters generate trivial gauge transformations, giving rise to novel symmetries for symmetries that are absent in their ungauged counterparts.
In this paper complexes of generalized Verma modules over the infinite-dimensional exceptional Lie superalgebras $E (3,8)$ and $E(5,10)$ are constructed and studied.
In this paper we define the degree of a morphism between (generalized) Verma modules over a graded Lie superalgebra and construct series of morphisms of various degrees between (generalized) Verma modules over the exceptional infinite-dimensional linearly compact simple Lie superalgebra E(5,10). We prove that all such morphisms of degree 1 are found.
We show that the existence of exceptional polynomials leads to the presence of non-trivial supersymmetry. The existence of these polynomials reveals several distinct isospectral potentials for the Schrödinger equation. All Schrödinger equations having Laguerre and Jacobi polynomials as their solutions, have non-trivial supersymmetric partners with corresponding exceptional polynomials as solutions.
We give a general construction of realizations of the contact superconformal algebras $K(2)$ and $\hat{K}'(4)$, and the exceptional superconformal algebra $CK_6$ as subsuperalgebras of matrices over a Weyl algebra of size $2^N\times 2^N$, where $N = 1, 2$ and $3$. We show that there is no such a realization for $K(2N)$, if $N\geq 4$.
In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the…
Recently one of the authors obtained a classification of simple infinite-dimensional Lie superalgebras of vector fields which extends the well-known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple equations, and of several exceptional superalgebras, among them E(3,6). In the article we study irreducible representations of the exceptional Lie superalgebra E(3,6). This superalgebra has s\ell(3)\times s\ell(2)\times g\ell(1) as the zero degree component of its consistent \ZZ-grading which leads us to believe that its representation theory ha…
This note is devoted to a more detailed description of one of the five simple exceptional Lie superalgebras of vector fields, cvect(0|3)_*, a subalgebra of vect(4|3). We derive differential equations for its elements, and solve these equations. Hence we get an exact form for the elements of cvect(0|3)_*. Moreover we realize cvect(0|3)_* by "glued" pairs of generating functions on a (3|3)-dimensional periplectic (odd symplectic) supermanifold and describe the bracket explicitly.
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable.
Dipartimento di Matematica, Universitá degli Studi di Roma “Tor Vergata”（罗马托维加塔大学数学系）；Department of Mathematics and Statistics, UiT The Arctic University of Norway（挪威北极大学UiT数学与统计系）
We continue the study of irreducible representations of the exceptional Lie superalgebra E(3,6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sl(3)\times sl(2)\times gl(1) as the zero degree component of its consistent ZZ-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3,6), completing thereby the classification and construction of all irreducible E(3,6)- modules that are L_0-locally finite.
We list defining relations for the four of the five exceptional simple Lie superalgebras some of which, as David Broadhurst conjectured and Kac demonstrated, may pertain to The Standard Model or Grand unified theories of elementary particles. For the fifth superalgebra the result is not final: there might be infinitely many relations. Contrariwise, for the same Lie superalgebra with Laurent polynomials as coefficients there are only finitely many relations.
We present the deformed (for the presence of Calogero potential terms) one-dimensional quantum oscillator with the exceptional Lie superalgebra $F(4)$ as spectrum-generating superconformal algebra. The Hilbert space is given by a $16$-ple of square-integrable functions. The energy levels are $\frac{2}{3}+n$, with $n=0,1,2,\ldots$. The ground state is $7$ times degenerate. The excited states are $8$ times degenerate. The $(7,8,8,8,\ldots )$ semi-infinite tower of states is recovered from the $(7;8;1)$ supermultiplet of the ${\cal N}=8$ worldline supersymmetry. The model is unique, up to similar…
New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial $g$. The cases where $g$ is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials…
Let $g$ be an exceptional Lie superalgebra, and let $p$ be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic Verma module in the parabolic category $O^p$, it is shown that the Jantzen filtration is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan-Lusztig polynomials. An explicit description of the submodule lattices of the parabolic Verma modules is given, and formulae for characters and dimensions of the finit…
Four $\ZZ_+$-bigraded complexes with the action of the exceptional infinite-dimensional Lie superalgebra E(3,6) are constructed. We show that all the images and cokernels and all but three kernels of the differentials are irreducible E(3,6)-modules. This is based on the list of singular vectors and the calculation of homology of these complexes. As a result, we obtain an explicit construction of all degenerate irreducible E(3,6)-modules and compute their characters and sizes. Since the group of symmetries of the Standard Model $SU(3) \times SU(2) \times U(1)$ (divided by a central subgroup of …
By using the technique of supersymmetric quantum mechanics, we study a quasi exactly solvable extension of the N-particle rational Calogero model with harmonic confining interaction. Such quasi exactly solvable many particle system, whose effective potential in the radial direction yields a supersymmetric partner of the radial harmonic oscillator, is constructed by including new long-range interactions to the rational Calogero model. An infinite number of bound state energy levels are obtained for this system under certain conditions. We also calculate the corresponding bound state wave functi…
Degenerate modules of the exceptional infinite-dimensional simple Lie superalgebras vle(3|6), ksle(5|10) and mb(3|8) have recently been constructed by Kac and Rudakov, and by Grozman, Leites and Shchepochkina. I rederive their results using a formalism which is contragredient to theirs; instead of finding singular vectors in induced modules, I build reducible tensor modules ("forms") from elementary differentials. There is a discrepancy between my result for ksle(5|10) and Kac' and Rudakov's one. Since the grade zero subalgebra of vle(3|6) and mb(3|8) is sl(3)+sl(2)+gl(1), gauge theories based…
We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK6, for which E(1, 6) is the annihilation algebra.
