Session 11
Supersymmetry and Quantum Groups
Chairs:
Vladimir Dobrev (Bulgarian Academy of Sciences, Bulgaria)
Alexey Isaev (Bogoliubov Laboratory of Theoretical Physics, JINR, Russia)
Titles and Abstracts
Naruhiko Aizawa (Osaka Prefecture University, Japan)
Title: Galilean superconformal algebras with central extension and their realizations
Abstract: We consider supersymmetric extensions of spin $ \ell $ class Galilean
conformal algebras with central extension. It is shonw that for two spatial dimension,
there exist exotic supersymmetric extension. We also present a boson-fermion
realization of the Galilean superconformal algebras for any $ \ell. $ This will reveal an
interesting connection of the superalgebras and a simple oscillator Hamiltonian.
Xavier Bekaert (Universite Francois Rabelais de Tours, France)
Title: Singletons and their maximal symmetry algebras
Abstract: Singletons are those unitary irreducible modules of the Poincare or (anti)
de Sitter group that can be lifted to unitary modules of the conformal group.
Higher-spin algebras are the corresponding realizations of the universal enveloping
algebra of the conformal algebra on these modules. Higher-spin algebras are
equivalently defined as maximal symmetry algebra of singletons. The manifest
realization of this relation will be reviewed.
I.L. Buchbinder (Tomsk State Pedagogical University, Russia)
Title: Background field formalism and construction of effective action for N=2, d=3
supersymmetric gauge theories
Abstract: The background field method for three-dimensional Yang-Mills and
Chern-Simons models in N=2 superspace is developed. Superfield proper time (heat
kernel) techniques are formulated and exact expressions of heat kernels for constant
backgrounds are presented. The background field method and heat kernel techniques
are applied for evaluating the low-energy effective actions in N=2supersymmetric
Yang-Mills and Chern-Simons models as well as in N=4 and N=8 SYM theories.
Vladimir Dobrev (Bulgarian Academy of Sciences, Bulgaria)
Title: Invariant Differential Operators for Euclidean Jordan Groups
Abstract: In the present talk we review the progress of the project of systematic
construction of invariant differential operators for non-compact Lie groups on the
example of the class of Euclidean Jordan groups. The latter are called also Hermitian
symmetric spaces of tube type, and also conformal Lie groups. We show more details
for the non-compact Lie groups SO(n,2), SU(n,n), Sp(n,R), where we give the main
multiplets of indecomposable elementary representations, including the necessary
data for all relevant invariant differential operators.
Sergey Fedoruk (Bogoliubov Laboratory of Theoretical Physics, JINR, Russia)
Title: Gauged N=4 supersymmetric mechanics with spin supermultiplets
Abstract: We present basics of the gauged superfield approach to constructing
N-extended supersymmetric quantum mechanics developed in arXiv:0812.4276,
arXiv:0905.4951, arXiv:0912.3508, arXiv:1112.1947 and arXiv:1204.4474. This
approach is illustrated by the one- and multi-particle systems, possessing N=4
Poincare and N=4 conformal supersymmetries.
Gaetano Fiore (Universita Federico II and INFN, Napoli, Italy)
Title: Twisting, second quantization on noncommutative spaces and "finite"
spacetime symmetry transformations
Abstract: We first review our application of the twist-induced star-deformation
procedure to second quantization on a noncommutative space(time). The procedure
deforms, in a coordinated way, the space(time) algebra and its symmetries, the
wave-mechanical description of a system of n bosons/fermions, the algebra of
creation and annihilation operators and also the commutation relations of the latter
with functions of spacetime; our key requirement is the mode-decomposition
independence of the quantum field. We then point out that the correct way to describe
"finite" space-time transformations is based on the coaction of the (deformed) Hopf
algebra H of functions on the group G of space(time) symmetries, rather than the
action of the Hopf algebra H', dual to H, deformation of the Universal Enveloping
Algebra of Lie(G). The noncommutativity of H, i.e. of the parameters of a change of
reference frames, hints at the quantum nature of the latter. We finally discuss the case
of the quantum theory of a scalar field on a Groenewold-Moyal-Weyl
noncommutative space(time).
Dimitri Gurevich (Valenciennes University, France)
Title: Reflection Equation Algebra and its applications to Noncommutative Geometry
Abstract: Reflection Equation Algebra is an algebra related to Quantum Groups. For
A_n series it can be considered as a "braided deformation" of the enveloping algebra
U(gl(n)). It admits a differential calculus, which in the limit q=1 turns into calculus
defined on the enveloping algebra U(gl(n)). This gives rise to a new type of
SU(n)-equivariant Noncommutative Geometry. In the frameworks of this Geometry it
is possible to quantize some dynamical models and integrable systems. The case
$n=2$ will be considered in detail.
Alexey Isaev (Bogoliubov Laboratory of Theoretical Physics, JINR, Russia)
Title: Conformal group: Yang-Baxter R-operator and star-triangle relation
Abstract: We construct R-operator which is solution of the Yang-Baxter equation and
acts in tensor product of two representations of 4-dimensional conformal group
labelled by lorentz spins ℓ and ℓ˙ and scaling dimension Δ. The R-operator is
constructed as composite operator from simper buildings blocks S1,S2 and S3 which
obey Coxeter relations S1S2S1=S2S1S2, S2S3S2=S3S2S3 which are equivalent to
the general integral star-triangle relation. The operators S1 and S3 have nice group
theoretical meaning–they are intertwining operators for representations of conformal
group. The work was done in collaboration with D. Chicherin and S. Derkachev.
Sergei Khoroshkin (Institute of Theoretical and Experimental Physics, Russia)
Title: Rational representations of Yangians
Abstract: Rational representations of Y(gl(N) are subquotients of tensor products of
defining vectors representations of Y(gl(N)) and of their duals. We classify irreducible
rational representations of Y(gl(N)), present their explicit construction with a special
attention to irregular representations.
Anatol Kirillov (Kyoto University, Japan)
Title: Some algebraic and combinatorial properties of Dunkl operators at critical level
Abstract: The Dunkl operators have been introduced in the later part of 80's of the
last century by Charles Dunkl [Du], [Du1] as a powerful mean to study of harmonic
and orthogonal polynomials related with finite Coxeter groups.
In my talk I introduce and study a certain class of nonhomogeneous quadratic
algebras together with the distinguish set of mutually commuting elements inside of
each, the so-called (universal) Dunkl elements. I describe relations among the Dunkl
elements. This result is a further extension and generalization of analogous results
obtained in [FK],[P] and [KM]. As an application we describe explicitly the set of
relations among the Gaudin elements in the group ring of the symmetric group, cf.
[MTV].
Yet another objective of my talk is to describe several combinatorial properties of
some special elements, the so-called Coxeter element and the
longest element,
in the associative quasi-classical Yang--Baxter algebra. In the case of Coxeter element
we relate the corresponding reduced polynomials introduced in [ST], with the
β-Grothendieck polynomials [FK1] for some special permutations π_{k}^{(n)}.
Moreover, we show that the specialization Θ_{π_{k}^{(n)}}^{(β)}(1) of the
β-Grothendieck polynomial Θ_{π_{k}^{(n)}}^{(β)}(X_n) counts the number of
k-dissections of a convex (n+k+1)-gon according to the number of diagonals involved.
When the number of diagonals in a k-dissection is the maximal possible, we recover
the well-known fact that the number of k-triangulations of a convex (n+k+1)-gon is
equal to the value of a certain Catalan-Hankel determinant, see e.g. [SS]. We also
show that for a certain 5-parameters family of vexillary permutations, the
specialization x_i=1, \forall i \ge 1, of the corresponding β-Schubert polynomials
Σ_{w}^{(β)}(1) coincides with some Fuss-Narayana polynomials and their
generalizations. As an example we show that the reduced polynomial corresponding
to a monomial x_{12}^{n} x_{23}^{m} counts the number of (n,m)-Delannoy
paths according to the number of NE-steps.
We also point out on a conjectural connection between the sets of maximal compatible
sequences for the permutation σ_{n,2n,2,0} and that σ_{n,2n+1,2,0} from one side,
and the set of VSASM(n) and that of CSTCPP(n) correspondingly, from the other.
Finally, we introduce and study a multiparameter generalization of reduced
polynomials introduced in [ST], as well as that of the Catalan, Narayana and (small)
Schröder numbers.
In the case of the longest element we relate the corresponding reduced polynomial
with the Ehrhart polynomial of the Chan--Robbins polytope.
[Du] C. Dunkl, Differential--difference operators associated to reflection groups,
Trans. AMS 311 (1989), 167--183.
[Du1] C. Dunkl, Harmonic polynomials and peak sets of reflection groups,
Geometriae Dedicata 32 (1989), 157--171.
[FK] S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert
calculus, Advances in Geometry, (J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan
and P. Xu, eds.), Progress in Math. 172 (1995), 147--182, Birkh\"auser Boston,
Boston, MA.
[FK1] S. Fomin and A.N. Kirillov, Yang-Baxter equation, symmetric functions and
Grothendieck polynomials, preprint arXiv:hep-th/9306005.
[KM] A.N. Kirillov and T. Maeno, A note on quantum K-theory of flag varieties,
preprint.
[KM3] A.N. Kirillov and T. Maeno, Extended quadratic algebra and a model of the
equivariant cohomology ring of flag varieties, Algebra i Analiz 22 (2010), no. 3,
155--176; translation in St. Petersburg Math. J. 22 (2011), no. 3, 447--462.
[MTV] E. Mukhin, V. Tarasov and A. Varchenko, Bethe subalgebras of the group
algebra of the symmetric group, preprint arXiv:1004.4248.
[P] A. Postnikov, On a quantum version of Pieri's formula, Advances in Geometry,
(J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu, eds.), Progress in Math.
172 (1995), 371--383, Birkhäuser Boston, MA.1999.
[SS] L. Serrano and C. Stump, Generalized triangulations, pipe dreams, and simplicial
spheres, FPSAC--2011, DMTCS (2011), 885--896.`
[ST] R. Stanley, {\it Catalan addendum}, preprint, version of 30 April 2011.
Petr Kulish (St. Petersburg Department of Steklov Mathematical Institute, Russia)
Title: Integrable Spin Systems and Representation Theory
Abstract: Quantum integrable spin chains are related to classical and quantum Lie
(super-) algebras and their finite and infinite representations. It is pointed out that with
appropriate boundary conditions corresponding algebra can be indentified with the
symmetry algebra of integrals of motion (transfer-matrix). According to a generalized
Schur-Weyl dulaity this yields a multiplet character of the spectrum and invariant
subspaces.
Jerzy Lukierski (Wroclaw University, Poland)
Title: Quantum deformations of D=4 superPoincare algebras and their Euclidean
counterparts
Abstract: We describe large class of D=4 Euclidean and supersymmetric (Poincare
and Euclidean) classical r-matrices. In our considerations we start from the
complexification of Zakrzewski table of D=4 Poincare classical r-matrices, which is
the most complete in present literature. Some of these r-matrices have supersymmetric
N=1 Poincare or Euclidean extensions which provide noncommutative superspaces
covariant under the Hopf-algebraic quantum supersymmetries. In particular covariant
supersymmetric extension of kappa-Minkowski space-time has been obtained.
The results based on the paper by A.Borowiec. J.Lukierski, M.Mozrzymas and
V.N.Tolstoy (arXiv:1112.1936v2, JHEP(2012))
Alexander Molev (University of Sydney, Australia)
Title: Sugawara operators, commutative subalgebras and Gaudin Hamiltonians
Abstract: For each simple Lie algebra g consider the corresponding affine vertex
algebra V(g) at the critical level. The center of this vertex algebra is a commutative
associative algebra whose structure was described by a remarkable theorem of Feigin
and Frenkel about two decades ago. The field corresponding to any element of the
Feigin-Frenkel center is a Sugawara operator. We give explicit formulas for
generators of the center and hence for Sugawara operators for all classical Lie
algebras g. Our approach is based on the Schur-Weyl duality and leads to explicit
constructions of commutative subalgebras of the universal enveloping algebras U(g[t])
and U(g) and to explicit formulas for higher Gaudin Hamiltonians.
Oleg Ogievetsky (Marseille University, France)
Title: Half-quantum linear algebra
Abstract: Spectrum of half-quantum matrices is discussed. This is a joint work with
A. Isaev.
Michel Rausch de Traubenberg (Strasbourg University and Institut Pluridisciplinaire
Hubert Curien, France)
Title: Higher order extensions of the Poincare algebra
Abstract: Lie algebras of order F (or F-Lie algebras) are possible generalisations of
Lie algebras (F=1) and Lie superalgebras (F=2). An F-Lie algebra admits a
ZF-gradation, the zero-graded part being a Lie algebra. An F-fold symmetric product
(playing the role of the anticommutator in the case F=2) expresses the zero graded
part in terms of the non-zero graded part. This structure enables us to define various
non-trivial extensions of the Poincare algebra. These extensions are study more
preciselly in two different contexts. The first algebra we are considering is shown to
be an (infinite dimensional) extension of the Poincare algebra in (1+2)-dimensions
and turns out to induce a symmetry which connects relativistics anyons. The second
extension we are studing is related to a specific finite dimensional Lie algebras of
order F in any space-time dimensions and induces a symmetry on p-forms. We then
summarized some of the main results obtained in that context. Finally, we show that
one is able to associate a group to these structures.
Dmitri Sorokin (Universita' degli Studi di Padova, Italy)
Title: Covariant Actions for Models with Non-Linear Twisted Self-Duality (with P.
Pasti and M. Tonin)
Abstract: We shall describe a systematic way of the generalization, to models with
non-linear duality, of the space-time covariant and duality-invariant formulation of
duality-symmetric theories in which the covariance and the consistency of the
construction are ensured by the presence of a single auxiliary scalar field and two
local symmetries. Application of this method to the study of the role of
duality-invariant counterterms in the UV behavior of quantum supergravities will be
discussed.
Yucai Su (Tongji University, China)
Title: Character and dimension formulas for queer Lie superalgebras
Abstract: Using Brundan's algorithm, we obtain closed character and dimension
formulas for queer Lie superalgebras. This is a recent joint work with R.B.Zhang.
Anne Taormina (University of Durham, UK)
Title: K3 elliptic genus: glimpse of a new Moonshine
Abstract: This contribution starts with a review of how ideas stemming from string
theory recently led to a remarkable and still unexplained connection between the
elliptic genus of a K3 surface and the sporadic group M24 (Mathieu 24). It has
sparked the interest of Number Theorists, Algebraic Geometers, Group Theorists and
Mathematical Physicists worldwide over the last year or so, and its explanation may
require novel ideas. M24 has been linked to the classical geometry of K3 surfaces by
Mukai, who classified the finite groups of symplectic automorphisms of all K3
surfaces in 1988 and showed that they are all subgroups of M23, each having at least
5 orbits on a set S of 24 elements. M23 is the stabiliser in M24 of an element in the
set S, when viewing M24 as the group of permutations of 24 elements preserving the
extended binary Golay code G24. Using lattice techniques, one may highlight the
existence of an `overarching symmetry' in this context. The discovery of this
overarching symmetry may provide a first step in understanding the connection
between the K3 elliptic genus and M24.
Valeriy Tolstoy (Lomonosov Moscow State University, Russia)
Title: Algebraic Structure of Mixed Para-statistics
Abstract: Para-Bose and para-Fermi statistics are generalizations of the usual Bose
and Fermi statistics. Quantum fields corresponding to these para-statistics satisfy
Green's trilinear relations (H.S. Green, Phys. Rev. 90, 270 (1953)). It was proved that
the trilinear relations are related with an orthogonal Lie algebra so(2m+1) (in the
case of the para-Fermi statistic) and with a Z_2-graded orthosymplectic Lie
superalgebra osp(1|2n)) (in the case of the para-Bose statistic). Here m (n) denotes a
number of parafermionic (para-bosonic) degrees of freedom. There are also two
mixed cases of the para-Bose and para-Fermi systems which are called "relative
para-Fermi sets" P_{FB} and "relative para-Bose sets" P_{BF} (O.W.
Greenberg and A.M.L. Messian, Phys. Rev. 138, 1155 (1965)). It was established that
the relative para-Fermi sets P_{FB} are associated with the Z_2-graded superalgebra
osp(2m+1|2n). In the case of the relative para-Bose sets $P_{BF} a similar
connection was not know up to now. In this talk we prove a new result that the case of
the relative para-Bose sets P_{BF} is associated with the Z_2 xZ_2 -graded
superalgebra osp(2m+1|2n).
Hans Wenzl (University of California, San Diego, USA)
Title: Applications of U'qso_n
Abstract: We show how the nonstandard $q$-deformation $U'_qso_n$ can be used to
decompose the $n$-fold tensor power of a spinor representation of the Drinfeld-Jimbo
quantum group $U_qso_N$. As a second application, we use $U'_qso_n$ and its
centralizer on certain tensor powers of vector representations to construct categorical
analogs of symmetric spaces for certain fusion tensor categories.