Wen-Li Yang, Yao-Zhong Zhang, Xin Liu
We construct the free field representation of the affine currents, energy-momentum tensor and screening currents of the first kind of the current superalgebra $gl(m|n)_k$ uniformly for $m=n$ and $m\neq n$. The energy-momentum tensor is given by a linear combination of two Sugawara tensors associated with the two independent quadratic Casimir elements of $gl(m|n)$.
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This paper generalizes Weyl realization to a class of Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ satisfying $[\mathfrak{g}_1,\mathfrak{g}_1]=\{0\}$. First, we give a novel proof of the Weyl realization of a Lie algebra $\mathfrak{g}_0$ by deriving a functional equation for the function that defines the realization. We show that this equation has a unique solution given by the generating function for the Bernoulli numbers. This method is then generalized to Lie superalgebras of the above type.
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.
By applying Niederer--like transformation, we construct a representation of the N=2 l-conformal Newton-Hooke superalgebra for the case of a negative cosmological constant in terms of linear differential operators as well as its dynamical realization. Another variant of N=2 supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies is proposed. The advantages of such realizations as compared to their analogues introduced in [J. Math. Phys. 53 (2012) 072904], [J. Math. Phys. 56 (2015) 022902] are discussed.
The Holstein-Primakoff and the Dyson realizations of the Lie superalgebra $gl(n/m)$ are generalized to the class of the quantum superalgebras $U_q[gl(n/m)]$ for any $n$ and $m$. It is shown how the elements of $U_q[gl(n/m)]$ can be expressed via $n-1$ pairs of Bose creation and annihilation operators and $m$ pairs of Fermi creation and annihilation operators.
A bosonization of the quantum affine superalgebra $U_q(\widehat{sl}(M|N))$ is presented for an arbitrary level $k \in {\bf C}$.The Wakimoto realization is given by using $ξ-η$ system. The screening operators that commute with $U_q(\widehat{sl}(M|N))$ are presented for the level $k \neq -M+N$. New bosonization of the affine superalgebra $\widehat{sl}(M|N)$ is obtained in the limit $q \to 1$.
Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of gl(1|1), especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules.
We give the realizations of the extended Weyl-Heisenberg (WH) algebra and the Rubakov-Spiridonov (RS) superalgebra in terms of anyons, characterized by the statistical parameter $ν\in[0,1]$, on two-dimensional lattice. The construction uses anyons defined from usual fermionic oscillators (Lerda-Sciuto construction). The anyonic realization of the superalgebra $sl(1/1)$ is also presented.
An analogue of the Holstein-Primakoff and of the Dyson realization for the Lie superalgebra $sl(1/n)$ is written down. The expressions are formally the same as for the Lie algebra $sl(n+1)$, however in the latter the Bose operators have to be replaced with Fermi operators.
The mathematical structure of a mixed paraparticle system (combining both parabosonic and parafermionic degrees of freedom) commonly known as the Relative Parabose Set, will be investigated and a braided group structure will be described for it. A new family of realizations of an arbitrary Lie superalgebra will be presented and it will be shown that these realizations possess the valuable representation-theoretic property of transferring invariably the super-Hopf structure. Finally two classes of virtual applications will be outlined: The first is of interest for both mathematics and mathemati…
We construct a free field realization of the quantum affine superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k \in {\mathbb C}$.
Super coherent states are useful in the explicit construction of representations of superalgebras and quantum superalgebras. In this contribution, we describe how they are used to construct (quantum) boson-fermion realizations and representations of (quantum) superalgebras. We work through a few examples: $osp(1|2)$ and its quantum version $U_t[osp(1|2)]$, $osp(2|2)$ in the non-standard and standard bases and $gl(2|2)$ in the non-standard basis. We obtain free boson-fermion realizations of these superalgebras. Applying the boson-fermion realizations, we explicitly construct their finite-dimens…
In this paper free field realizations of affine current superalgebras are considered. Based on quantizing differential operator realizations of the corresponding basic Lie superalgebras, general and simple expressions for both the bosonic and the fermionic currents are provided. Screening currents of the first kind are also presented. Finally, explicit free field realizations of primary fields with general, possibly non-integer, weights are worked out. A formalism is used where the (generally infinite) multiplet is replaced by a generating function primary operator. The results allow setting u…
The new extensions of the Poincaré superalgebra recently found in ten and eleven dimensions are shown to admit a linear realization. The generators of the nonlinear and linear group transformations are shown to fall into equivalent representations of the superalgebra. The parametrization of the coset space $G/H$, with $G$ a given extended supergroup and $H$ the Lorentz subgroup, that corresponds to the linear transformations is presented.
Based on the particular orderings introduced for the positive roots of finite dimensional basic Lie superalgebras, we construct the explicit differential operator representations of the $osp(2r|2n)$ and $osp(2r+1|2n)$ superalgebras and the explicit free field realizations of the corresponding current superalgebras $osp(2r|2n)_k$ and $osp(2r+1|2n)_k$ at an arbitrary level $k$. The free field representations of the corresponding energy-momentum tensors and screening currents of the first kind are also presented.
We give a realization of the quantum affine Lie superalgebras U_q(A(M,N))^(1) in terms of anyons defined on a one or two-dimensional lattice, the deformation parameter q being related to the statistical parameter $ν$ of the anyons by q = exp(iπν). The construction uses anyons contructed from usual fermionic oscillators and deformed bosonic oscillators. As a byproduct, realization deformed in any sector of the quantum superalgebras U_q(A(M,N)) is obtained.
The known Holstein-Primakoff and Dyson realizations for the Lie algebras $gl(n+1),\;n=1,2,\ldots$ in terms of Bose operators are generalized to the class of the Lie superalgebras $gl(m/n+1)$ for any $n$ and $m$. Formally the expressions are the same as for $gl(m+n+1)$, however both Bose and Fermi operators are involved.
Dynamical systems which are invariant under N=1 supersymmetric extension of the l-conformal Galilei algebra are constructed. These include a free N=1 superparticle which is governed by higher derivative equations of motion and an N=1 supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies. A Niederer-like transformation which links the models is proposed.
We use superalgebras to realize the 3-algebras used to construct N=6, 8 Chern-Simons-matter (CSM) theories. We demonstrate that the superalgebra realization of the 3-algebras provides a unified framework for classifying the gauge groups of the N \geq 5 theories based on 3-algebras. Using this realization, we rederive the ordinary Lie algebra construction of the general N=6 CSM theory from its 3-algebra counterpart, and reproduce all known examples as well. In particular, we explicitly construct the Nambu 3-bracket in terms of a double graded commutator of PSU(2|2). The N = 8 theory of Bagger, …
We give a bosonization of the quantum affine superalgebra $U_q(\widehat{sl}(N|1))$ for an arbitrary level $k \in {\bf C}$. The bosonization of level $k \in {\bf C}$ is completely different from those of level $k=1$. From this bosonization, we induce the Wakimoto realization whose character coincides with those of the Verma module. We give the screening that commute with $U_q(\widehat{sl}(N|1))$. Using this screening, we propose the vertex operator that is the intertwiner among the Wakimoto realization and typical realization. We study non-vanishing property of the correlation function defined …
In this paper we consider all consistent extensions of the AdS_5 x S^5 superalgebra, psu(2,2|4), to incorporate brane charges by introducing both bosonic and fermionic (non)central extensions. We study the Inonu-Wigner contraction of the extended psu(2,2|4) under the Penrose limit to obtain the most general consistent extension of the plane-wave superalgebra and compare these extensions with the possible BPS (flat or spherical) brane configurations in the plane-wave background. We give an explicit realization of some of these extensions in terms of the Tiny Graviton Matrix Theory (TGMT)[hep-th…
We suggest a programming realization of an algorithm for verifying a given set of algebraic relations in the form of a supercommutator multiplication table for the Verma module, which is constructed according to a generalized Cartan procedure for a quadratic superalgebra and whose elements are realized as a formal power series with respect to non-commuting elements. To this end, we propose an algebraic procedure of Verma module construction and its realization in terms of non-commuting creation and annihilation operators of a given Heisenberg--Weyl superalgebra. In doing so, we set up a proble…
We consider asymptotic limits of q-oscillator (or Heisenberg) realizations of Verma modules over the quantum superalgebra $U_{q}(gl(M|N))$, and obtain q-oscillator realizations of the contracted algebras proposed in [arXiv:1205.1471]. Instead of factoring out the invariant subspaces, we make reduction on generators of the q-oscillator algebra, which gives a shortcut to the problem. Based on this result, we obtain explicit q-oscillator representations of a Borel subalgebra of the quantum affine superalgebra $U_{q}(\hat{gl}(M|N))$ for Baxter Q-operators.
We present new superalgebra for $\mathcal{N}=2$ $D=3,4$ supergravity theory endowed with the $U(1)$ generator. The superalgebra is rooted in the so-called Soroka-Soroka algebra and spanned by the Lorentz $J_{ab}$ and Lorentz-like $Z_{ab}$, translation $P_a$ and $T$ generators, as well as two supercharges $Q^I_α$. It is the only possible realization for a given generator content. We construct a corresponding 3D Chern-Simons supergravity realization of the superalgebra and discuss its relevance.
A representation of the quantum superalgebra Uq(sl(M+1|N+1)) is constructed based on the q-differential operators acting on the coherent states parameterized by coordinates. These coordinates correspond to the local ones of the flag manifold. This realization provides us with a guide to construct the free field realization for the quantum affine superalgebra Uq^(sl(M+1|N+1)) at arbitrary level.
We present SuperMaxwell algebra: an N=1, D=4 algebra with two Majorana supercharges, obtained as the minimal enlargement of superPoincare containing the Maxwell algebra as a subalgebra. The new superalgebra describes the supersymmetries of generalized N=1, D=4 superspace in the presence of a constant Abelian SUSY field strength background. Applying the techniques of non-linear coset realization to the Maxwell supergroup we propose a new kappa-invariant massless superparticle model providing a dynamical realization of the SuperMaxwell algebra.
A loop-algebraic presentation is given for toroidal Lie superalgebras of classical types. Based on the loop superalgebra presentation free field realizations of toroidal Lie superalgebras are constructed for types $A(m,n)$, $B(m,n)$, C(n) and $D(m,n)$.
The coherent state method has proved to be useful in quantum physics and mathematics. This method, more precisely, the vector coherent state method, has been used by some authors to construct representations of superalgebras but almost, to our knowledge, it has not yet been extended to quantum superalgebras, except $U_q[osp(1|2)]$, one of the smallest quantum superalgebras. In this article the method is applied to a bigger quantum superalgebra, namely $U_q[gl(2|1)]$, in constructing $q$--boson-fermion realizations and finite-dimensional representations which, when irreducible, are classified i…
The superalgebra $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra have ``small'' irreducible representations in the superspaces $V^μ = t^μ\C[t, t^{-1}]\otimesΛ(N)$, where N = 2 and 3, respectively. For ${μ\in \C\backslash \Z}$ they are associated to the embeddings of these superalgebras into the Lie superalgebras of pseudodifferential symbols on the supercircle S^{1|N}. In this work we describe $\hat{K}'(4)$ and the exceptional N = 6 superconformal algebra in terms of matrices over a Weyl algebra. Correspondingly, we obtain realizations of their representations in $V^μ$ for $μ= 0…
Drinfeld realization of the centrally extended $\mathfrak{psl}(2|2)$ Yangian algebra with the manifest coproducts
本文给出了中心扩展李超代数 $\mathfrak{psl}(2|2)$ 的 Yangian 代数的 Drinfeld 实现，并证明了其 Hopf 代数结构及显式余积的存在性。
We obtain a new free field realization of $N=2$ super $W_{3}$ algebra using the technique of quantum hamiltonian reduction. The construction is based on a particular choice of the simple root system of the affine Lie superalgebra $sl(3|2)^{(1)}$ associated with a non-standard $sl(2)$ embedding. After twisting and a similarity transformation, this $W$ algebra can be identified as the extended topological conformal algebra of non-critical $W_{3}$ string theory.
Supersymmetric BMS$_4$ Algebras Revisited: Electric/Magnetic Superalgebras and Free Field Realization
本文系统分类了BMS$_4$代数的超对称扩展，揭示了电/磁超代数的构造及R-对称性在超对称扩展中的作用。
In this work we investigate connections between superalgebras and their realizations in terms of particles, branes and field theory models. We start from Poincaré superalgebras with brane charges and study its representations. The existence of new supermultiplets in different dimensions including an ultra short supermultiplet in D=11 different from the supergravity multiplet is shown. Generalizations of superalgebras containing brane charges, including those in D$>$11 are considered. The realization of these algebras at the level of relativistic particle models and, upon quantization, at the l…
Free field representations of the affine superalgebra $A(1,0)^{(1)}$ at level $k$ are needed in the description of the noncritical $N=2$ string. The superalgebra admits two inequivalent choices of simple roots. We give the Wakimoto representations corresponding to each of these and derive the relation between the two at the quantum level.
Considering supergravity theory is a natural step in the development of gravity models. This paper follows the ``algebraic`` path and constructs possible extensions of the Poincaré and Anti-de-Sitter algebras, which inherit their basic commutation structure. Previously achieved results of this type are fragmentary and show only a limited fraction of possible algebraic realizations. Our paper presents the newly obtained symmetry algebras, evaluated within an efficient pattern-based computational method of generating the so-called 'resonating' algebraic structures. These supersymmetric extension…
We study systematically various extensions of the Poincaré superalgebra. The most general structure starting from a set of spinorial supercharges $Q_α$ is a free Lie superalgebra that we discuss in detail. We explain how this universal extension of the Poincaré superalgebra gives rise to many other algebras as quotients, some of which have appeared previously in various places in the literature. In particular, we show how some quotients can be very neatly related to Borcherds superalgebras. The ideas put forward also offer some new angles on exotic branes and extended symmetry structures in M-…
The Hamiltonian of the simplest super $p$-brane model preserving 3/4 of the D=4 N=1 supersymmetry in the centrally extended symplectic superspace is derived and its symmetries are described. The constraints of the model are covariantly separated into the first- and the second-class sets and the Dirac brackets (D.B.) are constructed. We show the D.B. noncommutativity of the super $p$-brane coordinates and find the D.B. realization of the $OSp(1|8)$ superalgebra. Established is the coincidence of the D.B. and Poisson bracket realizations of the $OSp(1|8)$ superalgebra on the constraint surface a…
We find the free field construction of the basic $W$-current and screening currents for the deformed $W$-superalgebra ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$ associated with Lie superalgebra of type $A(M,N)$. Using this free field construction, we introduce the higher $W$-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin-diagrams for the Lie superalgebra $A(M,N)$, though the screening currents are not. This allows us to define ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$ by generators and relations.
We study the quantum $N=2$ super-$W_{3}$ algebra using the free field realization, which is obtained from the supersymmetric Miura transformation associated with the Lie superalgebra $A(2|1)$. We compute the full operator product expansions of the algebra explicitly. It is found that the results agree with those obtained by the OPE method.
We give an explicit quantum super field construction of the N=2 super Casimir WA(n)-algebras, which is obtained from supersymmetric Miura transformation for the Lie superalgebra A(n,n-1). And also we give an extension of this algebra including a super vertex operator which depends on simple root system of A(n,n-1).
Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~$ω$. By considering the latter together with its partner corresponding to the replacement of~$ω$ by~$-ω$, we are ab…
We develop the method of the hamiltonian reduction of affine Lie superalgebras to obtain explicit and general expressions both for the classical and the quantum extended superconformal algebras. By performing the gauge transformation which connects the diagonal gauge with the Drinfeld-Sokolov gauge and considering the quantum corrections, we get generic expressions for the classical and quantum free field realizations of the algebras.
In this note we give a new construction of the N=2 superconformal algebra using currents of the affine superalgebra $\hat{sl}(2 | 1)$ and free bosonic fields, and also the N=4 superconformal algebra without central charge in terms of currents of $\hat{sl}(2 | 2)$ and free bosonic fields. Arguments on commutants and invariant subspaces are added.
The purpose of this paper is to show that the so-called Maxwell superalgebra in four dimensions, which naturally involves the presence of a nilpotent fermionic generator, can be interpreted as a hidden superalgebra underlying N = 1, D=4 supergravity extended to include a 2-form gauge potential associated to a 2-index antisymmetric tensor. In this scenario, the theory is appropriately discussed in the context of Free Differential Algebras (an extension of the Maurer-Cartan equations to involve higher-degree differential forms). The study is then extended to the Free Differential Algebra describ…
Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
本文研究中心荷不超过24的良好全纯VOSA的幺正性与强分次局部性，排除特定假设赝副本与例外情形，证明多数此类VOSA满足相关性质，可生成对应共形网。
We study the classical and quantum $G$ extended superconformal algebras from the hamiltonian reduction of affine Lie superalgebras, with even subalgebras $G\oplus sl(2)$. At the classical level we obtain generic formulas for the Poisson bracket structure of the algebra. At the quantum level we get free field (Feigin-Fuchs) representations of the algebra by using the BRST formalism and the free field realization of the affine Lie superalgebra. In particular we get the free field representation of the $sl(2)\oplus sp(2N)$ extended superconformal algebra from the Lie superalgebra $osp(4|2N)$. We …
We present a geometric formulation of super $p$--brane theories in which the Wess--Zumino term is $(p+1)$--th order in the supersymmetric currents, and hence is manifestly supersymmetric. The currents are constructed using a supergroup manifold corresponding to a generalization of a superalgebra which we found sometime ago. Our results generalize Siegel's analogous reformulation of the Green--Schwarz superstring. The new superalgebras we construct underly the free differential superalgebras introduced by de Azcárraga and Townsend a few years ago.
We conjecture that the category of permutation-twisted modules for a multi-fold tensor product vertex operator superalgebra and a cyclic permutation of even order is isomorphic to the category of parity-twisted modules for the underlying vertex operator superalgebra. This conjecture is based on our observations of the cyclic permutation-twisted modules for free fermions as we discuss in this work, as well as previous work of the first author constructing and classifying permutation-twisted modules for tensor product vertex operator superalgebras and a permutation of odd order. In addition, we …
Motivated by application of current superalgebras in the study of disordered systems such as the random XY and Dirac models, we investigate $gl(2|2)$ current superalgebra at general level $k$. We construct its free field representation and corresponding Sugawara energy-momentum tensor in the non-standard basis. Three screen currents of the first kind are also presented.
Space-time N=2 and N=4 superconformal algebras can be built using world-sheet free fields or, equivalently, free field representations of osp(1|2) and sl(2|1), respectively. A prescription for the calculation of space-time correlators is given. As applications we compute one and two-point correlation functions of Virasoro (superVirasoro) generators. We also present a possible scenario for taking into account of NS fivebranes in the framework of our construction for description of superstring on AdS_3. It is nothing but a simplified version of Green's ``world-sheets for world-sheets''.
We construct a bosonization of the quantum superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k$. We construct the screening that commutes with the quantum superalgebra for an arbitrary level $k \neq -N+1$. We propose a bosonization of the vertex operator that gives the intertwiner among the Wakimoto realization and the typical representation.
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