Nontrivial Central Extensions Research Articles

Conserved gravitational charges, locality and the holographic weyl anomaly: a fresh viewpoint

Since the proposal of the AdS/CFT correspondence, made by Maldacena and Witten, there has been some controversy about the definition of conserved Noether charges associated with asymptotic isometries in asymptotically AdS spacetimes, namely, whether they form an anomalous (i.e., a nontrivial central extension) representation of the Lie algebra of the conformal group in odd bulk dimensions or not. In the present work, we shall review the derivation of these charges by using covariant phase space techniques, emphasizing the principle of locality underlying it. We shall also comment on how these issues manifest themselves in the quantum setting.

Lie Superalgebra Structures in H? (g;g)

Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let “-” be the change of parity functor, C i and H i the space of i-chains and i-cohomology. The Nijenhuis bracket makes into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in L * and for any Lie superalgebra g. We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l * for some simple finite dimensional Lie superalgebras g and their “relatives” - the nontrivial central extensions or derivation algebras of the considered simple ones.

Poisson algebra of diffeomorphism generators in a spacetime containing a bifurcation

In this article we will analyze the possibility of a nontrivial central extension of the Poisson algebra of the diffeomorphism generators which respect certain boundary conditions on the black hole bifurcation. The origin of a possible central extension in the algebra is due to the existence of boundary terms in the canonical generators, which are necessary to make them differentiable. The existence of such boundary terms depends on the exact boundary conditions that one takes. We will check two possible boundary conditions on the black hole bifurcation: fixed metric and fixed surface gravity. In the case of a fixed metric of the bifurcation the action acquires a boundary term but this term is canceled in the Legendre transformation and is thus absent in the Hamiltonian, and so in this case the possibility of a central extension is ruled out. In the case of fixed surface gravity the boundary term in the action is absent but therefore present in the Hamiltonian. Also in this case we will see that there is no central extension, also if there exist boundary terms in the generators.

Higher Genus Affine Algebras of Krichever—Novikov Type

For higher genus multi-point current algebras of Krichever-Novikov type associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles are stud- ied. They yield as central extensions almost-graded higher genus affine Lie algebras. In case that the Lie algebra is reductive a complete classification is given. For a sim- ple Lie algebra, like in the classical situation, there is up to equivalence and rescaling only one non-trivial almost-graded central extension. The classification is extended to the algebras of meromorphic differential operators of order less or equal one on the currents algebra.

A spinor-like representation of the contact superconformal algebra K′(4)

In this work we construct an embedding of a nontrivial central extension of the contact superconformal algebra K′(4) into the Lie superalgebra of pseudodifferential symbols on the supercircle S1/2. Associated with this embedding is a one-parameter family of spinor-like tiny irreducible representations of K′(4) realized just on 4 fields instead of the usual 16.

The Howe duality and the projective representations of symmetric groups

The symmetric group S k \mathfrak {S}_{k} possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S k \mathfrak {S}_{k} itself, coincide with the irreducible representations of the algebra A k \mathfrak {A}_{k} generated by indeterminates τ i , j \tau _{i, j} for i ≠ j i\neq j , 1 ≤ i , j ≤ n 1\leq i, j\leq n subject to the relations τ i , j = − τ j , i , τ i , j 2 = 1 , τ i , j τ m , l = − τ m , l τ i , j if { i , j } ∩ { m , l } = ∅ ; τ i , j τ j , m τ i , j = τ j , m τ i , j τ j , m = − τ i , m for any i , j , l , m . \begin{gather*} \tau _{i, j}=-\tau _{j, i}, \quad \tau _{i, j}^{2}=1, \quad \tau _{i, j}\tau _{m, l}=-\tau _{m, l}\tau _{i, j}\text { if }\{i, j\}\cap \{m, l\}=\emptyset ;\ \tau _{i, j}\tau _{j, m}\tau _{i, j}=\tau _{j, m}\tau _{i, j}\tau _{j, m}=-\tau _{i, m}\; \; \text { for any } i, j, l, m. \end{gather*} Recently M. Nazarov realized irreducible representations of A k \mathfrak {A}_{k} and Young symmetrizers by means of the Howe duality between the Lie superalgebra q ( n ) \mathfrak {q}(n) and the Hecke algebra H k = S k ∘ C l k H_{k}=\mathfrak {S}_{k}\circ Cl_{k} , the semidirect product of S k \mathfrak {S}_{k} with the Clifford algebra C l k Cl_{k} on k k indeterminates. Here I construct one more analog of Young symmetrizers in H k H_{k} as well as the analogs of Specht modules for A k \mathfrak {A}_{k} and H k H_{k} .

Five exceptional simple Lie superalgebras of vector fields

The five simple exceptional complex Lie superalgbras of vector fields are described. One of them is new; the other four are explicitely described for the first time. All of the exceptional Lie superalgebras are obtained with the help of the Cartan prolongation or a generalized prolongation. The description of several of the exceptional Lie superalgebras is associated with the Lie superalgebra AS - the nontrivial central extension of the supertraceless subalgebra SPE(4) of the periplectic Lie superalgebra PE(4) that preserves the nondegenerate odd bilinear form on the (4|4)-dimensional superspace. (A nontrivial central extension of SPE(n) only exists for n=4.)

Generalization of the algebra of the matrix pseudodifferential operators

We introduce the new nonhomogeneous graded matrix algebra Matn(R) over the ring ℜ of the pseudodifferential operators of the arbitrary orders. This algebra forms the generalization of the algebra of the standard matrix pseudodifferential operators. We show that the Matn(R) algebra admits the nontrivial central extension. The case n=2 is considered in some detail. As the example we consider the Mat2(W1+∞), Mat2(Virasoro) and Mat2(sl(2,C)) algebras.

Central extensions of some Lie algebras

We consider three Lie algebras: D e r C ( ( t ) ) Der \mathbb {C}((t)) , the Lie algebra of all derivations on the algebra C ( ( t ) ) \mathbb {C}((t)) of formal Laurent series; the Lie algebra of all differential operators on C ( ( t ) ) \mathbb {C}((t)) ; and the Lie algebra of all differential operators on C ( ( t ) ) ⊗ C n . \mathbb {C}((t))\otimes \mathbb {C}^n. We prove that each of these Lie algebras has an essentially unique nontrivial central extension.

Domain walls in supersymmetric Yang-Mills theories

We present a detailed analysis of the domain walls in supersymmetric gluodynamics and SQCD. We use the (corrected) Veneziano-Yankielowicz effective Lagrangians to explicitly obtain the wall profiles and check recent results of Dvali and Shifman: (i) the BPS-saturated nature of the walls and (ii) the exact expressions for the wall energy density which depend only on global features of dynamics (the existence of a nontrivial central extension of $N=1$ superalgebra in the theories which admit wall-like solutions). If supersymmetry is softly broken by the gluino mass, the degeneracy of the distinct vacua is gone, and one can consider the decay rate of the ``false'' vacuum into the genuine one. We do this calculation in the limit of the small gluino mass. Finally, we comment on the controversy regarding the existence of $N$ distinct chirally asymmetric vacua in $\mathrm{SU}(N)$ SUSY gluodynamics.

A newN= 6 superconformal algebra

In this paper we construct a newN = 6 superconformal algebra which extends the Virasoro algebra by theSO 6 current algebra, by 6 odd primary fields of conformal weight 3/2 and by 10 odd primary fields of conformal weight 1/2. The commutation relations of this algebra, which we will refer to asCK 6, are represented by short distance operator product expansions (OPE). We constructCK 6, as a subalgebra of theSO(6) superconformal algebra K6, thus giving it a natural representation as first order differential operators on the circle withN = 6 extended symmetry. We show thatCK 6 has no nontrivial central extensions.

Remarks on the universal central extension of sdiff(T2)loc

In a paper by C. N. Pope and K. S. Stelle [Phys. Lett. B 226, 257–263 (1989)], it was shown that the algebra of local area‐preserving diffeomorphisms on the torus is equivalent to su(∞). In this paper we present a few remarks on the universal central extension of this infinite dimensional Lie algebra, and construct, for this case, in analogy to the situation in which the affine Lie algebra has an underlying finite‐dimensional Lie algebra, the nontrivial universal central extension.

Central extensions and realizations of one-dimensional Galilean systems and quantization

The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei group, (1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, (1 + 1). As an application we construct the Stratonovich - Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.

Splitting metaplectic covers of dual reductive pairs

The symplectic group Sp(N, F) over a local fieldF (other than ℂ) has a unique non-trivial twofold central extension. The inclusion of {±1} into the circle ℂ1 induces an extension\(1 \to C^1 \to Mp(n,F) \to Sp(n,F) \to \). In this paper, an explicit splitting of the restriction of this extension to a dual reductive pair (G, H) in Sp(n, F) is given in all cases in which it exists. Such an explicit splitting is often an essential technical ingredient in the study of the local theta correspondence for the dual pair [4].

The Landau-Lifshitz equation, elliptic curves and the Ward transform

The Landau-Lifshitz (LL) equation is studied from a point of view that is close to that of Segal and Wilson's work on KdV. The LL hierarchy is defined and shown to exist using a dressing transformation that involves parameters λ1, λ2, λ3 that live on an elliptic curve Σ. The crucial role of the groupK ≃ ℤ2 × ℤ2 of translations by the half-periods of Σ and its non-trivial central extension\(\tilde K\) is brought out and an analogue of Birkhoff factorisation for\(\tilde K\)-equivariant loops in Σ is given. This factorisation theorem is given two treatments, one in terms of the geometry of an infinite-dimensional Grassmannian, and the other in terms of the algebraic geometry of bundles over Σ. Further, a Ward-like transform between a class of holomorphic vector bundles on the total spaceZ of a line-bundle over Σ and solutions of LL is constructed. An appendix is devoted to a careful definition of the Grassmannian of the Frechet spaceC ∞(S 1).

On the conformal field theory of the two-dimensional Ising model

An explicit construction of a free and massless Majorana quantum field theory, which exists on a conformal superworld M is presented. Emphasis is placed on the investigation of the action of an infinite-dimensional group G of spacetime symmetries on M. Starting with a one-particle theory, this action induces a strongly continuous representation of Diffapproximately +(S1) on the single-particle Hilbert space. After second quantization the group of implementers turns out to be a non-trivial central extension of Diffapproximately +(S1) by U(1), and a Schwinger term occurs, which gives rise to the anomalous transformation law of the energy-momentum tensor of the theory. This transformation law is studied, and an interesting connection to geometry is established.

Kac-Moody-Malcev and super-Kac-Moody-Malcev algebras

We define infinite-dimensional Malcev algebras generalizing infinite-dimensional Kac-Moody type algebras and consider their central extensions. The Kac-Moody-Malcev algebras, associated with the simple non-Lie Malcev algebras, have trivial central extensions only and super-Kac-Moody-Malcev algebras, associated with the simple non-Lie Malcev algebras, have nontrivial central extensions for Grassmann generators only.

Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions

Explicit formulas for the central extensions for Lie algebras of differential and pseudodifferential operators on the supercircle S1∣N and the general n-dimensional manifold Mn are given. Possible applications of these formulas to gravity theories coupled with higher spin fields in two and higher dimensions are discussed.

Relativistic SU(6) wave functions as the basis of modern approaches to hadronic wave functions

We show that the SU(2) Wess-Zumino-Witten model can be studied in the framework of a zero level Kac-Moody algebra in a consistent way. The dynamics emerges from a globally consistent lagrangian. The canonical formulation leads to a non-Sugawara hamiltonian that can be exactly quantised to generate a Virasoro algebra w ith nontrivial central extension.

A canonical approach to the dynamics of the Wess-Zumino-Witten model

We show that the SU(2) Wess-Zumino-Witten model can be studied in the framework of a zero level Kac-Moody algebra in a consistent way. The dynamics emerges from a globally consistent lagrangian. The canonical formulation leads to a non-Sugawara hamiltonian that can be exactly quantised to generate a Virasoro algebra w ith nontrivial central extension.