symmetry of quantum W∞ gravity

Pope, C.N.; Shen, Xianfeng; Xu, K.W.; Yuan, Ke

doi:10.1016/0550-3213(92)90067-l

Cited by 9 publications

(21 citation statements)

References 24 publications

(10 reference statements)

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“…The quantization procedure deforms the classical algebra w to the quantum algebra W due to the presence of anomalies -deformations of Moyal type of Poisson and symplecticdiffeomorphism algebras caused essentially by normal order ambiguities (see bellow). Also, generalizing the SL(2, R) Kac-Moody hidden symmetry of Polyakov's induced gravity, there are SL(∞, R) and GL(∞, R) Kac-Moody hidden symmetries for W ∞ and W 1+∞ gravities, respectively [33]. Moreover, as already mentioned, the symmetry W 1+∞ appears to be useful in the classification of universality classes in the fractional quantum Hall effect.…”

Section: Tensor Operator Algebras Of Su(2) and Large-n Matrix Modelsmentioning

confidence: 87%

Generalized higher-spin algebras and symbolic calculus on flag manifolds

Calixto¹

2006

Journal of Geometry and Physics

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We study a new class of infinite-dimensional Lie algebras W ∞ (N + , N − ) generalizing the standard W ∞ algebra, viewed as a tensor operator algebra of SU (1, 1) in a grouptheoretic framework. Here we interpret W ∞ (N + , N − ) either as an infinite continuation of the pseudo-unitary symmetry U (N + , N − ), or as a "higher-U (N + , N − )-spin extension" of the diffeomorphism algebra diff(N + , N − ) of the N = N + + N − torus U (1) N . We highlight this higher-spin structure of W ∞ (N + , N − ) by developing the representation theory of U (N + , N − ) (discrete series), calculating higher-spin representations, coherent states and deriving Kähler structures on flag manifolds. They are essential ingredients to define operator symbols and to infer a geometric pathway between these generalized W ∞ symmetries and algebras of symbols of U (N + , N − )-tensor operators. Classical limits (Poisson brackets on flag manifolds) and quantum (Moyal) deformations are also discussed. As potential applications, we comment on the formulation of diffeomorphism-invariant gauge field theories, like gauge theories of higher-extended objects, and non-linear sigma models on flag manifolds.

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Section: Tensor Operator Algebras Of Su(2) and Large-n Matrix Modelsmentioning

confidence: 87%

Generalized higher-spin algebras and symbolic calculus on flag manifolds

Calixto¹

2006

Journal of Geometry and Physics

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“…A Killing symmetry reduction of 4D SD Gravity furnishes the 3D continuous SL(∞, R) Toda theory [1] which exhibits a w ∞ symmetry (a Killing symmetry reduction of a CP 1 loop algebra over w ∞ ). Conversely, the induced 2D quantum W ∞ gravity action in the light-cone gauge has a hidden SL(∞, R) Kac-Moody symmetry [27]. The classical geometry of 2D w ∞ gravity is linked to the 4D Self-Dual gravity associated with the 4D cotangent space of 2D Riemann surfaces [38] and admits a Fedosov deformation quantization.…”

Section: Non-critical W ∞ (Super) Strings and The Critical (Super) Mementioning

confidence: 99%

“…One should notice that one is adding the weights as vectors in a Hilbert space and not the values of λ, λ * . Going back to eq-(6.24), the n th mode component associated with the function f (t) = a n cos(nt) leads tô 27) and similarly for the other generatorŝ…”

Section: Correspondence Between Highest Weight Representations and Thmentioning

confidence: 99%

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W_∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

Castro

2008

Int. J. Mod. Phys. A

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It is shown how w∞, w1+∞ Gauge Field Theory actions in 2D emerge directly from 4D Gravity. Strings and Membranes actions in 2D and 3D originate as well from 4D Einstein Gravity after recurring to the nonlinear connection formalism of Lagrange-Finsler and Hamilton-Cartan spaces. Quantum Gravity in 3D can be described by a W∞ Matrix Model in D = 1 that can be solved exactly via the Collective Field Theory method. We describe why a quantization of 4D Gravity could be attained via a 2D Quantum W∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that non-critical W∞ (super) strings are devoid of BRST anomalies in dimensions D = 27 (D = 11), respectively, follows and which coincide with the critical (super) membrane dimensions D = 27 (D = 11). We establish the correspondence between the states associated with the quasi finite highest weights irreducible representations of W∞,W∞ algebras and the quantum states of the continuous Toda molecule. Schroedinger-like QM wave functional equations are derived and solutions are found in the zeroth order approximation. Since higher-conformal spin W∞ symmetries are very relevant in the study of 2D W∞ Gravity, the Quantum Hall effect, large N QCD, strings, membranes, ...... it is warranted to explore further the interplay among all these theories. In this introductory section we will review the work of [2] and afterwards we will discuss the recent work related the Hidden Symmetries of M theory. Keywords1.1 Gravity in D = m + n as an m-dim Gauge Theory of diffeomorphisms of an internal n-dim space and HolographySome time ago Park [1] showed that 4D Self Dual Gravity is equivalent to a WZNW model based on the group SU (∞). Namely, 4D Self Dual Gravity is the non-linear sigma model based in 2D whose target space is the "group manifold" of area-preserving diffs of another 2D-dim manifold. Roughly speaking, this means that the effective D = 4 manifold, where Self Dual Gravity is defined, is "spliced" into two 2D-submanifolds: one submanifold is the original 2D base manifold where the non-linear sigma model is defined. The other 2D submanifold is the target group manifold of area-preserving diffs of a two-dim sphere S 2 . The authors [2] went further and generalized this particular Self Dual Gravity case to the full fledged gravity in D = 2 + 2 = 4 dimensions, and in general, to any combinations of m + n-dimensions. Their main result is that m + n-dim Einstein gravity can be identified with an m-dimensional generally invariant gauge theory of Dif f s N , where N is an n-dim manifold. Locally the m + ndim space can be written as Σ = M × N and the metric G AB decomposes as: [62]. The decomposition (1.1) must not be confused with the Kaluza-Klein reduction where one imposes an isometry restriction on the γ AB that turns A a µ into a gauge connection associated with the gauge group G generated by isometry. Dropping the isometry restrictions allows all the fields to depend on all the coordinates x, y. Nevertheless A a µ (x, y) can still be identified as a conne...

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“…The second approach is to extract the anomalous quantum dynamics directly from the the anomalous Ward identities of the worldsheet symmetries. In this latter approach, non-trivial correlation functions arise at the quantum level, revealing in some cases hidden quantum symmetries such as the SL(2, IR) symmetry found by Polyakov for the bosonic string [4], or the SL(∞, IR) symmetry found in worldsheet W ∞ gravity [5,6] (which becomes a GL(∞, IR) symmetry for W 1+∞ gravity [6].) Because it reveals hidden symmetries, the approach of extracting dynamics from anomalous Ward identities is clearly of great importance for the non-critical theories.…”

Section: Introductionmentioning

confidence: 99%

“…Attempts in [7] to extend the approach of [4][5][6] to the W N gravity case ran into the difficulty that a consistent set of conditions to impose on the background gauge fields to eliminate the anomalies could not be derived owing to their off-diagonal structure. These difficulties are presumably related to our imperfect understanding of W 3 geometry.…”

Section: Introductionmentioning

confidence: 99%

Canonical BRST quantisation of worldsheet gravities

Mohayee

Pope²,

Stelle

et al. 1995

Nuclear Physics B

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We reformulate the BRST quantisation of chiral Virasoro and W 3 worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator Q, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator Q 2 , calculated using operator products, to the gauge fermion.

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symmetry of quantum W∞ gravity

Cited by 9 publications

References 24 publications

Generalized higher-spin algebras and symbolic calculus on flag manifolds

Generalized higher-spin algebras and symbolic calculus on flag manifolds

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W_∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

Canonical BRST quantisation of worldsheet gravities

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symmetry of quantum W∞ gravity

Cited by 9 publications

References 24 publications

Generalized higher-spin algebras and symbolic calculus on flag manifolds

Generalized higher-spin algebras and symbolic calculus on flag manifolds

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

Canonical BRST quantisation of worldsheet gravities

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STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W_∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY