Lorentz algebra and critical dimension for the bosonic membrane

Marquard, U.; Scholl, M.

doi:10.1016/s0370-2693(89)80027-9

Cited by 20 publications

(24 citation statements)

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“…The bosonic case yields D = 27. We have shown why non-critical W ∞ (super) strings are devoid of BRST anomalies in dimensions D = 27, 11, respectively, and which coincide with the critical (super) membrane dimensions D = 27, 11 found by [69], [70]. For this reason we believe that the quantum (super) membrane should contain non-critical (super) W ∞ strings; i.e.…”

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

confidence: 58%

“…The purpose of this subsection is to review our proof [66] that non-critical W ∞ (super) strings are devoid of BRST anomalies in dimensions D = 27, 11, respectively, and which coincide with the the critical (super) membrane dimensions D = 27, 11 found by [69], [70]. We deem this review important due to the relationship between the large N limit of Self-Dual SU (N ) Yang-Mills and Self Dual Gravity [71].…”

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

confidence: 78%

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

confidence: 58%

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

confidence: 78%

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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“…In string theory the critical dimension such as D = 26 is well-known, it is for example related to Lorentz invariance in the light cone gauge. The critical dimension of the membrane has mainly been discussed with regard to its spectrum [13,14], to the truncated versions of the BRST [15] or the Lorentz algebra [16], where the D = 27 emerges as a necessary condition. Notice however that, it is not clear wether the dimension of embedding space plays as crucial a role for membranes as it does for strings.…”

Section: Introductionmentioning

confidence: 99%

“…A natural way to find a critical dimension for the membrane is to relate this latter to the string theory via dimensional reduction [17]. In this string limit, it is natural to obtain D − 1 = 26 for the bosonic membrane, since one of the D dimension in the membrane is absorbed by the gauge freedom [16,[18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Low-energy parabosonic membrane: new critical dimensions and deformed noncommutativity

Khodja¹,

Belaloui²

2009

Braz. J. Phys.

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We study a classical perturbative membrane based on the string-limit model and we discuss the consistency of the theory where the closure of the classical constraints algebra is verified. We paraquantize the model (extended string) both in the covariant and the transverse approaches. From the generalized Poincaré algebra, the so-called Poincaré (para) algebra, we show that the space-time critical dimensions D are related to the order of the paraquantization Q by the relation D = 3 + 24/Q. The symplectic structure is generalized for the paraquantum case and applied to the parabosonic membrane coupled to a constant 3-form field. This leads to a deformed noncommutative relations at the ends of the membrane (extended string) describing a geometry which might be called a q-noncommutativity.

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“…In [15,16] these critical dimensions were found using Weyl-ordering and point-splitting regularization. In our case the critical dimensions arise from requirement that the (D − 1)-dimensional Lorentz symmetry is non-anomalous.…”

Section: Introductionmentioning

confidence: 99%

Stretched quantum membranes

Björnsson

Hwang

2005

Nuclear Physics B

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We study bosonic and space-time supersymmetric membranes with small tensions corresponding to stretched configurations. Using a generalized lightcone gauge, one may set up a perturbation theory around configurations having zero tension. We will show, by explicit construction to all orders in perturbation theory, that these membrane configurations are canonically equivalent, and thereby solvable, to string-like configurations with string excitations transverse to the stretched direction. At the quantum level, it is shown that there exist an ordering such that equivalence by unitary transformations is achieved. Consistency requires the critical dimensions 27 and 11 for the bosonic and supersymmetric cases, respectively. The mass spectrum is determined to any order and is discrete containing massless states. The ground state is purely string-like, whereas excited string-like states split through the perturbation into an infinite set of states with equal or lower energies.

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Lorentz algebra and critical dimension for the bosonic membrane

Cited by 20 publications

References 5 publications

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

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

Low-energy parabosonic membrane: new critical dimensions and deformed noncommutativity

Stretched quantum membranes

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Lorentz algebra and critical dimension for the bosonic membrane

Cited by 20 publications

References 5 publications

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

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

Low-energy parabosonic membrane: new critical dimensions and deformed noncommutativity

Stretched quantum membranes

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Product

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

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