Higher symmetries of the Laplacian

Eastwood, Michael

doi:10.4007/annals.2005.161.1645

Cited by 231 publications

(466 citation statements)

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“…It can be checked in a straightforward manner that this expression has the form of equation (11) in the indices cdef . Hence it represents an element of g ⊗ A im , see Remark 2 with the notation A im as introduced in the proof of Theorem 2.…”

Section: The Joseph-like Ideal For Osp(m|2n)mentioning

confidence: 99%

“…This corresponds to a super version of the ambient space method, used for the minimal representation and the Joseph ideal for so(m) in e.g. [1,11,17,23].…”

Section: The Corresponding Representation Of Osp(m|2n)mentioning

confidence: 99%

“…This generalizes the appearance of the Joseph ideal of so(m) as the annihilator ideal of the conformal representation on harmonic functions, see e.g. [1,11,20,23]. In Subsection 6.2, the super metaplectic representation of [6,33] is considered.…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by classical results on the symmetries of the Laplace operator, see [11,17], we touch analogous questions in the last Section 7. It follows that the quotient of the universal enveloping algebra of osp(m|2n) with respect to the Joseph-like ideal forms an algebra of symmetries.…”

Section: Introductionmentioning

confidence: 99%

“…[1,20,23] for explicit descriptions in several signatures. This ideal subsequently plays an essential role in the description of the symmetries of the Laplace operator, for which its kernel is exactly this minimal representation, see [11]. The minimal representation for sp(2n) is known as the metaplectic representation, Segal-Shale-Weil representation or the symplectic spinors, see [18,24,31].…”

Section: Introductionmentioning

confidence: 99%

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Joseph Ideals and Harmonic Analysis for

Coulembier

Somberg

Souček

2013

International Mathematics Research Notices

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The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R m−2 . The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra ⊗g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in ⊗g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n) = spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R m−2|2n and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n). As a side result we obtain the proof of a conjecture of M. Eastwood about the Cartan product of irreducible representations of semisimple Lie algebras made in [Bull.

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Section: The Joseph-like Ideal For Osp(m|2n)mentioning

confidence: 99%

“…This corresponds to a super version of the ambient space method, used for the minimal representation and the Joseph ideal for so(m) in e.g. [1,11,17,23].…”

Section: The Corresponding Representation Of Osp(m|2n)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Joseph Ideals and Harmonic Analysis for

Coulembier

Somberg

Souček

2013

International Mathematics Research Notices

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Higher spin gauge theories in various dimensions

Vasiliev

2004

Fortschritte der Physik

193

284

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Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications

Sleight

Taronna

2018

Fortschritte der Physik

153

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We study conformal partial waves (CPWs) in Mellin space with totally symmetric external operators of arbitrary integer spin. The exchanged spin is arbitrary, and includes mixed symmetry and (partially)-conserved representations. In a basis of CPWs recently introduced in arXiv:1702.08619, we find a remarkable factorisation of the external spin dependence in their Mellin representation. This property allows a relatively straightforward study of inversion formulae to extract OPE data from the Mellin representation of spinning 4pt correlators and in particular, to extract closed-form expressions for crossing kernels of spinning CPWs in terms of the hypergeometric function 4 F 3 . We consider numerous examples involving both arbitrary internal and external spins, and for both leading and sub-leading twist operators. As an application, working in general d we extract new results for O(1/N) anomalous dimensions of double-trace operators induced by double-trace deformations constructed from single-trace operators of generic twist and integer spin. In particular, we extract the anomalous dimensions of double-trace operators [O J ] n,l with O J a single-trace operator of integer spin J. 1 See however [48] for recent analytic results for the scalar-fermion bootstrap. 2 The Mellin space approach to the conformal bootstrap was initiated in [32,33,57] for external scalar operators. 1 of 77) www.advancedsciencenews.com www.fp-journal.org In §5.2.1 we apply our formalism to spinning correlators in arbitrary space-time dimensions, which also requires to consider mixed symmetry CPWs and their crossing kernels -which we derive. In particular, using the methods of §3 we first extract mean-field theory OPE coefficients for leading twist double-trace operators [O J ] 0,l built from a spin-J single-trace operator O J and a scalar operator . Combining the latter with the results for crossing kernels, we then obtain the anomalous dimensions of the doubletrace operators [O J ] 0,l induced by the double-trace flow (1.1) with l = 0. (3 The basis itself was originally obtained in the context of simplifying the kinematic map between bulk cubic couplings and 3pt conformal structures, so our proposed framework also lends itself nicely to studies of bulk physics. 4 Some of such nested formulas recently appeared in the literature in a different CFT basis. [62] For external scalars, such formulas were originally given in [29].

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Higher symmetries of the Laplacian

Abstract: We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.

Cited by 231 publications

References 44 publications

Joseph Ideals and Harmonic Analysis for

Joseph Ideals and Harmonic Analysis for

Higher spin gauge theories in various dimensions

Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications

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