Classification of Casimirs in 2D Hydrodynamics

Izosimov, Anton; Khesin, Boris

doi:10.17323/1609-4514-2017-17-4-699-716

Cited by 9 publications

(7 citation statements)

References 8 publications

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“…As shown in ref. [77], whenever there are two edges corresponding to the same range of values of w, the function w can be modified so that the measures on the individual edges of the Reeb graph are changed but their sum remains the same. This leads to the construction of a family of functions with different measured Reeb graphs, but the same generalized enstrophies.…”

Section: Jhep09(2021)008mentioning

confidence: 99%

Gravitational edge modes, coadjoint orbits, and hydrodynamics

Donnelly

Freidel

Moosavian

et al. 2021

J. High Energ. Phys.

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The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner’s famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

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Section: Jhep09(2021)008mentioning

confidence: 99%

Gravitational edge modes, coadjoint orbits, and hydrodynamics

Donnelly

Freidel

Moosavian

et al. 2021

J. High Energ. Phys.

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“…Table 1: 7 types of neighborhoods of singular points with corresponding Reeb graphs and asymptotics for the measure on a Reeb graph (figures are partially taken from [9]). In order to simplify notation we assume that f pvq " 0.…”

Section: VImentioning

confidence: 99%

“…In the recent work [18] the orbit method was applied to the symplectomorphism group of the two-sphere. The classification of generic coadjoint orbits was obtained in [9,10] for the case of closed surfaces. In [9] there is a list of open questions for the case of surfaces with boundary.…”

Section: Introductionmentioning

confidence: 99%

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Classification of coadjoint orbits for symplectomorphism groups of surfaces

Kirillov¹

2020

Preprint

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“…We expect that the result of the present paper will also be useful in 2D fluid dynamics. In particular, it gives a partial answer to Problem 5.6 from [6] on the asymptotical properties of measures on Reeb graphs. This paper is organised as follows.…”

Section: Introductionmentioning

confidence: 99%