Abstract
We continue analysis of [1] and study rigidity and stability of the \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_4 \) algebra and its centrally extended version \( \widehat{\mathfrak{bm}{\mathfrak{s}}_4} \). We construct and classify the family of algebras which appear as deformations of \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_4 \) and in general find the four-parameter family of algebras \( \mathcal{W} \)(a, b; \( \overline{a},\overline{b} \)) as a result of the stabilization analysis, where \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_4 \) = \( \mathcal{W} \)(−1/2, −1/2; −1/2, −1/2). We then study the \( \mathcal{W} \)(a, b; \( \overline{a},\overline{b} \)) algebra, its maximal finite subgroups and stability for different values of the four parameters. We prove stability of the \( \mathcal{W} \)(a, b; \( \overline{a},\overline{b} \)) family of algebras for generic values of the parameters. For special cases of (a, b) = (\( \overline{a},\overline{b} \)) = (0, 0) and (a, b) = (0, −1), (\( \overline{a},\overline{b} \)) = (0, 0) the algebra can be deformed. In particular we show that centrally extended \( \mathcal{W} \)(0, −1; 0, 0) algebra can be deformed to an algebra which has three copies of Virasoro as a subalgebra. We briefly discuss these deformed algebras as asymptotic symmetry algebras and the physical meaning of the stabilization and implications of our result.
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References
A. Farahmand Parsa, H.R. Safari and M.M. Sheikh-Jabbari, On Rigidity of 3d Asymptotic Symmetry Algebras, JHEP 03 (2019) 143 [arXiv:1809.08209] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
A. Ashtekar, J. Bičák and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev. D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
B. Oblak, BMS Particles in Three Dimensions, Ph.D. thesis, Brussels University, Belgium, 2016. arXiv:1610.08526 [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
G. Barnich and G. Compère, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 (2010) [Ann. U. Craiova Phys. 21 (2011) S11] [arXiv:1102.4632] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
G. Barnich, Centrally extended BMS4 Lie algebroid, JHEP 06 (2017) 007 [arXiv:1703.08704] [INSPIRE].
C. Troessaert, The BMS4 algebra at spatial infinity, Class. Quant. Grav. 35 (2018) 074003 [arXiv:1704.06223] [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
A. Ashtekar and A. Magnon, Asymptotically anti-de Sitter space-times, Class. Quant. Grav. 1 (1984) L39 [INSPIRE].
G. Compère, W. Song and A. Strominger, New Boundary Conditions for AdS3, JHEP 05 (2013) 152 [arXiv:1303.2662] [INSPIRE].
H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev. D 95 (2017) 106005 [arXiv:1611.09783] [INSPIRE].
H. Afshar, D. Grumiller, M.M. Sheikh-Jabbari and H. Yavartanoo, Horizon fluff, semi-classical black hole microstates — Log-corrections to BTZ entropy and black hole/particle correspondence, JHEP 08 (2017) 087 [arXiv:1705.06257] [INSPIRE].
D. Grumiller and M. Riegler, Most general AdS 3 boundary conditions, JHEP 10 (2016) 023 [arXiv:1608.01308] [INSPIRE].
D. Grumiller, W. Merbis and M. Riegler, Most general flat space boundary conditions in three-dimensional Einstein gravity, Class. Quant. Grav. 34 (2017) 184001 [arXiv:1704.07419] [INSPIRE].
L. Donnay, G. Giribet, H.A. González and M. Pino, Supertranslations and Superrotations at the Black Hole Horizon, Phys. Rev. Lett. 116 (2016) 091101 [arXiv:1511.08687] [INSPIRE].
D. Grumiller, A. Perez, M. Sheikh-Jabbari, R. Troncoso and C. Zwikel, Soft hair on black hole and cosmological horizons in any dimension, to appear.
D. Grumiller and M.M. Sheikh-Jabbari, Membrane Paradigm from Near Horizon Soft Hair, Int. J. Mod. Phys. D 27 (2018) 1847006 [arXiv:1805.11099] [INSPIRE].
D. Kapec, V. Lysov and A. Strominger, Asymptotic Symmetries of Massless QED in Even Dimensions, Adv. Theor. Math. Phys. 21 (2017) 1747 [arXiv:1412.2763] [INSPIRE].
V. Hosseinzadeh, A. Seraj and M.M. Sheikh-Jabbari, Soft Charges and Electric-Magnetic Duality, JHEP 08 (2018) 102 [arXiv:1806.01901] [INSPIRE].
A. Strominger, Magnetic Corrections to the Soft Photon Theorem, Phys. Rev. Lett. 116 (2016) 031602 [arXiv:1509.00543] [INSPIRE].
M. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations, JHEP 11 (2016) 012 [arXiv:1605.09677] [INSPIRE].
H. Afshar, E. Esmaeili and M.M. Sheikh-Jabbari, Asymptotic Symmetries in p-Form Theories, JHEP 05 (2018) 042 [arXiv:1801.07752] [INSPIRE].
D. Francia and C. Heissenberg, Two-Form Asymptotic Symmetries and Scalar Soft Theorems, Phys. Rev. D 98 (2018) 105003 [arXiv:1810.05634] [INSPIRE].
M. Gerstenhaber, On the deformation of rings and algebras: I, Annals Math. 59 (1964) 59.
M. Gerstenhaber, On the deformation of rings and algebras: II, Annals Math. 84 (1966) 1.
M. Gerstenhaber, On the deformation of rings and algebras: III, Annals Math. 88 (1968) 1.
M. Gerstenhaber, On the deformation of rings and algebras: IV, Annals Math. 99 (1974) 257.
A. Nijenhuis and R. Richardson, Deformations of lie algebra structures, J. Math. Mech. 17 (1967) 89.
M. Levy-Nahas, Deformation and contraction of Lie algebras, J. Math. Phys. 8 (1967) 1211.
J. Whitehead, Combinatorial homotopy. I, Bull. Am. Math. Soc. 55 (1949) 213.
J. Whitehead, Combinatorial homotopy. II, Bull. Am. Math. Soc. 55 (1949) 453.
G. Hochschild and J.-P. Serre, Cohomology of lie algebras, Annals Math. 57 (1953) 591.
E. Inönü and E.P. Wigner, On the Contraction of groups and their represenations, Proc. Nat. Acad. Sci. 39 (1953) 510 [INSPIRE].
R. Vilela Mendes, Deformations, stable theories and fundamental constants, J. Phys. A 27 (1994) 8091 [INSPIRE].
J.M. Figueroa-O’Farrill, Deformations of the Galilean Algebra, J. Math. Phys. 30 (1989) 2735 [INSPIRE].
C. Chryssomalakos and E. Okon, Generalized quantum relativistic kinematics: A stability point of view, Int. J. Mod. Phys. D 13 (2004) 2003 [hep-th/0410212] [INSPIRE].
J. Figueroa-O’Farrill, Classification of kinematical Lie algebras, arXiv:1711.05676 [INSPIRE].
J.M. Figueroa-O’Farrill, Kinematical Lie algebras via deformation theory, J. Math. Phys. 59 (2018) 061701 [arXiv:1711.06111] [INSPIRE].
J.M. Figueroa-O’Farrill, Higher-dimensional kinematical Lie algebras via deformation theory, J. Math. Phys. 59 (2018) 061702 [arXiv:1711.07363] [INSPIRE].
T. Andrzejewski and J.M. Figueroa-O’Farrill, Kinematical lie algebras in 2 + 1 dimensions, J. Math. Phys. 59 (2018) 061703 [arXiv:1802.04048] [INSPIRE].
J.M. Figueroa-O’Farrill, Conformal Lie algebras via deformation theory, arXiv:1809.03603 [INSPIRE].
J. Figueroa-O’Farrill and S. Prohazka, Spatially isotropic homogeneous spacetimes, JHEP 01 (2019) 229 [arXiv:1809.01224] [INSPIRE].
A. Fialowski and M. Penkava, Deformation Theory of Infinity Algebras, J. Algebra. 255 (2002) 59 [math/0101097].
A. Fialowski, Formal rigidity of the witt and virasoro algebra, J. Math. Phys. 53 (2012) 073501.
S. Gao, C. Jiang and Y. Pei, The derivations, central extensions and automorphism group of the lie algebra W, arXiv:0801.3911.
S. Gao, C. Jiang and Y. Pei, Low-dimensional cohomology groups of the lie algebras W(a, b), Commun. Algebra 39 (2011) 397.
J. Ecker and M. Schlichenmaier, The Vanishing of the Low-Dimensional Cohomology of the Witt and the Virasoro algebra, arXiv:1707.06106 [INSPIRE].
J. Ecker and M. Schlichenmaier, The Low-Dimensional Algebraic Cohomology of the Virasoro Algebra, arXiv:1805.08433 [INSPIRE].
G. Barnich, A. Gomberoff and H.A. González, Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D 87 (2013) 124032 [arXiv:1210.0731] [INSPIRE].
M. Henneaux, Asymptotically anti-de Sitter Universes in D = 3, 4 and higher dimensions, in 4th Marcel Grossmann Meeting on the Recent Developments of General Relativity, Rome, Italy, June 17–21, 1985, pp. 959–966.
M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter Spaces, Commun. Math. Phys. 98 (1985) 391 [INSPIRE].
A. Ashtekar and S. Das, Asymptotically Anti-de Sitter space-times: Conserved quantities, Class. Quant. Grav. 17 (2000) L17 [hep-th/9911230] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Higher-Dimensional Supertranslations and Weinberg’s Soft Graviton Theorem, arXiv:1502.07644 [INSPIRE].
S. Hollands, A. Ishibashi and R.M. Wald, BMS Supertranslations and Memory in Four and Higher Dimensions, Class. Quant. Grav. 34 (2017) 155005 [arXiv:1612.03290] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
D.B. Fuks, Cohomology of infinite-dimensional Lie algebras, Springer Science & Business Media, (2012).
S. Weinberg, The quantum theory of fields. Vol. 1: Foundations, Cambridge University Press, (1995).
M. Schlichenmaier, An elementary proof of the vanishing of the second cohomology of the witt and virasoro algebra with values in the adjoint module, Forum Math. 26 (2014) 913.
P. Salgado and S. Salgado, \( \mathfrak{so} \)(D − 1, 1) ⊗ \( \mathfrak{so} \)(D − 1, 2) algebras and gravity, Phys. Lett. B 728 (2014) 5 [INSPIRE].
J. Hartong and T. Ortín, Tensor Hierarchies of 5- and 6-Dimensional Field Theories, JHEP 09 (2009) 039 [arXiv:0906.4043] [INSPIRE].
P. Concha, N. Merino, O. Mišković, E. Rodríguez, P. Salgado-ReboLledó and O. Valdivia, Asymptotic symmetries of three-dimensional Chern-Simons gravity for the Maxwell algebra, JHEP 10 (2018) 079 [arXiv:1805.08834] [INSPIRE].
R. Caroca, P. Concha, E. Rodríguez and P. Salgado-ReboLledó, Generalizing the \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 \) and 2D-conformal algebras by expanding the Virasoro algebra, Eur. Phys. J. C 78 (2018) 262 [arXiv:1707.07209] [INSPIRE].
I.M. Gel’fand and D. Fuks, Cohomologies of Lie algebra of tangential vector fields of a smooth manifold, Funct. Anal. Appl. 3 (1969) 194.
J. Unterberger and C. Roger, The Schrödinger-Virasoro Algebra, Springer, Berlin, Germany, (2012).
D. Degrijse and N. Petrosyan, On cohomology of split lie algebra extensions, J. Lie Theory 22 (2012) 1 [arXiv:0911.0545].
A. Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].
J. Hartong, Holographic Reconstruction of 3D Flat Space-Time, JHEP 10 (2016) 104 [arXiv:1511.01387] [INSPIRE].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat Holography: Aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].
G. Barnich, P.-H. Lambert and P. Mao, Three-dimensional asymptotically flat Einstein-Maxwell theory, Class. Quant. Grav. 32 (2015) 245001 [arXiv:1503.00856] [INSPIRE].
M. Henkel and S. Stoimenov, Meta-conformal algebras in d spatial dimensions, arXiv:1711.05062 [INSPIRE].
G. Barnich and C. Troessaert, Finite BMS transformations, JHEP 03 (2016) 167 [arXiv:1601.04090] [INSPIRE].
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Safari, H.R., Sheikh-Jabbari, M.M. BMS4 algebra, its stability and deformations. J. High Energ. Phys. 2019, 68 (2019). https://doi.org/10.1007/JHEP04(2019)068
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DOI: https://doi.org/10.1007/JHEP04(2019)068