Lecture 2: 3d gravity as group theory Quantum Coulomb Solution

Transcription

1 The Second Mandelstam Theoretical Physics School University of the Witwatersrand 17/01/2018 Lecture 2: 3d gravity as group theory Quantum Coulomb Solution Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes

2 Overview Which of the successes of AdS3 gravity can be reproduced in the flat case? all of them! Collaborators C. Troessaert H. A. Gonzalez B. Oblak A. Gomberoff A. Maloney

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5 3d AdS & flat Symmetries and solutions residual symmetries general solution to EOM closed form

6 3d AdS & flat Symmetries and solutions Infinitesimal and finite transformations Schwarzian derivative, EM tensor

7 3d AdS & flat Charge algebra Bulk computation: ADM type charges NB: solution space is dual to symmetry algebra Bondi mass aspect = supermomentum Bondi angular momentum aspect = super angular momentum

8 Asymptotic symmetries BMS3 Group simply connected covering group of Virasoro group Bott-Thurston 2- cocycle centrally extended BMS3 group Schwarzian derivative

9 Solution space Coadjoint representation action of symmetry group on solution space coadjoint action energy-momentum tensor of CFT2 change in orbital part due to supertranslation

10 3d AdS & flat Charge algebra Fourier modes Dirac bracket algebra BMS3 algebra

11 3d flat Charge algebra contraction appropriate combination for the limit Virasoro algebra contracts to relation to similar to contraction between Virasoro factor: centrally non extended superrotations

12 3d AdS & flat Group theory math summary covariant phase space of 3d gravity 2 copies of coadjoint representation of at coadjoint representation of at Dirac bracket = Kirillov-Kostant bracket on useful? central

13 Solution space Coadjoint orbits solution space classified according to coadjoint orbits result Virasoro coadjoint orbit little group orbit well-known math problem Witten, Coadjoint orbits of the Virasoro group, CMP 88 Balog et al., HEP-TH/ coadjoint orbits are symplectic spaces, bi-hamiltonian integrable systems geometric quantization/ UIRREPS

14 Solution space Virasoro coadjoint orbits 2 types of Virasoro coadjoint orbit constant representative for "restframe" constant representative little group case 2 constant p Minkowski little group

15 Solution space Holographic positive energy theorem total energy study behaviour of energy functional on orbit Schwarzian derivative inequality allows to show that energy for orbits with constant representatives is bounded from below iff orbit has a constant representative that lies above Minkowski space time similar results in AdS3

16 AdS3 & 3d flat Zero mode solutions zero mode solutions in both cases BMS form ADM form repeat derivation of entropy of cosmological solutions from Cardy type formula

17 3d AdS & flat Dual 2d theories: From CS start from CS formulation of 3d gravity solve constraints with asymptotic condition 2 copies of SL(2,R) cwzw model = non chiral SL(2,R) WZW model iso(2,1) cwzw model Hamiltonian reduction (Drinfeld-Sokolov) Liouville theory with iso(2,1) current algebra

18 3d AdS & flat Dual 2d theories: From CS BMS Liouville with centrally extended global BMS3 symmetry algebra Hamiltonian form of Liouville BMS algebra as global symmetry algebra

19 3d AdS & flat Dual 2d theories: Effective Lagrangians

20 3d AdS & flat Dual 2d theories: Effective Lagrangians

21 3d AdS & flat Dual 2d theories: Effective Lagrangians G.B., H. Gonzalez, P. Salgado, Geometric actions for 3d gravity, ARXIV:

22 Quantum aspects BMS3 particles: Induced representations scattering theory between and particle : UIRREP of BMS3 structure: finite-dimensional Lie groups all UIRREPS : Wigner-Mackey cf. Poincaré group 1) determine characters of 2) determine orbits & little groups of 3) induce UIRREPS of out of UIRREPS of little group again classified by coadjoint orbits of Virasoro group reproduced by geometric quantisation of coadjoint orbits

23 Quantum aspects One-loop partition function aim: compute to 1-loop gravity partition function background : thermal spinning flat space loop expansion classical contribution determined from boundary term

24 Quantum aspects One-loop partition function 1-loop contribution compute using heat kernels in flat space massless symmetric traceless tensor method of images to compactify massless complex vector ghost massless scalar (trace) result agrees with vacuum BMS character as computed from Frobenius formula

25 Conclusion 3d gravity with non trivial asymptotics group theory for the diffeomorphism group on the circle

26 References + C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited arxiv: Aspects of the BMS/CFT correspondence arxiv: " Entropy of three-dimensional asymptotically flat cosmological solutions " arxiv: A. Gomberoff, H.A. Gonzalez, Flat limit of three dimensional asymptotically anti-de Sitter spacetimes arxiv: H.A. Gonzalez, Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity arxiv: B. Oblak, Notes on the BMS group in three dimensions. I. Induced representations arxiv: II. Coadjoint representation arxiv: Holographic positive energy theorems in three-dimensional gravity arxiv: H.A. Gonzalez, A. Maloney, B. Oblak, One-loop partition function of three-dimensional flat gravity arxiv:

27 Asymptotic symmetries Digression: Asymptotics and soft behaviour Gervais & Zwanziger 1980

28 Quantum Coulomb solution & black hole microstates

29 Overview Main claim microstates responsible for BH entropy related to non-proper gauge DoF rather than physical gravitons and should be quantised as such

30 Overview 1) Arguments linearised Schwarzschild solution involves longitudinal DoF no physical gravitons in 3d but BTZ black hole observables = ADM surface charges involve unphysical DoF physical toy-model: electric charge in electromagnetism 2) BRST quantisation of free Maxwell field 3) Quantum Coulomb solution 4) Future directions & perspectives

31 Arguments GR DoF linearised GR = massless spin 2 gauge field on Minkowski background Hamiltonian formulation orthogonal decomposition of symmetric rank 2 tensor # of comp D=4 D= ADM 1962, Dynamics of GR, gr-qc/

32 Arguments Linearised Schwarzschild solution canonical pairs D=3 D=4 coupling to a massive particle at rest only Hamiltonian constraint is affected all other variables 0 after spatial diffeo linearised Schwarzschild solution, no TT variables involved

33 Arguments Observable observable surface charge ADM mass only sees exactly like for electric charge study electromagnetism first, simpler but similar physics

34 Electromagnetism DoF & reduced quantisation Hamiltonian formulation first class constraints physical DoF unphysical DoF Quantisation reduced phase space: transverse DoF in positive definite Hilbert space with charged sources: quantise transverse fluctuations around classical charged solution quantum nature of this classical background solution?

35 Electromagnetism indefinite metric Hilbert space quantise all polarisations in indefinite metric Hilbert space a) Gupta-Bleuler physical state condition b) BRST quantisation BRST charge gauge fixation fermionic DoF null states decouple cancel contributions from longitudinal and temporal photons path integral eliminate momenta covariant Faddeev-Popov path integral HT 1992, Quantization of Gauge Systems

36 Electromagnetism BRST quantisation expand fields in terms of oscillators physical transverse unphysical bosonic (null) unphysical ghost BRST charge physical states in particular BRST exact states decouple vacuum state is physical quartet mechanism Hamiltonian ghosts and unphysical bosonic DoF drop out responsible for black body entropy

37 Electromagnetism Coupling to a source static charge at the origin only Gauss law modified modified BRST charge c-number Fourier transform of old vacuum no longer physical new vacuum in terms of old vacuum coherent state of null photons

38 Electromagnetism Quantum Coulomb solution unusual classical properties instead of Ehrenfest theorem NB: requires infrared regularisation interpretation : extrapolation of Aharonov-Bohm effect to quantised electromagnetic field

39 Perspectives & Speculations Partition function of charged black body (micro)-canonical entropy : compute number of such microstates between Q and Q +δq? (grand)-canonical entropy : compute at fixed potential instead of fixed macroscopic charge goes like the surface of the body BRST exact (except for zero mode) for standard fall-off conditions bulk transverse oscillators, standard BB partition function which DoF? boundary unphysical photons

40 Perspectives & Speculations Partition function of charged black body non trivial fall-off conditions 2+1d theory induced on the boundary chemical potential cte infrared subtleties: work in a finite volume because zero mode is important, cf Bose- Einstein condensation try to compute this partition function in a theory with null oscillators on the boundary, to be continued

41 Perspectives & Speculations Partition function of charged black body Consistency check improved according to the RT prescription, cf Gibbons-Hawking term non trivial fall-offs as new DoF also in

42 Perspectives & Speculations AdS3 (or flat 3d) gravity no bulk Hamiltonian role of Q played by ADM mass M 1+1d dual boundary theory Liouville or difference of 2 chiral bosons CHvD 1992, CQG 1995 to be quantised as gauge DoF?