Berry Phases of Boundary Gravitons

Transkript

1 Berry Phases of Boundary Gravitons Blagoje Oblak (ETH Zurich) February 2018 Based on arxiv (JHEP) Based on arxiv (JHEP) Based on arxiv (CQG)

2 MOTIVATION Gravity has rich asymptotic symmetries Virasoro for AdS 3 [Brown-Henneaux 1986] BMS for Minkowski [Bondi et al. 1962] Aspt symmetries act on space-time metric Boundary gravitons Observable quantities? Berry phases for loops in space of For definiteness : 3D gravity { metrics reference frames Generalize Thomas precession [Thomas 1926]

3 PLAN OF THE TALK 1. Berry phases in group reps 2. Virasoro Berry phases 3. Berry phases & asymptotic symmetries

4 1. Berry phases in group reps A. Berry phases B. Application to unitary reps

5 BERRY PHASES System with parameters p 1,..., p n Coordinates on manifold M Hamiltonian H(p) with p M Eigenvalue E(p), eigenvector φ(p) Adiabatic variation of parameters Path γ(t) in M Time-dependent Hamiltonian H(γ(t)) Solve Schrödinger with ψ(0) = φ(γ(0)) Adiabatic theorem : ψ(t) = e iθ(t) φ(γ(t))

6 BERRY PHASES T T θ(t) = dt E(γ(t)) + i dt φ(γ(t)) φ(γ(t)) 0 0 t }{{}}{{} Dynamical phase Geometric phase Due to parameter-dependence in φ( ) Independent of t parametrization Closed paths : γ(t) = γ(0) Berry phase : T B φ [γ] = i dt φ(γ(t)) φ(γ(t)) = i φ( ) d φ( ) 0 t γ }{{} Berry connection

7 BERRY PHASES & GROUPS Group G, algebra g Time translations = e tx for some X g Let U = unitary rep of G Evolution operator U[e tx ] = e t u[x] Hamiltonian H = i u[x] This relies on a choice of frame! Let f G be a change of frame Hamiltonian U[ f ]H U[ f ] 1 G space of parameters Berry phases?

8 BERRY PHASES & GROUPS Let φ = eigenstate of H Let f (t) = closed path in G States U[ f (t)] φ Berry phase : B φ [ f ] = i dt φ U[ f (t)] t U[ f (t)] φ = i dt φ u[ f 1 ḟ ] φ Maurer-Cartan form Note : Berry phase vanishes for f (t) in stabilizer of φ Parameter space is G/G φ

9 BERRY PHASES & GROUPS Example : G = SU(2), highest-weight state j Parameter space = SU(2)/S 1 = S 2 Closed path f (t) SU(2) Berry phase = j (enclosed area) How does this work for Virasoro?

10 2. Virasoro Berry phases A. Maurer-Cartan form of Diff S 1 B. Maurer-Cartan form of Virasoro C. Berry phases

11 DIFF S 1 Diff S 1 = group of circle diffeos Half of 2D conformal group Elements are fcts f (ϕ) with f (ϕ + 2π) = f (ϕ) + 2π and f (ϕ) > 0 Group operation is composition : ( f g ) (ϕ) = f ( g(ϕ) ) = ( f g ) (ϕ) Infinite-dimensional Lie group Lie algebra = Vect S 1 = Witt algebra l m e imϕ ϕ

12 DIFF S 1 Examples : Rotations f (ϕ) = ϕ + θ Boosts :Text e i f (ϕ) = e iϕ cosh(λ/2) + sinh(λ/2) e iϕ sinh(λ/2) + cosh(λ/2) λ = rapidity

13 DIFF S 1 Examples : Rotations f (ϕ) = ϕ + θ Superboosts : e in f (ϕ) = e inϕ cosh(λ/2) + sinh(λ/2) e inϕ sinh(λ/2) + cosh(λ/2) λ = rapidity

14 DIFF S 1 Maurer-Cartan form? Let f (t) = path in G Derivative ḟ (t) = τ f (τ) τ=t is tangent vector at f (t) f 1 ḟ = ( f (t) 1 τ=t f (τ)) τ g For Diff S 1 : path f (t, ϕ) ( f 1 ḟ ) (t, ϕ) = τ ( f 1( t, f (τ, ϕ) )) = ḟ (t, ϕ) f (t, ϕ)

15 VIRASORO Virasoro group = Central extension of Diff S 1 = Diff S 1 R Lie algebra = Vect S 1 R = Virasoro algebra Extra central entry in Maurer-Cartan form : ( f 1 ḟ ) Virasoro = ( ḟ f ϕ, 1 48π 2π 0 dϕ ḟ ( ) ) f f f We can compute Berry phases! [Alekseev-Shatashvili, Rai-Rodgers 1989]

16 BERRY PHASES Berry : B φ [ f ] = i dt φ u[ f 1ḟ ] φ Apply this to Virasoro! Central charge c Highest weight h : u[l 0 ] h = h h u[l n ] h = 0 if n > 0 Take φ = h Take f (t, ϕ) closed path Berry phase B h,c [ f (t, ϕ)]?

17 BERRY PHASES Berry : B h,c [ f ] = i dt h u[ f 1ḟ ] h Maurer-Cartan : f 1ḟ ( = ḟ 1 ḟ ( f ) ) f ϕ, 48π dϕ f f i i [( ḟ dt h u )] f ϕ, 0 h = h c/24 dt dϕ ḟ 2π f dt dϕ ḟ ( f f f [( dt h u 0, ḟ ( f ) )] f f h = c 48π B h,c [ f ] = 1 dt dϕ ḟ [ 2π f h c 24 + c 24 ( ) f ] f )

18 BERRY PHASES B h,c [ f ] = 1 dt dϕ ḟ [ 2π f h c 24 + c 24 ( ) f ] f Actual parameter space is G/G φ = Diff S 1 /G h Stabilizer of h : U(1) or SL(2, R) Diff S 1 /S 1 or Diff S 1 / SL(2, R) Coadjoint orbits of Virasoro Berry phase = symplectic flux

19 BERRY PHASES Circular path f (t, ϕ) = g(ϕ) + ωt Closes at t = 2π/ω Berry phase : [ dϕ B h,c [ f ] = g h c 24 + c 24 ( g g ) ] ( + 2π h c ) 24

20 3. Berry phases & asymptotic symmetries A. Berry phases in AdS 3 B. Berry phases in flat space

21 BERRY PHASES IN ADS 3 AdS 3 space-time Infinity = time-like cylinder Light-cone coordinates x ± = t l ± ϕ Include gravity Asymptotic symmetries : Diff S 1 Diff S 1 [Brown-Henneaux 1986] (x +, x ) ( f (x + ), f (x ) )

22 BERRY PHASES IN ADS 3 On-shell AdS 3 metrics : ds 2 = l2 ( r 2 dr2 rdx + 4Gl r (T, T) = CFT stress tensor Pure AdS 3 : T = T = c/24 Aspt symmetry tsfs : ( f T ) ( f (x + )) = T(x )dx )( rdx 4Gl r T(x+ )dx +) c = 3l 2G 1 [ ( f (x + )) 2 T(x + ) + c ] 12 { f ; x+ } Boundary gravitons around (T, T) : { ( O T, T = f T, f T) f, f } Diff S 1

23 BERRY PHASES IN ADS 3 UIRREP of Diff S 1 Diff S 1 with weights h, h Particle dressed with bdry gravitons Berry phases appear when particles undergo cyclic changes of frames : B total = B h,c [ f ] + B h, c [ f ] Symplectic fluxes of bdry gravitons Space-time interpretation? Let s turn to Minkowskian gravity!

24 BERRY PHASES IN FLAT SPACE 3D Minkowski space-time Future null infinity Bondi coordinates ϕ, u Aspt symmetries : BMS 3 = Diff S 1 Vect S 1 ϕ f (ϕ) superrotations u u + α(ϕ) supertranslations Conformal tsfs of celestial circles

25 BERRY PHASES IN FLAT SPACE BMS 3 UIRREPs = dressed particles in flat space Labelled by mass and spin [Barnich-B.O. 2014] Quantum states = wavefcts in supermomentum space dual to supertranslations Rest-frame state φ Consider a closed path ( f (t), α(t)) in BMS 3 superrotations supertranslations States U[( f (t), α(t))] φ Berry phases?

26 BERRY PHASES IN FLAT SPACE Path f (t) without supertranslations dt dϕ B spin = 2π [ ḟ f s c 24 + c ( f ) ] 24 f c 0 iff parity is broken Identical to Virasoro phase Bulk interpretation?

27 BERRY PHASES IN FLAT SPACE Circular path f (t, ϕ) = g(ϕ) + ωt Take g = boost Berry phase : B = 2π s (cosh λ 1) Thomas precession! [Thomas 1926]

28 BERRY PHASES IN FLAT SPACE Circular path f (t, ϕ) = g(ϕ) + ωt Take g = superboost Berry phase : ( B = 2π s + c ) 24 (n2 1) (cosh λ 1) Thomas precession for dressed particles [B.O. 2017]

29 BERRY PHASES IN FLAT SPACE Paths with supertranslations : B scalar = dt dϕ 2π α f f [ M 1 8G + 1 4G { f ; ϕ} ] [B.O. 2017] Involves Planck mass Non-zero even when M = 0! Total Berry phase = B spin + B scalar [B.O. 2017] 2

30 Conclusion

31 CONCLUSION Recap : Berry phases occur in all unitary reps For Virasoro / BMS 3 : Berry connections on -diml coadjoint orbits Berry phases can be computed explicitly! What now? Extend Thomas precession Relation to memory effect? Appearance in KdV equation? In shallow water? Observable in quantum Hall effect? [with P. Mao] [with T. Neupert & F. Schindler]

32 Thank you for listening!