Berry Phases of Boundary Gravitons Blagoje Oblak (ETH Zurich) February 2018 Based on arxiv 1703.06142 (JHEP) Based on arxiv 1710.06883 (JHEP) Based on arxiv 1711.05753 (CQG)
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]
PLAN OF THE TALK 1. Berry phases in group reps 2. Virasoro Berry phases 3. Berry phases & asymptotic symmetries
1. Berry phases in group reps A. Berry phases B. Application to unitary reps
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))
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
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?
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
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?
2. Virasoro Berry phases A. Maurer-Cartan form of Diff S 1 B. Maurer-Cartan form of Virasoro C. Berry phases
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
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
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
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)
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]
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)]?
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 )
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
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
3. Berry phases & asymptotic symmetries A. Berry phases in AdS 3 B. Berry phases in flat space
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 ) )
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
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!
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
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?
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?
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]
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]
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
Conclusion
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]
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