The Schwarzian and black hole physics

Transcription

1 The Schwarzian and black hole physics Thomas Mertens Ghent University Based on arxiv: with J. Engelsöy and H. Verlinde arxiv: with G.J. Turiaci and H. Verlinde arxiv: arxiv: with H. Lam, G.J. Turiaci and H. Verlinde arxiv: with A. Blommaert and H. Verschelde The Schwarzian and black hole physics Thomas Mertens 1 8

2 Motivation Schwarzian theory: S Schw = C dt {f, τ} where { } ( ) f, τ = f f 3 f 2 2 f the Schwarzian derivative The Schwarzian and black hole physics Thomas Mertens 2 8

3 Motivation Schwarzian theory: S Schw = C dt {f, τ} where { } ( ) f, τ = f f 3 f 2 2 f the Schwarzian derivative Appears in: SYK model at low energies Schwarzian theory describes regime of N and βj Jackiw-Teitelboim (JT) 2d dilaton gravity JT is holographically dual to Schwarzian theory Dynamics of wiggly boundary curve described by S Schw Compare to CS / WZW, 3d gravity / Liouville topological dualities The Schwarzian and black hole physics Thomas Mertens 2 8

4 Definition Main goal: Compute all correlation functions: O l1 O l2... β = 1 Z M [Df ]O l1 O l2... e C ({ ) β 0 dτ f, τ }+ 2π2 β 2 f 2 with M = Diff(S 1 )/SL(2, R), f (τ + β) = f (τ) + β The Schwarzian and black hole physics Thomas Mertens 3 8

5 Definition Main goal: Compute all correlation functions: O l1 O l2... β = 1 Z M [Df ]O l1 O l2... e C ({ ) β 0 dτ f, τ }+ 2π2 β 2 f 2 with M = Diff(S 1 )/SL(2, R), f (τ + β) = f (τ) + β ( ) l f Bilocal operators: O l (τ 1, τ 2 ) = (τ 1 )f (τ 2 ) β π sin2 π β [f (τ 1) f (τ 2 )] C + is probing semi-classical regime: gravitational physics in exact correlators The Schwarzian and black hole physics Thomas Mertens 3 8

6 Method: 2d Liouville CFT Consider 2d Liouville CFT on a cylinder between ZZ-branes Vl1 Vl2 ZZ ZZ ZZ ZZ Vl1 Vl2 The Schwarzian and black hole physics Thomas Mertens 4 8

7 Method: 2d Liouville CFT Consider 2d Liouville CFT on a cylinder between ZZ-branes Vl1 Vl2 ZZ ZZ ZZ ZZ Vl1 Vl2 Take Schwarzian double-scaling limit: c +, c radius C, fixed radius 0 The Schwarzian and black hole physics Thomas Mertens 4 8

8 Method: 2d Liouville CFT Consider 2d Liouville CFT on a cylinder between ZZ-branes Vl1 Vl2 ZZ ZZ ZZ ZZ Vl1 Vl2 Take Schwarzian double-scaling limit: c +, c radius C, fixed radius 0 Path-integral proof of link Liouville - Schwarzian Liouville-Schwarzian Dictionary: T (w) c 24π {F (σ), σ}, F ( ) V l e 2lφ F O l (τ 1, τ 2 ) 1 F 2 l (F 1 F 2 ) 2 ( ) tan πf (τ) β The Schwarzian and black hole physics Thomas Mertens 4 8

9 Method: 2d Liouville CFT Consider 2d Liouville CFT on a cylinder between ZZ-branes Vl1 Vl2 ZZ ZZ ZZ ZZ Vl1 Vl2 Take Schwarzian double-scaling limit: c +, c radius C, fixed radius 0 Path-integral proof of link Liouville - Schwarzian Liouville-Schwarzian Dictionary: T (w) c 24π {F (σ), σ}, F ( ) tan πf (τ) β ( ) V l e 2lφ F O l (τ 1, τ 2 ) 1 F 2 l (F 1 F 2 ) 2 Liouville amplitudes with V l s between ZZ s Schwarzian bilocal correlators The Schwarzian and black hole physics Thomas Mertens 4 8

10 Result: 2-point correlator Schwarzian diagram: O l (τ 1, τ 2 ) = τ 2 l τ 1 The Schwarzian and black hole physics Thomas Mertens 5 8

11 Result: 2-point correlator Schwarzian diagram: O l (τ 1, τ 2 ) = τ 2 l τ 1 = 1 Z(β) dµ(k 1 )dµ(k 2 )e τk2 1 (β τ)k2 2 Γ ( l ± i(k 1 ± k 2 ) ) 2 πγ(2l) dµ(k) dk 2 sinh(2πk) The Schwarzian and black hole physics Thomas Mertens 5 8

12 Result: 2-point correlator Schwarzian diagram: O l (τ 1, τ 2 ) = τ 2 l τ 1 = 1 Z(β) dµ(k 1 )dµ(k 2 )e τk2 1 (β τ)k2 2 Γ ( l ± i(k 1 ± k 2 ) ) 2 πγ(2l) dµ(k) dk 2 sinh(2πk) Interpretation: Semi-classical limit exhibits quasi-normal modes, JT M(T H ) relation Interpret as black hole that emits and reabsorbs excitation The Schwarzian and black hole physics Thomas Mertens 5 8

13 Result: OTO 4-point correlator Schwarzian diagram: O l1 (τ 1, τ 2 )O l2 (τ 3, τ 4 ) = l 2 l 1 k OTO t τ 2 τ 3 k 1 τ 1 k s τ 4 k 4 The Schwarzian and black hole physics Thomas Mertens 6 8

14 Result: OTO 4-point correlator Schwarzian diagram: O l1 (τ 1, τ 2 )O l2 (τ 3, τ 4 ) = l 2 l 1 k OTO t τ 2 τ 3 k 1 τ 1 k s = dµ(k i )e k2 1 (τ 3 τ 1 ) k 2 t (τ 3 τ 2 ) k 2 4 (τ 4 τ 2 ) k 2 s (β τ 4+τ 1 ) { } γ l1 (k 1, k s )γ l2 (k s, k 4 )γ l1 (k 4, k t )γ l2 (k t, k 1 ) l1 k 1 k s l 2 k 4 k t k 4 τ 4 6j-symbol of SL(2, R) represents crossing of lines (OTO) Descends from R-matrix in 2d CFT The Schwarzian and black hole physics Thomas Mertens 6 8

15 Application: Shockwaves from semiclassics Semiclassical regime: C dq + 0 dp Ψ 1 (q +)Φ 3 (p )S(p, q + )Ψ 2 (q + )Φ 4 (p ) The Schwarzian and black hole physics Thomas Mertens 7 8

16 Application: Shockwaves from semiclassics Semiclassical regime: C dq + 0 dp Ψ 1 (q +)Φ 3 (p )S(p, q + )Ψ 2 (q + )Φ 4 (p ) Ψ, Φ = Kruskal wavefunctions = bulk-to-boundary propagators ( ) S = exp iβ 4πC p q + the Dray- t Hooft shockwave S-matrix The Schwarzian and black hole physics Thomas Mertens 7 8

17 Application: Shockwaves from semiclassics Semiclassical regime: C dq + 0 dp Ψ 1 (q +)Φ 3 (p )S(p, q + )Ψ 2 (q + )Φ 4 (p ) Ψ, Φ = Kruskal wavefunctions = bulk-to-boundary propagators ( ) S = exp iβ 4πC p q + the Dray- t Hooft shockwave S-matrix Precise match with known semiclassics Shenker-Stanford 15 The Schwarzian and black hole physics Thomas Mertens 7 8

18 Alternative perspective: Wilson lines Alternatively, without resorting to 2d Liouville CFT Bulk perspective on bilocal operators as boundary-anchored Wilson lines in SL(2, R) BF theory Crossing of Wilson lines 6j-symbol of group The Schwarzian and black hole physics Thomas Mertens 8 8

19 Alternative perspective: Wilson lines Alternatively, without resorting to 2d Liouville CFT Bulk perspective on bilocal operators as boundary-anchored Wilson lines in SL(2, R) BF theory Crossing of Wilson lines 6j-symbol of group Thank you! The Schwarzian and black hole physics Thomas Mertens 8 8