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 The Schwarzian and black hole physics Thomas Mertens 1 22

2 Outline Motivation SYK JT dilaton gravity The Schwarzian path integral Embedding in 2d Liouville CFT Partition function 2-point function 4-point function OTO 4-point function Generalizations Conclusion The Schwarzian and black hole physics Thomas Mertens 2 22

3 Motivation: SYK at low energies SYK-model: N Majorana fermions with all-to-all random interactions of q fermions The Schwarzian and black hole physics Thomas Mertens 3 22

4 Motivation: SYK at low energies SYK-model: N Majorana fermions with all-to-all random interactions of q fermions IR symmetry of solution to SYK Dyson-Schwinger equations for two-point function T ψ i (τ 1 )ψ i (τ 2 ) : G(τ 1, τ 2 ) f (τ 1 ) f (τ 2 ) G(f (τ 1 ), f (τ 2 )), = 1/q The Schwarzian and black hole physics Thomas Mertens 3 22

5 Motivation: SYK at low energies SYK-model: N Majorana fermions with all-to-all random interactions of q fermions IR symmetry of solution to SYK Dyson-Schwinger equations for two-point function T ψ i (τ 1 )ψ i (τ 2 ) : G(τ 1, τ 2 ) f (τ 1 ) f (τ 2 ) G(f (τ 1 ), f (τ 2 )), = 1/q IR conformal symmetry is spontaneously broken (by specific solution G c ) and explicitly broken (by UV) The Schwarzian and black hole physics Thomas Mertens 3 22

6 Motivation: SYK at low energies SYK-model: N Majorana fermions with all-to-all random interactions of q fermions IR symmetry of solution to SYK Dyson-Schwinger equations for two-point function T ψ i (τ 1 )ψ i (τ 2 ) : G(τ 1, τ 2 ) f (τ 1 ) f (τ 2 ) G(f (τ 1 ), f (τ 2 )), = 1/q IR conformal symmetry is spontaneously broken (by specific solution G c ) and explicitly broken (by UV) Effective Action (master fields Σ, G) N suppressed at N, unless also βj Schwarzian action Path-integrate over low-energy reparametrizations f of the classical solution with Schwarzian action The Schwarzian and black hole physics Thomas Mertens 3 22

7 Motivation: SYK at low energies SYK-model: N Majorana fermions with all-to-all random interactions of q fermions IR symmetry of solution to SYK Dyson-Schwinger equations for two-point function T ψ i (τ 1 )ψ i (τ 2 ) : G(τ 1, τ 2 ) f (τ 1 ) f (τ 2 ) G(f (τ 1 ), f (τ 2 )), = 1/q IR conformal symmetry is spontaneously broken (by specific solution G c ) and explicitly broken (by UV) Effective Action (master fields Σ, G) N suppressed at N, unless also βj Schwarzian action Path-integrate over low-energy reparametrizations f of the classical solution with Schwarzian action G c has SL(2, R)-symmetry not to be path integrated gauge redundancy The Schwarzian and black hole physics Thomas Mertens 3 22

8 Motivation: 2d dilaton gravity Jackiw-Teitelboim (JT) gravity is model of 2d dilaton gravity JT appears in s-wave dim. red. from near-extremal black holes The Schwarzian and black hole physics Thomas Mertens 4 22

9 Motivation: 2d dilaton gravity Jackiw-Teitelboim (JT) gravity is model of 2d dilaton gravity JT appears in s-wave dim. red. from near-extremal black holes JT is holographically dual to Schwarzian theory The Schwarzian and black hole physics Thomas Mertens 4 22

10 Motivation: 2d dilaton gravity Jackiw-Teitelboim (JT) gravity is model of 2d dilaton gravity JT appears in s-wave dim. red. from near-extremal black holes JT is holographically dual to Schwarzian theory Derivation: The Schwarzian and black hole physics Thomas Mertens 4 22

11 Motivation: 2d dilaton gravity Jackiw-Teitelboim (JT) gravity is model of 2d dilaton gravity JT appears in s-wave dim. red. from near-extremal black holes JT is holographically dual to Schwarzian theory Derivation: S = 1 16πG d 2 x gφ(r + 2) + 1 8πG dτ γφbdy K Path integrate over Φ R = 2: Geometry fixed as AdS 2 The Schwarzian and black hole physics Thomas Mertens 4 22

12 Motivation: 2d dilaton gravity Jackiw-Teitelboim (JT) gravity is model of 2d dilaton gravity JT appears in s-wave dim. red. from near-extremal black holes JT is holographically dual to Schwarzian theory Derivation: S = 1 16πG d 2 x gφ(r + 2) + 1 8πG dτ γφbdy K Path integrate over Φ R = 2: Geometry fixed as AdS 2 Consider boundary curve (f (τ), z = ɛ f (τ)) as UV cutoff, carving out a shape from AdS, f = time reparametrization The Schwarzian and black hole physics Thomas Mertens 4 22

13 Motivation: 2d dilaton gravity Jackiw-Teitelboim (JT) gravity is model of 2d dilaton gravity JT appears in s-wave dim. red. from near-extremal black holes JT is holographically dual to Schwarzian theory Derivation: S = 1 16πG d 2 x gφ(r + 2) + 1 8πG dτ γφbdy K Path integrate over Φ R = 2: Geometry fixed as AdS 2 Consider boundary curve (f (τ), z = ɛ f (τ)) as UV cutoff, carving out a shape from AdS, f = time reparametrization Fix asymptotics of dilaton Φ bdy 1/ɛ Using K = 1 + ɛ 2 {f, τ} S = C dt {f, τ}, C G 0 The Schwarzian and black hole physics Thomas Mertens 4 22

14 Motivation: 2d dilaton gravity Jackiw-Teitelboim (JT) gravity is model of 2d dilaton gravity JT appears in s-wave dim. red. from near-extremal black holes JT is holographically dual to Schwarzian theory Derivation: S = 1 16πG d 2 x gφ(r + 2) + 1 8πG dτ γφbdy K Path integrate over Φ R = 2: Geometry fixed as AdS 2 Consider boundary curve (f (τ), z = ɛ f (τ)) as UV cutoff, carving out a shape from AdS, f = time reparametrization Fix asymptotics of dilaton Φ bdy 1/ɛ Using K = 1 + ɛ 2 {f, τ} S = C dt {f, τ}, C G 0 Compare to CS / WZW, 3d gravity / Liouville topological dualities The Schwarzian and black hole physics Thomas Mertens 4 22

15 The Schwarzian theory: goal Main goal: Compute all correlation functions: O l1 O l2... β = 1 Z with action S[f ] = C = C β 0 β 0 M [Df ]O l1 O l2... e C ({ ) β 0 dτ f, τ }+ 2π2 β 2 f 2 dτ { F, τ } ( ) πf (τ), F tan β ( {f } 2π 2 ) dτ, τ + β 2 f 2 where { f, τ } ( ) = f f 3 f 2 2 f the Schwarzian derivative The Schwarzian and black hole physics Thomas Mertens 5 22

16 The Schwarzian theory: goal Main goal: Compute all correlation functions: O l1 O l2... β = 1 Z with action S[f ] = C = C β 0 β 0 M [Df ]O l1 O l2... e C ({ ) β 0 dτ f, τ }+ 2π2 β 2 f 2 dτ { F, τ } ( ) πf (τ), F tan β ( {f } 2π 2 ) dτ, τ + β 2 f 2 where { f, τ } ( ) = f f 3 f 2 2 f the Schwarzian derivative with M = Diff(S 1 )/SL(2, R), f (τ + β) = f (τ) + β The Schwarzian and black hole physics Thomas Mertens 5 22

17 The Schwarzian theory: goal Main goal: Compute all correlation functions: O l1 O l2... β = 1 Z with action S[f ] = C = C β 0 β 0 M [Df ]O l1 O l2... e C ({ ) β 0 dτ f, τ }+ 2π2 β 2 f 2 dτ { F, τ } ( ) πf (τ), F tan β ( {f } 2π 2 ) dτ, τ + β 2 f 2 where { f, τ } ( ) = f f 3 f 2 2 f the Schwarzian derivative with M = Diff(S 1 )/SL(2, R), f (τ + β) = f (τ) + β In particular: finding gravitational physics in exact correlators The Schwarzian and black hole physics Thomas Mertens 5 22

18 The Schwarzian theory: Operator insertions Bilocal operators: O l (τ 1, τ 2 ) ( F (τ 1 )F ) ( (τ 2 ) l f (τ 1 )f (τ 2 ) (F (τ 1 ) F (τ 2 )) 2 β π sin2 π β [f (τ 1) f (τ 2 )] Think of this expression as two-point function O l (τ 1, τ 2 ) = O(τ 1 )O(τ 2 ) CFT of some 1D matter CFT at finite temperature coupled to the Schwarzian theory ) l The Schwarzian and black hole physics Thomas Mertens 6 22

19 Partition function: Duistermaat-Heckman Stanford and Witten demonstrated that Schwarzian partition function is one-loop exact Stanford-Witten 17: The Schwarzian and black hole physics Thomas Mertens 7 22

20 Partition function: Duistermaat-Heckman Stanford and Witten demonstrated that Schwarzian partition function is one-loop exact Stanford-Witten 17: ( ) Z(β) = π 3/2exp ( ) π 2 β β = + 0 dk 2 sinh(2πk)e βk2 (C = 1/2) Density of states ρ(e) = sinh(2π E) The Schwarzian and black hole physics Thomas Mertens 7 22

21 Partition function: Duistermaat-Heckman Stanford and Witten demonstrated that Schwarzian partition function is one-loop exact Stanford-Witten 17: ( ) Z(β) = π 3/2exp ( ) π 2 β β = + 0 dk 2 sinh(2πk)e βk2 (C = 1/2) Density of states ρ(e) = sinh(2π E) Cardy scaling at high energies ρ(e) e 2π E 2d CFT origin? What about correlators? Need other techniques The Schwarzian and black hole physics Thomas Mertens 7 22

22 Partition function: Vacuum character Observation: ( ) Tr 0 q L 0 χ0 (q) = q 1 c 24 (1 q) η(τ) =, q = e 2πt The Schwarzian and black hole physics Thomas Mertens 8 22

23 Partition function: Vacuum character Observation: ( ) Tr 0 q L 0 χ0 (q) = q 1 c 24 (1 q) η(τ) =, q = e 2πt Take double-scaling limit c, t = 12πβ/c 0 The Schwarzian and black hole physics Thomas Mertens 8 22

24 Partition function: Vacuum character Observation: ( ) Tr 0 q L 0 χ0 (q) = q 1 c 24 (1 q) η(τ) =, q = e 2πt Take double-scaling limit c, t = 12πβ/c 0 ( ) Z(β) π 3/2exp ( ) π 2 β β The Schwarzian and black hole physics Thomas Mertens 8 22

25 Partition function: Vacuum character Observation: ( ) Tr 0 q L 0 χ0 (q) = q 1 c 24 (1 q) η(τ) =, q = e 2πt Take double-scaling limit c, t = 12πβ/c 0 ( ) Z(β) π 3/2exp ( ) π 2 β β Schwarzian limit from 2d CFT: Cylinder radius 0 while c keeping product fixed From Liouville perspective: The Schwarzian and black hole physics Thomas Mertens 8 22

26 Partition function: Vacuum character Observation: ( ) Tr 0 q L 0 χ0 (q) = q 1 c 24 (1 q) η(τ) =, q = e 2πt Take double-scaling limit c, t = 12πβ/c 0 ( ) Z(β) π 3/2exp ( ) π 2 β β Schwarzian limit from 2d CFT: Cylinder radius 0 while c keeping product fixed From Liouville perspective: χ 0 (q) = Cyl. amplitude of Liouville between ZZ = ZZ q L 0 ZZ The Schwarzian and black hole physics Thomas Mertens 8 22

27 Partition function: Vacuum character Observation: ( ) Tr 0 q L 0 χ0 (q) = q 1 c 24 (1 q) η(τ) =, q = e 2πt Take double-scaling limit c, t = 12πβ/c 0 ( ) Z(β) π 3/2exp ( ) π 2 β β Schwarzian limit from 2d CFT: Cylinder radius 0 while c keeping product fixed From Liouville perspective: χ 0 (q) = Cyl. amplitude of Liouville between ZZ = ZZ q L 0 ZZ ZZ = 0 dp Ψ ZZ (P) P χ 0 = 0 dp Ψ ZZ (P) 2 e β P 2 b 2 The Schwarzian and black hole physics Thomas Mertens 8 22

28 Partition function: Vacuum character Observation: ( ) Tr 0 q L 0 χ0 (q) = q 1 c 24 (1 q) η(τ) =, q = e 2πt Take double-scaling limit c, t = 12πβ/c 0 ( ) Z(β) π 3/2exp ( ) π 2 β β Schwarzian limit from 2d CFT: Cylinder radius 0 while c keeping product fixed From Liouville perspective: χ 0 (q) = Cyl. amplitude of Liouville between ZZ = ZZ q L 0 ZZ ZZ = 0 dp Ψ ZZ (P) P χ 0 = 0 dp Ψ ZZ (P) 2 e β P 2 b 2 with density Ψ ZZ (P) 2 = sinh ( 2πbP ) ( ) 2πP sinh b In Schwarzian limit: P = bk, b 0 c = 1 + 6(b + 1 b )2 The Schwarzian and black hole physics Thomas Mertens 8 22

29 Path-integral link Liouville and Schwarzian One can prove via Liouville phase space path integral between ZZ-branes: φ(0)=φ(t ) [Dφ] [Dπ T φ] e 0 dτ dσ(iπ φ φ H(φ,πφ )) Field redefinition (φ, π φ ) (A, B) Gervais-Neveu 82: e φ = 8 A σb σ (A B) 2, π φ = A σσ A σ B σσ B σ 2 A σ + B σ (A B) The Schwarzian and black hole physics Thomas Mertens 9 22

30 Path-integral link Liouville and Schwarzian One can prove via Liouville phase space path integral between ZZ-branes: φ(0)=φ(t ) [Dφ] [Dπ T φ] e 0 dτ dσ(iπ φ φ H(φ,πφ )) Field redefinition (φ, π φ ) (A, B) Gervais-Neveu 82: e φ = 8 A σb σ (A B) 2, π φ = A σσ A σ B σσ B σ H = c 24π {A(σ, τ), σ} c 24π {B(σ, τ), σ} Schwarzian limit: π φ φ 0 2 A σ + B σ (A B) ZZ-brane boundary conditions: A, B written in terms of 1 doubled field F The Schwarzian and black hole physics Thomas Mertens 9 22

31 Path-integral link Liouville and Schwarzian One can prove via Liouville phase space path integral between ZZ-branes: φ(0)=φ(t ) [Dφ] [Dπ T φ] e 0 dτ dσ(iπ φ φ H(φ,πφ )) Field redefinition (φ, π φ ) (A, B) Gervais-Neveu 82: e φ = 8 A σb σ (A B) 2, π φ = A σσ A σ B σσ B σ H = c 24π {A(σ, τ), σ} c 24π {B(σ, τ), σ} Schwarzian limit: π φ φ 0 2 A σ + B σ (A B) ZZ-brane boundary conditions: A, B written in terms of 1 doubled field F Liouville-Schwarzian Dictionary: T (w) c 24π {F (σ), σ} e φ F 1 F 2 (F 1 F 2 ) 2 The Schwarzian and black hole physics Thomas Mertens 9 22

32 2-point correlator Idea: V l (τ 1, τ 2 ) e 2lφ(τ 1,τ 2 ) O l (τ 1, τ 2 ) The Schwarzian and black hole physics Thomas Mertens 10 22

33 2-point correlator Idea: V l (τ 1, τ 2 ) e 2lφ(τ 1,τ 2 ) O l (τ 1, τ 2 ) ZZ V l ZZ = dpdq Ψ ZZ (P)Ψ ZZ (Q) P V l Q The Schwarzian and black hole physics Thomas Mertens 10 22

34 2-point correlator Idea: V l (τ 1, τ 2 ) e 2lφ(τ 1,τ 2 ) O l (τ 1, τ 2 ) ZZ V l ZZ = dpdq Ψ ZZ (P)Ψ ZZ (Q) P V l Q Two ways of computing matrix element: Schwarzian limit: Q Q 3-point function on sphere large c limit of DOZZ formula The Schwarzian and black hole physics Thomas Mertens 10 22

35 2-point correlator Idea: V l (τ 1, τ 2 ) e 2lφ(τ 1,τ 2 ) O l (τ 1, τ 2 ) ZZ V l ZZ = dpdq Ψ ZZ (P)Ψ ZZ (Q) P V l Q Two ways of computing matrix element: Schwarzian limit: Q Q 3-point function on sphere large c limit of DOZZ formula Computation in minisuperspace regime of 2d Liouville CFT: P e (β τ)h e 2lφ e τh Q Explicitly evaluate minisuperspace integrals The Schwarzian and black hole physics Thomas Mertens 10 22

36 2-point correlator Idea: V l (τ 1, τ 2 ) e 2lφ(τ 1,τ 2 ) O l (τ 1, τ 2 ) ZZ V l ZZ = dpdq Ψ ZZ (P)Ψ ZZ (Q) P V l Q Two ways of computing matrix element: Schwarzian limit: Q Q 3-point function on sphere large c limit of DOZZ formula Computation in minisuperspace regime of 2d Liouville CFT: P e (β τ)h e 2lφ e τh Q Explicitly evaluate minisuperspace integrals Both agree: G β l (τ 1, τ 2 ) = 1 Z(β) dµ(k) dk 2 sinh(2πk) dµ(k 1 )dµ(k 2 )e τk2 1 (β τ)k2 2 Γ ( l ± i(k 1 ± k 2 ) ) 2 πγ(2l) The Schwarzian and black hole physics Thomas Mertens 10 22

37 Application: Semiclassics for light operators Semi-classical regime C, l C: k 1 k 2 1 Redefine k1 2 = M + ω, k2 2 = M, M ω G ± l 0 dme 2π M β 2C M dω 2π e±i τ 2C ω+π ω 2 Γ(l±i M ω 2 ) M Γ(2l) (2 M) 2l 1 The Schwarzian and black hole physics Thomas Mertens 11 22

38 Application: Semiclassics for light operators Semi-classical regime C, l C: k 1 k 2 1 Redefine k1 2 = M + ω, k2 2 = M, M ω G ± l 0 dme 2π M β 2C M dω 2π e±i τ 2C ω+π ω 2 Γ(l±i M ω 2 ) M Γ(2l) (2 M) 2l 1 Interpretation: M-integral has saddle: M 0 = 4π 2 C 2 /β 2, the JT black hole E(T )-relation Remaining ω-integral is done explicitly to yield: G ±,cl l (τ 1, τ 2 ) = ( π β sinh π β τ 12 ) 2l The Schwarzian and black hole physics Thomas Mertens 11 22

39 Application: Semiclassics for light operators Semi-classical regime C, l C: k 1 k 2 1 Redefine k1 2 = M + ω, k2 2 = M, M ω G ± l 0 dme 2π M β 2C M dω 2π e±i τ 2C ω+π ω 2 Γ(l±i M ω 2 ) M Γ(2l) (2 M) 2l 1 Interpretation: M-integral has saddle: M 0 = 4π 2 C 2 /β 2, the JT black hole E(T )-relation Remaining ω-integral is done explicitly to yield: G ±,cl l (τ 1, τ 2 ) = ( π β sinh π β τ 12 ) 2l Γ-functions give poles: l i 2 = n M Matches with quasi-normal modes of AdS 2 BH metric Keeler-Ng 14: ω = i 2π β (n + l) ω The Schwarzian and black hole physics Thomas Mertens 11 22

40 Application: Semiclassics for light operators Semi-classical regime C, l C: k 1 k 2 1 Redefine k1 2 = M + ω, k2 2 = M, M ω G ± l 0 dme 2π M β 2C M dω 2π e±i τ 2C ω+π ω 2 Γ(l±i M ω 2 ) M Γ(2l) (2 M) 2l 1 Interpretation: M-integral has saddle: M 0 = 4π 2 C 2 /β 2, the JT black hole E(T )-relation Remaining ω-integral is done explicitly to yield: G ±,cl l (τ 1, τ 2 ) = ( π β sinh π β τ 12 ) 2l Γ-functions give poles: l i 2 = n M Matches with quasi-normal modes of AdS 2 BH metric Keeler-Ng 14: ω = i 2π β (n + l) Quantum black hole M emits and reabsorbs excitation with mass l and energy ω The Schwarzian and black hole physics Thomas Mertens ω

41 Time-ordered 4-point function ZZ V l1 V l2 ZZ Evaluated using Conformal blocks The Schwarzian and black hole physics Thomas Mertens 12 22

42 Time-ordered 4-point function ZZ V l1 V l2 ZZ Evaluated using Conformal blocks Q Q G l1 l 2 = dp dq dp s Ψ ZZ(P)Ψ ZZ (Q) P s l 1 l 2 l 1 l 2 P s P P The Schwarzian and black hole physics Thomas Mertens 12 22

43 Time-ordered 4-point function ZZ V l1 V l2 ZZ Evaluated using Conformal blocks Q Q G l1 l 2 = dp dq dp s Ψ ZZ(P)Ψ ZZ (Q) P s l 1 l 2 l 1 l 2 P s Conformal blocks dominated by primary in the intermediate channel reduce to DOZZ OPE coefficients P P The Schwarzian and black hole physics Thomas Mertens 12 22

44 Time-ordered 4-point function ZZ V l1 V l2 ZZ Evaluated using Conformal blocks Q Q G l1 l 2 = dp dq dp s Ψ ZZ(P)Ψ ZZ (Q) P s l 1 l 2 l 1 l 2 P s Conformal blocks dominated by primary in the intermediate channel reduce to DOZZ OPE coefficients G β l 1 l 2 = dk 2 1 dk2 4 dk2 s sinh 2πk 1 sinh 2πk 4 sinh 2πk s e k2 1 (τ 2 τ 1 ) k 2 4 (τ 4 τ 3 ) k 2 s (β τ 2+τ 3 τ 4 +τ 1 ) Γ(l 2 ±ik 4 ±ik s) Γ(l 1 ±ik 1 ±ik s) Γ(2l 1 ) Γ(2l 2 ) P P The Schwarzian and black hole physics Thomas Mertens 12 22

45 Diagrammatic decomposition Rules: τ 2 k τ 1 = e k2 (τ 2 τ 1 ), k 2 k 1 l = γ l (k 1, k 2 ) = Γ(l±ik1 ±ik 2 ) Γ(2l) Integrate over intermediate momenta k i with measure dµ(k) = dk 2 sinh(2πk) The Schwarzian and black hole physics Thomas Mertens 13 22

46 Diagrammatic decomposition Rules: τ 2 k τ 1 = e k2 (τ 2 τ 1 ), k 2 k 1 l = γ l (k 1, k 2 ) = Γ(l±ik1 ±ik 2 ) Γ(2l) Integrate over intermediate momenta k i with measure dµ(k) = dk 2 sinh(2πk) Examples: Z(β) = O l (τ 1, τ 2 ) = τ 2 l τ 1 O l1 (τ 1, τ 2 ) O l2 (τ 3, τ 4 ) = τ3 τ 2 l 1 τ 1 τ 4 Note: non-perturbative in Schwarzian coupling C l 2 The Schwarzian and black hole physics Thomas Mertens 13 22

47 OTO four-point correlator (1) Swapping two operators in 4-point correlator, means the conformal block is dominated by its primary in a different channel: F Ps [ ](z ) = dp t R PsP t [ ] FPt [ ](1/z ) The Schwarzian and black hole physics Thomas Mertens 14 22

48 OTO four-point correlator (1) Swapping two operators in 4-point correlator, means the conformal block is dominated by its primary in a different channel: F Ps [ ](z ) = dp t R PsP t [ ] FPt [ ](1/z ) Blocks unknown in general, but R-matrix is known as quantum 6j-symbol of U q (SL(2, R)) Teschner-Ponsot 99 The Schwarzian and black hole physics Thomas Mertens 14 22

49 OTO four-point correlator (1) Swapping two operators in 4-point correlator, means the conformal block is dominated by its primary in a different channel: F Ps [ ](z ) = dp t R PsP t [ ] FPt [ ](1/z ) Blocks unknown in general, but R-matrix is known as quantum 6j-symbol of U q (SL(2, R)) Teschner-Ponsot 99 Schwarzian limit of quantum 6j-symbol is 6j-symbol with 4 continuous and 2 discrete SL(2, R) labels The Schwarzian and black hole physics Thomas Mertens 14 22

50 OTO four-point correlator (1) Swapping two operators in 4-point correlator, means the conformal block is dominated by its primary in a different channel: F Ps [ ](z ) = dp t R PsP t [ ] FPt [ ](1/z ) Blocks unknown in general, but R-matrix is known as quantum 6j-symbol of U q (SL(2, R)) Teschner-Ponsot 99 Schwarzian limit of quantum 6j-symbol is 6j-symbol with 4 continuous and 2 discrete SL(2, R) labels Explicitly: [ R 21 ] { } 3 PsPt 4 l1 k 2 k s l 3 k 4 k = t Γ(l1 +ik 2 ±ik s)γ(l 3 ik 2 ±ik t)γ(l 1 ik 4 ±ik t)γ(l 3 +ik 4 ±ik s) Γ(l 1 ik 2 ±ik s)γ(l 3 +ik 2 ±ik t)γ(l 1 +ik 4 ±ik t)γ(l 3 ik 4 ±ik s) i du Γ(u)Γ(u 2ik s)γ(u+ik 2+4 s+t )Γ(u ik s+t 2 4 )Γ(l 1 +ik s 2 u)γ(l 3 +ik s 4 u) 2πi Γ(u+l 1 ik s 2 )Γ(u+l 3 ik s 4 ) i The Schwarzian and black hole physics Thomas Mertens 14 22

51 OTO four-point correlator (2) At the Schwarzian level, this procedure is captured by the diagram: k 1 k t l 2 l 1 k s k 4 e k2 1 (τ 3 τ 1 ) kt 2(τ 3 τ 2 ) k4 2(τ 4 τ 2 ) ks 2(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 The Schwarzian and black hole physics Thomas Mertens 15 22

52 Application: Shockwaves from semiclassics ~ t 2 t 2 Time delay: t 2 t 2 e λ M(t 2 t 1 ), λ M = 2π β M t 1 The Schwarzian and black hole physics Thomas Mertens 16 22

53 Application: Shockwaves from semiclassics ~ t 2 t 2 Time delay: t 2 t 2 e λ M(t 2 t 1 ), λ M = 2π β M W 3 W 4 t 1 Shenker-Stanford: OTO-correlator V 1 W 3 V 2 W 4 in boundary theory captures this behavior Shenker-Stanford 15 V 1 V 2 The Schwarzian and black hole physics Thomas Mertens 16 22

54 Application: Shockwaves from semiclassics ~ t 2 t 2 Time delay: t 2 t 2 e λ M(t 2 t 1 ), λ M = 2π β M W 3 W 4 t 1 Shenker-Stanford: OTO-correlator V 1 W 3 V 2 W 4 in boundary theory captures this behavior Shenker-Stanford 15 V 1 V 2 V 1 W 3 V 2 W 4, written using shockwaves in the AdS 2 bulk as 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 16 22

55 Application: Shockwaves from semiclassics ~ t 2 t 2 Time delay: t 2 t 2 e λ M(t 2 t 1 ), λ M = 2π β M W 3 W 4 t 1 Shenker-Stanford: OTO-correlator V 1 W 3 V 2 W 4 in boundary theory captures this behavior Shenker-Stanford 15 V 1 V 2 V 1 W 3 V 2 W 4, written using shockwaves in the AdS 2 bulk as 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 16 22

56 Application: Shockwaves from the exact OTO correlator Large C limit of complete OTO 4-point function, with light l, gives full eikonal shockwave expressions Exact match Derived full shockwave in semiclassical regime! The Schwarzian and black hole physics Thomas Mertens 17 22

57 Summary: Structural link theories Holography 3d Gravity 2d Liouville Dim. Red. 2d JT Gravity 1d Schwarzian The Schwarzian and black hole physics Thomas Mertens 18 22

58 Generalization: Group models Holography 3d Chern-Simons 2d WZW Dim. Red. 2d BF Theory 1d particle on group The Schwarzian and black hole physics Thomas Mertens 19 22

59 Generalization: Group models Holography 3d Chern-Simons 2d WZW Dim. Red. 2d BF Theory 1d particle on group Relevant for SYK-type models with internal symmetries The Schwarzian and black hole physics Thomas Mertens 19 22

60 Generalization: Group models Holography 3d Chern-Simons 2d WZW Dim. Red. 2d BF Theory 1d particle on group Relevant for SYK-type models with internal symmetries Correlators determined using 2d WZW CFT techniques τ 2 jm τ 1 = e C j (τ 2 τ 1 ), j 1m 1 JM = ( ) j1 j 2 J m 1 m 2 M j 2m 2 The Schwarzian and black hole physics Thomas Mertens 19 22

61 Wilson line perpective JT is Hamiltonian reduction of SL(2, R) BF The Schwarzian and black hole physics Thomas Mertens 20 22

62 Wilson line perpective JT is Hamiltonian reduction of SL(2, R) BF with action S = dttrφf Boundary-anchored Wilson lines are bilocal operators: in progress Pe A = g(τ f )g 1 (τ i ) Reduce to Schwarzian bilocals for the constrained SL(2, R) case The Schwarzian and black hole physics Thomas Mertens 20 22

63 Wilson line perpective JT is Hamiltonian reduction of SL(2, R) BF with action S = dttrφf Boundary-anchored Wilson lines are bilocal operators: in progress Pe A = g(τ f )g 1 (τ i ) Reduce to Schwarzian bilocals for the constrained SL(2, R) case Crossing of Wilson lines 6j-symbol of group G The Schwarzian and black hole physics Thomas Mertens 20 22

64 Conclusion Schwarzian QM is relevant as Low-energy universal gravity sector of all SYK-type models Cfr. Liouville compared to holographic 2d CFT The Schwarzian and black hole physics Thomas Mertens 21 22

65 Conclusion Schwarzian QM is relevant as Low-energy universal gravity sector of all SYK-type models Cfr. Liouville compared to holographic 2d CFT holographically dual to JT dilaton gravity The Schwarzian and black hole physics Thomas Mertens 21 22

66 Conclusion Schwarzian QM is relevant as Low-energy universal gravity sector of all SYK-type models Cfr. Liouville compared to holographic 2d CFT holographically dual to JT dilaton gravity Theory is exactly solvable and displays a wide variety of black hole quantum physics: Virtual black hole intermediate states Quasinormal modes and shockwaves in the semi-classical regime The Schwarzian and black hole physics Thomas Mertens 21 22

67 Conclusion Schwarzian QM is relevant as Low-energy universal gravity sector of all SYK-type models Cfr. Liouville compared to holographic 2d CFT holographically dual to JT dilaton gravity Theory is exactly solvable and displays a wide variety of black hole quantum physics: Virtual black hole intermediate states Quasinormal modes and shockwaves in the semi-classical regime Naturally embedded within Liouville theory The Schwarzian and black hole physics Thomas Mertens 21 22

68 Thank you! The Schwarzian and black hole physics Thomas Mertens 22 22