The Schwarzian and black hole physics
Thomas Mertens
Ghent University

Based on arXiv:1606.03438 with J. Engelsöy and H. Verlinde
arXiv:1705.08408 with G.J. Turiaci and H. Verlinde
arXiv:1801.09605
arXiv:1804.09834 with H. Lam, G.J. Turiaci and H. Verlinde
arXiv:1806.07765 with A. Blommaert and H. Verschelde

The Schwarzian and black hole physics

Thomas Mertens

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Motivation

RSchwarzian theory: SSchw = −C dt {f , τ } where 00 2
000
the Schwarzian derivativef , τ = ff 0 − 32 ff 0

The Schwarzian and black hole physicsThomas Mertens

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Motivation

RSchwarzian theory: SSchw = −C dt {f , τ } where 00 2
000
the Schwarzian derivativef , τ = ff 0 − 32 ff 0Appears in:
I SYK model at low energies

Schwarzian theory describes regime of N → ∞ and βJ → ∞I Jackiw-Teitelboim (JT) 2d dilaton gravity

JT is holographically dual to Schwarzian theory→ Dynamics of wiggly boundary curve described by SSchw
Compare to CS / WZW, 3d gravity / Liouville topologicaldualities

The Schwarzian and black hole physics

Thomas Mertens

2 8Definition
Main goal:
Compute all correlation functions:hO`1 O`2 . . .iβ =I with

1ZZ
M[D f ]O`1 O`2 . . . e

M = Diff(S 1 )/SL(2, R),

The Schwarzian and black hole physics

CRβ0dτ



f ,τ

2+ 2π2 f 02



βf (τ + β) = f (τ ) + β

Thomas Mertens

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Definition
Main goal:
Compute all correlation functions:hO`1 O`2 . . .iβ =I with

1ZZ
M[D f ]O`1 O`2 . . . e

M = Diff(S 1 )/SL(2, R),
I Bilocal operators: O` (τ1 , τ2 ) =CRβ0dτf ,τ

2+ 2π2 f 02

βf (τ + β) = f (τ ) + β`f 0 (τ1 )f 0 (τ2 )β2 πsin β [f (τ1 )−f (τ2 )]πC → +∞ is probing semi-classical regime: gravitational physics inexact correlators

The Schwarzian and black hole physics

Thomas Mertens

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Method: 2d Liouville CFTConsider 2d Liouville CFT on a cylinder between ZZ-branes
ZZ

Vl1 Vl2

The Schwarzian and black hole physics

ZZZZ

Thomas Mertens

Vl1 Vl2ZZ

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Method: 2d Liouville CFTConsider 2d Liouville CFT on a cylinder between ZZ-branes
ZZ

Vl1 Vl2ZZ

ZZ

Vl1 Vl2ZZ

Take Schwarzian double-scaling limit:
I c → +∞
, c × radius ∼ C , fixed
I radius → 0

The Schwarzian and black hole physics

Thomas Mertens

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Method: 2d Liouville CFTConsider 2d Liouville CFT on a cylinder between ZZ-branes
ZZ

Vl1 Vl2ZZ

ZZ

Vl1 Vl2ZZ

Take Schwarzian double-scaling limit:
I c → +∞
, c × radius ∼ C , fixed
I radius → 0
Path-integral proof of link Liouville - Schwarzian
Liouville-Schwarzian Dictionary:
I T (w ) → − c {F (σ), σ},F ≡ tan πfβ(τ )24π
 0 0 `F1 F2I V` ≡ e 2`φ → O` (τ1 , τ2 ) ≡(F1 −F2 )2The Schwarzian and black hole physics

Thomas Mertens

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Method: 2d Liouville CFTConsider 2d Liouville CFT on a cylinder between ZZ-branes
ZZ

Vl1 Vl2ZZ

ZZ

Vl1 Vl2ZZ

Take Schwarzian double-scaling limit:
I c → +∞
, c × radius ∼ C , fixed
I radius → 0Path-integral proof of link Liouville - Schwarzian
Liouville-Schwarzian Dictionary:
I T (w ) → − c {F (σ), σ},F ≡ tan πfβ(τ )24π
 0 0 `F1 F2I V` ≡ e 2`φ → O` (τ1 , τ2 ) ≡(F1 −F2 )2Liouville amplitudes with V` ’s between ZZ’s → Schwarzian bilocalcorrelators
The Schwarzian and black hole physics

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Result: 2-point correlator
Schwarzian diagram:hO` (τ1 , τ2 ) i = τ2

The Schwarzian and black hole physics`τ1Thomas Mertens

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Result: 2-point correlator
Schwarzian diagram:hO` (τ1 , τ2 ) i = τ2
1=Z (β)`τ1Zdµ(k1 )dµ(k2 )e−τ k12 −(β−τ )k22 Γ ` ± i(k1 ± k2 )√
2 πΓ(2`)dµ(k) ≡ dk 2 sinh(2πk)

The Schwarzian and black hole physics

Thomas Mertens

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Result: 2-point correlator
Schwarzian diagram:hO` (τ1 , τ2 ) i = τ2
1=Z (β)`τ1

Zdµ(k1 )dµ(k2 )e−τ k12 −(β−τ )k22 Γ ` ± i(k1 ± k2 )√
2 πΓ(2`)dµ(k) ≡ dk 2 sinh(2πk)
Interpretation:
I Semi-classical limit exhibits quasi-normal modes, JT M(TH )relation
I Interpret as black hole that emits and reabsorbs excitation

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Thomas Mertens

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Result: OTO 4-point correlator
Schwarzian diagram:k1

τ3DO`1 (τ1 , τ2 )O`2 (τ3 , τ4 )EOTO= kt

τ1
`2

`1

τ2ks

τ4k4

The Schwarzian and black hole physics

Thomas Mertens

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Result: OTO 4-point correlator
Schwarzian diagram:k1

τ3DO`1 (τ1 , τ2 )O`2 (τ3 , τ4 )EOTO= ktτ1`2

`1

τ2ks

τ4k4Z

2222dµ(ki )e −k1 (τ3 −τ1 )−kt (τ3 −τ2 )−k4 (τ4 −τ2 )−ks (β−τ4 +τ1 )on×γ`1 (k1 , ks )γ`2 (ks , k4 )γ`1 (k4 , kt )γ`2 (kt , k1 ) × ``1 kk1 kks=24t

I 6j-symbol of SL(2, R) represents crossing of lines (OTO)
I Descends from R-matrix in 2d CFT
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Application: Shockwaves from semiclassics

Semiclassical regime: C → +∞
R +∞
R +∞
dq+ 0 dp− Ψ∗1 (q+ )Φ∗3 (p− )S(p− , q+ )Ψ2 (q+ )Φ4 (p− )0The Schwarzian and black hole physics

Thomas Mertens

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Application: Shockwaves from semiclassics

Semiclassical regime: C → +∞
R +∞
R +∞
dq+ 0 dp− Ψ∗1 (q+ )Φ∗3 (p− )S(p− , q+ )Ψ2 (q+ )Φ4 (p− )0I Ψ, Φ = Kruskal wavefunctions = bulk-to-boundary

propagators
iβ
I S = exp
pq4πC − + the Dray-’t Hooft shockwave S-matrix

The Schwarzian and black hole physics

Thomas Mertens

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Application: Shockwaves from semiclassics

Semiclassical regime: C → +∞
R +∞
R +∞
dq+ 0 dp− Ψ∗1 (q+ )Φ∗3 (p− )S(p− , q+ )Ψ2 (q+ )Φ4 (p− )0I Ψ, Φ = Kruskal wavefunctions = bulk-to-boundary

propagatorsiβ
I S = exp
pq4πC − + the Dray-’t Hooft shockwave S-matrix
Precise match with known semiclassics Shenker-Stanford ’15

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Alternative perspective: Wilson lines
Alternatively, without resorting to 2d Liouville CFT
Bulk perspective on bilocal operators as boundary-anchored Wilsonlines in SL(2, R) BF theory

Crossing of Wilson lines ⇒ 6j-symbol of group

The Schwarzian and black hole physics

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Alternative perspective: Wilson lines
Alternatively, without resorting to 2d Liouville CFT
Bulk perspective on bilocal operators as boundary-anchored Wilsonlines in SL(2, R) BF theory

Crossing of Wilson lines ⇒ 6j-symbol of group

Thank you!
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