Effective Field Theory of Large-c CFTs Felix Haehl UBC Vancouver - - PowerPoint PPT Presentation

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Effective Field Theory of Large-c CFTs Felix Haehl UBC Vancouver - - PowerPoint PPT Presentation

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Effective Field Theory of Large-c CFTs Felix Haehl UBC Vancouver (> IAS) Based on 1712.04963, 1808.02898 with M. Rozali , and work in progress with W. Reeves & M. Rozali Effective Field Theory of Large-c CFTs Felix Haehl UBC

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SLIDE 1

Effective Field Theory

  • f Large-c CFTs

Felix Haehl UBC Vancouver (—> IAS)

Based on 1712.04963, 1808.02898 with M. Rozali, and work in progress with W. Reeves & M. Rozali

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SLIDE 2

Effective Field Theory

  • f Large-c CFTs

Felix Haehl UBC Vancouver (—> IAS)

Based on 1712.04963, 1808.02898 with M. Rozali, and work in progress with W. Reeves & M. Rozali

poster!

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SLIDE 3

Introduction

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SLIDE 4
  • Consider 2d CFT at finite temperature
  • Conf. symmetry is spontaneously broken
  • I want to study the Goldstone mode associated with this

effect

Basic idea

  • In the “holographic regime” (large c etc.) there is a

systematic effective field theory for this mode τ σ z = τ + iσ

(z, ¯ z) → (f(z), ¯ f(¯ z))

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SLIDE 5
  • Describes universal physics of CFTs associated

with energy-momentum conservation (“gravity”)

  • For example: effective field theory description for

universal aspects of… … OTOC observables, related to quantum chaos … conformal blocks, kinematic space operators, …

  • In the “holographic regime” (large c etc.) there is a

systematic effective field theory for this mode

V(t) V(t) W(o) W(o)

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SLIDE 6

Basics

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SLIDE 7

Reparametrization modes

  • Consider 2d CFT at finite temperature and a small

reparametrization (z, ¯ z) → (z + ✏, ¯ z + ¯ ✏) SCF T − → SCF T + Z d2z ¯ @✏ T(z) + @¯ ✏ ¯ T(¯ z)

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SLIDE 8

Reparametrization modes

  • Consider 2d CFT at finite temperature and a small

reparametrization

  • For conformal transformations,

the associated conserved symmetry currents are (z, ¯ z) → (z + ✏, ¯ z + ¯ ✏) SCF T − → SCF T + Z d2z ¯ @✏ T(z) + @¯ ✏ ¯ T(¯ z) ¯ @✏ = 0 = @¯ ✏ (J, ¯ J) = (✏ T, ¯ ✏ ¯ T)

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SLIDE 9

Reparametrization modes

  • Conformal symmetry is spontaneously broken
  • Regard as the associated Goldstone modes

[Turiaci-Verlinde ’16] [FH-Rozali ’18]

  • Consider 2d CFT at finite temperature and a small

reparametrization (z, ¯ z) → (z + ✏, ¯ z + ¯ ✏) SCF T − → SCF T + Z d2z ¯ @✏ T(z) + @¯ ✏ ¯ T(¯ z)

(✏, ¯ ✏)

  • have an effective action determined by hTµν · · · Tρσi

(✏, ¯ ✏)

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SLIDE 10
  • have an effective action determined by hTµν · · · Tρσi

(✏, ¯ ✏)

fixed by conformal symmetry! => dynamics of is universal W2 = Z d2z1 d2z2 ¯ @✏1 ¯ @✏2 hT(z1)T(z2)i + (anti-holo.)

(✏, ¯ ✏)

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SLIDE 11
  • The effective action is actually local

… because:

  • have an effective action determined by hTµν · · · Tρσi

(✏, ¯ ✏)

W2 = Z d2z1 d2z2 ¯ @✏1 ¯ @✏2 hT(z1)T(z2)i + (anti-holo.) ¯ ∂1hT(z1)T(z2)i ⇠ δ(2)(z1 z2)

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SLIDE 12
  • The effective action is actually local

… because:

  • have an effective action determined by hTµν · · · Tρσi

(✏, ¯ ✏)

W2 = Z d2z1 d2z2 ¯ @✏1 ¯ @✏2 hT(z1)T(z2)i + (anti-holo.) ¯ ∂1hT(z1)T(z2)i ⇠ δ(2)(z1 z2) (z = τ + iσ) W2 = c⇡ 6 Z d⌧d ¯ @✏ (@3

τ + @τ)✏ + (anti-holo.)

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SLIDE 13
  • have an effective action determined by hTµν · · · Tρσi

(✏, ¯ ✏)

W2 = c⇡ 6 Z d⌧d ¯ @✏ (@3

τ + @τ)✏ + (anti-holo.)

  • Analogous to Schwarzian action in d=1
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SLIDE 14
  • Euclidean propagator:

h✏(⌧, )✏(0, 0)i ⇠ 1 c sin2 ✓⌧ + i 2 ◆ log ⇣ 1 e−sgn(σ)i(τ+iσ)⌘

[FH-Rozali ’18] [Cotler-Jensen ’18]

  • have an effective action determined by hTµν · · · Tρσi

(✏, ¯ ✏)

W2 = c⇡ 6 Z d⌧d ¯ @✏ (@3

τ + @τ)✏ + (anti-holo.)

  • Analogous to Schwarzian action in d=1
  • ( Lorentzian “Schwinger-Keldysh” version is available )
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SLIDE 15

Feynman rules

h✏(⌧, )✏(0, 0)i ⇠ 1 c sin2 ✓⌧ + i 2 ◆ log ⇣ 1 e−sgn(σ)i(τ+iσ)⌘

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SLIDE 16

Feynman rules

E

h✏(⌧, )✏(0, 0)i ⇠ 1 c sin2 ✓⌧ + i 2 ◆ log ⇣ 1 e−sgn(σ)i(τ+iσ)⌘

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SLIDE 17

Feynman rules

  • “Coupling” to pairs of other operators via reparametrization:

E

h✏(⌧, )✏(0, 0)i ⇠ 1 c sin2 ✓⌧ + i 2 ◆ log ⇣ 1 e−sgn(σ)i(τ+iσ)⌘

= hO(x)O(Y )i ⇢ 1+

  • ∆

" @✏(x) + @✏(y) − ✏(x) − ✏(y) tan x−y

2

  • #

+ (anti-holo.)

hO(x)O(y)i ! [@f(x) @f(y)]∆ hO(f(x)) O(f(y))i f(x) = x + ✏(x)

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SLIDE 18

Feynman rules

  • “Coupling” to pairs of other operators via reparametrization:

E

h✏(⌧, )✏(0, 0)i ⇠ 1 c sin2 ✓⌧ + i 2 ◆ log ⇣ 1 e−sgn(σ)i(τ+iσ)⌘ O(x)

O(y)

E

B(1)

∆ (x, y)

= hO(x)O(Y )i ⇢ 1+ = hO(x)O(Y )i ⇢ 1+

  • ∆

" @✏(x) + @✏(y) − ✏(x) − ✏(y) tan x−y

2

  • #

+ (anti-holo.)

hO(x)O(y)i ! [@f(x) @f(y)]∆ hO(f(x)) O(f(y))i f(x) = x + ✏(x)

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SLIDE 19

E

h✏(⌧, )✏(0, 0)i ⇠ 1 c sin2 ✓⌧ + i 2 ◆ log ⇣ 1 e−sgn(σ)i(τ+iσ)⌘

O(x) O(y)

E

B(1)

∆ (x, y)

≡

∆ " @✏(x) + @✏(y) − ✏(x) − ✏(y) tan x−y

2

  • #

+ (anti-holo.)

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SLIDE 20

E

h✏(⌧, )✏(0, 0)i ⇠ 1 c sin2 ✓⌧ + i 2 ◆ log ⇣ 1 e−sgn(σ)i(τ+iσ)⌘

O(x) O(y)

E

B(1)

∆ (x, y)

≡

  • “Feynman rules” for reparametrization Goldstone
  • At large c, this gives a systematic perturbation theory
  • f energy-momentum exchanges (“gravity channel”)

∆ " @✏(x) + @✏(y) − ✏(x) − ✏(y) tan x−y

2

  • #

+ (anti-holo.)

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SLIDE 21

Applications

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SLIDE 22

Out-of-time-order correlators

>> skip

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SLIDE 23
  • “Usual” QFT: time-ordered correlators (TOCs):

V (0) V (0) W(t) W(t) td ∼ β 2π dissipation time:

Out-of-time-order correlators

t hW(t)W(t)V (0)V (0)iβ ⇠ hWWihV V i + O(e−t/td)

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SLIDE 24
  • “Usual” QFT: time-ordered correlators (TOCs):

V (0) V (0) W(t) W(t) td ∼ β 2π dissipation time:

Out-of-time-order correlators

hW(t)W(t)V (0)V (0)iβ ⇠ hWWihV V i + O(e−t/td)

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SLIDE 25
  • “Usual” QFT: time-ordered correlators (TOCs):

V (0) V (0) W(t) W(t) td ∼ β 2π dissipation time:

Out-of-time-order correlators

hW(t)W(t)V (0)V (0)iβ ⇠ hWWihV V i + O(e−t/td)

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SLIDE 26
  • “Usual” QFT: time-ordered correlators (TOCs):

V (0) V (0) W(t) W(t)

Out-of-time-order correlators

  • OTOCs display exp.

“Lyapunov” growth (quantum chaos): hW(t)W(t)V (0)V (0)iβ ⇠ hWWihV V i + O(e−t/td) hW(t)V (0)W(t)V (0)iβ

⇠ hWWihV V i h 1 # eλL (t−t∗)i

t∗ ∼ β 2π log N scrambling time:

[Shenker-Stanford ’13] [Maldacena-Shenker-Stanford ‘15] [Kitaev ’15] …..

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SLIDE 27
  • “Usual” QFT: time-ordered correlators (TOCs):

V (0) V (0) W(t) W(t)

Out-of-time-order correlators

  • OTOCs display exp.

“Lyapunov” growth (quantum chaos): hW(t)W(t)V (0)V (0)iβ ⇠ hWWihV V i + O(e−t/td) hW(t)V (0)W(t)V (0)iβ

⇠ hWWihV V i h 1 # eλL (t−t∗)i

t∗ ∼ β 2π log N scrambling time:

[Shenker-Stanford ’13] [Maldacena-Shenker-Stanford ‘15] [Kitaev ’15] …..

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SLIDE 28
  • “Usual” QFT: time-ordered correlators (TOCs):

V (0) V (0) W(t) W(t)

Out-of-time-order correlators

  • OTOCs display exp.

“Lyapunov” growth (quantum chaos): hW(t)W(t)V (0)V (0)iβ ⇠ hWWihV V i + O(e−t/td) hW(t)V (0)W(t)V (0)iβ

⇠ hWWihV V i h 1 # eλL (t−t∗)i

t∗ ∼ β 2π log N scrambling time:

[Shenker-Stanford ’13] [Maldacena-Shenker-Stanford ‘15] [Kitaev ’15] …..

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SLIDE 29

V (0) V (0) W(t) W(t) hW(t)V (0)W(t)V (0)iβ

⇠ hWWihV V i h 1 # eλL (t−t∗)i

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SLIDE 30

V (0) V (0) W(t) W(t) hW(t)V (0)W(t)V (0)iβ

⇠ hWWihV V i h 1 # eλL (t−t∗)i

  • : Lyapunov growth is

described by an exchange of the reparametrization mode: hW(t)V (0)W(t)V (0)iβ ⇠ hB(1)

∆W (t, t) B(1) ∆V (0, 0)iβ

⇠ h✏(t)✏(0)i

λL = 2πT

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SLIDE 31

V (0) V (0) W(t) W(t) hW(t)V (0)W(t)V (0)iβ

⇠ hWWihV V i h 1 # eλL (t−t∗)i

  • : Lyapunov growth is

described by an exchange of the reparametrization mode: hW(t)V (0)W(t)V (0)iβ ⇠ hB(1)

∆W (t, t) B(1) ∆V (0, 0)iβ

⇠ h✏(t)✏(0)i

λL = 2πT

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SLIDE 32

V (0) V (0) W(t) W(t) hW(t)V (0)W(t)V (0)iβ

⇠ hWWihV V i h 1 # eλL (t−t∗)i

  • : Lyapunov growth is

described by an exchange of the reparametrization mode: hW(t)V (0)W(t)V (0)iβ ⇠ hB(1)

∆W (t, t) B(1) ∆V (0, 0)iβ

⇠ h✏(t)✏(0)i

  • A universal contribution to the OTOC, described by

the collective mode

[FH-Rozali ’18] [Blake-Lee-Liu ‘18]

✏

λL = 2πT

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SLIDE 33

2k-point OTOC

F2k(t1, . . . , tk) = ⌦ ↵

β

hV1V1i · · · hVkVki V1 V1 · · · V3 V3 V2 V2 Vk Vk Vk−1 [ , ][ , ][ , ] [ , ] V4

[FH-Rozali ’17 ‘18]

  • Higher-point generalisation of OTOC:
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SLIDE 34

2k-point OTOC

F2k(t1, . . . , tk) = ⌦ ↵

β

hV1V1i · · · hVkVki V1 V1 · · · V3 V3 V2 V2 Vk Vk Vk−1 [ , ][ , ][ , ] [ , ] V4

[FH-Rozali ’17 ‘18]

  • Higher-point generalisation of OTOC:
  • Maximally OTO
  • Maximally “braided” in

Euclidean time

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SLIDE 35

2k-point OTOC

F2k(t1, . . . , tk) = ⌦ ↵

β

hV1V1i · · · hVkVki V1 V1 · · · V3 V3 V2 V2 Vk Vk Vk−1 [ , ][ , ][ , ] [ , ] V4

[FH-Rozali ’17 ‘18]

  • Higher-point generalisation of OTOC:
  • Maximally OTO
  • Maximally “braided” in

Euclidean time

  • Computation involves

(k-1) -exchanges ✏

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SLIDE 36

F2k(t1, . . . , tk) = ⌦ ↵

β

hV1V1i · · · hVkVki V1 V1 · · · V3 V3 V2 V2 Vk Vk Vk−1 [ , ][ , ][ , ] [ , ] V4

[FH-Rozali ’17 ‘18]

  • Higher-point generalisation of OTOC:
  • Hierarchy of time scales associated with

scrambling of quantum information

  • Have calculated this in the Schwarzian theory and

in maximally chaotic 2d CFTs (—> additional spatial dependence) F2k ∼ eλL(t−(k−1)t∗) with t = t1 − tk

  • Result:
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SLIDE 37

Kinematic space interpretation

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SLIDE 38
  • Kinematic space is the space of

timelike separated pairs of points in the CFT:

  • = space of causal diamonds

Kinematic space

x y

(xµ, yµ)

[Czech-Lamprou-McCandlish-Mosk-Sully ‘16] [de Boer-FH-Heller-Myers ‘16] …

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SLIDE 39
  • Kinematic space is the space of

timelike separated pairs of points in the CFT:

  • = space of causal diamonds
  • Natural for studying bulk

emergence, entanglement, …

Kinematic space

x y

(xµ, yµ)

[Czech-Lamprou-McCandlish-Mosk-Sully ‘16] [de Boer-FH-Heller-Myers ‘16] …

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SLIDE 40
  • Kinematic space is the space of

timelike separated pairs of points in the CFT:

  • = space of causal diamonds
  • Natural for studying bulk

emergence, entanglement, …

Kinematic space

x y

(xµ, yµ)

[Czech-Lamprou-McCandlish-Mosk-Sully ‘16] [de Boer-FH-Heller-Myers ‘16] …

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SLIDE 41
  • Operator product expansion:
  • Kinematic space is the space of

timelike separated pairs of points in the CFT:

  • = space of causal diamonds
  • Natural for studying bulk

emergence, entanglement, …

Kinematic space

x y

(xµ, yµ) O(x)O(y) = X

Oi

COOOi

  • 1 + a1∂ + a2∂2 + . . .
  • Oi(x)

[Czech-Lamprou-McCandlish-Mosk-Sully ‘16] [de Boer-FH-Heller-Myers ‘16] …

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SLIDE 42
  • Operator product expansion:
  • Kinematic space is the space of

timelike separated pairs of points in the CFT:

  • = space of causal diamonds
  • Natural for studying bulk

emergence, entanglement, …

Kinematic space

x y

(xµ, yµ) O(x)O(y) = X

Oi

COOOi

  • 1 + a1∂ + a2∂2 + . . .
  • Oi(x)

[Czech-Lamprou-McCandlish-Mosk-Sully ‘16] [de Boer-FH-Heller-Myers ‘16] …

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SLIDE 43

OPE blocks

| {z }

⌘hO(x)O(y)i⇥B∆i(x,y)

O(x)O(y) = X

Oi

COOOi

  • 1 + a1∂ + a2∂2 + . . .
  • Oi(x)
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SLIDE 44

OPE blocks

| {z }

⌘hO(x)O(y)i⇥B∆i(x,y)

“OPE block”

O(x)O(y) = X

Oi

COOOi

  • 1 + a1∂ + a2∂2 + . . .
  • Oi(x)
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SLIDE 45

OPE blocks

| {z }

⌘hO(x)O(y)i⇥B∆i(x,y)

“OPE block”

O(x)O(y) = X

Oi

COOOi

  • 1 + a1∂ + a2∂2 + . . .
  • Oi(x)
  • OPE block = field on kinematic space
  • Smeared representation of OPE block

B∆i(x, y) = C∆i Z

♦(x,y)

ddξ I∆i(x, y; ξ) Oi

Z

O(x) O(y)

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SLIDE 46
  • OPE block = field on kinematic space
  • Smeared representation of OPE block

B∆i(x, y) = C∆i Z

♦(x,y)

ddξ I∆i(x, y; ξ) Oi

Z

O(x) O(y)

⇠ hO(x)O(y) e Oi(ξ)i

[Ferrara-Parisi ’72] [Dolan-Osborn ’12] [Simmons-Duffin ‘12] “shadow operator” formalism

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SLIDE 47

⇠ hO(x)O(y) e T(ξ)i

  • OPE block = field on kinematic space
  • Smeared representation of OPE block

B∆i(x, y) = C∆i Z

♦(x,y)

ddξ I∆i(x, y; ξ) Oi

Z

O(x) O(y)

for stress tensor: BT (x, y) = Cd Z

♦(x,y)

ddξ IT (x, y; ξ) T(ξ)

[Ferrara-Parisi ’72] [Dolan-Osborn ’12] [Simmons-Duffin ‘12] “shadow operator” formalism

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SLIDE 48
  • OPE block = field on kinematic space
  • Smeared representation of OPE block

B∆i(x, y) = C∆i Z

♦(x,y)

ddξ I∆i(x, y; ξ) Oi

  • Can show: this is equivalent to the

coupling to our reparametrization mode!

O(x) O(y)

E

Z

O(x) O(y)

for stress tensor: B(1)

∆ (x, y) ∝ BT (x, y)

B(1)

∆ (x, y) = ∆

1 d (@µ✏µ(x) + @µ✏µ(y)) − 2 (✏(x) − ✏(y))µ (x − y)µ (x − y)2

  • BT (x, y) = Cd

Z

♦(x,y)

ddξ IT (x, y; ξ) T(ξ)

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SLIDE 49

O(x) O(y)

E

Z

O(x) O(y)

B(1)

∆ (x, y) ∝ BT (x, y)

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SLIDE 50

O(x) O(y)

E

Z

O(x) O(y)

B(1)

∆ (x, y) ∝ BT (x, y)

  • For example, stress tensor 4-point conformal block:

hV (x1)V (x2)W(x3)W(x4)i = hV V ihWWi ⇥ CV V T CW W T ⌦ (T + desc.)(T + desc.) ↵

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SLIDE 51

O(x) O(y)

E

Z

O(x) O(y)

B(1)

∆ (x, y) ∝ BT (x, y)

  • For example, stress tensor 4-point conformal block:

hV (x1)V (x2)W(x3)W(x4)i = hV V ihWWi ⇥ [1 + hBT (x1, x2)BT (x3, x4)i + . . .]

W W V V

ZZ (. . .)hTTi

hV (x1)V (x2)W(x3)W(x4)i = hV V ihWWi ⇥ CV V T CW W T ⌦ (T + desc.)(T + desc.) ↵

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SLIDE 52

O(x) O(y)

E

Z

O(x) O(y)

B(1)

∆ (x, y) ∝ BT (x, y)

  • For example, stress tensor 4-point conformal block:
  • A local reformulation of OPE block techniques

hV (x1)V (x2)W(x3)W(x4)i = hV V ihWWi ⇥ [1 + hBT (x1, x2)BT (x3, x4)i + . . .]

W W V V

ZZ (. . .)hTTi

= hV V ihWWi ⇥ [1 + hB∆(x1, x2)B∆(x3, x4)i + . . .]

V V W W

h✏✏i

hV (x1)V (x2)W(x3)W(x4)i = hV V ihWWi ⇥ CV V T CW W T ⌦ (T + desc.)(T + desc.) ↵

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SLIDE 53

O(x) O(y)

E

Z

O(x) O(y)

B(1)

∆ (x, y) ∝ BT (x, y)

  • A local reformulation of OPE block techniques
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SLIDE 54

O(x) O(y)

E

Z

O(x) O(y)

B(1)

∆ (x, y) ∝ BT (x, y)

  • A local reformulation of OPE block techniques
  • No time for details… see [FH-Reeves-Rozali (to appear soon)] …

—> boundary cond. on distinguishes block vs. shadow block

@(µ✏ν) − 1 d ⌘µν (@.✏) ∼ e Tµν

stress tensor “shadow”

✏

  • Basic idea: close connection between

reparametrization modes and shadow operators

  • Proposal:
slide-55
SLIDE 55
  • No time for details… see [FH-Reeves-Rozali (to appear soon)] …
  • Proposal:

O(x) O(y)

E

Z

O(x) O(y)

B(1)

∆ (x, y) ∝ BT (x, y)

  • Seems to work in higher dimensions, as well:

effective field theory <-> shadow operator formalism

  • Conformal blocks can be computed systematically

from reparametrization mode perturbation theory

[Cotler-Jensen ‘18]

@(µ✏ν) − 1 d ⌘µν (@.✏) ∼ e Tµν

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SLIDE 56

Summary

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SLIDE 57
  • Theory of reparametrization modes in CFTs similar to

Schwarzian in d=1 (e.g. SYK)

  • Systematic effective field theory to study OPE,

shadow operators, conformal blocks, quantum chaos, etc.

  • Example A: 2k-point OTOCs have hierarchy of

scrambling timescales

  • Example B: stress tensor OPE block = coupling of

bilocals (kinematic space fields) to reparametrization mode

Summary

t(k)

∗

∼ (k − 1) × t∗