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

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 Vancouver (—> IAS) Based on 1712.04963, 1808.02898 with M. Rozali , and work in progress with W. Reeves & M. Rozali poster!

Introduction

Basic idea • Consider 2d CFT at finite temperature τ σ z = τ + i σ • Conf. symmetry is spontaneously broken z ) → ( f ( z ) , ¯ ( z, ¯ f (¯ z )) • I want to study the Goldstone mode associated with this effect • In the “holographic regime” (large c etc.) there is a systematic effective field theory for this mode

• In the “holographic regime” (large c etc.) there is a systematic effective field theory for this mode • 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, … V(t) W(o) V(t) W(o)

Basics

Reparametrization modes • Consider 2d CFT at finite temperature and a small reparametrization ( z, ¯ z ) → ( z + ✏ , ¯ z + ¯ ✏ ) Z � ¯ ✏ ¯ d 2 z → S CF T + @✏ T ( z ) + @ ¯ T (¯ z ) S CF T −

Reparametrization modes • Consider 2d CFT at finite temperature and a small ( z, ¯ z ) → ( z + ✏ , ¯ z + ¯ ✏ ) reparametrization Z � ¯ ✏ ¯ d 2 z → S CF T + @✏ T ( z ) + @ ¯ T (¯ z ) S CF T − • For conformal transformations, ¯ @✏ = 0 = @ ¯ ✏ the associated conserved symmetry ( J, ¯ ✏ ¯ J ) = ( ✏ T, ¯ T ) currents are

Reparametrization modes • Consider 2d CFT at finite temperature and a small reparametrization ( z, ¯ z ) → ( z + ✏ , ¯ z + ¯ ✏ ) Z � ¯ ✏ ¯ d 2 z → S CF T + @✏ T ( z ) + @ ¯ T (¯ z ) S CF T − • Conformal symmetry is spontaneously broken • Regard as the associated Goldstone modes ( ✏ , ¯ ✏ ) [Turiaci-Verlinde ’16] [FH-Rozali ’18] • have an effective action determined by h T µ ν · · · T ρσ i ( ✏ , ¯ ✏ )

• have an effective action determined by h T µ ν · · · T ρσ i ( ✏ , ¯ ✏ ) Z d 2 z 1 d 2 z 2 ¯ @✏ 1 ¯ W 2 = @✏ 2 h T ( z 1 ) T ( z 2 ) i + (anti-holo.) fixed by conformal symmetry! ( ✏ , ¯ ✏ ) => dynamics of is universal

• have an effective action determined by h T µ ν · · · T ρσ i ( ✏ , ¯ ✏ ) Z d 2 z 1 d 2 z 2 ¯ @✏ 1 ¯ W 2 = @✏ 2 h T ( z 1 ) T ( z 2 ) i + (anti-holo.) • The effective action is actually local ¯ ∂ 1 h T ( z 1 ) T ( z 2 ) i ⇠ δ (2) ( z 1 � z 2 ) … because:

• have an effective action determined by h T µ ν · · · T ρσ i ( ✏ , ¯ ✏ ) Z d 2 z 1 d 2 z 2 ¯ @✏ 1 ¯ W 2 = @✏ 2 h T ( z 1 ) T ( z 2 ) i + (anti-holo.) • The effective action is actually local ¯ ∂ 1 h T ( z 1 ) T ( z 2 ) i ⇠ δ (2) ( z 1 � z 2 ) … because: W 2 = c ⇡ Z d ⌧ d � ¯ @✏ ( @ 3 τ + @ τ ) ✏ + (anti-holo.) 6 ( z = τ + i σ )

• have an effective action determined by h T µ ν · · · T ρσ i ( ✏ , ¯ ✏ ) W 2 = c ⇡ Z d ⌧ d � ¯ @✏ ( @ 3 τ + @ τ ) ✏ + (anti-holo.) 6 • Analogous to Schwarzian action in d=1

• have an effective action determined by h T µ ν · · · T ρσ i ( ✏ , ¯ ✏ ) W 2 = c ⇡ Z d ⌧ d � ¯ @✏ ( @ 3 τ + @ τ ) ✏ + (anti-holo.) 6 • Analogous to Schwarzian action in d=1 • Euclidean propagator: ✓ ⌧ + i � ◆ h ✏ ( ⌧ , � ) ✏ (0 , 0) i ⇠ 1 ⇣ 1 � e − sgn( σ ) i ( τ + i σ ) ⌘ c sin 2 log 2 [FH-Rozali ’18] [Cotler-Jensen ’18] • ( Lorentzian “Schwinger-Keldysh” version is available )

Feynman rules ✓ ⌧ + i � ◆ h ✏ ( ⌧ , � ) ✏ (0 , 0) i ⇠ 1 ⇣ 1 � e − sgn( σ ) i ( τ + i σ ) ⌘ c sin 2 log 2

Feynman rules E ✓ ⌧ + i � ◆ h ✏ ( ⌧ , � ) ✏ (0 , 0) i ⇠ 1 ⇣ 1 � e − sgn( σ ) i ( τ + i σ ) ⌘ c sin 2 log 2

Feynman rules E ✓ ⌧ + i � ◆ h ✏ ( ⌧ , � ) ✏ (0 , 0) i ⇠ 1 ⇣ 1 � e − sgn( σ ) i ( τ + i σ ) ⌘ c sin 2 log 2 • “Coupling” to pairs of other operators via reparametrization: ! [ @ f ( x ) @ f ( y )] ∆ h O ( f ( x )) O ( f ( y )) i h O ( x ) O ( y ) i � f ( x ) = x + ✏ ( x ) " # ⇢ � @✏ ( x ) + @✏ ( y ) − ✏ ( x ) − ✏ ( y ) = h O ( x ) O ( Y ) i 1+ + (anti-holo.) ∆ � x − y � tan 2

Feynman rules E ✓ ⌧ + i � ◆ h ✏ ( ⌧ , � ) ✏ (0 , 0) i ⇠ 1 ⇣ 1 � e − sgn( σ ) i ( τ + i σ ) ⌘ c sin 2 log 2 • “Coupling” to pairs of other operators via reparametrization: ! [ @ f ( x ) @ f ( y )] ∆ h O ( f ( x )) O ( f ( y )) i h O ( x ) O ( y ) i � f ( x ) = x + ✏ ( x ) " # ⇢ � @✏ ( x ) + @✏ ( y ) − ✏ ( x ) − ✏ ( y ) = h O ( x ) O ( Y ) i 1+ + (anti-holo.) ∆ � x − y � tan 2 ⇢ � B (1) = h O ( x ) O ( Y ) i 1+ ∆ ( x, y ) O ( x ) E O ( y )

E ✓ ⌧ + i � ◆ h ✏ ( ⌧ , � ) ✏ (0 , 0) i ⇠ 1 ⇣ 1 � e − sgn( σ ) i ( τ + i σ ) ⌘ c sin 2 log 2 O ( x ) " # @✏ ( x ) + @✏ ( y ) − ✏ ( x ) − ✏ ( y ) E B (1) ∆ + (anti-holo.) ∆ ( x, y ) ≡ � x − y � tan 2 O ( y )

E ✓ ⌧ + i � ◆ h ✏ ( ⌧ , � ) ✏ (0 , 0) i ⇠ 1 ⇣ 1 � e − sgn( σ ) i ( τ + i σ ) ⌘ c sin 2 log 2 O ( x ) " # @✏ ( x ) + @✏ ( y ) − ✏ ( x ) − ✏ ( y ) E B (1) ∆ + (anti-holo.) ∆ ( x, y ) ≡ � x − y � tan 2 O ( y ) • “Feynman rules” for reparametrization Goldstone • At large c , this gives a systematic perturbation theory of energy-momentum exchanges (“gravity channel”)

Applications

Out-of-time-order correlators >> skip

Out-of-time-order correlators • “Usual” QFT: time-ordered correlators ( TOCs ): h W ( t ) W ( t ) V (0) V (0) i β ⇠ h WW ih V V i + O ( e − t/t d ) t d ∼ β V (0) W ( t ) dissipation time: 2 π t V (0) W ( t )

Out-of-time-order correlators • “Usual” QFT: time-ordered correlators ( TOCs ): h W ( t ) W ( t ) V (0) V (0) i β ⇠ h WW ih V V i + O ( e − t/t d ) t d ∼ β V (0) W ( t ) dissipation time: 2 π V (0) W ( t )

Out-of-time-order correlators • “Usual” QFT: time-ordered correlators ( TOCs ): h W ( t ) W ( t ) V (0) V (0) i β ⇠ h WW ih V V i + O ( e − t/t d ) t d ∼ β V (0) dissipation time: 2 π V (0) W ( t ) W ( t )

Out-of-time-order correlators • “Usual” QFT: time-ordered correlators ( TOCs ): h W ( t ) W ( t ) V (0) V (0) i β ⇠ h WW ih V V i + O ( e − t/t d ) V (0) • OTOCs display exp. V (0) “Lyapunov” growth W ( t ) ( quantum chaos ): W ( t ) h W ( t ) V (0) W ( t ) V (0) i β h 1 � # e λ L ( t − t ∗ ) i ⇠ h WW ih V V i [Shenker-Stanford ’13] t ∗ ∼ β [Maldacena-Shenker-Stanford ‘15] scrambling time: 2 π log N [Kitaev ’15] …..

Out-of-time-order correlators • “Usual” QFT: time-ordered correlators ( TOCs ): h W ( t ) W ( t ) V (0) V (0) i β ⇠ h WW ih V V i + O ( e − t/t d ) V (0) • OTOCs display exp. V (0) “Lyapunov” growth W ( t ) ( quantum chaos ): W ( t ) h W ( t ) V (0) W ( t ) V (0) i β h 1 � # e λ L ( t − t ∗ ) i ⇠ h WW ih V V i [Shenker-Stanford ’13] t ∗ ∼ β [Maldacena-Shenker-Stanford ‘15] scrambling time: 2 π log N [Kitaev ’15] …..

Out-of-time-order correlators • “Usual” QFT: time-ordered correlators ( TOCs ): h W ( t ) W ( t ) V (0) V (0) i β ⇠ h WW ih V V i + O ( e − t/t d ) V (0) • OTOCs display exp. “Lyapunov” growth ( quantum chaos ): W ( t ) V (0) W ( t ) h W ( t ) V (0) W ( t ) V (0) i β h 1 � # e λ L ( t − t ∗ ) i ⇠ h WW ih V V i [Shenker-Stanford ’13] t ∗ ∼ β [Maldacena-Shenker-Stanford ‘15] scrambling time: 2 π log N [Kitaev ’15] …..

V (0) h W ( t ) V (0) W ( t ) V (0) i β W ( t ) h 1 � # e λ L ( t − t ∗ ) i ⇠ h WW ih V V i V (0) W ( t )

V (0) h W ( t ) V (0) W ( t ) V (0) i β W ( t ) h 1 � # e λ L ( t − t ∗ ) i ⇠ h WW ih V V i V (0) • : Lyapunov growth is λ L = 2 π T W ( t ) described by an exchange of the reparametrization mode: h W ( t ) V (0) W ( t ) V (0) i β ⇠ h B (1) ∆ W ( t, t ) B (1) ∆ V (0 , 0) i β ⇠ h ✏ ( t ) ✏ (0) i

V (0) h W ( t ) V (0) W ( t ) V (0) i β W ( t ) h 1 � # e λ L ( t − t ∗ ) i ⇠ h WW ih V V i V (0) • : Lyapunov growth is λ L = 2 π T W ( t ) described by an exchange of the reparametrization mode: h W ( t ) V (0) W ( t ) V (0) i β ⇠ h B (1) ∆ W ( t, t ) B (1) ∆ V (0 , 0) i β ⇠ h ✏ ( t ) ✏ (0) i

V (0) h W ( t ) V (0) W ( t ) V (0) i β W ( t ) h 1 � # e λ L ( t − t ∗ ) i ⇠ h WW ih V V i V (0) • : Lyapunov growth is λ L = 2 π T W ( t ) described by an exchange of the reparametrization mode: h W ( t ) V (0) W ( t ) V (0) i β ⇠ h B (1) ∆ W ( t, t ) B (1) ∆ V (0 , 0) i β ⇠ h ✏ ( t ) ✏ (0) i • A universal contribution to the OTOC , described by the collective mode ✏ [FH-Rozali ’18] [Blake-Lee-Liu ‘18]

2k-point OTOC • Higher-point generalisation of OTOC: [FH-Rozali ’17 ‘18] ⌦ ↵ [ ][ ][ ] [ ] V 1 V 1 V 2 V 3 V 2 V 3 V k V 4 V k − 1 V k · · · , , , β , F 2 k ( t 1 , . . . , t k ) = h V 1 V 1 i · · · h V k V k i