East Asia Joint Workshop on Fields and Strings 2017 KEK Theory workshop 2017 @ KEK, Nov.13-17, 2017 Path-Integral Complexity, Liouville Action and AdS/CFT Tadashi Takayanagi Yukawa Institute for Theoretical Physics (YITP), Kyoto Univ. Based on 1703.00456 [Phys.Rev.Lett. 119 (2017)071602] 1706.07056 [To appear in JHEP] Collaborators: Pawel Caputa (YITP) Nilay Kundu (YITP) Masamichi Miyaji (YITP) Poster Kento Watanabe (YITP) Presentation
① Introduction Motivation 1 What is the basic mechanism of AdS/CFT ? [After 20 years from Maldacena’s discovery] ⇒ One intriguing idea is the conjectured interpretation of AdS/CFT as tensor networks (TNs). [Swingle 2009,….] Tensor Network = Network of Quantum entanglement ``Emergent space from Quantum Entanglement’’ Holographic Entanglement Entropy
Holographic Entanglement Entropy Bdy CFT d Γ Area( ) Bulk = − ρ ρ = A Tr log S AdS d+1 A A A 4 G N A M A [Ryu-TT 2006, Hubeny-Rangamani-TT 2007] [Derivation: Casini-Huerta-Myers 2009, Γ A Lewkowycz-Maldacena 2013] Entanglement Wedge Minimal surface Q. Which bulk region is dual to a given region A in CFT ? ⇒ Entanglement Wedge M A (note: we took a time slice) M A = A region surrounded by A and Γ A (on a time slice) ρ ⇔ ρ CFT Bulk [Czech et.al. 2012, Wall 2012, in CFT in AdS A M Headrick et.al. 2014, … ] A
Some advertisements [1] What is the CFT interpretation of EW cross section ? Σ AB Σ Area( AB = ) We found ?? A M AB B an answer ! 4 G N Koji Umemoto’s poster (on Wed.) [2] Can we distinguish CFTs by behaviors of EE ? ⇒ Time evolutions of EE can distinguish CFTs ! (i) Rational CFTs, (ii) Irrational but integrable CFTs, and (iii) Chaotic CFTs (Hol. CFTs). Yuya Kusuki’s poster (on Wed.)
Boundary Bulk=AdS d+1 =CFT d d-1 Area in the unit A of Planck length γ A : Minimal Area surface B Planck length ~ 1 qubit Spacetime in gravity = Collections of bits of entanglement ⇒ Emergent space via tensor networks (TNs) ? TN = A geometrical description of wave functions (Ansatz for variation principle to find ground states)
Example of TN: Matrix Product State (MPS) [DMRG: White 92,…, Rommer-Ostlund 95,..] α β ( σ α α α ) M α σ αβ 1 2 n 3 α = χ 1,2,..., , σ σ σ i σ =↑ ↓ 1 2 n or . i Spin chain ∑ Ψ = σ σ σ σ σ σ Tr[ ( ) ( ) ( ) ] , , , M M M 1 2 1 2 n n σ σ σ , , , n Spins 1 2 n ≤ = χ Min [# links ] 2 log S A ⇒ Entangleme nt is not enought to describe 2D CFTs
MERA [Vidal 05, …] [TN for AdS/CFT: Swingle 09,…] Coarse-graining = Isometry [ ] [ ] † = δ T abc T bcd ad b d a d a = c Disentangler = Unitary trf. ≤ Min [# links ] S A ∝ log L ⇒ agrees with results in 2d CFT !
A Basic Key Idea: Tensor Network of MERA = a time slice of AdS space Questions [see e.g. Beny 2011, Bao et.al. 2015, Czech et.al. 2015] (a) Special Conformal invariance ? (b) Non-isotropic tensors ? ( EW is not properly realized) (c) Why the EE bound is saturated ? (d) How to derive Einstein eq. ? (Sub AdS Scale Locality) Recent developments in lattice models ・ Improved TN models: ⇒ (a),(b),(c) [Perfect TN: Pastawski-Yoshida-Harlow-Preskill 15] [Random TN: Hayden-Nezami-Qi-Thomas-Walter-Yang 16] ⇒ Refer also to Arpan Bhattacharrya’s talk
Some of these problems may be due to lattice artifacts. Moreover, we want to eventually understand the genuine AdS/CFT in the continuum limit. We propose a new alternative approach based on path-integrals , related to a continuum limit of TNs. Our guiding principle 1 Eliminating unnecessary tensors in TN for a given state = Creating the most efficient TN (= Optimization of TN ) Solving the dynamics of Gravity (Einstein eq. etc.)
Motivation 2 How can we define complexity in CFTs ? Computational Complexity of State Ψ = ⋅ ⋅ ⋅ A given N qubit state: 0 0 0 0 = Acting (Local) Unitary Gates Quantum Circuit on a simple reference state Ψ = ⋅ ⋅ ⋅ 0 0 0 0 Computational Complexity of State C = min[ #(Gates) ] Ψ
For tensor network descriptions, Complexity = Min [# of Tensors] in TN Recently, holographic formulas of complexity have been proposed and actively studied: (i) Complexity = Max. volume in AdS [Stanford-Susskind 14] (ii) Complexity= Gravity action in WDW patch [Brown-Roberts-Susskind-Swingle-Zhao 15] (iii) Information Metric = Max. volume in AdS [Miyaji-Numasawa-Shiba-Watanabe-TT 15]
This motivates us to consider its QFT counterpart We introduce ``Path-integral Complexity’’ . Our guiding principle 2 ・ Lattice (tensor) structures in TN Background metric g ab in Euclid path-integral ・ O ptimization of TN for a state Ψ [ ] Minimizing Path-integral Complexity I Ψ g ab g w.r.t the metric ab
② AdS from Optimization of Path-Integrals (2-1) Formulation A Basic Rule: Simplify a path-integral s.t. it produces the correct UV wave functional. Consider 2D CFTs for simplicity . ( z=- Euclidean time, x=space) Deformation of discretizations in path-integral = Curved metric such that one cell (bit) = unit length. φ = + 2 2 ( , ) 2 2 x z ( ). ds e dx dz Note: The original flat metric is given by (ε is UV cutoff): − = ε ⋅ + 2 2 2 2 ( ). ds dx dz
Optimization of Path-Integral [Miyaji-Watanabe-TT 16] Space (x) Euclidean Tensor Network Renormalization Time (-z) (TNR) = Optimization of TN [Evenbly-Vidal 14, 15] Optimization of ε Lattice Path-integral Constant Hyperbolic Space = Time slice of AdS3
The wave functional for CFT vacuum is given by g ab (x,z): background metric [ ] ( ) ∫ ∏ − Φ Ψ Φ = Φ ⋅ δ Φ − Φ = ( ) S g ( ) ( , ) ( ) ( , 0 ) x D x z e x x z CFT UV < < ∞ 0 z − ∞ < < ∞ x In CFTs, owing to the Weyl invariance, we have [ ] [ ] . ( ) φ = δ 2 Ψ Φ = φ ⋅ Ψ Φ g e Flat ( ) exp [ ( , )] ( ) x I x z x ab ab UV UV Optimized wave func. Original wave func. Our Proposal (Optimization of Path-integral for CFTs): φ I φ ( , ) [ ( , )] x z x z Minimize w.r.t φ − = = ε 2 2 | . e with the boundary condition ε z
A Reason for Minimization The normalization N estimates repetitions of same operations of path-integration. → Minimize this ! ⇒ Our Complexity Formula: Ψ = φ Min [ [ ( , )]] C I x z Ψ φ ( , ) x z ≡ computatio nal complexity of C Ψ Ψ the quantum state
(2-2) Liouville Action as Complexity in 2D CFTs [Caputa-Kundu-Miyaji-Watanabe-TT 17] Ψ Liouville Action φ = δ 2 φ = = φ g e # of Isometries [ ] Log [ ], I ab S Ψ L [Czech 17] = δ g ab [ ] c ∫ φ φ = ∂ φ + ∂ φ + 2 2 2 [ ] ( ) ( ) S dxdz e π L x z # of Unitaries 24 [ ] ( ) c ∫ 2 φ = ∂ φ + ∂ φ + + 2 ( ) ( surface term) dxdz e π x z 24 1 Hyperbolic plane (H 2 ) φ ⇒ = 2 Minimum : . e = Time slice of AdS3 2 z = + L 2 2 2 2 ( ) / . ds dx dz z = φ ⋅ Min [ [ ]] ~ C S c Ψ ε L φ
A Sketch: Optimization of Path-Integral [ ] MERA [ ] φ Ψ = 2 Φ Ψ UV Φ g e Flat ∝ ( ) x ( ) x 0 0 0 0 0 z=0 UV 0 0 0 0 0 Optimize Time -z = Hyperbolic Space H2 z= ∞ UV modes k>1/z + 2 2 dx dz = 2 Space x ds are not important ! 2 z
(2-3) Thermofield Double of 2D CFT The TFD state at T=1/β is described as the path -integral [ ] Ψ Φ Φ ( ), ( ) x x 1 2 g β β [ φ ∝ ∫ ∏ ] S − Φ = Φ ⋅ δ Φ − Φ = δ Φ − Φ = − . ( ) S e ( , ) ( ) ( , ) ( ) ( , ) L D x z e x x z x x z CFT 1 2 4 4 − β < < β / 4 / 4 z − ∞ < < ∞ x CFT2 CFT1 φ Minimizati on of [ ( , )] S x z Φ Φ ( ) ( ) 1 x 2 x L π 2 4 1 φ ⇒ = ⋅ 2 ( ) z ) . e + β/4 - β/4 ( z β π β 2 2 cos 2 / z Optimization = Time slice of BTZ black hole. (i.e. Einstein-Rosen Bridge ） .
(2-4) Primary States and Back-reactions |w|=1 Vacuum state on a circle We optimize the path-integral on a disk with the unit radius. = Hyperbolic Disk 4 dwd w The solution of = 2 . ds (=Time slice of − 2 2 Liouville equation ( 1 | | ) w Global AdS3) Primary state on a circle |w|=1 ( , ) O w w We insert an operator at w=0. It has conformal dim. h L =h R =h. O(0) − 2 φ ⋅ ⇒ h ( ) ~ . O x e
φ − φ Ψ Ψ ∝ ⋅ [ ] 2 ( 0 ) S h / . e e Thus we minimize L = φ = 2 1 g g e π 1 6 h φ ∂ ∂ φ + δ = 2 2 ( ) 0 . e w w w 4 c ζ ζ 4 d d θ = ζ ≡ = Solution: 2 a i , ds w re − ζ 2 2 ( 1 | | ) ≡ − θ θ + π ⇒ Deficit angle: ( 1 12 / ). a h c ~ 2 . a = − 1 24 / . Note: the AdS/CFT predicts a h c Interestingly, if we consider the quantum Liouville CFT , γα then = − αγ = + ≡ γ + γ 2 ( / 2 .), 1 3 , ( 2 / ). h Q c Q Q 4 = − 1 24 / . ⇒ We get a h c
Recommend
More recommend
