SYK model, Coadjoint orbits and Liouville bulk dual

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1 SYK model, Coadjoint orbits and Liouville bulk dual Gautam Mandal String theory: past and present SpentaFest ICTS, Bangalore, 11 January 2017 Ongoing work with Pranjal Nayak and Spenta

2 Scanned by CamScanner Let s write the paper today!

3 SYK model 1D stat mech model with Majorana fermions and disorder Sachdev-Ye, Kitaev τ i ψ i (τ)ψ i (τ + a)+ i<j<k<l J ijkl J ijkl J 2 /N 3 J ijkl ψ i (τ)ψ j (τ)ψ k (τ)ψ l (τ) At large N, important observable is G(τ,τ ) = i ψ i(τ)ψ i (τ ) Schwinger-Dyson equations: 1 G(ω) = iω +Σ(ω), Σ(τ,τ ) = J 2 G(τ,τ ) 3 SYK literature: Polchinski-Rosenhaus, Maldacena-Stanford, Gross-Rosenhaus, Stanford, Berkooz-P.Narayan-Rozali-Simon, Verlinde, Polchinski-Shenker-... Variants (without disorder): Gurau, Witten, Klebanov-Tarnopolsky

4 At strong coupling, the SD equations exhibit reparametrization invariance under τ f(τ) (Diff) J 2 dτ G(τ,τ )G(τ,τ ) 3 = δ(τ τ ) (1) where G(τ,τ ), behaves as a tensor of weight = 1/4: G G f : G f (f(τ 1 ), f(τ 2 )) ( f (τ 1 )f (τ 2 ) ) = G(τ1,τ 2 ) UV βj=0 IR βj=, reparametrization invariance

5 Solution of (1): G(τ,τ ) = G 0 (τ τ ), G 0 (τ) (J τ ) 1/2 sgnτ spontaneously breaks Diff SL(2). Goldstones of Diff/SL(2), characterized by τ f(τ) have zero action. At large but finite J, one has explicit breaking of Diff. The Goldstones now pick up a finite action (equivalent to a pion mass term) given by the Schwarzian form S N dτ{f,τ} J At finite temperatures, S N J [ (f ) 2 dτ f ( ) ] 2π 2 (f ) 2 β

6 Features: Zero temperature entropy (black hole?) S 0 = N Universal low energy sector S N J ( ) 1 π2 log dτ{f,τ} Chaos (Liapunov exponent saturates the gravity bound) λ = 2π β Regge spectrum (different properties from O(N) or N = 4 SYM)...

7 Towards a bulk dual A 2D bulk dual that captures the universal low energy sector (and hopefully more) CFT 1 AdS 2 Sen and collaborators, Dabholkar and collaborators In 1D, T µ µ = 0 implies T ττ = 0 = H. Hence all states are ground states created by fermion zero modes of SUSY BHs. In order to have non-trivial dynamics, need to explicitly break conformal invariance ( Diff). this is done in the SYK model by turning on a small 1/J. For a bulk representation of this: embed AdS 2 as the near-horizon geometry of a higher dimensional near-extremal black hole. Effective theory= dilaton gravity in 2D (Jackiw-Teitelboim). Almheiri-Polchinski, Maldacena-Stanford-Yang

8 Our approach Our approach: Low energy configurations of SYK are generated by f(τ) Diff G 0 G f 0 : Gf (f(τ 1 ), f(τ 2 )) ( f (τ 1 )f (τ 2 ) ) = G0 (τ 1,τ 2 ) This is a space of zero modes (all flat directions). A potential is created by explicit breaking of Diff (a pion mass term): S = N J dτ{f,τ} We would like to find a bulk dual of this picture.

9 A bit of history: CFT 2 AdS 3 The Brown-Henneaux geometries Here f(z), f( z) comprise 2D conformal transformations. The metrics (in Euclidean signature) are explicitly given by Banados 1999, see GM-Sinha-Sorokhoibam for eternal BTZ ds 2 = dζ2 + dz d z ζ 2 + L(z)dz 2 + L( z)d z 2 +ζ 2 L(z) L( z)dzd z where L, L are the hologrphic boundary stress tensors given by the Schwarzians L(z) = {f, z}, L( z) = { f, z}

10 A bit of history: CFT 2 AdS 3 The Brown-Henneaux geometries Here f(z), f( z) comprise 2D conformal transformations. The metrics (in Euclidean signature) are explicitly given by Banados 1999, see GM-Sinha-Sorokhoibam for eternal BTZ ds 2 = dζ2 + dz d z ζ 2 + L(z)dz 2 + L( z)d z 2 +ζ 2 L(z) L( z)dzd z where L, L are the hologrphic boundary stress tensors given by the Schwarzians L(z) = {f, z}, L( z) = { f, z} Note that the orbit of AdS 3 (Poincare) includes black holes (BTZ).

11 Brown-Henneaux metrics as Coadjoint orbits According to Banados, the Vir Vir transformations correspond to SL 2 SL 2 (large) gauge transformations in the Chern Simons formulation of 3D gravity with negative cosmological constant.

12 Brown-Henneaux metrics as Coadjoint orbits According to Banados, the Vir Vir transformations correspond to SL 2 SL 2 (large) gauge transformations in the Chern Simons formulation of 3D gravity with negative cosmological constant. The space of Brown-Henneaux geometries, therefore, can be regarded as SL 2 SL 2 orbits (in fact, they are coadjoint orbits of the corresponding loop groups ).

13 Brown-Henneaux metrics as Coadjoint orbits According to Banados, the Vir Vir transformations correspond to SL 2 SL 2 (large) gauge transformations in the Chern Simons formulation of 3D gravity with negative cosmological constant. The space of Brown-Henneaux geometries, therefore, can be regarded as SL 2 SL 2 orbits (in fact, they are coadjoint orbits of the corresponding loop groups ). It is an independent fact that coadjoint orbits admit a natural symplectic form (Kirillov). For coadjoint orbits of loop groups G, the action principle that follows gives the WZW action, based on the group G! Rai and Rogers

14 Brown-Henneaux metrics as Coadjoint orbits According to Banados, the Vir Vir transformations correspond to SL 2 SL 2 (large) gauge transformations in the Chern Simons formulation of 3D gravity with negative cosmological constant. The space of Brown-Henneaux geometries, therefore, can be regarded as SL 2 SL 2 orbits (in fact, they are coadjoint orbits of the corresponding loop groups ). It is an independent fact that coadjoint orbits admit a natural symplectic form (Kirillov). For coadjoint orbits of loop groups G, the action principle that follows gives the WZW action, based on the group G! Rai and Rogers This, therefore, says that WZW action should describe the dynamics in the space of Brown-Henneau geometries. This is already anticipated in Witten s work: Chern-Simons WZW (this predates AdS/CFT by a decade).

15 Coajoint orbits of AdS 2 We have been able to construct the following coadjoint orbit of AdS 2 under Diff, given by the following exact metrics GM-Nayak-Wadia 1701.nnnn ds 2 ĝ f αβ dxα dx β = ( 1 (dζ 2 4πµζ 2 + dτ 2 1 ζ 2{f(τ),τ} ) ) 2 2

16 Properties: All metrics represent normalizable deformations, are in the Fefferman-Graham gauge and are obtained by pulling back a Diff f(τ) from boundary to bulk by the exact map τ = f(τ) 2ζ2 f (τ)f (τ) 2 4f (τ) 2 +ζ 2 f (τ) 2, ζ = 4ζf (τ) 3 4f (τ) 2 +ζ 2 f (τ) 2 There are horizons in the new geometry, determined by g ττ = ( 1 ζ 2{f(τ),τ} ) 2 = 0 2

17 The coadjoint orbit (Kirillov) action What is the action in the space of these metrics?

18 The coadjoint orbit (Kirillov) action What is the action in the space of these metrics? Since the Diff orbit is a coadjoint orbit, a natural action is given by the Kirillov action.

19 The coadjoint orbit (Kirillov) action What is the action in the space of these metrics? Since the Diff orbit is a coadjoint orbit, a natural action is given by the Kirillov action. It is known for some time Polyakov 1987, Rai-Rogers, Alexeev-Shatashvili that Kirillov action in the space of 2D metrics is the induced gravity action of Polyakov S[g] = 1 ( g 16πb 2 R 1 ) M R 16πµ γk 4πb 2 M R where we have added a boundary term similar to the Gibbons-Hawking term. b 2 = G N

20 The Liouville action Choose the conformal gauge g αβ = e 2φ ĝ αβ [ S = 1 ĝ(ĝ αβ αφ 4π b 2 β φ+ ˆRφ+4πµe 2φ 1 1 )+ ĝˆr ˆR M 16π b 2 M ˆ ] + 2 ˆγ ˆKφ+ ˆγˆn µ φ µφ M M The codjoint orbit metrics ĝ αβ = ĝ f αβ, with φ = 0, are classical solutions of the above action.

21 The Liouville action Choose the conformal gauge g αβ = e 2φ ĝ αβ [ S = 1 ĝ(ĝ αβ αφ 4π b 2 β φ+ ˆRφ+4πµe 2φ 1 1 )+ ĝˆr ˆR M 16π b 2 M ˆ ] + 2 ˆγ ˆKφ+ ˆγˆn µ φ µφ M M The codjoint orbit metrics ĝ αβ = ĝαβ f, with φ = 0, are classical solutions of the above action. To compute the potential in the space of these coadjoint orbits (Goldstones), we must compute the on-shell action. To do this, we need a regulator ζ = δ. The on-shell action, up to O(δ) terms, coincides with that of AdS 2. This reproduces the SYK result that the Goldstones have zero action.

22 Explicit symmetry breaking Liouville theory is itself a conformal theory: CFT 2, with Vir Vir symmetry. This leads to 2 sets of classical solutions for the Liouville mode. With ĝ αβ = AdS 2, the solutions are characterized by two functions g(z), ḡ( z) [ ] φ(z, z) = 1 2 log (z + z) 2 g ḡ (1 g(z)ḡ( z)) 2 Normalizable solution: g(z)ḡ( z) z+ z=0 = 1, Non-normalizable solution: g(z)ḡ( z) z+ z=0 = 1 The solutions are subject to two Virasoro constraints: T zz = 0 = T z z We now turn on a non-normalizable solution generated by g = z +δg(z). In AdS/CFT a non-normalizable mode represents breaking of the boundary CFT symmetry. We will now repeat the computation of the on-shell Kirillov action in the presence of such a non-normalizable mode.

23 If we turn on a non-normalizable Liouville solution characterized by a δg(z) on top of the metric gαβ f, the on-shell action becomes S on shell = 1 2b 2 dτδg(iτ){f,τ} By an appropriate identification of this τ-parameter with the boundary clock, we can make δg= constant S on shell = δg 2b 2 dτ{f,τ} If we identify δg = 1 J, 1/b2 = N we recover the Schwarzian Goldstone action ( pion mass term) of SYK from the Liouville bulk dual.

24 Thermodynamics To compute the thermodynamics from the Liouville model, let us go to Euclidean AdS 2 and apply to it the coadjoint orbit transformation f(τ) which compactifies real line into a circle f(τ) = tan πτ β [ ( )] This gives dŝf 2 = 1 dζ 2 + dτ 2 1 π 2 ζ2. 4πζ 2 β 2 The geometries are capped: g ττ vanishes at ζ = β/π.

25 Thermodynamics To compute the thermodynamics from the Liouville model, let us go to Euclidean AdS 2 and apply to it the coadjoint orbit transformation f(τ) which compactifies real line into a circle f(τ) = tan πτ β [ ( )] This gives dŝf 2 = 1 dζ 2 + dτ 2 1 π 2 ζ2. 4πζ 2 β 2 The geometries are capped: g ττ vanishes at ζ = β/π. The on-shell action for this geometry is [ 2 S on shell = log Z = βf = b 2 log(2) 3 2b ] 2b 2 βj which shows the same features as SYK.

26 Conclusion Identified coadjoint orbits of AdS 2 under Diff as the space of Goldstones of SYK.

27 Conclusion Identified coadjoint orbits of AdS 2 under Diff as the space of Goldstones of SYK. Identified 2D Liouville action as a coadjoint orbit action which governs these Goldstones.

28 Conclusion Identified coadjoint orbits of AdS 2 under Diff as the space of Goldstones of SYK. Identified 2D Liouville action as a coadjoint orbit action which governs these Goldstones. The on-shell action of AdS 2 + (Diff transformations) is the same as that of AdS 2 (they differ by a regulator term which vanishes in the continuum limit).

29 Conclusion Identified coadjoint orbits of AdS 2 under Diff as the space of Goldstones of SYK. Identified 2D Liouville action as a coadjoint orbit action which governs these Goldstones. The on-shell action of AdS 2 + (Diff transformations) is the same as that of AdS 2 (they differ by a regulator term which vanishes in the continuum limit). AdS 2 + Diff + Non-normalizable Liouville mode generate a non-zero on-shell gravity action. This coincides with the Schwarzian of SYK.

30 Conclusion Identified coadjoint orbits of AdS 2 under Diff as the space of Goldstones of SYK. Identified 2D Liouville action as a coadjoint orbit action which governs these Goldstones. The on-shell action of AdS 2 + (Diff transformations) is the same as that of AdS 2 (they differ by a regulator term which vanishes in the continuum limit). AdS 2 + Diff + Non-normalizable Liouville mode generate a non-zero on-shell gravity action. This coincides with the Schwarzian of SYK. This technique reproduces the features of the low temperature thermodynamics of SYK from the Liouville bulk dual.

31 To do Need to couple matter to Liouville to obtain non-trivial excitations.

32 To do Need to couple matter to Liouville to obtain non-trivial excitations. Analytic understanding of long time dynamics of SYK in progress with P. Nayak, R. Sinha, J. Sonner, S. Majumdar,...

33 To do Need to couple matter to Liouville to obtain non-trivial excitations. Analytic understanding of long time dynamics of SYK in progress with P. Nayak, R. Sinha, J. Sonner, S. Majumdar,... Lots more.

34 Happy Birthday, Spenta!