Lifshitz-like systems andads null deformations

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1 Lifshitz-like systems andads null deformations K. Narayan Chennai Mathematical Institute [ arxiv: , Koushik Balasubramanian, KN, arxiv: , KN.] Lifshitz points, gravity duals AdS/CFT with null deformations x + -compact:z = 2 Lifshitz systems x + -noncompact: anisotropic Lifshitz systems [see also AdS/CFT cosmo singularities: hep-th/ , , Das, Michelson, KN, Trivedi; arxiv: , Awad, Das, KN, Trivedi; arxiv: , Awad, Das, Nampuri, KN, Trivedi.] Related work: Hartnoll,Polchinski,Silverstein,Tong; Donos,Gauntlett; Gregory,Parameswaran,Tasinato,Zavala;... Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.1/22

2 AdS/condmat and Lifshitz systems Interesting to explore holography with reduced symmetries. Generalizations of AdS/CFT to nonrelativistic systems holographic condensed matter,... Son; Balasubramanian,McGreevy; Adams et al; Herzog et al; Maldacena et al;... Lifshitz points: arise in magnetic systems with antiferromagnetic interactions, dimer models, liquid crystals,... Symmetries: t,x i -translations, x i -rotations, scaling t λ z t, x i λx i [z: dynamical exponent]. Landau-Ginzburg action (z = 2): S = d 3 x(( t ϕ) 2 κ( 2 ϕ) 2 ). Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.2/22

3 AdS/condmat and Lifshitz systems Interesting to explore holography with reduced symmetries. Generalizations of AdS/CFT to nonrelativistic systems holographic condensed matter,... Son; Balasubramanian,McGreevy; Adams et al; Herzog et al; Maldacena et al;... Lifshitz points: arise in magnetic systems with antiferromagnetic interactions, dimer models, liquid crystals,... Symmetries: t,x i -translations, x i -rotations, scaling t λ z t, x i λx i [z: dynamical exponent]. Landau-Ginzburg action (z = 2): S = d 3 x(( t ϕ) 2 κ( 2 ϕ) 2 ). Lifshitz spacetime: ds 2 4 = dt2 r 2z + dx2 i +dr2 r 2. Kachru,Liu,Mulligan Scaling: t λ z t, x i λx i, r λr [z = 1: AdS]. [note: smaller than Galilean symmetries: e.g. Galilean boosts broken] Solution to 4-dim gravity with Λ < 0 and massive gauge field A dt r z (or alternatively gauge field + 2-form: dualize to get A-mass) Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.3/22

4 Lifshitz points in string theory Previous attempts: e.g.li d M 10 d or Li d M 11 d as a solution in string or M-theory, supported by extra fluxes in the compact space (incl warping) (Li,Nishioka,Takayanagi) various basic violations. z = 3 Lifshitz solutions from D3/D7-constructions (Azeyanagi,Li,Takayanagi): 2 necessarily anistropic, require scalar that breaks scaling symmetry. Other solutions: geometry Lifshitz but e.g. scalar breaks symmetry. Various interesting suggestions in Hartnoll,Polchinski,Silverstein,Tong. Alternative way to realize z = 2 Lifshitz [Balasubramanian,KN]: x + -DLCQ of relativistic N=4 SYM z = 2 nonrelativistic (Galilean) 2+1-dim system. Gauge coupling gym 2 (x+ ) varying in lightlike x + -direction breaks x + -shift reducing to 2+1-dim Lifshitz symmetries. Concrete bulk realization: null deformations of AdS 5 S 5 (more generally, AdS X) sourced by lightlike scalar (e.g. dilaton in IIB). Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.4/22

5 AdS null deformations and Lifshitz ds 2 Einst = 1 w 2 [ 2dx + dx +dx 2 i w2 ( + Φ) 2 (dx + ) 2 ]+ dw2 w 2 +dω 2 5. Lightlike dilaton: Φ = Φ(x + ) (also 5-form). [DMNT,ADNT,ADNNT] Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.5/22

6 AdS null deformations and Lifshitz ds 2 Einst = 1 w 2 [ 2dx + dx +dx 2 i w2 ( + Φ) 2 (dx + ) 2 ]+ dw2 w 2 +dω 2 5. Lightlike dilaton: Φ = Φ(x + ) (also 5-form). [DMNT,ADNT,ADNNT] Φ = const i.e. g ++ = 0: DLCQ x + of AdS in lightcone coordinates nonrelativistic, Schrodinger (Galilean) symmetries [ Goldberger, Barbon et al, Maldacena et al]. Regard x t (time), x + compact coordinate (g ++ (Φ ) 2 > 0). Strictly: x + = const null surfaces and x = const surfaces spacelike (g < 0). Symmetries: x,x i -translations, x i -rotations, z = 2 scaling x t λ 2 t, x i λx i, w λw (x + compact, no scaling). x + compact lightlike boosts broken. Galilean boosts x i x i v i x, x + x (2v ix i v 2 i x ) : broken by g ++. Also broken z = 2 special conformal symmetry. Nontrivial x + -dependence z = 2 Galilean broken to Lifshitz. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.6/22

7 AdS null deformations ds 2 Einst = 1 w 2 [ 2dx + dx +dx 2 i w2 ( + Φ) 2 (dx + ) 2 ]+ dw2 w 2 +dω 2 5, Φ = Φ(x+ ). The coord. transfmn. w = re f/2, x = y w2 f 4, gives ds 2 = 1 [e f(x+) ( 2dx + dy +dx 2 r 2 i )+dr2 ]+dω 2 5, Φ(x+ ), with the EOM constraint R ++ = 1 2 (f ) 2 f = 1 2 (Φ ) 2. (DMNT,ADNT) Function-worth of solutions, for anyφ(x + ). Can replace S 5 X 5 (Sasaki-Einstein). These solutions can in fact be generalized to a large family of solutions with axion-dilaton, 3-form field strength turned on (Donos,Gauntlett). Preserve half lightcone supersymmetry. There also exist AdS 4 X 7 null deformations in M-theory, with scalar arising from the G-flux on X 7 : presumably dual to lightlike deformations of Chern-Simons (ABJM-like) theories arising on M2-brane stacks. Similarly AdS 7 solutions likely. Also, null deformations of AdS 3 exist in WZW models. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.7/22

8 Dual field theory, correlators ds 2 = 1 w 2 [ 2dx + dx +dx 2 i w2 (Φ ) 2 (dx + ) 2 ]+ dw2 w 2 +dω 2 5, Φ = Φ(x+ ). (DMNT) d = 4 N=4 super Yang-Mills theory with gauge coupling lightlike-deformed as g 2 YM (x+ ) = e Φ(x+ ) DLCQ x +. Boundary metric ds 2 4 = lim w 0w 2 ds 2 5 manifestly flat. Lightlike (chiral) deformation no nonzero contraction exists involving metric and coupling alone (only + Φ nonvanishing) various physical observables (e.g. trace anomaly, anomalous dims) unaffected. 2-point correlation function: operators O dual to massive scalarsϕ. [In conformal coordsds 2 = 1 r 2 [e f(x+) ( 2dx + dy +dx 2 i )+dr2 ], λ = e f(x+) dx +. S d 3 xd 3 x dx + dx + e 3f(x + )/2 e 3f(x+ )/2 ϕ(x +, x)ϕ(x +, x ) ( λ x + ) 1 1 [( x) 2 ], ( x) 2 = 2( x + )( x )+ i=1,2 ( x i) 2, = 2+ 4+m 2. x + x, x i : approximate λ x + dλ dx + = e f, e f(x+) 1.] O(x i )O(x i ) 1 [ i ( x i) 2 ], and O(t)O(t ) 1 ( x ). Agrees with equal-time 2-pt fn of 2+1-dim Lifshitz theory (KLM). Equal time correlators: 2+1-Lifshitz 2D Eucl. CFT (Ardonne,Fendley,Fradkin) Calculation difficult in form with g ++ 0: scalar wave eqn not straightforward to solve. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.8/22

9 Bulk z = 2 Lif 4 from dim nal redux ds 2 = 1 w 2 [ 2dx + dx +dx 2 i w2 (Φ ) 2 (dx + ) 2 ]+ dw2 w 2 +dω 2 5, Φ = Φ(x+ ). Naive, standard Kaluza-Klein reduction along x + as: ds 2 = g mn dx m dx n = G µν dx µ dx ν +G dd (x d +A µ dx µ ) 2 : 1 w 2 [ 1 4 w2 (Φ ) 2 (dx + ) 2 2dx + dx ] = 1 4 (Φ ) 2 [dx + 4dx w 2 (Φ ) 2 ] 2 4(dx ) 2 w 4 (Φ ) 2. Long-wavelength bulk 4-dim metric: ds 2 = 4(dx ) 2 w 4 (Φ ) 2 + dx2 i w 2 + dw2 w 2. Remnant(Φ ) 2, nontrivial x + -dependence: not standard KK-redux. Minimal off-shell metric ansatz containing above metric ds 2 = N 2 (x + )K 2 (s i )dt N 2 (x + ) (dx+ +N 2 (x + )A) w 2 (ds i ) 2 N(x + ) controlsg ++ component, x t and s i = x i,w. Kaluza-Klein gauge field A A 0 Kdt (purely electric, A i = 0). On-shell solution: g tt = N 2 K 2 (1 A 2 0 ) = 0, N = 2 Φ, K = 1 w 2. Want off-shell lower dim eff. action: calculate 5d Ricci scalar, action for scalar Φ, retaink(s i ),A 0 (s i ) independently (separate lower dim gauge field from metric). Eff action lower dim metric. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.9/22

10 Bulk z = 2 Lif 4 from dim nal redux ds 2 = N 2 (x + )K 2 (s i )dt N 2 (x + ) (dx+ +N 2 (x + )A) w 2 (ds i ) 2 R (5) = R (4) 2(NN +(N ) 2 ) F2 0i +2(NN +(N ) 2 )A 2 0, with R (4) = 2 K (w2 2 i K w wk +3K). Suggests lower dim spacetime is: ds 2 = K 2 (s i )dt w 2 ds i2. On-shell solution: N = 2 Φ, K = 1 w 2, Lif 4 : ds 2 = dt2 w 4 + dx2 i +dw2 w 2. Gauge field mass from scalar kinetic terms agrees with that from lower dim system: 1 2 g++ ( + Φ) N2 (1 A 2 0 )(Φ ) N2 (Φ ) 2 A 2 0. Consistency: 5-d system is on-shell solution if( i A 0 +A i K 0 K )2 = 4 w 2, [from[00]-component], admitting solution K = 1 w 2, A = dt. w 2 Naive dim redux has Φ, disappears here perhaps surprising. Nontrivial x + -dependence complicates Wilsonian separation-of-scales argument: mixing with other modes? e.g. turn on KK vector potential A i dx i. No conclusive result here for consistent dim nal reduction: then R (5) has extraneous factors ofn(x + ) appearing in analogous calculation. Harder to interpret lower dim system. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.10/22

11 Scalar probes of Lifshitz Long wavelength geometry seen by bulk supergravity scalar? Scalar action S = 1 G 5 d 5 x g g µν µ ϕ ν ϕ : restrict to modes with no x + -dependence (i.e. + ϕ = 0) S 1 d [ 4 x G 5 w w 4 dx ( + (Φ ) )( ϕ) 2 +w 2 L( i ϕ) 2 +w 2 L( w ϕ) 2] = 1 G 4 d 4 x w 5 [ w 4 ( ϕ) 2 +w 2 ( i ϕ) 2 +w 2 ( w ϕ) 2 ], L: compactification size. G 4 = G 5 Rescale x x = L L, 4-dim Newton const. dx + (Φ ) 2 x scalar action in 4-dim z = 2 Lifshitz background ds 2 = dt2 w 4 + dxi2 w 2 + dw2 w 2. Eqn of motion: 4-dim Lifshitz geometry arises on scales large compared with the typical scale of variation (compactification size), i.e. effectively setting Φ const. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.11/22

12 Recap: x + -compact,z = 2 Lifshitz Lightlike deformation break Galilean symmetries Lifshitz. AdS 5 S 5 null deformations with lightlike dilaton DLCQ x +: ds 2 = 1 w 2 [ 2dx + dx +dx 2 i w2 (Φ ) 2 (dx + ) 2 ]+ dw2 w 2 +dω 2 5, Φ = Φ(x+ ). Symmetries: x,x i -translations, x i -rotations, z = 2 scaling x t λ 2 t, x i λx i, w λw (x + compact, no scaling). d = 4N=4 super Yang-Mills theory with gauge coupling lightlike-deformed as g 2 YM (x+ ) = e Φ(x+ ) DLCQ x +. 2-point correlation function: operator O dual to massless scalarϕ. O(x i )O(x i ) 1 [ i ( x i) 2 ], and O(t)O(t ) 1 ( x ). Agrees with equal-time 2-pt fn of 2+1-dim Lifshitz theory (KLM). Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.12/22

13 x + -noncompact, anisotropic Lifshitz ds 2 = 1 [ 2dx + dx +dx 2 w 2 i w2 (Φ ) 2 (dx + ) 2 ]+ dw2 +dω 2 w 2 5, Φ = Φ(Qx + ), Q parameter of mass dim one. Symmetries: time x -translations, spatial x i -translations/rotations (x + -translations broken by nontrivial x + -dependence), and anisotropic Lifshitz scaling w λw, x i λx i, x λ 2 x, x + λ 0 x +, i.e. z = 2 {x i,x }: x λ 2 x, x i λx i, z = {x +,x }: x λ 2 x, x + λ 0 x +. In addition, dilaton Φ(Qx + ) acts as spatial x + -potential. [Recallz = 0 Schrodinger systems: ds 2 = dt 2 + dx2 i +dtdξ+dr2 r 2. Different from present context: x time since const-x surface is spacelike crucial forz = scaling, and broken Galilean boosts.] Also recall D3-D7 anisotropic Lifshitz systems + radial scalars (Azeyanagi,Li,Takayanagi), and dilatonic Lifshitz-like black branes (Goldstein,Kachru,Prakash,Trivedi). Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.13/22

14 Linear dilaton, anisotropic Lifshitz ds 2 = 1 w 2 [ 2dx + dx +dx 2 i +w2 Q 2 (dx + ) 2 ]+ dw2 w 2, Φ = Φ 0 +2Qx +. Now metric also hasx + -translations: linear dilaton acts as spatial x + -potential. Reminiscent of Liouville-like walls in c = 1 string theory. z = 2 Lifshitz-like induced metric on e.g. D5-brane probe wrapping const-x + hypersurface (and S 2 S 5 )? Difficult to realize since const-x + hypersurface is null. Might be interesting to study D-brane probes in these geometries. Dual gauge theory: d=4 N=4 SYM theory living on flat spacetime (flat boundary metric), with gauge coupling g 2 YM (x+ ) = e Φ(x+) g s e 2Qx+ (with g s = e Φ 0 ). x + : weakly coupled gauge theory. Consistent: string frame metricds 2 str = e Φ/2 ds 2 degenerates here, highly curved. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.14/22

15 Linear dilaton, Lifshitz, correlators ds 2 = 1 w 2 [ 2dx + dx +dx 2 i +w2 Q 2 (dx + ) 2 ]+ dw2 w 2, Φ = Φ 0 +2Qx +. 2-point correlation function, operators dual to bulk scalars. Scalar mode functions ϕ(x) = e ik x +ik i x i +ik + x + ( ) R(w), with ) radial equation w 3 1 w w 3 w R(w) (k 2 + m2 w 2 w 2 Q 2 k 2 R(w) = 0, k 2 = 2k + k +ki 2, α = iqk, = 2+ 4+m 2 2 = 2+ν. Can be solved in terms of conf.hypergeom.fns: pick growing solution, demanding regularity in interior. Also note forq = 0, this should reduce to AdS lightcone calculation (w 2 K ν (kw)). Bulk-boundary propagator: a = 1 2 k2 8α = ν k2 4iQk, c = 1 = ν +1. G(k i,k +,k,w) = N(k)e ik ix i +ik x +ik + x + w e αw2 U(a,c, 2αw 2 ). Momentum space 2-point correlation fns: (ν / Z) O(k)O( k) ν2 ν α ν Γ( ν) Γ(ν) ν = 2 ( = 4): O(k)O( k) Γ( 1+ν 2 + k2 ) 4iQk. Γ( 1 ν 2 + k2 ) 4iQk ((k 2 i 2k +k ) 2 +4Q 2 k 2 ) (log(iqk )+ψ( k2 i 2k +k 4iQk )). Reminiscent of 2-pt correlation fns in Kachru,Liu,Mulligan, but also differences Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.15/22

16 Linear dilaton, Lifshitz, correlators = 4: O(k)O( k) ((k 2 i 2k +k ) 2 +4Q 2 k 2 )(log(iqk )+ψ( k2 i 2k +k 4iQk )). [Consistent with Lifshitz scaling: (x,x +,x i ) (λ 2 x,λ 0 x +,λx i ).] [ForQ 0, ψ() log(): recover AdS 2-point correlation fnk 4 logk 2.] To gain some insight, note: (k 2 i 2k +k ) 2 +4Q 2 k 2 = (k 2 i 2(k + iq)k )(k 2 i 2(k ++iq)k ) Effective x + -momentum shift: k + k + ±iq (e ik +x + e ik +x + e ±Qx+ ) reminiscent of Liouville-like wall in c = 1 string theory. Also, for k i = 0: (k 2 i 2k +k ) 2 +4Q 2 k 2 (k 2 + +Q 2 )k 2, i.e. effective mass-gap in x + -direction. Free SYM: S = d 4 x g 2 Y M (x+ ) TrF2 d 4 x e Φ(x+) [( j A i ) 2 2( + A i )( A i )]. Wave modes e ik +x + +ik x +ik i x i k 2 i +2(k + +iq)k = 0, i.e. k + = k2 i 2k iq. For generic k i,k,x + -momentum k + nonzero, i.e. generic waves move along x + -direction due to dilaton x + -potential. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.16/22

17 Aspects of some solutions Φ = 2Qtanh(Qx + ), i.e. Φ = Φ 0 +2logcosh(Qx + ), ds 2 = 1 w 2 [ 2dx + dx +dx 2 i +w2 Q 2 tanh 2 (Qx + )(dx + ) 2 +dw 2 ]. x + ± : g ++ (Φ ) 2 Q 2, x + 0 : g Akin to junction of two Lifshitz-like systems joined together with 1 Q -sizedads-schrodinger-like core about x+ = 0. Q large, core shrinks, junction becomes sharper. SYM gauge coupling: g 2 YM = eφ = g s cosh 2 (Qx + ), x + -potential for charged dyonic states (varying D-probe tension). Φ = Q cosh 2 (Qx + ), Φ = Φ 0 +tanhqx +, ds 2 = 1 w 2 [ 2dx + dx +dx 2 i +w2 Q 2 (dx + ) 2 cosh 4 (Qx + ) +dw2 ]. Large x + : g ++ 0, Schrodinger-like. x + = 0: earlier linear dilatonic system, Lifshitz-like behaviour. Interpolating solutions with radialw-dependence? Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.17/22

18 Linear dilaton, non-geom DLCQ I I 0 +2QL S duality I x + x = L x = 2L Compactify x + -direction? Linear dilaton not x + -periodic. Piecewise linear dilaton: Φ = Φ 0 +2Qx +, x + [0,L], Φ = Φ(L) 2Q(x + L), x + [L,2L],... [Einstein metric ds 2 = 1 w 2 [ 2dx + dx +dx 2 i +w2 Q 2 (dx + ) 2 +dw 2 ] smooth.] Φ(x + ) continuous, not periodic: note S-duality symmetry, τ 1, i.e. Φ Φ. τ Φ periodic upto S-duality if Φ(x + +L) = Φ(x + ),i.e. Φ(L) = Φ(0) Φ 0 +2QL = Φ 0, i.e. g s = e Φ 0 = e QL. Asymptotic string coupling fixed by this non-geometric construction: large string corrections? Lightlike deformations ofads 5 S 5 : no nonzero contractions (only + Φ nonzero,g ++ = 0), likely no higher derivative corrections. Also preserves half susy. Also note: dilaton bounded, no singularities anywhere. Contrast with e.g. F-theory, elliptic fiber degenerations, 7-brane sources,... Lower dim Lifshitz symmetries lightlike structure in 5-dim, possibly controlled corrections. non-geometric construction for dualn=4 SYM too. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.18/22

19 Solutions with lightlike axion Lightlike axion: c 0 = c 0 (x + ). ds 2 = 1 w 2 [ 2dx + dx +dx 2 i +w2 ( + c 0 ) 2 (dx + ) 2 +dw 2 ]+ds 2 5. Linear axion configuration: ds 2 = 1 w 2 [ 2dx + dx +dx 2 i +w2 Q 2 (dx + ) 2 +dw 2 ]+ds 2 5, c 0 = c Qx+. With x + -direction compact: c 0 c 0 +2QL asx + x + +L: equivalent to τ τ +1shift if QL = 1 2. D3-D7 interpretation? Axion θ-angle in dual gauge theory. Possibly solutions with nontrivial axion-dilaton exist too. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.19/22

20 AdS 4 Lifshitz and M-theory ds 2 = 1 r 2 ( g µν dx µ dx ν +dr 2 )+4dΩ 2 X 7, G 4 = 6vol(M 4 )+CdΦ Ω 3. C normalization const, X 7 Sasaki-Einstein 7-manifold admitting harmonic 3-formΩ 3. ScalarΦ(x µ ): no natural interpretation in 11-dim, arises from 4-form flux after compactification. Φ = const, g µν = η µν usual AdS 4 X 7 solution. Effective 3-dim system: Rµν = 1 2 µφ ν Φ, Φ = 0. 3-dim gravity g MN trivial: dynamics driven by scalar. Null: ds 2 = 1 w 2 [ 2dx + dx +dx 2 i w2 (Φ ) 2 (dx + ) 2 ]+ dw2 w 2. Dim nal reduction dim bulk z = 2 Lifshitz spacetime ds 2 = dt2 w 4 + dx2 w 2 + dw2 w 2. Field theory dual: 1+1-dim strongly coupled theory, dim nal redux of null deformation of Chern-Simons theories on M2-branes at conical (CY 4-fold) singularities. ABJM generalizations... Explore further. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.20/22

21 OtherAdS Lifshitz-like solutions AdS 5 null deformation: lifts to (11-dim) M-theory. Similar to earlier solution: ds 2 = 2dx+ dx +d x 2 +dw 2 +(dx + ) 2, ϕ(x + 2 e ) = ilx + w 2 l 2 R. Note: no x + -dep in metric, KK-reduction standard, gives z = 2Li 4 and gauge field. M-theory lift: AdS null 5 CP 2 S 1 S 1 supported by G-flux. Boundary theory: unclear, possibly M5-brane dual. Anisotropic radial Kasner-like solutions ds 2 = 1 r 2 [dr 2 r 2p 0 dt 2 + i r2p i(dx i ) 2 ]+dω 2 5, eφ = r α, p 0 + i p i = 0, p i p2 i = α2 2. With nontrivial scalar Φ (α 0): static solutions with spatially anistropic Lifshitz-like scaling, except scalar breaks scaling symmetry (α = 0 p 0,p i = 0). Time-dep AdS (cosmological) solutions with Lifshitz-like symmetries. Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.21/22

22 Conclusions, open questions AdS null deformations dim nal redux z = 2 Lifshitz (for AdS 5 ) dual to DLCQ of N=4 SYM with varying coupling. x + -noncompact spatially anisotropic Lifshitz-like systems with dilaton x + -potential. Interesting structure in holographic correlators. Finite temperature? Fermions? Interesting radial dependence of scalar/metric: holographic RG flows between AdS-lightcone (Schrodinger), Lifshitz, or Li z1 Li z2? Holographic superfluid ground states? Further exploration of field theory dual to Lifshitz: N=4 SYM with lightlike varying coupling, DLCQ... Further exploration of AdS 4 solutions: 1+1-dim field theory dual possibly deformation of Chern-Simons theories on M2-branes at singularities (generalizations of ABJM... ). Lifshitz-like systems and AdS null deformations, K. Narayan, CMI p.22/22