#### DMCA

## UvA-DARE (Digital Academic Repository) Lifshitz from AdS at finite temperature and top down model

### Citations

453 |
Holographic reconstruction of spacetime and renormalization
- Haro, Solodukhin, et al.
(Show Context)
Citation Context ...details. For use in subsequent sections, we give the renormalised action, the holographic one point functions and their Ward identities. Note that holographic renormalization for Lifshitz solutions was also studied previously in [60–65]. In particular, it was shown in [61], using the radial Hamiltonian formalism [66, 67], that Lifshitz models can be holographically renormalized for any z. Since these models are non-relativistic it is natural to work in the vielbein formalism [60](see also [9]) and this is indeed what was done in [61]. In the current context we use instead the metric formalism [68] as this is more natural when studying the theory from the perspective of the AdS critical point. The action under consideration is Sbare = 1 16πGd+1 ∫ dd+1x √ −G ( R+ d(d− 1)− 1 4 FµνF µν − 1 2 M2AµA µ ) + 1 8πGd+1 ∫ ddx √−γK, (2.1) with M2 = d − 1 + O(ǫ2), γ the induced boundary metric and K the trace of the second fundamental form. The associated field equations are DµF µν =M2Aν , (2.2) Rµν = −dGµν + M2 2 AµAν + 1 2 GρσFµρFνσ + 1 4(1− d)F σλFσλGµν . (2.3) Taking the trace of the Einstein equations and plugging back into (2.1) the onshell action becomes Sonshell = 1 16πGd+1 ∫ dd+1x √ −G ( − ... |

343 | Lectures on holographic methods for condensed matter physics,” arXiv:0903.3246 [hep-th - Hartnoll |

300 |
Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry,” Phys
- Son
(Show Context)
Citation Context ...ck brane solution 7 4 Thermodynamics 11 4.1 On-shell action 13 5 Relation to top down solutions 15 5.1 Lifshitz with z ∼ 1 16 5.2 Backreaction of massive vector field 18 5.3 Three-dimensional Lifshitz geometries 19 6 Conclusions 21 1 Introduction In recent years there has been considerable work on the use of holographic models to gain insights into strong coupling physics in condensed matter systems (see [1–5] for reviews). Gauge/gravity duality may be an important tool in understanding strongly interacting non-relativistic scale invariant systems and gravity solutions exhibiting Schrodinger [6, 7] and Lifshitz symmetry [8] have been constructed. While simple models can capture interesting phenomenology, it is important to understand the nature of the corresponding dual non-relativistic theories better, from first principles. In [9] (see also [10, 11]) it was shown that the field theories dual to Schrodinger geometries can be understood as specific deformations of relativistic conformal field theories by operators which are exactly marginal from the perspective of the Schrodinger group, but are irrelevant from the perspective of the conformal symmetry group and moreover break the rela... |

155 | Lectures on Holographic Superfluidity and Superconductivity - Herzog - 2009 |

150 | Holographic duality with a view toward many-body physics,” arXiv:0909.0518
- McGreevy
(Show Context)
Citation Context ...1 4 FµνF µν − 1 2 M2AµA µ ) + 1 8πGd+1 ∫ ddx √−γK, (2.1) with M2 = d − 1 + O(ǫ2), γ the induced boundary metric and K the trace of the second fundamental form. The associated field equations are DµF µν =M2Aν , (2.2) Rµν = −dGµν + M2 2 AµAν + 1 2 GρσFµρFνσ + 1 4(1− d)F σλFσλGµν . (2.3) Taking the trace of the Einstein equations and plugging back into (2.1) the onshell action becomes Sonshell = 1 16πGd+1 ∫ dd+1x √ −G ( − 2d− 1 2(d− 1)FµνF µν ) (2.4) + 1 8πGd+1 ∫ ddx √−γK. It is useful to parametrize the metric and the vector field as ds2 = dr2 + e2rgijdx idxj , gij(x, r; ǫ) = g[0]ij(x, r) + ǫ 2g[2]ij(x, r) + . . . (2.5) Ai(x, r; ǫ) = ǫe rA(0)i(x) + . . . . For the metric, the notation g[a]ij captures the order in ǫ. When one considers the asymptotic behaviour near the conformal boundary, each of these coefficients admits a radial expansion as well and the order in radial expansion will be denoted (as usual) by curved parentheses. For example, g[0]ij(x, r) = g[0](0)ij(x) + e −2rg[0](2)ij(x) + · · · (2.6) is the asymptotic radial expansion of the metric to leading order in ǫ. Below we summarize the most general asymptotic solution given g[0](0)ij and A(0)i as Dirichlet data. – 4 – J H E P... |

117 | Gravity duals for non-relativistic CFTs
- Balasubramanian, McGreevy
- 2009
(Show Context)
Citation Context ...ck brane solution 7 4 Thermodynamics 11 4.1 On-shell action 13 5 Relation to top down solutions 15 5.1 Lifshitz with z ∼ 1 16 5.2 Backreaction of massive vector field 18 5.3 Three-dimensional Lifshitz geometries 19 6 Conclusions 21 1 Introduction In recent years there has been considerable work on the use of holographic models to gain insights into strong coupling physics in condensed matter systems (see [1–5] for reviews). Gauge/gravity duality may be an important tool in understanding strongly interacting non-relativistic scale invariant systems and gravity solutions exhibiting Schrodinger [6, 7] and Lifshitz symmetry [8] have been constructed. While simple models can capture interesting phenomenology, it is important to understand the nature of the corresponding dual non-relativistic theories better, from first principles. In [9] (see also [10, 11]) it was shown that the field theories dual to Schrodinger geometries can be understood as specific deformations of relativistic conformal field theories by operators which are exactly marginal from the perspective of the Schrodinger group, but are irrelevant from the perspective of the conformal symmetry group and moreover break the rela... |

81 |
Thermodynamics of asymptotically locally AdS spacetimes
- Papadimitriou, Skenderis
- 2005
(Show Context)
Citation Context ...uss the thermodynamics of the black brane solution, working in Euclidean signature for convenience. To define the mass we need to take into account the fact that the stress-energy tensor is not conserved by itself, but satisfies a non-trivial Ward identity (2.18). Consider the current Qj = (〈Tij〉 − Ai 〈Jj〉)ξi, (4.1) where ξi is such that ∇0ξ0 = z, ∇aξb = δab, ∇aξ0 = ∇0ξa = 0. Using the Ward identity we get ∇jQj = − 〈 J j 〉 (ξi∇iAj +Ai∇jξi) + 〈Tij〉∇jξi (4.2) = −z 〈 J t 〉 At + z 〈 T tt 〉 + 〈 T ii 〉 , which is precisely the Ward trace identity. Therefore the current Qi is conserved and following [70] we can define the conserved mass as M = ∫ t=const √ g(〈Ttt〉 − At 〈Jt〉). (4.3) – 11 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 Note that expressions for the one-point functions (2.14) and (2.17) remain the same upon analytic continuation to Euclidean signature as explained in [71]. The horizon location at order ǫ2 is shifted to y0 = yh ( 1− 1 dy2h ǫ2A2(0)tch ) . (4.4) The Hawking temperature obtained from the requirement of no conical singularity in Euclidean signature is shifted to T = dyh 4π ( 1 + ǫ2A2(0)t [ (d− 3) d ch y2h + 1 d ∂y∆c(yh) yh + bh ]) (4.5) with bh = ∆b(y = yh) and the entropy defined as... |

80 | Gravity duals of Lifshitz-like fixed points,” Phys
- Kachru, Liu, et al.
(Show Context)
Citation Context ...ynamics 11 4.1 On-shell action 13 5 Relation to top down solutions 15 5.1 Lifshitz with z ∼ 1 16 5.2 Backreaction of massive vector field 18 5.3 Three-dimensional Lifshitz geometries 19 6 Conclusions 21 1 Introduction In recent years there has been considerable work on the use of holographic models to gain insights into strong coupling physics in condensed matter systems (see [1–5] for reviews). Gauge/gravity duality may be an important tool in understanding strongly interacting non-relativistic scale invariant systems and gravity solutions exhibiting Schrodinger [6, 7] and Lifshitz symmetry [8] have been constructed. While simple models can capture interesting phenomenology, it is important to understand the nature of the corresponding dual non-relativistic theories better, from first principles. In [9] (see also [10, 11]) it was shown that the field theories dual to Schrodinger geometries can be understood as specific deformations of relativistic conformal field theories by operators which are exactly marginal from the perspective of the Schrodinger group, but are irrelevant from the perspective of the conformal symmetry group and moreover break the relativistic symmetry. Recentl... |

74 | Aspects of holography for theories with hyperscaling violation, - Dong, Harrison, et al. - 2012 |

72 | Generalized holographic quantum criticality at finite density, - Gouteraux, Kiritsis - 2011 |

70 |
Lifshitz spacetimes from AdS null and cosmological solutions,
- Balasubramanian, Narayan
- 2010
(Show Context)
Citation Context ...∝ T d−1z . (1.1) The thermodynamic quantities are obtained analytically and the analytic solutions could be useful in extracting quasi-normal modes, studying correlation functions etc. While it is a useful bottom up model, the Einstein-Proca model has a disadvantage: string theory embeddings are known only for specific values of the dynamical exponent z, see for example [51], none of which are close to one. There are two main classes of string theory embeddings of Lifshitz solutions known. The first is that of z = 2 Lifshitz which can be obtained from reducing z = 0 Schrodinger over a circle [10, 23, 52]. This system can be embedded in supergravity [53–56] and the detailed holographic dictionary was obtained in [26, 57], reducing the results obtained in [58]. However the reduction circle becomes null at infinity which implies the dual theory should be related to the Discretized Light Cone Quantization (DLCQ) of the deformed CFT corresponding to the z = 0 Schrodinger solution, and thus this approach suffers from the well-known subtleties associated with DLCQ. These z = 2 Lifshitz solutions are not in the same universality class as the solutions discussed in this paper. – 2 – J H E P 1 1 ( 2 0... |

68 |
Quantum effective action from the AdS/CFT correspondence, Phys
- Skenderis, Solodukin
(Show Context)
Citation Context ...)i is left undetermined by the asymptotic analysis and is related to the expectation value of the dual operator. Note that the expansion coefficients depend locally on the zeroth order expectation value of the dual stress energy tensor 〈Tij〉[0] (which is related to g[0](d)ij as in (2.16)). At first sight this might appear problematic since this coefficient is in general non-locally related to g[0](0) which might lead to non-local divergences but as we review below there are in fact no non-local divergences: the counterterm action is local. For the metric the asymptotic expansion is as follows [12, 68, 69]: gij = ηij + ǫ 2rh[2](0)ij + e −dr ( ǫ2rh[2](d)ij + (g[0](d)ij + ǫ 2g[2](d)ij) ) , (2.9) where the metric g[0](0)ij is chosen to be flat and h[2](0)ij = −A(0)iA(0)j + 1 2(d− 1)A(0)kA k (0)ηij . (2.10) g[0](d)ij is traceless and divergenceless, while h[2](d)ij = d 4(d− 1)A(0)kA k (0)g[0](d)ij + 1 d Ak(0)g[0](d)klA l (0)ηij (2.11) − d− 1 d (A(0)ig[0](d)jk +A(0)jg[0](d)ik)A k (0), tr(g[2](d)) = 2 d A(0)iA i (d) − d2 − 2d+ 2 d2(d− 1) A i (0)g[0](d)ijA j (0), (2.12) and the divergence of g[2](d) vanishes. The part of g[2](d)ij which is undetermined by the asymptotic analysis is related to the expe... |

58 | Black Holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent,” arXiv:0905.3183 [hep-th - Bertoldi, Burrington, et al. |

56 | Black holes in asymptotically Lifshitz spacetime - Danielsson, Thorlacius |

54 | Universal hydrodynamics of non-conformal branes - Kanitscheider, Skenderis |

52 | Constructing Lifshitz solutions from AdS, - Cassani, Faedo - 2011 |

42 |
Holography for asymptotically locally Lifshitz spacetimes
- Ross
(Show Context)
Citation Context ...nsion d. 2 Summary of holographic dictionary In this section we briefly review the holographic dictionary between bulk Lifshitz spacetimes with dynamical exponent z = 1+ ǫ2 and the dual Lifshitz invariant field theory. We – 3 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 will follow the discussion in [12] to which we refer the reader for more details. For use in subsequent sections, we give the renormalised action, the holographic one point functions and their Ward identities. Note that holographic renormalization for Lifshitz solutions was also studied previously in [60–65]. In particular, it was shown in [61], using the radial Hamiltonian formalism [66, 67], that Lifshitz models can be holographically renormalized for any z. Since these models are non-relativistic it is natural to work in the vielbein formalism [60](see also [9]) and this is indeed what was done in [61]. In the current context we use instead the metric formalism [68] as this is more natural when studying the theory from the perspective of the AdS critical point. The action under consideration is Sbare = 1 16πGd+1 ∫ dd+1x √ −G ( R+ d(d− 1)− 1 4 FµνF µν − 1 2 M2AµA µ ) + 1 8πGd+1 ∫ ddx √−γK, (2.1) with M2 = d − 1 + O(ǫ2), γ the indu... |

38 | On Lifshitz scaling and hyperscaling violation in string theory, - Narayan - 2012 |

32 | Thermodynamics of black branes in asymptotically Lifshitz spacetimes,” arXiv:0907.4755 [hep-th - Bertoldi, Burrington, et al. |

31 |
Holography for Schrodinger backgrounds,
- Guica, Skenderis, et al.
- 2011
(Show Context)
Citation Context ...tion In recent years there has been considerable work on the use of holographic models to gain insights into strong coupling physics in condensed matter systems (see [1–5] for reviews). Gauge/gravity duality may be an important tool in understanding strongly interacting non-relativistic scale invariant systems and gravity solutions exhibiting Schrodinger [6, 7] and Lifshitz symmetry [8] have been constructed. While simple models can capture interesting phenomenology, it is important to understand the nature of the corresponding dual non-relativistic theories better, from first principles. In [9] (see also [10, 11]) it was shown that the field theories dual to Schrodinger geometries can be understood as specific deformations of relativistic conformal field theories by operators which are exactly marginal from the perspective of the Schrodinger group, but are irrelevant from the perspective of the conformal symmetry group and moreover break the relativistic symmetry. Recently an analogous interpretation for Lifshitz spacetimes with dynamical exponent z close to one was developed in [12]: a specific deformation of a d-dimensional CFT by a dimension d vector operator generically leads ... |

29 | Lifshitz/Schrodinger Dp-branes and dynamical exponents, - Singh - 2012 |

29 | On the IR completion of geometries with hyperscaling violation, - Bhattacharya, Cremonini, et al. - 2013 |

28 | Holography for Einstein-Maxwell-dilaton theories from generalized dimensional reduction, - Gouteraux, Smolic, et al. - 2012 |

25 | Holographic models for theories with hyperscaling violation, - Gath, Hartong, et al. - 2013 |

25 |
The F(4) gauged supergravity in six dimensions
- Romans
- 1986
(Show Context)
Citation Context ...ere we will discuss mostly the Lifshitz solutions in four bulk dimensions (henceforth denoted Li4) which are obtained as solutions of the Romans gauged supergravity in six dimensions since the four-dimensional case is phenomenologically more interesting and moreover corresponding finite temperature solutions were constructed in [50]. An analogous discussion holds for the Lifshitz solutions in three bulk dimensions found in [59] and we will summarise the properties of these solutions at the end of this section. We begin by reviewing the equations of motion for the six-dimensional Romans theory [74]. The bosonic field content of 6D Romans’ supergravity consists of the metric, gAB, a dilaton, φ, an anti-symmetric two-form field, BAB, and a set of gauge vectors, (A (i) A ,AA) for the gauge group SU(2)×U(1). The bosonic part of the action for this theory is S = ∫ d6x √−g6 [ 1 4 R6− 1 2 (∂φ)2 − e − √ 2φ 4 ( H2 + F (i)2 ) −e 2 √ 2φ 12 G2 (5.1) −1 8 εABCDEF BAB ( FCDFEF +mBCDFEF + m2 3 BCDBEF + F (i) CDF (i) EF ) + 1 8 ( g2e √ 2φ + 4gme− √ 2φ −m2e−3 √ 2φ ) ] , where g is the gauge coupling, m is the mass of the two-form field BAB, FAB is a U(1) gauge field strength, F (i) AB is a nonabelian SU... |

24 | Lifshitz-like space-time from intersecting branes in string/M theory, - Dey, Roy - 2012 |

23 |
Holographic renormalization of general dilaton-axion gravity,
- Papadimitriou
- 2011
(Show Context)
Citation Context ...tion functions etc. While it is a useful bottom up model, the Einstein-Proca model has a disadvantage: string theory embeddings are known only for specific values of the dynamical exponent z, see for example [51], none of which are close to one. There are two main classes of string theory embeddings of Lifshitz solutions known. The first is that of z = 2 Lifshitz which can be obtained from reducing z = 0 Schrodinger over a circle [10, 23, 52]. This system can be embedded in supergravity [53–56] and the detailed holographic dictionary was obtained in [26, 57], reducing the results obtained in [58]. However the reduction circle becomes null at infinity which implies the dual theory should be related to the Discretized Light Cone Quantization (DLCQ) of the deformed CFT corresponding to the z = 0 Schrodinger solution, and thus this approach suffers from the well-known subtleties associated with DLCQ. These z = 2 Lifshitz solutions are not in the same universality class as the solutions discussed in this paper. – 2 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 The second class of top down embeddings of Lifshitz solutions consist of uplifts of solutions to Romans gauged supergravity theories [59]. Lifsh... |

22 |
From D3-branes to Lifshitz space-times,
- Chemissany, Hartong
- 2011
(Show Context)
Citation Context ...acting quasi-normal modes, studying correlation functions etc. While it is a useful bottom up model, the Einstein-Proca model has a disadvantage: string theory embeddings are known only for specific values of the dynamical exponent z, see for example [51], none of which are close to one. There are two main classes of string theory embeddings of Lifshitz solutions known. The first is that of z = 2 Lifshitz which can be obtained from reducing z = 0 Schrodinger over a circle [10, 23, 52]. This system can be embedded in supergravity [53–56] and the detailed holographic dictionary was obtained in [26, 57], reducing the results obtained in [58]. However the reduction circle becomes null at infinity which implies the dual theory should be related to the Discretized Light Cone Quantization (DLCQ) of the deformed CFT corresponding to the z = 0 Schrodinger solution, and thus this approach suffers from the well-known subtleties associated with DLCQ. These z = 2 Lifshitz solutions are not in the same universality class as the solutions discussed in this paper. – 2 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 The second class of top down embeddings of Lifshitz solutions consist of uplifts of solutions to Romans g... |

21 | Condensed Matter and AdS/CFT - Sachdev - 2011 |

21 | Holography for chiral scale-invariant models,
- Costa, Taylor
- 2011
(Show Context)
Citation Context ... years there has been considerable work on the use of holographic models to gain insights into strong coupling physics in condensed matter systems (see [1–5] for reviews). Gauge/gravity duality may be an important tool in understanding strongly interacting non-relativistic scale invariant systems and gravity solutions exhibiting Schrodinger [6, 7] and Lifshitz symmetry [8] have been constructed. While simple models can capture interesting phenomenology, it is important to understand the nature of the corresponding dual non-relativistic theories better, from first principles. In [9] (see also [10, 11]) it was shown that the field theories dual to Schrodinger geometries can be understood as specific deformations of relativistic conformal field theories by operators which are exactly marginal from the perspective of the Schrodinger group, but are irrelevant from the perspective of the conformal symmetry group and moreover break the relativistic symmetry. Recently an analogous interpretation for Lifshitz spacetimes with dynamical exponent z close to one was developed in [12]: a specific deformation of a d-dimensional CFT by a dimension d vector operator generically leads to a theory with Li... |

21 |
Quantum critical lines in holographic phases with (un)broken symmetry,
- Gouteraux, Kiritsis
- 2013
(Show Context)
Citation Context ...ersality class. From the bulk perspective, the simplest realization of Lifshitz is the bottom up Einstein-Proca model introduced in [13]. Black hole/brane solutions with Lifshitz asymptotics are needed to study the corresponding dual field theories at non-zero temperature. However, only numerical black hole solutions are available for generic values of z [14–28]. Note that analytic asymptotically Lifshitz black hole solutions are readily available in Einstein-Dilaton-Maxwell (EDM) theories, see the earliest examples in [13, 29], with the interpretation of the running scalar being discussed in [30]. More recently there has been considerable interest in solutions of EDM theories exhibiting hyperscaling violation, see for example [30–42]. In this paper we will focus on pure Lifshitz solutions although it would certainly be interesting to understand whether EDM solutions can admit analogous dual interpretations in terms of deformations of relativistic theories and indeed whether EDM solutions can be related to Lifshitz solutions through generalized dimensional reduction [43–45]. Note that issues and open questions involving the IR behaviour of the Lifshitz theory, see [46] and [47], do not... |

20 | Bianchi attractors: a classification of extremal black brane geometries, - Iizuka - 2012 |

20 | Holographic renormalization for asymptotically Lifshitz spacetimes, - Mann, McNees - 2011 |

17 | Hyperscaling violation from supergravity, - Perlmutter - 2012 |

17 | Lifshitz-like black brane thermodynamics in higher dimensions, - Bertoldi, Burrington, et al. - 2011 |

16 |
Lifshitz solutions in supergravity and string theory,
- Gregory, Parameswaran, et al.
- 2010
(Show Context)
Citation Context ...ned in [58]. However the reduction circle becomes null at infinity which implies the dual theory should be related to the Discretized Light Cone Quantization (DLCQ) of the deformed CFT corresponding to the z = 0 Schrodinger solution, and thus this approach suffers from the well-known subtleties associated with DLCQ. These z = 2 Lifshitz solutions are not in the same universality class as the solutions discussed in this paper. – 2 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 The second class of top down embeddings of Lifshitz solutions consist of uplifts of solutions to Romans gauged supergravity theories [59]. Lifshitz geometries LiD(z) in D = d+1 bulk dimensions with generic dynamical exponent z can be realized in this way. The structure of these solutions is as follows: products of LiD(z) with two-dimensional hyperboloids solve the equations of Romans gauged supergravity theories in (D + 2) dimensions, for specific choices of the masses and couplings in these theories. Since there are Lifshitz solutions with z ∼ 1 in these top down models, it is interesting to explore whether these can also be understood in terms of deformations of conformal field theories. In section 5 we show that these soluti... |

16 | Hamilton-Jacobi renormalization for Lifshitz spacetime, - Baggio, Boer, et al. - 2012 |

16 |
Gauged N = 4 supergravities in five-dimensions and their magnetovac backgrounds,
- Romans
- 1986
(Show Context)
Citation Context ...∼ 1 Lifshitz solutions in string theory which are obtained from deformations of supersymmetric CFTs and which do not suffer from such instabilities. Note that the second branch of Lifshitz solutions found in [59] have dynamical exponents z > 1 and are not connected to the unstable z = 1 critical point; these have been argued to be the stable branch [50, 72]. 5.3 Three-dimensional Lifshitz geometries In this section we briefly summarize the interpretation of the Li3×H2 solutions of Romans N = 4 gauged supergravity in five dimensions found in [59]. The bosonic field content of the Romans theory [75] consists of the metric, gAB, a dilaton, φ, two antisymmetric tensors BαAB, and a set of gauge vectors, (A (i) A ,AA) for the gauge group SU(2)×U(1). The bosonic part of the action for this theory is S = ∫ d5x √−g5 [ 1 4 R5− 1 2 (∂φ)2 − ξ 2 4 ( Bα2 + F (i)2 ) −ξ −4 4 F2 (5.24) −1 4 εABCDE ( 1 g1 ǫαβB α AB∇CBβDE − F (i) ABF (i) CDAE ) + 1 8 g2 ( g2ξ −2 + 2 √ 2g1ξ ) ] , where we have defined ξ = e √ 2φ/ √ 3. Here g1 and g2 are the gauge couplings for U(1) and SU(2) respectively. FAB is a U(1) gauge field strength and F (i)AB is a nonabelian SU(2) gauge field strength. Spacetime indices A,B, . . ... |

15 | Intersecting D-branes and Lifshitz-like space-time, - Dey, Roy - 2012 |

15 | Extremal horizons with reduced symmetry: hyperscaling violation, stripes and a classification for the homogeneous case, - Iizuka - 2013 |

13 | Flows involving Lifshitz solutions,
- Braviner, Gregory, et al.
- 2011
(Show Context)
Citation Context ...(0)tydc ( ∆b(yc) + 1 2(1− d) (4.27) + 1−d d ( y0 yc )d log yc+(d−2) ( y0 yc )d(k(d) 2d2 − (2d 2 + d− 2) 4d2(d− 1) + log y0 2d ))] . (4.28) The remaining contributing counterterm is − 1 32πGd+1 ∫ ddx √ γAiA i= 1 32πGd+1 ∫ ddxǫ2A2(0)t [ ydc− 2(d−1) d yd0 log yc (4.29) − 2yd0 ( 3d2 + 4d− 4 4d2 − d− 1 d ( k(d) d + log y0 ))] . Putting all these terms together gives the free energy Son-shell = βF = −β V yd0 16πGd+1 [ 1 + ǫ2A2(0)t ( k(d) 2d + log y0 2 − 1 4(d− 1) )] , (4.30) where β = 1/T . It is a simple check that F =M − TS. – 14 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 5 Relation to top down solutions In [50, 59, 72] Lifshitz solutions of Romans gauged supergravity theories were constructed and then uplifted to ten dimensional supergravities.1 General dynamical exponents with z ≥ 1 were obtained. Here we will consider the limit of these solutions as z → 1 and interpret them from the perspective of the dual conformal field theory of the AdS z = 1 solution. Recently uplifts of the six-dimensional Romans theory to type IIB were found [73] and thus these solutions may also be viewed as solutions of type IIB. Here we will discuss mostly the Lifshitz solutions in four bulk dimensions (henceforth denoted Li4) wh... |

11 | Lifshitz topological black holes, - Mann - 2009 |

10 | An analytic Lifshitz black hole, - Balasubramanian, McGreevy - 2009 |

9 | Lifshitz black holes in string theory, - Amado, Faedo - 2011 |

8 | Analytic Lifshitz black holes in higher dimensions, - Ayon-Beato, Garbarz, et al. - 2010 |

7 | Correlation functions in holographic RG flows,
- Papadimitriou, Skenderis
- 2004
(Show Context)
Citation Context ...In this section we briefly review the holographic dictionary between bulk Lifshitz spacetimes with dynamical exponent z = 1+ ǫ2 and the dual Lifshitz invariant field theory. We – 3 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 will follow the discussion in [12] to which we refer the reader for more details. For use in subsequent sections, we give the renormalised action, the holographic one point functions and their Ward identities. Note that holographic renormalization for Lifshitz solutions was also studied previously in [60–65]. In particular, it was shown in [61], using the radial Hamiltonian formalism [66, 67], that Lifshitz models can be holographically renormalized for any z. Since these models are non-relativistic it is natural to work in the vielbein formalism [60](see also [9]) and this is indeed what was done in [61]. In the current context we use instead the metric formalism [68] as this is more natural when studying the theory from the perspective of the AdS critical point. The action under consideration is Sbare = 1 16πGd+1 ∫ dd+1x √ −G ( R+ d(d− 1)− 1 4 FµνF µν − 1 2 M2AµA µ ) + 1 8πGd+1 ∫ ddx √−γK, (2.1) with M2 = d − 1 + O(ǫ2), γ the induced boundary metric and K the trace of the second... |

6 | Lifshitz-like systems and AdS null deformations,
- Narayan
- 2011
(Show Context)
Citation Context ...∝ T d−1z . (1.1) The thermodynamic quantities are obtained analytically and the analytic solutions could be useful in extracting quasi-normal modes, studying correlation functions etc. While it is a useful bottom up model, the Einstein-Proca model has a disadvantage: string theory embeddings are known only for specific values of the dynamical exponent z, see for example [51], none of which are close to one. There are two main classes of string theory embeddings of Lifshitz solutions known. The first is that of z = 2 Lifshitz which can be obtained from reducing z = 0 Schrodinger over a circle [10, 23, 52]. This system can be embedded in supergravity [53–56] and the detailed holographic dictionary was obtained in [26, 57], reducing the results obtained in [58]. However the reduction circle becomes null at infinity which implies the dual theory should be related to the Discretized Light Cone Quantization (DLCQ) of the deformed CFT corresponding to the z = 0 Schrodinger solution, and thus this approach suffers from the well-known subtleties associated with DLCQ. These z = 2 Lifshitz solutions are not in the same universality class as the solutions discussed in this paper. – 2 – J H E P 1 1 ( 2 0... |

6 | Conformal Lifshitz gravity from holography, - Griffin, Horava, et al. - 2012 |

4 | Lifshitz as a deformation of anti-de Sitter,
- Korovin, Skenderis, et al.
- 2013
(Show Context)
Citation Context ...nd the nature of the corresponding dual non-relativistic theories better, from first principles. In [9] (see also [10, 11]) it was shown that the field theories dual to Schrodinger geometries can be understood as specific deformations of relativistic conformal field theories by operators which are exactly marginal from the perspective of the Schrodinger group, but are irrelevant from the perspective of the conformal symmetry group and moreover break the relativistic symmetry. Recently an analogous interpretation for Lifshitz spacetimes with dynamical exponent z close to one was developed in [12]: a specific deformation of a d-dimensional CFT by a dimension d vector operator generically leads to a theory with Lifshitz scaling invariance. For both Schrodinger and Lifshitz dualities, this perspective not only elucidates the nature of the non-relativistic theories realised holographically but also demonstrates that new classes of theories with non-relativistic symmetries can be obtained as deformations of relativistic conformal field theories. Since such deformations do not need to be realized – 1 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 holographically, these results are interesting for field t... |

4 | On charged Lifshitz black holes, - Pang - 2010 |

4 | Asymptotically Lifshitz wormholes and black holes for Lovelock gravity in vacuum, - Matulich, Troncoso - 2011 |

4 | Anomalous breaking of anisotropic scaling symmetry in the quantum Lifshitz model, - Baggio, Boer, et al. - 2012 |

3 |
Black holes and black branes in Lifshitz spacetimes,
- Tarrio, Vandoren
- 2011
(Show Context)
Citation Context ...string theory with dynamical exponents close to one indeed lie in this universality class. From the bulk perspective, the simplest realization of Lifshitz is the bottom up Einstein-Proca model introduced in [13]. Black hole/brane solutions with Lifshitz asymptotics are needed to study the corresponding dual field theories at non-zero temperature. However, only numerical black hole solutions are available for generic values of z [14–28]. Note that analytic asymptotically Lifshitz black hole solutions are readily available in Einstein-Dilaton-Maxwell (EDM) theories, see the earliest examples in [13, 29], with the interpretation of the running scalar being discussed in [30]. More recently there has been considerable interest in solutions of EDM theories exhibiting hyperscaling violation, see for example [30–42]. In this paper we will focus on pure Lifshitz solutions although it would certainly be interesting to understand whether EDM solutions can admit analogous dual interpretations in terms of deformations of relativistic theories and indeed whether EDM solutions can be related to Lifshitz solutions through generalized dimensional reduction [43–45]. Note that issues and open questions invol... |

3 |
Wrapped M5-branes, consistent truncations and AdS/CMT,
- Donos, Gauntlett, et al.
- 2010
(Show Context)
Citation Context ...tions expected for Lifshitz invariant theories [48–50]. In particular we show how the Ward identity due to Lifshitz invariance implies the existence of a conserved mass and we show that the entropy scales with temperature as S ∝ T d−1z . (1.1) The thermodynamic quantities are obtained analytically and the analytic solutions could be useful in extracting quasi-normal modes, studying correlation functions etc. While it is a useful bottom up model, the Einstein-Proca model has a disadvantage: string theory embeddings are known only for specific values of the dynamical exponent z, see for example [51], none of which are close to one. There are two main classes of string theory embeddings of Lifshitz solutions known. The first is that of z = 2 Lifshitz which can be obtained from reducing z = 0 Schrodinger over a circle [10, 23, 52]. This system can be embedded in supergravity [53–56] and the detailed holographic dictionary was obtained in [26, 57], reducing the results obtained in [58]. However the reduction circle becomes null at infinity which implies the dual theory should be related to the Discretized Light Cone Quantization (DLCQ) of the deformed CFT corresponding to the z = 0 Schrod... |

3 | Topologically massive gravity and the AdS/CFT correspondence. [arXiv:0906.4926
- Skenderis, Taylor, et al.
(Show Context)
Citation Context ...der the current Qj = (〈Tij〉 − Ai 〈Jj〉)ξi, (4.1) where ξi is such that ∇0ξ0 = z, ∇aξb = δab, ∇aξ0 = ∇0ξa = 0. Using the Ward identity we get ∇jQj = − 〈 J j 〉 (ξi∇iAj +Ai∇jξi) + 〈Tij〉∇jξi (4.2) = −z 〈 J t 〉 At + z 〈 T tt 〉 + 〈 T ii 〉 , which is precisely the Ward trace identity. Therefore the current Qi is conserved and following [70] we can define the conserved mass as M = ∫ t=const √ g(〈Ttt〉 − At 〈Jt〉). (4.3) – 11 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 Note that expressions for the one-point functions (2.14) and (2.17) remain the same upon analytic continuation to Euclidean signature as explained in [71]. The horizon location at order ǫ2 is shifted to y0 = yh ( 1− 1 dy2h ǫ2A2(0)tch ) . (4.4) The Hawking temperature obtained from the requirement of no conical singularity in Euclidean signature is shifted to T = dyh 4π ( 1 + ǫ2A2(0)t [ (d− 3) d ch y2h + 1 d ∂y∆c(yh) yh + bh ]) (4.5) with bh = ∆b(y = yh) and the entropy defined as the area of the horizon becomes S = V yd−1h 4Gd+1 ( 1− ǫ2A2(0)t (d− 1) d ch y2h ) , (4.6) with V being the regulated volume of the horizon. The constant ch is directly related to the position of the horizon, which is the only independent parameter characterizing the th... |

3 |
An alternative IIB embedding of F (4) gauged supergravity,
- Jeong, Kelekci, et al.
- 2013
(Show Context)
Citation Context ... 2d + log y0 2 − 1 4(d− 1) )] , (4.30) where β = 1/T . It is a simple check that F =M − TS. – 14 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 5 Relation to top down solutions In [50, 59, 72] Lifshitz solutions of Romans gauged supergravity theories were constructed and then uplifted to ten dimensional supergravities.1 General dynamical exponents with z ≥ 1 were obtained. Here we will consider the limit of these solutions as z → 1 and interpret them from the perspective of the dual conformal field theory of the AdS z = 1 solution. Recently uplifts of the six-dimensional Romans theory to type IIB were found [73] and thus these solutions may also be viewed as solutions of type IIB. Here we will discuss mostly the Lifshitz solutions in four bulk dimensions (henceforth denoted Li4) which are obtained as solutions of the Romans gauged supergravity in six dimensions since the four-dimensional case is phenomenologically more interesting and moreover corresponding finite temperature solutions were constructed in [50]. An analogous discussion holds for the Lifshitz solutions in three bulk dimensions found in [59] and we will summarise the properties of these solutions at the end of this section. We begin by ... |

2 | Theory of superconductivity, - Horowitz - 2011 |

2 | Lifshitz black hole in three dimensions, - Ayon-Beato, Garbarz, et al. - 2009 |

2 | Lifshitz black holes in Brans-Dicke theory, - Maeda, Giribet - 2011 |

2 | Hyperscaling-violation on probe D-branes, - Ammon, Kaminski, et al. - 2012 |

1 | A Lifshitz black hole in four dimensional R2 gravity, - Cai, Liu, et al. - 2009 |

1 | Charged Lifshitz black holes, - Dehghani, Mann, et al. - 2011 |

1 | Lifshitz singularities,
- Horowitz, Way
- 2012
(Show Context)
Citation Context ...ssed in [30]. More recently there has been considerable interest in solutions of EDM theories exhibiting hyperscaling violation, see for example [30–42]. In this paper we will focus on pure Lifshitz solutions although it would certainly be interesting to understand whether EDM solutions can admit analogous dual interpretations in terms of deformations of relativistic theories and indeed whether EDM solutions can be related to Lifshitz solutions through generalized dimensional reduction [43–45]. Note that issues and open questions involving the IR behaviour of the Lifshitz theory, see [46] and [47], do not play a role here. In the first part of this paper we consider Einstein-Proca models and construct black brane solutions with Lifshitz asymptotics for dynamical exponent z = 1 + ǫ2, with ǫ being a small expansion parameter. Our solutions are constructed analytically, working perturbatively in ǫ. Applying the holographic dictionary developed in [12] we obtain the one-point function of the dual energy-momentum tensor and check the various thermodynamic relations expected for Lifshitz invariant theories [48–50]. In particular we show how the Ward identity due to Lifshitz invariance implie... |

1 | Lifshitz black holes in IIA supergravity,
- Barclay, Gregory, et al.
- 2012
(Show Context)
Citation Context ... interesting to understand this branch of the solutions further. The plan of this paper is as follows. In the next section we summarise the key results from [12]. Assuming that the sources are position independent we extend the previous analysis to generic dimension. In section 3 we develop the perturbation theory in ǫ and obtain the black brane solutions in generic dimensions. In section 4 we discuss the thermodynamics of our solutions, given an argument for and verifying the Lifshitz scaling behaviour. In section 5 we demonstrate that the Lifshitz solutions and the corresponding black holes [50] of the top-down model [59] are in the same universality class as those considered in the present paper, i.e. they can be viewed as describing the ground state and a thermal state, respectively, of a relativistic CFT deformed by a vector of dimension d. 2 Summary of holographic dictionary In this section we briefly review the holographic dictionary between bulk Lifshitz spacetimes with dynamical exponent z = 1+ ǫ2 and the dual Lifshitz invariant field theory. We – 3 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 will follow the discussion in [12] to which we refer the reader for more details. For use in subs... |

1 | A note on supersymmetric type II solutions of Lifshitz type, - Petrini, Zaffaroni - 2012 |

1 |
Holographic renormalization for z
- Chemissany, Geissbuhler, et al.
- 2012
(Show Context)
Citation Context ...acting quasi-normal modes, studying correlation functions etc. While it is a useful bottom up model, the Einstein-Proca model has a disadvantage: string theory embeddings are known only for specific values of the dynamical exponent z, see for example [51], none of which are close to one. There are two main classes of string theory embeddings of Lifshitz solutions known. The first is that of z = 2 Lifshitz which can be obtained from reducing z = 0 Schrodinger over a circle [10, 23, 52]. This system can be embedded in supergravity [53–56] and the detailed holographic dictionary was obtained in [26, 57], reducing the results obtained in [58]. However the reduction circle becomes null at infinity which implies the dual theory should be related to the Discretized Light Cone Quantization (DLCQ) of the deformed CFT corresponding to the z = 0 Schrodinger solution, and thus this approach suffers from the well-known subtleties associated with DLCQ. These z = 2 Lifshitz solutions are not in the same universality class as the solutions discussed in this paper. – 2 – J H E P 1 1 ( 2 0 1 3 ) 1 2 7 The second class of top down embeddings of Lifshitz solutions consist of uplifts of solutions to Romans g... |