### Citations

615 |
Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity
- Brown, Henneaux
- 1986
(Show Context)
Citation Context ... by δγµν = ∇µξν +∇νξµ such that δγrr = δγra = 0. These are given by ξr = rξr(0) , (2.49) ξa = ξa(0) − ∫ dr r γab∂bξ r (0) . (2.50) These are also known as Penrose–Brown–Henneaux (PBH) transformations =-=[50, 53]-=-. Under these PBH transformations the r-component of Aµ transforms as δAr = ξ µ∂µAr +Aµ∂rξ µ = rξr(0)∂rAr + ξ a∂aAr + ξ r (0)Ar − 1 r Aa∂aξ r (0) . (2.51) Using (2.40) together with the fact that in r... |

527 |
A Stress tensor for Anti-de Sitter gravity, Commun.Math.Phys
- Balasubramanian, Kraus
- 1999
(Show Context)
Citation Context ...enormalization For all AlAdS3 space-times of AdS3 gravity without matter fields the following bulk plus Gibbons–Hawking plus counterterm action suffices for the purpose of holographic renormalization =-=[58, 28]-=- S = ∫ d3x √−g (R+ 2) + 2 ∫ d2ξ √ −h ( K − 1 + 1 2 R(h) log r ) , (6.1) where we use radial gauge with the boundary at r = 0 and where R(h) is the Ricci scalar of the metric hab = gab and where h = de... |

453 |
Holographic reconstruction of spacetime and renormalization
- Haro, Solodukhin, et al.
(Show Context)
Citation Context ...there is the additional motivation to learn about holography beyond the familiar AdS/CFT context. In this light since Schrödinger space-times are not asymptotically locally AdS (AlAdS) as defined in =-=[28, 29]-=-, while they have at the same time a well defined relation to AlAdS space-times they provide a nice case for developing geometric tools to perform basic holographic calculations beyond the familiar Ad... |

300 |
Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry,” Phys
- Son
(Show Context)
Citation Context ...ty. Thirdly, they feature in the context of potential applications of holography to strongly coupled systems with Schrödinger symmetries. In this setting Schrödinger space-times were introduced1 in =-=[21, 22]-=- in an attempt to holographically describe strongly coupled systems with Schrödinger symmetries such as observed in ultracold dilute gases of point-like interacting spin-1/2 fermions fine-tuned to in... |

247 |
Conformal invariants, Élie Cartan et les Mathmatiques d’Aujourd’hui, Asterisque
- Fefferman, Graham
- 1985
(Show Context)
Citation Context ... full class of AlSch space-times in 3 dimensions. The success of the supergravity approximation of the AdS/CFT correspondence depends to a large extent on the celebrated Fefferman–Graham (FG) theorem =-=[46]-=- which in AdS/CFT language states that: given certain regularity assumptions the most general solution to the Einstein equations (possibly with matter) with AlAdS boundary conditions can be written in... |

176 |
Comments on string theory backgrounds with non-relativistic conformal symmetry
- Maldacena, Martelli, et al.
(Show Context)
Citation Context ... space-times are generic solutions of supergravity [8] and when the dynamical critical exponent z = 2 they can be obtained from an AdS compactification via a TsT transformation or a Null Melvin twist =-=[9, 10, 11, 12, 13, 14, 15]-=-. See also [16, 17, 18, 19] for ways of obtaining Schrödinger space-times in supergravity. Thirdly, they feature in the context of potential applications of holography to strongly coupled systems wit... |

139 |
Volume and area renormalizations for conformally compact Einstein metrics. The
- Graham
- 1999
(Show Context)
Citation Context ...t near Ω = 0 Aµ∂µΩ = at most of order Ω. (2.40) Since Ω is a defining function for an AlAdS space-time we also have by definition Ω4 ( R(γ)µνρσ + γµργνσ − γµσγνρ ) |Ω=0 = 0 , (2.41) so that (see e.g. =-=[52]-=-) γµνΩ−2∂µΩ∂νΩ|Ω=0 = 1 . (2.42) It follows that gµνΩ−2∂µΩ∂νΩ|Ω=0 = 1 + ( e+µΩ−1∂µΩ )2 |Ω=0 , (2.43) so that the boundary at Ω = 0 is also timelike with respect to the AlSch metric. Using (2.31) and (2... |

117 | Gravity duals for non-relativistic CFTs
- Balasubramanian, McGreevy
- 2009
(Show Context)
Citation Context ...ty. Thirdly, they feature in the context of potential applications of holography to strongly coupled systems with Schrödinger symmetries. In this setting Schrödinger space-times were introduced1 in =-=[21, 22]-=- in an attempt to holographically describe strongly coupled systems with Schrödinger symmetries such as observed in ultracold dilute gases of point-like interacting spin-1/2 fermions fine-tuned to in... |

104 |
Heating up Galilean holography
- Herzog, Rangamani, et al.
(Show Context)
Citation Context ... space-times are generic solutions of supergravity [8] and when the dynamical critical exponent z = 2 they can be obtained from an AdS compactification via a TsT transformation or a Null Melvin twist =-=[9, 10, 11, 12, 13, 14, 15]-=-. See also [16, 17, 18, 19] for ways of obtaining Schrödinger space-times in supergravity. Thirdly, they feature in the context of potential applications of holography to strongly coupled systems wit... |

81 |
Thermodynamics of asymptotically locally AdS spacetimes
- Papadimitriou, Skenderis
- 2005
(Show Context)
Citation Context ...there is the additional motivation to learn about holography beyond the familiar AdS/CFT context. In this light since Schrödinger space-times are not asymptotically locally AdS (AlAdS) as defined in =-=[28, 29]-=-, while they have at the same time a well defined relation to AlAdS space-times they provide a nice case for developing geometric tools to perform basic holographic calculations beyond the familiar Ad... |

78 |
Schrodinger invariance in strongly anisotropic critical systems
- Henkel
- 1994
(Show Context)
Citation Context ...e would have to resort at least to asymptotically Schrödinger space-times that break particle number in the IR2. Nevertheless there are other strongly coupled Schrödinger invariant systems see e.g. =-=[25, 26]-=- for which holography may be of use (see for example [27] in the context of ageing systems). On top of these three reasons there is the additional motivation to learn about holography beyond the famil... |

78 |
1954]: ‘On Unitary Ray Representations of Continuous Groups
- Bargmann
(Show Context)
Citation Context ... at least a well defined field theory for scalar fields on a Schrödinger space-time that form UIRs. In fact for the free theory the momentum m plays the role of the Bargmann superselection parameter =-=[66]-=- that labels the different Hilbert spaces. By this we mean that in the free theory unitary time evolution of states of the form e−imV ψm(T, r) cannot change the value of m. 52 Nevertheless one can rea... |

74 |
Nonrelativistic conformal field theories,” Phys
- Nishida, Son
(Show Context)
Citation Context ...e referred to as dipole CFTs [6]. In the case of a Schrödinger space-time we can expect on symmetry grounds that these theories organize themselves as a non-relativistic 33 CFT. From the analysis of =-=[59]-=- we know that primary operators (as defined in [59]) are always in an eigenstate of particle number and hence have a fixed m-dependence. It would be very useful to understand better from the point of ... |

70 |
Lifshitz spacetimes from AdS null and cosmological solutions,
- Balasubramanian, Narayan
- 2010
(Show Context)
Citation Context ...ext than for example z = 2 AlLif space-time we recall that the latter can be uplifted to a certain class of AlAdS space-times that asymptote to a z = 0 Schrödinger space-time in one dimension higher3=-=[41, 42, 43, 44, 37, 36]-=-. This uplift applies to the full class of z = 2 AlLif space-times. In contrast z = 2 AlSch can be related via TsT transformations to AlAdS space-times of the same dimensionality but this does not app... |

62 |
Holographic stress tensor for non-relativistic theories,” arXiv:0907.1846 [hep-th
- Ross, Saremi
(Show Context)
Citation Context ...he Schrödinger asymptotics provide a mild and on the other hand significant departure from standard AdS/CFT lore, much more so than in the case of asymptotically locally Lifshitz space-times (AlLif) =-=[30, 31, 32, 33, 34, 35, 36]-=-. It therefore provides a great opportunity to learn more about holography for non-AdS space-times. To understand better that asymptotically locally z = 2 Schrödinger space-times form a more signific... |

55 | Introducing cadabra: A symbolic computer algebra system for field theory problems
- Peeters
(Show Context)
Citation Context ...for Gravitational Physics (AEI) in Potsdam for warm hospitality during various stages of this project. 34 Some of the calculations in this paper have been performed using the software package Cadabra =-=[60, 61]-=-. A Locally Schrödinger space-times We review the properties of locally Schrödinger space-times. Some of the material here can also be found in [45]. A.1 Defining a locally Schrödinger space-time I... |

52 |
Constructing Lifshitz solutions from AdS,
- Cassani, Faedo
- 2011
(Show Context)
Citation Context ...ext than for example z = 2 AlLif space-time we recall that the latter can be uplifted to a certain class of AlAdS space-times that asymptote to a z = 0 Schrödinger space-time in one dimension higher3=-=[41, 42, 43, 44, 37, 36]-=-. This uplift applies to the full class of z = 2 AlLif space-times. In contrast z = 2 AlSch can be related via TsT transformations to AlAdS space-times of the same dimensionality but this does not app... |

49 |
A field-theory motivated approach to symbolic computer algebra
- Peeters
(Show Context)
Citation Context ...for Gravitational Physics (AEI) in Potsdam for warm hospitality during various stages of this project. 34 Some of the calculations in this paper have been performed using the software package Cadabra =-=[60, 61]-=-. A Locally Schrödinger space-times We review the properties of locally Schrödinger space-times. Some of the material here can also be found in [45]. A.1 Defining a locally Schrödinger space-time I... |

46 |
Families of IIB duals for nonrelativistic CFTs
- Hartnoll, Yoshida
(Show Context)
Citation Context ...supergravity [8] and when the dynamical critical exponent z = 2 they can be obtained from an AdS compactification via a TsT transformation or a Null Melvin twist [9, 10, 11, 12, 13, 14, 15]. See also =-=[16, 17, 18, 19]-=- for ways of obtaining Schrödinger space-times in supergravity. Thirdly, they feature in the context of potential applications of holography to strongly coupled systems with Schrödinger symmetries. ... |

43 |
Supersymmetric solutions for non-relativistic holography
- Donos, Gauntlett
(Show Context)
Citation Context ...supergravity [8] and when the dynamical critical exponent z = 2 they can be obtained from an AdS compactification via a TsT transformation or a Null Melvin twist [9, 10, 11, 12, 13, 14, 15]. See also =-=[16, 17, 18, 19]-=- for ways of obtaining Schrödinger space-times in supergravity. Thirdly, they feature in the context of potential applications of holography to strongly coupled systems with Schrödinger symmetries. ... |

42 |
Holography for asymptotically locally Lifshitz spacetimes
- Ross
(Show Context)
Citation Context ...he Schrödinger asymptotics provide a mild and on the other hand significant departure from standard AdS/CFT lore, much more so than in the case of asymptotically locally Lifshitz space-times (AlLif) =-=[30, 31, 32, 33, 34, 35, 36]-=-. It therefore provides a great opportunity to learn more about holography for non-AdS space-times. To understand better that asymptotically locally z = 2 Schrödinger space-times form a more signific... |

41 |
Spinor and twistor methods in space-time geometry
- Vol
- 1986
(Show Context)
Citation Context ...s not vanish on ∂M. Furthermore, let there be a function Ω (known as the defining function) that is zero at the boundary ∂M with the property that ∂µΩ is nonzero at the locus Ω = 0. Following Penrose =-=[50]-=- the metric γ ′ µν is a conformal completion of the metric γµν defined onM if the two are related via γ′µν = Ω2γµν . Now let there be a different metric defined on M that is denoted by gµν . We say th... |

40 | Phenomenology of local scale invariance: from conformal invariance to dynamical scaling,” Nucl
- Henkel
(Show Context)
Citation Context ...e would have to resort at least to asymptotically Schrödinger space-times that break particle number in the IR2. Nevertheless there are other strongly coupled Schrödinger invariant systems see e.g. =-=[25, 26]-=- for which holography may be of use (see for example [27] in the context of ageing systems). On top of these three reasons there is the additional motivation to learn about holography beyond the famil... |

39 | On the spectrum of nonrelativistic AdS/CFT
- Barbon, Fuertes
(Show Context)
Citation Context ...led away. Allowing β to be zero we can compare with the AdS case. The Klein–Gordon equation φ = 0 reads β2∂2V φ+ 2r 2∂T∂V φ+ r 2∂2rφ− r∂rφ = 0 . (C.2) The most general solution is given by (see also =-=[21, 22, 64, 5]-=-) φ = a−0 (T ) + a + 0 (T )r 2 + ∫ dωdme−iωT e−imV φω,m(r) , (C.3) where a−0 (T ) and a + 0 (T ) are arbitrary functions of T and where φω,m(r) satisfies φ∗−ω,−m = φω,m , (C.4) as well as r2φ′′ω,m(r)−... |

35 |
Conformal invariance for non-relativistic field theory,” Phys
- Mehen, Stewart, et al.
- 2000
(Show Context)
Citation Context ...phically describe strongly coupled systems with Schrödinger symmetries such as observed in ultracold dilute gases of point-like interacting spin-1/2 fermions fine-tuned to infinite scattering length =-=[23]-=-. This has proven to be difficult because a Schrödinger space-time has particle number as one of its isometries while the groundstate of these ultracold gases are superfluids, meaning that particle n... |

33 | Coset Construction for Duals of Non-relativistic CFTs
- Schafer-Nameki, Yamazaki, et al.
(Show Context)
Citation Context ...) are those for which LKAµ = 0. This follows from the fact that a (d+3)-dimensional space-time that admits at least the Schrödinger group as its isometries is either AdS or a Schrödinger space-time =-=[62]-=-. 16The TsT transformation, as a solution generating technique, also applies when the null Killing direction is non-compact. 17One can also take the plus sign in the definition of gµν , i.e. write gµν... |

25 |
Schrodinger holography with and without hyperscaling violation,”
- Kim
- 2012
(Show Context)
Citation Context ...tions can be extended to uplift hyperscaling violating Lifshitz space-times to one dimension higher where they become (conformal) Schrödinger space-times with z < 1 obeying the null energy condition =-=[37, 38, 39, 40]-=-. 2 via TsT were discussed for the first time in 5-dimensions in [13]. Inspired by [13] we made more precise the notion of a Schrödinger boundary for locally Schrödinger space-times in [45]. The cur... |

25 |
Holographic models for theories with hyperscaling violation,
- Gath, Hartong, et al.
- 2013
(Show Context)
Citation Context ...tions can be extended to uplift hyperscaling violating Lifshitz space-times to one dimension higher where they become (conformal) Schrödinger space-times with z < 1 obeying the null energy condition =-=[37, 38, 39, 40]-=-. 2 via TsT were discussed for the first time in 5-dimensions in [13]. Inspired by [13] we made more precise the notion of a Schrödinger boundary for locally Schrödinger space-times in [45]. The cur... |

22 |
From D3-branes to Lifshitz space-times,
- Chemissany, Hartong
- 2011
(Show Context)
Citation Context ...ext than for example z = 2 AlLif space-time we recall that the latter can be uplifted to a certain class of AlAdS space-times that asymptote to a z = 0 Schrödinger space-time in one dimension higher3=-=[41, 42, 43, 44, 37, 36]-=-. This uplift applies to the full class of z = 2 AlLif space-times. In contrast z = 2 AlSch can be related via TsT transformations to AlAdS space-times of the same dimensionality but this does not app... |

21 | Holography for chiral scale-invariant models,
- Costa, Taylor
- 2011
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21 |
Quantum effective action from the AdS
- Skenderis, Solodukhin
- 2000
(Show Context)
Citation Context ... a null Killing vector in such an AdS coordinate system. 38 Consider locally AdS3 space-times. The complete solution to the equations defining a locally AdS3 space-time (A.20) in FG gauge is given by =-=[63]-=- ds2 = dr2 r2 + γabdx adxb = dr2 r2 + 1 r2 ( γ(0)ab + r 2γ(2)ab + r 4γ(4)ab ) dxadxb , (A.33) where γ(2)ab satisfies R(0)abcd = −γ(0)acγ(2)bd + γ(0)adγ(2)bc + γ(0)bcγ(2)ad − γ(0)bdγ(2)ac , (A.34) 0 = ... |

20 |
Holographic renormalization for asymptotically Lifshitz spacetimes,
- Mann, McNees
- 2011
(Show Context)
Citation Context ...he Schrödinger asymptotics provide a mild and on the other hand significant departure from standard AdS/CFT lore, much more so than in the case of asymptotically locally Lifshitz space-times (AlLif) =-=[30, 31, 32, 33, 34, 35, 36]-=-. It therefore provides a great opportunity to learn more about holography for non-AdS space-times. To understand better that asymptotically locally z = 2 Schrödinger space-times form a more signific... |

17 |
The Particle number
- Balasubramanian, McGreevy
(Show Context)
Citation Context ...he same dimensionality but this does not apply to the full class of z = 2 AlSch space-times. FG expansions for AlSch solutions obtained 1Schrödinger space-times also appear in the work of [20]. 2See =-=[24]-=- for other ways of realizing the Schrödinger symmetry and breaking particle number. 3These observations can be extended to uplift hyperscaling violating Lifshitz space-times to one dimension higher w... |

17 |
Anomalous Breaking of Anisotropic Scaling
- Baggio, Boer, et al.
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17 | Hyperscaling violation from supergravity,
- Perlmutter
- 2012
(Show Context)
Citation Context ...tions can be extended to uplift hyperscaling violating Lifshitz space-times to one dimension higher where they become (conformal) Schrödinger space-times with z < 1 obeying the null energy condition =-=[37, 38, 39, 40]-=-. 2 via TsT were discussed for the first time in 5-dimensions in [13]. Inspired by [13] we made more precise the notion of a Schrödinger boundary for locally Schrödinger space-times in [45]. The cur... |

16 |
A fefferman-graham-like expansion for null warped ads(3
- Guica
(Show Context)
Citation Context ...ling limit of space-like warped AdS3 [2] which in turn plays an important role in the Kerr/CFT correspondence as well as in topologically massive gravity [3]. For related work on null warped AdS3 see =-=[4, 5, 2, 6, 7]-=-. Secondly, Schrödinger space-times are generic solutions of supergravity [8] and when the dynamical critical exponent z = 2 they can be obtained from an AdS compactification via a TsT transformation... |

16 |
Non-relativistic metrics with extremal limits, JHEP 0909
- Imeroni, Sinha
- 2009
(Show Context)
Citation Context ... space-times are generic solutions of supergravity [8] and when the dynamical critical exponent z = 2 they can be obtained from an AdS compactification via a TsT transformation or a Null Melvin twist =-=[9, 10, 11, 12, 13, 14, 15]-=-. See also [16, 17, 18, 19] for ways of obtaining Schrödinger space-times in supergravity. Thirdly, they feature in the context of potential applications of holography to strongly coupled systems wit... |

15 | Holographic Renormalization for z=2 Lifshitz Space-Times from AdS”,
- Chemissany, Geissbuhler, et al.
- 2012
(Show Context)
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14 |
Universality and exactness of Schrodinger geometries
- Kraus, Perlmutter
(Show Context)
Citation Context .../CFT correspondence as well as in topologically massive gravity [3]. For related work on null warped AdS3 see [4, 5, 2, 6, 7]. Secondly, Schrödinger space-times are generic solutions of supergravity =-=[8]-=- and when the dynamical critical exponent z = 2 they can be obtained from an AdS compactification via a TsT transformation or a Null Melvin twist [9, 10, 11, 12, 13, 14, 15]. See also [16, 17, 18, 19]... |

10 |
dipole theories and nonrelativistic CFTs
- El-Showk, Guica, et al.
- 2011
(Show Context)
Citation Context ...ce [1]. In this context 3-dimensional Schrödinger space-times are often referred to as null warped AdS3. The null warped AdS3 space-times can be obtained as a scaling limit of space-like warped AdS3 =-=[2]-=- which in turn plays an important role in the Kerr/CFT correspondence as well as in topologically massive gravity [3]. For related work on null warped AdS3 see [4, 5, 2, 6, 7]. Secondly, Schrödinger ... |

9 |
The geometry of Schródinger Symmetry
- Duval, Hassaine, et al.
(Show Context)
Citation Context ...-times of the same dimensionality but this does not apply to the full class of z = 2 AlSch space-times. FG expansions for AlSch solutions obtained 1Schrödinger space-times also appear in the work of =-=[20]-=-. 2See [24] for other ways of realizing the Schrödinger symmetry and breaking particle number. 3These observations can be extended to uplift hyperscaling violating Lifshitz space-times to one dimensi... |

8 | Correlation functions for Schródinger backgrounds
- Rees
(Show Context)
Citation Context ...onstrained. We will make this statement more precise in section 4. So far in the literature two types of scenarios have been studied: • Linearized perturbations around a fixed Schrödinger background =-=[5, 49]-=-. • Constrained solutions: in all cases that have been considered these turn out to be V - independent solutions. We will not study linearized perturbations but the simpler case of a real massless sca... |

7 | Asymptotically anti-de Sitter space-times: symmetries and conservation laws revisited
- Barnich, Brandt
- 2004
(Show Context)
Citation Context ...fly discuss the effect of imposing stronger boundary conditions than just AlSch boundary conditions. Demanding that Aµ is an asymptotic Killing vector of the AlAdS metric γµν we need (see for example =-=[56]-=-) S (γ) ab = o(r −2) , (4.68) S(γ)ra = o(r −3) , (4.69) S(γ)rr = o(r −2) , (4.70) while Aa = O(1) and Ar = O(r). Computing S (γ) µν for the metric γ we find S (γ) TT = 2 r2 H(0)∂TV(0) + . . . , (4.71)... |

7 |
Universal time-dependent deformations of Schrödinger geometry
- Nakayama
- 2010
(Show Context)
Citation Context ... µν = 0 when we take the scalars constant. In particular they are solutions of the 3-dimensional supergravity action given in [15]. The AµA µ = 0 solutions with V(0) constant have been constructed in =-=[57]-=-. It would be interesting to study the full class of AµA µ = 0 solutions in more detail as they are exact and generic solutions that can possibly describe interesting time dependent phenomena. Related... |

6 |
Asymptotically Schroedinger Space-Times: TsT Transformations and Thermodynamics
- Hartong, Rollier
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3 |
Stringy Schroedinger truncations,” arXiv:1212.6792 [hep-th
- Detournay, Guica
(Show Context)
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2 |
Asymptotically Schródinger Space-Times,” Fortsch.Phys
- Hartong, Rollier
(Show Context)
Citation Context ..., 38, 39, 40]. 2 via TsT were discussed for the first time in 5-dimensions in [13]. Inspired by [13] we made more precise the notion of a Schrödinger boundary for locally Schrödinger space-times in =-=[45]-=-. The current paper extends this work to the full class of AlSch space-times in 3 dimensions. The success of the supergravity approximation of the AdS/CFT correspondence depends to a large extent on t... |