### Citations

445 |
Ordinary Differential Equations
- Arnold
- 1975
(Show Context)
Citation Context ...l Forbenius expansion of the solution is smooth with respect to the parameters of the differential equation provided the equation and the boundary conditions are smooth with respect to the parameters =-=[18]-=-. Thus φi and its radial derivative at the boundary must be smooth with respect to ω. Then from the definition of the Green’s function (2.30), the only locations at which it or its n-th order derivati... |

173 | Anti-de Sitter space and holography,” Adv.Theor.Math.Phys - Witten - 1998 |

152 | Non-extreme black holes of five-dimensional - Behrndt, Cvetič, et al. - 1999 |

139 | Petkou, “Implications of Conformal Invariance in Field Theories for General Dimensions,” Annals Phys - Osborn, C - 1994 |

85 |
Starinets, “Minkowski space correlators
- Son, O
- 2002
(Show Context)
Citation Context ...t the equation in (3.14) is a relation for the Euclidean propagator. The Euclidean propagator can be analytically continued to the retarded Minkowski propagator in the whole of the upper half ω plane =-=[25]-=- using the relation G(iω, q) = −GE(ω, q) (3.15) This leads to the following behaviour for the retarded two point function of interest in the limit ω → i∞ lim ω→i∞ GiT (ω)−GiT=0(ω) = 4π2C k̂ii 〈Ok̂(0)〉... |

69 |
Hydrodynamics with triangle anomalies,
- Son, Surowka
- 2009
(Show Context)
Citation Context ...st and present both at short distance and long distances in the theory. It will be interesting to determine spectral densities sensitive to the U(1)3 anomaly present in N = 4 Yang-Mills hydrodynamics =-=[31]-=- and obtain sum rules which capture this anomaly. Acknowledgments We thank Sachin Jain and R. Loganayagam for useful discussions. The work of J.R.D is partially supported by the Ramanujan fellowship D... |

54 |
Differential regularization and renormalization: A New method of calculation in quantum field theory, Nucl. Phys. B371
- Freedman, Johnson, et al.
- 1992
(Show Context)
Citation Context ...ν 1 s2 . (3.5) Note that the since Âµν(s) = O(1/s4) it is not well defined as a distribution in 4 dimensions. Thus we need to regularize this by introducing a delta function following the methods of =-=[20, 22, 23, 24]-=-. Therefore we consider Aµν(s) = − ( ∂µ∂ν 1 s2 + 4π2Kδµνδ 4(s) ) . (3.6) This expression is identical to Âµν for s 6= 0. We now determine the constant K using Ward identities. As a consequence of inv... |

51 |
Multitrace operators, boundary conditions
- Witten
(Show Context)
Citation Context ...it is easy to read out the expectation values of the operators O1 and O2 at finite chemical potential from the background given in (2.3) and (2.4) following the – 24 – procedure detailed in [26]. See =-=[27, 28]-=- for earlier work. Details of this has been repeated in [12]. Here we write down the results for the expectation values. 〈O1〉 = N 2 8π2 r2+ L4 2√ 6 (k1 + k2 − 2k3), (3.22) 〈O2〉 = N 2 8π2 r2+ L4 2√ 2 (... |

45 |
Boundary Conditions and New Dualities: Vector Fields
- Marolf, Ross
(Show Context)
Citation Context ...ectively. it is easy to read out the expectation values of the operators O1 and O2 at finite chemical potential from the background given in (2.3) and (2.4) following the – 24 – procedure detailed in =-=[26]-=-. See [27, 28] for earlier work. Details of this has been repeated in [12]. Here we write down the results for the expectation values. 〈O1〉 = N 2 8π2 r2+ L4 2√ 6 (k1 + k2 − 2k3), (3.22) 〈O2〉 = N 2 8π2... |

44 |
TASI lectures on branes, black holes and anti-de Sitter space
- Duff
- 2001
(Show Context)
Citation Context ... chemical potentials are given by µi = 4π T0 3 L 1 1 + ki √√√√ ki 4∏ i=1 (1 + ki). – 37 – M5-brane The metric and the gauge field for the R-charged M5-brane with all the charges turned on is given by =-=[32]-=- ds27 = 4(πT0L) 2 9u H1/5 ( − fHdt 2 + dx21 + · · ·+ dx24 + dz2 ) + L2 4fu2 H1/5 du2 , At = 2 3 πT0 √√√√2ki 2∏ i=1 (1 + ki) u2 Hi , Hi = 1 + kiu 2 , T0 = 3r+ 2πL2 , u = r2+ r2 (B.6) Hi = 1 + kiu 2 , H... |

26 |
Double trace’ deformations, boundary conditions and space-time singularities
- Berkooz, Sever, et al.
(Show Context)
Citation Context ...it is easy to read out the expectation values of the operators O1 and O2 at finite chemical potential from the background given in (2.3) and (2.4) following the – 24 – procedure detailed in [26]. See =-=[27, 28]-=- for earlier work. Details of this has been repeated in [12]. Here we write down the results for the expectation values. 〈O1〉 = N 2 8π2 r2+ L4 2√ 6 (k1 + k2 − 2k3), (3.22) 〈O2〉 = N 2 8π2 r2+ L4 2√ 2 (... |

22 |
Spectral sum rules for the quark-gluon plasma,” Phys
- Romatschke, Son
(Show Context)
Citation Context ...y provide useful information of the theory. For example the shear and bulk sum rules in QCD is a consequence of the hydrodynamic behaviour at large distances and asymptotic freedom at short distances =-=[1, 2, 3, 4, 5]-=-. Similarly the Ferrell-Glover sum rule in BCS theory plays an important role in determining the skin depth of superconductors [6, 7]. Recently there has been interest in modeling systems that are stu... |

18 |
Bulk viscosity of QCD matter near the critical temperature
- Kharzeev, Tuchin
(Show Context)
Citation Context ...y provide useful information of the theory. For example the shear and bulk sum rules in QCD is a consequence of the hydrodynamic behaviour at large distances and asymptotic freedom at short distances =-=[1, 2, 3, 4, 5]-=-. Similarly the Ferrell-Glover sum rule in BCS theory plays an important role in determining the skin depth of superconductors [6, 7]. Recently there has been interest in modeling systems that are stu... |

18 |
Universal properties of bulk viscosity near the QCD phase transition,” Phys
- Karsch, Kharzeev, et al.
(Show Context)
Citation Context ...y provide useful information of the theory. For example the shear and bulk sum rules in QCD is a consequence of the hydrodynamic behaviour at large distances and asymptotic freedom at short distances =-=[1, 2, 3, 4, 5]-=-. Similarly the Ferrell-Glover sum rule in BCS theory plays an important role in determining the skin depth of superconductors [6, 7]. Recently there has been interest in modeling systems that are stu... |

14 | Sum Rules from an Extra Dimension
- Gulotta, Herzog, et al.
(Show Context)
Citation Context ...rules for the strongly coupled dual can be cast into a problem of establishing the analytic behaviour of solutions of the differential equations which determine the relevant retarded Green’s function =-=[11]-=-. By studying properties of these differential equations, a proof for various sum rules including the shear sum rule for N = 4 Yang-Mills was provided in [11]. In [12], this proof for the shear sum ru... |

11 |
Universality of the operator product expansions
- Anselmi, Freedman, et al.
- 1997
(Show Context)
Citation Context ...ecise about the structure of the tensor Aµν . Consider the three point function of two currents with a scalar operator of dimension – 21 – 2. Conformal invariance constrains this to the form given by =-=[21]-=- 〈J iµ(x)J jν(y)Ok̂(z)〉 = Cijk̂ 4Iµν(x− y)− 2Iµρ(x− z)Iρν(z − y) (x− y)4(x− z)4 . (3.3) The tensor Iµν is defined in (3.2). The leading term in the short distance expansion when x→ y of this three poi... |

10 |
A shear spectral sum rule in a non-conformal gravity dual,” arXiv:1006.4667 [hep-th
- Springer, Gale, et al.
(Show Context)
Citation Context ...l sum rule was derived and verified numerically for the N = 4 theory at strong coupling. The shear and bulk sum rules in non-conformal theories dual to the Chambilin-Real backgrounds were obtained in =-=[8, 9]-=-. The R-charge sum rule for N = 4 Yang-Mills was derived holographically in [10]. Using the gauge/gravity framework, the derivation of the sum rules for the strongly coupled dual can be cast into a pr... |

9 |
Bulk spectral functions in single and multi-scalar gravity duals,” arXiv:1010.2760 [hep-th
- Springer, Gale, et al.
(Show Context)
Citation Context ...l sum rule was derived and verified numerically for the N = 4 theory at strong coupling. The shear and bulk sum rules in non-conformal theories dual to the Chambilin-Real backgrounds were obtained in =-=[8, 9]-=-. The R-charge sum rule for N = 4 Yang-Mills was derived holographically in [10]. Using the gauge/gravity framework, the derivation of the sum rules for the strongly coupled dual can be cast into a pr... |

8 | Systematic differential renormalization to all orders
- Latorre, Manuel, et al.
- 1994
(Show Context)
Citation Context ...ν 1 s2 . (3.5) Note that the since Âµν(s) = O(1/s4) it is not well defined as a distribution in 4 dimensions. Thus we need to regularize this by introducing a delta function following the methods of =-=[20, 22, 23, 24]-=-. Therefore we consider Aµν(s) = − ( ∂µ∂ν 1 s2 + 4π2Kδµνδ 4(s) ) . (3.6) This expression is identical to Âµν for s 6= 0. We now determine the constant K using Ward identities. As a consequence of inv... |

7 |
Holographic electrical and thermal conductivity in strongly coupled gauge theory with multiple chemical potentials
- Jain
- 2010
(Show Context)
Citation Context ... r4f 2 φi − 4r 6 + r8 3∏ j=1 (1 + kj) √ ki ∑ l (√ kl H2i f φl ) = 0. (2.47) Here we have rewritten mi and F in terms of their original definitions. For ω = 0 the above equations can be exactly solved =-=[17]-=-. The solution which is finite at r → r+ is given by φi = τijφ j ∞, τij ≡ δij − 1 2Hi √ kikju (2.48) – 11 – where φi∞ are the values of φ i at the boundary. We will now determine the behaviour of the ... |

7 |
Conformal symmetry and differential regularization of the three gluon vertex,” Annals Phys
- Freedman, Grignani, et al.
- 1992
(Show Context)
Citation Context ...ν 1 s2 . (3.5) Note that the since Âµν(s) = O(1/s4) it is not well defined as a distribution in 4 dimensions. Thus we need to regularize this by introducing a delta function following the methods of =-=[20, 22, 23, 24]-=-. Therefore we consider Aµν(s) = − ( ∂µ∂ν 1 s2 + 4π2Kδµνδ 4(s) ) . (3.6) This expression is identical to Âµν for s 6= 0. We now determine the constant K using Ward identities. As a consequence of inv... |

6 |
Lattice Gauge Theory Sum Rule for the Shear
- Meyer
(Show Context)
Citation Context |

6 |
R-charge thermodynamical spectral sum rule
- Baier
(Show Context)
Citation Context ...pling. The shear and bulk sum rules in non-conformal theories dual to the Chambilin-Real backgrounds were obtained in [8, 9]. The R-charge sum rule for N = 4 Yang-Mills was derived holographically in =-=[10]-=-. Using the gauge/gravity framework, the derivation of the sum rules for the strongly coupled dual can be cast into a problem of establishing the analytic behaviour of solutions of the differential eq... |

5 |
Conductivity of superconducting films: A sum rule
- Ferrell, Glover
- 1958
(Show Context)
Citation Context ...large distances and asymptotic freedom at short distances [1, 2, 3, 4, 5]. Similarly the Ferrell-Glover sum rule in BCS theory plays an important role in determining the skin depth of superconductors =-=[6, 7]-=-. Recently there has been interest in modeling systems that are studied in condensed matter physics as well as QCD using the gauge gravity duality. Clearly sum rules derived using the AdS/CFT framewor... |

4 |
The Bulk Channel
- Meyer
(Show Context)
Citation Context |

3 |
Determination of the superconducting skin depth from the energy gap and sum
- Tinkham, Ferrell
- 1959
(Show Context)
Citation Context ...large distances and asymptotic freedom at short distances [1, 2, 3, 4, 5]. Similarly the Ferrell-Glover sum rule in BCS theory plays an important role in determining the skin depth of superconductors =-=[6, 7]-=-. Recently there has been interest in modeling systems that are studied in condensed matter physics as well as QCD using the gauge gravity duality. Clearly sum rules derived using the AdS/CFT framewor... |

1 |
Shear sum rules at finite chemical potential
- David, Jain, et al.
(Show Context)
Citation Context ...levant retarded Green’s function [11]. By studying properties of these differential equations, a proof for various sum rules including the shear sum rule for N = 4 Yang-Mills was provided in [11]. In =-=[12]-=-, this proof for the shear sum rules was developed and extended to systems with chemical potential. It was noticed there that the shear sum rules are modified due to the short distance properties of t... |