Abstract
In real experiments, one often adds “probes” to a system to examine its response. Or one adds impurities to a system to see how they change the properties of the system. In this chapter, we discuss how to add probes in AdS/CFT. As a typical example, we add “quarks” to gauge theories as probes and see the behavior of quark potentials.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
From \(t_\text {E}= it\), the Lorentzian action \(\mathsf {S}_L\), the Euclidean action \(\mathsf {S}_E\), and the potential \(V\) are related to each other by \(i \mathsf {S}_L = i\int dt (-V) = -\int dt_\text {E}V = - \mathsf {S}_E\).
- 2.
- 3.
The cutoff AdS is a toy model for the confinement, but we discuss an explicit example in Appendix.
- 4.
Note the factor of the dilation \(e^{-\phi }\). The dilaton \(\phi \) and the string coupling constant \(g_s\) are related by \(g_s \simeq e^\phi \), so this factor means that the mass density of the D-brane is proportional to \(1/g_s\) [Eq. (5.58)].
- 5.
The string has the turning point at \(r=r_m\), so our gauge is not well-defined in reality. But this is no problem because it is enough to consider only the half of the string by symmetry. One normally takes the gauge \(\sigma ^0 = t\), \(\sigma ^1 = x\), and \(r=r(x)\) instead of Eq. (8.38). The computation is slightly easier in our gauge.
- 6.
See Ref. [18] for a more appropriate procedure.
References
L. Susskind, E. Witten, The Holographic bound in anti-de Sitter space. arXiv:hep-th/9805114
A.W. Peet, J. Polchinski, UV/IR relations in AdS dynamics. Phys. Rev. D 59, 065011 (1999). arXiv:hep-th/9809022
S.-J. Rey, J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity. Eur. Phys. J. C 22, 379 (2001). arXiv:hep-th/9803001
J.M. Maldacena, Wilson loops in large N field theories. Phys. Rev. Lett. 80, 4859 (1998). arXiv:hep-th/9803002
J.K. Erickson, G.W. Semenoff, K. Zarembo, Wilson loops in N=4 supersymmetric Yang-Mills theory. Nucl. Phys. B 582, 155 (2000). arXiv:hep-th/0003055
N. Drukker, D.J. Gross, An Exact prediction of N=4 SUSYM theory for string theory. J. Math. Phys. 42, 2896 (2001). arXiv:hep-th/0010274
J. Polchinski, M.J. Strassler, Hard scattering and gauge/string duality. Phys. Rev. Lett. 88, 031601 (2002). arXiv:hep-th/0109174
S.-J. Rey, S. Theisen, J.-T. Yee, Wilson-Polyakov loop at finite temperature in large N gauge theory and anti-de Sitter supergravity. Nucl. Phys. B 527, 171 (1998). arXiv:hep-th/9803135
A. Brandhuber, N. Itzhaki, J. Sonnenschein, S. Yankielowicz, Wilson loops in the large N limit at finite temperature. Phys. Lett. B 434, 36 (1998). arXiv:hep-th/9803137
D. Bak, A. Karch, L.G. Yaffe, Debye screening in strongly coupled N=4 supersymmetric Yang-Mills plasma. JHEP 0708, 049 (2007). arXiv:0705.0994 [hep-th]
T. Matsui, H. Satz, \(J/{\Psi }\) suppression by quark-gluon plasma formation. Phys. Lett. B178, 416 (1986)
H. Liu, K. Rajagopal, U.A. Wiedemann, Calculating the jet quenching parameter from AdS/CFT. Phys. Rev. Lett. 97, 182301 (2006). arXiv:hep-ph/0605178
C.P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, L.G. Yaffe, Energy loss of a heavy quark moving through N=4 supersymmetric Yang-Mills plasma. JHEP 0607, 013 (2006). arXiv:hep-th/0605158
J. Casalderrey-Solana, D. Teaney, Heavy quark diffusion in strongly coupled N=4 Yang-Mills. Phys. Rev. D 74, 085012 (2006). arXiv:hep-ph/0605199
S.S. Gubser, Drag force in AdS/CFT. Phys. Rev. D 74, 126005 (2006). arXiv:hep-th/0605182
S.S. Gubser, D.R. Gulotta, S.S. Pufu, F.D. Rocha, Gluon energy loss in the gauge-string duality. JHEP 0810, 052 (2008). arXiv:0803.1470 [hep-th]
P.M. Chesler, K. Jensen, A. Karch, Jets in strongly-coupled N = 4 super Yang-Mills theory. Phys. Rev. D 79, 025021 (2009). arXiv:0804.3110 [hep-th]
N. Drukker, D.J. Gross, H. Ooguri, Wilson loops and minimal surfaces. Phys. Rev. D 60, 125006 (1999). arXiv:hep-th/9904191
G.T. Horowitz, R.C. Myers, The AdS/CFT correspondence and a new positive energy conjecture for general relativity. Phys. Rev. D 59, 026005 (1998). arXiv:hep-th/9808079
Author information
Authors and Affiliations
Corresponding author
Appendix: A Simple Example of the Confining Phase
Appendix: A Simple Example of the Confining Phase
In the text, we discussed the cutoff AdS spacetime as a toy model of the confining phase. Here, as an explicit example, we discuss the \(S^1\)-compactified \({\fancyscript{N}}=4\) SYM and its dual geometry.
AdS soliton The SAdS\(_5\) black hole is given by
We now compactify the \(z\)-direction as \(0 \le z <l\).
However, the compactified SAdS\(_5\) black hole is not the only solution whose asymptotic geometry is \(\mathbb {R}^{1,2} \times S^1\). The “double Wick rotation”
of the black hole gives the metric
which has the same asymptotic structure \(\mathbb {R}^{1,2} \times S^1\). The geometry (8.60) is known as the AdS soliton [19].
As Euclidean geometries, they are the same, but they have different Lorentzian interpretations. The AdS soliton is not a black hole. Rather, because of the factor \(h\) in front of \(dz'^2\), the spacetime ends at \(r=r_0\) just like the Euclidean black hole. From the discussion in the text, this geometry describes a confining phase.
For the SAdS black hole, the imaginary time direction has the periodicity \(\beta = \pi L^2/r_0\) to avoid a conical singularity. Similarly, for the AdS soliton, \(z'\) has the periodicity \(l\) given by
Wilson loop Let us consider the quark potential in this geometry. Take the quark separation as \({\fancyscript{R}}\) in the \(x\)-direction. This corresponds to a Wilson loop on the \(t'-x\) plane. Since the geometry ends at \(r=r_0\), the formula (8.22) gives
which is a confining potential.
In Sect. 8.2, we considered the Wilson loop in the SAdS black hole and discussed the Debye screening. Here, we consider a Wilson loop in the same Euclidean geometry, but the Wilson loop here is different from the one in Sect. 8.2:
-
For the AdS soliton, we consider the Wilson loop on the \(t'\)-\(x\) plane (temporal Wilson loop), but as the black hole, this is a Wilson loop on the \(z\)-\(x\) plane or a spatial Wilson loop.
-
For the black hole, we considered the temporal Wilson loop on the \(t\)-\(x\) plane, but as the AdS soliton, this is a spatial Wilson loop on the \(z'\)-\(x\) plane.
At high temperature \(Tl>1\), the AdS soliton undergoes a first-order phase transition to the SAdS black hole (Sect. 14.2.1).
Rights and permissions
Copyright information
© 2015 Springer Japan
About this chapter
Cite this chapter
Natsuume, M. (2015). AdS/CFT—Adding Probes. In: AdS/CFT Duality User Guide. Lecture Notes in Physics, vol 903. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55441-7_8
Download citation
DOI: https://doi.org/10.1007/978-4-431-55441-7_8
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-55440-0
Online ISBN: 978-4-431-55441-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)