The open string membrane paradigm with external electromagnetic fields

Keun-Young Kim 0 3 Jonathan P. Shock 0 1 Javier Tarro 0 2 Open Access 0 Kavli Institute for Theoretical Physics China , CAS, Beijing 100190, China 1 Max-Planck-Institut fur Physik (Werner-Heisenberg-Institut) , Forhringer Ring 6, 80805 Munchen, Germany 2 Institute for Theoretical Physics, Universiteit Utrecht , 3584 CE, Utrecht, The Netherlands 3 School of Physics and Astronomy, University of Southampton , Southampton, SO17 1BJ, U.K We study the effective geometry felt by the fluctuations of open strings living on the worldvolume of probe D-branes in the presence of background electromagnetic fields. This is captured by an effective action consisting of a Maxwell term and a topological term, with the role of the metric played by the open string metric. Studying generalized Eddington-Finkelstein coordinates for stationary but non-static manifolds, we consider an open string membrane paradigm to obtain a generic formula for the DC transport coefficients, including the effect of external electromagnetic fields present on the worldvolume of the probe branes. We show that the previously studied singular shell, present when a critical electric field strength is turned on, behaves as a horizon for the open string degrees of freedom. The results of this analysis can be used to define a membrane paradigm for a very general class of spacetimes with non-diagonal metrics. ArXiv ePrint: 1103.5627 1 Introduction Metallic AdS/CFT 2.1 Gauge theories in the presence of background electric fields 2.2 D-branes in the presence of background electric fields 2.3 The induced current and the singular shell 2.4 Thermodynamics of the probe-brane system 3 The open string metric and fluctuations in external fields 3.1 The effective action for fluctuations of gauge fields on probe branes 3.2 The membrane paradigm in the presence of background fields (I) (A generalization to non-diagonal spatial metrics) 3.3 Conductivity with an external magnetic field at finite baryon density 4 The singular shell as a horizon and the electric membrane paradigm 4.1 The open string black hole temperature at zero baryon density 4.2 Finite baryon density and the drift of the electric membrane 4.3 The open string black hole temperature at finite baryon density 4.4 A new electric membrane paradigm 4.5 The membrane paradigm in the presence of background fields (II) (A generalization to non-static, stationary metrics) 4.6 Microscopic excitations and the electric membrane 5 Conclusions A Coupling to scalar 1 Introduction The most powerful results to come from the AdS/CFT correspondence, and thus arguably from string theory, are those of a universal nature. Although we are some way from finding the holographic dual of QCD, or indeed of any real-world condensed matter system, we have found that there are certain quantities which are invariant in a wide class of theories. It is clear that such universal quantities should not depend on the microscopic nature of the theory at hand and thus those which are of a low-energy nature or at critical points in the phase diagram are the most striking. Of these, one of the most celebrated is the ratio of the shear viscosity to entropy density of large N gauge theories [1]. The ratio /s = 1/4 was known well before the advent of the AdS/CFT correspondence [2], and is a famed result of the black hole membrane paradigm which shall be discussed in the following section. The link to holographic gauge theories was shown in [1] by considering fluctuations on top of a supergravity background described by a Maxwell action. Using the prescription given in [3], it was possible to obtain the shear viscosity from the hydrodynamic expansion of the retarded correlator. The connection between the membrane paradigm result and holographic transport coefficients was put on a solid footing in [4], where it was shown that at the level of linear response theory, the holographic properties of a strongly coupled thermal field theory are determined, in the low frequency and momentum limit, by the horizon geometry of the gravitational dual. With this understanding, the universality of the ratio /s has been restricted to gravitational theories described by isotropic Einstein gravity (plus a cosmological constant) [5, 6]. However, the results presented in [4] are not always applicable. The assumptions considered in that work fail to capture the physics of probe branes with non-trivial field strength components on their worldvolumes. Non-zero field strengths are frequently needed to capture the phenomenology of the systems modelled by the AdS/CFT correspondence. For example, it is essential to turn on the temporal component of an abelian gauge field to describe, via the holographic dictionary [7, 8], a chemical potential. It is also interesting to include the effects of external electromagnetic fields in the models describing both the Quark-Gluon Plasma and lower-dimensional condensed matter systems. In [9] some of us studied the generalization of the formula for the DC conductivity provided in [4], applied to probe Dp-branes at finite baryon density. The main difference between both results is that in [9] the role of the metric is played by the non-symmetric quantity mn = gmn + 2Fmn, where gmn is the pullback of the 10-dimensional metric onto the worldvolume of the probe branes, and Fmn is the field strength associated to the U(1) gauge field living in the probe branes. The formula for the DC conductivity given in [9] recovers previous results from the vanishing electric field limit of the result in [10], where the DC conductivity is found by demanding reality of the action, and hence no fluctuations are involved. We will refer to this last calculation as the macroscopic one, whereas that in which fluctuations are studied will be referred to as the microscopic result. The microscopic calculation in the case of an external electromagnetic field was studied in [11]. It was shown that with the imposition of the appropriate boundary conditions at a special position on the probe-brane worldvolume (the singular shell) was enough to recover the macroscopic results for the conductivity via a Kubo relation. The Kubo relation is understood to give the conductivity in linear response theory. In the case of finite background fields, the response to an infinitesimal electric field must be interpreted as an infinitesimal addition to a finite background value and thus the microscopic and macroscopic calculations agree, even when one is beyond the linear response regime. In the present work we show that the microscopic determination of the DC transport coefficients performed in [4] can be extended to a new form of membrane, associated with the interactions of open string states in an asymptotically AdS background. Through the use of the open-string metric it is possible to show that the degrees of freedom on a probe Dq-brane feel an induced horizon due to the introduction of certain background fields (in p (...truncated)