Effective Temperature in Steady-state Dynamics from Holography

We argue that, within the realm of gauge-gravity duality, for a large class of systems in a steady-state there exists an effective thermodynamic description. This description comes equipped with an effective temperature and a free energy, but no well-defined notion of entropy. Such systems are described by probe degrees of freedom propagating in a much larger background, e.g. $N_f$ number of ${\cal N} =2$ hypermultiplets in ${\cal N}=4$ $SU(N_c)$ super Yang-Mills theory, in the limit $N_f \ll N_c$. The steady-state is induced by exciting an external electric field that couples to the hypermultiplets and drives a constant current. With various stringy examples, we demonstrate that an open string equivalence principle determines a unique effective temperature for all fluctuations in the probe-sector. We further discuss various properties of the corresponding open string metric that determines the effective geometry which the probe degrees of freedom are coupled to. We also comment on the non-Abelian generalization, where the effective temperature depends on the corresponding sector of the fluctuation modes.


Introduction
Quantum Field Theory (QFT) provides us with an extremely rich framework in which we can formulate and analyze a microscopic description of various degrees of freedom in Nature, their interactions and emerging phenomenology. The standard treatment, such as through summing up Feynman diagrams etc. however, is completely perturbative and relies the existence of a suitable (dimensionless) small coupling in which the putative perturbation can be carried out.
Unfortunately, the existence of such a small coupling is far from guaranteed in physical systems. In fact, recent experimental endeavours have produced numerous examples of stronglycoupled systems across a wide range of energy-scales, such as the quark-gluon plasma (QGP) at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) at TeV-scale (see e.g. [1]), or the cold atoms at unitarity at eV-scale (see e.g. [2]). A plethora of strongcoupling physics data currently confronts us for a theoretical handle and understanding of the same from a first principle calculation.
Notwithstanding the non-perturbative formulations in QFT, the likes of Schwinger-Dyson equation or Ward-Takahashi identities that result from an underlying symmetry of the path integral, the so-called AdS/CFT correspondence [3] has emerged to be an excellent tool for qualitative understanding of strong-coupling physics. 1 The core of this correspondence is rooted in a more general gauge-string duality, for which a vast number of large N c , SU (N c )-type gauge theories can be explored.
Moreover, gauge-string duality turns out to be very crucial to analyze time-dependent (hence out-of-equilibrium) physics at strong coupling. Typically, we only have a handful of examples where such questions can be completely addressed within a QFT-framework, see e.g. recent works in [6]. Moreover, many of these efforts rely heavily on conformal field theory in (1 + 1)dimension, where an infinite dimensional conformal group strongly facilitates the analysis. Thus the governing principles and universal features, assuming they exist, in a time-dependent system remains elusive.
Time-dependent aspects in physics typically involve the physics of thermalization or a quench process, and there is already a vast literature on analysis from a holographic perspective. However, there is an intermediate state of a system, which is stationary but not static. This is normally known as the non-equilibrium steady-state (NESS). Such a state is described by a time-varying microscopic description, that yields a time-independent macroscopic observable.
We know that in equilibrium thermodynamics provides us with a remarkably successful macroscopic description of a system, in terms of a handful of extensive and intensive variables. It is therefore a natural question whether for a steady-state system an "effective" or "analogous" thermodynamic description exists. It is useful to note that an "effective" thermal description of an inherently non-equilibrium system has already been observed in [7]- [9], in systems at quantum criticality [10] or aging glass systems [11]. More recently, far from equilibrium physics in a strongly coupled quantum critical system has been addressed in [12].
In this article, which is a sequel to [13], we will use gauge-string duality to demonstrate that for a very large class of systems in a steady-state, an effective thermodynamic description holds. This effective thermodynamic description comes equipped with an effective temperature, i.e. the intensive variable, but may not come with an "effective entropy", i.e. the canonically conjugate extensive variable.
The prototype of our model is as follows: We consider a bath of a large number of degrees of freedom, and for the time being we will restrict ourselves to the adjoint sector of an SU (N c )gauge theory in the large N c limit. In this heat bath, we want to introduce a "probe" sector: typically this would be a fundamental matter sector with N f number of flavours. We will work in the limit N c N f . In the dual gravitational description this is equivalent to studying the embeddings of a probe Dq-brane in the background of a large number of Dp-branes. The steadystate can be induced in the probe sector by exciting gauge fields on the probe worldvolume; this in turn corresponds to applying a constant electric field and inducing a steady current. This is schematically shown in Fig. 1.  Thus we have a bath of O(N 2 c ) degrees of freedom kept at a temperature T and we are introducing an O(N f ) probe degrees of freedom which is in thermal contact with the bath. Now we excite a constant electric field in one of the gauge theory directions, which should drive a current through Schwinger pair production even at the absence of a charge density. Clearly, there will be a reverse energy-flux flowing from the probe sector to the heat bath that should eventually raise the temperature of the bath. However, in the limit N f N c , this energy transfer is N f /N c suppressed.
Note that, the Dp-brane background are solutions of supergravity which essentially arises from the closed string sector. The dynamics of the probe Dq-branes are described by the Dirac-Born-Infeld action which arises from the open string sector. We will argue, through various "topdown" stringy constructions, that an open string equivalence principle determines the effective thermodynamics.
This article is divided in the following parts: In section 2, we begin reviewing the dynamics of a probe D7-brane in the background of N c D3-branes. First we describe the classical physics and then move on to discuss the fluctuations on the probe. In this section we carefully derive all possible fluctuations of the D7-brane and argue that the fluctuations on the brane obey an open string equivalence principle and sense an "effective" temperature. In section 3, we sketch out how to construct more examples following a similar approach. In section 4, we abstract away from the specific stringy construction and discuss general properties of the open string metric. In the next section we demonstrate that the same structure remains true for Schrödinger-symmetric spacetime. In the following section we argue that a richer, but qualitatively similar physics emerges if we consider the non-Abelian generalization of the DBI-action. Finally we conclude in section 7. Some technical details have been relegated to two appendices.

A model calculation: D3-D7 brane construction
Let us begin by discussing the prototypical model for AdS/CFT, which is obtained by considering the near-horizon limit of a stack of N c D3-branes sitting at the tip of a singularity. Let us consider the singularity to be the tip of a cone whose base is a Sasaki-Einstein 5-manifold, henceforth denoted by SE 5 . When the SE 5 ≡ S 5 , the 10-dimensional geometry takes the form AdS 5 -Schwarzschild×S 5 and the dual field theory is the N = 4 super Yang-Mills (SYM) theory with an SU (N c ) gauge group.
The corresponding gravity background is given by 2 where t and x ≡ {x 1 , x 2 , x 3 } are the field theory space-time directions, r ∈ [b, ∞) is the AdSradial direction, R is the curvature of AdS and dΩ 2 5 is the metric on a unit 5-sphere. The only matter field is a self-dual five-form F (5) , for which the four-form potential is denoted by C (4) and C (4) , whereas denotes the Hodge dual operator. The unit sphere metric can be written as The parameter b sets the radius of the black hole horizon and correspondingly the black hole temperature Evidently, the zero temperature limit is obtained by setting b = 0. The radius of AdS sets the 't Hooft coupling for the dual field theory via R 4 = α 2 g 2 YM N c , where α is the string tension and g YM is the gauge theory coupling.
The matter content of N = 4 SYM is: the gauge field A µ , four adjoint fermions λ and three complex scalars Φ a (a = 1, 2, 3). This theory has an SU (4) ∼ SO(6) R-symmetry, which corresponds to the rotational symmetry of the S 5 in the dual gravitational description. To introduce flavours, we need to introduce open string degrees of freedom which can be done by introducing probe D-branes of various dimensions along the lines of [15]. In particular, we can add N f probe D7-branes -with the constraint N f N c to avoid any backreaction -along the Poincare directions {t, x} and wrapping the S 3 ⊂ S 5 . The D7-brane is a codimension 2 object and therefore the profile can be specified by {θ(r), φ(r)}. Using the isometry along the φ-direction, we can set φ(r) = 0 without any loss of generality and the probe brane profile will be entirely specified by θ(r).
Before proceeding further, let us offer some comments on the dual gauge theory side 3 : adding D7-branes amounts to adding an N = 2 hypermultiplets in the background of N = 4 SYM. The hypermultiplets consists of two Weyl fermions, denoted by ψ andψ and two complex scalars, denoted by q andq. Now, {ψ, q} transforms under the fundamental representation of SU(N c ) and {ψ,q} transforms under the anti-fundamental. The operators corresponding to the D7-brane profile functions θ and φ can be obtained to be where Φ 1 is a complex scalar field in the N = 4 supermultiplet and m q is the mass of the fundamental quark. The worldvolume theory is described by a N = 4 SYM coupled to N f N = 2 hypermultiplets. In the N = 1 language, the vector multiplet of N = 4 decomposes into the N = 1 vector multiplet, denoted by W α , and 3 chiral superfields, denoted by Φ I , with I = 1, 2, 3. The N = 2 matter sector can be written in terms of the N = 1 chiral multiplets denoted by Q r ,Q r , where r = 1, . . . , N f . The complete Lagrangian can be written in the N = 1 notation as More details on the degrees of freedom are summarized in table 1, where the SO(4) ≡ SU (2) Φ × SU (2) R symmetry corresponds to the symmetry of the S 3 wrapped by the D7-brane. The transverse SO(2) symmetry can be explicitly broken by separating the D7 from the D3-brane stack, which amounts to setting m q = 0. The action for the probe D7-brane is given by the Dirac-Born-Infeld (DBI) Lagrangian with an Wess-Zumino term 4 where P [G ab + B ab ] denotes the pull-back of the NS-NS sector fields: G denotes the closed-string background metric and B denotes the NS-NS field; f ab is the worldvolume U (1) gauge field on the probe. The four-form potentials C (4) andC (4) yield the self-dual five-form. The collective coordinates {ξ}-denote D7-brane worldvolume coordinates, T D7 denotes the D7-brane tension and is given by T D7 = µ 7 /g s , where g s is the string coupling constant. Finally N f denotes the number of probe branes and we need to impose N f N c to suppress backreaction 5 . To introduce a steady-state in the probe flavour sector, we will excite the gauge field 6 a 1 (r) = −Et + a(r) , (2.12) where the subscript 1 denotes the spatial direction x 1 . Thus our ansatz (2.12) breaks the SO(3) rotational invariance down to SO (2). The field E corresponds to the field strength: f 1t = E, which represents a constant electric field in the boundary theory. Note that, only the flavour degrees of freedom couple to this electric field. The function a(r), which is hitherto undetermined, plays a more subtle role. The resulting effective action for the probe D7-brane is given by where we have set (2πα ) = 1 for convenience and ≡ d/dr. Let us further define the following dimensionless tilde-variables: 14) The action in (2.13) contains two dynamical functions: θ(r) and a(r). The equation of motion for a(r) immediately results in a first integral of motioñ Here we are using the Lorentzian signature. 5 For all practical purposes though, we will set N f = 1. This simplifies our analysis, otherwise we may need to worry about the non-Abelian generalization of the DBI-action. We will comment on this later. 6 We use a convention in which all gauge fields on the worldvolume are denoted by small letters, and we absorb the factor of (2πα ) in it.
whereJ is an undetermined constant. The asymptotic behaviour of the function a(r) can be obtained from (2.15) lim r→∞ã (r) = 0 +J 2r 2 + . . . , (2.16) where the first term, which is the non-normalizable term in the near-boundary expansion of the functionã, is set to zero by hand. Using the derivation in [17], the constantJ is related to the vacuum expectation value of the flavour current which is induced by the applied electric field where V R 3 represents the volume of the R 3 . 7 Thus the physical meaning of the function a(r) becomes clear: it encodes the response current in the flavour sector, which is sourced by the applied constant electric field. We will momentarily come back to how this current is determined in terms of the applied electric field and the temperature of the background. Before discussing further, let us offer a few comments over the physical meaning of the profile function θ(r). From the action (2.13), it is clear that the equation of motion for θ is rather involved. Let us instead focus on the larger behaviour, where the equation of motion becomes d dr which yields θ(r) =m r +c r 3 + . . . , (2.19) wherem andc are two constants, which are subsequently identified as the bare quark mass and the quark condensate via [18,19,20] In the absence of any external electric field, this flavour sector undergoes a topology changing first order phase transition at finite temperature [19,20]. The phase transition is driven by tuning the dimensionless ratio (m q /T ), where T is the background temperature. The corresponding two phases -which in the dual theory describes a phase of bound mesons and a plasma one -are characterized by qualitatively different embeddings of the D7-brane: the so called Minkowski embedding and the black hole embedding, respectively. For more details on this, the reader is referred to [19,20]. 7 Here we have used that 2πα = 1 and N f = 1. Reinstating these factors, the relation becomes: Now we will discuss how the boundary current is determined dynamically. Let us consider eliminating the functionã in favour of the constantJ. This can be achieved by considering the following Legendre transformation Now, on physical grounds, we must requireĨ D7 remains real. Thus, the two factors inside the square root must change sign at the same location, denoted byr * and this results in the following algebraic conditions and subsequentlyJ (2.24) Thus we have finally obtained an Ohm's law with a conductivity which is a non-linear function of the electric field itself. The location r * , which is above the location of the background event horizon, plays a crucial role in the dynamics. We will henceforth call this a pseudo horizon.
A few comments are in order. Note that the current vanishes whenr 4 * = 1/4 or θ = π/2. The first case corresponds to having a vanishing electric field, where the pseudo horizon and the event horizon coincide. The second scenario corresponds to the case when the S 3 , wrapped by the D7-brane, shrinks to a zero size as observed from (2.4). The shrinking S 3 corresponds to the Minkowski embeddings and thus corresponds to the bound meson states in the dual field theory. Since there is no charge carrier in this phase, the current is identically zero.
For simplicity, and this will suffice for our purposes, let us consider the case when the black hole in the bulk geometry disappears, i.e. b = 0. The action in this case is given by From (2.25), the pragmatic role of the function a(r) becomes clear: If we set a(r) = const, then the action vanishes at r 2 * = R 2 E, which is the location of the pseudo horizon. However, there is nothing special about this point in the bulk and hence the probe branes cannot end there. By exciting a non-zero a , we are allowed to go beyond the pseudo horizon.
The boundary conditions for the profile function θ can be fixed following [21]. The pulled-back D7-brane metric contains the following terms dr 2 r 2 1 + r 2 θ 2 + cos 2 θdΩ 2 3 ⊂ ds 2 D7 , (2.26) which leads to an excess angle in the {cos θ, Ω 3 }-plane. The coefficient of this excess angle is proportional to r −2 θ −2 evaluated at r = r min , where r min denotes the minimum radius where the Minkowski embedding reaches. Evidently, r min ≥ r * . To kill off this excess angle, we need to impose Finally the boundary conditions for the pseudo horizon embeddings can be fixed from the equation of motion where the hatted variable is defined as r = R √ Er. Clearly, the qualitative difference between the Minkowski embedding and the pseudo horizon embedding is how the brane ends, i.e. without or with a conical singularity.
We remind the reader that as far as the qualitative behaviour of the probe embedding is concerned, there are characteristic similarities to the physics in a thermal background which was analyzed in [19,20] including the first order phase transition. The pseudo horizon plays a role qualitatively similar to a background event horizon in this respect. We will further argue that this feature is rather robust. In the next section we will study the fluctuations in the bosonic sector.

Fluctuations: bosonic sector
Let us study the fluctuation modes on the probe brane. These fluctuation modes will correspond to the meson operators in the dual N = 2 field theory. For example, the scalar meson operators are given by 8 with conformal dimensions ∆ = 3 + . Here σ A = (σ 1 , σ 2 ) denotes the Pauli matrices doublet, and X A V , q m , ψ i are defined in table 1. X denotes the symmetric traceless operator X {i 1 ...i } of adjoint scalars X i , for i = 4, 5, 6, 7. The mesonic operators, O A scalar transform in the 2 , 2 of the SO(4) and have +2 charge under the SO(2) ≡ U (1) R .
To obtain the fluctuation action, we need to expand the following probe action up to quadratic order 9 where E = P [G + B] + f , and with r 2 = 1 2 The fluctuations can be represented by ab are the classical fields and δθ(ξ a ), δφ(ξ a ), δf ab (ξ a ) denote the scalar and vector fluctuations. The fluctuation action is then obtained to be and a coupled term From here on, we will use δφ as the fluctuation corresponding to φ on the worldvolume, where φ is a generic field defined on the probe.
where we have defined where δ ar , δ br etc denote the Kronecker delta; G θθ = R 2 and G φφ = R 2 sin 2 θ denotes the corresponding metric components in the 10-dimensional geometry in (2.1). The metric G mn represents the one on the We will now focus on simple cases when the modes decouple. A remarkable simplification occurs if we focus on the θ 0 (r) = 0 embedding, which corresponds to classically introducing massless flavours. Furthermore, if we concentrate on fluctuations oscillating only along the noncompact directions, then from (2.40) we conclude that g (1) ab = 0 and thus S coupled = 0 identically. Furthermore, by noting that the only non-vanishing components of A are A tx = −A xt and A xu = −A ux , it can be checked that With these simplifications, the effective scalar and vector fluctuation actions take very simple forms The fluctuation for δφ is multiplied by an overall factor of sin 2 θ 0 , which vanishes for the massless case; hence this mode is absent from the above effective actions. 10 The effective action describes a free scalar and a massless vector mode propagating in a background geometry which is governed by S, the so-called open string metric.
Let us now explicitly write down the elements of S and A including the gauge field in (2.12). They can be represented as where S x 2 x 3 is identical to the metric components in that plane, S S 3 is identical to the metric components along the S 3 and A x 2 x 3 = 0, A S 3 = 0. The non-triviality arises in the {t, x 1 ≡ x, u}plane. The non-vanishing components are where now denotes taking derivative with respect to u. The non-vanishing components of the anti-symmetric tensor A are given by We also note that the gauge field is given by Let us further investigate the properties of S. It is clear from the form of the fluctuation action, S plays the role of an effective metric as far as the fluctuations are concerned. 11 Therefore the effective geometry in the {t, x, u}-plane is described by a metric 12 where we have redefined the time coordinate Rewriting the 10-dimensional background as we get (2.58) The geometry described by (2.55) has an event horizon at where u * is precisely the location of the pseudo-horizon. Furthermore, the constant J can be fixed by demanding that the coefficient in front of dx 2 remains positive, which will be equivalent to setting (J 2 R 6 − u 6 f cos 2 θ) = 0 where (2.59) holds. 13

Fluctuations: fermionic sector
Let us now discuss the fermionic fluctuations, which correspond to the supersymmetric partners of the mesonic operators. These fermionic meson operators are of two types 14 with conformal dimensions ∆ = 5 2 + and ∆ = 9 2 + . The fermionic part of the D7-brane action at the quadratic order is given by 15 ab +Γ (10) f ab , (2.62) 13 Note that, this purely geometric condition determines the flavour current in the boundary theory and is a special case of the open string metric membrane paradigm discussed in [23]. 14 Here also we will follow the notations used in [22], where the zero temperature fermionic meson spectrum has been worked out in details. 15 The general form of the quadratic action for a Dp-brane is given in appendix C in e.g. where ξ a corresponds to the worldvolume coordinates, which are identified with spacetime coordinates {t, x 1 , x 2 , x 3 , u, ψ, β, γ}; Ψ represents a doublet Majorana-Weyl spinor in 10-dimensions where Ψ 1,2 each has positive chirality. In appendix B, we have definedΓ (10) . The worldvolume gamma matrices Γ a are the pull back of the 10-dimensional gamma matrices: Γ a = (∂ a X µ ) Γ µ , where a = 0, . . . , 7 and µ = 0, . . . , 9. The operator P − is a kappa symmetry projection operator: where Γ p has been defined in (A.23). In particular, we have where f ab is the classical gauge field on the worldvolume of the probe. 16 Let us now closely follow the discussion in [24], where one proceeds by fixing the κ-symmetry on the worldvolume of the probe. One can use the covariant gauge-fixingΓ (10) Ψ = Ψ, i.e. set Ψ 2 = 0, and this will yield the following action for the fermionic fluctuations and W a is defined in (A.32). We can now interpret the fermionic fluctuation action in (2.66) as a Dirac action with an effective mass term where the background geometry is again dictated by the (g (0) + f ) −1 , which is precisely the combination that appeared for the bosonic fluctuations. The action in (2.66) can be further simplified. Note that the gamma matrices Γ µ generate the Clifford algebra associated with the worldvolume induced metric 68) 16 The above expression is written for a general gauge field on the worldvolume of the probe; but for the specific ansatz in (2.12), the last two terms above vanishes identically. Written explicitly, the operator P − in this case takes the following form where P [G] stands for the pull back of the background geometry. Let us now define which then yields With this, the fermionic action now becomes which results in the following equation of motion The above equation is not analytically solvable, so we will not discuss its solutions. However, it is evident that the effective geometry as

Some properties of the open string metric
In the previous two sections we have explicitly demonstrated that all the fluctuation modes on the probe experience the open string metric geometry, therefore an open string equivalence principle holds for these modes. In this section, we will further investigate some properties of this metric. Let us begin by writing the open string metric once more The metric in (2.73) is asymptotically AdS, and in the infrared (obtained by solving f − E 2 R 4 /u 4 = 0) it behaves like an AdS-BH background. The geometry in (2.73), however, has a curvature varying with the radial coordinate u with a singularity located at u = 0. Note that this singularity is present even when the closed string background does not have a black hole inside, i.e. in the case when u H = 0. For further discussion, let us focus on the case θ 0 = 0 (assuming u H = 0): the trivial embedding. In this case, we can forget about the S 3 -directions and analyze the effective 5dimensional geometry.
One may wonder, since the geometry described in (2.77) approaches AdS in the UV and AdS-BH in the IR, perhaps (2.77) can be obtained as a solution of an effective 5-dimensional Einstein gravity with a suitable matter field where κ 5 is a constant which is related to the effective 5-dimensional Newton's constant, G 5 denotes the metric, R 5 denotes the Ricci-scalar and Λ 5 = −6/R 2 denotes the cosmological constant. Finally, S source is the hypothetical matter field that sources the geometry (2.77). It can be checked easily that such a putative source-term will yield a null energy condition violating energy-momentum tensor. 18 This means that the metric in (2.77) cannot be obtained from Einstein gravity. 19 Nevertheless, an effective temperature can be read off from the metric in (2.77). By Euclidean continuation of τ → iτ E , periodically compactifying the τ E direction and demanding the absence of any conical singularity yields the following effective temperature Note that, there is no black hole in the closed string background and therefore the adjoint degrees of freedom do not experience any temperature. By virtue of the open string equivalence principle, all open string modes in the flavour sector experience an effective temperature given in (2.79). 18 In this particular case, one can choose a null vector n µ = (n τ , 1, 0, 0, 0) with n 2 τ = (−g xx /g τ τ ). With this one can check that T µν n µ n ν < 0. 19  For the sake of completeness, this effective temperature is general is given by 20 where T is the temperature of the close string black hole and hence of the adjoint sector in the dual field theory.
Let us now go back to (2.77). As with a black hole geometry, the metric components are singular at u = u * : the open string metric event horizon. As we approach u → u * , the radial null light cones tend to "close up" since dτ /du → ±∞. This is a coordinate artifact and can be tamed with introducing the Eddington-Finkelstein coordinates. To that end, we focus on the near-horizon geometry of (2.77) in the {τ, u}-subspace where γ xx and γ ⊥ are numerical constants and α → (2M ) + is the location of the horizon. Following the standard lore of black holes, let us first define the Regge-Wheeler coordinate In terms of the {U, V }-patch, the metric (2.81) takes the following form where ds 2 ⊥ represents all other directions which we are suppressing here. Correspondingly Kruskal-Szekres time and radial coordinates can be defined as: Evidently, the coordinate singularity at α → 2M + disappears and we can extend the geometry to its Kruskal patch. The corresponding causal structure will have similar behaviour as seen in a maximally extended AdS-Schwarzschild geometry [25] and subsequently results in a Penrose diagram as shown in Fig. 2.
The location of the event horizon α = 2M is now given by U V = 0, which implies either U = 0 or V = 0. This divides the Kruskal patch in four regions depending on the sign of U and V and there is a symmetry (U, V ) → (−U, −V ). We will now comment on the τ = const 20 Here we still assume θ 0 = 0, i.e. the flavours are massless. For massive flavours, the general formula is more complicated and we will not write the explicit form here. It can be found in e.g. [23]. , the corresponding open string metric can also be Kruskal-extended, but the resulting Penrose diagram does not take a "square" shape. More details will appear in [25].
hypersurfaces in the Kruskal patch. To do so, we introduce the following coordinate The τ = const hypersurfaces have topology: R β ⊗ R x ⊗ R 2 . Unlike the usual Schwarzschild case, the transverse space, described by ds 2 ⊥ is insensitive to β. Nevertheless, for each value of α, we have two values of β which are exchanged by the symmetry: β → M 2 /4β. The two regions, parametrized by the two roots of β, are connected by an Einstein-Rosen bridge whose "throat" has a constant size. Let us consider the background closed string geometry obtained by placing N c number of D3-branes on the tip of a cone, whose base is a 5-dimensional Sasaki-Einstein manifold M 5 and taking the near-horizon limit. This gives a 10-dimensional geometry of the form AdS 5 × M 5 . In general this will preserve N = 1 supersymmetry. To begin with, let us take M 5 ≡ S 5 , in which case N = 4 supersymmetry is preserved. To introduce flavour degrees of freedom, we will introduce probe branes of various dimensions in the background such that these flavours are classically stable. We have schematically shown these constructions in table 2. In table 2,  The probe D5's yield a (2 + 1)-dimensional hypermultiplet in the fundamental representation of the gauge group. More explicitly, the background closed string metric can be written as The embedding function can be parametrized by ψ(u), with the condition that P [Ω 2 ] = 0, wherẽ Ω 2 denotes the metric components which yields the line element dΩ 2 2 . Exactly as in the previous section, there will be a family of solutions for the probe D5 which are characterized by the mass of the fundamental flavour. It is easy to check that we will also have a massless case -represented by ψ = 0 -which corresponds to the "equatorial" embedding. In this case, the worldvolume geometry on the probe D5 is simply AdS 4 × S 2 . It can also be checked easily that we can excite the worldvolume gauge field as in the previous section and the ψ = 0 embedding will still remain. If we focus on this embedding for simplicity, the compact directions are merely spectators.
The open string metric can be obtained to be Note that, evaluated on the embedding ψ(u) = 0, the open string metric in (3.90) yields AdS 4 ×S 2 only when both u H = 0 and E = 0, i.e. asymptotically. In the IR, however, we do not recover AdS 4 -Schwarzschild geometry in the strict sense, since the power of u in e.g. f (u) is reminiscent of the AdS 5 structure of the closed string metric. As before, requiring the positivity of γ xx the constant j (which is related to the flavour conductivity in the dual field theory) can be determined in terms of E. 21 The effective temperature is given by It is straightforward to also check that the near-horizon geometry has a very similar structure to the one obtained in (2.81) and correspondingly admits a Kruskal patch. Note that the metric in (3.90) was studied in [26] and it was concluded that the vector fluctuations experience an effective fluctuation-dissipation relation involving the effective temperature T eff . Here we argue that, because of the open string equivalence principle, this result holds for all fluctuations in the probe flavour sector. We will not construct explicit examples of type (ii), but point the reader to e.g. Sakai-Sugimoto model considered in [27,28]. Such constructions can clearly be generalized for other non-conformal backgrounds.

Abstracting away from the brane construction
Motivated by the explicit examples studied in details in the previous sections, let us now "abstract away" from the explicit 10-dimensional construction and take a more "phenomenological" approach. Our motivation here is to explore various features, keeping our discussion as general as possible.

Event horizons: some generalities
In [13], we had carried out a general analysis and obtained a general formula for T eff , the effective temperature that the steady-state system records. One intriguing question is the hierarchy of the two temperatures that we have at hand: T and T eff . If u * and u H denote the open string and closed string metric horizons respectively, then it is straightforward to observe that u * > u H in the convention where u → ∞ is the boundary of the geometry. However, as pointed out in [29] this does not imply that T eff > T . For most cases with an explicit brane construction, it is found that T eff > T .
Let us now consider a more general case, where a bunch of other gauge modes are also excited on the putative probe worldvolume. In the dual gauge theory description, these modes can correspond to introducing a non-zero chemical potential and/or a constant magnetic field in addition to the constant electric field. 22 In general, we have two inequivalent combinations of the electric and the magnetic field: (i) parallel and (ii) perpendicular. For simplicity, let us take d = 3 and m = 2 which is sufficient to include the structures we want. We will now focus on the {t, x 1 , x 2 , x 3 , u}-submanifold of the closed string geometry in and we will assume an SO(3)-symmetry in the {x 1 , x 2 , x 3 }-plane.
Let us now consider the case when (i) E B. For this, the ansatz for the worldvolume gauge field is where a x (u) contains the information about the gauge theory current and a t (u) contains the information about the chemical potential. With this ansatz we preserve an SO(2) ⊂ SO(3) of the closed string background. The open string metric can be obtained as with and The location of the open string metric event-horizon is given by Here the constant j and d are defined as where L represents the DBI Lagrangian of the probe. The constant j is related to the boundary current andd is related to the chemical potential of the system. The constant j can be fixed by requiring that the coordinate X does not change sign. For completeness, let us now comment on the case when (ii) E ⊥ B. In this case, there will be an additional current due to the Hall effect, see e.g. [37]. The worldvolume gauge field ansatz will take the form The open string metric can be obtained as before and there is an event-horizon which can subsequently be obtained. We refrain from providing the explicit expressions since they are complicated looking and not particularly illuminating.

Ergoplane: chemical potential
Let us now focus on a special case, where the open string metric acquires an "ergoplane" and an event horizon. This was explicitly demonstrated in the D3-D7 system in [23]. Here we will argue that a similar structure persists more generally. To this end, we consider only an electric field along with the chemical potential. By setting B = 0 in (4.95), we get It is clear that the event horizon of this metric obtained from (G tt G xx + E 2 ) = 0, but there is an ergoplane in this geometry which is obtained by The first condition gives the open string metric event horizon, but the second condition yields a root, denoted by u erg henceforth, which is different from u * . We also note that j 2 is determined from the algebraic condition:d 2 e 2φ G tt + G tt G m xx + e 2φ j 2 G xx u * = 0.
If the background represents an AdS d+2 geometry, in which the dilaton φ = 0, then it can readily seen that where we have used the conditions to determine u * , j and u erg . Evidently, the event horizon and the ergoplane merge in the limit (d 2 /E d ) → 0 and/or (1/d) → 0. 23

Exactly solvable toy examples
We will now discuss a few cases where e.g. the gauge field fluctuations can be solved analytically. Towards that end, let us take a generic Lifshitz background of the following form: where x is a 2-dimensional vector and v is the radial coordinate (v → 0 corresponds to boundary and v → ∞ corresponds to the deep-IR). In this background we will consider a probe of the same bulk space-time dimensions whose action is given by the DBI action. 24 Moreover, to introduce the "steady-state" physics that we are interested in, we introduce a gauge field on the world volume of the probe given by (4.109) The equation of motion for the function h(v) is given by where e = (2πα )E and j is the constant related to the boundary current. The on-shell Lagrangian using this equation of motion is obtained to be (4.111) 23 This inequality u erg > u * can be shown to hold for the AdS d+2 -Schwarzschild geometry as well, and u erg → u * in the limitd → 0 and/or d → ∞. 24 For now, we will assume that the full 10 or 11-dimensional background (if it exists) allows for such a probe brane which has a constant profile along the compact directions. Thus the effective lower dimensional problem that we have to solve is done by simply considering the "reduced" DBI action of the appropriate dimensions.
From the above expression, imposing the reality condition for the on-shell action, we get j = e which simply yields L os = 1/v 3+z . Now, we can consider the gauge field fluctuations living on the world volume of the probe: a = a 0 + δa, where a 0 denotes the classical profile of the gauge field. The fluctuation modes will sense the OSM. This fact is reflected in the action for the quadratic fluctuation for the gauge fields which is given by where −det(G + f ) = L os and δf = dδa is the field strength of the gauge field fluctuations. This action results in the following equation of motion In the {t, v}-plane the OSM is non-diagonal and we can diagonalize it by introducing an ingoing Eddington-Finkelstein type coordinate where τ is the new time coordinate. This gives (4.115) Now, the vector fluctuations that we consider are of the following form It is straightforward to check that the temporal and the spatial fluctuations decouple and hence we study only the spatial fluctuations here. 25 The general equation for the spatial vector fluctuations take the following form Interestingly, we can solve the above equation analytically for z = 1/2, 1, 3/2, 2. It will be interesting to see what happens to the fluctuation-dissipation relation in all these cases. 25 Interestingly the vector fluctuations along both the x-direction and the y-direction obey the same equation.
Some of the explicit solutions take the following form: Here c x is a constant. Using (4.118)-(4.120), the asymptotic form of the gauge field can be easily determined where Λ (1) is a numerical constant. Using this asymptotic expansion, the retarded current-current correlator is easily obtained to be where {i, j} takes values {x, y}. Now, using the Schwinger-Keldysh method in the context of gauge-gravity duality, as pioneered in [38], we can easily arrive at where G ij sym denotes the symmetric Schwinger-Keldysh correlator. Clearly, we observe that the fluctuations indeed measure a temperature T eff . 26 In this case, we get

The probe limit and its validity
It is clear from the above discussion that, in order for us to have the above structure, we need to treat an additional set of D-branes in a background which is obtained by solving supergravity. Let us imagine that the following geometric data represents a particular solution of 10-dimensional supergravity in string frame Here all the symbols have their usual meaning, specially dΩ 2 8−p stands for an (8 − p)-dimensional compact manifold. Quite clearly, the putative geometry in (4.125) is dual to a (p+1)-dimensional gauge theory, which possess SO(p) rotational invariance and is also homogeneous in the spatial coordinates. Furthermore, by assumption, there is a global symmetry which is the isometry group of the manifold whose line element is represented by dΩ 2 8−p . Evidently, (4.125)-(4.126) may not characterize the full geometric data that can further contain various non-vanishing form-fields. However, for our purposes, these form fields will not play any role as we will argue momentarily.
Before going further, let us note that the background in (4.125) source an Einstein tensor Let us now imagine that we place a Dq-brane in the above background in the probe limit. Let us imagine that this probe brane extends along R m ⊂ R p , the radial direction u and wraps an M q−m−1 ⊂ M 8−p . The last assumption imposes: (p + q) ≤ (9 + m). Now, to describe the steady-state, we excite a gauge field along one of the spatial directions as before (4.128) On the worldvolume of the probe brane this breaks the SO(m) → SO(m − 1). The resulting action is given by We have further assumed that there is no Wess-Zumino term for the above configuration. The gauge field equation now gives where J is a constant.

Now, the probe limit condition is satisfied if
This condition is non-trivial, since it involves the radial coordinate u.
To flesh out the idea, let us now consider a set of examples. Imagine we start from the geometry which is obtained by taking a near-horizon limit of N c coincident Dp-branes. This, in the string frame, is given by [39] , (4.133) Here dΩ 8−p denotes the line element of an (8 − p)-sphere and ω 8−p denotes the corresponding volume form. This background preserves 16-supercharges. It is straightforward to check that for the above geometry, the Einstein tensor behaves as: Now, let us imagine introducing an N f -number of probe D(p + 4)-branes that extend along the {t, x p }-directions and wraps three-cycle X 3 ⊂ S 8−p . This, in general, preserves N = 1 supersymmetry. Furthermore, we excite a gauge field along e.g. x 1 -direction that breaks SO(p) → SO(p − 1): (4.136) It is straightforward now to check, using the formulae (B.7)-(B.11), that the probe energymomentum tensor yields and Comparing (4.135) with (4.137), we observe that near the UV, the probe limit is perfectly fine, except the case with p = 4. Similarly, comparing (4.135) with (4.139), we observe that the deep infra-red of the background will be modified by the probe sector. Further note that, T tt ∼ (E ·J), which amounts to the Ohmic dissipation term from the perspective of the boundary field theory.

A few words on the effective thermodynamics
We have argued in the previous sub-section that, the fluctuation modes will sense an effective temperature T eff , which is an intensive variable. The thermodynamically conjugate variable is entropy, which is extensive and dynamical. As argued in [40], only thermodynamic quantity that can be reliably calculated in the probe limit is the free energy, which is simply given by the on-shell action.
The basic idea of gauge-gravity duality equates the bulk gravity path integral with the dual field theory path integral, i.e.
(4.140) Thus, in the probe limit i.e. N f N c , the probe sector path integral will give the path integral for the fundamental sector in the dual field theory. In Euclidean signature, this statement becomes an equality of the partition functions of the two sides. However, as [40] has emphasized already, this is useful as long as we calculate the free energy. Now, a proposal to obtain the thermodynamic free energy was put forward in [41], which was subsequently generalized in [13]. According to this proposal, the Helmholtz free energy is given by (4.141) Here u * is the osm event-horizon, S DBI is the Euclidean DBI action. This is the only extensive quantity that we can define.
On the other hand, given the existence of an osm event-horizon, and the open string data {G s , S}, where G s is the open string coupling [42], there is an area associated and therefore a natural notion of some entropic quantity, given by However, it is not clear to us at present what this quantity physically corresponds to. One possibility could be this measures the entanglement entropy for the pair creation process that drives the steady-state current (similar to [43]), but we leave this for future work.

A dynamical horizon
So far, we have observed stationary space-time described by the open string metric. Here we will review an example where the open string metric yields a dynamical geometry following [9]. Let us imagine for some given closed string metric, the induced metric on the probe worldvolume takes the effective form where we have introduced the Eddington-Finkelstein ingoing coordinate . (4.144) The metric takes the usual black hole metric form in the {t, u}-patch. The corresponding DBI action takes the following schematic form where we have considered a generic field strength of the world volume f = f xu (u)dx ∧ du + f xV (u)dx ∧ dV , which is capable of capturing the physics we want. Subsequently, we will get the following equation of motion The second condition above arises from the Bianchi identity. A simple solution of the above equations is given by [9] which has an AdS-Vaidya form. Choosing E(V ) to be a smooth function interpolating from zero to a non-vanishing finite value, we can simulate the formation of an event horizon in the open string metric. In the dual field theory this process of horizon formation encodes turning on a time-dependent boundary current j(t) ∼ E(t). At each t = const slice, the geometry in (4.149) has an apparent horizon located at (G tt + E(V ) 2 /G xx ) = 0.

Schrodinger symmetric background
We will now briefly discuss one special case which does not fall under the metric class that we have discussed so far. This special case is the metric with Schrödinger isometry. This geometry can be obtained by a non-relativistic deformation of the relativistic AdS-background [44,45] in e.g. type IIB supergravity where z is the radial direction, X 5 is an Einstein manifold and A is a vector eigenfunction of the Laplacian on X 5 and n is determined by the corresponding eigenvalue. The metric function Ω obeys an inhomogeneous scalar Laplace equation on the internal manifold. When X 5 is also a Sasaki manifold, it is possible to preserve at least two real supercharges depending on the choice of A [46]. It is easy to see that (5.150) arises from a deformation of an underlying AdS-metric. Turning off Ω and A and identifying we recover an AdS 5 geometry, where X + and X − are identified with the null coordinates. The metric (5.150) is invariant under the Schrodinger group, denoted by Schr(d) in d-spatial dimensions. This group is an analogue of the relativistic conformal group and has the following generators: temporal translation, spatial translations, rotations, Galilean boosts, dilatation and a special conformal transformation when n = 2. The index n, which encodes the information of how spatial and the time directions in the dual field theory scale, is known as the dynamical exponent. Note that, the dual field theory lives in 2-dimensions less: the time coordinate is identified with X + and x 1 and x 2 are the spatial directions.
For concreteness we will now focus on the case where X 5 ≡ S 5 and n = 2 [47,48,49], which can be written as an U (1) fibration over a Kähler-Einstein base: CP 2 : where σ i , i = 1, 2, 3 are the SU (2) left-invariant one-forms (following notations of [50]) Also, the vector field is given by A = cos 2 θσ 3 . In this case, the background above can be obtained from an AdS 5 × S 5 geometry by performing the null Melvin twist (nMT) procedure.
Thus, an intuitive way to embed the D7-branes in the Schrödinger geometry will be to consider an embedding analogous to section 2 [50]. This means that the D7-brane wraps the following directions: {X + , X − , x 1 , x 2 , z, α 1 , α 2 , α 3 } and θ(z) and χ(z) are the transverse scalars. Without any loss of generality, we can set χ(z) = 0, since it has an U (1)-symmetry. Now, it can be easily checked that the DBI action yields exactly the same result as in the purely AdS 5 × S 5 case. 27 Hence, the simplest embedding -which corresponds to introducing massless flavors -is again given by: θ(z) = 0.
To excite the gauge field analogous to (2.12), we consider[50] Following the similar chain of arguments, we will now have the open string event horizon and the boundary flavor current and finally the open string metric where we have redefined As before, we observe that the open string metric acquires an event horizon at z * = 1/ √ E in a similar manner as before. As before, the fluctuation modes will be governed by an open string equivalence principle and an effective thermodynamic description will prevail.

Further generalization: non-Abelian DBI
Let us begin with the non-Abelian version of the Dirac-Born-Infeld action [51]. In much of what we present below, we will closely follow the notations and conventions of [52]. This reads where Furthermore, µ, ν = 0, . . . , 9; a, b = 0, . . . , 7 and i, j = 8, 9, which are all space-time indices. Also, here STr corresponds to "symmetrized trace" and Φ i denotes the profile function of the brane. Evidently, the flavour index is suppressed in the above expressions.
To simplify the situation, we imagine placing N f probe D7-branes in an AdS-background. Thus, we can make use of the rotational symmetry along the {8, 9}-directions to set Φ 9 = 0. Thus we get Q j i = δ j i . Furthermore, here E µν = G µν since the NS-NS anti-symmetric form B µν = 0. Thus, to evaluate (6.159), we need to compute only the pull-back of the background metric, which is given by Let us now focus on the case of N f = 2. In this case, the corresponding group U (2) is generated by a set of 4 C-valued 2 × 2 matrices: Here σ 0 is the identity matrix and σ i , i = 1, 2, 3 are Pauli matrices. Let us suppose that we excite a gauge field of the following form This yields the following field-strength Evidently, we have used the fact that σ 0,3 are diagonal and therefore commute with each other. Now, to simplify the pull-back in (6.162), we follow the arguments outlined in [52]: First, using the U (1) ⊂ U (2), we align the σ 0 flavour-direction parallel to the profile function Φ 8 . Subsequently, since we are exciting only an "isospin" gauge direction (the one along σ 3 ), the representation in the flavour space is completely diagonal. This implies that all commutators in (6.162) involving the embedding function Φ 8 and the gauge field A µ will vanish identically. Hence we completely decouple the fields Φ 8 and A µ .
Thus, we evaluate the action in (6.159) to be (6.169) In the above calculations, we have used the explicit forms of the Pauli matrices given in (6.163). Finally we get where a I 's are first introduced in (6.165). It is evident from (6.170) that the embedding profile Φ is contained entirely inside −detP [G] -piece of the action. Thus, if in the absence of E ± and a ± , a profile function satisfies P [G] = G, i.e. Φ = constant, then at any finite value of {E ± , a ± }, Φ = constant will also satisfy the equations of motion. This is what we normally call a trivial embedding. For now, we will restrict ourselves only to these trivial embeddings. It is also noteworthy that E ± can be interpreted as independent electric fields at the boundary theory and a ± should subsequently contain the information of the corresponding currents. Now, the equation of motion resulting from (6.170) are given by where j ± are the corresponding constants of integration. Due to the decoupling of the ±-modes, it is evident that there will be corresponding boundary currents Thus, classically, we are still describing a decoupled system driven with an external field E ± and having a response current j ± . Evaluated on an AdS-background, where G tt = − r 2 L 2 , G xx = r 2 L 2 = G −1 rr , the solution for the decoupled gauge-modes are Thus, the on-shell Lagrangian takes the following form Subsequently, we get two decoupled pseudo-horizons It is natural to expect that there will be two decoupled sets of fluctuations, corresponding to the ±-modes, that will sense the above pseudo-horizon as the corresponding open string metric event-horizon. Now, we discuss the physics of the fluctuation modes. To that end, let us imagine the following fluctuations 28 Φ 8,9 = Φ 8,9 0 + δϕ 8,9 , A a = A a class + δA a 0 σ 0 + δA a i σ i , (6.177) where Φ 8,9 0 describe the classical profile and A 0 class is given in (6.165). Thus, δϕ and δA correspond to fluctuation modes. Evidently, we can write the fluctuation modes as where As we have explained before, in this case the fluctuation action will be determined from the expansion tr ab a −1 a (1) 2 + 1 2 tr ab a −1 a (2) + . . . , (6.180) 28 We will restrict ourselves to the scalar and the vector fluctuations only.
where a ab = G ab + F ab , a (1) It is again clear that (6.180) involves the inversion of the a-matrix in order to determine the "effective metric" in the kinetic term for the fluctuations. Now let us focus on the scalar kinetic term. Using (6.180) and (B.19) we get Proceeding in a similar manner, we can also get The fluctuation action in (6.182) and (6.183) are highly suggestive: Within each spin-sector, there are 3 different modes (corresponding to "+", "−" and "12") that sense three different effective geometries, subsequently given by S + , S − and (S + + S − ). These distinct modes "see" three distinct event horizons, given by In AdS-space this yields For example, we have completely ignored the backreaction of the probe sector. This backreaction will introduce explicit time-dependence in the problem since the constant energy that gets pumped into the system because of Ohmic dissipation, will eventually start increasing the background event horizon. If we keep the constant electric field turned on for a long but finite amount of time, it is likely that the dynamics will merge the background event horizon with the initial open string metric event horizon. This will be an interesting merger process to study. Furthermore, as we have seen, backreaction effects grow in the deep IR, which can lead to non-trivial ground states for the system.
Moving away from explicit time-dependence, the identification of the area of osm event horizon is an intriguing problem. It is also noteworthy that, the osm naturally allows a Kruskal extension and henceforth an Einstein-Rosen bridge. Although this is simply obtained by maximal analytic extension of a particular coordinate system, and does not come from an underlying dynamical process, this has the same flavour as the ER=EPR conjecture [53], in the context of gauge-gravity duality [43]. It will be extremely interesting to pin this problem down.
We end on the following note: As far as exploring governing principles of non-equilibrium physics is concerned, we have analyzed a tiny and a very special corner of it. It seems necessary that one studies the entire evolution process in details and look carefully for any universal qualitative feature that may survive the details of the system and the dynamics. Thus a complete understanding of the dynamical process is an enticing avenue for further and future work.

Appendix A. Fluctuations of the classical profile
We will outline the steps towards obtaining action for the quadratic fluctuations. Without loss of generality, let us focus for a Dp-brane, where p ∈ [0, 9]. We will first discuss the bosonic part of the fluctuations. Let us denote the embedding of the probe brane by where X µ 0 denotes the classical profile of the probe, and Σ i denotes the transverse fluctuations. Here the indices µ, ν, ρ = 0, . . . 9; the indices i, j, k = (p + 1), . . . , 9 and the indices a, b, c = 0 . . . p. Clearly, the Greek indices µ, ν, ρ etc.are used for the ten dimensional background, the indices i, j, k etc. are used to represent the transverse fluctuations (which, for the D7-brane probes described in section 2 are denoted by δθ and δφ respectively), and finally the indices a, b, c are used to represent the world volume of the probe brane.
The background metric G µν can be expanded under (A.1) as Here G 0 µν denotes the background metric evaluated at the classical profile of the probe. Now, the induced metric (up to quadratic order in the transverse fluctuations) on the probe D-brane can be obtained from the following formula g ab ≡ P [G ab ] = G µν ∂ a X µ ∂ b X ν = g (0) ab + g (1) ab + g (2) ab , Now the action functional for the probe brane is given by where ξ represents the world volume coordinates, and the Wess-Zumino term depends on the dimensionality of the Dp-brane as well as the background fluxes.
To proceed further, let us now define ab + E Tr g (0) + f −1 g (1) + δf g (0) + f −1 g (1) + δf . (A.14) It is evident that inverting the matrix (g (0) + f ) is crucial in determining the quadratic action. Inversion of (g (0) + f ) will consist of two pieces: a symmetric and an anti-symmetric one, denoted by S and A respectively. Following [42], these pieces are obtained to be (A.17) Let us now discuss the fermionic part. The quadratic term is given by [54,55,24] S fermion probe = T p 2 d p+1 ξe −Φ −det (E (0) )ψ (1 − Γ p ) (Γ a D a − ∆ + L p ) ψ , (A. 18) where ψ is the fermionic fluctuation field. In type IIA, this is a 10 dimensional Majorana spinor and in type IIB this is a positive chirality doublet Majorana-Weyl spinor. The worldvolume gamma matrices Γ a are pull-back of the 10 dimensional gamma matrices Γ µ : Γ a = Γ µ (∂ a X µ ) = Γ µ e µ µ (∂ a X µ ) , (A. 19) where X µ represents the 10 dimensional spacetime, and the vielbein is given by whereas for type IIB (where p = 2n + 1 is odd-valued) we have is the covariant derivative.
Let us further note that, after making a few formal manipulations as outlined in [24], the operator L p can be rewritten as where we have defined Furthermore, let us also defineẼ Similarly, From (B.14), it is clear that Clearly, (B.14) and (B. 16) define the open string metric with upper and lower indices respectively. Let us now consider the case of SU (2). In this case, we have defined Proceeding along similar lines as outlined above, we can show To derive the last line, we have used the fact (σ 3 ) 2 = σ 0 and σ 3 σ 0 = σ 0 σ 3 = σ 3 . We can write down the open string metric (with indices lowered) as