Gauge Invariance and Holographic Renormalization

We study the gauge invariance of physical observables in holographic theories under the local diffeomorphism. We find that gauge invariance is intimately related to the holographic renormalisation: the local counter terms defined in the boundary cancel most of gauge dependences of the on-shell action as well as the divergences. There is a mismatch in the degrees of freedom between the bulk theory and the boundary one. We resolve this problem by noticing that there is a residual gauge symmetry(RGS). By extending the RGS such that it satisfies infalling boundary condition at the horizon, we can understand the problem in the context of general holographic embedding of a global symmetry at the boundary into the local gauge symmetry in the bulk.


Introduction
According to AdS/CFT correspondence, any global symmetry at the boundary theory is lifted to a local symmetry in the bulk [1,2]. The gauge symmetry is essential to reduce the degree of freedom which is enlarged by going into one higher dimension. The physical goal in holography is the boundary quantities which does not know the presence of higher dimension or gauge degrees of freedom, while we use the tools in the bulk theory. Therefore the gauge invariance of a physical quantity is a critical issue for the validity of the AdS/CFT. Also tracing the gauge invariance gives much intuition on the way how holography actually works.
One can find gauge invariant combinations of the fields, and express the physical quantities in terms of such master variables, it is not always easy to find such gauge invariant combination. Even in the case they are available, it is not very convenient to use such fields, especially if many fields are coupled, because the physical quantities are defined in terms of the field variables which are formally gauge dependent. For example [2], energy momentum tensor and chemical potential is defined in terms of metric/gauge field which are not gauge invariant. Similarly, heat currents can be related to the metric perturbation defined only in a specific gauge where time period has definite relation with temperature.
In a recent work [3,4], based on [5,6], we developed a systematic method to numerically calculate the Green functions and all AC transports quantities simultaneously for the case where many fields are coupled and there are constraints due to gauge symmetry. Although we have tested the validity of the procedure by showing the agreement of zero frequency limits of AC conductivities with the known analytic DC conductivities [7][8][9] we still think that we need to prove the gauge invariance of our procedure. This is because there are residual gauge transformations even after we fix the axial gauge g rx = 0. Furthermore this residual gauge transformation does not go away by writing the equation of motion in terms of the the gauge invariant master fields P h , P χ (3.8), because the on-shell action can not be expressed in terms of those. Actually it is gauge dependent. Then we might ask why optical conductivities which are proportional to the retarded green function are gauge invariant while their generating function is NOT invariant. The purpose of this letter is to answer this question. As a by product, we found that gauge invariance is intimately related to the holographic renormalisation. Although the local counter terms were introduced to kill the divergences, it also kills most of gauge dependence. The proof of residual gauge invariance will be established by giving the connecting formula between the momentum and source and show that connection matrix is source independent.

Action and background solution
Let us first briefly review the system we will discuss, which has been analysed in detail in [3,7,10]. The holographically renormalised action(S ren ) is given by is the usual action for charged black hole in AdS space(Λ < 0) with the Gibbons-Hawking term and is the action for two free massless scalars added for a momentum relaxation effect. S c is the counter term which is included to cancel the divergence in S EM + S ψ . Here we introduced η c to keep track of the effect of the counter term. At the end of the computation we will set η c = 1. The action (2.1) yields general equations of motion 1 which admit the following solutions These are reduced to AdS-Reissner-Nordstrom(AdS-RN) black brane solutions when β = 0.
Here we have taken special β Ii , which satisfies 1 2 2 I=1 β I · β I = β 2 for general cases. The solutions (2.7) -(2.10) are characterised by three parameters: r 0 , µ, and β. r 0 is the black brane horizon position(f (r 0 ) = 0) and can be replaced by temperature T for the dual field theory: Non-vanishing components of energy-momentum tensor and charge density read T tt = 2 T xx implies that charge carriers are still of massless character. From here we set r 0 = 1 not to clutter.

Gauge fixing and residual gauge transformation
To study electric, thermoelectric, and thermal conductivities we introduce small fluctuations around the background (2.7) -(2.10) The fluctuations are chosen to be independent of x and y. This is allowed since all the background fields appearing in the equations of motion turn out to be independent of x and y. The gauge field fluctuation(δA x (t, r)) sources metric(δg tx (t, r), δg rx (t, r)) and scalar field(δψ 1 (t, r)) fluctuation and vice versa and all the other fluctuations are decoupled. We will work in momentum space and h tx (ω, r) and h rx (ω, r) is defined so that it goes to constant as r goes to infinity.
By linearising the full equation of motion, we get four equations. However one of them can be obtained by the others. Thus we may consider following three equations: If we differentiate the third equation with respect to r, all equations can be written in terms of three variables, P χ , P h , and a x , where Therefore, h rx is a non-dynamical degree of freedom. Indeed, P χ , P h , and a x are invariant under a diffeomorphism generated by ξ µ = (0,ζ(r)e −iωt , 0, 0), under which the fields are transformed as follows: Using this gauge degree of freedom, one may set h rx = 0, which is so called the axial gauge. Our numerical calculation in [3] has been performed in this gauge. A question arises whether the resulting physical quantities are independent of such gauge fixing condition. Furthermore, even after we fix h rx = 0, one can still find a residual gauge transformation which is given by constantζ [11]. This residual diffeomorphism doesn't change the gauge fixing condition h rx = 0 and generates constant shift on h tx and χ, because the equations of motion contain only derivatives of h tx and χ and the linear combination of them, ωχ(r) − iβh tx (r), which is invariant under where h 0 is a constant. Thus there is one parameter constant solution given by which does not satisfy in-falling boundary condition so it is not a physical degree of free-dom 2 . Therefore, we can call it a residual degree of freedom. This kind of solution was first introduced in [12] Why should there be such a residual degree of freedom? It can be traced to the difference of the differential equation near horizon and those near boundary. Near the black hole horizon (r → 1) the solutions are expanded as where ν ± = ±i4ω/(−12 + 2β 2 + µ 2 ) = ∓iω/(4πT ) and the incoming boundary condition corresponds to ν = ν + . By inserting these to the equations of motion, one can easily find a linear relations between the zero-th modes: Notice that all other modes are generated by these. Thus there is a well defined constraint equation which reduces the degrees of freedom.
On the other hand, by inserting the expansion near the boundary (r → ∞) to the equations of motion, we can not get any relation between the zero-th modes a (0) x , h tx , and χ (0) , all of which are related to the higher modes. More explicitly, which are evolution equations in r-direction. Therefore, there is no constraint equation. Then there is a crisis of mismatch of degrees of freedom and this crisis is resolved by the effective residual degree of freedom described above. However, this residual gauge degree of freedom raises another issue of invariance of physics under this symmetry.

Holographic renormalization and gauge invariance
Now we come back to the question whether physical quantities are independent of the choice of the gauge condition h rx (r) = 0. We will show this by proving that the generating function of physical quantities, the on-shell action, is invariant even in the case with h rx (r) = 0.
The on-shell renormalised action to quadratic order in fluctuation fields, S ren , is where f (r) = r 2 − β 2 2 − m 0 r + µ 2 4r 2 . We dropped the boundary contribution from the horizon as a prescription for the retarded Green function [13]. Near boundary r → ∞, the fluctuation fields in momentum space, (3.1) -(3.4), may be expanded as and using the equations of motion, we can obtain a quadratic action as follows where the argument of the fields with(without) a bar is −ω(ω). V 2 denotes volume in x-y space.
The second line is proportional to a gauge invariant combination under (3.13). Furthermore, by the equation of motion, h (4) rx is expressed as which vanishes due to the Ward identity: In our case, for fluctuations, the left hand side of (4.5) reads which is proportional to right hand side of (4.4), so h (4) rx = 0. The terms proportional to (η c − 1) include the divergent terms with Λ, a regularisation parameter, and finite terms without Λ. A remarkable fact is that with the counter term of weight η c = 1, not only the divergent terms are cancelled, but also all the h rx dependent finite terms disappears from the on-shell action, as we claimed in the beginning of this section.

Gauge invariance under the residual gauge transformation
Therefore, our action (4.3) boils down to 3 1) which is independent of the choice of h rx but still dependent on residual gauge (3.13). Because it is just a constant shift of the solution Φ, its effects are only shifts of zero-th modes and Φ (r) and all of its modes, especially (a (1) x , h Even though the action (5.1) is not invariant under the residual gauge transformation J a → J a + S a 0 it can be shown that physical observables derived from it are invariant. The key observation is that Π a is a linear combination of J a , that is, Π a = C a b J b and C a b 's are independent of J a , thus invariant.
To see why this is the case, we need to look at the procedure to construct the solutions in more detail. To discuss in a more general setup, let us consider N fields Φ a (x, r), (a = 1, 2, · · · , N ), where the index a may include components of higher spin fields. For convenience, r p is multiplied such that the solution Φ a (k, r) goes to constant at boundary. In our case, (Φ 1 , Φ 2 , Φ 3 ) = (a x , h tx , χ) and p = 0 for Φ 1 , Φ 3 and p = 2 for Φ 2 . Near horizon(r = 1), solutions can be expanded as where a new subscript i is introduced to denote the solutions corresponding to a specific independent set of initial conditions. For example, ϕ a i may be chosen as where we used (3.16) and ν = −iω/(4πT ) as shown below (3.15) for incoming boundary condition to compute the retarded Green's function [13]. Due to incoming boundary condition, ϕ a i determinesφ a i through horizon-regularity condition so that we can determine the solution completely. Each initial value vector ϕ i yields a solution, denoted by Φ i (r), which is expanded as where S a i are the sources(leading terms) of i-th solution and O a i are the operator expectation values corresponding to sources(δ a ≥ 1).
Notice that we have only two solutions while we have a three dimensional vector space J of boundary values J a , a = 1, 2, 3. To fix such mismatch of degree of freedom, we introduce a constant solution Φ 0 (r) = S 0 = (0, 1, iβ/ω) along the gauge-orbit direction of the residual gauge transformation so that S a 1 , S a 2 , S a 0 form a basis of J. Now S and O are generic regular matrices of order 3.
The general solution is a linear combination of them: let with real constants c i 's. We can choose c i such that the combined source term matches the boundary value J a : which yields Φ a (k, r) = Φ a i (k, r)c i → J a + · · · + Π a r δa + · · · , (near boundary) (5.8) where, with (5.5) and (5.7), Notice that both Π a and C a b are invariant under the transformation where O a 0 = 0 since it is the sub-leading term of the constant solutions.
A general on-shell quadratic action in momentum space has the form of where A and B are regular matrices of order N .J a means J a (−k) and, in matrix notation, J a can be understood as a row matrix. For example, in our case, the effective action (5.1) reads With (5.9) the action (5.10) becomes where the range of ω is chosen to be positive following the prescription in [13]. Notice that is independent of the choice of the initial condition (5.4), because the different choice of initial value vectors are nothing but a linear transformation ϕ a i → ϕ a j R j i , which induces right multiplications in the solutions: S → SR, O → OR. Since A and B are also independent of J, G R ab is independent of J and manifestly gauge invariant, completing our proof.

Conclusion
We investigated the gauge invariance of physical observables in a holographic theory under the local diffeomorphism. We find that gauge invariance is closely related to the holographic renormalisation. Apart from the zero-th mode residual gauge dependence, gauge dependence is cancelled by the local counter terms defined in the boundary, However, due to the difference in the space-time structure between the near-horizon and near boundary regions, there are residual gauge structure near boundary. Though the on-shell action of the fluctuation is not residual gauge invariant, we prove the gauge invariance of Green's functions in the context of algorithm by which all AC transports are constructed simultaneously.
Note added: After this work is almost finished, the paper [14] appeared where residual gauge invariance was discussed using a different method.