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Higher-spin Gauge and Trace Anomalies in Two-dimensional Backgrounds

Two-dimensional quantum fields in electric and gravitational backgrounds can be described by conformal field theories, and hence all the physical (covariant) quantities can be written in terms of the corresponding holomorphic quantities. In this paper, we first derive relations between covariant and holomorphic forms of higher-spin currents in these backgrounds, and then, by using these relations, obtain higher-spin generalizations of the trace and gauge (or gravitational) anomalies up to spin 4. These results are applied to derive higher-moments of Hawking fluxes in black holes in a separate paper arXiv:0710.0456.

Hawking radiation is a universal quantum effect which arises in the background spacetime with event horizons [1,2]. Such universal behavior arises because fields in black hole backgrounds can be reduced to an infinite set of two-dimensional conformal fields near the horizon. The emergence of conformal symmetries near the horizon was first emphasized in [3] and used to derive the Hawking radiation based on gauge or gravitational anomalies [4,5]. The anomaly method has been applied to rotating black holes [6,7] and various others. Such conformal structure near the horizon is also used to derive the higher-spin (HS) currents of Hawking radiation [8,9] by examining conformal transformation properties of these HS currents.
The above derivation of the HS fluxes is based on the fact that two-dimensional quantum fields can be described by conformal field theories even in the presence of the electric and gravitational backgrounds, and hence all the physical quantities are written in terms of the conformal, i.e. holomorphic and anti-holomorphic, quantities. The HS currents used in [8,9] are the holomorphic currents. They are holomorphic functions and different from the (u · · · u)component of the ordinary covariant currents by some functions of the electric and gravitational backgrounds. These differences are responsible for the conformal transformation properties of the conformal currents. In the simplest case of the energy-momentum (EM) tensor in the gravitational background, it is well-known that we can define the holomorphic EM tensor t(u) from the original covariant EM tensor T uu by where c is the central charge and ϕ is the conformal factor of the gravitational background. This relation gives the transformation property of t(u) under conformal transformations.
In this paper, we generalize this relation to all the HS currents in electric and gravitational backgrounds. This gives a further justification of our analysis in [8] and [9].
In section 2, we first review the relation between covariant and holomorphic forms of the U (1) current and the EM tensor in electric and gravitational backgrounds. For the case of EM tensor, this relation can be obtained from the conservation equations for EM tensor ∇ µ T µ ν = F µν J µ and the trace anomaly T µ µ = cR/24π. Note that T µν denotes the matter EM tensor and therefore it is not conserved by itself in the electric background.
In section 3, we generalize them to HS currents. Here we construct higher-spin (W 1+∞ ) currents from two-dimensional fermion fields in the electric and gravitational backgrounds. Since we do not know either conservation equations or trace anomalies for HS currents at the be-ginning, we cannot start from these equations. Instead we will take the following procedure to obtain the relations between covariant and holomorphic HS currents. The original fermion field ψ transforms covariantly under gauge and local Lorentz transformations. We will construct covariant HS currents by regularizing the fermion bilinears ∂ n ψ † (x)∂ m ψ(x) in the covariant way under gauge and general coordinate transformations. On the other hand, we can define a new fermion field Ψ which is holomorphic in the electric and gravitational backgrounds, and by using it, we construct a holomorphic form of the HS currents. After defining these two types of HS currents, we give relations between the covariant and holomorphic forms of HS currents.
In section 4, by using the relations between covariant and conformal HS currents in section 3, we obtain conservation equations and trace anomalies for the HS currents. This is the inverse step compared to the derivations of the holomorphic U (1) current and EM tensor in section 2.
We show that the relations in section 3 and some assumptions for the currents are sufficient to determine the explicit forms of trace anomalies for HS currents. For the classically traceless spin µνλ , it acquires the following quantum correction: This is considered as a spin-3 generalization of the trace anomaly for EM tensor. For the spin 4 current J (4) µνρσ , it is classically traceless but it acquires the quantum anomaly given by A generalization to higher spins than 4 is also possible but the calculation becomes more complicated.
In section 5, we consider a chiral theory where the central charges in the left and right handed sectors are different. In this case, we can obtain a generalization of the gauge(or gravitational) anomalies for higher-spin currents. We first review how we get the gravitational anomaly from the relations obtained in section 3, and then generalize it to spin 3 and 4 currents. For the spin 3 current, the generalization of the gauge anomaly becomes Here · · · represents classical violation of the conservation equation for matter currents in the electric and gravitational background. +(−) corresponds to the right (left) handed fermion.
These results can be applied to derive the HS fluxes of Hawking radiation. The relations between covariant and conformal HS currents obtained in section 3 provide another derivation of fluxes of HS currents in Hawking radiation. These relations are equivalent to solving the conservation equations and trace anomaly equations for HS currents. Hence the derivation gives a generalization of the Christensen and Fulling's method [11], in which the conservation equation of the EM tensor and the trace anomaly equation are solved with the regularity condition at the horizon. On the other hand, as we see in section 5, these relations can be rewritten as a generalization of the gauge anomaly. By applying these anomaly equations to black holes, it gives a generalization of the anomaly method [4,5] (see also appendix of [13] and [14]). These two derivations also clarify some points which were obscure in the previous papers [8,9]. We will discuss these applications in a separate paper [15].
In appendix A, we summarize the relations between holomorphic and covariant HS currents up to spin 4.

U(1) current and EM tensor
In this section, we review a derivation of the holomorphic U (1) and EM tensor in the background of U (1) gauge and gravitational fields. These holomorphic quantities are obtained by solving conservation equations together with the anomaly equations.
Throughout this paper we employ the conformal gauge ds 2 = e ϕ dudv for the gravitational background and the Lorenz gauge ∇ µ A µ = 0 for the gauge field background.
First we derive the holomorphic U (1) current. The U (1) current J µ satisfies the conservation equation ∇ µ J µ = 0, (2.1) and the chiral anomaly for the chiral current J 5µ is given by Here the charge of the field is set e = 1. F µν is the field strength of the background gauge field and ǫ µν is the covariant antisymmetric tensor, ǫ uv = 2e −ϕ = g uv . The chiral current is related to the gauge current by J 5µ = ǫ µν J ν . From eqs. (2.1) and (2.2), we find where the gauge conditions are used. Hence we define the (anti-)holomorphic U (1) currents as follows: The holomorphic U (1) currents generate a combination of the holomorphic gauge transformation, which is a combination of gauge and chiral transformations. Note that these currents are not covariant under the U (1) gauge transformations.
Next we derive the holomorphic EM tensor. The conservation equation of the matter EM tensor is given by The r.h.s. represents dissipation of the energy in the matter sector to the background gauge field. The trace anomaly of the EM tensor is given by where c is the central charge of the matter field and R is the Ricci scalar From these equations, we obtain Thus we define the holomorphic energy-momentum tensor as The anti-holomorphic one is defined similarly. These currents play a central role in conformal field theories since they generate conformal transformations, which is a combination of the general coordinate, Weyl and chiral transformations.

Holomorphic and covariant HS currents
In the previous section the relations between holomorphic and covariant quantities are obtained in the cases of the U (1) current and energy-momentum tensor. In this section, we give a generalization of such relations to higher-spin (HS) currents. We consider fermionic fields in the gravitational and electric backgrounds, and construct holomorphic and covariant currents from them. Then we investigate the relations between these currents.

Holomorphic HS currents
In order to construct the holomorphic higher-spin currents from fermionic fields, let us recall some properties of the fermion in the two dimensions. The equation of motion for the righthanded fermion with unit charge is given by in the gravitational and electric backgrounds (ϕ, A µ ). In the Lorentz gauge, the gauge field can be written locally as where η(u, v) is a scalar field. Since gravitational fields and gauge fields are not generally holomorphic, ψ(u, v) is not holomorphic either. In order to construct holomorphic quantities from the fermion field, we define a new field Ψ as Then the equation (3.1) becomes ∂ v Ψ = 0 and hence Ψ is holomorphic. Similarly we can define so that Ψ † also becomes holomorphic.
Regularized holomorphic currents are constructed from these holomorphic fields. For example, the holomorphic U (1) current can be defined as where the point splitting regularization is used and Ψ has the following operator product expansion, Note that we have not attached a Wilson line phase in the regularization, since the gauge field is not holomorphic and the Wilson line phase breaks the holomorphy. As a result, the holomorphic U (1) current is not gauge invariant. We can also construct holomorphic currents : ∂ n u Ψ † ∂ m u Ψ(u) : in the same way.
In order to clarify the difference between the holomorphic current and the ordinary covariant current, let us consider the covariant U (1) current J u in the electric background. We here omit the gravitational background for simplicity. J u can be defined as In contrast with the holomorphic U (1) current, we have attached the Wilson line phase in the regularization, so this current is gauge invariant but not holomorphic. By using (3.3), (3.4) and the operator product expansion (3.6), the covariant U (1) current can be related to the holomorphic U (1) current (3.5) as follows, equations and anomalies for these HS currents. We will discuss it in the next section.
In the following subsections, we will consider a generalization of the relation (3.8) to the HS currents. Instead of considering each HS currents separately, it turns out that it is useful to introduce the following generating function of the holomorphic currents .
This should be understood as a formal power series in terms of the parameters a and b around the position u. This function is holomorphic but not gauge covariant. In the subsection 3.3, we will construct a generating function for the covariant currents, and then give a relation between these two functions.
We here comment on the transformation property of the fermion field Ψ under (holomorphic) gauge transformations. In the Lorentz gauge, there remains residual holomorphic gauge symmetry, Under this transformation, Ψ(u) transforms as a field with twice the charge of ψ, This clarifies a point which we did not explain explicitly in [9], i.e., we there used this transformation property for the holomorphic field under the holomorphic transformation connecting a suitable gauge at infinity and a suitable one near the horizon.

Covariant HS currents
Now we will define the (u · · · u)-component of the covariant HS currents constructed from the fermion ψ in the electric and gravitational backgrounds. Since these currents are covariant under holomorphic general coordinate transformations, u →ũ = f (u), it is convenient to define the coordinate which is invariant under these transformations.
In the rest of this section, we consider v to be a fixed coordinate and treat the system as a one-dimensional one with the coordinate u. Then, under the holomorphic general coordinate transformations, we can define an "invariant coordinate" x which satisfies ∂ x = e −ϕ ∂ u and regard u as a function of x, i.e. u = u(x). Since dx is invariant under the above holomorphic transformations, the point splitting regularization is also invariant if it is defined in the x coordinate, not in the u coordinate * . u(x + ǫ) is now expanded as a formal power series of ǫ as where we used only the relation ∂ x = e −ϕ ∂ u . It is important that the last expression does not explicitly depend on x. A field φ located at u(x + ǫ) is defined as the following expansion, Let's first consider the relation between the holomorphic and covariant EM tensor in the electric and gravitational backgrounds. As explained above, covariant regularization can be defined in the x coordinate as follows, (3.14) * Since the coordinate x is formally introduced as a function of u, v should be kept fixed if a formula contains x explicitly. A derivative with respect to v must be taken only after the x coordinate is removed. derivative is defined by Hence e − 5 4 ϕ ∇ u ψ † and e − 1 4 ϕ ψ † transform as scalars under holomorphic coordinate transformations. Therefore T uu transforms as a weight 2 tensor.
This EM tensor can be rewritten in terms of the holomorphic fields by using eqs. (3.3) and (3.4) as and the gauge field is written by the scalar field η, eq. (3.2). By using the following operator product expansion we find This relation is equivalent to eq. (2.8) with the following identification of the holomorphic EM tensor, Generalizing these definitions of the covariant currents, we define the covariant HS current J (n) u···u as follows, In these notations, we denote the U(1) currents and the EM tensors as J u = J (1) The explicit forms of the relations between the covariant and holomorphic currents with spin 3 and 4 will be given in section 4.

Generating functions of HS currents
Instead of studying each HS current separately, it is simpler and more systematic to consider a generating function of the HS currents. We define the following generating function of the covariant HS currents as a formal power series with respect to a parameter a, In substituting the definition of J (n+1) u···u , we use the following relation, (3.23) † Our definitions of the HS currents are different from those of the W1+∞ algebra in [12]. Their HS currents are given by combining our HS currents and the derivative of them.
where u 0 is a fixed value on the u coordinate. A similar calculation can be done for the term including ψ. Then G cov (a) can be represented as .
where the gauge field is represented by using η, eq.(3.2). In this expression, in order to distinguish the quantum contributions from the classical ones, we recovered the Planck's constant . By expanding the relation with respect to the parameter a, we can obtain equations which relate the covariant HS currents with a sum of the holomorphic currents in the electric and gravitational background.
In the next section, we investigate the relations for spin 3 and 4 currents, and discuss the conservation equations, trace anomalies and generalizations of gauge (or gravitational) anomalies.

Trace anomalies for HS currents
In section 2, we derived the relation between T uu and t(u) from the conservation equation (2.5) and the anomaly equation (2.6). In this section, we take an inverse step for the HS currents. We first provide relations between the holomorphic and covariant HS currents from equation (3.25), and then, by using these relations, evaluate their conservation equations and trace anomalies.
In order to fix the definitions of the currents, we impose the following assumptions about the currents: 1. Anomalies appear in the trace parts of the currents only.

The covariant currents are totally symmetric.
Under these assumptions, we will obtain the conservation equations and trace anomalies for the spin 3 and 4 currents. The derivation can be straightforwardly applied to general HS currents, though calculations become more complicated.
We use the following notations of the currents in this section. J

Trace anomaly for spin 3 current
We here consider the spin 3 current. From eq.(3.20), the covariant spin 3 current is given by The corresponding holomorphic spin 3 current is The following relation between these currents can be derived by expanding G cov (a) (3.25) and taking the a 2 terms as By taking the derivative of eq. (4.3) with respect to v, we find Here we have used the equations in appendix A to describe the holomorphic currents in terms of the covariant currents. Since, as mentioned above, we assume that anomalies arise only in the trace part of the currents, we regard the last term in eq. (4.4), which is a quantum contribution, as the covariant derivative of the trace anomaly, Thus the uu component of the conservation equation becomes Here we have multiplied (4.4) and (4.5) by g uv . From this equation we may naively guess the general components of the conservation equation as follows, where the indices ν, ρ are symmetrized. But this is not traceless at the classical level, which contradicts with the second assumption. Note that terms proportional to g νρ can be added to the conservation law without affecting eq. (4.6). Thus by using this freedom, we can make the r.h.s. of the conservation equation traceless with respect to ν and ρ, This satisfies the three conditions we require.
Next the trace anomaly J (3) vuu can be read from equation (4.5), This can be covariantized as In order to check the consistency with the conservation law (4.8), we calculate J is given by where L 0 is the Lagrangian for the free fermion in the electric and gravitational backgrounds, A µ and g µν , respectively. We also introduce the following effective action for these gauge fields, The expectation values of the HS currents in these backgrounds are given by (4.14) Then the conservation equation (4.8) indicates that the effective action is invariant under the following infinitesimal transformations of the background fields, where ξ µν is a symmetric traceless parameter.
This transformation law is valid only for the weak B (3) field limit, i.e. we have assumed that the rank 3 gauge field B (3) was originally absent. Since the OPE between spin 3 currents generate higher-spin currents, they no longer form a closed algebra, contrary to the spin 1 or spin 2 currents. Hence higher-spin gauge symmetries larger than 2 and their backgrounds must be considered as W ∞ gauge symmetry and gauge fields as a whole. This is beyond the scope of the present paper.

Trace anomaly for spin 4 current
The covariant spin 4 current is given from eq. (3.20) by (4.18) and the corresponding holomorphic current is The relation between these two currents are obtained from a 3 terms of the equation (3.25) as (4.20) First we derive the uuu component of the conservation equation from this equation by taking the derivative of (4.20) with respect to v and multiplying g uv , As in the case of the spin 3 current, we regard the last term as the contribution of the trace anomaly because it is proportional to and quantum. From the assumptions 1 and 2, J vuuu is given by From (4.21), we guess the covariant conservation equation as Next we add appropriate terms proportional to g µν so that this conservation equation becomes classically traceless. In general, one can construct a rank 3 traceless symmetric tensor from any rank 3 symmetric tensor B νρσ by subtracting the trace part (g νρ B µ µσ + g σρ B µ µν + g σν B µ µρ )/4.
We define C ν as the trace of (4.23), Note that C ν includes the traces of the covariant spin 2 and 3 currents which vanish classically but not at the quantum level due to the trace anomalies. According to our assumption 1, we treat such anomalous quantities as contributions of the trace anomaly. Therefore we defineC ν as C ν without the anomalous terms, and construct a new conservation equation, which are classically traceless, Next the uu component of the trace anomaly can be read from (4.22), Then general components of the trace anomaly have the following form, where A is not fixed from (4.27) only. We can determine A by imposing consistency of (4.28) with (4.26). The trace of (4.26) becomes whereF ≡ ǫ µν F µν /2 = g uv F uv . On the other hand, the divergence of the (4.28) is By comparing these two equations, A is determined as As a result, we obtain the trace anomaly of the spin 4 current, As in the case of the rank 3 current, we can evaluate the transformation of the background fields from the conservation equation (4.26), where ξ µνρ denotes a symmetric traceless parameter.

Higher-spin gauge anomalies
In the previous section, we have obtained the conservation equations and trace anomalies in the HS currents by considering non-chiral theories; i.e. the anomaly coefficients are the same between the holomorphic and anti-holomorphic sectors.
In this section, we consider a chiral fermionic theory where we have c L left-handed fermions and c R ( = c L ) right-handed fermions. In this case, the conservation equation becomes anomalous.
This is a generalization of the gauge or gravitational anomalies to the HS currents. If these HS currents are coupled to HS gauge fields, these violation of conservation equations lead to quantum violation of HS local symmetries.
We here remark that, in the presence of c R right-handed and c L left-handed fermions, the coefficients of the anomalous terms in the (u · · · u) sector are multiplied by c R , and those in (v · · · v) sector by c L .
In the following of this section, we will derive the anomalous conservation equations for the currents up to rank 4.

U(1) gauge and gravitational anomalies
In this subsection, we reproduce the gauge and gravitational anomalies from the relations between the (anti-) holomorphic and covariant U (1) and spin 2 currents.
First we consider the U (1) current. The relations in the present case with c L = c R become where we have used the Lorenz gauge condition ∂ u A v = −∂ v A u . They can be written in the covariant forms as Thus, if c L = c R , the gauge symmetry is broken by the anomaly.
Next we consider the energy-momentum tensor. Now the relation (2.8) is modified as We can also obtain a similar equation for the right-handed fermion. By taking derivatives of them, we obtain In the case of the non-chiral theory (c L = c R ), we can regard the anomalous terms as the contribution of the trace anomaly. However, in the case c L = c R , the terms proportional to (c L − c R ) cannot be regarded as the contribution of the trace anomaly. As a result, we obtain the following anomalous conservation equation and trace anomaly equation: The first equation reproduces the gravitational anomaly for the covariant EM tensor.

Spin 3 and 4 gauge anomalies
We have shown that our method reproduces the correct anomaly equations for the rank 1 and 2 currents in the chiral theory. We further consider a generalization to higher-spin currents.
First we study the rank 3 current. The equation (4.4) now becomes We also obtain a similar equation for J (3) vvv . As in the case of the energy-momentum tensor, we cannot regard the anomalous term proportional to (c R − c L ) as the contribution of the trace anomaly.
Equations consistent with (5.11) can be given as follows: This is the spin 3 generalization of the gauge or gravitational anomaly. Note that, the conservation equation has been modified, but the transformation properties of the background gauge fields (4.15) -(4.17) are not changed, since they are classical properties.
We can similarly obtain a generalization to the rank 4 current; g νρĈσ + g ρσĈν + g σνĈρ , (5.14) Here we have modifiedC ν toĈ ν including the anomalous terms as follows, This is the spin 4 generalization of the gauge and gravitational anomalies. The r.h.s. of (5.14) contains both of classical and quantum parts. The classical parts arises due to the same reason as in the non-chiral case in section 4. The quantum parts are the anomalies.
In this paper, we considered a two-dimensional theory of fermions in the electric and gravitational backgrounds and obtained a generalization of the gauge, gravitational and trace anomalies for higher-spin (HS) currents up to spin 4. In order to derive these anomalies, we started from the relation between holomorphic and covariant forms of HS currents in the electric and gravitational backgrounds.
These anomaly equations can be applied to derive the higher-spin fluxes of Hawking radiation. This will be discussed in a separate paper [15].
In the cases of spins 1 and 2, the forms of anomalies can be determined by the descent equations and they have nice geometrical meanings. It will be interesting to investigate higherspin anomalies than 4, and examine whether there are any systematic structures in the form of anomalies.
Finally we notice that the anomalies we obtained are specific to HS currents constructed from fermions. If they are constructed from bosons, their anomalies have different combinations with different coefficients.