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Stability of Antisymmetric Tensor Fields of Chern-Simons Type in AdS Spacetime

Stability of massive antisymmetric tensor fields with the Chern-Simons type action in anti de Sitter spacetime is studied. It is found that there exists a complete set of solutions whose energy is conserved and positive definite if the mass is positive. Scalar products of the solutions are shown to be well-defined and conserved. In contrast to the previously studied scalar field case there is no other set of stable solutions with a different kind of boundary condition.


Introduction
Anti de Sitter (AdS) spacetime is a maximally symmetric spacetime with a negative constant curvature. It naturally appears as a solution of Einstein equation with a negative cosmological constant. It also appears in compactifications of higher dimensional supergravities in the Kaluza-Klein theory [1]. Recently interest in field theories in AdS spacetime has been much increased due to their relevance to the AdS/CFT correspondence in the string/M theory [2,3,4] (For a review see ref. [5].). An important issue of field theories in AdS spacetime is their stability, which was previously discussed in refs. [6,7,8]. (For recent studies on the stability see ref. [9].) Another important issue is a choice of boundary conditions of fields at spatial infinity. In ref. [10] boundary conditions are chosen such that the Cauchy problem for field equations is well-defined by requiring the conservation of the scalar product of fields.
The purpose of this paper is to study the stability of free massive antisymmetric tensor fields of arbitrary rank n in AdS spacetime. The string/M theory and supergravities contain antisymmetric tensor fields, which play an important role. Therefore, their stability is an important issue. There are two types of theories of antisymmetric tensor fields. One type of theories have an action with the second order kinetic term of the Maxwell type and the other type of theories have an action with the first order kinetic term of the Chern-Simons type [11]. (Equivalent theories to the latter were studied in refs. [12,13].) Both types of theories appear in supergravities [14]. In this paper we only consider the Chern-Simons type theories, which are theories of n-th rank antisymmetric tensor fields in d = 2n + 1 dimensions. As a preparation for study of the stability we first obtain the general solution of the field equation. Then we obtain conditions for the stability by studying the conservation and the positivity of the energy. We also study scalar products of the solutions.
The conditions for the stability of scalar fields in AdS spacetime were obtained in refs. [7,8]. A free massive scalar field theory in d-dimensional AdS spacetime is stable if the mass m satisfies where a −1 is the radius of AdS spacetime. More precisely, there exists a complete set of solutions of the field equation whose energy is conserved and positive definite.
Furthermore, when the mass satisfies there exists another set of stable solutions satisfying a different kind of boundary condition at spatial infinity. In the latter case the coefficient of the improvement term in the energy-momentum tensor must take a particular value. Scalar fields can be stable even if the mass squared is negative due to a positive contribution from the kinetic term to the energy.
As in the scalar field case we find that there exists a complete set of solutions for antisymmetric tensor fields whose energy is conserved and positive definite if the mass is positive m > 0. There are three kinds of improvement terms in the energymomentum tensor for antisymmetric tensor fields of rank n ≥ 2. The coefficients of these terms can take arbitrary values although they do not contribute to the energy. The scalar products of the solutions are shown to be well-defined and conserved.
In contrast to the scalar field case, however, there is no other set of stable solutions. The conservation of the energy allows another set of solutions satisfying a different kind of boundary condition at spatial infinity as in the scalar field case but their energy turns out to be divergent. Therefore, only one kind of boundary condition is possible for antisymmetric tensor fields with the Chern-Simons type action. A Chern-Simons type theory of the second rank atisymmetric tensor fields in five-dimensional AdS spacetime was previously studied in ref. [15] in the context of the AdS/CFT correspondence. By using a different approach it was found there that only one kind of boundary condition is possible, which is consistent with our result.
In the next section we introduce the first order action of antisymmetric tensor fields in AdS spacetime and obtain the energy-momentum tensor. In sect. 3 the general solution of the field equation is obtained in terms of the hypergeometric functions. The conservation and the positivity of the energy are discussed in sects. 4 and 5 respectively. In sect. 6 we show that there exists a well-define and conserved scalar product for the solutions. In Appendix we give a construction and useful identities of spherical harmonics for antisymmetric tensors on the (d−2)-dimensional sphere S d−2 .

Antisymmetric tensor fields in AdS spacetime
We consider antisymmetric tensor fields in d-dimensional AdS spacetime. AdS spacetime is a maximally symmetric spacetime and has the metric where µ, ν = 0, 1, · · · , d − 1 are d-dimensional world indices and the constant a −1 is the radius of AdS spacetime. The time coordinate t has a range −∞ < t < ∞, which corresponds to considering the universal covering of AdS spacetime. The radial coordinate ρ has a range 0 ≤ ρ < π 2 with the spatial infinity at ρ = π 2 . θ a and h ab (a, b = 1, 2, · · · , d − 2) are coordinates and the metric of the (d − 2)-dimensional unit sphere S d−2 . Non-vanishing components of the Christoffel connection are where γ a bc is the Christoffel connection of S d−2 . The Riemann tensor is given by Our conventions for the curvature tensors are R µν For other properties of AdS spacetime see, e.g. ref. [5]. We consider a free theory of a complex antisymmetric tensor field B µ 1 ···µn of rank n in d-dimensional AdS spacetime for d = 2n + 1. The Chern-Simons type action [11] is where ǫ µ 1 ···µ 2n+1 is the totally antisymmetric tensor and * denotes the complex conjugation. The reality of the action requires that the mass m is real. When m = 0, the action consists of only the kinetic term, which do not depends on the metric but is invariant under general coordinate transformations. This is an action of a topological field theory. We do not discuss the stability of the m = 0 case since there is no local degrees of freedom and the energy is zero. The field equation derived from this action is We define the energy-momentum tensor of the theory as a variation of the action with respect to the metric. To do this we need an action for general metric, which reduces to the original action (2.4) for the AdS metric (2.1). We use the action where α, β and γ are arbitrary constant parameters. The parameter µ is chosen as so that this action coincides with eq. (2.4) for the AdS metric. The last three terms in eq. (2.6) containing the curvature tensors are generalizations of the wellknown Rφ 2 term in the scalar field theory. These terms give improvement terms in the energy-momentum tensor. Note that we do not need to introduce the term R µτ νσ B * µν µ 3 ···µn B τ σµ 3 ···µn since it is related to the last term in eq. (2.6) by the Bianchi identity of the Riemann tensor. From eq. (2.6) we obtain the energy-momentum tensor as There is no contribution from the kinetic term.
The energy is defined as follows [6]. The energy-momentum tensor (2.8) satisfies D µ T µν = 0 for arbitrary α, β and γ when the field equation is used. We can construct a conserved current for each Killing vector ξ µ satisfying D µ ξ ν + D ν ξ µ = 0. The energy is the charge of this current for a timelike Killing vector ξ µ = (1, 0, · · · , 0) where the integral is over (d − 1)-dimensional space.

Solutions of the field equation
We shall obtain the general solution of the field equation (2.5). It is more convenient to rewrite the field equation in another form. Applying ∂ µ 1 to eq. (2.5) we obtain a constraint where D µ is the covariant derivative in AdS spacetime. Applying the differential operation in the first term of eq. (2.5) to eq. (2.5) again and using eq. (2.5) in the second term we obtain the second order equation We first solve these equations and then substitute the solutions into the first order equation (2.5) to obtain further conditions. By the constraint (3.1) the components B ta 2 ···an and B tρa 3 ···an are not independent but can be expressed by B a 1 ···an and B ρa 2 ···an . The field equation (3.2) for [µ 1 · · · µ n ] = [a 1 · · · a n ], [ρa 2 · · · a n ] gives where ∇ a is the covariant derivative on S d−2 using the Christoffel connection γ a bc in eq. (2.2). The differential operators L 1 and L 2 are defined as It can be shown that when eqs. We decompose the antisymmetric tensor field into transverse and longitudinal modes by using spherical harmonics Y (l) a 1 ···an (θ) for antisymmetric tensor fields on The quantum number l takes values l = 0, 1, 2, · · · for n = 0 and l = 1, 2, 3, · · · for n ≥ 1. In the Appendix we sketch how to construct Y (l) a 1 ···an and give some useful identities. Using the spherical harmonics the components of the antisymmetric tensor field are decomposed as The factor sin ρ cos ρ in front of R 3 is for later convenience. Substituting eq. (3.5) into the constraint (3.1) for [µ 2 · · · µ n ] = [a 2 · · · a n ], [ρa 3 · · · a n ], R 5 , · · ·, R 8 are expressed in terms of R 1 , · · ·, R 4 as Other components of the constraint (3.1) are then automatically satisfied. R 1 , · · ·, R 4 , in turn, are determined by solving eq. (3.3). To solve eq. (3.3) it is convenient to change the variable as v = sin 2 ρ. (3.7) Let us first consider R 1 . We define the function f 1 (v) by 3.9) and N 1 andÑ 1 are complex constants. The transverse part of the first equation in We choose the parameters κ and λ as Then, eq. (3.10) becomes a hypergeometric equation where (3.14) The solution which gives B a 1 ···an regular at ρ = 0 is a hypergeometric function The equation for R 4 can be solved similarly. We put where The equations for R 2 and R 3 are slightly more complicated since they are coupled equations. The longitudinal part of the first equation and the transverse part of the second equation in (3.3) become These equations can be diagonalized by defining new functionsR 2 andR 3 aŝ these equations become hypergeometric equations on f 2 and f 3 . The solutions which give B a 1 ···an and B ρa 2 ···an regular at ρ = 0 are where a 2 = 1 2 (λ + l − n + 1 − ω 2 ), b 2 = 1 2 (λ + l − n + 1 + ω 2 ), c 2 = l + n + 1, By these relationsN 4 andN 3 are related toN 1 andN 2 respectively and independent degrees of freedom are reduced from four to two.

Conservation of the energy
We first consider the conservation of the energy as a condition for the stability. From eq. (2.9) the time derivative of the energy (2.10) is given by where the integral is over S d−2 . T ρt can be written as Here, ∇ a (· · ·) a represents total derivative terms on S d−2 , which vanish in the integral (4.1). Substituting eq. (3.5) into eq. (4.1) it is divided into three independent parts To evaluate the right hand side of eq. (4.1) we need boundary behaviors of the functions R i for ρ → π 2 . They can be obtained from the behavior of the hypergeometric function 2 F 1 (a, b, c; v) for v → 1. When λ − 2n is not an integer, we find that near the boundary R's behave as with a i , b i and c i given in eqs. (3.14), (3.19) and (3.27). We see that the value of λ determines boundary behaviors of the solutions. The case in which λ − 2n is an integer is discussed at the end of this section.
Let us first considerĖ 1 . We obtaiṅ (4.7) The last three terms represent higher order terms than the first three respectively. When we choose λ = λ + , the first term automatically vanishes. For the second and the third terms to vanish we have to require either (i) On the other hand, when we choose λ = λ − , the third term vanishes automatically and we have to require either (iii) A 1 = 0 or (iv) B 1 = 0, 2(2λ − 2n + 1)α + β = 0. It can be shown that the conditions (i) and (ii) are equivalent to (iii) and (iv) respectively. It is enough to consider two cases (i) and (iv) and set B 1 = 0. For B 1 = 0 only C term survives among the higher order terms.
The conditions forĖ 2 andĖ 3 to vanish can be obtained in a similar way. It first requires B 2 = B 3 = B 4 = 0. The remaining terms arė (4.8) For λ = λ + all these terms vanish automatically. For λ = λ − we have to require the coefficients to vanish, which fixes the parameters α, β and γ.
To summarize, the conservation of the energy leads to one of the two possibilities (I) λ = λ + , B i = 0, In the case (II) the coefficients of the improvement terms must be chosen as (4.10) The conditions B i = 0 require a i = 0 or b i = 0, which lead to the quantization of ω i , , , where k i are non-negative integers. Then, the hypergeometric functions in eqs.
The conservation of the energy requires B i = 0 as above, which restrict the values of k ′ i to k ′ i = N, N + 1, · · ·. Redefining k i = k ′ i − N = 0, 1, · · · we recover the values of ω i in eq. (4.11). Therefore, the results obtained for non-integer λ − 2n is also valid for integer λ − 2n.

Positivity of the energy
We next consider the second condition of stability, i.e., the positivity of the energy. The integrand of the energy (2.10) is We see that the contributions from the α, β and γ terms in the action (2.6) are total derivative. Substituting eq. (3.5) into eq. (5.1) it is divided into three independent parts (5.2) E 1 , E 2 and E 3 depend only on R 1 , (R 2 , R 3 ) and R 4 respectively.
where ∆E 1 is total derivative terms The integral of the bulk term in eq. (5.3) is convergent for λ = λ + as seen from the boundary behavior of R 1 in eq. (4.5). However, it is divergent for λ = λ − . Therefore, the choice λ = λ − is not allowed and only λ = λ + is possible. In the case of scalar fields discussed in refs. [7,8] the mass term in the energy is also divergent but it is canceled by a divergent contribution from the kinetic term. Both of λ = λ + and λ = λ − are possible in the scalar field theories. In the present theory there is no kinetic term in the energy since the kinetic term of the action is a topological term. For λ = λ + the bulk term in eq. (5.3) is obviously positive definite when m > 0. The boundary term ∆E 1 vanishes since it has a positive power of cos ρ.
Similarly, E 2 and E 3 are given by where the boundary terms are (5.6) For λ = λ + the bulk integrals in eq. (5.5) are convergent and are positive definite when m > 0. The boundary terms ∆E 2 and ∆E 3 vanish in the same way as ∆E 1 .
It is thus proved that the energy is well-defined and positive definite for the case (I) in eq. (4.9) if the additional condition m > 0 is satisfied. The case (II) in eq. (4.9) is not allowed since the energy is divergent. Therefore, there exists only one complete set of solutions corresponding to the case (I) for antisymmetric tensor fields with the Chern-Simons type action.

Scalar product
Finally we consider a scalar product of the fields. The scalar product of two solutions B 1µ 1 ···µn and B 2µ 1 ···µn is defined as The integral in eq. (6.1) is convergent since √ −gJ t = O (cos ρ) 2λ−4n+1 for ρ → π 2 and 2λ − 4n + 1 > −1 for λ = λ + and m > 0. The scalar product (6.1) is also conserved d dt where ∇ a is the covariant derivative on S d−2 . Therefore, restricting to the unit sphere they represent transverse tensors on the sphere. The relation between T i 1 ···in and T a 1 ···an is T a 1 ···an = r n ∂n i 1 ∂θ a 1 · · · ∂n in ∂θ an T i 1 ···in . (A. 4) In the Cartesian coordinates spherical harmonics for such transverse antisymmetric tensors for n ≥ 1 are given by Y (l) i 1 ···in (θ) = r −l C [i 1 ···inj 1 ](j 2 ···j l ) x j 1 x j 2 · · · x j l . (A.5) Here, C [i 1 ···inj 1 ](j 2 ···j l ) is a constant coefficient, which is antisymmetric in i 1 , · · · , i n , j 1 and symmetric in j 2 , · · · , j l , and is traceless with respect to any pair of the indices. Spherical harmonics for n = 0 are given by Y (l) (θ) = r −l C (j 1 ···j l ) x j 1 · · · x j l , (A. 6) where C (j 1 ···j l ) is symmetric in j 1 , · · · , j l and is traceless with respect to any pair of the indices. Note that l takes values l = 0, 1, 2, · · · for n = 0 and l = 1, 2, 3, · · · for n ≥ 1. One can easily check that these tensors indeed satisfy the conditions (A.2). Applying the Laplacian ∆ d−1 = δ ij ∂ i ∂ j in R d−1 we find On the other hand, in the polar coordinates we have There is a duality relation between Y (l) a 1 ···am and Y (l) a 1 ···a d−m−3 , which we use in sect. 3. By appropriately choosing the coefficients C's in eqs. (A.5) and (A.6) the relation can be written as One can easily check that both hand sides of this equation are transverse and have the same eigenvalue of ∇ a ∇ a . The normalization factor on the right hand side is determined by repeated applications of this relation.