Dynamical symmetry enhancement near IIA horizons

We show that smooth type IIA Killing horizons with compact spatial sections preserve an even number of supersymmetries, and that the symmetry algebra of horizons with non-trivial fluxes includes an sl(2,R) subalgebra. This confirms the conjecture of [1] for type IIA horizons. As an intermediate step in the proof, we also demonstrate new Lichnerowicz type theorems for spin bundle connections whose holonomy is contained in a general linear group.


Introduction
It has been conjectured in [1], following earlier work in [2] and [3], that • the number of Killing spinors N, N = 0, of Killing horizons in supergravity is given by where N − ∈ N >0 and D E is a Dirac operator twisted by a vector bundle E, defined on the spatial horizon section S, which depends on the gauge symmetries of the supergravity theory in question, and • that horizons with non-trivial fluxes and N − = 0 admit an sl(2, R) symmetry subalgebra.
This conjecture encompasses the essential features of (super)symmetry enhancement near black hole Killing horizons, and some features of the same phenomenon near brane horizons, previously obtained in the literature based on a case-by-case investigation [4,5,6]. Symmetry enhancement near black hole and brane horizons has been instrumental in the development of the AdS/CFT correspondence [7]. So far, this conjecture has been established for minimal 5-dimensional gauged supergravity, D=11 M-theory, and D=10 IIB supergravity [2,1,3].
The main purpose of this paper is to prove the above conjecture for Killing horizons in IIA supergravity. The proof is based on three assumptions. First, it is assumed that the Killing horizons admit at least one supersymmetry, second that the near horizon geometries are smooth and third that the spatial horizon sections are compact without boundary 1 . It turns out that for IIA horizons, the contribution from the index of D E in the expression for N in (1.1) vanishes and therefore one concludes that IIA horizons always preserve an even number of supersymmetries, i.e.
(1.2) Furthermore from the second part of the conjecture, one concludes that all supersymmetric IIA horizons with non-trivial fluxes admit an sl(2, R) symmetry subalgebra.
To prove the conjecture, we first adapt the description of black hole near horizon geometries of [8,9] to IIA supergravity. The metric and the remaining fields of IIA horizons are given in (2.9). We then decompose the Killing spinor as ǫ = ǫ + + ǫ − using the lightcone projectors Γ ± ǫ ± = 0 and integrate the Killing spinor equations (KSEs) of IIA supergravity along the two lightcone directions. These directions arise naturally in the description of near horizon geometries. As a result, the Killing spinors of IIA horizons can be written as ǫ = ǫ(u, r, η ± ), where the dependence on the coordinates u, r is explicit and η ± are spinors which depend only on the coordinates of the spatial horizon section S given by the equation u = r = 0.
As a key next step in the proof, we demonstrate that the remaining independent KSEs are those obtained from the KSEs of IIA supergravity after naively restricting them to S. In particular, we find after an extensive use of the field equations and Bianchi identities that all the integrability conditions that arise along the lightcone directions, and the mixed directions between the lightcone and the S directions, are automatically satisfied. The independent KSEs on S split into two sets {∇ (±) , A (±) } of two KSEs with each set acting on the spinors η ± distinguished by the choice of lightcone direction, where ∇ (±) are derived from the gravitino KSE of IIA supergravity and A (±) are associated to the dilatino KSE of IIA supergravity. In addition we demonstrate that if η − is a Killing spinor on S, then η + = Γ + Θ − η − also solves the KSEs, where Θ − depends on the fluxes and the spacetime metric.
To show that the number of Killing spinors of IIA horizons is even, it suffices to show that there are as many η + Killing spinors as η − Killing spinors. For this, we first identify the Killing spinors η ± with the zero modes of Dirac-like operators D (±) coupled to fluxes. These are defined as D (±) = D (±) + qA (±) , where D (±) is the Dirac operator constructed from ∇ (±) . It is then shown that for a suitable choice of q all zero modes of these Dirac-like operators are in 1-1 correspondence with the Killing spinors.
The proof of the above correspondence between zero modes and Killing spinors for the D (+) operator utilizes the Hopf maximum principle and relies on the formula (3.6). Incidentally, this also establishes that η + is constant. The proof for the D (−) operator uses the partial integration of the formula (3.9) and this is similar to the classical Lichnerowicz theorem for the Dirac operator. In both cases, the proofs rely on the smoothness of data and the assumption that S is compact without boundary.
Therefore, the number of Killing spinors of IIA horizons is N = N + + N − , where N ± are the dimensions of the kernels of the D (±) operators. On the other hand, one can show that the zero modes of D (−) are in 1-1 correspondence with the zero modes of the adjoint (D (+) ) † of D (+) . As a result N + − N − is the index of D (+) . This vanishes as it is equal to the index of the Dirac operator acting on the spinor bundle constructed from the 16 dimensional Majorana representation of Spin (8). As a result N + = N − and the number of supersymmetries preserved by IIA horizons is even, which proves the first part of the conjecture.
To prove that IIA horizons admit an sl(2, R) symmetry subalgebra, we use the fact that if η − is a Killing spinor then η + = Γ + Θ − η − is also a Killing spinor. To see this we demonstrate that if the fluxes do not vanish, the kernel of Θ − is {0}, and so η + = 0. Using the Killing spinors now constructed from η − and η + = Γ + Θ − η − , we prove that the spacetime admits three Killing vectors, which leave all the fields invariant, satisfying an sl(2, R) algebra. This completes the proof of the conjecture for IIA horizons.
The results presented above for horizons in IIA supergravity do not follow from those we have obtained for M-horizons in [3]. Although IIA supergravity is the dimensional reduction of 11-dimensional supergravity, the reduction, after truncation of Kaluza-Klein modes, does not always preserve all the supersymmetry of 11-dimensional solutions; for a detailed analysis of these issues see [10,11]. As a result, for example, it does not follow that IIA horizons preserve an even number of supersymmetries because M-horizons do as shown in [3]. However since we have shown that both IIA and M-theory horizons preserve an even number of supersymmetries, one concludes that if the reduction process breaks some supersymmetry, then it always breaks an even number of supersymmetries. This paper is organized as follows. In section 2, we identify the independent KSEs for IIA horizons. In section 3, we establish the equivalence between zero modes of D (±) and Killing spinors, and show that the number of supersymmetries preserved by IIA horizons is even. In section 4, we show that η + = Γ + Θ − η − = 0. In section 5, we prove that IIA horizons with non-trivial fluxes admit an sl(2, R) symmetry subalgebra and in section 6 we give our conclusions. In appendix A, we give a list of Bianchi identities and field equations that are implied by the (independent) ones listed in section 2. In appendix B, we identify the independent KSEs, and in appendix C we establish the formulae (3.6) and (3.9).

IIA fields and field equations
The bosonic field content of IIA supergravity [12,13,14,15] are the spacetime metric g, the dilaton Φ, the 2-form NS-NS gauge potential B, and the 1-form and the 3-form RR gauge potentials A and C, respectively. In addition, the theory has non-chiral fermionic fields consisting of a Majorana gravitino and a Majorana dilatino but these are set to zero in all the computations that follow. The bosonic field strengths of IIA supergravity in the conventions of [16] are (2.1) These lead to the Bianchi identities The bosonic part of the IIA action in the string frame is This leads to the Einstein equation the dilaton field equation and the 4-form field equation This completes the description of the dynamics of the bosonic part of IIA supergravity.

Horizon fields, Bianchi identities and field equations
The description of the metric near extreme Killing horizons as expressed in Gaussian null coordinates [8,9] can be adapted to include all IIA fields. In particular, one writes where we have introduced the frame e + = du, e − = dr + rh − 1 2 r 2 ∆du, e i = e i I dy I , (2.10) and the dependence on the coordinates u and r is explicitly given. Moreover Φ and ∆ are 0-forms, h, L and T are 1-forms, X, M andF are 2-forms, Y,H are 3-forms andG is a 4-form on the spatial horizon section S, which is the co-dimension 2 submanifold given by the equation r = u = 0, i.e. all these components of the fields depend only on the coordinates of S. It should be noted that one of our assumptions is that all these forms on S are sufficiently differentiable, i.e. we require at least C 2 differentiability so that all the field equations and Bianchi identities are valid. Substituting the fields (2.9) into the Bianchi identities of IIA supergravity, one finds that where d h θ ≡ dθ − h ∧ θ for any form θ. These are the only independent Bianchi identities, see appendix A. Similarly, substituting the horizon fields into the field equations of IIA supergravity, we find that the 2-form field equation (2.6) gives  14) and the 4-form field equation (2.8) gives where∇ is the Levi-Civita connection of the metric on S. In addition, the dilaton field equation (2.5) becomes It remains to evaluate the Einstein field equation. This gives Above we have only stated the independent field equations. In fact, after substituting the near horizon geometries into the IIA field equations, there are additional equations that arise. However, these are all implied from the above field equations and Bianchi identities. For completeness, these additional equations are given in appendix A.
To summarize, the independent Bianchi identities and field equations are given in (2.11)-(2.19).

Integration of KSEs along the lightcone
The KSEs of IIA supergravity are the vanishing conditions of the gravitino and dilatino supersymmetry variations evaluated at the locus where all fermions vanish. These can be expressed as where ǫ is the supersymmetry parameter which from now on is taken to be a Majorana, but not Weyl, commuting spinor of Spin (9,1). In what follows, we shall refer to the D operator as the supercovariant connection. Supersymmetric IIA horizons are those for which there exists an ǫ = 0 that is a solution of the KSEs. To find the conditions on the fields required for such a solution to exist, we first integrate along the two lightcone directions, i.e. we integrate the KSEs along the u and r coordinates. To do this, we decompose ǫ as where Γ ± ǫ ± = 0, and find that and and η ± depend only on the coordinates of the spatial horizon section S. As spinors on S, η ± are sections of the Spin(8) bundle on S associated with the Majorana representation. Equivalently, the Spin(9, 1) bundle S on the spacetime when restricted to S decomposes as S = S − ⊕ S + according to the lightcone projections Γ ± . Although S ± are distinguished by the lightcone chirality, they are isomorphic as Spin(8) bundles over S. We shall use this in the counting of supersymmetries of IIA horizons.

Independent KSEs
The substitution of the spinor (2.22) into the KSEs produces a large number of additional conditions. These can be seen either as integrability conditions along the lightcone directions, as well as integrability conditions along the mixed lightcone and S directions, or as KSEs along S. A detailed analysis, presented in appendix B, of the formulae obtained reveals that the independent KSEs are those that are obtained from the naive restriction of the IIA KSEs to S. In particular, the independent KSEs are and Evidently, ∇ (±) arise from the supercovariant connection while A (±) arise from the dilatino KSE of IIA supergravity as restricted to S . Furthermore, the analysis in appendix B reveals that if η − solves (2.26) then also solves (2.26). This is the first indication that IIA horizons admit an even number of supersymmetries. As we shall prove, the existence of the η + solution is also responsible for the sl(2, R) symmetry of IIA horizons.

Supersymmetry enhancement
To prove that IIA horizons always admit an even number of supersymmetries, it suffices to prove that there are as many η + Killing spinors as there are η − Killing spinors, i.e. that the η + and η − Killing spinors come in pairs. For this, we shall identify the Killing spinors with the zero modes of Dirac-like operators which depend on the fluxes and then use the index theorem to count their modes.

Horizon Dirac equations
We define horizon Dirac operators associated with the supercovariant derivatives following from the gravitino KSE as where However, it turns out that it is not possible to straightforwardly formulate Lichnerowicz theorems to identify zero modes of these horizon Dirac operators with Killing spinors.
To proceed, we shall modify both the KSEs and the horizon Dirac operators. For this first observe that an equivalent set of KSEs can be chosen by redefining the supercovariant derivatives from the gravitino KSE aŝ Similarly, one can modify the horizon Dirac operators as i . However, we shall not assume this in general. As we shall see, there is an appropriate choice of q and appropriate choices of κ such that the Killing spinors can be identified with the zero modes of D (±) .

A Lichnerowicz type theorem for D (+)
First let us establish that the η + Killing spinors can be identified with the zero modes of a D (+) . It is straightforward to see that if η + is a Killing spinor, then η + is a zero mode of D (+) . So it remains to demonstrate the converse. For this assume that η + is a zero mode of D (+) , i.e. D (+) η + = 0. Then after some lengthy computation which utilizes the field equations and Bianchi identities, described in appendix C, one can establish the equalitỹ provided that q = −1. It is clear that if the last term on the right-hand-side of the above identity is positive semi-definite, then one can apply the maximum principle on η + 2 as the fields are assumed to be smooth, and S compact. In particular, if then the maximum principle implies that η + are Killing spinors and η + = const.

A Lichnerowicz type theorem for D (−)
Next we shall establish that the η − Killing spinors can also be identified with the zero modes of a modified horizon Dirac operator D (−) . It is clear that all Killing spinors η − are zero modes of D (−) . To prove the converse, suppose that η − satisfies D (−) η − = 0. The proof proceeds by calculating the Laplacian of η − 2 as described in appendix C, which requires the use of the field equations and Bianchi identies. One can then establish the formula∇ The last term on the RHS of (3.9) is negative semi-definite if − 1 4 < κ < 0. Provided that this holds, on integrating (3.9) over S and assuming that S is compact and without boundary, one finds that∇ (−) η − = 0 and A (−) η − = 0. Therefore, we have shown that for q = −1 and − 1 4 < κ < 0, This concludes the relationship between Killing spinors and zero modes of modified horizon Dirac operators.

Supersymmetry enhancement
The analysis developed so far suffices to prove that IIA horizons preserve an even number of supersymmetries. Indeed, if N ± is the number of η ± Killing spinors, then the number of supersymmetries of IIA horizon is N = N + + N − . Utilizing the relation between the Killing spinors η ± and the zero modes of the modified horizon Dirac operators D (±) established in the previous two sections, we have that is the same as the principal symbol of the standard Dirac operator acting on Majorana but not-Weyl spinors, the index vanishes 2 [17]. As a result, we conclude that dim Ker D (+) = dim Ker (D (+) ) † , (3.13) where (D (+) ) † is the adjoint of D (+) . Furthermore observe that 14) and so Therefore, we conclude that N + = N − and so the number of supersymmetries of IIA horizons N = N + + N − = 2N − is even. This proves the first part of the conjecture (1.1) for IIA horizons.

Construction of η + from η − Killing spinors
In the investigation of the integrability conditions of the KSEs, we have demonstrated that if η − is a Killing spinor, then η + = Γ + Θ − η − is also a Killing spinor, see (2.30). Since we know that the η + and η − Killing spinors appear in pairs, the formula (2.30) provides a way to construct the η + Killing spinors from the η − ones. However, this is the case provided that η + = Γ + Θ − η − = 0. Here, we shall prove that for horizons with non-trivial fluxes and so the operator Γ + Θ − pairs the η − with the η + Killing spinors. We shall prove Ker Θ − = {0} using contradiction. For this assume that Θ − has a non-trivial kernel, i.e. there is η − = 0 such that If this is the case, then the last integrability condition in (B.1) gives that This in turn implies that and hence as η − is no-where vanishing. Next the gravitino KSE ∇ (−) η − = 0 implies that which can be simplified further using to yield∇ As η − is no-where zero, this implies that Substituting, ∆ = 0 and dh = 0 into (A.5), we find that (4.11) Applying the maximum principle on η − 2 we conclude that all the fluxes apart from the dilaton Φ andH vanish and η − is constant. The latter together with (4.8) imply that h = 0.
Next applying the maximum principle to the dilaton field equation (2.17), we conclude that the dilaton is constant andH = 0. Combining all the results so far, we conclude that all the fluxes vanish which is a contradiction to the assumption that not all of the fluxes vanish. This establishes (4.1).

Killing vectors
To begin, first note that the Killing spinor ǫ on the spacetime can be expressed in terms of η ± as which is derived after collecting the results of section 2.3. Since the η − and η + Killing spinors appear in pairs for supersymmetric IIA horizons, let us choose a η − Killing spinor. Then from the results of the previous section, horizons with non-trivial fluxes also admit η + = Γ + Θ − η − as a Killing spinors. Using η − and η + = Γ + Θ − η − , one can construct two linearly independent Killing spinors on the spacetime as To continue, it is known from the general theory of supersymmetric IIA backgrounds that for any Killing spinors ζ 1 and ζ 2 the dual vector field of the 1-form bilinear is a Killing vector and leaves invariant all the other fields of the theory. Evaluating, the 1-form bilinears of the Killing spinor ǫ 1 and ǫ 2 , we find that where we have set Moreover, we have used the identities which follow from the first integrability condition in (B.1), η + = const and the KSEs of η + .

The geometry of S
First suppose that V = 0. Then the conditions L Ka g = 0 and L Ka F = 0, a = 1, 2, 3, where F denotes collectively all the fluxes of IIA supergravity, imply that i.e. V is an isometry of S and leaves all the fluxes on S invariant. In addition, one also finds the useful identities which imply that L V φ − 2 = 0. There are further restrictions on the geometry of S which will be explored elsewhere.
A special case arises for V = 0 where the group action generated by K 1 , K 2 and K 3 has only 2-dimensional orbits. A direct substitution of this condition in (5.8) reveals that Since dh = 0 and h is exact such horizons are static and a coordinate transformation r → ∆r reveals that the horizon geometry is a warped product of AdS 2 with S, AdS 2 × w S.

sl(2, R) symmetry of IIA-horizons
To uncover the sl(2, R) symmetry of IIA horizons it remains to compute the Lie bracket algebra of the vector fields associated to the 1-forms K 1 , K 2 and K 3 . For this note that these vector fields can be expressed as where we have used the same symbol for the 1-forms and the associated vector fields. These expressions are similar to those we have obtained for M-horizons in [3] apart form the range of the index i which is different. Using the various identities we have obtained, a direct computation reveals that the Lie bracket algebra is which is isomorphic to sl(2, R). This proves the second part of the conjecture and completes the analysis.

Conclusions
We have demonstrated that smooth IIA horizons with compact spatial sections, without boundary, always admit an even number of supersymmetries. In addition, those with non-trivial fluxes admit an sl(2, R) symmetry subalgebra. The above result together with those obtained in [2,3] and [1] provide further evidence in support the conjecture of [1] regarding the (super)symmetries of supergravity horizons. It also emphases that the (super)symmetry enhancement that is observed near the horizons of supersymmetric black holes is a consequence of the smoothness of the fields.
Apart from exhibiting an sl(2, R) symmetry, IIA horizons are further geometrically restricted. This is because we have not explored all the restrictions imposed by the KSEs and the field equations of the theory -in this paper we only explored enough to establish the sl(2, R) symmetry. However, the understanding of the horizons admitting two supersymmetries is within the capability of the technology developed so far for the classification of supersymmetric IIA backgrounds [19] and it will be explored elsewhere. The understanding of all IIA horizons is a more involved problem. As such spaces preserve an even number of supersymmetries and there are no IIA horizons with non-trivial fluxes preserving 32 supersymmetries, which follows from the classification of maximally supersymmetric backgrounds in [18], there are potentially 15 different cases to examine. Of course, all IIA horizons preserving more than 16 supersymmetries are homogenous spaces as a consequence of the results of [20]. It is also very likely that there are no IIA horizons preserving 28 and 30 supersymmetries in analogy with a similar result in IIB [21]. However to prove this, it is required to extend the IIB classification results to IIA supergravity, see also [22].
We expect that our results on IIA horizons can be extended to massive IIA supergravity [15]. This will be reported elsewhere.

Appendix A Horizon Bianchi identities and field equations
We remark that there are a number of additional Bianchi identities, which are However, these Bianchi identities are implied by those in (2.11). There is also a number of additional field equations given by are implied by (2.17)

B Integrability conditions and KSEs
Substituting the solution of the KSEs along the lightcone directions (2.23) back into the gravitino KSE (2.20) and appropriately expanding in the r, u coordinates, we find that for the µ = ± components, one obtains the additional conditions Similarly the µ = i component of the gravitino KSEs gives where we have set All the additional conditions above can be viewed as integrability conditions along the lightcone and mixed lightcone and S directions. We shall demonstrate that upon using the field equations and the Bianchi identities, the only independent conditions are (2.26).

B.1 Dilatino KSE
Substituting the solution of the KSEs (2.23) into the dilatino KSE (2.21) and expanding appropriately in the r, u coordinates, one obtains the following additional conditions We shall show that the only independent ones are those in (2.26).

B.2 Independent KSEs
It is well known that the KSEs imply some of the Bianchi identities and field equations of a theory. Because of this, to find solutions it is customary to solve the KSEs and then impose the remaining field equations and Bianchi identities. However, we shall not do Then consider the following, where the first terms cancels from the definition of curvature, and The expression in (B.11) vanishes on making use of (B.7), as A 1 = 0 is equivalent to the + component of (B.7). However a non-trivial identity is obtained by using (B.9) in (B.10), and expanding out the A 1 terms. Then, on adding (B.10) to the LHS of (B.5), with τ + eliminated in favour of η + as described above, one obtains the following This vanishes identically on making use of the Einstein equation (2.19). Therefore it follows that (B.5) is implied by the + component of (B.4), (B.6) and (B.7), the Bianchi identities (2.11) and the gauge field equations (2.12)-(2.16).

B.2.2 The (B.8) condition
Let us define where A 2 equals the expression in (B.8). One obtains the following identity We have made use of the + component of (B.4) in order to evaluate the covariant derivative in the above expression. In addition we have made use of the Bianchi identities (2.11) and the field equations (2.12)-(2.17).

B.2.3 The (B.1) condition
In order to show that (B.1) is implied from the independent KSEs we can compute the following, Since φ + = η + + uΓ + Θ − η − , we must consider the part of the + component of (B.7) which is linear in u. On defining one finds that the u-dependent part of (B.7) is proportional to We have made use of the − component of (B.4) in order to evaluate the covariant derivative in the above expression. In addition we have made use of the Bianchi identities (2.11) and the field equations (2.12)-(2.17).

B.2.5 The (B.2) condition
In order to show that (B.2) is implied from the independent KSEs we will show that it follows from (B.1). First act on (B.1) with the Dirac operator Γ i∇ i and use the field equations (2.12) -(2.17) and the Bianchi identities to eliminate the terms which contain derivatives of the fluxes and then use (B.1) to rewrite the dh-terms in terms of ∆. Then use the conditions (B.4) and (B.5) to eliminate the ∂ i φ-terms from the resulting expression, some of the remaining terms will vanish as a consequence of (B.1). After performing these calculations, the condition (B.2) is obtained, therefore it follows from section B.2.3 above that (B.2) is implied by (B.4) and (B.7) together with the field equations and Bianchi identities mentioned above.

B.2.6 The (B.3) condition
In order to show that (B.3) is implied by the independent KSEs we can compute the following, and where we have made use of the − component of (B.4) to evaluate the covariant derivative terms. The resulting expression corresponds to the expression obtained by expanding out the u-dependent part of the + component of (B.4) by using the − component of (B.4) to evaluate the covariant derivative. We have made use of the Bianchi identities (2.11) and the field equations (2.12)-(2.16).
To evaluate this expression note that It follows that and also In order to simplify the expression for the Laplacian, we shall attempt to rewrite the third line in (C.5) as where F (±) is linear in the fields and W (±)i is a vector. This expression is particularly advantageous, because the first term on the RHS can be rewritten using the horizon Dirac equation, and the second term is consistent with the application of the maximum principle/integration by parts arguments which are required for the generalized Lichnerowicz theorems. In order to rewrite (C.6) in this fashion, note that One finds that (C.6) is only possible for q = −1 and thus we have We remark that † is the adjoint with respect to the Spin(8)-invariant inner product , . In order to compute the adjoints above we note that the Spin(8)-invariant inner product restricted to the Majorana representation is positive definite and real, and so symmetric. With respect to this the gamma matrices are Hermitian and thus the skew symmetric products Γ [k] of k Spin(8) gamma matrices are Hermitian for k = 0 (mod 4) and k = 1 (mod 4) while they are anti-Hermitian for k = 2 (mod 4) and k = 3 (mod 4). The Γ 11 matrix is also Hermitian since it is a product of the first 10 gamma matrices and we take Γ 0 to be anti-Hermitian. It also follows that Γ 11 Γ [k] is Hermitian for k = 0 (mod 4) and k = 3 (mod 4) and anti-Hermitian for k = 1 (mod 4) and k = 2 (mod 4). This also implies the following identities η + , Γ [k] η + = 0, k = 2 (mod 4) and k = 3 (mod 4) (C.10) and η + , Γ 11 Γ [k] η + = 0, k = 1 (mod 4) and k = 2 (mod 4) . (C.11) It follows that 1 2∇ (C.12) It is also useful to evaluateR using (2.19)  One obtains, upon using the field equations and Bianchi identities, (C.14) Note that with the exception of the final line of the RHS of (C.14), all terms on the RHS of the above expression give no contribution to the second line of (C.12), using (C.10) and (C.11), since all these terms in (C.14) are anti-Hermitian and thus the bilinears vanish. Furthermore, the contribution to the Laplacian of η + 2 from the final line of (C.14) also vanishes; however the final line of (C.14) does give a contribution to the second line of (C.12) in the case of the Laplacian of η − 2 . We proceed to consider the Laplacians of η ± 2 separately, as the analysis of the conditions imposed by the global properties of S differs slightly in the two cases.
(C. 15) This proves (3.6). The Laplacian of η − 2 is calculated from (C.12), on taking account of the contribution to the second line of (C.12) from the final line of (C.14). One obtains This proves (3.9) and completes the proof.
It should be noted that in the η − case, one does not have to set q = −1. In fact, a formula similar to (C.15) can be established for arbitrary q. However some terms get modified and the end result does not have the simplicity of (C.15). For example, the numerical coefficient in front of the A (−) η − 2 is modified to −2 − 4κq + 16κ 2 + 2q 2 and of course reduces to that of (C.15) upon setting q = −1.