The hessian in spin foam models

We fill one of the remaining gaps in the asymptotic analysis of the vertex amplitudes of the Engle-Pereira-Rovelli-Livine (EPRL) spin foam models: We show that the hessian is nondegenerate for the stationary points that corresponds to geometric nondegenerate $4$ simplices. Our analysis covers the case when all faces are spacelike.


Introduction
One of the central results of the research on spin foam models (defined in [1,2] and extended in [3]) is the asymptotic analysis of the vertex amplitude accomplished in [4,5,6] for the euclidean case and in [7,8,9] for the lorentzian case). The graviton propagator [10,11,12], the relation to Regge calculus and various semiclassical limits [13,14] are all based on this result. Let us mention that exactly the asymptotic analysis [15] of the vertex of the Barrett-Crane model [16] led to the discovery of nongeometric sectors [17] and in consequence to the invention of the EPRL model. However, it is important to keep in mind that the analysis of the vertex amplitude does not capture all properties of the model -as seen by so-called flatness problem [18,19] that is not visible in the asymptotics of a single vertex.
The proof of the asymptotic formula for various spin foam models is not completely water-tight because of a few issues. First of all, the proof is based on stationary phase method and typically integration is done over noncompact domains. It is not clear if there are any contributions from infinity or from boundary of the domain of integration. In the Hnybida-Conrady extension [3] it is even not known if the amplitude is finite at all. Secondly, the contribution from a stationary point depends on whether the point is nondegenerate (i.e. the hessian at that point has no zero eigenvectors, after gauge fixing) or not. These issues were summarized in our previous paper [8].
The current paper is devoted to the problem of whether or not the hessian is nondegenerate for a given stationary point. The only analytic result in this direction that we know about for 4d models is the result [20] for the Barrett-Crane model [16]. For the euclidean EPRL model it was checked for specific examples that the hessian is nondegenerate 1 so its determinant is nonzero for generic boundary data. However, the example of the Barrett-Crane model can serve as a warning, as in this case the hessian is degenerate for configurations where the map from lengths to areas of the 4-simplex is not locally invertible. The lorentzian models are more complicated. The number of integration variables makes the determination of the determinant of the hessian an almost intractable task.
In this paper we will show that for the EPRL models in both, euclidean and lorentzian signature (we consider also Hnybida-Conrady extension), with spacelike faces, the hessian is nondegenerate for every stationary point that corresponds to a nondegenerate 4-simplex (of either lorentzian or other signature).
We will first consider the euclidean EPRL model with Barbero-Immirzi parameter γ < 1, as it can be treated in a considerably simpler way. The crucial observation for our analysis of this case is the specific behavior of the hessian for actions satisfying a certain reality condition: If e iS denotes the integrand of the amplitude, then the imaginary part of the action is nonnegative, In order to extend our result to the case of lorentzian models we introduce a reduced action that is more closely related to the action of the euclidean model. The reduced action is defined in such a way that non-degeneracy of its hessian is equivalent to the non-degeneracy of that of the full action. We the reexpress the analysis of the euclidean amplitude in symplectic geometric terms. The geometric theory of such actions is based on positive lagrangeans that were introduced by [21]. This makes it applicable to the lorentzian case as well.
The main reference for our notation is [8]. There are a few departures from that notation, for which we refer the reader to Appendix A.

Euclidean EPRL model with γ < 1.
In the following, our terminology and, in particular what is real and what is imaginary is based on the convention that the integrand of the integral we are approximating is e iS , and we will cal S the action. We note that this is different from the convention of [7].
For a symmetric (or hermitian) form H we will use the notation We will say that the vector v annihilates H if For a real symmetric form I we write I ≥ 0 if for any real vector w This is equivalent to the condition that for any complex vector v 2. The following is true for the real and imaginary part of the vector v (v = ℜv + iℑv): Proof. Let us write v = v r + iv a where v r and v a are real. We have from the linearity of the forms thus Rv r = Iv a and Rv a = −Iv r . Moreover from the symmetry of R As I ≥ 0 we see that I(v a , v a ) = 0 and I(v r , v r ) = 0, thus and also Rv r = Rv a = 0.

Lemma 2.
Suppose that the symmetric real form I = α I α and I α ≥ 0.
Proof. We have I(v, v) = 0 thus α I α (v, v) = 0. All terms are positive, thus each of them needs to be zero, but due to positivity this implies that I α v = 0.

Hessian in euclidean EPRL
The manifold of integration is 4 i=1 Spin(4) and thus the vectors of the tangent space can be described by We will denote self-dual (anti-self-dual) part by v ± . The tensor of second derivatives of the action (the hessian) is given by [5] 2 Let us consider the self-dual part (the antiself-dual is analogous). We can write H + as where I ab are given by in terms of symmetric real forms I ′ ab : This form is j + ab 2 times the expectation value of the projector onto the space perpendicular to n + ab , so it is nonnegative ( j + ab 2 ≥ 0), thus also I ab ≥ 0. The real form R is given by where we use the convention that n ab = −n ba .
Proof. If det H = 0 then there exists a nonzero vector v ′ such that Hv ′ = 0 thus Lemma 1 assures that there exists a nonzero real vector v that is annihilated by ℑH. It needs to be annihilated by every I ab due to Lemma 2. The conditions 2 Published version.
Either v(a) = v(b) = 0 or n ab , n a5 , n b5 are linearly dependent. As this is true for all a, b we have either v = 0 (contradiction) or there exist a, b fulfilling the statement of the lemma. Theorem 1. The hessian for the euclidean EPRL model with γ < 1 is nondegenerate for any stationary point that corresponds to a nondegenerate 4-simplex.
Proof. If n ab , n a5 , n b5 are linearly dependent then the matrixG ab5 defined in equation (301) from [8] is degenerate, and lemma 28 from [8] (in its version for euclidean signature) tells us that there exists at most one stationary point (a single vector geometry or a degenerate 4-simplex).
For the case of euclidean EPRL just considered the integration is over the compact manifold, thus the nondegeneracy of the hessian was the only missing part of the asymptotic analysis. We will not consider euclidean case with γ > 1 because it can be treated in an analogous way to the lorentzian case. We will now describe the lorentzian case in detail.

Extension to the lorentzian EPRL amplitude
In the case of the lorentzian EPRL amplitude, integration is over many more variables and the hessian is more complicated. The action is a sum 3 Actions as well as measure factors are at least locally analytic. If we denote by [Þ ij ] elements of CP (i.e., equivalence classes of spinors) then the stationary points are discrete and we are interested in one of them where g 0 5 = 1. We will denote bivectors (see Section 4.1 and Appendix A for and we will write B 0 ′ ij = g −1 i B 0 ij for a bivector in the node frame. We will call it the fundamental stationary point.

Reduced action
The variables {g i } appear in many places, but for fixed ij the variables Þ ij and Þ ji ∈ CP are only found in the actionS ij . Let us denote the form of second derivatives with respect to the CP variables by H ÞÞ . It is block diagonal, with blocks corresponding to {Þ ij , Þ ji }. We will show later that this form is nondegenerate (in the neighbourhood of the fundamental stationary point).
Let us (locally) analytically extend the action in the Þ variables to the Let us notice thatS depends only on the CP variables [Þ C ij ] (equivalence classes of spinors). As the hessian H ÞÞ is nondegenerate at the fundamental stationary point we can (in the neighbourhood of g 0 i ) find a unique (in the Here ∂S

∂[Þ C
ij ] is a holomorphic derivative as the antiholomorphic one gives = 0 everywhere. Let us notice that due to the form of the action the solution has a specific dependence on Let us introduce a reduced action .
(29) The point g 0 i is a stationary point of this action and the hessian at this point is H red = H red ij . Let us notice that S red ij depends only on the group element We have the projection map on the complexified tangent space We can also introduce a cross section We also use these maps restricted to fixed ij sectors (Π ij and Ξ ij ).

Lemma 4. The following holds:
Also Proof. Due to the condition (27) on Þ C we have for W ij ∈ (T SL(2, C)) 2 × Let us notice that for V ij ∈ T SL(2, C) 2 . Summing over ij we get also the second equality.

Lemma 5. The hessian is degenerate if and only if the reduced hessian is.
Proof. Let suppose that HV = 0, then for any W thus HΞ(W ) = 0.

Definition 1. An extremal point of the action S is a point on the real manifold where ∂ℑS = 0 and the tensor of second derivatives of ℑS is nonnegative definite.
If the action S satisfies the reality condition (1) (ℑS ≥ 0) then points on the real manifold where ℑS = 0 are extremal. The fundamental sta- The following is a consequence: is nonnegative definite.
Proof. The maps Ξ ij and Π ij are compatible with complex conjugation thus because the imaginary part of the hessian H ij is nonnegative definite.
Let us summarize:

Symplectic geometry
We will adapt the theory of positive lagrangeans introduced in [21]. Let Ω be the symplectic form on T * M . It is the inverse to the Poisson bracket Let us consider an analytic function S : M → C (maybe defined only on an open set U ). The manifold is lagrangean, that is it extends analytically to an analytic lagrangean submanifold of T * M C in some neighbourhood of the real T * M . Here we denoted by θ the tautological form θ = p µ dx µ . Over real points of M the complex conjugation of the tangent space of the lagrangean T C L S is in itself the tangent space of the holomorphic lagrangean The tangent space of L S can be identified by projection π : T * M → M with the tangent space of M . We will denote this map by Π S : T C L → T C M . Now we will state and prove some important facts about extremal points: Lemma 8. The following holds for an extremal point x 0 of the action S is nonnegative definite and Proof. At an extremal point p = dℜS because derivatives of imaginary parts vanish. Let us use local coordinates p µ , x µ on T * M then Every vector tangent to L can be written as where f µ are some complex constants. Thus at the point From tensoriality of the second derivative at a point where ∂ℑS = 0 we get thus it is nonnegative definite. Let V ∈ T C L S be such that IV = 0 then We need some definition.

Definition 2.
We will say that the lagrangean L at the real point is nonnegative definite. We will say that it is strictly positive if additionally

has no zero vectors).
A lagrangean is strictly positive if and only if that is, the only real vector in T C (x 0 ,p 0 ) L is the trivial vector. Let f be an analytic function on T * M (it extends locally to T * M C ) that vanishes on L S . The complex vector field is tangent to L S . If at the real point the lagrangean is positive then

Symplectic theory of T * SL(2, C)
The left invariant vector field Ä(L) of the Lie algebra element L corresponds to the first order jets of g → ge tL . The right invariant vector field Ê(L) of the same Lie algebra element will be g → e −tL g (the sign is necessary for proper commutation relations).
With every point of the cotangent bundle T * SL(2, C) we can associate a left and a right coalgebra element p L and p R given by the formula Let us notice that at the base point g where g acts on the coalgebra by the co-adjoint action (if we identify the coalgebra with bivectors using the scalar product then the coadjoint action is the same as the adjoint action, see appendix A). For any Lie algebra element L, are functions on T * SL(2, C). We have and also Let us denote by δ L S (δ R S) the covectors identified by with coalgebra as follows We will use δ for the left version. We can use the standard scalar product (·, ·) on bivectors to make the further identification of δS with a bivector. 6 For any function S on the group we can now define a lagrangean submanifold

Symplectic theory of a coadjoint orbit
Let us recall that we can identify the space of bivectors (Lie algebra so(1, 3) = Λ 2 R 4 ) with the coalgebra using the natural scalar product (·, ·) on bivectors. Let us consider a coadjoint orbit where C 1 = (B, B) and C 2 = (B, * B) are two Casimirs (invariants). The Lorentz group acts transitively on X n,ρ . We have a natural Poisson bracket given, for a linear function H(L)(B) = (B, L), by This turns the coadjoint orbits into symplectic manifolds. Let us introduce an isomorphism from so(1, 3) to sl(2, C) (traceless matrices) by (see appendix A) We have identity For the matrix M(B) there exist two spinors Þ ± B (unique up to a constant each) such that We can thus define a projection

Definition 3. A function
Usually we cannot define such actions globally. Let us introduce the notation 7 (where the action is on Weyl spinors S + , see appendix A) Let us notice that δ Þ S is a well defined function on CP if S satisfies (78).

Lemma 11.
For any real function S of type (n, ρ) the map is a symplectic diffeomorphism from T * CP to X n,ρ . This map is compatible with the projection onto CP.
If B ∈ X n,ρ and π(B) = [Þ] then However φ [Þ] is a bijection onto Lie H 0 [Þ] . In order to check that it is a symplectomorphism we will show that Poisson brackets between generators of so(1, 3) are right. For L ∈ so(1, 3) let us consider the pull back of the Hamiltonian H(L) to T * CP. It is Let us notice that Ä S + (S) descents to a function [Ä S + (S)] CP on CP. We have thus for a given bivector a function on T * CP Let us notice that Moreover thus Therefore finally Because the Hamiltonian vector fields of functions span in every point the whole tangent space, Ω is the same as the canonical symplectic form on the cotangent bundle.
Let us now consider a complex action (locally defined) S of type (n, ρ). Let us notice that ℑS is a function on CP. In particular ∂ 2 [ℑS] CP is a tensor on CP.

Lemma 12. The space
is a complex lagrangean manifold in X n,ρ and on the real point B ∈ X n,ρ where v ∈ T L ′ S and [Þ] = π(B). Remark: We regard CP as a real manifold, thus π(v) ∈ T C CP and the conjugation is with respect to this additional complex structure. It can be translated into inner complex conjugation.
Proof. Let S aux be an auxiliary real action of type (n, ρ). The difference f = S − S aux is a well defined function on CP. Moreover using the local identification of X with CP we have Indeed this is equivalent to We know that thus the result.

Casimir reduction
Let us consider a symplectic reduction of T * SL(2, C) with respect to Casimirs. For SL(2, C) the moment map is nondegenerate except for bivectors equal to zero.
Lemma 13. Two points (g, p) and (g ′ , p ′ ) are connected by a flow of Casimirs in SL(2, C) if and only if there exists λ, λ ′ ∈ R such that and p L = p ′ L (or equivalently p R = p ′ R ).
Proof. Left covectors are preserved by Casimirs, thus we only need to find the vector field on the group. Let us denote the projection on the group manifolds of the Poisson vector fields of the Casimirs by V 1 and V 2 . We identify bivectors with the left covectors on SL(2, C) by the scalar product and then and thus g is changed from the right (because left invariant vector field) by p L . The second Casimir is related to the first by Hodge star, thus Together (they commute) we have the flow From preservation of left covectors we have p L = p ′ L .
The symplectic reduction with respect to the Casimirs is given by Let us denote We have a map π Cn,ρ : C n,ρ → S to the symplectic reduction. If L ′ ⊂ S is a real lagrangean then is also a lagrangean and it is a subset of C n,ρ . The other way around, if a real lagrangean L ⊂ T * SL(2, C) is such that L ⊂ C n,ρ , then as Casimir generated directions belong to L we have where L ′ is a lagrangean in S. The same holds for complex lagrangeans (in locally holomorphic extensions).

Explicit description
There is a direct description of this symplectic reduction that is an analog of Peter-Weyl theorem in group representation theory. Let us notice that the left and right invariant covectors Poisson commute with the Casimirs. Moreover the equation has a solution for g if p R and p L are of the same type (nonzero) and g is unique up to [·] equivalence. Thus the map is an isomorphism of symplectic spaces. 8

Symplectic theory of S ′ ij
Let us consider an actioñ Let us now assume that for every ij the lagrangean is strictly positive at the point corresponding to the fundamental stationary point (that is [(g 0 . We will prove this fact in section 5.3. Because the action S β is real, the imaginary part of the hessian with respect to Þ ij and Þ ij is block diagonal with respect to every Þ variable. From strict positivity of the lagrangean every block is strictly positive, thus by lemma 1, the form H ÞÞ is nondegenerate.
We can now consider It is well defined for g ij in the neighbourhood of g 0 ij . Lemma 14. The lagrangean manifold of the action S ′ ij is given by Proof. Left and right invariant derivatives of S ′ ij are equal to derivatives of S n ij ij and, respectively, S n ji ji with spinors equal to the stationary point because derivatives with respect to Þ vanish in the point [Þ C ](g i , g j ). We see from the type of the actions that L S ′ ij ⊂ C C 2 ij ρ ij , thus it is an inverse image of a complex lagrangean in X 2 ij ,ρ ij × X 2 ij ,ρ ij . We see also that and by comparing dimension it needs to be equal.

Proof. From the previous lemma
Thus V is in the space of the Casimirs' Poisson vector fields. Thus its projection onto the tangent space of the group as stated.

Simplicity constraints
Our goal in this section is to show that L ′ ij is strictly positive at the extremal point coming from the fundamental stationary point. In fact it is a simple computation of a two dimensional matrix. However it is useful to describe this lagrangean (in the neighbourhood of this point). Let us notice that from the reality condition of the action we know that the lagrangean is positive.

Conditions on the action
Let us suppose that we have a function of the form defined and analytic for Þ ∈ U .
We have an action of the group on spinors Þ, thus we can also consider an where L I are Lie algebra basis. We associate with this operator a symbol (a homogenous polynomial on the Lie coalgebra) where p are Lie coalgebra elements. Let us remind that we identify both Lie algebra and coalgebra with bivectors (thanks to the scalar product). Let p(λ) be a polynomial of order m with m-homogeneous coefficient a m such that for every N Then taking the leading term in the N expansion, we get for any Þ

Bivector decomposition
For the given normal N 0 i (see [8]) with the norm c i = |N 0 i | 2 ∈ {−1, 1} we can decompose the bivector B as follows where v, w ∈ N 0 i ⊥ and the two terms belong to We can now introduce maps They are explicitly given by We can identify so(N 0 where × is defined by The Casimirs can be writen in terms of these vectors as follows With the vector v ∈ N 0 i ⊥ we can associate two complex vectors k i s (v) (s = ±1) given by the conditions: where In the case of spacelike faces we choose C > 0. In this situation vectors k i ±1 (v) are complex and we assume With the choice of signature ( We assume that k i 1 (v)·k i −1 (v) = −1, and this fixes vectors up to a phase.

Lemma 16. We have
Proof. Let us notice that k i Thus and substituting v · v = −c i C 2 we get the result.
Let us notice that if a complex vector w ∈ N 0 We can regard v · L i and v · K i as linear maps on bivectors, thus we can compute Poisson brackets. In order to do it we need to find the associated by (the scalar product) bivectors thus we get

Simplicity constraints
The coherent states Φ n ij (Þ ij ) satisfies the following equations 1. Diagonal simplicity constraints, that for fixed spins means that the values of the Casimir operators are related to twisted simplicity constraints 9Ĉ where ρ = γn and n = 2 ij . 10 9 Quantisation of the action of the Lie algebra element L isL = 1 i Ä S + (L). 10 Our convention differs from [3] by a sign in C2 that can be seen from (142).

Cross simplicity constraints, that are implemented in the EPRL model by
3. The coherent state condition k s ij (v ij ) ·ˆ L i = 0, where s ij is fixed and v ij is constructed from n ij .
These conditions impose several conditions on S n ij ij . We can describe them in terms of L ′ ij . Namely B ∈ L ′ ij needs to satisfy 1. Diagonal simplicity constraints (B, B)

Cross simplicity constraints
3. Coherent state condition k s ij (v ij ) · L i = 0, where s ij is fixed and v ij is constructed from n ij .
In order to analyze the conatraints let us introduce a twisting map We can compute Similarly Let us denote B τ = τ −1 (B) and L τ The first two conditions mean 1. Diagonal simplicity conditions: 2. Cross simplicity: Thus we can write and the Casimir conditions means that We are interested in the fundamental stationary point, and then We also have and, from positivity of the lagrangean, the right hand side needs to be positive. Let us notice that we see that t ij = −c i . Let us consider now coherent state condition k s ij (v ij ) · L i = 0. It means that However, and λ 1 = ±1. As the phase space point corresponding to the fundamental stationary point is in the lagrangean we have in the neighbourhood of this stationary point λ 1 = 1. We can now compute at the fundamental stationary point it is equal to s ij c i  ij , thus s ij = c i . We can now describe tangent space to the lagrangean at B 0 ′ ij . The conditions for bivectors to be tangent directions to L ′ ij is that It is not hard to find all vectors satisfying these conditions. Every tangent bivector can be uniquely described by a pair The conditions on B are We can now summarize Proof. We need to prove that the real tangent vector (bivector) is zero. Tangent vectors satisfy and from reality However vectors k i ±1 (v ij ) are linearly independent thus λ s = λ t = 0.

Reduced hessian
Let us denote the tensor of second derivatives of ℑS ij (g 0 ij ) by I ′ ij . We are interested in the second derivatives ℑS red ij (g i , g j ) at {g 0 k } (we assume g 0 5 = 1). The tangent vectors to the manifold 4 i=1 SL(2, C) are given by For convenience we assumed v(5) = 0. We use here the right invariant vector fields to identify V tot with T 4 i=1 SL(2, C) .

Lemma 18. We have
Proof. Standard result about functions of the form f (g −1 j g i ). We use the right invariant vector fields, thus We have for left invariant vector fields We can now compute for v Let us compute second derivative of We used BCH formula and commuted (we use notation g · L = gLg −1 ) We use now ∂ℑS ′ ij = 0 to get so we found the desired result.

Lemma 19. Let us suppose that
then there exist 1 ≤ a < b ≤ 4 such that the bivectors are linearly dependent.
Proof. As all I ij are positive definite and I ′ ij has the kernel spanned by we have We see that from I i5 (v, v) = 0 it follows that v(i) ∈ span{B 0 i5 , * B 0 i5 } and thus as v is nonzero there exist i, j = 5 such that and also This means linear dependence.

Lemma 20.
If the reconstructed 4-simplex (in any signature) with spacelike faces is nondegenerate then are linearly independent for {a, b, 5} distinct.
Proof. Let us assume a = 3, b = 4. The bivectors B ∆ ij for i, j ∈ {3, 4, 5} can be written as where η ij ⊥ e 12 and e 12 is the edge vector connecting vertex 1 with 2 (this edge is spacelike). Moreover η ij are independent if the 4-simplex is nondegenerate. Let us notice that e 12 * B ∆ ij = 0 and e 12 B ∆ ij = −η ij |e 12 | 2 . Let us assume that there is a linear equation for the bivectors ij∈{3,4,5} Contracting it with e 12 we get ij∈{3,4,5} Taking the Hodge dual of the equation and then contracting with e 12 we get ij∈{3,4,5} Thus the bivectors are linearly independent if the reconstructed 4-simplex is nondegenerate. Proof. If Hv = 0 then we are in the situation from lemma 19. From positivity of the I ij it thus follows that are linearly dependent. Let us consider now separetely two cases: 1. If the stationary point corresponds to a lorentzian 4-simplex then and τ preserves the space (200). By lemma 20 we have a contradiction.
2. If the stationary point (+) corresponds to a 4-simplex solution with other signature then there is the second point (−) and and B ∆ ij has selfdual and antiselfdual parts given by τ −1 (B ± ij ). From (200) it follows that there exist constants λ ij , λ ′ ij such that ij∈{a,b,5}, i<j thus taking L τ i and K τ i parts we get ij∈{a,b,5}, i<j As some coefficients need to be nontrivial we get that v + ij and thus also B + ij ((i, j) ∈ {a, b, 5}) are linearly dependent. But this means that are linearly dependent and from lemma 20 we have a contradiction.
Independently of the signature of the reconstructed 4-simplex the hessian is nondegenerate.

Summary
We showed that the hessian in the EPRL and Conrady-Hnybida (spacelike surfaces case) is nondegenerate for any stationary point (corresponding to a nondegenerate 4-simplex of either lorentzian, euclidean or split singature). We also showed nondegeneracy for the euclidean γ < 1 case. Our method works fine also for γ > 1, but we have not provided the details in this case. However, the method does not extend immediately to the situation when some of the faces are timelike (the asymptotic of this case was considered recently in [9]). The action in this case is purely real, and as we based our proof on the properties of imaginary part of the action, this case cannot be covered with the tools used in our paper unless they will be properly modified. The issue deserves a separate treatment and we leave this topic for future research.
For two spinors Ù, Ú we denote where Ä and Ê are left and right derivatives. We also denote δ = δ L .
Similarly, the vector field of the action of SL(2, C) on S + are denoted by Ä S + (L) for L ∈ so(1, 3). They correspond to the curves t → e −tL Þ.
11. The definition of L i K i is in section 5.2. For the twisting map τ , and twisted versions L τ , K τ i see section 5.3. 12. The vectors k i ±1 (v) are defined in 5.2. 13. X n,ρ is a coadjoint orbit space defined in equation (70). 14. The projection from X n,ρ (coadjoint orbit) to CP is denoted by π. The function f that is constant along the fibers can be pushed forward to CP and such push forward is denoted by [f ] CP (see section 4.2). 15. S red , S red ij are defined in section 3.1. Their hessians are denoted by H red and H red ij . 16. S ′ ij and g ij = g −1 j g i is defined in section 3.1. 17. L denotes lagrangeans. The subscript denotes the (part of the) action generating the given lagrangean. We use ′ to indicate lagrangeans in the coadjoint orbit space. 18. The form I on the tangent space of the lagrangean at the real point is defined in equation (49).