Non-minimal couplings to U (1) -gauge fields and asymptotic symmetries

: We analyse the asymptotic symmetries of electromagnetism non-minimally coupled to scalar fields, with non-minimal couplings of the Fermi type that occur in extended supergravity models. Our study is carried out at spatial infinity where minimal and non-minimal couplings exhibit very different asymptotic properties: while the former generically cannot be neglected at infinity, the latter can. Electromagnetic non-minimal couplings are in that respect similar to gravitational minimal couplings, which are also asymptotically subdominant. Because the non-minimally interacting model is asymptotic to the free one, its asymptotic symmetries are the same as the ones of the free theory, i.e., described by angle-dependent u (1) gauge transformations. We also analyse the duality symmetry and show that it is broken to its compact subgroup by the asymptotic conditions. Finally, we consider logarithmic gauge transformations and use them to simplify the symmetry algebra.


Introduction
The asymptotic symmetries of electromagnetism in 4-dimensional Minkowski space form an infinite-dimensional group parametrized by angle-dependent u(1) transformations [1][2][3][4][5], which is somewhat analogous to the BMS group [6][7][8].While originally found at null infinity, the same symmetry group was exhibited later at spatial infinity [9][10][11].In particular, the null infinity matching conditions were proved in [10,11] to be equivalent to appropriate parity conditions on the leading orders of the Cauchy data in an expansion near spatial infinity.
The introduction of minimal couplings to charged massless fields leads to a more complicated state of affairs.The specific examples of the massless complex Klein-Gordon field or of a collection of spin-1 gauge fields interacting through the Yang-Mills mechanism were explicitly investigated with different results at null infinity and spatial infinity.While the infinite-dimensional symmetry survives at null infinity [2][3][4][5][12][13][14], there is a tension between this infinite-dimensional symmetry and Lorentz invariance at spatial infinity [15][16][17].Lorentz invariance actually restricts the asymptotic conditions in such a way that no internal asymptotic symmetry survives and only the Poincaré symmetry is present.
The difficulties originate from the fact that interactions of the minimal coupling type cannot be neglected at infinity, where the theory does not linearize, contrary to what happens for gravity.This is very easy to see by mere inspection of the covariant derivatives which read, in the example of a complex one-form v k where A i is the electromagnetic vector potential and Γ k ij is the Levi-Civita connection.With the standard boundary conditions that read at large radial distance r one finds Therefore, ∂ i v k and ieA i v k behave in the same way as r → ∞ while Γ k ij v k is by contrast subleading and disappears to leading order.Electromagnetic covariant derivatives do not reduce to ordinary derivatives at infinity while gravitational covariant derivatives do.If The same is true for the Yang-Mills curvatures where "minimal coupling" yields Here, the f a ij 's are the abelian (free) curvatures, f a ij = ∂ i A a j − ∂ j A a i .The coupling term and the abelian curvature are of same order O 1 r 2 , so that the Yang-Mills curvatures do not linearize at infinity and the interactions cannot be neglected.
It turns out that for massless charged fields, or the Yang-Mills field itself, the minimal coupling terms appear non trivially in the surface integral by which the boosts fail to be canonical transformations under the boundary conditions that allow angle-dependent u(1) transformations at infinity.This explains the situation described above that the Poincaré group and the angle-dependent u(1) group are incompatible [15][16][17].We explicitly review the argument for massless scalar electrodynamics in the light of the asymptotic nonlinearization property in Appendix A.
It is interesting to note that the same important difference between the asymptotic properties of gravity and Yang-Mills gauge models plays a striking role in at different (but also asymptotic) context, that of the boost problem of Christodoulou and O'Murchadha [18].As stressed by these authors, while the Einstein equations linearize at infinity, the Yang-Mills equations exhibit a radically different behaviour which does not guarantee that asymptotically flat initial data have a regular development which includes complete spacelike hypersurfaces boosted relative to the initial one.
There exist other types of couplings to abelian gauge fields that have been considered in the literature.These are of non-minimal Fermi-type and play for instance an important role in extended supergravity models [19][20][21][22][23][24].The purpose of this paper is to prove that contrary to minimal couplings, these couplings do not spoil the asymptotic infinite-dimensional angledependent u(1) symmetry at spatial infinity.Intuitively, this follows from the fact pointed out above that the equations linearize at infinity as the interaction terms are subdominant with respect to the free ones, and we establish that this intuition is indeed correct.
The supergravity models enjoy the further interesting feature of being invariant under a "hidden" duality symmetry [19][20][21], which we also include in our asymptotic analysis.
Our paper is organized as follows.We start with the case of a single Maxwell field coupled to scalar fields on the coset manifold SL(2, R)/SO(2), for which the full duality group is SL(2, R) (Section 2).We show that in the standard one-potential formulation, the difficulties with boost invariance that arise when one adopts asymptotic conditions allowing non-trivial angle-dependent u(1) asymptotic symmetries can be overcome by exactly the same method as in the pure Maxwell case [10].This is because the scalar fields do not contribute to the relevant asymptotic analysis.The theory is then both Poincaré invariant and invariant under asymptotic angle-dependent O(1) gauge transformations, as in the absence of the scalar fields.
In Section 3, we consider the manifestly duality invariant formalism, which involves two potentials with a corresponding doubling of the asymptotic angle-dependent u(1)symmetries [25].We show that the scalar fields do not spoil the asymptotic properties.We also observe that the boundary conditions break the duality group SL(2, R) to its maximal compact subgroup SO(2), which rotates the two angle-dependent u(1)'s.Section 4 extends then the model to n Maxwell fields non-minimally coupled to scalar fields parametrizing the coset space Sp(2n, R)/U (n), for which the full duality group is Sp(2n, R) [24].This duality group has been argued recently to play a key role in the quantum theory, even in models where it is only a subgroup of it that is explicitly realized (such as maximal supergravity for which it is E 7 ⊂ Sp(56)) [26].Similar results are obtained: 2n independent angle-dependent u(1) asymptotic symmetries, one for each electric potential and one for each magnetic potential; and breaking of the duality group Sp(2n, R) to its maximal compact subgroup U (n) by the boundary conditions.The global U (n) transformations act as expected on the angle-dependent asymptotic symmetries.
In Section 5, we extend the formalism in another direction, by relaxing the bound-ary conditions to allow gauge transformations that grow logarithmically at spatial infinity, following [27].We find again that the non-minimally coupled scalar fields do not modify the asymptotic properties, and that the new logarithmic gauge symmetries can be used to rewrite the symmetry algebra as a direct sum by appropriate nonlinear redefinitions of the generators.
Finally, Section 6 is devoted to concluding comments.Appendix A completes the analysis by contrasting the asymptotic features of minimal versus non-minimal couplings at spatial infinity, and the corresponding incompatibility versus compatibility of boost invariance with the relaxed boundary conditions allowing angle-dependent asymptotic symmetries.

SL(2, R)-case: One-potential formulation
We consider first a single Maxwell field coupled to two scalar fields ϕ and χ parametrizing the coset manifold SL(2, R)/SO(2), for which the full duality group is SL(2, R).

Action in Hamiltonian form
The manifestly covariant action describing this system is given by where In the scalar action S s , the field ϕ denotes the dilaton, while the scalar field χ stands for the axion.The vector action S v contains non-minimal couplings between the field strength F µν , associated to the gauge potential A µ , and the dilaton and the axion fields.These couplings are such that the theory is invariant under SL(2, R)-duality, which is manifest in the scalar sector, but more subtle (as a Lagrangian symmetry) in the vector sector [28,29].
In order to write the action in Hamiltonian form, we compute the conjugate momentum of each dynamical field, that is to say, ) Then, the Hamiltonian actions read where and where the scalar and vector Hamiltonians are given by2 respectively.The specific form of the boundary terms B s ∞ and B v ∞ depends on the boundary conditions.Variation of the action with respect to A 0 , which appears as a Lagrange multiplier, yields the Gauss constraint G ≈ 0.

Boundary conditions
In Cartesian coordinates, the fall-off of the dilaton and axion fields, together with their corresponding conjugate momenta is given by ) This 1/r-fall-off of the fields is characteristic of massless fields.Finiteness of the kinetic term in the Hamiltonian scalar action requires to impose parity conditions on the leading order fields under the antipodal map3 .As in [30], we choose the leading order of the scalar fields to be even under the antipodal map, namely, while the leading order fields of the conjugate momenta are chosen to be odd fields on the sphere: While the Lagrangian is invariant under the full SL(2, R) duality group, the boundary conditions imposed on the scalar fields clearly break this group to its SO(2) subgroup, which is the stability subroup of the zero field configuration (ϕ = 0, χ = 0) to which the scalar fields are required to tend asymptotically.
The fall-off of the vector potential and its conjugate momentum read In order to avoid logarithmic divergences in the symplectic structure, we impose the twisted parity conditions introduced in [11].In spherical polar coordinates where the fall-off behaviour reads, ) the radial components are requested to obey strict parity conditions while the angular components of the vector potential are twisted by a total derivative term where If one were to impose strict parity conditions on all components of the vector field [31], one would find that the only non-trival gauge transformations are the global (angleindependent) ones.Relaxing them by an angle-dependent gauge transformation that is O(1) at infinity and generates leading terms in the potential of opposite parity is the first step towards exhibiting the infinite-dimensional angle-dependent u(1) asymptotic symmetry at spatial infinity [10].
The set of boundary conditions is completed by demanding a faster fall-off of Gauss's constraint, i.e., which is also required by finiteness of the kinetic term [10].

Poincaré transformations
The above boundary conditions in which one allows explicitly an angle-dependent O(1) gauge transformation term in the asymptotic form of the fields have been tailored to incorporate an infinite-dimensional angle-dependent u(1) symmetry.A central difficulty with these relaxed boundary conditions, however, is that they conflict with Poincaré invariance.This occurs in a subtle way, in the sense that the boundary conditions themselves are Poincaré invariant.The difficulty has to do with an additional condition that a transformation should fulfill to define an "invariance", that of leaving the symplectic form strictly invariant.This is a subtle question in the present case because the invariance of the Lagrangian density under Poincaré transformations up to a divergence ∂ µ k µ guarantees the invariance of the symplectic form, but only up to a surface term at spatial infinity.The whole question is then to check that this surface term vanishes with the chosen boundary conditions.The point is that it does not with (2.15)- (2.18).
This can be cured in the pure Maxwell case.To understand why the above non-minimal couplings do not spoil the curing procedure, we go step-by-step over it, following closely [10,11].

Poincaré transformations of the fields
The components of the vector fields generating Poincaré transformations read, in spherical coordinates, ) ) where D A stands for the covariant derivative on the sphere with metric γ AB .The parameter b generating Lorentz boosts satisfies the equation The constant parameter T generates time translations, while the parameter that generates spatial translations W is subject to the same equation as b, Finally, Y A is the sphere Killing vector, i.e., it satisfies the Killing equation The Poincaré transformations in phase space can be obtained as follows.First, one takes the brackets of the canonical variables with where the vector field (ξ, ξ k ) is assumed to decrease sufficiently fast at infinity that H ξ,ξ k is a well-defined generator, with no extra surface term at infinity [32].Second, one observes that the variations of the canonical variables obtained in this manner are local in space, so that their variations at a point x do not depend on how the vector field (ξ, ξ k ) behaves at infinity.The same expressions hold then for Poincaré transformations, even though in that case H ξ,ξ k is a functional that still needs to be improved in order to be a well-defined generator.
One finds that the scalar fields and their corresponding conjugate momenta transform as (2.28) The transformation laws of the gauge potential and its conjugate momentum are given by These transformations laws can be easily checked to preserve the asymptotic conditions.Furthemore, they coincide to leading order with the transformations of the uncoupled fields.The contributions due to the interactions are easily found to be subleading at least by a power of 1/r.

Non-invariance of the symplectic form under Lorentz boosts
Because the contributions to the Poincaré transformations coming from the interactions are subleading, the surface term at infinity giving the variation of the symplectic form under boosts takes exactly the same form as in the uncoupled theory.One finds indeed that the Lie derivative of the symplectic form along the phase space vector X ξ associated with boosts is given by the surface term exactly as in [10], with no additional contributions coming from the interactions with the scalar sector.Because of this key property, one can rescue boost invariance in the same way as in the free case, as we will now show.This is in sharp contrast with minimal couplings, which modify the leading asymptotic terms in the variations of the fields, leading to obstructions, as explained in Appendix A.

Re-establishing boost invariance
To re-establish invariance under Lorentz boosts, the method of [10] introduces a surface degree of freedom Ψ at infinity and modifies the symplectic structure by a surface term that involves it, The new field Ψ can be extended into the bulk, by incorporating it as the leading order coefficient in the fall-off of the time component of the vector potential A 0 and adding its corresponding conjugate momentum π 0 , which must be weakly zero.This amounts to consider the Hamiltonian formulation of the theory as it comes from the strict application of the Dirac procedure, in which one keeps the "primary constraint" π 0 ≈ 0 and its Lagrange multiplier which we denote by λ (see e.g.[33,34]).The Hamiltonian action principle with π 0 and λ included is given by where S s H is the same as in (2.6) and the vector Hamiltonian action S v H now reads where We have modified for convenience the vector Hamiltonian by adding the constraint term −∂ i π 0 A i , which is of course permissible.Variation of the action principle with respect to the Lagrange multiplier λ enforces the constraints π 0 ≈ 0, while Gauss's law appears now as a "secondary constraint".We can further extend the formalism by introducing an independent Lagrange multiplier ψ for the secondary constraint G, yielding Dirac's extended formulation that makes all symmetries manifest [33,34].This amounts to modifying the vector action as The two formulations are physically equivalent and one goes from the extended formulation to the non-extended one by imposing the gauge condition ψ = 0.
The fall-off (in Cartesian coordinates) of the time component of the vector potential and its conjugate momentum are given by with parity conditions on the leading orders that read (2.39) The Poincaré transformation laws are modified in the vector sector of the theory by adding gauge transformations in such a way that the symplectic form is invariant (see next subsection).This is of course also permissible.Explicitly, we take

Poincaré charges
One can check that the new symplectic form is then invariant under Lorentz boosts, and thus under the whole Poincaré group, i.e., L ξ Ω = 0 . (2.45) The Poincaré canonical generator can then be obtained by direct application of the equation Then, we obtain that (2.47) where the energy and momentum densities read respectively, with

Angle-dependent u(1) asymptotic symmetries
The asymptotic conditions are also invariant under the gauge transformations generated by the parameters with ϵ(−n i ) = ϵ(n i ) and µ(−n i ) = −µ(n i ).Transformation laws of the fields read where the remaining fields do not transform under gauge transformations.The canonical generator of the asymptotic symmetries is then given by G µ,ϵ = ˆd3 x µπ 0 + ϵG + ˛d2 x ϵ π r − √ γ µA r . (2.54) The algebra of the asymptotic symmetries is then the semi-direct sum of Poincaré and the infinite-dimensional set of u(1) charges, that is to say, where the components of the Poincaré vector field transform as The gauge parameters are boosted and rotated as follows (2.60) 3 SL(2, R)-case: Duality-invariant formulation

Action principle and boundary conditions
We now turn to the description of the duality symmetry, broken to SO(2) by the asymptotic conditions.In order to exhibit it, we go to the two-potential formulation along the Hamiltonian lines of [29].The Gauss's constraint is solved, by introducing a new vector potential Z i [28], We assume the absence of sources, i.e., no electric or magnetic charges, which would otherwise need the introduction of electric and magnetic Dirac strings [35].
Then, up to boundary terms, the vector action takes the form where the index a takes the values (1, 2) and A a i = (A i , Z i ).The field strength associated with the double vector potential A a i is given by The matrix is such that the Hamiltonian in (3.2) is invariant under sl(2, R) duality transformations of the fields where α, β = ±, 0. The phase space vectors ξ α (ϕ, χ) = ξ ϕ α (ϕ, χ) ∂ ∂ϕ + ξ χ α (ϕ, χ) ∂ ∂χ and the matrices X α fulfill both the sl(2, R) algebra, Explicit expressions for these quantities can be found for instance in [29].
For the scalar fields and their conjugate momenta we will adopt the asymptotic (and parity) conditions previously written.For the gauge potential, we will take the obvious generalization of (2.16), i.e., This implies the following fall-off for the corresponding field strength where with √ γ e AB ≡ ϵ rAB .We will assume the twisted parity conditions of [25] where λ a (−n i ) = λ a (n i ), which guarantee the finiteness of the symplectic structure.Since only ∂ A λ a appears in the asymptotic conditions, one can assume that these gauge functions have no zero mode.

Poincaré invariance
Rewritten in the two-potential formulation, the Poincaré transformations of the fields explicitly read ) .16) and are easily verified to preserve the asymptotic conditions.The variation of the symplectic form under Poincaré transformations takes again the form with no contribution from the interaction terms, which are subleading.The resolution to the Lorentz boost problem -the fact that L ξ Ω ̸ = 0 -proceeds then as in [25].We consider first the Hamiltonian action principle with additional degrees of freedom Ψ a at infinity, The scalar action S s H is the same as in (2.6), but the vector Hamiltonian action S v H in (3.2) is modified as follows Ψb . (3.19) The transformations under Lorentz boosts of the canonical variables are adjusted by new contributions involving Ψ a , in such a way that the symplectic form corresponding to the extended action, (3.20) is invariant under Poincaré transformations.One finds that the transformation laws of the scalar fields remain unchanged.The ones of the vector potentials, however, are modified by gauge transformation terms (as always permissible): As previously, the fields A a 0 can be thought as extensions into the bulk of the asymptotic boundary degrees of freedom Ψ a , i.e., Its Poincaré variation reads from which it follows that The Poincaré canonical generators, which exist because L X ξ Ω = d V (ι X ξ Ω) now vanishes, are obtained from ι X ξ Ω = −d V P ξ,ξ i and found to be where the boundary term reads .26) and the energy and momentum densities are given by (3.28)

Asymptotic electric and magnetic angle-dependent symmetries
In the two-potential formulation, the asymptotic conditions are invariant under two independent sets of angle-dependent u(1) gauge transformations, one "electric" and one "magnetic".These are described by the parameters with ϵ a (−n i ) = ϵ a (n i ) and µ a (−n i ) = −µ a (n i ).Transformation laws of the fields read The canonical generators of the asymptotic symmetries are given by Because the divergence ∂ i B ai of both magnetic fields identically vanishes, the zero mode of the u(1) symmetries is pure gauge in the sourceless context considered here.This allows one to assume that the gauge parameters ϵ b have no zero mode.

Duality symmetry
Finally, we turn to the duality symmetry.While the Lagrangian is invariant under SL(2, R) duality transformations, the boundary conditions on the scalar fields are only preserved by the stability subgroup SO(2) of the configuration ϕ = χ = 0.These are parametrized by with ρ constant and read The canonical generator of SO(2)-duality rotations is given by

Symmetry algebra
The asymptotic symmetry algebra is the semi-direct sum of the Poincaré algebra and two sets of infinite-dimensional u(1) algebras, which transform under SO(2)-duality rotations.The non-vanishing Poisson brackets of the asymptotic symmetry algebra read explicitly where the transformed gauge parameters μ, ε are given by where the zero mode in bµ a can be projected out.The transformed Poincaré generators ξ, ξi are given by the standard expression.
4 Generalization to Sp(2n, R) Our results can be generalized to the system of 2n gauge fields A M i and scalar fields φ Γ with the appropiate non-minimal couplings that makes it invariant under Sp(2n, R).We give directly the final results without repeating the explicit derivations.We follow the notations of [29].
The asymptotic conditions are taken to be as in the Sp(2) case for each respective field.These boundary conditions break the duality symmetry group Sp(2n, R) to its U (n) compact subgroup, which is the stability subgroup of the origin (the scalar fields go to zero at infinity).
The symplectic form defined by the action is invariant under Poincaré transformations, i.e., 5 Log-gauge transformations -Rewriting the algebra as a direct sum 5.1 Asymptotic conditions, charges and algebra

Action principle with log-relaxed asymptotic conditions
A noticeable feature of the symmetry algebra is that the Lorentz generators are not invariant under asymptotic O(1) gauge symmetries (although they are of course invariant under gauge transformations that vanish at infinity).This is because the Poisson brackets of the corresponding generators with the Lorentz generators are non-zero, due to the fact that the gauge generators are in a non-trivial representation of the Lorentz algebra.Technically, the dependence under "improper" [36] gauge transformations of the Lorentz generators follows from the surface term that must be included to make them well-defined when the more flexible boundary conditions allowing O(1) gauge symmetries at infinity are imposed.This surface term involves the bare potentials.
The gauge-dependence of the Lorentz generators leads to an ambiguity in the values of the angular momentum and boost generators.In the standard single-potential formulation, one can redefine the Lorentz generators in a manner that makes them invariant under all (proper and improper) gauge transformations [27].This is done by adding to the Lorentz generators nonlinear terms given (roughly) by the product of the generators of the improper gauge symmetries by the asymptotic fields Ψ or Φ.These fields can furthermore be shown to be equal (on the constraint surface) to the generators of a new type of gauge transformations, namely, gauge transformations that grow logarithmically in r at infinity.
The approach can be straightforwardly extended to the duality invariant formulation with non-minimal couplings to scalar fields considered in this paper.Ultimately, this is again because the non-minimal couplings are negligeable (to leading order) at infinity, so that the asymptotic computations of the free theory can be repeated without change.
We will thus only sketch here the procedure to arrive at an ambiguity-free angular momentum, following [27], and considering only the case of a single Maxwell field (in the double potential formulation).This is sufficient to convey the key ideas.
To allow the possibility to perform gauge transformations that behave asymptotically as ln r ("log-gauge transformations"), we take as asymptotic conditions for the gauge potentials: This leads to the following fall-off for the field strengths: where We impose as in [27] the parity conditions and We can assume that the even functions Φ a have no zero mode, and hence also the even functions Ψ a log which are their conjugates in the action (5.10) below.The action principle is modified by a surface term involving the new fields Ψ a log , yielding the vector Hamiltonian action (5.10)

Symmetries and charges
The preservation under Poincaré transformations of the symplectic form implies that the asymptotic fields should transform as ) and with zero modes that can be projected out in δ b,Y A Φ a and δ b,Y A Ψ a log .The Poincaré canonical generators are then given by where the boundary term reads . (5.16) The asymptotic conditions are also invariant under gauge transformations generated by the parameters The transformation laws of the fields read The canonical generator of the asymptotic symmetries is then given by The zero mode gauge transformations are proper gauge symmetries with zero charge.We can therefore assume here also that the even gauge parameters ϵ a and µ a log have no zero mode and this will be done in the description of the symmetry algebra.
To conclude, we have four independent groups of angle-dependent U (1) symmetries, two in the original O(1)-sector as above, and two new ones in the log-sector (all of which with harmonic number ℓ > 0).
Finally, the canonical generator of SO(2)-duality rotations is given by (5.23)

Symmetry algebra
The computation of the Poisson bracket algebra of the generators is direct.The brackets of the u(1)-conserved charges and Poincaré charges read where ) ) (5.29) The brackets of the u(1)-conserved charges with the SO(2)-rotation generator are given by (5.33) The brackets between the u(1)-conserved charges give a centrally extended abelian algebra: where R commutes with Poincaré generators.Written in this way, the algebra takes exactly the form analysed in [37], where it was shown that the presence of an invertible central charge among a set of generators {q i , p j } enables one to decouple them from the rest of the algebra.More precisely, through (nonlinear) redefinitions, one can rewrite the algebra as a direct sum involving as one of its summands the subalgebra generated by the q i 's and p j 's.This method was adopted first in the context of gravity to provide a supertranslationindependent definition of the angular momentum [38].
For the model considered here, one must redefine the Lorentz and SO(2)-duality rotation generators as (5.45) It is interesting to note that in the enlarged context where logarithmic gauge transformations are included, the functional ´d3 x ξH + ξ i H i is well-defined as a generator, without surface terms.We can explicitly check that the only non-vanishing brackets of the canonical generators are given by Pξ,ξ i = ´d3 x ξH + ξ i H i is invariant under all gauge transformations, proper and improper.This automatically implies that it has vanishing brackets with the u(1) generators.(5.62) The vanishing of the Poisson brackets of the Poincaré generators with the generators of improper gauge symmetries is in particular obvious, since Pξ,ξ i is manifestly invariant under both proper and improper gauge transformations.

Concluding remarks
In this paper, we have investigated the asymptotic structure at spatial infinity of electromagnetism coupled to scalar fields through non-minimal couplings of Fermi type.We have shown that the rich structure found for the free Maxwell theory is unaffected by these couplings, contrary to minimal couplings which do have a non trivial impact asymptotically.
In the asymptotic context, non minimal electromagnetic couplings are similar to minimal gravitational couplings.
Our study also shows the interplay between asymptotic symmetries and duality symmetries, which arise as "hidden symmetries" in supergravity [20].It would be interesting to push the analysis further and consider (conjectured) infinite-dimensional hidden symmetries such as E 10 [39,40], which could be broken to its "compact" subgroup K(E 10 ) by the relevant boundary conditions.