Heisenberg symmetry and hypermultiplet manifolds

We study the emergence of Heisenberg (Bianchi II) algebra in hyper-K\"ahler and quaternionic spaces. This is motivated by the r\^ole these spaces with this symmetry play in $\mathcal{N}=2$ hypermultiplet scalar manifolds. We show how to construct related pairs of hyper-K\"ahler and quaternionic spaces under general symmetry assumptions, the former being a zooming-in limit of the latter at vanishing cosmological constant. We further apply this method for the two hyper-K\"ahler spaces with Heisenberg algebra, which is reduced to $U(1)\times U(1)$ at the quaternionic level. We also show that no quaternionic spaces exist with a strict Heisenberg symmetry -- as opposed to $\text{Heisenberg} \ltimes U(1)$. We finally discuss the realization of the latter by gauging appropriate $Sp(2,4)$ generators in $\mathcal{N}=2$ conformal supergravity.


Introduction
In string theory, the Heisenberg algebra appears within the universal hypermultiplet of type IIA compactification [1]. The dilaton is contained in the scalar manifold, which is a fourdimensional quaternionic space [2]. 1 At tree level, the latter is CP 2 equipped with the Kähler (non-compact) Fubini-Study metric of SU(1, 2)/U (2). Perturbative corrections break the large isometry group of this space to the Heisenberg group, generated by three shifts (NSNS axion and RR scalar) [3]. More precisely, it was observed that the residual symmetry is rather Heisenberg ⋉ U(1), and that this symmetry uniquely determines the quaternionic space. Non-perturbative corrections further break the Heisenberg symmetry down to U(1) × U(1) [4] (or generically to a discrete subgroup of the Heisenberg group -see for example [5]).
The corresponding scalar manifold is thus a quaternionic space with two commuting Killing vectors. Metrics on these manifolds have been characterized by Calderbank and Pedersen [6].
In the above framework, supersymmetry is locally realized and the scalar curvature of the quaternionic space is directly proportional to the gravitational constant k 2 = 8πM −2 Planck [2]. For hypermultiplets of global N = 2, the relevant sigma-model target spaces are hyper-Kähler [7]. These Kähler spaces are Ricci-flat and, in the four-dimensional case, Riemann self-dual i.e. they are gravitational instantons. There exists then plausibly a low-energy decoupling limit of gravity M Planck → ∞, which deforms the quaternionic geometry into a hyper-Kähler limit. Since any hyper-Kähler manifold can be coupled to supergravity in a quaternionic manifold, this limiting process must smoothly interpolate between both geometries, and its description requires care. It implies to simultaneously "zooming-in" with appropriate k factors in order to recover non-trivial hyper-Kähler geometries [8]. This procedure has been demonstrated for specific cases, involving the quaternionic quotient method [9,10].
Although, as pointed out previously, the Heisenberg algebra is uniquely realized at the quaternionic level as Heisenberg ⋉ U(1), two distinct hyper-Kähler spaces exist with Bianchi II symmetry, realized either as Heisenberg ⋉ U(1) (biaxial), or as strict Heisenberg (trixial) [11]. The former corresponds indeed to the infinite-M Planck limit of the quaternionic space sharing its isometry and describing the string perturbative corrections to the hypermultiplet manifold, whereas nothing is known about the latter.
The purpose of the present note is to elaborate on the hyper-Kähler space with strict Heisenberg isometry. This raises a number of interesting questions, some of which stand beyond Heisenberg symmetry: • Under which conditions a hyper-Kähler space of a prescribed isometry can give rise to a quaternionic ascendent with the same or less symmetry?
• Conversely, what is the general limiting procedure for reaching smoothly a non-trivial hyper-Kähler space from a given quaternionic one?
Examples are known, where a quaternionic space is constructed starting from a four-dimensional hyper-Kähler geometry via an eight-dimensional hyper-Kähler cone [12,13]. An interesting relationship can be further settled amongst quaternionic spaces with an isometry and hyper-Kähler spaces with a rotational symmetry, equipped with a hyper-holomorphic connection (i.e. whose curvature is (1, 1) with respect to all complex structures in the hyper-Kähler family). This was developed in [14,15] from a mathematical point of view, and in [16] in a more physical framework. We will here provide an alternative and systematic algebraic procedure for a direct uplift, together with a scaling descent method. These general tools make it possible for investigating the more specific case of Heisenberg symmetry, reaching the following conclusions: • There are no quaternionic spaces with triaxial Heisenberg isometry.
• The hyper-Kähler space with Heisenberg ⋉ U(1) symmetry admits yet another quaternionic ascendent with U(1) × U(1) isometry, besides the known one with Heisenberg ⋉ U(1) symmetry of which it is the low-energy limit.
• The known quaternionic space with Heisenberg ⋉ U(1) symmetry is the extended symmetry point of an Sp(2, 4) gauging producing a family of Calderbank-Pedersen spaces.
In contrast with the biaxial case which captures well-identified perturbative string contributions [3], neither for the triaxial-Heisenberg hyper-Kähler, nor for the two quaternionic ascendents with U(1) × U(1) symmetry is such an interpretation available, though, and this issue is left for future work.
In the following we will first summarize the results regarding the realization of Heisenberg symmetry in hyper-Kähler spaces, Sec. 1. We will then move to quaternionic spaces in Sec. 2, and discuss the realization of Heisenberg symmetry, in particular the obstruction to a strict Heisenberg isometry group. The general procedure for taking the gravity-decoupling limit will also be presented, along with the systematic method for building up quaternionic ascendents, based on the existence of a Boyer-Finley field representation in quaternionic and hyper-Kähler spaces. As mentioned earlier, there are alternative methods for scanning these spaces, based on gaugings, which will be exposed in Sec. 3. Two appendices complete the technical details, in particular regarding the gauging procedure.

Translational vs rotational isometries
A four-dimensional hyper-Kähler space is Ricci-flat with (anti-)self-dual Riemann tensor: (+ corresponds to self-duality). In the presence of an isometry generated by a Killing vector ξ = ξ µ ∂ µ , using the Bianchi identity for the Riemann tensor, it is known that and consequently we can prove that (1.1) is equivalent to the Killing ξ is a translational vector; it is otherwise rotational.
Using the Killing vector at hand, we can adapt a coordinate τ to it, ξ = ∂ τ , and write the metric as a fiber along this isometry: where we note the gauge invariance When ∂ τ is a translational Killing vector, one is allowed to use the Gibbons-Hawking frame [17], or in everyday's language ∇V = ± ∇ ∧ ω, (1.8) whose compatibility yields the condition When ∂ τ is rotational, we can rewrite the metric in the Boyer-Finley frame [18], where expression (1.5) now holds with dℓ 2 = dZ 2 + e Ψ dX 2 + dY 2 , (1.10) and V, ω are now given by The third component w Z vanishes by a gauge choice of the coordinate τ. Now the (anti-)selfduality condition requires vanishing of the Laplacian over dℓ 2 in (1.10): This equation is known as the continual Toda equation, found in the context of continuum Lie algebras [19]. Notice that V satisfies instead of condition (1.9).
Finally we note that the translational case has an alternative formulation. Start with a function Ψ solution of the (flat-space) Laplace equation ∇ 2 Ψ = ∂ 2 X + ∂ 2 Y + ∂ 2 Z Ψ = 0, and write (1.11), in the gauge ω Z = 0. Therefore the function Ψ generates the metric without appearing explicitly in it.

The two geometries with Heisenberg symmetry
In a systematic investigation of four-dimensional gravitational instantons with Bianchi isometry group, the general solution for Bianchi-II type (Heisenberg symmetry) was found in Ref. [11]. The corresponding Riemann self-dual metrics read: with ε 0 a continuous parameter. When non-zero, this parameter can be reabsorbed in a coordinate redefinition, hence this family contains only two members. Here are the Maurer-Cartan left-invariant forms of Bianchi II. These are here realized as They are invariant under the Killing fields which obey the Heisenberg algebra: While X and Z are always translational, Y is rotational when ε > 0, or translational if ε = 0. Besides the generators of the Heisenberg algebra, the vector field turns out to play a rôle. It has the following commutation relations with X , Y , Z : These define a semi-direct product of a U(1) with the Heisenberg algebra, Z being the center of the resulting four-dimensional algebra.
The hyper-Kähler metric (1.14) is invariant under the full Heisenberg algebra. For vanishing ε, it is also invariant under M , which turns out to be rotational. It is common to call triaxial the realization of strict Heisenberg isometry as it occurs for ε > 0, and biaxial the case where it is accompanied with an extra U(1).

Kähler coordinates
Several Kähler coordinate systems are available for the spaces (1.21), providing various realizations of the Heisenberg ⋉ U(1) algebra. We can use for example a set of Kähler coordinates Φ and T defined as ( 1.22) and the Kähler potential is given by In these coordinates, the Heisenberg algebra is realized as: under which the Kähler potential is invariant up to the following Kähler transformation: The extra U(1) generator (1.19) reads now: It completes the Heisenberg ⋉ U(1) algebra. Note that M is a linear combination of the Heisenberg group generators X , Y and Z with field-dependent non-holomorphic coefficients, and thus does not correspond to a holomorphic transformation in these coordinates, in contrast to X , Y , Z . Consequently, the variation of the line element cannot be derived from the variation of the Kähler potential, but instead by direct computation of its Lie derivative: This vanishes only for ε = 0, in which case M generates a symmetry. Thus, for vanishing ε, the vector M is the generator of an extra isometry, promoting the symmetry to Heisenberg ⋉ U(1) with Kähler potential given by the finite part of (1.23) in the limit ε → 0 (the divergent terms are harmonic functions and play no rôle as they can be reabsorbed by Kähler transformations): (1.28) In the ε = 0 case at hand, the isometry generator M acts as a rotation in the (x, y)-plane.
As already mentioned, this action is non-holomorphic on the Kähler coordinates (T, Φ), but alternative sets of complex fields exist, in which all isometries are holomorphically implemented. We may choose for example: in which case the ε = 0 Kähler potential is given by The generators of the four isometries become and Q is now invariant under all transformations. These are the coordinates used in Ref. [8] and the relations between the three coordinate sets introduced here are as follows: 2 Quaternionic uplifts

Przanowski-Tod and Calderbank-Pedersen spaces
In the framework of N = 2 supergravity, we are here interested in four-dimensional quaternionic spaces with at least one shift symmetry, and these are part of a wide web of geometries with remarkable properties. A four-dimensional quaternionic space is an Einstein space with R = −12k 2 (k defining an overall scale in Planck units) and self-dual Weyl tensor. This space is always determined by a solution of the continual Toda equation [20][21][22][23]. If ∂ τ is the Killing field generating the shift symmetry, the metric on this space reads: with dℓ 2 of the form (1.10), and Ψ satisfying the continual Toda equation (1.12), The form A obeys with integrability condition: whereas U and Ψ are constrained by The corresponding quaternionic space is commonly known as Przanowski-Tod.
The Calderbank-Pedersen metrics studied in Ref. [6] are the most general quaternionic spaces with two commuting isometries (as they appear e.g. in the Heisenberg algebra (1.20)).
Hence, they belong to the Przanowski-Tod family. The generic Calderbank-Pedersen metric is expressed in terms of a function F(ρ, η), where ρ and η are two coordinates and the other two, τ and ψ, support the two Killing vectors ∂ τ , ∂ ψ . The function F is an eigenfunction of the Laplacian operator on the hyperbolic two-plane with metric dρ 2 +dη 2 ρ 2 , for eigenvalue 3 /4.

From Boyer-Finley to Przanowski-Tod and back
The key observation regarding hyper-Kähler spaces with a rotational Killing vector, on the one hand, and quaternionic spaces with a symmetry, on the other, is that they share the isometries, if present in one, may or may not be realized in the other.
The relationship between the pair of spaces built around one solution of Toda equation is even more intimate. Indeed, the hyper-Kähler member turns out to be the k → 0 limit of the quaternionic one, the limit being taken in an appropriate zoom-in manner for avoiding the trivialization of the geometry into flat space. For that, consider the following transformation: Performed on the Przanowski-Tod metric (2.1), on the form (2.2) and on the constraint equation (2.4), these read: In the double-scaling limit we recover a Ricci-flat (anti-)self-dual instanton (1.5) in Boyer-Finley frame (1.10) with V, ω satisfying (1.11). Whenever the quaternionic space is the manifold of a hypermultiplet coupled to N = 2 supergravity, the gravitational constant is k 2 = 8πM −2 Planck [2], and the limit k → 0 corresponds to a hypermultiplet of global N = 2.

An obstruction for quaternionic spaces with triaxial Heisenberg symmetry
The two hyper-Kähler spaces with Heisenberg algebra discussed earlier in Sec. 1.2, may be uplifted to quaternionic. Indeed, in both cases, a rotational Killing vector exists, and a Boyer-Finley frame can be exhibited, with a solution to Toda equation. Before analyzing these two specific spaces, we would like to demonstrate a general property, which will be illustrated afterwards: no four-dimensional quaternionic space exists with strict Heisenberg isometry.
There are at least two ways to prove this statement. Firstly, using isomonodromic deformations, a method developed by Hitchin [25] and Tod [26]; secondly using foliations with Heisenberg isometry. Here we choose the most economical one with the tools at hand, which is the second.
A general four-dimensional geometry with Heisenberg symmetry can be realized as a Bianchi-II foliation. We consider foliations of the type: with σ i the Bianchi II Maurer-Cartan forms, given in (1.16) and obeying (1.15). Here a, b, c are functions of t and fully characterize the geometry, which is by construction invariant under the Heisenberg algebra.
Two remarks are in order. Firstly, we might have chosen g ij (t) instead of diag(a, b, c). For unimodular Bianchi groups, however, such a non-diagonal form can always be brought into a diagonal one, and our choice is not restrictive (for a systematic analysis of this issue, see [11]). Secondly, the metric under consideration has strict Heisenberg symmetry, as long as We now impose that (2.14) satisfies Einstein's equations: where the scalar curvature is constant: R = −12k 2 . We finḋ (the dot stands for the derivative with respect to t) where λ(t), µ(t), ν(t) are first integrals The constant scalar curvature further imposes a(λ + µν) + bλν + cλµ = 3k 2 abc . (2.18) The requirement of Weyl self-duality is the next step: Using (2.16) and (2.17), three distinct cases emerge: Differentiating the latter with respect to t and using Eqs. (2.17), we obtain The ratio b /c remains thus constant, and the Heisenberg algebra has a biaxial realization in the quaternionic space (2.14) at hand: an extra Killing vector field emerges. Picking up for convenience b = c and so µ = ν, we find the general solution of (2.16), (2.17): where t has been traded for a new coordinate ρ: (up to an irrelevant additive constant reabsorbed in a redefinition of t). Trading (x, y, z) for (η, ψ, τ), the quaternionic space (2.14) reached with (2.23), (2.24) and (2.25) is This is the N = 2 hypermultiplet scalar manifold that captures string one-loop perturbative corrections, found in [3]. The metric (2.26) has well-defined M Planck → ∞ limit, which coincides with the Heisenberg ⋉ U(1)-symmetric hyper-Kähler space (1.21) at ε = 0 (see [8]).
In conclusion, the Heisenberg symmetry is always biaxially realized at the quaternionic level, leading to the geometry (2.26): no triaxial Heisenberg quaternionic space exists. We will meet an instance of this obstruction in the following, where the pure Heisenberg symmetry present at the hyper-Kähler level is actually broken to its U(1) × U(1) subgroup when moving to the quaternionic. This general conclusion is in agreement with the result of Ref. [13], where the quaternionic space obtained with Heisenberg symmetry has automatically an additional U(1) isometry.

The ε = 0 uplift
The corresponding hyper-Kähler metric, in the form (1.21) with ε = 0, is Gibbons-Hawking, as the fiber is supported by Z , which is translational. In this case the realization of the Heisenberg symmetry is biaxial and the rotational Killing vector is the extra generator M . We have to adopt it for the fiber and define τ such as M = ∂ τ . Together with τ, we introduce new coordinates X, Y, Z, for which we trade t, x, y and z: Here σ is an arbitrary constant, showing that the quaternionic ascendent is rather a oneparameter family (this is actually a systematic feature in all of our constructions).
To summarize, the quaternionic uplift of the unique Heisenberg ⋉ U(1)-symmetric hyper-Kähler space (Eq. (1.21) with ε = 0) breaks the Heisenberg ⋉ U(1) symmetry to U(1) × U(1). This space is not the one found in [3]. The latter is a Calderbank-Pedersen space with Together with Y = ∂ τ , the Killing field Z is now adapted to the coordinate X and everything depends on (Y, Z) only. The invariance under X (combination of ∂ X , ∂ Y and ∂ Z ) is the result of a fine cancellation amongst the two terms in (1.5). This cancellation no longer occurs in the uplifted quaternionic space, where ω is traded for A as in (2.2), and V for U given in (2.4). Again, the Heisenberg symmetry is broken down to U(1) × U(1), generated by Y and Z . In this case, the breaking was expected since no strict Heisenberg isometry exists at the quaternionic level, as shown in the beginning of the present section.

Heisenberg algebras and gaugings
To summarize at this point, any hyper-Kähler space with a rotational Killing vector can be uplifted to a quaternionic space, by going to the Boyer-Finley frame and using the available solution of the Toda equation, Ψ(X, Y, Z). The original hyper-Kähler space appears as a vanishing-k 2 double-scaling limit of the quaternionic ascendent. In general however, this procedure does not respect the isometry content of the hyper-Kähler space and there could be, as in the example of the Heisenberg⋉U(1) isometry, another quaternionic ascendent with identical isometry. Using this construction, we have studied the quaternionic ascendents of the Heisenberg hyper-Kähler instantons (1.21) with ε = 0 and ε = 1. We found in both cases Calderbank-Pedersen spaces with U(1) × U(1) isometry, and no further extension. This is not surprising for ε = 1, as we have proven in Sec. 2.3 that no quaternionic space with triaxial Heisenberg symmetry exists. The isometry of the ε = 1 hyper-Kähler geometry could then only be broken in its quaternionic ascendent.
The above results raise several questions, that we would like to investigate, at least partially, in the remaining of this note. Besides the interpretation of the two quaternionic spaces at hand in terms of string corrections to the hypermultiplet manifold, which seems out of reach at present, we would like to question the possibility of realizing these geometries in terms of gaugings inside the invariance group of the N = 2 hypermultiplet supergravity couplings, Sp(2, 4). We may wonder whether this is possible, and which generators should be gauged for reaching the two quaternionic ascendents we presented, Eqs. (2.30) and (2.35), as well as the already known Heisenberg ⋉ U(1) space, Eq. (2.31). The way to proceed is to use hypermultiplet(s) coupled to local N = 2 superconformal symmetry [27,28] and to perform a quaternionic quotient [9,10] using non-propagating vector multiplet(s) and additional hypermultiplet(s).
We will not present an exhaustive analysis of all possible gaugings, but rather use this method in order to recover the quaternionic space missing in our previous approach of Sec. 2: the one with Heisenberg ⋉ U(1) symmetry discussed in [3] and displayed in Eq. (2.31). For this purpose, we need to gauge the generators (Y , Z ). We will not study here other concrete possibilities, as e.g. the (M , Z ) gauging. We nevertheless wish to illustrate the power of the gauging procedure and use it to show that no triaxial Heisenberg symmetry could be reached for quaternionic spaces by gauging SU (1, 2) generators. This is achieved by analysing the little group of SU(1, 2) and demonstrating the absence of strict Heisenberg orbit. Since the general proof of non-existence of triaxial Heisenberg symmetry in quaternionic spaces was already presented in Sec. 2.3, we leave this complementary exercise for App. B.
The (Y , Z ) gauging provides a two-parameter family with generic U(1) × U(1) generated by Y and Z , which contains a one-parameter subfamily with Heisenberg ⋉ U(1) [8]. This family, is the one describing the one-loop perturbative corrections [3]. After a tedious computation, which is sketched in App. A, we find a Calderbank-Pedersen space with where χ and σ are the two arbitrary parameters, associated respectively with the Y and Z gaugings. The line element (Eqs. (2.5), (2.6) and (2.7)) reads: The family of Calderbank-Pedersen spaces at hand possesses generically two commuting isometries generated by Y = ∂ ψ and Z = ∂ τ . For vanishing χ, the function G(ρ, η) in Eq. (3.1) matches that of Eq. (2.31), whereas the metric (3.2), coincides with (2.26). This is where the symmetry is extended to Heisenberg ⋉ U(1), with two extra Killing fields X , M . When σ also vanishes, the isometry is further enhanced to U(1, 2), and the metric is the non-compact Fubini-Study. All these properties are analysed in App. A from the gauging perspective.
Our next task is to find a convenient zooming-in limit when k → 0, leading to a hyper-Kähler space for the family (3.2). Such a limit requires the Kretschmann (K = R κλµν R κλµν ) scalar to remain finite. In this limit, however, the latter vanishes unless In order for the line element to remain regular when V 1 vanishes, we should perform the appropriate coordinate rescalings, keeping in particular the τ-fiber finite (see also [8]). We define new coordinates t, x, y, z as , τ = z k 4 /3 1 + 4χ 2 1 /6 . (3.5) Hence, the limit k → 0 amounts to zooming around the point 5 (ρ 2 0 , η 0 ) = (4χ 2 − 2σ, χ), and obtain This is the hyper-Kähler metric with Heisenberg ⋉ U(1), given in Eq. (1.21) with ε = 0.

Conclusions and outlook
An important result in the present note is the obstruction for a quaternionic space to host a strict (triaxial) Heisenberg algebra. This can be rigorously demonstrated in at least two ways, we have chosen to use the foliation technique in Sec. 2.3 and further showed in App. B how the gauging techniques operate in the same direction (excluding the strict Heisenberg orbit).
The above obstruction is illustrated when scanning over the landscape of hyper-Kähler and quaternionic spaces. As a tool for such a scanning, we introduced a method which allows to uplift hyper-Kähler geometries possessing rotational Killing vectors, to quaternionic spaces. Indeed, both hyper-Kähler spaces with a rotational symmetry and quaternionic spaces with a symmetry rely on a solution of the continual Toda equation, and this solution bridges the two geometries (conversely, a systematic k → 0 zooming-in in the quaternionic space enables us to recover the original hyper-Kähler geometry, and this is useful when the latter appears in the global N = 2 limit). In the course of the uplifting, part of the extra symmetries are usually lost. Applied to either of the two Bianchi II hyper-Kähler spaces, i.e. with biaxial (Heisenberg ⋉ U(1)), or triaxial (strict Heisenberg) realization of the symmetry, the proposed uplift leads to a Calderbank-Pedersen geometry with only U(1) × U(1) isometry.
In order to make contact with supergravity applications, it is useful to recover the quaternionic space known to capture the perturbative corrections in the type IIA hypermultiplet scalar manifold, Eq. (2.31). The global N = 2 limit of this space is again the unique biaxial Heisenberg ⋉ U(1) hyper-Kähler. However, the sought for quaternionic space is not obtained using the above uplifting procedure. Instead, performing a gauging in the appro-priate directions inside the Sp(2, 4) algebra, a full family of Calderbank-Pedersen spaces with U(1) × U(1) symmetry is reached, Eq. (3.1), which contains an extended-symmetry point where Heisenberg ⋉ U(1) is realized. This is the corrected type IIA hypermultiplet scalar manifold.
SU(2) R index i = 1, 2 and Sp(2, 4) index α = 1, . . . , 6. They transform in the representation (6, 2) of Sp(2, 4) × SU(2) R . Their conjugates are A i α = (A α i ) * = ε ij ρ αβ A β j with ρ αβ ρ βγ = −δ α γ and ε ij ε jk = −δ i k . We choose the Sp(2, 4)-invariant metric as At the tree-level, the universal dilaton hypermultiplet is mapped, after Poincaré duality, to the quaternionic and Kähler pseudo-Fubini-Study metric of the coset space [30]. At one-loop, the isometry is lessened to the Heisenberg subalgebra of SU(1, 2), which is generated by the following three elements: with [X , Y ] = Z . We will gaugeT where (χ, σ) are two arbitrary parameters. This gauging contains a one-parameter subfamily with Heisenberg ⋉ U(1), asT commutes with it, where the U(1) is generated by We now come to the N = 2 conformal supergravity Lagrangian. Eliminating the auxiliary fields from the gauge-fixing of the dilation symmetry in the Poincaré theory, we find [8]: (A.6) 6 Our choice of η incorporates the nature of the hypermultiplets, such that the constraints can be solved [8].
Here A is a complex-doublet vector of components A α i (and A i α for its complex-conjugate) subject to the constraints and T the Sp(2, 4) extension of (A.4): Working along the same lines of [8] we find the solution for the constraints: where K, ∆ are real quantities given by The scalar kinetic Lagrangian (A.6) obtained with (A.10) reads: The Lagrangian can be expressed as usual: where q = (∆ 2 , Re Φ, Im Φ, ImS), and we introduce τ, η and ψ as τ = ImS and η /2 + iψ = Φ.
The metric G ab = g ab/k 2 describes a Calderbank-Pedersen space with where ρ = ∆ 2 − 2χ η + 2σ − 4χ 2 . The corresponding line element reads:  (3)). In the non-compact SU (1,2) case, this result remains true, but the identification of the little groups is more subtle.
Let U be an element of the su(1, 2) algebra with invariant metric η = diag(−1, 1, 1). The first direction is then time-like. 7 We shall parameterize U as (B.1) The non-compact SU(1, 2) has four Jordan conjugacy classes [29]. Following the terminology of Ref. [29], we shall refer to them as elliptic, hyperbolic, one-step parabolic and two-step In the second case, the U(1) is generated by M (which is present in all four cases).
2. In the hyperbolic class, U has a space-like eigenvector with imaginary eigenvalue (C = B = 0) and null (light-like) eigenvectors (non-orthogonal in metric η, with eigenvalues A and −A). It takes the form: It is a linear combination of the two commuting generators of the little group: G(U ) = U(1) × U(1). One generator is M , the second is in the SU(1, 1), which commutes with M .
3. In the one-step parabolic class, U can be written as: with (λ, a, ϕ) ∈ R. It admits one space-like and one light-like eigenvector with imaginary eigenvalue iλ.
• For λ = 0, one can write and, since R is an SU(1, 2) element, U is in the same orbit as − a 2 Z which is known to commute, as one can explicitly verify, with the four-dimensional Heisenberg ⋉U(1) algebra. 4. In the two-step parabolic class, with triple zero eigenvalue and a light-like eigenvector. The little group is G(U ) = U(1) × U(1), generated by Y and Z .
We can recapitulate the above results, regarding the possible gaugings within SU (1, 2), as follows. The little groups of the SU(1, 2) adjoint representation can be either: There is no three-dimensional little group in SU(1, 2), hence the option of strict Heisenberg symmetry is not available.