Higher spins from Nambu-Chern-Simons theory

We propose a new theory of higher spin gravity in three spacetime dimensions. This is defined by what we will call a Nambu-Chern-Simons (NCS) action; this is to a Nambu 3-algebra as an ordinary Chern-Simons (CS) action is to a Lie (2-)algebra. The novelty is that the gauge group of this theory is simple; this stands in contrast to previously understood interacting 3D higher spin theories in the frame-like formalism. We also consider the $N=8$ supersymmetric NCS-matter model (BLG theory), where the NCS action originated: Its fully supersymmetric M2 brane configurations are interpreted as Hopf fibrations, the homotopy type of the (infinite) gauge group is calculated and its instantons are classified.


Introduction
It is by now well known that interacting 3D higher spin gravity can be formulated without any of the complications of its higher dimensional cousins. This is essentially related to the fact that 3D Einstein gravity (with cosmological constant Λ) propagates zero degrees of freedom on-shell, and its solutions are always locally flat or (A)dS 3 . In fact the Einstein action may be written as a Chern-Simons (CS) theory for the isometry group (this was first observed by Achúcarro and Townsend [1], see also Witten [2]). When the cosmological constant is nonzero, this is a CS theory for some real form of sl(2) ⊕ sl (2), with the details depending on the sign of Λ and whether we are considering Lorentzian or Euclidean signature.
The first consistent interacting higher spin theories were put forward by Blencowe [3] and Bergshoeff, Blencowe and Stelle [4]. These are CS theories defined by a choice of Lie algebra with an sl(2) ⊕ sl(2) subalgebra, and include the familiar sl(n) ⊕ sl(n) theory that describes a finite number of interacting higher spins 2, 3 . . . n, its infinite dimensional cousin sdiff(S 2 ) ⊕ sdiff(S 2 ) (where sdiff(S 2 ) is the Lie algebra of area-preserving vector fields on S 2 ) and various deformations and superextensions thereof. The common feature of this class of theories is that the Lie algebra involved is always a direct sum of the form hs ⊕ hs (1.1) where hs ⊃ sl(2) is a placeholder for any of the aforementioned algebras 1 . Let us note that such a CS theory is parity preserving as long as the action takes the form 2) where both A ± are hs-valued connection 1-forms defined to trade places (A + ↔ A − ) under a parity transformation 2 , and indeed the higher spin theories we have discussed so far are of this form.
In this paper we will propose a consistent theory of higher spin modes interacting with gravity in three dimensions whose gauge group (and algebra) is instead simple. That appears to be in tension with parity; indeed the action is not that of an ordinary Chern-Simons gauge theory, but is instead what we will call a Nambu-Chern-Simons (NCS) action. This can be seen as a CS theory for the Nambu 3-algebra of smooth functions on the 3-sphere S 3 with volume form ǫ, where the ternary Nambu 3-bracket is defined as {f, g, h} ≡ ǫ ijk ∂ i f ∂ j g∂ k h , f, g, h : S 3 → R . (1.3) While one can, of course, replace the 3-sphere in this definition with another 3manifold, we will only consider the S 3 case in this paper, excepting a few remarks in the discussion later. With this choice of 3-manifold we will refer to this theory as SDiff(S 3 ) NCS theory. The gauge group SDiff(S 3 ) is the (infinite-dimensional but otherwise ordinary) Lie group of volume-preserving diffeomorphisms of S 3 . This group is known mathematical literature to be simple [8] 3 ; it appears impos-1 For a more recent higher spin theory of this sort, see [5]. 2 This intrinsic parity assignment is on top of the usual transformation of the components of any 1-form under spacetime parity, see e.g. [6] where this point is discussed in more detail. This mechanism for parity preservation in Chern-Simons theory was originally introduced in [7]. 3 More precisely, the identity component of SDiff(S 3 ) is a simple group; whenever referring to diffeomorphism groups in this paper we will always mean the identity component. The simplicity of SDiff(S 3 ) follows from (5.1.3, ii) of the textbook [8] which asserts that the kernel of the "flux homomorphism" Sω is simple. For the 3-sphere that kernel is the entire group SDiff(S 3 ) because the 2nd de Rham cohomology group vanishes. The first proof of this theorem appears to be [9]. sible therefore to write an ordinary CS Lagrangian for this group that preserves parity, and for this reason the NCS Lagrangian cannot be written as an ordinary CS Lagrangian for the corresponding Lie algebra of divergence-free vector fields on S 3 , sdiff(S 3 ).
A critical role in our analysis will be played by the inclusion A celebrated theorem of Hatcher [10] asserts that this inclusion is a homotopy equivalence; the topology of SDiff(S 3 ) is in this sense dominated by that of its SO(4) subgroup 4 . We will find that this rather counterintuitive fact has a field theory counterpart: The SDiff(S 3 ) NCS Lagrangian consistently truncates-in a sense we will make precise later-to its SO(4) sector, which is an ordinary CS theory. This will allow for an easy proof that the NCS level takes integer values, circumventing the need to extend the arguments of Dijkgraaf and Witten [11] to the non-compact, infinite dimensional case. The group SO(4) ∼ = SU (2) × SU (2)/Z 2 is the gauge group corresponding to the spin 2 sector. Truncating to this sector gives Euclidean gravity on 3D de Sitter space [12]; likewise, su(n) ⊕ su(n) CS theory describes (Euclidean de Sitter) higher spin gauge theory (see appendix C of [13]). The SDiff(S 3 ) NCS theory therefore also describes higher spins on Euclidean de Sitter space, which we verify by calculating the spectrum of perturbations around this vacuum using harmonic analysis. The gauge field modes naturally organise themselves into an infinite pyramid, where each line is irreducible under SO (4): In this sense, the SDiff(S 3 ) NCS theory unifies all su(n) ⊕ su(n) CS higher spin theories at once. Our result therefore extends the older results of Bergshoeff, Blencowe and Stelle [4] on the smaller, area-preserving groups SDiff(M 2 ) × SDiff(M 2 ). The NCS action we will consider in this work was originally written down by Bagger and Lambert [14] and studied in the more general context of SDiff gauge theory by Bandos and Townsend [15] as part of the SDiff(S 3 ) NCS-matter model with maximal N = 8 supersymmetry proposed to describe an-infinite, in this case-collection of parallel M2 branes in M-theory. This lies in the more general class of OSp(8|4)-symmetric 5 models of Bagger-Lambert [16,14,17] and Gustavsson [18] (BLG). A BLG Lagrangian is defined by a choice of 3-algebra and a single dimensionless coupling constant, which appears in the action as a pre-factor. Assuming the 3-algebra is finite dimensional, the requirement of manifest unitarity uniquely fixes the model uniquely [19,20]; the Lagrangian is that of an ordinary su(2) ⊕ su(2) CS-matter theory. The coupling constant is identified with the CS level k and is quantized in the standard way; this integer, along with the choice of gauge group (e.g. SO(4) versus SU (2)×SU (2)), characterises the geometry probed by the M2 branes.
This conclusion was arrived at in the original references [21,22,23] (see [24] for a review) through an analysis of the space of vacuum configurations preserving all supersymmetries. For the su(2) ⊕ su(2) models, the most straightforward approach leads to a nonsensical result that both contradicts expectations from supersymmetry and fails to have an interpretation in terms of M2 branes on some spacetime [25]. The correct calculation appears to involve careful consideration of global features of the gauge group as well as the presence of instantons [21,22,23,24]. While we were able to prove that the NCS level k of the SDiff(S 3 ) theory is quantized, calculating the vacuum moduli space in the manner of [21,22,23,24]-and thereby relating k to the specifics of the geometry probed by the M2 brane condensate-faces a technical obstruction. However, the "naïve" approach goes through for the SDiff(S 3 ) BLG model, giving which is simply a 2-sphere's worth of indistinguishable planar M2 branes on flat R 2,1 ×R 8 times a nonzero integer; we will see that both factors can be understood together geometrically as embeddings of the base of the Hopf fibration S 3 → S 2 in R 8 , where the integer is the Hopf invariant. Finally, we give calculations in appendix B of certain topological data of SDiff(S 3 ) BLG theory, including the homotopy type of SDiff(S 3 ) as well as that of the unbroken gauge group H, and we use the results to obtain the topological classification of supersymmetric SDiff(S 3 ) instantons. We expect this information will be relevant in future studies of SDiff(S 3 ) BLG gauge theory in general, and its vacuum moduli space in particular.

The SDiff(S 3 ) Nambu-Chern-Simons Lagrangian. Harmonic expansion
The fundamental object of investigation in this paper is going to be the NCS action associated to the infinite 3-algebra of functions on a 3-sphere S 3 equipped with the Nambu bracket. This is a gauge theory for the group SDiff(S 3 ) of volume-preserving diffeomorphisms of S 3 . A detailed exposition of SDiff gauge theory may be found in [15]. Let us mention here that we will not actually use 3-algebra notation, and will instead follow [15] in writing down expressions adapted to SDiff(M n ). Another paper which treats SDiff gauge theory is the review [26], where 3-algebra expressions can be found.
To define SDiff(S 3 ), we introduce a volume form ǫ ijk on S 3 or (equivalently) a nowhere vanishing scalar density e. An infinitesimal SDiff(S 3 ) gauge transformation is then given by the action of an S 3 vector field (and spacetime scalar Higher spins from Nambu-Chern-Simons theory 5 field) ξ i that preserves the volume form, or (equivalently) is divergence-free: The vanishing divergence condition (2.1) can be solved for in favour of an unconstrained gauge parameter ω i defined by (ε ijk is the invariant tensor density) up to an S 3 -exact 1-form which will drop out of all expressions. As a side note, the requirement that we can write any divergence-free vector field in this way forces us to consider 3-manifolds of trivial 2nd de Rham cohomology, singling out the 3-sphere as the simplest candidate M 3 . The SDiff(S 3 ) gauge potential (where σ labels the S 3 coordinates and x the 3D spacetime ones) is an S 3 -vector-field-valued spacetime 1-form, and is also divergence-free in its S 3 index; we will solve the corresponding constraint in terms of an S 3 and spacetime 1-form A µi : We can now write down the SDiff(S 3 ) NCS Lagrangian. If we parameterise S 3 by coordinates σ i , i = 1, 2, 3 the Lagrangian can be given as an integral: This is invariant up to a spacetime exact term under the gauge transformation In these expressions we use form notation for spacetime and index notation for S 3 ; L SDiff(S 3 ) is an S 3 -independent spacetime 3-form. We emphasise that despite the ambiguity in the definition of A i , L SDiff(S 3 ) is well-defined. The action is of course given as the spacetime integral of the Lagrangian 3-form: This is parity invariant : a spacetime parity transformation (x 0 , x 1 , can be compensated by a corresponding internal S 3 parity transformation, e.g.
cancelling the sign change from spacetime parity.
This Lagrangian is Chern-Simons-like in that with F i the SDiff(S 3 ) field strength spacetime 2-form: (2.10) On-shell field configurations are therefore pure gauge, at least locally. This is an "exotic" gauge theory (in the terminology of [15]) as far as the Lie algebra of SDiff(S 3 ) is concerned, but perfectly ordinary from where the Nambu 3-algebra of functions with the bracket (1.3) is sitting: The spacetime 4-form dL SDiff(S 3 ) is the obvious generalisation of the invariant Chern-Weil polynomial to the 3algebra case. We refer to 15.2.3 of [26] for details on this point. A less compact-but more useful-form of the SDiff(S 3 ) NCS Lagrangian can be obtained by expanding the gauge field s i µ in harmonics. The appropriate set of S 3 harmonics is somewhat obscure but has been considered previously in the literature; it is the set of vector hyperspherical harmonics [27] which we will denote by V i α , following the conventions of [28] (but see [29] for a systematic derivation from representation theory). Let us expand The index α labels a state in a particular class of representations of so(4) ∼ = su(2) ⊕ su (2); it denotes the triple of indices where j α ≥ 1 is an integer labelling a representation of SO(3) ⊂ SU (2) × SU (2)/Z 2 and ǫ α = ±1 distinguishes between the "left" and "right" su(2) in so(4) ∼ = su(2) ⊕ su(2), while M α = (m α ,m α ) denotes a particular state in the irreducible representation of su(2) ⊕ su(2) specified by j α and ǫ α , see appendix A for more details. Now since d acts on x alone the SDiff(S 3 ) NCS Lagrangian in harmonics is 6 (2.13) and the gauge transformation of the gauge field mode s α is: (2.14) 7 E αβγ are the structure constants of SDiff(S 3 ) and are defined in appendix A. In this notation, an S 3 parity transformation swaps left and right su(2) factors [27]: 2.1. Truncation to su(2) ⊕ su(2). Quantization of the NCS level. We will now exhibit the ordinary su(2) ⊕ su(2) CS Lagrangian inside the SDiff(S 3 ) NCS Lagrangian. We will first use an abstract argument to see this. Consider a finite number of divergence-free vector fields U α that form a (finite dimensional) Lie subalgebra K of SDiff(S 3 ) under the Lie bracket, with structure constants C γ αβ : If we now write down the vector field U γ in terms of the curl of a 1-form W γ , or and given that for all divergence-free vector fields U, we get using Cartan's magic formula. Setting this equal to (2.16) implies for some S 3 scalar κ αβ , which will drop out in the following. This relation is necessary for proving gauge-invariance (under the subalgebra K) of the truncated SDiff(S 3 ) CS action. The infinitesimal gauge transformation If we use the truncated field A i (x, σ)| K ≡ A α (x)W iα (σ) then the infinitesimal variation with gauge parameter ξ i | K = ξ α U α closes (up to a total derivative which later drops out): The gauge variation of s i | K also closes (this is trivial since we assumed K is a Lie subalgebra of vector fields). Let us now split sdiff( where K is the (finite dimensional) subalgebra of vector fields ξ| K and M consists of the rest of the SDiff(S 3 ) generators. We truncate by setting the M components of the gauge fields to zero. The gauge invariance of the truncated Chern-Simons type action then follows from In the second line we set s| M = 0 in the field strength 2-form; we have already varied the full action so this is allowed. We have also dropped the total S 3 derivatives arising from the variation of A because F i is divergence-free in its S 3 index. The first term in the above equation is the gauge variation of the truncated action, therefore we need the second term to vanish. A sufficient condition for this is The coefficient E αβγ is totally antisymmetric and vanishes whenever This demonstrates that the SDiff(S 3 ) NCS action reduces to an SO(4) CS action. For our purposes we will need a more explicit formula; thankfully the mode expansion (2.11) is well-adapted to this purpose. In this notation, the truncation to SO(4) is realised by discarding all j α > 1. Then s ǫ=+ (resp. s ǫ=− ) transforms as a gauge connection in the spin 1 representation of the left (resp. right) SU (2) and is inert under the other factor. The reality condition s i µ = (s i µ ) * in conjunction with the mode expansion imply that s 1,2,3;+ defined as (resp. for s ǫ=− ) are real. Then an explicit calculation of E αβγ in (2.13) gives s ± = s 1;± σ 1 + s 2;± σ 2 + s 3;± σ 3 (2.31) and the σ i are the Pauli matrices. The right-hand side is immediately recognised as the CS action for gauge group G = SO(4) expressed in terms of its left and right SU (2) gauge fields, and is known to be gauge invariant when its prefactor is [22] k 4π , k ∈ Z . (2.32) We now come to the issue of the quantization of the Chern-Simons level of the SDiff(S 3 ) NCS action. Consider acting with a finite SO(4) gauge transformation on a field configuration where only the j α = 1 mode has been switched on. It is clear from formula (2.14) and the preceding discussion that this transformation will not switch on any modes of j α > 1. Therefore the SDiff(S 3 ) NCS action inherits the quantization condition of SO(4) CS theory. By the discussion above, the SDiff(S 3 ) action is properly normalised as and k ∈ Z is a necessary condition for invariance under finite SO(4) gauge transformations. Let us phrase this as a NCS Lagrangian 3-form as defined previously (with an S 3 of radius unity). Under a finite gauge transformation with gauge parameter g SO(4) taking values in (4) is the transformed gauge field A| SO(4) entering the S SDiff(S 3 ) action, and | SO (4) indicates that only the SO(4) modes are nonvanishing.
We stress again that we have only shown a necessary condition for gauge invariance under finite SDiff(S 3 ) transformations. In light of the fact that SO(4) ⊂ SDiff(S 3 ) is a homotopy equivalence (proven in appendix B) one naturally expects that this is also sufficient, but properly speaking one would have to study SDiff(S 3 ) NCS theory in the manner of Dijkgraaf and Witten [11] to prove this.

Interpretation as higher spin gravity.
For convenience, we reproduce the mode expansion of the SDiff(S 3 ) NCS action from before and also the gauge transformation of the gauge field mode s α : Could this describe a (Euclidean) theory of higher spin de Sitter gravity? The lowest su(2) spin (j α = 1) sector is certainly identical to ordinary Euclidean Einstein gravity in 3D de Sitter space [12]. To answer the question, we will follow Blencowe [3] and Campoleoni et al. [30] and examine whether the linearised theory around de Sitter space has a higher spin interpretation. Let us produce the full equations of motion: We first need to check that de Sitter space is a solution. Although this follows from the analysis of the preceding subsection, it is easy to check directly. Since we know that the j α = 1 action is that of Euclidean de Sitter 3D gravity, we need only check that solves the full field equations whenevers α solves the truncated equations. There are two cases to consider: j α = 1: We know that E αβγ is only nonzero when j β = j γ (see discussion around (A.21)). The j β > 1 contributions vanish; what remains is the truncated field equation whichs α solves by assumption.
j α > 1: The potentially problematic terms are the j β = 1 and/or j γ = 1 ones. By the previous remark and antisymmetry of E αβγ only one of j β , j γ can equal 1, but in this case (2.37) reduces to 0 = 0.
It remains to show that the action expanded to quadratic order around this background actually describes higher spin fields. Let us write where h α is the perturbation. The quadratic action is (for the SDiff(S 3 ) constants f γ αβ see (A.21) in appendix A): As one might have already guessed this action turns out to describe higher spins in a frame-like formulation, so we will give a very brief description of this formalism here, following [30]. For a spin s ≥ 2 field one has a generalised dreibein e These can be easily understood from the CS point of view. The generalised dreibein and spin connection are packaged into two independent SU (s) gauge fields A ± as A ± = ω ± e (2.43) and the higher spin action takes the form This is invariant under parity if we assign the parity transformation A + ↔ A − , implying the previous e and ω parity assignments. A mode h α of the SDiff(S 3 ) gauge field is not irreducible under the diagonal SO(3), but transforms as (see appendix A) suggesting that the SDiff(S 3 ) NCS action propagates a countable number of massless fields of all integer spins s ≥ 2 around its Euclidean de Sitter vacuum. There are two complications here however: The first is that the SDiff(S 3 ) NCS action clearly propagates zero degrees of freedom on-shell and it is not a priori clear what spin means in this context. This problem also comes up when we consider sl(n) ⊕ sl(n) CS higher spin gravity, and was resolved by Blencowe; the upshot of the discussion in [3] is that the 3D linearised equations of motion are matched to the dimensionally-reduced equations of free 4D higher spin fields. The other complication is that while expressions for the SDiff(S 3 ) constants f γ αβ certainly exist (in appendix A of this paper), they are in terms of Clebsch-Gordan coefficients and we could not work with them in generality. This can however be circumvented. The quadratic action (2.40) has a very particular Chern-Simons-like form, and we will show that the equations of motion derived from this action are fixed by the twin requirements of linearised "higher spin" gauge invariance (of the form δh α = dξ α + . . . ) and background gauge invariance under the diagonal SO (3), under which both the field perturbation h α and the background gauge fields α transform. Once this uniqueness is shown, one need only show that the resulting equations indeed describe higher spin gauge fields in the sense of Blencowe above; thankfully this has been thoroughly demonstrated by Campoleoni et al. for the AdS 3 case [30] and it also follows from the observation that su(n) ⊕ su(n) CS theory describes Euclidean de Sitter higher spins [13]. Consider a pair of 1-form field perturbations which have the same intrinsic parity assignments as e and ω and are valued in an arbitrary irreducible representation of the diagonal SO (3), and consider also the background dreibein and spin connection which we will collectively denote by E a . We are using different indices to stress that H A and E a transform differently under local SO (3). Invariance under local SO(3) corresponds to background gauge transformations, under which H A varies tensorially, but some components of E a (specifically, the spin connection) do not. Let us assume that H A has an ordinary gauge invariance of the form where N A aB is a spacetime-constant SO(3) tensor and Ξ A is a 0-form gauge parameter valued in the same representation as H A , and that H A satisfies a field equation are necessary and sufficient conditions for gauge invariance. The first equation fixes the field equation in terms of the gauge transformation, while the second is a condition on the background field alone, and should be equivalent to-or be implied by-the equation of motion for the background fields. Note that this second equation takes the same form as the field equations in the dreibein formalism: where the proportionality constant in the last formula is of course the cosmological constant. Therefore to specify the field equation for free higher spins we need three things: -An on-shell background dreibein and spin connection E a ; -a representation of the local "Lorentz" group SO (3) whereD is now the background exterior SO(3)-covariant derivative and T is an invariant tensor of SO(3). T can be further specified as follows: E b ∧ H B transforms in the 1 ⊗ j representation of SO(3) (where j + 1 is the spin j + 1 ≥ 2 of the free higher spin field H), which decomposes as SinceDH A by definition transforms in the j representation of SO (3), T is fixed up to a constant (itself fixed by (2.51)) by the requirement that it projects E b ∧ H B onto its j subrepresentation which only appears once in the sum, and is therefore some Clebsch-Gordan coefficient. To be more concrete, write Let also E A collectively denote a background dreibein e a and spin connection ω a , which are also spacetime 1-forms, parity eigenstates, and are valued in the 1 of SO (3), with ω a transforming as an SO (3) connection and e a transforming tensorially, which are also on-shell in the sense that they satisfy a (parity-preserving) equation This proposition was hinted at in [30].
Let us see how this applies to the quadratic action around the de Sitter vacuum of SDiff(S 3 ) gauge theory. The field equation derived from the action (2.40) is of the form (2.48) discussed above, as is the linearised gauge variation. The analysis in SO(4) harmonics implies, as outlined above, that an irreducible diagonal-SO(3) multiplet may be specified by a pair of integers (J, j) where J ≥ 1 is equal to the integer j α specifying an SO(4) irrep as before and 1 ≤ j ≤ J labels a representation in the decomposition (2.45) under the diagonal SO(3) of the corresponding mode h α of the field perturbation. Now suppose we keep just the (J, j) harmonics, setting all other h α to zero. This is an SO(3) and parity invariant condition which breaks all linearised higher spin gauge invariances apart from those of the form (2.47), i.e. with a gauge parameter Ξ α restricted to the (J, j) harmonic. From the preceding argument, the linearised equations of motion for the (J, j) harmonic mode should therefore be identical to the free higher spin equations of motion for spin j + 1, i.e. (2.58) and (2.59) once we combine the h α into parity eigenstates. This gives the pyramid of spins (1.5) in the Introduction, where each line is specified J = j α has maximum spin j α + 1.
We have verified that this is true for the (2, 2) harmonic by comparison to the (linearised) spin 3 action of Campoleoni et. al [30], where also the equivalence of this frame-like formalism to the one of Fronsdal is established. This is a straightforward but tedious and somewhat lengthy calculation which we will only summarise here. The relevant field perturbations h α are given in the (ǫ α = +) ↔ 1/2⊗3/2 and (ǫ α = −) ↔ 3/2⊗1/2 representations of SU (2)×SU (2)/Z 2 , both of which decompose as 2 ⊕ 1 (2.61) under SO(3). The modes in the 1 multiplet are superfluous and are set to zero. On the other hand, the generalised dreibein and spin connection in the usual formulation of free higher spins in the frame-like formalism are symmetric traceless 2-index tensors of SO (3), which means they appear as the 2 in the decomposition The two actions can be compared once both changes of basis (3/2 ⊗ 1/2 → 2) and (1 ⊗ 1 → 2) have been performed and the relevant SDiff(S 3 ) constants f α βγ have been tabulated; we performed this with the help of Mathematica routines.

The SDiff(S 3 ) BLG model
The The Lagrangian density can be written (suppressing Spin (8) indices where appropriate) where {φ I , φ J , φ K } is the Nambu 3-bracket: Details on γ µ and ρ IJ matrices, as well as proof that this Lagrangian enjoys OSp(8|4) superconformal symmetry can be found in the reference [15]. See also [31] for a reformulation with manifest supersymmetry. This theory has a 1/2 BPS smooth funnel solution describing M2 branes blowing up into a planar M5: Consider the configuration where the X a (σ) parameterise a unit 3-sphere as and all other fields are set to zero. This configuration describes a planar M5 brane at x 1 = 0 with an M2 brane "spike" of 3-sphere cross-section. As shown in [15], this solves the BPS equation This is the Basu-Harvey equation [32], which arises in all BLG models as a BPS condition. When the BLG gauge group G is SO(4) ⊂ SDiff(S 3 ), the corresponding solution (3.5) is interpreted as a fuzzy funnel; this parallels the better understood type IIB D1-D3 brane intersections which can be obtained from the M2-M5 intersections we are discussing by compactification followed by T-duality. There is a single, dimensionless, coupling constant in the SDiff(S 3 ) BLG model, which appears in the action as a prefactor, and may be identified with the invariant volume d 3 σ e of the three-sphere 7 . Invariance of the SDiff(S 3 ) NCS term under finite SO(4) gauge transformations forces it to take integer values, and the properly normalised BLG action takes the form Let us take a moment to comment. The first thing to notice is that with no continuous dimensionless couplings in the action, the SDiff(S 3 ) BLG theory must be conformally invariant to all orders in perturbation theory. The second, more puzzling observation, is that the theory appears to be weakly coupled when the Chern-Simons level k is large, but one would not naïvely expect any tunable parameters from the M2 brane picture. This was first pointed out for the su(2) ⊕ su(2) BLG Lagrangian (which also has a single integer parameter) by Raammsdonk [25]. Similar observations can be made for ABJM theory. In either case the resolution involves the relation between the M theory and type IIA pictures.
We will sketch the mechanism originally given by Townsend for the centre of mass modes [33] or equivalently for a single brane. A D2-brane spans 3 worldvolume dimensions; in a ten-dimensional spacetime its collective coordinates are 7 transverse scalars and a single U (1) gauge field. The gauge field can be dualised to a periodic scalar corresponding to the position of the brane in the 11th compact dimension; the periodicity comes about because the translation symmetry (properly, shift symmetry) of that scalar is broken by instantons in the 3D theory [34] (what we would call garden-variety magnetic monopoles for the U (1) gauge field in one dimension higher) whose charge is interpreted as momentum along the 11th compact dimension. The upshot is that the moduli space is given by R 7 × S 1 , where the S 1 radius is proportional to the coupling. In the infrared/strong coupling limit that radius blows up and we obtain the moduli space of a single M2 brane, which is simply R 8 . Weak coupling would therefore seem to be associated to a compactification limit.
The integer k of BLG/ABJM was ultimately found to be related to the spacetime geometry probed by the branes through analysis of the supersymmetric vacuum moduli space. For G = U (n) × U (n) ABJM theories at CS level k this is simply [35] M = (R 8 /Z k ) n /S n , (3.9) describing n indistinguishable M2 branes probing R 8 /Z k . The results for G = SO(4), SU (2)×SU (2) BLG theories generally fail to have such a straightforward interpretation except for the very lowest values of k, but it has been argued that they involve some sort of generalised orbifold. In either case, k → ∞ is a compactification limit, as expected. A crucial role in the analyses of [21,22,23,24,7 To compare with [15]: It was pointed out in that paper that one can redefine the dynamical fields such that the coupling g appears only outside the action. Under the simultaneous scaling e → λe, A → λA and (φ, ψ) → √ λ(φ, ψ) the action goes as S → λ 2 S, thus the coupling g of [15] is identified with the volume squared of the S 3 . The sign of the coupling does not appear to make a difference in our analysis so we only consider the positive case.

35] is played by G-instantons
, which break what would naïvely be a continuous U (1) identification of R 8 [25] down to a discrete Z k subgroup; only the latter quotient makes any sense.
In the next section we will perform a straightforward calculation of the vacuum moduli space, which-in contrast to the SO(4) model-turns out to have a nice interpretation. This is rather fortuitous since the more sophisticated approach of [21,22,23] (which gives the correct result for G = SO(4), SU (2) × SU (2)) appears to fail for G = SDiff(S 3 ). Unexpectedly, this is not due to SDiff(S 3 )-instantons, which turn out to be very well behaved.
3.1. Vacuum moduli space. The classical vacuum moduli space of the SDiff(S 3 ) BLG theory of [15] is defined to be the space of all on-shell bosonic field configurations that preserve all 16 supersymmetries, modulo gauge transformations. Inspection of the supersymmetry transformations (formula (4.4) of [15]) shows that this is equivalent to If we rephrase a bit, the last equality states that the maximum rank of the Jacobian matrix of the mapping is two. Therefore, a generic point in moduli space is given by a mapping into what is locally a 2-manifold M 2 .
Making the further assumption that M 2 is a manifold globally it is clear that M 2 is actually a 2-sphere 8 and these generic points in moduli space describe embeddings of the Hopf bundle Maps f : S 3 → S 2 are classified up to homotopy by the Hopf invariant (see e.g. the textbook [36]), which we will define. Let Ω be a volume 2-form on S 2 and consider its pullback f * Ω to S 3 . While Ω is by definition a closed but not exact 2-form on S 2 , its pullback to S 3 is necessarily exact (the relevant de Rham cohomology for S 3 is trivial). We can therefore define a 1-form ω on S 3 by dω ≡ f * Ω (3.14) and the Hopf invariant is given by The appropriately normalised Hopf invariant is always an integer. Hopf's original map h : S 3 → S 2 has Hopf invariant equal to 1. One can compose h with a map w k : S 3 → S 3 that winds k ∈ Z times to obtain a map of any Hopf invariant k, i.e. a representative in each homotopy class. Since a map w k , k = 1 fails to be volume-preserving, it does not lie in SDiff(S 3 ); hence the Hopf invariant also distinguishes between different branches of our moduli space. The upshot of the discussion so far is that generic points in SDiff(S 3 ) BLG moduli space are specified by 16) i.e. the non-vanishing Hopf invariant 9 and the embedding ι into R 8 . However, we have yet to quotient by the gauge group. If we let (z, θ), z ∈ (C∪∞) ∼ = S 2 , θ ∈ S 1 parameterise a local trivialisation of the Hopf fibration, then we can partially gauge fix SDiff(S 3 ) by setting There is a residual gauge group of fibre-preserving-in the sense that they map fibres to fibres-diffeomorphisms inside SDiff(S 3 ), which a priori fill out a subgroup of Diff(S 2 ). If we remove the volume-preservation restriction, it is certainly true that fibre-preserving transformations project down to the full diffeomorphism group of the S 2 base (this is the content of the "Projection theorem" of Cerf and Palais, see e.g. [37]). Whether this is still true if we restrict to fibre-preserving, total-space-volume-preserving diffeomorphisms is apparently unknown for nontrivial bundles but is clearly true for trivial ones. Assuming that this is true, the manifold of supersymmetric vacua is simply with the proviso that Diff(S 2 ) denotes the identity component; this avoids double counting. We have included the vanishing Hopf invariant sector here with the understanding that it contains all the nongeneric points of moduli space, which are not embeddings of S 2 . This moduli space can be interpreted as the space of configurations of a 2-sphere's worth of indistinguishable M2 branes, each fully localised on R 8 . Since M(SDiff(S 3 ) BLG) describes embeddings of (the base of) the Hopf fibration in the transverse coordinates, it seems tempting to interpret the vacuum moduli spaces of G = SU (2)×SU (2)(/Z 2 ) BLG theories as fuzzy Hopf fibrations, mirroring the interpretation of the 1/2 BPS funnel solutions of the previous section as "fuzzy funnels". In particular, a point on the moduli space should represent a fuzzy 2-sphere embedded in R 8 , and the fuzzy Hopf fibration should then be understood as the map from an abstract fuzzy S 3 (which has a transitive action of the gauge group G) to the fuzzy 2-sphere.
Let us compare with G = SU (2)×SU (2)(/Z 2 ) BLG. The corresponding moduli space can be obtained from the above discussion by replacing every instance of a diffeomorphism group with the corresponding isometry group, and the 2-sphere by the pair of points z ∈ C; the result is (R 8 ) 2 /O(2) [25]. This conflicts with expectations from supersymmetry-at the very least we would expect a manifold of even dimensionality-and also fails to admit an interpretation as the configuration space for some number of indistinguishable M2 branes fully localised on the transverse R 8 . The more sophisticated calculations of references [21,22,23] instead give (R 8 ) 2 /Γ k , where Γ k is a dihedral group whose order depends on k and on whether we include the Z 2 quotient in the gauge group G. The same methods also give the correct result for ABJM theory [35]. The essential step appears to involve carefully integrating out the unbroken gauge field A H (where the abelian H ⊂ G acts trivially on scalars in M) in the low-energy effective action describing motion in the moduli space. This is complicated because the gauge field associated to the broken, residual gauge group K (where K = O(2) for G = SO(4) or SU (2) × SU (2) and K = Diff(S 2 ) for G = SDiff(S 3 )) couples to it through a BF type term: (3.19) When K = O(2), F K is an abelian field strength. Integrating by parts produces an F H = dA H term which can be dualised into a scalar; this is a periodic scalar whose periodicity is controlled by and is ultimately responsible for breaking K = O(2) down to Γ k . When K = Diff(S 2 ) (G = SDiff(S 3 )), the field strength F K is nonabelian and this line of reasoning cannot proceed. However, this is the only obstacle, as the homotopy groups π 2 (SDiff(S 3 )/H), π 1 (H) (3.21) are in fact identical to the corresponding groups for G = SO(4)! We prove this assertion in appendix B. Let us conclude this section with a comparison to the SDiff(S 2 ) super Yang-Mills theory of [15]. This can be seen as an n → ∞ limit of SU (n) super Yang-Mills [38]: As observed in [39], under SU (2) while the expansion for SU (n) (under the adjoint action of SU (2)) terminates at spin (n − 1). SDiff(S 2 ) super Yang-Mills is an ordinary Yang-Mills theory, albeit with an infinite gauge group and written in different notation. Its vacuum moduli space is given by seven scalars φ I , I = 1, . . . 7 and the (dual of) the SDiff(S 2 ) gauge field, A, all functions on the 2-sphere. The vacuum is supersymmetric if all of these commute where the bracket is the Poisson bracket of functions on S 2 . The 8 functions (φ I , A) must therefore only depend on a single coordinate, tracing out a curve in R 7 × S 1 . The infrared limit (where S 1 blows up to R) thus fails to match any of the generic sectors (nonvanishing Hopf invariant) of the SDiff(S 3 ) BLG vacuum moduli space.

Discussion
We have demonstrated that SDiff(S 3 ) NCS theory is a higher-spin gauge theory on Euclidean de Sitter space in dimension three, whose gauge group is simple, and which may be seen as a unification of all su(n) ⊕ su(n) theories at once. While that might be an interesting fact in itself (e.g. for cosmological applications), one wonders whether a generalisation to (Euclidean) anti de Sitter exists. The obvious candidate is the NCS theory associated to hyperbolic 3-space The antisymmetry of E αβγ S 3 is most easily seen from its explicit expression as an integral over S 3 in appendix A. It is not a priori clear whether such formulae make sense as written for noncompact 3-manifolds like H 3 , but we expect that the issue will be clarified once the harmonic analysis of (divergence-free) vector fields on these spaces is worked out. H 3 in particular might be tractable since ISO(H 3 ) = SO(3, 1), which is the most familiar of the Lorentz groups.
It is reasonable to expect that this SDiff(H 3 ) NCS theory describes a "pyramid of spins" (1.5) around its AdS 3 vacuum. A-perhaps disturbing-feature of this spectrum is the infinite proliferation of spin 2 excitations. One suspects that these should not be interpreted as different kinds of gravitons. In 3D the difference between CS gauge theory and CS gravity is that in the latter case the dreibein (a particular combination of CS gauge fields) is assumed to always be invertible, and in particular the vacuum state is not the one where the gauge fields vanish. In section 2.2 we implicitly assumed that the SO(4)-irreducible spin 2 modes (at the top of the pyramid) indeed give rise to an invertible dreibein, but no such requirement was made of the other spin 2 modes, and indeed the argument of Campoleoni et al. in [30] which establishes the equivalence of the frame-like formalism to the Fronsdal formalism for free 3D higher spins only relies on the invertibility of the background dreibein. One expects that this issue will be clarified if the dual CFT is found; presumably that theory would be invariant under the SDiff(H 3 ) analogue of a W-algebra.
One could go on listing various potential applications of SDiff(H 3 ) NCS gravity. The most striking difference with the SDiff(S 3 ) model we have introduced in this paper is that the SDiff(H 3 ) theory must-if its details work out as expected-admit BTZ black hole solutions. It would be interesting to see whether the SDiff(H 3 ) gauge symmetry can resolve spacetime singularities.
Throughout this paper we have treated SDiff(S 3 ) gauge theory as a 3D gauge theory and not as some sort of rewriting of a 6D gauge theory. It is natural to expect that the SDiff(S 3 ) BLG action can be rewritten as the M5 brane action, and indeed this latter point of view has been taken in a number of works. To our knowledge the full SDiff(S 3 ) BLG model has not yet been rewritten in sixdimensional language; [40] did so for the quadratic theory while [41] achieved this for the interacting theory in a "Carrollian" limit. If this connection is true, our results could be relevant to the physics of M5 branes; one wonders about the interpretation of the quantization of the NCS level in this context.
Let us conclude with some further discussion of the G = SDiff(S 3 ) BLG model. While our calculation of the supersymmetric vacuum moduli space produced a result (3.18) which has a clear interpretation in terms of M2 brane physics, it is somewhat disconcerting that there is no dependence on the NCS level k, and in particular that k → ∞ appears not to be a compactification (which is to say, Yang-Mills) limit. For the G = SO(4) model one can easily obtain Yang-Mills theory through the "novel Higgs mechanism" of Mukhi and Papageorgakis [42]. This works for G = SO(4) ∼ = SU (2) × SU (2)/Z 2 because one can give a VEV to one of the scalars that preserves the diagonal SO(3) subgroup; this is not a generic point of moduli space. There is an obstruction to doing this in SDiff(S 3 ) BLG because there is no obvious choice of scalar VEV that gives some distinguished subgroup, and furthermore there is no obvious subgroup to choose (since SDiff(S 3 ) is simple and not a product). Furthermore the SDiff(S 3 ) BLG moduli space M does not appear to contain the moduli space of the G = SO(4) model except for k = 1 (where G = SO(4) describes two branes on R 8 ). One still wonders therefore where this theory fits in the greater M theory picture. Right now it is not clear whether e.g. one should expect an AdS 4 bulk dual or an AdS 7 one; the latter possibility is correct if the model turns out to describe an M5 brane. 10 We hope our "global" results of appendix B will be useful in sorting this out in the future.

Acknowledgements
I would like to thank the following for providing helpful correspondence, advice, feedback and/or a shoulder to cry on while this work was in preparation: Goro Ishiki, Tyler Kelly, David Tong, Nick Dorey, Carl Turner, James Gundry, Andrew Singleton, Alec Barns-Graham, Chris Blair, Rahul Jha, Ciaran Hughes, Ariana Stylianidi Christodoulou, Juan Maldacena, Will Jay, Emanuel Malek, Panos Betzios, Michele Zoccali and especially Christoph Wockel. Special thanks goes to Allen Hatcher for very patiently explaining his work and for providing general topological guidance, and to my Ph.D. supervisor Paul Townsend for initial collaboration and continuous good suggestions and support throughout, as well as reading a number of versions of this manuscript. Last but not least, I would like to thank the organisers of TASI 2015 as well as Will and Annie Jay for hospitality while part of this work was completed.
I acknowledge support from the UK Science and Technology Facilities Council (grant ST/L000385/1), from Clare Hall College, Cambridge, and from the Cambridge Trust.

A. S 3 harmonics
In this paper we make heavy use of the harmonic expansion for divergence-free vector fields on S 3 , following the notation and conventions of [28]. The harmonics will be defined in terms of the round S 3 metric of unit radius, and we will make no distinction between vector and 1-form harmonics as one can use the round S 3 metric to go from one to the other. We have implicitly assumed here we are using the volume form derived from the round metric, as we have been doing in the main text. This is not restrictive; Moser's theorem [44] states that any two volume forms (which integrate to the same total volume) may be mapped into each other by a diffeomorphism, so the different SDiff(S 3 ) groups in Diff(S 3 ) are all conjugate to each other.
A divergence-free vector harmonic V α ≡ V jαMαǫα;i is specified by three indices (j α M α ǫ α ) which we package into a single greek index α. The i index is an S 3 index and will usually be dropped for brevity. j α ∈ Z, j α ≥ 1 and ǫ α = ±1 specify a representation of the isometry group SO(4) ∼ = (SU (2) × SU (2))/Z 2 , and M α = (m α ,m α ) is a pair of SU (2) indices denoting a particular state in that representation. The pair of (half)integer SU (2) spins (Q α ,Q α ) specifying the representation are and M α = (m α ,m α ) takes values The vector harmonics are generally complex. Their conjugates are defined as We have the orthnormality conditions (Note: vol S 3 =2π 2 ): where we introduced the barred index notation Vᾱ ≡ (V α ) * . We also have the relations which is to say, the vector harmonics are divergence-free and are also eigenfunctions of the curl operator associated to the round S 3 metric. We will take a moment to define the coefficients E αβγ which will turn out to be the structure constants of SDiff: Alex S. Arvanitakis E αβγ are manifestly antisymmetric. They may be calculated from [28]: That is two Wigner 3j symbols times a reduced matrix element R 4 . The arguments of nonvanishing symbols satisfy triangle inequalities in the 3 elements of the first row: The numerical coefficient R 4 has an explicit expression: 4(x + 1)(y + 1)(z + 1) (ǫ x (x + 1) + ǫ y (y + 1) + ǫ z (z + 1) + 2)(ǫ x (x + 1) + ǫ y (y + 1) + ǫ z (z + 1) − 2) (A.12) The right-hand side of the equation above is defined to be non-zero only if the triangle inequality |x − z| ≤ y ≤ x + z (A. 13) holds, and if σ ′ ≡ (x + y + z + 1)/2 is an integer.
A.1. SDiff(S 3 ) structure constants. Consider two volume-preserving S 3 vector fields t and s. The Lie bracket [t, s] can be written This expression is clearly divergence-free. What remains is to calculate (t × s) jMǫ . The orthonormality relation gives To proceed we select t = V jtMtǫt , s = V jsMsǫs bending our convention on vector harmonic indices slightly to avoid clutter. Then using (A.3) and (A.8): and finally 19) or in the more compact notation using barred indices Therefore the coefficients E αβγ defined previously are the structure constants of the Lie algebra (not the 3-algebra) of SDiff(S 3 ). E αβγ with the minimal value of j α = 1 is especially interesting. When j α = 1, one of Q α ,Q α vanishes. Using the two triangle inequalities (A.11) and (A.13) it is then trivial to show that E αβγ with j α = 1 is only nonzero when j β = j γ and ǫ β = ǫ γ . To highlight these facts we will write E αβγ with j α = 1 as f α βγ is antisymmetric in its lower two indices (which is to say in M β and M γ ).

B. The topological classification of SDiff(S 3 ) instantons
In the calculation of the BLG vacuum moduli space for whatever gauge group G a crucial role is played by the periodicity of the scalar that is dual to the unbroken gauge field; integrating out the scalar along with the broken gauge field has the effect of orbifolding the directions transverse to the M2 branes. The periodicity of the scalar is determined by the quantization of magnetic charge carried by G-instantons, which is itself determined by the second homotopy group where G is the full gauge group and H is the unbroken gauge group, i.e. the stabiliser of any point of moduli space M(SDiff(S 3 )BLG) as defined in the main text. This much has been detailed in section 3.1 of this paper and the original references [21,22,23]. Here we will calculate π 2 (G/H) in the case G = SDiff(S 3 ).
Our main tool will be the long exact homotopy sequence of the fibration (fibre bundle) G → G/H, so we will begin by calculating the homotopy type of G = SDiff(S 3 ) and H. We will get a handle on the topology of SDiff(S 3 ) using the famous result of Hatcher [10] which states Theorem 1. The group Diff(S 3 ) of diffeomorphisms of the 3-sphere S 3 strongly deformation retracts onto its O(4) subgroup.
In particular, the two groups have the same homotopy type. This does not immediately descend to the group of volume-preserving diffeomorphisms SDiff(S 3 ) that we are interested in 11 . However, the relationship between them is known. The following result was originally proven by Ebin and Marsden in the context of geometric hydrodynamics: Theorem 2. (5.1) of [43]. For any compact orientable manifold M without boundary,

Diff(M )
is diffeomorphic to SDiff(M ) × V This theorem is perhaps most easily understood as the statement that the principal bundle Diff(M )/SDiff(M ) is trivial, and may be identified with the space V (Moser's theorem [44]). V is an affine space, whence contractible, yielding a (strong) deformation retraction of Diff(S 3 ) to its SDiff(S 3 ) subgroup. Combined with Hatcher's result, we see the following Let us reiterate here that we always consider the identity component of any diffeomorphism groups in this paper unless otherwise stated. This theorem implies in particular π 1 (SDiff(S 3 )) = Z 2 , (B.2) π 2 (SDiff(S 3 )) = 0 . (B.3) Let us give a less terse definition of the unbroken gauge group H here. If h : S 3 → S 2 is the Hopf map that makes S 3 into the total space of an U (1)bundle P over S 2 , we have It is not hard to see that this is the set of g ∈ SDiff(S 3 ) that act trivially on φ : S 3 → R 8 which make the 3-bracket vanish, i.e. points on M(SDiff(S 3 )BLG). The key point is that g ∈ H thus defined fixes the S 2 base and is volume-preserving on the S 3 total space, hence it is volume-preserving on each S 1 fibre. But this means it acts as a U (1) transformation on the fibres! H is thus equivalently described as the group of gauge transformations Gau(P) of the U(1)-bundle P, also known as vertical automorphisms in the mathematical literature. There exists a result by Wockel which yields information about the homotopy type of Gau(P), valid under rather general conditions: Lemma 1. (II.8.) of [45]. If P = (K, h : P → S m ) is a principal K-bundle over S m and K is locally contractible, there is a long exact homotopy sequence · · · → π n+1 (K) → π n+m (K) → π n (Gau(P)) → π n (K) → π n+m−1 (K) → . . .
We are now in a position to calculate the homotopy groups of interest. We have the long exact homotopy sequence associated to the principal fibre bundle (G = SDiff(S 3 )) H ι − → G → G/H , (B.10) viz.
· · · → π 2 (G) → π 2 (G/H) → π 1 (H) → π 1 (G) → . . . (B.11) Using the previous results this becomes · · · → 0 → π 2 (G/H) At this point we remind the reader that the arrows in an exact sequence are group homomorphisms such that the kernel of each arrow is the image of the preceding one. Therefore There is an ambiguity in that π 2 (G/H) could be either Z or 2Z, corresponding to whether σ is the zero map or not. This is easily resolved however: Consider the commutative square where d is the "diagonal" embedding of U (1) in SO(4) ∼ = (SU (2) × SU (2))/Z 2 , ι is the inclusion of H into G which induces the map σ between π 1 (H) and π 1 (G), and the horizontal arrows are also inclusions; i in particular is the U (1) action on S 3 that makes it into a fibre bundle. d clearly induces the zero map on the fundamental groups 12 . The inclusion i : U (1) → H induces an isomorphism between the fundamental groups; this is because the map ev x0 : H → U (1) which induces the sequence (B.5) of reference [45] also induces an isomorphism (we proved this below Lemma 1), and because ev x0 • i = id U(1) since ev x0 is evaluation of an element of H-seen as a space of equivariant mappings to the gauge group U (1) of the Hopf bundle-at some point x 0 on S 3 . Now the induced square on the fundamental groups π 1 also commutes and we therefore have σi * = 0 =⇒ σ = 0, (B.15) the zero map.