$D=6$, $\mathcal{N}=(2,0)$ and $\mathcal{N}=(4,0)$ theories

Using a convolutive field theoretic product, it is shown here that the"square"of an Abelian $D=6$, $\mathcal{N}=(2,0)$ theory yields the free $D=6$, $\mathcal{N}=(4,0)$ theory constructed by Hull, together with its generalised (super)gauge transformations. This offers a new perspective on the $(4, 0)$ theory and chiral theories of conformal gravity more generally, while at the same time extending the domain of the"gravity = gauge $\times$ gauge"paradigm.


I. INTRODUCTION
It was at one time thought that non-trivial conformal quantum field theories exist in at most D = 4 spacetime dimensions. This was somewhat at odds with Nahm's classification of admissible supersymmetries [1], which includes D = 6 superconformal algebras. Indeed, a remarkable prediction of M-theory [2][3][4], anticipated in [5,6], is the existence of non-trivial D = 6 quantum field theories with N = (2, 0) supersymmetry and OSp (8|4) superconformal symmetry, contradicting the received wisdom of the time while placing another feather in Nahm's cap. These "(2, 0) theories" are not only central to our understanding of M-theory; they have fundamental implications for lower-dimensional gauge theories more generally, from S-duality to the Alday-Gaiotto-Tachikawa (AGT) correspondence [7][8][9].
Of course, the consistency of a given superalgebra does not imply that a corresponding non-trivial quantum field theory necessarily exists. See for example [10]. However, taking confidence from the (2, 0) story it is tempting to speculate that the D = 6, N = (4, 0) multiplet with OSp (8|8) superconformal symmetry, a longstanding and enticing outpost of Nahm's taxonomy, should also correspond to a non-trivial quantum theory. Indeed, drawing on a range of analogies with the (2,0) theories Hull argued [11][12][13] that a non-trivial "(4, 0) theory" may arise in the large D = 5 Plank length, l 5 , limit of M-theory compactified on 6-torus, T 6 . As emphasised by Hull, the (4, 0) theory would constitute the maximally symmetric phase of M-theory. Moreover, it contains a self-dual "gravi-gerbe" field, suggestive of a D = 6 chiral theory of conformal gravity. Note, a local variational principle, breaking manifest covariance, for the free gravi-gerbe field was recently developed in [14]. Consequently, just as for the (2, 0) theories before it, establishing its existence would have profound implications for not only M-theory, but also gravity more broadly understood. It should be stressed that while there is a large body of strong evidence, originating from string/M-theory, for the (2,0) * leron@stp.dias.ie theories, there are at present no comparable arguments supporting the existence of the (4,0) theory and it remains highly conjectural. For a more nuanced discussion of the various possibilities, and the associated difficulties, the reader is referred to [11-13, 15, 16].
Here we re-examine the free (4, 0) theory introduced in [11][12][13] from another, a priori unrelated, but equally provocative, perspective: "gravity = gauge × gauge". While on face-value a radical proposal, this paradigm has been reinvigorated in recent years by the remarkable Bern-Carrasco-Johansson double-copy procedure [17][18][19]; the scattering amplitudes of (super)gravity are conjectured to be the "double-copy" of (super) Yang-Mills amplitudes to all orders in perturbation theory! These fascinating amplitude relations are both computationally expedient and conceptually suggestive, facilitating previously intractable calculations while probing profound questions regarding the deep structure of perturbative quantum gravity [20,21].
In this context D = 5, N = 8 supergravity, the low energy limit of M-theory on a 6-torus, is the double-copy of D = 5, N = 4 super Yang-Mills theory. Of course, D = 5 Yang-Mills theory is non-renormalisable and we expect new physics to enter for energies E ≥ 1/g 2 Y M . For instance, it can be regarded as the low-energy sector of the world-volume theory of a stack of D4-branes in string theory. Taking the strong-coupling limit the Yang-Mills theory uplifts to a (2, 0) theory compactified on a circle of radius R ∝ g 2 Y M , which in this setting constitutes the low-energy theory arising on a stack of M5-branes in Mtheory. This raises a challenging question: what happens to the double-copy in this limit? Might we expect some relation of the type (4, 0) = (2, 0) × (2, 0), morally the M-theory uplift of gravity = gauge × gauge?
The (4, 0) = (2, 0) × (2, 0) picture was proposed in [16], where the ultra-short (4, 0) supermultiplet of the six-dimensional conformal superalgebra OSp (8|8) was derived and shown to consistently factorise, with respect to the R-symmetry algebras USp(4) × USp(4) ⊂ USp(8), into the product of two (2, 0) tensor multiplets. However, as emphasised in [16] the intrinsically non-perturbative nature of the (2, 0) theories makes amplitude relations hard to formulate, although there exist some limited tests [16,22,23]. Here we avoid this hurdle altogether by appealing to a complementary and independent offshell field-theoretic realisation of gravity as the "square of Yang-Mills" developed in [24][25][26][27][28][29][30][31][32][33], which can be used to study the product of two gauge theories without reference to amplitudes, allowing one to derive various properties, such as curvatures, dynamics, off-shell local symmetries and duality relations, directly. For two gauge potentials belonging to two distinct Yang-Mills theories, referred to as the left (no tilde) and right (tilde) factors, with arbitrary gauge groups G andG, the product is given by [26]: The bi-adjoint "spectator" scalar field Φ allows for arbitrary and independent G andG, while the convolution reflects the fact that the amplitude relations are multiplicative in momentum space. Crucially, together they ensure that both the global and local symmetries of the two factors are consistently mapped into those of the corresponding gravitational theory, including general coordinate transformations [24][25][26][27][28][29]. To linear approximation the equations of motion of the factors then imply those of the gravity theory and classical solutions of the Yang-Mills factors are mapped into solutions of their product [26,30,31]. Extending this construction it is shown here that, by defining a field (7) and ghost field (9) dictionary, the product of two arbitrary Abelian (2, 0) theories generates, with no further input, the free (4, 0) theory first constructed by Hull [11]. This represents a new perspective on the (4, 0) theory that may be exploited to better understand its remarkable, as yet rather mysterious, properties, while at the same time extending the rapidly evolving domain  of the gravity = gauge × gauge paradigm.

II. STRONGLY COUPLED YANG-MILLS AND (2, 0) THEORIES
The free (2, 0) theory is described by the (2, 0) tensor multiplet consisting of an Abelian two-form gauge potential B µν with self-dual three-form field strength H = H, four symplectic Majorana-Weyl spinors χ and five scalars Φ, transforming respectively as the 1, 4 and 5 of the the rigid Spin(5) ∼ = USp(4) R-symmetry. The two-form gauge and gauge-for-gauge transforms are given by leaving 15 − 6 + 1 = 10 off-shell degrees of freedom. The equation of motion d H = 0 leaves six on-shell degrees of freedom in the (3, 1) + (1, 3) representation of the spacetime little group Sp(1) × Sp(1). The self-duality condition, which with the Bianchi identity dH = 0 implies the equation of motion, further reduces these to the chiral (3, 1) representation. Dimensionally reducing on a circle, S 1 , with radius R yields the maximally supersymmetric Abelian D = 5, N = 4 gauge theory, consisting of a one-form Abelian gauge potential A m , four symplectic Majorana spinors ψ and five scalars φ, with coupling constant g 2 ∝ R and the same USp(4) R-symmetry. Going beyond the free theories it has been conjectured [56,57] that the strong coupling limit of D = 5, N = 4 Yang-Mills theory is given by an interacting (2, 0) theory compactified on S 1 with g 2 Y M ∝ R. Crucial to this picture is the existence of 1/2-supersymmetric instantonic 0-branes in the D = 5, N = 4 Yang-Mills theory, which preserve the full USp(4) R-symmetry. They have mass ∝ |n|/g 2 Y M , where n is the instanton number, so that they become light in the strong coupling limit and can be matched to the Kaluza-Klein modes of the (2, 0) theory compactified on S 1 , which have mass ∝ n/R [56].  (8) R-symmetry and an exceptional non-compact global E 6(6) (R) symmetry [58] that is broken by quantum effects to the discrete subgroup E 6(6) (Z), corresponding to the U-duality group of M-theory compactified on T 6 [59]. Its massless fields include 27 one-form Abelian gauge potentials A m , transforming in the fundamental 27 of E 6(6) . Hull [11][12][13] considered a large l 5 limit under the assumption that the E 6(6) symmetry is preserved and all supersymmetric states are protected. Decomposing the N = 8 multiplet with respect to an N = 4 subalgebra, we obtain five N = 4 Abelian gauge multiplets with coupling constant g 2 = l 5 , each of which therefore lifts to an Abelian (2, 0) theory as l 5 → ∞, where g 2 = l 5 is identified with R as before. If the E 6(6) symmetry is to be preserved it follows that all 27 one-forms must lift to two-forms. Hence, if all supersymmetries survive the entire N = 8 supergravity multiplet must lift to a D = 6 theory, where l 5 is identified with R such that the l 5 → ∞ limit is conformal. We therefore require a superconformal gravitational theory in D = 6 dimensions, consistent with a global E 6(6) symmetry, that yields D = 5, N = 8 supergravity when compactified on a circle. According to Nahm's classification there is a unique candidate satisfying these criteria, the (4, 0) theory.
As for the (2, 0) theory, it is not possible to construct a conventional set of local covariant interactions, making the non-linear theory difficult to probe. Nonetheless, an analysis of the BPS spectrum analogous to that of the (2, 0) theory suggests that the identification of the strong coupling limit of D = 5, N = 8 supergravity as the full interacting (4, 0) theory compactified on S 1 is in principle consistent [11]. In particular, D = 5, N = 8 supergravity admits 1/2-supersymmetric gravitational instantonic solutions, which preserve the E 6(6) symmetry [62]. In analogy with the instantons appearing in the (2, 0) story, these are the uplift of Euclidean D = 4 self-dual gravitational instantons [62][63][64], which can be interpreted as 0-branes [62]. They carry mass ∝ |n|/l 5 and so become light in the l 5 → ∞ limit. The analysis of [2,56] indicates that these solutions may be regarded as the Kaluza-Klein modes of a D = 6 theory on a circle of radius R ∝ l 5 [11]. This proposal still requires many checks, but encouragingly, these 1/2-supersymmetric states sit in massive (4, 0) multiplets that have precisely the correct content to have originated from an S 1 compactification of the D = 6, (4, 0) theory: 27 massive self-dual two-forms and 42 massive scalars. A more detailed analysis of the D = 5, N = 8 and D = 6, N = (4, 0) supersymmetric multiplets paints a compelling picture. In particular, the 27 self-dual two-forms in D = 6 couple to self-dual supersymmetric strings, which yield the required D = 5 charged 0-branes and 1-branes transforming in the 27 and 27 of the global E 6(6) .

IV. THE (2, 0) THEORY SQUARED
In direct analogy with (1) we apply the product to a pair of self-dual two-forms belonging to left and right Abelian (2, 0) tensor multiplets, Adopting this dictionary we recover precisely the free (4, 0) theory. In particular, the generalised gauge transformations of the gravi-gerbe field (3) are generated by the local symmetries of the left and right (2, 0) factors. Since the supercharges of the left and right theories generate the supersymmetries of their product [26], the remaining fields of the (4, 0) multiplet and their transformations then follow essentially automatically.
That the full set of generalised gauge transformations is generated directly by the left/right factors is a nice feature of the construction. Let us now define the irreducible GL(6, R) representations, which transform as the 1 + 20 + 84, 15 and 15 + 45 + 45 of Spin(1, 5), respectively. First, G µνρσ has the symmetries of (3) and, directly from (8), the generalised gauge transformations given in (6), where we have identified the ghost field, ξ νρσ := C (10) νρσ + C (01) νρσ .
Before concluding we note, briefly, that by going first to physical gauge the equations of motion, Bianchi identities and self-dualities relations for the free (4, 0) theory follow straightforwardly from those of the (2, 0) factors. Recall, the on-shell degrees of freedom of a self-dual two-form are given by a symmetric bi-spinor B AB , A, B = 1, 2, in the (3, 1) of Sp(1) × Sp(1), where B AB = 0. Hence, for example, the symmetrized product G (ABCD) = B (AB •B CD) yields the (5, 1) representation satisfying G (ABCD) = 0, which corresponds to the gravi-gerbe field (3) in physical gauge, where G ijkl = G ijkl = G ijkl [11].

V. CONCLUSIONS
We have shown that the linear (4, 0) theory and its local symmetries follow from the square of Abelian (2,0) theories. This leaves a number of directions for future work. Perhaps most obvious is the need to understand the (4, 0) theory beyond the linear approximation. A natural setting for such a question is higher gauge theory [67]. For example, a number of higher gauge (2, 0) models were developed in [68][69][70][71] using superconformal twistors. However, the (4, 0) theory will require new structures, gravitational analogs of the (2, 0) models, and it is not a priori clear how to proceed. Here, however, we have an extra input to guide our considerations: the (4, 0) higher gauge theory will be required to be consistent with the square of the (2, 0) theory.
Irrespective, we can still test (4, 0) = (2, 0) × (2, 0) by considering its compatification, in the first instance, on a circle. Besides testing the expected amplitude relations [23], we anticipate a matching of classical solutions, at least in a weak-field approximation, using the methodology developed in [30,31]. In particular, it is natural to expect that the 1/2-supersymmetric gravitational instantonic solutions of D = 5, N = 8 supergravity, which must be identified with Kaluza-Klein modes of the would-be (4, 0) theory, are related to the "square" of the 1/2-supersymmetric instantonic 0-branes in the D = 5, N = 4 Yang-Mills theory, which are the Kaluza-Klein modes of the (2, 0) factors.
We conclude with some rather speculative comments regarding the strong/weak gravitational S-duality suggested by the (4, 0) theory [11][12][13]. First, note that the generalised gauge invariant curvature, self-duality relations and Bianchi identities for G follow directly from those of B µν andB ρσ . In particular, the generalised gauge invariant curvature is the product of the left and right three-form curvatures, It then follows immediately that the left/right two-form self-duality conditions, H = H,H = H , and Bianchi identities, dH = dH = 0 imply the self-duality relations, R = R = R , and the Bianchi identities, ∂ [µ R νρσ]τ λκ = ∂ [κ R |µνρ|στ λ] = 0, respectively. Now, recall that a D = 6 Abelian two-form with self-dual field strength, H = H, compactified on T 2 yields an SL(2, Z) doublet of D = 4 one-forms A i , i = 1, 2, which are related through F i = F j ε jk γ ki , where γ ki is the constant metric on T 2 . Since the gravi-gerbe field-strength originates from H •H, feeding this observation into the (2, 0) × (2, 0) construction we anticipate an SL(2, Z) triplet of D = 4 linearised Riemann tensors, obeying the duality constraint R (ij) = R (kj) ε jk γ ki . This is indeed the case: the free (4, 0) theory compactified on T 2 yields linear N = 8 supergravity, with an SL(2, Z) symmetry acting on a triplet of duality related gravitational field-strengths [11][12][13]. Here it is shown to be the "square" of the familiar SL(2, Z) of the Abelian (2, 0) multiplet compactified on T 2 . Of course, this symmetry is broken by interactions. This is not, however, necessarily an argument against its existence; it simply tells us that it is not a symmetry of classical N = 8 supergravity, just as S-duality is not a symmetry of classical N = 4 super Yang-Mills theory. While this picture is suggestive, it is highly speculative and will depend crucially on the non-linear structure of the complete (4, 0) theory. Clearly it may fail to materialise and a strong degree of scepticism is advised, but the lessons in gauge theory and gravity learnt on the journey will regardless return many insights. Even more speculatively, if the (4, 0) theory on M 6 = X × C, where C is a punctured Riemann surface, admits quantities that are protected as we vary the size of X or C, then one might expect a gravitational analog, or square, of the AGT correspondence.
We are grateful to Michael J. Duff for illuminating conversations. The work of LB is supported by a Schrödinger Fellowship.