All Tree Amplitudes of 6D $(2,0)$ Supergravity: Interacting Tensor Multiplets and the $K3$ Moduli Space

We present a twistor-like formula for the complete tree-level S matrix of 6D $(2,0)$ supergravity coupled to $21$ abelian tensor multiplets. This is the low-energy effective theory that corresponds to Type IIB superstring theory compactified on a $\mathrm{K}3$ surface. The formula is expressed as an integral over the moduli space of certain rational maps of the punctured Riemann sphere. By studying soft limits of the formula, we are able to explore the local moduli space of this theory, ${SO(5,21)\over SO(5)\times SO(21)}$. Finally, by dimensional reduction, we also obtain a new formula for the tree-level S matrix of 4D $\mathcal{N}=4$ Einstein-Maxwell theory.


INTRODUCTION
To describe scattering amplitudes of supersymmetric theories in higher dimensions, [1,2] introduced a sixdimensional rational map formalism in the spirit of [3][4][5]. Using this formalism, extremely compact formulas were found for tree-level amplitudes of a wide range of interesting theories, including maximally supersymmetric gauge theories and supergravity in diverse dimensions, as well as the world-volume theories of probe D-branes and the M5-brane in flat space. In the case of the M5-brane [1], which contains a chiral tensor field, the formalism circumvents a common difficulty in formulating a covariant action principle due to the self-duality constraint.
In this article, we continue to explore the utility of the 6D rational maps and spinor-helicity formalism and present the tree-level S matrix for the theory of 6D (2,0) supergravity. This chiral theory arises as the low-energy limit of Type IIB string theory compactified on a K3 surface [6] and is particularly interesting because it describes the interaction of self-dual tensors and gravitons.
To describe massless scattering in 6D, it is convenient to introduce spinor-helicity variables [7], Here, and throughout, i = 1, . . . , n labels the n particles, A = 1, 2, 3, 4 is a spinor index of the Spin(5, 1) Lorentz group, and a = 1, 2 is a left-handed index of the SU (2) L × SU (2) R massless little group. This is the only non-trivial little-group information that enters for chiral (2, 0) supersymmetry-the (2, 0) supergravity multiplet and a number of (2, 0) tensor multiplets, which contain a chiral tensor. The tensor multiplets transform as singlets of SU (2) R , whereas the gravity multiplet is a triplet; later we will introduce the doublet indexâ for SU (2) R . We also introduce a flavor index f i with i = 1, . . . , 21 to label the 21 tensor multiplets; this is the number that arises in 6D from compactification of the NS and R fields of Type IIB superstring theory on a K3 surface. It is also the unique number for which the gravitational anomalies cancel [8]. We assume that we are at generic points of the moduli space, where perturbative amplitudes are well-defined [9]. Interestingly one can explore the moduli space of the theory from the S matrix by studying soft limits [10]. Indeed, we derive new soft theorems from the formula we construct, which describe precisely the moduli space of 6D (2, 0) supergravity: (21) . In the rational-map formulation, amplitudes for n particles are expressed as integrals over the moduli space of rational maps from the n-punctured Riemann sphere to the space of spinor-helicity variables. In general, the amplitudes take the following form [1,2,11], where dµ 6D n is the measure encoding the 6D kinematics and the product I L I R is the integrand that contains the dynamical information of the theories, including supersymmetry. The measure is given by and n = 2m + 2 (we will discuss n = 2m + 1 later). The coordinates σ i label the n punctures, and V n = i<j σ ij , with σ ij = σ i − σ j . They are determined up to an overall SL(2, C) σ Möbius group transformation, whose "volume" is divided out in a standard way. The 6D scattering equations are given by These maps are given by degree-m polynomials ρ A a (σ) = m k=0 ρ A a,k σ k , which are determined up to an overall SL(2, C) ρ transformation, whose volume is divided out. This group is a complexification of SU (2) L . It is straightforward to see that (4) implies the on-shell conditions p 2 i = 0 and momentum conservation. Furthermore as shown in [1,2], this construction implies that the integrals are completely localized on the (n−3)! solutions, which are equivalent to those of the general dimensional scattering equations [11], As we will see shortly that, unlike the generaldimensional scattering equations, the use of the spinorhelicity coordinates and 6D scattering equations allows us to make supersymmetry manifest. Now consider n = 2m + 1, for which we have [2], i .
(6) The polynomials ρ A a (σ) now are given by and there is a shift symmetry T (α) acting on ω A : ω A → ω A + α ξ ρ A m−1 , which we also have to mod out. Here we review the integrand factors for 6D (2, 2) supergravity, since they will be relevant. For (2, 2) supergravity, we have, where S n is a n × n anti-symmetric matrix, with entries: [S n ] ij = pi·pj σij . This matrix has rank (n−2), and the reduced Pfaffian and determinant are defined as Here S ij ij means that the i-th and j-th rows and columns of S n are removed, and the result is i, j independent [12]. Ω (2,2) F is a fermionic function of Grassmann coordinates η Ia i ,ηÎâ i , which we use to package the supermultiplet of on-shell states into a 'superfield', where B ab and Bâb are self-dual and anti self-dual two forms, and G abâb is the graviton. Here I,Î = 1, 2 are the R-symmetry indices corresponding to a SU (2) × SU (2) subgroup of the full U Sp(4) × U Sp(4) R-symmetry. The fermionic function Ω imposes the conservation of supercharge, which may be viewed as a double copy: Ω The n × 2n matrices C a,k; which is independent of A, B, and satisfies det W i = j =i σ −1 ij . The matrix C a,k;i,b is a symplectic Grassmannian which was used in [2] as an alternative way to impose the 6D scattering equations. Ω , and the definition is identical, with the understanding that we use the right-handed variables, such as ηÎ a ,λÂâ,ρÂâ,ξâ, (W i )b a , etc.
For n = 2m + 1, the integrands take a slightly different form. For the fermionic part, we have whereas the n × n matrix S n is modified to an (n + 1) × (n + 1) matrix, which we denoteŜ.Ŝ n is defined in the same way as S n , but with i, j = 1, . . . , n, ⋆. Here σ ⋆ is a reference puncture, and p ⋆ is given by where q andq are arbitrary spinors.
The 6D (2, 0) supergravity theory contains 21 tensor multiplets and the graviton multiplet. The superfield of the tensor multiplet is a singlet of the little group, where a, b = 1, 2 are the SU (2) L little-group indices. The graviton multiplet transforms as a (1, 3) of the little group, so the superfield carries explicit SU (2) R indices, and Φâb(η) = Φbâ(η). We see that both the tensor multiplet and graviton multiplet can be obtained from the 6D (2, 2) superfield in (10) via SUSY reductions [13] [14], These integrals have the effect of projecting onto the right-handed U Sp(4) R-symmetry singlet sector, which reduces (2, 2) → (2, 0). Using the reduction, the amplitudes of (2, 0) supergravity with n 1 supergravity multiplets and n 2 tensor multiplets of the same flavor (n 1 + n 2 = n) can be obtained from the (2, 2) supergravity amplitude via , so the integration removes all η's. The fermionic integral can be performed using (8), and (11) (or (13) for odd n), and we obtain whereM n1n2 ab , which we will define shortly, is obtained by integrating out Ω (0,2) F . We begin with n even, as the odd-n case works in a similar fashion. Introducing the n × n matrix C2 ,m;1,â1C2,m;2,â2 · · ·C2 ,m;n,ân thenM n1n2 ab is given bỹ Note that hereâ andb denote sets of indices. The indicesâ i ,b i are contracted if i ∈ n 2 , whereas for j ∈ n 1 we symmetrizeâ j ,b j . This corresponds to constructing little-group singlets for tensor multiplets and triplets for graviton multiplets. After the contraction and symmetrization, the result of (20) simplifies drastically [15] M n1n2 where X n2 is a n 2 × n 2 anti-symmetric matrix given by andM n10 ab contains only the graviton multiplets. Let's remark that the simplification (21) (especially the appearance of PfX n2 ) will be crucial for the generalization to amplitudes with multiple tensor flavors which is more interesting and relevant for type IIB on K3.
At this point in the analysis, we have obtained the treelevel amplitudes of 6D (2, 0) supergravity with a single flavor of tensor multiplets: The factor PfX n2 requires the non-vanishing amplitudes to contain an even number n 2 of tensor multiplets, as expected. For odd n, the matrixMâ 1···ân is given bỹ recallξb is the right-hand version of ξ b in (7). Then the amplitudes take the same form

Multi-flavor tensor multiplets
As we have emphasized, the identity (21) is crucial for the generalization to multiple tensor flavors, which is required for the 6D (2, 0) supergravity. Indeed, the formula takes a form similar to that of a Einstein-Maxwell theory worked out by Cachazo, He and Yuan [12], especially the object PfX n2 . In that case, in passing from single-U (1) photons to multiple-U (1) ones, one simply replaced the matrix X n by X n [12], which allows the introduction of multiple distinct flavors: namely, f i , f j are flavor indices, and δ fifj = 1 if particles i, j are of the same flavor, otherwise δ fifj = 0. Inspired by this result, we are led to a proposal for the complete tree-level S matrix of 6D (2, 0) supergravity with multiple flavors of tensor multiplets: Again, the 6D scattering equations and integrands take different forms depending on whether n is even or odd [17]. Since n 2 is necessarily even, this is equivalent to distinguishing whether n 1 is even or odd.
Equation (26) is our main result, which is a localized integral formula that describes all tree-level superamplitudes of abelian tensor multiplets (with multiple flavors) coupled to gravity multiplets. We can verify that it has all the correct properties. For instance, due to the fact that all the building blocks of the formula come from either 6D (2, 2) supergravity or Einstein-Maxwell theory, they all behave properly in the factorization limits, and transform correctly under the symmetries: SL(2, C) σ , SL(2, C) ρ , etc. Also, as we will show later, when reduced to 4D the proposed formula produces (supersymmetric) Einstein-Maxwell amplitudes, which is another consistency check. Finally, it is straightforward to check that the formula gives correct low-point amplitudes, e.g. [16] A (2,0) 0,4 We symmetrizeâ 3 ,b 3 andâ 4 ,b 4 for the graviton multiplets, and [1â 1

THE K3 MODULI SPACE FROM SOFT LIMITS
Type IIB string theory compactified on K3 has a well studied moduli space described by the coset [18], The discrete group is invisible in the supergravity approximation, so we concern ourselves with the local form of the moduli space of supergravity theory, namely SO(5,21) SO(5)×SO (21) . It has dimension of 105, which corresponds precisely to the 105 scalars in the 21 tensor multiplets. These scalars are Goldstone bosons of the breaking of SO (5,21) to SO(5)× SO (21), which are the R-symmetry and flavor symmetry, respectively. Therefore, the scalars obey soft theorems, which are the tools to explore the structure of the moduli space directly from the S matrix [10].
We find that the amplitudes behave like pion amplitudes with "Adler's zero" [19] in the single soft limit. Indeed for p 1 → 0, we find, and the same for other scalars in the tensor multiplets. The commutator algebra of the coset space may be explored by considering double soft limits for scalars. Begin with the flavor symmetry, we find for p 1 , p 2 → 0 simultaneously where f i 's are flavor indices, and R f1 f2 i is a generator of the unbroken SO (21), which may be viewed as the result of the commutator of two broken generators. R f1 f2 i acts on superfields as where f 3 = f 1 , f 2 . Therefore, the generator exchanges tensor multiplets of flavor f 1 with ones of f 2 , and sends all others and the graviton multiplet to 0.
To study the SO(5) R-symmetry generators we take soft limits of two scalars which do not form a R-symmetry singlet. For instance Similarly, other choices of soft scalars lead to the remaining R-symmetry generators: Finally, we consider the cases where soft scalars carry different flavors and do not form an R-symmetry singlet. This actually leads to new soft theorems: and similarly for other R-symmetry generators. The results of the soft limits now contain both flavor and Rsymmetry generators, reflecting the direct product structure in SO(5,21) SO(5)×SO (21) . This is a new phenomenon that is not present in pure (2, 0) supergravity [10,20].
The above soft theorems may be obtained by analyzing how the integrand and the scattering equations behave in the limits. For instance, the vanishing of the amplitudes in the single-soft limits is due to and the rest remains finite. The double-soft theorems require more careful analysis along the lines of, e.g. [21]. The structures of double-soft theorems, however, are already indicated by knowing the four-point amplitudes given in (27), since important contributions are diagrams with a four-point amplitude on one side such that the propagator becomes singular in the limit [22]. Finally, we have also checked the soft theorems explicitly using our formula (26) for various examples.

4D N = 4 EINSTEIN-MAXWELL THEORY
One can dimensionally reduce 6D (2, 0) supergravity to obtain 4D N = 4 Einstein-Maxwell theory. The tree-level amplitudes of this theory capture the leading low-energy behavior of Type IIB (or Type IIA) superstring theory on K3 × T 2 .
We are now ready to perform the dimensional reduction on (26) [24]. First, the 6D measure reduces to where R(ρ), R(ρ) are the resultants of the polynomials with d +d = n − 2, and the 4D scattering equations are given by The 2 × 2 matrix (W i ) ab reduces to with t i = λ α i ρ α (σi) ,t i =λα ĩ ρα(σi) (independent of α,α), and t iti = j =i 1 σij . As for the integrand, the parts that reduce to 4D non-trivially arẽ Assume we have m 1 G + superparticles and m 2 G − , with m 1 + m 2 = n 1 [25], we findT n1 is given bỹ where V m1 = i<j σ ij for i, j ∈ m 1 , and similarly for V m2 . We therefore obtain a general formula for the amplitudes of 4D N = 4 Einstein-Maxwell theory: where Ω N =4 F implements the 4D N = 4 supersymmetry, arising as the reduction of Ω The formula should be understood as summing over d,d obeying d +d = n − 2. However, it is clear from the superfields that we should require recall n 2 is even. Therefore, for a given number of photon and graviton multiplets, the summation over sectors becomes a sum over different m 1 , m 2 . We have checked (44) against many explicit amplitudes, and also verified that the integrand is identical to that of [12] for certain component amplitudes.

DISCUSSION AND CONCLUSION
We have presented a formula for the tree-level S matrix of 6D (2, 0) supergravity. The formula for single-flavor tensor multiplets is constructed via a SUSY reduction of the one for (2, 2) supergravity. We observed important simplifications in deriving the formula, particularly the appearance of the object PfX n , crucially for the generalization to 21 flavors required for (2, 0) supergravity. By studying soft limits of the formula, we were able to explore the moduli space of the theory. Via dimensional reduction, we also deduced a new formula for amplitudes of 4D N = 4 Einstein-Maxwell. Since 6D (2, 0) supergravity has a UV completion as a string theory, it would be of interest to extend our formula to include α ′ corrections, perhaps along the lines of [26]. Also, a recent paper [27] introduces an alternative form of the scattering equations that treats even and odd points equally, but uses a different formalism for supersymmetry. It will be interesting to study our formula into this formalism.
Our results provide an S matrix confirmation of various properties of (2, 0) supergravity and the dimensionally reduced theory as predicted by string dualities. While the 10D theory has a dilaton that sets the coupling, in 6D this scalar is one of the 105 moduli fields, and appears equally with the other 104 scalars. If one considers the compactification on K3 × T 2 , standard U-dualities imply equivalence to the Type IIA superstring theory on the same geometry or the heterotic string theory compactified to 4D on a torus. The formulas discussed in this article apply to all these cases, at least at generic points of the moduli space.