(2,2) Scattering and the Celestial Torus
Alexander Atanasov, Adam Ball, Walker Melton, Ana-Maria Raclariu, Andrew Strominger
aa r X i v : . [ h e p - t h ] J a n (2 , Scattering and the Celestial Torus
Alexander Atanasov , Adam Ball , Walker Melton , Ana-Maria Raclariu and Andrew Strominger Center for the Fundamental Laws of Nature, Harvard University,Cambridge, MA 02138, USA Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
Abstract
Analytic continuation from Minkowski space to (2 ,
2) split signature spacetime hasproven to be a powerful tool for the study of scattering amplitudes. Here we showthat, under this continuation, null infinity becomes the product of a null interval witha celestial torus (replacing the celestial sphere) and has only one connected component.Spacelike and timelike infinity are time-periodic quotients of AdS . These three com-ponents of infinity combine to an S represented as a toric fibration over the interval.Privileged scattering states of scalars organize into SL (2 , R ) L × SL (2 , R ) R conformalprimary wave functions and their descendants with real integral or half-integral con-formal weights, giving the normally continuous scattering problem a discrete character. ontents H -primaries to L -primaries 15 Scattering amplitudes in quantum field theory are often defined by analytic continuationfrom (4 ,
0) Euclidean signature to (3 ,
1) Lorentzian signature. This provides an efficient pre-scription for the Feynman-diagram singularities encountered in perturbation theory. More-over, positivity properties in Euclidean space enable powerful non-perturbative instantonand axiomatic analyses. Euclidean methods have also proven effective in quantum gravity.In recent years, however, analytic continuation from Minkowski space to a split (2 , K , — has emerged as acomplementary and surprisingly effective tool in quantum field theory. An awkward featureof Euclidean space is that particles cannot be on-shell. Amplitudes are therefore representedas analytic continuations of sums of off-shell processes, which can both become inordinatelycomplicated and obscure the underlying physics. Dramatic simplifications have been foundin some on-shell descriptions in Klein space [1–8]. The group-theoretic reduction of the 4D(2 ,
2) Lorentz group to the product of two 1D conformal groups, the associated reality ofthe self duality condition [9], and the non-degeneracy of massless three-point scattering alsolead to significant simplifications.In quantum gravity in asymptotically flat spacetimes, there are yet further reasons toconsider Klein space. The paucity of generally covariant bulk observables — and moregenerally the holographic principle — suggests that any theory of quantum gravity should After the mathematician Felix Klein, who pioneered the study of these spaces in the Erlangen Program.
1e defined by boundary observables. In Euclidean space, the conformal boundary is just apoint. It seems challenging to formulate a holographic dual which encodes the richness ofasymptotically flat quantum gravity by observables at a zero-dimensional point. Here wefind that, in contrast, Klein space has a rich conformal boundary at infinity, providing asuitable potential home for a holographic dual.In section 2 we show that the conformal boundary at null infinity in Klein space, denoted I , is the product of a null interval with the Lorentzian signature celestial torus. Both spatialand timelike infinity i and i ′ are the product of a disk with a circle and are endowed with theconformal metric of AdS / Z . Here the Z -quotient makes the familiar AdS cylinder periodic.The gluing of the toroidal boundaries of these AdS / Z geometries to the celestial tori at thetwo ends of I trivializes different cycles of the latter, giving a toric representation of the I ∪ i ∪ i ′ infinity as S . Since I has only one connected component, observables are givenby an S -vector rather than an S -matrix. The fact that the continuation from Minkowski toKlein space leads to the replacement of the sphere with a torus will perhaps prove useful forsharpening the concept of a celestial conformal field theory.Section 3 reviews the SL (2 , R ) L × SL (2 , R ) R symmetry of Klein space. Expressions aregiven for L n , ¯ L n , n = − , , L ± ¯ L generate the compact spaceand time directions of the celestial torus, as well as for the finite group action on the celestialtorus. The group action preserves the AdS / Z hypersurfaces which are a fixed distance fromthe origin and foliate Klein space.Section 4 considers conformal basis wave functions for massless scalars. Single-valuednesson the celestial torus requires that the L and ¯ L eigenvalues are either both integer orboth half-integer. “ L -primary” solutions are found corresponding to highest-weight statesannihilated by L and ¯ L . More general solutions are then obtained by taking descendants.Convolutions of these wave functions with the bulk field operator create states which havean interpretation as L , ¯ L eigenstates of the 1+1D celestial CFT living on a spatial circle ofthe celestial torus. The fact that the time direction of the torus is periodic is not a problembecause L + ¯ L is quantized. We also find lowest-weight solutions annihilated by L − and¯ L − , as well as mixed solutions annihilated by L ± , ¯ L ∓ .A striking feature of this construction is that the solutions are labelled by three integers:namely the conformal weights and the levels of the left and right descendants, giving L -primary scattering a discrete character. This contrasts with dynamics on the celestial spherein Minkowski space, where the conformal basis solutions are labelled by three continuousparameters: a position on the sphere and a continuous complex conformal dimension. The2iscrete character of celestial scattering in Klein space resonates with several other recentdevelopments. Spacetime translations shift conformal weights by a half-integer [10], so theset of all L -primaries and their descendants associated to a given spacetime field form arepresentation of the Poincar´e group. In gauge theory and gravity, the infinite hierarchy ofsoft currents appears at negative integer weight, while the positive integer weights appearrelated to Goldstone bosons [13–16]. Poles at negative even integer conformal weights incelestial scattering amplitudes were recently shown [17] to encode the coefficents in theWilsonian effective action. These poles characterize much or all of the theory and may benaturally probed by scattering L -primaries.In section 5 we construct, as Mellin transforms of plane waves, modes correspondingto particles which emerge at a fixed point on the celestial torus. These correspond to “ H -primary” operators which are primary with respect to elements H , ¯ H leaving fixed the pointat which the particles emerge. Scattering of such particles takes the form of a correlationfunction on the celestial torus. We show that L -primary wave functions can be expressedas weighted integrals over the torus of H -primary wave functions with quantized weights.This is a version of the celestial state-operator correspondence. Hence L -primary scatteringamplitudes are weighted celestial integrals of Mellin transforms of plane wave scatteringamplitudes. We close with a few comments in section 6. In this section we conformally compactify K , and derive the conformal geometry of nullinfinity I , spatial infinity i and timelike infinity i ′ . The flat metric on K , is ds = dzd ¯ z − dwd ¯ w. (2.1)In polar coordinates z = re iφ and w = qe iψ , this becomes ds = − dq − q dψ + dr + r dφ . (2.2)Now define q − r = tan U , q + r = tan V , giving ds = 1cos U cos V Å − dU dV −
14 sin ( V + U ) dψ + 14 sin ( V − U ) dφ ã . (2.3) Unlike the continuous complex highest weight representations discussed in [11,12] which are restricted tohave the real part of the conformal weight equal to unity and cannot be put in representations of translations. i ′ I U V
Figure 1: Toric Penrose diagram for signature (2 ,
2) Klein space. 45 o lines are null as usual.A Lorentzian torus is fibered over every point in the diagram. The spacelike cycle of thetorus degenerates along the timelike line U = V , while the timelike cycle degenerates alongthe spacelike line U = − V . Neither cycle degenerates at null infinity I which is the interval − π < U < π , V = π . Spacelike infinity i is at ( U, V ) = ( − π , π ) and has the conformalgeometry of signature (1 ,
2) AdS / Z . Timelike infinity i ′ is at ( U, V ) = ( π , π ) and has theconformal geometry of signature (2 ,
1) AdS / Z . The blue lines are lines of constant w ¯ w − z ¯ z with τ = 0 at U = 0.The coordinate ranges are the solid triangle − π < U < π and | U | < V < π , as depictedin figure 1. Null infinity I is at V = π/ i ( i ′ ) is the boundary at U = − π ( U = π ).Note that, unlike the case of M , , null infinity has only one connected component. Thismeans we cannot define an S -matrix. Instead we have only an S -vector in the sense of [18].It is an amplitude for a collection of incoming particles on I to scatter into nothing —which they must as there is nowhere to go! This S -vector together with a suitable analyticcontinuation procedure can in principle be used to define an S -matrix in M , . I is parameterized by the null coordinate − π < U < π and the periodic coordinates ψ and φ . Taking V → π while rescaling (2.3) by cos V one finds the conformal metric on I to be the square, Lorentzian torus ds I = − dψ + dφ , ψ ∼ ψ + 2 π, φ ∼ φ + 2 π. (2.4)Hence I is the product of the celestial torus with a null interval.Now we turn to i , i ′ . Since the boundary of a boundary is nothing, we must be able toglue these to I to get an S which is the topological boundary of K , . S is topologically4epresented in toric geometry as a torus fibration over the interval in which one of the twotorus cycles shrinks to zero at one end of the interval, and the other at the other end. Thenthere are no non-contractible cycles. In order to complete I to S in this manner, the i , i ′ “caps” must both be topologically the product of a disk and a circle. We now show that thisis indeed the case and moreover that the conformal geometry on each cap is AdS / Z .Following procedures which are standard for M , [19–23], we resolve i ′ by taking the τ → ∞ limit of the signature (2 ,
1) surface z ¯ z − w ¯ w = − τ . (2.5)We denote the two regions of K , with positive or negative z ¯ z − w ¯ w by K , ± . Coordinatescovering the region K , − , which contains i ′ , are z = τ e iφ sinh ρ,w = τ e iψ cosh ρ. (2.6)The inverse relations are τ = √ w ¯ w − z ¯ z, tanh ρ = … z ¯ zw ¯ w ,e iφ = » z/ ¯ z, e iψ = » w/ ¯ w. (2.7)The Klein space metric in these coordinates is ds = − dτ + τ ds , (2.8)where ds = − cosh ρ dψ + sinh ρ dφ + dρ (2.9)is the conformal geometry of i ′ . We recognize it as the standard metric on AdS / Z , wherethe Z acts as the time-like quotient ψ → ψ + 2 π .A similar construction for i begins with (˜ τ , ˜ ρ, φ, ψ ) covering K , with z ¯ z − w ¯ w = +˜ τ : z = ˜ τ e iφ cosh ˜ ρ,w = ˜ τ e iψ sinh ˜ ρ. (2.10)5he inverse relations are ˜ τ = √ z ¯ z − w ¯ w, tanh ˜ ρ = … w ¯ wz ¯ z ,e iφ = » z/ ¯ z, e iψ = » w/ ¯ w. (2.11)One finds ds = d ˜ τ − ˜ τ ds , (2.12) ds = − cosh ˜ ρ dφ + sinh ˜ ρ dψ + d ˜ ρ . (2.13)We see that the non-contractible loop in the AdS / Z factor is now φ instead of ψ and spacelikeinstead of timelike in the K , ± embedding space. Hence gluing the conformal geometries ofthe two AdS / Z caps to I trivializes both cycles of the celestial torus and the full topologyof infinity is S . The “Lorentz group” of K , is SO (2 , ∼ = SL (2 , R ) L × SL (2 , R ) R Z , where the Z is generated by − L × − R . The spin group is the double cover SL (2 , R ) L × SL (2 , R ) R . The symmetry isgenerated on Klein space by (real combinations of) the six Killing vector fields L = ¯ z∂ w + ¯ w∂ z , ¯ L = z∂ w + ¯ w∂ ¯ z ,L = 12 ( z∂ z + w∂ w − ¯ z∂ ¯ z − ¯ w∂ ¯ w ) , ¯ L = 12 ( − z∂ z + w∂ w + ¯ z∂ ¯ z − ¯ w∂ ¯ w ) ,L − = − z∂ ¯ w − w∂ ¯ z , ¯ L − = − ¯ z∂ ¯ w − w∂ z . (3.1)In K , − we may also write L = 12 e − iψ − iφ ( ∂ ρ − i tanh ρ ∂ ψ − i coth ρ ∂ φ ) ,L = − i ∂ ψ + ∂ φ ) ,L − = 12 e iψ + iφ ( − ∂ ρ − i tanh ρ ∂ ψ − i coth ρ ∂ φ ) , ¯ L = 12 e − iψ + iφ ( ∂ ρ − i tanh ρ ∂ ψ + i coth ρ ∂ φ ) , ¯ L = − i ∂ ψ − ∂ φ ) , ¯ L − = 12 e iψ − iφ ( − ∂ ρ − i tanh ρ ∂ ψ + i coth ρ ∂ φ ) , (3.2)6hile on K , L = 12 e − iψ − iφ ( ∂ ˜ ρ − i coth ˜ ρ ∂ ψ − i tanh ˜ ρ ∂ φ ) ,L = − i ∂ ψ + ∂ φ ) ,L − = 12 e iψ + iφ ( − ∂ ˜ ρ − i coth ˜ ρ ∂ ψ − i tanh ˜ ρ ∂ φ ) , ¯ L = 12 e − iψ + iφ ( ∂ ˜ ρ − i coth ˜ ρ ∂ ψ + i tanh ˜ ρ ∂ φ ) , ¯ L = − i ∂ ψ − ∂ φ ) , ¯ L − = 12 e iψ − iφ ( − ∂ ˜ ρ − i coth ˜ ρ ∂ ψ + i tanh ˜ ρ ∂ φ ) . (3.3)In either case on the boundary at ρ → ∞ (or ˜ ρ → ∞ ) these reduce to the familiar circleaction L n = − i e − in ( ψ + φ ) ( ∂ ψ + ∂ φ ) , ¯ L n = − i e − in ( ψ − φ ) ( ∂ ψ − ∂ φ ) , (3.4)for n = − , ,
1. The L n obey (for all ρ or ˜ ρ ) the SL (2 , R ) Lie bracket algebra[ L n , L m ] = ( n − m ) L m + n , (3.5)and similarly for the ¯ L n .AdS / Z is the SL (2 , R ) group manifold which admits an SL (2 , R ) L × SL (2 , R ) R groupaction. The generators above leave fixed the AdS / Z hypersurfaces of constant w ¯ w − z ¯ z . L − ¯ L generates AdS rotations and L + ¯ L generates AdS global time translations for w ¯ w − z ¯ z = τ , while for z ¯ z − w ¯ w = ˜ τ it is the other way around. In either case because ofthe mod Z quotient, the eigenvalues of L ± ¯ L are separately quantized. This is standard in SL (2 , R ) representation theory, but differs from familiar string theory applications in whichone works on the simply-connected universal cover AdS of AdS / Z , and only L − ¯ L isquantized. SO (2 ,
2) acts faithfully on the celestial torus. We define the null angles x ± ≡ ψ ± φ. (3.6)While ψ, φ naturally parametrize the cycles of the celestial torus, the symmetry group actsmore simply on x ± . In particular SL (2 , R ) L acts only on x + , while SL (2 , R ) R acts only on x − . The price for working with x ± is that their periodicity properties are not independent.7ather one has ( x + , x − ) ∼ ( x + + 2 π, x − + 2 π ) ∼ ( x + + 2 π, x − − π ) . (3.7)Finite elements of SL (2 , R ) L act as M¨obius transformations on tan x + by sending x + → x + ′ such that tan x + ′ a tan x + + bc tan x + + d (3.8)with ad − bc = 1. Note that tan x + = tan x + +2 π despite the fact that ( x + , x − ) and ( x + +2 π, x − ) are distinct points, so tan x + is not a good coordinate on the whole torus. In this section we construct the highest- and lowest-weight conformal primary wave functionsfor a massless scalar.Solving the massless scalar wave equation (cid:3)
Φ = 0 in a conformal basis reduces to find-ing representations of SL (2 , R ) L × SL (2 , R ) R as functions on the SL (2 , R ) group manifoldAdS / Z . On K , − the equation separates as Φ( τ, ψ, φ, ρ ) = Φ ( τ )Φ ( ψ, φ, ρ ) and can bewritten 1 τ ( ∂ τ τ ∂ τ )Φ = K Φ , (4.1) ∇ Φ = K Φ , (4.2)where the separation constant K can be anything at this point. The first equation (4.1) hastwo power law solutions which depend on K . (4.2) is the wave equation for a scalar of mass m = K on AdS / Z . In a standard basis, L + ¯ L generates time translations, while L − ¯ L generates space rotations. Both must be integers, implying that L and ¯ L are either bothintegers or both half-integers. (4.2) can be rewritten in terms of either the SL (2 , R ) L or SL (2 , R ) R Casimirs on AdS / Z (4 ¯ L − L − ¯ L − L ¯ L − )Φ = (4 L − L − L − L L − )Φ = K Φ . (4.3)Here we consider conformal primary solutions in a basis of ( L , ¯ L ) eigenstates. These obey In the familiar case of
AdS , ψ is not periodically identified and L + ¯ L is not quantized. L Φ = h Φ , ¯ L Φ = ¯ h Φ (4.4)for integer or half-integer ( h, ¯ h ), as well as the highest-weight condition L Φ = ¯ L Φ = 0 . (4.5)We refer to these as “ L -primary”, to distinguish them from operator-type primaries discussedin the next section. Commuting L and ¯ L to the right where they annihilate Φ , the waveequation reduces to (4.5) together with K = 4 h ( h −
1) = 4¯ h (¯ h −
1) (4.6)for some integer or half-integer ( h, ¯ h ) eigenvalues of L , ¯ L . (4.1) then has two solutionsΦ = τ − h , e Φ = τ h − , (4.7)which are indirectly related by the shadow transform. Moreover the highest-weight condi-tions (4.5) imply h = ¯ h (4.8)together with ∂ ρ Φ + 2 h tanh ρ Φ = 0 . (4.9)This is solved by Φ ∝ e ihψ cosh h ρ . (4.10)Putting this together we have a pair of conformal primary solutions in K , − for every half-integer value of h , Φ ++ h = e ihψ cosh h ρ τ − h , e Φ ++ h = e ihψ cosh h ρ τ h − . (4.11)We can construct descendant solutions by acting with L − , ¯ L − on Φ ++ h . However we stillhave to match this to a solution on K , . For this purpose it is easiest to work in terms ofthe ( z, ¯ z, w, ¯ w ) coordinates. Then we findΦ ++ h = ¯ w − h , e Φ ++ h = ¯ w − h ( w ¯ w − z ¯ z ) h − . (4.12)9he equation of motion (cid:3) Φ = 4( ∂ z ∂ ¯ z − ∂ w ∂ ¯ w )Φ = 0 (4.13)has ∂ h ¯ w δ (2) ( w ) sources at w = 0 for positive h , which may be important or need regulationin some applications. The singularity along the light cone of the origin z ¯ z = w ¯ w can beregulated with a ± iε prescription, a choice of which may be necessary for example to definescattering amplitudes. Similar regulators are likely needed in solutions below but will notbe analyzed herein. Near I at V = π one findsΦ ++ h → ( π − V ) h e ihψ , e Φ ++ h → ( π − V ) e ihψ (2 tan U ) h − . (4.14)We could also consider lowest-weight solutions obeying L − Φ = ¯ L − Φ = 0 . (4.15)Inspection of (3.1) immediately reveals that the complex conjugatesΦ −− h = (Φ ++ h ) ∗ = w − h , e Φ −− h = ( e Φ ++ h ) ∗ = w − h ( w ¯ w − z ¯ z ) h − , (4.16)obey (4.15) and L Φ −− h = ¯ L Φ −− h = − h Φ −− h , L e Φ −− h = ¯ L e Φ −− h = − h e Φ −− h . (4.17)There are further mixed solutions obeying L Φ = ¯ L − Φ = 0 . (4.18)Again from (3.1) we see that under the exchange z ↔ w , we have L n ↔ L n , ¯ L n ↔ − ¯ L − n . (4.19)It follows that Φ + − h = ¯ z − h , e Φ + − h = ¯ z − h ( w ¯ w − z ¯ z ) h − , (4.20)obey (4.18) and L Φ + − h = − ¯ L Φ + − h = h Φ + − h , L e Φ + − h = − ¯ L e Φ + − h = h e Φ + − h . (4.21)10inally the other class of mixed solutions L − Φ = ¯ L Φ = 0 (4.22)is given by Φ − + h = z − h , e Φ − + h = z − h ( w ¯ w − z ¯ z ) h − . (4.23)The full SL (2 , R ) L × SL (2 , R ) R multiplets can for all cases be obtained by suitable actions of L ± , ¯ L ± . In the previous section we constructed wave functions whose convolutions with bulk fieldoperators create states in the (1 ,
1) celestial CFT on the Lorentzian torus. These are L -primary with respect to the standard SL (2 , R ) L × SL (2 , R ) R action, diagonalizing both timetranslations and space rotations.The (1 ,
1) CFT also contains local operators acting at points on the torus O h, ¯ h (ˆ x + , ˆ x − ) , ˆ x ± = ˆ ψ ± ˆ φ, (5.1)which are H -primary rather than L -primary [24]. They are annihilated by the raising oper-ators in the basis that diagonalizes boosts towards (ˆ x + , ˆ x − ). This basis is H ˆ x = 12 Ä e i ˆ x + L − e − i ˆ x + L − ä , H ˆ x ± = iL ∓ i Ä e i ˆ x + L + e − i ˆ x + L − ä , ¯ H ˆ x = 12 Ä e i ˆ x − ¯ L − e − i ˆ x − ¯ L − ä , ¯ H ˆ x ± = i ¯ L ∓ i Ä e i ˆ x − ¯ L + e − i ˆ x − ¯ L − ä . (5.2)These obey the commutation relations[ H n , H m ] = ( n − m ) H n + m , [ ¯ H n , ¯ H m ] = ( n − m ) ¯ H n + m . (5.3)Analogous primary operators were constructed in Minkowski space, where they live on thesphere rather than the torus, as Mellin transforms of momentum space field operators in [11].The construction is easily continued to Klein space. Let us write in ( z, ¯ z, w, ¯ w ) coordinates p = ω ˆ p (ˆ x ) = ω ( e i ˆ φ , e − i ˆ φ , e i ˆ ψ , e − i ˆ ψ ) , (5.4) X = ( re iφ , re − iφ , qe iψ , qe − iψ ) , (5.5)11o that p = 0 andˆ p (ˆ x ) · X = r cos Ä ˆ φ − φ ä − q cos Ä ˆ ψ − ψ ä = ( r − q ) cos ˆ x + − x + x − − x − r + q ) sin ˆ x + − x + x − − x − , (5.6)where ˆ x = (ˆ x + , ˆ x − ) . As usual the Mellin transform gives ϕ h ( X ; ˆ x ) = Z ∞ dωω h − e iω ˆ p · X = e − πih Γ(2 h )(ˆ p · X ) h . (5.7)These obey, by construction, the wave equation as well as H ˆ x ϕ h ( X ; ˆ x ) = ¯ H ˆ x ϕ h ( X ; ˆ x ) = 0 , (5.8) H ˆ x ϕ h ( X ; ˆ x ) = ¯ H ˆ x ϕ h ( X ; ˆ x ) = hϕ h ( X ; ˆ x ) , (5.9) H ˆ x − ϕ h ( X ; ˆ x ) = − ∂ + ϕ h ( X ; ˆ x ) , (5.10)¯ H ˆ x − ϕ h ( X ; ˆ x ) = − ∂ − ϕ h ( X ; ˆ x ) . (5.11)Scattering amplitudes of particles with these wave functions are Mellin transforms of planewave amplitudes, and are identified with conformal primary correlation functions on thecelestial torus. These wave functions have branch cuts for generic h and are periodic in boththe time and space directions ψ and φ . One may also consider the shadows of these solutions e ϕ h ( X ; ˆ x ) = ϕ h ( X ; ˆ x )( X ) h − , (5.12)which obey (5.8).We can put our (1 ,
1) CFT on the Lorentzian cylinder just as well as the Lorentziantorus, and it is instructive to see how they are related. In a conventional (1 ,
1) CFT on thecylinder there is a canonical map from primary operators at a point to operator modes onthe circle, given by an integral over a causal diamond O m,n = Z π d ˆ x + Z π d ˆ x − e − im ˆ x + − in ˆ x − O h, ¯ h (ˆ x + , ˆ x − ) , (5.13) In Minkowski space the ± iε prescription at ˆ p · X = 0 distinguishes ingoing and outgoing solutions. InKlein space changing the sign in front of ε is equivalent to changing the sign of ˆ p and so does not give anew solution, in accord with the fact that I has only one connected component. In the case when 2 h − h − ¯ h ∈ Z . Spatial periodicity requires m − n ∈ Z . In order that our modes createthe primary and descendant states associated with O h, ¯ h , we need m, n ∈ Z − h . When weinstead consider this mode expansion on the torus, the timelike periodicity further requires m + n ∈ Z , which means that the only consistent operators of the type (5.13) arise from H -primaries with h + ¯ h ∈ Z . Accordingly we henceforth restrict to h ± ¯ h ∈ Z .The analog of this map (on the torus now) at the level of the wave functions (5.7) isΦ m,n ( X ) = Z π d ˆ x + Z π d ˆ x − e − im ˆ x + − in ˆ x − ϕ h ( X ; ˆ x + , ˆ x − ) . (5.14)Using L = − i H ˆ x + H ˆ x − ) , L ± = e ∓ i ˆ x + iH ˆ x − iH ˆ x − ± H ˆ x ) , (5.15)together with (5.8), and integrating by parts with respect to ˆ x + one finds the standard moderelations L Φ m,n = ( h − − m )Φ m +1 ,n , (5.16) L Φ m,n = − m Φ m,n , (5.17) L − Φ m,n = (1 − h − m )Φ m − ,n . (5.18)Any solutions Φ constructed from linear combinations of ϕ h ( X ; ˆ x ) obey( L ( L − − L − L )Φ = ( L ( L + 1) − L L − )Φ = h ( h − . (5.19)Hence highest-weight solutions obeying L Φ = 0 have m = − h or m = h −
1, while lowest-weight solutions obeying L − Φ = 0 have m = h or m = 1 − h . Similar relations hold for the¯ L n . The highest-weight solution with m = n = − h isΦ − h, − h = Z π d ˆ x + Z π d ˆ x − e ih (ˆ x + +ˆ x − ) ϕ h ( X ; ˆ x + , ˆ x − ) (5.20)= 2 h π e iπh Γ(2 h ) ¯ w − h ∝ Φ ++ h , (5.21)in agreement with (4.12). The integral here is performed in appendix A.1. The poles atnegative half-integer h are inherited from the normalization of the wave functions (5.7) andcan be absorbed by a redefinition of the wave functions resulting in finite amplitudes [26–28].For h >
0, taking descendants of the primary generates the standard infinite-dimensionalunitary SL (2 , R ) L × SL (2 , R ) R representation. For h <
0, after taking 2 h descendants (on13ither left or right) we reach m = h and the representation terminates. This is a non-unitaryfinite-dimensional representation.Similarly for m = n = h we haveΦ h,h = Z π d ˆ x + Z π d ˆ x − e − ih (ˆ x + +ˆ x − ) ϕ h ( X ; ˆ x + , ˆ x − ) (5.22)= 2 h π e iπh Γ(2 h ) w − h ∝ Φ −− h . (5.23)This is a lowest-weight solution. The representations are filled out by acting with powers of L , ¯ L . Mixed primary solutions can also be obtained by taking ( m, n ) as ( h, − h ) or ( − h, h ). Celestial S -vector elements of particles with Klein space H -primary wave functions are givenby Mellin transforms of momentum space S -vector elements. The k th external particle islabeled by 3 continuous parameters: h k , x + k , x − k . They take the form of CFT correlationfunctions on a Lorentzian torus.Celestial S -vector elements of particles with L -primary wave functions, and their descen-dants, can also be computed from momentum space S -vector elements, with the additionalweighted integral over the celestial torus given in (5.13). The k th external particle is labeledby 3 discrete parameters: h k and the left and right levels of the descendant. It is interestingthat the L -primary scattering problem has a discrete character. This resonates with theresults of [17] where it was shown, for the Minkowskian four-particle amplitude, that theWilsonian coefficients are encoded in poles at discrete integral conformal weights. Theseare likely probed by L -primary scattering amplitudes. We defer a more detailed analysis ofproperties of the L -primary solutions to future work. Acknowledgments
We are grateful to Nick Agia, Nima Arkani-Hamed, Alfredo Guevara, Mina Himwich, NoahMiller, Monica Pate and E. Witten for useful conversations, as well as to Nima Arkani-Hamed for an interesting history lesson on Felix Klein. This work was supported by DOEgrant de-sc/0007870, by Gordon and Betty Moore Foundation and John Templeton Founda-tion grants via the Black Hole Initiative and by a Hertz Fellowship to A.A. A.R. is supportedby the Stephen Hawking Postdoctoral Fellowship at Perimeter Institute. Research at Perime-ter Institute is supported in part by the Government of Canada through the Department14f Innovation, Science and Industry Canada and by the Province of Ontario through theMinistry of Colleges and Universities.
A Mapping H -primaries to L -primaries In this appendix we evaluate the integral (5.20) mapping conformal H -primary solutions to L -primary ones. We start with I = Z π Z π d ˆ x + d ˆ x − e − im ˆ x + e − in ˆ x − e − iπh Γ(2 h )(ˆ p · X ) h . (A.1)This integral consists of points inside the causal diamond, covering half of the celestial torus.It can be related to an integral over the full torus by noticing that the integral over theother half is obtained from the integral over the diamond by shifting ˆ x + → ˆ x + + 2 π forfixed ˆ x − . This transformation amounts to taking ˆ p to its antipodal point − ˆ p . Under thistransformation, e − im ˆ x + − in ˆ x − Φ h (ˆ x + , ˆ x − ) → e − πih e − im ˆ x + − in ˆ x − e ± πih Φ h (ˆ x + , ˆ x − ) . (A.2)For h ∈ Z the phases on the RHS cancel in which case the integrals over the diamond andits complement are equal. The integral over the causal diamond can then be replaced byhalf the integral over the torus. The change of variables to ˆ ψ, ˆ φ then leads to I = Z T d ˆ ψd ˆ φ e − i ( m + n ) ˆ ψ e − i ( m − n ) ˆ φ e − iπh Γ(2 h ) î r cos Ä ˆ φ − φ ä − q cos Ä ˆ ψ − ψ äó h . (A.3)Setting m = n and w = e i ( ψ − ˆ ψ ) , z = e i ( ˆ φ − φ ) , (A.4)we find I = − h e − imψ I dww I dzz w m e − iπh Γ(2 h )[ r ( z + z − ) − q ( w + w − )] h , (A.5)15here the contours are the unit circles | z | = | w | = 1 . For fixed w and | r ( z + z − ) | < | q ( w + w − ) | we can expand the denominator I = − h e − imψ e − iπh Γ(2 h ) I dww I dzz w m ∞ X k =0 ( − q ) − h ( w + w − ) − h Å − rq z + z − w + w − ã k − hk ! . (A.6)and first evaluate the z integral, I dzz ( z + z − ) k = 2 πi Γ( k +1)Γ( k/ , k ∈ Z + , , else . (A.7)Then, upon a redefinition of the summation variable, I = − h e − imψ e − iπh Γ(2 h )2 πi ∞ X k =0 I dww w m ( w + w − ) − h − k ( − q ) − h Å − rq ã k × Γ( − h + 1)Γ(1 + k ) Γ( − h − k + 1) . (A.8)We now easily see that the remaining integral above simplifies and gives us the solutions in4.12. First set m = − h . The integral over w reduces to I dww ( w + 1) − h − k w k . (A.9)For k >
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