# Onℛ4 Terms and MHV Amplitudes in 𝒩=5,6 Supergravity Vacua of Type II Superstrings

Authors: Massimo Bianchi
Journal: Advances in High Energy Physics
 (2011)

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## Abstract

We compute one-loop threshold corrections toℛ4 terms in 𝒩=5,6 supergravity vacua of Type II superstrings. We then discuss nonperturbative corrections generated by asymmetric D-brane instantons. Finally we derive generating functions for MHV amplitudes at tree level in 𝒩=5,6 supergravities.


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## Body

## 1. Introduction

𝒩=5,6 supergravities in D=4 enjoy many of the remarkable properties of 𝒩=8 supergravity. Their massless spectra are unique and consist solely of the supergravity multiplets. Their R-symmetries are not anomalous [1]. Regular BH solutions can be found whereby the scalars are stabilized at the horizon by the attractor mechanism (for a recent review see,  e.g., [2]). It is thus tempting to conjecture that if pure 𝒩=8 supergravity turned out to be UV finite [3–7] then 𝒩=5,6 supergravities should be so, too.As shown in [8–10], Type II superstrings or M-theory accommodate 𝒩=8 supergravity in such a way as to include nonperturbative states that correspond to singular BH solutions in D=4. The same is true for 𝒩=5,6 supergravities. While the embedding of 𝒩=8 supergravity corresponds to simple toroidal compactifications, the embedding of 𝒩=5,6 supergravities, pioneered by Ferrara and Kounnas in [11] and recently reviewed in [12], requires asymmetric orbifolds [13, 14] or free fermion constructions [15–20].The inclusion of BPS states, whose possible singular behavior from a strict 4D viewpoint is resolved from a higher-dimensional perspective, generates higher derivative corrections to the low-energy effective action. In particular a celebratedℛ4 term appears that spoils the continuous noncompact symmetry of “classical” supergravity. Absence of such a term has been recently shown for pure 𝒩=8 supergravity in [21]. In superstring theory, the ℛ4 term receives contribution at tree level, one loop, and from nonperturbative effects associated to D-instantons [22] and other wrapped branes [23]. Proposals for the relevant modular form of the E7(7)(Z)   U-duality group have been recently put forward in [24–26] that seem to satisfy all the checks.In this paper we consider one-loop threshold corrections to the same kind of terms in superstring models with𝒩=5,6 supersymmetry in D=4 and 𝒩=6 in D=5. After excluding ℛ2 terms (ℛ3 terms cannot be supersymmetrized on shell when all particles are in the supergravity multiplet [21]), we will derive formulae for the “perturbative” threshold corrections. In D=4 we will also discuss other MHV amplitudes (for a recent review see,   e.g., [27]) that can be obtained by orbifold techniques from the generating function of 𝒩=8 supergravity amplitudes [28].Aim of the analysis is threefold. First, we would like to show that𝒩=5,6 supersymmetric models in D=4 behave very much as their common 𝒩=8 supersymmetric parent. The threshold corrections that we find may be taken as evidence that, as in the 𝒩=8 case, superstring calculations do not reproduce field theory results, where such ℛ4 corrections are absent as a result of the unbroken (anomaly free) continuous U-duality symmetry as in the 𝒩=8 case [1]. This is in line with the nondecoupling in Type II superstrings of BPS states that are singular from the strict 4-dimensional supergravity perspective [8–10].Second, (gauged)𝒩=5,6 supergravities have played a crucial role in the recent understanding of M2-brane dynamics [29–32], and nonperturbative tests may be refined by considering the effects of world-sheet instantons in CP3 [33–36] along the lines of our present (ungauged) analysis. Finally, in addition to world-sheet instantons, D-brane instantons corresponding to Euclidean bound states of “exotic” D-branes should contribute to generalize “standard” D-brane instanton calculus to Left-Right asymmetric backgrounds.Plan of the paper is as follows. In Section1, we briefly review 𝒩=5,6 supergravities in D=4,5 and their embedding in Type II superstrings. We then pass to consider in Section 2 a 4-graviton amplitude at one loop which allows to derive the “perturbative” threshold corrections to ℛ4 terms, thus excluding ℛ2 terms. For simplicity, we only give the explicit result for 𝒩=6 in D=5 in Section 3 and sketch how to complete the nonperturbative analysis by including asymmetric D-brane instantons [12] in Section 4. Finally, in Section 5 we consider MHV amplitudes in 𝒩=5,6 supergravities in D=4 and show how they can be obtained at tree level by orbifold techniques from the generating function for MHV amplitudes in 𝒩=8 supergravity [28]. Section 6 contains a summary of our results and directions for further investigation.


## 2. Type II Superstring Models with𝒩=5,6 in D=4,5

Let us briefly recall how𝒩=5,6 supergravities can be embedded in String Theory. The highest dimension where classical 𝒩=6 supergravity with 24 supercharges can be defined is D=6. However the resulting 𝒩=(2,1) theory is anomalous and thus inconsistent at the quantum level [37]. So we are led to consider D=5 and then reduce to D=4. 𝒩=5 supergravity with 20 supercharges can only be defined as D=4 and lower. Although we will only focus on ℛ4 terms in D=4 the parent D=5 theory is instrumental to the identification of the relevant BPS instantons.


### 2.1.𝒩=6=2L+4R Supergravity in D=5

The simplest way to embed𝒩=6 in Type II superstrings is to quotient a toroidal compactification T5=T4×S1 by a chiral Z2 twist of the L-movers (T-duality) on four internal directions  (2.1)XLi⟶-XLi,ΨLi⟶-ΨLi,i=6,7,8,9
accompanied by an order-two shift that makes twisted states massive. As a result half of the supersymmetries in the L-moving sector are broken. The perturbative spectrum is coded in the one-loop torus partition function.In the untwisted sector, one finds(2.2)Tu=12{(Qo+Qv)Q¯Λ5,5[00]+(Qo-Qv)(Xo-Xv)Q¯Λ1,5[01]},
where Xo-Xv=4η2/θ22 (with η denoting Dedekind's function and θ1,…4 denoting Jacobi's elliptic functions) describes the effect of the Z2 projection on four internal L-moving bosons, while  (2.3)Λl,r[ab]=∑pL,pReiπ[aLpL-aRpR]q(1/2)(pL+(1/2)bL)2q¯(1/2)(pR+(1/2)bR)2
are (shifted) Lorentzian lattice sums of signature (l,r) and Q=V8-S8, Qo=V4O4-S4S4, Qv=O4V4-C4C4, with On,Vn,Sn,Cn the characters of SO(n) at level κ=1 (for n odd Sn coincides with Cn and will be denoted by Σn).At the massless level, inD=5 notation with SO(3) little group, one finds  (2.4)(V3+O3-2Σ3)×(V¯3+5O¯3-4Σ3¯)⟶(g+b2+ϕ)NS-NS+6ANS-NS+5ϕNS-NS+8AR-R+8ϕR-R-Fermi
that form the 𝒩=6 supergravity multiplet in D=5(2.5)SGN=6D=5={gμν,6ψμ,15Aμ,20χ,14φ}.
The R-symmetry is Sp(6) while the “hidden” noncompact symmetry is SU*(6), of dimension 35 and rank 3 generated by 6×6 matrices of the form Z=(Z1,Z2;-Z¯2,Z¯1) with Tr(Z1+Z¯1)=0.For later purposes, let us observe that the 128 massless states of𝒩=6 supergravity in D=5 are given by the tensor product of the 8 massless states of 𝒩=2 SYM (for the Left-movers) and the 16 massless states of 𝒩=4 SYM (for the Right-movers), namely,  (2.6)SGN=6D=5=SYMN=2D=5⊗SYMN=4D=5={Aμ,2λ,ϕ}L⊗{Ãν,4λ̃,5ϕ̃}R.After dualizing all massless 2 forms into vectors, the15=7NS-NS+8R-R vectors transform according to the antisymmetric tensor of SU*(6). The 14=1NS-NS+5NS-NS+8R-R scalars parameterize the moduli space  (2.7)MN=6D=5=SU*(6)Sp(6).By world-sheet modular transformations (firstS and then T) one finds the contribution of the twisted sector  (2.8)Tt=12{(Qs+Qc)(Xs+Xc)Q¯Λ1,5[10]+(Qs-Qc)(Xs-Xc)Q¯Λ1,5[11]},
where Xs+Xc=4η2/θ42, Xs-Xc=4η2/θ32, Qs=O4S4-C4O4 (massless), Qc=V4C4-S4V4 (massive). Due to the (L-R symmetric) Z2 shift, the massless spectrum receives no contribution from the twisted sector. Nonperturbative states associated to L-R asymmetric bound states of D-branes were studied in [12]. There are several other ways to embed 𝒩=6 supergravity in Type II superstrings, reviewed in [12].


### 2.2.𝒩=6 Supergravities in D=4

Reducing on another circle with or without further shifts yields𝒩=6 supergravity in D=4 [11].The massless spectrum is given by(2.9)(V2+2O2-2S2-2C2)×(V¯2+6O¯2-4S¯2-4C¯2)⟶(g+b+ϕ)NS-NS+8ANS-NS+12ϕNS-NS+8AR-R+16ϕR-R-Fermi
and gives rise to the 𝒩=6 supergravity multiplet in D=4(2.10)SGN=6D=4={gμν,6ψμ,16Aμ,26χ,30φ}.For later purposes, let us observe that the 128 massless states of𝒩=6 supergravity in D=4 are given by the tensor product of the 8 massless states of 𝒩=2 SYM (for the Left-movers) and the 16 massless states of 𝒩=4 SYM (for the Right-movers),namely,    (2.11)SGN=6D=4=SYMN=2D=4⊗SYMN=4D=4={Aμ,2λ,2ϕ}L⊗{Ãν,4λ̃,6ϕ̃}R.The hidden noncompact symmetry isSO*(12), of dimension 66 and rank 3 generated by 12×12 matrices of the form Z=(Z1,Z2;-Z¯2,Z¯1) with Z1=-Z1t and Z2 hermitean. They satisfy L†𝒥L=𝒥 with 𝒥=-𝒥t=-𝒥† the symplectic metric in 12D. After dualizing all masseless 2 forms into axions, the 30=2NS-NS+12NS-NS+16R-R scalar parameterize the moduli space  (2.12)MN=6D=4=SO*(12)U(6).
The 16=8NS-NS+8R-R vectors together with their magnetic duals transform according to the 32-dimensional chiral spinor representation of SO*(12).Due to the (L-R symmetric) Z2 shift, the massless spectrum receives no contribution from the twisted sector. Nonperturbative states associated to L-R asymmetric bound states of D-branes were studied in [12].


### 2.3.𝒩=5=1L+4R Supergravity in D=4

The highest dimension where𝒩=5 supergravity exists is D=4. In D=5 because one cannot impose a symplectic Majorana condition on an odd number of spinors. A simple way to realize 𝒩=5=1L+4R supergravity in D=4 is to combine Z2L×Z2L twists, acting by T-duality along T67894 and T45894, with order two shifts, that eliminate massless twisted states. In [11], “minimal” 𝒩=5 superstring solutions of this kind have been classified into four classes which correspond to different choices of the basis sets of free fermions or inequivalent choices of shifts in the orbifold language.Due to the uniqueness of𝒩=5 supergravity in D=4, all models display the same massless spectrum  (2.13)SGN=5D=4={gμν,5ψμ,10Aμ,11χ,10ϕ}.For later purposes, let us observe that the 64 massless states of𝒩=5 supergravity in D=4 are given by the tensor product of the 4 massless states of 𝒩=1 SYM (for the Left-movers) and the 16 massless states of 𝒩=4 SYM (for the Right-movers), namely,  (2.14)SGN=5D=4=SYMN=1D=4⊗SYMN=4D=4={Aμ,λ}L⊗{Ãν,4λ̃,6ϕ̃}R.The massless scalars parameterize the moduli space(2.15)MN=5D=4=SU(5,1)U(5).
The graviphotons together with their magnetic duals transform according to the 20 complex (3-index totally antisymmetric tensor) representation of SU(5,1).


## 2.1.𝒩=6=2L+4R Supergravity in D=5

The simplest way to embed𝒩=6 in Type II superstrings is to quotient a toroidal compactification T5=T4×S1 by a chiral Z2 twist of the L-movers (T-duality) on four internal directions  (2.1)XLi⟶-XLi,ΨLi⟶-ΨLi,i=6,7,8,9
accompanied by an order-two shift that makes twisted states massive. As a result half of the supersymmetries in the L-moving sector are broken. The perturbative spectrum is coded in the one-loop torus partition function.In the untwisted sector, one finds(2.2)Tu=12{(Qo+Qv)Q¯Λ5,5[00]+(Qo-Qv)(Xo-Xv)Q¯Λ1,5[01]},
where Xo-Xv=4η2/θ22 (with η denoting Dedekind's function and θ1,…4 denoting Jacobi's elliptic functions) describes the effect of the Z2 projection on four internal L-moving bosons, while  (2.3)Λl,r[ab]=∑pL,pReiπ[aLpL-aRpR]q(1/2)(pL+(1/2)bL)2q¯(1/2)(pR+(1/2)bR)2
are (shifted) Lorentzian lattice sums of signature (l,r) and Q=V8-S8, Qo=V4O4-S4S4, Qv=O4V4-C4C4, with On,Vn,Sn,Cn the characters of SO(n) at level κ=1 (for n odd Sn coincides with Cn and will be denoted by Σn).At the massless level, inD=5 notation with SO(3) little group, one finds  (2.4)(V3+O3-2Σ3)×(V¯3+5O¯3-4Σ3¯)⟶(g+b2+ϕ)NS-NS+6ANS-NS+5ϕNS-NS+8AR-R+8ϕR-R-Fermi
that form the 𝒩=6 supergravity multiplet in D=5(2.5)SGN=6D=5={gμν,6ψμ,15Aμ,20χ,14φ}.
The R-symmetry is Sp(6) while the “hidden” noncompact symmetry is SU*(6), of dimension 35 and rank 3 generated by 6×6 matrices of the form Z=(Z1,Z2;-Z¯2,Z¯1) with Tr(Z1+Z¯1)=0.For later purposes, let us observe that the 128 massless states of𝒩=6 supergravity in D=5 are given by the tensor product of the 8 massless states of 𝒩=2 SYM (for the Left-movers) and the 16 massless states of 𝒩=4 SYM (for the Right-movers), namely,  (2.6)SGN=6D=5=SYMN=2D=5⊗SYMN=4D=5={Aμ,2λ,ϕ}L⊗{Ãν,4λ̃,5ϕ̃}R.After dualizing all massless 2 forms into vectors, the15=7NS-NS+8R-R vectors transform according to the antisymmetric tensor of SU*(6). The 14=1NS-NS+5NS-NS+8R-R scalars parameterize the moduli space  (2.7)MN=6D=5=SU*(6)Sp(6).By world-sheet modular transformations (firstS and then T) one finds the contribution of the twisted sector  (2.8)Tt=12{(Qs+Qc)(Xs+Xc)Q¯Λ1,5[10]+(Qs-Qc)(Xs-Xc)Q¯Λ1,5[11]},
where Xs+Xc=4η2/θ42, Xs-Xc=4η2/θ32, Qs=O4S4-C4O4 (massless), Qc=V4C4-S4V4 (massive). Due to the (L-R symmetric) Z2 shift, the massless spectrum receives no contribution from the twisted sector. Nonperturbative states associated to L-R asymmetric bound states of D-branes were studied in [12]. There are several other ways to embed 𝒩=6 supergravity in Type II superstrings, reviewed in [12].


## 2.2.𝒩=6 Supergravities in D=4

Reducing on another circle with or without further shifts yields𝒩=6 supergravity in D=4 [11].The massless spectrum is given by(2.9)(V2+2O2-2S2-2C2)×(V¯2+6O¯2-4S¯2-4C¯2)⟶(g+b+ϕ)NS-NS+8ANS-NS+12ϕNS-NS+8AR-R+16ϕR-R-Fermi
and gives rise to the 𝒩=6 supergravity multiplet in D=4(2.10)SGN=6D=4={gμν,6ψμ,16Aμ,26χ,30φ}.For later purposes, let us observe that the 128 massless states of𝒩=6 supergravity in D=4 are given by the tensor product of the 8 massless states of 𝒩=2 SYM (for the Left-movers) and the 16 massless states of 𝒩=4 SYM (for the Right-movers),namely,    (2.11)SGN=6D=4=SYMN=2D=4⊗SYMN=4D=4={Aμ,2λ,2ϕ}L⊗{Ãν,4λ̃,6ϕ̃}R.The hidden noncompact symmetry isSO*(12), of dimension 66 and rank 3 generated by 12×12 matrices of the form Z=(Z1,Z2;-Z¯2,Z¯1) with Z1=-Z1t and Z2 hermitean. They satisfy L†𝒥L=𝒥 with 𝒥=-𝒥t=-𝒥† the symplectic metric in 12D. After dualizing all masseless 2 forms into axions, the 30=2NS-NS+12NS-NS+16R-R scalar parameterize the moduli space  (2.12)MN=6D=4=SO*(12)U(6).
The 16=8NS-NS+8R-R vectors together with their magnetic duals transform according to the 32-dimensional chiral spinor representation of SO*(12).Due to the (L-R symmetric) Z2 shift, the massless spectrum receives no contribution from the twisted sector. Nonperturbative states associated to L-R asymmetric bound states of D-branes were studied in [12].


## 2.3.𝒩=5=1L+4R Supergravity in D=4

The highest dimension where𝒩=5 supergravity exists is D=4. In D=5 because one cannot impose a symplectic Majorana condition on an odd number of spinors. A simple way to realize 𝒩=5=1L+4R supergravity in D=4 is to combine Z2L×Z2L twists, acting by T-duality along T67894 and T45894, with order two shifts, that eliminate massless twisted states. In [11], “minimal” 𝒩=5 superstring solutions of this kind have been classified into four classes which correspond to different choices of the basis sets of free fermions or inequivalent choices of shifts in the orbifold language.Due to the uniqueness of𝒩=5 supergravity in D=4, all models display the same massless spectrum  (2.13)SGN=5D=4={gμν,5ψμ,10Aμ,11χ,10ϕ}.For later purposes, let us observe that the 64 massless states of𝒩=5 supergravity in D=4 are given by the tensor product of the 4 massless states of 𝒩=1 SYM (for the Left-movers) and the 16 massless states of 𝒩=4 SYM (for the Right-movers), namely,  (2.14)SGN=5D=4=SYMN=1D=4⊗SYMN=4D=4={Aμ,λ}L⊗{Ãν,4λ̃,6ϕ̃}R.The massless scalars parameterize the moduli space(2.15)MN=5D=4=SU(5,1)U(5).
The graviphotons together with their magnetic duals transform according to the 20 complex (3-index totally antisymmetric tensor) representation of SU(5,1).


## 3. Four-Graviton One-Loop Amplitude

Since𝒩=5,6 supergravities can be obtained as asymmetric orbifolds of tori, tree-level scattering amplitudes of untwisted states such as gravitons are identical to the corresponding amplitudes in the parent 𝒩=8 theory. In particular, denoting by fℛ4𝒩=5,6(φ) the moduli dependent coefficient function of the ℛ4 term, one has  (3.1)fR4N=5,6=2nζ(3)V(Td)gs2ls2+Id,dN=8nls2+⋯,
where n is the order of the orbifold group, that reduces the volume of Td with d=5,6 to the volume of the orbifold, ℓs2=α′ and ⋯ stands for nonperturbative terms. The one-loop threshold integral is given by  (3.2)Id,dN=8=(2π)d∫Fd2ττ22[τ2d/2Γd,d(G,B;τ)-τ2d/2]=2π2-d/2Γ(d2-1)Ev=2d,s=d/2-1SO(d,d∣Z),
where  (3.3)Ev=2d,s=d/2-1SO(d,d∣Z)=∑m⃗,n⃗:m⃗⋅n⃗=0[(m⃗+Bn⃗)tG-1(m⃗+Bn⃗)+n⃗tGn⃗]-d+2
is a constrained Epstein series that encodes the contribution of perturbative 1/2 BPS, states   that is, those satisfying m⃗·n⃗=0. The subtraction eliminates IR divergences,   that is the terms with m⃗=n⃗=0. For 𝒩=5,6 the contribution of the (r,s)=(0,0) “untwisted” sector is up to a factor 1/n the same as in toroidal Type II compactifications with restricted metric Gij and antisymmetric tensor Bij.In the following we will focus on the contribution of the “twisted” sectors (we write “twisted” in quotes, since the terminology includes projections of the untwisted sector,   i.e., amplitudes withr=0 and s=1,…,n-1) with (r,s)≠(0,0).Recall that the partition function reads(3.4)Z=Q¯1n∑r,s0,n-1∑αθα(0)η3∏I=13θα(ursI)θ1(ursI)Γ[rs],
where ursI encode the effect of the Left-moving twist on the three complex internal directions, while Γ[sr] denote the twisted and shifted lattice sums.Following the analysis in [38] for one-loop scattering of vector bosons in unoriented D-brane worlds and exploiting the “factorization” of world-sheet correlation functions one has  (3.5)A4h=1n∑r,s0,n-1∫d2ττ22Γ[rs]C4vLC4vR.Since in both𝒩=5,6 cases the orbifold projection only acts by a shift of the lattice on the Left-movers,   that is, preserves all four space-time supersymmetries, their contribution is simply  (3.6)C4vL=const
after summing over spin structures. In the terminology of [38] only terms with 4 fermion pairs contribute. Recall that the graviton vertex in the q=0 superghost picture reads  (3.7)Vh=hμν(∂Xμ+ikψψμ)(∂¯X̃ν+ikψ̃ψ̃ν)eikX,
and, for fixed graviton helicity (henceforth we use D=4 notation but the analysis is valid in D=5 too), one can exploit “factorization” of the physical polarization tensor  (3.8)hμν(2σ)=aμ(σ)aν(σ)
in terms of photon polarization vectors.In theR-moving sector however, the orbifold projection breaks 1/2 (𝒩=6) or 3/4 (𝒩=5) of the original four space-time supersymmetries. Correlation functions of two and three fermion bilinears will be nonvanishing, too.For two fermion bilinears one has [38]  (3.9)〈∂Xμ1∂Xμ2k3ψψμ3k4ψψμ4〉[ημ1μ2∂1∂2G12-∑i≠1kiμ1∂1G1i∑j≠2kjμ2∂2G1j]=[k3k4ημ3μ4-k3μ4k4μ3],
where 𝒢ij denotes the scalar propagator on the torus (with α′=2)  (3.10)Gz,w=-log⁡|θ1(z-w)||θ1′(0)|-πIm⁡(z-w)2Im⁡τ.Similarly, for three fermion bilinears, one finds [38]  (3.11)〈∂Xμ1k2ψψμ2k3ψψμ3k4ψψμ4〉=∑i≠1kiμ1∂1G1i[k2k3k4μ2ημ3μ4-⋯]ω234
with ω234=∂logθ1(z23)+∂logθ1(z34)+∂logθ1(z42).For four fermion bilinears, disconnected contractions yield [38]  (3.12)〈k1ψψμ1k2ψψμ2k3ψψμ3k4ψψμ4〉disc={[k1k2ημ1μ2-k1μ2k2μ1][k1k2ημ1μ2-k1μ2k2μ1][k3k4ημ3μ4-k3μ4k4μ3]×(℘12+℘34-Δrs)+⋯[k1k2ημ1μ2-k1μ2k2μ1]},
where ℘ is Weierstrass function  (3.13)℘(z)=1z2+∑m,n′1(z+n+mτ)2-1(n+mτ)2=-∂z2log⁡θ1(z)-2η1=-2∂z2G(z,z¯)-π23Ê2
with η1=-θ1′′′/6θ1′ and Ê2 the nonholomorphic modular form of weight 2 (Eisenstein series). Weierstrass function satisfies ℘(1/2)=e1, ℘(τ/2)=e2, ℘(1/2+τ/2)=e3 with e1+e2+e3=0.Finally, connected contractions of four fermion bilinears yield [38]  (3.14)〈k1ψψμ1k2ψψμ2k3ψψμ3k4ψψμ4〉conn=[k1μ4k2μ1k3μ2k4μ3±⋯](℘13-ω123ω143+Δrs),
where, for 𝒩=6,(3.15)Δrs=℘(urs)
while, for 𝒩=5,(3.16)Δrs=3η1+16H′′′(urs)H′(urs)
with ℋ′/ℋ=∑I∂logθ1(ursI), which is clearly moduli independent, since no NS-NS moduli survive except for the axio-dilaton. Dependence on R-R moduli and the axio-dilaton is expected to be generated by L-R asymmetric bound states of Euclidean D-branes and NS5-branes.


### 3.1. World-Sheet Integrations

Worldsheet integrations can be performed with the help of∫d2z∂z𝒢zw=0=∫d2z∂z2𝒢zw as well as of  (3.17)∫d2zd2w(∂zGzw)2=-τ2Ê2π23,∫d2zd2w[ημ1μ2∂1∂2G12k1k2G12-∑i≠1kiμ1∂1G1i∑j≠2kjμ2∂2G1j]=-τ2Ê2π23[ημ1μ2k1k2-k1μ2k2μ1].For𝒩=6=4L+2R, setting fμνL/R=kμaνL/R-kνaμL/R, one has  (3.18)Lefftwist=1n∑r,s′∫d2τΓ[rs]〈f1f2f3f4〉LMHV×{4[(f1f2)(f3f4)+⋯]Rπ23Ê2+[(f1f2)(f3f4)+⋯]R(-2π23Ê2+℘(urs))+[(f1f2f3f4)+⋯]R(-2π23Ê2-℘(urs))},
where, including all permutations,  (3.19)〈f1f2f3f4〉MHV=(f1f2f3f4)+(f1f3f4f2)+(f1f4f2f3)-2(f1f2)(f3f4)-2(f1f3)(f4f2)-2(f1f4)(f2f3)
is the structure that appears in 4-pt vector boson amplitudes, that are necessarily MHV (Maximally Helicity Violating) in D=4 (in D=5 there is more than one “helicity,” but the tensor structure has the same form [39, 40]).Combining theR-moving contributions one eventually finds  (3.20)Lefftwist=〈R1R2R3R4〉MHV1n∑r,s′∫d2τΓ[rs](+2π23Ê2-℘(urs)),
where ℛi denote the linearized Riemann tensors of the four gravitons and  (3.21)〈R1R2R3R4〉MHV=〈f1f2f3f4〉LMHV〈f1f2f3f4〉RMHV
reproduces the expected ℛ4 structure, which is MHV in D=4, and no lower derivative ℛ2 and/or ℛ3 terms [21].For𝒩=5=4L+1R in D=4 one gets similar results with ℰ𝒩=2R=Γ[sr] replaced by ℰ𝒩=1R=ℐabℋ′/ℋ(α′τ2)-2 which is moduli independent.Henceforth we will focus on the𝒩=6=4L+2R case and explore NS-NS moduli dependence of the one-loop threshold in D=5.


## 3.1. World-Sheet Integrations

Worldsheet integrations can be performed with the help of∫d2z∂z𝒢zw=0=∫d2z∂z2𝒢zw as well as of  (3.17)∫d2zd2w(∂zGzw)2=-τ2Ê2π23,∫d2zd2w[ημ1μ2∂1∂2G12k1k2G12-∑i≠1kiμ1∂1G1i∑j≠2kjμ2∂2G1j]=-τ2Ê2π23[ημ1μ2k1k2-k1μ2k2μ1].For𝒩=6=4L+2R, setting fμνL/R=kμaνL/R-kνaμL/R, one has  (3.18)Lefftwist=1n∑r,s′∫d2τΓ[rs]〈f1f2f3f4〉LMHV×{4[(f1f2)(f3f4)+⋯]Rπ23Ê2+[(f1f2)(f3f4)+⋯]R(-2π23Ê2+℘(urs))+[(f1f2f3f4)+⋯]R(-2π23Ê2-℘(urs))},
where, including all permutations,  (3.19)〈f1f2f3f4〉MHV=(f1f2f3f4)+(f1f3f4f2)+(f1f4f2f3)-2(f1f2)(f3f4)-2(f1f3)(f4f2)-2(f1f4)(f2f3)
is the structure that appears in 4-pt vector boson amplitudes, that are necessarily MHV (Maximally Helicity Violating) in D=4 (in D=5 there is more than one “helicity,” but the tensor structure has the same form [39, 40]).Combining theR-moving contributions one eventually finds  (3.20)Lefftwist=〈R1R2R3R4〉MHV1n∑r,s′∫d2τΓ[rs](+2π23Ê2-℘(urs)),
where ℛi denote the linearized Riemann tensors of the four gravitons and  (3.21)〈R1R2R3R4〉MHV=〈f1f2f3f4〉LMHV〈f1f2f3f4〉RMHV
reproduces the expected ℛ4 structure, which is MHV in D=4, and no lower derivative ℛ2 and/or ℛ3 terms [21].For𝒩=5=4L+1R in D=4 one gets similar results with ℰ𝒩=2R=Γ[sr] replaced by ℰ𝒩=1R=ℐabℋ′/ℋ(α′τ2)-2 which is moduli independent.Henceforth we will focus on the𝒩=6=4L+2R case and explore NS-NS moduli dependence of the one-loop threshold in D=5.


## 4. One-Loop Threshold Integrals

One-loop threshold integrals for toroidal compactifications have been briefly reviewed above and shown to represent the contribution of the(r,s)≠(0,0) untwisted sector. For (r,s)≠(0,0) the threshold integrals involve shifted lattice sums as in heterotic strings with Wilson lines [41–45].For simplicity let us discuss here the case of𝒩=6 in D=5. For definiteness we consider n=2 (Z2 shift orbifold) and start at the special point in the moduli space where T5=TSO(8)4×S1. Later on we will include off-diagonal moduli that effectively behave as Wilson lines.In the “twisted”[10] sector, the relevant threshold integral is of the form  (4.1)I1,5N=6[01]=(2π)5∫Fd2ττ2τ25/2Γ1,1[01](R)O¯8[2π23Ê2+℘(12)]=(2π)5∫Fd2ττ2τ22Rα′∑m,ne-|2m+(2n+1)τ|2πR2/4α′τ2O¯8[2π23Ê2+℘(12)].
Setting (2m,2n+1)=(2ℓ+1)(2m′,2n′+1) and using invariance of O¯8 under τ→τ+2 allow to unfold the integral to the double strip  (4.2)(2π)5Rα′∫-1+1dτ1∫0∞dτ2∑le-(2l+1)2πR2/4α′τ2∑N,N¯dN¯q¯N¯cNq¯N,
where O¯8=∑N=|r|2/2dNq¯N and (2π2/3)Ê2+℘(1/2)=∑NcNqN. Performing the trivial integral over τ1 (level matching N¯=N) and the less trivial integral over τ2 by means of  (4.3)∫0∞dyyν-1e-cy-b/y=(bc)ν/2Kν(bc),
where Kν(z) is a Bessel function of second kind, finally yields  (4.4)I1,5N=6[01](R,Ai=0)=(2π)5(Rα′)3/2∑l=0∞∑N=1∞(2l+1)NdNσ1(N)K1(4π(2l+1)NRα′),
where  (4.5)σ1(N)=∑d∣N=ψ(N)-ψ(1)=cNN
from the expansion of Ê2 in powers of q.The result can be easily generalized to the other sectors of theZ2 orbifold under consideration as well as to different (orbifold) constructions that give rise to different shifted lattice sums. Manifest SO(1,5∣Z) symmetry can be achieved turning on off-diagonal components of B and G (subject to restrictions). Denoting by 2Ai=Gi5+Bi5 and observing that Gi5-Bi5=0 by construction, one finds  (4.6)I1,5N=6[rs](R,Ai)=(2π)5R2α′∑l=0∞∑w⃗∈Γ[sr]cw⃗22∫0∞dy(2l+1)e-(2l+1)2πR2/4α′y-2πw⃗⋅w⃗+2πiw⃗⋅A⃗=(2π)5(Rα′)3/2∑l=0∞∑w⃗∈Γ[sr]σ1(w⃗22)(2l+1)×w⃗22e2πir⃗⋅A⃗K1(4π(2l+1)w⃗2R2α′).Summing up the contributions of the various sectors,   that is, various shifted lattice sums, yields the complete one-loop threshold correction to theℛ4 terms for 𝒩=6 superstring vacua in D=5. Clearly only NS-NS moduli (except the dilaton) appear that expose SO(1,5)    T-duality symmetry.The analysis is rather more involved inD=4 where one-loop threshold integrals receive contribution from trivial, degenerate, and nondegenerate orbits [46, 47]. Alternative methods for unfolding the integrals over the fundamental domain have been proposed [48, 49].Explicit computation is beyond the scope of the present investigation. It proceeds along the lines above and presents close analogy with threshold computations in𝒩=2 heterotic strings sectors in the present of Wilson lines [41, 42, 44] or, equivalently, 𝒩=4 heterotic strings in D=8 [50]. Rather than focussing on this interesting but rather technical aspect of the problem, let us turn our attention onto the nonperturbative dependence on the other R-R moduli as well as dilaton. This is brought about by the inclusion of asymmetric D-brane instantons.


## 5. Low-Energy Action andU-Duality

In [12] the conserved charges coupling to the surviving R-R and NS-NS graviphotons were identified as combinations of those appearing in toroidal compactifications. In the case of maximal 𝒩=8 supergravity, the 12 NS-NS graviphotons couple to windings and KK momenta. Their magnetic duals couple to wrapped NS5-branes (H-monoples) and KK monopoles. The 32 R-R graviphotons (including magnetic duals) couple to 6 D1-, 6 D5-, and 20 D3-branes in Type IIB and to 1 D0-, 15 D2-, 15 D4-, and 1 D6-branes in Type IIA.An analogous statement applies to Euclidean branes inducing instanton effects. In toroidal compactifications with𝒩=8 supersymmetry, one has 15 kinds of worldsheet instantons (EF1), 1 D(−1), 15 ED1, 15 ED3 and one each of EN5, ED5, EKK5 for Type IIB. For Type IIA superstrings one finds 6 ED0, 20 ED2, 6 ED4 and one each of EN5 and EKK5.In a series of paper [24, 25], a natural proposal has been made for the nonperturbative completion of the modular form of Ed+1(Z) that represent the scalar dependence of the ℛ4 and higher derivative terms in 𝒩=8 superstring vacua. The explicit formulae are rather simple and elegant. In particular  (5.1)fR4N=8(Φ)=E[10d],3/2E(d+1∣Z)(Φ),
where ℰ[10d],3/2E(d+1∣Z)(Φ) is an Einstein series of the relevant U-duality group. The above proposal satisfies a number of consistency checks including perturbative string limit   that is small string coupling in which E(d+1∣Z)→SO(d,d∣Z) and [10⋯0]→2d, large radius limit in which E(d+1∣Z)→E(d∣Z) and [10⋯0]→[10⋯0], and M-theory limit in which E(d+1∣Z)→SL(d+1∣Z) and [10⋯0]→[10⋯0]′. Moreover fℛ4 only receives contribution from 1/2 BPS states as expected for a supersymmetric invariant that can be written as an integral over half of (on-shell) superspace.An independent but not necessarily inequivalent proposal has been made in [26].We expect similar results forℛ4 terms in 𝒩=5,6 superstring vacua with the following caveats. First, in 𝒩=5,6 superspace ℛ4 terms are 1/5 and 1/3 BPS, respectively, since they require integrations over 16 Grassman variables. Indeed we have explicitly seen that one-loop threshold correction involves the left-moving sector, in which supersymmetry is partially broken, in an essential way. Second, the U-duality group is not of maximal rank, and the same applies to the T-duality subgroup, present in the 𝒩=6 case. Third, 𝒩=5,6 only exist in D≤5 or D≤4. Some decompactification limits should produce 𝒩=8 vacua in D=10.Let us try and identify the relevant 1/3 or 1/5 BPS Euclidean D-brane bound states.


### 5.1.𝒩=6 ED-Branes

In the Type IIB description, the chiralZ2 projection (T-duality) from 𝒩=8 to 𝒩=6 yields the Euclidean D-brane bound states of the form  (5.2)D(-1)+ED3T̂4,ED1T2+ED5T2×T̂4,ED1S1×Ŝ1+ED3S1×T̂⊥3,ED1T̂2+ED1T̂⊥2,ED3T2×T̂2+ED3T2×T̂⊥2.The above bound states of Euclidean D-branes are 1/3 BPS since they preserve 8 supercharges out of the 24 supercharges present in the background.A similar analysis applies to world-sheet and ENS5 instantons.There are several other superstring realizations of𝒩=6 supergravity in D=4. Given the uniqueness of the low-energy theory, they all share the same massless spectrum but the massive spectrum and the relevant (Euclidean) D-brane bound-states depend on the choice of model.


### 5.2. Nonperturbative Threshold Corrections

By analogy with𝒩=8 one would expect fℛ4=ΘG,   that is, an automorphic form of the U-duality group G,   that is, G=SO*(12) (SU*(6)) for 𝒩=6 in D=4 (D=5) and G=SU(5,1) for 𝒩=5 in D=4. The relevant “instantons” should be associated to BPS particles in one higher dimension (when possible).For𝒩=6, in the decompactification limit the relevant decomposition under SO*(12)→SU(5,1)×R+ is  (5.3)66⟶350+10+15+2+15′-2
so that the 15 particle charges in D=6 satisfy 15 1/3 BPS “purity” conditions in D=5(5.4)∂I3∂Q[ij]=0,
where ℐ3𝒩=6,D=5=ɛijklmn𝒬[ij]𝒬[kl]𝒬[mn]. The moduli space decomposes according to  (5.5)SO*(12)U(6)⊃SU(5,1)Sp(6)×R15×R+.
More precisely the 15 charges decompose under SO(1,5) into a 15-dim irrep. The “purity” conditions include det𝒬=0, viewed as a 6×6 antisymmetric matrix.For𝒩=6, in the string theory limit the relevant decomposition under SO*(12)→SO(2,6)×SL(2)S is  (5.6)32⟶(8v,2)NS-NS+(8s,1)R-R+(8c,1)R-R
that yields  (5.7)66⟶(28,1)+(1,3)+(8s,2)+(8c,2)+3(1,1).
The moduli space decomposes according to  (5.8)SO*(12)U(6)⊃SO(6,2)SO(6)×SO(2)×SL(2)U(1)×R16.
Further decomposition under SL(2)T×SL(2)U×SL(2)S should allow to get the “non-Abelian” part of the automorphic from from the “Abelian” one by means of SL(2)U=τ≡SL(2)B. In particular the action for a (T-duality invariant) bound state of ED5 and three ED1's into the action of EN5 and EF1's, while the action of (T-duality invariant) bound state of ED(−1) and three ED3's is invariant (singlet). Clearly further detailed analysis is necessary.


### 5.3.𝒩=5 ED-Branes

In the Type IIB description, the two chiralZ2 projections (“T-duality” on T12344 and T34564) from 𝒩=8 to 𝒩=5 yield Euclidean D-brane bound states of the form
(5.9)D(-1)+ED3T̂12344+ED3T̂34564+ED3T̂12564,ED(-1)12+ED5123456+ED134+ED156,ED1i1i2+ED3i1j2k3l3+ED3j1i2k3l3+ED1j1j2,ED1i1i3+ED3i1j2k2l3+ED3j1j2k2k3+ED1j1k3,ED1i2i3+ED1j2j3+ED3i1j1j2i3+ED3i1j1i2j3.Bound states of Euclidean D-branes carrying the above charges are 1/5 BPS since they preserve 4 supercharges out of the 20 supercharges present in the background.As in the𝒩=6 case, a different analysis applies to BPS states carrying KK momenta or windings or their magnetic duals. However, at variant with the 𝒩=6, the three massive gravitini cannot form a single complex 2/5 BPS multiplet. One of them, together with its superpartners, should combine with string states which are degenerate in mass at the special rational point in the moduli space where the chiral Z2×Z2 projection is allowed.


## 5.1.𝒩=6 ED-Branes

In the Type IIB description, the chiralZ2 projection (T-duality) from 𝒩=8 to 𝒩=6 yields the Euclidean D-brane bound states of the form  (5.2)D(-1)+ED3T̂4,ED1T2+ED5T2×T̂4,ED1S1×Ŝ1+ED3S1×T̂⊥3,ED1T̂2+ED1T̂⊥2,ED3T2×T̂2+ED3T2×T̂⊥2.The above bound states of Euclidean D-branes are 1/3 BPS since they preserve 8 supercharges out of the 24 supercharges present in the background.A similar analysis applies to world-sheet and ENS5 instantons.There are several other superstring realizations of𝒩=6 supergravity in D=4. Given the uniqueness of the low-energy theory, they all share the same massless spectrum but the massive spectrum and the relevant (Euclidean) D-brane bound-states depend on the choice of model.


## 5.2. Nonperturbative Threshold Corrections

By analogy with𝒩=8 one would expect fℛ4=ΘG,   that is, an automorphic form of the U-duality group G,   that is, G=SO*(12) (SU*(6)) for 𝒩=6 in D=4 (D=5) and G=SU(5,1) for 𝒩=5 in D=4. The relevant “instantons” should be associated to BPS particles in one higher dimension (when possible).For𝒩=6, in the decompactification limit the relevant decomposition under SO*(12)→SU(5,1)×R+ is  (5.3)66⟶350+10+15+2+15′-2
so that the 15 particle charges in D=6 satisfy 15 1/3 BPS “purity” conditions in D=5(5.4)∂I3∂Q[ij]=0,
where ℐ3𝒩=6,D=5=ɛijklmn𝒬[ij]𝒬[kl]𝒬[mn]. The moduli space decomposes according to  (5.5)SO*(12)U(6)⊃SU(5,1)Sp(6)×R15×R+.
More precisely the 15 charges decompose under SO(1,5) into a 15-dim irrep. The “purity” conditions include det𝒬=0, viewed as a 6×6 antisymmetric matrix.For𝒩=6, in the string theory limit the relevant decomposition under SO*(12)→SO(2,6)×SL(2)S is  (5.6)32⟶(8v,2)NS-NS+(8s,1)R-R+(8c,1)R-R
that yields  (5.7)66⟶(28,1)+(1,3)+(8s,2)+(8c,2)+3(1,1).
The moduli space decomposes according to  (5.8)SO*(12)U(6)⊃SO(6,2)SO(6)×SO(2)×SL(2)U(1)×R16.
Further decomposition under SL(2)T×SL(2)U×SL(2)S should allow to get the “non-Abelian” part of the automorphic from from the “Abelian” one by means of SL(2)U=τ≡SL(2)B. In particular the action for a (T-duality invariant) bound state of ED5 and three ED1's into the action of EN5 and EF1's, while the action of (T-duality invariant) bound state of ED(−1) and three ED3's is invariant (singlet). Clearly further detailed analysis is necessary.


## 5.3.𝒩=5 ED-Branes

In the Type IIB description, the two chiralZ2 projections (“T-duality” on T12344 and T34564) from 𝒩=8 to 𝒩=5 yield Euclidean D-brane bound states of the form
(5.9)D(-1)+ED3T̂12344+ED3T̂34564+ED3T̂12564,ED(-1)12+ED5123456+ED134+ED156,ED1i1i2+ED3i1j2k3l3+ED3j1i2k3l3+ED1j1j2,ED1i1i3+ED3i1j2k2l3+ED3j1j2k2k3+ED1j1k3,ED1i2i3+ED1j2j3+ED3i1j1j2i3+ED3i1j1i2j3.Bound states of Euclidean D-branes carrying the above charges are 1/5 BPS since they preserve 4 supercharges out of the 20 supercharges present in the background.As in the𝒩=6 case, a different analysis applies to BPS states carrying KK momenta or windings or their magnetic duals. However, at variant with the 𝒩=6, the three massive gravitini cannot form a single complex 2/5 BPS multiplet. One of them, together with its superpartners, should combine with string states which are degenerate in mass at the special rational point in the moduli space where the chiral Z2×Z2 projection is allowed.


## 6. Generating MHV Amplitudes in𝒩=5,6 SG's

Very much like, tree-level amplitudes in𝒩=8 supergravity in D=4 can be identified with “squares” of tree-level amplitudes in 𝒩=4 SYM theory [3, 4], tree-level amplitudes in 𝒩=5,6 supergravity in D=4 can be identified with “products” of tree-level amplitudes in 𝒩=4 and 𝒩=1,2 SYM theory.As previously observed, a first step in this direction is to show that the spectra of𝒩=5,6 supergravity are simply the tensor products of the spectra of 𝒩=4 and 𝒩=1,2 SYM theory.The second step is to work in the helicity basis and focus on MHV amplitudes (for a recent review see,   e.g., [27]). In 𝒩=4 SYM the generating function for (colour-ordered) n-point MHV amplitudes is given by [51]  (6.1)FMHVN=4SYM(ηia,uiα)=δ8(∑iηiauiα)〈u1u2〉〈u2u3〉⋯〈unu1〉,
where ηia with i=1,…n and a=1,…4 are auxiliary Grassmann variables and ui are commuting left-handed spinors, such that pi=uiu¯i.Individual amplitudes are obtained by taking derivatives with respect to the Grassman variablesη's according to the rules  (6.2)A+⟶1,λa+⟶∂∂ηa,…,A-⟶14!ɛabcd∂4∂ηa⋯∂ηd.
The intermediate derivatives representing scalars (φ~∂2/∂η2) and right-handed gaugini (λ-~∂3/∂η3).One can reconstruct all tree-level amplitudes, be they MHV or not, from MHV amplitudes using factorization, recursion relation or otherwise, see   for example [27].One can easily derive (super)gravity MHV amplitudes by simply taking the product of the generating functions for SYM amplitudes(6.3)GMHVN=8SG(ηiA,uiα)=C(ui)δ16(∑iηiAuiα)〈u1u2〉2〈u2u3〉2⋯〈unu1〉2=C(ui)FMHV,LN=4SYM(ηiaL,uiα)FMHV,RN=4SYM(ηiaR,uiα),
where ηA=(ηiaL,ηiaR) with A=1,…8 and the correction factor 𝒞(ui) is only a function of the spinors ui, actually of the massless momenta pi=uiu¯i [28].The relevant dictionary would read(6.4)h+⟶1,ψA+⟶∂∂ηA,…,h-⟶1N!∂8∂η8.In principle one can reconstruct all tree-level amplitudes, be they MHV or not, from MHV amplitudes using factorization, recursion relations, or otherwise, see   for example [27]. Unitary methods allow to extend the analysis beyond tree level. If all 𝒩=8 supergravity amplitudes were expressible in terms of squares of 𝒩=4 SYM amplitudes, UV finiteness of the latter would imply UV finiteness of the former. Although support to this conjecture at the level of 4-graviton amplitudes, which are necessarily MHV, seems to exclude the presence of ℛ4 corrections, which are 1/2 BPS saturated, it would be crucial to explicitly test the absence of D8ℛ4 corrections, the first that are not BPS saturated.Going back to the problem of expressing MHV amplitudes in𝒩=5,6 supergravities in terms of SYM amplitudes, one has to resort to “orbifold” techniques.In the𝒩=6 case, half of the 4 η's (say ηL3 and ηL4) of the “left” 𝒩=4 SYM factor are to be projected out,   that is, “odd” under a Z2 involution. As a result the generating function is the same as in 𝒩=8 supergravity but the dictionary gets reduced to  (6.5)h+⟶1,ψA′+⟶∂∂ηA′,A0+=∂2∂ηL3∂ηL4,AA′B′+=∂2∂ηA′∂ηB′,…,h-=16!ɛA1′⋯A6′∂2+6∂ηL3∂ηL4∂ηA1′⋯∂ηA6′,
where A′=1,…6.Further reduction is necessary for𝒩=5 case; 3 of the 4 η's of the “left” 𝒩=4 SYM factor are to be projected out. For instance, they may acquire a phase ω=exp(i2π/3) under a Z3 projection.The same projections should be implemented on the intermediate states flowing around the loops. Although tree-level amplitudes in𝒩=5,6 supergravity are simply a subset of the ones in 𝒩=8 supergravity, naive extension of the argument at loop order does not immediately work [52–54]. Several cancellations are not expected to take place despite the residual supersymmetry of the left SYM factor. However, in view of the recent observations on the factorization of 𝒩=4 SYM into a kinematical part and a group theory part, where the latter satisfies identities similar to the former [55–57] and can thus be consistently replaced with the former giving rise to consistent and UV finite 𝒩=8 SG amplitudes, it may well be the case that a similar decomposition can be used to produce, possibly UV finite, 𝒩=5,6 SG amplitudes. Our results on ℛ4 lend some support to this viewpoint.


## 7. Conclusions

Let us summarize our results. We have shown that the first higher derivative corrections to the low-energy effective action around superstring vacua with𝒩=5,6 supersymmetry are ℛ4 terms as in 𝒩=8. Contrary to 𝒩≤4, no ℛ2 terms appear. In this respect 𝒩=5,6 supersymmetric models in D=4, having no massless matter multiplets to add, behave similarly to their common 𝒩=8 supersymmetric parent. It is worth stressing again that such nonvanishing threshold corrections confirm that, as in the 𝒩=8 case, superstring calculations do not reproduce field theory results. As in 𝒩=8 supergravity, it is known that ℛ4 corrections are absent in 𝒩=5,6 supergravity due to the anomaly free continuous duality symmetry [1].Relying on previous results on vector boson scattering at one loop in unoriented D-brane worlds [38], we have studied four graviton scattering amplitudes and derived explicit formulae for the one-loop threshold corrections in asymmetric orbifolds that realize the above vacua. In addition to a term 1/n×f𝒩=8×cℛ4, coming from the (0,0) sector, contributions from nontrivial sectors of the orbifold to f𝒩=5,6×cℛ4 display a close similarity with heterotic threshold corrections in the presence of Wilson lines [41, 42, 44]. For illustrative purposes, we have computed the relevant integrals for 𝒩=6 in D=5 exposing the expected SO(1,5)   T-duality symmetry. The analysis in D=4 is technically more involved and will be performed elsewhere. We have also identified the relevant 1/3 or 1/5 BPS bound states of Euclidean D-branes that contribute to the nonperturbative dependence of the thresholds on R-R scalars and on the axio-dilaton. By analogy with 𝒩=8 it is natural to conjecture the possible structure of the automorphic form of the relevant U-duality group. A more detailed analysis of this issue is however necessary. Finally, in view of the potential UV finiteness of 𝒩=5,6 supergravities, we have discussed how to compute tree-level MHV amplitudes using generating function and orbifolds techniques [28]. All other tree-level amplitudes should follow from factorization and in fact should coincide with 𝒩=8 amplitudes involving only 𝒩=5 or 𝒩=6 supergravity states in the external legs. Loop amplitudes require a separate investigation. In particular no generalization of the KLT relations is known beyond tree level [58].



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*Source: 479038-2011-09-20.xml*