We compute one-loop threshold corrections to terms in 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 supergravities.

1. Introduction

supergravities in enjoy many of the remarkable properties of 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 supergravity turned out to be UV finite [37] then supergravities should be so, too.

As shown in [810], Type II superstrings or M-theory accommodate supergravity in such a way as to include nonperturbative states that correspond to singular BH solutions in . The same is true for supergravities. While the embedding of supergravity corresponds to simple toroidal compactifications, the embedding of supergravities, pioneered by Ferrara and Kounnas in [11] and recently reviewed in [12], requires asymmetric orbifolds [13, 14] or free fermion constructions [1520].

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 celebrated term appears that spoils the continuous noncompact symmetry of “classical” supergravity. Absence of such a term has been recently shown for pure supergravity in [21]. In superstring theory, the 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    U-duality group have been recently put forward in [2426] 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 with supersymmetry in and in . After excluding terms ( 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 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 supergravity amplitudes [28].

Aim of the analysis is threefold. First, we would like to show that supersymmetric models in behave very much as their common supersymmetric parent. The threshold corrections that we find may be taken as evidence that, as in the case, superstring calculations do not reproduce field theory results, where such corrections are absent as a result of the unbroken (anomaly free) continuous U-duality symmetry as in the 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 [810].

Second, (gauged) supergravities have played a crucial role in the recent understanding of M2-brane dynamics [2932], and nonperturbative tests may be refined by considering the effects of world-sheet instantons in [3336] 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 Section 1, we briefly review supergravities in 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 terms, thus excluding terms. For simplicity, we only give the explicit result for in 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 supergravities in and show how they can be obtained at tree level by orbifold techniques from the generating function for MHV amplitudes in supergravity [28]. Section 6 contains a summary of our results and directions for further investigation.

2. Type II Superstring Models with in

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

2.1. Supergravity in

The simplest way to embed in Type II superstrings is to quotient a toroidal compactification by a chiral twist of the L-movers (T-duality) on four internal directions 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 where (with denoting Dedekind's function and denoting Jacobi's elliptic functions) describes the effect of the projection on four internal L-moving bosons, while are (shifted) Lorentzian lattice sums of signature and , , , with the characters of at level (for odd coincides with and will be denoted by ).

At the massless level, in notation with little group, one finds that form the supergravity multiplet in The R-symmetry is while the “hidden” noncompact symmetry is , of dimension 35 and rank 3 generated by matrices of the form with .

For later purposes, let us observe that the 128 massless states of supergravity in are given by the tensor product of the 8 massless states of SYM (for the Left-movers) and the 16 massless states of SYM (for the Right-movers), namely,

After dualizing all massless 2 forms into vectors, the vectors transform according to the antisymmetric tensor of . The scalars parameterize the moduli space

By world-sheet modular transformations (first and then ) one finds the contribution of the twisted sector where , , (massless), (massive). Due to the (L-R symmetric) 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 supergravity in Type II superstrings, reviewed in [12].

2.2. Supergravities in

Reducing on another circle with or without further shifts yields supergravity in [11].

The massless spectrum is given by and gives rise to the supergravity multiplet in

For later purposes, let us observe that the 128 massless states of supergravity in are given by the tensor product of the 8 massless states of SYM (for the Left-movers) and the 16 massless states of SYM (for the Right-movers),namely,

The hidden noncompact symmetry is , of dimension 66 and rank 3 generated by matrices of the form with and hermitean. They satisfy with the symplectic metric in 12D. After dualizing all masseless 2 forms into axions, the scalar parameterize the moduli space The vectors together with their magnetic duals transform according to the 32-dimensional chiral spinor representation of .

Due to the (L-R symmetric) 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. Supergravity in

The highest dimension where supergravity exists is . In because one cannot impose a symplectic Majorana condition on an odd number of spinors. A simple way to realize supergravity in is to combine twists, acting by T-duality along and , with order two shifts, that eliminate massless twisted states. In [11], “minimal” 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 of supergravity in , all models display the same massless spectrum

For later purposes, let us observe that the 64 massless states of supergravity in are given by the tensor product of the 4 massless states of SYM (for the Left-movers) and the 16 massless states of SYM (for the Right-movers), namely,

The massless scalars parameterize the moduli space The graviphotons together with their magnetic duals transform according to the 20 complex (3-index totally antisymmetric tensor) representation of .

3. Four-Graviton One-Loop Amplitude

Since 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 theory. In particular, denoting by the moduli dependent coefficient function of the term, one has where is the order of the orbifold group, that reduces the volume of with to the volume of the orbifold, and stands for nonperturbative terms. The one-loop threshold integral is given by where is a constrained Epstein series that encodes the contribution of perturbative 1/2 BPS, states that is, those satisfying . The subtraction eliminates IR divergences, that is the terms with . For the contribution of the “untwisted” sector is up to a factor the same as in toroidal Type II compactifications with restricted metric and antisymmetric tensor .

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 with and ) with .

Recall that the partition function reads where encode the effect of the Left-moving twist on the three complex internal directions, while 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

Since in both 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 after summing over spin structures. In the terminology of [38] only terms with 4 fermion pairs contribute. Recall that the graviton vertex in the superghost picture reads and, for fixed graviton helicity (henceforth we use notation but the analysis is valid in too), one can exploit “factorization” of the physical polarization tensor in terms of photon polarization vectors.

In the R-moving sector however, the orbifold projection breaks 1/2 () or 3/4 () 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] where denotes the scalar propagator on the torus (with )

Similarly, for three fermion bilinears, one finds [38] with .

For four fermion bilinears, disconnected contractions yield [38] where is Weierstrass function with and the nonholomorphic modular form of weight 2 (Eisenstein series). Weierstrass function satisfies , , with .

Finally, connected contractions of four fermion bilinears yield [38] where, for , while, for , with , 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 as well as of

For , setting , one has where, including all permutations, is the structure that appears in 4-pt vector boson amplitudes, that are necessarily MHV (Maximally Helicity Violating) in (in there is more than one “helicity,” but the tensor structure has the same form [39, 40]).

Combining the R-moving contributions one eventually finds where denote the linearized Riemann tensors of the four gravitons and reproduces the expected structure, which is MHV in , and no lower derivative and/or terms [21].

For in one gets similar results with replaced by which is moduli independent.

Henceforth we will focus on the case and explore NS-NS moduli dependence of the one-loop threshold in .

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 untwisted sector. For the threshold integrals involve shifted lattice sums as in heterotic strings with Wilson lines [4145].

For simplicity let us discuss here the case of in . For definiteness we consider ( shift orbifold) and start at the special point in the moduli space where . Later on we will include off-diagonal moduli that effectively behave as Wilson lines.

In the “twisted” sector, the relevant threshold integral is of the form Setting and using invariance of under allow to unfold the integral to the double strip where and . Performing the trivial integral over (level matching ) and the less trivial integral over by means of where is a Bessel function of second kind, finally yields where from the expansion of in powers of .

The result can be easily generalized to the other sectors of the orbifold under consideration as well as to different (orbifold) constructions that give rise to different shifted lattice sums. Manifest symmetry can be achieved turning on off-diagonal components of and (subject to restrictions). Denoting by and observing that by construction, one finds

Summing up the contributions of the various sectors, that is, various shifted lattice sums, yields the complete one-loop threshold correction to the terms for superstring vacua in . Clearly only NS-NS moduli (except the dilaton) appear that expose     T-duality symmetry.

The analysis is rather more involved in 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 in heterotic strings sectors in the present of Wilson lines [41, 42, 44] or, equivalently, heterotic strings in [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 and -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 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 with 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 that represent the scalar dependence of the and higher derivative terms in superstring vacua. The explicit formulae are rather simple and elegant. In particular where 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 and , large radius limit in which and , and M-theory limit in which and . Moreover 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 for terms in superstring vacua with the following caveats. First, in superspace 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 case. Third, only exist in or . Some decompactification limits should produce vacua in .

Let us try and identify the relevant 1/3 or 1/5 BPS Euclidean D-brane bound states.

5.1. ED-Branes

In the Type IIB description, the chiral projection (T-duality) from to yields the Euclidean D-brane bound states of the form

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 of supergravity in . 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 with one would expect , that is, an automorphic form of the U-duality group , that is, () for in () and for in . The relevant “instantons” should be associated to BPS particles in one higher dimension (when possible).

For , in the decompactification limit the relevant decomposition under is so that the 15 particle charges in satisfy 15 1/3 BPS “purity” conditions in where . The moduli space decomposes according to More precisely the 15 charges decompose under into a 15-dim irrep. The “purity” conditions include , viewed as a antisymmetric matrix.

For , in the string theory limit the relevant decomposition under is that yields The moduli space decomposes according to Further decomposition under should allow to get the “non-Abelian” part of the automorphic from from the “Abelian” one by means of . 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. ED-Branes

In the Type IIB description, the two chiral projections (“T-duality” on and ) from to yield Euclidean D-brane bound states of the form

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 the case, a different analysis applies to BPS states carrying KK momenta or windings or their magnetic duals. However, at variant with the , 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 projection is allowed.

6. Generating MHV Amplitudes in SG's

Very much like, tree-level amplitudes in supergravity in can be identified with “squares” of tree-level amplitudes in SYM theory [3, 4], tree-level amplitudes in supergravity in can be identified with “products” of tree-level amplitudes in and SYM theory.

As previously observed, a first step in this direction is to show that the spectra of supergravity are simply the tensor products of the spectra of and 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 SYM the generating function for (colour-ordered) -point MHV amplitudes is given by [51] where with and are auxiliary Grassmann variables and are commuting left-handed spinors, such that .

Individual amplitudes are obtained by taking derivatives with respect to the Grassman variables 's according to the rules The intermediate derivatives representing scalars () and right-handed gaugini ().

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 where with and the correction factor is only a function of the spinors , actually of the massless momenta [28].

The relevant dictionary would read

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 supergravity amplitudes were expressible in terms of squares of 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 corrections, which are 1/2 BPS saturated, it would be crucial to explicitly test the absence of corrections, the first that are not BPS saturated.

Going back to the problem of expressing MHV amplitudes in supergravities in terms of SYM amplitudes, one has to resort to “orbifold” techniques.

In the case, half of the 4 's (say and ) of the “left” SYM factor are to be projected out, that is, “odd” under a involution. As a result the generating function is the same as in supergravity but the dictionary gets reduced to where .

Further reduction is necessary for case; 3 of the 4 's of the “left” SYM factor are to be projected out. For instance, they may acquire a phase under a projection.

The same projections should be implemented on the intermediate states flowing around the loops. Although tree-level amplitudes in supergravity are simply a subset of the ones in supergravity, naive extension of the argument at loop order does not immediately work [5254]. 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 SYM into a kinematical part and a group theory part, where the latter satisfies identities similar to the former [5557] and can thus be consistently replaced with the former giving rise to consistent and UV finite SG amplitudes, it may well be the case that a similar decomposition can be used to produce, possibly UV finite, SG amplitudes. Our results on 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 with supersymmetry are terms as in . Contrary to , no terms appear. In this respect supersymmetric models in , having no massless matter multiplets to add, behave similarly to their common supersymmetric parent. It is worth stressing again that such nonvanishing threshold corrections confirm that, as in the case, superstring calculations do not reproduce field theory results. As in supergravity, it is known that corrections are absent in 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 , coming from the sector, contributions from nontrivial sectors of the orbifold to 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 in exposing the expected    T-duality symmetry. The analysis in 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 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 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 amplitudes involving only or 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].


Discussions with C. Bachas, M. Cardella, S. Ferrara, F. Fucito, M. Green, E. Kiritsis, S. Kovacs, L. Lopez, J. F. Morales, N. Obers, R. Poghossyan, R. Richter, M. Samsonyan, and A. V. Santini are kindly acknowledged. This work was partially supported by the ERC Advanced Grant no. 226455 “Superfields” and by the Italian MIUR-PRIN Contract 2007-5ATT78 “Symmetries of the Universe and of the Fundamental Interactions”.