We argue that N=4 supergravity is 3-loop UV finite because the relevant supersymmetric candidate counterterm is known to be SL(2, R)x SO(6) invariant, which violates the Noether-Gaillard-Zumino current conservation. Analogous arguments, based on the universality properties of groups of type E7, also apply to N=5,6,8 in 4,5,7 loops, respectively, since the 1/N BPS invariants break duality symmetry between Bianchi identities and quantum corrected vector field equations.
We reconsider the one loop divergence of ${\cal N}=8$ supergravity in four dimensions. We compute the finite effective potential of ${\cal N}=8$ anti-deSitter supergravity and interpret it as logarithmic running of the cosmological constant. We find that quantum inequivalence between fields that are classically dual is due to boundary modes in AdS$_4$. Some subtleties are traced to the difference between the Euler characteristic of global and thermal AdS$_4$.
We compute the coefficient of the potential three-loop divergence in pure N=4 supergravity and show that it vanishes, contrary to expectations from symmetry arguments. The recently uncovered duality between color and kinematics is used to greatly streamline the calculation. We comment on all-loop cancellations hinting at further surprises awaiting discovery at higher loops.
We investigate the occurrence of divergences in maximal supergravity in various dimensions from the point of view of supersymmetry constraints on the U-duality invariant threshold functions defining the higher derivative couplings in the effective action. Our method makes use of tensorial differential equations that constrain these couplings. We study in detail the Fourier modes and wave-front sets of the associated automorphic functions and find that they are always associated to special nilpotent orbits.
We determine the minimum dimension at which divergences first occur in half-maximal 16-supercharge supergravity at three loops. For the four-point amplitudes, we find that the critical dimension is D=14/3 and give the explicit form of the divergence. We also give the divergence of an additional piece that gives us access to theories with higher degrees of supersymmetry, in particular 20- and 32-supercharge supergravities. We give the divergences of half-maximal supergravity when matter vector multiplets are included as well. Explicit forms of various divergences at one and two loops are also g…
We review the question of quantum consistency of N=4 conformal supergravity in 4 dimensions. The UV divergences and anomalies of the standard ("minimal") conformal supergravity where the complex scalar $ϕ$ is not coupled to the Weyl graviton kinetic term can be cancelled by coupling this theory to N=4 super Yang-Mills with gauge group of dimension 4. The same turns out to be true also for the "non-minimal" N=4 conformal supergravity with the action (recently constructed in arXiv:1609.09083) depending on an arbitrary holomorphic function $f(ϕ)$. The special case of the "non-minimal" conformal s…
We demonstrate how duality invariance of the low energy expansion of the four-graviton amplitude in type II string theory determines the precise coefficients of multiloop logarithmic ultraviolet divergences of maximal supergravity in various dimensions. This is illustrated by the explicit moduli-dependence of terms of the form D^{2k}R^4, with k<= 3, in the effective action. Furthermore, we show that in the supergravity limit the perturbative contributions are swamped by an accumulation of non-perturbative effects of zero-action instantons.
We consider super Yang-Mills Theory in $N=1$ conformal supergravity. Using the background field method and the Feddeev-Popov procedure, the quantized action of the theory is presented. Its one-loop effective action is studied using the heat kernel method. We shall develop a non-iterative scheme, generalizing the non-supersymmetric case, to obtain the super heat kernel coefficients. In particular, the first three coefficients, which govern the one-loop divergence, will be calculated. We shall also demonstrate how to schematically derive the higher order coefficients. The method presented here c…
The ultraviolet divergences of amplitude diagrams in maximal supergravity are investigated using the pure spinor superfield formalism in maximal supergravity, with maximally supersymmetric Yang-Mills theory for reference. We comment on the effects of the loop regularisation in relation to the actual absence of high powers (within the degrees of freedom) of the non-minimal variable r. The absence affects previous results of the field theory description, which is examined more closely (with a new b-ghost) with respect to the limit on the dimension for finiteness of the theory, dependent on the n…
We show that the double-copy structure of gravity forbids divergences in pure half-maximal (16 supercharge) supergravity at four and five points at one loop in D<8 and at two loops in D<6. We link the cancellations that render these supergravity amplitudes finite to corresponding ones that eliminate forbidden color factors from the divergences of pure nonsupersymmetric Yang-Mills theory. The vanishing of the two-loop four-point divergence in D=5 half-maximal supergravity is an example where a valid counterterm satisfying the known symmetries exists, yet is not present. We also give explicit fo…
Using the duality between color and kinematics, we construct the two- and three-loop amplitudes of half-maximal supergravity with matter multiplets and show that new divergences occur in D=4 and D=5. Bossard, Howe and Stelle have recently conjectured the existence of 16-supercharge off-shell harmonic superspaces in order to explain the ultraviolet finiteness of pure half-maximal supergravity with no matter multiplets in D=4 at three loops and in D=5 at two loops. By assuming the required superspace exists in D=5, they argued that no new divergences should occur at two loops even with the addit…
We apply the heat kernel method (using Avramidi's non-recursive technique) to the study of the effective action of chiral matter in a complex representation of an arbitrary gauge sector coupled to background U(1) supergravity. This generalizes previous methods, which restricted to 1) real representations of the gauge sector in traditional Poincaré supergravity or 2) vanishing supergravity background. In this new scheme, we identify a classical ambiguity in these theories which mixes the supergravity U(1) with the gauge U(1). At the quantum level, this ambiguity is maintained since the effectiv…
The ultraviolet divergences of amplitude diagrams in maximal supergravity are characterised by a first possible divergence at seven loops for the 4-point amplitude (logarithmic) and, in its absence, at eight loops. We revisit the pure spinor superfield theory results of [arXiv:1412.5983], stating the absence of the divergence originating in the 4-point 7-loop amplitude as well as those of more than seven loops. The analysis, performed in terms of the one-particle irreducible loop structures giving rise to the divergences, is extended, especially with respect to the limits on the dimension for …
We demonstrate that pure N=4 supergravity is ultraviolet divergent at four loops. The form of the divergence suggests that it is due to the rigid U(1) duality-symmetry anomaly of the theory. This is the first known example of an ultraviolet divergence in a pure ungauged supergravity theory in four dimensions. We use the duality between color and kinematics to construct the integrand of the four-loop four-point amplitude, whose ultraviolet divergence is then extracted by standard integration techniques.
At low energies, interactions of massless particles in type II strings compactified on a torus $T^d$ are described by an effective Wilsonian action $\mathcal{S}(Λ)$, consisting of the usual supergravity Lagrangian supplemented by an infinite series of higher-derivative vertices, including the much studied $\nabla^{4p+6q} \mathcal{R}^4$ gravitational interactions. Using recent results on the asymptotics of the integrands governing four-graviton scattering at genus one and two, I determine the $Λ$-dependence of the coefficient of the above interaction, and show that the logarithmic terms appeari…
The invariants in D=4, N=4 supergravity are discussed up to the three-loop order (where one expects a general R^4 structure). Because there is an anomaly in the rigid SL(2,R) symmetry of this theory, the analysis of possible restrictions on three-loop divergences due to duality needs careful treatment. We show that this anomalous symmetry is still strong enough at the three-loop order to require duality invariance of candidate counterterms. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L \ge 2 loops wou…
We demonstrate that the two point function of the supercurrent dual to the gravitino in the four-dimensional extremal anti-de Sitter Reissner-Nordstrom black hole does not exhibit a Fermi surface singularity. In our analysis, we utilize the ingoing Eddington-Finkelstein coordinate system, which enables us to bypass certain complications in the determination of the allowed near horizon behavior of the gravitino field at zero frequency. We check that our method agrees with previous results for the massless charged Dirac field.
On-shell Pauli-Villars regularization of the one-loop divergences of supergravity theories is used to study the anomaly structure of supergravity and the cancellation of field theory anomalies under a $U(1)$ gauge transformation and under the T-duality group of modular transformations in effective supergravity theories with three Kähler moduli $T^i$ obtained from orbifold compactification of the weakly coupled heterotic string. This procedure requires constraints on the chiral matter representations of the gauge group that are consistent with known results from orbifold compactifications. Paul…
The invariants in half-maximal supergravity theories in D=4,5 are discussed in detail up to dimension eight (e.g. R^4). In D=4, owing to the anomaly in the rigid SL(2,R) duality symmetry, the restrictions on divergences need careful treatment. In pure N=4 supergravity, this anomalous symmetry still implies duality invariance of candidate counterterms at three loops. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L>1 loops would also have to be writable as full-superspace integrals. At the three-loop orde…
We derive a new non-singular tree-level KLT relation for the $n=5$-point amplitudes, with manifest $2(n-2)!$ symmetry, using information from one-loop amplitudes and IR divergences, and speculate how one might extend it to higher n-point functions. We show that the subleading-color ${\cal N}=4$ SYM 5-point amplitude has leading IR divergence of $1/ε$, which is essential for the applications of this paper. We also propose a relation between the subleading-color ${\cal N}=4$ SYM and ${\cal N}=8$ supergravity 1-loop 5-point amplitudes, valid for the IR divergences and possibly for the whole ampli…
Recent calculations have confirmed the belief that the ultraviolet behaviour of N-extended supergravities improves with increasing N. We give a comprehensive review.
The $D^4\mathcal{R}^4$ and $D^6\mathcal{R}^4$ coefficient functions in the effective action of type II string theory compactified on $T^d$ contain terms of the form $\mathcal{E}_1 {\rm ln}g_d$ and $\mathcal{E}_2 ({\rm ln}g_d)^2$ in specific dimensions, where $g_d$ is the T--duality invariant string coupling, and $\mathcal{E}_1$ and $\mathcal{E}_2$ are U--duality invariant coefficient functions. We derive these non--analytic terms from nested ultraviolet divergences in two and three loop maximal supergravity. For the $D^4\mathcal{R}^4$ coupling, the contribution involves $\mathcal{E}_{\mathcal{…
${{\cal N}=4}$ supergravity is understood to contain a $U(1)$ anomaly which manifests itself via the nonvanishing of loop-level scattering amplitudes that violate a tree-level charge conservation rule. In this letter we provide detailed evidence that at one loop such anomalous amplitudes can be set to zero by the addition of a finite local counterterm. We show that the same counterterm also cancels evanescent contributions which play an important role in the analysis of ultraviolet divergences in dimensionally regularized gravity. These cancellations call for a reanalysis of the four-loop ultr…
We consider classical type II supergravity with arbitrary fluxes, Dirichlet branes and orientifold planes and prove that a four-dimensional compactification of this theory can never correspond to a metastable de Sitter state.
We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of N=4 super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitude's integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding N=8 supergravity integrand, whose graph numerators are double copies (squares) of the N=4 super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the dualit…
We use the recently developed generalized double-copy construction to obtain an improved representation of the five-loop four-point integrand of $N = 8$ supergravity whose leading ultraviolet behavior we analyze using state of the art loop-integral expansion and reduction methods. We find that the five-loop critical dimension where ultraviolet divergences first occur is $D_c=24/5$, corresponding to a $D^8 R^4$ counterterm. This ultraviolet behavior stands in contrast to the cases of four-dimensional $N = 4$ supergravity at three loops and $N = 5$ supergravity at four loops whose improved ultra…
We examine the NS-NS corrections to the type II supergravity given by Gross and Sloan, under the linear T-duality transformations. We show that the gravity and the B-field couplings are invariant under off-shell T-duality, whereas the dilaton couplings are invariant under on-shell T-duality. Requiring the compatibility of the dilaton couplings with off-shell T-duality as a guiding principle, we extend them to include the appropriate Ricci curvature and the divergence of the B-field strength.
A first quantised approach to loop amplitudes based on the pure spinor particle is applied to the systematics of four-particle amplitudes in maximally supersymmetric field theories. Counting of fermionic zero modes allows the identification of momentum factors multiplying R**4 in the case of supergravity (and F**4 in the Yang--Mills case) thereby making manifest their ultraviolet properties as a function of dimension, D. For L=2,3,4 loops the leading supergravity divergence is in D=4+6/L dimensions and proportional to d*2L R**4, in line with earlier field theory calculations. However, at five …
According to the recent pure spinor analysis of the UV divergences by Karlsson, there are no divergent 1PI structures beyond 6 loops in D=4 N=8 supergravity. In combination with the common expectation that the UV divergences do not appear at less than 7 loops, this may imply that the 4-point amplitude in D=4 N=8 supergravity is all-loop finite. This differs from the result of the previous studies of pure spinors, which suggested that there is a UV divergence at 7 loops in D=4. Therefore an independent investigation of the pure spinor formalism predictions is desirable, as well as continuation …
Supersymmetric integrands for the two-loop five-point amplitudes in ten-dimensional super Yang--Mills and type II supergravity are proposed. The kinematic numerators are manifestly local and satisfy the duality between color and kinematics described by Bern, Carrasco and Johansson. Our results are expected to reproduce the integrated two-loop amplitudes in dimensions $D<7$. The UV divergence in the critical dimension $D=7$ matches the low-energy limit of the corresponding superstring amplitudes and is written in terms of SYM tree amplitudes.
We present the full two-loop four-graviton amplitudes in N=4,5,6 supergravity. These results were obtained using the double-copy structure of gravity, which follows from the recently conjectured color-kinematics duality in gauge theory. The two-loop four-gluon scattering amplitudes in N=0,1,2 supersymmetric gauge theory are a second essential ingredient. The gravity amplitudes have the expected infrared behavior: the two-loop divergences are given in terms of the squares of the corresponding one-loop amplitudes. The finite remainders are presented in a compact form. The finite remainder for N=…
We show that the four-loop four-point amplitudes of N = 5 supergravity are ultraviolet finite in four dimensions, contrary to expectations based on supersymmetry and duality-symmetry arguments. We explain why the diagrams of any covariant local formalism cannot manifestly exhibit the necessary cancellations for finiteness but instead require a new type of nontrivial ultraviolet cancellation that we call "enhanced cancellations". We also show that the three-loop four-point amplitudes in N = 4 and N = 5 supergravity theories display enhanced cancellations. To construct the loop integrand, we use…
We consider the four-graviton amplitudes in CHL constructions providing four-dimensional N=4 models with various numbers of vector multiplets. We show that in these models the two-loop amplitude has a prefactor of d^2R^4. This implies a non-renormalisation theorem for the R^4 term, which forbids the appearance of a three-loop ultraviolet divergence in four dimensions in the four-graviton amplitude. We connect the special nature of the R^4 term to the U(1) anomaly of pure N=4 supergravity.
We consider the D^6 R^4 term which is the leading contribution in the low momentum expansion of the three loop, four graviton amplitude in maximal supergravity. We calculate the moduli dependent coefficient of this term in 11 dimensional supergravity compactified on T^2. Only diagrams that involve the Mercedes skeleton contribute resulting in a compact expression involving an integral over 6 Schwinger parameters. We express this integral as an integral over the moduli of an auxiliary T^3. This includes an SL(3,Z) invariant integral over the shape moduli of the T^3. We discuss the renormalizati…
