We present and prove a formula for the MHV scattering amplitude of n gravitons at tree level.Some of the more interesting features of the formula, which set it apart as being significantly different from many more familiar formulas, include the absence of any vestigial reference to a cyclic ordering of the gravitons-making it in a sense a truly gravitational formula, rather than a recycled Yang-Mills result, and the fact that it simultaneously manifests both S n−2 symmetry as well as large-z behavior that is O(1/z 2 ) term-by-term, without relying on delicate cancellations.The formula is seemingly related to others by an enormous simplification provided by O(n n ) iterated Schouten identities, but our proof relies on a complex analysis argument rather than such a brute force manipulation. We find that the formula has a very simple link representation in twistor space, where cancellations that are non-obvious in physical space become manifest.

Evidence has recently emerged for a hidden symmetry of planar scattering amplitudes in N 4 superYang-Mills theory called dual conformal symmetry. At weak coupling the presence of this symmetry has been observed through five loops, while at strong coupling the symmetry has been shown to have a natural interpretation in terms of a T-dualized AdS 5 . In this paper we study dual conformally invariant off-shell four-point Feynman diagrams. We classify all such diagrams through four loops and evaluate 10 new offshell integrals in terms of Mellin-Barnes representations, also finding explicit expressions for their infrared singularities.

We show that for a class of model Hamiltonians for which certain trial quantum Hall wavefunctions are exact ground states, there is a single spectral density function which controls all two-point correlation functions of density, current and stress tensor components. From this we show that the static structure factors of these wavefunctions behaves at long wavelengths as s4k 4 where the coefficient s4 is directly related to the shift: s4 = (S − 1)/8.

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