Scattering Amplitudes and Wilson Loops in Twistor Space

A Twistorial Approach to Integrability in N=4 SYM

“…We mostly follow the notation and conventions used in [3,4], where further details about the formalism, for example the construction of the twistor action, can be found. One starts with the complexification of conformally compactified Minkowski space CM # , which admits a description as a complex quadric surface in CP 5 .…”

“…The function h(x, λ) is a holomorphic frame that depends smoothly on the line X, see for instance (2.32ff.) of [3]. One can define an action for the supertwistor field A(Z) as the sum of a holomorphic Chern-Simons action, introduced in [6],…”

“…Note that the Graßmann variables θ αa l span an independent Graß-mann algebra at each site. The parallel propagators U (Z l , Z l+1 ) are given by concatenating two frames as H(x, θ, λ l )H −1 (x, θ, λ l+1 ), where the H are supersymmetric generalizations of the frames h of (14), see (2.54ff ) of [3]. These parallel propagators U can be expressed as path-ordered exponentials of A and serve to guarantee the gauge invariance of the operator O I , see (4.10) of [4].…”

“…In our calculation this 2 One way of obtaining the coefficients A, B and C is to set the flavor indices to specific values (for example, (abcd) = (1213) and (a ′ b ′ c ′ d ′ ) = (3424) will yield −B). 3 We stress that whenever we refer to 1/0-like divergences we are referring to the unphysical divergences that arise from evaluating 1/(ρ1 − ρ2)(ρ3 − ρ4) on the support of the delta functions on the fibers and not to the UV divergences that arises from z → x1 or z → x2. amounts to making the following replacements…”

Twistors, harmonics and holomorphic Chern-Simons

“…In ref. [17] it was shown using the loop equations satisfied by the supersymmetric Wilson loops that their integrand satisfies the same recursion relation as the loop level extension of the BCFW recursion presented in ref. [13].…”