Multiloop SYM and supergravity amplitudes in pure spinor superspace

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We present a string-inspired method to construct supersymmetric loop integrands for scattering amplitudes in ten-dimensional SYM and supergravity. The kinematic dependence is described in pure spinor superspace and determined from the BRST-symmetry of the underlying pure spinor superstring. We focus on five-point examples at one- and two-loop order which satisfy the BCJ duality between color and kinematics. Hence, the supergravity integrands follow straightforwardly from a double copy construction of the cubic diagrams in SYM. Preliminary examples and open questions at three-loops are mentioned.