Amplitudes in the N=4 supersymmetric Yang-Mills theory from quantum geometry of momentum space

We discuss multiloop maximally helicity violating amplitudes in the N=4 supersymmetric Yang-Mills theory in terms of effective gravity in the momentum space with IR regulator branes as degrees of freedom. Kinematical invariants of external particles yield the moduli spaces of complex or Kahler structures which are the playgrounds for the Kodaira-Spencer or Kahler type gravity. We suggest fermionic representation of the loop maximally helicity violating amplitudes in the N=4 supersymmetric Yang-Mills theory assuming the identification of the IR regulator branes with Kodaira-Spencer fermions in the B model and Lagrangian branes in the A model. The two-easy mass box diagram is related to the correlator of fermionic currents on the spectral curve in the B model or hyperbolic volume in the A model and it plays the role of a building block in the whole picture. The Bern-Dixon-Smirnov-like ansatz has the interpretation as the semiclassical limit of a fermionic correlator. It is argued that fermionic representation implies a kind of integrability on the moduli spaces. We conjecture the interpretation of the reggeon degrees of freedom in terms of the open strings stretched between the IR regulator branes.