Amplitudes in N = 4 SYM from the Quantum geometry of the Momentum Space

We discuss loop MHV amplitudes in the N = 4 SYM theory in terms of the effective gravity in the momentum space with the IR regulator branes as degrees of freedom. Rapidities of external particles yield the moduli space of complex structures providing the playground for the Kodaira-Spencer(KS) type gravity. We suggest the fermionic representation for the loop MHV amplitudes in the N = 4 SYM theory assuming the identification of KS fermions with the IR regulator branes in the B model. The two-easy mass box diagram is treated as the four fermion correlator on the spectral curve and it plays the role of a building block in the whole picture. The BDS anzatz has the interpretation as semiclassical limit of a fermionic correlator. It is argued that fermionic representation implies integrability on the moduli spaces which fixes the dependence of the amplitudes on the cross-ratios of the external momenta.