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

TL;DR

This paper explores the geometric and fermionic structures underlying multiloop MHV amplitudes in N=4 SYM theory, proposing a novel interpretation involving quantum geometry, fermionic representations, and integrability on moduli spaces.

Contribution

It introduces a fermionic representation of loop amplitudes, linking IR regulator branes to Kodaira-Spencer fermions and Lagrangian branes, and suggests a new geometric framework for understanding amplitudes.

Findings

Fermionic representation implies integrability on moduli spaces.

Two-mass box diagram related to fermionic currents and hyperbolic volume.

BDS-like ansatz interpreted as semiclassical fermionic correlator limit.

Abstract

We discuss multiloop MHV amplitudes in the N=4 SYM 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(KS) or Kahler type gravity. We suggest fermionic representation of the loop MHV amplitudes in the N=4 SYM theory assuming the identification of the IR regulator branes with KS fermions in the B model and Lagrangian branes in A model. The two-easy mass box diagram is related to the correlator of fermionic currents on the spectral curve in B model or hyperbolic volume in the A model and it plays the role of a building block in the whole picture. The BDS-like anzatz 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.