Loop Equation in D=4, N=4 SYM and String Field Equation on AdS_5 \times S^5

TL;DR

This paper derives a loop equation in 4D N=4 SYM involving scalar, gaugino, and gauge fields, and compares its most singular part with the string Hamiltonian constraint on AdS_5 x S^5, linking gauge theory and string theory.

Contribution

It formulates a detailed loop equation in N=4 SYM including fermionic fields and connects it to string equations on AdS_5 x S^5, advancing the gauge-string duality understanding.

Findings

Derived the loop equation including scalar and fermionic fields.

Matched the singular part of the loop equation with string Hamiltonian constraint.

Provided a quadratic order approximation in fermionic coordinates.

Abstract

We consider the loop equation in four-dimensional N=4 SYM, which is a functional differential equation for the Wilson loop W(C) and expresses the propagation and the interaction of the string C. Our W(C) consists of the scalar and the gaugino fields as well as the gauge field. The loop C is specified by six bosonic coordinates y^i(s) and two fermionic coordinates \zeta(s) and \eta(s) besides the four-dimensional spacetime coordinates x^\mu(s). We have successfully determined, to quadratic order in \zeta and \eta, the parameters in W(C) and the loop differential operator so that the equation of motion of SYM can be correctly reproduced to give the non-linear term of W(C). We extract the most singular and linear part of our loop equation and compare it with the Hamiltonian constraint of the string propagating on AdS_5 \times S^5 background.