The hexagon in the mirror: the three-point function in the SoV representation

TL;DR

This paper derives an integral formula for three-point functions in N=4 super Yang-Mills theory using the separation of variables approach, revealing new symmetries and connections to the quantum spectral curve.

Contribution

It introduces a novel integral expression for three-point functions in the su(2) sector, utilizing a hexagonal boundary mapping and the SoV method, with new insights into the SoV basis and overlaps.

Findings

Derived a multiple-integral expression for three-point functions.

Uncovered symmetry under 90° rotation in the six-vertex model.

Constructed explicit SoV basis with twisted boundary conditions.

Abstract

We derive an integral expression for the leading-order type I-I-I three-point functions in the $\mathfrak{su}(2) $-sector of $\mathcal{N}=4$ super Yang-Mills theory, for which no determinant formula is known. To this end, we first map the problem to the partition function of the six vertex model with a hexagonal boundary. The advantage of the six-vertex model expression is that it reveals an extra symmetry of the problem, which is the invariance under 90$^{\circ}$ rotation. On the spin-chain side, this corresponds to the exchange of the quantum space and the auxiliary space and is reminiscent of the mirror transformation employed in the worldsheet S-matrix approaches. After the rotation, we then apply Sklyanin's separation of variables (SoV) and obtain a multiple-integral expression of the three-point function. The resulting integrand is expressed in terms of the so-called Baxter polynomials, which is closely related to the quantum spectral curve approach. Along the way, we also derive several new results about the SoV, such as the explicit construction of the basis with twisted boundary conditions and the overlap between the orginal SoV state and the SoV states on the subchains.