Exact overlaps for all integrable two-site boundary states of $\mathfrak{gl}(N)$ symmetric spin chains

TL;DR

This paper derives explicit formulas for overlaps between Bethe eigenstates and integrable boundary states in $rak{gl}(N)$ symmetric spin chains, covering all two-site boundary states and applicable to models in high-energy physics.

Contribution

It provides the first comprehensive set of overlap formulas for all integrable two-site boundary states in $rak{gl}(N)$ spin chains, including new conjectures for $rak{sp}(N)$ cases.

Findings

Derived formulas for $rak{gl}(M)igoplusrak{gl}(N-M)$ boundary states.

Conjectured formulas for $rak{sp}(N)$ boundary states.

Applicable to $SO(6)$ and $SU(4)$ spin chains in super Yang-Mills and ABJM theories.

Abstract

We find closed formulas for the overlaps of Bethe eigenstates of $\mathfrak{gl}(N)$ symmetric spin chains and integrable boundary states. We derive the general overlap formulas for $\mathfrak{gl}(M)\oplus\mathfrak{gl}(N-M)$ symmetric boundary states and give a well-established conjecture for the $\mathfrak{sp}(N)$ symmetric case. Combining these results with the previously derived $\mathfrak{so}(N)$ symmetric formula, now we have the overlap functions for all integrable boundary states of the $\mathfrak{gl}(N)$ spin chains which are built from two-site states. The calculations are independent from the representations of the quantum space therefore our formulas can be applied for the $SO(6)$ and the alternating $SU(4)$ spin chains which describe the scalar sectors of $\mathcal{N}=4$ super Yang-Mills and ABJM theories which are important application areas of our results.