Cutting the cylinder into squares: the square form factor

Abstract

In this article we present a method for constructing two-point functions in the spirit of the hexagon proposal, which leads us to propose a "square form factor". Since cutting the square gives us two squares, we can write a consistency condition that heavily constrains such form factors. In particular, we are able to use this constraint to reconstruct the Gaudin through the forest expansion of the determinant appearing in its definition. We also use this procedure to compute the norm of off-shell Bethe states for some simple cases.

Norms and scalar products forAdS3

“…or one can follow [38] to compute on-shell norms by expanding the norms into different powers of the length of the spin chain…”

“…In this section we adapt the method described in [88] and [89] to the present case. As the expressions appearing during the rest of this article usually involve several products, we will use the following shorthand notations when need to multiply over sets of rapidities…”

“…One can use equation (3.2) in two different ways, one can either follow [88] to compute off-shell scalar products by expanding the scalar products into different powers of α and δ functions (3.5) or one can follow [89] to compute on-shell norms by expanding the norms into different powers of the length of the spin chain…”

“…In this appendix we are going to use brute force and the method described in [89] to compute off-shell norms for states with a low number of magnons and a general on-shell norm for any number of magnons, both in the pure R-R case. As described in section 4, we have to distinguish here between symmetric or asymmetric norms depending on the imaginary part of the inhomogeneities θ i , i = 1, ..., L, corresponding to the rapidity of the so-called frame particles (see section 2).…”

“…Appendix B collects some intermediate steps in the computation of the scalar product. Appendix C contains a computation of the norm of on-shell Bethe states using a variation of the composite model trick developed in [38].…”