Colour-twist operators. Part I. Spectrum and wave functions

Abstract

We introduce a new class of operators in any theory with a 't Hooft large-N limit that we call colour-twist operators. They are defined by twisting the colourtrace with a global symmetry transformation and are continuously linked to standard, untwisted single-trace operators. In particular, correlation functions between operators that are twisted by an R-symmetry of N = 4 SYM extend those in the γ-deformed theory. The most general deformation also breaks the Lorentz symmetry but preserves integrability in the examples we consider. In this paper, we focus on colour-twist operators in the fishnet model. We exemplify our approach for the simplest colour-twist operators with one and two scalar fields, which we study non-perturbatively using field-theoretical as well as integrability methods, finding a perfect match. We also propose the quantisation condition for the Baxter equation appearing in the integrability calculation in the fishnet model. The results of this paper constitute a crucial step towards building the separation of variable construction for the correlation functions by means of the Quantum Spectral Curve approach.

Separation of variables in AdS/CFT: functional approach for the fishnet CFT

“…single-trace operator CFT wave function Q-functions (1.3) For the simplest family of nontrivial operators, those with length one in the presence of twists [48], the map between wave functions and Q-functions was found by the present authors with A. Sever in [49].…”

“…single-trace operator CFT wave function Q-functions (1.3) For the simplest family of nontrivial operators, those with length one in the presence of twists [48], the map between wave functions and Q-functions was found by the present authors with A. Sever in [49]. Generalising this result to all states is currently an important open problem, but the general methods of [4,7,20] give a clear indication of how to construct the SoV transformation at least formally.…”

“…Another important technical ingredient used in this paper are quasi-periodic, or twisted, boundary conditions along the spin chain, which break the conformal symmetry of local operators [48]. The twists are crucial for the SoV, since they lift the degeneracies of the spectrum and guarantee that solutions of the QSC are identified one-to-one with the eigenstates of the spin chain.…”

“…The twists are crucial for the SoV, since they lift the degeneracies of the spectrum and guarantee that solutions of the QSC are identified one-to-one with the eigenstates of the spin chain. We use the construction of the colour-twist [48], which is particularly simple for the fishnet model, and introduces the twists directly at the level of Feynman graphs.…”

Operator Product Expansion for Form Factors

“…The starting point of this approach is the dual formulation of FFs in terms of wrapped polygonal Wilson loops [3,[6][7][8][9], as shown in figure 1.1. These Wilson loops are periodic and are defined in the planar theory with a periodicity constraint that twists the color trace by a spacetime translation, see [8,12]. The OPE for these wrapped polygonal Wilson loops mirrors the pentagon operator product Figure 1.1: In the planar limit, an MHV form factor is equal to the expectation value of a wrapped polygonal Wilson loop, multiplied by the tree-level form factor.…”