Spectral determinants for twist field correlators

Twist fields were introduced a few decades ago as a quantum counterpart to classical kink configurations and disorder variables in low dimensional field theories. In recent years they received a new incarnation within the framework of geometric entropy and strong coupling limit of four-dimensional scattering amplitudes. In this paper, we study their two-point correlation functions in a free massless scalar theory, namely, twist-twist and twist-antitwist correlators. In spite of the simplicity of the model in question, the properties of the latter are far from being trivial. The problem is reduced, within the formalism of the path integral, to the study of spectral determinants on surfaces with conical points, which are then computed exactly making use of the zeta function regularization. We also provide an insight into twist correlators for a massive complex scalar by means of the Lifshitz-Krein trace formula.

Figure 1

Generic correlation function of twist operators at insertion points $z_j$ and discontinuity contours $C[z_j,\infty ]$, with eigenfunctions of the corresponding Laplace equation developing twisted periodicity conditions (2.9).

Figure 2

Cut structure from the insertion of twist-twist (left panel) and twist-antitwist operators (right panel) in the complex plane. Below each case is a graphical representation under the regularized form of the map (3.2), when the twist operator at $z=x$ moves to infinity and explicitly exhibited twisted periodicity conditions.