Abstract
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--anti-twist 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.
Highlights
Two-dimensional conformal field theory on a Riemann surface R can be reformulated as a theory on a branched covering of the complex plane, where each branch point zi corresponds to an insertion of a conformal primary field VðziÞ [1,2,3], known as the branch point twist field
(with a cut emanating to infinity) on the covering of the original Riemann surface with N consecutive sheets enumerated by l 1⁄4 1; ...; N and parametrized by the single-valued coordinate ζ, ζ 1⁄4 pN ffizffiffiffi−ffiffiffiffizffiffijffi: ð1:1Þ
The scattering amplitudes as strong coupling are given by the (L − 5)point correlation function of branch point twist fields [15,17,18], A2L g21⁄4→∞ hVðzL−5Þ...Vðz1ÞVð0Þi: ð1:12Þ
Summary
Two-dimensional conformal field theory on a (say, hyperelliptic) Riemann surface R can be reformulated as a theory on a branched covering of the complex plane, where each branch point zi corresponds to an insertion of a conformal primary field VðziÞ [1,2,3], known as the branch point twist field. One subtle point that one has to keep in mind is that the introduction of twist fields requires operating with complex rather than real fields so one effectively doubles the number of degrees of freedom to construct a microscopic definition of pentagon operators This is a common trick, successfully employed in many different circumstances, e.g., in the Ising model [16]. Taking the square root at the end reduces this number in half In this manner, the (square of the) scattering amplitudes as strong coupling are given by the (L − 5)point correlation function of branch point twist fields [15,17,18], A2L g21⁄4→∞ hVðzL−5Þ...Vðz1ÞVð0Þi: ð1:12Þ. The appendix contains information on elementary mathematical aspects of q-series relevant to proper analysis of twist-antitwist functions
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