Abstract
We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.
Highlights
We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup
These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators
Our main result is the derivation of new correlation functions of exponentials of the layering operators in the BLS
Summary
Our main result is the derivation of new correlation functions of exponentials of the layering operators in the BLS. We obtain the two-point function of these operators in the upper half-plane with the boundary condition that any loop intersecting the real axis is erased, and the four-point function in the full plane. The four-point function of these operators in the full plane C (section 5) is given by. All n-point functions in the full plane vanish unless a (periodic) charge conservation condition is satisfied:. We find an apparently infinite new set of primary operators of integer spin in the BLS, with conformal dimensions. Weights for Brownian loops to encircle various subsets of points in the plane or upper half-plane can be found in (B.1), (B.3), and (5.8)
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