Scalar Conformal Primary Fields in the Brownian Loop Soup

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The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity \(\lambda >0\), with central charge \(c=2 \lambda \). Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary “layering vertex operators” \(\mathcal {O}_{\beta }\) with dimensions \((\Delta , \Delta )\), with \(\Delta = \frac{\lambda }{10}(1-\cos \beta )\), that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions \((\Delta + k/3, \Delta + k'/3)\), for all non-negative integers \(k, k'\) satisfying \(|{k-k'}| = 0 \mod 3\). In this paper we introduce the edge counting field \({\mathcal {E}}(z)\) that counts the number of loop boundaries that pass close to the point z. We rigorously prove that the n-point functions of \({\mathcal {E}}\) are well defined and behave as expected for a conformal primary field with dimensions (1/3, 1/3). We analytically compute the four-point function \(\left\langle {\mathcal {O}_{\beta }(z_1) \mathcal {O}_{-\beta }(z_2) \mathcal {E}(z_3) \mathcal {E}(z_4)}\right\rangle \) and analyze its conformal block expansion. The operator product expansions of \(\mathcal {E} \times \mathcal {E}\) and \(\mathcal {E} \times \mathcal {O}_{\beta }\) contain higher-order edge operators with “charge” \(\beta \) and dimensions \((\Delta + k/3, \Delta + k/3)\). Hence, we have explicitly identified all scalar primary operators among the new set mentioned above. We also re-compute the central charge by an independent method based on the operator product expansion and find agreement with previous methods.