Scalar conformal primary fields in the Brownian loop soup

Abstract

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 $(\Delta +k/3,\Delta +k^{\prime }/3)$, for all non-negative integers $k,k^{\prime }$ satisfying $|k-k^{\prime }|=0$ 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 $⟨{\mathcal{O}}_{\beta }(z_1){\mathcal{O}}_{-\beta }(z_2)\mathcal{E}(z_3)\mathcal{E}(z_4)⟩$ and analyze its conformal block expansion. The operator product expansions of $\mathcal{E}\times \mathcal{E}$ and $\mathcal{E}\times {\mathcal{O}}_{\beta }$ produce 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.

40 pages, 2 figures, clarified the relation to the scaling limit of critical percolation, corrected definition (2.3) and equations depending on it, corrected proof of Lemma 2.2, added Lemma A.2

References in corpus (7)

• Two-Dimensional Critical Percolation: The Full Scaling Limit
• Two-dimensional O(n) models and logarithmic CFTs
• Conformal Correlation Functions in the Brownian Loop Soup
• Twist operator correlation functions in O(n) loop models
• Random walk loop soups and conformal loop ensembles
• Exact Correlation Functions in the Brownian Loop Soup
• New Recipes for Brownian Loop Soups

Cited by in corpus (1)

• The Brownian loop soup stress-energy tensor