# Crossing kernel Write a general correlation function of $n$ operators on a genus-$g$ surfaces (in 2d CFT) as$\begin{aligned} G_{g, n}\left(\left\{q_i\right\}\right) & =\sum_{\left\{\mathcal{O}_j\right\}} \mathcal{C}_{\left\{\mathcal{O}_j\right\}} \mathcal{F}\left(\left\{P_j\right\} |\left\{q_i\right\}\right) \\ & \equiv \int\left[d P_j\right] \rho\left(\left\{P_j\right\}\right) \mathcal{F}\left(\left\{P_j\right\}|\left\{q_i\right\}\right),\end{aligned}$where $\{\mathcal{O}_j\}$ are the internal operators that contribute, $\mathcal{C}_{\{\mathcal{O}_j\}}$ are the corresponding products of OPE coefficients and $\mathcal{F}\left(\left\{P_j\right\}|\left\{q_i\right\}\right)$ (known as [[0031 Conformal block|conformal blocks]]) encode the contribution of all descendants of the operators $\{\mathcal{O}_j\}$. This can be expanded in a different channel,$\begin{aligned} G_{g, n}\left(\left\{q_i\right\}\right) & =\sum_{\left\{\mathcal{O}_k\right\}} \widetilde{\mathcal{C}}_{\left\{\mathcal{O}_k\right\}} \tilde{\mathcal{F}}\left(\left\{R_k\right\} |\left\{\tilde{q}_i\right\}\right) \\ & =\int\left[d R_k\right] \tilde{\rho}\left(\left\{R_k\right\}\right) \widetilde{\mathcal{F}}\left(\left\{R_k\right\}|\left\{\tilde{q}_i\right\}\right),\end{aligned}$and they must be equal to each other! Defining the *crossing kernel* to via$\mathcal{F}\left(\left\{P_j\right\} |\left\{q_i\right\}\right)=\int\left[d R_k\right] \mathbb{K}_{\left\{R_k\right\}\left\{P_j\right\}} \tilde{\mathcal{F}}\left(\left\{R_k\right\}|\left\{\tilde{q}_j\right\}\right),$crossing symmetry then implies$\tilde{\rho}\left(\left\{R_k\right\}\right)=\int\left[d P_j\right] \mathbb{K}_{\left\{R_k\right\}\left\{P_j\right\}} \rho\left(\left\{P_j\right\}\right).$ The discussion so far has been general. But as argued in [[1989#Moore, Seiberg]], modular covariance of one-point function on the torus and four-point crossing on the plane/sphere are enough to ensure all crossing symmetry. So the elementary crossing kernels that one really needs to consider are *fusion kernels* (corresponding to four-point crossing the sphere) and *modular kernels* (corresponding to one-point modular transformation on the torus). The benefit of considering crossing kernels is that there is no need to know the conformal blocks explicitly, which can be a great advantage when dealing with some abstract problems. ## Refs - [[1999#Ponsot, Teschner]]: explicit integral fusion kernel for $c \in \mathbb{C} \backslash(-\infty, 1]$ - [[2000#Ponsot, Teschner]]: an explicit integral transformation involving a distributional kernel - [[2012#Teschner, Vartanov]]: new integral representation - [[2013#Iorgov, Lisovyy, Tykhyy]]: $c=1$ non-integral formula - [[2023#Ribault, Tsiares]]: $c=25$ and conjecture for $c<1$ - [[2024#Roussillon]]: series representation of the Virasoro kernels for all irrational $c$ (different expressions for $c \in \mathbb{C} \backslash(-\infty, 1]$ and $c<1$) ## Related - [[0602 Moore-Seiberg construction]] - [[0406 Cardy formula]] \[*Note: most notations and reviews come from from [[2019#Collier, Maloney, Maxfield, Tsiares]].*\]