Geodesic bulk diagrams on the Bruhat–Tits tree

Geodesic bulk diagrams were recently shown to be the geometric objects which compute global conformal blocks. We show that this duality continues to hold in $p$-adic $\mathrm{AdS}/\mathrm{CFT}$, where the bulk is replaced by the Bruhat–Tits tree, an infinite regular graph with no cycles, and the boundary is described by $p$-adic numbers, rather than reals. We apply the duality to evaluate the four-point function of scalar operators of generic dimensions using tree-level bulk diagrams. Relative to standard results from the literature, we find intriguing similarities as well as significant simplifications. Notably, all derivatives disappear in the conformal block decomposition of the four-point function. On the other hand, numerical coefficients in the four-point function as well as the structure constants take surprisingly universal forms, applicable to both the reals and the $p$-adics when expressed in terms of local zeta functions. Finally, we present a minimal bulk action with nearest neighbor interactions on the Bruhat–Tits tree, which reproduces the two-, three-, and four-point functions of a free boundary theory.

Figure 1

Boundary points $x_i$ in the s-channel configuration. Solid lines are geodesics on the Bruhat–Tits tree, tracing the path joining the four points together. Bulk points $c_1$, $c_2$ are uniquely fixed once $x_i$ are specified. The conformal cross-ratio in (30) is given by $u=p^{-d(c_1,c_2)}$ where $d(c_1,c_2)$ is the graph distance between points $c_1$ and $c_2$. In the $s$-channel configuration, $u<1$.

Figure 2

A geodesic subway diagram. Bulk point $a$ runs along the geodesic joining $x_1$ with $x_2$, and $b$ runs along the geodesic joining $x_3$ with $x_4$. Colors differentiate the individual propagators in (56).

Figure 3

Geodesic subway diagram ${\mathcal{W}}_{\mathrm{\Delta }}^T$, with exchange of a scalar in the (13)(24) channel. The bulk point $a$ runs along the geodesic joining $x_1$ with $x_3$, and $b$ runs along the geodesic joining $x_2$ with $x_4$, while a scalar of dimension $\mathrm{\Delta }$ is exchanged between $a$ and $b$. Colors differentiate the individual propagators found in ${\mathcal{W}}_{\mathrm{\Delta }}^T$ in (81).

Figure 4

A graphical method of representing terms in (d5)–(d6). Here and in Fig. 5, a solid line means that the amplitude should include a factor of the propagator between the endpoints of that line, whereas a dashed line means that we are dividing by that propagator. When a combination of solid and dashed lines is labeled with a power $\delta$, like $\mathrm{\Delta }-{\mathrm{\Delta }}_{12}$, it means that each step along this combinations of lines is weighted by a factor of $p^{-\delta }$. To improve readability we use $u^{\delta }$ instead of $\delta$ to label combined lines between $c_1$ and $c_2$.

Figure 5

Subway diagrams leading to summands in (d7).