Celestial OPE in self-dual gravity

In this paper we compute the celestial operator product expansion between two outgoing positive helicity gravitons in the self-dual gravity. It has been shown that the self-dual gravity is a $w_{1+\infty }$-invariant theory whose scattering amplitudes are one-loop exact with all positive helicity gravitons. Celestial $w_{1+\infty }$ symmetry is generated by an infinite tower of (conformally soft) gravitons which are holomorphic conserved currents. We find that at any given order only the descendants of a finite number of $w_{1+\infty }$ currents contribute to the operator product expansion. This is somewhat surprising but, this is consistent with our earlier analysis based on $w_{1+\infty }$ symmetry alone. The phenomenon of truncation also suggests that in some (unknown) formulation the spectrum of conformal dimensions in the dual two dimensional theory can be bounded from below.

Figure 1

The figure shows the soft currents. The rows and the columns are indexed by the $\overline{s{l}_2(R)}$ weights and the dimension $(\mathrm{\Delta }=k=1,0,-1,-2,\dots )$ of the conformally soft graviton $H^k(z,\overline{z})$, which generates the currents sitting in a row, respectively. $\overline{s{l}_2(R)}$ acts horizontally along a row and $s{l}_2(R{)}_V$ acts vertically along a column. In this way they generate the whole symmetry algebra starting from the current $H_{\frac{1}{2}}^1(z)$ on the top-left corner.