Commutation relations for two-sided radial SLE

We study the commutation relation for 2-radial SLE in the unit disc starting from two boundary points. We follow the framework introduced by Dub\'{e}dat. Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE$_\kappa$ for $\kappa\in (0,8)$: it is either a two-sided radial SLE$_\kappa$ with spiral of constant spiraling rate or a chordal SLE$_\kappa$ weighted by a power of the conformal radius of its complement. Namely, for fixed $\kappa$ and starting points, we have exactly two one-parameter continuous families of locally commuting 2-radial SLE. Two-sided radial SLE with spiral is a generalization of two-sided radial SLE (without spiral) and satisfies the resampling property. We also discuss the semiclassical limit of the commutation relation as $\kappa \to 0$. In particular, we show that the limit for the second family with an appropriately chosen power of conformal radius is a chord that minimizes a modified chordal Loewner energy, which is unique only when the endpoints are not antipodal.