Open intersection numbers and free fields

A complete set of the Virasoro and W-constraints for the Kontsevich-Penner model, which conjecturally describes intersections on moduli spaces of open curves, was derived in our previous work. Here we show that these constraints can be described in terms of free bosonic fields with twisted boundary conditions, which gives a modification of the well-known construction of the $W^{(3)}$ algebra in conformal field theory. This description is natural from the point of view of the spectral curve description, and should serve as a new important ingredient of the topological recursion/Givental decomposition.