Identify Sym^k(D_n) with the Landau–Ginzburg sector of W_{n,k}

Determine whether the Liouville sector Sym^k(D_n), where D_n = {z in C | Re(z^n) ≤ 10} is the "disk with n stops" and Sym^k(D_n) denotes the sector inside the k-th symmetric power constructed in this work, is isomorphic (as a Liouville sector) to the Landau–Ginzburg sector associated to the superpotential W_{n,k}(z_1, ..., z_k) = ∑_{i=1}^k z_i^n on Sym^k(C).

Background

In the calculations section, the authors consider the covering map w = z^n and define D_n := {Re(z^n) ≤ 10} as a model of a disk with n stops, endowed with a strictly plurisubharmonic potential. For the symmetric power Sym^k(C), they introduce the superpotential W_{n,k}(z_1, ..., z_k) = ∑ z_i^n and use the Hamiltonian flow of Im(W_{n,k}) to displace Lagrangians in Sym^k(D_n), establishing vanishing results for the relevant Fukaya categories.

This setup suggests a Landau–Ginzburg viewpoint in which Sym^k(D_n) could coincide with the Liouville sector determined by the superpotential W_{n,k}. However, the authors explicitly note that they have not proved this identification, leaving open the precise relationship between the sector constructed via their symmetric power/gluing method and the Landau–Ginzburg sector for W_{n,k}.