Reconciling single- and double-wall Hilbert spaces in the large-circle limit

Establish how locality renders the physics near a single Maldacena–Ludwig wall independent of the number of walls on a circle by explicitly demonstrating the mechanism that reconciles the two Hilbert space constructions—H_{two walls} = H_{\psi,\text{NS}} \oplus H_{\psi,\text{R}} and H_{one wall} = H_{\psi_C,1} \oplus H_{\psi_C,2}—in the limit of infinite circumference.

Background

The paper primarily analyzes a system on a circle with two Maldacena–Ludwig walls, for which the Hilbert space and state interpretation are straightforward. Appendix A discusses the alternative of putting a single wall on the circle, where the currents obey a twisted boundary condition and the fermionic description involves fields \psi_C with boundary conditions leading to the total Hilbert space H_{\text{one wall}} = H_{\psi_C,1} \oplus H_{\psi_C,2}.

Although locality suggests that, in the large-circumference limit, local physics near a single wall should not depend on whether there is one or two walls on the circle, the authors note that the two Hilbert space constructions appear very different. They explicitly state that it is not immediately clear how this expected independence is achieved, highlighting an unresolved technical issue in reconciling the two descriptions.