Associate a full 2d CFT to any 4d N=2 SCFT

Establish a general construction that associates to every four-dimensional N=2 superconformal field theory T a non-chiral two-dimensional conformal field theory C[T] whose chiral algebra coincides with the vertex operator algebra V[T] provided by the SCFT/VOA correspondence, thereby upgrading the map T ↦ V[T] to T ↦ C[T].

Background

The SCFT/VOA correspondence canonically associates a vertex operator algebra V[T] to any 4d N=2 SCFT T, capturing a protected sector via cohomological reduction. While this has proved powerful, it only supplies a chiral algebra, not a full non-chiral 2d CFT. The authors motivate and develop a 2d/2d correspondence in which compactification of T on S^2 yields a non-unitary 2d CFT C[T] whose chiral algebra is V[T], and compactification on a Riemann surface produces a unitary 2d (2,2) theory whose S^2 partition function matches correlators of C[T] on that surface.

The open question asks for a universal, purely two-dimensional construction that, for any 4d N=2 SCFT, produces the full CFT C[T] upgrading the SCFT/VOA correspondence from a chiral algebra to a complete non-chiral theory.