Hilbert-space and operator interpretation of three-dimensional QFTs

Determine the Hilbert space in which a three-dimensional $\mathcal{N}=2$ quantum field theory is represented as a state or operator, and explain precisely how this operator interpretation arises from the associated four-dimensional quantum field theory.

Background

The Sb3S^3_b partition function of a three-dimensional N=2\mathcal{N}=2 theory is interpreted as a wave function, and a theory with two global symmetries is correspondingly proposed to define an operator on a Hilbert space. The text then explicitly identifies unresolved conceptual questions concerning the nature of that Hilbert space and the meaning of treating a QFT itself as an operator rather than merely as a theory acting on a Hilbert space.

References

The considerations so far have been purely mathematical, and many questions remain unanswered. First of all, what is this ``certain Hilbert space'', and how does it arise? What does it mean for a 3d QFT to be a state (an operator) in a Hilbert space?

— Geometries of Quantum Field Theories  (2609.28210 - Yamazaki, 23 Sep 2026) in Chapter 5, Section “3d Theories as Domain Walls,” subsection “Domain Wall Operator”