Field Theory of Primaries in W_N Minimal Models

TL;DR

This work constructs a nonlinear field theory for primary operators in $W_N$ minimal models at large $N$, using a collective Hamiltonian with a $1/N$ interaction $G=1/N$ and an emergent winding-number extension to reproduce the full primary spectrum and generate multi-trace primaries within a Fock-space framework. It identifies a subsector where primaries correspond to Schur polynomials in winding variables, fixes interaction coefficients by matching three-point functions, and demonstrates that diagonalization of the Hamiltonian yields exact conformal dimensions for the primaries. A close connection to matrix-vector models is uncovered, with a geometric interpretation in terms of loop joining/splitting and an emergent extra dimension that renders the theory local in the winding-coordinate conjugate. This framework enables a systematic $1/N$ expansion and provides a concrete step toward reconstructing bulk higher-spin dynamics from CFT data, while leaving open issues on derivatives, descendants, null states, and extended geometries for future work.

Abstract

For W_N minimal model CFT's at Large N, we formulate a nonlinear field theory of primary operators. A classification of single-trace operators is given first based on which an interacting field theory operating in Fock space is built. A hamiltonian is constructed with the property that it reproduces exactly the spectrum of conformal dimensions of all the primaries. This field theory is characterized by cubic (and quartic) interactions with G=1/N as an interaction parameter. It is seen that the 1/N nonlinear representation contains the interactions and structure known from Matrix-vector models.

Figures (8)

• Figure 1: Young tableau $\Lambda=(\overline{R},S)$
• Figure 2: The left and right figure are an open loop with winding number 4 and a closed loop with winding number 3, respectively. For $H_3$, we can cut the open loop only at point 4. In addition, the closed loop can be cut at points 1, 2 and 3. On the other hands, for $H_4$, the open loop can be cut at point 1, 2, 3 and 4. Moreover, we can cut the closed loop at point 1, 2 and 3.
• Figure 3: Two figure are open loops with winding number 4 and 2, respectively. They have two distinct ends, $A$ and $B$. For $H_6$, we can cut these open loops at the white points. e.g. points 1, 2, 3 and 4 for the left open loop.
• Figure 4: Young tableau
• Figure 5: Clusters