Free box^k Scalar Conformal Field Theory

TL;DR

The paper analyzes non-unitary free scalar CFTs defined by L ~ φ†□^kφ in d≥2, focusing on the singlet sector of U(N) or O(N). It uncovers a rich representation-theoretic structure, including zero-norm primary/descendant states and extension operators that couple to standard Verma modules, and shows how these features shape the OPE and conformal blocks. For □^2, it provides a detailed account of two Regge towers, their conservedness, and the emergence of extended modules in d=3 and d=6, then generalizes to □^k with hs_k symmetry, predicting module linkings and finite-theory regimes in even dimensions d≤2k. The results illuminate non-unitary CFT structures with potential AdS/dS duals and motivate further work on bulk duals, curvature couplings, and possible Liouville-like or log-CFT variants. Overall, the work broadens the landscape of higher-derivative conformal theories, offering detailed spectra, OPE structures, and symmetry algebras that may inform quantum gravity and holography in non-unitary settings.

Abstract

We consider the generalizations of the free U(N) and O(N) scalar conformal field theories to actions with higher powers of the Laplacian box^k, in general dimension d. We study the spectra, Verma modules, anomalies and OPE of these theories. We argue that in certain d and k, the spectrum contains zero norm operators which are both primary and descendant, as well as extension operators which are neither primary nor descendant. In addition, we argue that in even dimensions d <= 2k, there are well-defined operator algebras which are related to the box^k theories and are novel in that they have a finite number of single-trace states.

Figures (8)

• Figure 1: Spectrum of single trace primaries in the case $d>6$ for the $\square^2$ theory. The two Regge trajectories are clearly visible, with $b=0$ on the bottom, $b=1$ on the top. Unfilled circles are the operators satisfying conservation conditions, with the top trajectory being singly conserved and the bottom trajectory being triply conserved. Filled circles are the operators satisfying no conservation condition. Blue means the two point function has positive norm, red means it has negative norm. The dotted line is the unitarity bound (not shown is the $s=0$ bound $\Delta\geq {d\over 2}-1$). The case $d=5$ looks the same except that the $b=1$ scalar now has positive norm. The cases $d=2,4$ are discussed in section \ref{['sectionfinitebox2']}, the $d=3,6$ cases in section \ref{['sec:nuances']}.
• Figure 2: Spectrum of single trace primaries in the finite case $d=4,2$ for the $\square^2$ theory. Blue states have positive norm, red states have negative norm, and green states are zero norm null states which are projected out. (Unfilled circles are the operators satisfying conservation conditions, filled circles are the operators satisfying no conservation condition, and the dotted line is the unitarity bound.)
• Figure 3: A diagram of the generic scalar Verma modules in the $\square^2$ theory. There are two separate Verma modules corresponding to $j_{0}^{(0)}$ (left) and $j_{0}^{(1)}$ (right) in the theory. The blue states are the highest-weight states and are primary; they are annihilated by $K$. One can move around the module by raising with a $P$ (red arrow) or lowering with a $K$ (blue arrow). Of course, the module continues up and to the right.
• Figure 4: A diagram of the scalar Verma modules in the $d=6$$\square^2$ theory. There is one generalized Verma module with an extension, the extension being a projective Verma module as shown on the right. The single Verma module corresponds to $j_{0}^{(0)}$ (on the left). The blue state is the highest-weight state and is primary; it is annihilated by $K$. One can move around the module by raising with a $P$ (red arrow) or lowering with a $K$ (blue arrow). The state in green is the null state $\square j_{0}^{(0)}$; it is both a primary and a descendant. The state in red is an extension state $\tilde{j}_{0}^{(1)}$; it is neither a primary nor a descendant. Of course, the modules continue up and to the right.
• Figure 5: Spectrum of primaries in the cases $d=6$ (left) and $d=3$ (right) where modules join in the $\square^2$ theory. Blue states have positive norm, red states have negative norm, and green states are zero-norm states which are both primary and descendant and get paired with extension states. The mixing of two modules is denoted by an arrow. (The bottom row's unfilled circles are the operators satisfying triple conservation conditions, the top row's unfilled circles are the operators satisfying single conservation conditions, filled circles are the operators satisfying no conservation condition, and the dotted line is the unitarity bound.)