On Higher-dimensional Carrollian and Galilean Conformal Field Theories

TL;DR

<3-5 sentence high-level summary>This work examines Carrollian and Galilean conformal field theories in dimensions $d>2$, developing highest-weight representations (HWR) for the Carrollian and Galilean conformal algebras (CCA and GCA) and revealing that finite-dimensional rotation representations are indecomposable multiplets joined by boosts. It provides a comprehensive classification of chain representations and analyzes two- and three-point correlators via Ward identities, uncovering novel features such as time dependence in CCFT correlators, multi-level structures, and a lack of universal selection rules due to indecomposability. The authors further relate covariant and contravariant tensor representations and connect scale-spin constructions to the multiplet framework, with implications for non-Lorentzian bootstrap, OPE, and quantization in higher dimensions. These results illuminate the rich representation theory and constrained dynamics of CCFT/GCFT beyond $d=2$, and lay groundwork for future holographic and bootstrap studies in non-relativistic and ultra-relativistic conformal systems.

Abstract

In this paper, we study the Carrollian and Galilean conformal field theories (CCFT and GCFT) in $d>2$ dimensions. We construct the highest weight representations (HWR) of Carrollian and Galilean conformal algebra (CCA and GCA). Even though the two algebras have different structures, their HWRs share similar structure, because their rotation subalgebras are isomorphic. In both cases, we find that the finite dimensional representations are generally reducible but indecomposable, and can be organized into the multiplets. Moreover, it turns out that the multiplet representations in $d>2$ CCA and GCA carry not only the simple chain structure appeared in logCFT or $2d$ GCFT, but also more generally the net structures. We manage to classify all the allowed chain representations. Furthermore we discuss the two-point and three-point correlators by using the Ward identities. We mainly focus on the two-point correlators of the operators in chain representations. Even in this relative simple case, we find some novel features: multiple-level structure, shortage of the selection rule on the representations, undetermined 2-pt coefficients, etc.. We find that the non-trivial correlators could only appear for the representations of certain structure, and the correlators are generally polynomials of time coordinates for CCFT (spacial coordinates for GCFT), whose orders depend on the levels of the correlators.

Figures (21)

• Figure 1: Structure of Galilean conformal algebra.
• Figure 2: The vector representation of CCA rotation group. The meaning of this Young diagram structure will be introduced in section \ref{['subsec:TensorRepsYoungTableau']}.
• Figure 3: The vector representation of CCA by taking the limit from vector representation of $SO(4)$.
• Figure 4: The rank-2 tensor representation of CCA.
• Figure 5: The rank-3 tensor representation of CCA. The upper panel shows the decomposition in terms of Young diagram, with the number in the box representing the spacial indices and temporal indices respectively. The lower panel represents the same decomposition, but now the subsectors are labeled by the representations of $SO(3)$ directly, where $(j)$ labels the spin-$j$ representation of dimension $2j+1$. The last summand $(0)\to (1)$ is isomorphic to the dual vector $V^{\vee}$. Since $ISO(d-1)$ is not in $SU(d)$, the dual of $V$ is not isomorphic to $V$ itself.

Theorems & Definitions (1)

• Theorem 1