Partition function of free conformal fields in 3-plet representation

TL;DR

This work analyzes free conformal fields in the 3-plet representation of a global $U(N)$ symmetry, formulating the singlet partition function on $S^1\times S^{d-1}$ and probing its large-$N$ behavior. The authors derive exact large-$N expressions for the low-temperature expansion and show that, unlike vector or adjoint cases, the 3-plet expansion becomes asymptotic due to rapid growth in the number of singlet operators, yielding a phase transition at $T_c \sim 1/\log N$ that vanishes as $N\to\infty$. A detailed eigenvalue-density analysis reveals a cubic (in Fourier modes) effective action that drives a first-order transition, with a high-temperature phase scaling as $\mathcal{O}(N^2)$ and a low-temperature phase of $\mathcal{O}(N^0)$. The results generalize to higher $p$-plets and tensor theories, providing insight into the AdS dual spectrum, which in the 3-plet case suggests an abundance of massive modes akin to a tensionless membrane structure. These findings have potential implications for the AdS/CFT correspondence beyond vector and adjoint representations, including connections to the $(2,0)$ tensor multiplet and $AdS_7$ holography.

Abstract

Simplest examples of AdS/CFT duality correspond to free CFTs in d dimensions with fields in vector or adjoint representation of an internal symmetry group dual in the large N limit to a theory of massless or massless plus massive higher spins in AdS_{d+1}. One may also study generalizations when conformal fields belong to higher dimensional representations, i.e. carry more than two internal symmetry indices. Here we consider the case of the 3-fundamental ("3-plet") representation. One motivation is a conjectured connection to multiple M5-brane theory: heuristic arguments suggest that it may be related to an (interacting) CFT of 6d (2,0) tensor multiplets in 3-plet representation of large N symmetry group that has an AdS_7 dual. We compute the singlet partition function Z on S^1 x S^{d-1} for a free field in 3-plet representation of U(N) and analyse its novel large N behaviour. The large N limit of the low temperature expansion of Z which is convergent in the vector and adjoint cases here is only asymptotic, reflecting the much faster growth of the number of singlet operators with dimension, indicating a phase transition at very low temperature. Indeed, while the critical temperatures in the vector (T_c ~ N^a, a >0) and adjoint (T_c ~ 1) cases are finite, we find that in the 3-plet case T_c ~ 1/ log N, i.e. it approaches zero at large N. We discuss some details of large N solution for the eigenvalue distribution. Similar conclusions apply to higher p-plet representations of U(N) or O(N) and also to the free p-tensor theories invariant under [U(N)]^p or [O(N)]^p starting with p=3.

Figures (5)

• Figure 1: Exact eigenvalues for $N=40$, $M=4$ and three values of $\beta$ well below, just below and just above the transition. The right graph represents the plot of the maximal eigenvalue that demonstrates that it jumps at a certain temperature from a value $\simeq{ \pi\over 2}$ to $\pi$.
• Figure 2: Comparison between the edge of the exact eigenvalue distribution at $N=40$ in the case of one harmonic in the potential term in (\ref{['4.7']}) (solid circles) with the solution of (\ref{['5.17']}) (solid line). We also show the high temperature approximation (\ref{['5.18']}) in the one-harmonic case (dashed line).
• Figure 3: The action $S$ in (\ref{['4.3']}) for the $U(2)$ theory as a function of the eigenvalue $\alpha=\alpha_{1}=-\alpha_{2}$. The left plot is for one and the right plot is for ten harmonics included in (\ref{['4.3']}).
• Figure 4: Plot of the measure term $S_M$ in (\ref{['D.5']}) and the potential term $V$ in (\ref{['D.11']}) in the eigenvalue density action at $a_{1}= 3\,N\,z_\Phi(e^{-\beta})=4$. The black line is the total action that has a minimum at the position predicted by (\ref{['5.11']}), i.e. at $u\simeq 0.145$.
• Figure 5: Solution of the system (\ref{['F.7']}) for $u$ as a function of variable $z\equiv z_\Phi$ for $N=40$.