The large N limit of M2-branes on Lens spaces

TL;DR

The paper derives the exact localized partition function for 3d $\mathcal{N}=2$ ABJM theory on Lens spaces $L(p,1)=S^3/\mathbb{Z}_p$, revealing a matrix-model structure with a sum over flat connections. In the large-$N$ M-theory limit with fixed Chern-Simons level $k$, a uniform flat-connection sector dominates, the long-range forces cancel, and the remaining local action yields a universal eigenvalue distribution independent of $p$, giving a large-$N$ free energy $F=\frac{1}{p}F_{S^3}=\frac{\pi\sqrt{2k}}{3p}N^{3/2}$, in agreement with the gravity dual AdS$_4/\mathbb{Z}_p$. Numerical saddle-point studies support the analytic saddle results and clarify the eigenvalue structure, including symmetries and density. The work establishes a concrete connection between field theory localization on curved quotients and holographic expectations, and provides explicit expressions for the matter potentials and the full partition function on $L(p,1)$.

Abstract

We study the matrix model for N M2-branes wrapping a Lens space L(p,1) = S^3/Z_p. This arises from localization of the partition function of the ABJM theory, and has some novel features compared with the case of a three-sphere, including a sum over flat connections and a potential that depends non-trivially on p. We study the matrix model both numerically and analytically in the large N limit, finding that a certain family of p flat connections give an equal dominant contribution. At large N we find the same eigenvalue distribution for all p, and show that the free energy is simply 1/p times the free energy on a three-sphere, in agreement with gravity dual expectations.

Figures (4)

• Figure 1: Distribution of eigenvalues for $p=2$, $N=100$.
• Figure 2: Distribution of eigenvalues for $p=1,2,3$ and $N=100$.
• Figure 3: Distribution of eigenvalues for $N=100,~p=2$ and $n=3$, $10$, $20$, $30$, $50$, $70$, $80$, $90$, $97$ from left to right and top to bottom.
• Figure 4: Distribution of eigenvalues for $N=100,~p=3$ and ${\bf m} = {\bf \tilde{m}} = (0,\ldots,0, 1,\ldots, 1, 2,\ldots, 2)$. We see that the eigenvalues distribute in three segments.