Large-$N$ Free Energy of Chiral $\mathcal{N}=2$ Chern-Simons-Matter Theories

Abstract

We present the first successful large-$N$ computation of the $S^3$ free energy in chiral $\mathcal{N}=2$ Chern-Simons-matter theories, long believed to evade the universal M2-brane scaling $F_{S^3}\!\sim\!N^{3/2}$. Using a stable numerical continuation method that directly solves the saddle-point equations, we obtain convergent large-$N$ solutions for benchmark chiral quivers, including the holographic dual to AdS$_4\times Q^{1,1,1}/\mathbb{Z}_k$ and $D_3$ models. The resulting free energies precisely match the holographic predictions, resolving a decade-old puzzle in AdS$_4$/CFT$_3$ and establishing a practical computational framework for precision holography in strongly coupled gauge theories.

Figures (4)

• Figure 1: Complex eigenvalue distributions for the $Q^{1,1,1}/\mathbb{Z}_k$ quiver at $N=27$ and $N=54$. Colors label the gauge nodes: $\mathrm{U}(N)_k$ ($\mu_i$), $\mathrm{U}(N)_{-k}$ ($\nu_i$), $\mathrm{U}(N)_0^{(T)}$ ($\alpha_i$), and $\mathrm{U}(N)_0^{(B)}$ ($\beta_i$). Each node forms a smooth curve; real parts scale as $\sqrt{N}$, while imaginary parts remain $\mathcal{O}(1)$ and cluster into node-dependent bands. The distributions satisfy $\nu_i = \overline{\mu_i}$ and $\beta_i = \overline{\alpha_i}$, indicating complex conjugation symmetry across the quiver.
• Figure 2: Diagnostics of the $Q^{1,1,1}/\mathbb{Z}_k$ saddle. (a) Normalized density $\rho(x)$ of real parts for the zero-level node ($\alpha$) in the rescaled variable $x=\Re\lambda/\sqrt{N}$; the $\beta$ node shows an identical profile, and the $\pm k$ nodes ($\mu,\nu$) follow the same envelope within numerical precision. (b) Imaginary-part differences along quiver links versus the average real part. The link phases show $x$-dependent structure in the bulk with tail plateaux near multiples of $\pi/3$: $\Im(\mu-\alpha) \simeq -2\pi/3$ (left tail), $\Im(\nu-\alpha) \simeq -2\pi/3$ (right tail), while $\Im(\alpha-\beta)$ varies across the support. The separations are $\mathcal{O}(1)$ and $N$-independent.
• Figure 3: Real part of the $S^3$ free energy $F(N)$ for $Q^{1,1,1}/\mathbb{Z}_k$ (points) with least-squares fit (line). The fitted $N^{3/2}$ coefficient agrees with $c_{3/2}^{\text{hol}} = 2.418$ from \ref{['Q111:c32:an']}.
• Figure 4: Diagnostics for the non-canonical $D_3$ saddle. (a) Normalized density $\rho(x)$ of real parts showing $\sqrt{N}$ broadening in $x=\Re\lambda/\sqrt{N}$. (b) Imaginary-part differences along quiver links versus the average real part. The saddle remains smooth and stable across the continuation range.