Calculating the partition function of N=2 Gauge theories on $S^3$ and AdS/CFT correspondence

TL;DR

This work tests AdS$_4$/CFT$_3$ for ${\cal N}=2$ quiver Chern-Simons-matter theories by computing the three-sphere partition function $Z$ via localization, reducing it to a matrix model, and analyzing the large-$N$ limit. Focusing on non-chiral theories dual to ${\rm AdS}_4\times X^7$ with $X^7\in\{Q^{1,1,1}, V^{5,2}\}$, the authors derive saddle-point equations for eigenvalues and obtain explicit $N^{3/2}$ scaling of the free energy $F=-\log Z$, with coefficients that match the volumes of the internal manifolds: ${\rm Vol}(Q^{1,1,1})$ and ${\rm Vol}(V^{5,2})$. Through detailed treatment of adjoint and fundamental matter in three examples, they show how marginality and $F$-extremization constrain R-charges to reproduce gravity results, providing strong evidence for the AdS$_4$/CFT$_3$ dictionary in these ${\cal N}=2$ settings. The work also highlights the tractability of non-chiral models and discusses limitations and future directions for chiral theories.

Abstract

We test the AdS/CFT correspondence by computing the partition function of some $\mathcal{N}=2$ quiver Chern-Simons-matter theories on three-sphere. The M-theory backgrounds are of the Freund-Rubin type with the seven-dimensional internal space given as Sasaki-Einstein manifolds $Q^{1,1,1}$ or $V^{5,2}$. Localization technique reduces the exact path integral to a matrix model, and we study the large-N behavior of the partition function. For simplicity we consider only non-chiral models which have a real-valued partition function. The result is in full agreement with the prediction of the gravity duals, i.e. the free energy is proportional to $N^{3/2}$ and the coefficient matches correctly the volume of $Q^{1,1,1}$ and $V^{5,2}$.

Figures (9)

• Figure 1: The quiver diagram for Chern-Simons dual of $AdS_4 \times V^{5,2}$
• Figure 2:
• Figure 3: These figures show the density of roots $\rho(x)$ for $V^{5,2}$. The dots are numerical data and the red lines show the analytical predictions.
• Figure 4: The quiver diagram for Chern-Simons dual of $AdS_4 \times Q^{1,1,1}$
• Figure 5: The distribution of roots for $Q^{1,1,1}$ theory with different R charges. The roots are plotted with black and blue dots for $N=20$ and $100$, respectively. The real part of the roots scales as $\sqrt{N}$. The maximum values of imaginary part of $\lambda$ are bounded by $\pi\Delta_B$, which are represented by green lines.