Calculating the partition function of $ \mathcal{N} = 2 $ gauge theories on S 3 and AdS/CFT correspondence

Abstract

We test the AdS/CFT correspondence by computing the partition function of some 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 .

Localization in supergravity and quantum AdS 4 /CFT3 holography

“…The free energy exhibits a scaling of N 3/2 where N is the rank of the gauge group with the exact numerical prefactor matching the regularized classical action of 11d SUGRA on AdS 4 × S 7 /Z k . Generalizations of this large N behavior were found for a large class of theories and the prefactor consistently matched the classical supergravity action of the dual M-theory on AdS 4 × X 7 , which is proportional to the volume of the compact X 7 [25][26][27][28][29][30].…”

The free energy of $ \mathcal{N} = 2 $ supersymmetric AdS4 solutions of M-theory

“…In [7,8,9] the leading large N contribution to the free energy of Chern-Simonsmatter theories on S 3 was computed for large classes of N = 2 theories, and succesfully matched to the gravity prediction in a class of Sasaki-Einstein geometries. This remarkable matching was first obtained in [10] for the ABJM theory and then in [11] for several N = 3 examples.…”

N=2supersymmetric theories on squashed three-sphere

“…It was later extended to theories with N = 2 superconformal theories [5,6]. It is used to check dualities among 3d field theories [7,8,9,10,11,12,13] and gauge/gravity correspondence [14,15,16,17,18]. The superconformal index is also used to check these dualities [19,20,21,22,13].…”