Exact results on ABJ theory and the refined topological string

Abstract

Abstract:We study the partition function of the ABJ theory, which is the N = 6 superconformal Chern-Simons matter theory with gauge group U(N ) × U(N + M ) and Chern-Simons levels (k, −k). We exactly compute the ABJ partition function on a three sphere for various k, M and N via the Fermi gas approach. By using these exact data, we show that the ABJ partition function is completely determined by the refined topological string on local P 1 × P 1 , including membrane instanton effects in the M-theory dual.

Instanton effects in rank deformed superconformal Chern-Simons theories from topological strings

“…This observation is consistent JHEP08(2017)003 with the Yukawa coupling [19,48] and would be an additional positive evidence for the unification. Also, it seems that the coefficient…”

“…Indeed ρABJM is positive definite which is manifest after the similarity transformation ρABJM → e − M p 2k ρABJMe M p 2k [19]. The density matrix for Z ABJM k (N, N + M ) without taking the absolute values is given as e − πiM 2 ρABJM.…”

“…as in the case of the ABJM theory [19]. For the membrane instanton the multi-covering structure (2.40) is kept unchanged, while each component…”

“…Furthermore, the reduced grand potentials are given in appendix C.4.2 and appendix C.4.3, where µ eff is determined as 19) for k = 2, 4, 6 while 20) for (k, M ) = (3, 1). We find that the comments found in the last subsection also apply here.…”

“…Finally, the non-perturbative effects of the matrix model in the grand canonical ensemble are given explicitly by the free energy of the topological string theory on the local P 1 × P 1 geometry [8,9,17]. Although this result was originally found for the case of equal ranks N 2 = N 1 , it was soon generalized into the case of different ranks N 2 = N 1 [18,19].…”