Numerical Studies of the Abjm Theory for Arbitrary N at Arbitrary Coupling Constant

Abstract

We show that the ABJM theory, which is an N = 6 superconformal U(N ) × U(N ) Chern-Simons gauge theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility of probing the quantum aspects of M-theory and testing the AdS 4 /CF T 3 duality at the quantum level. Here we calculate the free energy, and confirm the N 3/2 scaling in the M-theory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion. The method can be easily generalized to the calculations of BPS operators and to other theories that reduce to matrix models. *

Symmetry breaking in quantum curves and super Chern-Simons matrix models

“…In [6] it was found that the grand canonical partition function of the M2-branes on the background C 4 /Z k can be rewritten as a spectral determinant of a quantum-mechanical operator associated with the geometry P 1 × P 1 , following preceding works [7][8][9][10][11]. After further studies of the spectral determinant in [12][13][14][15][16][17] finally it was conjectured [18] that the grand potential of the M2-branes is expressed as the free energy of the topological string theory on the local P 1 × P 1 geometry. The rank deformations with the inclusion of fractional M2-branes [19,20] were studied in [21][22][23][24] and found to match the conjectured topological string free energy.…”

“…and the BPS indices forming the representations of the D 5 algebra are broken to representations of D 4 (see table 1 for the split of the BPS indices). Furthermore, for the (2, 2) model with the rank deformations which is connected to the (1, 1, 1, 1) model without rank deformations at (M I , M II ) = (k/2, k/2) through the Hanany-Witten effect, the Kähler parameters are 12) and the BPS indices in the representations of the D 4 algebra are further split into representations of the (A 1 ) 3 algebra (see table 2 for the further split of the BPS indices). Hence, from table 1 it was found that the symmetry for the (2, 2) model without rank deformations is broken to D 4 while from table 2 the symmetry for the (1, 1, 1, 1) model is further broken to (A 1 ) 3 .…”

ABJM matrix model and 2D Toda lattice hierarchy

“…Fourthly, though we have mainly focused on the open string side so far, the story on the closed string side is interesting as well. The large z expansion of the grand canonical partition function was studied in a series of works [13,25,[33][34][35][36][37][38][39][40][41][42] and it was found to be expressed by the free energy of the closed topological string theory on local P 1 ×P 1 . Interestingly enough, in the analysis [25,38,39,43,44] an integrable structure [45,46] originating from the polymer matrix model [47] was utilized.…”

Two-point functions in ABJM matrix model

“…For simplicity in discussing the numerical results, we always consider the case of M ≥ 0. We have computed the two-point function s YsZ k (N, N + M) of 2 ≤ |Y | + |Z| ≤ 5 and k = 3, 4, 6, 8, 12 up to N = N max with (k, N max ) = (3, 7), (4,13), (6,8), (8,4), (12,5) for M within the range of convergence.…”

“…Another interesting progress is the study of this matrix model. After the study of the large N limit in the 't Hooft expansion [9][10][11], where the degrees of freedom N 3/2 of the M2-branes were reproduced, it was found that all of the perturbative corrections in the large N limit are summed up to the Airy function [12,13]. These studies further lead beautifully to an unexpected description of the Fermi gas formalism [14] where the partition function is reexpressed as that of a non-interacting Fermi gas system with a non-trivial one-particle Hamiltonian and the Chern-Simons level k is identified as the Planck constant.…”