ABJ fractional brane from ABJM Wilson loop

Abstract

We present a new Fermi gas formalism for the ABJ matrix model. This formulation identifies the effect of the fractional M2-brane in the ABJ matrix model as that of a composite Wilson loop operator in the corresponding ABJM matrix model. Using this formalism, we study the phase part of the ABJ partition function numerically and find a simple expression for it. We further compute a few exact values of the partition function at some coupling constants. Fitting these exact values against the expected form of the grand potential, we can determine the grand potential with exact coefficients. The results at various coupling constants enable us to conjecture an explicit form of the grand potential for general coupling constants. The part of the conjectured grand potential from the perturbative sum, worldsheet instantons and bound states is regarded as a natural generalization of that in the ABJM matrix model, though the membrane instanton part contains a new contribution.

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

“…Unlike the situation with the general R charges [38], the FI terms completely cancel in these deformations. By using the technique [18] called the open string formalism, we can factorize the grand canonical partition function of these models as the product of the grand partition function of M = 0 and the determinant of a matrix whose matrix elements are expressed with a generalization of the two functions φ m (q) and ψ m (q) introduced in [23]. Although these functions were originally introduced from a technical reason to compute the grand canonical partition function of M = 0, it is interesting to find that these functions appear naturally in the rank deformations as well.…”

“…Most of the contents in this section are reviews, though in section 2.1.3 and section 2.2.3 we will also raise some questions and clarify some points which we believe are not so trivial even to the experts but important for our later analysis. The main references of the review part are [17,18,23] (see also [40,41]). …”

“…One of the standard techniques to study this ABJM model with generally non-equal ranks M ≥ 0 is to utilize the following two essentially same determinant formulas [18] 4) which are obtained by combining the Vandermonde determinant and the Cauchy determinant. By multiplying these two determinant formulas, the extra exponential factors e ± M 2 m µm and e ± M 2 n νn cancel among themselves and we can perform the ν integrations by using a determinant formula in [18] (which is further generalized in appendix B)…”

“…By multiplying these two determinant formulas, the extra exponential factors e ± M 2 m µm and e ± M 2 n νn cancel among themselves and we can perform the ν integrations by using a determinant formula in [18] (which is further generalized in appendix B)…”

“…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].…”