Notes on planar resolvents of Chern-Simons-matter matrix models

Abstract

We revisit planar resolvents of matrix models corresponding to N ≥ 3 Chern-Simons-matter theories with the gauge groups of the form U(N 1 ) × U(N 2 ) coupled to any number of bi-fundamental hypermultiplets. We find that the derivative of a suitably defined planar resolvent can be written explicitly. From this resolvent, we derive the explicit formula for (a linear combination of) the vevs of BPS Wilson loops. As a non-trivial check, we show that the formula reproduces the perturbative expansion of the vevs of the BPS Wilson loops.

Strong coupling limit of a family of Chern-Simons-matter theories

“…which can be obtained by using the method developed in [13]. This result was reproduced in [30] from the resolvent v(z) reviewed above by using the formulas (2.19)(2.20) for cases when N 1 = N 2 and k 1 = ±k 2 are satisfied. This can be regarded as a non-trivial check of the validity of the resolvent v(z) obtained in [30].…”

“…This paper is organized as follows. In section 2, we show that the above-mentioned results for the Chern-Simons-matter theories are derived from the resolvent obtained in [30]. Possible gravity duals for the theories are briefly reconsidered in section 3, taking into account the results obtained in the previous section.…”

“…Various quantities can be calculated perturbatively when the 't Hooft couplings t α are small. For example, the vevs W α are given as [30]…”

Linear Chern-Simons-matter theories in the planar limit

“…The equations (A.7) are not convenient for determining v a (z) since log x in the left-hand side requires us to handle the log branch cuts. To avoid this difficulty, we take the derivative of the equations [14]. As a result, we obtain…”

Toward the construction of the general multi-cut solutions in Chern-Simons matrix models

“…Ui−Ũj . The advantage of the U i variable [48] is that it makes equations symmetric under U → 1/U corresponding to the symmetry u → −u in the original variable (1.1). on the interval [A, B] near Z = 1, where A and B (|A| < |B|) will be fixed soon.…”