Dualities in ABJM matrix model from closed string viewpoint

Abstract

We propose a new formalism to study the ABJM matrix model. Contrary to expressing the fractional brane background with the Wilson loops in the open string formalism, we formulate the Wilson loop expectation value from the viewpoint of the closed string background. With this new formalism, we can prove some duality relations in the matrix model.

Hanany-Witten transition in quantum curves

“…This was named the open string formalism in [29] since the spectral operator seems to reflect the closed string background after expanding the Fredholm determinant with traces and the correction by expectation values is reminiscent of the idea of taking care of the deformations by adding open string fluctuations to a fixed closed string background. Another generalization by correcting the spectral operator (2.18) while keeping the expression of the Fredholm determinant (2.17) fixed was also proposed in [30][31][32][33][34][35]. This was named the closed string formalism since now we try to take care of the deformations by changing the spectral operator for the closed string background.…”

“…In this appendix we review the Fermi gas formalism for the super Chern-Simons matrix models (2.11) without rank deformations, propose a generalization with some rank deformations and relate the result to spectral operators. The techniques are mostly taken from many previous works including [5,18,20,28,33,35,46], though we stress that our piecewise derivation is now much clearer.…”

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

“…Our strategy to derive the closed string formalism for the partition function of the (2, 2) model is completely parallel to that of the ABJM theory in [57,58,65], only except that we have to repeat it twice. Namely, we utilize the following two techniques.…”

“…The closed string formalism for the ABJM theory was first conjectured in [63] and later proved in [64]. The derivation was improved largely later and generalized to the partition function of the orientifold ABJM theory with the orthosymplectic gauge group in [57,58] and the vacuum expectation values of the half-BPS Wilson loop in the ABJM theory in [65]. We shall follow these simplified derivations in the following.…”

“…where the normalization is 2) and the density matrix is given as the "square" of that of the ABJM theory Note that the hyperbolic tangent function in [65] appears after we take the trace of ρ ABJM .…”