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

Abstract

In our previous work [1], we pointed out that various multi-cut solutions exist in the Chern-Simons (CS) matrix models at large-N due to a curious structure of the saddle point equations. In the ABJM matrix model, these multi-cut solutions might be regarded as the condensations of the D2-brane instantons. However many of these multi-cut solutions including the ones corresponding to the condensations of the D2brane instantons were obtained numerically only. In the current work, we propose an ansatz for the multi-cut solutions which may allow us to derive the analytic expressions for all these solutions. As a demonstration, we derive several novel analytic solutions in the pure CS matrix model and the ABJM matrix model. We also develop the argument for the connection to the instantons. *

Chern-Simons theory on Seifert manifold and matrix model

“…Chern-Simons theory on S 3 does not allow any no gap solution 20 but admits multi-gap (more than one) solutions [14,44]. A two gap solution, in particular, was explicitly studied in [14].…”

“…One needs to continue the sum over R beyond integrable representations for λ > λ * [42,43] using the symmetries of summand. Chern-Simons theory on S 3 does not allow any no gap solution 20 but admits multi-gap (more than one) solutions [14,44]. A two gap solution, in particular, was explicitly studied in [14].…”

“…Although the underlying meaning of these multi-gap phases (condensation of D2 brane instantons ? [44]) is not very clear yet, we attempt to show that such phases can be engineered with free energy same as that of one-gap phase. Understanding of physical meaning of these multi-gap phases (in topological string theory side) is an interesting avenue to pursue.…”

“…Supported by numerical calculations of[44] 23. Though numerically challenging, but one can certainly look for such equi-free energy solutions for higher gap phases as well.…”