Abstract
It was known that the U(N )4 super Chern-Simons matrix model describing the worldvolume theory of D3-branes with two NS5-branes and two (1, k)5-branes in IIB brane configuration (dual to M2-branes after taking the T-duality and the M-theory lift) corresponds to the D5 quantum curve. For deformations of these two objects, on one hand the super Chern-Simons matrix model has three degrees of freedom (of relative rank defor- mations interpreted as fractional branes in brane configurations), while on the other hand the D5 curve has five degrees of freedom (characterized by point configurations of asymp- totic values). To identify the three-dimensional parameter space of brane configurations in the five-dimensional space of point configurations, we propose the necessity to cut the compact T-duality circle (or the circular quiver diagram) open, which is similar to the idea of “fixing a reference frame” or “fixing a local chart”. Since the parameter space of curves enjoys the D5 Weyl group beautifully, we are naturally led to conjecture that M2-branes are not only deformed by fractional branes but more obscure geometrical backgrounds.
Highlights
The simplest super Chern-Simons matrix model describing M2-branes is the ABJM matrix model
To identify the three-dimensional parameter space of brane configurations in the five-dimensional space of point configurations, we propose the necessity to cut the compact T-duality circle open, which is similar to the idea of “fixing a reference frame” or “fixing a local chart”
After seeing that the spectral operators without rank deformations (2.20) fall into the D5 curve and some rank deformations still correspond to the topological string theory on local del Pezzo D5, it is natural to consider that the expression of the Fredholm determinant (2.17) is still valid when we introduce rank deformations for the matrix model
Summary
We review brane configurations in type IIB string theory, super Chern-Simons matrix models obtained from the brane configurations by the localization technique and quantum curves obtained in the analysis of the matrix models. In reviewing each topic we emphasize that we have often unconsciously taken the idea of fixing a reference frame for granted. We believe that the importance of fixing a frame in discussing the correspondence was not pointed out explicitly previously and we try to explain our idea carefully through the reviews of various aspects
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