Hanany-Witten transition in quantum curves

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 D 5 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 deformations interpreted as fractional branes in brane configurations), while on the other hand the D 5 curve has five degrees of freedom (characterized by point configurations of asymptotic 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 D 5 Weyl group beautifully, we are naturally led to conjecture that M2-branes are not only deformed by fractional branes but more obscure geometrical backgrounds.

Fermi gas approach to general rank theories and quantum curves

“…The matrix models associated with these gauge theories also can be written as the partition functions of ideal Fermi gases when all the nodes have equal ranks, and much studied in the Fermi gas description [3,[27][28][29][30][31][32] as with the ABJ(M) theory. However, when the ranks are different, so far the Fermi gas formalism is applied only to special cases such as theories with gauge groups consisting of four unitary groups [33,34].…”

“…It was revealed that some types of quantum curves obey the Weyl group symmetry of SO (10) and the exceptional groups [35,36]. Furthermore, it was found in [34] by connecting the SO (10) quantum curve to the rank deformed theories with four nodes without using the Fermi gas formalism that the Hanany-Witten transition is realized as a portion of the Weyl group symmetry of SO (10). This fact implies that quantum curves play an important role in understanding M2-brane physics.…”

“…Therefore, this procedure can be applied to a wide class of rank deformations. This procedure first appeared in [34] though they computed the deformed factor associated with the brane configuration consisting of only one NS5-brane and one (1, k)5-brane. In this paper, we focus on brane configurations consisting of an NS5-brane at the center, any number of (1, k)5-branes on both side of the NS5-brane and D3-branes stretched between two 5-branes such that the number monotonically increases with approaching the NS5-brane (see figure 1 in section 3.1).…”

“…After that, it was gradually realized that this type of computations can be performed more systematically by using operator formalism [10,33,[42][43][44]. In [34], by using this formalism, they computed the deformed factor consisting of one NS5-brane and one (1, k)5-brane, which is closely related to the ABJ theory. In this section we review this computation for the ease of understanding the main computation of this paper in section 3.2 where we will apply the Fermi gas formalism to more complicated deformed factors.…”

“…can be written as the specific form which can be termed the quantum curve or not. The progress for the ABJ theory gave a positive answer [23] (see also appendix A.3 in [34])…”