Brane Transitions from Exceptional Groups

Abstract

It is a well-known result by Hanany and Witten that, when two five-branes move across each other, D3-branes stretching between them are generated. Later the same brane configurations played a crucial role in understanding the worldvolume theory of multiple M2-branes. Recently the partition function of multiple M2-branes was transformed to the spectral determinant for quantum algebraic curves, where the characteristic 3/2 power law of degrees of freedom is reproduced and the determinant enjoys a large symmetry given by exceptional Weyl groups. The large exceptional Weyl group reproduces the Hanany-Witten brane transitions and, besides, contains brane transitions unknown previously. Aiming at understanding the new brane transitions better, we generalize our previous study on the D 5 quantum curve to the E 7 case. Our result suggests a "local" rule for the brane transitions.

Duality Cascades and Affine Weyl Groups

“…In relating parameters between the brane configurations and the D 5 quantum curve in [3], the brane configurations without rank differences (but with various orders of 5-branes) were considered as markers. These brane configurations appear as vertices of the fundamental domain Interestingly, it was found in [3] and further explained in [4] that the brane configurations enjoy symmetries of the B 3 Weyl group. As recapitulated in appendix A.1, the appearance of the B 3 Weyl group can be understood from the Z 2 folding of the affine Dynkin diagram of D 5 .…”

“…We first consider brane configurations corresponding to the D 5 quantum curve, which consist of two NS5-branes and two (1, k)5-branes. As found in [3,4] and recapitulated in appendix A.1, without further deformations, the brane configurations are parameterized by three relative ranks and enjoy symmetries of the B 3 Weyl group, which is an invariant subgroup of the original D 5 Weyl group.…”

“…We may gain new insights on both brane transitions and quantum curves by relating them directly. Indeed, in studying the partition functions of the generalized ABJM theories, recently new types of brane transitions were proposed [3][4][5]. In this paper we shall turn to subsequent applications of the HW transitions in a circle known as duality cascades and study the processes with Weyl groups.…”

“…We also gain insights for brane transitions from the quantum curves. Especially for partition functions, we can relate directly the space of rank deformations in brane configurations to the parameter space of quantum curves and translate symmetries of the Weyl group into brane transitions [3,4]. Namely, for brane configurations, even starting from the partition functions without rank deformations, we can apply the HW transitions to exchange 5-branes into the standard order and read off the rank deformations.…”

“…Due to this reason, the D 5 and E 7 Weyl groups are subject to the Z 2 folding and reduce respectively to the B 3 and F 4 Weyl groups. Then, we introduce the deformation of FI parameters for full parameters of quantum curves and study the same problem of the 0 1 2 6 3 7 4 8 5 9 [4,8] θ [5,9] θ This paper is organized as follows. We study the cases without deformations of FI parameters in the next section, while turn to the deformation in section 3.…”