Uncovering the symmetries on $[p,q]$ 7-branes: Beyond the Kodaira classification

Abstract

We begin a classification of the symmetry algebras arising on configurations of type IIB [p, q] 7-branes. These include not just the Kodaira symmetries that occur when branes coalesce into a singularity, but also algebras associated to other physically interesting brane configurations that cannot be collapsed. We demonstrate how the monodromy around the 7-branes essentially determines the algebra, and thus 7-brane gauge symmetries are classified by conjugacy classes of the modular group SX(2,Z). Through a classic map between the modular group and binary quadratic forms, the monodromy fixes the asymptotic charge form which determines the representations of the various (p, q) dyons in probe D3-brane theories. This quadratic form also controls the change in the algebra during transitions between different brane configurations. We give a unified description of the brane configurations extending the DJV? EN an d Argyres-Douglas H^ series beyond the Kodaira cases. We anticipate the appearance of affine and indefinite infinite-dimensional algebras, which we explore in a sequel paper.

Geometric constraints on the space of N $$ \mathcal{N} $$ = 2 SCFTs. Part II: construction of special Kähler geometries and RG flows

“…Furthermore, the ordering of the JHEP02(2018)002 singularities is arbitrary, and can be changed by moving cuts across neighboring singularities (therefore passing them through neighboring cuts). Upon dragging the monodromy paths, one finds that neighboring monodromies can get interchanged or conjugated by each other, giving an action of the braid group on Z strands on SL(2, Z) matrices [18,19]. We will describe this action in more detail below.…”

“…Taking the trace of (2.12) using (2.13) and the definition (2.7) of the charge inner product, it follows that the trace of the total monodromy is [19] …”

“…It is clear (e.g., by taking the path arbitrarily close to H α ) that the motion only rearranges the subset of the I 1 singularities which collide at H α . 4 This subset of I 1 singularities and their pattern of rearrangement can be encoded in a branched path (i.e., topologically a tree) in the Coulomb branch connecting these singularities [18,19]. We call these branched paths "string webs" since that is how they appear in F-theory constructions.…”

“…In particular we carefully define and discuss maximal and sub-maximal deformations of Kodaira singularities. The string web description of neutral BPS state is reviewed [18,19] and the operation of coalescence of singularities is introduced. These are the necessary ingredients to go from the deformation pattern associated to each N = 2 SCFTs to the explicit form of the SW curve.…”

Brane Solutions of Gravity-Dilaton-Axion Systems

“…For the other two conjugacy class the quantization conditions are much more difficult to work out. The general solution can be found in [14,15]. The two solutions given in table 2 corresponding to the SO(2) conjugacy class are especially interesting in the context of the AdS/CFT correspondence since after reduction over x they lead to 1/2 supersymmetric domain wall solutions in SO(2) gauged supergravity theories.…”