Maruyoshi-Song Flows and Defect Groups of $D_p^b(G)$ Theories

TL;DR

This paper investigates the defect groups of $D_p^b(G)$ theories using geometric engineering and BPS quivers, establishing a connection with Maruyoshi-Song flows and conjecturing their structure for various cases.

Contribution

It extends the understanding of defect groups in $D_p^b(G)$ theories by relating them to $G^{(b)}[k]$ theories through geometric and quiver analysis.

Findings

Defect groups are compatible with Maruyoshi-Song flows when $b=h^8(G)$.

Conjecture that defect groups of $D_p^b(G)$ are given by those of $G^{(b)}[k]$ theories.

Cross-checked results for $A_n, E_6, E_8$ using BPS quivers and intersection matrices.

Abstract

We study the defect groups of $D_p^b(G)$ theories using geometric engineering and BPS quivers. In the simple case when $b=h^\vee (G)$, we use the BPS quivers of the theory to see that the defect group is compatible with a known Maruyoshi-Song flow. To extend to the case where $b\neq h^\vee (G)$, we use a similar Maruyoshi-Song flow to conjecture that the defect groups of $D_p^b(G)$ theories are given by those of $G^{(b)}[k]$ theories. In the cases of $G=A_n, \;E_6, \;E_8$ we cross check our result by calculating the BPS quivers of the $G^{(b)}[k]$ theories and looking at the cokernel of their intersection matrix.