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

Saghar S. Hosseini, Robert Moscrop

TL;DR

The paper develops a dual approach to determine the defect groups of D_p^b(G) theories, tying geometric-engineering data to higher-form symmetries. By combining Orlik's algorithm with BPS-quiver methods and leveraging Maruyoshi-Song flows, the authors show that the defect groups of D_p^b(G) can be inferred from those of G^{(b)}[k], with cross-checks in A_n, E_6, and E_8. They demonstrate that MS flows preserve higher-form symmetry structure and provide explicit computations for both the torsion and free components of the defect groups, including detailed analyses for A_n^{(n)}[k] and the E6/E8 families. These results unify geometric-engineering perspectives with quiver techniques, clarifying the global structure and screening patterns of these 4d N=2 theories.

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.

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

TL;DR

The paper develops a dual approach to determine the defect groups of D_p^b(G) theories, tying geometric-engineering data to higher-form symmetries. By combining Orlik's algorithm with BPS-quiver methods and leveraging Maruyoshi-Song flows, the authors show that the defect groups of D_p^b(G) can be inferred from those of G^{(b)}[k], with cross-checks in A_n, E_6, and E_8. They demonstrate that MS flows preserve higher-form symmetry structure and provide explicit computations for both the torsion and free components of the defect groups, including detailed analyses for A_n^{(n)}[k] and the E6/E8 families. These results unify geometric-engineering perspectives with quiver techniques, clarifying the global structure and screening patterns of these 4d N=2 theories.

Abstract

We study the defect groups of theories using geometric engineering and BPS quivers. In the simple case when , 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 , we use a similar Maruyoshi-Song flow to conjecture that the defect groups of theories are given by those of theories. In the cases of we cross check our result by calculating the BPS quivers of the theories and looking at the cokernel of their intersection matrix.

Paper Structure

This paper contains 13 sections, 1 theorem, 58 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Higher form symmetries from geometric engineering
  3. Geometric origin
  4. Defect groups from BPS quivers
  5. Defect group from Orlik's conjecture
  6. Defect groups of $D_p(G)$ theories
  7. Summary of $D_p(G)$ theories
  8. Defect group and Maruyoshi-Song flows
  9. Defect groups of $D_p^b(G)$ theories
  10. Orlik's algorithmic approach
  11. BPS quivers for $A_n^{(n)}[k]$ theories
  12. Extension to $E_6$ and $E_8$ cases
  13. A quick review of the polar curve method

Key Result

Theorem A.1

(Gabrielov gabrielov). Let $f$ be an isolated singularity depending on a linear function $z$ and $\{\varphi_i : i=1,\ldots, \mu(f|_{z=0})\}$ be an admissible system of distinguished paths connecting 0 to the critical values of $f|_{z=\varepsilon}$. There exists a distinguished basis of vanishing cyc and if $|m-k|>1$ or $|m-k|=1$ with $(m-k)(i-j)>0$. Here $\{\Delta_i\}$ is a distinguished basis of

Figures (1)

  • Figure 1: A visualisation of how the covering of $\mathbb{C}$ by the sets $T_m$ changes under $k\mapsto k+1$. Specifically we see that $m\mapsto m+1$ for all but exactly one critical point, where we get $m\mapsto m+2$. Here we have taken $n=6$ and each critical point of $f|_{z=\varepsilon}$ is denoted by a blue dot.

Theorems & Definitions (1)

  • Theorem A.1