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∨(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 ≠ h∨(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 = An, E6, E8 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.
Highlights
As studied in [8], given the above type of the geometry, we expect some of these 4d theories to have 1-form symmetries
We study the defect groups of Dpb(G) theories using geometric engineering and BPS quivers
As we will see when we do dimensional reduction to 4d, only a commuting subset of the flux operators in 10d can be realised in 4d to give rise to the operators generating the 1-form symmetries in 4d
Summary
We give a short review on how to find the discrete higher form symmetries of geometric engineered field theories following the methods developed in [8]. The operators φσ, with σ a torsional class in the first K-theory group of the boundary K1(∂M10), measuring the torsional part of flux expectation values for the IIB supergravity fields at infinity on a cycle Poincaré dual to σ do not commute [17, 18]. Fixing a boundary condition at infinity for the fluxes corresponds to picking a specific state in the Hilbert space. Not all theories have a maximally isotropic subgroup L that is invariant under large diffeomorphisms of the boundary ∂M10 This means the partition vector Z| = i zi |σi, L0 ∗ defined in terms of the dual basis vectors |σi, L0 ∗ of the Hilbert space, and the partition function which depend on the choice of L, are not invariant under such transformations
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