Abstract
Abstract In previous work we have shown that the equivariant index of multi-centered $ \mathcal{N}=2 $ black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription — the Coulomb branch formula — for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke’s result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincaré polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package “CoulombHiggs.m” implementing the Coulomb and Higgs branch formulae.
Highlights
This comparison is complicated by the fact that on the macroscopic side, contributions to the index Ω(γ) originate from single centered black holes with charge γ, and from multi-centered black holes with constituents carrying charges {αi} such that γ = αi [4,5,6,7]
In previous work we have shown that the equivariant index of multi-centered N = 2 black holes localizes on collinear configurations along a fixed axis
We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription — the Coulomb branch formula — for computing the cohomology of the moduli space of quiver representations
Summary
We establish a recursive algorithm for computing the Coulomb index gCoulomb({α1, · · · αn}; {c1, · · · cn}; y) for general charge configurations αi, away from the walls of marginal stability in the space of FI parameters ci. For this we first deform all the vanishing αij’s to non-zero values in such a way that the deformed quiver does not have any oriented loop To see that this is always possible, let us carry out the deformation one link at a time. If so let us flip the sign of αij of the deformed link In this case C together with the added link no longer forms an oriented loop. We can repeat the argument and show that all the vanishing αij’s can be made non-zero and for appropriate choice of sign of the deformed αij’s the new quiver does not have any oriented loop.
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