Abstract
In quiver quantum mechanics with 4 supercharges, supersymmetric ground states are known to be in one-to-one correspondence with Dolbeault cohomology classes on the moduli space of stable quiver representations. Using supersymmetric localization, the refined Witten index can be expressed as a residue integral with a specific contour pre- scription, originally due to Jeffrey and Kirwan, depending on the stability parameters. On the other hand, the physical picture of quiver quantum mechanics describing interactions of BPS black holes predicts that the refined Witten index of a non-Abelian quiver can be expressed as a sum of indices for Abelian quivers, weighted by ‘single-centered invariants’. In the case of quivers without oriented loops, we show that this decomposition naturally arises from the residue formula, as a consequence of applying the Cauchy-Bose identity to the vector multiplet contributions. For quivers with loops, the same procedure produces a natural decomposition of the single-centered invariants, which remains to be elucidated. In the process, we clarify some under-appreciated aspects of the localization formula. Part of the results reported herein have been obtained by implementing the Jeffrey-Kirwan residue formula in a public Mathematica code.
Highlights
In the non-relativistic limit, the interactions between the centers can be described by a supersymmetric quantum mechanics of n particles in R3, interacting through electromagnetic, scalar exchange and possibly gravitational forces [5,6,7,8,9]
In quiver quantum mechanics with 4 supercharges, supersymmetric ground states are known to be in one-to-one correspondence with Dolbeault cohomology classes on the moduli space of stable quiver representations
The physical picture of quiver quantum mechanics describing interactions of BPS black holes predicts that the refined Witten index of a non-Abelian quiver can be expressed as a sum of indices for Abelian quivers, weighted by ‘single-centered invariants’
Summary
The quiver quantum mechanics associated to (Q, γ) is a 0+1 dimensional gauge theory with four supercharges [7] It includes vector multiplets for the gauge group G =. The group SO(3) associated to physical rotations in R3 acts on the cohomology of the Higgs branch via the Lefschetz action generated by contraction and wedge product with the natural Kahler form on MQ, induced from the flat Kahler form on the ambient space ⊕eabA∈AQCNa ⊗ CNb. Na and all chiral multiplets as well as off-diagonal vector multiplets are massive. In the case where the dimension vector γ is primitive and the superpotential W is generic, MQ is compact, so Ω(γ, z, y) is a symmetric Laurent polynomial in y, which can be viewed as the character of the Lefschetz action of SO(3) on the cohomology of the quiver moduli space MQ. The advantage is that χQ({Na, ζa)}, y) satisfies a much simpler wall-crossing formula than χQ({Na, ζa)}, y) [20, 21, 24]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
