Abstract
We consider a special class of N=4 quiver quantum mechanics relevant in the description of BPS states of D4D0 branes in type II Calabi-Yau compactifications and the corresponding 4-dimensional black holes. These quivers have two abelian nodes in addition to an arbitrary number of non-abelian nodes and satisfy some simple but stringent conditions on the set of arrows, in particular closed oriented loops are always present. The Higgs branch can be described as the vanishing locus of a section of a vector bundle over a product of a projective space with a number of Grassmannians. The Lefschetz-Sommese theorem then allows to separate induced from intrinsic cohomology which leads to the notion of pure-Higgs states. We compute explicit formulae for an index counting these pure-Higgs states and prove — for this special class of quivers — some previously stated conjectures about them.
Highlights
Spatial bound states of dyonic centers [1, 13, 14]
We consider a special class of N=4 quiver quantum mechanics relevant in the description of BPS states of D4D0 branes in type II Calabi-Yau compactifications and the corresponding 4-dimensional black holes
The index counting pure-Higgs states, a quiver invariant, coincides with the single center black hole contributions to the Coulomb branch formula [22, 23] and it is known that the number of pure-Higgs states grows exponentially in the charges [15, 16] leading to a non-negligible macroscopic entropy
Summary
The geometric interpretation mentioned above is that V is the vanishing locus of a section of a complex vector bundle E over a compact Kähler manifold M More colloquially this means that V is a subset of M, formed by the solutions to a number of equations defined on M. The Lefschetz-Sommese theorem does not determine the amount of intrinsic cohomology, i.e. it leaves the β(p,q) indeterminate apart from being non-negative integers. From a physics point of view this is nothing but a classification by angular momentum, while from the geometric perspective this corresponds to the Lefschetz SU(2) action on the cohomology of the Kähler manifold K. It is generated by the Kähler form J as follows. As we’ll review Iind(V, t) contains all information on the SU(2) representation content, at least for the special class of Kähler manifolds that we will consider
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