Ab initio wall-crossing

Abstract

We derive supersymmetric quantum mechanics of n BPS objects with 3n position degrees of freedom and 4n fermionic partners with SO(4) R-symmetry. The potential terms, essential and sufficient for the index problem for nonthreshold BPS states, are universal, and 2(n − 1) dimensional classical moduli spaces M n emerge from zero locus of the potential energy. We emphasize that there is no natural reduction of the quantum mechanics to M n , contrary to the conventional wisdom. Nevertheless, via an index-preserving deformation that breaks supersymmetry partially, we derive a Dirac index on M n as the fundamental state counting quantity. This rigorously fills a missing link in the "Coulomb phase" wall-crossing formula in literature. We then impose Bose/Fermi statistics of identical centers, and derive the general wall-crossing formula, applicable to both BPS black holes and BPS dyons. Also explained dynamically is how the rational invariant ∼ Ω(β)/p 2 , appearing repeatedly in wall-crossing formulae, can be understood as the universal multiplicative factor due to p identical, coincident, yet unbound, BPS particles of charge β. Along the way, we also clarify relationships between field theory state countings and quantum mechanical indices. 1 hykim@phya.snu.ac.kr 2 jaemo@postech.ac.kr 3 zlwang@kias.re.kr 4 piljin@kias.re.kr #1 These development were, however, restricted to weakly coupled theories, even though it generalized greatly the previous decades of monopole/dyon studies in the fully supersymmetric setting of N = 2 and N = 4 Yang-Mills theories. Much of these findings, and their relation to the more conventional monopole dynamics from 1980's and 1990's, was summarized in a review article [12]. #4 This same numerical factor 1/p 2 had appeared before in the context of the D-brane bound state problems of 1990's [26, 27], where identical nature of the D-branes were also of some importance.

The Coulomb Branch Formula for Quiver Moduli Spaces

“…Wall-crossing. It was shown in §4.1 and §5.4 of [9] that the Coulomb branch formula satisfies the wall-crossing formula given in [14] (and further elaborated in [19][20][21]), provided the single-centered invariants Ω S (γ) stay constant across the wall.…”

Deuteron transverse densities in holographic QCD

“…Many other interesting works have been done in nuclear physics using AdS/QCD formalism (see Refs. [93,94,95,96] for detailed reviews).…”

Exact results in quiver quantum mechanics and BPS bound state counting

“…The wall crossing formula is physically interpreted and rederived in the Coulomb branch of the quiver quantum mechanics [7,8]. There are many important developments around the wall crossing formula [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. (See also the review of the developments [23].…”