Wall crossing from Boltzmann black hole halos

Jan Manschot 2 Boris Pioline 0 Ashoke Sen 1 0 Laboratoire de Physique Theorique et Hautes Energies , CNRS UMR 7589, Universite Pierre et Marie Curie , 4 place Jussieu, 75252 Paris cedex 05, France 1 Harish-Chandra Research Institute , Chhatnag Road, Jhusi, Allahabad 211019, India 2 Institut de Physique Theorique , CEA Saclay, CNRS-URA 2306, 91191 Gif sur Yvette, France A key question in the study of N = 2 supersymmetric string or field theories is to understand the decay of BPS bound states across walls of marginal stability in the space of parameters or vacua. By representing the potentially unstable bound states as multi-centered black hole solutions in N = 2 supergravity, we provide two fully general and explicit formulae for the change in the (refined) index across the wall. The first, Higgs branch formula relies on Reineke's results for invariants of quivers without oriented loops, specialized to the Abelian case. The second, Coulomb branch formula results from evaluating the symplectic volume of the classical phase space of multi-centered solutions by localization. We provide extensive evidence that these new formulae agree with each other and with the mathematical results of Kontsevich and Soibelman (KS) and Joyce and Song (JS). The main physical insight behind our results is that the Bose-Fermi statistics of individual black holes participating in the bound state can be traded for Maxwell-Boltzmann statistics, provided the (integer) index () of the internal degrees of freedom carried by each black hole is replaced by an effective (rational) index () = Pm| (/m)/m2. A similar map also exists for the refined index. This observation provides a physical rationale for the appearance of the rational Donaldson-Thomas invariant () in the works of KS and JS. 1 Introduction and summary 2 3 4 Boltzmannian view of the wall-crossing 2.1 BPS states in N = 2 supergravity 2.2 Wall crossing: preliminaries 2.3 Bose-Fermi statistics to Maxwell-Boltzmann statistics 2.4 General wall-crossing formula and charge conservation 2.5 Semi-primitive wall-crossing from Boltzmann gas of black hole molecules 2.6 Refined index in supergravity Multi-black hole bound states and quiver quantum mechanics 3.1 Higgs branch analysis 3.2 Coulomb branch analysis 3.3 Comparison of the results of Higgs branch and Coulomb branch analysis Wall crossing from the Kontsevich-Soibelman formula 4.1 The KS formula 4.2 Charge conservation from KS formula 4.3 Primitive wall-crossing 4.4 Generic 3-body and 4-body contributions 4.5 Semi-primitive wall-crossing formulae and generalizations 4.5.1 Order one 4.5.2 Order two 4.5.3 Order three 4.6 12 > 0 case 4.7 Refined wall-crossing and motivic invariants 4.8 Semi-primitive refined wall-crossings and its generalizations 4.9 KS vs. supergravity Wall-crossing from the Joyce-Song formula 5.1 Statement of the JS formula 5.2 Index of supersymmetric bound states from the JS formula 5.3 Generic 2-body, 3-body and 4-body contributions 5.4 Semi-primitive wall-crossing formula from JS A Wall crossing formulae in special cases B D6-D0 bound states C Seiberg-Witten spectra and generalizations D U(N) quiver quantum mechanics from Boltzmann black hole halos Introduction and summary In quantum field theories and string theory vacua with extended supersymmetry, the spectrum of BPS states can sometimes be determined exactly in a weakly coupled region of the space of parameters (or vacua). In extrapolating the BPS spectrum to another point in parameter space, one must be wary of two issues: BPS states may pair up and disappear, and single particle states may decay into the continuum of multi-particle states. The first issue can be evaded by considering a suitable index (; ta), where is the vector of electric and magnetic charges carried by the state and ta parametrizes the value of the couplings (or moduli), designed such that contributions from long multiplets cancel. The index (; ta) is then a piecewise constant function of the parameters ta. To deal with the second problem, it is important to understand how (; ta) changes across certain codimension-one subspaces of the parameter space, known as walls of marginal stability, where a single-particle BPS state becomes marginally unstable against decay into two (or more) BPS states [16]. Initial progress in this direction for four-dimensional string vacua came from supergravity, where BPS states are represented by (in general multi-centered) classical black hole solutions. Since the class of multi-centered solutions potentially unstable at a certain wall of marginal stability exists only on one side of the wall [79], the discontinuity () in (, ta) is equal to the index of the multi-centered solutions with total charge , up to a sign depending whether one enters or leaves the side on which these solutions exist [10]. Based on this physical picture, one easily finds that the jump of the index in the simplest case, where the only configuration that ma (...truncated)