Zhao-Long Wang

Quiver invariants from intrinsic Higgs states

A bstractIn study of four-dimensional BPS states, quiver quantum mechanics plays a central role. The Coulomb phases capture the multi-centered nature of such states, and are well understood in the context of wall-crossing. The Higgs phases are given typically by F-term-induced complete intersections in the ambient D-term-induced varieties, and the ground states can be far more numerous than the Coulomb phase counterparts. We observe that the Higgs phase BPS states are naturally and geometrically grouped into two parts, with one part given by the pulled-back cohomology from the D-term-induced ambient space. We propose that these pulled-back states are in one-to-one correspondence with the Coulomb phase states. This also leads us to conjecture that the index associated with the rest, intrinsic to the Higgs phase, is a fundamental invariant of quivers, independent of branches. For simple circular quivers, these intrinsic Higgs states belong to the middle cohomology and thus are all angular momentum singlets, supporting the single-center black hole interpretation.

Ab initio wall-crossing

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 non-threshold BPS states, are universal, and 2(nu2009−u20091) dimensional classical moduli spaces

BPS States, Refined Indices, and Quiver Invariants

{mathcal{M}_n}

f(R) gravities, Killing spinor equations, “BPS” domain walls and cosmology

emerge from zero locus of the potential energy. We emphasize that there is no natural reduction of the quantum mechanics to

Pseudo-supersymmetry, consistent sphere reduction and Killing spinors for the bosonic string

{mathcal{M}_n}

Fermi surfaces and analytic Green’s functions from conformal gravity

, contrary to the conventional wisdom. Nevertheless, via an index-preserving deformation that breaks supersymmetry partially, we derive a Dirac index on Mn 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 invariantu2009∼u2009Ω(β)/p2, 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.

Supersymmetric asymptotic AdS and Lifshitz solutions in Einstein-Weyl and conformal supergravities

A bstractFor Du2009=u20094 BPS state construction, counting, and wall-crossing thereof, quiver quantum mechanics offers two alternative approaches, the Coulomb phase and the Higgs phase, which sometimes produce inequivalent counting. The authors have proposed, in arXiv:1205.6511, two conjectures on the precise relationship between the two, with some supporting evidences. Higgs phase ground states are naturally divided into the Intrinsic Higgs sector, which is insensitive to wall-crossings and thus an invariant of quiver, plus a pulled-back ambient cohomology, conjectured to be an one-to-one image of Coulomb phase ground states. In this note, we show that these conjectures hold for all cyclic quivers with Abelian nodes, and further explore angular momentum and R-charge content of individual states. Along the way, we clarify how the protected spin character of BPS states should be computed in the Higgs phase, and further determine the entire Hodge structure of the Higgs phase cohomology. This shows that, while the Coulomb phase states are classified by angular momentum, the Intrinsic Higgs states are classified by R-symmetry.

Exact Greenʼs function and Fermi surfaces from conformal gravity

A bstractWe derive the condition on f(R) gravities that admit Killing spinor equations and construct explicit such examples. The Killing spinor equations can be used to reduce the fourth-order differential equations of motion to the first order for both the domain wall and FLRW cosmological solutions. We obtain exact “BPS” domain walls that describe the smooth Randall-Sundrum II, AdS wormholes and the RG flow from IR to UV. We also obtain exact smooth cosmological solutions that describe the evolution from an inflationary starting point with a larger cosmological constant to an ever-expanding universe with a smaller cosmological constant. In addition, We find exact smooth solutions of pre-big bang models, bouncing or crunching universes. An important feature is that the scalar curvature R of all these metrics is varying rather than a constant. Another intriguing feature is that there are two different f(R) gravities that give rise to the same “BPS” solution. We also study linearized f(R) gravities in (A)dS vacua.

Gauged Kaluza-Klein AdS pseudo-supergravity

Abstract Certain supergravity theories admit a remarkable consistent dimensional reduction in which the internal space is a sphere. Examples include type IIB supergravity reduced on S 5 , and eleven-dimensional supergravity reduced on S 4 or S 7 . Consistency means that any solution of the dimensionally-reduced theory lifts to give a solution in the higher dimension. Although supersymmetry seems to play a role in the consistency of these reductions, it cannot be the whole story since consistent sphere reductions of non-supersymmetric theories are also known, such as the reduction of the effective action of the bosonic string in any dimension D on either a 3-sphere or a ( D − 3 ) -sphere, retaining the gauge bosons of SO ( 4 ) or SO ( D − 2 ) respectively. We show that although there is no supersymmetry, there is nevertheless a natural Killing spinor equation for the D-dimensional bosonic string. A projection of the full integrability condition for these Killing spinors gives rise to the bosonic equations of motion (just as happens in the supergravity examples). Thus it appears that by extending the notion of supersymmetry to “pseudo-supersymmetry” in this way, one may be able to obtain a broader understanding of a relation between Killing spinors and consistent sphere reductions.

Pseudo-Killing spinors, pseudo-supersymmetric p-branes, bubbling and less-bubbling AdS spaces

A bstractWe construct T2-symmetric charged AdS black holes in conformal gravity. The most general solution up to an overall conformal factor contains three non-trivial parameters: the mass, electric charge and a quantity that can be identified as the massive spin-2 hair. We study the Dirac equation for the charged massless spinor in this background. The equation can be solved in terms of the general Heun’s function for generic frequency ω and wave number k. This allows us to obtain the analytic Green’s function G(ω, k) for both extremal and non-extremal black holes. For some special choice of back hole parameters, we find that the Green’s function reduces to simpler hypergeometric or confluent hypergeometric functions. We study the Fermi surfaces associated with the poles of the Green’s function with vanishing ω. We find examples where the Fermi surfaces for non-Fermi liquids as well as the characteristic Fermi ones can arise. We illustrate the non-trivial differences in the Green’s function and Fermi surfaces between the extremal and non-extremal black holes.