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**Geometric engineering of (framed) BPS states.**
*(English)*
Zbl 1365.81092

This long and diverse paper covers a very wide range of topics and problems in mathematical physics and algebraic geometry, but the main theme is the study of what happens to BPS states under “geometric engineering.” The latter is a construction introduced by S. Katz et al. [Nucl. Phys., B 497, No. 1–2, 173–195 (1997; Zbl 0935.81058)], which constructs an \(\mathcal N=2\) supersymmetric \(4d\) gauge theory out of a (noncompact) toric Calabi-Yau \(3\)-fold with singularities. The authors construct from these Calabi-Yau varieties certain collections of “fractional branes” and their \(\text{Ext}^1\)-quiver \(\mathcal Q\). In the IIA field theory limit the fractional branes are BPS particles and their central mass is computed. The gauge theory BPS states are the bound states of the fractional branes, and the derived category of the Calabi-Yau \(3\)-fold is related to a derived category associated to \(\mathcal Q\). Various conjectures about BPS indices and their relations to Donaldson-Thomas invariants are discussed.

The authors also study the IIB field theory limit and predict a complete geometric construction for the weak coupling BPS spectrum of the gauge theory. They study the “large radius” BPS spectrum and the analogous weak coupling BPS spectrum for the gauge theory, especially for gauge groups \(\mathrm{SU}(2)\) and \(\mathrm{SU}(3)\). A number of useful appendices discuss: (A) the explicit quivers that arise; (B) a review of the theory of motives and motivic measures; (C) Kronecker modules for the quiver with two nodes and two arrows from the first node to the second node; (D) background material on extensions; and (E) classification of fixed points.

The authors also study the IIB field theory limit and predict a complete geometric construction for the weak coupling BPS spectrum of the gauge theory. They study the “large radius” BPS spectrum and the analogous weak coupling BPS spectrum for the gauge theory, especially for gauge groups \(\mathrm{SU}(2)\) and \(\mathrm{SU}(3)\). A number of useful appendices discuss: (A) the explicit quivers that arise; (B) a review of the theory of motives and motivic measures; (C) Kronecker modules for the quiver with two nodes and two arrows from the first node to the second node; (D) background material on extensions; and (E) classification of fixed points.

Reviewer: Jonathan Rosenberg (College Park)

### MSC:

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

16G20 | Representations of quivers and partially ordered sets |

14C15 | (Equivariant) Chow groups and rings; motives |