Article

# Derived categories of small toric Calabi-Yau 3-folds and counting invariants

09/2008;

Source: arXiv

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**ABSTRACT:**The following is an expanded write-up of my short lecture series "Noncommutative Resolutions" given to the MSRI Graduate Student Workshop "Noncommutative Algebraic Geometry" during June 2012. These notes include the lecture material (Sections 1-6), background notes on quivers (Appendix A) and all the exercise sheets (Appendix B). The accompanying video lectures can be found at http://www.msri.org/web/msri/scientific/workshops/show/-/event/Wm921310/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we apply recent methods of localized GLSMs to make predictions for Gromov-Witten invariants of noncommutative resolutions, as defined by e.g. Kontsevich, and use those predictions to examine the connectivity of the SCFT moduli space. Noncommutative spaces, in the present sense, are defined by their sheaves, their B-branes. Examples of abstract CFT's whose B-branes correspond with those defining noncommutative spaces arose in examples of abelian GLSMs describing branched double covers, in which the double cover structure arises nonperturbatively. This note will examine the GLSM for P^7[2,2,2,2], which realizes this phenomenon. Its Landau-Ginzburg point is a noncommutative resolution of a (singular) branched double cover of P^3. Regardless of the complex structure of the large-radius P^7[2,2,2,2], the Landau-Ginzburg point is always a noncommutative resolution of a singular space, which begs the question of whether the noncommutative resolution is connected in SCFT moduli space by a complex structure deformation to a smooth branched double cover. Using recent localization techniques, we make a prediction for the Gromov-Witten invariants of the noncommutative resolution, and find that they do not match those of a smooth branched double cover, telling us that these abstract CFT's are not continuously connected to sigma models on smooth branched double covers.Journal of Geometry and Physics 12/2012; · 1.06 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with "exotic" SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.01/2013;

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