scholarly journals Quasimaps and stable pairs

2021 ◽  
Vol 9 ◽  
Author(s):  
Henry Liu
Keyword(s):  
Mirror Symmetry ◽  
Crepant Resolution ◽  
Box Counting ◽  
Stable Pairs ◽  
Crepant Resolution Conjecture

Abstract We prove an equivalence between the Bryan-Steinberg theory of $\pi $ -stable pairs on $Y = \mathcal {A}_{m-1} \times \mathbb {C}$ and the theory of quasimaps to $X = \text{Hilb}(\mathcal {A}_{m-1})$ , in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on Y arising from 3D mirror symmetry for quasimaps to X, including the Donaldson-Thomas crepant resolution conjecture.

scholarly journals THE COHOMOLOGICAL CREPANT RESOLUTION CONJECTURE FOR ℙ(1,3,4,4)

2009 ◽  
Vol 20 (06) ◽  
pp. 791-801 ◽  
Author(s):  
S. BOISSIÈRE ◽  
E. MANN ◽  
F. PERRONI
Keyword(s):  
Projective Space ◽  
Cohomology Ring ◽  
Weighted Projective Space ◽  
Crepant Resolution ◽  
Crepant Resolution Conjecture

We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space ℙ(1,3,4,4). To compute the quantum corrected cohomology ring, we combine the results of Coates–Corti–Iritani–Tseng on ℙ(1,1,1,3) and our previous results.

Where to Go from Here

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Mirror Symmetry ◽  
Derived Categories ◽  
Derived Category ◽  
Vector Spaces ◽  
Crepant Resolution ◽  
Hilbert Schemes ◽  
Mckay Correspondence ◽  
Homological Mirror Symmetry ◽  
Twisted Sheaves ◽  
A Finite Group

This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the equivariant-derived category of a variety endowed with the action of a finite group and the derived category of a crepant resolution of the quotient. This chapter gives the results from Bridgeland, King, and Reid for a special crepant resolution provided by Hilbert schemes and of Bezrukavnikov and Kaledin for symplectic vector spaces. A brief discussion of Kontsevich's homological mirror symmetry is included, as well as a discussion of stability conditions on triangulated categories. Twisted sheaves and their derived categories can be dealt with in a similar way, and some of the results in particular for K3 surfaces are presented.

scholarly journals Crepant resolutions and open strings

2019 ◽  
Vol 2019 (755) ◽  
pp. 191-245 ◽  
Author(s):  
Andrea Brini ◽  
Renzo Cavalieri ◽  
Dustin Ross
Keyword(s):  
Vector Space ◽  
Crepant Resolution ◽  
Complete Proof ◽  
Open Strings ◽  
Witten Theory ◽  
Crepant Resolutions ◽  
Symplectic Vector Space ◽  
Crepant Resolution Conjecture ◽  
Minimal Resolutions ◽  
Type Statement

AbstractIn the present paper, we formulate a Crepant Resolution Correspondence for open Gromov–Witten invariants (OCRC) of toric Lagrangian branes inside Calabi–Yau 3-orbifolds by encoding the open theories into sections of Givental’s symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan–Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates–Corti–Iritani–Tseng and Ruan, we furthermore propose (1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and (2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold {A_{n}}-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov–Witten theory of this family of targets.

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