scholarly journals On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

2019 ◽  
Vol 19 (6) ◽  
pp. 2715-2762
Author(s):  
Fabio Perroni ◽  
Alexander Rahm
Keyword(s):  
Crepant Resolution ◽  
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.

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.

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.

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