scholarly journals CHEN–RUAN COHOMOLOGY OF ADE SINGULARITIES

2007 ◽  
Vol 18 (09) ◽  
pp. 1009-1059 ◽  
Author(s):  
FABIO PERRONI
Keyword(s):  
Cohomology Ring ◽  
Crepant Resolution ◽  
The Family ◽  
Crepant Resolution Conjecture

We study Ruan's cohomological crepant resolution conjecture [41] for orbifolds with transversal ADE singularities. In the An-case, we compute both the Chen–Ruan cohomology ring [Formula: see text] and the quantum corrected cohomology ring H*(Z)(q1,…,qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between [Formula: see text] and H*(Z)(-1) in the A1-case, verifying Ruan's conjecture. In the An-case, the family H*(Z)(q1,…,qn) is not defined for q1 = ⋯ = qn = -1. This implies that the conjecture should be slightly modified. We propose a new conjecture in the An-case (Conjecture 1.9). Finally, we prove Conjecture 1.9 in the A2-case by constructing an explicit isomorphism.

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 Crepant resolutions and open strings II

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Andrea Brini ◽  
Renzo Cavalieri
Keyword(s):  
Three Dimensional ◽  
Type A ◽  
Projective Planes ◽  
Closed String ◽  
The Body ◽  
The Other ◽  
Crepant Resolution ◽  
Open Strings ◽  
The Family ◽  
Crepant Resolutions

We recently formulated a number of Crepant Resolution Conjectures (CRC) for open Gromov-Witten invariants of Aganagic-Vafa Lagrangian branes and verified them for the family of threefold type A-singularities. In this paper we enlarge the body of evidence in favor of our open CRCs, along two different strands. In one direction, we consider non-hard Lefschetz targets and verify the disk CRC for local weighted projective planes. In the other, we complete the proof of the quantized (all-genus) open CRC for hard Lefschetz toric Calabi-Yau three dimensional representations by a detailed study of the G-Hilb resolution of $[C^3/G]$ for $G=\mathbb{Z}_2 \times \mathbb{Z}_2$. Our results have implications for closed-string CRCs of Coates-Iritani-Tseng, Iritani, and Ruan for this class of examples. Comment: v2: typos fixed, minor changes. v3: some minor points have been clarified, further typos fixed. v4: version accepted for publication on EPIGA

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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