Young person’s guide to canonical singularities

“…I would like to thank S. Shenker and H. Verlinde for raising questions that stimulated this study, D. Morrison for comments and for pointing out reference [20], H. Ooguri and S. Sethi for discussions, A. Karch and B. Kol for pointing out an error in an earlier version of section 5, and the Aspen Center for Physics for its hospitality.…”

Section: Acknowledgmentsmentioning

confidence: 99%

New “gauge” theories in six dimensions

Witten¹

1998

Advances in Theoretical and Mathematical Physics

111

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More general constructions are given of six-dimensional theories that look at low energy like six-dimensional super Yang-Mills theory. The constructions start with either parallel fivebranes in Type IIB, or M-theory on (C 2 x § 1 )/r for T a suitable finite group. Via these constructions, one can obtain six-dimensional theories with any simple gauge group, and SU(r) theories with any rational theta angle. A matrix construction of these theories is also possible.

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“…The singular locus of its coarse moduli space |P(1, 3, 4, 4)| is the disjoint union of an isolated singularity of type (1/3)(1, 1, 1) (we use Reid's notation in [17]) and a transversal A 3 -singularity. Using toric methods, we construct a crepant resolution Z of |P(1, 3, 4, 4)|.…”

Section: §1 Introductionmentioning

confidence: 99%

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

Boissière¹,

Mann²,

Perroni³

2011

Nagoya Math. J.

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Abstract. We prove a formula computing the Gromov-Witten invariants of genus zero with three marked points of the resolution of the transversal A3-singularity of the weighted projective space P(1, 3, 4, 4) using the theory of deformations of surfaces with An-singularities. We use this result to check Ruan's conjecture for the stack P(1, 3, 4, 4). §1. IntroductionThe main results of this paper concern the weighted projective space P(1, 3, 4, 4). The singular locus of its coarse moduli space |P(1, 3, 4, 4)| is the disjoint union of an isolated singularity of type (1/3)(1, 1, 1) (we use Reid's notation in [17]) and a transversal A 3 -singularity. Using toric methods, we construct a crepant resolution Z of |P(1, 3, 4, 4)|. In Theorem 3.3.1, we determine a formula for certain Gromov-Witten invariants of Z over the A 3 -singularity using the theory of deformations of surfaces with rational double points and the deformation invariance property of Gromov-Witten invariants. We then apply this result in Theorem 5.2.1 to construct a ring isomorphism-predicted in [18] by Ruan's cohomological crepant resolution conjecture -between the quantum corrected cohomology ring of Z and the Chen-Ruan orbifold cohomology of P(1, 3, 4, 4), after evaluating the quantum parameters related to the transversal A 3 -singularity to a fourth root of the unity and putting the last parameter to zero. This last evaluation is quite surprising (in [2], we show that this parameter can be evaluated to 1). To confirm this property, we show in Proposition 5.3.1 that, for all weighted projective spaces P(1, . . . , 1, n) with n weights equal to 1 which have only

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Young person’s guide to canonical singularities

Cited by 458 publications

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Complex projective 3-fold with non-negative canonical Euler-Poincare characteristic

Complex projective 3-fold with non-negative canonical Euler-Poincare characteristic

New “gauge” theories in six dimensions

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

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