Crepant Resolutions of ${\mathbb{C}}^3/{\mathbb{Z}}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)

Abstract

As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold ${\mathbb{C}}^3/{\mathbb{Z}}_4$, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface ${\mathbb{F}}_2$. We determine the explicit Kähler geometry of the smooth resolved manifold $Y$, which is the total space of the canonical bundle of ${\mathbb{F}}_2$. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet-Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space $\mathbb{P}[1,1,2]$, while the other is the product of the ALE space $A_1$ by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.

arXiv comment71 pages, 18 figures, 11 tables. v2: 72 pages, revised after the referees' remarks, corrections in the metadata

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Cited by in corpus (3)

• Mass Deformations of Unoriented Quiver Theories
• Resolution a la Kronheimer of ${\mathbb{C}}^3/\Gamma$ singularities and the Monge-Ampere equation for Ricci-flat Kaehler metrics in view of D3-brane solutions of supergravity
• D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler-Einstein surfaces