The Kähler Quotient Resolution of ${\mathbb{C}}^3/\Gamma$ singularities, the McKay correspondence and D=3 $\mathcal{N}=2$ Chern-Simons gauge theories

Abstract

We advocate that a generalized Kronheimer construction of the Kähler quotient crepant resolution ${\mathcal{M}}_{\zeta }⟶{\mathbb{C}}^3/\Gamma$ of an orbifold singularity where $\Gamma \subset \mathrm{SU}(3)$ is a finite subgroup naturally defines the field content and interaction structure of a superconformal Chern-Simons Gauge Theory. This is supposedly the dual of an M2-brane solution of $D=11$ supergravity with $\mathbb{C}\times {\mathcal{M}}_{\zeta }$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $p\wedge p=0$ which provides the Kähler analogue of the holomorphic sector in the hyperKähler moment map equations canonically defines the structure of a universal superpotential in the CS theory. The kernel of the above equation can be described as the orbit with respect to a quiver Lie group ${\mathcal{G}}_{\Gamma }$ of a locus $L_{\Gamma }\subset {\mathrm{Hom}}_{\Gamma }(\mathcal{Q}\otimes R,R)$ that has also a universal definition. We discuss the relation between the coset manifold ${\mathcal{G}}_{\Gamma }/{\mathcal{F}}_{\Gamma }$, the gauge group ${\mathcal{F}}_{\Gamma }$ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the tautological vector bundles that are in a one-to-one correspondence with the nontrivial irreps of $\Gamma$. These first Chern classes provide a basis for the cohomology group $H^2({\mathcal{M}}_{\zeta })$. We discuss the relation with conjugacy classes of $\Gamma$ and provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of ${\mathbb{C}}^2/\Gamma$ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.

120 pages, 7 figures. v2: 121 pages, a few minor changes. v3: references added. v4: 122 pages, 9 figures, minor changes in the presentation. Final version to be published in Commun. Math. Phys

References in corpus (5)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• D-brane Instantons on the T^6/Z_3 orientifold
• Dynamical supersymmetry breaking from unoriented D-brane instantons
• Supersymmetric M2-branes with Englert Fluxes and the simple group PSL(2,7)
• The Integral Form of D=3 Chern-Simons Theories Probing ${\mathbb{C}}^n/\Gamma$ Singularities

Cited by in corpus (5)

• 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
• Moduli spaces of $\mathbb{Z}/k\mathbb{Z}$-constellations over ${\mathbb{A}}^2$
• Crepant Resolutions of ${\mathbb{C}}^3/{\mathbb{Z}}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)
• D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler-Einstein surfaces