English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We advocate that a generalized Kronheimer construction of the K\"ahlerquotient crepant resolution $\mathcal{M}_\zeta \longrightarrow\mathbb{C}^3/\Gamma$ of an orbifold singularity where $\Gamma\subset\mathrm{SU(3)}$ is a finite subgroup naturally defines the field content andinteraction structure of a superconformal Chern-Simons Gauge Theory. This issupposedly the dual of an M2-brane solution of $D=11$ supergravity with$\mathbb{C}\times\mathcal{M}_\zeta$ as transverse space. We illustrate anddiscuss many aspects of this of constructions emphasizing that the equation$\pmb{p}\wedge\pmb{p}=0$ which provides the K\"ahler analogue of theholomorphic sector in the hyperK\"ahler moment map equations canonicallydefines the structure of a universal superpotential in the CS theory. Thekernel of the above equation can be described as the orbit with respect to aquiver Lie group $\mathcal{G}_\Gamma$ of a locus $L_\Gamma \subset\mathrm{Hom}_\Gamma(\mathcal{Q}\otimes R,R)$ that has also a universaldefinition. 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 mapequations and the first Chern classes of the tautological vector bundles thatare 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 emphasizingthe role of a generalized McKay correspondence. The case of the ALE manifoldresolution of $\mathbb{C}^2/\Gamma$ singularities is utilized as a comparisonterm and new formulae related with the complex presentation of Gibbons-Hawkingmetrics are exhibited.

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