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The Kähler Quotient Resolution of \({{\mathbb{C}}^3/ \Gamma}\) Singularities, the McKay Correspondence and \({D = 3\,\,\mathcal{N} = 2}\) Chern–Simons Gauge Theories

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Abstract

We advocate that the generalized Kronheimer construction of the Kähler quotient 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 and the interaction structure of a superconformal Chern–Simons gauge theory. This latter 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 type of constructions emphasizing that the equation p\(\wedge\)p = 0which 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. Furthermore the kernel \({\mathcal{D}_\Gamma}\) of the above equation can be described as the orbit with respect to a quiver Lie group \({\mathcal{G}_\Gamma}\) of a special locus \({L_\Gamma \subset \mathrm{Hom}_\Gamma(\mathcal{Q}\otimes R,R)}\) that has also a universal definition. We provide an extensive discussion of 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 so named tautological vector bundles that are in one-to-one correspondence with the nontrivial irreps of \({\Gamma}\). These first Chern classes are represented by (1,1)-forms on \({\mathcal{M}_\zeta}\) and provide a basis for the cohomology group \({H^2(\mathcal{M}_\zeta)}\). We also discuss the relation with conjugacy classes of \({\Gamma}\) and we 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.

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Acknowledgements

We acknowledge important clarifying discussions with our long time collaborators and friends Pietro Antonio Grassi, Dimitri Markushevich, Aleksander Sorin and Mario Trigiante. U.B.’s research is partially supported by PRIN 2015 “Geometria delle varietà algebriche” and INdAM-GNSAGA. This work was completed while U.B. was visiting the Instituto deMatemática e Estatística of the University of São Paulo, Brazil, supported by the FAPESP grant 2017/22091-9. He likes to thank FAPESP for providing support and his hosts, in particular P. Piccione, for their hospitality.

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Authors and Affiliations

  1. SISSA, Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136, Trieste, Italy

    Ugo Bruzzo

  2. INFN – Sezione di Trieste, Trieste, Italy

    Ugo Bruzzo

  3. Dipartimento di Matematica G. Peano, Universitá di Torino, Via Carlo Alberto 10, 10123, Turin, Italy

    Anna Fino

  4. Dipartimento di Fisica, INFN – Sezione di Torino, Universitá di Torino, Via P. Giuria 1, 10125, Turin, Italy

    Pietro Fré

  5. National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, Moscow, Russia, 115409

    Pietro Fré

  6. Arnold-Regge Center, Via P. Giuria 1, 10125, Turin, Italy

    Ugo Bruzzo, Anna Fino & Pietro Fré

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  1. Ugo Bruzzo
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Correspondence to Ugo Bruzzo.

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Communicated by N. Nekrasov

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Bruzzo, U., Fino, A. & Fré, P. The Kähler Quotient Resolution of \({{\mathbb{C}}^3/ \Gamma}\) Singularities, the McKay Correspondence and \({D = 3\,\,\mathcal{N} = 2}\) Chern–Simons Gauge Theories. Commun. Math. Phys. 365, 93–214 (2019). https://doi.org/10.1007/s00220-018-3203-z

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