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

Bruzzo, Ugo; Fino, Anna; Fré, Pietro

doi:10.1007/s00220-018-3203-z

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

Ugo Bruzzo
(
- SISSA, Trieste and
- INFN, Trieste
)
,
Anna Fino
(
- Turin U.
)
,
Pietro Fré
(
- INFN, Turin and
- Moscow Phys. Eng. Inst. and
- Turin U.
)

Oct 3, 2017

122 pages

Published in:

Commun.Math.Phys. 365 (2019) 1, 93-214

Published: Jul 30, 2018

e-Print:

1710.01046 [hep-th]

DOI:

10.1007/s00220-018-3203-z

Report number:

ARC-17-6

View in:

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Abstract: (Springer)

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.

Note:

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

orbifold: singularity
group: Lie
gauge field theory: quiver
Chern-Simons term
superpotential
supergravity: 1
holomorphic
cohomology
supersymmetry: 2
Kaehler

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The Kähler Quotient Resolution of C3/Γ{{\mathbb{C}}^3/ \Gamma}C3/Γ Singularities, the McKay Correspondence and D=3 N=2{D = 3\,\,\mathcal{N} = 2}D=3N=2 Chern–Simons Gauge Theories

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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