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Register a new userThe Kahler 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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We advocate that a generalized Kronheimer construction of the Kahler quotient crepant resolution $mathcal{M}_zeta longrightarrow mathbb{C}^3/Gamma$ of an orbifold singularity where $Gammasubset mathrm{SU(3)}$ 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 $mathbb{C}timesmathcal{M}_zeta$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $pmb{p}wedgepmb{p}=0$ which provides the Kahler analogue of the holomorphic sector in the hyperKahler 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.
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In supersymmetric (SUSY) field theory, there exist configurations which formally satisfy SUSY conditions but are not on original path integral contour. We refer to such configurations as complexified supersymmetric solutions (CSS). In this paper we discuss that CSS provide important information on large order behavior of weak coupling perturbative series in SUSY field theories. We conjecture that CSS with a bosonic (fermionic) free parameter give poles (zeroes) of Borel transformation of perturbative series whose locations are uniquely determined by actions of the solutions. We demonstrate this for various SUSY observables in 3d $mathcal{N}=2$ SUSY Chern-Simons matter theories on sphere. First we construct infinite number of CSS in general 3d $mathcal{N}=2$ SUSY theory with Lagrangian where adjoint scalar in vector multiplet takes a complex value and matter fields are nontrivial. Then we compare their actions with Borel transformations of perturbative expansions by inverse Chern-Simons levels for the observables and see agreement with our conjecture. It turns out that the CSS explain all the Borel singularities for this case.
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