Geometry & Physics Seminar - Talk by P. Fré

[Ugo Bruzzo]

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GEOMETRY & PHYSICS SEMINAR
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Prof. Pietro Fré (Università di Torino) will give a talk on

The Kähler Quotient Resolution of C3/Gamma singularities,
 the McKay correspondence and D=3 N=2 Chern-Simons gauge theories

on Thursday, October 19th, at 2:30 pm in room 004, SISSA building A.

Abstract: We advocate that a generalized Kronheimer construction of the Kähler
quotient crepant resolution M_z —> C3/Gamma of an orbifold singularity, where \Gamma\subset
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
C x M_z  as transverse space. We illustrate and
discuss many aspects of this construction emphasizing that the equation
p \wedge p = 0 which provides the Kähler analogue of the
holomorphic sector in the hyperK\"ahler 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 
G_Gamma of a locus  L_Gamma \subset Hom_\Gamma(QxR,R)
that has also a universal definition. We discuss the relation between the coset manifold
G_Gamma/\F_Gamma, the gauge group 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
H2(M_z). We discuss the relation with the 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 C2/Gamma singularities is utilized as a comparison
term and new formulae related with the complex presentation of Gibbons-Hawking
metrics are exhibited. Joint work with U. Bruzzo and A. Fino.