The Kahler Quotient Resolution of C-3/Gamma Singularities, the McKay Correspondence and D=3N=2 Chern-Simons Gauge Theories

Author(s)

Bruzzo, Ugo ^{[1, 2, 3]} ; Fino, Anna ^{[4, 2]} ; Fre, Pietro ^{[5, 2, 6]}

Affiliation

^[1] INFN, Sez Trieste, Trieste - Italy

^[2] Arnold Regge Ctr, Via P Giuria 1, I-10125 Turin - Italy

^[3] SISSA, Scuola Int Super Studi Avanzati, Via Bonomea 265, I-34136 Trieste - Italy

^[4] Univ Torino, Dipartimento Matemat G Peano, Via Carlo Alberto 10, I-10123 Turin - Italy

^[5] Natl Res Nucl Univ MEPhI, Moscow Engn Phys Inst, Kashirskoye Shosse 31, Moscow 115409 - Russia

^[6] Univ Torino, INFN, Sez Torino, Dipartimento Fis, Via P Giuria 1, I-10125 Turin - Italy

• Document type: Journal article
• Source: Communications in Mathematical Physics; v. 365, n. 1, p. 93-214, JAN 2019.
• Web of Science Citations: 1

Abstract

We advocate that the generalized Kronheimer construction of the Kahler quotient crepant resolution M?C3/ of an orbifold singularity wherSU(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 CxM as transverse space. We illustrate and discuss many aspects of this type of constructions emphasizing that the equation p perpendicular to p = 0which 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. Furthermore the kernel D of the above equation can be described as the orbit with respect to a quiver Lie group G of a special locus LHom(Q circle times R,R) that has also a universal definition. We provide an extensive discussion of the relation between the coset manifold G/F, the gauge group F 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 . These first Chern classes are represented by (1,1)-forms on M and provide a basis for the cohomology group H2(M). We also discuss the relation with conjugacy classes of 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 C2/ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited. (AU)

• FAPESP's process: 17/22091-9 - Cohomology of Lie algebroids in the holomorphic and algebraic settings: theory and applications
• Grantee: Paolo Piccione
• Support Opportunities: Research Grants - Visiting Researcher Grant - International