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On framed quivers, BPS invariants and defects. (English) Zbl 1394.14012

Summary: In this note, we review some of the uses of framed quivers to study BPS invariants of Donaldson-Thomas type. We will mostly focus on non-compact Calabi-Yau threefolds. In certain cases, the study of these invariants can be approached as a generalized instanton problem in a six dimensional cohomological Yang-Mills theory. One can construct a quantum mechanics model based on a certain framed quiver which locally describes the theory around a generalized instanton solution. The problem is then reduced to the study of the moduli spaces of representations of these quivers. Examples include the affine space and noncommutative crepant resolutions of orbifold singularities. In the second part of the survey, we introduce the concepts of defects in physics and argue with a few examples that they give rise to a modified Donaldson-Thomas problem. We mostly focus on divisor defects in six dimensional Yang-Mills theory and their relation with the moduli spaces of parabolic sheaves. In certain cases also, this problem can be reformulated in terms of framed quivers.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
81T13 Yang-Mills and other gauge theories in quantum field theory
32Q25 Calabi-Yau theory (complex-analytic aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
16G20 Representations of quivers and partially ordered sets
81T60 Supersymmetric field theories in quantum mechanics
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References:

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