On Framed Quivers, BPS Invariants and Defects

Abstract

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.

A note on discrete dynamical systems in theories of class S

“…These theories admit supersymmetric line defects which are associated to the spectral problem of computing framed BPS states [25]. In many cases such a problem can be approached by localization on quiver moduli spaces [9,[12][13][14][15], or on the moduli spaces of semiclassical configurations [5,32], by using cluster algebras [3,10,11,41], or via BPS graphs or spectral networks [21,22,27,34].…”

Quantum line defects and refined BPS spectra

“…In this note we have modified this construction to deal with a particular class of quivers, which describe the IR coupling of a line defect to a quantum field theory of class S. To ensure that the enumerative problem is well defined we have limited ourselves to quivers which have already been studied in [9,10,11]. For these quivers (and those which can be obtained by decoupling limits) the formalism can be applied to determine the spin content of the framed BPS spectra and therefore the quantum line operators.…”

“…Here we are using the physical requirement that V f ≃ C. Therefore we will loosely talk of cyclic modules generated by v f ∈ V f . For a more in depth discussion we refer the reader to [11,10].…”

“…In this note we will adapt the K-theoretic approach to the counting of BPS states developed by Nekrasov and Okounkov in [32,35] to study quantum line defects in class S supersymmetric quantum field theories. The formalism of framed BPS quivers associates to a line operator a certain enumerative problem of Donaldson-Thomas type, the counting of certain stable quiver representations [9,10,11,16]. In certain cases this problem can be solved combinatorially via virtual localization with respect to a natural toric action.…”

Defects, nested instantons and comet shaped quivers

“…5 which describes a flag manifold with extra adjoint hypers at each node. We notice that also the T SU [N ] quivers for defects studied [48][49][50][51][52][53] are based on flag manifold quivers but display a different field content. It should be possible to compare the two kind of defect gauge theories in suitable limits by finding an appropriate dictionary.…”