Quivers, Line Defects and Framed BPS Invariants

Abstract

A large class of N = 2 quantum field theories admits a BPS quiver description and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modelled by framed quivers. The associated spectral problem characterises the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution of this problem. We detail our formalism with several explicit examples.Once again to construct the pyramid arrangement we have to look at the relevant terms in the Jacobian algebra J W = ⊕ n≥0 J W,n :Now we write down the Jacobian algebra elements corresponding to cyclic modules (6.32) and summands with higher grading do not contribute to the classification of the fixed points due to the condition dim V f = 1 and equations (6.15) and (6.18) evaluated at B =B = 0. To write down J W,4 we have usedthanks to (6.16) and (6.24) respectively, andWe can write the cyclic elements of the Jacobian algebra as A 2ψ A 1 C 3 v} . (6.44) the cyclic vector v. As usual this is graded J W = n≥0 J W,n and the first few terms are is obtained from the analog situation for SU (3), equation (6.14). To build the pyramid arrangement we have to study modules generated by the action of the Jacobian algebra on a single vector v ∈ V f , with V f one dimensional. The Jacobian algebra thus generated is naturally graded J W = n≥0 J W,n , and the first few terms are easy to work outAt the next levels we have four independent vectorswhere we have used the F-term relations obtained from (7.16) to show7.19) 56 Similar arguments give J W,7 = {ψà 2 ψ A 3 φà 1C v ,ψ A 2 ψ A 3 φà 1C v} , J W,8 = {A 3ψÃ2 ψ A 3 φà 1C v ,à 3ψÃ2 ψ A 3 φà 1C v} , J W,9 = {φ A 3ψÃ2 ψ A 3 φà 1C v ,φà 3ψÃ2 ψ A 3 φà 1C v} , J W,10 = {à 1φ A 3ψÃ2 ψ A 3 φà 1C v , A 1φ A 3ψÃ2 ψ A 3 φà 1C v} . (7.20)

Quantum line defects and refined BPS spectra

“…Determining W L can be in general difficult; a few examples where discussed in [11]. The main difficulty is that when discussing a BPS quiver one has already taken a Wilsonian limit, by integrating out heavy degrees of freedom.…”

“…As a consequence the coupling of the quiver to the framing node cannot in general be derived from known BPS quivers by sending the mass of a particle to infinity. Such a coupling can be determined by string theory engineering [9] or by indirect methods [11].…”

“…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.…”

“…In certain cases, eventually by taking a scaling limit, such functions correspond to refined BPS degeneracies. The starting point of our construction are the results of [11] which establish a formalism to compute framed BPS spectra associated with line defects using equivariant localization. This note improves on this result by showing how one can extend the localization computation to include spin/refined information.…”