Quantum line defects and refined BPS spectra

We consider the q-nonabelianization map, which maps links L in a 3-manifold M to combinations of links $$ \tilde{L} $$ L ˜ in a branched N -fold cover $$ \tilde{M} $$ M ˜ . In quantum field theory terms, q-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional (2, 0) superconformal field theory of type $$ \mathfrak{gl} $$ gl (N ) on M × ℝ2,1, and we consider surface defects placed on L × {x4 = x5 = 0}; in the IR we have the (2, 0) theory of type gl (1) on $$ \tilde{M} $$ M ˜ × ℝ2,1, and put the defects on $$ \tilde{L} $$ L ˜ × {x4 = x5 = 0}. In the case M = ℝ3, q-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group U(N ). In the case M = C × ℝ, when the projection of L to C is a simple non-contractible loop, q-nonabelianization computes the protected spin character for framed BPS states in 4d $$ \mathcal{N} $$ N = 2 theories of class S. In the case N = 2 and M = C × ℝ, we give a concrete construction of the q-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering $$ \tilde{C}\to C $$ C ˜ → C .

Section: Jhep09(2020)153mentioning

confidence: 99%

q-nonabelianization for line defects

Neitzke

Yan

2020

“…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].…”

Section: Introduction and Discussionmentioning

confidence: 99%

“…In both cases the problem has a combinatorial solution for a large class of quivers. One of the main results of [14] is the identification of the protected spin character (2.22) with refined Donaldson-Thomas invariants of the moduli space of framed quiver representations Moving along the Coulomb branch B it will generically happen that the core charge will change and the framed BPS quiver mutate accordingly. This process can occur at anti-walls, loci where Z(γ)/ξ ∈ − i R + , with γ the charge of an ordinary BPS state.…”

Section: Jhep05(2021)224mentioning

confidence: 99%

“…Having fixed a positive basis {e i } n i=1 at a point of B, any stable framed BPS state has charge of the form γ = e f + i∈Q 0 d i e i , where e f is the core charge and d i ∈ Z ≥0 . We identify the BPS degeneracies with (eventually refined) Donaldson-Thomas invariants of framed quivers, assuming cyclic stability conditions, as in [9,12,14]. Therefore for a line defect labelled by α in the UV, we can write…”

Section: Framed Quivers and Cluster Transformationsmentioning

confidence: 99%

“…Having established what is the smallest physical fractional monodromy we can use it to define our dynamical system R. Now to construct the full set of allowed line operators we start from a chamber with a basis {e i } of Γ and construct core charges in an extension of Γ, with the condition that they have integral pairing with the element of the basis and between each other. From each of this core charges, assuming we can compute its line operator (for example with localization techniques [12,14]) we iterate the map R to generate a full orbit. The relation between R and wall-crossing (or equivalently the rules of quiver mutations) ensures that each element of the orbit is an allowed line operator, whose core charge obeys the Dirac quantization condition.…”

Section: Line Defects Quantum Monodromies and R-symmetriesmentioning

confidence: 99%

See 2 more Smart Citations

A note on discrete dynamical systems in theories of class S

Cirafici

2021

Self Cite

In this note we consider the set of line operators in theories of class S. We show that this set carries the action of a natural discrete dynamical system associated with the BPS spectrum. We discuss several applications of this perspective; the relation with global properties of the theory; the set of constraints imposed on the spectrum generator, in particular for the case of SU(2) $$ \mathcal{N} $$ N = 2*; and the relation between line defects and certain spherical Double Affine Hecke Algebras.

The quantum UV-IR map for line defects in $$ \mathfrak{gl} $$(3)-type class S theories

Neitzke

Yan

2022

We consider the quantum UV-IR map for line defects in class S theories of $$ \mathfrak{gl} $$ gl (3)-type. This map computes the protected spin character which counts framed BPS states with spin for the bulk-defect system. We give a geometric method of computing this map motivated by the physics of five-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric Yang-Mills theory, and compute it explicitly in various examples. As a spin-off we propose a new way of computing a certain specialization of the HOMFLY polynomial for links in ℝ3, as a sum over BPS webs attached to the link.