Framed BPS states

2010

380 citations recorded by the source corpus.

We consider a class of line operators in d = 4,N = 2 supersymmetric field theories which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call “framed BPS states.” These include halo bound states similar to those of d = 4,N = 2 supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the KontsevichSoibelman wall-crossing formula for the ordinary BPS particles, by reducing it to the semiprimitive wall-crossing formula. After reducing on S1, the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the “Darboux coordinates” on the moduli spaceM of the theory. Moreover, we introduce a “protected spin character” which keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of protected spin characters admit a multiplication which defines a deformation of the algebra of holomorphic functions on M. As an illustration of these ideas, we consider the six-dimensional (2, 0) field theory of A1 type compactified on a Riemann surface C. Here we show (extending previous results) that line operators are classified by certain laminations on a suitably decorated version of C, and we compute the spectrum of framed BPS states in several explicit examples. Finally we indicate some interesting connections to the theory of cluster algebras. ar X iv :1 00 6. 01 46 v2 [ he pth ] 6 O ct 2 01 2

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7f88b6d11f3c6db936554638d6c7c920edf01316

Citations in the corpus

• Witten index and wall crossing (2014, neutral): “A given quiver theory is a low energy dynamics of particular charge sector, and our index computes the relevant protected spin character [58] Tr [ (−1)3y33 ]”
• Four-dimensional SCFTs from M5-branes (2012, neutral): “This approach has been taken first in [2, 3] and then in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], often with remarkable efficacy.”
• Knot Homology from Refined Chern-Simons Theory (2011, neutral): “In some special cases, it can happen that the index and the Poincaré polynomials agree (for a famous example, see [39], and for more recent examples [40]).”
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions (2022, neutral): “Similar fusion rules have also recently been emphasized in [19] and echo the fusion categories characterizing topological lines in 1+1d as well as the operator product expansions of supersymmetric line defects explored for instance in [50, 51].”
• Wall-crossing in coupled 2d-4d systems (2011, neutral): “(As in [3], we refer to the BPS states of the original four-dimensional theory without defects as “vanilla.”
• On the Defect Group of a 6D SCFT (2015, neutral): “For example, in the context of four-dimensional theories, the spectrum of line operators and surface operators must be specified [1–6].”
• R-Twisting and 4d/2d Correspondences (2010, neutral): “(These five functions also play a privileged role in the physics of the corresponding N = 2 theories, namely they correspond to five distinguished line operators [56].”
• General Argyres-Douglas theory (2012, neutral): “The BPS spectrum and wall crossing behavior of above two class of theories are studied in [35, 3, 36, 37] , extended object like line operators and surface operators are studied in [38, 39].”
• Wall crossing from Boltzmann black hole halos (2010, neutral): “In supersymmetric gauge theories one can define an alternative version of this index as [21], Ωref(γ, y) = TrH(γ)(2J3)(−1)(−y) ≡ ∑”
• Generalized global symmetries (2014, neutral): “For a recent review see [44, 45] and references therein.”
• Schur indices, BPS particles, and Argyres-Douglas theories (2015, neutral): “For each fixed charge γ ∈ Γ the spectrum of one-particle BPS states is captured by an index, the protected spin character [18].”
• Spectral Networks (2012, neutral): “13) to those of [5, 21], we find that the enhanced BPS degeneracies reviewed in §3.”
• The Coulomb Branch of 3d N = 4 Theories (2015, neutral): “This same basic idea was used in [23] and later [24] to quantize algebras of line operators in four-dimensional theories, by forcing the line operators to lie in a common plane (see also the recent review [25]).”
• Fivebranes and Knots (2011, neutral): “A related road to a physical interpretation of Khovanov homology has appeared much more recently in a study of supersymmetric line operators in four-dimensional gauge theories with N = 2 supersymmetry [43].”