BPS Graphs: From Spectral Networks to BPS Quivers

In this note, using Nekrasov's gauge origami framework, we study two different versions of the the BPS/CFT correspondence - first, the standard AGT duality and, second, the quiver W algebra construction which has been developed recently by Kimura and Pestun. The gauge origami enables us to work with both dualities simultaneously and find exact matchings between the parameters. In our main example of an A-type quiver gauge theory, we show that the corresponding quiver qW-algeb...

We apply the techniques provided by the recent works Gaiotto, Moore and Neitzke, to derive the most general spectrum generating functions for coupled 2d-4d $A_1$ theories of class ${\cal S}$, in presence of surface and line defects. As an application of the result, some well-known BPS spectra are reproduced. Our results apply to a large class of coupled 2d-4d systems, the corresponding spectrum generating functions can be easily derived from our general expressions.

For D=4 BPS state construction, counting, and wall-crossing thereof, quiver quantum mechanics offers two alternative approaches, the Coulomb phase and the Higgs phase, which sometimes produce inequivalent counting. The authors have proposed, in arXiv:1205.6511, two conjectures on the precise relationship between the two, with some supporting evidences. Higgs phase ground states are naturally divided into the Intrinsic Higgs sector, which is insensitive to wall-crossings and t...

We exactly evaluate the partition function (index) of N=4 supersymmetric quiver quantum mechanics in the Higgs phase by using the localization techniques. We show that the path integral is localized at the fixed points, which are obtained by solving the BRST equations, and D-term and F-term conditions. We turn on background gauge fields of R-symmetries for the chiral multiplets corresponding to the arrows between quiver nodes, but the partition function does not depend on the...

The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also deriv...

We study the matrix models that describe the BPS bound states of branes arising from the quiver picture of the derived category. These theories have a topological partition function that localizes to the Euler character of the anti-ghost bundle over the classical BPS moduli space. We examine the effective internal geometry of D6/D2 bound states in the local vertex geometry, using BPS 0-brane probes. The Kahler blowups of the Calabi-Yau that we find utilizing these quiver theo...

The BPS spectrum of certain N=2 supersymmetric field theories can be determined algebraically by studying the representation theory of BPS quivers. We introduce methods based on BPS quivers to study line defects. The presence of a line defect opens up a new BPS sector: framed BPS states can be bound to the defect. The defect can be geometrically described in terms of laminations on a curve. To a lamination we associate certain elements of the Leavitt path algebra of the BPS q...

Bipartite graphs, especially drawn on Riemann surfaces, have of late assumed an active role in theoretical physics, ranging from MHV scattering amplitudes to brane tilings, from dimer models and topological strings to toric AdS/CFT, from matrix models to dessins d'enfants in gauge theory. Here, we take a brief and casual promenade in the realm of brane tilings, quiver SUSY gauge theories and dessins, serving as a rapid introduction to the reader.

We study the basic features of BPS quiver mutations in 4D $\mathcal{N}=2$ supersymmetric quantum field theory with $G=ADE$ gauge symmetries.\ We show, for these gauge symmetries, that there is an isotropy group $\mathcal{G}_{Mut}^{G}$ associated to a set of quiver mutations capturing information about the BPS spectra. In the strong coupling limit, it is shown that BPS chambers correspond to finite and closed groupoid orbits with an isotropy symmetry group $\mathcal{G}_{strong...

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of $N$ D-brane probes for both $N \to \infty$ and finite $N$. The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``Plethystic Exponential'' provides a simple bridge between...