Toric Quiver Asymptotics and Mahler Measure: $\mathcal{N}=2$ BPS States

We define "BPS graphs" on punctured Riemann surfaces associated with $A_{N-1}$ theories of class $\mathcal{S}$. BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementa...

We study the growth of the ground state degeneracy in the Kronecker model of quiver quantum mechanics. This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in the context of BPS state counting in four-dimensional N=2 systems. For large ranks, the ground state degeneracy is exponential with slope a modular function that we are able to compute at integral values of its argument. We also observe that the exponential of the sl...

We provide a formalism using the $q$-Cartan matrix to compute the instanton partition function of quiver gauge theory on various manifolds. Applying this formalism to eight dimensional setups, we introduce the notion of double quiver gauge theory characterized by a pair of quivers. We also explore the BPS/CFT correspondence in eight dimensions based on the $q$-Cartan matrix formalism.

We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour in...

In this note we review a construction of a BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that this space contains elements of an equivariant generalized cohomology theory $E_G^{*}(-)$ of the quiver representation moduli space giving concretely Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial slice is compactified to a point, a circle or a torus respectively, and something more amorphous in...

We evaluate the large-$N$ behavior of the superconformal indices of toric quiver gauge theories, and use it to find the entropy functions of the dual electrically charged rotating $\mathrm{AdS}_5$ black holes. To this end, we employ the recently proposed Bethe Ansatz method, and find a certain set of solutions to the Bethe Ansatz Equations of toric theories. This, in turn, allows us to compute the large-$N$ behavior of the index for these theories, including the infinite fami...

We construct a statistical model that reproduces the BPS partition function of D4-D2-D0 bound states on a class of toric Calabi-Yau three-folds. The Calabi-Yau three-folds we consider are obtained by adding a compact two-cycle to $A_{N-1}$-ALE $\times \mathbb{C}$. We show that in the small radii limit of the Calabi-Yau the D4-D2-D0 partition function is correctly reproduced by counting the number of triangles and parallelograms.

The quiver Yangians were originally defined for the quiver and superpotential from string theory on general toric Calabi-Yau threefolds, and serve as BPS algebras of these systems. Their characters reproduce the unrefined BPS indices, which correspond to classical Donaldson-Thomas (DT) invariants. We generalize this construction in two directions. First, we show that this definition extends to arbitrary quivers with potentials. Second, we explain how to define the characters ...

We study the partition functions of BPS vortices and magnetic monopole operators, in gauge theories describing $N$ M2-branes. In particular, we explore two closely related methods to study the Cardy limit of the index on $S^2\times\mathbb{R}$. The first method uses the factorization of this index to vortex partition functions, while the second one uses a continuum approximation for the monopole charge sums. Monopole condensation confines most of the $N^2$ degrees of freedom e...

We consider large N U(N)^M thermal N=2 quiver gauge theories on S^1 x S^3. We obtain a phase diagram of the theory with R-symmetry chemical potentials, separating a low-temperature/high-chemical potential region from a high-temperature/low-chemical potential region. In close analogy with the N=4 SYM case, the free energy is of order O(1) in the low-temperature region and of order O(N^2 M) in the high-temperature phase. We conclude that the N=2 theory undergoes a first order H...