Let R be an E 2 ring spectrum with zero odd dimensional homo-topy groups. Every map of ring spectra M U → R is represented by a map of E 2 ring spectra. If 2 is invertible in π 0 R, then every map of ring spectra M SO → R is represented by a map of E 2 ring spectra.

We compute the elliptic genera of two-dimensional N = (2, 2) and N = (0, 2) gauge theories via supersymmetric localization, for rank-one gauge groups. The elliptic genus is expressed as a sum over residues of a meromorphic function whose argument is the holonomy of the gauge field along both the spatial and the temporal directions of the torus. We illustrate our formulas by a few examples including the quintic Calabi-Yau, N = (2, 2) SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric N = (0, 2) model.

The superconformal index of the quiver gauge theory dual to type IIB string theory on the product of an arbitrary smooth Sasaki-Einstein manifold with five-dimensional AdS space is calculated both from the gauge theory and gravity viewpoints. We find complete agreement. Along the way, we find that the index on the gravity side can be expressed in terms of the Kohn-Rossi cohomology of the Sasaki-Einstein manifold and that the index of a quiver gauge theory equals the Euler characteristic of the cyclic homology of the Ginzburg dg algebra associated to the quiver.

We explore various aspects of the correspondence between dimer models and integrable systems recently introduced by Goncharov and Kenyon. Dimer models give rise to relativistic integrable systems that match those arising from 5d N = 1 gauge theories studied by Nekrasov. We apply the correspondence to dimer models associated to the Y p,0 geometries, showing that they give rise to the relativistic generalization of the periodic Toda chain originally studied by Ruijsenaars. The correspondence reduces the calculation of all conserved charges to a straightforward combinatorial problem of enumerating non-intersecting paths in the dimer model. We show how the usual periodic Toda chain emerges in the non-relativistic limit and how the Lax operator corresponds to the Kasteleyn matrix of the dimer model. We discuss how the dimer models for general Y p,q manifolds give rise to other relativistic integrable systems, generalizing the periodic Toda chain and construct the integrable systems for general Y p,p explicitly. The impurities introduced in the construction of Y p,q quivers are identified with impurities in twisted sl(2) XXZ spin chains. Finally we discuss how the physical concept of higgsing a dimer model provides an efficient method for producing new integrable systems starting from known ones. We illustrate this idea by constructing the integrable systems for higgsings of Y 4,0 .

Abstract:We study the geometric description of BPS states in supersymmetric theories with eight supercharges in terms of geodesic networks on suitable spectral curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from gauge theory to local Calabi-Yau threefolds and related models. The differential is multi-valued on the covering curve and features a new type of logarithmic singularity in order to account for D0-branes and non-compact D4-branes, respectively. We describe local rules for the three-way junctions of BPS trajectories relative to a particular framing of the curve. We reproduce BPS quivers of local geometries and illustrate the wall-crossing of finite-mass bound states in several new examples. We describe first steps toward understanding the spectrum of framed BPS states in terms of such "exponential networks".

The superconformal index of the quiver gauge theory dual to type IIB string theory on the product of an arbitrary smooth Sasaki-Einstein manifold with five-dimensional AdS space is calculated both from the gauge theory and gravity viewpoints. We find complete agreement. Along the way, we find that the index on the gravity side can be expressed in terms of the Kohn-Rossi cohomology of the Sasaki-Einstein manifold and that the index of a quiver gauge theory equals the Euler characteristic of the cyclic homology of the Ginzburg dg algebra associated to the quiver.

The low energy effective theory on a stack of D3-branes at a Calabi-Yau singularity is an N = 1 quiver gauge theory. The AdS/CFT correspondence predicts that the strong coupling dynamics of the gauge theory is described by weakly coupled type IIB supergravity on AdS 5 × L 5 , where L 5 is a Sasaki-Einstein manifold. Recent results on Calabi-Yau algebras efficiently determine the Hilbert series of any superconformal quiver gauge theory. We use the Hilbert series to determine the volume of the horizon manifold in terms of the fields of the quiver gauge theory. One corollary of the AdS/CFT conjecture is that the volume of the horizon manifold L 5 is inversely proportional to the a-central charge of the gauge theory. By direct comparison of the volume determined from the Hilbert series and the a-central charge, this prediction is proved independently of the AdS/CFT conjecture.

Abstract:We derive the superconformal index of the world-volume theory on M2-branes probing the cone over an arbitrary Sasaki-Einstein seven-manifold. The index is expressed in terms of the cohomology groups of the cone. We match our supergravity results with known results from gauge theory. Along the way we derive the spectrum of short KaluzaKlein multiplets on generic Sasaki-Einstein seven-manifolds.