Properly ordered dimers, RR-charges, and an efficient inverse algorithm

The $\mathcal{N}=1$ superconformal field theories that arise in AdS-CFT from placing a stack of D3-branes at the singularity of a toric Calabi-Yau threefold can be described succinctly by dimer models. We present an efficient algorithm for constructing a dimer model from the geometry of the Calabi-Yau. Since not all dimers produce consistent field theories, we perform several consistency checks on the field theories produced by our algorithm: they have the correct number of gauge groups, their cubic anomalies agree with the Chern-Simons coefficients in the AdS dual, and all gauge invariant chiral operators satisfy the unitarity bound. We also give bounds on the ratio of the central charge of the theory to the area of the toric diagram. To prove these results, we introduce the concept of a properly ordered dimer.Comment: 33 pages, 19 figures, some corrections and clarification

Bounding Selmer groups for the Rankin--Selberg convolution of Coleman families

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$.Comment: Final version. To appear in Canadian Jour. Mat

Gluon energy loss in the gauge-string duality

We estimate the stopping length of an energetic gluon in a thermal plasma of strongly coupled N=4 super-Yang-Mills theory by representing the gluon as a doubled string rising up out of the horizon.Comment: 33 pages, 8 figures. v2: minor improvement

Matrix Models for Supersymmetric Chern-Simons Theories with an ADE Classification

We consider N=3 supersymmetric Chern-Simons (CS) theories that contain product U(N) gauge groups and bifundamental matter fields. Using the matrix model of Kapustin, Willett and Yaakov, we examine the Euclidean partition function of these theories on an S^3 in the large N limit. We show that the only such CS theories for which the long range forces between the eigenvalues cancel have quivers which are in one-to-one correspondence with the simply laced affine Dynkin diagrams. As the A_n series was studied in detail before, in this paper we compute the partition function for the D_4 quiver. The D_4 example gives further evidence for a conjecture that the saddle point eigenvalue distribution is determined by the distribution of gauge invariant chiral operators. We also see that the partition function is invariant under a generalized Seiberg duality for CS theories.Comment: 20 pages, 3 figures; v2 refs added; v3 conventions in figure 3 altered, version to appear in JHE

Vanishing theorems for Shimura varieties at unipotent level

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite $\Gamma_1(p^\infty)$-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of the Shimura datum splits at $p$. This generalizes and strengthens the vanishing result proved in "Shimura varieties at level $\Gamma_1(p^\infty)$ and Galois representations". As an application of this vanishing theorem, we prove a result on the codimensions of ordinary completed homology for the same groups, analogous to conjectures of Calegari--Emerton for completed (Borel--Moore) homology.Comment: 38 pages, minor revisions to improve expositio

The ABCDEF's of Matrix Models for Supersymmetric Chern-Simons Theories

We consider N = 3 supersymmetric Chern-Simons gauge theories with product unitary and orthosymplectic groups and bifundamental and fundamental fields. We study the partition functions on an S^3 by using the Kapustin-Willett-Yaakov matrix model. The saddlepoint equations in a large N limit lead to a constraint that the long range forces between the eigenvalues must cancel; the resulting quiver theories are of affine Dynkin type. We introduce a folding/unfolding trick which lets us, at the level of the large N matrix model, (i) map quivers with orthosymplectic groups to those with unitary groups, and (ii) obtain non-simply laced quivers from the corresponding simply laced quivers using a Z_2 outer automorphism. The brane configurations of the quivers are described in string theory and the folding/unfolding is interpreted as the addition/subtraction of orientifold and orbifold planes. We also relate the U(N) quiver theories to the affine ADE quiver matrix models with a Stieltjes-Wigert type potential, and derive the generalized Seiberg duality in 2 + 1 dimensions from Seiberg duality in 3 + 1 dimensions.Comment: 30 pages, 5 figure

From Necklace Quivers to the F-theorem, Operator Counting, and T(U(N))

The matrix model of Kapustin, Willett, and Yaakov is a powerful tool for exploring the properties of strongly interacting superconformal Chern-Simons theories in 2+1 dimensions. In this paper, we use this matrix model to study necklace quiver gauge theories with {\cal N}=3 supersymmetry and U(N)^d gauge groups in the limit of large N. In its simplest application, the matrix model computes the free energy of the gauge theory on S^3. The conjectured F-theorem states that this quantity should decrease under renormalization group flow. We show that for a simple class of such flows, the F-theorem holds for our necklace theories. We also provide a relationship between matrix model eigenvalue distributions and numbers of chiral operators that we conjecture holds more generally. Through the AdS/CFT correspondence, there is therefore a natural dual geometric interpretation of the matrix model saddle point in terms of volumes of 7-d tri-Sasaki Einstein spaces and some of their 5-d submanifolds. As a final bonus, our analysis gives us the partition function of the T(U(N)) theory on S^3.Comment: 3 figures, 41 pages; v2 minor improvements, refs adde

Operator Counting and Eigenvalue Distributions for 3D Supersymmetric Gauge Theories

We give further support for our conjecture relating eigenvalue distributions of the Kapustin-Willett-Yaakov matrix model in the large N limit to numbers of operators in the chiral ring of the corresponding supersymmetric three-dimensional gauge theory. We show that the relation holds for non-critical R-charges and for examples with {\mathcal N}=2 instead of {\mathcal N}=3 supersymmetry where the bifundamental matter fields are nonchiral. We prove that, for non-critical R-charges, the conjecture is equivalent to a relation between the free energy of the gauge theory on a three sphere and the volume of a Sasaki manifold that is part of the moduli space of the gauge theory. We also investigate the consequences of our conjecture for chiral theories where the matrix model is not well understood.Comment: 27 pages + appendices, 5 figure

Sum Rules from an Extra Dimension

Using the gravity side of the AdS/CFT correspondence, we investigate the analytic properties of thermal retarded Green's functions for scalars, conserved currents, the stress tensor, and massless fermions. We provide some results concerning their large and small frequency behavior and their pole structure. From these results, it is straightforward to prove the validity of various sum rules on the field theory side of the duality. We introduce a novel contraction mapping we use to study the large frequency behavior of the Green's functions.Comment: v2: 23 pages (plus appendix), revised presentation, discussion of branch cuts moved to appendix, and some minor changes; v1: 24 pages (plus appendix