Brane Tilings and M2 Branes

Brane tilings are efficient mnemonics for Lagrangians of N=2 Chern-Simons-matter theories. Such theories are conjectured to arise on M2-branes probing singular toric Calabi-Yau fourfolds. In this paper, a simple modification of the Kasteleyn technique is described which is conjectured to compute the three dimensional toric diagram of the non-compact moduli space of a single probe. The Hilbert Series is used to compute the spectrum of non-trivial scaling dimensions for a selected set of examples.Comment: 47 pages, 23 figure

Counting Chiral Operators in Quiver Gauge Theories

We discuss in detail the problem of counting BPS gauge invariant operators in the chiral ring of quiver gauge theories living on D-branes probing generic toric CY singularities. The computation of generating functions that include counting of baryonic operators is based on a relation between the baryonic charges in field theory and the Kaehler moduli of the CY singularities. A study of the interplay between gauge theory and geometry shows that given geometrical sectors appear more than once in the field theory, leading to a notion of "multiplicities". We explain in detail how to decompose the generating function for one D-brane into different sectors and how to compute their relevant multiplicities by introducing geometric and anomalous baryonic charges. The Plethystic Exponential remains a major tool for passing from one D-brane to arbitrary number of D-branes. Explicit formulae are given for few examples, including C^3/Z_3, F_0, and dP_1.Comment: 75 pages, 22 figure

Gravity duals to deformed SYM theories and Generalized Complex Geometry

We analyze the supersymmetry conditions for a class of SU(2) structure backgrounds of Type IIB supergravity, corresponding to a specific ansatz for the supersymmetry parameters. These backgrounds are relevant for the AdS/CFT correspondence since they are suitable to describe mass deformations or beta-deformations of four-dimensional superconformal gauge theories. Using Generalized Complex Geometry we show that these geometries are characterized by a closed nowhere-vanishing vector field and a modified fundamental form which is also closed. The vector field encodes the information about the superpotential and the type of deformation - mass or beta respectively. We also show that the Pilch-Warner solution dual to a mass-deformation of N =4 Super Yang-Mills and the Lunin-Maldacena beta-deformation of the same background fall in our class of solutions.Comment: LaTex, 29 page

Comments on the non-conformal gauge theories dual to Ypq manifolds

We study the infrared behavior of the entire class of Y(p,q) quiver gauge theories. The dimer technology is exploited to discuss the duality cascades and support the general belief about a runaway behavior for the whole family. We argue that a baryonic classically flat direction is pushed to infinity by the appearance of ADS-like terms in the effective superpotential. We also study in some examples the IR regime for the L(a,b,c) class showing that the same situation might be reproduced in this more general case as well.Comment: 48 pages, 27 figures; updated reference

A Note on Supersymmetric Type II Solutions of Lifshitz Type

We discuss a class of supersymmetric type II non-relativistic solutions with exact or asymptotic scale invariance. As already emerged from previous investigations, we find a clear correspondence between anisotropic d-dimensional vacua and relativistic solutions in (d + 1)-dimensions. We will show that supersymmetric four-dimensional Poincare' invariant backgrounds in type IIB can descend to analogous solutions with anisotropic scaling in t and (x, y). This result can be applied to scale invariant theories, domain walls interpolating between four-dimensional Lifshitz vacua and more general solutions with only asymptotic or approximate scaling behaviour.Comment: Added subsection on hyperscaling violation example

Towards Supergravity Duals of Chiral Symmetry Breaking in Sasaki-Einstein Cascading Quiver Theories

We construct a first order deformation of the complex structure of the cone over Sasaki-Einstein spaces Y^{p,q} and check supersymmetry explicitly. This space is a central element in the holographic dual of chiral symmetry breaking for a large class of cascading quiver theories. We discuss a solution describing a stack of N D3 branes and M fractional D3 branes at the tip of the deformed spaces.Comment: 28 pages, no figures. v2: typos, references and a note adde

Anomalies and O-plane charges in orientifolded brane tilings

We investigate orientifold of brane tilings. We clarify how the cancellations of gauge anomaly and Witten's anomaly are guaranteed by the conservation of the D5-brane charge. We also discuss the relation between brane tilings and the dual Calabi-Yau cones realized as the moduli spaces of gauge theories. Two types of flavor D5-branes in brane tilings and corresponding superpotentials of fundamental quark fields are proposed, and it is shown that the massless loci of these quarks in the moduli space correctly reproduce the worldvolume of flavor D7-branes in the Calabi-Yau cone dual to the fivebrane system.Comment: 46 pages, 19 figure

SQCD: A Geometric Apercu

We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties.Comment: 49 pages, 1 figur

Linear Sigma Models with Torsion

Gauged linear sigma models with (0,2) supersymmetry allow a larger choice of couplings than models with (2,2) supersymmetry. We use this freedom to find a fully linear construction of torsional heterotic compactifications, including models with branes. As a non-compact example, we describe a family of metrics which correspond to deformations of the heterotic conifold by turning on H-flux. We then describe compact models which are gauge-invariant only at the quantum level. Our construction gives a generalization of symplectic reduction. The resulting spaces are non-Kahler analogues of familiar toric spaces like complex projective space. Perturbatively conformal models can be constructed by considering intersections.Comment: 40 pages, LaTeX, 1 figure; references added; a new section on supersymmetry added; quantization condition revisite

Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics

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 (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.Comment: 59+1 pages, 7 Figure