Anomaly Cancellations in Brane Tilings

We re-interpret the anomaly cancellation conditions for the gauge symmetries and the baryonic flavor symmetries in quiver gauge theories realized by the brane tilings from the viewpoint of flux conservation on branes.Comment: 10 pages, LaTeX; v2: minor corrections, a note on the zero-form flux adde

Counting Orbifolds

We present several methods of counting the orbifolds C^D/Gamma. A correspondence between counting orbifold actions on C^D, brane tilings, and toric diagrams in D-1 dimensions is drawn. Barycentric coordinates and scaling mechanisms are introduced to characterize lattice simplices as toric diagrams. We count orbifolds of C^3, C^4, C^5, C^6 and C^7. Some remarks are made on closed form formulas for the partition function that counts distinct orbifold actions.Comment: 69 pages, 9 figures, 24 tables; minor correction

Calabi-Yau Orbifolds and Torus Coverings

The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to the counting of orbifolds of toric Calabi-Yau singularities, with particular attention to Abelian orbifolds of C^D. By doing so, the work introduces a novel analytical method for counting Abelian orbifolds, verifying previous algorithm results. One identifies a p-fold cover of the torus T^{D-1} with an Abelian orbifold of the form C^D/Z_p, for any dimension D and a prime number p. The counting problem leads to polynomial equations modulo p for a given Abelian subgroup of S_D, the group of discrete symmetries of the toric diagram for C^D. The roots of the polynomial equations correspond to orbifolds of the form C^D/Z_p, which are invariant under the corresponding subgroup of S_Ds. In turn, invariance under this subgroup implies a discrete symmetry for the corresponding quiver gauge theory, as is clearly seen by its brane tiling formulation.Comment: 33 pages, 5 figures, 7 tables; version published on JHE

R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization

We conjecture a general formula for assigning R-charges and multiplicities for the chiral fields of all gauge theories living on branes at toric singularities. We check that the central charge and the dimensions of all the chiral fields agree with the information on volumes that can be extracted from toric geometry. We also analytically check the equivalence between the volume minimization procedure discovered in hep-th/0503183 and a-maximization, for the most general toric diagram. Our results can be considered as a very general check of the AdS/CFT correspondence, valid for all superconformal theories associated with toric singularities.Comment: 43 pages, 17 figures; minor correction

Brane Tilings and Specular Duality

We study a new duality which pairs 4d N=1 supersymmetric quiver gauge theories. They are represented by brane tilings and are worldvolume theories of D3 branes at Calabi-Yau 3-fold singularities. The new duality identifies theories which have the same combined mesonic and baryonic moduli space, otherwise called the master space. We obtain the associated Hilbert series which encodes both the generators and defining relations of the moduli space. We illustrate our findings with a set of brane tilings that have reflexive toric diagrams.Comment: 42 pages, 16 figures, 5 table

Phases of M2-brane Theories

We investigate different toric phases of 2+1 dimensional quiver gauge theories arising from M2-branes probing toric Calabi-Yau 4 folds. A brane tiling for each toric phase is presented. We apply the 'forward algorithm' to obtain the toric data of the mesonic moduli space of vacua and exhibit the equivalence between the vacua of different toric phases of a given singularity. The structures of the Master space, the mesonic moduli space, and the baryonic moduli space are examined in detail. We compute the Hilbert series and use them to verify the toric dualities between different phases. The Hilbert series, R-charges, and generators of the mesonic moduli space are matched between toric phases.Comment: 60 pages, 28 figures, 6 tables. v2: minor correction

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

Symmetries of Abelian Orbifolds

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of multiplicative sequences, one for each cycle in the Cycle Index of the permutation group. A multiplicative sequence is controlled by its values on prime numbers and their pure powers. Therefore, we pay particular attention to orbifolds of the form C^D/Gamma where the order of Gamma is p^alpha. We propose a generalization of these sequences for any D and any p.Comment: 75 pages, 13 figures, 30 table