arXiv subjects
Publications and source records attributed to Amihay Hanany.
Connections between different M2-brane theories are established via the Higgs mechanism, which can be most efficiently studied on brane tilings. This leads to several M2-brane models, with brane tilings or Chern-Simons levels which have not been considered so far. The moduli spaces of these models are identified and examined in detail. The toric diagrams are constructed using Kasteleyn matrices and the forward algorithm.
We analyze the action of Toric (Seiberg) duality on the combined mesonic and baryonic moduli space of quiver gauge theories obtained from D3 branes at Calabi-Yau singularities. We analyze in particular the structure of the master space, the complete moduli space for one brane, for different toric phases of a given singularity. We show that the Hilbert Series for the largest component of the master space of different phases is the same, when refined with all the non anomalous charges. This reflects the fact that the quiver gauge theories associated with different phases are related by Seiberg duality when the number of branes is greater than one.
We present the complete classification of smooth toric Fano threefolds, known to the algebraic geometry literature, and perform some preliminary analyses in the context of brane-tilings and Chern-Simons theory on M2-branes probing Calabi-Yau fourfold singularities. We emphasise that these 18 spaces should be as intensely studied as their well-known counter-parts: the del Pezzo surfaces.
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.
We initiate a systematic investigation of the space of 2+1 dimensional quiver gauge theories, emphasising a succinct "forward algorithm". Few "order parametres" are introduced such as the number of terms in the superpotential and the number of gauge groups. Starting with two terms in the superpotential, we find a generating function, with interesting geometric interpretation, which counts the number of inequivalent theories for a given number of gauge groups and fields. We demonstratively list these theories for some low numbers thereof. Furthermore, we show how these theories arise from M2-branes probing toric Calabi-Yau 4-folds by explicitly obtaining the toric data of the vacuum moduli space. By observing equivalences of the vacua between markedly different theories, we see a new emergence of "toric duality".
A new infinite class of Chern-Simons theories is presented using brane tilings. The new class reproduces all known cases so far and introduces many new models that are dual to M2 brane theories which probe a toric non-compact CY 4-fold. The master space of the quiver theory is used as a tool to construct the moduli space for this class and the Hilbert Series is computed for a selected set of examples.
Partition functions for M2-brane theories in various backgrounds are computed. We consider in particular configurations of membranes at orbifold singularities preserving N=5 or N=6 supersymmetry. The worldvolume membrane theory for some of these configurations has been recently constructed in terms of N=6 Chern-Simons theories. The detailed structure of the partition functions as well as their transformation rules under the R-symmetry are explicitly computed using the Plethystic Programme.
We construct several examples of (2+1) dimensional N=2 supersymmetric Chern-Simons theories, whose moduli space is given by non-compact toric Calabi-Yau four-folds, which are not derivable from any (3+1) dimensional CFT. One such example is the gauge theory associated with the cone over Q^{111}. For several examples, we explicitly confirm the matter content, superpotential interactions and RG flows suggested by crystal models. Our results provide additional support to the idea that crystal models are relevant for describing the structure of these CFTs.
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.
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.
We use the plethystic programme and the Molien-Weyl fomula to compute generating functions, or Hilbert Series, which count gauge invariant operators in SQCD with the SO and Sp gauge groups. The character expansion technique indicates how the global symmetries are encoded in the generating functions. We obtain the full character expansion for each theory with arbitrary numbers of colours and flavours. We study the orientifold action on SQCD with the SU gauge group and examine how it gives rise to SQCD with the SO and Sp gauge groups. We establish that the classical moduli space of SQCD is not only irreducible, but is also an affine Calabi-Yau cone over a weighted projective variety.
The full moduli space M of a class of N=1 supersymmetric gauge theories is studied. For gauge theories living on a stack of D3-branes at Calabi-Yau singularities X, M is a combination of the mesonic and baryonic branches, the former being the symmetric product of X. In consonance with the mathematical literature, the single brane moduli space is called the master space F. Illustrating with a host of explicit examples, we exhibit many algebro-geometric properties of the master space such as when F is toric Calabi-Yau, behaviour of its Hilbert series, its irreducible components and its symmetries. In conjunction with the plethystic programme, we investigate the counting of BPS gauge invariants, baryonic and mesonic, using the geometry of F and show how its refined Hilbert series not only engenders the generating functions for the counting but also beautifully encode ``hidden'' global symmetries of the gauge theory which manifest themselves as symmetries of the complete moduli space M for arbitrary number of branes.
Supersymmetric gauge theories have an important but perhaps under-appreciated notion of a master space, which controls the full moduli space. For world-volume theories of D-branes probing a Calabi-Yau singularity X the situation is particularly illustrative. In the case of one physical brane, the master space F is the space of F-terms and a particular quotient thereof is X itself. We study various properties of F which encode such physical quantities as Higgsing, BPS spectra, hidden global symmetries, etc. Using the plethystic program we also discuss what happens at higher number N of branes. This letter is a summary and some extensions of the key points of a longer companion paper arXiv:0801.1585.
We introduce new techniques based on brane tilings to investigate D3-branes probing orientifolds of toric Calabi-Yau singularities. With these new tools, one can write down many orientifold models and derive the resulting low-energy gauge theories living on the D-branes. Using the set of ideas in this paper one recovers essentially all orientifolded theories known so far. Furthermore, new orientifolds of non-orbifold toric singularities are obtained. The possible applications of the tools presented in this paper are diverse. One particular application is the construction of models which feature dynamical supersymmetry breaking as well as the computation of D-instanton induced superpotential terms.
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.
We propose a programme for systematically counting the single and multi-trace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for world-volume quiver gauge theories of D-branes probing Calabi-Yau singularities, an illustrative case to which the programme is not limited, though in which a full intimate web of relations between the geometry and the gauge theory manifests herself. We can also use generalisations of Hardy-Ramanujan to compute the entropy of gauge theories from the plethystic exponential. In due course, we also touch upon fascinating connections to Young Tableaux, Hilbert schemes and the MacMahon Conjecture.
We show how it is possible to use the plethystic program in order to compute baryonic generating functions that count BPS operators in the chiral ring of quiver gauge theories living on the world volume of D branes probing a non compact CY manifold. Special attention is given to the conifold theory and the orbifold C^2/Z_2 times C, where exact expressions for generating functions are given in detail. This paper solves a long standing problem for the combinatorics of quiver gauge theories with baryonic moduli spaces. It opens the way to a statistical analysis of quiver theories on baryonic branches. Surprisingly, the baryonic charge turns out to be the quantized Kahler modulus of the geometry.
This note is presenting the generating functions which count the BPS operators in the chiral ring of a N=2 quiver gauge theory that lives on N D3 branes probing an ALE singularity. The difficulty in this computation arises from the fact that this quiver gauge theory has a moduli space of vacua that splits into many branches -- the Higgs, the Coulomb and mixed branches. As a result there can be operators which explore those different branches and the counting gets complicated by having to deal with such operators while avoiding over or under counting. The solution to this problem turns out to be very elegant and is presented in this note. Some surprises with "surgery" of generating functions arises.