42 research outputs found
Brane Tilings and Quiver Gauge Theories
This work presents recent developments on brane tilings and their vacuum moduli
spaces.
Brane tilings are bipartite periodic graphs on the torus and represent 4d N = 1
supersymmetric worldvolume theories living on D3-branes probing Calabi-Yau 3-fold
singularities. The graph and combinatorial properties of brane tilings make the set
of supersymmetric quiver theories represented by them one of the largest and richest
known so far. The aim of this work is to give a concise pedagogical introduction to brane
tilings and a summary on recent exciting advancement on their classification, dualities
and construction.
At first, particular focus is given on counting distinct Abelian orbifolds of the form
C3/[gamma]. The presented counting of Abelian orbifolds of C3 and in more general of CD gives a first insight on the rich combinatorial nature of brane tilings. Following the classification theme, the work proceeds with the identification of all brane tilings whose
mesonic moduli spaces as toric Calabi-Yau 3-folds are represented by reflexive polygons.
There are 16 of these special convex lattice polygons. It is shown that 30 brane tilings
are associated with them. Some of these brane tilings are related by a correspondence
known as toric duality.
The classification of brane tilings with reflexive toric diagrams led to the discovery
of a new correspondence between brane tilings which we call specular duality. The
new correspondence identifies brane tilings with the same master space - the combined
mesonic and baryonic moduli space. As a by-product, the new correspondence paves
the way for constructing brane tilings which are not confined to the torus but are on
Riemann surfaces with arbitrary genus. We give the first classification of genus 2 brane
tilings, illustrate the corresponding supersymmetric quiver theories and analyse their
vacuum moduli spaces.Open Acces
On the Master Space for Brane Brick Models
We systematically study the master space of brane brick models that represent
a large class of 2d (0,2) quiver gauge theories. These 2d (0,2) theories are
worldvolume theories of D1-branes that probe singular toric Calabi-Yau 4-folds.
The master space is the freely generated space of chiral fields subject to the
J- and E-terms and the non-abelian part of the gauge symmetry. We investigate
several properties of the master space for abelian brane brick models with U(1)
gauge groups. For example, we calculate the Hilbert series, which allows us by
using the plethystic programme to identify the generators and defining
relations of the master space. By studying several explicit examples, we also
show that the Hilbert series of the master space can be expressed in terms of
characters of irreducible representations of the full global symmetry of the
master space.Comment: 48 pages, 5 figures, 10 table
Machine Learning Regularization for the Minimum Volume Formula of Toric Calabi-Yau 3-folds
We present a collection of explicit formulas for the minimum volume of
Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric
Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an
infinite class of 4d N=1 supersymmetric gauge theories, which are realized as
worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under
the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is
inversely proportional to the central charge of the corresponding 4d N=1
superconformal field theories. The presented formulas for the minimum volume
are in terms of geometric invariants of the toric Calabi-Yau 3-folds. These
explicit results are derived by implementing machine learning regularization
techniques that advance beyond previous applications of machine learning for
determining the minimum volume. Moreover, the use of machine learning
regularization allows us to present interpretable and explainable formulas for
the minimum volume. Our work confirms that, even for extensive sets of toric
Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with
remarkable accuracy.Comment: 15 pages, 9 figures, 4 table
Brane Tilings and Reflexive Polygons
Reflexive polygons have attracted great interest both in mathematics and in
physics. This paper discusses a new aspect of the existing study in the context
of quiver gauge theories. These theories are 4d supersymmetric worldvolume
theories of D3 branes with toric Calabi-Yau moduli spaces that are conveniently
described with brane tilings. We find all 30 theories corresponding to the 16
reflexive polygons, some of the theories being toric (Seiberg) dual to each
other. The mesonic generators of the moduli spaces are identified through the
Hilbert series. It is shown that the lattice of generators is the dual
reflexive polygon of the toric diagram. Thus, the duality forms pairs of quiver
gauge theories with the lattice of generators being the toric diagram of the
dual and vice versa.Comment: 142 pages, 50 figures, 61 tables; version to be published in
Fortschritte der Physi
Brane brick models, toric Calabi-Yau 4-folds and 2d (0,2) quivers
We introduce brane brick models, a novel type of Type IIA brane congurations consisting of D4-branes ending on an NS5-brane. Brane brick models are T-dual to D1-branes over singular toric Calabi-Yau 4-folds. They fully encode the innite class of 2d (generically) N = (0; 2) gauge theories on the worldvolume of the D1-branes and streamline their connection to the probed geometries. For this purpose, we also introduce new combinatorial procedures for deriving the Calabi-Yau associated to a given gauge theory and vice versa
Brane Brick Models for the Sasaki-Einstein 7-Manifolds Y^{p,k}(CP^1 x CP^1) and Y^{p,k}(CP^2)
The 2d (0,2) supersymmetric gauge theories corresponding to the classes of
Y^{p,k}(CP^1 x CP^1) and Y^{p,k}(CP^2) manifolds are identified. The complex
cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau
4-folds. These infinite families of geometries are the largest ones for
Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions
are known explicitly. This work therefore presents the largest classification
of 2d (0,2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds
with known metrics.Comment: 42 pages, 21 figures, 1 tabl
New directions in bipartite field theories
We perform a detailed investigation of Bipartite Field Theories (BFTs), a general class of 4d N = 1 gauge theories which are defined by bipartite graphs. This class of theories is considerably expanded by identifying a new way of assigning gauge symmetries to graphs. A new procedure is introduced in order to determine the toric Calabi-Yau moduli spaces of BFTs. For graphs on a disk, we show that the matroid polytope for the corresponding cell in the Grassmannian coincides with the toric diagram of the BFT moduli space. A systematic BFT prescription for determining graph reductions is presented. We illustrate our ideas in infinite classes of BFTs and introduce various operations for generating new theories from existing ones. Particular emphasis is given to theories associated to non-planar graphs
Fano 3-Folds, Reflexive Polytopes and Brane Brick Models
Reflexive polytopes in n dimensions have attracted much attention both in mathematics and theoretical physics due to their connection to Fano n-folds and mirror symmetry. This work focuses on the 18 regular reflexive polytopes corresponding to smooth Fano 3-folds. For the first time, we show that all 18 regular reflexive polytopes have corresponding 2d (0, 2) gauge theories realized by brane brick models. These 2d gauge theories can be considered as the worldvolume theories of D1-branes probing the toric Calabi-Yau 4-singularities whose toric diagrams are given by the associated regular reflexive polytopes. The generators of the mesonic moduli space of the brane brick models are shown to form a lattice of generators due to the charges under the rank 3 mesonic flavor symmetry. It is shown that the lattice of generators is the exact polar dual reflexive polytope to the corresponding toric diagram of the brane brick model. This duality not only highlights the close relationship between the geometry and 2d gauge theory, but also opens up pathways towards new discoveries in relation to reflexive polytopes and brane brick models
Brane brick models in the mirror
Brane brick models are Type IIA brane configurations that encode the 2d N = (0; 2) gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We use mirror symmetry to improve our understanding of this correspondence and to provide a systematic approach for constructing brane brick models starting from geometry. The mirror configuration consists of D5-branes wrapping 4-spheres and the gauge theory is determined by how they intersect. We also explain how 2d (0; 2) triality is realized in terms of geometric transitions in the mirror geometry. Mirror symmetry leads to a geometric unification of dualities in different dimensions, where the order of duality is n - 1 for a Calabi-Yau n-fold. This makes us conjecture the existence of a quadrality symmetry in 0d. Finally, we comment on how the M-theory lift of brane brick models connects to the classification of 2d (0; 2) theories in terms of 4-manifolds
Mass Deformations of Brane Brick Models
We investigate a class of mass deformations that connect pairs of 2d (0,2)
gauge theories associated to different toric Calabi-Yau 4-folds. These
deformations are generalizations to 2d of the well-known Klebanov-Witten
deformation relating the 4d gauge theories for the C^2/Z_2 x C orbifold and the
conifold. We investigate various aspects of these deformations, including their
connection to brane brick models and the relation between the change in the
geometry and the pattern of symmetry breaking triggered by the deformation. We
also explore how the volume of the Sasaki-Einstein 7-manifold at the base of
the Calabi-Yau 4-fold varies under deformation, which leads us to conjecture
that it quantifies the number of degrees of freedom of the gauge theory and its
dependence on the RG scale.Comment: 45 pages, 19 figures, 3 table