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Tilings, Dimers and Quiver Gauge Theories
No full textHanany, Amihay. (2012). Tilings, Dimers and Quiver Gauge Theories. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/130028
Brane Tilings: NSVZ Beta Function
No full textHanany, Amihay. (2009). Brane Tilings: NSVZ Beta Function. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/53195
Brane Tilings, M2-branes and Orbifolds
Full text linkBrane Tilings represent one of the largest classes of superconformal theories
with known gravity duals in 3+1 and also 2+1 dimensions. They provide a
useful link between a large class of quiver gauge theories and their moduli
spaces, which are the toric Calabi-Yau (CY) singularities.
This thesis includes a discussion of an algorithm that can be used to
generate all brane tilings with any given number of superpotential terms.
All tilings with at most 8 superpotential terms have been generated using
an implementation of this method.
Orbifolds are a subject of central importance in string theory. It is widely
known that there may be two or more orbifolds of a space by a finite group.
Abelian Calabi-Yau orbifolds of the form C³/Γ can be counted according to
the size of the group |Γ|. Three methods of counting these orbifolds will be
given.
A brane tiling together with a set of Chern Simons levels is sufficient to define a quiver Chern-Simons theory which describes the worldvolume theory
of the M2-brane probe. A forward algorithm exists which allows us to easily
compute the toric data associated to the moduli space of the quiver Chern-Simons theory from knowledge of the tiling and Chern-Simons levels. This
forward algorithm will be discussed and illustrated with a few examples. It
is possible that two different Chern-Simons theories have the same moduli-space.
This effect, sometimes known as 'toric duality' will be described
further. We will explore how two Chern-Simons theories (corresponding to
brane tilings) can be related to each other by the Higgs mechanism and how
brane tilings (with CS levels) that correspond to 14 fano 3-folds have been
constructed.
The idea of 'child' and 'parent' brane tilings will be introduced and we
will discuss how it has been possible to count 'children' using the symmetry
of the 'parent' tiling
Magnetic quivers – a new perspective on supersymmetric gauge theories
Full text linkWe focus on supersymmetric gauge theories with eight superchrages in spacetime dimensions
d = 3, 4, 5, 6. These theories have very rich vacuum structures so our focus will be on their
moduli spaces of vacua. For d =, 4, 5, 6, we look at the Higgs branch moduli space. The
usual story is that the Higgs branch is a classical object that can be easily computed from
its Lagrangian. However, non-perturbative contributions can enhance the Higgs branch and a
classical description no longer works. In 6d N = (1, 0) and 5d N = (1, 0), these contributions
originate from tensionless BPS-strings and massless gauge instantons respectively as we tune
gauge coupling(s) to infinity. For 4d N = 2 theories, many gauge theories, and in particular
superconformal field theories (SCFTs), do not even have a Lagrangian description. We offer
a unifying solution to these problems in the form of magnetic quivers. These are 3d N = 4
gauge theories whose Coulomb branch is the same as the Higgs branch of the higher dimensional
theories. Using brane systems of D_d−D_(d+2)−NS5, with the possible inclusion of Od orientifold
planes, we show how the magnetic quivers of these theories can be extracted. Then, a) using
the monopole formula we study the moduli space as an algebraic variety by computing its
Hilbert series and b) using the new concept of Quiver subtraction we extract the phase diagram
(Hasse diagram) of these moduli spaces. Examples we explore include 5d SQCD theories at UV
fixed point, 4d rank one SCFTs, class S theories, S-fold theories etc. For the second outcome
of the thesis, we focus on new features of gauge theories with orthosymplectic gauge groups
such as discrete subgroups and non-simply laced edges, leading to a general classification of
such theories. For the final outcome, we study gauge theories with a mixture of unitary and
special unitary gauge groups which lead to a slew of new gauge theories related by 3d mirror
symmetry.Open Acces
SQFT - branes - moduli, symplectic singularities in physics
No full textIn this thesis quantum field theories with 8 supercharges, and especially their moduli spaces, are studied. A collection of methods are used to study these singular moduli spaces, such as counting gauge invariant operators via means of a Hilbert series, brane realisations, and computation of their singularity structure in terms of a Hasse diagram. A particular focus lies on an explanation of this Hasse diagram as a geometric realisation of the Higgs mechanism. The moduli space of quantum chromodynamics with 8 super- charges is discussed in great detail. After this more complicated theories are discussed, in particular ones, where the Higgs branch consists of multiple cones.Open Acces
Group symmetries and the moduli space structures of SUSY quiver gauge theories
Full text linkThis thesis takes steps towards the development of a systematic account of the relationships between SUSY quiver gauge theories and the structures of their moduli spaces. Highest Weight Generating functions (“HWGs”), which concisely encode the field content of a moduli space, are introduced and developed to augment the established plethystic techniques for the construction and analysis of Hilbert series (“HS”). HWGs are shown to provide a faithful means of decoding and describing HS in terms of their component fields, which transform in representations of Classical and/or Exceptional symmetry groups. These techniques are illustrated in the context of Higgs branch quiver theories for SQCD and instanton moduli spaces, as a prelude to an account of the quiver theory constructions for the canonical class of moduli spaces represented by the nilpotent orbits of Classical and Exceptional symmetry groups. The known Higgs and/or Coulomb branch quiver theory constructions for nilpotent orbits are systematically extended to give a complete set of Higgs branch quiver theories for Classical group nilpotent orbits and a set of Coulomb branch constructions for near to minimal orbits of Classical and Exceptional groups. A localisation formula (“NOL Formula”) for the normal nilpotent orbits of Classical and Exceptional groups based on their Characteristics is proposed and deployed. Dualities and other relationships between quiver theories, including A series 3d mirror symmetry, are analysed and discussed. The use of nilpotent orbits, for example in the form of T(G) quiver theories, as building blocks for the systematic (de)construction of moduli spaces is illustrated. The roles of orthogonal bases, such as characters and Hall Littlewood polynomials, in providing canonical structures for the the analysis of quiver theories is demonstrated, along with their potential use as building blocks for more general families of quiver theories.Open Acces
Iter geometria - vacuum moduli spaces of supersymmetric quiver gauge theories with 8 supercharges
Full text linkSupersymmetric gauge theories have been at the heart of research in theoretical physics for
the past ve decades. The study of these theories holds not only the promise for the ultimate
understanding of Nature, but also for discoveries of new mathematical constructions and
phenomena. The latter results from a highly geometrical nature of these theories, which is
prominently carried by the moduli space of vacua.
This thesis is dedicated to the study of moduli spaces of supersymmetric quiver gauge theories
with 8 supercharges. Typically, such moduli spaces consist of two branches, known as
the Coulomb branch and the Higgs branch. Following a review of the Higgs and the Coulomb
branch computational techniques in the rst part of this thesis, it is shown how the study
of Coulomb branches of 3d N = 4 minimally unbalanced theories is used for developing a
classification of singular hyperK ahler cones with a single Lie group isometry. As another application,
three-dimensional Coulomb branches are used to study Higgs branches of a stack of n
M5 branes on an A-type orbifold singularity. Analysis of such systems gives rise to a discrete
gauging phenomenon with importance for both physics and mathematics. From the physics perspective,
discrete gauging solves the problem of understanding the non-classical Higgs branch
phases of the corresponding 6d N = (1; 0) world-volume theory, even when coincident subsets
of M5 branes introduce tensionless BPS strings into the spectrum. From the mathematical
perspective, discrete gauging provides a new method for constructing non-Abelian orbifolds
with certain global symmetry. The thesis also includes an investigation of theories associated
with non-simply laced quivers. Remarkably, the formalized analysis of the so-called ungauging
schemes and the corresponding Coulomb branches reproduce orbifold relations amid closures
of nilpotent orbits of Lie algebras studied by Kostant and Brylinski. Finally, three-dimensional
Coulomb branches are employed to understand the Higgs mechanism in supersymmetric gauge
theories with 8 supercharges in 3; 4; 5, and 6 dimensions. It is illustrated that the physical
phenomenon of partial Higgsing is directly related to the mathematical structure of the moduli
space, and in particular, to the geometry of its singular points. The developed techniques provide
a new set of algorithmic methods for computing the geometrical structure of symplectic
singularities in terms of Hasse diagrams.Open Acces
Counting gauge invariant operators in supersymmetric theories using Hilbert series
Full text linkIn this thesis, the problem of counting gauge invariant operators in certain
supersymmetric theories is discussed.
These objects have a very important role in supersymmetric gauge theories,
since they can be used to describe the space of zero-energy solutions,
called moduli space, of such theories. In order to approach the counting
problem, a technique is used based on a function known in Algebraic Geometry
as the Hilbert series. For the examined theories, this can be considered
a a partition function counting gauge invariant operators in the field theory
according to their charges under quantum global symmetries.
In the first part of the thesis, particular focus will be given to the application
of the Hilbert series to conformal Chern-Simons theories living on the
world-volume of M2-branes probing different toric Calabi-Yau 4-fold singularities.
It will be shown how the Hilbert series can be combined with the
brane tiling formalism to characterise the mesonic moduli space of vacua of
a given theory through its generators and the relations they satisfy. Then,
toric duality for these theories will be presented, with special attention to
the role played by Hilbert series in making such feature manifest between
two or more theories. Finally, Chern-Simons theories living on M2-branes
probing cones over smooth toric Fano 3-folds and their mesonic Hilbert
series will be presented.
In the second part, it will be shown how the Hilbert series can be applied
to counting gauge invariant operators in supersymmetric generalisations of
Quantum Chromodynamics, known as SQCD theories. The discussion will
hinge on a specific class of theories, with N multiplets transforming in the
fundamental and anti-fundamental and one in the adjoint representation of
the gauge group. For each classical group, the Hilbert series of the moduli
space will be used to determine the dimension on the spaces, their generators
and to argue that they are all Calabi-Yau manifolds
Brane Tilings and Quiver Gauge Theories
Full text linkThis 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
Properties of moduli spaces of supersymmetric quiver gauge theories with 8 supercharges
No full textThe thesis focuses on the study of moduli spaces of 3d N = 4 supersymmetric field theories.
Two aspects are emphasized. Firstly, discrete quotients of the Coulomb branch are studied.
Secondly, the Hasse diagram for Higgs branches are studied. Both aspects are given by
diagrammatic operations on the quiver diagrams.
In the Introduction and Background part, the framework of supersymmetry and the nec
essary mathematics underlying the following two chapters are introduced at a pedagogical
level.
In the second part, it is shown that two families of quivers, the quivers with complete
graphs and the quivers with multiple adjoint loops, have Coulomb branches related by
a quotient of a permutation symmetry. Quotient of cyclic groups is also studied. The
two operations can be combined to generate a quotient by a semi-direct product group of
permutation and cyclic groups. The quotient relations are demonstrated by the Molien
sum and Abelionization process. Examples are included to demonstrate the operations.
In the third part, a bottom to up quiver subtraction algorithm is introduced. The algorithm
can generate the whole Hasse diagram for the Higgs branch of a single-laced unitary quiver.
The interesting feature of the algorithm is that it gives the monodromy of slices around
the leaves. It also calculates the Namikawa Weyl group.Open Acces
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