273 research outputs found
Highest Weight Generating Functions for Hilbert Series
We develop a new method for representing Hilbert series based on the highest
weight Dynkin labels of their underlying symmetry groups. The method draws on
plethystic functions and character generating functions along with Weyl
integration. We give explicit examples showing how the use of such highest
weight generating functions (HWGs) permits an efficient encoding and analysis
of the Hilbert series of the vacuum moduli spaces of classical and exceptional
SQCD theories and also of the moduli spaces of instantons. We identify how the
HWGs of gauge invariant operators of a selection of classical and exceptional
SQCD theories result from the interaction under symmetrisation between a
product group and the invariant tensors of its gauge group. In order to
calculate HWGs, we derive and tabulate character generating functions for low
rank groups by a variety of methods, including a general character generating
function, based on the Weyl Character Formula, for any classical or exceptional
group.Comment: 76 page
Counting BPS Operators in the Chiral Ring of N=2 Supersymmetric Gauge Theories or N=2 Braine Surgery
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.Comment: 24 pages LaTe
Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits
We approach the topic of Classical group nilpotent orbits from the
perspective of their moduli spaces, described in terms of Hilbert series and
generating functions. We review the established Higgs and Coulomb branch quiver
theory constructions for A series nilpotent orbits. We present systematic
constructions for BCD series nilpotent orbits on the Higgs branches of quiver
theories defined by canonical partitions; this paper collects earlier work into
a systematic framework, filling in gaps and providing a complete treatment. We
find new Coulomb branch constructions for above minimal nilpotent orbits,
including some based upon twisted affine Dynkin diagrams. We also discuss
aspects of 3d mirror symmetry between these Higgs and Coulomb branch
constructions and explore dualities and other relationships, such as
HyperKahler quotients, between quivers. We analyse all Classical group
nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert
series and the Highest Weight Generating functions for their decompositions
into characters of irreducible representations and/or Hall Littlewood
polynomials.Comment: 67 pages, 11 figure
Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics
We propose an explanation via string theory of the correspondence between the
Coulomb branch of certain three-dimensional supersymmetric gauge theories and
certain moduli spaces of magnetic monopoles. The same construction also gives
an explanation, via duality of Type IIB superstrings, of the
recently discovered ``mirror symmetry'' in three dimensions. New phase
transitions in three dimensions as well as new infrared fixed points and even
new coupling constants not present in the known Lagrangians are predicted from
the string theory construction. An important role in the construction is played
by a novel aspect of brane dynamics in which a third brane is created when two
branes cross.Comment: 46 pages, 20 figures, A discussion on continuation past infinite
coupling is adde
Monopoles in String Theory
A realization of E_{n+1} monopoles in string theory is given. The NS five
brane stuck to an Orientifold eight plane is identified as the 't Hooft
Polyakov monopole. Correspondingly, the moduli space of many such NS branes is
identified with the moduli space of SU(2) monopoles. These monopoles transform
in the spinor representation of an SO(2n) gauge group when n D8 branes are
stacked upon the orientifold plane. This leads to a realization of E_{n+1}
monopole moduli spaces. Charge conservation leads to a dynamical effect which
does not allow the NS branes to leave the orientifold plane. This suggests that
the monopole moduli space is smooth for n8 obeys a similar condition.
Using a chain of dualities, we also connect our system to an Heterotic
background with Kaluza-Klein monopoles.Comment: 26 pages, 3 figure
Conformal Manifolds for the Conifold and other Toric Field Theories
In the space of couplings of the 4D N=1 gauge theory associated to D3 branes
probing Calabi-Yau singularities, there is a manifold over which superconformal
invariance is preserved. The AdS/CFT correspondence is valid precisely for this
"conformal manifold". We identify the conformal manifold for all the Y^{p,q}
toric singularities, paying special attention to the case of the conifold,
Y^{1,0}. For a general Y^{p,q} the conformal manifold is three dimensional,
while for the conifold it is five dimensional. There is always an exactly
marginal deformation, analogous to the beta-deformation of N=4 SYM, which
involves fluxes in the dual gravity description. This beta-deformation exists
for any toric Calabi-Yau singularity.Comment: 30 pages, 5 figures; V2: References added, minor change
Quiver Subtractions
We study the vacuum structure of gauge theories with eight supercharges. It
has been recently discovered that in the Higgs branch of and SQCD
theories with eight supercharges, the new massless states, arising when the
gauge coupling is taken to infinity, can be described in terms of Coulomb
branches of quiver gauge theories. The description of this
new phenomenon draws from the ideas discovered in the analysis of nilpotent
orbits as Higgs and Coulomb branches of theories and promotes the Higgs
mechanism known as the Kraft-Procesi transition to the status of a new
operation between quivers. This is the quiver subtraction. This paper
establishes this operation formally and examines some immediate consequences.
One is the extension of the physical realization of Kraft-Procesi transitions
from the classical to the exceptional Lie algebras. Another result is the
extension from special nilpotent orbits to non-special ones. One further
consequence is the analysis of the effect in SQCD of
integrating out a massive quark while the gauge coupling remains infinite. In
general, the subtraction of quivers sheds light on the different types of
singularities within the Coulomb branch and the structure of the massless
states that arise at those singular points; including the nature of the new
Higgs branches that open up. This allows for a systematic analysis of mixed
branches of quivers that do not necessarily have a simple
embedding in string theory. The subtraction of two quivers is an extremely
simple resource for the theoretical physicist interested in the vacuum
structure of gauge theories, and yet its power is so remarkable that is bound
to play a crucial role in the coming discoveries of new and exciting physics in
and dimensions.Comment: 21 page
(0,4) brane box models
Two-dimensional supersymmetric quiver gauge theories are
realized as D3-brane box configurations (two dimensional intervals) which are
bounded by NS5-branes and intersect with D5-branes. The periodic brane
configuration is mapped to D1-D5-D5 brane system at orbifold singularity via
T-duality. The matter content and interactions are encoded by the
quiver diagrams which are determined by the brane
configurations. The Abelian gauge anomaly cancellation indicates the presence
of Fermi multiplets at the NS-NS junction. We also discuss the brane
construction of supersymmetric boundary conditions in 3d
gauge theories involving two-dimensional boundary degrees of
freedom that cancel gauge anomaly.Comment: 42 pages, 23 figure
Coulomb branches for rank 2 gauge groups in 3d N=4 gauge theories
The Coulomb branch of 3-dimensional N=4 gauge theories is the space of bare
and dressed BPS monopole operators. We utilise the conformal dimension to
define a fan which, upon intersection with the weight lattice of a GNO-dual
group, gives rise to a collection of semi-groups. It turns out that the unique
Hilbert bases of these semi-groups are a sufficient, finite set of monopole
operators which generate the entire chiral ring. Moreover, the knowledge of the
properties of the minimal generators is enough to compute the Hilbert series
explicitly. The techniques of this paper allow an efficient evaluation of the
Hilbert series for general rank gauge groups. As an application, we provide
various examples for all rank two gauge groups to demonstrate the novel
interpretation.Comment: v2: 98 pages, 30 figures, 34 tables, 1 appendix, matches JHEP versio
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