83 research outputs found
Permutations of Massive Vacua
We discuss the permutation group G of massive vacua of four-dimensional gauge
theories with N=1 supersymmetry that arises upon tracing loops in the space of
couplings. We concentrate on superconformal N=4 and N=2 theories with N=1
supersymmetry preserving mass deformations. The permutation group G of massive
vacua is the Galois group of characteristic polynomials for the vacuum
expectation values of chiral observables. We provide various techniques to
effectively compute characteristic polynomials in given theories, and we deduce
the existence of varying symmetry breaking patterns of the duality group
depending on the gauge algebra and matter content of the theory. Our examples
give rise to interesting field extensions of spaces of modular forms.Comment: 44 pages, 1 figur
The Conformal Characters
We revisit the study of the multiplets of the conformal algebra in any
dimension. The theory of highest weight representations is reviewed in the
context of the Bernstein-Gelfand-Gelfand category of modules. The
Kazhdan-Lusztig polynomials code the relation between the Verma modules and the
irreducible modules in the category and are the key to the characters of the
conformal multiplets (whether finite dimensional, infinite dimensional, unitary
or non-unitary). We discuss the representation theory and review in full
generality which representations are unitarizable. The mathematical theory that
allows for both the general treatment of characters and the full analysis of
unitarity is made accessible. A good understanding of the mathematics of
conformal multiplets renders the treatment of all highest weight
representations in any dimension uniform, and provides an overarching
comprehension of case-by-case results. Unitary highest weight representations
and their characters are classified and computed in terms of data associated to
cosets of the Weyl group of the conformal algebra. An executive summary is
provided, as well as look-up tables up to and including rank four.Comment: 41 pages, many figure
Duality and Modularity in Elliptic Integrable Systems and Vacua of N=1* Gauge Theories
We study complexified elliptic Calogero-Moser integrable systems. We
determine the value of the potential at isolated extrema, as a function of the
modular parameter of the torus on which the integrable system lives. We
calculate the extrema for low rank B,C,D root systems using a mix of analytical
and numerical tools. For so(5) we find convincing evidence that the extrema
constitute a vector valued modular form for a congruence subgroup of the
modular group. For so(7) and so(8), the extrema split into two sets. One set
contains extrema that make up vector valued modular forms for congruence
subgroups, and a second set contains extrema that exhibit monodromies around
points in the interior of the fundamental domain. The former set can be
described analytically, while for the latter, we provide an analytic value for
the point of monodromy for so(8), as well as extensive numerical predictions
for the Fourier coefficients of the extrema. Our results on the extrema provide
a rationale for integrality properties observed in integrable models, and embed
these into the theory of vector valued modular forms. Moreover, using the data
we gather on the modularity of complexified integrable system extrema, we
analyse the massive vacua of mass deformed N=4 supersymmetric Yang-Mills
theories with low rank gauge group of type B,C and D. We map out their
transformation properties under the infrared electric-magnetic duality group as
well as under triality for N=1* with gauge algebra so(8). We find several
intriguing properties of the quantum gauge theories.Comment: 35 pages, many figure
A limit for large -charge correlators in theories
Using supersymmetric localization, we study the sector of chiral primary
operators with large -charge in
four-dimensional superconformal theories in the weak coupling
regime , where is kept fixed as
, representing the gauge theory coupling(s). In this limit,
correlation functions of these operators behave in a simple way, with
an asymptotic behavior of the form , modulo
corrections, with for a gauge
algebra and a universal function . As a
by-product we find several new formulas both for the partition function as well
as for perturbative correlators in gauge theory
with fundamental hypermultiplets
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