7,753 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 Galois group of a stable homotopy theory
To a "stable homotopy theory" (a presentable, symmetric monoidal stable
-category), we naturally associate a category of finite \'etale algebra
objects and, using Grothendieck's categorical machine, a profinite group that
we call the Galois group. We then calculate the Galois groups in several
examples. For instance, we show that the Galois group of the periodic
-algebra of topological modular forms is trivial and that
the Galois group of -local stable homotopy theory is an extended version
of the Morava stabilizer group. We also describe the Galois group of the stable
module category of a finite group. A fundamental idea throughout is the purely
categorical notion of a "descendable" algebra object and an associated analog
of faithfully flat descent in this context.Comment: 93 pages. To appear in Advances in Mathematic
Galois Groups in Rational Conformal Field Theory
It was established before that fusion rings in a rational conformal field
theory (RCFT) can be described as rings of polynomials, with integer
coefficients, modulo some relations. We use the Galois group of these relations
to obtain a local set of equation for the points of the fusion variety. These
equations are sufficient to classify all the RCFT, Galois group by Galois
group. It is shown that the Galois group is equivalent to the pseudo RCFT
group. We prove that the Galois groups encountered in RCFT are all abelian,
implying solvability by radicals of the modular matrix.Comment: 24 pages. Typos correcte
Some aspects of semi-abelian homology and protoadditive functors
In this note some recent developments in the study of homology in
semi-abelian categories are briefly presented. In particular the role of
protoadditive functors in the study of Hopf formulae for homology is explained.Comment: 7 page
Protoadditive functors, derived torsion theories and homology
Protoadditive functors are designed to replace additive functors in a
non-abelian setting. Their properties are studied, in particular in
relationship with torsion theories, Galois theory, homology and factorisation
systems. It is shown how a protoadditive torsion-free reflector induces a chain
of derived torsion theories in the categories of higher extensions, similar to
the Galois structures of higher central extensions previously considered in
semi-abelian homological algebra. Such higher central extensions are also
studied, with respect to Birkhoff subcategories whose reflector is
protoadditive or, more generally, factors through a protoadditive reflector. In
this way we obtain simple descriptions of the non-abelian derived functors of
the reflectors via higher Hopf formulae. Various examples are considered in the
categories of groups, compact groups, internal groupoids in a semi-abelian
category, and other ones
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