5,426 research outputs found
Hodge structures associated to SU(p,1)
Let A be an abelian variety over C such that the semisimple part of the Hodge
group of A is a product of copies of SU(p,1) for some p>1. We show that any
effective Tate twist of a Hodge structure occurring in the cohomology of A is
isomorphic to a Hodge structure in the cohomology of some abelian variety
BPS states in (2,0) theory on R x T5
We consider theory on a space-time of the form , where
the first factor denotes time, and the second factor is a flat spatial
five-torus. In addition to their energy, quantum states are characterized by
their spatial momentum, 't Hooft flux, and -symmetry
representation. The momentum obeys a shifted quantization law determined by the
't Hooft flux. By supersymmetry, the energy is bounded from below by the
magnitude of the momentum. This bound is saturated by BPS states, that are
annihilated by half of the supercharges. The spectrum of such states is
invariant under smooth deformations of the theory, and can thus be studied by
exploiting the interpretation of theory as an ultra-violet completion
of maximally supersymmetric Yang-Mills theory on . Our main
example is the -series of theories, where such methods allow us to
study the spectrum of BPS states for many values of the momentum and the 't
Hooft flux. In particular, we can describe the -symmetry transformation
properties of these states by determining the image of their
representation in a certain quotient of the representation ring.Comment: 22 page
Arithmeticity vs. non-linearity for irreducible lattices
We establish an arithmeticity vs. non-linearity alternative for irreducible
lattices in suitable product groups, such as for instance products of
topologically simple groups. This applies notably to a (large class of)
Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as
we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page
On abstract commensurators of groups
We prove that the abstract commensurator of a nonabelian free group, an
infinite surface group, or more generally of a group that splits appropriately
over a cyclic subgroup, is not finitely generated.
This applies in particular to all torsion-free word-hyperbolic groups with
infinite outer automorphism group and abelianization of rank at least 2.
We also construct a finitely generated, torsion-free group which can be
mapped onto Z and which has a finitely generated commensurator.Comment: 13 pages, no figur
On the cohomology of some exceptional symmetric spaces
This is a survey on the construction of a canonical or "octonionic K\"ahler"
8-form, representing one of the generators of the cohomology of the four
Cayley-Rosenfeld projective planes. The construction, in terms of the
associated even Clifford structures, draws a parallel with that of the
quaternion K\"ahler 4-form. We point out how these notions allow to describe
the primitive Betti numbers with respect to different even Clifford structures,
on most of the exceptional symmetric spaces of compact type.Comment: 12 pages. Proc. INdAM Workshop "New Perspectives in Differential
Geometry" held in Rome, Nov. 2015, to appear in Springer-INdAM Serie
Palinomorfos fúngicos del Pleistoceno-Holoceno en el Valle del arroyo Chasicó, provincia de Buenos Aires, Argentina
Bound states in N = 4 SYM on T^3: Spin(2n) and the exceptional groups
The low energy spectrum of (3+1)-dimensional N=4 supersymmetric Yang-Mills
theory on a spatial three-torus contains a certain number of bound states,
characterized by their discrete abelian magnetic and electric 't Hooft fluxes.
At weak coupling, the wave-functions of these states are supported near points
in the moduli space of flat connections where the unbroken gauge group is
semi-simple. The number of such states is related to the number of normalizable
bound states at threshold in the supersymmetric matrix quantum mechanics with
16 supercharges based on this unbroken group. Mathematically, the determination
of the spectrum relies on the classification of almost commuting triples with
semi-simple centralizers. We complete the work begun in a previous paper, by
computing the spectrum of bound states in theories based on the
even-dimensional spin groups and the exceptional groups. The results satisfy
the constraints of S-duality in a rather non-trivial way.Comment: 20 page
Distinction of representations via Bruhat-Tits buildings of p-adic groups
Introductory and pedagogical treatmeant of the article : P. Broussous
"Distinction of the Steinberg representation", with an appendix by Fran\c{c}ois
Court\`es, IMRN 2014, no 11, 3140-3157. To appear in Proceedings of Chaire Jean
Morlet, Dipendra Prasad, Volker Heiermann Ed. 2017. Contains modified and
simplified proofs of loc. cit. This article is written in memory of
Fran\c{c}ois Court\`es who passed away in september 2016.Comment: 33 pages, 4 figure
The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
Let k be a global field and let k_v be the completion of k with respect to v,
a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected
algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let
G=\mathbf{G}(k_v). Let \Gamma be an arithmetic lattice in G and let C=C(\Gamma)
be its congruence kernel. Lubotzky has shown that C is infinite, confirming an
earlier conjecture of Serre. Here we provide complete solution of the
congruence subgroup problem for \Gamm$ by determining the structure of C. It is
shown that C is a free profinite product, one of whose factors is
\hat{F}_{\omega}, the free profinite group on countably many generators. The
most surprising conclusion from our results is that the structure of C depends
only on the characteristic of k. The structure of C is already known for a
number of special cases. Perhaps the most important of these is the
(non-uniform) example \Gamma=SL_2(\mathcal{O}(S)), where \mathcal{O}(S) is the
ring of S-integers in k, with S=\{v\}, which plays a central role in the theory
of Drinfeld modules. The proof makes use of a decomposition theorem of
Lubotzky, arising from the action of \Gamma on the Bruhat-Tits tree associated
with G.Comment: 27 pages, 5 figures, to appear in J. Reine Angew. Mat
A tour on Hermitian symmetric manifolds
Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous
and such that every point has a symmetry preserving the Hermitian structure.
The aim of these notes is to present an introduction to this important class of
manifolds, trying to survey the several different perspectives from which
Hermitian symmetric manifolds can be studied.Comment: 56 pages, expanded version. Written for the Proceedings of the
CIME-CIRM summer course "Combinatorial Algebraic Geometry". Comments are
still welcome
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