1,828 research outputs found
Remarks on E11 approach
We consider a few topics in approach to superstring/M-theory: even
subgroups ( orbifolds) of , n=11,10,9 and their connection to
Kac-Moody algebras; subgroup of and coincidence of one of
its weights with the weight of , known to contain brane charges;
possible form of supersymmetry relation in ; decomposition of
w.r.t. the and its square root at first few levels; particle orbit
of . Possible relevance of coadjoint orbits method is
noticed, based on a self-duality form of equations of motion in .Comment: Two references adde
Eisenstein series and automorphic representations
We provide an introduction to the theory of Eisenstein series and automorphic
forms on real simple Lie groups G, emphasising the role of representation
theory. It is useful to take a slightly wider view and define all objects over
the (rational) adeles A, thereby also paving the way for connections to number
theory, representation theory and the Langlands program. Most of the results we
present are already scattered throughout the mathematics literature but our
exposition collects them together and is driven by examples. Many interesting
aspects of these functions are hidden in their Fourier coefficients with
respect to unipotent subgroups and a large part of our focus is to explain and
derive general theorems on these Fourier expansions. Specifically, we give
complete proofs of the Langlands constant term formula for Eisenstein series on
adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic
spherical Whittaker function associated to unramified automorphic
representations of G(Q_p). In addition, we explain how the classical theory of
Hecke operators fits into the modern theory of automorphic representations of
adelic groups, thereby providing a connection with some key elements in the
Langlands program, such as the Langlands dual group LG and automorphic
L-functions. Somewhat surprisingly, all these results have natural
interpretations as encoding physical effects in string theory. We therefore
also introduce some basic concepts of string theory, aimed toward
mathematicians, emphasising the role of automorphic forms. In particular, we
provide a detailed treatment of supersymmetry constraints on string amplitudes
which enforce differential equations of the same type that are satisfied by
automorphic forms. Our treatise concludes with a detailed list of interesting
open questions and pointers to additional topics which go beyond the scope of
this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with
highlighted applications to string theory. v2: 375 pages. Substantially
extended and small correction
Representations of G+++ and the role of space-time
We consider the decomposition of the adjoint and fundamental representations
of very extended Kac-Moody algebras G+++ with respect to their regular A type
subalgebra which, in the corresponding non-linear realisation, is associated
with gravity. We find that for many very extended algebras almost all the A
type representations that occur in the decomposition of the fundamental
representations also occur in the adjoint representation of G+++. In
particular, for E8+++, this applies to all its fundamental representations.
However, there are some important examples, such as An+++, where this is not
true and indeed the adjoint representation contains no generator that can be
identified with a space-time translation. We comment on the significance of
these results for how space-time can occur in the non-linear realisation based
on G+++. Finally we show that there is a correspondence between the A
representations that occur in the fundamental representation associated with
the very extended node and the adjoint representation of G+++ which is
consistent with the interpretation of the former as charges associated with
brane solutions.Comment: 45 pages, 9 figures, 9 tables, te
Sugawara-type constraints in hyperbolic coset models
In the conjectured correspondence between supergravity and geodesic models on
infinite-dimensional hyperbolic coset spaces, and E10/K(E10) in particular, the
constraints play a central role. We present a Sugawara-type construction in
terms of the E10 Noether charges that extends these constraints infinitely into
the hyperbolic algebra, in contrast to the truncated expressions obtained in
arXiv:0709.2691 that involved only finitely many generators. Our extended
constraints are associated to an infinite set of roots which are all imaginary,
and in fact fill the closed past light-cone of the Lorentzian root lattice. The
construction makes crucial use of the E10 Weyl group and of the fact that the
E10 model contains both D=11 supergravity and D=10 IIB supergravity. Our
extended constraints appear to unite in a remarkable manner the different
canonical constraints of these two theories. This construction may also shed
new light on the issue of `open constraint algebras' in traditional canonical
approaches to gravity.Comment: 49 page
A reduction principle for Fourier coefficients of automorphic forms
In this paper we analyze a general class of Fourier coefficients of
automorphic forms on reductive adelic groups
and their covers. We prove that any such
Fourier coefficient is expressible through integrals and sums involving
'Levi-distinguished' Fourier coefficients. By the latter we mean the class of
Fourier coefficients obtained by first taking the constant term along the
nilradical of a parabolic subgroup, and then further taking a Fourier
coefficient corresponding to a -distinguished nilpotent orbit in
the Levi quotient. In a follow-up paper we use this result to establish
explicit formulas for Fourier expansions of automorphic forms attached to
minimal and next-to-minimal representations of simply-laced reductive groups.Comment: 35 pages. v2: Extended results and paper split into two parts with
second part appearing soon. New title to reflect new focus of this part. v3:
Minor corrections and updated reference to the second part that has appeared
as arXiv:1908.08296. v4: Minor corrections and reformulation
E10 and SO(9,9) invariant supergravity
We show that (massive) D=10 type IIA supergravity possesses a hidden rigid
SO(9,9) symmetry and a hidden local SO(9) x SO(9) symmetry upon dimensional
reduction to one (time-like) dimension. We explicitly construct the associated
locally supersymmetric Lagrangian in one dimension, and show that its bosonic
sector, including the mass term, can be equivalently described by a truncation
of an E10/K(E10) non-linear sigma-model to the level \ell<=2 sector in a
decomposition of E10 under its so(9,9) subalgebra. This decomposition is
presented up to level 10, and the even and odd level sectors are identified
tentatively with the Neveu--Schwarz and Ramond sectors, respectively. Further
truncation to the level \ell=0 sector yields a model related to the reduction
of D=10 type I supergravity. The hyperbolic Kac--Moody algebra DE10, associated
to the latter, is shown to be a proper subalgebra of E10, in accord with the
embedding of type I into type IIA supergravity. The corresponding decomposition
of DE10 under so(9,9) is presented up to level 5.Comment: 1+39 pages LaTeX2e, 2 figures, 2 tables, extended tables obtainable
by downloading sourc
A group model for stable multi-subject ICA on fMRI datasets
Spatial Independent Component Analysis (ICA) is an increasingly used
data-driven method to analyze functional Magnetic Resonance Imaging (fMRI)
data. To date, it has been used to extract sets of mutually correlated brain
regions without prior information on the time course of these regions. Some of
these sets of regions, interpreted as functional networks, have recently been
used to provide markers of brain diseases and open the road to paradigm-free
population comparisons. Such group studies raise the question of modeling
subject variability within ICA: how can the patterns representative of a group
be modeled and estimated via ICA for reliable inter-group comparisons? In this
paper, we propose a hierarchical model for patterns in multi-subject fMRI
datasets, akin to mixed-effect group models used in linear-model-based
analysis. We introduce an estimation procedure, CanICA (Canonical ICA), based
on i) probabilistic dimension reduction of the individual data, ii) canonical
correlation analysis to identify a data subspace common to the group iii)
ICA-based pattern extraction. In addition, we introduce a procedure based on
cross-validation to quantify the stability of ICA patterns at the level of the
group. We compare our method with state-of-the-art multi-subject fMRI ICA
methods and show that the features extracted using our procedure are more
reproducible at the group level on two datasets of 12 healthy controls: a
resting-state and a functional localizer study
Generalised BPS conditions
We write down two E11 invariant conditions which at low levels reproduce the
known half BPS conditions for type II theories. These new conditions contain,
in addition to the familiar central charges, an infinite number of further
charges which are required in an underlying theory of strings and branes. We
comment on the application of this work to higher derivative string
corrections
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