230,831 research outputs found
Group dualities, T-dualities, and twisted K-theory
This paper explores further the connection between Langlands duality and
T-duality for compact simple Lie groups, which appeared in work of Daenzer-Van
Erp and Bunke-Nikolaus. We show that Langlands duality gives rise to
isomorphisms of twisted K-groups, but that these K-groups are trivial except in
the simplest case of SU(2) and SO(3). Along the way we compute explicitly the
map on induced by a covering of compact simple Lie groups, which is
either 1 or 2 depending in a complicated way on the type of the groups
involved. We also give a new method for computing twisted K-theory using the
Segal spectral sequence, giving simpler computations of certain twisted
K-theory groups of compact Lie groups relevant for D-brane charges in WZW
theories and rank-level dualities. Finally we study a duality for orientifolds
based on complex Lie groups with an involution.Comment: 29 pages, mild revisio
A new algorithm for fast generalized DFTs
We give an new arithmetic algorithm to compute the generalized Discrete
Fourier Transform (DFT) over finite groups . The new algorithm uses
operations to compute the generalized DFT over
finite groups of Lie type, including the linear, orthogonal, and symplectic
families and their variants, as well as all finite simple groups of Lie type.
Here is the exponent of matrix multiplication, so the exponent
is optimal if . Previously, "exponent one" algorithms
were known for supersolvable groups and the symmetric and alternating groups.
No exponent one algorithms were known (even under the assumption )
for families of linear groups of fixed dimension, and indeed the previous
best-known algorithm for had exponent despite being the focus
of significant effort. We unconditionally achieve exponent at most for
this group, and exponent one if . Our algorithm also yields an
improved exponent for computing the generalized DFT over general finite groups
, which beats the longstanding previous best upper bound, for any .
In particular, assuming , we achieve exponent , while the
previous best was
Integral group actions on symmetric spaces and discrete duality symmetries of supergravity theories
For a split, simply connected, semisimple Lie group of rank
and the maximal compact subgroup of , we give a method for computing
Iwasawa coordinates of using the Chevalley generators and the Steinberg
presentation. When is a scalar coset for a supergravity theory in
dimensions , we determine the action of the integral form
on . We give explicit results for the action of the
discrete --duality groups and on the
scalar cosets and
for type IIB supergravity
in ten dimensions and 11--dimensional supergravity in dimensions,
respectively. For the former, we use this to determine the discrete U--duality
transformations on the scalar sector in the Borel gauge and we describe the
discrete symmetries of the dyonic charge lattice. We determine the
spectrum--generating symmetry group for fundamental BPS solitons of type IIB
supergravity in dimensions at the classical level and we propose an
analog of this symmetry at the quantum level. We indicate how our methods can
be used to study the orbits of discrete U--duality groups in general
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