43 research outputs found
On vector configurations that can be realized in the cone of positive matrices
Let ,..., be vectors in an inner product space. Can we find a
natural number and positive (semidefinite) complex matrices ,...,
of size such that for all
? For such matrices to exist, one must have
for all . We prove that if then this trivial necessary
condition is also a sufficient one and find an appropriate example showing that
from this is not so --- even if we allowed realizations by positive
operators in a von Neumann algebra with a faithful normal tracial state.
The fact that the first such example occurs at is similar to what one
has in the well-investigated problem of positive factorization of positive
(semidefinite) matrices. If the matrix has a positive
factorization, then matrices ,..., as above exist. However, as we
show by a large class of examples constructed with the help of the Clifford
algebra, the converse implication is false.Comment: 8 page
Conformal covariance and the split property
We show that for a conformal local net of observables on the circle, the
split property is automatic. Both full conformal covariance (i.e.
diffeomorphism covariance) and the circle-setting play essential roles in this
fact, while by previously constructed examples it was already known that even
on the circle, M\"obius covariance does not imply the split property.
On the other hand, here we also provide an example of a local conformal net
living on the two-dimensional Minkowski space, which - although being
diffeomorphism covariant - does not have the split property.Comment: 34 pages, 3 tikz figure
Systems of mutually unbiased Hadamard matrices containing real and complex matrices
We use combinatorial and Fourier analytic arguments
to prove various non-existence results on systems of real and com-
plex unbiased Hadamard matrices. In particular, we prove that
a complete system of complex mutually unbiased Hadamard ma-
trices (MUHs) in any dimension cannot contain more than one
real Hadamard matrix. We also give new proofs of several known
structural results in low dimensions
From vertex operator algebras to conformal nets and back
We consider unitary simple vertex operator algebras whose vertex operators
satisfy certain energy bounds and a strong form of locality and call them
strongly local. We present a general procedure which associates to every
strongly local vertex operator algebra V a conformal net A_V acting on the
Hilbert space completion of V and prove that the isomorphism class of A_V does
not depend on the choice of the scalar product on V. We show that the class of
strongly local vertex operator algebras is closed under taking tensor products
and unitary subalgebras and that, for every strongly local vertex operator
algebra V, the map W\mapsto A_W gives a one-to-one correspondence between the
unitary subalgebras W of V and the covariant subnets of A_V. Many known
examples of vertex operator algebras such as the unitary Virasoro vertex
operator algebras, the unitary affine Lie algebras vertex operator algebras,
the known c=1 unitary vertex operator algebras, the moonshine vertex operator
algebra, together with their coset and orbifold subalgebras, turn out to be
strongly local. We give various applications of our results. In particular we
show that the even shorter Moonshine vertex operator algebra is strongly local
and that the automorphism group of the corresponding conformal net is the Baby
Monster group. We prove that a construction of Fredenhagen and J\"{o}rss gives
back the strongly local vertex operator algebra V from the conformal net A_V
and give conditions on a conformal net A implying that A= A_V for some strongly
local vertex operator algebra V.Comment: Minor correction