138 research outputs found
On Soliton Automorphisms in Massive and Conformal Theories
For massive and conformal quantum field theories in 1+1 dimensions with a
global gauge group we consider soliton automorphisms, viz. automorphisms of the
quasilocal algebra which act like two different global symmetry transformations
on the left and right spacelike complements of a bounded region. We give a
unified treatment by providing a necessary and sufficient condition for the
existence and Poincare' covariance of soliton automorphisms which is applicable
to a large class of theories. In particular, our construction applies to the
QFT models with the local Fock property -- in which case the latter property is
the only input from constructive QFT we need -- and to holomorphic conformal
field theories. In conformal QFT soliton representations appear as twisted
sectors, and in a subsequent paper our results will be used to give a rigorous
analysis of the superselection structure of orbifolds of holomorphic theories.Comment: latex2e, 20 pages. Proof of Thm. 3.14 corrected, 2 references added.
Final version as to appear in Rev. Math. Phy
Galois extensions of braided tensor categories and braided crossed G-categories
We show that the author's notion of Galois extensions of braided tensor
categories [22], see also [3], gives rise to braided crossed G-categories,
recently introduced for the purposes of 3-manifold topology [31]. The Galois
extensions C \rtimes S are studied in detail, and we determine for which g in G
non-trivial objects of grade g exist in C \rtimes S.Comment: Some comments and references added. Final version, to appear in J.
Alg. latex2e, ca. 25 p., requires diagrams.te
Orbifold aspects of the Longo-Rehren subfactors
In this article, we will prove that the subsectors of -induced
sectors for forms a modular category, where is the crossed product of by the group dual of a
finite group . In fact, we will prove that it is equivalent to M\"uger's
crossed product. By using this identification, we will exhibit an orbifold
aspect of the quantum double of (not necessarily non-degenerate)
obtained from a Longo-Rehren inclusion under certain
assumptions.
We will apply the above description of the quantum double of to the
Reshetikhin-Turaev topological invariant of closed 3-manifolds, and we obtain a
simpler formula, which is a degenerate version of Turaev's theorem that the
Reshetikhin-Turaev invariant for the quantum double of a modular category
is the product of Reshetikhin-Turaev invariant of
and its complex conjugate.Comment: 19 page
Superselection Structure of Massive Quantum Field Theories in 1+1 Dimensions
We show that a large class of massive quantum field theories in 1+1
dimensions, characterized by Haag duality and the split property for wedges,
does not admit locally generated superselection sectors in the sense of
Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1+1
dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories.
Even charged representations which are localizable only in wedge regions are
ruled out. Furthermore, Haag duality holds in all locally normal
representations. These results are applied to the theory of soliton sectors.
Furthermore, the extension of localized representations of a non-Haag dual net
to the dual net is reconsidered. It must be emphasized that these statements do
not apply to massless theories since they do not satisfy the above split
property. In particular, it is known that positive energy representations of
conformally invariant theories are DHR representations.Comment: latex2e, 21 pages. Final version, to appear in Rev. Math. Phys. Some
improvements of the presentation, but no essential change
Monoids, Embedding Functors and Quantum Groups
We show that the left regular representation \pi_l of a discrete quantum
group (A,\Delta) has the absorbing property and forms a monoid
(\pi_l,\tilde{m},\tilde{\eta}) in the representation category Rep(A,\Delta).
Next we show that an absorbing monoid in an abstract tensor *-category C gives
rise to an embedding functor E:C->Vect_C, and we identify conditions on the
monoid, satisfied by (\pi_l,\tilde{m},\tilde{\eta}), implying that E is
*-preserving. As is well-known, from an embedding functor E: C->\mathrm{Hilb}
the generalized Tannaka theorem produces a discrete quantum group (A,\Delta)
such that C is equivalent to Rep_f(A,\Delta). Thus, for a C^*-tensor category C
with conjugates and irreducible unit the following are equivalent: (1) C is
equivalent to the representation category of a discrete quantum group
(A,\Delta), (2) C admits an absorbing monoid, (3) there exists a *-preserving
embedding functor E: C->\mathrm{Hilb}.Comment: Final version, to appear in Int. Journ. Math. (Added some references
and Subsection 1.2.) Latex2e, 21 page
Weakly group-theoretical and solvable fusion categories
We introduce two new classes of fusion categories which are obtained by a
certain procedure from finite groups - weakly group-theoretical categories and
solvable categories. These are fusion categories that are Morita equivalent to
iterated extensions (in the world of fusion categories) of arbitrary,
respectively solvable finite groups. Weakly group-theoretical categories have
integer dimension, and all known fusion categories of integer dimension are
weakly group theoretical. Our main results are that a weakly group-theoretical
category C has the strong Frobenius property (i.e., the dimension of any simple
object in an indecomposable C-module category divides the dimension of C), and
that any fusion category whose dimension has at most two prime divisors is
solvable (a categorical analog of Burnside's theorem for finite groups). This
has powerful applications to classification of fusion categories and
semsisimple Hopf algebras of a given dimension. In particular, we show that any
fusion category of integer dimension <84 is weakly group-theoretical (i.e.
comes from finite group theory), and give a full classification of semisimple
Hopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes.Comment: 28 pages, latex; added many references and details in proof
The Mathieu conjecture for reduced to an abelian conjecture
We reduce the Mathieu conjecture for to a conjecture about moments of
Laurent polynomials in two variables with single variable polynomial
coefficients.Comment: Considerably revised and simplified. Now 5 pages. To appear in
Indagationes Mathematica
On the moments of a polynomial in one variable
Let be a non-zero polynomial with complex coefficients and define
. We use ideas of Duistermaat and van der Kallen to
prove . In particular, for infinitely many .Comment: 4 pages, no figure
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