18,740 research outputs found
Structure and Classification of Superconformal Nets
We study the general structure of Fermi conformal nets of von Neumann
algebras on the circle, consider a class of topological representations, the
general representations, that we characterize as Neveu-Schwarz or Ramond
representations, in particular a Jones index can be associated with each of
them. We then consider a supersymmetric general representation associated with
a Fermi modular net and give a formula involving the Fredholm index of the
supercharge operator and the Jones index. We then consider the net associated
with the super-Virasoro algebra and discuss its structure. If the central
charge c belongs to the discrete series, this net is modular by the work of F.
Xu and we get an example where our setting is verified by considering the
Ramond irreducible representation with lowest weight c/24. We classify all the
irreducible Fermi extensions of any super-Virasoro net in the discrete series,
thus providing a classification of all superconformal nets with central charge
less than 3/2.Comment: 49 pages. Section 8 has been removed. More details concerning the
diffeomorphism covariance are give
N=2 superconformal nets
We provide an Operator Algebraic approach to N=2 chiral Conformal Field
Theory and set up the Noncommutative Geometric framework. Compared to the N=1
case, the structure here is much richer. There are naturally associated nets of
spectral triples and the JLO cocycles separate the Ramond sectors. We construct
the N=2 superconformal nets of von Neumann algebras in general, classify them
in the discrete series c<3, and we define and study an operator algebraic
version of the N=2 spectral flow. We prove the coset identification for the N=2
super-Virasoro nets with c<3, a key result whose equivalent in the vertex
algebra context has seemingly not been completely proved so far. Finally, the
chiral ring is discussed in terms of net representations.Comment: 42 pages. Final version to be published in Communications in
Mathematical Physic
U-duality and Network Configurations of Branes
We explicitly write down the invariant supersymmetry conditions for branes
with generic values of moduli and U-duality charges in various space-time
dimensions . We then use these results to obtain new BPS states,
corresponding to network type structure of such branes.Comment: 26 pages, Latex, Title and Text modified (minor modifications), to
appear in Int. Jour. Mod. Phys.
Representation theory in chiral conformal field theory: from fields to observables
This article develops new techniques for understanding the relationship
between the three different mathematical formulations of two-dimensional chiral
conformal field theory: conformal nets (axiomatizing local observables), vertex
operator algebras (axiomatizing fields), and Segal CFTs. It builds upon
previous work which introduced a geometric interpolation procedure for
constructing conformal nets from VOAs via Segal CFT, simultaneously relating
all three frameworks. In this article, we extend this construction to study the
relationship between the representation theory of conformal nets and the
representation theory of vertex operator algebras. We define a correspondence
between representations in the two contexts, and show how to construct
representations of conformal nets from VOAs. We also show that this
correspondence is rich enough to relate the respective 'fusion product'
theories for conformal nets and VOAs, by constructing local intertwiners (in
the sense of conformal nets) from intertwining operators (in the sense of
VOAs). We use these techniques to show that all WZW conformal nets can be
constructed using our geometric interpolation procedure.Comment: 79 pages. v2: minor revisions and update
Fermion condensation and super pivotal categories
We study fermionic topological phases using the technique of fermion
condensation. We give a prescription for performing fermion condensation in
bosonic topological phases which contain a fermion. Our approach to fermion
condensation can roughly be understood as coupling the parent bosonic
topological phase to a phase of physical fermions, and condensing pairs of
physical and emergent fermions. There are two distinct types of objects in
fermionic theories, which we call "m-type" and "q-type" particles. The
endomorphism algebras of q-type particles are complex Clifford algebras, and
they have no analogues in bosonic theories. We construct a fermionic
generalization of the tube category, which allows us to compute the
quasiparticle excitations in fermionic topological phases. We then prove a
series of results relating data in condensed theories to data in their parent
theories; for example, if is a modular tensor category containing
a fermion, then the tube category of the condensed theory satisfies
.
We also study how modular transformations, fusion rules, and coherence
relations are modified in the fermionic setting, prove a fermionic version of
the Verlinde dimension formula, construct a commuting projector lattice
Hamiltonian for fermionic theories, and write down a fermionic version of the
Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted
to three detailed examples of performing fermion condensation to produce
fermionic topological phases: we condense fermions in the Ising theory, the
theory, and the theory, and compute the
quasiparticle excitation spectrum in each of these examples.Comment: 161 pages; v2: corrected typos (including 18 instances of "the the")
and added some reference
Normalisation Control in Deep Inference via Atomic Flows
We introduce `atomic flows': they are graphs obtained from derivations by
tracing atom occurrences and forgetting the logical structure. We study simple
manipulations of atomic flows that correspond to complex reductions on
derivations. This allows us to prove, for propositional logic, a new and very
general normalisation theorem, which contains cut elimination as a special
case. We operate in deep inference, which is more general than other syntactic
paradigms, and where normalisation is more difficult to control. We argue that
atomic flows are a significant technical advance for normalisation theory,
because 1) the technique they support is largely independent of syntax; 2)
indeed, it is largely independent of logical inference rules; 3) they
constitute a powerful geometric formalism, which is more intuitive than syntax
Vector lattices with a Hausdorff uo-Lebesgue topology
We investigate the construction of a Hausdorff uo-Lebesgue topology on a
vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal,
and what the properties of the topologies thus obtained are. When the vector
lattice has an order dense ideal with a separating order continuous dual, it is
always possible to supply it with such a topology in this fashion, and the
restriction of this topology to a regular sublattice is then also a Hausdorff
uo-Lebesgue topology. A regular vector sublattice of
for a semi-finite measure falls into this
category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is
then the convergence in measure on subsets of finite measure. When a vector
lattice not only has an order dense ideal with a separating order continuous
dual, but also has the countable sup property, we show that every net in a
regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology
always contains a sequence that is uo-convergent to the same limit. This
enables us to give satisfactory answers to various topological questions about
uo-convergence in this context.Comment: 37 pages. Minor changes; a few references added. Final version, to
appear in J. Math. Anal. App
Loop groups and noncommutative geometry
We describe the representation theory of loop groups in terms of K-theory and
noncommutative geometry. This is done by constructing suitable spectral triples
associated with the level l projective unitary positive-energy representations
of any given loop group . The construction is based on certain
supersymmetric conformal field theory models associated with LG in the setting
of conformal nets. We then generalize the construction to many other rational
chiral conformal field theory models including coset models and the moonshine
conformal net.Comment: Revised versio
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