48 research outputs found
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
The twisted Drinfeld double (or quasi-quantum double) of a finite group with
a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid
is the loop (or inertia) groupoid of the original group and the twisting is
shown geometrically to be the loop transgression of the 3-cocycle. The twisted
representation theory of finite groupoids is developed and used to derive
properties of the Drinfeld double, such as representations being classified by
their characters.
This is all motivated by gerbes and 3-dimensional topological quantum field
theory. In particular the representation category of the twisted Drinfeld
double is viewed as the `space of sections' associated to a transgressed gerbe
over the loop groupoid.Comment: 25 pages, 10 picture
Superconformal Coset Equivalence from Level-Rank Duality
We construct a one-to-one map between the primary fields of the N=2
superconformal Kazama-Suzuki models G(m,n,k) and G(k,n,m) based on complex
Grassmannian cosets, using level-rank duality of Wess-Zumino-Witten models. We
then show that conformal weights, superconformal U(1) charges, modular
transformation matrices, and fusion rules are preserved under this map,
providing strong evidence for the equivalence of these coset models.Comment: 25 pages, harvmac, no figures, added referenc
Central Invariants and Frobenius-Schur Indicators for Semisimple Quasi-Hopf Algebras
In this paper, we obtain a canonical central element for each
semi-simple quasi-Hopf algebra over any field and prove that is
invariant under gauge transformations. We show that if is algebraically
closed of characteristic zero then for any irreducible representation of
which affords the character , takes only the values 0, 1 or
-1, moreover if is a Hopf algebra or a twisted quantum double of a finite
group then is the corresponding Frobenius-Schur Indicator. We
also prove an analog of a Theorem of Larson-Radford for split semi-simple
quasi-Hopf algebra over any field . Using this result, we establish the
relationship between the antipode , the values of , and certain
associated bilinear forms when the underlying field is algebraically closed
of characteristic zero.Comment: 32 pages (version 3
Implications of an arithmetical symmetry of the commutant for modular invariants
We point out the existence of an arithmetical symmetry for the commutant of
the modular matrices S and T. This symmetry holds for all affine simple Lie
algebras at all levels and implies the equality of certain coefficients in any
modular invariant. Particularizing to SU(3)_k, we classify the modular
invariant partition functions when k+3 is an integer coprime with 6 and when it
is a power of either 2 or 3. Our results imply that no detailed knowledge of
the commutant is needed to undertake a classification of all modular
invariants.Comment: 17 pages, plain TeX, DIAS-STP-92-2
On parity functions in conformal field theories
We examine general aspects of parity functions arising in rational conformal
field theories, as a result of Galois theoretic properties of modular
transformations. We focus more specifically on parity functions associated with
affine Lie algebras, for which we give two efficient formulas. We investigate
the consequences of these for the modular invariance problem.Comment: 18 pages, no figure, LaTeX2
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups
We establish braided tensor equivalences among module categories over the
twisted quantum double of a finite group defined by an extension of a group H
by an abelian group, with 3-cocycle inflated from a 3-cocycle on H. We also
prove that the canonical ribbon structure of the module category of any twisted
quantum double of a finite group is preserved by braided tensor equivalences.
We give two main applications: first, if G is an extra-special 2-group of width
at least 2, we show that the quantum double of G twisted by a 3-cocycle w is
gauge equivalent to a twisted quantum double of an elementary abelian 2-group
if, and only if, w^2 is trivial; second, we discuss the gauge equivalence
classes of twisted quantum doubles of groups of order 8, and classify the
braided tensor equivalence classes of these quasi-triangular quasi-bialgebras.
It turns out that there are exactly 20 such equivalence classes.Comment: 27 pages, LateX, a few of typos in v2 correcte
Scaling functions from q-deformed Virasoro characters
We propose a renormalization group scaling function which is constructed from
q-deformed fermionic versions of Virasoro characters. By comparison with
alternative methods, which take their starting point in the massive theories,
we demonstrate that these new functions contain qualitatively the same
information. We show that these functions allow for RG-flows not only amongst
members of a particular series of conformal field theories, but also between
different series such as N=0,1,2 supersymmetric conformal field theories. We
provide a detailed analysis of how Weyl characters may be utilized in order to
solve various recurrence relations emerging at the fixed points of these flows.
The q-deformed Virasoro characters allow furthermore for the construction of
particle spectra, which involve unstable pseudo-particles.Comment: 31 pages of Latex, 5 figure
Poincare Polynomials and Level Rank Dualities in the Coset Construction
We review the coset construction of conformal field theories; the emphasis is
on the construction of the Hilbert spaces for these models, especially if fixed
points occur. This is applied to the superconformal cosets constructed by
Kazama and Suzuki. To calculate heterotic string spectra we reformulate the
Gepner con- struction in terms of simple currents and introduce the so-called
extended Poincar\'e polynomial. We finally comment on the various equivalences
arising between models of this class, which can be expressed as level rank
dualities. (Invited talk given at the III. International Conference on
Mathematical Physics, String Theory and Quantum Gravity, Alushta, Ukraine, June
1993. To appear in Theor. Math. Phys.)Comment: 14 pages in LaTeX, HD-THEP-93-4
Automorphisms of the affine SU(3) fusion rules
We classify the automorphisms of the (chiral) level-k affine SU(3) fusion
rules, for any value of k, by looking for all permutations that commute with
the modular matrices S and T. This can be done by using the arithmetic of the
cyclotomic extensions where the problem is naturally posed. When k is divisible
by 3, the automorphism group (Z_2) is generated by the charge conjugation C. If
k is not divisible by 3, the automorphism group (Z_2 x Z_2) is generated by C
and the Altsch\"uler--Lacki--Zaugg automorphism. Although the combinatorial
analysis can become more involved, the techniques used here for SU(3) can be
applied to other algebras.Comment: 21 pages, plain TeX, DIAS-STP-92-4
Levodopaâinduced dyskinesia are mediated by cortical gamma oscillations in experimental Parkinsonism
Background Levodopa is the most efficacious drug in the symptomatic therapy of motor symptoms in Parkinson's disease (PD); however, longâterm treatment is often complicated by troublesome levodopaâinduced dyskinesia (LID). Recent evidence suggests that LID might be related to increased cortical gamma oscillations. Objective The objective of this study was to test the hypothesis that cortical highâgamma network activity relates to LID in the 6âhydroxydopamine model and to identify new biomarkers for adaptive deep brain stimulation (DBS) therapy in PD. Methods We recorded and analyzed primary motor cortex (M1) electrocorticogram data and motor behavior in freely moving 6âOHDA lesioned rats before and during a daily treatment with levodopa for 3âweeks. The results were correlated with the abnormal involuntary movement score (AIMS) and used for generalized linear modeling (GLM). Results Levodopa reverted motor impairment, suppressed beta activity, and, with repeated administration, led to a progressive enhancement of LID. Concurrently, we observed a highly significant stepwise amplitude increase in finely tuned gamma (FTG) activity and gamma centroid frequency. Whereas AIMS and FTG reached their maximum after the 4th injection and remained on a stable plateau thereafter, the centroid frequency of the FTG power continued to increase thereafter. Among the analyzed gamma activity parameters, the fraction of longest gamma bursts showed the strongest correlation with AIMS. Using a GLM, it was possible to accurately predict AIMS from cortical recordings. Conclusions FTG activity is tightly linked to LID and should be studied as a biomarker for adaptive DBS