22 research outputs found
Mapping Class Group Actions on Quantum Doubles
We study representations of the mapping class group of the punctured torus on
the double of a finite dimensional possibly non-semisimple Hopf algebra that
arise in the construction of universal, extended topological field theories. We
discuss how for doubles the degeneracy problem of TQFT's is circumvented. We
find compact formulae for the -matrices using the canonical,
non degenerate forms of Hopf algebras and the bicrossed structure of doubles
rather than monodromy matrices. A rigorous proof of the modular relations and
the computation of the projective phases is supplied using Radford's relations
between the canonical forms and the moduli of integrals. We analyze the
projective -action on the center of for an
-st root of unity. It appears that the -dimensional
representation decomposes into an -dimensional finite representation and a
-dimensional, irreducible representation. The latter is the tensor product
of the two dimensional, standard representation of and the finite,
-dimensional representation, obtained from the truncated TQFT of the
semisimplified representation category of .Comment: 45 page
The Bethe-Ansatz for N=4 Super Yang-Mills
We derive the one loop mixing matrix for anomalous dimensions in N=4 Super
Yang-Mills. We show that this matrix can be identified with the Hamiltonian of
an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz
to find a recipe for computing anomalous dimensions for a wide range of
operators. We give exact results for BMN operators with two impurities and
results up to and including first order 1/J corrections for BMN operators with
many impurities. We then use a result of Reshetikhin's to find the exact
one-loop anomalous dimension for an SO(6) singlet in the limit of large bare
dimension. We also show that this last anomalous dimension is proportional to
the square root of the string level in the weak coupling limit.Comment: 35 pages, 3 figures, LaTeX; v2 references added, typos corrected,
\Lambda fixed; v3 expanded discussion of higher loops in conclusion, matches
published versio
Spiders for rank 2 Lie algebras
A spider is an axiomatization of the representation theory of a group,
quantum group, Lie algebra, or other group or group-like object. We define
certain combinatorial spiders by generators and relations that are isomorphic
to the representation theories of the three rank two simple Lie algebras,
namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider
for A1. Among other things, they yield bases for invariant spaces which are
probably related to Lusztig's canonical bases, and they are useful for
computing quantities such as generalized 6j-symbols and quantum link
invariants.Comment: 33 pages. Has color figure
Sums over Graphs and Integration over Discrete Groupoids
We show that sums over graphs such as appear in the theory of Feynman
diagrams can be seen as integrals over discrete groupoids. From this point of
view, basic combinatorial formulas of the theory of Feynman diagrams can be
interpreted as pull-back or push-forward formulas for integrals over suitable
groupoids.Comment: 27 pages, 4 eps figures; LaTeX2e; uses Xy-Pic. Some ambiguities
fixed, and several proofs simplifie
Zero modes' fusion ring and braid group representations for the extended chiral su(2) WZNW model
The zero modes' Fock space for the extended chiral WZNW model gives
room to a realization of the Grothendieck fusion ring of representations of the
restricted quantum universal enveloping algebra (QUEA) at an even
(-th) root of unity, and of its extension by the Lusztig operators. It is
shown that expressing the Drinfeld images of canonical characters in terms of
Chebyshev polynomials of the Casimir invariant allows a streamlined
derivation of the characteristic equation of from the defining relations of
the restricted QUEA. The properties of the fusion ring of the Lusztig's
extension of the QUEA in the zero modes' Fock space are related to the braiding
properties of correlation functions of primary fields of the extended
current algebra model.Comment: 36 pages, 1 figure; version 3 - improvements in Sec. 2 and 3:
definitions of the double, as well as R- (and M-)matrix changed to fit the
zero modes' one
Higher Algebraic Structures and Quantization
We derive (quasi-)quantum groups in 2+1 dimensional topological field theory
directly from the classical action and the path integral. Detailed computations
are carried out for the Chern-Simons theory with finite gauge group. The
principles behind our computations are presumably more general. We extend the
classical action in a d+1 dimensional topological theory to manifolds of
dimension less than d+1. We then ``construct'' a generalized path integral
which in d+1 dimensions reduces to the standard one and in d dimensions
reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory
the path integral over the circle is the category of representations of a
quasi-quantum group. In this paper we only consider finite theories, in which
the generalized path integral reduces to a finite sum. New ideas are needed to
extend beyond the finite theories treated here.Comment: 62 pages + 16 figures (revised version). In this revision we make
some small corrections and clarification
q-Deformed Superalgebras
The article deals with q-analogs of the three- and four-dimensional Euclidean
superalgebra and the Poincare superalgebra.Comment: 38 pages, LateX, no figures, corrected typo
A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
We construct an invariant J_M of integral homology spheres M with values in a
completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at
each root of unity \zeta gives the the SU(2) Witten-Reshetikhin-Turaev
invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2)
Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is
an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on
\zeta behaves like an ``analytic function'' defined on the set of roots of
unity. That is, the \tau_\zeta(M) for all roots of unity are determined by a
"Taylor expansion" at any root of unity, and also by the values at infinitely
many roots of unity of prime power orders. In particular, \tau_\zeta(M) for all
roots of unity are determined by the Ohtsuki series, which can be regarded as
the Taylor expansion at q=1.Comment: 66 pages, 8 figure
Asymptotic shapes with free boundaries
We study limit shapes for dimer models on domains of the hexagonal lattice with free boundary conditions. This is equivalent to the large deviation phenomenon for a random stepped surface over domains fixed only at part of the boundary