98 research outputs found
Higher Order Terms in the Melvin-Morton Expansion of the Colored Jones Polynomial
We formulate a conjecture about the structure of `upper lines' in the
expansion of the colored Jones polynomial of a knot in powers of (q-1). The
Melvin-Morton conjecture states that the bottom line in this expansion is equal
to the inverse Alexander polynomial of the knot. We conjecture that the upper
lines are rational functions whose denominators are powers of the Alexander
polynomial. We prove this conjecture for torus knots and give experimental
evidence that it is also true for other types of knots.Comment: 21 pages, 1 figure, LaTe
A Rational Logarithmic Conformal Field Theory
We analyse the fusion of representations of the triplet algebra, the
maximally extended symmetry algebra of the Virasoro algebra at c=-2. It is
shown that there exists a finite number of representations which are closed
under fusion. These include all irreducible representations, but also some
reducible representations which appear as indecomposable components in fusion
products.Comment: 10 pages, LaTe
Topological quantum field theory and four-manifolds
I review some recent results on four-manifold invariants which have been
obtained in the context of topological quantum field theory. I focus on three
different aspects: (a) the computation of correlation functions, which give
explicit results for the Donaldson invariants of non-simply connected
manifolds, and for generalizations of these invariants to the gauge group
SU(N); (b) compactifications to lower dimensions, and relations with
three-manifold topology and with intersection theory on the moduli space of
flat connections on Riemann surfaces; (c) four-dimensional theories with
critical behavior, which give some remarkable constraints on Seiberg-Witten
invariants and new results on the geography of four-manifolds.Comment: 10 pages, LaTeX. Talk given at the 3rd ECM, Barcelona, July 2000;
references adde
On Integrable Quantum Group Invariant Antiferromagnets
A new open spin chain hamiltonian is introduced. It is both integrable
(Sklyanin`s type matrices are used to achieve this) and invariant under
transformations in nilpotent irreps for
. Some considerations on the centralizer of nilpotent
representations and its representation theory are also presented.Comment: IFF-5/92, 13 pages, LaTex file, 8 figures available from author
A Local Logarithmic Conformal Field Theory
The local logarithmic conformal field theory corresponding to the triplet
algebra at c=-2 is constructed. The constraints of locality and crossing
symmetry are explored in detail, and a consistent set of amplitudes is found.
The spectrum of the corresponding theory is determined, and it is found to be
modular invariant. This provides the first construction of a non-chiral
rational logarithmic conformal field theory, establishing that such models can
indeed define bona fide conformal field theories.Comment: 29 pages, LaTeX, minor changes, reference adde
The conformal current algebra on supergroups with applications to the spectrum and integrability
We compute the algebra of left and right currents for a principal chiral
model with arbitrary Wess-Zumino term on supergroups with zero Killing form. We
define primary fields for the current algebra that match the affine primaries
at the Wess-Zumino-Witten points. The Maurer-Cartan equation together with
current conservation tightly constrain the current-current and current-primary
operator product expansions. The Hilbert space of the theory is generated by
acting with the currents on primary fields. We compute the conformal dimensions
of a subset of these states in the large radius limit. The current algebra is
shown to be consistent with the quantum integrability of these models to
several orders in perturbation theory.Comment: 45 pages. Minor correction
Matrix Model as a Mirror of Chern-Simons Theory
Using mirror symmetry, we show that Chern-Simons theory on certain manifolds
such as lens spaces reduces to a novel class of Hermitian matrix models, where
the measure is that of unitary matrix models. We show that this agrees with the
more conventional canonical quantization of Chern-Simons theory. Moreover,
large N dualities in this context lead to computation of all genus A-model
topological amplitudes on toric Calabi-Yau manifolds in terms of matrix
integrals. In the context of type IIA superstring compactifications on these
Calabi-Yau manifolds with wrapped D6 branes (which are dual to M-theory on G2
manifolds) this leads to engineering and solving F-terms for N=1 supersymmetric
gauge theories with superpotentials involving certain multi-trace operators.Comment: harvmac, 54 pages, 13 figure
Remodeling the B-model
We propose a complete, new formalism to compute unambiguously B-model open
and closed amplitudes in local Calabi-Yau geometries, including the mirrors of
toric manifolds. The formalism is based on the recursive solution of matrix
models recently proposed by Eynard and Orantin. The resulting amplitudes are
non-perturbative in both the closed and the open moduli. The formalism can then
be used to study stringy phase transitions in the open/closed moduli space. At
large radius, this formalism may be seen as a mirror formalism to the
topological vertex, but it is also valid in other phases in the moduli space.
We develop the formalism in general and provide an extensive number of checks,
including a test at the orbifold point of A_p fibrations, where the amplitudes
compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our
formalism to predict the disk amplitude for the orbifold C^3/Z_3.Comment: 83 pages, 9 figure
Classical BV theories on manifolds with boundary
In this paper we extend the classical BV framework to gauge theories on
spacetime manifolds with boundary. In particular, we connect the BV
construction in the bulk with the BFV construction on the boundary and we
develop its extension to strata of higher codimension in the case of manifolds
with corners. We present several examples including electrodynamics, Yang-Mills
theory and topological field theories coming from the AKSZ construction, in
particular, the Chern-Simons theory, the theory, and the Poisson sigma
model. This paper is the first step towards developing the perturbative
quantization of such theories on manifolds with boundary in a way consistent
with gluing.Comment: The second version has many typos corrected, references added. Some
typos are probably still there, in particular, signs in examples. In the
third version more typoes are corrected and the exposition is slightly
change
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
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