133 research outputs found
Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology
For a fixed parabolic subalgebra p of gl(n,C) we prove that the centre of the
principal block O(p) of the parabolic category O is naturally isomorphic to the
cohomology ring of the corresponding Springer fibre. We give a diagrammatic
description of O(p) for maximal parabolic p and give an explicit isomorphism to
Braden's description of the category Perv_B(G(n,n)) of perverse sheaves on
Grassmannians. As a consequence Khovanov's algebra H^n is realised as the
endomorphism ring of some object from Perv_B(G(n,n)) which corresponds under
localisation and the Riemann-Hilbert correspondence to a full
projective-injective module in the corresponding category . From there
one can deduce that Khovanov's tangle invariants are obtained from the more
general functorial invariants involving category O by restriction.Comment: 39 pages, 9 figures, added a few remark
A BGG-type resolution for tensor modules over general linear superalgebra
We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct
sums of Kac modules for the finite-dimensional irreducible tensor
representations of the general linear superalgebra. As a consequence it follows
that the unique maximal submodule of a corresponding reducible Kac module is
generated by its proper singular vector.Comment: 11pages, LaTeX forma
Critical Excitation Spectrum of Quantum Chain With A Local 3-Spin Coupling
This article reports a measurement of the low-energy excitation spectrum
along the critical line for a quantum spin chain having a local interaction
between three Ising spins and longitudinal and transverse magnetic fields. The
measured excitation spectrum agrees with that predicted by the (D, A)
conformal minimal model under a nontrivial correspondence between translations
at the critical line and discrete lattice translations. Under this
correspondence, the measurements confirm a prediction that the critical line of
this quantum spin chain and the critical point of the 2D 3-state Potts model
are in the same universality class.Comment: 7 pages, 2 figure
Singular Vectors of the Virasoro Algebra
We give expressions for the singular vectors in the highest weight
representations of the Virasoro algebra. We verify that the expressions ---
which take the form of a product of operators applied to the highest weight
vector --- do indeed define singular vectors. These results explain the
patterns of embeddings amongst Virasoro algebra highest weight representations.Comment: 15 p
Loop Variables and the Virasoro Group
We derive an expression in closed form for the action of a finite element of
the Virasoro Group on generalized vertex operators. This complements earlier
results giving an algorithm to compute the action of a finite string of
generators of the Virasoro Algebra on generalized vertex operators. The main
new idea is to use a first order formalism to represent the infinitesimal group
element as a loop variable. To obtain a finite group element it is necessary to
thicken the loop to a band of finite thickness. This technique makes the
calculation very simple.Comment: 23 pages, PSU/T
Torus Knot and Minimal Model
We reveal an intimate connection between the quantum knot invariant for torus
knot T(s,t) and the character of the minimal model M(s,t), where s and t are
relatively prime integers. We show that Kashaev's invariant, i.e., the
N-colored Jones polynomial at the N-th root of unity, coincides with the
Eichler integral of the character.Comment: 10 page
Homogeneous components in the moduli space of sheaves and Virasoro characters
The moduli space of framed torsion free sheaves on the
projective plane with rank and second Chern class equal to has the
natural action of the -dimensional torus. In this paper, we look at the
fixed point set of different one-dimensional subtori in this torus. We prove
that in the homogeneous case the generating series of the numbers of the
irreducible components has a beautiful decomposition into an infinite product.
In the case of odd these infinite products coincide with certain Virasoro
characters. We also propose a conjecture in a general quasihomogeneous case.Comment: Published version, 19 page
Form factors of descendant operators in the massive Lee-Yang model
The form factors of the descendant operators in the massive Lee-Yang model
are determined up to level 7. This is first done by exploiting the conserved
quantities of the integrable theory to generate the solutions for the
descendants starting from the lowest non-trivial solutions in each operator
family. We then show that the operator space generated in this way, which is
isomorphic to the conformal one, coincides, level by level, with that implied
by the -matrix through the form factor bootstrap. The solutions we determine
satisfy asymptotic conditions carrying the information about the level that we
conjecture to hold for all the operators of the model.Comment: 23 page
Coset Constructions in Chern-Simons Gauge Theory
Coset constructions in the framework of Chern-Simons topological gauge
theories are studied. Two examples are considered: models of the types
with and
coprime integers, and . In the latter
case it is shown that the Chern-Simons wave functionals can be identified with
t he characters of the minimal unitary models, and an explicit representation
of the knot (Verlinde) operators acting on the space of characters is
obtained.Comment: 15 page
Character decomposition of Potts model partition functions. I. Cyclic geometry
We study the Potts model (defined geometrically in the cluster picture) on
finite two-dimensional lattices of size L x N, with boundary conditions that
are free in the L-direction and periodic in the N-direction. The decomposition
of the partition function in terms of the characters K\_{1+2l} (with
l=0,1,...,L) has previously been studied using various approaches (quantum
groups, combinatorics, transfer matrices). We first show that the K\_{1+2l}
thus defined actually coincide, and can be written as traces of suitable
transfer matrices in the cluster picture. We then proceed to similarly
decompose constrained partition functions in which exactly j clusters are
non-contractible with respect to the periodic lattice direction, and a
partition function with fixed transverse boundary conditions.Comment: 21 pages, 4 figure
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