6,201 research outputs found
Stabilization phenomena in Kac-Moody algebras and quiver varieties
Let X be the Dynkin diagram of a symmetrizable Kac-Moody algebra, and X_0 a
subgraph with all vertices of degree 1 or 2. Using the crystal structure on the
components of quiver varieties for X, we show that if we expand X by extending
X_0, the branching multiplicities and tensor product multiplicities stabilize,
provided the weights involved satisfy a condition which we call ``depth'' and
are supported outside . This extends a theorem of Kleber and Viswanath.
Furthermore, we show that the weight multiplicities of such representations
are polynomial in the length of X_0, generalizing the same result for A_\ell by
Benkart, et al.Comment: final version, to appear in International Math Research Notices. 17
pages, 4 figure
Entanglement of Solitons in the Frenkel-Kontorova Model
We investigate entanglement of solitons in the continuum-limit of the
nonlinear Frenkel-Kontorova chain. We find that the entanglement of solitons
manifests particle-like behavior as they are characterized by localization of
entanglement. The von-Neumann entropy of solitons mixes critical with
noncritical behaviors. Inside the core of the soliton the logarithmic increase
of the entropy is faster than the universal increase of a critical field,
whereas outside the core the entropy decreases and saturates the constant value
of the corresponding massive noncritical field. In addition, two solitons
manifest long-range entanglement that decreases with the separation of the
solitons more slowly than the universal decrease of the critical field.
Interestingly, in the noncritical regime of the Frenkel-Kontorova model,
entanglement can even increase with the separation of the solitons. We show
that most of the entanglement of the so-called internal modes of the solitons
is saturated by local degrees of freedom inside the core, and therefore we
suggest using the internal modes as carriers of quantum information.Comment: 16 pages, 22 figure
Kazhdan-Lusztig tensoring and Harish-Chandra categories
We use the Kazhdan-Lusztig tensoring to define affine translation functors,
describe annihilating ideals of highest weight modules over an affine Lie
algebra in terms of the corresponding VOA, and to sketch a functorial approach
to ``affine Harish-Chandra bimodules''.Comment: 22 pages late
Quantum Teichm\"uller space from quantum plane
We derive the quantum Teichm\"uller space, previously constructed by Kashaev
and by Fock and Chekhov, from tensor products of a single canonical
representation of the modular double of the quantum plane. We show that the
quantum dilogarithm function appears naturally in the decomposition of the
tensor square, the quantum mutation operator arises from the tensor cube, the
pentagon identity from the tensor fourth power of the canonical representation,
and an operator of order three from isomorphisms between canonical
representation and its left and right duals. We also show that the quantum
universal Teichm\"uller space is realized in the infinite tensor power of the
canonical representation naturally indexed by rational numbers including the
infinity. This suggests a relation to the same index set in the classification
of projective modules over the quantum torus, the unitary counterpart of the
quantum plane, and points to a new quantization of the universal Teichm\"uller
space.Comment: 41 pages, 9 figure
Gaudin models with irregular singularities
We introduce a class of quantum integrable systems generalizing the Gaudin
model. The corresponding algebras of quantum Hamiltonians are obtained as
quotients of the center of the enveloping algebra of an affine Kac-Moody
algebra at the critical level, extending the construction of higher Gaudin
Hamiltonians from hep-th/9402022 to the case of non-highest weight
representations of affine algebras. We show that these algebras are isomorphic
to algebras of functions on the spaces of opers on P^1 with regular as well as
irregular singularities at finitely many points. We construct eigenvectors of
these Hamiltonians, using Wakimoto modules of critical level, and show that
their spectra on finite-dimensional representations are given by opers with
trivial monodromy. We also comment on the connection between the generalized
Gaudin models and the geometric Langlands correspondence with ramification.Comment: Latex, 72 pages. Final version to appear in Advances in Mathematic
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