6,313 research outputs found
A Q-operator for the quantum transfer matrix
Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is
constructed employing the representation theory of quantum groups. The spectrum
of this Q-operator is discussed and novel functional relations which describe
the finite temperature regime of the XXZ spin-chain are derived. For
non-vanishing magnetic field the previously known Bethe ansatz equations can be
replaced by a system of quadratic equations which is an important advantage for
numerical studies. For vanishing magnetic field and rational coupling values it
is argued that the quantum transfer matrix exhibits a loop algebra symmetry
closely related to the one of the classical six-vertex transfer matrix at roots
of unity.Comment: 20 pages, v2: some minor style improvement
From Quantum B\"acklund Transforms to Topological Quantum Field Theory
We derive the quantum analogue of a B\"acklund transformation for the
quantised Ablowitz-Ladik chain, a space discretisation of the nonlinear
Schr\"odinger equation. The quantisation of the Ablowitz-Ladik chain leads to
the -boson model. Using a previous construction of Baxter's Q-operator for
this model by the author, a set of functional relations is obtained which
matches the relations of a one-variable classical B\"acklund transform to all
orders in . We construct also a second Q-operator and show that it is
closely related to the inverse of the first. The multi-B\"acklund transforms
generated from the Q-operator define the fusion matrices of a 2D TQFT and we
derive a linear system for the solution to the quantum B\"acklund relations in
terms of the TQFT fusion coefficients.Comment: 29 pages,4 figures (v3: published version
The su(n) WZNW fusion ring as integrable model: a new algorithm to compute fusion coefficients
This is a proceedings article reviewing a recent combinatorial construction
of the su(n) WZNW fusion ring by C. Stroppel and the author. It contains one
novel aspect: the explicit derivation of an algorithm for the computation of
fusion coefficients different from the Kac-Walton formula. The discussion is
presented from the point of view of a vertex model in statistical mechanics
whose partition function generates the fusion coefficients. The statistical
model can be shown to be integrable by linking its transfer matrix to a
particular solution of the Yang-Baxter equation. This transfer matrix can be
identified with the generating function of an (infinite) set of polynomials in
a noncommutative alphabet: the generators of the local affine plactic algebra.
The latter is a generalisation of the plactic algebra occurring in the context
of the Robinson-Schensted correspondence. One can define analogues of Schur
polynomials in this noncommutative alphabet which become identical to the
fusion matrices when represented as endomorphisms over the state space of the
integrable model. Crucial is the construction of an eigenbasis, the Bethe
vectors, which are the idempotents of the fusion algebra.Comment: 33 pages, 8 figures; published in conference proceedings (Kokyuroku
Bessatsu) of the workshop "Infinite Analysis 10, Developments in Quantum
Integrable Systems", Research Institute for Mathematical Sciences, Kyoto,
June 14-16, 2010; v2: some typos in Section 4 fixe
Dimers, crystals and quantum Kostka numbers
We relate the counting of honeycomb dimer configurations on the cylinder to the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We show that these dimer configurations yield the quantum Kostka numbers of the small quantum cohomology ring of the Grassmannian, i.e. the expansion coefficients when multiplying a Schubert class repeatedly with different Chern classes. This allows one to derive sum rules for Gromov-Witten invariants in terms of dimer configurations
A Combinatorial Derivation of the Racah-Speiser Algorithm for Gromov-Witten invariants
Using a finite-dimensional Clifford algebra a new combinatorial product
formula for the small quantum cohomology ring of the complex Grassmannian is
presented. In particular, Gromov-Witten invariants can be expressed through
certain elements in the Clifford algebra, this leads to a q-deformation of the
Racah-Speiser algorithm allowing for their computation in terms of Kostka
numbers. The second main result is a simple and explicit combinatorial formula
for projecting product expansions in the quantum cohomology ring onto the sl(n)
Verlinde algebra. This projection is non-trivial and amounts to an identity
between numbers of rational curves intersecting Schubert varieties and
dimensions of moduli spaces of generalised theta-functions.Comment: 24 pages, 3 figure
Quantum Integrability and Generalised Quantum Schubert Calculus
We introduce and study a new mathematical structure in the generalised
(quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert
calculus to a quantum integrable system known in the physics literature as the
asymmetric six-vertex model. Our approach offers a new perspective on already
established and well-studied special cases, for example equivariant K-theory,
and in addition allows us to formulate a conjecture on the so-far unknown case
of quantum equivariant K-theory.Comment: 57 pages, 10 figures; v2: some references added and some minor
changes; v3: abstract shortened, some typos corrected and a discussion of the
Bethe roots for the non-equivariant case added; v4: accepted versio
Cylindric Reverse Plane Partitions and 2D TQFT
The ring of symmetric functions carries the structure of a Hopf algebra. When
computing the coproduct of complete symmetric functions one arrives
at weighted sums over reverse plane partitions (RPP) involving binomial
coefficients. Employing the action of the extended affine symmetric group at
fixed level we generalise these weighted sums to cylindric RPP and define
cylindric complete symmetric functions. The latter are shown to be
-positive, that is, their expansions coefficients in the basis of complete
symmetric functions are non-negative integers. We state an explicit formula in
terms of tensor multiplicities for irreducible representations of the
generalised symmetric group. Moreover, we relate the cylindric complete
symmetric functions to a 2D topological quantum field theory (TQFT) that is a
generalisation of the celebrated -Verlinde algebra
or Wess-Zumino-Witten fusion ring, which plays a prominent role in the context
of vertex operator algebras and algebraic geometry.Comment: 13 pages, 1 figure, accepted conference proceedings article for
FPSAC2018 (Hanover
Equivariant quantum cohomology and Yang-Baxter algebras
There are two intriguing statements regarding the quantum cohomology of
partial flag varieties. The first one relates quantum cohomology to the
affinisation of Lie algebras and the homology of the affine Grassmannian, the
second one connects it with the geometry of quiver varieties. The connection
with the affine Grassmannian was first discussed in unpublished work of
Peterson and subsequently proved by Lam and Shimozono. The second development
is based on recent works of Nekrasov, Shatashvili and of Maulik, Okounkov
relating the quantum cohomology of Nakajima varieties with integrable systems
and quantum groups. In this article we explore for the simplest case, the
Grassmannian, the relation between the two approaches. We extend the definition
of the integrable systems called vicious and osculating walkers to the
equivariant setting and show that these models have simple expressions in a
particular representation of the affine nil-Hecke ring. We compare this
representation with the one introduced by Kostant and Kumar and later used by
Peterson in his approach to Schubert calculus. We reveal an underlying quantum
group structure in terms of Yang-Baxter algebras and relate them to Schur-Weyl
duality. We also derive new combinatorial results for equivariant Gromov-Witten
invariants such as an explicit determinant formula.Comment: 60 pages, 11 figures; v2: statement about Schur-Weyl duality added
and introduction slightly rewritten, see last paragraph on page 2 and first
paragraph on page 3 as well as Theorem 1.3 in the introductio
PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
We investigate the PT-symmetry of the quantum group invariant XXZ chain. We
show that the PT-operator commutes with the quantum group action and also
discuss the transformation properties of the Bethe wavefunction. We exploit the
fact that the Hamiltonian is an element of the Temperley-Lieb algebra in order
to give an explicit and exact construction of an operator that ensures
quasi-Hermiticity of the model. This construction relys on earlier ideas
related to quantum group reduction. We then employ this result in connection
with the quantum analogue of Schur-Weyl duality to introduce a dual pair of
C-operators, both of which have closed algebraic expressions. These are novel,
exact results connecting the research areas of integrable lattice systems and
non-Hermitian Hamiltonians.Comment: 32 pages with figures, v2: some minor changes and added references,
version published in JP
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