2,114 research outputs found
Gaudin Model, Bethe Ansatz and Critical Level
We propose a new method of diagonalization of hamiltonians of the Gaudin
model associated to an arbitrary simple Lie algebra, which is based on Wakimoto
modules over affine algebras at the critical level. We construct eigenvectors
of these hamiltonians by restricting certain invariant functionals on tensor
products of Wakimoto modules. In conformal field theory language, the
eigenvectors are given by certain bosonic correlation functions. Analogues of
Bethe ansatz equations naturally appear as Kac-Kazhdan type equations on the
existence of certain singular vectors in Wakimoto modules. We use this
construction to expalain a connection between Gaudin's model and correlation
functions of WZNW models.Comment: 40 pages, postscript-file (references added and corrected
A class of Baker-Akhiezer arrangements
We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh–Veselov Baker–Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker–Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasi-invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero–Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1
A note on the relationship between rational and trigonometric solutions of the WDVV equations
Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds. As an application it is shown that certain trigonometric and rational solutions of the WDVV equations are related by such a twisted Legendre transform
Quantum Algebraic Approach to Refined Topological Vertex
We establish the equivalence between the refined topological vertex of
Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of
type W_{1+infty} introduced by Miki. Our construction involves trivalent
intertwining operators Phi and Phi^* associated with triples of the bosonic
Fock modules. Resembling the topological vertex, a triple of vectors in Z^2 is
attached to each intertwining operator, which satisfy the Calabi-Yau and
smoothness conditions. It is shown that certain matrix elements of Phi and
Phi^* give the refined topological vertex C_{lambda mu nu}(t,q) of
Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined
topological vertex C_{lambda mu}^nu(q,t) of Awata-Kanno. The gluing factors
appears correctly when we consider any compositions of Phi and Phi^*. The
spectral parameters attached to Fock spaces play the role of the K"ahler
parameters.Comment: 27 page
Quantum W-algebras and Elliptic Algebras
We define quantum W-algebras generalizing the results of Reshetikhin and the
second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated
to sl_N is an associative algebra depending on two parameters. For special
values of parameters it becomes the ordinary W-algebra of sl_N, or the
q-deformed classical W-algebra of sl_N. We construct free field realizations of
the quantum W-algebras and the screening currents. We also point out some
interesting elliptic structures arising in these algebras. In particular, we
show that the screening currents satisfy elliptic analogues of the Drinfeld
relations in U_q(n^).Comment: 26 pages, AMSLATE
Feigin-Frenkel center in types B, C and D
For each simple Lie algebra g consider the corresponding affine vertex
algebra V_{crit}(g) at the critical level. The center of this vertex algebra is
a commutative associative algebra whose structure was described by a remarkable
theorem of Feigin and Frenkel about two decades ago. However, only recently
simple formulas for the generators of the center were found for the Lie
algebras of type A following Talalaev's discovery of explicit higher Gaudin
Hamiltonians. We give explicit formulas for generators of the centers of the
affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of
types B, C and D. The construction relies on the Schur-Weyl duality involving
the Brauer algebra, and the generators are expressed as weighted traces over
tensor spaces and, equivalently, as traces over the spaces of singular vectors
for the action of the Lie algebra sl_2 in the context of Howe duality. This
leads to explicit constructions of commutative subalgebras of the universal
enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the
Gaudin model associated with each Lie algebra g. We also introduce analogues of
the Bethe subalgebras of the Yangians Y(g) and show that their graded images
coincide with the respective commutative subalgebras of U(g[t]).Comment: 29 pages, constructions of Pfaffian-type Sugawara operators and
commutative subalgebras in universal enveloping algebras are adde
Factorizable ribbon quantum groups in logarithmic conformal field theories
We review the properties of quantum groups occurring as Kazhdan--Lusztig dual
to logarithmic conformal field theory models. These quantum groups at even
roots of unity are not quasitriangular but are factorizable and have a ribbon
structure; the modular group representation on their center coincides with the
representation on generalized characters of the chiral algebra in logarithmic
conformal field models.Comment: 27pp., amsart++, xy. v2: references added, some other minor addition
Solutions of the Knizhnik - Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models
In the spirit of the quantum Hamiltonian reduction we establish a relation
between the chiral -point functions, as well as the equations governing
them, of the WZNW conformal theory and the corresponding Virasoro
minimal models. The WZNW correlators are described as solutions of the Knizhnik
- Zamolodchikov equations with rational levels and isospins. The technical tool
exploited are certain relations in twisted cohomology. The results extend to
arbitrary level and isospin values of the type , $ \
2j, 2j' \in Z\!\!\!Z_+$.Comment: 40 page
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