114 research outputs found
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
The Integrals of Motion for the Deformed W-Algebra II: Proof of the commutation relations
We explicitly construct two classes of infinitly many commutative operators
in terms of the deformed W-algebra , and give proofs of the
commutation relations of these operators. We call one of them local integrals
of motion and the other nonlocal one, since they can be regarded as elliptic
deformation of local and nonlocal integrals of motion for the algebra.Comment: Dedicated to Professor Tetsuji Miwa on the occasion on the 60th
birthda
Logarithmic intertwining operators and W(2,2p-1)-algebras
For every , we obtained an explicit construction of a family of
-modules, which decompose as direct sum of simple Virasoro
algebra modules. Furthermore, we classified all irreducible self-dual
-modules, we described their internal structure, and
computed their graded dimensions. In addition, we constructed certain hidden
logarithmic intertwining operators among two ordinary and one logarithmic
-modules. This work, in particular, gives a mathematically
precise formulation and interpretation of what physicists have been referring
to as "logarithmic conformal field theory" of central charge
. Our explicit construction can be
easily applied for computations of correlation functions. Techniques from this
paper can be used to study the triplet vertex operator algebra
and other logarithmic models.Comment: 22 pages; v2: misprints corrected, other minor changes. Final version
to appear in Journal of Math. Phy
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
Lie superalgebras and irreducibility of A_1^(1)-modules at the critical level
We introduce the infinite-dimensional Lie superalgebra and
construct a family of mappings from certain category of -modules
to the category of A_1^(1)-modules of critical level. Using this approach, we
prove the irreducibility of a family of A_1^(1)-modules at the critical level.
As a consequence, we present a new proof of irreducibility of certain
Wakimoto modules. We also give a natural realizations of irreducible quotients
of relaxed Verma modules and calculate characters of these representations.Comment: 21 pages, Late
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
Free-Field Realization of D-dimensional Cylindrical Gravitational Waves
We find two-dimensional free-field variables for D-dimensional general
relativity on spacetimes with D-2 commuting spacelike Killing vector fields and
non-compact spatial sections for D>4. We show that there is a canonical
transformation which maps the corresponding two-dimensional dilaton gravity
theory into a two-dimensional diffeomorphism invariant theory of the free-field
variables. We also show that the spacetime metric components can be expressed
as asymptotic series in negative powers of the dilaton, with coefficients which
can be determined in terms of the free fields.Comment: 15 pages, Late
On Vertex Operator Construction of Quantum Affine Algebras
We describe the construction of the quantum deformed affine Lie algebras
using the vertex operators in the free field theory. We prove the Serre
relations for the quantum deformed Borel subalgebras of affine algebras, namely
the case of is considered in detail. We provide some
formulas for generators of affine algebra.Comment: LaTeX, 9 pages; typos corrected, references adde
Cyclotomic Gaudin models: construction and Bethe ansatz
This is a pre-copyedited author produced PDF of an article accepted for publication in Communications in Mathematical Physics, Benoit, V and Young, C, 'Cyclotomic Gaudin models: construction and Bethe ansatz', Commun. Math. Phys. (2016) 343:971, first published on line March 24, 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2601-3 © Springer-Verlag Berlin Heidelberg 2016To any simple Lie algebra and automorphism we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case . We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.Peer reviewe
Note on the Algebra of Screening Currents for the Quantum Deformed W-Algebra
With slight modifications in the zero modes contributions, the positive and
negative screening currents for the quantum deformed W-algebra W_{q,p}(g) can
be put together to form a single algebra which can be regarded as an elliptic
deformation of the universal enveloping algebra of \hat{g}, where g is any
classical simply-laced Lie algebra.Comment: LaTeX file, 9 pages. Errors in Serre relation corrected. Two
references to Awata,H. et al adde
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