281 research outputs found
Gaudin model and Deligne's category
We show that the construction of the higher Gaudin Hamiltonians associated to
the Lie algebra admits an interpolation to any complex .
We do this using the Deligne's category , which is a formal
way to define the category of finite-dimensional representations of the group
, when is not necessarily a natural number.
We also obtain interpolations to any complex of the no-monodromy
conditions on a space of differential operators of order , which are
considered to be a modern form of the Bethe ansatz equations. We prove that the
relations in the algebra of higher Gaudin Hamiltonians for complex are
generated by our interpolations of the no-monodromy conditions.
Our constructions allow us to define what it means for a pseudo-deifferential
operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the
Gaudin model associated with the Lie superalgebra ,
we show that a ratio of monodromy-free differential operators is a
pseudo-differential operator without monodromy.Comment: 35 page
On the fundamental group of the complement of a complex hyperplane arrangement
We construct two combinatorially equivalent line arrangements in the complex
projective plane such that the fundamental groups of their complements are not
isomorphic. The proof uses a new invariant of the fundamental group of the
complement to a line arrangement of a given combinatorial type with respect to
isomorphisms inducing the canonical isomorphism of the first homology groups.Comment: 12 pages, Latex2e with AMSLaTeX 1.2, no figures; this last version is
almost the same as published in Functional Analysis and its Applications 45:2
(2011), 137-14
Limits of Gaudin Systems: Classical and Quantum Cases
We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new ''Gaudin'' algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties of ''Manin matrices'' to provide explicit generators of the Gaudin Algebras in the quantum case
Manin matrices and Talalaev's formula
We study special class of matrices with noncommutative entries and
demonstrate their various applications in integrable systems theory. They
appeared in Yu. Manin's works in 87-92 as linear homomorphisms between
polynomial rings; more explicitly they read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). We claim
that such matrices behave almost as well as matrices with commutative elements.
Namely theorems of linear algebra (e.g., a natural definition of the
determinant, the Cayley-Hamilton theorem, the Newton identities and so on and
so forth) holds true for them.
On the other hand, we remark that such matrices are somewhat ubiquitous in
the theory of quantum integrability. For instance, Manin matrices (and their
q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and
the so--called Cartier-Foata matrices. Also, they enter Talalaev's
hep-th/0404153 remarkable formulas: ,
det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show
that theorems of linear algebra, after being established for such matrices,
have various applications to quantum integrable systems and Lie algebras, e.g
in the construction of new generators in (and, in general,
in the construction of quantum conservation laws), in the
Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also
discuss applications to the separation of variables problem, new Capelli
identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints
e.g. in Newton id-s fixed, normal ordering convention turned to standard one,
refs. adde
SU(3) Richardson-Gaudin models: three level systems
We present the exact solution of the Richardson-Gaudin models associated with
the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for
any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For
the rational case additional cubic integrals of motion are obtained, whose
number is added to that of the quadratic ones to match, as required from the
integrability condition, the number of quantum degrees of freedom of the model.
We discuss different SU(3) physical representations and elucidate the meaning
of the parameters entering in the formalism. By considering a bosonic mapping
limit of one of the SU(3) copies, we derive new integrable models for three
level systems interacting with two bosons. These models include a generalized
Tavis-Cummings model for three level atoms interacting with two modes of the
quantized electric field.Comment: Revised version. To appear in Jour. Phys. A: Math. and Theo
A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories
We give an explicit differential equation which is expected to determine the
instanton partition function in the presence of the full surface operator in
N=2 SU(N) gauge theory. The differential equation arises as a quantization of a
certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and
Tsuda.Comment: 15 pages, v2: typos corrected and references added, v3: discussion,
appendix and references adde
Integrable Models From Twisted Half Loop Algebras
This paper is devoted to the construction of new integrable quantum
mechanical models based on certain subalgebras of the half loop algebra of
gl(N). Various results about these subalgebras are proven by presenting them in
the notation of the St Petersburg school. These results are then used to
demonstrate the integrability, and find the symmetries, of two types of
physical system: twisted Gaudin magnets, and Calogero-type models of particles
on several half-lines meeting at a point.Comment: 22 pages, 1 figure, Introduction improved, References adde
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