326 research outputs found
HERA-B Framework for Online Calibration and Alignment
This paper describes the architecture and implementation of the HERA-B
framework for online calibration and alignment. At HERA-B the performance of
all trigger levels, including the online reconstruction, strongly depends on
using the appropriate calibration and alignment constants, which might change
during data taking. A system to monitor, recompute and distribute those
constants to online processes has been integrated in the data acquisition and
trigger systems.Comment: Submitted to NIM A. 4 figures, 15 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
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
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
A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating
4-dimensional super-symmetric gauge theory for a gauge group G with certain
2-dimensional conformal field theory. This conjecture implies the existence of
certain structures on the (equivariant) intersection cohomology of the
Uhlenbeck partial compactification of the moduli space of framed G-bundles on
P^2. More precisely, it predicts the existence of an action of the
corresponding W-algebra on the above cohomology, satisfying certain properties.
We propose a "finite analog" of the (above corollary of the) AGT conjecture.
Namely, we replace the Uhlenbeck space with the space of based quasi-maps from
P^1 to any partial flag variety G/P of G and conjecture that its equivariant
intersection cohomology carries an action of the finite W-algebra U(g,e)
associated with the principal nilpotent element in the Lie algebra of the Levi
subgroup of P; this action is expected to satisfy some list of natural
properties. This conjecture generalizes the main result of arXiv:math/0401409
when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the
works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of
certain shifted Yangians.Comment: minor change
Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
Gaudin algebras form a family of maximal commutative subalgebras in the
tensor product of copies of the universal enveloping algebra U(\g) of a
semisimple Lie algebra \g. This family is parameterized by collections of
pairwise distinct complex numbers . We obtain some new commutative
subalgebras in U(\g)^{\otimes n} as limit cases of Gaudin subalgebras. These
commutative subalgebras turn to be related to the hamiltonians of bending flows
and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of
spectrum in the Gaudin model for some new cases.Comment: 11 pages, 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
Combinatorial Hopf algebras in quantum field theory I
This manuscript stands at the interface between combinatorial Hopf algebra
theory and renormalization theory. Its plan is as follows: Section 1 is the
introduction, and contains as well an elementary invitation to the subject. The
rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf
algebra theory and examples, in ascending level of complexity. Part II turns
around the all-important Faa di Bruno Hopf algebra. Section 7 contains a first,
direct approach to it. Section 8 gives applications of the Faa di Bruno algebra
to quantum field theory and Lagrange reversion. Section 9 rederives the related
Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf
algebras of Feynman graphs and, more generally, to incidence bialgebras. In
Section10 we describe the first. Then in Section11 we give a simple derivation
of (the properly combinatorial part of) Zimmermann's cancellation-free method,
in its original diagrammatic form. In Section 12 general incidence algebras are
introduced, and the Faa di Bruno bialgebras are described as incidence
bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us
to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next,
the general algebraic-combinatorial proof of the cancellation-free formula for
antipodes is ascertained; this is the heart of the paper. The structure results
for commutative Hopf algebras are found in Sections 14 and 15. An outlook
section very briefly reviews the coalgebraic aspects of quantization and the
Rota-Baxter map in renormalization.Comment: 94 pages, LaTeX figures, precisions made, typos corrected, more
references adde
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