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: $[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]$ (e.g. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). 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: $det(\partial_z-L_{Gaudin}(z))$, 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 $Z(U(\hat{gl_n}))$ (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

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

Spectral Duality in Integrable Systems from AGT Conjecture

We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl(N) Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrodinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2x2 and NxN representations of the Toda chain and the famous AHH duality

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL(k) Heisenberg chain is dual to the special reduced k+2-points gl(N) Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.Comment: 36 page

Electronic structure and dynamics of optically excited single-wall carbon nanotubes

We have studied the electronic structure and charge-carrier dynamics of individual single-wall carbon nanotubes (SWNTs) and nanotube ropes using optical and electron-spectroscopic techniques. The electronic structure of semiconducting SWNTs in the band-gap region is analyzed using near-infrared absorption spectroscopy. A semi-empirical expression for $E_{11}^{\rm S}$ transition energies, based on tight-binding calculations is found to give striking agreement with experimental data. Time-resolved PL from dispersed SWNT-micelles shows a decay with a time constant of about 15 ps. Using time-resolved photoemission we also find that the electron-phonon ({\it e-ph}) coupling in metallic tubes is characterized by a very small {\it e-ph} mass-enhancement of 0.0004. Ultrafast electron-electron scattering of photo-excited carriers in nanotube ropes is finally found to lead to internal thermalization of the electronic system within about 200 fs.Comment: 10 pages, 10 figures, submitted to Applied Physics

Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions

We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction.Comment: 27 pages LaTe

Miniband-related 1.4–1.8 μm luminescence of Ge/Si quantum dot superlattices

The luminescence properties of highly strained, Sb-doped Ge/Si multi-layer heterostructures with incorporated Ge quantum dots (QDs) are studied. Calculations of the electronic band structure and luminescence measurements prove the existence of an electron miniband within the columns of the QDs. Miniband formation results in a conversion of the indirect to a quasi-direct excitons takes place. The optical transitions between electron states within the miniband and hole states within QDs are responsible for an intense luminescence in the 1.4–1.8 µm range, which is maintained up to room temperature. At 300 K, a light emitting diode based on such Ge/Si QD superlattices demonstrates an external quantum efficiency of 0.04% at a wavelength of 1.55 µm

Клинико-морфологическое наблюдение ребенка с врожденным пневмоцистозом

The article presents an interesting case of congenital infection pneumocystis conclusively proven and confirmed by various methods of laboratory diagnostics, including morphological with a special color. In addition, parents were surveyed, which were the source of infection. Congenital pneumonia, especially in preterm infants, are a direct threat to life. In this regard, the timely examination of the mother and child, as well as the appointment of etiotropic antibacterial drugs is an important task to prevent the fatal outcome of the disease.В статье представлен случай врожденной пневмоцистной инфекции, убедительно доказанной и подтвержденной различными методами лабораторной диагностики, в т.ч. морфологическими со специальной окраской. Помимо этого, были обследованы родители, которые явились источниками инфицирования. Врожденные пневмонии, особенно у недоношенных новорожденных, представляют собой непосредственную угрозу жизни. В связи с этим своевременное обследование матери и ребенка, а также назначение этиотропных антибактериальных препаратов являются важной задачей по предотвращению фатальных исходов данного заболевания