Erratum to: Classical $${mathcal {W}}$$ W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

Communications in Mathematical Physics -     

$\ensuremath{\mathcal{P}}$$\ensuremath{\mathcal{T}}$ Symmetry vs. Hermiticity

The relation between the P\ensuremath{\mathcal{P}} T\ensuremath{\mathcal{T}} symmetry and Hermiticity is discussed. In the finite-dimensional linear space, any Hermitian matrix is a special case of P\ensuremath{\mathcal{P}} T\ensuremath{\mathcal{T}}-symmetric matrices. Explicit results in 2×2 are shown. The early belief that the P\ensuremath{\mathcal{P}} T\ensuremath{\mathcal{T}}-symmetric quantum mechanics is a generalization of the conventional Hermitian quantum mechanics is confirmed.     

Time-Dependent Vacuum Energy Induced by {\mathcal{D}}–Particle Recoil

We consider cosmology in the framework of a material reference system of particles, including the effects of quantum recoil induced by closed-string probe particles. We find a time-dependent contribution to the cosmological vacuum energy, which relaxes to zero as 1/t ² for large times t. If this energy density is dominant, the Universe expands with a scale factor R(t)t ². We show that this possibility is compatible with recent observational constraints from high–redshift supernovae, and may also respect other phenomenological bounds on time variation in the vacuum energy imposed by early cosmology.     

Conformal classical Yang–Baxter equation,S-equation and $${\mathcal {O}}$$O-operators

Letters in Mathematical Physics - Conformal classical Yang–Baxter equation and S-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In...     

Integrable sigma-models and Drinfeld–Sokolov hierarchies

Local commuting charges in sigma-models with classical Lie groups as target manifolds are shown to be related to the conserved quantities appearing in the Drinfeld–Sokolov (generalized mKdV) hierarchies. Conversely, the Drinfeld–Sokolov construction can be used to deduce the existence of commuting charges in these and in wider classes of sigma-models, including those whose target manifolds are exceptional groups or symmetric spaces. This establishes a direct link between commuting quantities in integrable sigma-models and in affine Toda field theories.     

Semi-Classical Twists for {\mathfrak{sl}}_{3} and {\mathfrak{sl}}_{4} Boundary r-Matrices of Cremmer–Gervais Type

We obtain explicit formulas for the semi-classical twists deforming the coalgebraic structure of $U({\mathfrak{sl}}_{3})$ and $U({\mathfrak{sl}}_{4})$ . In rank 2 and 3 the corresponding universal R-matrices quantize the boundary r-matrices of Cremmer–Gervais type defining Lie Frobenius structures on the maximal parabolic subalgebras in ${\mathfrak{sl}}_{n}$ .     

The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Scaling symmetry of -type Drinfel’d–Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n + 1) regular singularities. Especially in the case of n = 3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painlevé VI equation. We also discuss Weyl group symmetry of the hierarchy.      

Traces of Intertwiners for Quantum Affine {\mathfrak{sl}}_2 and Felder–Varchenko Functions

We show that the traces of \({U_q({\widehat{\mathfrak{sl}}}_2)}\)-intertwiners of [ESV02] valued in the three-dimensional evaluation representation converge in a certain region of parameters and give a representation-theoretic construction of Felder–Varchenko’s hypergeometric solutions to the q-KZB heat equation given in [FV02]. This gives the first proof that such a trace function converges and resolves the first case of the Etingof–Varchenko conjecture of [EV00]. As applications, we prove a symmetry property for traces of intertwiners and prove Felder–Varchenko’s conjecture in [FV04] that their elliptic Macdonald polynomials are related to the affine Macdonald polynomials defined as traces over irreducible integrable \({U_q({\widehat{\mathfrak{sl}}}_2)}\)-modules in [EK95]. In the trigonometric and classical limits, we recover results of [EK94,EV00]. Our method relies on an interplay between the method of coherent states applied to the free field realization of the q-Wakimoto module of [Mat94], convergence properties given by the theta hypergeometric integrals of [FV02], and rationality properties originating from the representation-theoretic definition of the trace function.     

The MV formalism for $${\mathrm{IBL}}_\infty $$- and $${\mathrm{BV}}_\infty $$BV∞-algebras

We develop a new formalism for the quantum master equation \(\Delta e^{S/\hbar } = 0\) and the category of \(\mathrm{IBL}_\infty \)-algebras and simplify some homotopical algebra arising in the context of oriented surfaces with boundary. We introduce and study a category of MV-algebras, which, on the one hand, contains such important categories as those of \(\mathrm{IBL}_\infty \)-algebras and \(\mathtt{L}_\infty \)-algebras and, on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation. We also present geometric interpretation of our results.     

W-algebras and the equivalence of bihamiltonian,Drinfeld–Sokolov and Dirac reductions

We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld–Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic subspace. In addition, using this result we prove again that the transverse Poisson structure to a nilpotent orbit is polynomial and we better clarify the relation between classical and finite $W$-algebras.     

Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions

A Darboux transformation for the Drinfeld–Sokolov–Satsuma–Hirota system of coupled equations is constructed with the aid of gauge transformations between the Lax pairs. As an application, several types of solutions of the Drinfeld–Sokolov–Satsuma–Hirota system are obtained, including soliton solutions, periodic solutions, rational solutions and others.     

Structural, thermal and transport properties of {\text{B}}{{\text{i}}_4}{{\text{V}}_{2 - {\text{x}}}}{\text{G}}{{\text{a}}_{\text{x}}}{{\text{O}}_{11 - \delta }} (0 ≤ x ≤ 0.4)

Solid electrolytes are mostly used in solid oxide fuel cells (SOFC). In the present study, gallium-substituted compounds ( $ {\text{B}}{{\text{i}}_4}{{\text{V}}_{2 - {\text{x}}}}{\text{M}}{{\text{e}}_{\text{x}}}{{\text{O}}_{11 - \delta }} $ ; Me?=?Ga³⁺; 0≤x≤0.4) were prepared by solid-state reaction technique for its use as an electrolyte in SOFC. Structural and conductivity behavior was studied as a function of the Ga³⁺ substitution on vanadium site. The compounds remain in the orthorhombic α-phase for x?=?0 and 0.1 whereas higher concentration of dopant leads to β-phase stabilization. The highest and lowest ionic conductivity were observed in x?=?0.2 and x?=?0.4 samples, respectively. The prepared samples were studied by using alternating current conductivity, differential thermal analysis and X-ray diffraction techniques. The results are discussed on the basis of formation of oxygen vacancy and its ordering.     

A complete $\mathcal{O}(\alpha_{\mathrm{S}}^{2})$ calculation of the signal–background interference for the Higgs diphoton decay channel

We analyze a noncommutative model of BTZ spacetime based on deformation of the standard symplectic structure of phase space, i.e., a modification of the standard commutation relations among coordinates and momenta in phase space. We find a BTZ-like solution that is nonperturbative in the non-trivial noncommutative structure. It is shown that the use of deformed commutation relations in the modified non-canonical phase space eliminates the horizons of the standard metric.     

Inclusion relations for the spaces L(ℝ), \mathcal{H}\mathcal{K} (ℝ) ∩ \mathcal{B}\mathcal{V} (ℝ), and L 2(ℝ)

The inclusion relations for the spaces $ \mathcal{H}\mathcal{K} $ (I), L(I), $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I), and L ²(I) are found. On unbounded intervals, functions in $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I) need not be Lebesgue integrable.     

The Local Gromov–Witten Theory of $${\mathbb{C}\mathbb{P}^1}$$ and Integrable Hierarchies

In this paper we begin the study of the relationship between the local Gromov–Witten theory of Calabi–Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full-descendent genus zero theory. Our main tool is the application of Dubrovin’s formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of the resolved conifold in the equivariantly Calabi–Yau case. For this example the relevant dispersionless system turns out to be related to the long-wave limit of the Ablowitz–Ladik lattice. This identification provides us with a complete procedure to reconstruct the dispersive hierarchy which should conjecturally be related to the higher genus theory of the resolved conifold. We give a complete proof of this conjecture for genus g ≤ 1; our methods are based on establishing, analogously to the case of KdV, a “quasi-triviality” property for the Ablowitz–Ladik hierarchy at the leading order of the dispersive expansion. We furthermore provide compelling evidence in favour of the resolved conifold/Ablowitz–Ladik correspondence at higher genus by testing it successfully in the primary sector for g = 2.     

Top polarization and exploration of \mathbf{\emph{t}} \boldsymbol{\bar t} forward–backward asymmetry?

The large forward?Cbackward asymmetry in top pair-production observed at the Tevatron has generated much excitement in recent times. Many different models have been proposed as possible explanations for this anomaly. We discuss how the measurement of the polarization of the top quark can be used to establish the role of new physics in generating the observed asymmetry and to characterize and discriminate among various models.     

$${\mathcal{N}=2}$$ Superconformal Algebra and the Entropy of Calabi–Yau Manifolds

We use the representation theory of \({\mathcal{N}=2}\) superconformal algebra to study the elliptic genera of Calabi–Yau (CY) D-folds. We compute the entropy of CY manifolds from the growth rate of multiplicities of the massive (non-BPS) representations in the decomposition of their elliptic genera. We find that the entropy of CY manifolds of complex dimension D behaves differently depending on whether D is even or odd. When D is odd, CY entropy coincides with the entropy of the corresponding hyperKähler (D ? 3)-folds due to a structural theorem on Jacobi forms. In particular, we find that the Calabi–Yau 3-fold has a vanishing entropy. At D > 3, using our previous results on hyperKähler manifolds, we find \({S_{CY_D}\sim 2\pi \sqrt{\frac{(D-3)^2}{2(D-1)}n}}\). When D is even, we find the behavior of CY entropy behaving as \({S_{CY_D}\sim 2 \pi\sqrt{\frac{D-1}{2}n}}\). These agree with Cardy’s formula at large D.     

Neutron Pairing Correlations in an {\alpha}–{n}–{n} Three-Cluster Model of the {^{6}}He Nucleus

An \({\alpha}\)–n–n three-cluster model of the \({^6}\)He nucleus is studied by solving the Faddeev equations, where the cluster potential between \({\alpha}\) and n takes into account the Pauli exclusion correction, using the Fish-Bone Optical Model (Schmid in Z Phys A 297:105, 1980). The resulting binding energy of the ground state (\({0^+}\)) is 0.831 MeV and the resonance energy of the first excited state (\({2^+}\)), 0.60–i0.012 MeV, is extracted from the three-cluster break-up threshold. These theoretical values are in reasonable agreement with the experimental data: 0.973 MeV and 0.824–i0.056 MeV, respectively. In order to investigate the structure of these states, we calculate the angle density matrix for the \({\angle n_1 \alpha n_2}\) angle in the triangle formed by the three clusters. The angle density matrix of the ground state has two peaks and the configuration of \({0^+}\) wave function corresponding to the peaks constitutes a mixture of an acute-angled triangle structure and an obtuse-angled one. This finding is consistent with the former result from a variational approach (Hagino and Sagawa in Phys Rev C 72:044321, 2005). On the other hand, in the case of \({2^+}\) state only a single peak is obtained.     

The 3–3–1 Model with RH Neutrinos and Associated ZH Production at High-Energy ${\bf \textit{e}}^{+}{\bf \textit{e}}^{-}$ Collider

In the context of SU(3) _C ?? SU(3) _L ?? U(1) _X (3–3–1) model with right-handed neutrinos, we study the Higgsstrahlung process e ^?+? e ^???→ZH and calculate the cross section of this process at leading order. Our numerical results showed that the production cross sections for this process can be significantly large as \(M_{Z'}\approx \sqrt{s}\). With reasonable values of the Z′ mass M _Z′, Z′ exchange can generate large corrections to the cross sections of this process, which might be detected in the future high-energy linear e ^?+? e ^??? collider experiments.     

Backbending Phenomena in Even–Even $${}^{\boldsymbol{162{-}172}}$$ Hf Isotopes

Physics of Atomic Nuclei - This study probes the backbending phenomena in even–even $${}^{162{-}172}$$ Hf isotopes. Experimental ground-state rotational energies up to $$J^{\pi}=36^{+}$$...