Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

Bertrand curves in the three-dimensional sphere

A curve α$\alpha$ immersed in the three-dimensional sphere S³${\mathbb{S}}^3$ is said to be a Bertrand curve if there exists another curve β$\beta$ and a one-to-one correspondence between α$\alpha$ and β$\beta$ such that both curves have common principal normal geodesics at corresponding points. The curves α$\alpha$ and β$\beta$ are said to be a pair of Bertrand curves in S³${\mathbb{S}}^3$. One of our main results is a sort of theorem for Bertrand curves in S³${\mathbb{S}}^3$ which formally agrees with the classical one: “Bertrand curves in S³${\mathbb{S}}^3$ correspond to curves for which there exist two constants λ≠0$\lambda \ne 0$ and μ$\mu$ such that λκ+μτ=1$\lambda \kappa +\mu \tau =1$”, where κ$\kappa$ and τ$\tau$ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S³${\mathbb{S}}^3$ as the only twisted curves in S³${\mathbb{S}}^3$ having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S³${\mathbb{S}}^3$ and (1,3)-Bertrand curves in R⁴${\mathbb{R}}^4$.     

On the nuclear modification factor at RHIC and LHC

We show that pQCD factorization incorporated with pre-hadronization energy-loss effect naturally leads to flatness of the nuclear modification factor RAA$R_{\mathit{AA}}$ for produced hadrons at high transverse momentum pT$p_T$. We consider two possible scenarios for the pre-hadronization: In scenario 1, the produced gluon propagates through dense QCD medium and loses energy. In scenario 2, all gluons first decay to quark–antiquark pairs and then each pair loses energy as propagating through the medium. We show that the estimates of the energy-loss in these two different models lead to very close values and is able to explain the suppression of high-pT$p_T$ hadrons in nucleus–nucleus collisions at RHIC. We show that the onset of the flatness of RAA$R_{\mathit{AA}}$ for the produced hadron in central collisions at midrapidity is about pT≈15$p_T\approx 15$ and 25 GeV at RHIC and the LHC energies, respectively. We show that the smallness (RAA<0.5$R_{\mathit{AA}}<0.5$ ) and the high-pT$p_T$ flatness of RAA$R_{\mathit{AA}}$ obtained from the kT$k_T$ factorization supplemented with the Balitsky–Kovchegov (BK) equation is rather generic and it does not strongly depend on the details of the BK solutions. We show that energy-loss effect reduces the nuclear modification factor obtained from the kT$k_T$ factorization about 30–50% at moderate pT$p_T$.