Narrow Results

The Gukov–Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of “strange identities” for positive braid knots.

In this paper a list of R-matrices on a certain coupled Lie algebra is obtained. With one of these R-matrices, we construct infinitely many bi-Hamiltonian structures for each of the two-component BKP and the Toda lattice hierarchies. We also show that, when such two hierarchies are reduced to their subhierarchies, these bi-Hamiltonian structures are reduced correspondingly.

We construct a family of $q$ deformations of $\mathrm{\text{E}}(2)$ group for nonzero complex parameters $\left|q\right|<1$ as locally compact braided quantum groups over the circle group $𝕋$ viewed as a quasitriangular quantum group with respect to the unitary $R$-matrix $\mathrm{\text{R}}(m,n):={(q/\bar{q})}^{mn}$ for all $m,n\in \mathbb{Z}$. For real $0<\left|q\right|<1$, the deformation coincides with Woronowicz’s ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups. As an application, we study the braided analogue of the contraction procedure between ${\mathrm{\text{SU}}}_{\mathrm{\text{q}}}(2)$ and ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups by contracting ${\mathrm{\text{U}}}_{\mathrm{\text{q}}}(2)$ groups.

The ${}^{12}$C(${}^{20}$Ne, ${}^{16}$O)${}^{16}$O reaction has been used for the first time to determine ANC of ${}^{16}$O levels. The ${}^{12}$C(${}^{20}$Ne, ${}^{16}$O)${}^{16}$O $\alpha$-transfer angular distributions are measured at $E_{\mathrm{\text{lab}}}=150$MeV and used to determine the ANC of the ground, 6.05MeV ($0^{+}$), 6.13MeV ($3^{-}$), 6.92MeV ($2^{+}$) and 7.12MeV ($1^{-}$) states of ${}^{16}$O. The ANC values are $C^2$$(0^{+})=(22.2\pm 7.1)\times 10^4$${\mathrm{\text{fm}}}^{-1}$, $C^2$$(0^{+})=(1.40\pm 0.54)\times 10^6{\mathrm{\text{fm}}}^{-1}$, $C^2$$(3^{-})=(2.30\pm 0.92)\times 10^4{\mathrm{\text{fm}}}^{-1}$, $C^2$$(2^{+})=(1.07\pm 0.32)\times 10^{10}{\mathrm{\text{fm}}}^{-1}$ and $C^2$$(1^{-})=(32.29\pm 10.14)\times 10^{28}{\mathrm{\text{fm}}}^{-1}$. Astrophysical S-factors at 300keV are extracted $\{S_{E1}^{300}=(79.18\pm 26.60$) keV b and $S_{E2}^{300}=(52.51\pm 24.3)$ keV b$\}$ using these ANC values by R-matrix theory.

The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The κ-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are infinitely many such realizations related by similarity transformations. For a given realization, we construct corresponding coproducts of commutative coordinates and momenta (bialgebroid structure). The κ-deformed phase space has twisted Hopf algebroid structure. General method for the construction of twist operator (satisfying cocycle and normalization condition) corresponding to deformed coalgebra structure is presented. Specially, twist for natural realization (classical basis) of κ-Minkowski space–time is presented. The cocycle condition, κ-Poincaré algebra and R-matrix are discussed. Twist operators in arbitrary realizations are constructed from the twist in the given realization using similarity transformations. Some examples are presented. The important physical applications of twists, realizations, R-matrix and Hopf algebroid structure are discussed.

The reaction ${}^{13}$C${(\alpha ,n)}^{16}$O is an important astrophysical process producing neutrons for $s$-process nucleosynthesis of nuclei heavier than iron in low-mass AGB stars. The reaction rate of ${}^{13}$C${(\alpha ,n)}^{16}$O at relevant energies is dominated by a ${1/2}^{+}$ state in ${}^{17}$O near the $\alpha$-threshold. The adopted value for the excitation energy of the state is −3keV below the threshold at 6359keV. However, recent investigations have indicated that the state is, instead of a sub-threshold state, an above threshold resonance state. The observation is also corroborated by neutron scattering studies from ${}^{16}$O. The location of the state and its implication on the low energy behavior of the astrophysical $S$-factor are of definite interest. The aim of this work is to ascertain the energy location of ${1/2}^{+}$ state in the excitation spectrum of ${}^{17}$O and to estimate its neutron and $\alpha$-partial widths. Subsequently, we look into the effect of the resultant set of parameters of the state on the $S$-factor and reaction rate at important astrophysical energies. A multilevel, multichannel $R$-matrix analysis has been performed for the existing ${}^{16}$O$(n,n)$, ${}^{13}$C$(\alpha ,n)$, ${}^{13}$C$(\alpha ,\alpha )$ and ${}^{16}$O$(n,\alpha )$ data. The excitation energy of the $1/2+$ state in ${}^{17}$O is found to be at 6371.9keV, about 12.9keV above the threshold. The resulting $S$-factor around the Gamow energy of 190keV $(T_9\sim 0.1)$ is $(1.91\pm 0.27)\times 10^6$$\mathrm{\text{MeV}}\cdot \mathrm{\text{b}}$. Relatively, higher $S$-factor value yields a larger reaction rate for ${}^{13}$C($\alpha ,n$)${}^{16}$O at the required temperature window. Consequently, larger number of neutrons relevant to $S$-process nucleosynthesis will be produced from ${}^{13}$C${(\alpha ,n)}^{16}$O reaction.

The isotope ${}^{16}\mathrm{\text{O}}$ is one of the most abundant light isotopes, and therefore the process ${}^{16}\mathrm{\text{O}}{(n,\gamma )}^{17}\mathrm{\text{O}}$ reaction plays a critical role in stellar nucleosynthesis. Reaction cross-sections and rates for neutron capture by ${}^{16}\mathrm{\text{O}}$ at astrophysical energies have been calculated with special attention to low-lying, narrow resonances while implementing $R$-matrix formalism during integration.

We prove that the singularities of the $R$-matrix $R(k)$ of the minimal quantization of the adjoint representation of the Yangian $Y(𝔤)$ of a finite dimensional simple Lie algebra $𝔤$ are the opposite of the roots of the monic polynomial $p(k)$ entering in the OPE expansions of quantum fields of conformal weight $3/2$ of the universal minimal affine $W$-algebra at level $k$ attached to $𝔤$.

In this paper we continue our studies of Hitchin systems on singular curves (started in [1]). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec {f ∈ ℂ[z] : f(A(∊)) = f(B(∊)); ∊^N = 0}, where A(∊), B(∊) are arbitrary polynomials). The most simple examples are the generalized cusp curves which are projectivizations of Spec {f ∈ ℂ[z] : f′(0) = f″(0) = ⋯ f^N-1(0) = 0}. We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials A(∊), B(∊) defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop r-matrix formalism for the functions on the truncated loop group GL_n(ℂ[z]), z^N = 0. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space.