A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

Generic Rigidity for Circle Diffeomorphisms with Breaks

We prove that \({C^r}\)-smooth (\({r > 2}\)) circle diffeomorphisms with a break, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, are generically, i.e., for almost all irrational rotation numbers, not \({C^{1+\varepsilon}}\)-rigid, for any \({\varepsilon > 0}\). This result complements our recent proof, joint with Khanin (Geom Funct Anal 24:2002–2028, 2014), that such maps are generically \({C^1}\)-rigid. It stands in remarkable contrast to the result of Yoccoz (Ann Sci Ec Norm Sup 17:333–361, 1984) that \({C^r}\)-smooth circle diffeomorphisms are generically \({C^{r-1-\varkappa}}\)-rigid, for any \({\varkappa > 0}\).     

Universal Components of Random Nodal Sets

We give, as L grows to infinity, an explicit lower bound of order \({L^{\frac{n}{m}}}\) for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of P with eigenvalues below L. Here, P denotes an elliptic self-adjoint pseudo-differential operator of order \({m > 0}\), bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed n-dimensional manifold M equipped with some Lebesgue measure. In fact, for every closed hypersurface \({\Sigma}\) of \({\mathbb{R}^n}\), we prove that there exists a positive constant \({p_\Sigma}\) depending only on \({\Sigma}\), such that for every large enough L and every \({x \in M}\), a component diffeomorphic to \({\Sigma}\) appears with probability at least \({p_\Sigma}\) in the vanishing locus of a random section and in the ball of radius \({L^{-\frac{1}{m}}}\) centered at x. These results apply in particular to Laplace–Beltrami and Dirichlet-to-Neumann operators.     

Trigonometric and Elliptic Ruijsenaars–Schneider Systems on the Complex Projective Space

We present a direct construction of compact real forms of the trigonometric and elliptic \({n}\)-particle Ruijsenaars–Schneider systems whose completed center-of-mass phase space is the complex projective space \({{\mathbb{CP}}^{n-1}}\) with the Fubini–Study symplectic structure. These systems are labeled by an integer \({p\in\{1,\ldots,n-1\}}\) relative prime to \({n}\) and a coupling parameter \({y}\) varying in a certain punctured interval around \({p\pi/n}\). Our work extends Ruijsenaars’s pioneering study of compactifications that imposed the restriction \({0 < y < \pi/n}\), and also builds on an earlier derivation of more general compact trigonometric systems by Hamiltonian reduction.     

Theoretical analysis of somatostatin receptor 5 with antagonists and agonists for the treatment of neuroendocrine tumors

We report on SSTR5 receptor modeling and its interaction with reported antagonist and agonist molecules. Modeling of the SSTR5 receptor was carried out using multiple templates with the aim of improving the precision of the generated models. The selective SSTR5 antagonists, agonists and native somatostatin SRIF-14 were employed to propose the binding site of SSTR5 and to identify the critical residues involved in the interaction of the receptor with other molecules. Residues Q2.63, D3.32, Q3.36, C186, Y7.34 and Y7.42 were found to be highly significant for their strong interaction with the receptor. SSTR5 antagonists were utilized to perform a 3D quantitative structure–activity relationship study. A comparative molecular field analysis (CoMFA) was conducted using two different alignment schemes, namely the ligand-based and receptor-based alignment methods. The best statistical results were obtained for ligand-based (\({q}^{2} = 0.454\), \({r}^{2}\) = 0.988, noc = 4) and receptor-guided methods (docked mode 1:\({q}^{2} = 0.530\), \({r}^{2} = 0.916\), noc = 5), (docked mode 2:\({q}^{2}\) = 0.555, \({r}^{2 }= 0.957\), noc = 5). Based on CoMFA contour maps, an electropositive substitution at \(\hbox {R}^{1}\), \(\hbox {R}^{2}\) and \(\hbox {R}^{4}\) position and bulky group at \(\hbox {R}^{4}\) position are important in enhancing molecular activity.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Weyl type N solutions with null electromagnetic fields in the Einstein–Maxwell <Emphasis Type="Italic">p</Emphasis>-form theory

We consider d-dimensional solutions to the electrovacuum Einstein–Maxwell equations with the Weyl tensor of type N and a null Maxwell \((p+1)\)-form field. We prove that such spacetimes are necessarily aligned, i.e. the Weyl tensor of the corresponding spacetime and the electromagnetic field share the same aligned null direction (AND). Moreover, this AND is geodetic, shear-free, non-expanding and non-twisting and hence Einstein–Maxwell equations imply that Weyl type N spacetimes with a null Maxwell \((p+1)\)-form field belong to the Kundt class. Moreover, these Kundt spacetimes are necessarily \({ CSI}\) and the \((p+1)\)-form is \({ VSI}\). Finally, a general coordinate form of solutions and a reduction of the field equations are discussed.     

Positive Casimir and Central Characters of Split Real Quantum Groups

We consider the one parameter family \({\alpha \mapsto T_{\alpha}}\) (\({\alpha \in [0,1)}\)) of Pomeau-Manneville type interval maps \({T_{\alpha}(x) = x(1+2^{\alpha} x^{\alpha})}\) for \({x \in [0,1/2)}\) and \({T_{\alpha}(x)=2x-1}\) for \({x \in [1/2, 1]}\), with the associated absolutely continuous invariant probability measure \({\mu_{\alpha}}\). For \({\alpha \in (0,1)}\), Sarig and Gouëzel proved that the system mixes only polynomially with rate \({n^{1-1/{\alpha}}}\) (in particular, there is no spectral gap). We show that for any \({\psi \in L^{q}}\), the map \({\alpha \to \int_0^{1} \psi\, d \mu_{\alpha}}\) is differentiable on \({[0,1-1/q)}\), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For \({\alpha \ge 1/2}\) we need the \({n^{-1/{\alpha}}}\) decorrelation obtained by Gouëzel under additional conditions.     

T-Duality Simplifies Bulk-Boundary Correspondence

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice \({{{\mathbb{Z}}}^{4}}\), for the weakly coupled n-component \({|\varphi|^{4}}\) spin model for all \({n \ge 1}\), and for the continuous-time weakly self-avoiding walk. For the \({|\varphi|^{4}}\) model, we prove that the critical two-point function has |x|^?2 (Gaussian) decay asymptotically, for \({n \ge 1}\). We also determine the asymptotic decay of the critical correlations of the squares of components of \({\varphi}\), including the logarithmic corrections to Gaussian scaling, for \({n \ge 1}\). The above extends previously known results for n = 1 to all \({n \ge 1}\), and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the “watermelon” network consisting of p weakly mutually- and self-avoiding walks, for all \({p \ge 1}\), including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove the existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the \({|\varphi|^{4}}\) model, and provides a unified treatment of both models, and of all the above results.     

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.     

Spherical T-Duality

We introduce spherical T-duality, which relates pairs of the form (P, H) consisting of a principal SU(2)-bundle \({P \rightarrow M}\) and a 7-cocycle H on P. Intuitively spherical T-duality exchanges H with the second Chern class c ₂(P). Unless \({dim(M) \leq 4}\), not all pairs admit spherical T-duals and the spherical T-duals are not always unique. Nonetheless, we prove that all spherical T-dualities induce a degree-shifting isomorphism on the 7-twisted cohomologies of the bundles and, when \({dim(M) \leq 7}\), also their integral twisted cohomologies and, when \({dim(M) \leq 4}\), even their 7-twisted K-theories. While spherical T-duality does not appear to relate equivalent string theories, it does provide an identification between conserved charges in certain distinct IIB supergravity and string compactifications.     

Minimizers of the Lawrence–Doniach Functional with Oblique Magnetic Fields

We study minimizers of the Lawrence–Doniach energy, which describes equilibrium states of superconductors with layered structure, assuming Floquet-periodic boundary conditions. Specifically, we consider the effect of a constant magnetic field applied obliquely to the superconducting planes in the limit as both the layer spacing s → 0 and the Ginzburg–Landau parameter \({\kappa = \epsilon^{-1} \to \infty}\), under the hypotheses that \({s=\epsilon^\alpha}\) with 0 < α < 1. By deriving sharp matching upper and lower bounds on the energy of minimizers, we determine the lower critical field and the orientation of the flux lattice, to leading order in the parameter \({\epsilon}\). To leading order, the induced field is characterized by a convex minimization problem in \({\mathbb {R}^3}\). We observe a “flux lock-in transition”, in which flux lines are pinned to the horizontal direction for applied fields of small inclination, and which is not present in minimizers of the anisotropic Ginzburg–Landau model. The energy profile we obtain suggests the presence of “staircase vortices”, which have been described qualitatively in the physics literature.     

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

Let \({T=\mathbb R^d}\) . Let a function \({QT^2\to\mathbb C}\) satisfy \({Q(s,t)=\overline{Q(t,s)}}\) and \({|Q(s,t)|=1}\). A generalized statistics is described by creation operators \({\partial_t^\dagger}\) and annihilation operators ? _t , \({t\in T}\), which satisfy the Q-commutation relations: \({\partial_s\partial^\dagger_t = Q(s, t)\partial^\dagger_t\partial_s+\delta(s, t)}\) , \({\partial_s\partial_t = Q(t, s)\partial_t\partial_s}\), \({\partial^\dagger_s\partial^\dagger_t = Q(t, s)\partial^\dagger_t\partial^\dagger_s}\). From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which Q(s, t) is equal to q if s < t, and to \({\bar q}\) if s > t. Here \({q\in\mathbb C}\) , |q| = 1. We start the paper with a detailed discussion of a Q-Fock space and operators \({(\partial_t^\dagger,\partial_t)_{t\in T}}\) in it, which satisfy the Q-commutation relations. Next, we consider a noncommutative stochastic process (white noise) \({\omega(t)=\partial_t^\dagger+\partial_t+\lambda\partial_t^\dagger\partial_t}\) , \({t\in T}\) . Here \({\lambda\in\mathbb R}\) is a fixed parameter. The case λ = 0 corresponds to a Q-analog of Brownian motion, while λ ≠ 0 corresponds to a (centered) Q-Poisson process. We study Q-Hermite (Q-Charlier respectively) polynomials of infinitely many noncommutatative variables \({(\omega(t))_{t\in T}}\) . The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define Q-cumulants of a field \({(\xi(t))_{t\in T}}\). This allows us to define a Q-Lévy process as a field \({(\xi(t))_{t\in T}}\) whose values at different points of T are Q-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a Q-Lévy process, and derive a Nualart–Schoutens-type chaotic decomposition for such a process.     

Blow-up of Critical Besov Norms at a Potential Navier–Stokes Singularity

We prove that if an initial datum to the incompressible Navier–Stokes equations in any critical Besov space \({\dot B^{-1+\frac 3p}_{p,q}({\mathbb {R}}^{3})}\), with \({3 < p, q < \infty}\), gives rise to a strong solution with a singularity at a finite time \({T > 0}\), then the norm of the solution in that Besov space becomes unbounded at time T. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003) concerning suitable weak solutions blowing up in \({L^{3}({\mathbb R}^{3})}\). Our proof uses profile decompositions and is based on our previous work (Gallagher et al., Math. Ann. 355(4):1527–1559, 2013), which provided an alternative proof of the \({L^{3}({\mathbb R}^{3})}\) result. For very large values of p, an iterative method, which may be of independent interest, enables us to use some techniques from the \({L^{3}({\mathbb R}^{3})}\) setting.     

The Einstein–Maxwell Equations and Conformally Kähler Geometry

Page’s Einstein metric on \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\) is conformally related to an extremal Kähler metric. Here we construct a family of conformally Kähler solutions of the Einstein–Maxwell equations that deforms the Page metric, while sweeping out the entire Kähler cone of \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\). The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein–Maxwell equations on the smooth 4-manifolds \({S^2 \times S^2}\) and \({{\mathbb{CP}}_2\#\overline{\mathbb{CP}}_2}\).     

Disappearance of Mott Oscillations in Sub-barrier Elastic Scattering of Identical Nuclei and Atomic Ions

The scattering of identical nuclei at low energies exhibits conspicuous Mott oscillations which can be used to investigate the presence of components in the predominantly Coulomb interaction arising from several physical effects. It is found that at a certain critical value of the Sommerfeld parameter the Mott oscillations disappear and the cross section becomes quite flat. We call this effect Transverse Isotropy (TI). The critical value of the Sommerfeld parameter at which TI sets in is found to be \({\eta_{c} = \sqrt{3s + 2}}\), where s is the spin of the nuclei participating in the scattering. No TI is found in the Mott scattering of identical Fermionic nuclei. The critical center of mass energy corresponding to \({\eta_c}\) is found to be \({E_c = 0.40}\) MeV for \({\alpha + \alpha}\) (s = 0), 1.2 MeV for \({^{6}}\)Li + \({^{6}}\)LI (s = 1) and 7.1 MeV for \({^{10}}\)B + \({^{10}}\)B (s = 3). We further found that the inclusion of the nuclear interaction induces a significant modification in the TI. We suggest measurements at these sub-barrier energies for the purpose of extracting useful information about the nuclear interaction between light heavy ions. We also suggest extending the study of the TI to the scattering of identical atomic ions.     

Character and Dimension Formulae for Queer Lie Superalgebra

We study transmission problems with free interfaces from one random medium to another. Solutions are required to solve distinct partial differential equations, \({{\rm L}_{+}}\) and \({{\rm L}_{-}}\), within their positive and negative sets respectively. A corresponding flux balance from one phase to another is also imposed. We establish existence and \({L^{\infty}}\) bounds of solutions. We also prove that variational solutions are non-degenerate and develop the regularity theory for solutions of such free boundary problems.     

On the Convexity of the KdV Hamiltonian

Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), extends to a real analytic function on the positive quadrant \({\ell^{2}_{+}(\mathbb{N})}\) of \({\ell^{2}(\mathbb{N})}\) and is strictly concave near \({0}\). As a consequence, the differential of \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), admit global KdV-Birkhoff coordinates. In particular, it means that \({\ell^{2}_+(\mathbb{N})}\) is the space of action variables of the underlying phase space \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).     

Decay Behaviors of the <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript> Hadronic Molecules

In this proceeding, we present our recent work on decay behaviors of the P_c hadronic molecules, which can help to disentangle the nature of the two P_c pentaquark-like structures. The results turn out that the relative ratio of the decays of P _c ⁺ (4380) to \({\bar D *}{\Lambda _c}\) and J/ψp is very different for P_c being a \({\bar D *}{\Sigma _c}\) or \(\bar D\Sigma _c *\) bound state with \({J^P} = \frac{{{3 - }}}{2}\) And from the total decay width, we find that P_c(4380) being a \(\bar D\Sigma _c *\) molecule state with \({J^P} = \frac{{{3 - }}}{2}\) and P_c(4450) being a \({\bar D *}{\Sigma _c}\) molecule state with \({J^P} = \frac{{{5 + }}}{2}\) is more favorable to the experimental data.     

Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.