Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra \mathfrakpgl(p+1|q){\mathfrak{pgl}(p+1|q)} is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of \mathfrakpgl(n|n)\not @ \mathfraksl(n|n){\mathfrak{pgl}(n|n)\not\cong \mathfrak{sl}(n|n)}), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.     

Noncommutative Independence from the Braid Group $${\mathbb{B}_{\infty}}$$

We introduce ‘braidability’ as a new symmetry for infinite sequences of noncommutative random variables related to representations of the braid group \({\mathbb{B}_{\infty}}\) . It provides an extension of exchangeability which is tied to the symmetric group \({\mathbb{S}_{\infty}}\) . Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem [Kös08]. This endows the braid groups \({\mathbb{B}_{n}}\) with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law. Furthermore we use the concept of product representations of endomorphisms [Goh04] with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of \({\mathbb{B}_{\infty}}\) and the irreducible subfactor with infinite Jones index in the non-hyperfinite I I ₁-factor L \({(\mathbb{B}_{\infty})}\) related to it. Our investigations reveal a new presentation of the braid group \({\mathbb{B}_{\infty}}\) , the ‘square root of free generator presentation’ \({\mathbb{F}^{1/2}_{\infty}}\) . These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory [GJS07]; and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level.     

A Centre-Stable Manifold for the Focussing Cubic NLS in $${\mathbb{R}}^{1+3}$$

Consider the focussing cubic nonlinear Schrödinger equation in \({\mathbb{R}}^3\) :

$i\psi_t+\Delta\psi = -|\psi|^2 \psi. \quad (0.1) $

It admits special solutions of the form e ^itα ?, where \(\phi \in {\mathcal{S}}({\mathbb{R}}^3)\) is a positive (? > 0) solution of

$-\Delta \phi + \alpha\phi = \phi^3. \quad (0.2)$

The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional manifold that consists of functions of the form \(e^{i(v \cdot + \Gamma)} \phi(\cdot - y, \alpha)\) . We prove that any solution starting sufficiently close to a standing wave in the \(\Sigma = W^{1, 2}({\mathbb{R}}^3) \cap |x|^{-1}L^2({\mathbb{R}}^3)\) norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that \({\mathcal{N}}\) is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones [BatJon]. The proof is based on the modulation method introduced by Soffer and Weinstein for the L ²-subcritical case and adapted by Schlag to the L ²-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in \({\mathbb{R}}^3\) for the nonselfadjoint Schrödinger operator obtained by linearizing (0.1) around a standing wave solution. All results in this paper depend on the standard spectral assumption that the Hamiltonian

$\mathcal H = \left ( \begin{array}{cc}\Delta + 2\phi(\cdot, \alpha)^2 - \alpha &;\quad \phi(\cdot, \alpha)^2 \\ -\phi(\cdot, \alpha)^2 &;\quad -\Delta - 2 \phi(\cdot, \alpha)^2 + \alpha \end{array}\right ) \quad (0.3)$

has no embedded eigenvalues in the interior of its essential spectrum \((-\infty, -\alpha) \cup (\alpha, \infty)\) .

     

Dimensional Crossover in Anisotropic Percolation on $${\mathbb {Z}}^{d+s}$$

We consider bond percolation on \({\mathbb {Z}}^d\times {\mathbb {Z}}^s\) where edges of \({\mathbb {Z}}^d\) are open with probability \(p<p_c({\mathbb {Z}}^d)\) and edges of \({\mathbb {Z}}^s\) are open with probability q, independently of all others. We obtain bounds for the critical curve in (p, q), with p close to the critical threshold \(p_c({\mathbb {Z}}^d)\). The results are related to the so-called dimensional crossover from \({\mathbb {Z}}^d\) to \({\mathbb {Z}}^{d+s}\).     

Global Well-Posedness of the Energy-Critical Defocusing NLS on $${\mathbb{R} \times \mathbb{T}^3}$$

We prove global well-posedness in H ¹ for the energy-critical defocusing initial-value problem \({(i\partial_t+\Delta_x)u=u|u|^2,\quad u(0)=\phi,}\) in the semiperiodic setting \({x\in\mathbb{R} \times \mathbb{T}^3}\) .     

{\mathbb{Z}}-Actions on AH Algebras and {\mathbb{Z}^2}-Actions on AF Algebras

We consider \mathbbZ{\mathbb{Z}}-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two \mathbbZ{\mathbb{Z}}-actions with the Rohlin property on such a C*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer \mathbbZ²{\mathbb{Z}^2}-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants.     

Study of Inelastic $$\boldsymbol{A}$$ ( $$\boldsymbol{p,p^{\prime}}$$ ) $$\boldsymbol{X}$$ Reaction with $${}^{\mathbf{9}}$$ Be and $${}^{\mathbf{90}}$$ Zr Nuclei at 1 GeV

Physics of Atomic Nuclei - The secondary proton polarization and differential cross sections of the ( $$p,p^{\prime}$$ ) inelastic reaction on nuclei $${}^{9}$$ Be and $${}^{90}$$ Zr at the initial...     

Thomae Formula for General Cyclic Covers of $${{\mathbb {CP}}^1}$$

Let X be a general cyclic cover of \mathbbCP¹{\mathbb{CP}^{1}} ramified at m points, λ₁... λ _m . we define a class of non-positive divisors on X of degree g −1 supported in the pre images of the branch points on X, such that the Riemann theta function does not vanish on their image in J(X). We generalize the results of Bershadsky and Radul (Commun Math Phys 116:689–700, 1988), Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) and Enolskii and Grava (Lett Math Phys 76(2–3):187–214, 2006) and prove that up to a certain determinant of the non-standard periods of X, the value of the Riemann theta function at these divisors raised to a high enough power is a polynomial in the branch point of the curve X. Our approach is based on a refinement of Accola’s results for 3 cyclic sheeted cover (Accola, in Trans Am Math Soc 283:423–449, 1984) and a generalization of Nakayashiki’s approach explained in Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) for general cyclic covers.     

Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local $${\mathbb{C}\mathbb{P}^2}$$

The holomorphic anomaly equations describe B-model closed topological strings in Calabi–Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local \({\mathbb{C}\mathbb{P}^2}\) toric Calabi–Yau background, and by addressing the associated (resurgent) large-order analysis of both perturbative and multi-instanton sectors. In particular, analyzing the asymptotic growth of the perturbative free energies, one finds contributions from three different instanton actions related by \({\mathbb{Z}_3}\) symmetry, alongside another action related to the Kähler parameter. Resurgent transseries methods then compute, from the extended holomorphic anomaly equations, higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies, as dictated by resurgence. The asymptotic large-order growth of the one-instanton sector unveils the presence of resonance, i.e., each instanton action is necessarily joined by its symmetric contribution. The structure of different resurgence relations is extensively checked at the numerical level, both in the holomorphic limit and in the general nonholomorphic case, always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations. The resurgence relations further imply that the string free energy displays an intricate multi-branched Borel structure, and that resonance must be properly taken into account in order to describe the full transseries solution.     

$$\mathbb{C}^2/\mathbb{Z}_{n}$$ Fractional Branes and Monodromy

We construct geometric representatives for the fractional branes in terms of branes wrapping certain exceptional cycles of the resolution. In the process we use large radius and conifold-type monodromies, and also check some of the orbifold quantum symmetries. We find the explicit Seiberg-duality which connects our fractional branes to the ones given by the McKay correspondence. We also comment on the Harvey-Moore BPS algebras.     

Ionization of Coulomb Systems in {\mathbb{R}^3} by Time Periodic Forcings of Arbitrary Size

We analyze the long time behavior of solutions of the Schrödinger equation ${i\psi_t=(-\Delta-b/r+V(t,x))\psi}We analyze the long time behavior of solutions of the Schr?dinger equation iy_t=(-D-b/r+V(t,x))y{i\psi_t=(-\Delta-b/r+V(t,x))\psi}, x ? \mathbbR³{x\in\mathbb{R}^3}, r =  |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) =  V(t +  2π/ω, x) with zero time average.     

Investigation of Excitation of $${K}$$ Isomers $${}^{{179m2}}$$ Hf and $${}^{{180m}}$$ Hf in ( $${\gamma,\gamma^{\prime}}$$ ) Reactions

Physics of Atomic Nuclei - The weighted-average cross sections for $${}^{179m2}$$ Hf and $${}^{180m}$$ Hf population in ( $$\gamma,\gamma^{\prime}$$ ) reactions induced by gamma rays of...     

Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation

We consider an aggregation equation in , n ≥ 2 with fractional dissipation, namely, , where 0 ≤ γ < 1 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e ^−|x|. We prove that for a class of smooth initial data, the solutions develop blow-up in finite time.     

Analysis of Angular $$\boldsymbol{t{-}\gamma}$$ Correlations in the Reaction $${}^{\boldsymbol{27}}$$ Al $$\boldsymbol{(\alpha,t)^{28}}$$ Si(2 $${}^{\boldsymbol{+}}$$ )

Physics of Atomic Nuclei - Angular $$t$$ – $$\gamma$$ correlations measured earlier in the reaction $${}^{27}$$ Al $$(\alpha,t)^{28}$$ Si(2 $${}^{+}$$ ) occurring at $$E_{\alpha}=30.3$$ MeV...     

A Chiang-type lagrangian in $$\mathbb {CP}^2$$

We analyse a monotone lagrangian in \(\mathbb {CP}^2\) that is hamiltonian isotopic to the standard lagrangian \(\mathbb {RP}^2\), yet exhibits a distinguishing behaviour under reduction by one of the toric circle actions, namely it intersects transversally the reduction level set and it projects one-to-one onto a great circle in \(\mathbb {CP}^1\). This lagrangian thus provides an example of embedded composition fitting work of Wehrheim–Woodward and Weinstein.     

Cohomology of {\mathcal {K}(2)} Acting on Linear Differential Operators on the Superspace {\mathbb{R}^{1|2}}

We compute the first differential cohomology of the Lie superalgebra ${\mathcal{K}(2)}$ of contact vector fields on the (1, 2)-dimensional real superspace with coefficients in the superspace of linear differential operators between the superspaces of weighted densities—a superisation of a result by Feigin and Fuchs. We give explicit expressions of 1-cocycles generating these spaces.