Authors:David Sutter; Mario Berta; Marco Tomamichel Pages: 37 - 58 Abstract: Abstract We prove several trace inequalities that extend the Golden–Thompson and the Araki–Lieb–Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden–Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2778-5 Issue No:Vol. 352, No. 1 (2017)

Authors:Rolf Gohm; Florian Haag; Burkhard Kümmerer Pages: 59 - 94 Abstract: Abstract We introduce a notion of universal preparability for a state of a system, more precisely: for a normal state on a von Neumann algebra. It describes a situation where from an arbitrary initial state it is possible to prepare a target state with arbitrary precision by a repeated interaction with a sequence of copies of another system. For \({\mathcal{B}(\mathcal{H})}\) we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes. PubDate: 2017-05-01 DOI: 10.1007/s00220-017-2851-8 Issue No:Vol. 352, No. 1 (2017)

Authors:Arthur Jaffe; Zhengwei Liu Pages: 95 - 133 Abstract: Abstract We define a planar para algebra, which arises naturally from combining planar algebras with the idea of \({\mathbb{Z}_{N}}\) para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects that are invariant under para isotopy. For each \({\mathbb{Z}_{N}}\) , we construct a family of subfactor planar para algebras that play the role of Temperley–Lieb–Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras, which one can use in the study of quantum information. An important ingredient in planar para algebra theory is the string Fourier transform (SFT), which we use on the matrix algebra generated by the Pauli matrices. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita–Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivity by relating the two reflections through the string Fourier transform. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2779-4 Issue No:Vol. 352, No. 1 (2017)

Authors:Jirui Guo; Zhentao Lu; Eric Sharpe Pages: 135 - 184 Abstract: Abstract In this paper we study the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle. Quantum sheaf cohomology is a (0,2) deformation of the ordinary quantum cohomology ring, realized as the OPE ring in A/2-twisted theories. Quantum sheaf cohomology has previously been computed for abelian gauged linear sigma models (GLSMs); here, we study (0,2) deformations of nonabelian GLSMs, for which previous methods have been intractable. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. We also utilize recent advances in supersymmetric localization to compute A/2 correlation functions and check the general result in examples. In this paper we focus on physics derivations and examples; in a companion paper, we will provide a mathematically rigorous derivation of the classical sheaf cohomology ring. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2763-z Issue No:Vol. 352, No. 1 (2017)

Authors:Ezra Getzler Pages: 185 - 199 Abstract: Abstract We extend our previous calculation of the BV cohomology of the spinning particle with a flat target to the general case, in which the target carries a non-trivial pseudo-Riemannian metric and a magnetic field. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2764-y Issue No:Vol. 352, No. 1 (2017)

Authors:Julien Guillod Pages: 201 - 214 Abstract: Abstract The Navier–Stokes equations in a two-dimensional exterior domain are considered. The asymptotic stability of stationary solutions satisfying a general hypothesis is proven under any L 2-perturbation. In particular, the general hypothesis is valid if the steady solution is the sum of the critically decaying flux carrier with flux \({\left \Phi \right < 2 \pi}\) and a small subcritically decaying term. Under the central symmetry assumption, the general hypothesis is also proven for any critically decaying steady solutions under a suitable smallness condition. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2794-5 Issue No:Vol. 352, No. 1 (2017)

Authors:Charles Collot; Frank Merle; Pierre Raphaël Pages: 215 - 285 Abstract: Abstract We consider the energy critical semilinear heat equation $$\partial_tu = \Delta u + u ^{\frac{4}{d-2}}u, \quad x \in \mathbb{R}^d$$ and give a complete classification of the flow near the ground state solitary wave $$Q(x) = \frac{1}{\left(1+\frac{ x ^2}{d(d-2)}\right)^{\frac{d-2}{2}}}$$ in dimension \({d \ge 7}\) , in the energy critical topology and without radial symmetry assumption. Given an initial data \({Q + \varepsilon_0}\) with \({\ \nabla \varepsilon_0\ _{L^2} \ll 1}\) , the solution either blows up in the ODE type I regime, or dissipates, and these two open sets are separated by a codimension one set of solutions asymptotically attracted by the solitary wave. In particular, non self similar type II blow up is ruled out in dimension \({d \ge 7}\) near the solitary wave even though it is known to occur in smaller dimensions (Schweyer, J Funct Anal 263(12):3922–3983, 2012). Our proof is based on sole energy estimates deeply connected to Martel et al. (Acta Math 212(1):59–140, 2014) and draws a route map for the classification of the flow near the solitary wave in the energy critical setting. A by-product of our method is the classification of minimal elements around Q belonging to the unstable manifold. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2795-4 Issue No:Vol. 352, No. 1 (2017)

Authors:Paolo Aschieri; Pierre Bieliavsky; Chiara Pagani; Alexander Schenkel Pages: 287 - 344 Abstract: Abstract We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf–Galois extensions as twists of Hopf–Galois extensions. A sheaf approach is also considered, and examples presented. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2765-x Issue No:Vol. 352, No. 1 (2017)

Authors:Xiaosen Han; Gabriella Tarantello Pages: 345 - 385 Abstract: Abstract In this paper we study the existence of vortex-type solutions for a system of self-dual equations deduced from the mass-deformed Aharony–Bergman–Jafferis–Maldacena (ABJM) model. The governing equations, derived by Mohammed, Murugan, and Nastse under suitable ansatz involving fuzzy sphere matrices, have the new feature that they can support only non-topological vortex solutions. After transforming the self-dual equations into a nonlinear elliptic \({2\times 2}\) system we prove first an existence result by means of a perturbation argument based on a new and appropriate scaling for the solutions. Subsequently, we prove a more complete existence result by using a dynamical analysis together with a blow-up argument. In this way we establish that any positive energy level is attained by a 1-parameter family of vortex solutions, which also correspond to (constraint) energy minimizers. In other words, we register the exceptional fact in a BPS-setting that, neither a “quantization” effect nor an energy gap is induced upon the system by the rigid “critical” coupling of the self-dual regime. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2817-2 Issue No:Vol. 352, No. 1 (2017)

Authors:Riccardo Adami; Enrico Serra; Paolo Tilli Pages: 387 - 406 Abstract: Abstract We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with L 2-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo–Nirenberg inequalities and by estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2797-2 Issue No:Vol. 352, No. 1 (2017)

Authors:Michael Freedman; Matthew Headrick Pages: 407 - 438 Abstract: Abstract The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, we invoke the notion of a “flow”, defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness “bit threads”. The entanglement entropy of a boundary region is given by the maximum flux out of it of any flow, or equivalently the maximum number of bit threads that can emanate from it. The threads thus represent entanglement between points on the boundary, and naturally implement the holographic principle. As we explain, this new picture clarifies several conceptual puzzles surrounding the RT formula. We give flow-based proofs of strong subadditivity and related properties; unlike the ones based on minimal surfaces, these proofs correspond in a transparent manner to the properties’ information-theoretic meanings. We also briefly discuss certain technical advantages that the flows offer over minimal surfaces. In a mathematical appendix, we review the max flow-min cut theorem on networks and on Riemannian manifolds, and prove in the network case that the set of max flows varies Lipshitz continuously in the network parameters. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2796-3 Issue No:Vol. 352, No. 1 (2017)

Authors:Zhiyuan Zhang; Zhiyan Zhao Pages: 877 - 921 Abstract: Abstract For the solution u(t) to the discrete Schrödinger equation $$ {\rm i} \frac{d}{dt}u_n(t)=-(u_{n+1}(t)+u_{n-1}(t))+V(\theta + n\alpha)u_n(t), \quad n\in\mathbb{Z},$$ with \({\alpha\in\mathbb{R}{\setminus} \mathbb{Q}}\) and \({V\in C^\omega(\mathbb{T},\mathbb{R})}\) , we consider the growth rate with t of its diffusion norm \({\langle u(t)\rangle_{p}:=\left(\sum_{n\in\mathbb{Z}}(n^{p}+1) u_n(t) ^2\right)^\frac12}\) , and the (non-averaged) transport exponents $$\beta_u^{+}(p) := \limsup_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}, \quad \beta_u^{-}(p):= \liminf_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}.$$ We will show that, if the corresponding Schrödinger operator has purely absolutely continuous spectrum, then \({\beta_{u}^{\pm}(p)=1}\) , provided that u(0) is well localized. PubDate: 2017-05-01 DOI: 10.1007/s00220-017-2848-3 Issue No:Vol. 351, No. 3 (2017)

Authors:Giuseppe Genovese; Giambattista Giacomin; Rafael Leon Greenblatt Pages: 923 - 958 Abstract: Abstract We consider a certain infinite product of random \({2 \times 2}\) matrices appearing in the solution of some 1 and 1 + 1 dimensional disordered models in statistical mechanics, which depends on a parameter \({\varepsilon > 0}\) and on a real random variable with distribution \({\mu}\) . For a large class of \({\mu}\) , we prove the prediction by Derrida and Hilhorst (J Phys A 16:2641, 1983) that the Lyapunov exponent behaves like \({C \epsilon^{2 \alpha}}\) in the limit \({\epsilon \searrow 0}\) , where \({\alpha \in (0,1)}\) and \({C > 0}\) are determined by \({\mu}\) . Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small \({\epsilon}\) . We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense that implies a suitable control of the Lyapunov exponent. PubDate: 2017-05-01 DOI: 10.1007/s00220-017-2855-4 Issue No:Vol. 351, No. 3 (2017)

Authors:Ovidiu Costin; Roland Donninger; Irfan Glogić Pages: 959 - 972 Abstract: Abstract We consider co-rotational wave maps from Minkowski space in d + 1 dimensions to the d-sphere. Recently, Bizoń and Biernat found explicit self-similar solutions for each dimension \({d\geq 4}\) . We give a rigorous proof for the mode stability of these self-similar wave maps. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2776-7 Issue No:Vol. 351, No. 3 (2017)

Authors:Marc Herzlich Pages: 973 - 992 Abstract: Abstract In this paper, we develop a general study of contributions at infinity of Bochner–Weitzenböck-type formulas on asymptotically flat manifolds, inspired by Witten’s proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreducible natural bundle and a very large choice of first-order operators may lead to a positive mass theorem along the same lines if the necessary curvature conditions are satisfied. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2777-6 Issue No:Vol. 351, No. 3 (2017)

Authors:S. V. Agapov; M. Bialy; A. E. Mironov Pages: 993 - 1007 Abstract: Abstract For a magnetic geodesic flow on the 2-torus the only known integrable example is that of a flow integrable for all energy levels. It has an integral linear in momenta and corresponds to a one parameter group preserving the Lagrangian function of the magnetic flow. In this paper the problem of integrability on a single energy level is considered. Then, in addition to the example mentioned above, a few other explicit examples with quadratic in momenta integrals can be constructed by means of the Maupertuis’ principle. Recently we proved that such an integrability problem can be reduced to a remarkable semi-Hamiltonian system of quasi-linear PDEs and to the question of the existence of smooth periodic solutions for this system. Our main result of the present paper states that any Liouville metric with the zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with a small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of an integral quadratic in momenta. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2822-5 Issue No:Vol. 351, No. 3 (2017)

Authors:Jonathan Brundan; Alexander P. Ellis Pages: 1045 - 1089 Abstract: Abstract This work is a companion to our article “Super Kac–Moody 2-categories,” which introduces super analogs of the Kac–Moody 2-categories of Khovanov–Lauda and Rouquier. In the case of \({\mathfrak{sl}_2}\) , the super Kac–Moody 2-category was constructed already in [A. Ellis and A. Lauda, “An odd categorification of \({U_q(\mathfrak{sl}_2)}\) ”], but we found that the formalism adopted there became too cumbersome in the general case. Instead, it is better to work with 2-supercategories (roughly, 2-categories enriched in vector superspaces). Then the Ellis–Lauda 2-category, which we call here a \({\Pi}\) -2-category (roughly, a 2-category equipped with a distinguished involution in its Drinfeld center), can be recovered by taking the superadditive envelope then passing to the underlying 2-category. The main goal of this article is to develop this language and the related formal constructions in the hope that these foundations may prove useful in other contexts. PubDate: 2017-05-01 DOI: 10.1007/s00220-017-2850-9 Issue No:Vol. 351, No. 3 (2017)

Authors:Svetlana Jitomirskaya; Fan Yang Pages: 1127 - 1135 Abstract: Abstract We prove that Schrödinger operators with meromorphic potentials \({(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n}\) have purely singular continuous spectrum on the set \({\{E: L(E) < \delta{(\alpha, \theta)}\}}\) , where \({\delta}\) is an explicit function and L is the Lyapunov exponent. This extends results of Jitomirskaya and Liu (Arithmetic spectral transitions for the Maryland model. CPAM, to appear) for the Maryland model and of Avila,You and Zhou (Sharp Phase transitions for the almost Mathieu operator. Preprint, 2015) for the almost Mathieu operator to the general family of meromorphic potentials. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2823-4 Issue No:Vol. 351, No. 3 (2017)

Authors:Yuki Arano Pages: 1137 - 1147 Abstract: Abstract We determine a substantial part of the unitary representation theory of the Drinfeld double of a q-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every q-deformation of a higher rank compact Lie group has central property (T). We also determine the unitary dual of \({SL_q(n,\mathbb{C})}\) . PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2704-x Issue No:Vol. 351, No. 3 (2017)

Authors:Ilia Binder; Damir Kinzebulatov; Mircea Voda Pages: 1149 - 1175 Abstract: Abstract We consider continuous one-dimensional multifrequency Schrödinger operators, with analytic potential, and prove Anderson localization in the regime of positive Lyapunov exponent for almost all phases and almost all Diophantine frequencies. PubDate: 2017-05-01 DOI: 10.1007/s00220-016-2723-7 Issue No:Vol. 351, No. 3 (2017)