Authors:Boris Feigin; Michio Jimbo; Tetsuji Miwa; Eugene Mukhin Pages: 2543 - 2579 Abstract: Abstract We introduce and study a category \(\mathcal {O}^\mathrm{fin}_{\mathfrak {b}}\) of modules of the Borel subalgebra \(U_q\mathfrak {b}\) of a quantum affine algebra \(U_q\mathfrak {g}\) , where the commutative algebra of Drinfeld generators \(h_{i,r}\) , corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional \(U_q\mathfrak {g}\) modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in \(\mathcal {O}^\mathrm{fin}_{\mathfrak {b}}\) . Among them, we find the Baxter \(Q_i\) operators and \(T_i\) operators satisfying relations of the form \(T_iQ_i=\prod _j Q_j+ \prod _k Q_k\) . We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the \(Q_i\) operators acting in an arbitrary finite-dimensional representation of \(U_q\mathfrak {g}\) . PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0577-y Issue No:Vol. 18, No. 8 (2017)

Authors:Kohei Iwaki; Olivier Marchal Pages: 2581 - 2620 Abstract: Abstract The goal of this article is to prove that the determinantal formulas of the Painlevé 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary nonzero monodromy parameter. The result is established for a WKB expansion of two different Lax pairs associated with the Painlevé 2 system, namely the Jimbo–Miwa Lax pair and the Harnad–Tracy–Widom Lax pair, where a small parameter \(\hbar \) is introduced by a proper rescaling. The proof is based on showing that these systems satisfy the topological type property introduced in Bergère et al. (Ann Henri Poincaré 16:2713, 2015), Bergère and Eynard (arxiv:0901.3273, 2009). In the process, we explain why the insertion operator method traditionally used to prove the topological type property is currently incomplete and we propose new methods to bypass the issue. Our work generalizes similar results obtained from random matrix theory in the special case of vanishing monodromies (Borot and Eynard in arXiv:1011.1418, 2010; arXiv:1012.2752, 2010). Explicit computations up to \(g=3\) are provided along the paper as an illustration of the results. Eventually, taking the time parameter t to infinity we observe that the symplectic invariants \(F^{(g)}\) of the Jimbo–Miwa and Harnad–Tracy–Widom spectral curves converge to the Euler characteristic of moduli space of genus g Riemann surfaces. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0576-z Issue No:Vol. 18, No. 8 (2017)

Authors:Mansur I. Ismailov Pages: 2621 - 2639 Abstract: Abstract The inverse scattering problem of recovering the matrix coefficient of a first-order system on the half-line from its scattering matrix is considered. In the case of a triangular structure of the matrix coefficient, this system has a Volterra-type integral transformation operator at infinity. Such a transformation operator allows to determine the scattering matrix on the half-line via the matrix Riemann–Hilbert factorization in the case, where the contour is real line, the normalization is canonical, and all the partial indices are zero. The ISP on the half-line is solved by reducing it to an ISP on the whole line for the considered system with the coefficients that are extended to the whole line by zero. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0575-0 Issue No:Vol. 18, No. 8 (2017)

Authors:Athanasios Chatzistavrakidis; Andreas Deser; Larisa Jonke; Thomas Strobl Pages: 2641 - 2692 Abstract: Abstract We study the propagation of bosonic strings in singular target space-times. For describing this, we assume this target space to be the quotient of a smooth manifold M by a singular foliation \({{\mathcal {F}}}\) on it. Using the technical tool of a gauge theory, we propose a smooth functional for this scenario, such that the propagation is assured to lie in the singular target on-shell, i.e., only after taking into account the gauge-invariant content of the theory. One of the main new aspects of our approach is that we do not limit \({{\mathcal {F}}}\) to be generated by a group action. We will show that, whenever it exists, the above gauging is effectuated by a single geometrical and universal gauge theory, whose target space is the generalized tangent bundle \(TM\oplus T^*M\) . PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0580-3 Issue No:Vol. 18, No. 8 (2017)

Authors:Alexander Schenkel; Jochen Zahn Pages: 2693 - 2714 Abstract: Abstract We formulate an algebraic criterion for the presence of global anomalies on globally hyperbolic space-times in the framework of locally covariant field theory. We discuss some consequences and check that it reproduces the well-known global SU(2) anomaly in four space-time dimensions. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0590-1 Issue No:Vol. 18, No. 8 (2017)

Authors:Christian Gérard; Michał Wrochna Pages: 2715 - 2756 Abstract: Abstract We consider the massive Klein–Gordon equation on a class of asymptotically static spacetimes (in the long-range sense) with Cauchy surface of bounded geometry. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0573-2 Issue No:Vol. 18, No. 8 (2017)

Authors:Oran Gannot Pages: 2757 - 2788 Abstract: Abstract This paper establishes the existence of quasinormal frequencies converging exponentially to the real axis for the Klein–Gordon equation on a Kerr–AdS spacetime when Dirichlet boundary conditions are imposed at the conformal boundary. The proof is adapted from results in Euclidean scattering about the existence of scattering poles generated by time-periodic approximate solutions to the wave equation. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0568-z Issue No:Vol. 18, No. 8 (2017)

Authors:Paul T. Allen; Iva Stavrov Allen Pages: 2789 - 2814 Abstract: Abstract We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0565-2 Issue No:Vol. 18, No. 8 (2017)

Authors:Marcus Khuri; Naqing Xie Pages: 2815 - 2830 Abstract: Abstract We establish inequalities relating the size of a material body to its mass, angular momentum, and charge, within the context of axisymmetric initial data sets for the Einstein equations. These inequalities hold in general without the assumption of the maximal condition and use a notion of size which is easily computable. Moreover, these results give rise to black hole existence criteria which are meaningful even in the time-symmetric case, and also include certain boundary effects. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0582-1 Issue No:Vol. 18, No. 8 (2017)

Authors:Johannes Kellendonk Pages: 2251 - 2300 Abstract: Abstract The notion of a topological phase of an insulator is based on the concept of homotopy between Hamiltonians. It therefore depends on the choice of a topological space to which the Hamiltonians belong. We advocate that this space should be the \(C^*\) -algebra of observables. We relate the symmetries of insulators to graded real structures on the observable algebra and classify the topological phases using van Daele’s formulation of K-theory. This is related but not identical to Thiang’s recent approach to classify topological phases by K-groups in Karoubi’s formulation. PubDate: 2017-07-01 DOI: 10.1007/s00023-017-0583-0 Issue No:Vol. 18, No. 7 (2017)

Authors:Alex Clark; Lorenzo Sadun Pages: 2301 - 2326 Abstract: Abstract Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free \(\mathbb {Z}^d\) actions on Cantor sets admit “small cocycles.” These represent classes in \(H^1\) that are mapped to small vectors in \(\mathbb {R}^d\) by the Ruelle–Sullivan (RS) map. We show that there exist \(\mathbb {Z}^2\) actions where no such small cocycles exist, and where the image of \(H^1\) under RS is \(\mathbb {Z}^2\) . Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of “virtual eigenvalues,” i.e., elements of \(\mathbb {R}^d\) that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles. PubDate: 2017-07-01 DOI: 10.1007/s00023-017-0579-9 Issue No:Vol. 18, No. 7 (2017)

Authors:Abel Klein; Son T. Nguyen; Constanza Rojas-Molina Pages: 2327 - 2365 Abstract: Abstract We extend to the two-particle Anderson model the characterization of the metal–insulator transport transition obtained in the one-particle setting by Germinet and Klein. We show that, for any fixed number of particles, the slow spreading of wave packets in time implies the initial estimate of a modified version of the bootstrap multiscale analysis. In this new version, operators are restricted to boxes defined with respect to the pseudo-distance in which we have the slow spreading. At the bottom of the spectrum, within the regime of one-particle dynamical localization, we show that this modified multiscale analysis yields dynamical localization for the two-particle Anderson model, allowing us to obtain a characterization of the metal–insulator transport transition for the two-particle Anderson model at the bottom of the spectrum. PubDate: 2017-07-01 DOI: 10.1007/s00023-017-0578-x Issue No:Vol. 18, No. 7 (2017)

Authors:Martin Fraas; Lisa Hänggli Pages: 2447 - 2465 Abstract: Abstract We consider a driven open system whose evolution is described by a Lindbladian. The Lindbladian is assumed to be dephasing and its Hamiltonian part to be given by the Landau–Zener Hamiltonian. We derive a formula for the transition probability which, unlike previous results, extends the Landau–Zener formula to open systems. PubDate: 2017-07-01 DOI: 10.1007/s00023-017-0567-0 Issue No:Vol. 18, No. 7 (2017)

Authors:Fumio Hiai; Robert König; Marco Tomamichel Pages: 2499 - 2521 Abstract: Abstract We show that recent multivariate generalizations of the Araki–Lieb–Thirring inequality and the Golden–Thompson inequality (Sutter et al. in Commun Math Phys, 2016. doi:10.1007/s00220-016-2778-5) for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regard to logarithmic integral averages of vectors of singular values. PubDate: 2017-07-01 DOI: 10.1007/s00023-017-0569-y Issue No:Vol. 18, No. 7 (2017)

Authors:Kota Uriya Pages: 2523 - 2542 Abstract: Abstract In this paper, we are concerned with the asymptotic behavior of the solution to systems of cubic nonlinear Schrödinger equations in one dimension. It is known that mass transition phenomenon occurs for a system of quadratic nonlinear Schrödinger equations in two dimensions under the mass resonance condition. We show that mass transition phenomenon also occurs for systems with cubic nonlinearity under the corresponding mass resonance conditions. PubDate: 2017-07-01 DOI: 10.1007/s00023-017-0581-2 Issue No:Vol. 18, No. 7 (2017)

Authors:Gregory J. Galloway; Eric Ling Abstract: Abstract The existence, established over the past number of years and supporting earlier work of Ori (Phys Rev Lett 68(14):2117–2120, 1992), of physically relevant black hole spacetimes that admit \(C^0\) metric extensions beyond the future Cauchy horizon, while being \(C^2\) -inextendible, has focused attention on fundamental issues concerning the strong cosmic censorship conjecture. These issues were recently discussed in the work of Sbierski (The \({C}^0\) -inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry. arXiv:1507.00601v2, (to appear in J. Diff. Geom.), 2015), in which he established the (nonobvious) fact that the Schwarzschild solution in global Kruskal–Szekeres coordinates is \(C^0\) -inextendible. In this paper, we review aspects of Sbierski’s methodology in a general context and use similar techniques, along with some new observations, to consider the \(C^0\) -inextendibility of open FLRW cosmological models. We find that a certain special class of open FLRW spacetimes, which we have dubbed ‘Milne-like,’ actually admits \(C^0\) extensions through the big bang. For spacetimes that are not Milne-like, we prove some inextendibility results within the class of spherically symmetric spacetimes. PubDate: 2017-07-14 DOI: 10.1007/s00023-017-0602-1

Authors:Ram Band; Guillaume Lévy Abstract: Abstract A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs. PubDate: 2017-07-13 DOI: 10.1007/s00023-017-0601-2

Authors:Hugo Bringuier Abstract: Abstract Open quantum walks (OQWs), originally introduced in Attal et al. (J Stat Phys 147(4):832–852, 2012), are quantum generalizations of classical Markov chains. Recently, natural continuous time models of OQW have been developed in Pellegrini (J Stat Phys 154(3):838–865, 2014). These models, called continuous time open quantum walks (CTOQWs), appear as natural continuous time limits of discrete time OQWs. In particular, they are quantum extensions of continuous time Markov chains. This article is devoted to the study of homogeneous CTOQW on \(\mathbb {Z}^d\) . We focus namely on their associated quantum trajectories which allow us to prove a central limit theorem for the “position” of the walker as well as a large deviation principle. PubDate: 2017-07-13 DOI: 10.1007/s00023-017-0597-7

Authors:Benjamin Dodson; Nishanth Gudapati Abstract: Abstract We consider the Cauchy problem of \(2+1\) equivariant wave maps coupled to Einstein’s equations of general relativity and prove that two separate (nonlinear) subclasses of the system disperse to their corresponding linearized equations in the large. Global asymptotic behavior of \(2+1\) Einstein-wave map system is relevant because the system occurs naturally in \(3+1\) vacuum Einstein’s equations. PubDate: 2017-07-12 DOI: 10.1007/s00023-017-0599-5

Authors:Adrian P. C. Lim Abstract: Abstract A hyperlink is a finite set of non-intersecting simple closed curves in \(\mathbb {R} \times \mathbb {R}^3\) . Let S be an orientable surface in \(\mathbb {R}^3\) . The dynamical variables in general relativity are the vierbein e and a \(\mathfrak {su}(2)\times \mathfrak {su}(2)\) -valued connection \(\omega \) . Together with Minkowski metric, e will define a metric g on the manifold. Denote \(A_S(e)\) as the area of S, for a given choice of e. The Einstein–Hilbert action \(S(e,\omega )\) is defined on e and \(\omega \) . We will quantize the area of the surface S by integrating \(A_S(e)\) against a holonomy operator of a hyperlink L, disjoint from S, and the exponential of the Einstein–Hilbert action, over the space of vierbeins e and \(\mathfrak {su}(2)\times \mathfrak {su}(2)\) -valued connections \(\omega \) . Using our earlier work done on Chern–Simons path integrals in \(\mathbb {R}^3\) , we will write this infinite dimensional path integral as the limit of a sequence of Chern–Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between L and S. By assigning an irreducible representation of \(\mathfrak {su}(2)\times \mathfrak {su}(2)\) to each component of L, the area operator gives the total net momentum impact on the surface S. PubDate: 2017-07-10 DOI: 10.1007/s00023-017-0600-3