Authors:J. Wang; T. Jäger Pages: 1 - 36 Abstract: Abstract We study the phenomenon of mode-locking in the context of quasiperiodically forced non-linear circle maps. As a main result, we show that under certain \({\mathcal{C}^1}\) -open condition on the geometry of twist parameter families of such systems, the closure of the union of mode-locking plateaus has positive measure. In particular, this implies the existence of infinitely many mode-locking plateaus (open Arnold tongues). The proof builds on multiscale analysis and parameter exclusion methods in the spirit of Benedicks and Carleson, which were previously developed for quasiperiodic \({{\rm SL}(2,\mathbb{R})}\) -cocycles by Young and Bjerklöv. The methods apply to a variety of examples, including a forced version of the classical Arnold circle map. PubDate: 2017-07-01 DOI: 10.1007/s00220-017-2870-5 Issue No:Vol. 353, No. 1 (2017)

Authors:Boele Braaksma; Gérard Iooss; Laurent Stolovitch Pages: 37 - 67 Abstract: Abstract This paper establishes the existence of quasipatterns solutions of the Swift–Hohenberg PDE. In a former approach we avoided the use of Nash–Moser scheme, but our proof contains a gap. The present proof of existence is based on the works by Berti et al related to the Nash–Moser scheme. For solving the small divisor problem, we need to introduce a new free parameter related to the freedom in the choice of parameterization of the bifurcating solution. Thanks to a transversality condition, the result gives only a bifurcating set, located in a small hornlike region centered on a curve, with the origin at the bifurcation point. PubDate: 2017-07-01 DOI: 10.1007/s00220-017-2878-x Issue No:Vol. 353, No. 1 (2017)

Authors:Matthew Daws; Adam Skalski; Ami Viselter Pages: 69 - 118 Abstract: Abstract We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with low duals, Property (T) is shown to be equivalent to Property (T)1,1 of Bekka and Valette. This is used to extend to this class of quantum groups classical theorems on ‘typical’ representations (due to Kerr and Pichot), and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Finally, we discuss in the Appendix equivalent characterisations of the notion of a quantum group morphism with dense image. PubDate: 2017-07-01 DOI: 10.1007/s00220-017-2862-5 Issue No:Vol. 353, No. 1 (2017)

Authors:Dmitry Beliaev; Bertrand Duplantier; Michel Zinsmeister Pages: 119 - 133 Abstract: Abstract We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated in Beliaev and Smirnov (Commun Math Phys 290(2):577–595, 2009), as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied in Duplantier et al. (Ann Henri Poincaré 16(6):1311–1395, 2014. arXiv:1211.2451) and Loutsenko and Yermolayeva (J Stat Mech P04007, 2013), a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of whole-plane SLE of Beliaev and Smirnov, a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum. PubDate: 2017-07-01 DOI: 10.1007/s00220-017-2868-z Issue No:Vol. 353, No. 1 (2017)

Authors:Hans Lindblad Pages: 135 - 184 Abstract: Abstract We give asymptotics for the Einstein vacuum equations in wave coordinates with small asymptotically flat data. We show that the behavior is wave like at null infinity and homogeneous towards time like infinity. We use the asymptotics to show that the outgoing null hypersurfaces approach the Schwarzschild ones for the same mass and that the radiated energy is equal to the initial mass. PubDate: 2017-07-01 DOI: 10.1007/s00220-017-2876-z Issue No:Vol. 353, No. 1 (2017)

Authors:Guido Franchetti; Bernd J. Schroers Pages: 185 - 228 Abstract: Abstract We define and compute the L 2 metric on the framed moduli space of circle invariant 1-instantons on the 4-sphere. This moduli space is four dimensional and our metric is \({SO(3) \times U(1)}\) symmetric. We study the behaviour of generic geodesics and show that the metric is geodesically incomplete. Circle-invariant instantons on the 4-sphere can also be viewed as hyperbolic monopoles, and we interpret our results from this viewpoint. We relate our results to work by Habermann on unframed instantons on the 4-sphere and, in the limit where the radius of the 4-sphere tends to infinity, to results on instantons on Euclidean 4-space. PubDate: 2017-07-01 DOI: 10.1007/s00220-016-2769-6 Issue No:Vol. 353, No. 1 (2017)

Authors:Jacopo Bellazzini; Nabile Boussaïd; Louis Jeanjean; Nicola Visciglia Pages: 229 - 251 Abstract: Abstract We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are L 2-supercritical; in particular, we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC. PubDate: 2017-07-01 DOI: 10.1007/s00220-017-2866-1 Issue No:Vol. 353, No. 1 (2017)

Authors:Ilan Hirshberg; Gábor Szabó; Wilhelm Winter; Jianchao Wu Pages: 253 - 316 Abstract: Abstract We introduce a notion of Rokhlin dimension for one parameter automorphism groups of \({C^*}\) -algebras. This generalizes Kishimoto’s Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing \({C^*}\) -algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative \({C^*}\) -algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product \({C^{*}}\) -algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological \({K}\) -theory together with the space of invariant probability measures and a natural pairing given by the Ruelle–Sullivan map). PubDate: 2017-07-01 DOI: 10.1007/s00220-016-2762-0 Issue No:Vol. 353, No. 1 (2017)

Authors:Lotte Hollands; Andrew Neitzke Pages: 317 - 351 Abstract: Abstract We use the method of spectral networks to compute BPS state degeneracies in the Minahan-Nemeschansky \({E_6}\) theory, on its Coulomb branch, without turning on a mass deformation. The BPS multiplicities come out in representations of the \({E_6}\) flavor symmetry. For example, along the simplest ray in electromagnetic charge space, we give the first 14 numerical degeneracies, and the first 7 degeneracies as representations of \({E_6}\) . We find a complicated spectrum, exhibiting exponential growth of multiplicities as a function of the electromagnetic charge. There is one unexpected outcome: the spectrum is consistent (in a nontrivial way) with the hypothesis of spin purity, that if a BPS state in this theory has electromagnetic charge equal to n times a primitive charge, then it appears in a spin- \({\frac{n}{2}}\) multiplet. PubDate: 2017-07-01 DOI: 10.1007/s00220-016-2798-1 Issue No:Vol. 353, No. 1 (2017)

Authors:D. Bambusi Pages: 353 - 378 Abstract: Abstract We study the Schrödinger equation on \({\mathbb{R}}\) with a potential behaving as \({x^{2l}}\) at infinity, \({l \in [1, + \infty)}\) and with a small time quasiperiodic perturbation. We prove that if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible. PubDate: 2017-07-01 DOI: 10.1007/s00220-016-2825-2 Issue No:Vol. 353, No. 1 (2017)

Authors:Sergei Alexandrov; Sibasish Banerjee; Jan Manschot; Boris Pioline Pages: 379 - 411 Abstract: Abstract We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi–Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the M5-brane elliptic genus. We extend this analysis to the two-instanton level, where wall-crossing phenomena start playing a role. We focus on the contact potential, an analogue of the Kähler potential which must transform as a modular form under S-duality. We show that it can be expressed in terms of a suitable modification of the partition function of D4-D2-D0 BPS black holes, constructed out of the generating function of MSW invariants (the latter coincide with Donaldson–Thomas invariants in a particular chamber). Modular invariance of the contact potential then requires that, in the case where the D3-brane wraps a reducible divisor, the generating function of MSW invariants must transform as a vector-valued mock modular form, with a specific modular completion built from the MSW invariants of the constituents. Physically, this gives a powerful constraint on the degeneracies of BPS black holes. Mathematically, our result gives a universal prediction for the modular properties of Donaldson–Thomas invariants of pure two-dimensional sheaves. PubDate: 2017-07-01 DOI: 10.1007/s00220-016-2799-0 Issue No:Vol. 353, No. 1 (2017)

Authors:Catherine Meusburger Pages: 413 - 468 Abstract: Abstract We prove that Kitaev’s lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern–Simons theory for the Drinfeld double D(H). This shows that Kitaev models are a special case of the older and more general combinatorial models. This equivalence is an analogue of the relation between Turaev–Viro and Reshetikhin–Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories. We first prove that the triangle operators of a Kitaev model form a module algebra over a Hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism. Both algebras can be viewed as the algebra of functions on gauge fields in a Hopf algebra gauge theory. The isomorphism between them induces an algebra isomorphism between their subalgebras of invariants, which are interpreted as gauge invariant functions or observables. It also relates the curvatures in the two models, which are given as holonomies around the faces of the lattice. This yields an isomorphism between the subalgebras obtained by projecting out curvatures, which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models. PubDate: 2017-07-01 DOI: 10.1007/s00220-017-2860-7 Issue No:Vol. 353, No. 1 (2017)

Authors:Bruce K. Driver; Franck Gabriel; Brian C. Hall; Todd Kemp Pages: 967 - 978 Abstract: Abstract We prove the Makeenko–Migdal equation for two-dimensional Euclidean Yang–Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces. PubDate: 2017-06-01 DOI: 10.1007/s00220-017-2857-2 Issue No:Vol. 352, No. 3 (2017)

Authors:Andrey Gogolev Pages: 439 - 455 Abstract: Abstract We establish a strong form of local rigidity for hyperbolic automorphisms of the 3-torus with real spectrum. Namely, let \({L:\mathbb{T}^3\to\mathbb{T}^3}\) be a hyperbolic automorphism of the 3-torus with real spectrum and let f be a C 1 small perturbation of L. Then f is smoothly ( \({C^\infty}\) ) conjugate to L if and only if obstructions to C 1 conjugacy given by the eigenvalues at periodic points of f vanish. By combining our result and a local rigidity result of Kalinin and Sadovskaya (Ergod Theory Dyn Syst 29:117–136, 2009) for conformal automorphisms this completes the local rigidity program for hyperbolic automorphisms in dimension 3. Our work extends de la Llave–Marco–Moriyón 2-dimensional local rigidity theory (Commun Math Phys 109:368–378, 1987; Ergod Theory Dyn Syst 17(3):649–662, 1997; Commun Math Phys 109(4):681–689, 1987). PubDate: 2017-06-01 DOI: 10.1007/s00220-017-2863-4 Issue No:Vol. 352, No. 2 (2017)

Authors:Péter Vrana; Matthias Christandl Pages: 621 - 627 Abstract: Abstract We study the problem of converting a product of Greenberger–Horne–Zeilinger (GHZ) states shared by subsets of several parties in an arbitrary way into GHZ states shared by every party. Such a state can be described by a hypergraph on the parties as vertices and with each hyperedge corresponding to a GHZ state shared among the parties incident with it. Our result is that if SLOCC transformations are allowed, then the best asymptotic rate is the minimum of bipartite log-ranks of the initial state, which in turn equals the minimum cut of the hypergraph. This generalizes a result by Strassen on the asymptotic subrank of the matrix multiplication tensor. PubDate: 2017-06-01 DOI: 10.1007/s00220-017-2861-6 Issue No:Vol. 352, No. 2 (2017)

Authors:Sabine Bögli Pages: 629 - 639 Abstract: Abstract We study Schrödinger operators \({H=-\Delta + V}\) in \({L^{2}(\Omega)}\) where \({\Omega}\) is \({\mathbb{R}^d}\) or the half-space \({{\mathbb {R}_{+}^{d}}}\) , subject to (real) Robin boundary conditions in the latter case. For \({p > d}\) we construct a non-real potential \({V \in L^{p}(\Omega) \cap L^{\infty}(\Omega)}\) that decays at infinity so that H has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum \({\sigma_{\rm ess}(H)=[0,\infty)}\) . This demonstrates that the Lieb–Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case. PubDate: 2017-06-01 DOI: 10.1007/s00220-016-2806-5 Issue No:Vol. 352, No. 2 (2017)

Authors:Roland Grinis Pages: 641 - 702 Abstract: Abstract In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere \({\mathbb{S}^{n-1} \subset\mathbb{R}^{n}}\) with \({n\geq2}\) , and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Côte, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Côte, Commun. Pure Appl. Math. 69(4):609–612, 2016; Côte, Am. J. Math. 137(1):139–207, 2015; Côte et al., Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Rivière (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010). PubDate: 2017-06-01 DOI: 10.1007/s00220-016-2766-9 Issue No:Vol. 352, No. 2 (2017)

Authors:Christos Mantoulidis; Pengzi Miao Pages: 703 - 718 Abstract: Abstract We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consists of 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi–Tam and Wang–Yau on the quasi-local mass problem in general relativity. In turn, we define a variational analog of Brown–York quasi-local mass without assuming that the boundary 2-sphere has positive Gauss curvature. PubDate: 2017-06-01 DOI: 10.1007/s00220-016-2767-8 Issue No:Vol. 352, No. 2 (2017)

Authors:Mihály Weiner Pages: 759 - 772 Abstract: Abstract Let c, h and \({c,\tilde{h}}\) be two admissible pairs of central charge and highest weight for \({{\rm Diff}^+(S^1)}\) . It is shown here that the positive energy irreducible projective unitary representations \({U_{c,h}}\) and \({U_{c,\tilde{h}}}\) of the group \({{\rm Diff}^+(S^1)}\) are locally equivalent. This means that for any \({I\Subset S^1}\) open proper interval, there exists a unitary operator W I such that \({W_I U_{c,h}(\gamma)W_I^* = U_{c,\tilde{h}}(\gamma)}\) for all \({\gamma \in {\rm Diff}^+(S^1)}\) which act identically on \({I^c\equiv S^1{\setminus} I}\) (i.e., which can “displace” or “move” points only in I). This result extends and completes earlier ones that dealt with only certain regions of the “c, h-plane”, and closes the gap in the full classification of superselection sectors of Virasoro nets. PubDate: 2017-06-01 DOI: 10.1007/s00220-016-2824-3 Issue No:Vol. 352, No. 2 (2017)