Authors:Xifeng Su; Rafael de la Llave Pages: 459 - 475 Abstract: We show that a foliation of equilibria (a continuous family of equilibria whose graph covers all the configuration space) in ferromagnetic transitive models are ground states. The result we prove is very general, and it applies to models with long range and many-body interactions. As an application, we consider several models of networks of interacting particles including models of Frenkel–Kontorova type on \({\mathbb{Z}^d}\) and one-dimensional quasi-periodic media. The result above is an analogue of several results in the calculus of variations (fields of extremals) and in PDE’s. Since the models we consider are discrete and long range, new proofs need to be given. We also note that the main hypothesis of our result (the existence of foliations of equilibria) is the conclusion (using KAM theory) of several recent papers. Hence, we obtain that the KAM solutions recently established are minimizers when the interaction is ferromagnetic and transitive (these concepts are defined later). PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2913-y Issue No:Vol. 354, No. 2 (2017)

Authors:Yang-Hui He; Vishnu Jejjala; Luca Pontiggia Pages: 477 - 524 Abstract: We explore the distribution of topological numbers in Calabi–Yau manifolds, using the Kreuzer–Skarke dataset of hypersurfaces in toric varieties as a testing ground. While the Hodge numbers are well-known to exhibit mirror symmetry, patterns in frequencies of combination thereof exhibit striking new patterns. We find pseudo-Voigt and Planckian distributions with high confidence and exact fit for many substructures. The patterns indicate typicality within the landscape of Calabi–Yau manifolds of various dimension. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2907-9 Issue No:Vol. 354, No. 2 (2017)

Authors:M. Bertola; A. Tovbis Pages: 525 - 547 Abstract: Finite-gap (algebro-geometric) solutions to the focusing Nonlinear Schrödinger Equation (fNLS) 0-1 $$i \psi_t + \psi_{xx} + 2 \psi ^2\psi=0,$$ are quasi-periodic solutions that represent nonlinear multi-phase waves. In general, a finite-gap solution for (0-1) is defined by a collection of Schwarz symmetrical spectral bands and of real constants (initial phases), associated with the corresponding bands. In this paper we prove an interesting new formula for the maximal amplitude of a finite-gap solution to the focusing Nonlinear Schrödinger equation with given spectral bands: the amplitude does not exceed the sum of the imaginary parts of all the endpoints in the upper half plane. In the case of the straight vertical bands, that amounts to the half of the sum of the length of all the bands. The maximal amplitude will be attained for certain choices of the initial phases. This result is an important part of a criterion for the potential presence of the rogue waves in finite-gap solutions with a given set of spectral endpoints, obtained in Bertola et al. (Proc R Soc A, 2016. doi:10.1098/rspa.2016.0340). A similar result was also obtained for the defocusing Nonlinear Schrödinger equation. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2895-9 Issue No:Vol. 354, No. 2 (2017)

Authors:Joscha Diehl; Massimiliano Gubinelli; Nicolas Perkowski Pages: 549 - 589 Abstract: We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Gonçalves and Jara (Arch Ration Mech Anal 212(2):597–644, 2014) and the corresponding uniqueness result of Gubinelli and Perkowski (Energy solutions of KPZ are unique, 2015). PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2918-6 Issue No:Vol. 354, No. 2 (2017)

Authors:Alex Blumenthal; Lai-Sang Young Pages: 591 - 619 Abstract: We prove the absolute continuity of stable foliations for mappings of Banach spaces satisfying conditions consistent with time-t maps of certain classes of dissipative PDEs. This property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space; it is also used in proofs of ergodicity. Absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory. On Banach spaces, the absence of nice geometric properties poses some additional difficulties. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2912-z Issue No:Vol. 354, No. 2 (2017)

Authors:Matthias Ludewig Pages: 621 - 640 Abstract: We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimensional path spaces of piecewise geodesics subordinated to increasingly fine partitions of the time interval. We consider a subclass of mixed boundary conditions which includes standard Dirichlet and Neumann boundary conditions. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2915-9 Issue No:Vol. 354, No. 2 (2017)

Authors:N. Arrizabalaga; L. Le Treust; N. Raymond Pages: 641 - 669 Abstract: This paper is devoted to the spectral investigation of the MIT bag model, that is, the Dirac operator on a smooth and bounded domain of \({\mathbb{R}^3}\) with certain boundary conditions. When the mass m goes to \({\pm\infty}\) , we provide spectral asymptotic results. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2916-8 Issue No:Vol. 354, No. 2 (2017)

Authors:Mathew Bullimore; Tudor Dimofte; Davide Gaiotto Pages: 671 - 751 Abstract: We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory with \({\mathcal{N}=4}\) supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2903-0 Issue No:Vol. 354, No. 2 (2017)

Authors:J. Avan; L. Frappat; E. Ragoucy Pages: 753 - 773 Abstract: We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 1990s. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the gl(2) case) a Liouville formula. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2909-7 Issue No:Vol. 354, No. 2 (2017)

Authors:Stefano Marò; Alfonso Sorrentino Pages: 775 - 808 Abstract: In this article we develop an analogue of Aubry–Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. Besides describing their structure and their dynamical significance, we shall analyze their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system. PubDate: 2017-09-01 DOI: 10.1007/s00220-017-2900-3 Issue No:Vol. 354, No. 2 (2017)

Authors:Vladimir Mitev; Matthias Staudacher; Zengo Tsuboi Pages: 1 - 30 Abstract: The S-matrix of the \({AdS_5\times S^5}\) string theory is a tensor product of two centrally extended su \({(2 2)\ltimes \mathbb{R}^2}\) S-matrices, each of which is related to the R-matrix of the Hubbard model. The R-matrix of the Hubbard model was first found by Shastry, who ingeniously exploited the fact that, for zero coupling, the Hubbard model can be decomposed into two XX models. In this article, we review and clarify this construction from the AdS/CFT perspective and investigate the implications this has for the \({AdS_5\times S^5}\) S-matrix. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2905-y Issue No:Vol. 354, No. 1 (2017)

Authors:Charles Bordenave; Alice Guionnet Pages: 115 - 159 Abstract: We prove that the eigenvectors associated to small enough eigenvalues of a heavy-tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity. The delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of \({L^4}\) and \({L^2}\) -norms of vectors. In contrast, as a consequence of a previous result, for random matrices with sufficiently heavy tails, the eigenvectors associated to large enough eigenvalues are localized according to the same criterion. The proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2914-x Issue No:Vol. 354, No. 1 (2017)

Authors:Kenji Nakanishi Pages: 161 - 212 Abstract: Consider the focusing nonlinear Schrödinger equation with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We classify the global dynamics into nine sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2902-1 Issue No:Vol. 354, No. 1 (2017)

Authors:Dongho Chae; Jörg Wolf Pages: 213 - 230 Abstract: We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on \({\mathbb{R}^2}\) . We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2908-8 Issue No:Vol. 354, No. 1 (2017)

Authors:Jonathan Engle; Maximilian Hanusch; Thomas Thiemann Pages: 231 - 246 Abstract: We show that the standard representation of homogeneous isotropic loop quantum cosmology (LQC) is the GNS-representation that corresponds to the unique state on the reduced quantum holonomy-flux *-algebra that is invariant under residual diffeomorphisms—both when the standard algebra is used as well as when one uses the extended algebra proposed by Fleischhack. More precisely, we find that in both situations the GNS-Hilbert spaces coincide, and that in the Fleischhack case the additional algebra elements are just mapped to zero operators. In order for the residual diffeomorphisms to have a well-defined action on the quantum algebra, we have let them act on the fiducial cell as well as on the dynamical variables, thereby recovering covariance. Consistency with Ashtekar and Campilgia in the Bianchi case is also shown. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2881-2 Issue No:Vol. 354, No. 1 (2017)

Authors:Mariana Haragus; Jin Li; Dmitry E. Pelinovsky Pages: 247 - 268 Abstract: We present a general counting result for the unstable eigenvalues of linear operators of the form J L in which J and L are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator K such that the operators J L and J K commute, we prove that the number of unstable eigenvalues of J L is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev–Petviashvili) equation. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2898-6 Issue No:Vol. 354, No. 1 (2017)

Authors:Vadim Gorin Pages: 317 - 344 Abstract: We prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains. The result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in Gelfand–Tsetlin bases of tensor products of representations of unitary groups. In a weaker form our theorem also applies to random domino tilings. PubDate: 2017-08-01 DOI: 10.1007/s00220-016-2801-x Issue No:Vol. 354, No. 1 (2017)

Authors:Anton Nedelin; Fabrizio Nieri; Maxim Zabzine Pages: 1059 - 1102 Abstract: We study partition functions of 3d \({\mathcal{N}=2}\) \({{\rm U}(N)}\) gauge theories on compact manifolds which are S 1 fibrations over S 2. We show that the partition functions are free field correlators of vertex operators and screening charges of the q-Virasoro modular double, which we define. The inclusion of supersymmetric Wilson loops in arbitrary representations allows us to show that the generating functions of Wilson loop vacuum expectation values satisfy two \({{\rm SL}(2,\mathbb{Z})}\) -related commuting sets of q-Virasoro constraints. We generalize our construction to 3d \({\mathcal{N}=2}\) unitary quiver gauge theories and as an example we give the free boson realization of the ABJ(M) model. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2882-1 Issue No:Vol. 353, No. 3 (2017)

Authors:Andrew Schopieray Pages: 1103 - 1127 Abstract: The Witt group of nondegenerate braided fusion categories \({{\mathcal{W}}}\) contains a subgroup \({{\mathcal{W}}_{\rm un}}\) consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite dimensional simple Lie algebra \({{\mathfrak{g}}}\) and positive integer k there exists a pseudo-unitary category \({{\mathcal{C}}({\mathfrak{g}},k)}\) consisting of highest weight integrable \({{\hat{g}}}\) -modules of level k where \({\hat{{\mathfrak{g}}}}\) is the corresponding affine Lie algebra. Relations between the classes \({[{\mathcal{C}}({\mathfrak{sl}}_2,k)]}\) , \({k\geq1}\) have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes \({[{\mathcal{C}}({\mathfrak{sl}}_3,k)]}\) , \({k\geq1}\) with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras \({{\mathfrak{g}}}\) and positive integer levels k. PubDate: 2017-08-01 DOI: 10.1007/s00220-017-2831-z Issue No:Vol. 353, No. 3 (2017)

Authors:Masoumah Al-Ali; Andrew R. Linshaw Pages: 1129 - 1150 Abstract: The Zamolodchikov \({\mathcal{W}_3}\) -algebra \({\mathcal{W}^c_3}\) with central charge c has full automorphism group \({\mathbb{Z}_2}\) . It was conjectured in the physics literature over 20 years ago that the orbifold \({(\mathcal{W}^c_3)^{\mathbb{Z}_2}}\) is of type \({\mathcal{W}(2,6,8,10,12)}\) for generic values of c. We prove this conjecture for all \({c \neq \frac{559 \pm 7 \sqrt{76657}}{95}}\) , and we show that for these two values, the orbifold is of type \({\mathcal{W}(2,6,8,10,12,14)}\) . This paper is part of a larger program of studying orbifolds and cosets of vertex algebras that depend continuously on a parameter. Minimal strong generating sets for orbifolds and cosets are often easy to find for generic values of the parameter, but determining which values are generic is a difficult problem. In the example of \({(\mathcal{W}^c_3)^{\mathbb{Z}_2}}\) , we solve this problem using tools from algebraic geometry. PubDate: 2017-08-01 DOI: 10.1007/s00220-016-2812-7 Issue No:Vol. 353, No. 3 (2017)