Authors:Jean-Christophe Mourrat; Hendrik Weber Pages: 673 - 753 Abstract: We prove an a priori bound for the dynamic \({\Phi^4_3}\) model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows one to construct invariant measures via the Krylov–Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean \({\Phi^4_3}\) field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2997-4 Issue No:Vol. 356, No. 3 (2017)

Authors:Frédéric Faure; Tobias Weich Pages: 755 - 822 Abstract: We consider a \({\mathbb{R}}\) -extension of one dimensional uniformly expanding open dynamical systems and prove a new explicit estimate for the asymptotic spectral gap. To get these results, we use a new application of a “global normal form” for the dynamical system, a “semiclassical expression beyond the Ehrenfest time” that expresses the transfer operator at large time as a sum over rank one operators (each is associated to one orbit). In this paper we establish the validity of the so-called “diagonal approximation” up to twice the local Ehrenfest time. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-3000-0 Issue No:Vol. 356, No. 3 (2017)

Authors:Tri Lai; Gregg Musiker Pages: 823 - 881 Abstract: Given one of an infinite class of supersymmetric quiver gauge theories, string theorists can associate a corresponding toric variety (which is a Calabi–Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface d P 3. In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables that are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the d P 3 brane tiling for these formulas in most cases. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2993-8 Issue No:Vol. 356, No. 3 (2017)

Authors:Jürg Fröhlich; Antti Knowles; Benjamin Schlein; Vedran Sohinger Pages: 883 - 980 Abstract: We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions \({d =1,2,3}\) . The many-body quantum thermal states that we consider are the grand canonical ensemble for d = 1 and an appropriate modification of the grand canonical ensemble for \({d =2,3}\) . In dimensions d = 2, 3, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2994-7 Issue No:Vol. 356, No. 3 (2017)

Authors:Terry Gannon; Scott Morrison Pages: 981 - 1015 Abstract: We compute the modular data (that is, the S and T matrices) for the centre of the extended Haagerup subfactor [BMPS12]. The full structure (i.e., the associativity data, also known as 6-j symbols or F matrices) still appears to be inaccessible. Nevertheless, starting with just the number of simple objects and their dimensions (obtained by a combinatorial argument in [MW14]) we find that it is surprisingly easy to leverage knowledge of the representation theory of \({SL (2, \mathbb{Z})}\) into a complete description of the modular data. We also investigate the possible character vectors associated with this modular data. This is the published version of arXiv:1606.07165. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-3003-x Issue No:Vol. 356, No. 3 (2017)

Authors:Matthew Buican; Andrey Gromov Pages: 1017 - 1056 Abstract: Motivated by the three-dimensional topological field theory/two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, which can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic “topological symmetries” that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry/topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2995-6 Issue No:Vol. 356, No. 3 (2017)

Authors:Dominic W. Berry; Andrew M. Childs; Aaron Ostrander; Guoming Wang Pages: 1057 - 1081 Abstract: We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-3002-y Issue No:Vol. 356, No. 3 (2017)

Authors:Dominique Malicet Pages: 1083 - 1116 Abstract: In this paper, we study random walks \({g_n=f_{n-1}\ldots f_0}\) on the group Homeo (S 1) of the homeomorphisms of the circle, where the homeomorphisms f k are chosen randomly, independently, with respect to a same probability measure \({\nu}\) . We prove that under the only condition that there is no probability measure invariant by \({\nu}\) -almost every homeomorphism, the random walk almost surely contracts small intervals. It generalizes what has been known on this subject until now, since various conditions on \({\nu}\) were imposed in order to get the phenomenon of contractions. Moreover, we obtain the surprising fact that the rate of contraction is exponential, even in the lack of assumptions of smoothness on the f k ’s. We deduce various dynamical consequences on the random walk (g n ): finiteness of ergodic stationary measures, distribution of the trajectories, asymptotic law of the evaluations, etc. The proof of the main result is based on a modification of the Ávila-Viana’s invariance principle, working for continuous cocycles on a space fibred in circles. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2996-5 Issue No:Vol. 356, No. 3 (2017)

Authors:Can Ozan Oğuz; Michael Reeks Pages: 1117 - 1154 Abstract: We show that the trace decategorification, or zeroth Hochschild homology, of the twisted Heisenberg category defined by Cautis and Sussan is isomorphic to a quotient of \({W^-}\) , a subalgebra of \({W_{1+\infty}}\) defined by Kac, Wang, and Yan. Our result is a twisted analogue of that by Cautis, Lauda, Licata, and Sussan relating \({W_{1+\infty}}\) and the trace decategorification of the Heisenberg category. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2992-9 Issue No:Vol. 356, No. 3 (2017)

Authors:Otis Chodosh; Yakov Shlapentokh-Rothman Pages: 1155 - 1250 Abstract: We construct one-parameter families of solutions to the Einstein–Klein–Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein–Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein–Klein–Gordon equations. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2998-3 Issue No:Vol. 356, No. 3 (2017)

Authors:Alexis Drouot Pages: 357 - 396 Abstract: Kinetic Brownian motion on the cosphere bundle of a Riemannian manifold \({\mathbb{M}}\) is a stochastic process that models the geodesic equation perturbed by a random white force of size \({\varepsilon}\) . When \({\mathbb{M}}\) is compact and negatively curved, we show that the L 2-spectrum of the infinitesimal generator of this process converges to the Pollicott–Ruelle resonances of \({\mathbb{M}}\) as \({\varepsilon}\) goes to 0. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2956-0 Issue No:Vol. 356, No. 2 (2017)

Authors:Estelle Basor; Pavel Bleher Pages: 397 - 425 Abstract: We obtain an asymptotic formula, as \({n\to\infty}\) , for the monomer–monomer correlation function \({K_2(n)}\) in the classical dimer model on a triangular lattice, with the horizontal and vertical weights \({w_h=w_v=1}\) and the diagonal weight \({w_d=t > 0}\) , between two monomers at vertices q and r that are n spaces apart in adjacent rows. We find that \({t_c=\frac{1}{2}}\) is a critical value of t. We prove that in the subcritical case, \({0 < t < \frac{1}{2}}\) , as \({n\to\infty, K_2(n)=K_2(\infty)\left[1-\frac{e^{-n/\xi}}{n}\,\Big(C_1+C_2(-1)^n+ \mathcal{O}(n^{-1})\Big) \right]}\) , with explicit formulae for \({K_2(\infty), \xi, C_1}\) , and \({C_2}\) . In the supercritical case, \({\frac{1}{2} < t < 1}\) , we prove that as \({n\to\infty, K_2(n)=K_2(\infty)\Bigg[1-\frac{e^{-n/\xi}}{n}\, \Big(C_1\cos(\omega n+\varphi_1)+C_2(-1)^n\cos(\omega n+\varphi_2)+ C_3+C_4(-1)^n + \mathcal{O}(n^{-1})\Big)\Bigg]}\) , with explicit formulae for \({K_2(\infty), \xi,\omega}\) , and \({C_1, C_2, C_3, C_4, \varphi_1, \varphi_2}\) . The proof is based on an extension of the Borodin–Okounkov–Case–Geronimo formula to block Toeplitz determinants and on an asymptotic analysis of the Fredholm determinants in hand. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2985-8 Issue No:Vol. 356, No. 2 (2017)

Authors:N. Martynchuk; K. Efstathiou Pages: 427 - 449 Abstract: The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií as a generalization of standard (‘integer’) monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows one to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2988-5 Issue No:Vol. 356, No. 2 (2017)

Authors:Sergey Bravyi; David Gosset Pages: 451 - 500 Abstract: We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of n fermionic modes and has a Hamiltonian \({H=H_0+H_{imp}}\) , where H 0 is quadratic in creation–annihilation operators and H imp is an arbitrary Hamiltonian acting on a subset of O(1) modes. We show that the ground energy of H can be approximated with an additive error \({2^{-b}}\) in time \({n^3 \exp{[O(b^3)]}}\) . Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of \({\exp{[O(b^3)]}}\) fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of H 0. A key ingredient of our proof is Zolotarev’s rational approximation to the \({\sqrt{x}}\) function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2976-9 Issue No:Vol. 356, No. 2 (2017)

Authors:Matthew D. Blair; Christopher D. Sogge Pages: 501 - 533 Abstract: We prove a Kakeya–Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors (Blair and Sogge in Anal PDE 8:747–764, 2015) and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper (Blair and Sogge in J Differ Geom, 2015), it will be seen that these sharpened estimates yield improved L q (M) bounds on eigenfunctions in the presence of nonpositive curvature when \({2 < q < \frac{2(d+1)}{d-1}}\) . PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2977-8 Issue No:Vol. 356, No. 2 (2017)

Authors:Nicolai Reshetikhin; Ananth Sridhar Pages: 535 - 565 Abstract: The main result of this paper is the construction of infinitely many conserved quantities (corresponding to commuting transfer-matrices) for the limit shape equation for the six vertex model on a cylinder. This suggests that the limit shape equation is an integrable PDE with gradient constraints. At the free fermionic point this equation becomes the complex Burgers equation. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2983-x Issue No:Vol. 356, No. 2 (2017)

Authors:Philip Candelas; Xenia de la Ossa; Jock McOrist Pages: 567 - 612 Abstract: Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in \({\alpha^{\backprime}}\) , in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kähler, as is required by supersymmetry. Checking the metric is Kähler is intricate and the anomaly cancellation equation for the H field plays an essential role. The Kähler potential nevertheless takes a remarkably simple form: it is the Kähler potential of special geometry with the Kähler form replaced by the \({\alpha^{\backprime}}\) -corrected hermitian form. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2978-7 Issue No:Vol. 356, No. 2 (2017)

Authors:Takuro Mochizuki; Masaki Yoshino Pages: 613 - 625 Abstract: We study singular monopoles on open subsets in the 3-dimensional Euclidean space. We give two characterizations of Dirac type singularities. One is given in terms of the growth order of the norms of sections which are invariant by the scattering map. The other is given in terms of the growth order of the norms of the Higgs fields. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2981-z Issue No:Vol. 356, No. 2 (2017)

Authors:Indranil Biswas; Viktoria Heu; Jacques Hurtubise Pages: 627 - 640 Abstract: For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this universal deformation. In a previous work we proved that the vector bundle corresponding to a general parameter of this family is stable. Here we prove that the vector bundle corresponding to a general parameter is in fact very stable, meaning it does not admit any nonzero nilpotent Higgs field. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2987-6 Issue No:Vol. 356, No. 2 (2017)

Authors:Farzad Fathizadeh; Matilde Marcolli Pages: 641 - 671 Abstract: We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson–Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes. PubDate: 2017-12-01 DOI: 10.1007/s00220-017-2991-x Issue No:Vol. 356, No. 2 (2017)