Authors:Qian Wang Pages: 827 - 850 Abstract: We derive a first order representation formula for the tensorial wave equation \({\Box_\mathbf{g} \phi^I=F^I}\) in globally hyperbolic Lorentzian spacetimes \({(\mathcal{M}^{2+1}, \mathbf{g})}\) by giving a geometric formulation of the method of descent which is applicable for any dimension. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3151-7 Issue No:Vol. 360, No. 3 (2018)

Authors:Alberto De Sole; Victor G. Kac; Daniele Valeri Pages: 851 - 918 Abstract: We prove that any classical affine W-algebra \({\mathcal{W}}\) ( \({\mathfrak{g}, f)}\) , where \({\mathfrak{g}}\) is a classical Lie algebra and f is an arbitrary nilpotent element of \({\mathfrak{g}}\) , carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3142-8 Issue No:Vol. 360, No. 3 (2018)

Authors:Shamit Kachru; Arnav Tripathy Pages: 919 - 933 Abstract: We show that BPS jumping loci–loci in the moduli space of string compactifications where the number of BPS states jumps in an upper semi-continuous manner—naturally appear as Fourier coefficients of (vector space-valued) automorphic forms. For the case of T2 compactification, the jumping loci are governed by a modular form studied by Hirzebruch and Zagier, while the jumping loci in K3 compactification appear in a story developed by Oda and Kudla–Millson in arithmetic geometry. We also comment on some curious related automorphy in the physics of black hole attractors and flux vacua. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3090-3 Issue No:Vol. 360, No. 3 (2018)

Authors:Razvan Gurau Pages: 985 - 1007 Abstract: It is well known that tensor models for a tensor with no symmetry admit a 1/N expansion dominated by melonic graphs. This result relies crucially on identifying jackets, which are globally defined ribbon graphs embedded in the tensor graph. In contrast, no result of this kind has so far been established for symmetric tensors because global jackets do not exist. In this paper we introduce a new approach to the 1/N expansion in tensor models adapted to symmetric tensors. In particular we do not use any global structure like the jackets. We prove that, for any rank D, a tensor model with two symmetric tensors and interactions the complete graph K D+1 admits a 1/N expansion dominated by melonic graphs. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3055-y Issue No:Vol. 360, No. 3 (2018)

Authors:Michael Brannan; David Kerr Pages: 1043 - 1059 Abstract: For second countable discrete quantum groups, and more generally second countable locally compact quantum groups with trivial scaling group, we show that property (T) is equivalent to every weakly mixing unitary representation not having almost invariant vectors. This is a generalization of a theorem of Bekka and Valette from the group setting and was previously established in the case of low dual by Daws, Skalski, and Viselter. Our approach uses spectral techniques and is completely different from those of Bekka–Valette and Daws–Skalski–Viselter. By a separate argument we furthermore extend the result to second countable nonunimodular locally compact quantum groups, which are shown in particular not to have property (T), generalizing a theorem of Fima from the discrete setting. We also obtain quantum group versions of characterizations of property (T) of Kerr and Pichot in terms of the Baire category theory of weak mixing representations and of Connes and Weiss in terms of the prevalence of strongly ergodic actions. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3037-0 Issue No:Vol. 360, No. 3 (2018)

Authors:Louis Funar; Pierre Lochak Pages: 1061 - 1082 Abstract: Using quantum representations of mapping class groups, we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3126-8 Issue No:Vol. 360, No. 3 (2018)

Authors:Jason Miller Pages: 1083 - 1119 Abstract: Suppose that h is a Gaussian free field (GFF) on a planar domain. Fix \({\kappa \in (0,4)}\) . The \({{\rm SLE}_\kappa}\) light cone L \({(\theta)}\) of h with opening angle \({\theta \in [0,\pi]}\) is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field \({e^{ih/\chi}}\) , \({\chi = \tfrac{2}{\sqrt{\kappa}} - \tfrac{\sqrt{\kappa}}{2}}\) , with angles in \({[-\tfrac{\theta}{2},\tfrac{\theta}{2}]}\) . We derive the Hausdorff dimension of L \({(\theta)}\) . If \({\theta =0}\) then L \({(\theta)}\) is an ordinary \({{\rm SLE}_{\kappa}}\) curve (with \({\kappa < 4}\) ); if \({\theta = \pi}\) then L \({(\theta)}\) is the range of an \({{\rm SLE}_{\kappa'}}\) curve ( \({\kappa' = 16/\kappa > 4}\) ). In these extremes, this leads to a new proof of the Hausdorff dimension formula for \({{\rm SLE}}\) . We also consider \({{\rm SLE}_\kappa(\rho)}\) processes, which were originally only defined for \({\rho > -\,2}\) , but which can also be defined for \({\rho \leq -2}\) using Lévy compensation. The range of an \({{\rm SLE}_\kappa(\rho)}\) is qualitatively different when \({\rho \leq -2}\) . In particular, these curves are self-intersecting for \({\kappa < 4}\) and double points are dense, while ordinary \({{\rm SLE}_\kappa}\) is simple. It was previously shown (Miller and Sheffield in Gaussian free field light cones and \({{\rm SLE}_\kappa(\rho)}\) , 2016) that certain \({{\rm SLE}_\kappa(\rho)}\) curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of \({{\rm SLE}_\kappa(\rho)}\) ... PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3109-9 Issue No:Vol. 360, No. 3 (2018)

Authors:D. Dragičević; G. Froyland; C. González-Tokman; S. Vaienti Pages: 1121 - 1187 Abstract: We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form \({T_{\sigma^{n-1} \omega} \circ\cdots\circ T_{\sigma\omega}\circ T_\omega}\) . An important aspect of our results is that we only assume ergodicity and invertibility of the random driving \({\sigma:\Omega\to\Omega}\) ; in particular no expansivity or mixing properties are required. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3083-7 Issue No:Vol. 360, No. 3 (2018)

Authors:Iris Cong; Meng Cheng; Zhenghan Wang Pages: 1189 - 1190 Abstract: There were two errors in the original publication. First, the term B K in Eq. (2.20) was not well-defined in the case of non-normal subgroups K. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3134-8 Issue No:Vol. 360, No. 3 (2018)

Authors:Jian Ding; Li Li Pages: 523 - 553 Abstract: We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory, and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3140-x Issue No:Vol. 360, No. 2 (2018)

Authors:Matti Lassas; Gunther Uhlmann; Yiran Wang Pages: 555 - 609 Abstract: We consider inverse problems in space–time (M, g), a 4-dimensional Lorentzian manifold. For semilinear wave equations \({\square_g u + H(x, u) = f}\) , where \({\square_g}\) denotes the usual Laplace–Beltrami operator, we prove that the source-to-solution map \({L: f \rightarrow u _V}\) , where V is a neighborhood of a time-like geodesic \({\mu}\) , determines the topological, differentiable structure and the conformal class of the metric of the space–time in the maximal set, where waves can propagate from \({\mu}\) and return back. Moreover, on a given space–time (M, g), the source-to-solution map determines some coefficients of the Taylor expansion of H in u. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3135-7 Issue No:Vol. 360, No. 2 (2018)

Authors:Gregory J. Galloway; Eric Ling Pages: 611 - 617 Abstract: We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant. More specifically, for 3 + 1 dimensional spacetimes which satisfy the null energy condition and contain a future expanding compact Cauchy surface, we establish a precise connection between the topology of the Cauchy surfaces and the occurrence of past singularities. In addition to the Penrose singularity theorem, the proof makes use of some recent advances in the topology of 3-manifolds and of certain fundamental existence results for minimal surfaces. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3020-9 Issue No:Vol. 360, No. 2 (2018)

Authors:Yaiza Canzani; Jeffrey Galkowski; John A. Toth Pages: 619 - 637 Abstract: Let (M, g) be a compact, smooth, Riemannian manifold and \({\{ \phi_h \}}\) an L2-normalized sequence of Laplace eigenfunctions with defect measure \({\mu}\) . Let H be a smooth hypersurface with unit exterior normal \(\nu\) . Our main result says that when \(\mu\) is not concentrated conormally to H, the eigenfunction restrictions to H satisfy $$\int_H \phi_h d\sigma_H = o(1) \quad {\rm and} \quad \int_H h D_{\nu} \phi_h d\sigma_H = o(1),$$ \({h \to 0^+}\) . PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3081-9 Issue No:Vol. 360, No. 2 (2018)

Authors:Giacomo De Palma; Dario Trevisan Pages: 639 - 662 Abstract: We prove the conditional Entropy Power Inequality for Gaussian quantum systems. This fundamental inequality determines the minimum quantum conditional von Neumann entropy of the output of the beam-splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies. We also prove that, for any couple of values of the quantum conditional entropies of the two inputs, the minimum of the quantum conditional entropy of the output given by the conditional Entropy Power Inequality is asymptotically achieved by a suitable sequence of quantum Gaussian input states. Our proof of the conditional Entropy Power Inequality is based on a new Stam inequality for the quantum conditional Fisher information and on the determination of the universal asymptotic behaviour of the quantum conditional entropy under the heat semigroup evolution. The beam-splitter and the squeezing are the central elements of quantum optics, and can model the attenuation, the amplification and the noise of electromagnetic signals. This conditional Entropy Power Inequality will have a strong impact in quantum information and quantum cryptography. Among its many possible applications there is the proof of a new uncertainty relation for the conditional Wehrl entropy. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3082-8 Issue No:Vol. 360, No. 2 (2018)

Authors:Manuel Bärenz; John Barrett Pages: 663 - 714 Abstract: A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. A state sum formula for the invariant is constructed via the chain-mail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the Crane-Yetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the Crane-Yetter invariant for nonmodular categories is stronger than signature and Euler invariant. A special case is the four-dimensional untwisted Dijkgraaf–Witten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in Walker-Wang models. Relations to different approaches to quantum gravity such as Cartan geometry and teleparallel gravity are also discussed. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3012-9 Issue No:Vol. 360, No. 2 (2018)

Authors:N. J. B. Aza; J.-B. Bru; W. de Siqueira Pedra Pages: 715 - 726 Abstract: We supplement the determinantal and Pfaffian bounds of Sims and Warzel (Commun Math Phys 347:903–931, 2016) for many-body localization of quasi-free fermions, by considering the high dimensional case and complex-time correlations. Our proof uses the analyticity of correlation functions via the Hadamard three-line theorem. We show that the dynamical localization for the one-particle system yields the dynamical localization for the many-point fermionic correlation functions, with respect to the Hausdorff distance in the determinantal case. In Sims and Warzel (2016), a stronger notion of decay for many-particle configurations was used but only at dimension one and for real times. Considering determinantal and Pfaffian correlation functionals for complex times is important in the study of weakly interacting fermions. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3121-0 Issue No:Vol. 360, No. 2 (2018)

Authors:Xenia de la Ossa; Magdalena Larfors; Eirik E. Svanes Pages: 727 - 775 Abstract: Heterotic string compactifications on integrable G 2 structure manifolds Y with instanton bundles \({(V,A), (TY,\tilde{\theta})}\) yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative \({\mathcal{D}}\) and show that it is equivalent to a heterotic G 2 system encoding the geometry of the heterotic string compactifications. This operator \({\mathcal{D}}\) acts on a bundle \({\mathcal{Q}=T^*Y \oplus {\rm End}(V) \oplus {\rm End}(TY)}\) and satisfies a nilpotency condition \({\check{{\mathcal{D}}}^2=0}\) , for an appropriate projection of \({\mathcal D}\) . Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group \({\check H^1_{\check{{\mathcal{D}}}}(\mathcal{Q})}\) . We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the \({\alpha'}\) expansion. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3013-8 Issue No:Vol. 360, No. 2 (2018)

Authors:A. Alexandrov; G. Chapuy; B. Eynard; J. Harnad Pages: 777 - 826 Abstract: A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2D Toda \({\tau}\) -functions of hypergeometric type, which serve as generating functions for weighted single and double Hurwitz numbers; the Baker function, which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements; the multipair correlators and the multicurrent correlators. Multiplicative recursion relations and a linear differential system are deduced for the adapted bases and their duals, and a Christoffel–Darboux type formula is derived for the pair correlator. The quantum and classical spectral curves linking this theory with the topological recursion program are derived, as well as the generalized cut-and-join equations. The results are detailed for four special cases: the simple single and double Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of coverings, the strongly monotone case, corresponding to Belyi curves and the simplest version of quantum weighted Hurwitz numbers. PubDate: 2018-06-01 DOI: 10.1007/s00220-017-3065-9 Issue No:Vol. 360, No. 2 (2018)

Authors:Jue Liu; Junbin Li Abstract: Published in 1999, Christodoulou proved that the naked singularities of a self-gravitating scalar field are not stable in spherical symmetry and therefore the cosmic censorship conjecture is true in this context. The original proof is by contradiction and sharp estimates are obtained strictly depending on spherical symmetry. In this paper, appropriate a priori estimates for the solution are obtained. These estimates are more relaxed but sufficient for giving another robust argument in proving the instability, in particular not by contradiction. In a companion paper, we are able to prove certain instability theorems of the spherically symmetric naked singularities of a scalar field under gravitational perturbations without symmetries. The argument given in this paper plays a central role. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3157-1

Authors:Wei Qian; Wendelin Werner Abstract: We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero-level lines (these signs are alternating for the zero-boundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE3 with its own symmetric image, one obtains the CLE4-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain. PubDate: 2018-06-01 DOI: 10.1007/s00220-018-3159-z