Authors:Nathanaël Berestycki; Benoît Laslier; Gourab Ray Pages: 427 - 462 Abstract: Abstract In this paper we consider random planar maps weighted by the self-dual Fortuin–Kasteleyn model with parameter \({q \in (0,4)}\) . Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the annealed critical exponent associated with the length of cluster interfaces, which is shown to be $$\frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8},$$ where \({\kappa' }\) is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop, which is shown to be 1 for all values of \({q \in (0,4)}\) . Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2933-7 Issue No:Vol. 355, No. 2 (2017)

Authors:James E. Tener Pages: 463 - 518 Abstract: Abstract In this article, we provide a detailed construction and analysis of the mathematical conformal field theory of the free fermion, defined in the sense of Graeme Segal. We verify directly that the operators assigned to disks with two disks removed correspond to vertex operators, and use this to deduce analytic properties of the vertex operators. One of the main tools used in the construction is the Cauchy transform for Riemann surfaces, for which we establish several properties analogous to those of the classical Cauchy transform in the complex plane. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2959-x Issue No:Vol. 355, No. 2 (2017)

Authors:Guillaume Dauphinais; David Poulin Pages: 519 - 560 Abstract: Abstract While topological quantum computation is intrinsically fault-tolerant at zero temperature, it loses its topological protection at any finite temperature. We present a scheme to protect the information stored in a system supporting non-cyclic anyons against thermal and measurement errors. The correction procedure builds on the work of Gács (J Comput Syst Sci 32:15–78, 1986. doi:10.1145/800061.808730) and Harrington (Analysis of quantum error-correcting codes: symplectic lattice codes and toric code, 2004) and operates as a local cellular automaton. In contrast to previously studied schemes, our scheme is valid for both abelian and non-abelian anyons and accounts for measurement errors. We analytically prove the existence of a fault-tolerant threshold for a certain class of non-Abelian anyon models, and numerically simulate the procedure for the specific example of Ising anyons. The result of our simulations are consistent with a threshold between \({10^{-4}}\) and \({10^{-3}}\) . PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2923-9 Issue No:Vol. 355, No. 2 (2017)

Authors:Varghese Mathai; Guo Chuan Thiang Pages: 561 - 602 Abstract: Abstract The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the connections between Weyl points. Dually, a topological semimetal can be represented by Euler chains from which its surface Fermi arc connectivity can be deduced. These dual pictures, and the link to topological invariants of insulators, are organised using geometric exact sequences. We go beyond Dirac-type Hamiltonians and introduce new classes of semimetals whose local charges are subtle Atiyah–Dupont–Thomas invariants globally constrained by the Kervaire semicharacteristic, leading to the prediction of torsion Fermi arcs. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2965-z Issue No:Vol. 355, No. 2 (2017)

Authors:Takeo Kojima Pages: 603 - 644 Abstract: Abstract A bosonization of the quantum affine superalgebra \({U_q(\widehat{sl}(M N))}\) is presented for an arbitrary level \({k \in {\bf C}}\) . Screening operators that commute with \({U_q(\widehat{sl}(M N))}\) are presented for the level \({k \neq -M+N}\) . PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2957-z Issue No:Vol. 355, No. 2 (2017)

Authors:Iris Cong; Meng Cheng; Zhenghan Wang Pages: 645 - 689 Abstract: Abstract We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev’s quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2960-4 Issue No:Vol. 355, No. 2 (2017)

Authors:Radu A. Ionaş Pages: 691 - 740 Abstract: Abstract We describe a quaternionic-based Ansatz generalizing the Gibbons–Hawking Ansatz to a class of hyperkähler metrics with hidden symmetries. We then apply it to obtain explicit expressions for gravitational instanton metrics of type D k . PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2852-7 Issue No:Vol. 355, No. 2 (2017)

Authors:Chris M. Ormerod; Eric M. Rains Pages: 741 - 766 Abstract: Abstract We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at \({4m+12}\) points for \({m \geq 1}\) , which appear in pairs due to a symmetry condition. We parameterize this linear system in terms of a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which m = 1 with the elliptic Painlevé equation; hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2934-6 Issue No:Vol. 355, No. 2 (2017)

Authors:Andrew Granville; Igor Wigman Pages: 767 - 802 Abstract: Abstract We study the small scale distribution of the L 2-mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300, 2017) showed the existence of a density one subsequence whose L 2-mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the L 2-mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2953-3 Issue No:Vol. 355, No. 2 (2017)

Authors:Konstantin Khanin; Ke Zhang Pages: 803 - 837 Abstract: Abstract We show that for a large class of randomly kicked Hamilton–Jacobi equations, the unique global minimizer is almost surely hyperbolic. Furthermore, we prove that the unique forward and backward viscosity solutions, though in general only Lipshitz, are smooth in a neighborhood of the global minimizer. Related results in the one-dimensional case were obtained by E, Khanin et al. (Ann Math (2) 151(3):877–960, 2000). However, the methods in the above paper are purely one-dimensional and cannot be extended to the case of higher dimensions. Here we develop a completely different approach. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2919-5 Issue No:Vol. 355, No. 2 (2017)

Authors:Sergei Gukov; Du Pei Pages: 1 - 50 Abstract: Abstract We study complex Chern–Simons theory on a Seifert manifold M 3 by embedding it into string theory. We show that complex Chern–Simons theory on M 3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern–Simons theory on \({\Sigma\times S^1}\) and (4) index of a spin c Dirac operator on the moduli space of flat connections to a new set of relations between (1) the “equivariant Verlinde algebra” for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern–Simons theory on \({\Sigma \times S^1}\) and (4) the equivariant index of a spin c Dirac operator on the moduli space of Higgs bundles. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2931-9 Issue No:Vol. 355, No. 1 (2017)

Authors:Alexandre Eremenko; Andrei Gabrielov Pages: 51 - 95 Abstract: Abstract We study real solutions of a class of Painlevé VI equations. To each such solution we associate a geometric object, a one-parametric family of circular pentagons. We describe an algorithm that permits to compute the numbers of zeros, poles, 1-points and fixed points of the solution on the interval \({(1,+\infty)}\) and their mutual position. The monodromy of the associated linear equation and parameters of the Painlevé VI equation are easily recovered from the family of pentagons. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2921-y Issue No:Vol. 355, No. 1 (2017)

Authors:Kwokwai Chan; Naichung Conan Leung; Qin Li Pages: 97 - 144 Abstract: Abstract We investigate the perturbative aspects of Rozansky–Witten’s 3d \({\sigma}\) -model (Rozansky and Witten in Sel Math 3(3):401–458, 1997) using Costello’s approach to the Batalin–Vilkovisky (BV) formalism (Costello in Renormalization and effective field theory, American Mathematical Society, Providence, 2011). We show that the BV quantization (in Costello’s sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of regularization due to Kontsevich (First European congress of mathematics, Paris, 1992) and Axelrod–Singer (J Differ Geom 39(1):173–213, 1994). We also study the factorization algebra structure of quantum observables following Costello–Gwilliam (Factorization algebras in quantum field theory, Cambridge University Press, Cambridge 2017). In particular, we show that the cohomology of local quantum observables on a genus g handle body is given by \({H^*(X, (\wedge^*T_X)^{\otimes g})}\) (where X is the target manifold), and we prove that the partition function reproduces the Rozansky–Witten invariants. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2924-8 Issue No:Vol. 355, No. 1 (2017)

Authors:Tarek Elgindi; Wenqing Hu; Vladimír Šverák Pages: 145 - 159 Abstract: Abstract We consider various questions about the 2d incompressible Navier–Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2877-y Issue No:Vol. 355, No. 1 (2017)

Authors:Lydia Bieri; Shuang Miao; Sohrab Shahshahani; Sijue Wu Pages: 161 - 243 Abstract: Abstract We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler–Poisson system in moving bounded simply-connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data that are small perturbations of size \({\epsilon}\) of these static states, measured in appropriate Sobolev spaces, the solution exists and the perturbation remains of size \({\epsilon}\) on a time interval of length at least \({c\epsilon^{-2},}\) where c is a constant independent of \({\epsilon.}\) This should be compared with the lifespan \({O(\epsilon^{-1})}\) provided by local well-posedness. The key ingredient of our proof is finding a two-step nonlinear transformation which removes quadratic terms from the nonlinearity. Compared with the gravity water wave problem, besides the different geometry of the bounded moving domain, an important difference is that the gravity in water waves is a constant vector, while the self-gravity in the Euler–Poisson system depends nonlinearly on the interface. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2884-z Issue No:Vol. 355, No. 1 (2017)

Authors:Shamit Kachru; Arnav Tripathy Pages: 245 - 259 Abstract: Abstract We perform a refined count of BPS states in the compactification of M-theory on \({K3 \times T^2}\) , keeping track of the information provided by both the \({SU(2)_L}\) and \({SU(2)_R}\) angular momenta in the SO(4) little group. Mathematically, this four variable counting function may be expressed via the motivic Donaldson–Thomas counts of \({K3 \times T^2}\) , simultaneously refining Katz, Klemm, and Pandharipande’s motivic stable pairs counts on K3 and Oberdieck–Pandharipande’s Gromov–Witten counts on \({K3 \times T^2}\) . This provides the first full answer for motivic curve counts of a compact Calabi–Yau threefold. Along the way, we develop a Hodge-elliptic genus for Calabi–Yau manifolds—a new counting function for BPS states that interpolates between the Hodge polynomial and the elliptic genus of a Calabi–Yau. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2910-1 Issue No:Vol. 355, No. 1 (2017)

Authors:Virginia Agostiniani; Lorenzo Mazzieri Pages: 261 - 301 Abstract: Abstract In this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity. In the case where the static potential is bounded, we introduce new quantities which are proven to be monotone along the level set flow of the potential function. We then show how to use these properties to detect the rotational symmetry of the static solutions, deriving a number of sharp inequalities. Among these, we prove the validity—without any dimensional restriction—of the Riemannian Penrose Inequality, as well as of a reversed version of it, in the class of asymptotically flat static metrics with connected horizon. As a consequence of our analysis, a simple proof of the classical 3-dimensional Black Hole Uniqueness Theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2922-x Issue No:Vol. 355, No. 1 (2017)

Authors:Seung-Yeop Lee; Meng Yang Pages: 303 - 338 Abstract: Abstract We consider the orthogonal polynomials, \({\{P_n(z)\}_{n=0,1,\ldots}}\) , with respect to the measure $$ z-a ^{2c} e^{-N z ^2}dA(z)$$ supported over the whole complex plane, where \({a > 0}\) , \({N > 0}\) and \({c > -1}\) . We look at the scaling limit where n and N tend to infinity while keeping their ratio, n/N, fixed. The support of the limiting zero distribution is given in terms of certain “limiting potential-theoretic skeleton” of the unit disk. We show that, as we vary c, both the skeleton and the asymptotic distribution of the zeros behave discontinuously at c = 0. The smooth interpolation of the discontinuity is obtained by the further scaling of \({c=e^{-\eta N}}\) in terms of the parameter \({\eta\in[0,\infty).}\) PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2888-8 Issue No:Vol. 355, No. 1 (2017)

Authors:Tomoyuki Arakawa; Thomas Creutzig; Kazuya Kawasetsu; Andrew R. Linshaw Pages: 339 - 372 Abstract: Abstract Let \({\mathfrak{g}}\) be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of \({\mathfrak{s}\mathfrak{l}_2}\) inducing the minimal gradation on \({\mathfrak{g}}\) . The corresponding minimal \({\mathcal{W}}\) -algebra \({\mathcal{W}^k(\mathfrak{g}, e_{-\theta})}\) introduced by Kac and Wakimoto has strong generators in weights \({1,2,3/2}\) , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra \({V^{k'}(\mathfrak{g}^{\natural})}\) , where \({\mathfrak{g}^{\natural} \subset \mathfrak{g}}\) denotes the centralizer of \({\mathfrak{s}\mathfrak{l}_2}\) . Therefore, \({\mathcal{W}^k(\mathfrak{g}, e_{-\theta})}\) has an action of a connected Lie group \({G^{\natural}_0}\) with Lie algebra \({\mathfrak{g}^{\natural}_0}\) , where \({\mathfrak{g}^{\natural}_0}\) denotes the even part of \({\mathfrak{g}^{\natural}}\) . We show that for any reductive subgroup \({G \subset G^{\natural}_0}\) , and for any reductive Lie algebra \({\mathfrak{g}' \subset \mathfrak{g}^{\natural}}\) , the orbifold \({\mathcal{O}^k = \mathcal{W}^k(\mathfrak{g}, e_{-\theta})^{G}}\) and the coset \({\mathcal{C}^k = {\rm Com}(V(\mathfrak{g}'),\mathcal{W}^k(\mathfrak{g}, e_{-\theta}))}\) are strongly finitely generated for generic values of \({k}\) . Here \({V(\mathfrak{g}')}\) denotes the affine vertex algebra associated to \({\mathfrak{g}'}\) . We find explicit minimal strong generating sets for \({\mathcal{C}^k}\) when \({\mathfrak{g}' = \mathfrak{g}^{\natural}}\) and \({\mathfrak{g}}\) is either \({\mathfrak{s}\mathfrak{l}_n}\) , \({\mathfrak{s}\mathfrak{p}_{2n}}\) , \({\mathfrak{s}\mathfrak{l}(2 n)}\) for \({n \neq 2}\) , \({... PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2901-2 Issue No:Vol. 355, No. 1 (2017)

Authors:Milán Mosonyi; Tomohiro Ogawa Pages: 373 - 426 Abstract: Abstract We determine the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity. Our form of the exponent is an exact analogue of Arimoto’s, given as a transform of the Rényi capacities with parameters \({\alpha > 1}\) . It is important to note that, unlike in the classical case, there are many inequivalent ways to define the Rényi divergence of states, and hence the Rényi capacities of channels. Our exponent is in terms of the Rényi capacities corresponding to a version of the Rényi divergences that has been introduced recently in Müller-Lennert et al. (J Math Phys 54(12):122203, 2013. arXiv:1306.3142), and Wilde et al. (Commun Math Phys 331(2):593–622, 2014. arXiv:1306.1586). Our result adds to the growing body of evidence that this new version is the natural definition for the purposes of strong converse problems. PubDate: 2017-10-01 DOI: 10.1007/s00220-017-2928-4 Issue No:Vol. 355, No. 1 (2017)