Abstract: Abstract We consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago we proved that, if the null geodesic generators of such a horizon were all closed curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy–Kowaleski theorem one could establish the existence of infinite dimensional, analytic families of such ‘generalized Taub-NUT’ spacetimes and show that, generically, they admitted only the single (horizon-generating) Killing field alluded to above. In this article we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon generating null geodesic densely fills a 2-torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as ‘ergodic’, the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the conjectures that every such spacetime with an ergodic horizon is trivially constructable from the flat Kasner solution by making certain ‘irrational’ toroidal compactifications and that degenerate compact Cauchy horizons do not exist in the analytic case. PubDate: 2019-09-18

Abstract: Abstract We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra that provides an additional “representation constraint”). Every Leibniz algebra gives rise to a Lie n-algebra in a canonical way (for every \(n\in \mathbb {N}\cup \{ \infty \}\) ). It is the gauging of this \(L_\infty \) -algebra that explains the tensor hierarchy of the bosonic sector of gauged supergravity theories. The tower of p-from gauge fields corresponds to Lyndon words of the universal enveloping algebra of the free Lie algebra of an odd vector space in this construction. Truncation to some n yields the reduced field content needed in a concrete spacetime dimension. PubDate: 2019-09-17

Abstract: Abstract We study the graph isomorphism game that arises in quantum information theory. We prove that the non-commutative algebraic notion of a quantum isomorphism between two graphs is same as the more physically motivated one arising from the existence of a perfect quantum strategy for graph isomorphism game. This is achieved by showing that every algebraic quantum isomorphism between a pair of (quantum) graphs X and Y arises from a certain measured bigalois extension for the quantum automorphism groups \(G_X\) and \(G_Y\) of X and Y. In particular, this implies that the quantum groups \(G_X\) and \(G_Y\) are monoidally equivalent. We also establish a converse to this result, which says that a compact quantum group G is monoidally equivalent to the quantum automorphism group \(G_X\) of a given quantum graph X if and only if G is the quantum automorphism group of a quantum graph that is algebraically quantum isomorphic to X. Using the notion of equivalence for non-local games, we apply our results to other synchronous games, including the synBCS game and certain related graph homomorphism games. PubDate: 2019-09-14

Abstract: Abstract We introduced in a previous paper a general notion of asymptotic morphism of a given local net of observables, which allows to describe the sectors of a corresponding scaling limit net. Here, as an application, we illustrate the general framework by analyzing the Schwinger model, which features confined charges. In particular, we explicitly construct asymptotic morphisms for these sectors in restriction to the subnet generated by the derivatives of the field and momentum at time zero. As a consequence, the confined charges of the Schwinger model are in principle accessible to observation. We also study the obstructions, that can be traced back to the infrared singular nature of the massless free field in \(d=2\) , to perform the same construction for the complete Schwinger model net. Finally, we exhibit asymptotic morphisms for the net generated by the massive free charged scalar field in four dimensions, where no infrared problems appear in the scaling limit. PubDate: 2019-09-14

Abstract: Abstract We introduce a space of distributional 1-forms \(\Omega ^1_\alpha \) on the torus \(\mathbf {T}^2\) for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an \(\Omega ^1_\alpha \) -valued random variable A for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang–Mills measure in the sense of Lévy (Mem Am Math Soc 166(790), 2003). It holds furthermore that \(\Omega ^1_\alpha \) embeds into the Hölder–Besov space \(\mathcal {C}^{\alpha -1}\) for all \(\alpha \in (0,1)\) , so that A has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations. PubDate: 2019-09-14

Abstract: Abstract We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential. PubDate: 2019-09-13

Abstract: Abstract We show that every (graded) derivation on the algebra of free quantum fields and their Wick powers in curved spacetimes gives rise to a set of anomalous Ward identities for time-ordered products, with an explicit formula for their classical limit. We study these identities for the Koszul–Tate and the full BRST differential in the BV–BRST formulation of perturbatively interacting quantum gauge theories, and clarify the relation to previous results. In particular, we show that the quantum BRST differential, the quantum antibracket and the higher-order anomalies form an \(L_\infty \) algebra. The defining relations of this algebra ensure that the gauge structure is well-defined on cohomology classes of the quantum BRST operator, i.e., observables. Furthermore, we show that one can determine contact terms such that also the interacting time-ordered products of multiple interacting fields are well defined on cohomology classes. An important technical improvement over previous treatments is the fact that all our relations hold off-shell and are independent of the concrete form of the Lagrangian, including the case of open gauge algebras. PubDate: 2019-09-10

Abstract: Abstract This paper, the third in a series, completes our description of all (radial) solutions on \({\mathbb {C}}^*\) of the tt*-Toda equations \(2(w_i)_{ {t{\bar{t}}} }=-e^{2(w_{i+1}-w_{i})} + e^{2(w_{i}-w_{i-1})}\) , using a combination of methods from p.d.e., isomonodromic deformations (Riemann–Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth near 0. For such solutions, we compute explicitly the Stokes data and connection matrix of the associated meromorphic system, in the resonant cases as well as the non-resonant case. This allows us to give a complete picture of the monodromy data, holomorphic data, and asymptotic data of the global solutions. PubDate: 2019-09-10

Abstract: Abstract We exclude Type I blow-up, which occurs in the form of atomic concentrations of the \(L^2\) norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale. PubDate: 2019-09-10

Abstract: Abstract We prove asymptotic stability of the Couette flow for the 2D Euler equations in the domain \(\mathbb {T}\times [0,1]\) . More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey space) of the Couette flow, then the velocity field converges strongly to a nearby shear flow. Our solutions are defined on the compact set \(\mathbb {T}\times [0,1]\) (“the channel”) and therefore have finite energy. The vorticity perturbation, which is initially assumed to be supported in the interior of the channel, will remain supported in the interior of the channel at all times, will be driven to higher frequencies by the linear flow, and will converge weakly to another shear flow as \(t\rightarrow \infty \) . PubDate: 2019-09-07

Abstract: Abstract This paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic Lorentzian manifolds. Functorial constructions that map between these two types of theories in both directions are developed under certain natural hypotheses, including suitable variants of the local constancy and descent axioms. The main result is an equivalence theorem between (Cauchy constant and additive) algebraic quantum field theories and (Cauchy constant, additive and time-orderable) prefactorization algebras. A concept of \(*\) -involution for the latter class of prefactorization algebras is introduced via transfer. This involves Cauchy constancy explicitly and does not extend to generic (time-orderable) prefactorization algebras. PubDate: 2019-09-05

Abstract: Abstract We study the Stokes phenomenon of the generalized Knizhnik–Zamolodchikov (gKZ) equations, and prove that their Stokes matrices satisfy the Yang–Baxter equations. In particular, the monodromy of the gKZ equations defines a family of braid groups representations via the Stokes matrices. PubDate: 2019-09-04

Abstract: Abstract Let (M, g) be an \((n+1)\) -dimensional asymptotically locally hyperbolic manifold with a conformal compactification whose conformal infinity is \((\partial M,[\gamma ])\) . We will first observe that \({\mathcal Ch}(M,g)\le n\) , where \({\mathcal Ch}(M,g)\) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded from below by \(-n\) and its scalar curvature approaches \(-n(n+1)\) fast enough at infinity, then \({\mathcal Ch}(M,g)= n\) if and only \({\mathcal Y}(\partial M,[\gamma ])\ge 0\) , where \({\mathcal Y}(\partial M,[\gamma ])\) denotes the Yamabe invariant of the conformal infinity. This gives an answer to a question raised by Lee (Commun. Anal. Geom. 2:253–271, 1995). PubDate: 2019-09-04

Abstract: Abstract The q-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra \(\mathfrak {osp}_q(1\vert 2)\) . It turned out to be isomorphic to the Askey-Wilson algebra. In the present paper these results will be extended to higher rank. The rank \(n-2\) q-Bannai-Ito algebra \(\mathcal {A}_n^q\) , which by the established isomorphism also yields a higher rank version of the Askey-Wilson algebra, is constructed in the n-fold tensor product of \(\mathfrak {osp}_q(1\vert 2)\) . An explicit realization in terms of q-shift operators and reflections is proposed, which will be called the \(\mathbb {Z}_2^n\) q-Dirac–Dunkl model. The algebra \(\mathcal {A}_n^q\) is shown to arise as the symmetry algebra of the constructed \(\mathbb {Z}_2^n\) q-Dirac–Dunkl operator and to act irreducibly on modules of its polynomial null-solutions. An explicit basis for these modules is obtained using a q-deformed \(\mathbf {CK}\) -extension and Fischer decomposition. PubDate: 2019-09-04

Abstract: Abstract This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schrödinger operator involving an Aharonov–Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller–Lieb–Thirring inequality. We prove that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field. The method relies on the study of the magnetic kinetic energy of the wave function and amounts to study the symmetry properties of the optimal functions in a magnetic Hardy–Sobolev interpolation inequality. We give a quantified range of symmetry by a non-perturbative method. To establish the symmetry breaking range, we exploit the coupling of the phase and of the modulus and also obtain a quantitative result. PubDate: 2019-09-03

Abstract: Abstract The generalized double semion (GDS) model, introduced by Freedman and Hastings, is a lattice system similar to the toric code, with a gapped Hamiltonian whose definition depends on a triangulation of the ambient manifold M, but whose space of ground states does not depend on the triangulation, but only on the underlying manifold. In this paper, we use topological quantum field theory (TQFT) to investigate the low-energy limit of the GDS model. We define and study a functorial TQFT \(Z_{\mathrm {GDS}}\) in every dimension n such that for every closed \((n-1)\) -manifold M, \(Z_{\mathrm {GDS}}(M)\) is isomorphic to the space of ground states of the GDS model on M; the isomorphism can be chosen to intertwine the actions of the mapping class group of M that arise on both sides. Throughout this paper, we compare our constructions and results with their known analogues for the toric code. PubDate: 2019-09-03

Abstract: Abstract Based on an identity of Jacobi, we prove a simple formula that computes the pushforward of analytic functions of the exceptional divisor of a blowup of a projective variety along a smooth complete intersection with normal crossing. We use this pushforward formula to derive generating functions for Euler characteristics of crepant resolutions of singular Weierstrass models given by Tate’s algorithm. Since the Euler characteristic depends only on the sequence of blowups and not on the Kodaira fiber itself, several distinct Tate models have the same Euler characteristic. In the case of elliptic Calabi–Yau threefolds, using the Shioda–Tate–Wazir theorem, we also compute the Hodge numbers. For elliptically fibered Calabi–Yau fourfolds, our results also prove a conjecture of Blumenhagen, Grimm, Jurke, and Weigand based on F-theory/heterotic string duality. PubDate: 2019-09-03

Abstract: Abstract We construct a Borcherds–Kac–Moody (BKM) superalgebra on which the Conway group \(\hbox {Co}_0\) acts faithfully. We show that the BKM algebra is generated by the physical states (BRST cohomology classes) in a chiral superstring theory. We use this construction to produce denominator identities for the chiral partition functions of the Conway module \(V^{s\natural }\) , a supersymmetric \(c=12\) chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to \(\hbox {Co}_0\) . In particular, these functions satisfy a genus zero property analogous to that of monstrous moonshine. Finally, we suggest how one may promote the denominators to spacetime BPS indices in type II string theory, which might thus furnish a physical explanation of the genus zero property of Conway moonshine. PubDate: 2019-09-01

Abstract: Abstract A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. We define and prove magnetic criticality of a large family of Ising models on planar graphs whose dual is a circle pattern. Our construction includes as a special case the critical isoradial Ising models of Baxter. PubDate: 2019-08-08