Abstract: A shaped triangulation is a finite triangulation of an oriented pseudo-three-manifold where each tetrahedron carries dihedral angles of an ideal hyperbolic tetrahedron. To each shaped triangulation, we associate a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3–2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges through the Neumann–Zagier Poisson bracket. Similarly to Turaev–Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions in a way that they enjoy a manifest tetrahedral symmetry. We conjecture that for shaped triangulations of closed three-manifolds, our partition function is twice the absolute value squared of the partition function of Techmüller TQFT defined by Andersen and Kashaev. This is similar to the known relationship between the Turaev–Viro and the Witten–Reshetikhin–Turaev invariants of three-manifolds. We also discuss interpretations of our construction in terms of three-dimensional supersymmetric field theories related to triangulated three-dimensional manifolds. PubDate: 2016-05-01

Abstract: We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in
\({\hbar}\)
, and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string. PubDate: 2016-05-01

Abstract: We prove a formula for the global gravitational anomaly of the self-dual field theory in the presence of background gauge fields, assuming the results of arXiv:1110.4639. Along the way, we also clarify various points about the self-dual field theory. In particular, we give a general definition of the theta characteristic entering its partition function and settle the issue of its possible metric dependence. We treat the cohomological version of type IIB supergravity as an example of the formalism. We show the apparent existence of a mixed gauge-gravitational global anomaly, occurring when the B-field and Ramond–Ramond two-form gauge fields have non-trivial Wilson lines, and suggest a way in which it could cancel. PubDate: 2016-05-01

Abstract: For the fundamental representations of the simple Lie algebras of type B
n
, C
n
and D
n
, we derive the braiding and fusion matrices from the generalized Yang–Yang function and prove that the corresponding knot invariants are Kauffman polynomial. PubDate: 2016-05-01

Abstract: We propose a new family of matrix models whose 1/N expansion captures the all-genus topological string on toric Calabi–Yau threefolds. These matrix models are constructed from the trace class operators appearing in the quantization of the corresponding mirror curves. The fact that they provide a non-perturbative realization of the (standard) topological string follows from a recent conjecture connecting the spectral properties of these operators, to the enumerative invariants of the underlying Calabi–Yau threefolds. We study in detail the resulting matrix models for some simple geometries, like local
\({\mathbb{P}^2}\)
and local
\({\mathbb{F}_2}\)
, and we verify that their weak ’t Hooft coupling expansion reproduces the topological string free energies near the conifold singularity. These matrix models are formally similar to those appearing in the Fermi-gas formulation of Chern–Simons matter theories, and their 1/N expansion receives non-perturbative corrections determined by the Nekrasov–Shatashvili limit of the refined topological string. PubDate: 2016-05-01

Abstract: We consider the Schrödinger equation with a Hamiltonian given by a second-order difference operator with nonconstant growing coefficients, on the half one-dimensional lattice. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of noncommutative field theory. We prove pointwise in time decay estimates with the decay rate
\({t^{-1}\log^{-2}t}\)
, which is optimal with the chosen weights and appears to be so generally. We use a novel technique involving generating functions of orthogonal polynomials to achieve this estimate. PubDate: 2016-05-01

Abstract: The van der Waals–London’s law, for a collection of atoms at large separation, states that their interaction energy is pairwise attractive and decays proportionally to one over their distance to the sixth. The first rigorous result in this direction was obtained by Lieb and Thirring (Phys Rev A 34(1):40–46, 1986), by proving an upper bound which confirms this law. Recently the van der Waals–London’s law was proven under some assumptions by Anapolitanos and Sigal (arXiv:1205.4652v2). Following the strategy of Anapolitanos and Sigal (arXiv:1205.4652v2) and reworking the approach appropriately, we prove estimates on the remainder of the interaction energy. Furthermore, using an appropriate test function, we prove an upper bound for the interaction energy, which is sharp to leading order. For the upper bound, our assumptions are weaker, the remainder estimates stronger and the proof is simpler. The upper bound, for the cases it applies, improves considerably the upper bound of Lieb and Thirring. Their bound holds in a much more general setting, however. Here we consider only spinless Fermions. PubDate: 2016-05-01

Abstract: In this note, we study a fractional Poisson–Nernst–Planck equation modeling a semiconductor device. We prove several decay estimates for the Lebesgue and Sobolev norms in one, two and three dimensions. We also provide the first term of the asymptotic expansion as
\({t\rightarrow\infty}\)
. PubDate: 2016-04-29

Abstract: This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. In the case of real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region. PubDate: 2016-04-26

Abstract: The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L1-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence. PubDate: 2016-04-26

Abstract: We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class
\({\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}\)
, we recover isomorphism classes of geometric string structures on Spin
n
-bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class
\({\frac{1}{2} p_1}\)
together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory. PubDate: 2016-04-22

Abstract: We consider a system of N bosons confined to a thin waveguide, i.e. to a region of space within an
\({\epsilon}\)
-tube around a curve in
\({\mathbb{R}^3}\)
. We show that when taking simultaneously the NLS limit
\({N \to \infty}\)
and the limit of strong confinement
\({\epsilon \to 0}\)
, the time-evolution of such a system starting in a state close to a Bose–Einstein condensate is approximately captured by a non-linear Schrödinger equation in one dimension. The strength of the non-linearity in this Gross–Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the “bending” and the “twisting” of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl (On the time-dependent Gross–Pitaevskii-and Hartree equation. arXiv:0808.1178, 2008). PubDate: 2016-04-22

Abstract: The scattering of photons and heavy classical Coulomb interacting particles, with realistic particle–photon interaction (without particle recoil) is studied adopting the Koopman formulation for the particles. The model is translation invariant and allows for a complete control of the Dollard strategy devised by Kulish–Faddeev and Rohrlich (KFR) for QED: in the adiabatic formulation, the Møller operators exist as strong limits and interpolate between the dynamics and a non-free asymptotic dynamics, which is a unitary group; the S-matrix is non-trivial and exhibits the factorization of all the infrared divergences. The implications of the KFR strategy on the open questions of the LSZ asymptotic limits in QED are derived in the field theory version of the model, with the charged particles described by second quantized fields: i) asymptotic limits of the charged fields,
\({\Psi_{{\rm out}/{\rm in}}(x)}\)
, are obtained as strong limits of modified LSZ formulas, with corrections given by a Coulomb phase operator and an exponential of the photon field; ii) free asymptotic electromagnetic fields,
\({B_{{\rm out}/{\rm in}}(x)}\)
, are given by the massless LSZ formula, as in Buchholz approach; iii) the asymptotic field algebras are a semidirect product of the canonical algebras generated by
\({B_{{\rm out}/{\rm in}}}\)
,
\({\Psi_{{\rm out}/{\rm in}}}\)
; iv) on the asymptotic spaces, the Hamiltonian is the sum of the free (commuting) Hamiltonians of
\({B_{{\rm out}/{\rm in}}}\)
,
\({\Psi_{{\rm out}/{\rm in}}}\)
and the same holds for the generators of the space translations. PubDate: 2016-04-21

Abstract: We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local
\({{\mathbb{P}}^2}\)
, local
\({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\)
and local
\({{\mathbb{F}}_1}\)
. In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background-independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry. PubDate: 2016-04-09

Abstract: We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl–Dirac operators on
\({\mathbb{R}^3}\)
. In particular, we are interested in those operators
\({\mathcal{D}_B}\)
for which the associated magnetic field
\({B}\)
is given by pulling back a two-form
\({\beta}\)
from the sphere
\({\mathbb{S}^2}\)
to
\({\mathbb{R}^3}\)
using a combination of the Hopf fibration and inverse stereographic projection. If
\({\int_{\mathbb{s}^2} \beta \neq 0}\)
, we show that
$$\sum_{0 \leq t \leq T} {\rm dim Ker} \mathcal{D}{tB}=\frac{T^2}{8\pi^2}\,\Big \int_{\mathbb{S}^2}\beta\Big \,\int_{\mathbb{S}^2} {\beta} +o(T^2)$$
as
\({T\to+\infty}\)
. The result relies on Erdős and Solovej’s characterisation of the spectrum of
\({\mathcal{D}_{tB}}\)
in terms of a family of Dirac operators on
\({\mathbb{S}^2}\)
, together with information about the strong field localisation of the Aharonov–Casher zero modes of the latter. PubDate: 2016-04-09

Abstract: The dissipative wave equation with a critical quintic non-linearity in smooth bounded three-dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown. PubDate: 2016-04-07

Abstract: We prove exponential decay of correlations for a class of
\({C^{1+\alpha}}\)
uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing. PubDate: 2016-04-07

Abstract: De Rham cohomology with spacelike compact and timelike compact supports has recently been noticed to be of importance for understanding the structure of classical and quantum Maxwell theory on curved spacetimes. Similarly, causally restricted cohomologies of different differential complexes play a similar role in other gauge theories. We introduce a method for computing these causally restricted cohomologies in terms of cohomologies with either compact or unrestricted supports. The calculation exploits the fact that the de Rham–d’Alembert wave operator can be extended to a chain map that is homotopic to zero and that its causal Green function fits into a convenient exact sequence. As a first application, we use the method on the de Rham complex, then also on the Calabi (or Killing–Riemann–Bianchi) complex, which appears in linearized gravity on constant curvature backgrounds. We also discuss applications to other complexes, as well as generalized causal structures and functoriality. PubDate: 2016-04-05

Abstract: We provide estimates on the Bartnik mass of constant mean curvature surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton’s modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton’s modified Ricci flow with prescribed scalar curvature. Such manifolds were first constructed by the first author in her dissertation conducted under the supervision of M. T. Wang. We make a further study of this class of manifolds which we denote Ham3, bounding the ADM masses of such manifolds and analyzing the rigid case when the Hawking mass of the inner surface of the manifold agrees with its ADM mass. PubDate: 2016-04-05

Abstract: We study the linear relaxation Boltzmann equation, a simple semiclassical kinetic model. We provide a resolvent estimate for an associated non-selfadjoint operator as well as an estimate on the return to equilibrium. This is done using a scaling argument and non-semiclassical hypocoercive estimate. PubDate: 2016-04-01