A construction of Frobenius manifolds with logarithmic poles and applications

A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a generalization of the reconstruction theorem of Kontsevich and Manin for projective smooth varieties with convergent Gromov-Witten potential. A second application is a construction of Frobenius manifolds out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled.Comment: 46 page

A canonical Frobenius structure

We show that it makes sense to speak of THE Frobenius manifold attached to a convenient and nondegenerate Laurent polynomialComment: 24 page

tt*-geometry on the big phase space

The big phase space, the geometric setting for the study of quantum cohomology with gravitational descendents, is a complex manifold and consists of an infinite number of copies of the small phase space. The aim of this paper is to define a Hermitian geometry on the big phase space. Using the approach of Dijkgraaf and Witten, we lift various geometric structures of the small phase space to the big phase space. The main results of our paper state that various notions from tt*-geometry are preserved under such liftings

Interval total colorings of graphs

A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An \emph{interval total $t$-coloring} of a graph $G$ is a total coloring of $G$ with colors $1,2,\...,t$ such that at least one vertex or edge of $G$ is colored by $i$, $i=1,2,\...,t$, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this paper we investigate some properties of interval total colorings. We also determine exact values of the least and the greatest possible number of colors in such colorings for some classes of graphs.Comment: 23 pages, 1 figur

RCS043938-2904.9: A New Rich Cluster of Galaxies at z=0.951

We present deep I, J_s, K_s imaging and optical spectroscopy of the newly discovered Red-Sequence Cluster Survey cluster RCS043938-2904.9. This cluster, drawn from an extensive preliminary list, was selected for detailed study on the basis of its apparent optical richness. Spectroscopy of 11 members places the cluster at z=0.951 +- 0.006, and confirms the photometric redshift estimate from the (R-z) color-magnitude diagram. Analysis of the infrared imaging data demonstrates that the cluster is extremely rich, with excess counts in the Ks-band exceeding the expected background counts by 9 sigma. The properties of the galaxies in RCS043938-2904.9 are consistent with those seen in other clusters at similar redshifts. Specifically, the red-sequence color, slope and scatter, and the size-magnitude relation of these galaxies are all consistent with that seen in the few other high redshift clusters known, and indeed are consistent with appropriately evolved properties of local cluster galaxies. The apparent consistency of these systems implies that the rich, high-redshift RCS clusters are directly comparable to the few other systems known at z ~ 1, most of which have been selected on the basis of X-ray emission.Comment: 12 pages, 1 color figure. Accepted for publication on The ApJ Letter

Macroscopic conductivity of free fermions in disordered media

We conclude our analysis of the linear response of charge transport in lattice systems of free fermions subjected to a random potential by deriving general mathematical properties of its conductivity at the macroscopic scale. The present paper belongs to a succession of studies on Ohm and Joule's laws from a thermodynamic viewpoint starting with [1-3]. We show, in particular, the existence and finiteness of the conductivity measure μΣ for macroscopic scales. Then we prove that, similar to the conductivity measure associated to Drude's model, μΣ converges in the weak*-topology to the trivial measure in the case of perfect insulators (strong disorder, complete localization), whereas in the limit of perfect conductors (absence of disorder) it converges to an atomic measure concentrated at frequency ν = 0. However, the AC-conductivity μΣ| ℝ\{0} does not vanish in general: We show that μΣ(ℝ\{0}) > 0, at least for large temperatures and a certain regime of small disorder

AC-Conductivity Measure from Heat Production of Free Fermions in Disordered Media

We extend (Bru et al. in J Math Phys 56:051901-1-51, 2015) in order to study the linear response of free fermions on the lattice within a (independently and identically distributed) random potential to a macroscopic electric field that is time- and space-dependent. We obtain the notion of a macroscopic AC-conductivity measure which only results from the second principle of thermodynamics. The latter corresponds here to the positivity of the heat production for cyclic processes on equilibrium states. Its Fourier transform is a continuous bounded function which is naturally called (macroscopic) conductivity. We additionally derive Green–Kubo relations involving time-correlations of bosonic fields coming from current fluctuations in the system. This is reminiscent of non-commutative central limit theorems

Microscopic Conductivity of Lattice Fermions at Equilibrium - Part I: Non-Interacting Particles

We consider free lattice fermions subjected to a static bounded potential and a time- and space-dependent electric field. For any bounded convex region $\mathcal{R}\subset \mathbb{R}^{d}$ ($d\geq 1$) of space, electric fields $\mathcal{E}$ within $\mathcal{R}$ drive currents. At leading order, uniformly with respect to the volume $\left| \mathcal{R}\right|$ of $\mathcal{R}$ and the particular choice of the static potential, the dependency on $\mathcal{E}$ of the current is linear and described by a conductivity distribution. Because of the positivity of the heat production, the real part of its Fourier transform is a positive measure, named here (microscopic) conductivity measure of $\mathcal{R}$, in accordance with Ohm's law in Fourier space. This finite measure is the Fourier transform of a time-correlation function of current fluctuations, i.e., the conductivity distribution satisfies Green-Kubo relations. We additionally show that this measure can also be seen as the boundary value of the Laplace-Fourier transform of a so-called quantum current viscosity. The real and imaginary parts of conductivity distributions satisfy Kramers-Kronig relations. At leading order, uniformly with respect to parameters, the heat production is the classical work performed by electric fields on the system in presence of currents. The conductivity measure is uniformly bounded with respect to parameters of the system and it is never the trivial measure $0\,\mathrm{d}\nu$. Therefore, electric fields generally produce heat in such systems. In fact, the conductivity measure defines a quadratic form in the space of Schwartz functions, the Legendre-Fenchel transform of which describes the resistivity of the system. This leads to Joule's law, i.e., the heat produced by currents is proportional to the resistivity and the square of currents

Quantum deformations of associative algebras and integrable systems

Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton equations too. In particular, such deformations for the three-dimensional and infinite-dimensional algebras are described by the Boussinesq equation and KP hierarchy, respectively.Comment: Numeration of the formulas is correcte