arXiv subjects
Publications and source records attributed to Takashi Ichikawa.
We associate each quiver with soliton solutions of nonlinear integrable systems containing the KP and Toda hierarchies. We give an explicit and computable formula of these soliton solutions which are regarded as universal ones obtained by degenerating quasi-periodic solutions.
We study dimer models on infinite minimal graphs with Fock's weights for degenerating families of M-curves of any genus based on works of Boutillier-Cimasoni-de Tilière and Bobenko A.I.- Bobenko N.-Suris for a fixed M-curve. We show that these dimer models for families of M-curves behave consistently under their degeneration shrinking real circles to points, and that they can be calculated as power series in the associated deformation parameters which are regarded as the perturbation of Kenyon's critical dimer models.
By using Schottky uniformization theory of degenerating algebraic curves, we describe the tropical convergence of harmonic amoebas of pointed Riemann surfaces to tropical curves which are not necessarily simple. We extend Lang's results on the simple tropical convergence based on the Frenchel-Nielsen coordinates to the nonsimple case. Our results are hoped to give contributions in compactifying the moduli space of pointed Riemann surfaces with tropical curves, and in studying crystallization of general dimer models.
We study the variation of the Enriquez connection for higher genus polylogarithms under degenerations of Riemann surfaces with marked points, and show that this connection becomes the connection constructed by the author for degenerating families of pointed Riemann surfaces. Therefore, we have an important application that the higher genus polylogarithms derived from the Enriquez connection can be described explicitly as power series in deformation parameters and their logarithms associated with the families whose coefficients are expressed by multiple zeta values.
Real and regular soliton solutions of the KP hierarchy have been classified in terms of the totally nonnegative (TNN) Grassmannians. These solitons are referred to as KP solitons, and they are expressed as singular (tropical) limits of shifted Riemann theta functions. In this talk, for each element of the TNN Grassmannian, we construct a Schottky group, which uniformizes the Riemann surface associated with a real finite-gap solution. Then we show that the KP solitons are obtained by degenerating these finite-gap solutions.
In order to generalize a program by Agostini and others producing new KP solutions, we construct families of quasi-periodic KP solutions which are derived from degenerating Riemann surfaces associated with tropical curves having nontrivial weights. By taking the regularized tropical limits of these solutions, we obtain formulas of general KP solutions which are expressed by mixtures of solitons and Riemann theta functions.
We construct polylogarithms on families of pointed Riemann surfaces of any genus which describe monodromies of meromorphic connections with simple poles. Furthermore, we show that the polylogaritms are computable as power series in deformation parameters and their logarithms associated with the families.
Using abelian differentials and periods of the universal Mumford curve, we study the universal expression and asymptotic behavior of tau functions defined for stably degenerating families of algebraic curves with additional data. Furthermore, we apply this study to constructing solutions to the KP hierarchy which are expressed by nonarchimedean theta functions and by mixtures of quasi-periodic solutions with solitons containing real solutions defined on families of M-curves.
We construct the universal Mumford curve of given genus as a family of Mumford curves over the deformation space of degenerate curves in the category of arithmetic formal geometry. Furthermore, we give explicit formulas of abelian differentials and their periods of the universal Mumford curve.
A near-infrared telescope with an effective aperture diameter of thirty millimeters has been developed. The primary objective of the development is to observe northern bright stars in the $J$, $H$, and $K_{\rm s}$ bands and provide accurate photometric data on those stars. The second objective is to repeatedly observe a belt-like region along the northern Galactic plane ($|b| \le 5^\circ$ and $δ\ge -30^\circ$) to monitor bright variable stars there. The telescope has been in use since December 2016. The purpose of this paper is to describe the design and operational performances of the telescope, photometric calibration methods, and our scientific goals. We show that the telescope has the ability to provide photometry with an uncertainty of less than 5\% for stars brighter than 7, 6.5, and 6~mag in the $J$, $H$, and $K_{\rm s}$ bands, respectively. The repeatability of the photometric measurements for the same star is better than 1\% for bright stars. Our observations will provide accurate photometry on bright stars that are lacking in the Two Micron Sky Survey and the Two Micron All-Sky Survey. Repeated observations at a good cadence will also reveal their nature of the variability in the near-infrared.
We describe iterated integrals as unipotent periods on families of marked elliptic curves in terms of multiple zeta values and elliptic multiple zeta values.
We introduce a notion of generalized modular functors with Hilbert spaces of infinite dimension in general, and show that a generalized modular functor with data of conformal dimensions determines uniquely wave functions as its flat sections. Furthermore, we study an example of generalized modular functors derived from the Liouville conformal field theory. In this example, wave functions are seen to be gluing conformal blocks which are identified with eigenfunctions in the quantum Teichmüller theory.
A generalized Tate curve is a universal family of curves with fixed genus and degeneration data which becomes Schottky uniformized Riemann surfaces and Mumford curves by specializing moduli and deformation parameters. By considering each generalized Tate curve as a family of degenerating Riemann surfaces, we give explicit formulas of the period isomorphism between its de Rham and Betti cohomology groups, and of the associated objects: Gauss-Manin connection, variation of Hodge structure and monodromy weight filtration. A remarkable fact is that similar formulas hold also for families of Mumford curves. Furthermore, we show that for a generalized Tate with maximally degenerate closed fiber, its local unipotent periods can be expressed as power series in the deformation parameters whose coefficients are multiple polylogarithm functions. This p-adic version is also given.
We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented by an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. By this formula, we have a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves.
As a higher genus version of universal mixed elliptic motives by Hain and Matsumoto, we consider mixed Teichmüller motives as certain motivic local systems on the moduli space of pointed curves. We show that the category of mixed Teichmüller motives is equivalent to a full subcategory of a certain product category of mixed Tate motives over Z and universal mixed elliptic motives. Furthermore, we show that unipotent fundamental torsors for the universal open curve give rise to a pro-object in the category of mixed Teichmüller motives. Our results can give a realization of motivic correlators proposed by Goncharov.
We study the gluing theory of Riemann surfaces using formal algebraic geometry, and give computable relations between the associated parameters for different gluing processes. As its application to the Liouville conformal field theory, we construct the sheaf of tempered conformal blocks on the moduli space of pointed Riemann surfaces which satisfies the factorization property and has a natural action of the Teichmüller groupoid.
Using the arithmetic Schottky uniformization theory, we show the arithmeticity of $PSL_{2}({\mathbb C})$ Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Riemann-Roch isomorphism as Zograf-Mcintyre-Takhtajan's infinite product for families of algebraic curves. By this formula, we determine the unknown constant which appears in the holomorphic factorization formula of determinant of Laplacians on Riemann surfaces via the classical Liouville action. As an application, we show the rationality of Ruelle zeta values for Schottky uniformized $3$-manifolds.
To study the accretion phase for local massive galaxies, we search accreting satellites around a massive compact galaxy (M_*~3.9x10^10Msun), spectroscopically confirmed (z_spec-1.9213) in the eXtreme Deep Field, which has been originally reported in Szomoru et al. We detect 1369 satellite candidates within the projected virial radius (rvir~300 kpc) of the compact galaxy in the all-combined ACS image with 5sigma-limiting magnitude of mACS~30.6 ABmag, which corresponds to ~1.6x10^7M_sun at the redshift. The photometric redshift measured with 12 multi-band images confirms 34 satellites out of the candidates. Most of the satellites are found to have the rest-frame colors consistent with star forming galaxies. We investigate the relation between stellar mass and star formation rate (the star formation main sequence), and find the steeper slope at the low-mass end (<10^8M_sun), while more massive satellites are consistently on the sequence reported in previous studies. Within the uncertainties of star formation and photometric redshift, we conjecture possible scenarios for the compact galaxy which evolves to a local massive galaxy by way of significant size and mass growth. While merging of the existing total stellar mass of the satellites is not enough to explain the mass growth predicted by observations and simulations, the contribution by in-situ star formation in the satellites would compensate the deficit. Provided that most satellites keep the observed in-situ star formation and then quench before they accrete by, e.g., environmental quenching, the compact galaxy would become a massive early-type galaxy consistent with the local size-mass relation.