arXiv subjects

Tadashi Udagawa

Publications and source records attributed to Tadashi Udagawa.

6 recordsLinked to original sources

arXiv2026

Noncompact Iwasawa factorization and translationally equivariant hyperbolic affine spheres

We establish the noncompact Iwasawa factorization for a Delaunay-type potential associated with affine spheres on the complex plane away from countably many lines. Using the DPW method, we give an explicit description of the factorization in terms of Weierstrass elliptic functions via a reduction to a linear system related to the Tzitzéica equation. As an application, we construct explicit translationally equivariant hyperbolic affine spheres and classify them according to their slice curves. In particular, we show that every such affine sphere is equiaffinely equivalent to one whose slice curve is a circle, hyperbola, or parabola, consistent with the Calabi correspondence between hyperbolic affine spheres and proper convex cones.

math.DG↗

arXiv2026

On tt*-structures from $ADE$-type Stokes data

Cecotti and Vafa introduced the topological anti-topological fusion (tt*)-equation, whose solutions describe massive deformations of supersymmetric conformal field theories. We provide a rigorous analytic formulation of the $ADE$ classification of tt*-structures. Under natural structural assumptions, a tt*-structure over $\mathbb{C}^*$ can be described via isomonodromic deformations with upper unitriangular real Stokes matrices. Two fundamental issues arise: the ambiguities of Stokes matrices, governed by an action of a group $\tilde{Br}_n$, which is generated by reordering operations, and the solvability of the associated Riemann-Hilbert problem. Our first main result shows that the classification reduces to admissible Stokes matrices modulo $\tilde{Br}_n$-action, and that the $\tilde{Br}_n$-orbit of a Stokes matrix determines a tt*-structure over $\mathbb{C}^*$. Our second main result establishes that upper unitriangular matrices whose symmetrizations coincide with Cartan matrices of type $A_n, D_n, E_6, E_7,$ or $E_8$ give rise to tt*-structures over $\mathbb{C}^*$. This provides a direct analytic realization of the $ADE$ classification and clarifies the interplay between Stokes phenomena, $\tilde{Br}_n$-symmetry, and positivity of Cartan-type matrices.

math.DG↗

arXiv2025

The tt*-structure for the quantum cohomology of complex Grassmannian

The tt*-equation (topological-anti-topological fusion equation) was introduced by S. Cecotti and C. Vafa for describing massive deformation of supersymmetric conformal field theories. B. Dubrovin formulated the tt*-equation as a flat bundle, called tt*-structure. In this paper, we construct a tt*-structure for the quantum cohomology of the Grassmannian of complex $k$-plane and obtain global solutions to the tt*-equation, following the idea of Bourdeau. We give a precise mathematical formulation and a description of the solutions by using p.d.e. theory and the harmonic map theory developed by J. Dorfmeister, F. Pedit and H. Wu (the DPW method). Furthermore, we give an isomorphism between tt*-structure for the $k$-th exterior product of tt*-structure for the quantum cohomology of the complex projective space and the tt*-structure for the quantum cohomology of the Grassmannian.

hep-th↗

arXiv2025

Classification of Toda-type tt*-structures and $\mathbb{Z}_{n+1}$-fixed points

We classify Toda-type tt*-structures in terms of the anti-symmetry condition. A Toda-type tt*-structure is a flat bundle whose flatness condition is the tt*-Toda equation (Guest-Its-Lin). We show that the Toda-type tt*-structure can be described as a fixed point of $e^{\sqrt{-1}\frac{2π}{n+1}}$-multiplication and this ``intrinsic'' description reduces the possibilities of the anti-symmetry condition to only two cases. We give an application to the relation between tt*-Toda equations and representation theory.

math.DG↗

arXiv2024

Globality of the DPW construction for Smyth potentials in the case of SU(1,1)

We construct harmonic maps into SU(1,1)/U(1) starting from Smyth potentials ξ, by the DPW method, In this method, harmonic maps are obtained from the Iwasawa factorization of a solution L of L^{-1} dL = ξ. However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that L can be expressed in terms of Bessel functions and from the asymptotic expansion of Bessel functions we solve a Riemann-Hilbert problem to give a global Iwasawa factorization. In this way we give a more direct proof of the globality of our solution than in the work of Dorfmeister-Guest-Rossman (2010), while avoiding the general isomonodromy theory used by Guest-Its-Lin (2015).

math.DG↗

arXiv2023

Solutions of the tt*-equations constructed from the SU(2)$_k$-fusion ring, and Smyth potentials

Cecotti and Vafa introduced the tt*-equation (topological-antitopological fusion equation), whose solutions describe massive deformations of supersymmetric conformal field theories. We describe some solutions of the tt*-equation constructed from the SU(2)$_k$-fusion algebra. The idea of the construction is due to Cecotti and Vafa, but we give a precise mathematical formulation and a description of the "holomorphic data" corresponding to the solutions by using the DPW method. Furthermore, we give a relation between the solutions and the representations of SU(2). As a special case, we consider the solutions corresponding to the supersymmetric A$_k$-minimal model.

math-ph↗