The aim of my research is to construct tt*-structures by using isomonodromy theory and to provide explicitly solvable solutions to the tt*-equations by using the DPW method. Only a few concrete examples of solvable tt*-equations are known, such as the sinh–Gordon equation and the tt*-Toda equation.During this year, three papers were accepted.1. T. Udagawa, The Iwasawa factorization with rotationally symmetric parts and the Lame equation, Tohoku Mathematical Journal (to appear).2. T. Udagawa, Classification of Toda-type tt*-structures and Zn-fixed points, Journal of Mathematical Physics, 2025, 66, no. 12, Paper No. 123509.3. T. Udagawa, Solutions of the tt*-equations constructed from the (SU2)k-fusion ring, and Smyth potentials, Tokyo Journal of Mathematics, 2025, 48, no. 2.In Paper 1, we proved that the Iwasawa factorization for the Delaunay potential can be carried out on the complex plane except countably many lines. We expressed the factorization by the Lamé equation and Weierstrass functions and derived explicit parametrizations of constant mean curvature surfaces in three-dimensional Minkowski space using the DPW method. We also showed that the resulting surfaces are rotationally symmetric and that each cross-section is an ellipse or a hyperbola on a hyperboloid.In Paper 2, we classified tt*-structures under the anti-symmetry condition for the tt*-Toda equations. We characterized Toda-type tt*-structures intrinsically as fixed points of multiplication by n-th roots of unity, and we showed that the anti-symmetry condition essentially reduces to two types. This classification is organized by a parameter l = 0 or 1 in the anti-symmetry condition. We also established a correspondence between the classification of the tt*-Toda equations and representation theory.In Paper 3, we provided a precise mathematical formulation of the tt*-structure constructed from the (SU(2))ₖ-fusion ring, and we construct a solution to the corresponding tt*-equation. We also gave a description of the “holomorphic data” corresponding to the solutions. Furthermore, we showed that equivalence relations among SU(2) representations correspond to gauge equivalence of harmonic maps. The paper also analyzes the supersymmetric Ak minimal model as a special case. In addition, the following two papers are currently under submission:4. T. Udagawa, A Lie-theoretic description of the tt*-equation constructed from the (SU(N))k-fusion ring.5. T. Udagawa, The tt*-structure for the quantum cohomology of complex Grassmannian.There were four presentations.

The final goal of my research is to solve tt*-equations. In my research, I use two methods for solving the tt*-equations. In the first way, I try to construct a new solution to the tt*-equation from global solutions to the tt*-Toda equation (Guest-Its-Lin). Regarding the first way, I investigated intrinsic properties of the tt*-Toda equation. I gave an intrinsic description of the tt*-Toda equation as a flat bundle, and I classified the flat bundles. From the classification, I gave an equivalence relation on restrictive conditions called "anti-symmetry conditions" of the tt*-Toda equation, and we can show that there are essentially only two types of these conditions. As an application, we observed the relation between the tt*-Toda equations and representations of the vertex algebra following the results of Fredrickson and Neizke. By using the global solutions to the tt*-Toda equation, I also gave a solution to the tt*-equation on the quantum cohomology ring of the Grassmannian. The solution is given as the exterior product of solutions to the tt*-Toda equation. In the second way, I consider an isomonodromic deformation of the tt*-equation with certain conditions and I try to solve the tt*-equation by using the Vanishing Lemma. In this method, the existence of solutions to the tt*-equation is equivalent to the existence of a Riemann-Hilbert problem and there is a correspondence between the solutions and upper unitriangular matrices. However, upper triangular matrices do not always correspond to solvable solutions. Physicists claimed that the ADE-type Cartan matrices give solvable solutions to the tt*-equations. We gave the mathematical proof for the claim regarding upper triangular matrices by using Vanishing Lemma. I expect that this method could be applied to more general cases. In physics, it was claimed that upper unitriangular matrices with certain conditions give solvable solutions. Regarding papers, one paper was accepted, and one paper is in application. I am preparing papers about my results above. Regarding presentations, I gave 4 presentations (3 talks and 1 poster presentation). Regarding other research activities, I visited Taiwan from 2/27 to 3/4, and I discussed with Prof. Martin Guest, who is the visitor to the National Taiwan University. We mainly discussed the paper about intrinsic properties of the tt*-Toa equation.