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.