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

Abstract: 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.