Some $tt$* structures and their integral Stokes data

Abstract

In [16] a description was given of all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpretation in quantum cohomology, and enumerate those solutions with integral Stokes data. This leads to a characterization of quantum Dmodules for certain complete intersections of Fano type in weighted projective spaces. 1. The tt*-Toda equations The tt* (topological—anti-topological fusion) equations were introduced by S. Cecotti and C. Vafa in their work on deformations of quantum field theories with N=2 supersymmetry (section 8 of [3], and also [4],[5]). This has led to the development of an area known as tt* geometry ([3],[11],[19]), a generalization of special geometry. Solutions of the tt* equations can be interpreted as pluriharmonic maps with values in the noncompact real symmetric space GLnR/On, or as pluriharmonic maps with values in a certain classifying space of variations of polarized (finite or infinite-dimensional) Hodge structure. Frobenius manifolds with real structure, e.g. quantum cohomology algebras, provide a very special class of solutions “of geometric origin” (see [11]). These special solutions lie at the intersection of p.d.e. theory, integrable systems, and (differential, algebraic, and symplectic) geometry. However, very few concrete examples have been worked out in detail, and their study is just beginning. It is relatively straightforward to obtain local solutions, but these special solutions have (or are expected to have) global properties, and these properties are hard to establish. In [17], [16] a family of global solutions was constructed by relatively elementary p.d.e. methods. In this article we shall describe the special solutions in terms of their holomorphic data. This allows us to obtain — in a very restricted situation — an a fortiori characterization of 2000 Mathematics Subject Classification. Primary 81T40; Secondary 53D45, 35J60.