Workshop on:

tt*-geometry

Supported in part by JSPS grant (A) 21244004 "Exploitation of new relations between differential geometry and quantum cohomology in the context of integrable systems"

Despite their very different origins, the theories of (1) Gauss maps of CMC immersions (more generally, harmonic or pluriharmonic maps into symmetric spaces) (2) quantum cohomology (more generally, Frobenius manifolds) are very closely linked by means of certain families of flat connections. These "integrable systems" provide links between several important fields: differential geometry, singularity theory, variations of Hodge structures, Gromov-Witten invariants, and so on. The aim of this workshop is to exchange ideas and techniques from these fields, focusing on the meromorphic connections which arise from harmonic maps and quantum cohomology.

PROGRAM

Tuesday 17 August

10:30-12:00 Martin Guest (TMU), "Harmonic bundles and harmonic maps"

Abstract:The "loop group approach" to the theory of harmonic maps from surfaces to symmetric spaces is a well-established framework for effective calculations. Nevertheless, many unsolved problems in harmonic map theory (in particular, the theory of CMC surfaces) involve global questions which are difficult to treat in this framework. A more intrinsic approach should focus on bundles. without making particular choices of trivializations, as in the theory of harmonic bundles. This theory has been developed and extended in recent years, with new examples coming from singularity theory and mirror symmetry. We shall explain this framework and its relation with loop groups.

13:30-15:00 Josef Dorfmeister (TU Munich), "Generalized isotropic harmonic maps"

Abstract: In surface theory one is frequently interested in surfaces with symmetries, i.e. in surfaces that are invariant under certain rigid motions like rotations, translations or specific combinations thereof. Clearly, symmetries induce on the surface isometries relative to the induced metric. In many cases, however, surfaces have isometries that can not be induced by rigid motions of the surounding space (usually a space form). A particularly interesting special case of symmetries are 1-parameter groups of symmetries. For CMC surfaces in R^3 exactly the Delaunay surfaces admit such a group of symmetries. Similarly special surfaces have been determined in all other integrable surface classes. It therefore seems to be of interest to investigate more generally surfaces admitting 1-parameter groups of self-isometries. For the CMC surfaces in R^3 these are the "Smyth surfaces". In this talk we consider more generally harmonic maps from surfaces to symmetric spaces admitting a 1-parameter group of self-transformations, the generalized isotropic harmonic maps. We will characterize the potentials that will produce such harmonic maps.

15:15-16:45 Claus Hertling (Mannheim), "(Sign) harmonic maps and CMC surfaces"

Abstract: CMC surfaces in Euclidean space are related to rank 2 bundles which are almost harmonic, though their hermitian metric is indefinite and they carry extra structure, a flat quaternionic structure and a non-flat involution. CMC surfaces in Minkowski space are related to true harmonic bundles of rank 2 with extra structure. These statements will be explained in the talk. They form the first step of a project in which we shall apply recent work of T. Mochizuki on tame and wild harmonic bundles and meromorphic connections to CMC surfaces.

17:00-18:30 Sanae Kurosu (Tokyo University of Science), "A tt*-bundle structure associated with a harmonic map into unit sphere and a (DC\tilde{C})-structure"

Abstract: We construct a tt^*-bundle structure and a DC\tilde{C}-structure from a harmonic map into a unit sphere. We also study a relation between a tt^*-bundle structure and a DC\tilde{C}-structure on a complex vector bundle.