I n t r o d u c t i o n Ideas from quantum field theory and string theory have had an enormous im- pact on geometry over the last two decades. One extremely fruitful source of new mathematical ideas can be traced back to works of Cecotti, Vafa, and their coau- thors around 1991 on the Geometry of Topological Field Theory. The motivation for their "tt*-geometry" came from physics, but the work turned out to unify ideas that had been developing in such separate branches of mathematics as singularity theory, Hodge theory, integrable systems, matrix models, and Hurwitz spaces. From one point of view, the geometry developed by Cecotti, Vafa, et al in- volved a refinement of the notion of Higgs bundle. This geometric structure had been previously introduced and studied by Hitchin. The refinement was closely related to what have since been codified by Dubrovin as Frobenius manifolds. Es- sentially the same objects had been previously introduced, on moduli spaces ass- ociated to quasihomogeneous singularities, by Kyoji Saito. They appeared also in Hodge theoretic work of Morihiko Saito and in numerous other approaches. In the papers by Cecotti, Vafa et al, these purely holomorphic structures were com- bined with a generalization of the Gauss -Manin connection, a generalization which nowadays is described as variations of harmonic Higgs bundles. Following ideas of Hitchin, Deligne and Simpson, the entire structure is most conveniently described by introducing a twistor line and a new, flat connection on the resulting space. The flatness equations for this connection are the tt*-equations of Cecotti and Vafa. The twistor structures capture the essence of the Corlette-Simpson non-Abelian Hodge theory, which establishes the equivalence of flat connections and Higgs bun- dles. The tt*-geometry combines algebro-geometric notions also with symplectic ones. Such a combination turns out to be tailor made for investigations of mirror symmetry and Gromov-Witten invariants. These structures and equations have since appeared in many other branches of geometry and physics: Integrable systems have been closely related to the topic ever since Hitchin's works on moduli spaces of stable bundles, and Dubrovin has proved integrability of the tt*-equations. Almost simultaneously with the origi- nal work by Cecotti and Vafa, Simpson developed his notion of harmonic bundles. These have since evolved into mixed twistor structures, an essential contritubion to the field. Another key ingredient is Mochizuki's ground breaking work on tame -and more recently also wild- harmonic bundles. Hertling and collaborators have clarified and further developed the precise mathematical counterparts of the fun- damental structures used in the physics papers of Cecotti and Vafa, introducing "TERP structures". On the physics side, the tt*-equations have been understood as governing tree level amplitudes in topological quantum field theories. Their generalization to higher-genus amplitudes by Bershadsky-Cecotti-Ooguri-Vafa is
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