Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa

Abstract

Using nonlinear pde techniques, we construct a new family of globally smooth tt* structures. This includes tt* structures associated to the (orbifold) quantum cohomology of a finite number of complex projective spaces and weighted projective spaces. The existence of such "magical solutions" of the tt* equations, namely smooth solutions characterized by asymptotic boundary conditions, was predicted by Cecotti and Vafa. In our situation, the tt* equations belong to a class of equations which we call the tt*-Toda lattice. Solutions of the tt*-Toda lattice are harmonic maps which have dual interpretations as Frobenius structures or variations of (semi-infinite) Hodge structures.

arXiv comment35 pages

References in corpus (7)

• Geometry and Integrability of Topological-Antitopological Fusion
• Exact Results for Supersymmetric Sigma Models
• Real and integral structures in quantum cohomology I: toric orbifolds
• Multidimensional Toda type systems
• tt*-geometry in quantum cohomology
• Orbifold quantum D-modules associated to weighted projective spaces
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Cited by in corpus (4)

• The tt* structure of the quantum cohomology of CP^1 from the viewpoint of differential geometry
• Generic Ends of the Moduli Space of $SL(n,\mathbb{C})$-Higgs Bundles
• Isomonodromy aspects of the tt* equations of Cecotti and Vafa II. Riemann-Hilbert problem
• Some tt* structures and their integral Stokes data