Integrable systems and quantum cohomology

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.

In the present notes we explain the relationship between Calabi-Yau integrable systems and Hitchin systems based on work by Diaconescu-Donagi-Pantev and the author. Besides a review of these integrable systems, we highlight related topics, for example variations of Hodge structures, cameral curves and Slodowy slices, along the way.

We give a way of constructing real variations of mixed Hodge structures over compact Kähler manifolds by using mixed Hodge structures on Sullivan’s 1-minimal models of certain differential graded algebras associated with real variations of Hodge structures.

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS ‘span’ the space of all infinitesimal VHS; and (3) show that the cohomology classes dual to the Schubert VHS form a basis of the invariant characteristic cohomology associated with the infinitesimal period relation (a.k.a. Griffiths’ transversality).

This is a very interesting collection of introductory and review articles on the theory and applications of classical and quantum integrable systems. The book reviews several integrable systems such as the KdV equation, vertex models, RSOS and IRF models, spin chains, integrable differential equations, discrete systems, Ising, Potts and other lattice models and reaction--diffusion processes, as well as outlining major methods of solving integrable systems. These include Lax pairs, Bäcklund and Miura transformations, the inverse scattering method, various types of the Bethe Ansatz, Painlevé methods, the dbar method and fusion methods to mention just a few.

In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi–;Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the rigidity of families. We then apply the criterion to obtain that some important and typical families of Calabi–Yau varieties are rigid, for examples., Lefschetz pencils of Calabi–Yau varieties, strongly degenerated families (not only for families of Calabi–Yau varieties), families of Calabi–Yau varieties admitting a degeneration with maximal unipotent monodromy.

We address the question of whether integrable models allow for -symmetric deformations which preserve their integrability. For this purpose we carry out the Painlevé test for -symmetric deformations of Burgers and the Korteweg–De Vries equations. We find that the former equation allows for infinitely many deformations which pass the Painlevé test. For a specific deformation we prove the convergence of the Painlevé expansion and thus establish the Painlevé property for these models, which are therefore thought to be integrable. The Korteweg–De Vries equation does not allow for deformations which pass the Painlevé test in complete generality, but we are able to construct a defective Painlevé expansion.

OF THE DISSERTATION On the Limiting Behavior of Variations of Hodge Structures by Genival Francisco Fernandes da Silva Jr. Doctor of Philosophy in Mathematics Washington University in St. Louis, 2016 Professor Matt Kerr, Chair In this dissertation we will address three results concerning the limiting behavior of variations of Hodge structures. The first chapter introduces the main concepts involved and fixes some notation. In chapter two we discuss extension classes representing LMHS, compute them for a class of toric families and introduce an alternative method for the computation of VHS arising from middle convolution. The next chapter is concerned with the so called Apery constants; we provide a method of computing such constants by using higher normal functions coming from geometry. Finally, in the last chapter we analyze a family of surfaces with geometric monodromy group G2, and discuss the generic global Torelli theorem for such a family.

We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0$$\\end{document} that is a Calabi–Yau cone, extending the work of Székelyhidi (Duke Math J 168(14):2651–2700, 2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by C×V0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}\ imes V_0$$\\end{document}. This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.

In a previous paper, the author introduced an integral structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation of toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of variations of integral Hodge structure for a mirror pair of Calabi-Yau hypersurfaces (Batyrev’s mirror).

Let Σ \Sigma be a closed surface of genus at least two and ρ : π 1 ( Σ ) → G \rho \colon \pi _1(\Sigma ) \to G a Hitchin representation into G = P S L ( n , R ) G=PSL(n,\mathbb {R}) , P S p ( 2 n , R ) PSp(2n,\mathbb {R}) , P S O ( n , n + 1 ) PSO(n,n+1) or G 2 G_2 . We consider the energy functional E E on the Teichmüller space of Σ \Sigma which assigns to each point in T ( Σ ) \mathcal {T}(\Sigma ) the energy of the associated ρ \rho -equivariant harmonic map. The main result of this paper is that E E is strictly plurisubharmonic. As a corollary we obtain an upper bound of 3 ⋅ genus ( Σ ) − 3 3 \cdot \text {genus}(\Sigma ) -3 on the index of any critical point of the energy functional.

We first construct compatible actions of the product of the unit interval and the unit circle as a monoid on a semi-stable degeneration of pairs and on the associated log topological spaces. Then we show that the log topological family is locally trivial in piecewise smooth category over the base, i.e., the associated log topological family recovers the vanishing cycles of the original semi-stable degeneration in the most naive sense. Using this result together with the log Riemann-Hilbert correspondence, we introduce two types of integral structure of the variation of mixed Hodge structure associated to a semi-stable degeneration of pairs.

Variation of mixed Hodge structure. II

We introduce a multiple generalization of the tame symbol, called {\em polysymbols}, associated to meromorphic functions on a Riemann surface as the Massey products in Deligne cohomology, and also give a geometric construction of polysymbols using Chen's iterated integrals. We then deduce some basic properties of polysymbols using our holonomy formula, and show that trivializations of polysymbols give variations of mixed Hodge structure.

A guide to unipotent variations of mixed hodge structure