We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt ∗-Toda equations) which were introduced by Cecotti and Vafa. It is known from Guest and Lin (J. Reine Angew. Math. 689, 1–32 2014) Guest et al. (It. Math. Res. Notices 2015, 11745–11784 2015) and Mochizuki (2013, 2014) that these solutions can be parametrized by monodromy data of a certain flat SLn+ 1ℝ-connection. Using Boalch’s Lie-theoretic description of Stokes data, and Steinberg’s description of regular conjugacy classes of a linear algebraic group, we express this monodromy data as a convex subset of a Weyl alcove of SUn+ 1.

In Guest et al. arXiv:1209.2045(part I) we computed the Stokes data for the smooth solutions of the tt*-Toda equations whose existence we had previously established by p.d.e. methods. Here we formulate the existence problem as a Riemann-Hilbert problem, based on this Stokes data, and solve it under certain conditions (Theorem 5.4). In the process, we compute the connection matrix for all smooth solutions, thus completing the computation of the monodromy data (Theorem 5.5). We also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions (Theorem 4.1), clarifying the region of validity of the formulae established earlier by Tracy and Widom. Finally, we resolve some conjectures of Cecotti and Vafa concerning the positivity of S + S (t) (where S is the Stokes matrix) and the unimodularity of the eigenvalues of the monodromy matrix (Theorem 5.6). In particular, we show that "unitarity implies regularity" for the tt*-Toda equations.

We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations or equations for harmonic maps into SLnR/SOn) in terms of (1) asymptotic data, (2) holomorphic data, and (3) monodromy data, and we compute all of this data explicitly. This allows us, in particular, to find all solutions with integral Stokes data. These include solutions associated to non-linear sigma models (quantum cohomology) or Landau-Ginzburg models (unfoldings of singularities), as conjectured by Cecotti and Vafa in the 1990s.

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 characterised 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. © De Gruyter 2014.

We construct in an abstract fashion (without using Gromov-Witten invariants) the orbifold quantum cohomology of weighted projective space, starting from a certain differential operator. We obtain the product, grading, and intersection form by making use of the associated self-adjoint D-module and the Birkhoff factorization procedure. The method extends in principle to the more difficult case of Fano hypersurfaces in weighted projective space, where Gromov-Witten invariants have not yet been computed, and we illustrate this by means of an example originally studied by A. Corti. In contrast to the case of weighted projective space itself or the case of a Fano hypersurface in projective space, a "small cell" of the Birkhoff decomposition plays a role in the calculation.

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 paper, 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 D-modules for certain complete intersections of Fano type in weighted projective spaces.

We propose a new point of view on quantum cohomology, motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is a D-module which "quantizes" a commutative algebra associated to the (uncompactified) space of rational curves. Under appropriate conditions, we show that the associated flat connection may be gauged to the flat connection underlying quantum cohomology. This method clarifies the role of the Birkhoff factorization in the "mirror transformation", and it gives a new algorithm (requiring construction of a Groebner basis and solution of a system of o.d.e.) for computation of the quantum product. (C) 2004 Elsevier Ltd. All rights reserved.

We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kähler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. We derive a simple and explicit "differential operator formula" for the necessary quantum products, which applies both to the finite-dimensional and to the infinite-dimensional situations.

The space of holomorphic maps from S-2 to a complex algebraic variety X, i.e. the space of parametrized rational curves on X, arises in several areas of geometry. It is a well known problem to determine an integer n(D) such that the inclusion of this space in the corresponding space of continuous maps induces isomorphisms of homotopy groups up to dimension n(D), where D denotes the homotopy class of the maps. The solution to this problem is known for an important but special class of varieties, the generalized flag manifolds: such an integer may be computed, and n(D) --> infinity as D --> infinity. We consider the problem for another class of varieties, namely, toric varieties. For smooth toric varieties and certain singular ones, n(D) may be computed, and n(D) --> infinity as D --> infinity. For other singular toric varieties, however, it turns out that n(D) cannot always be made arbitrarily large by a suitable choice of D.

We describe a general framework in which subgroups of the loop group LAMBDA-Gl(n)C act on the space of harmonic maps from S2 to Gl(n)C. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of LAMBDA-Gl(n)C, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action ("completion") gives rise to her fundamental process of "adding a uniton". It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from S2 to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton".

For equivariant maps from a compact homogeneous space into an adjoint orbit of a compact Lie group, it is shown that the energy function is the restriction of a quadratic function on the Lie algebra, providing the orbit has the metric induced from the Lie algebra. This is related to similar functions studied by R. Bott [2] and by F. C. Kirwan [7]. One obtains a simple version of the harmonic map equation, and an identity relating the energy and the square of the norm of the moment map. Several applications are given, including an example which illustrates how a change of metric in a flag manifold affects the harmonicity of equivariant maps from the two-sphere to the flag manifold.