arXiv subjects

Martin Guest

Publications and source records attributed to Martin Guest.

5 recordsLinked to original sources

arXiv2018

Kostant, Steinberg, and the Stokes matrices of the tt*-Toda equations

We propose a Lie-theoretic definition of the tt*-Toda equations for any complex simple Lie algebra $\mathfrak{g}$, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes data of a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. Exploiting a framework introduced by Boalch, we show that this data has a remarkable structure, which can be described using Kostant's theory of Cartan subalgebras in apposition and Steinberg's theory of conjugacy classes of regular elements. A by-product of this is a convenient visualization of the orbit structure of the roots under the action of a Coxeter element. As an application, we compute canonical Stokes data of certain solutions of the tt*-Toda equations in terms of their asymptotics.

math.DG↗

arXiv2018

A Lie-theoretic description of the solution space of the tt*-Toda equations

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 [GL] [GIL1] [M1] [M2] that these solutions can be parametrized by monodromy data of a certain flat $SL_{n+1}\mathbb{R}$-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 $SU_{n+1}$.

math.DG↗

arXiv2010

The tt* structure of the quantum cohomology of CP^1 from the viewpoint of differential geometry

The quantum cohomology of CP^1 provides a distinguished solution of the third Painleve equation. S. Cecotti and C. Vafa discovered this from a physical viewpoint. We derive it from a differential geometric viewpoint, using the theory of harmonic maps and in particular the generalized Weierstrass representation (DPW representation) for spacelike surfaces of constant mean curvature in Minkowski space. The nontrivial aspects are the characterization of the solution, and its global behaviour. From our point of view, the latter property says that the extended harmonic map remains entirely within a single Iwasawa orbit of the appropriate loop group.

math.DG↗

arXiv2001

Morse theory in the 1990's

This is a survey article on Morse theory based on lectures to graduate students and advanced undergraduates. After a brief review of standard material, mostly without proofs, the Morse theory of complex Grassmannian manifolds is worked out in detail. In contrast to standard treatments, gradient flow lines and their structure are emphasized. This leads to a convenient interpretation of Schubert calculus via the momentum map of a torus action, and, more generally, to a still-unfolding toric/combinatoric manifestation of Morse theory. The latter provides a miniature version of the "field-theoretic point of view" of Cohen-Jones-Segal, Betz-Cohen, and Fukaya.

math.DG↗

arXiv1996

Harmonic two-spheres in compact symmetric spaces, revisited

Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic 2-spheres in a Lie group G and proved that when G=SU(n) the uniton number cannot exceed n-1. In this paper, using new methods inspired by Morse Theory, we explain this result and extend it to an arbitrary compact group G. The same methods also yield Weierstrass formulae for these harmonic maps and simple proofs of most of the known classification theorems for harmonic 2-spheres in symmetric spaces.

dg-ga↗