arXiv subjects

Martin A. Guest

Publications and source records attributed to Martin A. Guest.

At least 19 recordsLinked to original sources

arXiv2026

Geometry of the tt*-Toda equations I: universal centralizer and symplectic groupoids

We investigate the geometry of a certain space of meromorphic connections with irregular singularities, and prove in particular that it is a (real) symplectic Lie groupoid. The connections have a physical meaning: they correspond to certain solutions of the topological-antitopological fusion (tt*) equations of Cecotti and Vafa, and hence to deformations of supersymmetric quantum field theories. The groupoid structure arises because we restrict ourselves to the tt* equations of Toda type, whose monodromy data has a Lie theoretic description. To obtain these results, we show first that the universal centralizer of a Lie group is a holomorphic symplectic groupoid over the Steinberg cross section.

math.SG↗

arXiv2025

Connection formulae for the radial Toda equations I

This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type $A_n$. The principal issue is the connection formulae between the asymptotic parameters describing the behavior of the general solution at zero and infinity. To reach this goal we are using a fusion of the PDE analysis and the Riemann-Hilbert nonlinear steepest descent method of Deift and Zhou which is applicable to 2D Toda in view of its Lax integrability. A principal technical challenge is the extension of the nonlinear steepest descent analysis to Riemann-Hilbert problems of matrix rank greater than $2$. In this paper, we meet this challenge for the case $n=2$ (the rank $3$ case) and it already captures the principal features of the general $n$ case.

math-ph↗

arXiv2025

Connection formulae for the radial Toda equations II

This is a continuation of [arXiv:2309.16550] in which we computed the asymptotics near $x = \infty$ of all solutions of the radial Toda equation. In this article, we compute the asymptotics near $x = \infty$ of all solutions of a "partner" equation. The equations are related in the sense that their respective monodromy data constitute connected components of the same "monodromy manifold". While all solutions of the radial Toda equation are smooth, those of the partner equation have infinitely many singularities, and this makes the Riemann-Hilbert nonlinear steepest descent method (and the asymptotics of solutions) more involved.

math-ph↗

arXiv2023

The tt*-Toda equations of A_n type

In previous articles we have studied the A_n tt*-Toda equations (topological-antitopological fusion equations of Toda type) of Cecotti and Vafa, giving details mainly for n=3. Here we give a proof of the existence and uniqueness of global solutions for any n, and a new treatment of their asymptotic data, monodromy data, and Stokes data.

math.DG↗

arXiv2022

Quantum cohomology: is it still relevant?

This article, intended for a general mathematical audience, is an informal review of some of the many interesting links which have developed between quantum cohomology and "classical" mathematics. It is based on a talk given at the Autumn Meeting of the Mathematical Society of Japan in September 2021.

math.DG↗

arXiv2022

Positive energy representations of affine algebras and Stokes matrices of the affine Toda equations

We give a construction which produces a positive energy representation of the affine Lie algebra of type A_n from the Stokes data of a solution of the tt*-Toda equations of type A_n. The construction appears to play a role in conformal field theory. We illustrate this with several examples: the fusion ring, W-algebra minimal models (Argyres-Douglas theory), as well as topological-antitopological fusion itself. (Minor typographical changes for this version.)

math.DG↗

arXiv2021

Polytopes, supersymmetry, and integrable systems

We review some links between Lie-theoretic polytopes and field theories in physics, which were proposed in the 1990's. A basic ingredient is the Coxeter Plane, whose relation to integrable systems and the Stokes Phenomenon has only recently come to light. We use this to give a systematic mathematical treatment, which gives further support to the physical proposals. This article is based on a talk which was scheduled to be given at the workshop "Representations of Discrete Groups and Geometric Topology on Manifolds", Josai University, 12-13 March 2020.

math.DG↗

arXiv2020

Topological-antitopological fusion and the quantum cohomology of Grassmannians

We suggest an explanation for the part of the Satake Correspondence which relates the quantum cohomology of complex Grassmannians and the quantum cohomology of complex projective space, as well as their respective Stokes data, based on the original physics approach using the tt* equations. We also use the Stokes data of the tt* equations to provide a Lie-theoretic link between particles in affine Toda models and solitons in certain sigma-models. Along the way, we illustrate some (well known) relations between the tt* equations, the non-abelian Hodge Correspondence, and quantum cohomology.

math.DG↗

arXiv2018

Isomonodromy aspects of the tt* equations of Cecotti and Vafa III. Iwasawa factorization and asymptotics

This paper, the third in a series, completes our description of all (radial) solutions on C* of the tt*-Toda equations, using a combination of methods from p.d.e., isomonodromic deformations (Riemann-Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth near 0. For such solutions, we compute explicitly the Stokes data and connection matrix of the associated meromorphic system, in the resonant cases as well as the non-resonant case. This allows us to give a complete picture of the monodromy data of the global solutions.

math.DG↗

arXiv2015

The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles

The paper is about a Painlevé III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation, which we denote by P_{III}(0,0,4,-4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics. Complex multi-valued solutions on C^* are the natural context for most of the paper, but in the last three chapters real solutions on the positive real line (with or without singularities) are addressed. Results about the asymptotics of real solutions near 0 and near infinity are combined with results on the global geometry of the moduli spaces of initial data and monodromy data. This leads to a new global picture of all zeros and poles of all real solutions on the positive real line.

math.AG↗

arXiv2014

Orbifold quantum D-modules associated to weighted projective spaces

We construct in an abstract fashion the orbifold quantum cohomology (quantum orbifold cohomology) of weighted projective space, starting from the orbifold quantum 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 to the more difficult case of Fano hypersurfaces in weighted projective space. However, in contrast to the case of weighted projective space itself or a Fano hypersurface in projective space, a "small Birkhoff cell" can appear in the construction; we give an example of this phenomenon.

math.AG↗

arXiv2014

Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data

We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii) holomorphic data, and (iii) monodromy data. This allows us to find all solutions with integral Stokes data. These include solutions associated to nonlinear sigma models (quantum cohomology) or Landau-Ginzburg models (unfoldings of singularities), as conjectured by Cecotti and Vafa.

math.DG↗

arXiv2013

Isomonodromy aspects of the tt* equations of Cecotti and Vafa II. Riemann-Hilbert problem

In Part I (arXiv:1209.2045) we computed the Stokes data, though not the "connection matrix", for the smooth solutions of the tt*-Toda equations whose existence we established by p.d.e. methods. Here we give an alternative proof of the existence of some of these solutions by solving a Riemann-Hilbert problem. In the process, we compute the connection matrix for all smooth solutions, thus completing the computation of the monodromy data. We also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions, clarifying the region of validity of the formulae established earlier by Tracy and Widom. Finally, for the tt*-Toda equations, 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.

math.DG↗

arXiv2012

Some tt* structures and their integral Stokes data

In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (arxiv:1209.2045) we described 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 D-modules for certain complete intersections of Fano type in weighted projective spaces.

math.DG↗

arXiv2010

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

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.

math-ph↗

arXiv2009

Differential equations aspects of quantum cohomology

The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of view as part of a framework which accommodates the KdV equation and other well known integrable systems. In the case of quantum cohomology, the theory is remarkably effective in packaging geometric information; we illustrate this with reference to simple examples of Gromov-Witten invariants, variations of Hodge structure, the Reconstruction Theorem, and the Crepant Resolution Conjecture. Based on lectures given at the summer school "Geometric and Topological Methods for Quantum Field Theory", Villa de Leyva, 2007.

math.DG↗

arXiv2004

Quantum cohomology via D-modules

We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a commutative algebra associated to the (uncompactified) space of rational curves. A standard loop group factorization procedure converts the D-module to Givental's D-module and the commutative algebra to the quantum cohomology algebra. We apply this only to the small quantum cohomology of full flag manifolds and semi-positive toric manifolds, but even in these cases the method is effective. In particular it gives an algorithm (requiring construction of a Groebner basis and solution of a system of o.d.e.) for the quantum product.

math.DG↗

arXiv2001

An update on harmonic maps of finite uniton number, via the zero curvature equation

This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from the two-sphere to a compact Lie group or symmetric space. Extended solutions are equivalent to harmonic maps, but the advantage of extended solutions is the fact that they are solutions to a zero curvature equation. The closely related complex extended solutions (or complex extended frames) are in fact even more advantageous, as has been pointed out by Dorfmeister, Pedit and Wu. We use them to review the existing theory and also to prove some new results concerning harmonic maps into the unitary group.

math.DG↗