### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 337-380 |

Number of pages | 44 |

Journal | Communications in Mathematical Physics |

Volume | 336 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*336*(1), 337-380. https://doi.org/10.1007/s00220-014-2280-x

**Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa II : Riemann–Hilbert Problem.** / Guest, Martin; Its, Alexander R.; Lin, Chang Shou.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 336, no. 1, pp. 337-380. https://doi.org/10.1007/s00220-014-2280-x

}

TY - JOUR

T1 - Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa II

T2 - Riemann–Hilbert Problem

AU - Guest, Martin

AU - Its, Alexander R.

AU - Lin, Chang Shou

PY - 2015

Y1 - 2015

N2 - 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 + St (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.

AB - 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 + St (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.

UR - http://www.scopus.com/inward/record.url?scp=84925003937&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925003937&partnerID=8YFLogxK

U2 - 10.1007/s00220-014-2280-x

DO - 10.1007/s00220-014-2280-x

M3 - Article

AN - SCOPUS:84925003937

VL - 336

SP - 337

EP - 380

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -