dorsal/arxiv
View SchemaSpectral Curves and Whitham Equations in Isomonodromic Problems of Schlesinger Type
| Authors | Kanehisa Takasaki |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9704004 |
| URL | https://arxiv.org/abs/solv-int/9704004 |
| Journal | Asian J.Math. 4 (2) (1998), 1049-1078 |
Abstract
It has been known since the beginning of this century that isomonodromic problems --- typically the Painlev\'e transcendents --- in a suitable asymptotic region look like a kind of ``modulation'' of isospectral problem. This connection between isomonodromic and isospectral problems is reconsidered here in the light of recent studies related to the Seiberg-Witten solutions of $N = 2$ supersymmetric gauge theories. A general machinary is illustrated in a typical isomonodromic problem, namely the Schlesinger equation, which is reformulated to include a small parameter $\epsilon$. In the small-$\epsilon$ limit, solutions of this isomonodromic problem are expected to behave as a slowly modulated finite-gap solution of an isospectral problem. The modulation is caused by slow deformations of the spectral curve of the finite-gap solution. A modulation equation of this slow dynamics is derived by a heuristic method. An inverse period map of Seiberg-Witten type turns out to give general solutions of this modulation equation. This construction of general solution also reveals the existence of deformations of Seiberg-Witten type on the same moduli space of spectral curves. A prepotential is also constructed in the same way as the prepotential of the Seiberg-Witten theory.
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"abstract": "It has been known since the beginning of this century that isomonodromic\nproblems --- typically the Painlev\\\u0027e transcendents --- in a suitable\nasymptotic region look like a kind of ``modulation\u0027\u0027 of isospectral problem.\nThis connection between isomonodromic and isospectral problems is reconsidered\nhere in the light of recent studies related to the Seiberg-Witten solutions of\n$N = 2$ supersymmetric gauge theories. A general machinary is illustrated in a\ntypical isomonodromic problem, namely the Schlesinger equation, which is\nreformulated to include a small parameter $\\epsilon$. In the small-$\\epsilon$\nlimit, solutions of this isomonodromic problem are expected to behave as a\nslowly modulated finite-gap solution of an isospectral problem. The modulation\nis caused by slow deformations of the spectral curve of the finite-gap\nsolution. A modulation equation of this slow dynamics is derived by a heuristic\nmethod. An inverse period map of Seiberg-Witten type turns out to give general\nsolutions of this modulation equation. This construction of general solution\nalso reveals the existence of deformations of Seiberg-Witten type on the same\nmoduli space of spectral curves. A prepotential is also constructed in the same\nway as the prepotential of the Seiberg-Witten theory.",
"arxiv_id": "solv-int/9704004",
"authors": [
"Kanehisa Takasaki"
],
"categories": [
"solv-int",
"hep-th",
"math.QA",
"nlin.SI",
"q-alg"
],
"journal_ref": "Asian J.Math. 4 (2) (1998), 1049-1078",
"title": "Spectral Curves and Whitham Equations in Isomonodromic Problems of Schlesinger Type",
"url": "https://arxiv.org/abs/solv-int/9704004"
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