تقرير
Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the $\mathfrak{sl}_2$ case
| العنوان: | Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the $\mathfrak{sl}_2$ case |
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| المؤلفون: | Marchal, Olivier, Orantin, Nicolas |
| سنة النشر: | 2019 |
| المجموعة: | Mathematics High Energy Physics - Theory Mathematical Physics Nonlinear Sciences |
| مصطلحات موضوعية: | Mathematical Physics, High Energy Physics - Theory, Nonlinear Sciences - Exactly Solvable and Integrable Systems |
| الوصف: | In this paper, we show that it is always possible to deform a differential equation $\partial_x \Psi(x) = L(x) \Psi(x)$ with $L(x) \in \mathfrak{sl}_2(\mathbb{C})(x)$ by introducing a small formal parameter $\hbar$ in such a way that it satisfies the Topological Type properties of Berg\`ere, Borot and Eynard. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce $\hbar$. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expressed in terms of intersections of tautological classes in the Deligne-Mumford compactification of the moduli space of surfaces. We present a few examples including any Fuchsian system of $\mathfrak{sl}_2(\mathbb{C})(x)$ as well as some elements of Painlev\'e hierarchies. Comment: 39 pages |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1063/5.0002260 |
| URL الوصول: | http://arxiv.org/abs/1901.04344 |
| رقم الأكسشن: | edsarx.1901.04344 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1063/5.0002260 |
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