Hyperbolic $2$-spheres with conical singularities, accessory parameters and Kähler metrics on $\mathcal {M}_{0,n}$
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- by Leon Takhtajan and Peter Zograf PDF
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Abstract:
We show that the real-valued function $S_\alpha$ on the moduli space ${\mathcal {M}}_{0,n}$ of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic $2$-sphere with $n\geq 3$ conical singularities of arbitrary orders $\alpha =\{\alpha _1,\dots , \alpha _n\}$, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on ${\mathcal {M}}_{0,n}$ parameterized by the set of orders $\alpha$, explicitly relate accessory parameters to these metrics, and prove that the functions $S_\alpha$ are their Kähler potentials.References
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Additional Information
- Leon Takhtajan
- Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
- Email: leontak@math.sunysb.edu
- Peter Zograf
- Affiliation: Steklov Mathematical Institute, St. Petersburg, 191011 Russia
- Email: zograf@pdmi.ras.ru
- Received by editor(s): March 12, 2002
- Published electronically: December 9, 2002
- Additional Notes: Research of the first author was partially supported by the NSF grant DMS-9802574
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 1857-1867
- MSC (2000): Primary 14H15; Secondary 30F45, 81T40
- DOI: https://doi.org/10.1090/S0002-9947-02-03243-9
- MathSciNet review: 1953529