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Sixth Painlev\'e Equation, Universal Elliptic Curve, and Mirror of $\bold{P}^2$
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An algebro-geometric setting for the study of the Painlev\'e VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two. Relations with the theory of elliptic functions and the quantum cohomology of projective plane are discussed.
Forward citations
Cited by 2 Pith papers
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Modular transformations of tau functions and conformal blocks on the torus
The paper derives the modular connection constant for tau functions on the one-punctured torus and obtains an exact closed formula for the c=1 Virasoro modular kernel.
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Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlev\'e VI distribution
The full persistence distribution in 1D Ising coarsening equals a Pfaffian Fredholm determinant of the sech kernel and is controlled by a Painlevé VI equation that is the mean curvature of a Bonnet surface.
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