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I'm reading about the Gauss hypergeometric differential equation: \[ x(1-x)y'' + (c - (a + b + 1)x)y' - aby =0 \]

Apparently this is a kind of Fuchsian differential equation: \[ \left(\begin{array}{c} \frac{dy}{dx} \\\\ \frac{dz}{dx} \end{array} \right) = \left(\begin{array}{cc} 0 & 1 \\\\ -ab & \frac{c_1}{x} + \frac{c_2}{x-1}\end{array} \right) \left(\begin{array}{c} y \\\\z \end{array} \right) \]

whose sections define a local system over a sphere with 3 punctures, $\mathbb{C}\backslash \{0,1 \infty \}$. We can worry about the monodromy of ODE solutions about these three points $0,1,\infty$.

The paper I'm reading says Fuchsian equations may be is solved by a Selberg-like integral.

\[ y(x)= \int_\Delta s^{c-b}(s-1)^{c-a-1} (s-x)^{-b} \, ds \]

Where the integral is over a "twisted cycle" $\Delta$ in $H_1(\mathbb{C}\backslash\{ 0,1,x\}, L^* )$.


My trouble is understanding what these twisted cycles look like. The local system $L^*$ is dual to the local system $L$ generated by sections $s \mapsto s^{c-b}(s-1)^{c-a-1} (s-x)^{-b} $, which is multivalued. The values form a discrete set with (possibly countably) many fibers over a point in $\mathbb{C}\backslash\{ 0,1,x\}$ so $L$ should be dual to a branched cover over the three-punctured sphere possibly with more information. Then maybe $H_1(\mathbb{C}\backslash\{ 0,1,x\}, L^*)$ is the collection of loops in this branched cover.


How is $H_1(\mathbb{C}\backslash\{ 0,1,x\}, L^*)$ different from $H_1(\mathbb{C}\backslash\{ 0,1,x\})$ ?

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