DOZZ structure constant in the probabilistic Liouville convention

Fix $0<\gamma <2$, $\mu >0$, and $Q=2\mathrm{/}\gamma +\gamma \mathrm{/}2$ as in the probabilistic Liouville sphere model. Let $\mathrm{\Gamma }$ be Euler's gamma function, defined on $\mathrm{Re}z>0$ by $\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}dt$ and elsewhere by its meromorphic continuation, and let $\mathrm{\ell }(z)=\mathrm{\Gamma }(z)\mathrm{/}\mathrm{\Gamma }(1-z)$. For $0<\mathrm{Re}z<Q$, set

${\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}(z):=\exp \text{\hspace{0.17em}⁣}\left({\int }_0^{\mathrm{\infty }}((Q\mathrm{/}2-z{)}^2{e}^{-t}-\frac{{\sinh }^2((Q\mathrm{/}2-z)t\mathrm{/}2)}{\sinh (\gamma t\mathrm{/}4)\sinh (t\mathrm{/}\gamma )})\frac{dt}{t}\right)\mathrm{.}$

The defining integral converges absolutely on this strip, locally uniformly in $z$, by Absolute convergence of the Liouville Upsilon defining integral. Use the source's entire analytic continuation from this strip, fixed by the following shift relations:

${\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}(z+\gamma \mathrm{/}2)=\mathrm{\ell }(\gamma z\mathrm{/}2)(\gamma \mathrm{/}2{)}^{1-\gamma z}{\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}(z),{\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}(z+2\mathrm{/}\gamma )=\mathrm{\ell }(2z\mathrm{/}\gamma )(\gamma \mathrm{/}2{)}^{4z\mathrm{/}\gamma -1}{\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}(z)\mathrm{.}$

For any complex power of the positive real base in the following expression, use $a^w:=\exp (w\log a)$ with the ordinary real logarithm of $a>0$. For $\bar{\alpha }={\alpha }_1+{\alpha }_2+{\alpha }_3$, the DOZZ structure constant in this convention is the meromorphic function

$C_{\gamma ,\mu }^{\mathrm{D}\mathrm{O}\mathrm{Z}\mathrm{Z}}({\alpha }_1,{\alpha }_2,{\alpha }_3)={(\pi \mu \mathrm{\ell }({\gamma }^2\mathrm{/}4)(\gamma \mathrm{/}2{)}^{2-{\gamma }^2\mathrm{/}2})}^{(2Q-\bar{\alpha })\mathrm{/}\gamma }\frac{{\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}^{\mathrm{\prime }}(0){\prod }_{i=1}^3{\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}({\alpha }_i)}{{\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}(\bar{\alpha }\mathrm{/}2-Q){\prod }_{i=1}^3{\mathrm{{\rm Y}}}_{\gamma \mathrm{/}2}(\bar{\alpha }\mathrm{/}2-{\alpha }_i)}\mathrm{.}$

This is a definition of a special function. It does not assert that a probabilistic three-point correlation equals this function.

Probabilistic Liouville CFT on the sphere

Absolute convergence of the Liouville Upsilon defining integral