The price index is given by $c(p_L, p_H,1)$ which is the minimal cost of producing 1 unit of output. It is given by:
$$
\min p_L Y_L + p_H Y_H \text{ s.t. } \left[\gamma_L Y_L^{(\varepsilon -1)/\varepsilon} + \gamma_H Y_H^{(\varepsilon-1)/\varepsilon}\right]^{\frac{\varepsilon}{1 - \varepsilon}} = 1.
$$
The first order conditions give:
$$
\begin{align*}
&p_L = \lambda Y^{-1} \gamma_L Y_H^{-1/\varepsilon}\\
&p_H = \lambda Y^{-1} \gamma_H Y_L^{-1/\varepsilon}.
\end{align*}
$$
This gives:
$$
Y_H = \left(\frac{p_L}{p_H}\right)^\varepsilon \left(\frac{\gamma_H}{\gamma_L}\right)^\varepsilon Y_L.
$$
Substituting into the constraint gives:
$$
Y_L = \gamma_L^{\varepsilon} p_L^{-\varepsilon} \left[\gamma_L^\varepsilon p_L^{1- \varepsilon} + \gamma_H^\varepsilon p_H^{1 - \varepsilon} \right]^{-\frac{\varepsilon}{\varepsilon-1}}.\\
Y_H = \gamma_H^{\varepsilon} p_H^{-\varepsilon} \left[\gamma_L^\varepsilon p_L^{1- \varepsilon} + \gamma_H^\varepsilon p_H^{1 - \varepsilon} \right]^{-\frac{\varepsilon}{\varepsilon-1}}.
$$
As such,
$$
c(p_L, p_H, 1) = p_L Y_L + p_H Y_H = \left[\gamma_L^\varepsilon p_L^{1 - \varepsilon} + \gamma_H^\varepsilon p_H^{1 - \varepsilon}\right]^{\frac{1}{\varepsilon - 1}}.
$$