Reconstructing Atiyah-Hitchin manifold in the generalized Legendre transform

Abstract

We revisit the Atiyah-Hitchin manifold using the generalized Legendre transform approach. Originally it is examined by Ivanov and Roček, and it has been further explored by Ionaș, with a particular focus on calculating the explicit forms of the Kähler potential and the Kähler metric. Notably, there exists a distinction between the former study and the latter. In the framework of the generalized Legendre transform approach, a Kähler potential is formulated through the contour integration of a specific function with holomorphic coordinates. It’s essential to note that the choice of the contour in the latter differs from that in the former. This discrepancy in contour selection may result in variations in both the Kähler potential and, consequently, the Kähler metric. Our findings demonstrate that the former exclusively yields the real Kähler potential, aligning with its defined properties. In contrast, the latter produces a complex Kähler potential. We present the derivation of the Kähler potential and metric for the Atiyah-Hitchin manifold in terms of holomorphic coordinates, considering the contour specified by Ivanov and Roček.