Revisiting Atiyah-Hitchin manifold in the generalized Legendre transform

We revisit construction of the Atiyah-Hitchin manifold in the generalized Legendre transform approach. This was originally studied by Ivanov and Roček [I. T. Ivanov, M. Roček, Commun. Math. Phys. 182, 291 (1996)] and subsequently investigated more by Ionas [R. A. Ionas, arXiv:0712.3598], in the latter of which the explicit forms of the Kähler potential and the Kähler metric are calculated. There is a difference between the former and the latter. In the generalized Legendre transform approach, a Kähler potential is constructed from the contour integration of one function with holomorphic coordinates. The choice of the contour in the latter is different from that in the former; this difference may yield a discrepancy in the Kähler potential and eventually in the Kähler metric. We show that the former only gives the real Kähler potential, which is consistent with its definition, while the latter yields the complex one. We derive the Kähler potential and the metric for the Atiyah-Hitchin manifold in terms of holomorphic coordinates for the contour considered by Ivanov and Roček for the first time.