We revisit construction of the Atiyah-Hitchin manifold in the generalized Legendre transform approach. This is originally studied by Ivanov and Rocek and is subsequently investigated more by Ionas, 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 the former's one, whose 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.
The Atiyah-Hitchin manifold is the moduli space of parity inversion symmetric charge two SU(2) monopoles in Euclidean space. Here a hyperbolic analogue is presented, by calculating the boundary metric on the moduli space of parity inversion symmetric charge two SU(2) monopoles in hyperbolic space. The calculation of the metric is performed using a twistor description of the moduli space and the result is presented in terms of standard elliptic integrals.
We construct a new class of stationary exact solutions to five-dimensional Einstein-Gauss-Bonnet gravity. The solutions are based on four-dimensional self-dual Atiyah-Hitchin geometry. We find analytical solutions to the five-dimensional metric function that are regular everywhere. We find some constraints on the possible physical solutions by investigating the solutions numerically. We also study the behavior of the solutions in the extremal limits of the Atiyah-Hitchin geometry. In the extremal limits, the Atiyah-Hitchin metric reduces to a bolt structure and Euclidean Taub-NUT space, respec…
我们概述了荷为2的SU（2）BPS单极子的模空间上的Atiyah-Hitchin度规的构造，首先作为C^3中遵循Donaldson的代数曲线，然后作为Toda场方程在连续大N极限下的解。我们采用扭量方法来解决基本的一致化问题，通过广义Legendre变换得到度规的Kahler坐标和Kahler势。我们还评论了扭曲和量子模空间的塞伯格-威滕构造之间的联系，因为它们出现在三维超对称规范理论中，并简要讨论了在大N Toda理论的背景下C^3中代数曲线的一致化。（根据1998年9月在Buckow举行的关于基本粒子理论的第32届Ahrenshoop国际研讨会、在Crete举行的关于量子场论的第21次三角会议和在Corfu举行的关于规范理论的量子方面的TMR会议、超对称性和统一性的讲话;将在Fortschritte der Physik的会议录中发表。
We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For higher $n$, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on of one the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperkähler m…
We obtain $D_k$ ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as non-linear superpositions of the moduli space of centred $SU(2)$ monopoles, equipped with the Atiyah-Hitchin metric, and $k$ copies of the Taub-NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we…
We study Heterotic supergravity on Hyper-Kahler manifolds in the presence of non-trivial warping and three form flux with Abelian bundles in the large charge limit. We find exact, regular solutions for multi-centered Gibbons-Hawking spaces and Atiyah-Hitchin manifolds. In the case of Atiyah-Hitchin, regularity requires that the circle at infinity is of the same order as the instanton number, which is taken to be large. Alternatively there may be a non-trivial density of smeared five branes at the bolt.
The Atiyah-Drinfeld-Hitchin-Manin (ADHM) transform and its various generalizations are examples of non-linear integral transforms between finite-dimensional moduli spaces. This note describes a natural infinite-dimansional generalization, where the transform becomes a map from boundary data to a family of solutions of the self-duality equations in a domain.
We construct a noncommutative Kähler manifold based on a non-linear perturbations of Moyal integrable deformations of $D=4$ self-dual gravity. The deformed Kähler manifold preserves all the properties of the commutative one, and we obtain the associated noncommutative Kähler potential using the Moyal deformed gravity approach. We apply this construction to the Atiyah-Hitchin metric and its Kähler potential, which is useful in the description of interactions among magnetic monopoles at low energies.
