Research interests
My research interests are in Differential Geometry and Geometric Analysis, in special geometries on Riemannian manifolds and gauge theory. In particular: G_2 geometry, SU(3) geometry, instantons, cohomogeneity one manifolds, heterotic systems.
Publications and preprints
[4] Gauge theory on T^*CP^2: explicit Sp(2)-instantons, HYM connections, and Spin(7)-instantons, I. Alonso, J. Madnick, E.A. Windes
We construct and classify SU(3)-invariant primitive Hermitian Yang-Mills connections and Sp(2)-instantons with gauge groups S = S^1 and S = SO(3) over the Calabi manifold X = T^*CP^2, the unique non-flat, complete, cohomogeneity-one hyperkahler 8-manifold. Moreover, in the case of S = S^1, we also classify the SU(3)-invariant Spin(7)-instantons over X in the following sense. Letting ϕ_I, ϕ_J, ϕ_K denote the Spin(7)-structures on X induced from the complex structures I,J,K in the hyperkahler triple, we prove that on each invariant S^1-bundle E_k→X, k∈ℤ, the space of invariant Spin(7)-instantons with respect to ϕ_L forms a one-parameter family modulo gauge. Moreover, every pair of one-parameter families of ϕ_I-, ϕ_J-, and ϕ_K-Spin(7)-instantons intersects only at the unique invariant Sp(2)-instanton on E_k, which is non-flat when k≠0.
[3] New examples of G_2-instantons on R^4xS^3, I. Alonso
Journal of Geometry and Physics 206, 105292 (2024) doi.org/10.1016/j.geomphys.2024.105292 arxiv
We study the existence of SU(2)^2-invariant G_2-instantons on R^4xS^3 with the coclosed G_2-structures found on [arXiv:2209.02761]. We find an explicit 1-parameter family of SU(2)^3-invariant G_2-instantons on the trivial bundle on R^4xS^3 and study its ''bubbling'' behaviour. We prove the existence a 1-parameter family on the identity bundle. We also provide existence results for locally defined SU(2)^2-invariant G_2-instantons.
[2] Coclosed G_2-structures on SU(2)^2-invariant cohomogeneity one manifolds, I. Alonso
Annals of Global Analysis and Geometry 65, 1 (2025) doi.org/10.1007/s10455-024-09981-w arxiv
We consider two different SU(2)^2-invariant cohomogeneity one manifolds, one non-compact M=R^4xS^3 and one compact M=S^4xS^3, and study the existence of coclosed SU(2)^2-invariant G_2-structures constructed from half-flat SU(3)-structures. For each of these manifolds, we prove the existence of a family of coclosed G_2-structures (but not necessarily torsion-free) which is given by three smooth functions satisfying certain boundary conditions and a non-zero parameter. Moreover, any coclosed G_2-structure constructed from half-flat SU(3)-structures is in this family.
[1] On the existence of balanced metrics on six-manifolds of cohomogeneity one, I. Alonso, F. Salvatore
Annals of Global Analysis and Geometry 61, 309–331 (2022) doi.org/10.1007/s10455-021-09807-z arxiv
We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced SU(n)-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull–Strominger system. In this paper, we provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.