We numerically construct the Sasaki--Einstein metric on the link of the cone over the second del Pezzo surface $\mathrm{dP}_2$ and two primitive harmonic basic $(1,1)$-forms at its irregular volume-minimizing Reeb vector. Being toric, the metric in symplectic coordinates is encoded in a single convex function on a polygon. We approximate the correction to the canonical Guillemin potential in two ways: polynomial expansion and neural networks. The polynomial fit achieves a held-out mean-squared Monge--Amp\`ere residual below $10^{-13}$, in contrast to the $10^{-2}$ plateau for the non-volume-minimizing regular Reeb vector. We validate our method against closed-form metrics of $Y^{p,q}$ using curvature invariants, and also against the numerical result of Doran et al. (2007) for the K\"ahler--Einstein metric of $\mathrm{dP}_3$ using the Laplacian spectrum of low torus-invariant modes. Our data for the metric and harmonic forms can be used to study warped non-conformal holographic IIB backgrounds, the analogues of the Klebanov--Tseytlin solution on the conifold.
The compact four-dimensional manifold $\mathbb{CP}_2\# 3\overline{\mathbb{CP}_2}$ is known to admit a toric K\"ahler-Einstein metric $g_{\text{KE}}$, but the metric is not known in closed form, which makes it difficult to draw conclusions about its geometry. In this article, we use a combination of analytic and compute...
Timothy Buttsworth, William C. Hadden, Elli Heyes et al.· 0 citations
Let $X$ be a compact Riemann surface of genus $g\geq2$, and let $\M$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $$ \Vol_{g...
Adapting a construction of Kurihara in the setting of Drinfeld modular forms, we define Poincar\'e series on the Drinfeld half-plane $\Omega$. These series are built from products of meromorphic $1$-forms that are naturally associated to geodesics on the Bruhat-Tits tree. We establish convergence under a finiteness con...
Let $(M^n,g)$ carry a conformal product structure whose adapted Weyl connection $D$ preserves orthogonal distributions of ranks $p,q\ge2$. We express the Faraday form $d\theta$ directly in terms of the Weyl curvature of $g$. If $S$ is the orthogonal involution determined by the splitting and \[ \mathcal K_W(X)=\sum_i W...
We study the coordinate-plane Kac walk on $\mathrm{SO}(n)$ with independent uniform rotation angles. For the unnormalized Hilbert-Schmidt Riemannian distance, we prove that the Wasserstein-$2$ Lipschitz coefficient of a block of $2\binom n2$ steps is at most $2n^{-1/200}$ for sufficiently large $n$. This gives a fixed-...
We find Sasaki-Einstein metrics on rational homology $(4n-1)$-spheres for $n>1$ built from cyclic polynomials of index 1 cutting out rational varieties. The Einstein metrics found here are inequivalent to the ones found by Boyer and Galicki in arXiv:math/0311355. Our findings are consequence of an improvement, for hype...
Jaime Cuadros Valle, Joe Lope Vicente· 0 citations
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