Numerical metrics, curvature expansions and Calabi-Yau manifolds

Abstract

We discuss the extent to which numerical techniques for computing approximations to Ricci-flat metrics can be used to investigate hierarchies of curvature scales on Calabi-Yau manifolds. Control of such hierarchies is integral to the validity of curvature expansions in string effective theories. Nevertheless, for seemingly generic points in moduli space it can be difficult to analytically determine if there might be a highly curved region localized somewhere on the Calabi-Yau manifold. We show that numerical techniques are rather efficient at deciding this issue.

Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning

“…This is important as it shows that the numerical approximations that are being obtained are close to full solutions to the system, rather than just being metrics which approximate the desired properties in a system which admits no exact solution. In particular, the method utilizing extremization of an energy functional [9,13] can rest on Yau's theorem [1], and Donaldson's approach [4][5][6][7] can use certain results pertaining to balanced metrics and the algebraic ansatz for the Kähler potential [4,33]. In the case of more general SU(3) structures, there are, to our knowledge, no such existence theorems available.…”

“…Unfortunately, Ricci-flat metrics on CY three-folds, being high-dimensional structures with no continuous isometries, are seemingly prohibitively hard to find analytically. This has led to the development of a number of numerical, and other, approaches to computing these and related quantities in the literature [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. One common feature which is seen in such work is that each computation of a metric is performed at one point in moduli space at a time.…”

Chern-Simons invariants and heterotic superpotentials

“…Although it is not guaranteed, it is also not unreasonable to think that the connection on this bundle might also extend holomorphically to the ambient space. Indeed, the ansatze that are used to describe fiber metrics in numerical work [48][49][50][51][52][53][54][55][56][57][58] are…”

Generalized vanishing theorems for Yukawa couplings in heterotic compactifications

“…Unfortunately, this would require the inclusion of normalization considerations arising from the matter field Kähler potential and this object is extremely difficult to compute. Some approximations to the Kähler potential do exist in the literature [17][18][19][20][21], and its computation is the ultimate goals of much of the work on numerical approaches to Ricci-flat metrics and gauge bundle connections on Calabi-Yau manifolds [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Nevertheless, it is fair to say that there is still very little known about the physically normalized Yukawa couplings.…”