A plethora of K3 metrics

Abstract

We extend our recent study of K3 metrics near the T 4 /Z 2 orbifold locus to the other torus orbifold loci. In particular, we provide several new constructions of K3 surfaces as hyper-Kähler quotients, which yield new formulae for K3 metrics. We then relate these to the construction of [1]. As a corollary, we derive infinitely many constraints on the (as yet unknown) BPS spectra of the Minahan-Nemeschansky SCFTs with E n global symmetry. Specifically, we find linear combinations of E n characters (evaluated at different points) hiding within K3 metrics and we compute their second order Taylor expansions. We also find novel strong relationships between the BPS spectra of these SCFTs, as well as with that of the SU (2) N f = 4 SCFT. Finally, we provide a new derivation of the class S constructions of these SCFTs and state some experimental observations regarding their BPS spectra.

A Panorama Of Physical Mathematics c. 2022

“…The main difficulty in this approach is that of determining the relevant spectral networks and BPS spectrum, since the D3 probe theory is not a standard quantum field theory. Recently this idea has been pushed further in the context of little string theory in [1009,1010]. It would be a major achievement of the mathematics-physics dialogue if these developments led to a tractable and exact formula for the metric on any smooth nondegenerate K3 surfaces.…”

Calabi-Yau metrics, CFTs and random matrices

“…Further work includes extending our analysis to the non-scalar spectrum, understanding the interplay of the explicit spectrum with the conformal [5][6][7][8][9][10][11] and geometric [12][13][14][15] bootstrap programmes, exploring whether RMT sheds light on interacting CFTs in an averaged sense or if it can be used to understand the "typical" properties of Calabi-Yau compactifications [16][17][18][19][20][21][22] 1 At least in six dimensions and higher -see [1][2][3] for progress on computing K3 metrics.…”

Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning

“…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.…”