Physics of infinite complex structure limits in eight dimensions

Abstract

We investigate infinite distance limits in the complex structure moduli space of F-theory compactified on K3 to eight dimensions. While this is among the simplest possible arenas to test ideas about the Swampland Distance Conjecture, it is nevertheless non-trivial enough to improve our understanding of the physics for these limiting geometries, including phenomena of emergence. It also provides a perspective on infinite distance limits from the viewpoint of open strings. The paper has two quite independent themes. In the main part we show that all degenerations of elliptic K3 surfaces at infinite distance as analysed in the companion paper [1] can be interpreted as (partial) decompactification or emergent string limits in F-theory, in agreement with the Emergent String Conjecture. We present a unified geometric picture of the possible towers of states that can become light and illustrate our general claims via the connection between Kulikov models of degenerating K3 surfaces and the dual heterotic string. As an application we classify the possible maximal non-abelian Lie algebras and their Kac-Moody and loop extensions that can arise in the infinite distance limits. In the second part we discuss the infinite distance behaviour of certain exact quartic gauge couplings. We encounter a tension with the hypothesis that effective couplings should be fully generated by integrating out massive states. We show that by appropriately renormalizing the string coupling, at least partial emergence can be achieved.

Elliptic K3 surfaces at infinite complex structure and their refined Kulikov models

“…With this motivation in mind, this article characterises the Weierstrass models for elliptic K3 surfaces at infinite distance in complex structure moduli space, in a form suitable for a physics interpretation in the companion paper [1]. Our results have been obtained independently of the recent analysis of [39][40][41][42][43], but to the best of our understanding are compatible with them wherever the two approaches overlap.…”

“…We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in the companion paper [1]. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra.…”

“…A chain of blowups can always be found, possibly up to base change, to remove the non-minimal singularity at the cost of degenerating the surface to one with several components, which turn out to be exactly of the form characterised in sections 3.1 and 3.2. For the interpretation of the physics in [1] it will be very important that the intersection loci of these surfaces can be assumed to be disjoint from the special fibers of the Weierstrass model. This can be achieved, if necessary, by additional blowups allowed in turn by an appropriate base change.…”

“…Examples of this type are related to fibers of non-minimal Kodaira type over loci of codimension two on the base of an elliptic Calabi-Yau threefold. 1 The present work is dedicated to the geometry of complex structure deformations at infinite distance. Our focus will be on such degenerations for elliptically fibered K3 surfaces.…”

“…The physics motivation for analysing infinite distance degenerations stems from the desire to understand the boundaries of moduli space for theories of quantum gravity, such as M-or F-theory compactified on the K3 surface [1]. The Swampland approach to quantum gravity [11], reviewed in [12][13][14][15], makes specific predictions for the behaviour of a consistent gravity theory at infinite distance in its moduli space.…”