Elliptic K3 Surfaces at Infinite Complex Structure and their Refined Kulikov models

Abstract

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. 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.Email: seungjoolee at ibs.re.kr, timo.weigand at desy.de

On Towers of Light States at Infinite Distance

“…More recently, elliptic Kulikov models of Type III have also been refined in the same vein into Type III.a and III.b models [46]. 8 For a concise description of this refinement, let us first consider the Kulikov Weierstrass model Y by blowing down an elliptic Kulikov model X along the fiber, i.e., by contracting all the exceptional fibral curves.…”

“…or else generic singular fibers in codimension zero (in absence of non-minimal fibers in codimension one). It has indeed been shown [46] that the former can lead to degenerations of Type II.a (if γ = 0) or of Type III (if γ > 0), 9 and the latter, to those of Type II.b. As it turns out, however, the refinement of elliptic Type III Kulikov models is more naturally seen from the central element Y 0 (as opposed to Ŷ0 ), of which decomposition (3.2) can be proven to take a chain form.…”

“…The refined classification is then motivated by the types of the generic fibers over the base components B p , which are necessarily of Kodaira types I np for X to be a Kulikov model. 11 One can then show for any Type III Kulikov Weierstrass models that n p > 0 for the middle components, and hence, elliptic Type III models are naturally refined by the degeneration types of the end components Y 0 and Y P as follows [46].…”

“…To be precise, γ > 0 together with α = β = 0 guarantee that the associated Kulikov model is of Type III. If α, β, and γ are all strictly positive, even a Type I Kulikov model might arise, as exemplified in[46].…”

Membrane Limits in Quantum Gravity

“…The Emergent String Conjecture [20] is a refinement of the SDC proposing that infinite distance limits in moduli space are either pure decompactification limits or signal a transition to a duality frame determined by a unique emergent critical weakly coupled string such that T str ∼ M 2 KK . The Emergent String Conjecture has been scrutinized in the context of the Kähler moduli space of F/M/IIA-theory in 6D/5D/4D in [20][21][22], in the complex structure moduli space of F-theory in 8D in [23,24], and for the 4D N = 2 hypermultiplet moduli space in [25,26]. The case of M-theory on G 2 manifolds was treated in [27], while 4D N = 1 F-theory was studied in [28], with the leading quantum corrections included in [29].…”