Moduli spaces of Calabi-Yau d-folds as gravitational-chiral instantons

Abstract

Motivated by the swampland program, we show that the Weil-Petersson geometry of the moduli space of a Calabi-Yau manifold of complex dimension d ≤ 4 is a gravitational instanton (i.e. a finite-action solution of the Euclidean equations of motion of gravity with matter). More precisely, the moduli geometry of Calabi-Yau d-folds (d ≤ 4) describes instantons of (E)AdS Einstein gravity coupled to a standard chiral model.From the point of view of the low-energy physics of string/M-theory compactified on the Calabi-Yau X, the various fields propagating on its moduli space are the couplings appearing in the effective Lagrangian "Image missing".

Weak gravity bounds in asymptotic string compactifications

“…Now we prepare for a more general approach that uses asymptotic Hodge theory and does not rely on the knowledge of a prepotential. Being a vast subject, we will only introduce the JHEP06(2021)162 tools we need and refer the interested reader to [55,56] for the mathematical literature or [4,8,9,[12][13][14][15][16][17][18] for recent applications in the Swampland programme. We begin by briefly reviewing the concept of Hodge structure and introduce the nilpotent orbit theorem which allows for a first approximation of the holomorphic three-form Ω.…”

“…As described by (4.7) we can obtain these states fromã 0 by applying lowering operators, where we denote the new lowering operator associated with s by N − II . 18 From the discrete data d i we can infer that each lowering operator N − II , N − III , N − IV can be applied once onã 0 , resulting in eight different elementary charges for Q G in total. Their properties have been 18 The lowering operators N − III and N − IV are not precisely the same matrices as in the previous example, since the procedure to construct them out of the log-monodromy matrices Nt and Nu changes when the saxion s is also involved in the limit.…”

“…More recently, it was also suggested in[17,18] that a certain structural principle of Hodge theory should be translated to a Swampland criterium.…”

Swampland geometry and the gauge couplings

“…The tricky case is M vector , which is a special Kähler manifold. That the Einstein equations (6.7) hold in this case was shown in [28]. Finally the last statement follows from the fact that evaluated on the appropriate classical solution the action density is proportional to the volume form [28], so that finite action is equivalent to finite volume of M (i) , which is one of the swampland conjectures.…”

“…More in detail: for N ≥ 3 SUGRA item 2. is shown in section 4.9 of [36] and for N = 2 SUGRA in refs. [15,28]. In N = 1 SUGRA the gauge coupling τ ij is a holomorphic map between the scalars' Kähler manifold and the Siegel variety Sp(2h, Z)\Sp(2h, R)/U(h).…”