Moduli space holography and the finiteness of flux vacua

Thomas W. Grimm

doi:10.1007/jhep10(2021)153

Abstract

A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2, ℂ)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to a Poincaré metric with Sl(2, ℝ) isometry. The crucial part of the bulk theory on the moduli space is a sigma model for group-valued matter fields. It is discussed how this might be coupled to a two-dimensional gravity theory. The classical bulk-boundary matching is then given by the proof of the famous Sl(2) orbit theorem of Hodge theory, which is reformulated in a more physical language. Applying this correspondence to the flux landscape in Calabi-Yau fourfold compactifications it is shown that there are no infinite tails of self-dual flux vacua near any co-dimension one boundary. This finiteness result is a consequence of the constraints on the near boundary expansion of the bulk solutions that match to the boundary data. It is also pointed out that there is a striking connection of the finiteness result for supersymmetric flux vacua and the Hodge conjecture.

Highlights

Motivated by the recent advances in uncovering quantum gravity constraints on effective theories, we argued for a holographic approach to study the field spaces and vacua of valid effective theories
Several of the swampland conjectures, such as the distance conjecture, are constraining the behaviour of effective theories when moving to the asymptotic regions in the scalar field space
In string theory compactifications the complex structure moduli space of Calabi-Yau manifolds provides a very general example of a field space arising in consistent effective theories

Summary

Motivation using asymptotic Hodge theory

We provide the motivation for the construction of the bulk theory and the bulk-boundary matching by introducing some results from asymptotic Hodge theory. While many of the described facts are true for general Kähler manifolds, we will restrict our attention to complex D-dimensional Calabi-Yau manifolds YD. In this cases, the geometry of the complex structure moduli space M of YD can be encoded by the moduli dependence of the (D, 0)-form Ω. We explain how the Hodge decomposition near the boundary can always be encoded by a expansion that is polynomial in the moduli and is best described by a so-called nilpotent orbit. Crucial for developing the bulk theory is the fact that the nilpotent orbits satisfy a set of differential equations We introduce these equations, point out their relation to Nahm’s equations, and discuss an associated action principle.

Hodge norm and Hodge decomposition in the bulk

Near-bounday expansions and nilpotent orbits

Differential equations and constraints from nilpotent orbits

Metric on the moduli space and its near boundary expansion

The bulk theory on the moduli space

Bulk action for matter fields

On the coupling to gravity

Boundary eigenstates

Boundary operators and the phase operator

Classification of boundary theories

Bulk-boundary correspondence

Bulk theory solutions as series expansions