Moduli Space Holography and the Finiteness of Flux Vacua

TL;DR

This paper proposes a holographic framework for understanding string moduli spaces, demonstrating that flux vacua are finite near boundaries, and connects this to Hodge theory and the Hodge conjecture.

Contribution

It develops a boundary-bulk correspondence for moduli spaces in string theory, providing a new proof of flux vacua finiteness and linking it to Hodge theory.

Findings

No infinite tails of flux vacua near boundaries in Calabi-Yau fourfolds.

Boundary data characterized by Hilbert space and SL(2,C) algebra.

Connection between flux vacua finiteness and Hodge conjecture.

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 Poincare 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.