Bulk Reconstruction in Moduli Space Holography

TL;DR

This paper develops a holographic method to reconstruct solutions in a non-linear sigma-model related to moduli space, revealing connections to Calabi-Yau periods and boundary conditions in string theory compactifications.

Contribution

It introduces a holographic approach to solve a sigma-model without geometric assumptions, deriving unique near-boundary solutions linked to Calabi-Yau moduli.

Findings

Explicit solutions for boundary data modeling Calabi-Yau periods

Demonstration of unique near-boundary reconstruction with a single matching condition

Connection of solutions to conifold, large complex structure, and Tyurin degeneration points

Abstract

It was recently suggested that certain UV-completable supersymmetric actions can be characterized by the solutions to an auxiliary non-linear sigma-model with special asymptotic boundary conditions. The space-time of this sigma-model is the scalar field space of these effective theories while the target space is a coset space. We study this sigma-model without any reference to a potentially underlying geometric description. Using a holographic approach reminiscent of the bulk reconstruction in the AdS/CFT correspondence, we then derive its near-boundary solutions for a two-dimensional space-time. Specifying a set of $ Sl(2,\mathbb{R})$ boundary data we show that the near-boundary solutions are uniquely fixed after imposing a single bulk-boundary matching condition. The reconstruction exploits an elaborate set of recursion relations introduced by Cattani, Kaplan, and Schmid in the proof of the $Sl(2)$-orbit theorem. We explicitly solve these recursion relations for three sets of simple boundary data and show that they model asymptotic periods of a Calabi--Yau threefold near the conifold point, the large complex structure point, and the Tyurin degeneration.