Geometries from Young Diagrams

TL;DR

This paper explores how the geometry of extra dimensions in holographic string theory can be understood through Young diagrams in the dual super Yang-Mills theory, revealing a combinatorial encoding of bulk geometry.

Contribution

It demonstrates that the structure of Young diagrams naturally encodes the emergent holographic dimensions and local physics in the dual gauge theory.

Findings

Young diagrams represent bulk geometry features.

Localization of gravitons corresponds to specific Young diagram configurations.

The approach provides a combinatorial perspective on holographic emergence.

Abstract

Type IIB string theory on spacetimes that are asymptotically AdS$_5\times$S$^5$ can be defined using four dimensional ${\cal N}=4$ super Yang-Mills theory. Six of the dimensions of the string theory are holographically reconstructed in the Yang-Mills theory. In this article we study how these dimensions and local physics in these dimensions emerge. We reorganize the dynamics of the ${1\over 2}$ BPS sector of the field theory by rewriting it in terms of Schur polynomials. The Young diagram labeling of these polynomials can be viewed as a book keeping device which summarizes how the operator is constructed. We show that aspects of the geometry of the extra holographic dimensions are captured very naturally by the Young diagram. Gravitons which are localized at a specific position in the bulk correspond to boxes added at a specific location on the Young diagram.