Code subspaces for LLM geometries

TL;DR

This paper explores the use of code subspaces in effective field theory around LLM geometries, extending previous work beyond infinite N, and discusses how to extract topology and the ambiguities in defining a global metric operator.

Contribution

It extends the code subspace framework to finite N LLM geometries and analyzes topology extraction and metric operator ambiguities.

Findings

Code subspaces effectively describe nearby states in LLM geometries.

Topology can be inferred from uncertainty and entanglement entropy beyond infinite N.

Global metric operator definition depends on the choice of reference state.

Abstract

We consider effective field theory around classical background geometries with a gauge theory dual, in the class of LLM geometries. These are dual to half-BPS states of $\cal{N}=$ 4 SYM. We find that the language of code subspaces is natural for discussing the set of nearby states, which are built by acting with effective fields on these backgrounds. This work extends our previous work by going beyond the strict infinite $N$ limit. We further discuss how one can extract the topology of the state beyond $N\rightarrow\infty$ and find that uncertainty and entanglement entropy calculations still provide a useful tool to do so. Finally, we discuss obstructions to writing down a globally defined metric operator. We find that the answer depends on the choice of reference state that one starts with. Therefore there is ambiguity in trying to write an operator that describes the metric globally.