Gauging the bulk: generalized gauging maps and holographic codes

TL;DR

This paper introduces a generalized gauging map that transforms global symmetries into local gauge symmetries in holographic codes, enabling new insights into holography, dualities, and symmetry constraints.

Contribution

It develops a generalized gauging map that is an isometry on all sectors, and applies it to create and analyze holographic codes with gauge and global symmetries, revealing new dualities and constraints.

Findings

The generalized gauging map preserves duality between bulk and boundary systems.

Constructed holographic codes with gauge symmetries dual to boundary global symmetries.

Demonstrated codes with arbitrary transversal gate sets for any compact Lie group.

Abstract

Gauging is a general procedure for mapping a quantum many-body system with a global symmetry to one with a local gauge symmetry. We consider a generalized gauging map that does not enforce gauge symmetry at all lattice sites, and show that it is an isometry on the full input space including all charged sectors. We apply this generalized gauging map to convert global-symmetric bulk systems of holographic codes to gauge-symmetric bulk systems, and vice versa, while preserving duality with a global-symmetric boundary. We separately construct holographic codes with gauge-symmetric bulk systems by directly imposing gauge-invariance constraints onto existing holographic codes, and show that the resulting bulk gauge symmetries are dual to boundary global symmetries. Combining these ideas produces a toy model that captures several interesting features of holography - it exhibits a rudimentary sort of dynamical duality, can be modified to demonstrate the relationship between metric fluctuations and approximate error-correction, and serves as an illustration for certain no-go theorems concerning symmetries in holography. Finally, we apply the generalized gauging map to construct codes with arbitrary transversal gate sets - for any compact Lie group, we use a symmetry-preserving truncation scheme to construct covariant finite-dimensional approximate holographic codes.