[Paper Review] Cornering Relative Symmetry Theories

Mirjam Cvetič, Ron Donagi+3 more|arXiv (Cornell University)|Aug 22, 2024

Origins and Evolution of Life4 citations

TL;DR

This paper introduces a nested framework of relative symmetry theories to describe d-dimensional quantum field theories (QFTs) localized at corners of higher-dimensional bulk systems, where symmetry data is encoded in a (d+1)-dimensional symmetry topological field theory (SymTFT). By analyzing string-theoretic constructions with non-gapped bulk sectors, the authors show how filtering through scaling limits leads to a hierarchy of relative QFTs and boundary conditions, with the original QFT emerging as a corner mode in a D=d+m dimensional system, unifying top-down and bottom-up perspectives on global symmetries in QFTs with finite and continuous symmetries.

ABSTRACT

The symmetry data of a $d$-dimensional quantum field theory (QFT) can often be captured in terms of a higher-dimensional symmetry topological field theory (SymTFT). In top down (i.e., stringy) realizations of this structure, the QFT in question is localized in a higher-dimensional bulk. In many cases of interest, however, the associated $(d+1)$-dimensional bulk is not fully gapped and one must instead consider a filtration of theories to reach a gapped bulk in $D = d+m$ dimensions. Overall, this leads us to a nested structure of relative symmetry theories which descend to coupled edge modes, with the original QFT degrees of freedom localized at a corner of this $D$-dimensional bulk system. We present a bottom up characterization of this structure and also show how it naturally arises in a number of string-based constructions of QFTs with both finite and continuous symmetries.

Motivation & Objective

• To unify top-down string-theoretic constructions of QFTs with bottom-up symmetry theory frameworks by extending the SymTFT paradigm to non-gapped, non-topological bulk systems.
• To address the challenge of describing QFTs with finite and continuous global symmetries when the associated (d+1)-dimensional bulk is not fully gapped, requiring a filtration of theories.
• To characterize the emergence of d-dimensional QFTs as corner modes in a D=d+m dimensional system, where symmetry data is encoded through a hierarchy of relative QFTs and boundary conditions.
• To formalize the role of scaling limits in extracting gapped or free symmetry theories from non-gapped bulk systems, generalizing previous work on free-field SymTh.

Proposed method

• Introduces a nested structure of relative symmetry theories, where a d-dimensional QFT arises as a corner mode in a D=d+m dimensional bulk system with multiple layers of boundary conditions and defect operators.
• Uses a top-down approach based on string compactifications on Calabi-Yau cones, where the radial direction of the cone corresponds to a flow from a bulk symmetry theory to the boundary QFT.
• Applies scaling limits to non-gapped bulk theories S_{d+1}(g) to extract effective symmetry theories (SymTh), allowing for both gapped and gapless regimes.
• Characterizes the decomposition of the original QFT as a corner of a quiver-like structure involving multiple relative QFTs T_d, boundary conditions B_d, and symmetry theories S_{d+1}, S_{d+2}, etc.
• Employs formal topological field theory techniques with non-compact gauge groups to describe the resulting SymTh, generalizing earlier frameworks based on finite or compact symmetries.
• Demonstrates that the corner QFT is recovered via a sequence of boundary conditions and defect operators, with the partition function computed as a path integral over the full D-dimensional system.

Experimental results

Research questions

• RQ1 How can the symmetry data of a d-dimensional QFT be systematically captured in a higher-dimensional bulk theory when the bulk is not fully gapped?
• RQ2 What is the role of scaling limits in extracting a gapped or free symmetry theory from a non-gapped bulk system in string-theoretic constructions?
• RQ3 How does the corner QFT emerge as a relative theory in a nested hierarchy of symmetry and boundary theories in D=d+m dimensions?
• RQ4 In what way do string-based constructions naturally lead to a filtration of relative symmetry theories rather than a single SymTFT?
• RQ5 How can the formalism of SymTFT be generalized to include continuous symmetries and non-compact gauge sectors in the bulk?

Key findings

• The d-dimensional QFT emerges as a corner mode in a D=d+m dimensional system, with symmetry data encoded in a nested hierarchy of relative QFTs and boundary conditions.
• Scaling limits of non-gapped bulk theories S_{d+1}(g) yield effective symmetry theories (SymTh), generalizing previous frameworks based on free fields or compact gauge groups.
• String-theoretic constructions naturally realize this nested structure, with singularities in the extra-dimensional geometry corresponding to relative QFTs and bulk modes to non-topological boundary conditions.
• The corner QFT is recovered via a sequence of topological and non-topological boundary conditions, with the partition function independent of the interval length due to the topological nature of the SymTFT.
• The framework unifies top-down and bottom-up perspectives, showing that the same corner QFT can be derived from both string compactifications and formal symmetry theory constructions.
• The formalism accommodates both finite and continuous global symmetries, with continuous symmetries described via non-compact gauge groups in the bulk, extending prior work on finite symmetries.