M-theoretic Genesis of Topological Phases

TL;DR

This work establishes a novel bridge between M-theory and (2+1)d topological phases by encoding topological order data in non-hyperbolic 3-manifolds via wrapped M5-branes. It provides a systematic algorithm to extract modular data (S,T, GSD) from 3-manifold topology, reproducing all unitary bosonic topological orders up to rank 4 and extending to fermionic and non-unitary theories through nontrivial $H_1(M,\mathbb{Z}_2)$ via 1-form symmetry gauging. By linking Dehn surgery and Seifert-fibered manifolds to explicit 3d ${\cal N}=2$ gauge theories, the approach yields UV-complete descriptions whose IR physics is a TFT[M], with a flat-connection–loop-operator dictionary that recovers the anyon data. The framework suggests a broad program: all (2+1)d topological phases may be geometrically realized by M5-branes on suitable non-hyperbolic 3-manifolds, with extensions to higher rank and non-unitary/fermionic theories and potential new mathematics of non-hyperbolic manifolds. The work also notes avenues beyond modular data and conjectures connecting 3-manifold invariants to universal properties of topological phases, inviting further cross-pollination between physics and 3-manifold topology.

Abstract

We present a novel M-theoretic approach of constructing and classifying anyonic topological phases of matter, by establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds. In this construction, the topological phases emerge as macroscopic world-volume theories of M5-branes wrapped around certain types of non-hyperbolic 3-manifolds. We devise a systematic algorithm for identifying the emergent topological phases from topological data of the internal wrapped 3-manifolds. As a benchmark of our approach, we reproduce all the known unitary bosonic topological orders up to rank 4. Remarkably, our construction is not restricted to an unitary bosonic theory but it can also generate fermionic and/or non-unitary topological phases in an equivalent fashion. Hence, we pave a new route toward the classification of topological phases of matter.

Figures (1)

• Figure 1: Dehn surgery presentation of the Seifert fiber manifold $S^2(\frac{p_1}{q_1},\frac{p_2}{q_2},\frac{p_3}{q_3})$ using a link with 4 components.