### 2-Groups in Condensed Matter Physics

#### Posted by John Baez

This blog was born in 2006 when a philosopher, a physicist and a mathematician found they shared an interest in categorification — and in particular, categorical groups, also known as 2-groups. So it’s great to see 2-groups showing up in theoretical condensed matter physics. From today’s arXiv papers:

- J.P. Ang and Abhishodh Prakash, Higher categorical groups and the classification of topological defects and textures.

Abstract.Sigma models effectively describe ordered phases of systems with spontaneously broken symmetries. At low energies, field configurations fall into solitonic sectors, which are homotopically distinct classes of maps. Depending on context, these solitons are known as textures or defect sectors. In this paper, we address the problem of enumerating and describing the solitonic sectors of sigma models. We approach this problem via an algebraic topological method – combinatorial homotopy, in which one models both spacetime and the target space with algebraic objects which are higher categorical generalizations of fundamental groups, and then counts the homomorphisms between them. We give a self-contained discussion with plenty of examples and a discussion on how our work fits in with the existing literature on higher groups in physics.

The fun will really start when people actually synthesize materials described by these materials! Condensed matter physicists are doing pretty well at realizing theoretically possible phenomena in the lab, so I’m optimistic. But I don’t think it’s happened yet.

My friend Chenchang Zhu, a mathematician, has also been working on these things with two physicists. The abstract only briefly mentions 2-groups, but they play a fundamental role in the paper:

- Chenchang Zhu, Tian Lan and Xiao-Gang Wen, Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all (3+1)d topological orders for boson systems.

Abstract.A discrete non-linear $\sigma$-model is obtained by triangulate both the space-time $M^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the complex homomorphisms $\phi \colon M^{d+1} \to K$, with an partition function that is independent of space-time triangulation, then the corresponding non-linear $\sigma$-model will be called a topological non-linear $\sigma$-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms $\phi$) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If $K$ is a space with only non-trivial first homotopy group $G$ which is finite, those topological non-linear $\sigma$-models can realize all $(3+1)d$ bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group $\pi_1(K)=G$. Here, we show that the $(3+1)d$ bosonic topological orders with emergent fermions can be realized by topological non-linear σ-models with $\pi_1(K) =$ finite groups, $\pi_2(K)=\mathbb{Z}_2$, and $\pi_{n > 2}(K)=0$. A subset of those topological non-linear $\sigma$-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of $(3+1)$d bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions.

The cobordism hypothesis, too, is getting into the act in the last sentence!

## Re: 2-Groups in Condensed Matter Physics

Yes, we didn’t do too badly with our original choices. Categorified geometry is flourishing, as is categorified logic.