# Chiral Fermion Problem and the String Net Model

In Xiao-Gang Wen's book "Quantum Field Theory of Many-Body Systems", he mentions that

(the string-net condensation picture)...has a problem: we do not yet know how to produce the $SU(2)$ part of the standard model due to the chiral fermion problem.

I have been trying to research the chiral fermion problem and its connection to string-net condensation, but I can't find any good resources/descriptions on any sites or articles. A summary or link(s) on what this problem is and how it relates to the string-net model would be greatly appreciated.

The string-net condensation is a general construction to obtain gauge fields and fermions. The chiral fermion problem refers to the fact that in the Standard Model (SM), the SU(2) gauge field only couples to the left-handed fermions but not the right-handed fermions. However in the (early version of) string-net condensation, the emergent gauge field will always couple to both handed fermions, which is not the chiral coupling as expected in the SM. So if we wish to understand the SM from the string-net condensation approach, we need to resolve this chiral fermion problem. The major obstacle was to construct a chiral gauge theory without gauge anomaly (no matter perturbative or global anomaly). If one can find a chiral gauge theory that is free of anomaly, then it can be readily put on the lattice and formulated as a string-net condensation.

Recently, substantial progress has been made to solve the chiral fermion problem, enlighten by the theory of symmetry protected topological (SPT) state. It is realized that the gauge anomaly are actually classified by the SPT orders [Wen, 2013a], and the SPT classification can be reduced by the interaction [Fidkowski, Kitaev, 2009; 2010]. This means under certain conditions, the perturbative gauge anomaly can be removed by the fermion interaction (at the UV cut-off energy scale). Along this line of though, Prof. Wen argued that the SO(10) chiral gauge theory (with $16n$ chiral fermions) is actually anomaly free under interaction. Therefore it has a non-perturbative lattice definition [Wen, 2013b], which can be further formulated as string-net condensation. Because the SO(10) chiral gauge theory is a Grand Unification Theory (GUT) which includes the SM, so this result (if verified) would solve the chiral fermion problem and eventually unify all matters and gauge forces under the string-net condensation framework. There are also several following-up works [Wang, Wen, 2013; You, BenTov, Xu, 2014; You, Xu, 2014] in support of Prof. Wen's result using different approaches.