# Topological defects when breaking chiral $U(3)_L\otimes U(3)_R$ to $U(1)\otimes U(1)\otimes U(1)$

Consider a chiral fermion flavor symmetry, $U(3)_L\otimes U(3)_R$, such as the flavor symmetry of the approximately massless up, down, and strange quarks. In QCD, this symmetry is spontaneously broken down to $U(3)_V$. Now let us assume that the symmetry $U(3)_L\otimes U(3)_R$ could get completely broken down to $U(1)\otimes U(1)\otimes U(1)$ (from which one axial $U(1)$ symmetry is anomalous).

What kinds of topological defects would we expect? Strings and domain walls will be created, if the anomalous axial $U(1)$ symmetry is explicitly broken down to $Z_N$ and $Z_N$ is spontaneously broken down to nothing, similar to what happens in QCD axion scenarios. But what other topological defects will emerge when breaking the entire large symmetry group? Textures, monopoles, or even more?

• You have to calculate homotopic groups $$\pi_{3}(U_{L}(3)\times U_{R}(3)/U_{V}(1)\times U_{A}(1)) \simeq \pi_{3}(SU_{L}(3)\times SU_{R}(3))$$ (I don't know what is the extra $U(1)$ You've written above). – Name YYY Dec 2 '16 at 9:54
• Thanks for your comment! I am not a mathematician and thus would prefer to find a paper reference instead of calculating the result on my own. I know that $\pi_3(SU(3))=Z$ (textures) and $\pi_2(SU(3)/U(1)^2)=Z^2$ or $\pi_2(SU(2)/U(1))=Z$ (monopoles). However, I haven't seen these kinds of computations anywhere for more complicated groups, i.e. for $SU(3)\otimes SU(3)$ or $SU(2)\otimes SU(2)$ - do you know if this has been done? – Thomas Dec 8 '16 at 16:01
• The issue is solved, since the fundamental group functor takes products to products, i.e. knowing the homotopy groups of $SU(3)$ is sufficient. Thanks again for your help! – Thomas Dec 8 '16 at 17:04