We know that the Spin group is quite a useful concept in physics. For example, Spin$(3)=SU(2)$ (and Spin$(6)=SU(4)$) that describe gauge groups in the Standard model and the isospin symmetry in the quarks or in the mesons or even approximately in the baryon (proton/neutron). Spin$(3)$ is also a double cover of $SO(n)$.

question: Now, what is the physical use/application of Postnikov tower [constructing a topological space from its homotopy groups]? And what is the physical application the topological groups arise in Postnikov tower, say the Fivebrane($n$) and String($n$)?

There is a short exact sequence of topological groups $$ 0\rightarrow K(Z,2)\rightarrow \text{String}( n)\rightarrow \text{Spin}( n)\rightarrow 0 $$

The spin group appears in a Postnikov tower anchored by the orthogonal group: $$ { \ldots \rightarrow {\text{Fivebrane}}(n)\rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)} $$ The string group is an entry in the Postnikov tower for the orthogonal group, preceded by the fivebrane group in the tower.

  • ${\text{Spin}(n)}$ is obtained from ${\text{SO}(n)}$ by killing $\pi _{1}$

  • String($n$) is obtained by killing the $\pi _{3}$ homotopy group for ${\text{Spin}(n)}$.

Thus,

  • Fivebrane($n$) is obtained by killing the $\pi _{5}$ homotopy group for ${\text{String}(n)}$, correct? The tower is constructed by killing higher homotopy group $\pi _{2k+1}$?

The structures associated to the higher Postnikov groups naturally generalize the anomaly cancellation interpretation of the spin structure for higher-dimensional objects (see the linked nLab articles for references):

  • A spin structure on a manifold is the condition that the worldline anomaly of a superparticle moving on it (where "moving on it" means we are considering the $\sigma$-model with the manifold as the target space) cancels, i.e. it is a condition for the quantum theory of the superparticle to be well-defined.

  • A string structure on a manifold is the condition that the worldsheet anomaly of a superstring moving on it cancels.

  • A fivebrane structure on a manifold is the condition that the worldvolume anomaly of a fivebrane moving on it cancels.

  • Thanks, +1 ACuriousMind, there are some subquestions of mine that you don't touch. – wonderich Sep 28 '17 at 15:05
up vote -1 down vote accepted

Later I found that there are more extensive, rich and detailed answers for the physical/math applications in this MO post https://mathoverflow.net/questions/59772/what-do-whitehead-towers-have-to-do-with-physics

In addition, String(n) itself if the 3-connected cover of Spin(n), which is itself is the simply connected cover of the special orthogonal group SO(n), which in turn is the connected component (of the identity) of the orthogonal group O(n).

Fivebrane(n) is one element in the Whitehead tower of O(n). Fivebrane(n) is defined to be, as a topological group, the 7-connected cover of the String group String(n), for any n $\in $ natural positive numbers. Thus Fivebrane(n) should be obtained by killing the $\pi_7$ homotopy group for String(n).[So the OP is incorrect in the last sentence.]

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