Recall the following table of Clifford algebras: $$\begin{array}{ccc} n & Cl_n & M_n/i^{*}M_{n+1}\\ 1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\ 2 & \mathbb{H} & \mathbb{Z}/2\mathbb{Z} \\ 3 & \mathbb{H}+\mathbb{H} & 0 \\ 4 & \mathbb{R}(4) & \mathbb{Z} \\ 5 & \mathbb{C}(4) & 0 \\ 6 & \mathbb{R}(8) & 0 \\ 7 & \mathbb{R}(8)+\mathbb{R}(8) & 0 \\ 8 & \mathbb{R}(16) & \mathbb{Z} \\ \end{array}$$

Here $Cl_n=T(\mathbb{R}^{n})/(v\otimes v-q(v))$ where $q$ is the quadratic function associated to the standard inner product on $\mathbb{R}^{n}$. $M_k$ denotes the free abelian group on irreducible $\mathbb{Z}/2\mathbb{Z}$ graded modules (these are easy to work out as this group is isomorphic to the free abelian group of ungraded irreducible modules on the algebra of one dimension lower and these are all matrix algebras). Finally $i:Cl_n\rightarrow Cl_{n+1}$ is the inclusion. For example (using the above isomorphism) the first two groups correspond to $\mathbb{C}$ splitting as two copies of $\mathbb{R}$ as a real representation and $\mathbb{H}$ splitting as two copies of $\mathbb{C}$ as a complex representation. From here the table is periodic mod 8 (up to graded Morita equivalence).

The last column certainly looks eerily familiar, it is the "Bott song" i.e. the coefficients of $\mathrm{KO}$. In their classic paper "Clifford Modules", Atiyah, Bott and Shaprio construct a class in $KO(B^n,S^{n-1})$ from a graded irreducible $Cl_n$ module using the difference construction where the trivialisation on the boundary is given by Clifford multiplication. They use this as well as knowledge of $KO_*$ to show that the aforementioned map gives a ring isomorphism:

$$ \bigoplus_{k\geq 0} M_k/M_{k+1}\rightarrow KO_{*} $$

Here the ring structure on the left is given by the super tensor product of modules. This is all rather spectacular. It seems reasonable from the perspective of the construction of the ABS map (i.e. by taking the difference bundle of a $\mathbb{Z}/2\mathbb{Z}$ graded vector bundle) that some aspect of the representation theory of superalgebras over $\mathbb{R}$ (in particular the Brauer--Wall group) should be closely conceptually related to real K-theory. Another basic observation is that, via the clutching construction, one identifies the coefficient ring of real K-theory with the (shifted) homotopy groups of the stable orthogonal group. The group $Pin$ may be constructed as the subgroup generated by the unit sphere in $V$ inside the group of units of $Cl_n(V)$ and this is the universal cover of the orthogonal group. It is conceivable that these groups and their representation theory are of more fundamental importance in understanding the rôle Clifford algebras play.

In Karoubi's book "K-theory: An Introduction" he provides an explicit construction of real and complex K-theory in terms of the K-theory of the Banach category of vector bundles with a $Cl_n$ action (this K-theory group is very similar in spirit to the above quotients $M_k/M_{k+1}$) and uses this to prove real Bott periodicity. Others have followed the same path and constructed K-theory classes in terms of something like bundles of $\mathbb{Z}/2\mathbb{Z}$ graded Hilbert spaces with a $Cl_{n}$ action and a self adjoint operator switching degrees. Unfortunately I find these accounts rather technical and don't really intuit why such constructions are reasonable to expect.

**Question**: Is there a conceptual explanation for the tight relationship (or some aspect thereof) between $KO$ and the representation theory of Clifford algebras?

Finally the work of Douglas and Henriques aims to replicate the above for $TMF$. Here they replace Clifford algebras with conformal nets, the spin groups with the string groups etc. If there is any general philosophical perspective on what drives such a generalisation and in particular on which aspects are important in the classical picture, I would also be extremely interested.