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Karoubi K-theory

Karoubi defined K-theory classes given by Clifford-algebra module bundles, where a ${\mathrm{Cl}}_n$-module respresents a class in $K^n$ and represents the trivial class if it extends to a ${\mathrm{Cl}}_{n+1}$-module.

Idea

We have a sequence of Clifford algebras ${\mathrm{Cl}}_n$ which are generated by $n$ anticommuting square roots of $\pm 1$. The sequence is periodic up to Morita equivalence; ${\mathrm{Cl}}_8$ is $ℝ(16)$, the algebra of $16\times 16$ real matrices, which is Morita equivalent to $ℝ$, and from then on it repeats every 8 with extra matrix dimensions thrown in.

Here we treat Clifford algebras as ${\mathbb{Z}}_2$-graded algebras: while ${\mathrm{Cl}}_6$ is Morita equivalent to $ℝ(8)$ as an algebra, it is not so as a graded algebra.

It turns out that $K^n(X)$ can be represented geometrically by ‘bundles of Clifford modules’ over $X$. Start with $K^0$; we know that elements of $K^0(X)$ are ‘formal differences’ $V-W$ of vector bundles over $X$. We can model the formal difference $V-W$ with an honest geometric object by using the ${\mathbb{Z}}_2$-graded vector bundle $V\oplus W$, where $V$ is even and $W$ is odd. Such a thing should represent the zero class in K-theory just when $V$ and $W$ are isomorphic; this can be rephrased as saying that there exists an odd operator $e$ on $V\oplus W$ (hence, taking $V$ to $W$ and vice versa) such that $e^2=1$. But this just says that $V\oplus W$ has an action of the first Clifford algebra ${\mathrm{Cl}}_1$.

More generally, Karoubi proved that for any $n$, $K^{-n}X$ can be represented by ${\mathrm{Cl}}_n$-module bundles on $X$ modulo those such that the ${\mathrm{Cl}}_n$-action extends to a ${\mathrm{Cl}}_{n+1}$-action. When $n=0$ this is what we had above, since a ${\mathrm{Cl}}_0$-module is just a vector space. This allows us to deduce Bott periodicity for K-groups from the algebraic periodicity (up to Morita equivalence) of Clifford algebras.

Remarks

Chris Douglas et al. are proposing that this description of K-theory has a good categorification that might be relevant for tmf.

Here is a report by Mike Shulman on a talk by Chris Douglas on this topic, from which also part of the above text is taken.

References

• Karoubi, Twisted K-theory old and new, (pdf)

• Karoubi, Donovan, Graded Brauer groups and $K$-theory with local coefficients (pdf)