nLab Karoubi K-theory

Karoubi defined K-theory classes given by Clifford module bundles, where a $Cl_n$-module represents a class in $K^n$ and represents the trivial class if it extends to a $Cl_{n+1}$-module.

(Over the point this is the Atiyah-Bott-Shapiro isomorphism.)

Idea

We have a sequence of Clifford algebras $Cl_n$ which are generated by $n$ anticommuting square roots of $\pm 1$. The sequence is periodic up to Morita equivalence; $Cl_8$ is $\mathbb{R}(16)$, the algebra of $16 \times 16$ real matrices, which is Morita equivalent to $\mathbb{R}$, and from then on it repeats every 8 with extra matrix dimensions thrown in – this is Bott periodicity. (Notice that here we treat Clifford algebras as $\mathbb{Z}_2$-graded algebras: while $Cl_6$ is Morita equivalent to $\mathbb{R}(8)$ as an algebra, it is not so as a graded algebra.)

It turns out that $K^n(X)$ can be represented geometrically by Clifford module bundles over $X$. Start with $K^0$; we know that elements of $K^0(X)$ are ‘formal differences’ $V - W$ of vector bundles over $X$ (virtual vector bundles). 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 $Cl_1$.

More generally, Karoubi proved that for any $n$, $K^{-n}X$ can be represented by $Cl_n$-module bundles on $X$ modulo those such that the $Cl_n$-action extends to a $Cl_{n+1}$-action. When $n = 0$ this is what we had above, since a $Cl_0$-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.

Categorification

Chris Douglas and André Henriques have proposed that this description of K-theory has a good categorification that might be relevant for tmf (DougHen).

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.

Related concepts

• topological K-theory

• Kasparov K-theory

• conformal nets

• mass term

• K-theory classification of topological phases of matter

References

• Michael Atiyah, Isadore Singer, Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’IHÉS, Tome 37 (1969) 5-26 $[$numdam:PMIHES_1969__37__5_0$]$

• Max Karoubi, Espaces Classifiants en K-Théorie, Transactions of the American Mathematical Society 147 1 (Jan., 1970) 75-115 $[$doi:10.2307/1995218$]$

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

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

• Max Karoubi, K-theory: An introduction. Springer Science & Business Media, 2009. (doi).

Relation to twisted K-theory:

• Max Karoubi, Clifford modules and twisted K-theory, Proceedings of the International Conference on Clifford algebras (ICCA7) (arXiv:0801.2794)

Categorification using conformal nets:

• Chris Douglas, André Henriques, Topological modular forms and conformal nets, arXiv:1103.4187