nLab Karoubi K-theory

Karoubi 1978 §III.4 defined topological K-theory classes equivalently by Clifford module bundles, where a $Cl_n$-module represents a class in $K^n$ and represents the trivial class if, roughly, 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 $Cl_1$.

The idea is now to generalize this to K-theory in other degrees.

Definition

By default, let the ground field be the complex numbers.

Consider

• $X$ a suitable topological space,

• $Vec_X$ the category of topological vector bundles over $X$,

• $p,q \in \mathbb{N}$,

• $Cl^{p,q}$ the Clifford algebra with that signature,

• $Cl^{p,q}Mod_X \,\coloneqq\, Cl^{p,q}Mod\big(Vec_X\big)$ the category of module objects,

• $u \colon Cl^{p,q+1}_X \xrightarrow{\;} Cl^{p,q}_X$ the forgetful functor.

Then Karoubi 78 Def. 4.11 defines the K-theory group $K^{p,q}(X)$ to be the relative Grothendieck group of this forgetful functor.

This means [K78 §2.13] that $K^{p,q}(X)$ is given by equivalence classes $[E,F,\alpha]$ of triples consisting of

• $E, F \,\in\, ClMod^{p,q+1}_X$

• $\alpha \,\colon\, u(E) \xrightarrow{\sim} u(F)$ an isomorphism

subject to the following equivalence relations:

1. an isomorphism of such triples $(E_i, F_i, \alpha_i)$ is a pair of isomorphism $f \colon E_1 \to E_2$, $g \colon F_1 \to F_2$ which intertwine $\alpha$ in that the following diagram commutes:

   $$\begin{array}{ccc} u(E_1) &\overset{u(f)}{\longrightarrow}& u(E_2) \\ \mathllap{{}^{\alpha_1}} \Big\downarrow && \Big\downarrow \mathrlap{{}^{\alpha_2}} \\ u(F_1) &\underset{u(g)}{\longrightarrow}& u(F_2) \end{array}$$

2. an equivalence of such triples is an isomorphism between their direct sum with any triples $(E,F,\alpha)$ for which $E \simeq F$ and $\alpha$ is homotopic, via $Cl^{p,q}$-isomorphisms, to the identity morphism (“elementary triples”).

According to K78 Thm. 4.12 this definition reproduces the ordinary definition of topological K-theory groups in degree $p-q$.

Categorification

Douglas & Henriques 2011 have proposed 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.

Related concepts

• topological K-theory

• Kasparov K-theory

• conformal nets

• mass term

• K-theory classification of topological phases of matter

References

The above definition is due to

• Max Karoubi, section III.4 of: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer (1978) [doi:10.1007%2F978-3-540-79890-3, pdf]

Related results and discussion:

• 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:10.1007/978-3-540-79890-3]

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]

A categorification using conformal nets:

• Chris Douglas, André Henriques: Topological modular forms and conformal nets, in Mathematical Foundations of Quantum Field and Perturbative String Theory, Proc. Sympos. Pure Math. 83, AMS (2011) 341-354 [arXiv:1103.4187]