\( \newcommand{\sAlg}{\mathsf{sAlg}} \newcommand{\sVect}{\mathsf{sVect}} \newcommand{\Pic}{\operatorname{Pic}} \newcommand{\Cl}{\operatorname{Cl}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\grotimes}{\hat{\otimes}} \newcommand{\fib}{\operatorname{fib}} \newcommand{\Mod}{\operatorname{Mod}} \)
A long time ago in a category
far, far away....

SUPER
K-THEORY

FOR SUPER ALGEBRAS



DAVID ARETZ
MPIM Bonn
Higher Differential Geometry, Greifswald


EPISODE MMXXVI
MAY THE 4TH BE WITH YOU

Main Result

Theorem (A.-Stehouwer, in preparation)

Graded \( K \)-theory defines a symmetric monoidal functor

\( K^{gr}: \) \( \Pic(\sAlg_2) \) \( \longrightarrow \) \( \Pic(KO) \) \( \,. \)

which induces an isomorphism on \( \pi_0, \pi_1 \) and \( \pi_2 \), and \( \pi_i\Pic(\sAlg_2) \) vanishes for \( i > 2 \).

Super (Easy) Algebras

  • Super Vector Spaces: \( (\sVect_{\R},\grotimes) \) symm. monoidal, \( V=V_0\oplus V_1 \).
  • Koszul Braiding: \[ v \otimes w \longmapsto (-1)^{|v||w|} w \otimes v \]
  • Super Algebras: \( (\sAlg_2,\grotimes) \) — symmetric monoidal 2-category; 1-morphisms are super bimodules \( {}_A M_B \) — super Morita equivalence.
  • Clifford Algebras \( \Cl_{p,q} \): odd generators \( e_1,\dots,e_p,f_1,\dots,f_q \) with \( e_i^2=+1 \), \( f_j^2=-1 \), pairwise anticommute. \[ \Cl_{p,q} \grotimes \Cl_{p',q'} \cong \Cl_{p+p',q+q'} \,. \]

Clifford Algebras

\( \Cl_0 \) \( \Cl_{-1} \) \( \Cl_{-2} \) \( \Cl_{-3} \) \( \Cl_{-4} \) \( \Cl_{-5} \) \( \Cl_{-6} \) \( \Cl_{-7} \) \( \Cl_{-8} \)
\( \R \) \( \C \) \( \mathbb{H} \) \( \mathbb{H} \oplus \mathbb{H} \) \( M_2(\mathbb{H}) \) \( M_4(\C) \) \( M_8(\R) \) \( M_8(\R)^{\oplus 2} \) \( M_{16}(\R) \)

Fundamental Facts

\( \Cl_{n+8}\simeq \Cl_n \), \( \Cl_{1,1}\simeq \R \), and these are Morita invertible: \( \Cl_n\grotimes \Cl_{-n}\simeq \R \).

The Picard Groupoid: \( \Pic(\sAlg_2) \)

\[ \begin{aligned} \pi_0 \Pic(\sAlg_2) &= \{\text{Morita eq. classes of invertible super algebras}\} \\ &= \{\Cl_0,\dots,\Cl_7\} \cong \Z/8 \end{aligned} \] (Super Brauer group)

\[ \begin{aligned} \pi_1 \Pic(\sAlg_2) &= \{\text{invertible } \R\text{-}\R\text{-super bimodules}\} \\ &= \{\R, \Pi \R \} \cong \Z/2 \end{aligned} \]

\[ \begin{aligned} \pi_2 \Pic(\sAlg_2) &= \{\text{isometries of bimodules } \R \to \R\} \\ &= \{\pm \operatorname{id} \} \cong \Z/2 \end{aligned} \]

Topological \( K \)-theory

  • \( KO \)-Theory: A multiplicative generalized cohomology theory formed by the group completion of vector bundles.
  • Bott Periodicity: \( KO^{n+8}(X) \cong KO^n(X) \).
  • \( KO \) is a commutative ring spectrum, making \( (\Mod(KO), \otimes_{KO}) \) symmetric monoidal.

The Picard group \( \Pic(KO) \):

\[ \begin{aligned} \pi_0 \Pic(KO) &= \{\otimes_{KO}\text{-invertible } KO\text{-modules}\} \\ &= \{KO, \dots, \Sigma^7 KO\} \cong \Z/8 \end{aligned} \]

\[ \begin{aligned} \pi_1 \Pic(KO) &= \{\text{invertible elements in } \pi_0 KO\} \\ &= \{\R, -\R\} \cong \Z/2 \end{aligned} \]

\[ \begin{aligned} \pi_{2}\Pic(KO) &\cong \Z/2 \\ \pi_i \Pic(KO) &\cong \pi_{i-1} KO \end{aligned} \]

Graded \( K \)-theory

For an ungraded Banach algebra \( A \):

\[ k_A = k(\Mod_A)= \text{group completion of }(\Mod_A^{fgp,\cong},\oplus) \]

Example: \( ko = k_{\R} \), \( k_{C(X;\R)}\simeq ko(X) \).

Definition: Connective graded \( K \)-theory

For a super Banach algebra \( A \), we define graded \( K \)-theory as the fiber:

\[ k^{gr}_A := \fib\left(k(\Mod^{graded}_A) \longrightarrow k(\Mod_{A}^{ungraded})\right) \,. \]
  • Spectral refinement of Karoubi \( K \)-theory.
  • Upgrades to a lax symmetric monoidal functor \( \sAlg_2 \to \mathrm{Sp} \).
  • Extends topological \( K \)-theory.

Proof

Theorem (A.-Stehouwer)

Graded \( K \)-theory defines a symmetric monoidal functor

\( K^{gr}: \) \( \Pic(\sAlg_2) \) \( \longrightarrow \) \( \Pic(KO) \) \( \,. \)

which induces an isomorphism on \( \pi_0, \pi_1 \) and \( \pi_2 \), and \( \pi_i\Pic(\sAlg_2) \) vanishes for \( i > 2 \).

Key Steps:

(\( \pi_0=\Z/8 \)) \( K^{gr}_{\Cl_n}\simeq \Sigma^n KO \).

(\( \pi_1=\Z/2 \)) The operation \( V \mapsto V \grotimes \Pi \R \) corresponds to multiplication by \( -1 \in \pi_0 KO \).

(\( \pi_2=\Z/2 \)) The bimodule map \( -\operatorname{id}:\R \to \R \) corresponds to the Moebius bundle over \( S^1 \).

Episode VIII: The Outlook Continues...

  • Direct link: Rep theory \( \longleftrightarrow \) topological \( K \)-theory
  • \( \mathbb{E}_\infty \)-\( \mathrm{Spin} \) orientation \( M\mathrm{Spin} \to KO \)
  • Twisted equivariant \( K \)-theory
  • \( K \)-homology (The Dark Side of the pairing)
  • Differential refinements(?)
  • \( KK \)-theory(?) (A New Hope)

Thank You!

May the Fourth be with you.