Differential twist and twisted differential K-theory

Let $E$ be a commutative ring spectrum and let a differential refinement consist of a real commutative differential graded algebra $A$ and an equivalence $\rho :E\wedge H\mathbb{R}\simeq HA$ of commutative $H\mathbb{R}$-algebra spectra, preserving the unit, multiplication and their coherent homotopies. For a smooth manifold $M$, let $\mathrm{\Omega }A={\mathrm{\Omega }}_M^{\ast }{\otimes }_{\mathbb{R}}A$ be the sheaf of differential graded algebras of forms with coefficients in $A$. Write $\underset{‾}{A}$ for the locally constant coefficient sheaf. The de Rham map $H\underset{‾}{A}\to H\mathrm{\Omega }A$, induced by locally constant degree-zero forms, is an equivalence of sheaves of commutative $H\mathbb{R}$-algebra spectra: on coordinate balls the radial Poincaré homotopy, tensored with $A$ with the graded signs, contracts the form direction to constants. Compose the sheafification of $\rho$ with this de Rham equivalence to obtain ${\rho }_M:\underset{‾}{E\wedge H\mathbb{R}}\simeq H\mathrm{\Omega }A$. Whenever $(E_{\tau }{)}_{\mathbb{R}}$ is regarded as an $H\mathrm{\Omega }A$-module below, its scalar action is transported along this specified algebra equivalence (equivalently, use derived extension of scalars along ${\rho }_M$). A differential twist consists of a sheaf $E_{\tau }$ of $E$-module spectra on $M$ that is locally equivalent to the constant sheaf $E$ as an $E$-module, a module ${\mathcal{M}}_{\tau }$ over $\mathrm{\Omega }A$, and an equivalence $c:(E_{\tau }{)}_{\mathbb{R}}\simeq H({\mathcal{M}}_{\tau })$ of sheaves of $H\mathrm{\Omega }A$-modules, where $(E_{\tau }{)}_{\mathbb{R}}:=E_{\tau }\wedge H\mathbb{R}$. The module ${\mathcal{M}}_{\tau }$ is required to be invertible, meaning that there is an $\mathrm{\Omega }A$-module $\mathcal{N}$ with an actual isomorphism ${\mathcal{M}}_{\tau }{\otimes }_{\mathrm{\Omega }A}\mathcal{N}\cong \mathrm{\Omega }A$ of sheaves of differential graded modules; it is $K$-flat, meaning that tensoring it with a quasi-isomorphism of $\mathrm{\Omega }A$-modules gives a quasi-isomorphism; and it is weakly locally constant, meaning that its image after localization at quasi-isomorphisms is locally constant. Morphisms consist of equivalences of the topological twists and actual isomorphisms of the differential graded form modules, compatible with $c$ through a specified homotopy identifying the comparison maps. Write $\kappa :E_{\tau }\to (E_{\tau }{)}_{\mathbb{R}}$ for realification. Define the associated differential-function spectrum by the homotopy pullback

where the map from $E_{\tau }$ to $H({\mathcal{M}}_{\tau })$ is $c\circ \kappa$, the other map is induced by inclusion of the naive degreewise nonnegative truncation, and $H$ is the Eilenberg–Mac Lane spectrum of a differential graded module. Use cohomological grading for the differential-form module: ${\mathcal{M}}_{\tau }[q{]}^j={\mathcal{M}}_{\tau }^{j+q}$ with differential $(-1{)}^{q}d$ and shifted action $a{\cdot }_{[q]}m=(-1{)}^{q\mathrm{\mid }a\mathrm{\mid }}am$ for homogeneous $a\in \mathrm{\Omega }A$, so $H({\mathcal{M}}_{\tau }[q])\simeq {\mathrm{\Sigma }}^{q}H({\mathcal{M}}_{\tau })$. For each integer $q$, apply the same pullback prescription to the shifted triple $({\mathrm{\Sigma }}^q{E}_{\tau },{\mathcal{M}}_{\tau }[q],c[q])$ and define ${\widehat{E}}_{\tau }^q(M)={\pi }_0\mathbf{R}\mathrm{\Gamma }(M;\mathrm{Diff}({\mathrm{\Sigma }}^q{E}_{\tau },{\mathcal{M}}_{\tau }[q],c[q]))$, where $c[q]$ is the suspended comparison equivalence. The naive truncation is applied after this shift. In particular all cohomological degrees are not defined by homotopy groups of one fixed unshifted truncation. For a $U(1)$-bundle gerbe with connection, a degree-three complex $K$-twist is specified only after choosing a comparison from its Dixmier–Douady class and differential data to such a full twist; the gerbe presentation alone is not that comparison.

Differential extension of a generalized cohomology theory

U(1)-bundle gerbe with connection and curving

An isomorphism of twists consists of an equivalence of the topological module spectra and an actual isomorphism of sheaves of differential graded form modules, together with a homotopy identifying the induced real comparison maps. A quasi-isomorphism of form modules alone is not a permitted change of presentation: naive truncation need not preserve such a quasi-isomorphism. Automorphisms and their homotopies remain part of the twist data.

The twist includes the topological class, differential-form module, real comparison and their equivalences. It is not defined by a curvature three-form alone. No map from every gerbe to a complex $K$-theory twist is asserted without the specified comparison data.