Higher U(1)-bundles with connection via Deligne hypercocycles

For $k\ge 2$, a degree-$k$ higher $U(1)$-bundle with connection on a finite-dimensional Hausdorff second-countable smooth manifold is a cohomology class represented on a good cover by a total-degree $(k-1)$ cocycle in the Deligne sheaf complex $\mathcal{T}\stackrel{d_{\mathcal{T}}}{\to }{\mathrm{\Omega }}^1\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\mathrm{\Omega }}^{k-1}$, where $\mathcal{T}(V)=C^{\mathrm{\infty }}(V,\mathbb{R}\mathrm{/}\mathbb{Z})$ and $d_{\mathcal{T}}g=(2\pi i{)}^{-1}(e^{2\pi ig}{)}^{-1}d(e^{2\pi ig})$. Its degree is aligned with ${\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^k(M;\mathbb{Z})$ by Smooth Deligne cohomology model.

Smooth Deligne cohomology model

Exponential-sequence equivalence of Deligne models

Write ${\check{C}}^p(\mathcal{U};{\mathcal{F}}^q)$ for alternating Cech $p$-cochains with values in the sheaf ${\mathcal{F}}^q$. The total differential is $D={\delta }_{\mathrm{C}\mathrm{e}\mathrm{c}\mathrm{h}}+(-1{)}^p{d}_{\mathrm{D}\mathrm{e}\mathrm{l}}$ on bidegree $(p,q)$, where $d_{\mathrm{D}\mathrm{e}\mathrm{l}}$ is $d_{\mathcal{T}}$ in degree zero and exterior derivative in later degrees. A representative has components $g\in {\check{C}}^{k-1}(\mathcal{U};\mathcal{T})$, $A_1\in {\check{C}}^{k-2}(\mathcal{U};{\mathrm{\Omega }}^1),\dots ,A_{k-2}\in {\check{C}}^1(\mathcal{U};{\mathrm{\Omega }}^{k-2})$, and $B\in {\check{C}}^0(\mathcal{U};{\mathrm{\Omega }}^{k-1})$, and the cocycle equations are exactly $D(g,A_1,\dots ,B)=0$. Two representatives define the same object here when their difference is a total coboundary, with circle-valued addition interpreted in $\mathbb{R}\mathrm{/}\mathbb{Z}$.

The local curvature forms $d{B}_i$ agree on overlaps by the top component of the cocycle equation; they therefore define a global closed $k$-form $R$. The integral characteristic class $I$ is the connecting class of the Deligne cocycle under $0\to \mathbb{Z}\to {\mathrm{\Omega }}^0\to \mathcal{T}\to 0$. We use “higher bundle” for this cocycle class only; this definition does not specify a higher category of morphisms.

For $k=2$ these are line bundles with connection; for $k=3$ they are bundle gerbes with connection and curving. Equivalence to the smooth-singular character model is conditional on Deligne and Cheeger–Simons models agree under geometric cycle inputs.