Bundle gerbe

Recall for a smooth manifold ${\displaystyle M}$ the p-th Deligne cohomology groups are defined by the hypercohomology of the complex

$${\displaystyle \mathbb {Z} (q)_{D}^{\infty }={\underline {\mathbb {Z} }}(q)\to {\mathcal {A}}_{M,\mathbb {C} }^{0}\xrightarrow {d} {\mathcal {A}}_{M,\mathbb {C} }^{1}\xrightarrow {d} \cdots \xrightarrow {d} {\mathcal {A}}_{M,\mathbb {C} }^{q-1}}$$

called the weight q Deligne complex, where ${\displaystyle {\mathcal {A}}_{M,\mathbb {C} }^{k}}$ is the sheaf of germs of smooth differential k-forms tensored with ${\displaystyle \mathbb {C} }$. So, we write

$${\displaystyle \mathbb {H} _{D}^{*}(M,\mathbb {Z} (q)_{D}^{\infty })}$$

for the Deligne-cohomology groups of weight ${\displaystyle q}$. In the case ${\displaystyle q=3}$ the Deligne complex is then

$${\displaystyle {\underline {\mathbb {Z} }}(3)\to {\mathcal {A}}_{M,\mathbb {C} }^{0}\xrightarrow {d} {\mathcal {A}}_{M,\mathbb {C} }^{1}\xrightarrow {d} {\mathcal {A}}_{M,\mathbb {C} }^{2}}$$

We can understand the Deligne cohomology groups by looking at the Cech resolution giving a double complex. There is also an associated short exact sequence[1]^: 7

$${\displaystyle 0\to {\frac {{\mathcal {A}}_{\mathbb {C} ,M}^{2}(M)_{cl}}{{\mathcal {A}}_{\mathbb {C} ,M}^{2}(M)_{cl,0}}}\to \mathbb {H} ^{3}(M,\mathbb {Z} (3)_{D}^{\infty })\to H^{3}(M,\mathbb {Z} )\to 0}$$

where ${\displaystyle {\mathcal {A}}_{\mathbb {C} ,M}^{2}(M)_{cl}}$ are the closed germs of complex valued 2-forms on ${\displaystyle M}$ and ${\displaystyle {\mathcal {A}}_{\mathbb {C} ,M}^{2}(M)_{cl,0}}$ is the subspace of such forms where period integrals are integral. This can be used to show ${\displaystyle H^{3}(M,\mathbb {Z} )}$ are the isomorphism classes of ${\displaystyle \mathbb {C} ^{*}}$ bundle-gerbes on a smooth manifold ${\displaystyle M}$, or equivalently, the isomorphism classes of ${\displaystyle B\mathbb {C} ^{*}}$-bundles on ${\displaystyle M}$.

• WZW branes and strings