U(1)-bundle gerbe with connection and curving

Let $M$ and $Y$ be finite-dimensional Hausdorff second-countable smooth manifolds and let $\pi :Y\to M$ be a surjective smooth submersion admitting local smooth sections. For $q\ge 2$, define $Y^{[q]}=\{(y_1,\dots ,y_q)\in Y^q:\pi (y_1)=\cdots =\pi (y_q)\}$, and write $P_{ab}$ for the pullback of a line bundle $P\to Y^{[2]}$ to the pair $(y_a,y_b)$.

A Hermitian complex line bundle is a smooth rank-one complex vector bundle with a smoothly varying positive Hermitian metric; its unit-vector bundle is a smooth principal right $U(1)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$-bundle as in Smooth principal $G$-bundle, and the associated line uses the relation $[pg,z]=[p,gz]$ for the standard scalar representation, as in Associated bundles. A unitary connection $\mathrm{\nabla }$ on $P$ is a complex-linear connection satisfying the Leibniz rule and preserving this metric, in the sense of Connection on a smooth vector bundle. In a unit frame $e$ write $\mathrm{\nabla }e=Ae$; metric compatibility gives $A\in i{\mathrm{\Omega }}^1$. A unit vector is $p=e(x)u$ with $u\in U(1)$. Along a curve $t\mapsto e(x(t))u(t)$, its covariant derivative is $e(x(t))(\dot{u}+A(\dot{x})u)$, so the horizontal condition is $u^{-1}\dot{u}+A(\dot{x})=0$. Thus the principal connection form on the unit-vector bundle is $\omega =u^{-1}du+A$, with the convention of Principal connection 1-form from a horizontal distribution. Equivalently, sections correspond to functions $f$ with $f(pg)=g^{-1}f(p)$, and their covariant derivative is represented by differentiating along the horizontal distribution. Its curvature $F_{\mathrm{\nabla }}$ is the imaginary-valued two-form defined by Curvature of a vector-bundle connection; in this unit frame, $F_{\mathrm{\nabla }}=dA+A\wedge A=dA$ because $A$ is scalar-valued, so $F_{\mathrm{\nabla }}\mathrm{/}(2\pi i)$ is real-valued.

A $U(1)$-bundle gerbe with connection and curving consists of this $\pi$, a Hermitian line bundle $P\to Y^{[2]}$, a smooth unitary bundle isomorphism $m:P_{12}\otimes P_{23}\to P_{13}$ over $Y^{[3]}$, and a unitary connection $\mathrm{\nabla }$ on $P$ preserved by $m$. The product is associative on $Y^{[4]}$ and has unit identifications $P_{y,y}\cong \mathbb{C}$ and inverse identifications $P_{y_2,y_1}\cong P_{y_1,y_2}^{\ast }$, all preserving the induced connections. A curving is a real smooth two-form $B\in {\mathrm{\Omega }}^2(Y)$ satisfying ${\mathrm{pr}}_2^{\ast }B-{\mathrm{pr}}_1^{\ast }B=F_{\mathrm{\nabla }}\mathrm{/}(2\pi i)\text{on }{Y}^{[2]}\mathrm{.}$ The descended closed three-form $H$ with ${\pi }^{\ast }H=dB$ is supplied by Bundle-gerbe three-curvature descends and is closed.

Cheeger–Simons differential character

Smooth Deligne cohomology model

For each $(y_1,y_2,y_3,y_4)\in Y^{[4]}$, associativity means $m(m(p_{12}\otimes p_{23})\otimes p_{34})=m(p_{12}\otimes m(p_{23}\otimes p_{34}))$. Product compatibility means $m$ preserves the tensor-product connection ${\mathrm{\nabla }}_{12}\otimes 1+1\otimes {\mathrm{\nabla }}_{23}$. The Dixmier–Douady cocycle is obtained from local sections $s_i:U_i\to Y$, unit sections ${\sigma }_{ij}$ of $(s_i,s_j{)}^{\ast }P$, and the functions $g_{ijk}:U_{ijk}\to U(1)$ determined by $m({\sigma }_{ij}\otimes {\sigma }_{jk})={\sigma }_{ik}{g}_{ijk}$.

The underlying topological gerbe is trivial when there is a smooth Hermitian line bundle $Q\to Y$ and a product-preserving unitary isomorphism $P\cong {\mathrm{pr}}_1^{\ast }{Q}^{\ast }\otimes {\mathrm{pr}}_2^{\ast }Q$. The product on the latter line bundle contracts the middle factors $Q_{y_2}\otimes Q_{y_2}^{\ast }$ by evaluation.

For two underlying gerbes $(P,Y,M)$ and $(P^{\mathrm{\prime }},Y^{\mathrm{\prime }},M)$, a common refinement consists of a surjective smooth submersion $\rho :Z\to M$ and surjective smooth submersions $f:Z\to Y$, $f^{\mathrm{\prime }}:Z\to Y^{\mathrm{\prime }}$ over $M$, with $\pi f=\rho ={\pi }^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$. One such refinement is $Z=Y{\times }_M{Y}^{\mathrm{\prime }}$ with its projections. Put $Z^{[q]}=Z{\times }_M\cdots {\times }_{M}Z$ ($q$ factors), let $f^{[q]}$ and $f^{\mathrm{\prime }[q]}$ apply these maps in each factor, and put $L=(f^{[2]}{)}^{\ast }P$, $L^{\mathrm{\prime }}=(f^{\mathrm{\prime }[2]}{)}^{\ast }{P}^{\mathrm{\prime }}$. The quotient gerbe on this refinement has line bundle $D=L^{\ast }\otimes L^{\mathrm{\prime }}\to Z^{[2]}$. For $\alpha \in L_{12}^{\ast }$ and $\beta \in L_{23}^{\ast }$ define $\alpha \star \beta =(\alpha \otimes \beta )\circ m_L^{-1}\in L_{13}^{\ast }$. Its product is $m_D((\alpha \otimes p^{\mathrm{\prime }})\otimes (\beta \otimes q^{\mathrm{\prime }}))=(\alpha \star \beta )\otimes m_{L^{\mathrm{\prime }}}(p^{\mathrm{\prime }}\otimes q^{\mathrm{\prime }})\mathrm{.}$ Here $m_L,m_{L^{\mathrm{\prime }}}$ are the pulled-back products; the ordinary symmetry of tensor products of complex lines reorders the factors without a sign. This defines a smooth unitary associative product, with units and inverses induced from those of $L,L^{\mathrm{\prime }}$. The two gerbes are stably equivalent when on some common refinement there is a smooth Hermitian line bundle $Q\to Z$ and a product-preserving unitary isomorphism $D\cong {\mathrm{pr}}_1^{\ast }{Q}^{\ast }\otimes {\mathrm{pr}}_2^{\ast }Q$. Stable equivalence forgets connection and curving data. It is an equivalence relation: the identity refinement gives reflexivity, dualizing the trivializing line gives symmetry, and for transitivity the fibre product of two refinements over their shared intermediate gerbe submersion is a common refinement; tensoring the pulled-back trivializing line bundles contracts the shared middle gerbe line.

Bundle-gerbe three-curvature descends and is closed

The two-form interpretation used above follows directly in rank one from the curvature operator. In a unit frame $e$, ${\mathrm{\nabla }}_X(fe)=(Xf+A(X)f)e$. Expanding $R^{\mathrm{\nabla }}(X,Y)(fe)$ cancels all derivatives of $f$ and gives $f(X(A(Y))-Y(A(X))-A([X,Y]))e=fdA(X,Y)e$. Under $e^{\mathrm{\prime }}=eg$, with $g$ circle-valued, $A^{\mathrm{\prime }}=A+g^{-1}dg$ and $d(g^{-1}dg)=0$, as follows locally from $g=e^{2\pi it}$. Thus the local $dA$ glue to the global imaginary-valued curvature two-form used in the curving equation.

The diagram displays the required associativity. Every arrow $m_{ab,c}$ is the stated product map; the equality follows from the definition and is part of the gerbe data.

The differential-character convention is additive $U(1)=\mathbb{R}\mathrm{/}\mathbb{Z}$. The exponential identifies a holonomy value $\theta \in \mathbb{R}\mathrm{/}\mathbb{Z}$ with $e^{2\pi i\theta }$. Raw bundle-gerbe presentations depend on the submersion and are not themselves classified by $H^3(M;\mathbb{Z})$.