M2 Smooth Family Wess Zumino Transgression

For every $r\ge 2$, define $q_{!}:{\mathrm{\Omega }}^r(S)\to {\mathrm{\Omega }}^{r-2}(C)$ by inserting the $r-2$ lifted base vectors before the two vertical arguments and integrating over the oriented fibre; define it to be zero for $r<2$. Changing a lift inserts three vertical arguments, which vanish on a two-dimensional fibre. A finite submersion-chart cover of a compact fibre, with a partition of unity, gives smooth coefficients by differentiation under compactly supported integrals. This constructs the operation on the three-, four-, and two-forms used here, not just on $c$.

For a smooth map $g:M\to C$ from a compact oriented manifold of dimension $m$ (a path, loop, disk, or cylinder suffices), the pullback $S_M=M{\times }_{C}S$ is a smooth compact oriented manifold of dimension $m+2$: local submersion coordinates prove smoothness and properness proves compactness. Its boundary is $S_{\mathrm{\partial }M}$ with base-before-fibre orientation. Triangulate it using [F3], orient its top simplices, and sum their smooth characteristic maps with integral coefficients. Interior faces cancel in pairs, leaving the induced triangulated boundary. Composing with $S_M\to S\to Y$ gives a finite smooth integral chain on $Y$, with precisely this boundary. Integrals of forms on the chain equal their manifold integrals, since the simplex interiors partition the manifold and their faces have measure zero. Thus the inverse images used below really do provide chains in the domain of [F1].

For a form $\eta$ of degree $m+2$, the definition in step 1.1 and integration in submersion coordinates give ${\int }_M{g}^{\ast }{q}_{!}\eta ={\int }_{S_M}\eta \mathrm{.}$ Apply [F2] to the chains of step 1.2 over an oriented coordinate disk $D$ of dimension $r-1$, for $\eta \in {\mathrm{\Omega }}^r(S)$. It gives ${\int }_{\mathrm{\partial }D}{q}_{!}\eta ={\int }_{S_D}d\eta ={\int }_D{q}_{!}d\eta \mathrm{.}$ Ordinary Stokes on $D$ and shrinking coordinate disks show $d{q}_{!}\eta =q_{!}d\eta$: a nonzero coefficient of the difference would have nonzero integral on a sufficiently small oriented coordinate disk. The even fibre dimension and base-before-fibre convention give the displayed positive sign. For $\eta =f^{\ast }c$ this proves $dA=2\pi q_{!}{f}^{\ast }h$. For $\eta =f^{\ast }{\lambda }_{ij}$ it proves $q_{!}{f}^{\ast }d{\lambda }_{ij}=d(q_{!}{f}^{\ast }{\lambda }_{ij})$, and hence the transition formula. In the latter case $D$ is a path and $q_{!}{\lambda }_{ij}$ is a function.

For a smooth loop $\gamma :S^1\to C$, step 1.2 produces a smooth integral three-cycle $z_{\gamma }$ on $Y$. Its value under $x$ is independent of the chosen triangulation: triangulate $S_{S^1}\times [0,1]$ relative to the two end triangulations using [F3]. Its projection to $S_{S^1}$ has rank at most three, so the pullback of the four-form $f^{\ast }h$ is zero. The boundary law [F1] on this four-chain gives equality of the two character values. Compatible subdivisions cause the same zero-curvature comparison, so the assertion also covers boundary subdivisions required by [F3]. For a smooth disk in $C$, its pullback four-chain has boundary the swept loop chain, and [F1] gives $\exp (2\pi ix(z_{\mathrm{\partial }D}))=\exp \text{\hspace{0.17em}⁣}\left(2\pi i{\int }_D{q}_{!}{f}^{\ast }h\right)=\exp \text{\hspace{0.17em}⁣}\left(i{\int }_{\mathrm{\partial }D}A\right)\mathrm{.}$ The last equality uses step 2.1 on a local potential chart. This establishes the stated curvature law for the supplied character on actual swept cycles; it constructs neither a shifted character nor a universal chain-transfer operation.

Replacing $c$ by $T{C}_3$ for fixed $T>0$ changes none of steps 1.1–3.1, provided $d(T{C}_3)=2\pi h$ with the stated normalization. This is an algebraic conditional substitution, not a derivation of a physical normalization or quantum phase.