The quantum field theory generalisation of Noether's theorem about symmetries and conservation laws is the WardTakahashi identity. What is a suitable treatment of this in the context of differential geometry and a modern setting? Thanks

From a path integral point of view, the WardTakahashi identity is a straightforward consequence of the fundamental theorem of calculus. Let $\delta$ be a vector field on the space $\mathcal{F}$ of fields which expresses the action of the group of gauge symmetries. Suppose that $d\phi$ is a measure which is invariant under these gauge symmetries. The fundamental theorem of calculus tells us that the integral of the total derivative $\int_{\mathcal{F}} \delta(g) d\phi = 0,$ for any $g$ such that the contribution from the boundary of $\mathcal{F}$ is $0$. In the special case where $g$ is the product $\mathcal{O}e^{S}$ of a gaugeinvariant observable and the exponential of a gaugeinvariant action, we get a constraint on the path integral (expectation with respect to $e^{S(\phi)}d\phi$): $0 = \int_{\mathcal{F}} \delta(\mathcal{O}(\phi) e^{S(\phi)})d\phi = \int_{\mathcal{F}}\[\delta(\mathcal{O})  \mathcal{O} \delta(S) \]e^{S(\phi)}d\phi = \langle \delta \mathcal{O}\rangle  \langle \mathcal{O} \delta(S) \rangle.$ If we use the fact that the variation of the action is the spacetime integral of the divergence of the current $\delta(S) = \int_\Sigma \nabla \dot{} J$, then we get the usual WardTakahashi identity: $\langle \delta \mathcal{O}\rangle = \int_\Sigma \langle \mathcal{O} \nabla \dot{} J\rangle.$ Everything I've written here is only obviously true for finitedimensional integrals. However, any path integral should be arbitrarily well approximated by such finitedimensional integrals, so one can at least hope to transport the final result, even if the steps themselves can not be carried over. 

