Geometric Construct for Integrating Symmetric Tensors?

Question

I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds.

The motivation comes from gauge theories, commonly used in theoretical physics. The simplest example is a vector field $A_i$ defined on a Euclidean space $R^m$, where the vector field is only defined up to addition of a gradient, $A_i \sim A_i + \partial_i \alpha$, for some scalar $\alpha$. The meaningful quantities are the integrals of $A_i$ around closed curves, which eliminates any contribution from the gradient term. More generally, allow $A$ to be an $n$-form defined on $R^m$ ($m>n$), defined only up to addition of an exact form, $A\sim A+d\alpha$. The meaningful quantities are the integrals of $A$ over closed $n$-dimensional submanifolds of $R^m$, which kill the contribution from the exact form.

Now, to the main question. Allow the gauge field to be a symmetric tensor $A_{ij}$ defined on the Euclidean space, but only up to a second derivative, $A_{ij}\sim A_{ij}+\partial_i\partial_j \alpha$ for arbitrary scalar $\alpha$. I'm wondering what the appropriate geometric construct is for integrating symmetric tensors which would allow me to obtain meaningful quantities, independent of $\alpha$.

I should mention that I have indeed been able to find appropriate "objects" of integration when the theory is defined on a lattice approximation to $R^m$, so I believe the concept to be well-defined. I just need a better way to mathematically characterize these objects in order to make meaningful statements in the continuum limit. Based on the lattice analysis, it seems that the "objects" of integration have well-defined dimension (based on notions of self-similarity, as in fractals), but do not appear to be conventional manifolds. Furthermore, the "dimension" of the objects can actually exceed $m$ in certain cases, leading to a picture in which higher-dimensional objects have been embedded in a lower-dimensional Euclidean space. But these are merely conjectures at this point.

Does anyone have any ideas as to what mathematical construct I am looking for?

Answer 1

Here is one way to construct all "local" gauge-invariant quantities out of a symmetric tensor $A_{ij}$. It would be up to you to decide how it meshes with the intuition you gained from your investigations on the lattice.

First, define $R[A]_{ij:kl} = \partial_i\partial_k A_{jl} - \partial_j\partial_k A_{il} - \partial_i\partial_l A_{jk} + \partial_j\partial_l A_{ik}$ and $C[A]_{ij:k} = \partial_i A_{jk} - \partial_j A_{ik}$. You might notice that $R[A]$ is the linearized Riemann tensor, applied to a perturbation $\eta_{ij} \mapsto \eta_{ij} + A_{ij}$ of the flat Euclidean metric (or of whatever signature you like), while $C[A]$ is not exactly the linearized Christoffel connection, but a certain projection of it.

It is a straight forward exercise to show that a pure gauge field configuration $A_{ij} = \partial_i \partial_j \alpha$ is annihilated by both $R[A]$ and $C[A]$. But it is also true that any $A_{ij}$ annihilated by both $R[A]$ and $C[A]$ must be of that form, at least locally. First, $R[A] = 0$ implies that $A_{ij} = \partial_i v_j + \partial_j v_i$ for some $v_i$ (at least locally), which is essentially an infinitesimal restatement of the well-known fact that any metric that has zero Riemann curvature admits coordinates in which it takes the standard Euclidean form. Next, for $A_{jk} = \partial_j v_k + \partial_k v_j$ we have $C[A]_{ij:k} = \partial_k (\partial_i v_j - \partial_j v_i)$. By applying the Poincaré lemma, $C[A] = 0$ means that (locally) $v_i = \partial_i \alpha + B_{ij} x^j$, where $B_{ij} = B_{[ij]}$ is constant and the $x^j$ are Euclidean coordinates. But that last term is irrelevant, since we have found that our original $A_{ij} = \partial_i v_j + \partial_j v_i = \partial_i \partial_j (2\alpha)$ (at least locally).

Now consider any integral of the form \begin{equation} F(A) = \int ( f^{ijkl}(x) R[A]_{ij:kl} + g^{ijk}(x) C[A]_{ij:k} ) \, dx . \end{equation} For example, the tensors $f$ and $g$ could be arbitrary test functions (smooth and compactly supported), but could also be distributions supported on lower dimensional submanifolds, whatever makes sense in your context. The point is that all (reasonable) linear gauge-invariant functionals of the $A_{ij}$ field can be written in the above form.