Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$. Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral $k$-currents in $M$ and write ${\cal D}^{\mathit{int}}_{k}(M)_{\partial\xi}$ for the space of integral $k$-currents in $M$ with boundary $\partial\xi$, so ${\cal D}^{\mathit{int}}_{k}(M)_{0}$ is the space of integral $k$-cycles. Write ${\cal D}^{\mathit{flat}}_{k}(M)$ for the space of flat $k$-currents in $M$.
The Gromov-Hausdorff tangent space to ${\cal
D}^{\mathit{int}}_{k}(M)_{0}$ at 0 can be identified with a linear
space of flat $(k{+}1)$-currents
$$
T_{0} {\cal D}^{\mathit{int}}_{k}(M)_{0} = {\cal V}_{k+1}\subset {\cal D}^{\mathit{flat}}_{k+1}(M)
$$
as follows.
Suppose $\epsilon \mapsto \xi(\epsilon)$ is a path in
${\cal D}^{\mathit{int}}_{k}(M)_{0}$ starting at $\xi(0)=0$.
The integral $(k{+}1)$-current $\xi_{S}(\epsilon)$
that solves the Plateau problem $\partial\xi_{S}(\epsilon) = \xi(\epsilon)$
is well-defined to first order in $\epsilon$, independent of the
riemannian metric on $M$, so the tangent vector
to the path at $\epsilon =0$ can be identified with the flat $(k{+}1)$-current
$$
\dot \xi = \lim_{\epsilon\rightarrow 0}
\frac{\xi_{S}(\epsilon)}{\epsilon}
\,.
$$
Alternatively, think of $\xi\colon[0,\epsilon]\rightarrow {\cal
D}^{\mathit{int}}_{k}(M)_{0}$ as an integral 1-current in
${\cal D}^{\mathit{int}}_{k}(M)_{0}$, which maps naturally to an
integral 1-current in ${\cal D}^{\mathit{int}}_{k+1}(M)$.
The Hodge $*$-operator acting in the middle dimension, on $n$-forms and on $n$-currents, depends only on the conformal structure of $M$. The question is, does the Hodge $*$-operator preserve the tangent space $T_{0} {\cal D}^{\mathit{int}}_{n-1}(M)_{0}$, i.e., does $$ * {\cal V}_{n} = {\cal V}_{n} \;? $$
The question arises in my recent theoretical physics paper Quantum field theories of extended objects at the start of a project to construct a general class of quantum field theories on the spaces ${\cal D}^{\mathit{int}}_{n-1}(M)$. Assuming an affirmative answer, I argue in the paper that each of the spaces ${\cal D}^{\mathit{int}}_{n-1}(M)_{\partial\xi}$ of relative $(n{-}1)$-cycles in $M$ is a quasi Riemann surface --- a space with the structure of the space of integral currents in a Riemann surface. They form a fiber bundle ${\cal Q}(M)$ of quasi Riemann surfaces naturally associated to $M$. I conjecture that quasi Riemann surfaces are classified up to isomorphism by their homology data --- the middle-dimension homology group as a lattice in a finite dimensional Hilbert space. Assuming the conjecture, the fiber bundle ${\cal Q}(M)$ is naturally embedded in a universal homogeneous bundle of quasi Riemann surfaces. The homogeneity is with respect to groups that are described only as abstract automorphism groups associated to the homology data. This universal homogeneous bundle of quasi Riemann surfaces is the prospective geometric setting for the quantum field theories of extended objects.
None of this is done rigorously. The theoretical physics paper might be regarded as a series of mathematical questions following on the basic one posed here. The mathematical questions are listed in section 20 of the paper. I would be grateful for any advice about their mathematical coherence and feasibility, and even more grateful for answers. I imagine there might be mathematical applications, if all can be made solid.
Appendix A of the paper contains what I believe is the germ of a pedestrian proof of an affirmative answer to the posted question. A path of integral $1$-cycles in ${\mathbb R}^{4}$ is constructed whose tangent vector is the flat 2-current $$ d^{4}x\;\delta(x_{1})\delta(x_{2})\theta_{[0,1]}(x_{3})\delta(x^{4}) \hat e^{1}\wedge \hat e^{2} $$ where $\theta_{[0,1]}(x_{3})$ is the characteristic function of the unit interval in the 3-axis. From this construction, and its generalization to ${\mathbb R}^{2n}$, I argue that, for any manifold $M$, the tangent space ${\cal V}_{n}$ consists of all the flat $n$-currents that are supported on the support sets of the integral $(n{-}1)$-currents, which subspace of flat $n$-currents is manifestly closed under the Hodge $*$-operator. I would be grateful for this argument to be turned into a rigorous proof, or for advice about the possibility of doing so, or for a more elegant proof.
Finally, I am not at all sure what are the appropriate tags for this question.
