in the Lagrange multipliers $\lambda_a$. These charges descend from the brane charges in ten or eleven dimensions, and, in the appropriate units, are expected to be quantized.

The Euclidean space we shall concentrate on is actually flat space with a countable number of points, the locations of the instantons, excised[^4], $$\label{space}
  {\mathcal{M}}= \fieldR^D - \cup_i\, \{\vec{x}_i\}\ ,$$ such that non-trivial cycles with corresponding charges exist. Stated differently, in the supergravity approximation it will typically not be possible to find regular solutions at the locations of the instantons, as we will explicitly see below. The only singularity which can still lead to a finite action is a logarithmic singularity in $\phi$ at the origin, $$\label{small-r}
  \phi \rightarrow c\, \ln r\ ,$$ for some constant $c$. In our examples below, $\chi$ will tend to a constant $\chi_0$, and the tensors have the same $1/r^3$ behaviour such that the charges stay the same when the $H^a$ are integrated around an infinitesimal sphere around the origin.

The Bogomol’nyi bound
---------------------

The Euclidean double-tensor multiplet action is positive semi-definite (apart from the Einstein-Hilbert term). In fact, we can derive a lower bound by writing it as $${\mathcal{L}}_\mathrm{DT}^E = {\mathrm{d}}^Dx\, \sqrt{g\,} R + {\tfrac{1}{2}}*\! \big( N\! 