  We develop novel first-kind boundary integral equations for Euclidean Dirac
operators in 3D Lipschitz domains comprising square-integrable potentials and
involving only weakly singular kernels. Generalized Garding inequalities are
derived and we establish that the obtained boundary integral operators are
Fredholm of index zero. Their finite dimensional kernels are characterized and
we show that their dimension is equal to the number of topological invariants
of the domain's boundary, in other words to the sum of its Betti numbers. This
is explained by the fundamental discovery that the associated bilinear forms
agree with those induced by the 2D surface Dirac operators for H-1/2 surface de
Rham Hilbert complexes whose underlying inner-products are the non-local inner
products defined through the classical single-layer boundary integral operators
for the Laplacian. Decay conditions for well-posedness in natural energy spaces
of the Dirac system in unbounded exterior domains are also presented.
