Background field dependence from the Slavnov-Taylor identity in (non-perturbative) Yang-Mills theory

We show that in Yang-Mills theory the Slavnov-Taylor (ST) identity, extended in the presence of a background gauge connection, allows to fix in a unique way the dependence of the vertex functional on the background, once the 1-PI amplitudes at zero background are known. The reconstruction of the background dependence is carried out by purely algebraic techniques and therefore can be applied in a non-perturbative scheme (e.g. on the lattice or in the Schwinger-Dyson approach), provided that the latter preserves the ST identity. The field-antifield redefinition, which replaces the classical background-quantum splitting when quantum corrections are taken into account, is considered on the example of an instanton background in SU(2) Yang-Mills theory.