  We prove an energy identity for anti-self-dual connections on the product
C\times\Sigma of the complex plane and a Riemann surface. The energy is a
multiple of a basic constant that is determined from the values of a
corresponding Chern-Simons functional on flat connections and its ambiguity
under gauge transformations. For SU(2)-bundles this identity supports the
conjecture that the finite energy anti-self-dual instantons correspond to
holomorphic bundles over CP^1\times\Sigma. Such anti-self-dual instantons on
SU(n)- and SO(3)-bundles arise in particular as bubbles in adiabatic limits
occurring in the context of mirror symmetry and the Atiyah-Floer conjecture.
Our identity proves a quantization of the energy of these bubbles that
simplifies and strengthens the involved analysis.
