Energy identity for anti-self-dual instantons on ${\mathbb C}\times\Sigma$

We establish 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 considerably.

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Published 1 January 2006