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# Mathematical Research Letters

## Volume 13 (2006)

### Number 1

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

Pages: 161 – 166

DOI: http://dx.doi.org/10.4310/MRL.2006.v13.n1.a12

#### Author

#### Abstract

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.