Energy identity for anti-self-dual instantons on C\timesΣ

Details Energy identity for anti-self-dual instantons on C\timesΣ Energy identity for anti-self-dual instantons on C\timesΣ

2005-03-16

arxiv.org/abs/math/0503341v1

Mathematics

Differential Geometry

6 pages, short version of an earlier note on my homepage

Scientific paper

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.

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