The ADHM Construction and Anselmi's Topological Anomalies

Abstract

We examine the anomalies arising in instanton calculus as detailed by Damiano Anslemi in 1994. Whereas Anselmi uses BRST theory, we use the ADHM construction to arrive at the same conclusions from a differential-geometric way. We observe that Anselmi's TQFT is similar to Donaldson Theory applied to charge 1 instantons on the 4-sphere,although the latter is really only used for 4-manifolds with non-zero second positive betti number. We demonstrate why the anomalies occur in the case of charge 1 instantons and move to show that similar anomalies cannot occur for instantons of higher charge. To do this, we use an equivariant integration theory for hyperKahler manifolds and apply it to the hyperKahler geometry involved in the ADHM construction.