The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is usually presented for the lowest few $k=1,2,3,4,5,\ldots$ in papers, reviews and books. For simplicity let's stick to $k>1$ and $x=y$.
Generally the calculation goes like this: the general formula for $b_{2k}(x,x)$ is obtained for a general $A_\mu(x)$ gauge field, so various combinations of $F_{\mu\nu}$ and covariant derivatives and traces, and then the $q=1$ solution is substituted into these. Clearly, for general gauge field $A_\mu$ we can't do much better, as $k$ increases the complexity in terms of $F_{\mu\nu}$, etc. grows quickly.
But since we are only interested in the instanton case can't we derive $b_{2k}(x,x)$ in closed form for any $k$? We don't need the generic form with arbitrary $A_\mu$. Can't we use this to go much beyond the generic case?
Actually, some simplification can be achieved in the general case if one assumes the gauge field solves the equations of motion (so-called on-shell heat kernel coefficients). But I'd think one should be able to go further in the instanton case because those are very special solutions. And currently it seems $b_6$ is known for on-shell coefficients and also only $b_6$ is known for instantons. Why is it that one can't go higher for instantons?
