12-10 Notes
In this final lecture, we wrap up our discussion of the construction of \(k\)-instantons over \(S^4\).
Let \(C,D\) two be \((n+k) \times k\) matrices. Consider the matrix \(Cx+Dy\), with \(x,y \in \mathbb{H}\). Its columns determine for us a quaternionic k-plane in \(\mathbb{H}^{n+k}\). We get a bundle \(E\) over \(\mathbb{PH}^1 = S^4\), where the fiber \(E|_{[x:y]}\) is the orthogonal complement of \(Cx + Dy\) in \(H^{n+k}\). We get a connection on \(E\) induced from the product connection on \(\mathbb{H^{n+k}} \times S^4\).
Write \(v(x,y) = Cx+Dy\). Then \(Q = v(v^*v)^{-1}v^*\) is the orthogonal projection onto the span of \(Cx+Dy\). Set \(y = 1\). Let \(P = I - Q\) be the complementary projection. The curvature our connection can be written as \[F = PC dx(Cx+D)^*(Cx+D)^{-1}\wedge d\bar x C^*)P.\] Since \(dx \wedge d\bar x\) is self-dual, then \(F\) is self-dual if and only if \[(Cx+D)^*(Cx+D)^{-1} \textrm{\;is\;real.}\]
More explicitly, let \(C = \begin{pmatrix}C_0 \\ C_1\end{pmatrix}\), \(D = \begin{pmatrix}D_0 \\ D_1\end{pmatrix}\), where \(C_0,D_0\) are \(n \times k\) and \(C_1,D_1\) are \(k\times k\). Observe that the above reality is condition is preserved under left of multiplication of \(Cx + D\) by an element of \(Sp(n+k)\), or right multiplication by an element of \(O(k)\). One can therefore check that without loss of generality, we can take \(C_0 = O, C_1 = -I.\) write \(D = \begin{pmatrix}\Lambda \\ B\end{pmatrix}\). Then the expression in (1) becomes \(\Lambda^*\Lambda + B^*B - (\bar x B + B^* x) + |x|^2I\). Thus, for this expression to be real for all \(x\), we need (i) \(\Lambda^*\Lambda + B^*B\) to be real (ii) \(\bar x B + B^*x\) is real \(\iff B^T = B\).
Suppose \(n=1\), \(\Lambda = [\lambda_1, \ldots, \lambda_n]\). If we do a formal dimension count, then \(\dim \Lambda = 4k\), \(\dim B = 4k(k+1)/2 = 2k^2 + 2k\), condition (i) imposes \(3k(k-1)/2\) constraints, and the dimension of the symmetry group \(O(k) \times Sp(1)\) is \(k(k-1)/2 + 3\). So overall, the formal dimension of the moduli space of \(SU(2)\) \(k\)-instantons is \[d_k = (4k + 2k^2+2k) - (3k(k-1)/2 + k(k-1)/2 + 3) = 8k-3.\] This is the correct answer in fact as can be shown rigorously using the Atiyah-Singer index theorem.
