Degeneracy loci of families of Dirac operators (Q2844738)

Denote by \(E^w_k \to X\) a complex, rank-two vector bundle over a smooth, closed, oriented, four-manifold, with \(c_1(E^w_k) =w\) and \(k =-(1/4)\langle p_1(\mathfrak{su}(E)),[X]\rangle\). The Donaldson invariants are defined by integrating cohomology classes, the \(\mu\)-classes, over the moduli space \(M^w_k\) of projectively anti-self-dual connections on \(E^w_k\). If \((W,\rho)\) is a \(\mathrm{spin}^c\) structure on \(X\), then each unitary connection \(A\) on \(E^w_k\) defines a Dirac operator on the \(\mathrm{spin}^u\) structure \(\mathfrak{t}=(W\otimes E, \rho \otimes id_E)\) of index \(n_\alpha (\mathfrak{t})\). If \(n_\alpha (\mathfrak{t})\leq 0\) and we denote \(\Lambda =c_1(W^+)+c_1(E)\), then the subspace \(J_k^{\Lambda,w}\subset M^w_k\), of connections whose Dirac operator has one-dimensional kernel is, for generic perturbations, a smooth submanifold of codimension \(2(1-n_\alpha(\mathfrak{t}))\).NEWLINENEWLINE The author obtains generalizations of results from [\textit{W.-M. R. Leung}, On \(\mathrm{spin}^c\) invariants of four-manifolds. Oxford: Oxford University (PhD thesis) (1995)], by computing the Poincaré dual of \(J_k^{\Lambda,w}\):NEWLINENEWLINE Theorem 1.1. Let \(E^w_k\to X\) be a complex, rank-two vector bundle over a smooth, closed, oriented, four-manifold with \(b_1(X)=0\). Let \(M^w_k\) be the moduli space of projectively anti-self-dual connections on \(E^w_k\) and let \(K^w_k\subset M^w_k\) be any compact codimension-zero submanifold. Let \(\mathfrak{t}\) be a \(\mathrm{spin}^u\) structure on \(X\) with \(p_1(\mathfrak{t})=k\), \(c_1(\mathfrak{t})=\Lambda\), and \(n_\alpha (\mathfrak{t})\leq 0\). Let \(J_k^{\Lambda ,w}\subset M^w_k\) be the degeneracy locus of the \(\mathrm{spin}^u\) structure \(\mathfrak{t}\). Then, as an element of rational cohomology, the Poincaré dual of \(J_k^{\Lambda,w}\) in \(K^w_k\) is NEWLINE\[NEWLINE(-1)^{1-n_{\alpha}(\mathfrak{t})}\sum_{i+2j+2k =1-n_{\alpha}(\mathfrak{t})}f_{i,2j,2k}\mu(\mathfrak{t})^{i}\cup \Omega^{j} \cup \mu(x)^{k},NEWLINE\]NEWLINE where \(\mu(\mathfrak{t})\), \(\Omega\) are \(\mu\)-classes and \(f_{i,j,k}\) are coefficients determined by a certain recursion relation, obtained from the function NEWLINE\[NEWLINEF(x,y,z)=\exp ((1/2)xJ_1(z)+(1/4)y^2J_2(z)+J_3(z)).NEWLINE\]NEWLINE The functions \(J_i(z)\) are defined by NEWLINE\[NEWLINE\begin{aligned} & J_1(z)=z^{-1}\tan^{-1}(z),\\ & J_2(z)=z^{-3}(z-\tan^{-1}(z)),\\ & J_3(z)=-(1/2)n_{\alpha} (\mathfrak{t})\ln(1+z^2)+ k (z^{-1}\tan^{-1}(z)-1).\end{aligned}NEWLINE\]