Donaldson invariants of product ruled surfaces and two-dimensional gauge theories (Q5953624)

\textit{G. Moore} and \textit{E. Witten} in [Adv. Theor. Math. Phys. 1, No. 2, 298-387 (1997; Zbl 0899.57021)] showed some unexpected perspectives of Donaldson invariants from the physics approach. The relation between Donaldson invariants and Seiberg-Witten invariants is explored by the \(u\)-plane integral. The \(u\)-plane integration technique can be extended to obtain Donaldson invariants for non-simply connected 4-manifolds and general wall-crossing formulae. For a smooth 4-manifold \(X\) with \(b_1(X)\) even and \(b^+_2(X) = 1\) (for example \(X = \Sigma_g \times S^2\), where \(\Sigma_g\) is a Riemann surface of genus \(g\)), let \(\Omega\) be a symplectic 2-form on the Jacobian torus \(J(X)\) of \(X\). Then the \(u\)-plane integral is given by \[ Z_u = - 4 \pi i \int_{{\Gamma}^0(4)\setminus H}\frac{dx dy}{y^{1/2}} \int_{J(X)} h_{\infty} f_{\infty} \Psi(S) \cdot \frac{\Omega^{b_1/2}}{(b_1/2)!} \] where \(x+iy\) is a coordinate of the upper half-plane \(H\), \({\Gamma}^0(4)\) is a principal congruence subgroup of level \(4\), \(h_{\infty} = \theta_2 \theta_3/2\) with \(\theta_i\) Jacobi theta functions, \(f_{\infty}\) is a differential form on \(J(X)\) (almost holomorphic modular form) and \(\Psi(S)\) is a Siegel-Narain theta function. The integral \(Z_u\) has discontinuous variation at the cusps (located at \(\tau = i \infty, \tau = 0, \tau = 2\)) of \({\Gamma}^0(4)\setminus H\). In Moore and Witten's paper, the authors showed that the discontinuous variation of \(Z_u\) at \(\tau = 0\) and \(\tau = 2\) (for Donaldson invariant contribution) cancel the contribution to wall-crossing from \(\tau = i \infty \) (for Seiberg-Witten invariants). The main result is \(Z_u = Z_D - Z_{SW}\) as the \(u\)-plane integral measures the difference of the contribution from \(\tau = 0\) and \(\tau = 2\) and the contribution from \(\tau = i \infty \). The index calculation from the heat kernel can be expressed as a contribution at \(t = 0\) and a contribution at \(t = \infty\) from linear operator theory. The \(u\)-plane integral is a generalization of 1-real parameter of linear theory into 1-complex parameter of nonlinear theory, in section 3 of the paper under review, the \(u\)-plane integral, the Donaldson and Seiberg-Witten invariants for the smooth 4-manifold \(\Sigma_g \times S^2\) are explicitly calculated. Using the identification of the moduli space of anti-self-dual connections (charge zero) on \(\Sigma_g \times S^2\) with the moduli space of flat connections on \(\Sigma_g\), the authors derive the first application of their results to the intersection pairing, recursive relation and Verlinde's formula in section 4. When singularities occur in the moduli spaces, the paper only indicates certain modifications needed without further pursuit. Their second application is to derive the eigenvalue spectrum of the Fukaya-Floer cohomology of \(\Sigma_g \times S^1\) (which is related to results of Mu\(\widetilde{n}\)oz reviewed earlier). These applications based on the authors' viewpoint are examples to preferably use either ``electric'' expressions or ``magnetic'' expressions (but not both) in some physical meanings. It would be interesting to know if the physics approach in this paper can be extended to \(X^4 \times S^2\) for symplectic 6-manifolds, i.e., if the dimension reduction, \(q\)-plane integral (for a quaternion parameter \(q\)) can be extended to relate the Donaldson invariants and Gromov-Witten invariants by \(Z_q = Z_D - Z_{GW}\).