I will describe constraints on the topology of Lefschetz fibrations with 4-dimensional fibers (in the smooth category), obtained from Seiberg—Witten theory. Our main result relates the index of a Dirac operator on the total space with the monodromy of the fibration, and is reminiscent of the Grothendieck—Riemann—Roch formula in the holomorphic setting. As topological applications, we construct closed simply-connected symplectic 4-manifolds which admit Torelli symplectic diffeomorphisms that are not isotopic to products of squared Dehn twists on Lagrangian spheres, answering a question of Donaldson in the negative. Based on joint work with Hokuto Konno, Jianfeng Lin, and Anubhav Mukherjee.