I am trying to understand the correspondence between Donaldson invariants and different correlation functions in certain topological quantum field theories. To be exact, among others I am reading Witten's "Supersymmetric Yang-Mills Theory on a Four-Manifold". However, topic seems to be a difficult one for me, and so I wonder, whether there are some (hopefully easier) examples of the same concept. Namely, are there some analogs of Donaldson Invariants in 2 dimensions which can be alternatively computed from the side of physics?
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$\begingroup$ I don’t understand your question fully, but at the least you should look up “Nahm’s equations”. $\endgroup$– Chris GerigCommented Dec 7, 2019 at 1:50
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It depends what you mean by the Donaldson invariants. The smooth structure on a two manifold is unique, so there aren't any analogues of the Donaldson invariant in that sense. But there are 2d physics analogues of the Donaldson invariants, in that there are two-dimensional topologically twisted quantum field theories some of whose correlation functions reduce to intersection numbers of moduli spaces. The most direct analogy is twisted two-dimensional gauge theory, which computes some of the cohomology of the moduli space of bundles on a Riemann surface. A more famous example is the A-type topologically twisted string theory, which gives Gromov-Witten invariants.
An old reference that sketches out these analogies is Witten's Introduction to Cohomological Field Theories. Also worth time if you have enough of it is Cordes, Moore, & Ramgoolam's Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories.
