A differential form with logarithmic singularities is a meromorphic differential form on some space $X$ which is a holomorphic differential form on a suitably dense open subspace with at most logarithmic singularities at the boundary.
These are the differential forms on spaces in logarithmic geometry. They form the logarithmic generalization of the holomorphic de Rham complex.
The moduli spaces of connections with logarithmic singularity at prescribed points are for instance the object of interest in “ramified” geometric Langlands duality. (e.g. Witten 08, section 4)
A 't Hooft operator is a gauge field configuration with certain logarithmic differential form singularities.
The brief idea is well described in
Further details are discussed in
Arthur Ogus, Chapter IV of Lectures on logarithmic algebraic geometry, TeXed notes, 2001, pdf
Arthur Ogus, slides 31 ff in Logarithmic geometry, talk slides 2009 (pdf)
José Ignacio Burgos, A $C^\infty$-logarithmic Dolbeault complex, Compositio Math. 92 (1994), no. 1, 61-86. MR 1275721 (95g:32056)
Claire Voisin, section 8.2.2 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3
Discussion in the context of geometric Langlands duality includes
In the context of differential algebraic K-theory
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