In the AQFT formalization of quantum field theory a system is characterized by its local net of observables on spacetime, which is in particular a copresheaf of algebras. Accordingly, one can consider notions of cohomology with coefficients in such a local net.
The concept was first poposed around
John Roberts, Mathematical Aspects of Local Cohomology talk at Colloqiumon Operator Algebras and their Applications to Mathematical Physics, Marseille 20-24 June, (1977)
Here the idea was put forward that local nets of observables should carry a notion of cohomology – or rather of nonabelian cohomology – with coefficients in some kind of ∞-category. Motivated by this John Roberts was one of the first to consider strict ω-categories. He conjectured that these are characterized by their ω-nerves being complicial sets. This led Ross Street to develop the notion of orientals and eventually to prove this conjecture. An account of this development is on pages 9-10 of
More comments on the role of cohomology in AQFT are in
John Roberts, A survey of local cohomology Mathematical Problems in Theoretical Physics Lecture Notes in Physics, (1978) Volume 80/1978
John Roberts, Net cohomology and its applications to field theory, Quantum fields-algebras, processes (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978), pp. 239-268, Springer, Vienna (1980).
John Roberts, The Search for Quantum Differential Geometry Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, (1982) Volume 153/1982, 374-379
John Roberts, The cohomology and homology of quantum field theory at Quantum fields and quantum space time (Cargèse, 1996), 357-368, NATO Adv. Sci. Inst. Ser. B Phys., 364, Plenum, New York, (1997)
The description of DHR superselection theory in terms of net cohomology was given in
Its generalization to general spacetimes (curved and with nontrivial topology) is discussed in
The trivial Sectors of the Massless scalar free field in 1 + 3 dimensions was discussed in
Buchholz, Doplicher, Longo, Roberts (1992)
Fabio Ciolli, Massless scalar free Field in dimensions, II: Net Cohomology and Completeness of Superselection Sectors (arXiv:0811.4673)
Fabio Ciolli, Net Cohomology and the Construction of Physical Models , talk at Vietri (2009) (pdf)