nLab cohomology in a local net of observables

Idea

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.

One such notion was introduced in (Roberts 76), there called local cohomology or net cohomology. It has been shown to encode the DHR superselection theory of local nets (Roberts 90).

References

Original articles

The concept was first poposed around

• John RobertsLocal cohomology and superselection structure Comm. Math. Phys. Volume 51, Number 2 (1976), 107-119 (EUCLID)

• John Roberts, Mathematical Aspects of Local Cohomology talk at Colloqiumon Operator Algebras and their Applications to Mathematical Physics, Marseille 20-24 June, (1977)

• P. Leyland, John Roberts, The cohomology of nets over Minkowski space (EUCLID)

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 ∞-categories. He conjectured that these are characterized by their ∞-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

• Ross Street, An Australian conspectus of higher category theory (pdf)

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

• John Roberts, (1990)

Its generalization to general spacetimes (curved and with nontrivial topology) is discussed in

• Giuseppe Ruzzi, Homotopy of posets, net-cohomology and superselection sectors in globally hyperbolic spacetimes (arXiv:math-ph/0412014)

• Romeo Brunetti, Giuseppe Ruzzi, Superselection Sectors and General Covariance I (arXiv:gr-qc/0511118)

• Romeo Brunetti, Giuseppe Ruzzi, Quantum charges and spacetime topology: The emergence of new superselection sectors (arXiv:0801.3365)

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 $1+1$ dimensions, II: Net Cohomology and Completeness of Superselection Sectors (arXiv:0811.4673)

Talks

• John Roberts, Cohomology in the service of AQFT , talk at Göttingen, AQFT, the first 50 years (2009) (pdf)

• Giuseppe Ruzzi, Net cohomology and local charges , talk at Vietri (2009) (pdf)

• Fabio Ciolli, Net Cohomology and the Construction of Physical Models , talk at Vietri (2009) (pdf)