INTRODUCTION 5
II. Systems of algebras of observables
(i) Nets of algebras. In the form of the Haag-Kastler axioms, the description of
QFT through its local algebras of observables had been given a clean mathematical
formulation [HaM¨ u06] a long time ago [HaKa64]. This approach had long pro-
duced fundamental structural results about QFT, such as the PCT theorem and
the spin-statistics theorem (cf. [StWi00]). Only recently has it finally been shown
in detail [BDF09] how examples of AQFT nets can indeed be constructed along
the lines of perturbation theory and Wilsonian effective field theory, thus connect-
ing the major tools of practicing particle physicists with one of the major formal
axiom systems. Using an operadic variant of Haag-Kastler nets in the case of Eu-
clidean (“Wick rotated”) QFT called factorization algebras a similar discussion
is sketched in [CoGw]. At the same time, the original axioms have been found
to naturally generalize from Minkowski spacetime to general (globally hyperbolic)
curved and topologically nontrivial spacetimes [BFV01].
(ii) Boundaries and defects. The Haag-Kastler axioms had been most fruitful
in the description of 2 dimensional and conformal field theory (“conformal nets”),
where they serve to classify chiral 2d CFTs [KaLo03][Ka03], construct integral
2d QFTs [Le06] and obtain insights into boundary field theories (open strings)
[LoRe04]. Remarkably, the latter has recently allowed a rigorous re-examination
[LoWi10] of old arguments about the background-independence of string field the-
ory. (The contribution by Douglas-Henriques in this volume presents a modern
version of the Haag-Kastler axioms for conformal nets and extends the discussion
from boundary field theory to field theory with defects.)
(iii) Higher chiral algebras. The geometric reformulation of vertex operator
algebras in terms of chiral algebras [BeDr04] has proven to be fruitful, in par-
ticular in its higher categorical generalizations [Lur11] by factorizable cosheaves
of ∞-algebras. While the classical AQFT school restricted attention to QFT over
trivial topologies, it turns out that also topological QFTs can be described and
constructed by local assignments of algebras “of observables”. In [Lur09b] n-
dimensional extended TQFTs are constructed from En-algebras algebras over the
little n-cubes operad by a construction called topological chiral homology, which
is a grand generalization of Hochschild homology over arbitrary topologies. (The
contribution by Weiss in this volume discusses the theory of homotopy algebras
over operads involved in these constructions.)
This last work is currently perhaps the most formalized and direct bridge be-
tween the two axiom systems, the functorial and the algebraic one. This indicates
the closure of a grand circle of ideas and makes the outline of a comprehensive
fundamental formalization of full higher-genus QFT visible.
III. Quantization of classical field theories
While a realistic axiomatization is the basis for all mathematical progress in
QFT, perhaps even more important in the long run for physics is that with the sup-
posed outcome of the (path integral) quantization process thus identified precisely
by axioms for QFT, it becomes possible to consider the nature of the quantization
process itself. This is particularly relevant in applications of QFT as worldvolume
theories in string theory, where one wishes to explicitly consider QFTs that arise as
the quantization of sigma-models with specified gauge background fields. A good
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