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

5