INTRODUCTION 3

While these axiomatizations were known and thought of highly by a few select

researchers who worked on them, they were mostly happily ignored by the quantum

ﬁeld theory and string theory community at large, and to a good degree rightly so:

nobody should trust an axiom system that has not yet proven its worth by providing

useful theorems and describing nontrivial examples of interest. But neither the

study of cobordism representations nor that of systems of algebras of observables

could for a long time – apart from a few isolated exceptions – claim to add much

to the world-view of those who value formal structures in physics, but not a priori

formal structures in mathematics. It is precisely this that is changing now.

Major structural results have been proven about the axioms of functorial quan-

tum ﬁeld theory (FQFT) in the form of cobordism representations and dually those

of local nets of algebras (AQFT) and factorization algebras. Furthermore, classes of

physically interesting examples have been constructed, ﬁlling these axiom systems

with life. We now provide a list of such results, which, while necessarily incom-

plete, may serve to give an impression of the status of the ﬁeld, and serve to put

the contributions of this book into perspective.

I. Cobordism representations

(i) Topological case. The most foundational result in TQFT is arguably the

formulation and proof [Lur09b] of the cobordism hypothesis [BaDo95] which

classiﬁes extended (meaning: “fully local”) n-dimensional TQFT by the “fully

dualizability”-structure on the “space” of states (an object in a symmetric monoidal

(∞, n)-category) that it assigns to the point. (In this volume the contribution by

Bergner surveys the formulation and proof of the cobordism hypothesis). This

hugely facilitates the construction of interesting examples of extended n-dimensional

TQFTs. For instance

• recently it was understood that the state-sum constructions of 3d TQFTs

from fusion categories (e.g. [BaKi00]) are subsumed by the cobordism

hypothesis-theorem and the fact [DSS11] that fusion categories are the

fully dualizable objects in the (∞, 3)-category of monoidal categories with

bimodule categories as morphisms;

• the Calabi-Yau A∞-categories that Kontsevich conjectured [Ko95] en-

code the 2d TQFTs that participate in homological mirror symmetry

have been understood to be the “almost fully dualizable” objects (Calabi-

Yau objects) that classify extended open/closed 2-dimensional TQFTs on

cobordisms with non-empty outgoing boundary with values in the (∞, 1)-

category of chain complexes (“TCFTs” [Cos07a], [Lur09b]);

In this context crucial aspects of Witten’s observation in [Wi92] have

been made precise [Cos07b], relating Chern-Simons theory to the eﬀec-

tive target space theory of the A- and B-model topological string, thus

providing a rigorous handle on an example of the eﬀective background

theory induced by a string perturbation series over all genera.

(ii) Conformal case. A complete classiﬁcation of rational full 2d CFTs on cobor-

disms of all genera has been obtained in terms of Frobenius algebra objects in

modular tensor categories [FRS06]. While the rational case is still “too simple”

for the most interesting applications in string theory, its full solution shows that

already here considerably more interesting structure is to be found than suggested

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