2 HISHAM SATI AND URS SCHREIBER

including gravity – without much more of a structural guidance than the folklore

of the path integral, however useful that had otherwise proven to be.

While everyone involved readily admitted that nobody knew the full answer to

What is string theory?

perhaps it was gradually forgotten that nobody even knew the full answer to

What is quantum ﬁeld theory?

While a huge discussion ensued on the “landscape” moduli space of backgrounds

for string theory, it was perhaps forgotten that nobody even had anything close to

a full answer to

What is a string theory background?

or even to what should be a simpler question:

What is a classical string theory background?

which in turn is essentially the question:

What is a full 2-dimensional σ-model conformal ﬁeld theory?

Most of the literature on 2-dimensional conformal ﬁeld theory (2d CFT) describes

just what is called chiral conformal ﬁeld theory, formalized in terms of vertex oper-

ator algebras or local conformal nets. But this only captures the holomorphic and

low-genus aspect of conformal ﬁeld theory and is just one half of the data required

for a full CFT, the remaining piece being the full solution of the sewing constraints

that makes the theory well deﬁned for all genera.

With these questions – fundamental as they are for perturbative string theory –

seemingly too hard to answer, a plethora of related model and toy model quantum

ﬁeld theoretic systems found attention instead. A range of topological (quantum)

ﬁeld theories (T(Q)FTs) either approximates the physically relevant CFTs as in

the topological A-model and B-model, or encodes these holographically in their

boundary theory as for Chern-Simons theory and its toy model, the Dijkgraaf-

Witten theory.

In this way a wealth of worldvolume QFTs appears that in some way or an-

other is thought to encode information about string theory. Furthermore, in each

case what really matters is the full worldvolume QFT: the rule that assigns corre-

lators to all possible worldvolume cobordisms, because this is what is needed even

to write down the corresponding second quantized perturbation series. However,

despite this urgent necessity for understanding QFT on arbitrary cobordisms, the

tools to study or even formulate this precisely were for a long time largely unavail-

able. Nevertheless, proposals for how to make these questions accessible to the

development of suitable mathematical machinery already existed.

Early on it was suggested, based on topological examples, that the path integral

and the state-propagation operators that it is supposed to yield are nothing but

a representation of a category of cobordisms [At88]. It was further noticed that

this prescription is not restricted to TQFTs, and in fact CFTs were proposed to

be axiomatized as representations of categories of conformal cobordisms [Se04]. In

parallel to this development, another school developed a dual picture, now known

as local or algebraic quantum ﬁeld theory (AQFT) [Ha92], where it is not the

state-propagation – the Schr¨ odinger picture – of QFT that is axiomatized and

made accessible to high-powered machinery, but rather the assignment of algebras

of observables – the Heisenberg picture of QFT.

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