6 HISHAM SATI AND URS SCHREIBER

understanding of this quantization step is one of the links between the worldvol-

ume theory and the target space theory and hence between the abstract algebraic

description of the worldvolume QFT and the phenomenological interpretation of

its correlators in its target space, ultimately connecting theory to experiment. We

now indicate some of the progress in mathematically understanding the process of

quantization in general and of sigma-models in particular.

(i) Path integral quantization. It has been suggested (e.g. [Fre06]) that the

path integral is to be understood abstractly as a pull-push operation – an integral

transform – acting on states in the form of certain cocycles, by ﬁrst pulling them up

to the space of worldvolume conﬁgurations along the map induced by the incom-

ing boundary, and then pushing forward along the map induced by the outgoing

boundary. This is fairly well understood for Dijkgraaf-Witten theory [FrQu93]. In

[FHLT10] it is claimed that at least for all the higher analogs of Dijkgraaf-Witten

theory (such as the Yetter model [MaPo07]) a formal pull-push path integral quan-

tization procedure exists in terms of colimits of n-categorical algebras, yielding fully

extended TQFTs.

A more geometric example for which pull-push quantization is well understood

is Gromov-Witten theory [Ka06]. More recently also Chas-Sullivan’s string topol-

ogy operations have been understood this way, for strings on a single brane in

[Go07] and recently for arbitrary branes in [Ku11]. In [BZFNa11] it is shown

that such integral transforms exist on stable ∞-categories of quasicoherent sheaves

for all target spaces that are perfect derived algebraic stacks, each of them thus

yielding a 2-dimensional TQFT from background geometry data.

(ii) Higher background gauge ﬁelds. Before even entering (path integral)

quantization, there is a fair bit of mathematical subtleties involved in the very

deﬁnition of the string’s action functional in the term that describes the coupling

to the higher background gauge ﬁelds, such as the Neveu-Schwarz (NS) B-ﬁeld

and the Ramond-Ramond (RR) ﬁelds. All of these are recently being understood

systematically in terms of generalized diﬀerential cohomology [HS05].

Early on it had been observed that the string’s coupling to the B-ﬁeld is glob-

ally occurring via the higher dimensional analog of the line holonomy of a circle

bundle: the surface holonomy [GaRe02][FNSW09] of a circle 2-bundle with con-

nection [Sch11]: a bundle gerbe with connection, classiﬁed by degree-3 ordinary

diﬀerential cohomology. More generally, on orientifold target space backgrounds it

is the nonabelian (Z2//U(1))-surface holomomy [ScWa08][Ni11] over unoriented

surfaces [SSW05].

After the idea had materialized that the RR ﬁelds have to be regarded in K-

theory [MoWi00] [FrHo00], it eventually became clear [Fre01] that all the higher

abelian background ﬁelds appearing in the eﬀective supergravity theories of string

theory are properly to be regarded as cocycles in generalized diﬀerential cohomology

[HS05] – the RR-ﬁeld being described by diﬀerential K-theory [BuSch11] – and

even more generally in twisted such theories: the presence of the B-ﬁeld makes the

RR-ﬁelds live in twisted K-theory (cf. [BMRZ08]).

A perfectly clear picture of twisted generalized cohomology theory in terms

of associated E∞-module spectrum ∞-bundle has been given in [ABG10]. This

article in particular identiﬁes the twists of tmf-theory, which are expected [Sa10]

[AnSa11] to play a role in M-theory in the higher analogy of twisted K-theory in

string theory.

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