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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