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.
Previous Page Next Page