Each single contribution is in general not a globally well defined function on the space of string configurations, instead each is a section of a possibly non-trivial line bundle over the configuration space (the last one for instance of the Pfaffian bundle). Therefore the total action functional is a section of the tensor product of these three line bundles.
The non-triviality of this tensor product line bundle (as a line bundle with connection) is the Freed-Witten-Kapustin quantum anomaly. The necessary conditions for this anomaly to vanish, hence for this line bundle to be trivializable, is the Freed-Witten anomaly cancellation condition.
More precisely, the naive holonomy of an ordinary Chan-Paton principal connection would be globally well defined. But in order to cancel the anomaly contribution form the other two factors, one may take the Chan-Paton bundle to be a twisted bundle, the twist being the B-field restricted to the brane. Then its holonomy becomes anomalous, too, but there are then interesting configurations where the product of all three anomalies cancels. This refined argument has been made precise by Kapustin, and so one should probably speak of the Freed-Witten-Kapustin anomaly cancellation.
We interpret the Freed-Witten-Kapustin mechanism in terms of push-forward in generalized cohomology in topological K-theory interpreted in terms of KK-theory with push-forward maps given by dual morphisms between Poincaré duality C*-algebras (based on Brodzki-Mathai-Rosenberg-Szabo 06, section 7, Tu 06):
If we redefine the twist on to absorb this “quantum correction” as then this is
is called (the K-class of) a Chan-Paton gauge field on the D-brane satisfying the Freed-Witten-Kapustin anomaly cancellation mechanism. (The orginal Freed-Witten anomaly cancellation assumes given by a twisted line bundle in which case it exhibits a twisted spin^c structure on .) Finally its push-forward
is called the corresponding D-brane charge.
| 1 | complex K-theory | third integral SW class | spin^c-structure | K-theoretic geometric quantization | Freed-Witten anomaly | | 2 | EO(n) | Stiefel-Whitney class | | | | | 2 | integral Morava K-theory | seventh integral SW class | | | Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation |
The generalization to the case that the two classes differ by a torsion class was considered in
and section 10 of
(which discusses twists as (infinity,1)-module bundles).
and generalized to equivariant KK-theory in
A clean formulation and review is provided in
Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, Classifying A-field and B-field configurations in the presence of D-branes (arXiv:0810.4291)
Fabio Ferrari Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)
Fabio Ferrari Ruffino, Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes (arXiv:1104.2798)
Raffaele Savelli, On Freed-Witten Anomaly and Charge/Flux quantization in String/F Theory, Phd thesis (2011) (pdf)
In (Laine) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).
Lecture notes along these lines are in Lagrangians and Action functionals – 3d Chern-Simons theory of
The KK-theory-description of the FEK anomaly used above is discussed in