group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The effective bosonic exponentiated action functional of the type II superstring sigma-model for open strings ending on D-branes has three factors:
the higher holonomy of the background B-field over the string 2-dimensional worldsheet;
the ordinary holonomy of the Chan-Paton bundle on the D-brane along the boundary of the string;
the Berezinian path integral over the fermions.
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 from 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):
Let $i \colon Q \to X$ be a map of compact manifolds and let $\chi \colon X \to B^2 U(1)$ modulate a circle 2-bundle regarded as a twist for K-theory. Then forming twisted groupoid convolution algebras yields a KK-theory morphism of the form
with notation as in this definition. By this proposition the dual morphism is of the form
If we redefine the twist on $X$ to absorb this “quantum correction” as $\chi \mapsto \frac{1}{\chi \otimes W_3(T X)}$ then this is
where now we may interpret $\frac{W_3(i^\ast \tau_X)}{W_3(\tau_Q)}$ as the third integral Stiefel-Whitney class of the normal bundle $N Q$ of $i$ (see Nuiten).
Postcomposition with this map in KK-theory now yields a map from the $i^\ast \chi \otimes W_3(N Q)$-twisted K-theory of $Q$ to the $\chi$-twisted K-theory of $X$:
If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a D-brane worldvolume, then $\chi$ would be the class of the background B-field and an element
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 $\xi$ given by a twisted line bundle in which case it exhibits a twisted spin^c structure on $Q$.) Finally its push-forward
is called the corresponding D-brane charge.
The special case where the class of the restriction of the B-field to the D-brane equals the third integral Stiefel-Whitney class of the D-brane was discussed in
The generalization to the case that the two classes differ by a torsion class was considered in
Aspects of the interpretation of this by push-forward in generalized cohomology in twisted K-theory are formalized in
and section 10 of
(which discusses twists as (infinity,1)-module bundles).
The formulation by postcomposition with dual morphisms in KK-theory which we use above is based on the observations in section 7 of
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)
and
In (Laine) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).
A discussion from the point of view of higher geometric quantization or extended prequantum field theory is at the end of
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