cohomology

# Contents

## Idea

The effective bosonic exponentiated action functional of the type II superstring sigma-model for open strings ending on D-branes has three factors:

1. the higher holonomy of the backgroun B-field over the string 2-dimensional worldsheet;

2. the ordinary holonomy of the Chan-Paton bundle on the D-brane along the boundary of the string;

3. 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 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.

## Details

… for the moment see the discussion at twisted spin^c structure.

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory $\mathrm{KU}$third integral SW class ${W}_{3}$spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class ${w}_{4}$
2integral Morava K-theory $\stackrel{˜}{K}\left(2\right)$seventh integral SW class ${W}_{7}$Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

## References

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

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

• Kim Laine, Geometric and topological aspects of Type IIB D-branes, Master thesis (arXiv:0912.0460)

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

Revised on June 17, 2013 17:33:58 by Urs Schreiber (82.113.99.37)