Critical string models
In the context of string theory, the background gauge field for the open string sigma-model over a D-brane in bosonic string theory or type II string theory is a unitary principal bundle with connection, or rather, by the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation mechanism, a twisted unitary bundle, whose twist is the restriction of the ambient B-field to the D-brane.
The first hint for the existence of such background gauge fields for the open string 2d-sigma-model comes from the fact that the open string’s endpoint can naturally be taken to carry labels . Further analysis then shows that the lowest excitations of these -strings behave as the quanta of a -gauge field, the -excitation being the given matrix element of a -valued connection 1-form .
This original argument goes back work by Chan and Paton. Accordingly one speaks of Chan-Paton factors and Chan-Paton bundles .
We discuss the Chan-Paton gauge field and its quantum anomaly cancellation in extended prequantum field theory.
Throughout we write Smooth∞Grpd for the cohesive (∞,1)-topos of smooth ∞-groupoids.
The -field as a prequantum 2-bundle
For a type II supergravity spacetime, the B-field is a map
If is a Lie group, this is the prequantum 2-bundle of -Chern-Simons theory. Viewed as such we are to find a canonical ∞-action of the circle 2-group on some , form the corresponding associated ∞-bundle and regard the sections of that as the prequantum 2-states? of the theory.
The Chan-Paton gauge field is such a prequantum 2-state.
The Chan-Paton gauge field
We discuss the Chan-Paton gauge fields over D-branes in bosonic string theory and over -D-branes in type II string theory.
We fix throughout a natural number , the rank of the Chan-Paton gauge field.
For a smooth manifold and modulating a circle 2-group-principal 2-bundle, maps
in the slice (∞,1)-topos , hence diagrams of the form
in are equivalently rank- unitary twisted bundles on , with the twist being the class .
There is a further differential refinement
where is the universal moduli 2-stack of circle 2-bundles with connection (bundle gerbes with connection).
for the differential smooth universal Dixmier-Douady class of prop. 4, regarded as an object in the slice (∞,1)-topos over .
be an inclusion of smooth manifolds or of orbifolds, to be thought of as a D-brane worldvolume inside an ambient spacetime .
Then a field configuration of a B-field on together with a compatible rank- Chan-Paton gauge field on the D-brane is a map
in the arrow (∞,1)-topos , hence a diagram in of the form
This identifies a twisted bundle with connection on the D-brane whose twist is the class in of the bulk B-field.
This relation is the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation for the bosonic string or else for the type II string on D-branes. (FSS)
The open string sigma-model
The moduli stack of such field configurations is the homotopy pullback
The anomaly-free open string coupling to the Chan-Paton gauge field
For a smooth manifold with boundary of dimension and for a circle n-bundle with connection on some , then the transgression of to the mapping space yields a section of the complex line bundle associated to the pullback of the ordinary transgression over the mapping space out of the boundary: we have a diagram
The operation of forming the holonomy of a twisted unitary connection around a curve fits into a diagram in of the form
It follows that on the moduli space of the open string sigma-model of prop. 5 above there are two -valued action functionals coming from the bulk field and the boundary field
Neither is a well-defined -valued function by itself. But by pasting the above diagrams, we see that both these constitute sections of the same complex line bundle on the moduli stack of fields:
Therefore the product action functional is a well-defined function
This is the Kapustin anomaly-free action functional of the open string.
In the traditional physicist’s string theory introductions one finds Chan-Paton bundles discussed for instance
section 2.4 of
or around p. 66 of
These lectures tend to ignore most of the subtleties though. For traditional discussion of the Freed-Witten-Kapustin anomaly, see there. The above account in terms of higher geometry and extended prequantum field theory is due to section 5.4 of
Lecture notes along these lines are at