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
These lectures tend to ignore most of the global 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
Discussion of the derivation of the Yang-Mills theory on the D-brane from open string scattering amplitudes/string field theory includes
- David Gross, Edward Witten, Superstring modifications of Einstein’s equations, Nuclear Physics B Volume 277, 1986, Pages 1-10
and for the non-abelian case:
- Semyon Klevtsov, Yang-Mills theory from String field theory on D-branes (pdf)