Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
An orientifold is a background for string sigma-models that combines aspects of -orbifolds with orientation reversal on the worldsheet (therefore the name): it consists of a bundle gerbe on a space with a -action that satisfies a peculiar twisted equvariance condition with respect to this action.
Such orientifold gerbes with connection are the right structure for the definition of surface holonomy of unoriented surfaces. Therefore they serve for defining the gauge part of the action functional for unoriented strings.
More precisely, the gauge fields that constitute the background for a string -model, such as the Kalb-Ramond field and the RR-fields are modeled as cocycles in the differential cohomology of the target space, and an orientifold is the data given by an orbifold spacetime that involves the group and equipped with certain [classes in its( twisted) differential cohomology that is suitably -equivariant.]
Bosonic string: orientifold circle -bundles with connection
We discuss the notion of circle n-bundles with connection over double covering spaces with orientifold structure.
The smooth automorphism 2-group of the circle group is that corresponding to the smooth crossed module
where the differential is trivial (constant on the neutral element) and where the action of on is given under the identification of with the unit circle in the plane by reversal of the sign of the angle.
This is an extension of smooth ∞-groups
The nature of is clear. Let be the evident inclusion. We have to show that its delooping is the homotopy fiber of .
For this it is sufficient to show that is equivalent to the ordinary pullback of simplicial presheaves of the 2-universal principal bundle.
This pullback is the 2-groupoid whose
objects are elements of ;
morphisms are labeled by such that ;
all 2-morphisms are endomorphisms, labeled by ;
vertical composition of 2-morphisms is given by the group operation in ,
horizontal composition of 1-morphisms with 1-morphisms is given by the group operation in
horizontal composition of 1-morphisms with 2-morphisms (whiskering) is given by the action of on .
This 2-groupoid has vanishing , and has . The inclusion of into this pullback is the obvious one, including elements in as endomorphisms of the trivial element in . This is manifestly an isomorphism on and trivially an isomorphism on all other homotopy groups, hence is a weak equivalence.
A -gerbe in the full sense Giraud (as opposed to a -bundle gerbe in the sense of Murray) is equivalent to an -principal 2-bundle, not in general to a circle 2-bundle, wich is only a special case.
More generally we have:
For every the automorphism -group of is given by the crossed complex
with in degree and acting by automorphisms.
This is an extension of cohesive -groups
For a double cover is a -principal bundle.
For , , an orientifold circle -bundle (with connection) is an -principal ∞-bundle (with ∞-connection) on that extends with respect to the extension def. 1 of by .
This means that if is the cocycle for the double cover , this factors as
where is the cocycle for the given -principal ∞-bundle.
Every orientifold circle -bundle (with connection) on induces an ordinary circle n-bundle (with connection) on the given double cover such that restricted to any fiber of this is equivalent to .
This is a special case of a general statement about extensions of -bundles, discussed at cohesive (infinity,1)-topos here.
Orientifold circle 2-bundles (with connection) over smooth manifold are equivalent to the Jandl gerbes (with connection) discussed in (SSW05)
The name Jandl gerbe refers to the poem lichtung by Ernst Jandl.
The geometric realization
of is a homotopy 3-type with homotopy groups
and nontrivial action of on .
By the theorem discussed here at ETop∞Grpd we have that
where on the right we have the ordinary classifying spaces going by these names;
generally geometric realization preserves fiber sequences of nice enough objects, such as those under consideration, so that we have a fiber sequence
Since and and all other homotopy groups of these two spaces are trivial, the homotopy groups of follow by the long exact sequence of homotopy groups associated to our fiber sequence.
Superstring: Twisted differential -cohomology
The B-field background gauge field in superstring theory is not actually quite a connection on a line 2-bundle, but on a supergeometry line 2-bundle (DFM I, DFM II, Freed). See at Line 2-bundle – Examples – Super line 2-bundle for details.
Therefore for superstring orientifolds the above discussion is to be refined from line 2-bundles to such super line 2-bundles. See around (DFM II, supposition 3.6) for the discussion of the relation to the above.
A definition and study of orientifold bundle gerbes, modeling the B-field background for the bosonic string, is in
A more encompassing formalization in terms of differential cohomology in general and twisted differential K-theory in particular that also takes the spinorial degrees of freedom into account is being announced in
A summary talk on this is
More details are in
Related lecture notes / slides include
A formulation of some of the relevant aspects of orientifolds in terms of the differential nonabelian cohomology with coefficients in the 2-group coming from the crossed module is indicated in
More on this in section 3.3.10 of
The “higher orientifold” appearing in Horava-Witten theory with circle 2-bundles replaced by the circle 3-bundles of the supergravity C-field is discussed towards the end of