nLab orientifold

Context

Differential cohomology

differential cohomology

Contents

Idea

An orientifold is a background for string sigma-models that combines aspects of ${ℤ}_{2}$-orbifolds with orientation reversal on the worldsheet (therefore the name): it consists of a bundle gerbe on a space with a ${ℤ}_{2}$-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 $\sigma$-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 ${ℤ}_{2}$ and equipped with certain classes in its( twisted) differential cohomology that is suitably ${ℤ}_{2}$-equivariant.

Bosonic string: orientifold circle $n$-bundles with connection

We discuss the notion of circle n-bundles with connection over double covering spaces with orientifold structure.

Proposition

The smooth automorphism 2-group of the circle group $U\left(1\right)$ is that corresponding to the smooth crossed module

$\mathrm{Aut}\left(BU\left(1\right)\right)\simeq \left[U\left(1\right)\to {ℤ}_{2}\right]\phantom{\rule{thinmathspace}{0ex}},$Aut(\mathbf{B}U(1)) \simeq [U(1) \to \mathbb{Z}_2] \,,

where the differential $U\left(1\right)\to {ℤ}_{2}$ is trivial (constant on the neutral element) and where the action of ${ℤ}_{2}$ on $U\left(1\right)$ is given under the identification of $U\left(1\right)$ with the unit circle in the plane by reversal of the sign of the angle.

This is an extension of smooth ∞-groups

$BU\left(1\right)\to \mathrm{Aut}\left(BU\left(1\right)\right)\to {ℤ}_{2}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{B}U(1) \to Aut(\mathbf{B}U(1)) \to \mathbb{Z}_2 \,.
Proof

The nature of $\mathrm{Aut}\left(BU\left(1\right)\right)$ is clear. Let $BU\left(1\right)\to \mathrm{Aut}\left(BU\left(1\right)\right)$ be the evident inclusion. We have to show that its delooping is the homotopy fiber of $B\mathrm{Aut}\left(BU\left(1\right)\right)\to B{ℤ}_{2}$.

For this it is sufficient to show that ${B}^{2}U\left(1\right)$ is equivalent to the ordinary pullback of simplicial presheaves $B\mathrm{Aut}\left(BU\left(1\right)\right){×}_{B{ℤ}_{2}}E{ℤ}_{2}$ of the $ℤ$2-universal principal bundle.

This pullback is the 2-groupoid whose

• objects are elements of ${ℤ}_{2}$;

• morphisms ${\sigma }_{1}\to {\sigma }_{2}$ are labeled by $\sigma \in {ℤ}_{2}$ such that ${\sigma }_{2}=\sigma {\sigma }_{1}$;

• all 2-morphisms are endomorphisms, labeled by $c\in U\left(1\right)$;

• vertical composition of 2-morphisms is given by the group operation in $U\left(1\right)$,

• horizontal composition of 1-morphisms with 1-morphisms is given by the group operation in ${ℤ}_{2}$

• horizontal composition of 1-morphisms with 2-morphisms (whiskering) is given by the action of ${ℤ}_{2}$ on $U\left(1\right)$.

This 2-groupoid has vanishing ${\pi }_{1}$, and has ${\pi }_{2}=U\left(1\right)$. The inclusion of ${B}^{2}U\left(1\right)$ into this pullback is the obvious one, including elements in $U\left(1\right)$ as endomorphisms of the trivial element in ${ℤ}_{2}$. This is manifestly an isomorphism on ${\pi }_{2}$ and trivially an isomorphism on all other homotopy groups, hence is a weak equivalence.

Observation

A $U\left(1\right)$-gerbe in the full sense Giraud (as opposed to a $U\left(1\right)$-bundle gerbe in the sense of Murray) is equivalent to an $\mathrm{Aut}\left(BU\left(1\right)\right)$-principal 2-bundle, not in general to a circle 2-bundle, wich is only a special case.

More generally we have:

Observation

For every $n\in ℕ$ the automorphism $\left(n+1\right)$-group of ${B}^{n}U\left(1\right)$ is given by the crossed complex

$\mathrm{Aut}\left({B}^{n+1}U\left(1\right)\right)\simeq \left[U\left(1\right)\to 0\to \cdots \to 0\to {ℤ}_{2}\right]$Aut(\mathbf{B}^{n+1} U(1)) \simeq [U(1) \to 0 \to \cdots \to 0 \to \mathbb{Z}_2]

with $U\left(1\right)$ in degree $n+1$ and ${ℤ}_{2}$ acting by automorphisms.

This is an extension of cohesive $\infty$-groups

${B}^{n+1}U\left(1\right)\to \mathrm{Aut}\left({B}^{n+1}U\left(1\right)\right)\to {ℤ}_{2}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{B}^{n+1} U(1) \to Aut(\mathbf{B}^{n+1} U(1)) \to \mathbb{Z}_2 \,.
Definition

For $X\in \mathrm{Smooth}\infty \mathrm{Grpd}$ a double cover $\stackrel{^}{X}\to X$ is a ${ℤ}_{2}$-principal bundle.

For $n\in ℕ$, $n\ge 1$, an orientifold circle $n$-bundle (with connection) is an $\mathrm{Aut}\left({B}^{n}U\left(1\right)\right)$-principal ∞-bundle (with ∞-connection) on $X$ that extends $\stackrel{^}{X}\to X$ with respect to the extension def. 1 of ${ℤ}^{2}$ by $\mathrm{AUT}\left({B}^{n}U\left(1\right)\right)$.

This means that if $X\to B{ℤ}^{2}$ is the cocycle for the double cover $\stackrel{^}{X}$, this factors as

$X\stackrel{g}{\to }B\mathrm{AUT}\left({B}^{n-1}U\left(1\right)\right)\to B{ℤ}^{2}$X \stackrel{g}{\to} \mathbf{B} AUT(\mathbf{B}^{n-1} U(1)) \to \mathbf{B}\mathbb{Z}^2

where $g$ is the cocycle for the given $\mathrm{AUT}\left({B}^{n}U\left(1\right)\right)$-principal ∞-bundle.

Proposition

Every orientifold circle $n$-bundle (with connection) on $X$ induces an ordinary circle n-bundle (with connection) $\stackrel{^}{P}\to \stackrel{^}{X}$ on the given double cover $\stackrel{^}{X}$ such that restricted to any fiber of $\stackrel{^}{X}$ this is equivalent to $\mathrm{AUT}\left({B}^{n-1}U\left(1\right)\right)\to {ℤ}_{2}$.

This is a special case of a general statement about extensions of $\infty$-bundles, discussed at cohesive (infinity,1)-topos here.

Observation

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.

Proof

By the general discussion at Euclidean-topological ∞-groupoid and smooth ∞-groupoid we have that $\left[U\left(1\right)\to mathb{Z}_{2}\right]$-principal ∞-bundles on $X$ are given by Cech cocycles relative to any good open cover of $X$ with coefficients in the sheaf of 2-groupoids $B\left[U\left(1\right)\to {ℤ}_{2}\right]$. Writing this out in components it is straightforward to check that this coincides with the data of a Jandl gerbe (with connection) locally trivialized with over this cover.

Remark

Orientifold circle $n$-bundles are not ${ℤ}_{2}$-equivariant circle $n$-bundles: in the latter case the orientation reversal acts by an automorphism between the bundle and its pullback along the orientation reversal, whereas for an orientifold circle $n$-bundle the orientation reversal acts by an equivalence to the dual of the pulled-back bundle.

Observation
$≔\mid B\left[U\left(1\right)\to {ℤ}_{2}\right]\mid$$\tilde R$ \coloneqq |\mathbf{B}[U(1) \to \mathbb{Z}_2]|

of $B\left[U\left(1\right)\to ℤ\right]$ is a homotopy 3-type with homotopy groups

${\pi }_{0}\left(\stackrel{˜}{R}\right)=0\phantom{\rule{thinmathspace}{0ex}};$\pi_0(\tilde R) = 0 \,;
${\pi }_{1}\left(\stackrel{˜}{R}\right)={ℤ}_{2}\phantom{\rule{thinmathspace}{0ex}};$\pi_1(\tilde R) = \mathbb{Z}_2 \,;
${\pi }_{2}\left(\stackrel{˜}{R}\right)=0\phantom{\rule{thinmathspace}{0ex}};$\pi_2(\tilde R) = 0 \,;
${\pi }_{3}\left(R\prime \right)=ℤ$\pi_3(R') = \mathbb{Z}

and nontrivial action of ${\pi }_{1}$ on ${\pi }_{3}$.

Proof

By the theorem discussed here at ETop∞Grpd we have that

1. specifically

1. $\mid B{ℤ}_{2}\mid \simeq B{ℤ}_{2}$;

2. $\mid {B}^{2}U\left(1\right)\mid \simeq {B}^{2}U\left(1\right)\simeq K\left(ℤ;3\right)$;

where on the right we have the ordinary classifying spaces going by these names;

2. generally geometric realization preserves fiber sequences of nice enough objects, such as those under consideration, so that we have a fiber sequence

$K\left(ℤ,3\right)\to \stackrel{˜}{R}\to B{ℤ}_{2}$K(\mathbb{Z},3) \to \tilde R \to B \mathbb{Z}_2

in Top.

Since ${\pi }_{3}\left(K\left(ℤ\right),3\right)\simeq ℤ$ and ${\pi }_{1}\left(B{ℤ}_{2}\right)\simeq {ℤ}_{2}$ and all other homotopy groups of these two spaces are trivial, the homotopy groups of $\stackrel{˜}{R}$ follow by the long exact sequence of homotopy groups associated to our fiber sequence.

Superstring: Twisted differential $R$-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.

References

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 $\mathrm{AUT}\left(U\left(1\right)\right)$ coming from the crossed module $\left[U\left(1\right)\to {ℤ}_{2}\right]$ is indicated in

More on this in section 3.3.10 of

Revised on December 13, 2012 15:57:44 by Urs Schreiber (71.195.68.239)