nLab
orientifold

Idea

An orientifold is a background for string sigma-models that combines aspects of ${\mathbb{Z}}_2$-orbifolds with orientation reversal on the worldsheet (therefore the name): it consists of a bundle gerbe on a space with a ${\mathbb{Z}}_2$-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 ${\mathbb{Z}}_2$ and equipped with certain classes in its( twisted) differential cohomology that is suitably ${\mathbb{Z}}_2$-equivariant.

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 $U(1)$ is that corresponding to the smooth crossed module

$$\mathrm{AUT}(U(1))\simeq [U(1)\to {\mathbb{Z}}_2],$$AUT(U(1)) \simeq [U(1) \to \mathbb{Z}_2] \,,

where the differential $U(1)\to {\mathbb{Z}}_2$ is trivial and where the action of ${\mathbb{Z}}_2$ on $U(1)$ is given under the identification of $U(1)$ with the unit circle in the plane by reversal of the sign of the angle.

This is an extension of smooth ∞-groups

$$BU(1)\to \mathrm{AUT}(U(1))\to {\mathbb{Z}}_2.$$\mathbf{B}U(1) \to AUT(U(1)) \to \mathbb{Z}_2 \,.

Proof

The nature of $\mathrm{AUT}(U(1))$ is clear. Let $BU(1)\to \mathrm{AUT}(U(1))$ be the evident inclusion. We have to show that its delooping is the homotopy fiber of $B\mathrm{AUT}(U(1))\to B{\mathbb{Z}}_2$.

For this it is sufficient to show that $B^{2}U(1)$ is equivalent to the ordinary pullback of simplicial presheaves $B\mathrm{AUT}(U(1)){\times }_{B{\mathbb{Z}}_2}E{\mathbb{Z}}_2$ of the $\mathbb{Z}$2-universal principal bundle.

This pullback is the 2-groupoid whose

• objects are elements of ${\mathbb{Z}}_2$;

• morphisms ${\sigma }_1\to {\sigma }_2$ are labeled by $\sigma \in {\mathbb{Z}}_2$ such that ${\sigma }_2=\sigma {\sigma }_1$;

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

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

• horizontal composition of 1-morphisms with 1-morphisms is given by the group operation in ${\mathbb{Z}}_2$

• horizontal composition of 1-morphisms with 2-morphisms (whiskering) is given by the action of ${\mathbb{Z}}_2$ on $U(1)$.

This 2-groupoid has vanishing ${\pi }_1$, and ${\pi }_2=U(1)$. The inclusion of $B^{2}U(1)$ into this pullback is the obvious one, includion elements in $U(1)$ as endomorphisms of the trivial element in ${\mathbb{Z}}_2$. This is manifestly an isomorphism on ${\pi }_2$ and trivially an isomorphism on all other homotopy groups, hence is a weak equivalence.

Observation

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

More generally we have:

Observation

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

$$\mathrm{AUT}(B^{n}U(1))\simeq [U(1)\to 0\to \cdots \to 0\to {\mathbb{Z}}_2]$$AUT(\mathbf{B}^n U(1)) \simeq [U(1) \to 0 \to \cdots \to 0 \to \mathbb{Z}_2]

with $U(1)$ in degree $n+1$ and ${\mathbb{Z}}_2$ acting by automorphisms.

This is an extension of cohesive $\mathrm{\infty }$-groups

$$B^{n+1}U(1)\to \mathrm{AUT}(B^{n}U(1))\to {\mathbb{Z}}_2.$$\mathbf{B}^{n+1} U(1) \to AUT(\mathbf{B}^n U(1)) \to \mathbb{Z}_2 \,.

Definition

For $X\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ a double cover $\hat{X}\to X$ is a ${\mathbb{Z}}_2$-principal bundle.

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

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

$$X\stackrel{g}{\to }B\mathrm{AUT}(B^{n-1}U(1))\to B{\mathbb{Z}}^2$$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 $\mathrm{AUT}(B^{n}U(1))$-principal ∞-bundle.

Proposition

Every orientifold circle $n$-bundle (with connection) on $X$ induces an ordinary circle n-bundle (with connection) $\hat{P}\to \hat{X}$ on the given double cover $\hat{X}$ such that restricted to any fiber of $\hat{X}$ this is equivalent to $\mathrm{AUT}(B^{n-1}U(1))\to {\mathbb{Z}}_2$.

This is a special case of a general statement about extensions of $\mathrm{\infty }$-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 $[U(1)\to mathb{Z}_2]$-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 $B[U(1)\to {\mathbb{Z}}_2]$. 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 ${\mathbb{Z}}_2$-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

The geometric realization

$$\tilde{R}:=\mid B[U(1)\to {\mathbb{Z}}_2]\mid$$\tilde R := |\mathbf{B}[U(1) \to \mathbb{Z}_2]|

of $B[U(1)\to \mathbb{Z}]$ is the homotopy 3-type with homotopy groups

$${\pi }_0(\tilde{R})=0;$$\pi_0(\tilde R) = 0 \,;

$${\pi }_1(\tilde{R})={\mathbb{Z}}_2;$$\pi_1(\tilde R) = \mathbb{Z}_2 \,;

$${\pi }_2(\tilde{R})=0;$$\pi_2(\tilde R) = 0 \,;

$${\pi }_3(R\prime )=\mathbb{Z}$$\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. $\mid B{\mathbb{Z}}_2\mid \simeq B{\mathbb{Z}}_2$;

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

   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

   $$K(\mathbb{Z},3)\to \tilde{R}\to B{\mathbb{Z}}_2$$K(\mathbb{Z},3) \to \tilde R \to B \mathbb{Z}_2

   in Top.

Since ${\pi }_3(K(\mathbb{Z}),3)\simeq \mathbb{Z}$ and ${\pi }_1(B{\mathbb{Z}}_2)\simeq {\mathbb{Z}}_2$ and all other homotopy groups of these two spaces are trivial, the homotopy groups of $\tilde{R}$ follow by the long exact sequence of homotopy groups associated to our fiber sequence.

Finally, since the action of ${\mathbb{Z}}_2$ in the crossed module is nontrivial, ${\pi }_1(\tilde{R})$ must act notriviall on ${\pi }_3(\mathbb{Z})$. It can only act nontrivial in a single way, up to homotopy.

Remark

The space

$$R:={\mathbb{Z}}_2\times \tilde{R}$$R := \mathbb{Z}_2 \times \tilde R

is taken to be the coefficient object for orientifold (differential) cohomology as appearing in string theory in (DFM I). More details are in (DFM II).

References

A definition and study of orientifold bundle gerbes, modeling the Kalb-Ramond field, is in

• Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes (arXiv)

• Krzysztof Gawedzki, Rafal R. Suszek, Konrad Waldorf, Bundle Gerbes for Orientifold Sigma Models (arXiv)

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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv, slides)

A summary talk on this is

• Greg Moore, The RR-charge of an orientifold (ppt)

More details are in

• Jacques Distler, Dan Freed, Greg Moore, Spin structures and superstrings (arXiv:1007.4581)

A formulation of some of the relevant aspects of orientifolds in terms of the differential nonabelian cohomology with coefficients in the 2-group $\mathrm{AUT}(U(1))$ coming from the crossed module $[U(1)\to {\mathbb{Z}}_2]$ is indicated in

• Urs Schreiber, talk Background fields in twisted differential nonabelian cohomology at Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology

More on this in section 3.3.10 of

• Urs Schreiber, differential cohomology in a cohesive topos