Ore extension

### Idea

An Ore extension of a unital ring $R$ is certain generalization of the ring $R[T]$ of polynomials in one variable $T$ with coefficients in $R$. While keeping the left $R$-module structure intact, unlike in the polynomial ring, the coefficients in $R$ and the indeterminate $T$ do not need to commute, but rather commute up to a skew-derivation. A skew-polynomial ring is a special case.

### Definition

Given an endomorphism $\sigma: R\to R$, a __$\sigma$-derivation__ $d: R\to R$ is an additive map satisfying the $\sigma$-twisted Leibniz rule

$$
d(r s) = d(r) s + \sigma(r) d(s),\,\,\,\,\,\forall r,s\in R.
$$

If $\sigma$ is an injective endomorphism of $R$, and $d$ a $\sigma$-derivation $d$ then the free left $R$-module underlying the ring of polynomials in one variable $R[T]$ is equipped with the unique multiplication rule which is making it into a unital ring, extends $R = R 1\subset R[T]$ and such that
$$
T \cdot r = \sigma(r) T + d(r), \,\,\,\,\forall r\in R.
$$
$R[T]$ with this ring structure is called the Ore extension of $R$.
If $d = 0$ identically, then we say that $R[T]$ is a __skew polynomial ring__.

### Literature

* K. R. Goodearl, R. B. Warfield, _An introduction to noncommutative Noetherian rings_, London Math. Society Student Texts __61__, Camb. Univ. Press.
* Louis H. Rowen, _Ring theory_, student edition, Acad. Press 1991, sec. 1.6
* wikipedia [Ore extension](http://en.wikipedia.org/wiki/Ore_extension)

[[!redirects skew-polynomial ring]]


Ore localization

#Contents#
* table of contents
{:toc}

## Definition

If $S\subset R$ is a left [[Ore set]] in a [[monoid]] (or a [[ring]]) $R$, then we call the pair $(j,S^{-1}R)$ where $j:R\to S^{-1}R$ is a morphism of monoids (rings) the __left Ore localization__ of $R$ with respect to $S$ if it is the  universal object in the category $C = C(R,S)$
whose objects are the pairs $(f,Y)$ where $f : R \rightarrow Y$ is a morphism of rings from $R$ into a
ring $Y$ such that the image $f(S)$ of $S$ consists of units (=multiplicatively invertible elements), and
the morphisms $\alpha : (f,Y) \rightarrow (f',Y')$
are maps of rings $\alpha : Y \rightarrow Y'$ such that $\alpha \circ f = f'$.

The definition of $C$ makes sense even if $S\subset R$ is not left Ore; the universal object in $C$ may then exist when $S$ is not left Ore, for example this is the case when $S$ is right Ore, while not left Ore. In fact, the universal object is a left Ore localization (i.e. $S$ is left Ore) iff it lies in the full subcategory $C^l$ of $C$ whose objects $(f,Y)$
satisfy two additional conditions:

(i) $f(S)^{-1}f(R) = \{(f(s))^{-1}f(r)\,|\, s \in S, r\in R\}$ is a subring in $Y$,

(ii) $ker\,f = I_S$.

Hence $(j,S^{-1}R)$ is universal in $C^l$, and that characterizes it, but the universality in $C$, although not characteristic, appears to be more useful in practice.

For every left Ore set $S\subset R$ in a monoid or ring $R$, the left Ore localization exists and it can be defined as follows. As a set, $S^{-1}R := S\times R/\sim$, where
$\sim$ is the following relation of equivalence:
\[ (s,r) \sim (s',r') \,\,\Leftrightarrow\,\,
(\exists \tilde s\in S \,\,\exists \tilde{r}\in R) \,\,
(\tilde{s}s' = \tilde{r}s\,\,and\,\,\tilde{s}r' = \tilde{r}r).\]
A class of equivalence of $(s,r)$ is denoted $s^{-1}r$ and
called a left fraction. The multiplication is defined by
$s_1^{-1}r_1\cdot s_2^{-1}r_2 = (\tilde{s}s_1)^{-1} (\tilde{r}r_2)$ where
$\tilde{r} \in R, \tilde{s} \in S$
satisfy $\tilde{r}s_2 = \tilde{s}r_1$
(one should think of this, though it is not yet
formally justified at this point, as
$s^{-1}\tilde{r} = r_1 s_2^{-1}$, what enables to put inverses one
next to another and then the multiplication rule is obvious).
If the monoid $R$ is a ring then we can extend the addition to $S^{-1}R$ too. Suppose we are given two fractions
with representatives $(s_1,r_1)$ and $(s_2,r_2)$. Then by the left Ore
condition we find $\tilde{s} \in S$, $\tilde{r}\in R$ such that
$\tilde{s} s_1 = \tilde{r} s_2$. The sum is then defined
\[ s_1^{-1} r_1 + s_2^{-1} r_2 \,:=\,
 (\tilde{s}s_1)^{-1} (\tilde{s}r_1 + \tilde{r}r_2) \]
It is a long and at points tricky to work out all the details of this definition. One has to show that $\sim$ is indeed relation of equivalence, that the operations are well defined, and that $S^{-1}R$ is indeed
a ring. Even the commutativity of the addition needs work (there is an alternative definition of addition in which $\tilde{s}$ above is not required to be in $S$ but the  product $\tilde{s}r_1$ is in $S$; this approach is manifestly commutative but it has some other drawbacks).
At the end, one shows that the map $j = j_S : R \rightarrow S^{-1}R$ given by $i(r) = 1^{-1}r$ is a homomorphism of rings, which is 1-1 iff the 2-sided ideal $I_S = \{ n \in R \,|\,\exists s \in S,\, sn = 0\}$ is zero.

Basic property of Ore localization is flatness: $S^{-1}R$ is flat $R$-bimodule.
The left Ore localization $j : R\to S^{-1}R$ induced a flat localization functor for the category of left $R$-modules or the category of right $R$-modules. The localization functor is the extension of scalars along $l_S: R\to S^{-1}R$, that is (in the case of left $R$-modules) $M\mapsto S^{-1}R\otimes_R M$ is the flat localization functor $Q^*_S:{}_R Mod\to {}_{S^{-1}R}Mod$ and the restriction of scalars is its right adjoint $Q_{S*}$ which is fully faithful and it has its own right adjoint (the localization functor is affine).

## Related concepts

* [[Ore domain]]

* [[localization of a ring]]

* [[Cohn localization]]

## References

* K. R. Goodearl, Robert B. Warfield, _An introduction to noncommutative Noetherian rings_, London Math. Soc. Student Texts __16__ (1st ed,), 1989, xviii+303 pp.; or __61__ (2nd ed.), 2004, xxiv+344 pp.

* [[Zoran Škoda]], _Noncommutative localization in noncommutative geometry_,  London Math. Society Lecture Note Series 330 ([pdf](http://www.maths.ed.ac.uk/~aar/books/nlat.pdf)), ed.  A. Ranicki; pp. 220--313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276).

[[!redirects Ore localizations]]


Ore set

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#Contents#
* table of contents
{:toc}

## Idea

A multiplicative closed subset $S\subset R$ containing a unit element in a [[monoid]] $R$ is a **left Ore set** if it satisfies the

* (left cancellability) If $n s = m s$ for $n,m\in R$ and $s\in S$, then $\exists s'\in S$ such that $s' n = s' m$.

* (left Ore condition) For any $r\in R$ and $s\in S$ there
$\exists r'\in R$, $\exists s'\in S$ such that $s'r = r's$.

Sometimes both conditions are called "Ore conditions".
Notice that in both conditions the new elements whose existence is required are on the left.

If $S$ is a left Ore set in a monoid than there is a well-defined [[equivalence relation]] $\sim$ on pairs $(s,r)\in S\times R$ such that the set of equivalence classes, which are denoted by $s^{-1}r := [(s,r)]\in S^{-1}R:=S\times R/\sim$ becomes a monoid together with a monoid map $j_S: R\to S^{-1}R$ given by $r\mapsto 1^{-1}r$ is a homomorphism of monoids; moreover this monoid map satisfies a universal property, see [[Ore localization]]. The Ore localization of monoids has been generalized to categories, see [[category of fractions]].

If $R$ is a (unital) ring and $S\subset R$ is left Ore in a multiplicative monoid underlying $R$, then the addition on $S^{-1}R$ is also well defined, commutative and associative (checking all this is rather complicated) on $S^{-1}R$ such that the localization map is the map of rings and satisfies the universal property for the Ore localization of *rings*.

A **right Ore (sub)set** in a monoid or $R$ is a subset $S\subset R$ such that $S$ is left Ore subset in the opposite ring $R^{op}$.

An **Ore set** is a subset $S\subset R$ which is simultaneously left and right Ore subset. If $S\subset R\subset R'$ where $R$ and $R'$ are rings is a multiplicative subset then the satisfaction of Ore conditions in $R$ and Ore conditions in $R'$ are independent in general: the reason is that in a bigger ring one has simultaneously more conditions, but also a bigger set of possible solutions for the conditions. In general it is not sufficient to check the Ore condition on generators. If $S,T\subset R$ are two left Ore sets, it is not true in general that the image $i_T(S)$ in $T^{-1}R$ is left Ore; if it is then automatically $i_S(T)$ is left Ore in $S^{-1}R$ (mutually compatible left Ore sets) and $(i_S(T))^{-1}S^{-1} R$ is a ring canonically isomorphic to $(i_T(S))^{-1}T^{-1}T$.

The left and right Ore conditions for rings were introduced by [[Øystein Ore]] in 1931 in order to study the linear equations class of rings which are now called [[Ore domain]]s.

(nlab note: there are many results on Ore conditions which are independent from the study of Ore localization; thus the entries should be separated)

## Related concepts

* [[localization of a ring]]

* [[Cohn localization]]

* [[Ore localization]]

## References

* Wikipedia, *[Ore condition](https://en.wikipedia.org/wiki/Ore_condition)*

[[!redirects Ore sets]]

[[!redirects Ore subset]]
[[!redirects Ore subsets]]


Oren Ben-Bassat

[home page](https://sites.google.com/site/orenbenbassat/)

category: people


Oren Bergman

* [Institute page](https://phys.technion.ac.il/en/people/faculty/person/60)

## Selected writings

On [[duality between type I and heterotic string theory]]:

* [[Oren Bergman]], [[Matthias Gaberdiel]], G. Lifschytz, _String Creation and Heterotic-Type I' Duality_, Nucl. Phys. B524 (1998) 524-544 ([arXiv:hep-th/9711098](https://arxiv.org/abs/hep-th/9711098))

Understanding [[solitonic brane|solitonic]] (non-[[singular brane|singular]]) [[D-branes]] and their [[T-duality]] in terms of the [[D-brane charge quantization in K-theory]]:

* {#BergmanGimonHořava99} [[Oren Bergman]], [[Eric G. Gimon]], [[Petr Hořava]], *Brane Transfer Operations and T-Duality of Non-BPS States*, JHEP 9904 (1999) 010 \[doi:10.1088/1126-6708/1999/04/010, [arXiv:hep-th/9902160](https://arxiv.org/abs/hep-th/9902160)\]

On the [[ABJM model]]:

* {#ABJM08} [[Ofer Aharony]], [[Oren Bergman]], Daniel Louis Jafferis, [[Juan Maldacena]], _$N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_, JHEP 0810:091 (2008) [DOI:10.1088/1126-6708/2008/10/091](http://iopscience.iop.org/article/10.1088/1126-6708/2008/10/091/meta;jsessionid=FCE6764D4E19F3038C9530E50B057A56.c3.iopscience.cld.iop.org) ([arXiv:0806.1218](http://arxiv.org/abs/0806.1218))

* {#AharonyBergmanJafferis08} [[Ofer Aharony]], [[Oren Bergman]], Daniel Louis Jafferis, _Fractional M2-branes_, JHEP 0811:043,2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))

On [[D-brane charge]] in [[K-theory]] on [[orientifolds]]:

* {#BergmanGimonSugimoto01} [[Oren Bergman]], [[Eric Gimon]], [[Shigeki Sugimoto]], _Orientifolds, RR Torsion, and K-theory_, JHEP 0105:047, 2001 ([arXiv:hep-th/0103183](https://arxiv.org/abs/hep-th/0103183))

On [[D4-brane]] [[ADE-singularity|ADE]]-[[orientifold]] models in [[type I' string theory]]:

* [[Oren Bergman]], Diego Rodriguez-Gomez, _5d quivers and their $AdS_6$ duals_, J. High Energ. Phys. (2012) 2012: 171 ([arXiv:1206.3503](https://arxiv.org/abs/1206.3503))

## Related entries

* [[ABJM model]], [[3d superconformal gauge field theory]]

* [[orientifold]]

category: people


Organization of the nLab > history


Ori Ganor

* [webpage](https://physics.berkeley.edu/people/faculty/ori-ganor)

## Selected writings

On [[small instantons]] as [[NS5-branes]]/[[M5-branes]] in [[E8]]-[[heterotic string theory]]/[[heterotic M-theory]]:

* {#GanorHanany96} [[Ori Ganor]], [[Amihay Hanany]], _Small $E_8$ Instantons and Tensionless Non-critical Strings_, Nucl. Phys. B474 (1996) 122-140 ([arXiv:hep-th/9602120](https://arxiv.org/abs/hep-th/9602120))

On the [[D=6 N=(1,0) SCFT]] on [[heterotic M5-branes]] and their [[KK-compactifications]]:

* [[Ori Ganor]], [[David Morrison]], [[Nathan Seiberg]], _Branes, Calabi-Yau Spaces, and Toroidal Compactification of the $N=1$ Six-Dimensional $E_8$ Theory_, Nucl. Phys. B487:93-127, 1997 ([arXiv:hep-th/9610251](https://arxiv.org/abs/hep-th/9610251))

category: people


oriental

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#### Higher category theory
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#Contents#
* table of contents
{:toc}

## Idea

_Orientals_ are "oriented simplices": the $n$-th oriental is the [[simplicial set|simplicial]] $n$-[[simplex]] turned into a [[globular set|globular]] simplex, hence equipped with source and target relations, assigning to each $k$-face a set of $(k-1)$-faces called its source and a set of $(k-1)$-faces called its target, subject to some natural axioms. Thus an oriental is a translation from [[simplicial set|simplicial]] to _[[globular set|globular]]_ [[geometric shapes for higher structures]]. (For more discussion of this point see also at _[[Kan complex]]_ the section _[As models for ∞-groupoids](Kan%20complex#AsGrpds)_.)

Each oriental freely generates (see below) a structure of a [[strict omega-category]] $O(\Delta^n)$, such that $k$-morphisms in $O(\Delta^n)$ are [[pasting diagrams]] of $k$-faces in $\Delta^n$.

One of the axioms is a [[globular set|globularity axiom]], which says that the source of a source (that is, the union of sources of all $(k-1)$-faces in the source of a $k$-face) equals the source of the target, and similarly that the target of a source equals the target of the target. Thus, orientals mediate between the [[simplicial set|simplicial]] and the [[globular set|globular]] world of [[higher category theory|infinity-categories]].

[[!include the first few orientals]]

The construction of orientals is designed to be compatible with face and degeneracy maps. Therefore the orientals arrange themselves into a [[cosimplicial object|cosimplicial]] [[∞-category]], i.e., a functor
$$
  O : \Delta \to \omega Cat
$$
from the [[simplex category]] $\Delta$ to the category of [[strict omega-category|strict omega-categories]].

## Relation to other concepts

### Relation to $\omega$-groupoids

Regarding the standard $n$-simplex as a filtered space with the standard filtering, and denoting for $X$ a filtered space by $\Pi_\omega(X)$ the fundamental filter-respecting $\omega$-groupoid of $X$, we obtain a cosimplicial [[omega-groupoid|omega-groupoid]]

$$
  \Pi_\omega(\Delta^{-})
  :
  \Delta \to \omega Grpd
$$

+-- {: .query}

It should be true that $\Pi_\omega(\Delta^n)$ is the _free $\omega$-groupoid_ on $O(\Delta^n)$. Is that right?

=--

### Relation to cyclic polytopes

There is a convex-geometric perspective on orientals outlined in [Kapranov-Voevodsky](#KapranovVoevodsky)  We will try to explain the punch-line first, then go into details.

In $O(\Delta^n)$, denote the source of the $n$-morphism corresponding to the $n$-face in $\Delta^n$ by $\hat 0_{n-1}$, and denote its target by $\hat 1_{n-1}$.  Since
these are source and target of the same morphism, they both have the same
source, and both have the same target.  Write $\hat 0_{n-2}$ for this source,
and $\hat 1_{n-2}$ for this target.  Inductively define $\hat 0_k$ and $\hat 1_k$
for all $0\leq k \leq n-1$.

The promised punchline is that the $(k+1)$-morphisms from $\hat 0_k$ to $\hat 1_k$ are in natural correspondence to the triangulations of the $(k+1)$-dimensional cyclic polytope with $n+1$ vertices.

For example, when $k=1$, we have $\hat 0_1$ is the 1-[[morphism]] $0\rightarrow 1\rightarrow \dots \rightarrow n$, and $\hat 1_1$ is the 1-morphism $0\rightarrow n$.  The [[2-morphism]]s from $\hat 0_1$ to $\hat 1_1$ consist of [[triangulation]]s of the $(n+1)$-gon.  See the diagrams above for examples.

#### Details

We are going to embed the [[simplex]] $\Delta^n$ inside $\mathbb {R}^n$, so we can think (convex-)geometrically about it.

Let is get the main convex-geometric definitions out of the way at the beginning.

The _moment curve_ in $\mathbb {R}^d$ is defined parametrically by $p(t)=(t,t^2,\dots,t^d)$.

A _cyclic [[polytope]]_ of [[dimension]] $d$ with $n$ vertices
is the [[convex hull]] of $n$ points on the moment curve in $\mathbb {R}^d$.  For many purposes, the choice of the $n$ points is irrelevant; for example, the face lattice of the polytope does not depend on which points were chosen.

A _[[triangulation]]_ of a polytope is a subdivision of the polytope into
maximal-dimensional simplices, which overlap only on their boundaries, and
all of whose vertices are vertices of the original polytope.

To fix an embedding of $\Delta^n$ into $\mathbb {R}^n$, choose real numbers $a_0\lt a_1\lt\dots\lt a_{n}$, and embed $\Delta^n$ inside $\mathbb {R}^n$ so that its $i$-th vertex is at $p(a_i)$.

Now, as already described above, we have that $\hat 0_{n-1}$ and $\hat 1_{n-1}$  are compositions of the morphisms corresponding to some $n-1$-faces, so that the morphism  coming from the unique $n$-face of $\Delta_n$ goes from $\hat 0_{n-1}$ to $\hat 1_{n-1}$.  Which faces belong to which $(n-1)$-morphism?

The answer is very simple: $\hat 0_{n-1}$ is the composition of the _lower_ faces, while $\hat 1_{n-1}$ is the composition of the _upper_ faces, where lower and upper are always taken with respect to the final co-ordinate.

Now, project into $\mathbb {R}^{n-1}$ by forgetting the last co-ordinate.  The
image of $\Delta^n$ inside $\mathbb {R}^{n-1}$ is the convex hull of $n+1$ points
on the moment curve in $\mathbb {R}^{n-1}$, that is to say, it is a cyclic polytope
with $n+1$ vertics in dimension $n-1$.

$\hat 0_{n-2}$ consists of the lower faces of this cyclic polytope (where, as always, we take upper/lower with respect to the final co-ordinate, though note that in this case, the final co-ordinate is the $n-1$-st co-ordinate, since we have already forgotten the $n$-th co-ordinate).  Similarly, $\hat 1_{n-2}$ consists of the upper faces of this cyclic polytope.

The projections of $\hat 0_{n-1}$ and $\hat 1_{n-1}$, meanwhile, are triangulations
of this cyclic polytope.   The cyclic polytope of dimension $n-1$ with $n+1$ vertices
has only two triangulations, so these are all of them.

In general, $\hat 0_{k}$ consists of the lower faces of the image of the
projection of $\Delta_n$ into $\mathbb {R}^{k+1}$ by forgetting the last $n-k+1$ coordinates, while $\hat 1_k$ consists of the upper faces of the image.  And then $\hat 0_{k+1}$, $\hat 1_{k+1}$ correspond to particular triangulations of this image and, indeed, the $(k+1)$-morphisms from $\hat 0_k$ to $\hat 1_k$ correspond bijectively to triangulations of the image (which is a cyclic polytope).

## Applications

### $\omega$-Nerve

The $\omega$-nerve $N(C)$ of an [[omega-category]] $C$ is a simplicial set which generalizes the [[nerve]] of an ordinary category: the collection of $k$-simplices in $N(C)$ is the collection of images of the $k$-th oriental $O_k$ in $C$, i.e.

$$
  (N(C))_k := Hom_{\omega Cat}(O([k]), C)
  \,.
$$

This naturally extends to a functor

$$
  N : \omega Cat \stackrel{Hom_{\omega Cat}(O([-_2]),-_1)}{\to}
  SimplicialSets
  \,.
$$

The nerve functor is faithful. This means that omega categories can be regarded as simplicial sets equipped with extra structure. The precise nature of this structure was identified by Dominic Verity in terms of [[complicial set]]s in his work on the [Doplicher-Roberts conjecture](http://golem.ph.utexas.edu/string/archives/000711.html).

### Free $\omega$-Category on a simplicial set

The $\omega$-nerve has a [[adjoint functor|left adjoint]], the free $\omega$category on a simplicial set

$$
  F : SimplicialSets \to \omega Cat
$$

given by the [[end|coend]]

$$
  F(S_\bullet) := \int^{[n] \in \Delta} S_n \cdot
   O([n])
  \,.
$$

(Here one uses that $\omega Cat$ is naturally tensored over $Sets$: the notation $S_n \cdot O([n])$ refers to a coproduct of an $S_n$-indexed collection of copies of $O([n])$. See also [[enriched category theory]].)

Because the nerve functor is faithful, the counit of the adjunction $F \dashv N$,

$$\varepsilon_C: F N C \to C,$$

is an epimorphism for omega categories $C$.

+-- {: .query}

Is $F$ faithful? It seems to be... If not, how does it fail to be faithful?

=--

### Descent and codescent

The orientals $O(\Delta^{(n)})$, as well as the $\Pi_\omega(\Delta^{(n)})$ play a central role in the description of [[descent|descent and codescent]] in [[omega-category|omega-categorical terms]].

## References {#lit}

The original article:

* {#Street87} [[Ross Street]], *The algebra of oriented simplexes*, J. Pure Appl. Algebra **49** (1987) 283-335;  $[$doi:10.1016/0022-4049(87)90137-X, [pdf](http://www.math.mq.edu.au/~street/aos.pdf), [[Street-AlgebraOfOrientedSimplexes.pdf:file]] MR89a:18019$]$

The relation to cyclic polytopes is discussed in

* [[Mikhail Kapranov]] and [[Vladimir Voevodsky]], _Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results)_.
International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990).
Cahiers Topologie Géom. Différentielle Catég. 32 (1991), no. 1, 11–27.
([pdf](http://archive.numdam.org/article/CTGDC_1991__32_1_11_0.pdf)).
{#KapranovVoevodsky}

The $\omega$-groupoids $\Pi_\omega(\Delta^{n})$ are discussed in detail in

* [[Ronnie Brown]], Philip J. Higgins and Rafael Sivera, _Nonabelian algebraic topology_ ([pdf](http://www.bangor.ac.uk/~mas010/pdffiles/rbrsbookb-e040609.pdf))

An introductory survey of the role of orientals in Street's definition of an $\omega$-category is given in section 6 of

* [[Eugenia Cheng]], [[Aaron Lauda]], _Higher dimensional categories: an illustrated guidebook_ (2004) ([pdf](https://eugeniacheng.com/wp-content/uploads/2017/02/cheng-lauda-guidebook.pdf))

[[!redirects orientals]]


oriental > Delta2

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oriental > Delta3

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oriental > Delta4

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orientation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
#### Integration theory
+-- {: .hide}
[[!include integration theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definitions

The basic concept is for [[vector spaces]], and the remainder are defined in terms of that.

+-- {: .num_defn #basis}
###### Definition

**_(orientation of a vector space)_**
Given an [[ordered field]] $K$ and a [[vector space]] $V$ over $K$ of dimension $n$ (a [[natural number]]), an __orientation__ of $V$ is a choice of one of the two [[equivalence classes]] of [[ordered basis|ordered bases]] of $V$, where two bases are considered equivalent if the [[transformation matrix]] from one to the other has [[positive number|positive]] [[determinant]].
=--

In the case $n = 0$, the only ordered basis is the [[empty list]], but we still declare there to be two orientations by fiat, usually called positive and negative.  We can make the definition seamless by taking the elements of the equivalence class to be pairs consisting of an ordered basis and a nonzero [[sign]] (positive or negative), with $(B_1, s_1) \sim (B_2, s_2)$ iff $\sgn \det I^{B_1}_{B_2} = s_1/s_2$.  This is redundant except in dimension $0$, where now each equivalence class has a single element, $(*, +)$ for the positive orientation and $(*, -)$ for the negative orientation (where $*$ is the empty list).

In any case, this ensures that if $\omega$ is an orientation, then there is also an __opposite orientation__ $-\omega$.

A fancier way to say the same is
+-- {: .num_defn #bigwedge}
###### Definition

**_(orientation of a vector space)_**
For $V$ a vector space of dimension $n$, an __orientation__ of $V$ is an equivalence class of nonzero elements of the [[line]] $\bigwedge^n V$, the $n$th [[alternating power]] of $V$, where two such elements are considered equivalent when either (hence each) is a positive multiple of the other.
=--

Note that by both definitions, an orientation of a [[line]] (with $n = 1$) is an equivalence class of nonzero elements.

Assuming that $K$ is the field of [[real numbers]] or something like it, we can generalize from vector spaces to vector bundles:

+-- {: .num_defn #bundle}
###### Definition

**_(orientation of a vector bundle)_**
For $X$ a [[manifold]] and $V \to X$ a [[vector bundle]] of [[rank]] $k$, an __orientation__ on $V$ is an [[equivalence class]] of [[trivialization|trivializations]] of the [[line bundle]] $\bigwedge^k V$ that is obtained by associating to each [[fiber]] of $V$ its $k$th [[alternating power]].
=--

Equivalently for a [[smooth manifold]] this is an equivalence class of an everywhere non-vanishing element of $\bigwedge^k_{C^\infty(X)} \Gamma(V)$, which may be considered the [[sign]] of the element.

+-- {: .num_defn #manifold}
###### Definition

**_(orientation of a manifold)_**
For $X$ a [[manifold]] of dimension $n$, an __orientation__ of $X$ is an orientation of the [[tangent bundle]] $T X$ (or [[cotangent bundle]] $T^* X$).
=--

This is equivalently a choice of everywhere non-vanishing [[differential form]] on $X$ of degree $n$; the orientation may be considered the [[sign]] of the $n$-form (and the $n$-form\'s [[absolute value]] is a [[pseudoform|pseudo]]-$n$-form).

A vector space always has an orientation, but a manifold or bundle may not.  If an orientation exists, $V$ (or $X$) is called __orientable__.  If $X$ is a [[connected space]] and $V$ (or $X$) is orientable, then there are exactly $2$ orientations; more generally, the entire bundle is orientable iff the restriction to each [[connected component]] is orientable, and then the number of orientations is $2^k$, where $k$ is the number of orientable components.  (Or we can always say that the number of orientations is $2^k 0^m$, where now $m$ is also the number of nonorientable components.)

## Properties

### In terms of lifting through Whitehead tower

An orientation on a [[Riemannian manifold]] $X$ is equivalently a lift $\hat g$ of the classifying map $g : X \to \mathcal{B}O(n)$ of its [[tangent bundle]] through the fist step $S O(n) \to O(n)$ in the [[Whitehead tower]] of $X$:

$$
  \array{
    && \mathcal{B}S O(n)
    \\
    & {}^{\hat g}\nearrow & \downarrow
    \\
    X &\stackrel{g}{\to}& \mathcal{B} O(n)
  }
  \,.
$$

From this perspective a choice of orientation is the first in a series of special structures on $X$ that continue with

* **orientation**

* [[spin structure]]

* [[string structure]], [[differential string structure]]

* [[fivebrane structure]], [[differential fivebrane structure]]

### In terms of orientation in generalized cohomology

For $R$ an [[E-∞ ring]] spectrum, there is a general notion of $R$-orientation of vector bundles. This is described at

* [[orientation in generalized cohomology]].

For $R = H(\mathbb{R})$ be the [[Eilenberg-MacLane spectrum]]
for the [[discrete group|discrete]] [[abelian group]] $\mathbb{R}$
of [[real number]]s, orientation in $R$-cohomology is equivalent to the
ordinary notion of orientation described above.

## Related concepts

* [[orientation double cover]]

[[!include higher spin structure - table]]

[[!redirects orientation]]
[[!redirects orientations]]

[[!redirects oriented]]
[[!redirects orientedness]]

[[!redirects orientable]]
[[!redirects orientability]]
[[!redirects nonorientable]]
[[!redirects nonorientability]]
[[!redirects non-orientable]]
[[!redirects non-orientability]]

[[!redirects orientation structure]]
[[!redirects orientation structures]]

[[!redirects oriented manifold]]
[[!redirects orientable manifold]]
[[!redirects nonorientable manifold]]
[[!redirects non-orientable manifold]]


orientation in differential cohomology

[[!redirects differential orientation]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[orientation in generalized cohomology]] is a certain cohomology class. An __orientation in differential cohomology__ (possibly shortened to somewhat cryptic _differential orientation_) is not just the orientation in the underlying generalized cohomology theory, but it has more structure, namely it is a refinement of the ordinary orientation class to a cocycle in the corresponding [[differential cohomology]].

## Examples

* For [[ordinary differential cohomology]] a differential orientation is a [[differential Thom class]]. See there for details.

## References

Around def. 4.19 in

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_


orientation in generalized cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Generally, for $E$ an [[E-∞ ring]] [[spectrum]], and $P \to X$ a [[sphere spectrum]]-bundle, an _$E$-orientation_ of $P$ is a trivialization of the [[associated bundle|associated]] $E$-bundle.

Specifically, for $P = Th(V)$ the [[Thom space]] of a [[vector bundle]] $V \to X$, an $E$-orientation of $V$ is an $E$-orientation of $P$.

More generally, for $A$ an $E$-algebra spectrum, an $E$-bundle is $A$-orientable if the associated $A$-bundle is trivializable. For more on this see [[(∞,1)-vector bundle]].

The existence of an $E$-orientation is necessary in order to have a notion of [[fiber integration]] in $E$-cohomology.

## Definition

### Concretely
 {#TraditionalDefinition}

#### Orientation of a vector bundle

+-- {: .num_defn #EOrientationOfAVectorBundle}
###### Definition

Let $E$  be a [[multiplicative cohomology theory]] and let $V \to X$ be a topological [[vector bundle]] of [[rank]] $n$. Then an **$E$-orientation** or **$E$-[[Thom class]]** on $V$ is an element of degree $n$

$$
  u \in \tilde E^n(Th(V))
$$

in the [[reduced cohomology|reduced]] $E$-[[cohomology ring]] of the [[Thom space]] of $V$, such that for every point $x \in X$ its [[pullback in cohomology|restriction]] $i_x^* u$ along

$$
  i_x
    \;\colon\;
  S^n \simeq Th(\mathbb{R}^n) \overset{Th(j_x)}{\longrightarrow} Th(V)
$$

(for $\mathbb{R}^n \overset{fib_x}{\hookrightarrow} V$ the [[fiber]] of $V$ over $x$) is a _generator_, in that it is of the form

$$
  i^\ast u = \epsilon \cdot \gamma_n
$$

for

* $\epsilon \in \tilde E^0(S^0) $ a [[unit]] in $E^\bullet$;

* $\gamma_n \in \tilde E^n(S^n)$ the image of the multiplicative unit
  under the [[suspension isomorphism]] $\tilde E^0(S^0) \stackrel{\simeq}{\to}\tilde E^n(S^n)$.

=--

(e.g. [Kochman 96, def. 4.3.4](#Kochman96))

#### Orientation of a manifold

+-- {: .num_defn #EOrientationOfAManifold}
###### Definition

Let $E$  be a [[multiplicative cohomology theory]] and let $X$ be a [[manifold]], possibly [[manifold with boundary|with boundary]], of [[dimension]] $n$. An **$E$-orientation** of $X$ is a class in the $E$-[[generalized homology]]

$$
  \iota \in E_n(X,\partial X)
$$

with the property that for each point $x \in Int(X)$ in the [[interior]], it maps to a generator of $E_\bullet(\ast)$ under the map

$$
  E_\bullet(X,\partial X)
    \longrightarrow
  E_\bullet(X,\; X - \{x\})
    \simeq
  E_\bullet(D^n, S^{n-1})
    \simeq
  E_{\bullet-n}(\ast)
  \,,
$$

where the isomorphism is the excision isomorphism ([def.](generalized+homology#excision)) for the complement of a closed [[n-ball]] around $x$.

=--

(e.g.  [Kochman 96, p. 134](#Kochman96))

+-- {: .num_prop}
###### Proposition

$E$-orientations of manifolds (def. \ref{EOrientationOfAManifold}) are equivalent to $E$-orientations of their stable [[normal bundle]] (def. \ref{EOrientationOfAVectorBundle}).

=--

(e.g. [Rudyak 98, chapter V, theorem 2.4](#Rudyak98)) (also [Kochman 96, prop. 4.3.5](#Kochman96), but maybe that proof needs an extra argument)

#### Universal orientation of vector bundles
 {#UniversalOrientationOfVectorBundles}

+-- {: .num_remark}
###### Remark

Recall that a _[[(B,f)-structure]]_ $\mathcal{B}$ is a system of [[Serre fibrations]] $B_n \overset{f_n}{\longrightarrow} B O(n)$ over the [[classifying spaces]] for [[orthogonal structure]] equipped with maps

$$
  j_n \;\colon\; B_n \longrightarrow B_{n+1}
$$

covering the canonical inclusions of classifying spaces. For instance for $G_n \to O(n)$ a compatible system of [[topological group]] [[homomorphisms]], then the $(B,f)$-structure given by the [[classifying spaces]] $B G_n$ (possibly suitably resolved for the maps $B G_n \to B O(n)$ to become Serre fibrations) defines _[[G-structure]]_.

Given a $(B,f)$-structure, then there are the [[pullbacks]] $V^{\mathcal{B}}_n \coloneqq f_n^\ast (E O(n)\underset{O(n)}{\times}\mathbb{R}^n)$ of the [[universal vector bundles]] over $B O(n)$, which are the _universal vector bundles equipped with $(B,f)$-structure_

$$
  \array{
    V^{\mathcal{B}}_n &\longrightarrow& E O(n)\underset{O(n)}{\times} \mathbb{R}^n
    \\
    \downarrow &(pb)& \downarrow
    \\
    B_n & \underset{f_n}{\longrightarrow} & B O(n)
  }
  \,.
$$

Finally recall that there are canonical morphisms ([prop.](Thom+spectrum#PullbackOfUniversalOnBundleUnderCoordinateRestriction))

$$
  \phi_n
    \;\colon\;
  \mathbb{R} \oplus V^{\mathcal{B}}_n
    \longrightarrow
  V^{\mathcal{B}}_{n+1}
$$

=--

+-- {: .num_defn #EOrientationOfABfStructure}
###### Definition

Let $E$  be a [[multiplicative cohomology theory]] and let $\mathcal{B}$ be a multiplicative [[(B,f)-structure]]. Then a **universal $E$-orientation for vector bundles with $\mathcal{B}$-structure** is an $E$-orientation, according to def. \ref{EOrientationOfAVectorBundle}, for each rank-$n$ universal vector bundle with $\mathcal{B}$-structure:

$$
  \xi_n
    \in
  \tilde E^n(Th(V_n^{\mathcal{B}}))
  \;\;\;\;
  \forall n \in \mathbb{N}
$$

such that these are compatible in that

1. for all $n \in \mathbb{N}$ then

   $$
     \xi_n = \phi_n^\ast \xi_{n+1}
     \,,
   $$

   where

   $$
     \xi_n
      \in
     \tilde E^n(Th(V_n))
       \simeq
     \tilde E^{n+1}(\Sigma Th(V_n))
        \simeq
     \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))
   $$

   (with the first isomorphism is the [[suspension isomorphism]] of $E$ and the second exhibiting the [[homeomorphism]] of Thom spaces $Th(\mathbb{R} \oplus V)\simeq \Sigma Th(V)$ ([prop.](Thom+space#SuspensionOfThomSpaces))) and where

   $$
     \phi_n^\ast
     \;\colon\;
      \tilde E^{n+1}(Th(V_{n+1}))
      \longrightarrow
     \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))
   $$

   is pullback along the canonical $\phi_n \colon \mathbb{R}\oplus V_n \to V_{n+1}$ ([prop.](Thom+spectrum#PullbackOfUniversalOnBundleUnderCoordinateRestriction)).

1. for all $n_1, n_2 \in \mathbb{N}$ then

   $$
     \xi_{n+1} \cdot \xi_{n+2} = \xi_{n_1 + n_2}
     \,.
   $$

=--

+-- {: .num_prop #UniversalEOrientationsAreEquivalentlyMorphismsOfRingSpectra}
###### Proposition

A universal $E$-orientation, in the sense of def. \ref{EOrientationOfABfStructure}, for vector bundles with [[(B,f)-structure]] $\mathcal{B}$, is equivalently (the homotopy class of) a homomorphism of [[homotopy-commutative ring spectra]]

$$
  \xi \;\colon\; M\mathcal{B} \longrightarrow E
$$

from the universal $\mathcal{B}$-[[Thom spectrum]] to a spectrum which, via the [[Brown representability theorem]], represents the given [[generalized (Eilenberg-Steenrod) cohomology theory]] $E$ (and which we denote by the same symbol).

=--

+-- {: .proof}
###### Proof

The [[Thom spectrum]] $M\mathcal{B}$ has a standard structure of a [[CW-spectrum]]. Let now $E$ denote a [[sequential spectrum|sequential]] [[Omega-spectrum]] representing the multiplicative cohomology theory of the same name. Since, in the standard [[model structure on topological sequential spectra]], [[CW-spectra]] are cofibrant ([prop.](Introduction+to+Stable+homotopy+theory+--+1#CellSpectraAreCofibrantInModelStructureOnTopologicalSequentialSpectra)) and Omega-spectra are fibrant ([thm.](Introduction+to+Stable+homotopy+theory+--+1#StableModelStructureOnSequentialSpectraIsModelCategory)) we may represent all morphisms in the [[stable homotopy category]] ([def.](Introduction+to+Stable+homotopy+theory+--+1#TheStableHomotopyCategory)) by actual morphisms

$$
  \xi
   \;\colon\;
  M \mathcal{B}
    \longrightarrow
  E
$$

of sequential spectra (due to [this lemma](Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory)).

Now by definition ([def.](Introduction+to+Stable+homotopy+theory+--+1#SequentialSpectra)) such a homomorphism is precissely a sequence of base-point preserving [[continuous functions]]

$$
  \xi_n
    \;\colon\;
  (M\mathcal{B})_n
   =
  Th(V_n^{\mathcal{B}})
    \longrightarrow
  E_n
$$

for $n \in \mathbb{N}$, such that they are compatible with the structure maps $\sigma_n$ and equivalently with their $(S^1 \wedge(-)\dashv Maps(S^1,-)_\ast)$-[[adjuncts]] $\tilde \sigma_n$, in that these diagrams commute:

$$
  \array{
    S^1 \wedge Th(V^{\mathcal{B}}_n)
     &\overset{S^1 \wedge \xi_n}{\longrightarrow}&
    S^1 \wedge E_n
    \\
    {}^{\mathllap{\sigma^{M\mathcal{B}}_n}}\downarrow
      &&
    \downarrow^{\mathrlap{\sigma^E_n}}
    \\
    Th(V^{\mathcal{B}}_{n+1})
      &\underset{\xi_{n+1}}{\longrightarrow}&
    E_{n+1}
  }
  \;\;\;\;\;\;\;\;\;
  \leftrightarrow
  \;\;\;\;\;\;\;\;\;
  \array{
     Th(V^{\mathcal{B}}_n)
       &\overset{\xi_n}{\longrightarrow}&
     E_n
     \\
     {}^{\mathllap{\tilde \sigma^{M\mathcal{B}}_n}}\downarrow
       &&
     \downarrow^{\mathrlap{\tilde \sigma^E_n}}
     \\
     Maps(S^1,Th(V^{\mathcal{B}}_{n+1}))
       &\underset{Maps(S^1,\xi_{n+1})_\ast}{\longrightarrow}&
     Maps(S^1, E_{n+1})_{\ast}
  }
$$

for all $n \in \mathbb{N}$.

First of all this means (via the identification given by the [[Brown representability theorem]], see [this prop.](Introduction+to+Stable+homotopy+theory+--+S#AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum)) that the components $\xi_n$ are equivalently representatives of elements in the [[cohomology groups]]

$$
  \xi_n \in \tilde E^n(Th(V^{\mathcal{B}}_n))
$$

(which we denote by the same symbol, for brevity).

Now by the definition of universal [[Thom spectra]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#UniversalThomSpectrum), [def.](Introduction+to+Stable+homotopy+theory+--+S#UniversalThomSpectrumForBfStructure)), the structure map $\sigma_n^{M\mathcal{B}}$ is just the map $\phi_n \colon \mathbb{R}\oplus Th(V^{\mathcal{B}}_n)\to Th(V_{n+1}^{\mathcal{B}})$ from above.

Moreover, by the [[Brown representability theorem]], the [[adjunct]]   $\tilde \sigma_n^E \circ \xi_n$ (on the right) of $\sigma^E_n \circ S^1 \wedge \xi_n$ (on the left) is what represents (again by [this prop.](Introduction+to+Stable+homotopy+theory+--+S#AdditiveReducedCohomologyTheoryRepresentedByOmegaSpectrum)) the image of

$$
  \xi_n \in E^n(Th(V^{\mathcal{B}}_n))
$$

under the [[suspension isomorphism]]. Hence the [[commutative square|commutativity]] of the above squares is equivalently the first compatibility condition from def. \ref{EOrientationOfABfStructure}: $\xi_n \simeq \phi_n^\ast \xi_{n+1}$ in $\tilde E^{n+1}(Th(\mathbb{R}\oplus V_n^{\mathcal{B}}))$

Next,  $\xi$ being a homomorphism of [[ring spectra]] means equivalently (we should be modelling $M\mathcal{B}$ and $E$ as [[structured spectra]] ([here.](Introduction+to+Stable+homotopy+theory+--+1#DiagramSpectra)) to be more precise on this point, but the conclusion is the same) that for all $n_1, n_2\in \mathbb{N}$ then

$$
  \array{
    Th(V_{n_1}^{\mathcal{B}}) \wedge Th(V_{n_2}^{\mathcal{B}})
      &\overset{}{\longrightarrow}&
    Th(V_{n_1 + n_2})
    \\
    {}^{\mathllap{\xi_{n_1} \wedge \xi_{n_2}}}\downarrow
      &&
    \downarrow^{\mathrlap{\xi_{n_1 + n_2}}}
    \\
    E_{n_1} \wedge E_{n_2}
      &\underset{\cdot}{\longrightarrow}&
    E_{n_1 + n_2}
  }
  \,.
$$

This is equivalently the condition $\xi_{n_1} \cdot \xi_{n_2} \simeq \xi_{n_1  + n_2}$.

Finally, since $M\mathcal{B}$ is a [[ring spectrum]], there is an essentially unique multiplicative homomorphism from the [[sphere spectrum]]

$$
  \mathbb{S}
    \overset{e}{\longrightarrow}
  M\mathcal{B}
  \,.
$$

This is given by the component maps

$$
  e_n
    \;\colon\;
  S^n
    \simeq
  Th(\mathbb{R}^n)
    \longrightarrow
  Th(V_{n}^{\mathcal{B}})
$$

that are induced by including the fiber of $V_{n}^{\mathcal{B}}$.

Accordingly the composite

$$
  \mathbb{S}
    \overset{e}{\longrightarrow}
  M\mathcal{B}
    \overset{\xi}{\longrightarrow}
  E
$$

has as components the restrictions $i^\ast \xi_n$ appearing in def. \ref{EOrientationOfAVectorBundle}. At the same time, also $E$ is a ring spectrum, hence it also has an essentially unique multiplicative morphism $\mathbb{S} \to E$, which hence must agree with $i^\ast \xi$, up to homotopy. If we represent $E$ as a [[symmetric ring spectrum]], then the canonical such has the required property: $e_0$ is the identity element in degree 0 (being a unit of an ordinary ring, by definition) and hence $e_n$ is necessarily its image under the suspension isomorphism, due to compatibility with the structure maps and using the above analysis.

=--

### Abstractly

Let $E$ be a [[E-∞ ring]] [[spectrum]]. Write $\mathbb{S}$ for the [[sphere spectrum]].

For the following see also May,Sigurdsson: _[[Parametrized Homotopy Theory]]_ ([MO comment](http://mathoverflow.net/a/107837/381))

#### $GL_1(R)$-principal $\infty$-bundles

Write $R^\times$ or $GL_1(R)$ for the [[general linear group]] of the $E_\infty$-ring $R$: it is the subspace of the degree-0 space $\Omega^\infty R$ on those points that map to multiplicatively invertible elements in the ordinary ring $\pi_0(R)$.

Since $R$ is $E_\infty$, the space $GL_1(R)$ is itself an [[infinite loop space]]. Its one-fold [[delooping]] $B GL_1(R)$ is the [[classifying space]] for $GL_1(R)$-[[principal ∞-bundle]]s (in [[Top]]): for $X \in Top$ and $\zeta : X \to B GL_1(R)$ a map, its [[homotopy fiber]]

$$
  \array{
    GL_1(R) &\to& P &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    * &\stackrel{x}{\to}& X
   &\stackrel{\zeta}{\to}&
   B GL_1(R)
  }
$$

is the $GL_1(R)$-principal $\infty$-bundle $P \to X$ classified by that map.

+-- {: .num_example}
###### Example

For $R = \mathbb{S}$ the [[sphere spectrum]], we have that $B GL_1(\mathbb{S})$ is the [[classifying space]] for spherical fibrations.

=--

+-- {: .num_example}
###### Example

There is a canonical morphism

$$
  B O \to B GL_1(\mathbb{S})
$$

from the classifying space of the [[orthogonal group]] to that of the [[infinity-group of units]] of the [[sphere spectrum]], called the _[[J-homomorphism]]_. Postcomposition with this sends real [[vector bundle]]s $V \to X$ to sphere bundles. This is what is modeled by the [[Thom space]] construction

$$
  J : V \mapsto S^V
$$

which sends each fiber to its [[one-point compactification]].

=--

#### $GL_1(R)$-associated $\infty$-bundles

For $P \to X$ a $GL_1(R)$-[[principal ∞-bundle]] there is canonically the corresponding [[associated ∞-bundle]] with fiber $R$. Precisely, in the [[stable (∞,1)-category]] $Stab(Top)$ of [[spectra]], regarded as the [[stabilization]] of the [[(∞,1)-topos]] [[Top]]

$$
  Stab(Top) \simeq Spectra
   \stackrel{\overset{\Sigma^\infty}{\leftarrow}}{\underset{\Omega^\infty}{\to}}
  Top
$$

the associated bundle is the [[smash product]] over $\Sigma^\infty GL_1(R)$

$$
  X^\zeta := \Sigma^\infty P \wedge_{\Sigma^\infty GL_1(R)} R
  \,.
$$

This is the generalized **[[Thom spectrum]]**. For $R = K O$ the real [[K-theory spectrum]] this is given by the ordinary [[Thom space]] construction on a [[vector bundle]] $V \to X$.

An $E$-orientation of a vector bundle $V \to X$ is a trivialization of the $E$-module bundle $E \wedge S^V$, where we fiberwise form the [[smash product]] of $E$ with the [[Thom space]] of $V$.

+-- {: .num_prop}
###### Proposition

For $f : R \to S$ a morphism of $E_\infty$-rings, and $\zeta : X \to B GL_1(R)$ the classifying map for an $R$-bundle, the corresponding associated $S$-bundle classified by the composite

$$
  X \stackrel{\zeta}{ \to } B GL_1(R) \stackrel{f}{\to} B GL_1(S)
$$

is given by the [[smash product]]

$$
  X^{f \circ \zeta} \simeq X^\zeta \wedge_R S
  \,.
$$

=--

This appears as ([Hopkins, bottom of p. 6](#Hopkins)).

#### $R$-Orientations

For $X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})$ a sphere bundle, an **$R$-orientation** on $X^\zeta$ is a trivialization of the associated $R$-bundle $X^\zeta \wedge R$, hence a trivialization (null-homotopy) of the classifying morphism

$$
  X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
  \,,
$$

where the second map comes from the unit of $E_\infty$-rings $\mathbb{S} \to R$ (the [[sphere spectrum]] is the [[initial object]] in $E_\infty$-rings).

Specifically, for $V : X \to B O$  a [[vector bundle]], an $E$-orientation on it is a trivialization of the $R$-bundle associated to the associated [[Thom space]] sphere bundle, hence a trivialization of the morphism

$$
  \array{
  B O &\stackrel{J}{\to}& B GL_1(\mathbb{S})
    &\stackrel{\iota}{\to}&
   B GL_1(R)
   \\
   {}^{\mathllap{V}}\uparrow & \nearrow_{\mathrlap{\zeta}}
   \\
   X
  }
  \,.
$$

This appears as ([Hopkins, p.7](#Hopkins)).

A natural $R$-orientation of _all_ vector bundles is therefore a trivialization of the morphism

$$
  B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
 \,.
$$

Similarly, an $R$-orientation of _all_ [[spinor bundle]]s is a trivialization of

$$
  B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
$$

and an $R$-orientation of all [[string group]]-bundles a trivialization  of

$$
  B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
$$

and so forth, through the  [[Whitehead tower]] of $B O$.

Now, the [[Thom spectrum]] [[MO]] is the [[spherical fibration]] over $B O$ [[associated infinity-bundle|associated]] to the $O$-[[universal principal bundle]]. In generalization of the way that a trivialization of an ordinary $G$-principal bundle $P$ is given by a $G$-equivariant map $P \to G$, one finds that trivializations of the morphism

$$
  B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
$$

correspond to $E_\infty$-maps

$$
  M O \to R
$$

from the [[Thom spectrum]] to $R$. Similarly trivialization of

$$
  B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
$$

corresponds to morphisms

$$
  M Spin \to R
$$

and trivializations of

$$
  B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S})
    \stackrel{\iota}{\to}
   B GL_1(R)
$$

to morphisms

$$
  M String \to R
$$

and so forth.

This is the way orientations in generalized cohomology often appear in the literature.

+-- {: .num_example}
###### Example

The construction of the [[string orientation of tmf]], hence a morphism

$$
  M String \to tmf
$$

is discussed in ([Hopkins, last pages](#Hopkins)).

=--

## Properties

### Relation to top Chern class

\begin{prop}
  The [[top Chern class]] of a [[complex vector bundle]] $\mathcal{V}_X$ equals the [[pullback in cohomology|pullback]] of any [[Thom class]]
$th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big)$ on $\mathcal{V}_X$ along the [[zero-section]]:

$$
  \mathcal{V}_X
  \;
  \text{has complex rank}\;n
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  c_n
  \big(
    \mathcal{V}_X
  \big)
  \;\;\;
    =
  \;\;\;
  (0_X)^\ast
  (th)
  \;\;
  \in
  \;
  H^{2n}
  \big(
    X ;
    \,
    \mathbb{Z}
  \big)
$$
\end{prop}

For [[ordinary cohomology]] this is [Bott-Tu 82, Prop. 12.4](top+Chern+class#BottTu82). For [[Whitehead-generalized cohomology]] see at _[[universal complex orientation on MU]]_.

### Relation to the Todd class

+-- {: .num_prop #RationalToddClassIsChernCharacterOfThomClass}
###### Proposition
**([[rational Todd class is Chern character of Thom class]])**

Let $V \to X$ be a [[complex vector bundle]] over a [[compact topological space]]. Then the [[Todd class]] $Td(V) \,\in\, H^{ev}(X; \mathbb{Q})$ of $V$ in [[rational cohomology]] equals the [[Chern character]] $ch$ of the [[Thom class]] $th(V) \,\in\, K\big( Th(V) \big)$ in the [[complex topological K-theory]] of the [[Thom space]] $Th(V)$, when both are compared via the [[Thom isomorphisms]] $\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big)$:

$$
  \phi_{H\mathbb{Q}}
  \big(
    Td(V)
  \big)
  \;=\;
  ch\big( th(V) \big)
  \,.
$$

More generally , for $x \in K(X)$ any class, we have

$$
  \phi_{H\mathbb{Q}}
  \big(
    ch(x)
    \cup
    Td(V)
  \big)
  \;=\;
  ch\big( \phi_{K}(x) \big)
  \,,
$$

which specializes to the previous statement for $x = 1$.

=--

([Karoubi 78, Chapter V, Theorem 4.4](rational+Todd+class+is+Chern+character+of+Thom+class#Karoubi78))

### Relation to trivializations

The relation (equivalence) between choices of [[Thom classes]] and trivializations of [[(∞,1)-line bundles]] is
discussed e.g. in [Ando-Hopkins-Rezk 10, section 3.3](#AndoHopkinsRezk10)

### Relation to genera
 {#RelationToGenera}

Let $G$ be a [[topological group]] equipped with a [[homomorphism]] to the [[stable orthogonal group]], and write $B G \to B O$ for the corresponding map of [[classifying spaces]]. Write $J \colon B O \longrightarrow B GL_1(\mathbb{S})$ for the [[J-homomorphism]].

For $E$ an [[E-∞ ring]], there is a canonical homomorphism $B GL_1(\mathbb{S}) \to B GL_1(E)$ between the [[deloopings]] of the [[∞-groups of units]]. A trivialization of the total composite

$$
  B G \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to B GL_1(E)
$$

is a universal $E$-orientation of [[G-structures]]. Under [[(∞,1)-colimit]] in $E Mod$ this induces a homomorphism of $E$-[[∞-modules]]

$$
  \sigma \;\colon\; M G \to E
$$

from the universal $G$-[[Thom spectrum]] to $E$.

If here $G \to GL_1(\mathbb{S})$ is the $\Omega^\infty$-component of a map of [[spectra]] then this $\sigma$ is a homomorphism of [[E-∞ rings]] and in this case there is a [[bijection]] between universal orientations and such $E_\infty$-ring homomorphisms ([Ando-Hopkins-Rezk 10, prop. 2.11](#AndoHopkinsRezk10)).

The latter, on passing to [[homotopy groups]], are [[genera]] on manifolds with [[G-structure]].

### Relation to cubical structures

For $E$ a [[multiplicative cohomology theory|multiplicative]] [[weakly periodic cohomology theory|weakly periodic]] [[complex orientable cohomology theory]] then $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of [[cubical structure on a line bundle|cubical structures]] on the trivial line bundle over the [[formal group]] of $E$.

In particular, [[homotopy classes]] of maps of [[E-infinity ring]] spectra $MU\langle 6\rangle \to E$ from the [[Thom spectrum]] to $E$, and hence universal $MU\langle 6\rangle$-[[orientation in generalized cohomology|orientations]] (see there) of $E$ are in natural bijection with these cubical structures.

See at _[[cubical structure]]_ for more details and references.
This way for instance the [[string orientation of tmf]] has been constructed. See there for more.

## Examples

### Orientations related to genera and indices

[[!include genera and partition functions - table]]

### Complex orientation

An $E_\infty$ [[complex oriented cohomology theory]] $E$ is indeed equipped with a universal complex orientation given by an $E_\infty$-ring homomorphism $MU \to E$, see [here](complex+oriented+cohomology+theory#InTermsOfGenera).

## Related concepts

* [[differential Thom class]]

* [[differential orientation]], [[fiber integration in differential cohomology]]

* [[(infinity,1)-vector bundle]]

## References

Discussion in terms of [[Thom classes]]:

* [[Frank Adams]], part III, section 10 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Kochman96} [[Stanley Kochman]], section 4.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Rudyak98} [[Yuli Rudyak]], _In Thom spectra, Orientability and Cobordism_, Springer 1998 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf))

* [[Jacob Lurie]], lecture 5 of _[[Chromatic Homotopy Theory]]_, 2010 ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf))

* [[eom]], _[Orientation](http://eom.springer.de/o/o070200.htm)_

* [[Manifold Atlas]], _[Orientation of manifolds in generalized cohomology theories](http://www.map.mpim-bonn.mpg.de/Orientation_of_manifolds_in_generalized_cohomology_theories)_

In the generality of [[equivariant vector bundles]] and [[equivariant complex oriented cohomology theory]]:

* [[Steven Costenoble]], [[Peter May]], [[Stefan Waner]], _Equivariant orientation theory_, Homology Homotopy Appl.
Volume 3, Number 2 (2001), 265-339 ([euclid:hha/1139840256](https://projecteuclid.org/euclid.hha/1139840256))

A comprehensive account of the general abstract perspective (with predecessors in [Ando-Hopkins-Rezk 10](#AndoHopkinsRezk10)) is in

* {#ABGHR} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], section 6 of _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

Lecture notes on this include

* {#Hopkins} [[Mike Hopkins]] (notes by [[André Henriques]]), _The String orientation of tmf_  ([arXiv:0805.0743](http://arxiv.org/abs/0805.0743))

which are motivated towards constructing the [[string orientation of tmf]], based on

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

For an informal exposition in terms of spectra, see

* Mattia Coloma, [[Domenico Fiorenza]], Eugenio Landi, _An exposition of the topological half of the Grothendieck-Hirzebruch-Riemann-Roch theorem in the fancy language of spectra_, ([arXiv:1911.12035](https://arxiv.org/abs/1911.12035))

[[!redirects Thom class]]
[[!redirects Thom classes]]

[[!redirects orientations in generalized cohomology]]

[[!redirects E-orientation]]
[[!redirects E-orientations]]


orientations in higher quantization - table

| [[chromatic level]] | [[generalized cohomology theory]] / [[E-∞ ring]] |  [[obstruction]] to [[orientation in generalized cohomology]] | [[orientation in generalized cohomology|generalized orientation]]/[[polarization]] | [[quantization]] | incarnation as [[quantum anomaly]] in [[higher gauge theory]]  |
|-----|--------------------------------------------------|---|--|---|---|
|  1  | [[complex K-theory]] $KU$ | [[integral Stiefel-Whitney class|third integral SW class]] $W_3$ | [[spin^c-structure]] | [[geometric quantization by push-forward|K-theoretic geometric quantization]]  | [[Freed-Witten anomaly]] |
|  2  | [[EO(n)]]  |  [[Stiefel-Whitney class]] $w_4$ |  |  |   |
|  2  | [[integral Morava K-theory]] $\tilde K(2)$ | [[integral Stiefel-Whitney class|seventh integral SW class]] $W_7$ |  |  | [[Diaconescu-Moore-Witten anomaly]] in [Kriz-Sati interpretation](Diaconescu-Moore-Witten+anomaly#ReferencesInterpretationInSecondMoravaKTheory) |


oriented graph

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[graph theory]], an **orientation** of an undirected [[graph]] is a choice of one of two possible directions for every [[edge]]. An **oriented graph** is an undirected graph equipped with an orientation.

Since the precise definition of "undirected graph" can vary depending on context (notably, whether one allows loops or multiple edges), the precise definition of "oriented graph" is likewise context dependent.

## Definitions

### Oriented simple graphs

In [[combinatorics]], the most common sense of "oriented graph" is "simple graph equipped with an orientation".

More formally: let $\binom{V}{2}$ denote the set of two-element subsets of $V$,
and $\Delta \hookrightarrow V^2 = V \times V$ the diagonal $\{(v, v): v \in V\}$. Then there is a quotient map $q: V^2 \setminus \Delta \to \binom{V}{2}$, realizing $\binom{V}{2}$ as the set of orbits of the fixed-point free involution $(x, y) \mapsto (y, x)$ on $V^2 \setminus \Delta$. Then an oriented graph consists of a graph $(V, E \hookrightarrow \binom{V}{2}$ together with a section of the quotient $p$ identified in the pullback diagram

$$\array{
F & \to & V^2 \setminus \Delta \\
\mathllap{p} \downarrow & & \downarrow \mathrlap{q} \\
E & \hookrightarrow & \binom{V}{2}.
}$$

Equivalently, an oriented graph consists of a set $V$ and a subset $E \hookrightarrow V^2$ such that $(x, y) \in E$ implies $(y, x) \notin E$. We may depict a pair $(x, y)$ in $E$ as an arrow from $x$ to $y$. Of course the condition rules out loops (arrows from a vertex to itself), and evidently the notion of oriented graph is very different from the notion of [[digraph]] where 2-cycles are allowed (an arrow from $x$ to $y$ and another back from $y$ to $x$).

### More general graph orientations

If the underlying undirected graph is allowed to have loops and multiple edges, then an oriented graph is essentially the same thing as a [[directed graph]] (i.e., [[quiver]]).

Formally, let an undirected graph $G$ be given by a quiver $s,t : E \rightrightarrows V$ together with a [[fixed point]] free [[involution]] $i : E \to E$ such that $s\circ i = t$ and $t \circ i = s$ (in the style of [Serre 1977](#Serre1977)).
Then an orientation of $G$ consists of a subset $E^+ \subseteq E$ such that $E$ is the [[disjoint union]] $E = E^+ \uplus i(E^+)$.

Any such orientation induces a corresponding quiver $G^+ = E^+ \rightrightarrows V$.
Conversely, any quiver $E \rightrightarrows V$ can be seen as an orientation of the undirected Serre graph defined by adding a formal inverse to each edge.

Although the terms "oriented graph" and "directed graph" become essentially equivalent under this reading, in many situations it is useful to consider an oriented graph as one of the many possible orientations of a given undirected graph.
For example, various [[graph invariants]] (such as the [[flow polynomial]], or [[W. T. Tutte|Tutte]]'s original definition of the [[Tutte polynomial]]) can be defined relative to an arbitrary orientation of a graph, but are independent of the choice of orientation.

## References

* [[W. T. Tutte]], A contribution to the theory of chromatic polynomials. Canad. J. Math. 6:80–91, 1954. [(doi)](http://dx.doi.org/10.4153/CJM-1954-010-9)

* Frank Harary (1969), _Graph Theory_, Addison-Wesley.
* {#Serre1977} [[Jean-Pierre Serre]] (1977), _Trees_, Springer.
* [[W. T. Tutte]] (1984), _Graph Theory_, Addison-Wesley.


oriented link diagram

_your text_


oriented matroid

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[matroid]]

## References

* Wikipedia [oriented matroid](http://en.wikipedia.org/wiki/Oriented_matroid)

* A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler,  _Oriented matroids_, Cambridge Univ. Press 1993, 2000, [view at reslib.com](http://reslib.com/book/Oriented_Matroids)


oriented submanifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By an _oriented submanifold_ one may mean

1. a [[submanifold]] with [[orientation structure]] on the [[tangent bundle]] of its underlying abstract [[differentiable manifold]];

1. a [[submanifold]] of a [[differentiable manifold]] with [[orientation structure]] on its [[normal bundle]].

## Related concepts

* [[Thom's theorem]], [[MSO]]

* [[framed submanifold]]

## References

Normally oriented submanifolds:

* {#Thom54} [[René Thom]], p. 67 in: _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

[[!redirects normally oriented submanifold]]

[[!redirects normally oriented submanifold]]
[[!redirects normally oriented submanifolds]]


orientifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _orientifold_ ([Dai-Leigh-Polchinski 89, p. 12](#DaiLeighPolchinski89)) is a [[target space|target]] [[spacetime]] for [[string theory|string]] [[sigma-models]] that combines aspects of $\mathbb{Z}_2$-[[orbifold]]s with _[[orientation]] reversal_ on the worldsheet, whence the name.

In [[type II string theory]] orientifold backgrounds (inducing e.g. [[type I string theory]], and the [[Sugimoto string]]) with $\mathbb{Z}_2$-[[fixed points]] -- called _[[O-planes]]_ (see there for more) --  are required for [[RR-field tadpole cancellation]]. This is a key consistency condition in particular for [[intersecting D-brane models]] used in [[string phenomenology]].

Where generally ([[higher gauge field|higher gauge]]) [[field (physics)|fields]] in [[physics]]/[[string theory]] are [[cocycles]] in ([[differential cohomology theory|differential]]) [[cohomology theory]] and typically in [[complex oriented cohomology theory]], fields on orientifolds are cocycles in genuinely $\mathbb{Z}_2$-[[equivariant cohomology]] and typically in [[real-oriented cohomology theory]]. For instance, the [[B-field]], which otherwise is a (twisted) cocycle in ([[ordinary differential cohomology|ordinary]]) [[differential cohomology]], over an orientifold is a cocycle in (twisted) [[HZR-theory]], and the [[RR-fields]], which usually are cocycles in ([[twisted cohomology|twisted]] [[differential K-theory|differential]]) [[K-theory]], over an orientifold are cocycles in [[KR-theory]] ([Witten 98](#Witten98)).

An explicit model for [[B-fields]] for the [[bosonic string]]  on orientifolds (differential [[HZR-theory]]) is given in ([Schreiber-Schweigert-Waldorf 05](#Jandl)) and examples are analyzed in this context in ([Gawedzki-Suszek-Waldorf 08](#GawedzkiSuszekWaldorf08)). See also ([HMSV 16](#HMSV16), [HMSV 19](#HMSV19)).

The claim that for the [[superstring]] the B-field is more generally a cocycle with coefficients in the [[Picard infinity-group]] of [[complex K-theory]] ([[super line 2-bundles]]) and a detailed discussion of the orientifold version of this can be found in ([Distler-Freed-Moore 09](#Precis), [Distler-Freed-Moore 10](#DistlerFreedMooreII)) with details in ([Freed 12](#Freed)).

The quadratic pairing entering the [[11d Chern-Simons theory]] that governs the [[RR-field]] here as a [[self-dual higher gauge field]] is given in ([DFM 10, def. 6](#Precis)).

## Properties

### Orientifold backreaction?
 {#OrientifoldBackreactionOrNot}

Essentially all existing results on orientifolds (such as [[O-plane]]-charges and [[RR-field tadpole cancellation]]) are derived for string [[sigma-models]] on [[flat orbifold]] ([[toroidal orbifold]]) or orientifolded [[Calabi-Yau manifold]] [[target spacetimes]], or for orientifoldings of algebraically defined [[rational CFT]] [[string vacua]] ([[Gepner models]], [[non-geometric vacua]]).

#### No string-theory results on back-reacted orientifolds
 {#NoResultOnBackreactedOrientifolds}

There are to date no results in actual [[string theory]] for what one might expect to be the curved back-reacted geometry of [[orientifolds]], analogous to the curved [[near horizon geometry]] that is well-known for the case of [[black brane|black]] [[D-branes]].

From [Banks-van den Broeck 06](#BanksBroeck06):

> The discussion of these $[$orientifold$]$ compactifications is generally carried out in low energy [[effective field theory]] $[$[3a](https://arxiv.org/abs/hep-th/0412277), [3b](https://arxiv.org/abs/hep-th/0506066) $]$ $[$[4](https://arxiv.org/abs/hep-th/0412129)$]$, despite the fact that they all contain [[orientifold]] [[singularities]]. Further, there is no [[perturbative string theory|perturbative world sheet treatment]] of these backgrounds.

> $[...]$

> The inevitable orientifold of [[flux compactifications]] is one potential barrier to an [[effective field theory]] treatment $[$footnote 1 : [[Greg Moore|G. Moore]] and S. Ramanujam have emphasized to us the problems with the back reaction of the orientifold $[...$] $]$

From [Cordova-de Luca-Tomasiello 19](#CordovaLucaTomasiello19) (p. 2 and p. 30):

> To assess the validity  of these $[$assumed effective orientifold$]$ solutions in [[string theory]], one should  ideally use the full string theory action, or  switch to a [[duality in string theory|dual]] description. Unfortunately neither of  these options is available

> $[...]$

> the presence of [[O-planes]] is inferred by comparison with their flat-space behavior. Since the $[$spacetimes with orientifolds considered$]$ have strong curvature and coupling, [[higher curvature corrections|stringy corrections]] come into play, and it is impossible to decide with [[supergravity]] alone whether the solutions are valid.

> It is important to stress that this will be so for any solution with [[O-planes]]. It would be important, then,to develop techniques to decide  whether a solution with O-planes will survive in full string theory.  In other words, it would be important to understand what conditions one needs  to impose near the [[O-plane]] [[singularities]].

> $[...]$

>  We clearly need alternative procedures that are better justified physically.

#### A popular assumption in effective field theory
 {#AnAssumptionInEffectiveFieldTheory}

Nevertheless, motivated from the fact that the computations for [[flat orbifolds]] show that the [[O-planes]] there are similar to (while clearly different from!) [[D-branes]] (*not* in their back-reacted form as [[black branes]], though!) with [[negative number|negative]] tension (i.e. negative [[energy]] density) it may seem plausible that the low energy [[effective field theory]] of [[perturbative string theory vacua]] including [[O-planes]] is a modification of [[supergravity]] where negative-energy source-terms are added to the [[equations of motion]] much like one might add [[black brane|black]] [[D-brane]]-contributions, but simply equipped with a negative sign prefactor.

This _ad hoc_ effective field theory picture of orientifold backgrounds has been advocated, seminally, in [Giddings-Kachru-Polchinski 01](#GiddingsKachruPolchinski01), in a one-sentence argument (below (2.19)):

> String theory does have such $[$negative tension$]$ objects, and so evades the no-go theorem $[$which rules out certain warped solutions of supergravity$]$.

There this is followed by reference to a presumed example considered earlier in [Verlinde 00](#Verlinde00), where the statement is introduced in a similar manner (above (9)):

> In a more complete treatment, we must also include the backreaction of the 64 orientifold planes. These have a negative tension equal to −1/4 times the D3-brane tension, which need to be taken into account. We can write an explicit form for the background metric $[$just as for D-branes but with a minus sign included$]$.

One may feel this is plausible -- and it might even be right, sometimes -- but there does not exist a derivation of this statement from actual [[perturbative string theory]] (see [above](#NoResultOnBackreactedOrientifolds)), beyond the hand-waving leap of faith extrapolating from perturbative string theory on flat orientifolds to curved backreacted orientifold throat geometries (if such indeed exist).

#### Its use in the landscape/swampland literature
 {#UseInLandscapeSwamplandLiterature}

Nevertheless, in the wake of the discussion of the [[landscape of string theory vacua|landscape]] (or not) of [[de Sitter spacetime|de Sitter]] [[string theory vacua]] and of the "[[swampland conjectures]]" it became popular to rely on the handwaving argument of [Giddings-Kachru-Polchinski 01](#GiddingsKachruPolchinski01) and behave as if it is established that questions about [[effective field theory|low energy effective]] [[orientifold]] [[string theory vacua]] may be answered using a modification of [[supergravity]] where the [[equations of motion]] are changed -- by hand -- simply by including negative-tension source terms of some form.

This step happens for instance around (2.2) in [Junghans 20](#Junghans20) where it is justified, without references, by the words "as is standard in the literature" (footnote 5 in [Junghans 20](#Junghans20)).

\linebreak

### Lifts to M-theory F-theory

Lifts of orientifold background from [[type II string theory]] to [[F-theory]] go back to ([Sen 96](#Sen96), [Sen 97a](#Sen97a)).

Lifts of [[type IIA string theory]] orientifolds of [[D6-branes]] to [[ADE singularities|D-type ADE singularities]] in [[M-theory]] (through the [[duality between M-theory and type IIA string theory]]) goes back to ([Sen 97b](#Sen97b)). See at _[[heterotic M-theory on ADE-orbifolds]]_.

A more general scan of possible lifts of type IIA orientifolds to M-theory is indicated in ([Hanany-Kol 00, around (3.2)](#HananyKol00)), see ([Huerta-Sati-Schreiber 18, Prop. 4.7](#HuertaSatiSchreiber18)) for details.

For instance the [[O4-plane]] lifts to the [[MO5-plane]].

(...)

## Related concepts

* [[fractional D-brane]]

  [[permutation D-brane]]

* [[O-plane]],

  [[O-plane charge]],

  [[RR-field tadpole cancellation]]

* [[worldsheet parity operator]]

* [[type I string theory]]

* [[Sugimoto string]]

* [[real-oriented cohomology theory]]

* [[higher orientifold]]

## References
 {#References}

### General

The concept originates around

* {#DaiLeighPolchinski89} Jin Dai, [[Robert Leigh]], [[Joseph Polchinski]], p. 12 of _New Connections Between String Theories_, Mod. Phys. Lett. A4 (1989) 2073-2083 ([spire:25758](http://inspirehep.net/record/25758), [pdf scan](https://lib-extopc.kek.jp/preprints/PDF/1989/8905/8905564.pdf))

Early accounts include

* {#Mukhi97} [[Sunil Mukhi]], _Orientifolds: The Unique Personality Of Each Spacetime Dimension_, [Workshop on Frontiers of Field Theory, Quantum Gravity and String Theory, Puri, India, 12 - 21 Dec 1996](http://cds.cern.ch/record/323857) ([arXiv:hep-th/9710004](https://arxiv.org/abs/hep-th/9710004), [cern:335233](http://cds.cern.ch/record/335233))

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv. Theor. Math. Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

Traditional lecture notes include

* [[Atish Dabholkar]], _Lectures on Orientifolds and Duality_, In *Trieste 1997, High energy physics and cosmology* 128-191 ([arXiv:hep-th/9804208](https://arxiv.org/abs/hep-th/9804208), [spire:454332](https://inspirehep.net/literature/454332))

* Carlo Angelantonj, [[Augusto Sagnotti]], _Open Strings_, Phys. Rept. 371:1-150,2002; Erratum ibid.376:339-405, 2003 ([arXiv:hep-th/0204089](https://arxiv.org/abs/hep-th/0204089))

Textbook discussion is in

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], Section 15.3 of: *Basic Concepts of String Theory*, Theoretical and Mathematical Physics, Springer (2013) [[doi:10.1007/978-3-642-29497-6](https://doi.org/10.1007/978-3-642-29497-6)]

and specifically in the context of [[intersecting D-brane models]] with an eye towards [[string phenomenology]] in

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], sections 5.3.4 and 10.1.3 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

Exposition:

* Marcus Berg, _Introduction to Orientifolds_ ([pdf](https://tp.hotell.kau.se/marcus/physics/talks/orienti_short.pdf), [[BergOrientifolds.pdf:file]])

The original observation that [[D-brane charge]] for orientifolds should be in [[KR-theory]] is due to

* {#Witten98} [[Edward Witten]], section 5 of _D-branes and K-theory_, J. High Energy Phys., 1998(12):019, 1998 ([arXiv:hep-th/9810188](http://arxiv.org/abs/hep-th/9810188))

and was re-amplified in

* {#Gukov00} [[Sergei Gukov]], _K-Theory, Reality, and Orientifolds_, Commun. Math. Phys. **210** (2000) 621-639 [[arXiv:hep-th/9901042](http://arxiv.org/abs/hep-th/9901042), [doi:10.1007/s002200050793](https://doi.org/10.1007/s002200050793)]

### In terms of KO-theory

Discussion of orbi-orienti-folds in terms of [[equivariant K-theory|equivariant]] [[KO-theory]] is in

* N. Quiroz, [[Bogdan Stefanski]], _Dirichlet Branes on Orientifolds_, Phys.Rev. D66 (2002) 026002 ([arXiv:hep-th/0110041](https://arxiv.org/abs/hep-th/0110041))

* [[Volker Braun]], [[Bogdan Stefanski]], _Orientifolds and K-theory_, Braun, Volker. "Orientifolds and K-theory." Progress in String, Field and Particle Theory. Springer, Dordrecht, 2003. 369-372 ([arXiv:hep-th/0206158](https://arxiv.org/abs/hep-th/0206158))

* H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, _D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory_, JHEP 0812:007, 2008 ([arXiv:0809.4238](https://arxiv.org/abs/0809.4238))

A definition and study of orientifold [[bundle gerbes]], modeling the [[B-field]] [[background gauge field|background]] for the [[bosonic string]] (differential [[HZR-theory]]), is in

* {#Jandl} [[Urs Schreiber]], [[Christoph Schweigert]], [[Konrad Waldorf]], _Unoriented WZW models and Holonomy of Bundle Gerbes_, Communications in Mathematical Physics August 2007, Volume 274, Issue 1, pp 31-64 ([arXiv:hep-th/0512283](http://arxiv.org/abs/hep-th/0512283))

* {#GawedzkiSuszekWaldorf08} [[Krzysztof Gawedzki]], Rafal R. Suszek,  [[Konrad Waldorf]], _Bundle Gerbes for Orientifold Sigma Models_ Adv. Theor. Math. Phys. 15(3), 621-688 (2011) ([arXiv:0809.5125](http://arxiv.org/abs/0809.5125))

see also

* {#HMSV16} [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Real bundle gerbes, orientifolds and twisted KR-homology_ ([arXiv:1608.06466](http://arxiv.org/abs/1608.06466))

* {#HMSV19} [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Sign choices for orientifolds_, Commun. Math. Phys. 378, 1843–1873 (2020) ([arXiv:1905.06041](https://arxiv.org/abs/1905.06041))

An elaborate formalization in terms of [[differential cohomology]] in general and [[twisted K-theory|twisted]] [[differential K-theory]] in particular that also takes the spinorial degrees of freedom into account is briefly sketched out in

* {#Precis} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

based on stuff like

* {#BergmanGimonSugimoto01} [[Oren Bergman]], [[Eric Gimon]], [[Shigeki Sugimoto]], _Orientifolds, RR Torsion, and K-theory_, JHEP 0105:047, 2001 ([arXiv:hep-th/0103183](https://arxiv.org/abs/hep-th/0103183))

Details on the computation of [[string scattering amplitudes]] in such a background:

* {#DistlerFreedMooreII} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Spin structures and superstrings_, Surveys in Differential Geometry, Volume 15 (2010) ([arXiv:1007.4581](http://arxiv.org/abs/1007.4581), [doi:10.4310/SDG.2010.v15.n1.a4](http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4))

Related lecture notes / slides include

* [[Daniel Freed]], _Dirac charge quantization, K-theory, and orientifolds_, talk at a workshop _Mathematical methods in general relativity and quantum field theories_, Paris, November 2009 ([pdf](http://www.ma.utexas.edu/users/dafr/paris_nt.pdf), [[FreedK09.pdf:file]])

* {#Moore10} [[Greg Moore]], _The RR-charge of an orientifold_, Oberwolfach talk 2010 ([pdf](https://www.physics.rutgers.edu/~gmoore/Oberwolfach_June2010_FINAL.pdf), [[MooreOrientifold2010.pdf:file]], [ppt](http://www.physics.rutgers.edu/~gmoore/AnnArbor_Feb2010_FINAL.ppt))

* {#Freed} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_, lecures at _[K-Theory and Quantum Fields](http://www.esi.ac.at/activities/events/2012/k-theory-and-quantum-fields)_, ESI 2012 ([[FreedESI2012.pdf:file]])

A detailed list of examples of [[KR-theory]] of orientifolds and their [[T-duality]] is in

* {#DMR13} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _T-duality For Orientifolds and Twisted KR-theory_ ([arXiv:1306.1779](http://arxiv.org/abs/1306.1779))

* {#DMR14} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _String theory on elliptic curve orientifolds and KR-theory_ ([arXiv:1402.4885](http://arxiv.org/abs/1402.4885))

A formulation of some of the relevant aspects of (bosonic) orientifolds in terms of the  [[schreiber:differential cohomology in an (∞,1)-topos -- survey|differential nonabelian cohomology]] with coefficients in the [[2-group]] $AUT(U(1))$ coming from the [[crossed module]] $[U(1) \to \mathbb{Z}_2]$ is indicated in

* [[Urs Schreiber]],  talk [[schreiber:Background fields in twisted differential nonabelian cohomology|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]], _[[schreiber:differential cohomology in a cohesive topos]]_

### Examples and Models
 {#ReferencesExamplesAndModels}

Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIB string theory]], hence for [[D9-branes]] and [[D5-branes]]:

* [[Eric Gimon]], [[Joseph Polchinski]], Section 3.2 of: _Consistency Conditions for Orientifolds and D-Manifolds_, Phys. Rev. D54: 1667-1676, 1996 ([arXiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

* {#GimonJohnson96} [[Eric Gimon]], [[Clifford Johnson]], _K3 Orientifolds_, Nucl. Phys. B477: 715-745, 1996 ([arXiv:hep-th/9604129](https://arxiv.org/abs/hep-th/9604129))

* {#BuchelShiuTye99} Alex Buchel, [[Gary Shiu]], S.-H. Henry Tye, _Anomaly Cancelations in Orientifolds with Quantized B Flux_, Nucl.Phys. B569 (2000) 329-361 ([arXiv:hep-th/9907203](https://arxiv.org/abs/hep-th/9907203))

* P. Anastasopoulos, A. B. Hammou, _A Classification of Toroidal Orientifold Models_, Nucl. Phys. B729:49-78, 2005 ([arXiv:hep-th/0503044](https://arxiv.org/abs/hep-th/0503044))

Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIA string theory]], hence for [[D8-branes]] and [[D4-branes]]:

* {#ParkUranga98} [[Jaemo Park]], [[Angel Uranga]], _A Note on Superconformal $\mathcal{N}=2$ theories and Orientifolds_, Nucl. Phys. B542:139-156, 1999 ([arXiv:hep-th/9808161](https://arxiv.org/abs/hep-th/9808161))

* {#AFIRU00a} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _D=4 Chiral String Compactifications from Intersecting Branes_, J. Math. Phys. 42:3103-3126, 2001 ([arXiv:hep-th/0011073](https://arxiv.org/abs/hep-th/0011073))

* {#AFIRU00b} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _Intersecting Brane Worlds_, JHEP 0102:047, 2001 ([arXiv:hep-ph/0011132](https://arxiv.org/abs/hep-ph/0011132))

* {#KataokaShimojo01} H. Kataoka, M. Shimojo, _$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes_, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 ([arXiv:hep-th/0112247](https://arxiv.org/abs/hep-th/0112247), [doi:10.1143/PTP.107.1291](https://doi.org/10.1143/PTP.107.1291))

> The $\mathbb{Z}_N$ action with even $N$ contains an order 2 element $[ ...]$  Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use of the same dimensional D-branes, we restrict ourselves to the case that the order $N$ of $\mathbb{Z}_N$ is odd. ([p. 4](https://arxiv.org/pdf/hep-th/0112247.pdf#page=4))

* {#Honecker01} [[Gabriele Honecker]], _Non-supersymmetric Orientifolds with D-branes at Angles_, Fortsch.Phys. 50 (2002) 896-902 ([arXiv:hep-th/0112174](https://arxiv.org/abs/hep-th/0112174))

* {#Honecker02a} [[Gabriele Honecker]], _Intersecting brane world models from D8-branes on $(T^2 \times T^4/\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds_, JHEP 0201 (2002) 025 ([arXiv:hep-th/0201037](https://arxiv.org/abs/hep-th/0201037))

* {#Honecker02b} [[Gabriele Honecker]], _Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes_, thesis 2002 ([[HoneckerIntersectingDBraneModels02.pdf:file]])

The [[Witten-Sakai-Sugimoto model]] on [[D4-D8-brane bound states]] for [[QCD]] with [[orthogonal group|orthogonal]] [[gauge groups]] on O-planes:

* {#ImotoSakaiSugimoto09} Toshiya Imoto, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _$O(N)$ and $USp(N)$ QCD from String Theory_, Prog.Theor.Phys.122:1433-1453, 2010 ([arXiv:0907.2968](https://arxiv.org/abs/0907.2968))

* Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, _5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry_, JHEP 1210 (2012) 142 ([arXiv:1206.6781](https://arxiv.org/abs/1206.6781))

Specifically D5 brane models [[T-duality|T-dual]] to D6/D8 models:

* [[Angel Uranga]], _A New Orientifold of $\mathbb{C}^2/\mathbb{Z}_N$ and Six-dimensional RG Fixed Points_, Nucl. Phys. B577:73-87, 2000 ([arXiv:hep-th/9910155](https://arxiv.org/abs/hep-th/9910155))

* {#FengHeKarchUranga01} Bo Feng, [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], _Orientifold dual for stuck NS5 branes_, JHEP 0106:065, 2001 ([arXiv:hep-th/0103177](https://arxiv.org/abs/hep-th/0103177))

Specifically for [[D6-branes]]:

* {#IshiharaKataokaSato99} S. Ishihara, H. Kataoka, Hikaru Sato, _$D=4$, $N=1$, Type IIA Orientifolds_, Phys. Rev. D60 (1999) 126005 ([arXiv:hep-th/9908017](https://arxiv.org/abs/hep-th/9908017))

* [[Mirjam Cvetic]], Paul Langacker, Tianjun Li, Tao Liu, _D6-brane Splitting on Type IIA Orientifolds_, Nucl. Phys. B709:241-266, 2005 ([arXiv:hep-th/0407178](https://arxiv.org/abs/hep-th/0407178))

Specifically for [[D3-branes]]/[[D7-branes]]:

* [Feng-He-Karch-Uranga 01](#FengHeKarchUranga01)

Specifically 2d [[toroidal orbifold|toroidal]] orientifolds:

2d toroidal [[orientifolds]]:

* Dongfeng Gao, [[Kentaro Hori]], Section 7.3 of: _On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds_ ([arXiv:1004.3972](https://arxiv.org/abs/1004.3972))

* {#DMR14} [[Charles Doran]], Stefan Mendez-Diez, [[Jonathan Rosenberg]], _String theory on elliptic curve orientifolds and KR-theory_ ([arXiv:1402.4885](http://arxiv.org/abs/1402.4885))

Various:

* [[Dieter Lüst]], S. Reffert, E. Scheidegger, S. Stieberger, _Resolved Toroidal Orbifolds and their Orientifolds_, Adv.Theor.Math.Phys.12:67-183, 2008 ([arXiv:hep-th/0609014](https://arxiv.org/abs/hep-th/0609014))

### Orientifold Gepner models

Orientifolds of [[Gepner models]]

* Brandon Bates, [[Charles Doran]], Koenraad Schalm, _Crosscaps in Gepner Models and the Moduli space of $T^2$ Orientifolds_, Advances in Theoretical and Mathematical Physics, Volume 11, Number 5, 839-912, 2007 ([arXiv:hep-th/0612228](https://arxiv.org/abs/hep-th/0612228))

Specifically [[string phenomenology]] and the [[landscape of string theory vacua]] of Gepner model [[orientifold]] compactifications:

* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))

* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))

### Lift to M-theory
 {#ReferencesInMTheory}

Lifts of orientifolds to [[M-theory]] ([[MO5]], [[MO9]]) and [[F-theory]] are discussed in

* {#Sen96} [[Ashoke Sen]], _F-theory and Orientifolds_ ([arXiv:hep-th/9605150](http://arxiv.org/abs/hep-th/9605150))

* {#Sen97a} [[Ashoke Sen]], _Orientifold Limit of F-theory Vacua_ ([arXiv:hep-th/9702165](http://arxiv.org/abs/hep-th/9702165))

* {#Sen97b} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

* {#Hori98} [[Kentaro Hori]], _Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold_, Nucl. Phys. B539:35-78, 1999 ([arXiv:hep-th/9805141](https://arxiv.org/abs/hep-th/9805141))

* {#Gimon98} [[Eric Gimon]], _On the M-theory Interpretation of Orientifold Planes_ ([arXiv:hep-th/9806226](https://arxiv.org/abs/hep-th/9806226), [spire:472499](http://inspirehep.net/record/472499))

* {#AKY98} Changhyun Ahn, Hoil Kim, Hyun Seok Yang, _$SO(2N)$ $(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$_, Phys.Rev. D59 (1999) 106002 ([arXiv:hep-th/9808182](https://arxiv.org/abs/hep-th/9808182))

* {#HananyKol00} [[Amihay Hanany]], [[Barak Kol]], section 4 of _On Orientifolds, Discrete Torsion, Branes and M Theory_, JHEP 0006 (2000) 013 ([arXiv:hep-th/0003025](https://arxiv.org/abs/hep-th/0003025))

* {#ArgyresMaimonPelland02} [[Philip Argyres]], [[Ron Maimon]], Sophie Pelland, _The M theory lift of two O6 planes and four D6 branes_, JHEP 0205 (2002) 008 ([arXiv:hep-th/0204127](https://arxiv.org/abs/hep-th/0204127))

  following

* [[Edward Witten]], _Solutions Of Four-Dimensional Field Theories Via M Theory_, ([arXiv:hep-th/9703166](https://arxiv.org/abs/hep-th/9703166))

The [[MO5]] is originally discussed in

* Keshav Dasgupta, [[Sunil Mukhi]], _Orbifolds of M-theory_, Nucl. Phys. B465 (1996) 399-412 ([arXiv:hep-th/9512196](https://arxiv.org/abs/hep-th/9512196))

* {#Witten95} [[Edward Witten]], _Five-branes And M-Theory On An Orbifold_, Nucl. Phys. B463:383-397, 1996 ([arXiv:hep-th/9512219](https://arxiv.org/abs/hep-th/9512219))

See also:

* [[Andreas Braun]], _M-Theory and Orientifolds_ ([arXiv1912.06072](https://arxiv.org/abs/1912.06072))

The classification in [Hanany-Kol 00 (3.2)](#HananyKol00) also appears, with more details, in Prop. 4.7 of

* {#HuertaSatiSchreiber18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_ ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

The "higher orientifold" appearing in [[Horava-Witten theory]] with circle 2-bundles replaced by the [[circle n-bundle with connection|circle 3-bundles]] of the [[supergravity C-field]] is discussed towards the end of

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field|The E8 moduli 3-stack of the C-field in M-theory]]_ ([arXiv:1202.2455](http://arxiv.org/abs/1202.2455))

### Orientifold backreaction

* {#Verlinde00} [[Herman Verlinde]], _Holography and Compactification_, Nucl. Phys. B580 (2000) 264-274 ([arXiv:hep-th/9906182](https://arxiv.org/abs/hep-th/9906182))

* {#GiddingsKachruPolchinski01} [[Steven Giddings]], [[Shamit Kachru]], [[Joseph Polchinski]], _Hierarchies from Fluxes in String Compactifications_, Phys. Rev. D66:106006, 2002 ([arXiv:hep-th/0105097](https://arxiv.org/abs/hep-th/0105097))

* {#BanksBroeck06} [[Tom Banks]], K. van den Broek, _Massive IIA flux compactifications and U-dualities_, JHEP 0703:068, 2007 ([arXiv:hep-th/0611185](https://arxiv.org/abs/hep-th/0611185))

* {#CordovaLucaTomasiello19} [[Clay Cordova]], G. Bruno De Luca, [[Alessandro Tomasiello]], _New de Sitter Solutions in Ten Dimensions and Orientifold Singularities_ ([arXiv:1911.04498](https://arxiv.org/abs/1911.04498))

* {#Junghans20} Daniel Junghans, _O-plane Backreaction and Scale Separation in Type IIA Flux Vacua_ ([arXiv:2003.06274](https://arxiv.org/abs/2003.06274))

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