WISC

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### Context
#### Foundations
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[[!include foundations - contents]]
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# Contents
* table of contents
{: toc}

## Idea

The assumption that every [[set]] has a **Weakly Initial Set of Covers**, or $WISC$, is a weak form of the [[axiom of choice]].  Like the [[axiom of multiple choice]] and the axiom of [[small violations of choice]] (which both imply it), it says intuitively that "$AC$ fails to hold only in a small way" (i.e. not in a [[proper class|proper-class]] way).

## Statement (for Sets)

Precisely, $WISC$ is the statement that for any [[set]] $X$, the [[full subcategory]] $(Set/X)_{surj}$ of the [[slice category]] $Set/X$ consisting of the [[surjections]] has a [[weakly initial set]].  In other words, there is a family of surjections $\{f_i\colon P_i \twoheadrightarrow X\}_{i\in I}$ such that for any surjection $Q\twoheadrightarrow X$, there exists some $f_i$ which factors through $Q$.

## Relationships to other axioms

* WISC is implied by [[COSHEP]], since any surjection $P\twoheadrightarrow X$ such that $P$ is projective is necessarily a weakly initial (singleton) set in $(Set/X)_{surj}$.

* WISC is also implied by the [[axiom of multiple choice]] (which is in turn implied by COSHEP).  For if $X$ is in some collection family $\{D_c\}_{c\in C}$, then the family of all surjections of the form $D_c \twoheadrightarrow X$ is weakly initial in $(Set/X)_{surj}$.

* A [[ΠW-pretopos]] satisfying WISC is a _[[predicative topos]]_.

* Since [[Michael Rathjen]] proves that [[SVC]] implies [[AMC]] (at least in [[ZF]]), SVC therefore also implies WISC.

* WISC also follows from the assertion that the [[free exact completion]] of $Set$ is [[well-powered category|well-powered]], which in turn follows from assertion that $Set$ has a [[generic proof]] (so that $Set_{ex/lex}$ is a [[topos]]).  Both of these can also be regarded as saying that choice is only violated "in a small way."

* WISC implies that the category of [[anafunctors]] between any two [[small categories]] is [[essentially small category|essentially small]]; see [here](/nlab/show/anafunctor#SizeQuestions), or below.

* WISC implies (in [[ZF]]) that there exist arbitrarily large [[regular cardinals]].  Therefore, WISC is not provable in ZF, as Moti Gitik constructed a model of ZF with only one regular cardinal, using large cardinal assumptions. A proof without large cardinals was given in  ([Karagila](#Karagila14)).

### Applications

#### Local smallness of anafunctor categories

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######Proposition
WISC implies the local essential smallness of $Cat_ana$, the bicategory of categories and [[anafunctors]].
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######Proof
Let $X,Y$ be small categories and consider the category $Cat_{ana}(X,Y)$, with objects which are spans
$$
(j,f) : X \stackrel{j}{\leftarrow} X[U] \stackrel{f}{\to} Y
$$
where $X[U] \to X$ is a surjective-on-objects, [[fully faithful]] functor. The underlying map on object sets is $U \to X_0$. By WISC there is a surjection $V \to X_0$ and a map $V\to U$ over $X_0$. We can thus define a commuting triangle of functors
$$
  \array{
    X[V] &  \to  &  X[U]    \\
    & k \searrow & \downarrow j\\
    && X
  }
$$
where $X[V] \to X$ is the canonical fully faithful functor arising from $V\to X_0$ (the arrows of $X[V]$ are given by $V^2 \times_{X_0^2} X_1$). This gives rise to a transformation from $(j,f)$ to a span with left leg $k$. Thus $Cat_{ana}(X,Y)$ is equivalent to the [[full subcategory]] of anafunctors where the left leg has as object component an element of the weakly initial set of surjections. Since there is only a set of functors $X[V] \to Y$ for each $V\to X_0$, this subcategory is small.
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#### Existence of higher inductive types

[Swan](#Swan18) showed that WISC implies the existence of [[W-types with reductions]], a kind of simple [[higher inductive type]].

## In other sites - external version

Let $(C,J)$ be a [[site]] with a singleton [[Grothendieck pretopology]] $J$. It makes sense to consider a version of WISC for $(C,J)$, along the lines of the following: Let $(C/a)_{cov}$ be the full subcategory of the slice category $C/a$ consisting of the covers. WISC then states that

* For all objects $a$ of $C$, $(C/a)_{cov}$ has a weakly initial set.

This definition is called _external_ because it refers to an external category of sets. This is to be contrasted with the _internal_ version of WISC, discussed below.

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###### Example
Assuming AC for $Set$, the category $Top$ with any of its usual pretopologies satisfies \'internal WISC\'.  Consider, for instance, the pretopology in which the covers are the maps admitting local sections, i.e. those $p\colon Y\to X$ such that for any $x\in X$ there exist an open set $U\ni x$ such that $p^{-1}(U)\to U$ is split epic.  If $Set$ satisfies AC, then a weakly initial set in $Top/_{cov}X$ is given by the set of all maps $\coprod_{U\in \mathcal{U}} U \to X$ where $\mathcal{U}\subset \mathcal{P}(X)$ is an open cover of $X$.  For if $p\colon Y\to X$ admits local sections, then for each $x\in X$ we can choose an $U_x \ni x$ over which $p$ has a section, resulting in an open cover $\mathcal{U} = \{U_x \mid x\in X\}$ of $X$ for which $\coprod_{U\in \mathcal{U}} U \to X$ factors through $p$.  (If $Set$ merely satisfies WISC itself, then a more involved argument is required.)
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And now a non-example

+-- {: .num_example}
###### Example
The category of [[affine schemes]] can be equipped with the [[fpqc topology]] (so this is the [[fpqc site]] over $Spec(\mathbb{Z})$). This does not satisfy WISC. Namely, given any set of fpqc covers of $Spec(R)$, there is a surjective fpqc map which is refined by none of the given covering families ([Stacks Project Tag 0BBK](#Stacks0BBK)).
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More generally, for a non-singleton pretopology on $C$, we can reformulate WISC along the lines of \'there is a set of covering families weakly initial in the category of all covering families of any object\'.

Given a site $(C,J)$ with $J$ subcanonical, and $C$ finitely complete, we can define a (weak) 2-category $Ana(C,J)$ of internal categories, anafunctors and transformations. If WISC holds for $(C,J)$, then $Ana(C,J)$ is locally essentially small.

## In other categories - internal version

To consider an _internal_ version of WISC, which doesn't refer to an external notion of set, one needs to assume that the ambient category $C$ has a strong enough [[internal logic]], such as a pretopos (this is the context in which van den Berg and Moerdijk work). Then the ordinary statement of WISC in set can be written in the internal logic, using the [[stack semantics]], as a statement about the objects and arrows of $C$. It is in this form that WISC is useful as a replacement choice principle in intuitionistic, constructive or predicative set theory, as these are modelled on various topos-like categories (or in the case of [van den Berg and Moerdijk](#vdBM12), a [[algebraic set theory|category of classes]], although this is not necessary for the approach).

Importantly, the internal form of WISC is stable under many more categorical constructions than other forms of choice.  For instance, any [[Grothendieck topos]] or [[realizability topos]] inherits WISC from its base topos of sets (even if the latter is constructive or even predicative); see [van den Berg](#vdBerg12).  This is in contrast to nearly all other choice principles, including weaker forms such as [[COSHEP]] and [[SVC]], which fail in at least some Grothendieck toposes even when the base is a model of ZFC.

## References
 {#References}

* {#vdBM12} [[Benno van den Berg]], [[Ieke Moerdijk]], *The Axiom of Multiple Choice and Models for Constructive Set Theory*, [arXiv](http://arxiv.org/abs/1204.4045).  In this paper WISC is called the "axiom of multiple choice".

* [[Thomas Streicher]], *Realizability Models for $CZF + \neg Pow$*, [unpublished note](http://www.mathematik.tu-darmstadt.de/~streicher/CIZF/rmczfnp.pdf).  In this note WISC is called $TTCA_f$ ($TTCA$ stands for "type-theoretic collection axiom).

* [[Benno van den Berg]], *WISC is independent from ZF*, [PDF](http://www.staff.science.uu.nl/~berg0002/papers/WISC.pdf)

* {#vdBerg12} [[Benno van den Berg]], _Predicative toposes_ ([arXiv:1207.0959](http://arxiv.org/abs/1207.0959)).  In this paper WISC is called the "axiom of multiple choice".

* {#Swan18} [[Andrew Swan]], *W-Types with Reductions and the Small Object Argument*, 2018, [arxiv:1802.07588](https://arxiv.org/abs/1802.07588)

The following two papers give models of set theory (without large cardinals) in which WISC fails.

* {#Karagila14} Asaf Karagila, *Embedding Orders Into Cardinals With $DC_\kappa$*, Fund. Math. 226 (2014), 143-156, doi:[10.4064/fm226-2-4](http://dx.doi.org/10.4064/fm226-2-4), [arXiv:1212.4396](http://arxiv.org/abs/1212.4396).
{#Karagila}

* {#Roberts15} [[David Roberts]], _The weak choice principle WISC may fail in the category of sets_, [Studia Logica](http://link.springer.com/journal/11225) Volume 103 (2015) Issue 5, pp 1005-1017, doi:[10.1007/s11225-015-9603-6](http://dx.doi.org/10.1007/s11225-015-9603-6) [arXiv:1311.3074](http://arxiv.org/abs/1311.3074)

The Stacks Project shows how to construct a counterexample to WISC from any set of fpqc covers of an affine scheme.

* {#Stacks0BBK} [[The Stacks Project]], [Tag 0BBK](http://stacks.math.columbia.edu/tag/0BBK)

category: foundational axiom

[[!redirects WISC]]
[[!redirects WISCs]]
[[!redirects weakly initial set of covers]]
[[!redirects weakly initial sets of covers]]
[[!redirects axiom of weakly initial sets of covers]]

[[!redirects SOSHWIS]]


Witold Hurewicz

* [Wikipedia entry](http://en.wikipedia.org/wiki/Witold_Hurewicz)

## Selected writings

Introducing the [[Hurewicz theorem]]:

* {#Hurewicz35} [[Witold Hurewicz]], *Beiträge zur Topologie der Deformationen (I. Höherdimensionale Homotopiegruppen)*, [[Koninklijke Akademie van Wetenschappen]] **38** (1935) 112–119.  $[$[pdf](https://dwc.knaw.nl/DL/publications/PU00016667.pdf)$]$

Introducing [[compactly generated topological spaces]]:

* unrecorded lecture introducing [[compactly generated topological spaces]], Princeton 1948-49 ([communicated](compactly+generated+topological+space#fnref:2) by [[David Gale]])

Origin of the notion of [[Hurewicz fibrations]]:

* [[Witold Hurewicz]], _On the concept of fiber space_, Proc. Nat. Acad. Sci. USA __41__ (1955) 956--961; ([doi:10.1073%2Fpnas.41.11.956](https://dx.doi.org/10.1073%2Fpnas.41.11.956), [jstor:89187](https://www.jstor.org/stable/89187), [pdf](http://www.pnas.org/content/41/11/956.full.pdf). MR0073987)

## Related $n$Lab entries

* [[Hurewicz theorem]]

* [[Hurewicz cofibration]]

* higher [[homotopy groups]]

* [[long exact sequence of homotopy groups]]

category: people

[[!redirects Hurewicz]]


Witt cohomology

# Contents
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{: toc}

## Idea

Serre decided to try taking coefficients in the Witt vectors as an early attempt at a [[Weil cohomology theory]]. Ultimately, it wasn't successful for this purpose, but has been generalized in several ways for other purposes with great success.

## Sheaf of Witt vectors

Let $X$ be a [[scheme]] over a [[perfect field]] $k$ of positive [[characteristic]] $p$. Let $W$ and $W_n$ be the functors of [[Witt vectors]] and truncated Witt vectors respectively. The functorial nature allows us to define a [[sheaf]] of Witt vectors $\mathcal{W}$ and $\mathcal{W}_n$ just by taking Witt vectors of the rings of sections of $\mathcal{O}_X$.

Note that as a sheaf of sets $\mathcal{W}_n$ is just $\mathcal{O}_X^n$. The ring structure is just the addition and multiplication of the Witt vectors. The operations on the Witt vectors sheafify as well. When $n\geq m$ we have the exact sequence $0\to \mathcal{W}_m\stackrel{V}{\to} \mathcal{W}_n\stackrel{R}{\to}\mathcal{W}_{n-m}\to 0$. If we take $m=1$, then we get the sequence $0\to \mathcal{O}_X\to \mathcal{W}_n\to \mathcal{W}_{n-1}\to 0$

## Definition

The sheaf of Witt vectors is an abelian sheaf, so we just define cohomology $H^q(X, \mathcal{W}_n)$ as the standard sheaf cohomology (on the [[Zariski site]] of $X$). Let $\Lambda=W(k)$, then since $\mathcal{W}_n$ are $\Lambda$-modules annihilated by $p^n\Lambda$, we get that $H^q(X, \mathcal{W}_n)$ are also $\Lambda$-modules annihilated by $p^n\Lambda$.

In fact, all of our old operators $F$, $V$, and $R$ still act on $H^q(X, \mathcal{W}_n)$. They are easily seen to satisfy the formulas $F(\lambda w)=F(\lambda)F(w)$, $V(\lambda w)=F^{-1}(\lambda)V(w)$, and $R(\lambda w)=\lambda R(w)$ for $\lambda\in \Lambda$. If $X$ is projective then $H^q(X, \mathcal{W}_n)$ is a finite $\Lambda$-module.

What is usually referred to with Witt cohomology is $H^q(X, \mathcal{W})$ which is defined to be $lim H^q(X, \mathcal{W}_n)$. Note that even if $X$ is projective, this limit does not have to be a finite type $\Lambda$-module.

## References

* J.P. [[Serre]], Sur la topologie des variétés algébriques en caractéristique p, Symposium de Topologie Algébrique, Mexico (August, 1956)

[[!redirects Witt cohomology]]
[[!redirects Witt Cohomology]]


Witt ring

There are two different notions called *Witt ring*:

* [[ring of Witt vectors]]

* [[Witt ring of equivalence classes of bilinear forms]]


Witt ring of equivalence classes of bilinear forms

## References

* T.Y. Lam, Introduction To Quadratic Forms Over Fields, Graduate Studies in Mathematics, 2004

* M. Marshall, Abstract Witt Rings, Queens Papers in Pure and Applied Mathematics, 1980

* Baptiste Calmès, Jens Hornbostel, push-forwards for Witt groups of schemes, [
arXiv:0806.0571v2
] (http://arxiv.org/abs/0806.0571v2)


Witt vectors > history

< [[Witt vectors]]


Witt-Burnside functor

[[!redirects Witt-Burnside functors]]
##Idea

Dress and Siebeneicher ([DS88](#DS88)) generalized the [[Witt vector]] construction to the
case of pro-$p$ groups for prime integer $p$. These functors, $W_G$, were originally defined for all profinite groups, $G$, and are now called **Witt-Burnside functors** due to the fact that they generalize both the $p$-typical (recovered when $G = \mathbb{Z}_p$ as an additive group) and 'big' Witt vector construction (recovered when $G = \hat{\mathbb{Z}}$), as well as Burnside functors (recovered as $W_G(\mathbb{Z})$).

Witt-Burnside functors have been used to study equivariant ring spectra as they arise as a left adjoint for [[Tambara functors]].

##References

* {#DS88} A. Dress, C. Siebeneicher, _The Burnside ring of profinite groups and
the Witt vector construction_, Adv. Math., 70, (1988), 87–132.
*  Lance Edward Miller, _On the structure of Witt-Burnside rings attached to pro-p groups_, ([ arXiv:1103.4644](http://arxiv.org/abs/1103.4644))

[[!redirects Witt-Burnside functors]]


Witten conjecture

#Contents#
* table of contents
{:toc}

## Idea

There are many [[conjectures]] due to [[Edward Witten]]; but by _the Witten conjecture_  one usually refers to the statement proven in ([Kontsevich 92](#Kontsevich92)) with new proofs due to [Mirzakhani 07a](#Mirzakhani07a), [Mirzakhani 07b](#Mirzakhani07b) and others.

This is about a [[generating function]] for [[intersection pairing]] numbers of Mumford-Morita-Miller stable classes on the compactification of a [[moduli space of punctured curves]]. The conjecture relates them to the [[Korteweg-de Vries integrable hierarchy]]. The conjecture was formulated by [[Edward Witten]] based on the conjectured equivalence of the [[partition function]] of two models of 2-dimensional [[quantum gravity]] (i.e. [[string]] [[worldsheet]] [[field theories]]), which are not manifestly equivalent.

## References

Reviews include

* Claude Itzykson in [MR93e:32027](http://www.ams.org/mathscinet-getitem?mr=93e:32027)

* Ali Ulas Ozgur Kisisel,  _Integrable systems and Gromov-Witten theory_,   p. 135--161 in: Topics in cohomological studies of algebraic varieties,
Impanga lecture notes. Edited by Piotr Pragacz. Trends Math., Birkhäuser, Basel, 2005, [MR2006d:14066](http://www.ams.org/mathscinet-getitem?mr=2006d:14066)

* {#DijkgraafWitten18} [[Robbert Dijkgraaf]], [[Edward Witten]], _Developments in Topological Gravity_ ([arXiv:1804.03275](https://arxiv.org/abs/1804.03275))

The original proof is due to

* {#Kontsevich92} [[Maxim Kontsevich]], _Intersection theory on the moduli space of curves and the matrix Airy function_, Comm. Math. Phys. __147__ (1992), no. 1, 1--23, [euclid](http://projecteuclid.org/euclid.cmp/1104250524), [author's pdf](http://193.51.104.7/~maxim/TEXTS/intersection_theory_6.pdf)

A celebrated new proof is due to

* {#Mirzakhani07a} [[Maryam Mirzakhani]], _Simple Geodesics and Weil-Petersson Volumes Of Moduli Spaces of Bordered Riemann Surfaces_, Invent. Math. 167 (2007) 179-222

* {#Mirzakhani07b} [[Maryam Mirzakhani]], _Weil-Petersson Volumes And Intersection Theory On The Moduli Space Of Curves_, Journal of the American Mathematical Society 20 (2007) 1-23

Some of the other newer approaches to the proof:

* M. E. Kazarian, S. K. Lando, _An algebro-geometric proof of Witten's conjecture_

and also versions for higher spin

* Tyler J.Jarvis, Takashi Kimura, Arkady Vaintrob, _Moduli spaces of higher spin curves and integrable hierarchies_, Compositio Math. __126__ (2001),  no. 2, 157--212.

Gregory Naber's lectures on Witten conjecture (this is another Witten conjecture, about Seiberg-Witten invariants) can be found at his [homepage](http://www.pages.drexel.edu/~gln22).

See also

* Wikipedia, _[Witten conjecture](https://en.wikipedia.org/wiki/Witten_conjecture)_

[[!redirects Witten's conjecture]]


Witten genus

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###Context###
#### Index theory
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[[!include index theory - contents]]
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#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
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#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Witten genus_ is a [[genus]] with [[coefficients]] in [[power series]] in one variable, playing the role of a universal [[elliptic genus]].
This arises ([Witten 87](#Witten87)) as the [[large volume limit]] of the [[partition function]] of the [[superstring]] (hence in the string worldsheet [[perturbation theory]] about constant [[worldsheet]] configurations). Specifically, for the [[type II superstring]] this reproduces the universal [[elliptic genus]] as previously introduced by [[Serge Ochanine]], while for the [[heterotic string]] it yields what is now called the Witten genus proper.
Concretely, as Witten argued, this is a formal [[power series]] in string oscillation modes of the [[A-hat genus]] of the symmetric [[tensor powers]] of the [[tangent bundle]] that these modes take values in.

In ([Witten 86](#Witten86)) it is suggested, by regarding the [[superstring]] [[sigma-model]] as [[quantum mechanics]] on the [[smooth loop space]] of its [[target space]], that the Witten genus may be thought of as the [[large volume limit]] of an $S^1$-equivariant [[A-hat genus]] on [[smooth loop space]], hence the [[index]] of the [[Dirac-Ramond operator]] in that limit.  (Ever since this suggestion people have tried to make precise the concept of Dirac operator on a [[smooth loop space]] (e.g. [Alvarez-Killingback-Mangano-Windey 87](#AlvarezKillingbackManganoWindey87)). But notice that, by the above, only the [[formal loop space]] and the [[Dirac-Ramond operator]] really appears in the definition of the Witten genus.)

A priori the [[coefficients]] of the Witten genus as a genus on [[orientation|oriented]] manifolds are [[formal power series]] over the [[rational numbers]]

$$
  w \;\colon\; M SO_\bullet \longrightarrow \mathbb{Q}[ [ q ] ]
  \,.
$$

In the construction from [[string theory|string physics]] this map is interpreted as sending a [[target space|target]] [[spacetime]] $X$ of the [[superstring]] to the function $w_X(q) = w_X(e^{2 \pi i \tau})$ which to each modulus $\tau \in \mathbb{C}$ characterizing a toroidal [[Riemann surface]] assigns the [[partition function]] of the [[superstring]] with [[worldsheet]] the [[torus]] $\mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau)$ and propagating on [[target space]] $X$.

On manifolds with [[spin structure]] the genus refines to integral power series (via the integrality of the [[A-hat genus]] ([Chudnovsky-Chudnovsky 88](#ChudnovskyChudnovsky88), [Kreck-Stolz 93](#KreckStolz93), [Hovey 91](#Hovey91)). Moreover on manifolds with rational string structure it takes values in [[modular forms]] ([Zagier 86](#Zagier86)) and crucially, on manifolds with [[string structure]] it takes values in [[topological modular forms]]

$$
  \array{
     M String_\bullet &\longrightarrow&
     tmf_\bullet
       \\
     \downarrow && \downarrow
     \\
     \Omega_\bullet^{String, rat} &\longrightarrow& MF_\bullet
     \\
     \downarrow && \downarrow
     \\
     M Spin_\bullet &\longrightarrow& \mathbb{Z}[[ q ] ]
     \\
     \downarrow && \downarrow
     \\
     M SO_\bullet &\stackrel{w}{\longrightarrow}& \mathbb{Q}[ [ q ] ]
  }
  \,.
$$

(On the left is the image under forming [[Thom spectra]]/[[cobordism rings]] of the first stages in the [[Whitehead tower]] of $BO$, see also at _[[higher spin structure]]_.)

Observe here that [[topological modular forms]] are the [[coefficient]] [[ring]]
of the [[E-∞ ring]] [[spectrum]] known as _[[tmf]]_. By the general way in which [[genera]] (see there) tend to appear as [[decategorifications]] of [[homomorphisms]] of [[E-∞ rings]] out of a [[Thom spectrum]], this suggests that the Witten genus is the value on [[homotopy groups]] of a homomorphism of [[E-∞ rings]] of the form

$$
 \sigma \colon M String \longrightarrow tmf
$$

from the [[Thom spectrum]] of [[String bordism]] to the [[tmf]]-spectrum.  This lift of the Witten genus to a universal [[orientation in generalized cohomology|orientation]] in universal [[elliptic cohomology]] indeed exists and is called the _[[sigma-orientation]]_, or the _[[string orientation of tmf]]_.

This construction has been the central motivation behind the search for and construction of [[tmf]] ([Hopkins 94](#Hopkins94)). A construction of the [[string orientation of tmf]] is given in ([Ando-Hopkins-Rezk 10](#AndoHopkinsRezk)) and it is shown that indeed it refines the Witten genus ([Ando-Hopkins-Rezk 10, prop. 15.3](#AndoHopkinsRezk)).

It is maybe noteworthy that [[tmf]] (and hence its universal string orientation) also arises canonically from just studying [[chromatic homotopy theory]] (see [Mazel-Gee 13](#tmf#MazelGee13) for a nice survey of this) a fundamental topic in [[stable homotopy theory]], hence a fundamental topic in [[mathematics]]. Therefore in the Witten genus some very fundamental pure mathematics happens to equivalently incarnate as some conjecturally very fundamental [[physics]] ([[string theory]]).

## Properties

### Characteristic series
 {#CharacteristicSeries}

The [characteristic series](genus#LogarithmAndCharacteristicSeries) of the Witten genus as a [[power series]] in $z$ with [[coefficients]] in formal power series in $q$ over $\mathbb{Q}$ is

$$
  \begin{aligned}
    K_w(z)(q)
    & =
    \frac{z}{\exp_w(z)(q)}
    \\
    & =
    \frac{z}{\sigma_L(z)(q)}
    \\
    & =
    \frac{z/2}{sinh(z/2)}
    \prod_{n \geq 1}
    \frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})}
      \\
      & =
      \exp\left(
        \sum_{k \geq 2} 2G_k(q) \frac{z^k}{k!}
      \right)
   \end{aligned}
  \,,
$$

where

* $\sigma_L$ is the [[Weierstrass sigma-function]] (see e.g. [Ando Basterra 00, section 5.1](#AndoBasterra00));

* $G_k$ are the [[Eisenstein series]] ([Zagier 86, equation (14)](#Zagier86), [Ando-Hopkins-Rezk 10, prop. 10.9](#AndoHopkinsRezk10)).

This is a [[modular form]] with respect to the variable $q$, see also the the discussion below at _[Integrality and modularity](#IntegralityAndModularity)_ . Such functions which are power series of two variables $z$ and $q$ with elliptic nature in $z$ and modular nature in $q$ are called _[[Jacobi forms]]_ ([Zagier 86, p. 8](#Zagier86), [Ando-French-Ganter 08](#AndoFrenchGanter08)).

There are various further ways to equivalently re-express the above in terms of other special [[modular forms]]. Here are some:

#### In terms of Kac-Weyl characters
 {#CharacteristicSeriesInTermsOfKacWeylCharacters}

The Witten genus has a close relation to the [[Kac-Weyl character]] of [[loop group representations]].

Consider of four [[irreducible representations|irreducible]] [[level]]-1  positive energy [[Spin group|Spin]]$(2k)$-[[loop group representation]] the one denoted

$$
  \tilde S_+ - \tilde S_- \in Rep(\tilde L Spin(2k))
$$

and write its [[Kac-Weyl character]] as

$$
  \chi(\tilde S_+ - \tilde S_-) \in Rep(Spin(2k))[ [ q^{1/12} ] ]
  \,.
$$

Under passing to [[group characters]] this is ([Brylinski 90, p. 7(467)](#Brylinski90), reviewed in [KL 96, section 1.2](#KL96)) equivalently

$$
  \chi(\tilde S_+ - \tilde S_-) = \frac{\prod_{1}^k \theta}{\eta^k}
 \,,
$$

where on the right we have the [[Jacobi theta-function]] $\theta$ divided by the [[Dedekind eta-function]] $\eta$.

Comparison shows that in terms of this the exponential series of the Witten genus is equivalently (by the [[splitting principle]] the $k$-fold products are left implicit):

$$
  \exp_w = z/K_w = \eta^2 \, \chi(\tilde S_+ - \tilde S_-)
  \,.
$$

Notice that, by the relation (see [here](equivariant+elliptic+cohomology#RelationToLoopGroupRepresentations)) between [[equivariant elliptic cohomology]] and [[loop group representations]], over the complex numbers $\chi(\tilde S_+ - \tilde S_-)$ may be regarded as an element of the $Spin(2k)$-[[equivariant elliptic cohomology]] of the point (at the [[Tate curve]], see at [[twisted ad-equivariant Tate K-theory]]).

### Integrality and modularity
 {#IntegralityAndModularity}

A priori, the Witten genus has [[coefficients]] the [[power series]] ring $\mathbb{Q}[ [q] ]$ over the [[rational numbers]]. But under suitable conditions ([[quantum anomaly cancellation]]) it takes values in more interesting subrings.

#### For the type II superstring
 {#ModularityForTypeIISuperstring}

The genus obtained from the [[type II superstring]] in the NS-R sector is a [[modular form]] for the [[congruence subgroup]] $\Gamma_2(2)$. ([Witten 87a, below (13)](Witten87a)) See at _[congruence subgroup -- Relation to spin structures](level+structure+on+an+elliptic+curve#RelationToSpinStructures)_ for more.

Hence, with suitable normalization, the universal Witten-Ochanine genus takes values in the subring $MF_\bullet^{\mathbb{Q}}(\Gamma_0(2)) \hookrightarrow \mathbb{Q}[ [q] ]$ of [[modular forms]] for $\Gamma_0(2)\subset SL_2(\mathbb{Z})$ with rational coefficients ([Zagier 86, item d) on page 2](#Zagier86) based on [Chudnovsky-Chudnovsky 88](#ChudnovskyChudnovsky88)).

#### For the heterotic superstring
 {#ModularityForHeteroticString}

On manifolds with [[spin structure]] the heterotic string Witten genus has integral coeffcients, hence in the ring $\mathbb{Z}[ [ q ] ]$ ([Chudnovsky-Chudnovsky 88](#ChudnovskyChudnovsky88), [Landweber 88](#Landweber88)), see also ([Kreck-Stolz 93](#KreckStolz93), [Hovey 91](#Hovey91)).

On manifolds with rational [[string structure]] (meaning spin structure and the [[first fractional Pontryagin class]] is at most [[torsion subgroup|torsion]]), then the Witten genus takes values in actual [[modular forms]] $MF_\bullet$ ([Zagier 86, page 6](#Zagier86)).

On manifolds with actual [[string structure]], finally, the Witten genus factors through [[topological modular forms]] ([Hopkins 94](#Hopkins94), [Ando-Hopkins-Rezk 10](#AndoHopkinsRezk)).

### Relation to Dirac operators and supersymmetric QM on loop space
 {#RelationToSuSyQMOnLoopSpace}

Originally in ([Witten 87a](#Witten87a)) the elliptic genus was derived as the [[large volume limit]] of the [[index]] of the [[supercharge]] of the [[superstring]] [[worldsheet]] [[2d SCFT]]. Here the "large volume limit" is what restricts the oscillations of the string to be "small". But then in ([Witten87b](#Witten87b)) it was observed that if this supercharge -- the [[Dirac-Ramond operator]] -- would really behave like a [[Dirac operator on smooth loop space]], then the elliptic genus would be the $S^1$-equivariant [[index of a Dirac operator]], where $S^1$ acts by rigid rotationl of the parameterization of the loops, and by analogy standard formulas for equivariant indices in K-theory would imply the localization to the tangent spaces to the space of constant loops.

Notice that the would-be [[Dirac operator on smooth loop space]] is what would realize the [[superstring]] quantum dynamics as [[supersymmetric quantum mechanics]] on [[smooth loop space]]. This observation was the original motivation for the study of [[supersymmetric quantum mechanics]] in ([Witten 82](#Witten82), [Witten 85](#Witten85)) in the presence of a given [[Killing vector field]] (correspinding to the $S^1$-action on loop space ).

### Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models
 {#TwistedWittenGenus}

If the [[superstring]] in question is the [[heterotic string]] then generally there is a "[[twisted differential string-structure|twist]]" of its [[background fields]] by a [[gauge field]], hence by a $G$-[[principal bundle]] for $G$ some simply connected [[compact Lie group]] (notably [[E8]]). The partition function in this case is a "twisted Witten genus" ([Witten 87, equations (30), (31)](#Witten87), [Brylinski 90](#Brylinski90), [KL 95](#KL95)). The modularity condition then is no longer just that the [[tangent bundle]] has [[string structure]], but that together with the gauge bundle it has [[twisted string structure]], hence [[string^c 2-group|String^c]]-structure for $c$ the $G$-[[second Chern class]] (explicitly identified as such in ([Chen-Han-Zhang 10](#ChenHanZhang10)).

An elegant formulation of twisted Witten genera (and proof of their rigidity) in terms of highest weight [[loop group representations]] is given in ([KL 95](#KL95)) along the lines of ([Brylinski 90](#Brylinski90)). In ([Distler-Sharpe 07](#DistlerSharpe07)), following suggestions around ([Ando 07](#Ando07)) this is interpreted geometrically in terms of fiberwise [[index|indices]] of [[parameterized WZW models]] [[associated bundle|associated]] to the given [[string 2-group|String]]-[[principal 2-bundle]].

What should be a concrete computation of the twisted Witten genus specifically for $G =$ [[E8]] is in ([Harris 12, section 4](#Harris12)).

### As the global character of sheaves of vertex operator algebras

For $U \subset \mathbb{C}$ an [[open subset]] of the [[complex plane]] then the space $\mathcal{D}^{ch}(U)$ of [[chiral differential operators]] on $U$ is naturally a [[super vertex operator algebra]]. For $X$ a [[complex manifold]] such that its [[first Chern class]] and [[second Chern class]] vanish over the [[rational numbers]], then this assignment gives a [[sheaf of vertex operator algebras]] $\mathcal{D}^{ch}_X(-)$ on $X$. Its [[cochain cohomology]] $H^\bullet(\mathcal{D}^{ch}_X)$ is itself a [[super vertex operator algebra]] and its super-[[Kac-Weyl character]] is proportional to the [[Witten genus]] $w(X)$ of $X$:

$$
  char H^\bullet(\mathcal{D}^{ch}_X)\propto w(X)
  \,.
$$

Physically this result is understood by observing that $\mathcal{D}^{ch}_X$is the sheaf of [[quantum observables]] of the [[topological twist|topologically twisted]] [[2d (2,0)-superconformal QFT]] (see there for more on this) of which the [[Witten genus]] is (the [[large volume limit]] of) the [[partition function]].

As highlighted in ([Cheung 10, p. 2](#Cheung10)), there is a [[resolution]]  by the [[chiral Dolbeault complex]] which gives a precise sense in which over a [[complex manifold]] the Witten genus is a stringy analog of the [[Todd genus]]. See ([Cheung 10](#Cheung10)) for a brief review, where furthermore the problem of generalizing of this construction to [[sheaves of vertex operator algebras]] over more general [[string structure]] manifolds is addressed.

### Stolz conjecture

The _[[Stolz conjecture]]_ due to ([Stolz 96](#Stolz96)) asserts that if $X$ is a [[closed manifold]] with [[String structure]] which furthermore admits a [[Riemannian metric]] with [[positive number|positive]] [[Ricci curvature]], then its Witten genus vanishes.

### Relation to BPS state counting on target space
 {#RelationToBPSStateCounting}

By [[supersymmetry]] and by the same argument that controls the expression of the [[index of a Dirac operator]] in terms of [[supersymmetric quantum mechanics]], the Witten genus may be thought of as counting those string states on which the left moving [[supercharge]] acts trivially. In terms of the [[target space]] theory these are the [[BPS states]]. (reviews include [Dijkgraaf 98](#Dijkgraaf98)).

Therefore the Witten genus may also be used as a generating function for BPS state counting. As such it has for instance been used in the microscopic explanation of [[Bekenstein-Hawking entropy]] of [[black holes]], see at _[[black holes in string theory]]_.

### Relation to Cayley plane bundles

The rational Witten genus vanishes on total spaces of [[Cayley plane]]-[[fiber bundles]], and is indeed characterized by this property ([McTague 10](#McTague10), [McTague 11](#McTague11)).

## Related concepts

* [[Witten index]]

* [[Tate K-theory]]

* [[elliptic Chern character]]

* [[M5-brane elliptic genus]]

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

## References
 {#References}

[[!include elliptic cohomology -- references]]

[[!redirects Witten genera]]

[[!redirects twisted Witten genus]]
[[!redirects twisted Witten genera]]


Witten index

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#Contents#
* table of contents
{:toc}

## Idea

The _Witten index_ or _supersymmetric index_ of a [[supersymmetry|supersymmetric]] [[quantum field theory]] is its [[partition function]] with the [[trace]]-operation replaced by the [[super-trace]].

## Examples

The Witten index of the [[superstring]] is the [[Witten genus]] of the corresponding [[target space]].

## References

### General

See also

* Wikipedia, _[Witten index](#https://en.m.wikipedia.org/wiki/Witten_index)_

### For $D=2$

For the [[superstring]] [[D=2 CFT]]:

See at _[[Witten genus]]_.

### For $D=3$ $\mathcal{N}=1$ SYM

For [[D=3 N=1 super Yang-Mills theory]]:

* [[Edward Witten]], _Supersymmetric Index Of Three-Dimensional Gauge Theory_, in: [[The many faces of the superworld]]_ ([arXiv:hep-th/9903005](https://arxiv.org/abs/hep-th/9903005))

### For $D=3$ $\mathcal{N}=2$ SYM

For [[D=3 N=2 super Yang-Mills theory]]:

* [[Mathew Bullimore]], Andrea E. V. Ferrari, Heeyeon Kim, _The 3d Twisted Index and Wall-Crossing_ ([arXiv:1912.09591](https://arxiv.org/abs/1912.09591))

### For $D=3$ $\mathcal{N}=4$ SYM

For [[D=3 N=4 super Yang-Mills theory]]:

* [[Mathew Bullimore]], Andrea E.V. Ferrari, Heeyeon Kim, _Twisted Indices of 3d N=4 Gauge Theories and Enumerative Geometry of Quasi-Maps_ ([arXiv:1812.05567](https://arxiv.org/abs/1812.05567))

* [[Davide Gaiotto]], Tadashi Okazaki, _Sphere correlation functions and Verma modules_ ([arXiv:1911.11126](https://arxiv.org/abs/1911.11126))

using discussion in

* [[Andrei Okounkov]], Section 3.4 of: _Lectures on K-theoretic computations in enumerative geometry_ ([arXiv:1512.07363](https://arxiv.org/abs/1512.07363))

### For $D=4$ $\mathcal{N}=1$ SYM

For [[D=4 N=1 super Yang-Mills theory]]:

* [[Leonardo Rastelli]], Shlomo S. Razamat, _The supersymmetric index in four dimensions_, Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 44 ([arXiv:1608.02965](https://arxiv.org/abs/1608.02965))

[[!redirects Witten indices]]

[[!redirects supersymmetric index]]
[[!redirects supersymmetric indices]]


Witten's Dark Fantasy

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#Contents#
* table of contents
{:toc}

## Idea

What has been called _Witten's Dark Fantasy_ (in [Heckmann-Lawrie-Lin-Zoccarato 19, Section 8](#HeckmannLawrieLinZoccarato19)) is an argument, going back to [Witten 95a](#Witten95a), [Witten 95b, Sec. 3](#Witten95b), [Witten 00, p. 7](#Witten00), for the existence of [[non-perturbative effect|non-perturbative]] _non-_[[supersymmetry|supersymmetric]] 4d [[string vacua]]/[[string phenomenology]] with fundamentally _vanishing_ [[cosmological constant]] (i.e. vanishing "[[dark energy]]").

The original idea was formulated in terms of 3d [[M-theory on 8-manifolds]] decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]).

Based on an observation of [Vafa 96, Section 4.3](#Vafa96) that the argument should have a natural realization in 4d [[F-theory on Spin(7)-manifolds]] ([[T-duality|T-dual]] to the previous perspective), a detailed construction was finally laid out in [Bonetti-Grimm-Pugh 13](#BonettiGrimmPugh13), [Heckmann-Lawrie-Lin-Zoccarato1 18](#HeckmannLawrieLinZoccarato118), [Heckman-Lawrie-Lin-Sakstein-Zoccarato 19](#HeckmanLawrieLinSaksteinZoccarato19).

## Properties

### $1/2$-Supersymmetry

The key technical point is the claim that a careful analysis of [[D=4 N=1 supergravity]] obtained after [[KK-compactification]] of  [[F-theory on Spin(7)-manifolds]] ([[T-duality|T-dual to]] [[M-theory on Spin(7)-manifolds]]) reveals, in contrast to the [[number of supersymmetries|N=1 supersymmetry]] of [[F-theory on CY4-folds]], an "$N= 1/2$ supersymmetry", where:

1. the [[vacuum state]] is [[supersymmetry|supersymmetric]] and hence has vanishing [[cosmological constant]];

1. but no [[finite number|finite]]-[[energy]]-excitation of the vacuum appears supersymmetrically,

   hence [[fermions]] and [[bosons]] in the model do _not_ appear in supersymmetric spectra.

([Vafa 96, Sec. 4.3](#Vafa96) [BGP 13](#BonettiGrimmPugh13), [HLLZ 18, Sec. 4](#HeckmannLawrieLinZoccarato118))

### Closed spatial slices
 {#ClosedSpatialSlices}

The concrete realization of [[Witten's Dark Fantasy]] in the [[F-theory]] [[model (in theoretical physics)|model]] of [Heckmann-Lawrie-Lin-Zoccarato1 18](#HeckmannLawrieLinZoccarato118) is a [[cosmology]] where spatial slices are [[closed manifold|closed]] and in fact of the [[topology]] of the [[3-sphere]]:

> graphics from [Heckman-Lawrie-Lin-Sakstein-Zoccarato 19](#HeckmanLawrieLinSaksteinZoccarato19)

### Phenomenology
 {#Phenomenology}

When the idea of "Witten's Dark Fantasy" was proposed in [Witten 95a](#Witten95a), [Witten 95b](#Witten95b), [Witten 00](#Witten00) it was right before observation of [[red shifts]] of [[supernovae]] convinced cosmologists, in 2001, of a relatively small but [[positive number|positive]] [[cosmological constant]]. When this result became enshrined in what is now the [[standard model of cosmology]], the idea of vanishing cosmological constant in [[string theory]] fell out of favor, and a vocal sub-community instead embarked on arguing that [[de Sitter spacetime]]-[[string vacua]] with [[positive number|positive]] [[cosmological constant]] had to be searched at random in a large [[landscape of string theory vacua]].

However, debate remains over whether the apparently observed [[cosmological constant]] is actually real:

1. The authors of [KLKCR 19](#KLKCR19) claim that temporal evolution of [[supernovae]] luminosity had been underappreciated, which makes the apparent evidence for a [[positive number|positive]] [[cosmological constant]] completely go away. Earlier, [NGS 16](#NielsenGuffantiSarkar16) had pointed out that even with the established interpretation of the data, a vanishing [[cosmological constant]] is not excluded by the data.

1. Since around 2000 authors have argued that the apparent [[cosmological constant]] may be an artifact of the usual [[FRW model]]-cosmologies not taking sizeable backreaction of [[inhomogeneous cosmology|cosmic inhomogeneities]] into account. The situation with this debate currently remains open (see at _[[inhomogeneous cosmology]]_).

   While it is uncontroversial that [[inhomogeneous cosmology|cosmic inhomogeneity]] does have a measurable effect on cosmic expansion, the general current consensus seems to be that it is too small to explain all of the dark energy of the [[standard model of cosmology]]. But in view of the first item above, this would be a moot point.

> from [KLKCR 19](#KLKCR19)

In the extreme case, if re-analysis of the data, combined with effects of [[inhomogeneous cosmology|cosmic inhomogeneity]], and possibly combined with [[higher curvature corrections]] to [[gravity]] (such as control the observationally preferred [[Starobinsky model of cosmic inflation]]), would explain all of the apparently observed [[cosmological constant]], then _Witten's dark fantasy_ would again appear to be viable [[string phenomenology]].

It is then interesting to notice that also the [[closed manifold|closed]] spatial slices found in the model [above](#ClosedSpatialSlices) have recently been argued to be preferred by observational data ([VMS 19](#VMS19)).

That available experimental data supports neither a cosmological constant nor open spatial slices is further argued in [Di Valentino-Melchiorri-Silk 20](#ValentinoMelchiorriSilk20).

## Related concepts

* [[cosmology]], [[inhomogeneous cosmology]]

* [[M-theory]], [[landscape of string theory vacua]]

## References

### Theory

#### General idea

The idea in rough form goes back to

* {#Witten95a} [[Edward Witten]], _Strong coupling and the cosmological constant_, Mod. Phys. Lett. A 10:2153-2156, 1995 ([arXiv:hep-th/9506101](https://arxiv.org/abs/hep-th/9506101))

* {#Witten95b} [[Edward Witten]], Section 3 of _Some Comments On String Dynamics_, talk at [Strings95](https://cds.cern.ch/record/305869) ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))

* {#Witten00} [[Edward Witten]], p. 7 of: _The Cosmological Constant From The Viewpoint Of String Theory_, lecture at [DM2000](http://inspirehep.net/record/972507), in: David Kline (ed.) _Sources and detection of dark matter and dark energy in the universe 2000_, Springer 2001. 27-36. ([arXiv:hep-ph/0002297](https://arxiv.org/abs/hep-ph/0002297), [doi:10.1007/978-3-662-04587-9](https://link.springer.com/book/10.1007/978-3-662-04587-9))

The observation that the idea should naturally embed in [[F-theory]], namely as [[F-theory on Spin(7)-manifolds]] is due to

* {#Vafa96} [[Cumrun Vafa]], Section 4.3 of: _Evidence for F-Theory_, Nucl. Phys. B469:403-418, 1996 ([arxiv:hep-th/9602022](https://arxiv.org/abs/hep-th/9602022))

#### Detailed implementation
  {#DetailedImplementation}

A detailed implementation of the idea in [[F-theory on Spin(7)-manifolds]] is developed in:

* {#BonettiGrimmPugh13} [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))

* {#BonettiGrimmPugh13b} [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP 02 (2014) 076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

* {#HeckmannLawrieLinZoccarato118} [[Jonathan Heckman]], Craig Lawrie, Ling Lin, [[Gianluca Zoccarato]], _F-theory and Dark Energy_, Fortschritte der Physik  ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))

* {#HeckmanLawrieLinSaksteinZoccarato19} [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, [[Gianluca Zoccarato]], _Pixelated Dark Energy_, Fortsch. d. Physik ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489), [doi:10.1002/prop.201900071]( https://doi.org/10.1002/prop.201900071))

### Phenomenology
 {#ReferencesPhenomenology}

The [[standard model of cosmology]], as per 2020, with its [[positive number|positive]] [[dark energy]]-density and open spatial slices contradicts the vanishing [[cosmological constant]] and preferred [[closed manifold|closed]] ([[3-sphere|spherical]]) spatial slices of [[Witten's Dark Fantasy]].

It may very well be that [[Witten's Dark Fantasy]] is [[phenomenology|phenomenologicaly]] unviable. But it is interesting to notice that there is recent and very recent astrophysical analysis which claims problems with exactly these two aspects of the [[standard model of cosmology]]. If these contrarian authors are actually right, then [[Witten's Dark Fantasy]] is exactly the kind of model needed to match observation.

#### Dark energy or not?
 {#ReferencesDarkEnergy}

Argument that the observed type Ia [[supernovae]] are actually consistent with a vanishing [[cosmological constant]]:

* {#NielsenGuffantiSarkar16} J. T. Nielsen, A. Guffanti, [[Subir Sarkar]], _Marginal evidence for cosmic acceleration from Type Ia supernovae_, Nature Scientific Reports volume 6, Article number: 35596 (2016) ([arXiv:1506.01354](https://arxiv.org/abs/1506.01354), [web discussion](https://4gravitons.wordpress.com/2016/11/11/a-response-from-nielsen-guffanti-and-sarkar/))

* {#ColinMoymhayeeRamezSarkar19} Jacques Colin, Roya Mohayaee, [[Mohamed Rameez]], [[Subir Sarkar]], _A response to Rubin & Heitlauf: "Is the expansion of the universe accelerating? All signs still point to yes"_ ([arXiv:1912.04257](https://arxiv.org/abs/1912.04257))

Stronger argument that the observed type Ia [[supernovae]] in fact _prefer_ a vanishing [[cosmological constant]] (due to time-dependency of SN brightness that had been missed):

* {#KLKCR19} Yijung Kang, Young-Wook Lee, Young-Lo Kim, Chul Chung, Chang Hee Ree, _Early-type Host Galaxies of Type Ia Supernovae. II. Evidence for Luminosity Evolution in Supernova Cosmology_, Astrophysical Journal  ([arXiv:1912.04903](https://arxiv.org/abs/1912.04903))

  exposition:

  _[New evidence shows that the key assumption made in the discovery of dark energy is in error](https://phys.org/news/2020-01-evidence-key-assumption-discovery-dark.amp?__twitter_impression=true)_

#### Open spatial slices or not?
 {#ReferencesOpenSlices}

Arguments that the [[Planck collaboration|PLANCK satellite]] data actually prefers a [[closed manifold|closed]] spatial slices (contrary to the assumption in the current [[standard model of cosmology]]):

* {#Handley19} Will Handley, _Curvature tension: evidence for a closed universe_ ([arXiv:1908.09139](https://arxiv.org/abs/1908.09139), [spire:1751120](http://inspirehep.net/record/1751120))

* {#VMS19} Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, _Planck evidence for a closed Universe and a possible crisis for cosmology_, Nature Astronomy 2019 ([arXiv:1911.02087](https://arxiv.org/abs/1911.02087),
[doi:s41550-019-0906-9](https://www.nature.com/articles/s41550-019-0906-9))

A critique of these arguments is given in

* George Efstathiou, Steven Gratton, _The evidence for a spatially flat Universe_ ([arXiv:2002.06892](https://arxiv.org/abs/2002.06892))

but this critique again rests on just the combination [[Planck collaboration]] & baryon acoustic peak (BAO) & [[supernova]]-data which the above references argue cannot sensibly be combined.

From the abstract of [Handley 19](#Handley19):

> The curvature parameter tension between Planck 2018, cosmic microwave background lensing, and baryon acoustic oscillation data is measured using the suspiciousness statistic to be 2.5 to 3σ. Conclusions regarding the spatial curvature of the universe which stem from the combination of these data should therefore be viewed with suspicion. Without CMB lensing or BAO, Planck 2018 has a moderate preference for closed universes, with Bayesian betting odds of over 50:1 against a flat universe, and over 2000:1 against an open universe.

#### Both or neither?

Combined argument that experimental data disfavours both a [[cosmological constant]] (favoring instead effective [[phantom dark energy]]) as well as an open universe:

* {#ValentinoMelchiorriSilk20} Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, _Cosmic Discordance: Planck and luminosity distance data exclude LCDM_ ([arXiv:2003.04935](https://arxiv.org/abs/2003.04935))

[[!redirects Witten's dark fantasy]]

[[!redirects Witten's Dark Phantasy]]


WKB method

## Idea

A method for [[semiclassical approximation]]...

## References

A general survey is in

* Wikipedia, _[WKB approximation](http://en.wikipedia.org/wiki/WKB_approximation)_

An introduction to the formulation of the WKB method in [[symplectic geometry]]/[[geometric quantization]] is in the first section of

* Sean Bates, [[Alan Weinstein]], _Lectures on the geometry of quantization_, [pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)

[[!redirects WKB approximation]]


Wlodzimierz M. Tulczyjew

* [Research Gate page](https://www.researchgate.net/scientific-contributions/Wlodzimierz-M-Tulczyjew-77896531)

## Selected writings

On [[symplectic manifold|symplectic]] [[phase spaces]] of [[classical field theories]]:

* [[Jerzy Kijowski]], [[Wlodzimierz M. Tulczyjew]], *A Symplectic Framework for Field Theories*, Lecture Notes in Physics **107** (1979, 2005) [[doi:10.1007/3-540-09538-1](https://doi.org/10.1007/3-540-09538-1)]

category: people

[[!redirects Wlodzimierz Tulczyjew]]


Wojciech Chachólski

[[!redirects Wojciech Chacholski]]

* [webpage](http://www.math.kth.se/~wojtek/)

## Selected writings

On [[homotopy pullbacks]]:

* {#CPS05} [[Wojciech Chacholski]], Wolfgang Pitsch, and [[Jerome Scherer]], _Homotopy pullback squares up to localization_, in [[Dominique Arlettaz]], [[Kathryn Hess]] (eds.) _An Alpine Anthology of Homotopy Theory_ ([arXiv:math/0501250](http://arxiv.org/abs/math/0501250), [pdf](http://sma.epfl.ch/~jscherer/articles/hopullbacks2.pdf), [doi:10.1090/conm/399](http://dx.doi.org/10.1090/conm/399))

On [[posets]] in [[persistent homology theory]]:

* [[Wojciech Chacholski]], Alvin Jin, Francesca Tombari,, *Realisations of posets and tameness* $[$[arXiv:2112.12209](https://arxiv.org/abs/2112.12209), [talk](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#ChacholskiTalkAtHomotopicalPerspectivesOnTDA2022)$]$

On [[topological data analysis]]:

* Jens Agerberg, [[Wojciech Chacholski]], Ryan Ramanujam, *Data, geometry and homology* $[$[arXiv:2203.08306](https://arxiv.org/abs/2203.08306)$]$

On [[persistent homology]]:

* [[Wojciech Chachólski]], Alessandro De Gregorio, Nicola Quercioli, Francesca Tombari: *Symmetries of data sets and functoriality of persistent homology*, Theory and Applications of Categories **39** 23 (2023) 667-686 [[tac:39/23](http://www.tac.mta.ca/tac/volumes/39/23/39-23abs.html)]

## Related entries

* [[homotopy theory]]

* [[homotopy pullback]]

* [[topological data analysis]]

category:people


Wojciech De Roeck

* [GoogleScholar page](https://scholar.google.be/citations?user=72_fCE0AAAAJ&hl=en)

## Selected writings

On the [[quantum adiabatic theorem]] in [[condensed matter theory]] with an eye towards [[topological phases of matter]]:

* [[Sven Bachmann]], Spyridon Michalakis, [[Bruno Nachtergaele]], Robert Sims, *Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems*, Commun. Math. Phys. **309** (2012) 835-871 ([arXiv:1102.0842](https://arxiv.org/abs/1102.0842), [doi:10.1007/s00220-011-1380-0](https://doi.org/10.1007/s00220-011-1380-0))

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *Adiabatic Theorem for Quantum Spin Systems*, Phys. Rev. Lett. **119** (2017) 060201 ([doi:10.1103/PhysRevLett.119.060201](https://doi.org/10.1103/PhysRevLett.119.060201))

* [[Sven Bachmann]], [[Wojciech De Roeck]], [[Martin Fraas]], *The adiabatic theorem in a quantum many-body setting*, in: *Analytic Trends in Mathematical Physics*, Contemporary Mathematics **741** (2020) 43-58 ([arXiv:1808.09985](https://arxiv.org/abs/1808.09985), [ISBN:978-1-4704-5388-6](https://bookstore.ams.org/conm-741))

category: people


Wojciech Zurek

* [Wikipedia entry](www.winedt.com/registration.html)

## Selected writings

Introducing the [[no-cloning theorem]] in [[quantum information theory]]:

* [[William Wooters]], [[Wojciech Zurek]], *A single quantum cannot be cloned*, Nature **299** (1982) 802-803 [[doi:10.1038/299802a0](https://doi.org/10.1038/299802a0)]

Mathematical argument that [[quantum computation]] with [[quantum error correction]] is practically feasible:

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojciech H. Zurek]], *Resilient Quantum Computation: Error Models and Thresholds*, Proceedings of the Royal Society A **454** 1969 (1998) [[arXiv:quant-ph/9702058](https://arxiv.org/abs/quant-ph/9702058)]

Realization of [[quantum error correction]] in [[experiment]]:

* D. G. Cory, M. D. Price, W. Maas, [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojchiek H. Zurek]], T. F. Havel, and S. S. Somaroo, *Experimental Quantum Error Correction*, Phys. Rev. Lett. 81, 2152 (1998) ([doi:10.1103/PhysRevLett.81.2152](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.2152))

On [[decoherence]] in [[quantum  physics]] and the [[interpretation of quantum mechanics]]:

* [[Wojciech Zurek]], *Decoherence, einselection, and the quantum origins of the classical*, Rev. Mod. Phys. **75** (2003) 715-775 [[quant-ph/0105127](http://arxiv.org/abs/quant-ph/0105127), [doi:10.1103/RevModPhys.75.715](http://dx.doi.org/10.1103/RevModPhys.75.715)]

## Related $n$Lab entries

* [[no-cloning theorem]]

* [[einselection]]

category: people

[[!redirects Wojchiek H. Zurek]]
[[!redirects Wojciech H. Zurek]]


Wolf space

+-- {: .rightHandSide}
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#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[space]] which is both a [[quaternion-Kähler manifold]] as well as a [[symmetric space]]. Also known as a _Wolf space_.

## Examples

* [[quaternionic projective plane]]

## Properties

### As positive quaternion-Kähler manifolds

> under construction

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

Every [[Wolf space]] is a [[positive quaternion-Kähler manifold]].

=--

In fact the [[Wolf spaces]] are the only known examples  of [[positive quaternion-Kähler manifold]] (which is not hyper-Kähler ?!), as of today (e.g. [Salamon 82, Section 5](#Salamon82)).

This leads to the **conjecture** that un every dimension, the [[Wolf spaces]] are the only [[positive quaternion-Kähler manifolds]].

The conjecture has been proven for the following [[dimensions]]

* $d = 4$ (Hitchin)

* $d = 8 $ ([Poon-Salamon 91](#PoonSalamon91), [LeBrun-Salamon 94](#LeBrunSalamon94))

## References

* [[Joseph K. Wolf]], _Complex homogeneous contact manifolds and quaternionic symmetric spaces_, Journal of Math. and Mech., 14 (1965), p. 166 ([jstor:24901319](https://www.jstor.org/stable/24901319))

* {#Salamon82} [[Simon Salamon]], _Quaternionic Kähler manifolds_, Invent Math (1982) 67: 143. ([doi:10.1007/BF01393378](https://doi.org/10.1007/BF01393378))

* {#PoonSalamon91} Y. S. Poon, [[Simon Salamon]], _Quaternionic Kähler 8-manifolds with positive scalar curvature_, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 ([euclid:1214446322](https://projecteuclid.org/euclid.jdg/1214446322))

* {#LeBrunSalamon94} Claude LeBrun, [[Simon Salamon]], _Strong rigidity of positive quaternion Kähler manifolds_, Inventiones Mathematicae 118, 1994, 109–132 ([dml:144231](https://eudml.org/doc/144231), [doi:10.1007/BF01231528](https://doi.org/10.1007/BF01231528))

* {#Amann09} Amann, _Positive Quaternion Kähler Manifolds_, 2009 ([pdf](https://d-nb.info/996176438/34))

See also

* Wikipedia, _[Quaternion-Kähler symmetric space](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space)_

[[!redirects Wolf spaces]]

[[!redirects quaternion-Kähler symmetric space]]
[[!redirects quaternion-Kähler symmetric spaces]]

[[!redirects quaternion-Kaehler symmetric space]]
[[!redirects quaternion-Kaehler symmetric spaces]]

[[!redirects quaternion Kähler symmetric space]]
[[!redirects quaternion Kähler symmetric spaces]]

[[!redirects quaternion Kaehler symmetric space]]
[[!redirects quaternion Kaehler symmetric spaces]]


Wolfgang Altmannshofer

* [webpage](https://www.perimeterinstitute.ca/people/wolfgang-altmannshofer)

## Selected writints

On the [[flavour problem]] and [[flavour anomalies]]:

* [[Wolfgang Altmannshofer]], _The Flavor Puzzle_, Aspen 2014 ([pdf](https://www.aspenphys.org/science/colloquia/2014_slides/altmannshofer.pdf), [[AltmannshoferFlavorProblem.pdf:file]])

On [[flavour anomalies]]:

* Jason Aebischer, [[Wolfgang Altmannshofer]], Diego Guadagnoli, Meril Reboud, [[Peter Stangl]], David M. Straub, _B-decay discrepancies after Moriond 2019_ ([arXiv:1903.10434](https://arxiv.org/abs/1903.10434))

* {#AltmannshoferStangl21} [[Wolfgang Altmannshofer]], [[Peter Stangl]], _New Physics in Rare B Decays after Moriond 2021_ ([arXiv:2103.13370](https://arxiv.org/abs/2103.13370))

On [[Z' bosons]] in view of the [[flavour anomalies]]:

* [[Wolfgang Altmannshofer]], Joe Davighi, Marco Nardecchia, _Gauging the accidental symmetries of the Standard Model, and implications for the flavour anomalies_ ([arXiv:1909.02021](https://arxiv.org/abs/1909.02021))

category: people


Wolfgang Dür

* [Institute page](https://www.uibk.ac.at/th-physik/staff/duer/)

* [GoogleScholar page](https://scholar.google.at/citations?user=YZZSOsoAAAAJ&hl=en)

## Selected writings

On [[measurement-based quantum computation]]:

* [[Hans J. Briegel]], [[Dan E. Browne]], [[Wolfgang Dür]], [[Robert Raussendorf]], [[Maarten Van den Nest]], *Measurement-based quantum computation*, Nature Physics **5** 1 (2009) 19-26 [[arXiv:0910.1116](https://arxiv.org/abs/0910.1116), [doi:10.1038/nphys1157](https://doi.org/10.1038/nphys1157)]

category: people


Wolfgang L. Reiter

* [ResearchGate page](https://www.researchgate.net/profile/Wolfgang-Reiter-3)

## Selected writings

On [[experiment|experimental]] [[quantum physics]] and its [[epistemology|epistemological]] [[interpretation of quantum mechanics|interpretation]]:

* [[Daniel Greenberger]], [[Wolfgang L. Reiter]], [[Anton Zeilinger]], *Epistemological and Experimental Perspectives on Quantum Physics*, Vienna Circle Institute Yearbook (VCIY) **7** (1999) [[doi:10.1007/978-94-017-1454-9](https://doi.org/10.1007/978-94-017-1454-9)]

category: people

[[!redirects Wolfgang Reiter]]


Wolfgang Lerche

* [webpage](http://ph-dep-th.web.cern.ch/ph-dep-th/?site=php/viewprofile.php&id=73)

## Selected writings

Relation of [[Niemeier lattices]] to [[heterotic string theory]]:

* [[Wolfgang Lerche]], [[Dieter Lüst]], [[Adrian Norbert Schellekens]], *Ten-dimensional heterotic strings from Niemeier lattices*, Physics Letters B **181** 1–2 (1986) 71-75 [doi:10.1016/0370-2693(86)91257-8, [inspire:233203](https://inspirehep.net/literature/233203)]

On the [[partition function]] of the [[superstring]] ([[heterotic string theory|heterotic string]] and [[type II string theory|type II string]]) as a [[modular form]] with values in the [[Chern character]] of the [[background field|background]] [[field strengths]] ("character-valued partition function", then also called the *[[elliptic genus]]*/*[[Witten genus]]*) and relation to [[Green-Schwarz anomaly cancellation]]:

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], *Heterotic string-loop calculation of the anomaly cancelling term*, Nuclear Physics B Volume 289, 1987, Pages 609-627 (doi:10.1016/0550-3213(87)90397-X)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]],  [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)

On the [[duality between F-theory and heterotic string theory]]:

* {#Lerche99} [[Wolfgang Lerche]], _On the Heterotic/F-Theory Duality in Eight Dimensions_, In: Baulieu L., Green M., Picco M., Windey P. (eds.) _Progress in String Theory and M-Theory_, NATO Science Series (Series C: Mathematical and Physical Sciences), vol 564. Springer 2001 ([arXiv:hep-th/9910207](https://arxiv.org/abs/hep-th/9910207), [doi:10.1007/978-94-010-0852-5_2](https://doi.org/10.1007/978-94-010-0852-5_2))

On [[D-branes]] at an [[ADE-singularity]] and their [[open string]] [[worldsheet]] [[2d CFT]]s (see also [here](ADE+singularity#ReferencesTypeIIStringsOnADE)):

* [[Wolfgang Lerche]], [[Carsten Andrew Lütken]], [[Christoph Schweigert]], *D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories*, Nucl.Phys. B **622** (2002) 269-278  ([doi:10.1016/S0550-3213%2801%2900613-7](https://doi.org/10.1016/S0550-3213%2801%2900613-7), [arXiv:hep-th/0006247](https://arxiv.org/abs/hep-th/0006247))

On how this $\widehat{\mathfrak{su}(2)}^{\kappa-2}$-CFT encodes the [[BPS states]] of [[special unitary group|$SU(\kappa)$]]-[[SYM]] on [[D3-branes]] transverse to the singularity:

* [[Wolfgang Lerche]], *On a Boundary CFT Description of Nonperturbative $\mathcal{N} = 2$ Yang-Mills Theory* ([arXiv:hep-th/0006100](https://arxiv.org/abs/hep-th/0006100))

category: people


Wolfgang Lueck

* [webpage](http://www.him.uni-bonn.de/lueck/)

* [webpage at Hausdorff Centre](https://www.hcm.uni-bonn.de/people/profile/wolfgang-lueck/)

## Selected writings

On [[equivariant homotopy theory]] and [[equivariant algebraic K-theory]]:

* {#Lueck89} [[Wolfgang Lück]], _Transformation Groups and Algebraic K-Theory_, Lecture Notes in Mathematics **1408** (Springer 1989) ([doi:10.1007/BFb0083681](https://doi.org/10.1007/BFb0083681))

On [[equivariant K-theory]] as a [[Spectra]]-valued [[presheaf]] on the [[orbit category]] (a [[naive G-spectrum]]):

* [[James Davis]], [[Wolfgang Lück]], _Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory_, K-Theory 15:201–252, 1998 ([pdf](https://jfdmath.sitehost.iu.edu/teaching/m721F19/assembly.pdf))

On [[equivariant principal bundles]] and their [[classifying spaces]]:

* [[Wolfgang Lück]], *Survey on Classifying Spaces for Families of Subgroups*,  In: *Infinite Groups: Geometric, Combinatorial and Dynamical Aspects* Progress in Mathematics, **248** Birkhäuser (2005) ([arXiv:math/0312378](https://arxiv.org/abs/math/0312378), [doi:10.1007/3-7643-7447-0_7](https://doi.org/10.1007/3-7643-7447-0_7))

* [[Bernardo Uribe]], [[Wolfgang Lück]], _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

On [[Euler characteristics]] of [[categories]] and [[homotopy colimits]]:

* [[Thomas M. Fiore]], [[Wolfgang Lück]], [[Roman Sauer]], *Euler Characteristics of Categories and Homotopy Colimits* [[arXiv:1007.3868](http://arxiv.org/abs/1007.3868)]

On [[proper equivariant homotopy theory]]:

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], _Proper equivariant stable homotopy theory_ ([arXiv:1908.00779](https://arxiv.org/abs/1908.00779))

On the [[equivariant Chern character]] in [[equivariant K-theory]]:

* [[Wolfgang Lück]], [[Bob Oliver]], _Chern characters for the equivariant K-theory of proper G-CW-complexes_, In: Aguadé J., Broto C., Casacuberta C. (eds.) _Cohomological Methods in Homotopy Theory_ Progress in Mathematics, vol 196. Birkhäuser 2001 ([doi:10.1007/978-3-0348-8312-2_15](https://doi.org/10.1007/978-3-0348-8312-2_15))

On the [[equivariant Chern-Dold character]] in generalized [[equivariant cohomology]]:

* [[Wolfgang Lück]], _Chern characters for proper equivariant homology theories and applications to K- and L-theory_, Journal für die reine und angewandte Mathematik, Volume 2002: Issue 543 ([doi:10.1515/crll.2002.015](https://doi.org/10.1515/crll.2002.015), [pdf](https://www.him.uni-bonn.de/lueck/data/eh.pdf))

* [[Wolfgang Lück]], _Equivariant Cohomological Chern Characters_, International Journal of Algebra and Computation, Vol. 15, No. 05n06, pp. 1025-1052 (2005) ([arXiv:math/0401047](https://arxiv.org/abs/math/0401047), [doi:10.1142/S0218196705002773](https://doi.org/10.1142/S0218196705002773))

* [[Wolfgang Lück]], _Equivariant Chern characters_, 2006 ([pdf](https://www.him.uni-bonn.de/lueck/data/goettingen_equi_Chern061130trans.pdf), [[LuckEquivariantChernCharacters.pdf:file]])

On [[equivariant stable cohomotopy]] and the [[Burnside ring]]:

* {#Lueck05} [[Wolfgang Lück]], _The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups_, Pure Appl. Math. Q.  1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 479--541 ([arXiv:math/0504051](https://arxiv.org/abs/math/0504051))

## Related entries

* [[global equivariant stable homotopy theory]]

* [[equivariant stable cohomotopy]]

* [[Atiyah-Hirzebruch spectral sequence]]

* [[smash product]]

category: people

[[!redirects Wolfgang Lück]]


Wolfgang Pauli

* [Wikipedia entry](http://en.wikipedia.org/wiki/Wolfgang_Pauli)

## Selected writings

Introducing the [[Pauli exclusion principle]]

* {#Pauli25} [[Wolfgang Pauli]], *Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren*, Zeitschrift für Physik, **31** 1 (1925) 765-783 [[doi:10.1007/BF02980631](https://doi.org/10.1007/BF02980631)]

and the [[Born-Pauli rule]]:

* [[Wolfgang Pauli]], *Über Gasentartung und Paramagnetismus*, Zeitschrift für Physik: A Hadrons and nuclei **41** (1927) 81–102 [[doi:10.1007/BF01391920](https://doi.org/10.1007/BF01391920)]

## Concepts

* [[Pauli exclusion principle]]

category: people


Wolfgang Polak

* [GoogleScholar page](https://scholar.google.com/citations?user=91RtF7oAAAAJ&hl=en)

## Selected writings

On [[quantum computing]]:

* [[Eleanor Rieffel]], [[Wolfgang Polak]],  *An Introduction to Quantum Computing for Non-Physicists*, ACM Comput. Surveys **32** (2000) 300-335 [[arXiv:quant-ph/9809016](https://arxiv.org/abs/quant-ph/9809016), [doi:10.1145/367701.367709](https://dl.acm.org/doi/10.1145/367701.367709)]

* [[Eleanor Rieffel]], [[Wolfgang Polak]], *Quantum Computing -- A gentle introduction*, MIT Press (2011) [[ISBN:9780262526678](https://mitpress.mit.edu/9780262526678/quantum-computing/), [pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608150244118.pdf)]

category: people


Wolfgang Rindler

* [Wikipedia entry](https://en.wikipedia.org/wiki/Wolfgang_Rindler)

## related $n$Lab entries

* [[Lorentzian manifold]]

* [[spinor]]

category: people


Wolfgang Rump

Wolfgang Rump is a mathematician working at the University of Stuttgart.

He got his PhD degree in 1978 from the University of Stuttgart, advised by Klaus Wilhelm Roggenkamp.

## Selected writings

On [[quasi-abelian categories]]:

* [[Wolfgang Rump]], _Almost abelian categories_, Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 3 (2001) 163--225 [[numdam:CTGDC_2001__42_3_163_0](http://www.numdam.org/item?id=CTGDC_2001__42_3_163_0)]

On [[brace]]s, [[$L$-algebra]]s and related algebraic structures:

* W. Rump, _Braces, radical rings, and the quantum Yang-Baxter equation_, J. Algebra __307__ (2007), no. 1, 153--170
* Wolfgang Rump, _Generalized radical rings, unknotted biquandles, and quantum groups_, Colloquium Mathematicum __109__ (2007), 85-100 [doi](https://doi.org/10.4064/cm109-1-7)
* W. Rump, _The brace of a classical group_, Note di Matematica __34__ (2014) n.1, 115--144 [doi](https://doi.org/10.1285/i15900932v34n1p115)
* W. Rump, _Construction of finite braces_, Ann. Comb. 23, 391--416 (2019) [doi](https://doi.org/10.1007/s00026-019-00430-1)
* W. Rump, _The category of $L$-algebras_, Theory App. Cat. __39__ (2023), n. 21, 598--624 [pdf](http://www.tac.mta.ca/tac/volumes/39/21/39-21.pdf)

category: people


Wolfgang Soergel

[[!redirects Wolgang Soergel]]

* [webpage](http://home.mathematik.uni-freiburg.de/soergel/)

category: people


Wolfgang Steinicke

* [webpage](http://www.klima-luft.de/steinicke/)

## related $n$Lab entries

* [[gravitational waves]]

category: people


Wolfgang Ziller

* [webpage](https://www.math.upenn.edu/~wziller/)

## Selected writings

On [[Hopf fibrations]] and notably the [[quaternionic Hopf fibration]]:

* {#GluckWarnerZiller86} [[Herman Gluck]], [[Frank Warner]], [[Wolfgang Ziller]], _The geometry of the Hopf fibrations_, L'Enseignement Mathématique, **32** (1986), 173-198 [[ResearchGate](https://www.researchgate.net/publication/266548925_The_geometry_of_the_Hopf_fibrations), [[GluckWarnerZiller-HopfFibrations.pdf:file]]]

On [[equivariant differential topology]]:

* {#Ziller13} [[Wolfgang Ziller]], _Group actions_, 2013 ([pdf](https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf), [[ZillerGroupActions.pdf:file]])

category: people


Wolfhart Zimmermann

* [Wikipedia entry](https://en.wikipedia.org/wiki/Wolfhart_Zimmermann)

## Selected writings

Introducing the [[LSZ reduction formula]]:

* {#LehmannSymanzikZimmermann55} [[Harry Lehmann]], [[Kurt Symanzik]], [[Wolfhart Zimmermann]], _Zur Formulierung quantisierter Feldtheorien_, Nuovo Cimento 1(1), 205 (1955) ([doi:10.1007/BF02731765](https://doi.org/10.1007/BF02731765))

## Related $n$Lab entries

* [[retarded product]]

* [[LSZ reduction formula]]

category: people


Wolfram Schwabhäuser

* [webpage](https://de.wikipedia.org/wiki/Wolfram_Schwabhäuser)

## related $n$Lab entries

* [[Euclidean geometry]], [[synthetic geometry]]

category: people


Wolfram Weise

* [webpage](https://www.professoren.tum.de/en/weise-wolfram)

## Selected writings

On the equivalence between [[hidden local symmetry]]- and [[massive Yang-Mills theory]]-description of [[Skyrmion]] [[quantum hadrodynamics]]:

* [[Atsushi Hosaka]], H. Toki, [[Wolfram Weise]], _Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 ([spire:24079](http://inspirehep.net/record/24079))

On [[nucleons]] as [[Skyrmions]] in [[quantum hadrodynamics]] with [[pions]], [[omega-mesons]] and [[rho-mesons]] ($\pi$-$\rho$-$\omega$-model):

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], [[Wolfram Weise]], _Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties_, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], Andreas Wirzba, [[Wolfram Weise]], _Skyrmions with $\rho$ and $\omega$ Mesons as Dynamical Gauge Bosons_, Phys. Rev. Lett. 57, 1676 (1986) ([doi:10.1103/PhysRevLett.57.1676](https://doi.org/10.1103/PhysRevLett.57.1676))

On [[vector meson dominance]]:

* {#PillerWeise90} G. Piller, [[Wolfram Weise]], _Vector meson dominance: Selected topics_ 1990 ([spire310958](https://inspirehep.net/literature/310958), [[PillerWeiseVMD.pdf:file]])

category: people


wonderful compactification

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#Contents#
* table of contents
{:toc}

## Idea

A certain kind of [[compactification]] of a [[variety]] used for [[resolution of singularities]].

## Related concepts

* [[Fulton-MacPherson compactification]]

## References

See also

* Wikipedia _[Wonderful compactification](https://en.wikipedia.org/wiki/Wonderful_compactification)_

In generalization of [[Fulton-MacPherson compactifications]] of [[configuration spaces of points]]:

* {#deConciniProcesi95} [[Corrado de Concini]], [[Claudio Procesi]], _Wonderful models of subspace arrangements_, Selecta Mathematica, New Series Vol. 1, No. 3, 1995 ([pdf](https://link.springer.com/content/pdf/10.1007/BF01589496.pdf))

see also

* {#Feichtner05} Eva Maria Feichner, _De Concini–Procesi Wonderful Arrangement Models:  A Discrete Geometer’s Point of View_, Math. Sci. Res. Inst. Publ 52, 2005 ([pdf](http://library.msri.org/books/Book52/files/17feich.pdf))

In relation to [[Feynman amplitudes on compactified configuration spaces of points]]:

* [[Marko Berghoff]], _Wonderful compactifications in quantum field theory_,  Communications in Number Theory and Physics Volume 9 (2015) Number 3 ([arXiv:1411.5583](https://arxiv.org/abs/1411.5583))

[[!redirects wonderful compactifications]]


word

\tableofcontents

\section{Introduction}

A _word_ in the [[elements]] of a [[set]] is, roughly speaking, a concatenation of elements of that set. To make this precise, one typically uses the machinery of [[free object|free algebraic structures]]. One may allow in the concatenation certain canonical elements of the free algebraic gadget constructed from the elements of the set one started with, such as inverses.

By extension, one may also refer to elements of any algebraic structure, at least when described in terms of generators and relations (i.e. explicitly as a quotient of a free algebraic structure), as words.

\section{Prototypical example}

The prototypical algebraic structure with which to make sense of the notion of a word is that of a [[monoid]]. If 'word' is used in a context where the intended algebraic structure is not made clear, use of [[free monoids]] is likely intended.

\begin{defn} Let $X$ be a set. A _word_ in the elements of $X$ is an element of the [[free monoid]] on $X$. \end{defn}

\begin{rmk} A free monoid has in particular an identity element, which is the _empty word_. \end{rmk}

\begin{rmk} We do not assume commutativity. \end{rmk}

\begin{example} Let $X = \{a, b\}$ be a set. Examples of words in $X$ are the empty word, $a$, $b$, $a b$, $a^{5}$, $a b^{3}$, $a b a b a b$, $b^{3}a^{2}b^{5}$ and so on. \end{example}

\section{In groups}

Another common case is that in which the algebraic structure is that of groups.

\begin{defn} Let $X$ be a set. A _word_ in the free group on the elements of $X$ is an element of the [[free group]] on $X$. \end{defn}

\begin{rmk} As for monoids, a free group has in particular an identity element, which is the _empty word_. \end{rmk}

\begin{rmk} As for monoids, we do not assume commutativity. \end{rmk}

\begin{example} Let $X = \{a, b\}$ be a set. Examples of words in the free group on $X$ are the empty word, $a$, $b$, $a b$, $a^{5}$, $a^{-5}$, $a b^{3}$, $a b a b a b$, $a^{-1}b^{-1}a^{-1}b^{-1}a^{-1}b^{-1}$, $a b^{-1}$, $b^{3}a^{2}b^{5}$, $a^{-3}b^{2}a^{7}b^{-2}$, and so on. \end{example}

## Related concepts

* [[free monoid]]

* [[string (computer science)]]

* [[syntax]]

* [[language]]

[[!redirects words]]


world sheets for world sheets

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#Contents#
* table of contents
{:toc}

## Idea

A mechanism in the context of [[string theory]]:

before [[gauge fixing]], the [[worldsheet]] [[QFT]] describing the first-quantized [[string]] (see also at _[What is a particle?](particle#WhatIsAParticle)_ for conceptual background is a [[2d quantum gravity]] coupled to [[scalar fields]] (the embedding fields that say how the string sits in [[target space|target]] [[spacetime]]). Being [[quantum gravity]], even if in the comparatively extremely simple case of only two dimensions, one may ask if this arises as the [[effective field theory]]/[[second quantization]] of _another_ kind of string, one whose target spacetime is the [[worldsheet]] of the original string.

This idea is known as "world sheets for world sheets", going back to ([Green87](#Green87)).

This may be compared to how the [[worldvolume]] theory of the [[topological membrane]] which is a [[Chern-Simons theory]], arises as the second quantization of [[topological strings]] ([[A-model]]) for which that 3d worldvolume of the membrane is their [[target space]] ([Witten 92](#Witten92)), see at _[TCFT – Worldsheet and effective background theories](TCFT#ActionFunctionals)_ for more on this.

## References

* {#Green87} [[Michael Green]], _World Sheets for World Sheets_, Nucl.Phys. B293 (1987) 593-611

  **Abstract** In this paper I suggest that the two-dimensional world sheet underlying the dynamics of relativistic strings may itself emerge from an approximation to a two-dimensional string theory. Coordinates that represent space-time and internal symmetries should emerge as wave functions of the massless states of that theory. This is motivated by studying various string theories with two-dimensional target spaces and internal symmetry which are related to generalized nonlinear sigma models.

See also

* [[Paul Ginsparg]], [[Gregory Moore]], _Lectures on 2D gravity and 2D string theory (TASI 1992)_ ([arXiv:hep-th/9304011](http://arxiv.org/abs/hep--th/9304011))

* [[Edward Witten]], _Chern-Simons Gauge Theory As A String Theory_, Prog.Math. 133 (1995) 637-678 ([arXiv:hep-th/9207094](http://arxiv.org/abs/hep-th/9207094))
 {#Witten92}

*  Eliezer Rabinovici, _World sheets for world sheets revisited_ ([talk](http://pirsa.org/09070008/), [slides](http://pirsa.org/pdf/files/91367ab3-1691-4b7d-a5d8-10bcae327dc0.pdf))


worldline formalism

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## Idea

Traditionally, the [[Feynman perturbation series]]/[[scattering amplitudes]] in [[perturbative quantum field theory]] are defined -- given a [[free field theory|free field]] [[vacuum]] and a [[local observable|local]] [[interaction]] [[action functional]] -- by applying the _[[Feynman rules]]_ ([this prop.](A+first+idea+of+quantum+field+theory#FeynmanPerturbationSeriesAwayFromCoincidingPoints)) to the monomial terms in the [[interaction]] [[Lagrangian density]] and deriving from that *[[Feynman rules]]* for how to weigh each [[Feynman diagram]] by a [[probability amplitude]] -- its _[[Feynman amplitude]]_ -- subject to [[renormalization]] choices.

In contrast, in what is called the _worldline formalism_ of [[perturbative quantum field theory]] this assignment is obtained instead more conceptually as the [[correlators]]/[[n-point functions]] of a 1-dimensional QFT that lives _on_ the Feynman graphs, namely the [[worldline]] theory (usually a [[sigma-model]] into the given [[target|target]] [[spacetime]]), which may be thought of as being the ([[relativistic particle|relativistic]]) [[quantum mechanics]] of the [[particles]] that are the quanta of the [[field (physics)|fields]] in the field theory.

One may think of this as realizing the intuition that the [[edges]] in a [[Feynman diagram]] correspond to worldlines of [[virtual particles]].

Mathematically, the key step here is a [[Mellin transform]] -- introducing a "[[Schwinger parameter]]" -- which realizes the [[Feynman propagator]] $\Delta_F(x,y)$ as a [[path integral]] for a [[relativistic particle]] travelling from $y$ to $x$.

This worldline formalism is equivalent to the traditional formulation. It has the conceptual advantage that it expresses the [[Feynman perturbation series]] of [[perturbative  quantum field theory]] manifestly as a [[second quantization]] of its particle content given explicitly as the [[superposition]] of all 1-particle processes, and the calculational advantage of automatically summing over subsets of Feynman diagrams related by exchange of external legs, thus maintaining [[permutation]] [[symmetry]] and explicit [[gauge invariance]] of [[on-shell]] [[scattering amplitudes]].

The worldline formulation of QFT has an evident generalization to higher dimensional [[worldvolumes]]: in direct analogy one can consider summing the [[correlators]]/[[n-point functions]] over [[worldvolume]] [[theory (physics)|theories]] of "higher dimensional particles" ("[[branes]]") over all possible worldvolume geometries. Indeed, for 2-[[dimensional]] branes this is precisely the way in which [[perturbative string theory]] is defined: the [[string scattering amplitudes]] are given by the analogous "worldsheet formalism" known as the [[string perturbation series]] as the sum over all surfaces of the [[correlators]]/[[n-point functions]] of of a 2d [[SCFT]] of central charge 15.

[[!include second quantization -- table]]

Indeed, after decades of [[Feynman rules]], the worldline formalism for QFT was found only _via_ [[string theory]] in ([Bern-Kosower 91](#BernKosower91)), by looking at the point particle limit of [[string scattering amplitudes]].

> graphics grabbed from [Jurke 10](https://benjaminjurke.com/content/articles/2010/string-theory/)

> graphics grabbed from [Schubert 96](#Schubert96)

Then ([Strassler 92](#Strassler92), [Strassler 93](#Strassler93)) observed that generally the worldline formlism is obtained from the correlators of the 1d QFT of [[relativistic particles]] on their [[worldline]].

> graphics grabbed from ([Schmidt-Schubert 94](#SchmidtSchubert94))

The first calculuation along these lines was actually done earlier in ([Metsaev-Tseytlin 88](#MetsaevTseytlin88)) where the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] was obtained as the point-particle limit of the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in ([Nepomechie 83](#Nepomechie83)) that when computed via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].

## Related entries

* [[virtual particle]]

* [string theory FAQ -- What is the relationship between quantum field theory and string theory?](string+theory+FAQ#RelationshipBetweenQuantumFieldTheoryAndStringTheory)

## References

Precursor observations include

* [[Richard Feynman]], Appendix of _Mathematical formulation of the quantum theory of electromagnetic interaction_, Physics Review Volume 80 (1950) 440.

* {#Nepomechie83} R.I. Nepomechie, _Remarks on quantized Yang-Mills theory in 26 dimensions_, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 (doi:10.1016/0370-2693(83)90385-4)

* {#MetsaevTseytlin88} [[Ruslan Metsaev]], [[Arkady Tseytlin]], _On loop corrections to string theory effective actions_, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (doi:10.1016/0550-3213(88)90306-9)

The worldline formalism as such was first derived from the point-particle limit of [[string scattering amplitudes]] in

* {#BernKosower91} [[Zvi Bern]], D. Kosower, _Efficient calculation of one-loop QCD amplitudes_ Phys. Rev. D 66 (1991),([journal](http://prl.aps.org/abstract/PRL/v66/i13/p1669_1))

* {#BernKosower92} [[Zvi Bern]], D. Kosower, _The computation of loop amplitudes in gauge theories_, Nucl. Phys. B379 (1992)  ([journal](http://ccdb4fs.kek.jp/cgi-bin/img_index?9108076))

Then it was related to actual worldline quantum field theory in

* {#Strassler92} [[Matthew Strassler]], _Field Theory Without Feynman Diagrams: One-Loop Effective Actions_, Nucl. Phys. B385:145-184,1992 ([arXiv:hep-ph/9205205](http://arxiv.org/abs/hep-ph/9205205))

* {#Strassler93} [[Matthew Strassler]], _The Bern-Kosower Rules and Their Relation to Quantum Field Theory_, PhD thesis, Stanford 1993 ([spires](http://inspirehep.net/record/364079?ln=de))

Review:

* {#SchmidtSchubert94} M. G. Schmidt, [[Christian Schubert]], _The Worldline Path Integral Approach to Feynman Graphs_ ([arXiv:hep-ph/9412358](http://arxiv.org/abs/hep-ph/9412358))

* {#Schubert96} [[Christian Schubert]], _An Introduction to the Worldline Technique for Quantum Field Theory Calculations_, Acta Phys. Polon.B27:3965-4001, 1996 ([arXiv:hep-th/9610108](https://arxiv.org/abs/hep-th/9610108))

* [[Christian Schubert]], _Perturbative Quantum Field Theory in the String-Inspired Formalism_, Phys.Rept. 355 (2001) 73-234 ([arXiv:hep-th/0101036](https://arxiv.org/abs/hep-th/0101036))

* Olindo Corradini, [[Christian Schubert]], James P. Edwards, Naser Ahmadiniaz, *Spinning Particles in Quantum Mechanics and Quantum Field Theory* ([arXiv:1512.08694](https://arxiv.org/abs/1512.08694))

  > (with emphasis on [[spinning particles]])

* James P. Edwards, C. Moctezuma Mata, U. Müller, [[Christian Schubert]], *New techniques for worldline integration*, SIGMA 17 (2021), 065, 19 pages ([arXiv:2106.12071](https://arxiv.org/abs/2106.12071))

* James P. Edwards, [[Christian Schubert]],
*Quantum mechanical path integrals in the first quantised approach to quantum field theory*, ([arXiv:1912.10004](https://arxiv.org/abs/1912.10004))

Exposition with an eye towards [[quantum gravity]] is in

* {#Witten11} [[Edward Witten]], from 30:40 on in _Quantum Gravity_, Solomon Lefschetz Memorial Lecture Series, November 2011 ([video](https://www.youtube.com/watch?v=uRdCJMYc2Ds#t=15))

* {#Witten15} [[Edward Witten]], _What every physicist should know about string theory_, talk at [Strings2015](https://strings2015.icts.res.in)  ([pdf](https://strings2015.icts.res.in/talkDocuments/26-06-2015-Edward-Witten.pdf), [[WittenWorldlineFormalism.pdf:file]], [video recording](https://www.youtube.com/watch?v=H0jLD0PphTY))

See also:

* [[Thanu Padmanabhan]], *World-line Path integral for the Propagator expressed as an ordinary integral: Concept and Applications*, Found Phys., 51 35 (2021) ([arXiv:2104.07041](https://arxiv.org/abs/2104.07041), [doi:10.1007/s10701-021-00447-8]( https://doi.org/10.1007/s10701-021-00447-8))

Discussion of the [[Schwinger effect]] via [[worldline formalism]]:

* [[Ian Affleck]], [[Orlando Alvarez]], [[Nicholas S. Manton]], _Pair Production At Strong Coupling In Weak External Fields_, Nuclear Physics B Volume 197 (1982) 509 (doi:10.1016/0550-3213(82)90455-2)

* [[Gerald Dunne]], [[Christian Schubert]], _Worldline Instantons and Pair Production in Inhomogeneous Fields_, Phys. Rev. D72 (2005) 105004 ([arXiv:hep-th/0507174](https://arxiv.org/abs/hep-th/0507174))

* [[Christian Schubert]], *The worldline formalism in strong-field QED* [[arXiv:2304.07404](https://arxiv.org/abs/2304.07404)]

Discussion for [[QCD]] with emphasis of [[2d QCD]] and [[AdS/QCD]]:

* Adi Armoni, Oded Mintakevich, _Comments on Mesonic Correlators in the Worldline Formalism_,  Nuclear Physics B, Volume 852, Issue 1, 1 November 2011, Pages 61-70 ([arxiv:1102.5318](https://arxiv.org/abs/1102.5318))

Further development:

* M. G. Schmidt, [[Christian Schubert]], _Worldline Green Functions for Multiloop Diagrams_, Phys.Lett. B331 (1994) 69-76 ([arXiv:hep-th/9403158](https://arxiv.org/abs/hep-th/9403158))

* [[Eric D'Hoker]], Darius G. Gagne, _Worldline Path Integrals for Fermions with General Couplings_, Nucl.Phys. B467 (1996) 297-312 ([arXiv:hep-th/9512080](https://arxiv.org/abs/hep-th/9512080))

* C. Alexandrou, R. Rosenfelder, A. W. Schreiber, _Worldline path integral for the massive Dirac propagator: A four-dimensional approach_, Phys. Rev. A59 (1999) 1762-1776 ([arXiv:hep-th/9809101](http://arxiv.org/abs/hep-th/9809101))

* Fiorenzo Bastianelli, Olindo Corradini, Andrea Zirotti, _BRST treatment of zero modes for the worldline formalism in curved space_, JHEP 0401 (2004) 023 ([arXiv:hep-th/0312064](https://arxiv.org/abs/hep-th/0312064))

* Peng Dai, [[Warren Siegel]], _Worldline Green Functions for Arbitrary Feynman Diagrams_, Nucl.Phys.B770:107-122,2007 ([arXiv:hep-th/0608062](https://arxiv.org/abs/hep-th/0608062))

* S. A. Franchino-Viñas, S. Mignemi, _Worldline Formalism in Snyder Spaces_ ([arXiv:1806.11467](https://arxiv.org/abs/1806.11467))

* Olindo Corradini, Maurizio Muratori, _String-inspired Methods and the Worldline Formalism in Curved Space_ ([arXiv:1808.05401](https://arxiv.org/abs/1808.05401))

* James P. Edwards, Urs Gerber, [[Christian Schubert]], Maria Anabel Trejo, Thomai Tsiftsi, Axel Weber, _Applications of the worldline Monte Carlo formalism in quantum mechanics_,  ([arXiv:1903.00536](https://arxiv.org/abs/1903.00536))

* Olindo Corradini, Maurizio Muratori, _A Monte Carlo Approach to the Worldline Formalism in Curved Space_ ([arXiv:2006.02911](https://arxiv.org/abs/2006.02911))

* Naser Ahmadiniaz, Victor Miguel Banda Guzman, Fiorenzo Bastianelli, Olindo Corradini, James P. Edwards, [[Christian Schubert]], _Worldline master formulas for the dressed electron propagator, part 1: Off-shell amplitudes_ ([arXiv:2004.01391](https://arxiv.org/abs/2004.01391))

* Naser Ahmadiniaz, Victor Miguel Banda Guzman, Fiorenzo Bastianelli, Olindo Corradini, James P. Edwards, [[Christian Schubert]], _Worldline master formulas for the dressed electron propagator, part 2: On-shell amplitudes_ ([arXiv:2107.00199](https://arxiv.org/abs/2107.00199))

* [[Christian Schubert]] et al., *Summing Feynman diagrams in the worldline formalism* [[arXiv:2208.06585](https://arxiv.org/abs/2208.06585) ]

As a means of constructing [[UV-completions]]:

* Steven Abel, Nicola Andrea Dondi, _UV Completion on the Worldline_ ([arXiv:1905.04258](https://arxiv.org/abs/1905.04258))

For [[black hole]] [[scattering]]:

* {#MogullPlefkaSteinhoff20} Gustav Mogull, [[Jan Plefka]], Jan Steinhoff, _Classical black hole scattering from a worldline quantum field theory_ ([arXiv:2010.02865](https://arxiv.org/abs/2010.02865))

and for photon-graviton scattering:

* Fiorenzo Bastianelli, Francesco Comberiati, Leonardo de la Cruz, *Light bending from eikonal in worldline quantum field theory* ([arXiv:2112.05013](https://arxiv.org/abs/2112.05013))

A list of more literature is at

* The Tangent Bundle, _[QFT Worldline formalism](http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/worldline_formalism)_

[[!redirects worldline formalisms]]

[[!redirects worldline theory]]
[[!redirects worldline theories]]

[[!redirects worldline method]]
[[!redirects worldline methods]]


worldsheet instanton

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## Idea

In [[perturbative string theory]], _worldsheet instantons_ are contributions to the [[string scattering amplitudes]] of [[sigma-model]] (i.e. "geometric") [[perturbative string theory vacua]] which correspond to [[worldsheet]] configurations that may not be connected continuously to configurations constant on a point, hence which [[wrapped brane|wrap]] some 2-[[cycle]] in the [[target spacetime]].

Worldsheet instantons contribute [[non-perturbative effects]] to the [[string scattering amplitudes]] with respect to the [[string length scale]] $\ell_s$, see [there](non-perturbative+effect#WorldsheetAndBraneInstantons).

In contrast, [[non-perturbative effects|non-perturbative contributions]] to [[string scattering amplitudes]] $g_s$ in the [[string coupling constant]] come from [[D-brane instantons]].

These two kinds of stringy non-perturbative effects unify under the [[duality between M-theory and type IIA string theory]] to [[membrane instantons]], depending on whether the [[M2-brane]] configuration [[wrapped brane|wraps]] the M-theory circle fiber $S^1_{10}$ or not, see [there](non-perturbative+effect#WorldsheetAndBraneInstantons).

## Yukawa couplings in intersecting D-brane models

In [[intersecting D-brane models]] [[Yukawa couplings]] are encoded by [[worldsheet instantons]] of open strings stretching between the [[brane intersection|intersecting]] [[D-branes]] (see [Marchesano 03, Section 7.5](#Marchesano03)). Mathematically this is encoded by  [[derived hom-spaces]] in a [[Fukaya category]] (see [Marchesano 03, Section 7.5](#Marchesano03)).

\begin{center}
<img src="https://ncatlab.org/nlab/files/YukawaFukaya.jpg" width="600">
\end{center}

> table grabbed from [Marchesano 03](#Marchesano03)

## References

In [[intersecting D-brane models]]:

* {#Marchesano03} [[Fernando Marchesano]], section 7.1.1 of _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))

[[!redirects worldsheet instantons]]


worldsheet parity operator

In [[orientifold]] [[string theory]] [[vacua]] the [[real space]]-structure of [[target space]] is accompanied by a corresponding [[involution]] of the [[worldsheet]]. This is known as the _worldsheet parity operator_.

[[!redirects worldsheet parity]]


worldvolume

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## Idea

In a [[sigma-model]] [[quantum field theory]] a [[field history]] is a [[morphism]] $\phi \colon \Sigma \to X$ for $\Sigma$ an $n$-[[dimension|dimensional]] [[manifold]] or similar. One is to think of this as being the _trajectory: of an $(n-1)$-[[brane]] propagating in the [[target space]] $X$.

For the case $n = 1$ (for instance the [[relativistic particle]], a 0-[[brane]]) the term **worldline** for $\Sigma$ has a long tradition. Accordingly one calls $\Sigma$ the **worldvolume** of the given $(n-1)$-[[brane]] when $n \gt 1$. For the case $n=2$ (the case of relevance in [[string theory]]) one also says **worldsheet**.

Hence generally for any [[field theory]] defined on a [[worldvolume]] or [[spacetime]] $\Sigma$, and with type of fields determined by a [[field bundle]] $E \overset{fb}{\to} \Sigma$, one may think of a [[section]] of the field bundle as a _field trajectory_.

The [[space]] of all these is the _[[space of trajectories]]_ (space of histories).

## Related concepts

* [[worldline formalism]] (for [[scattering amplitudes]] in [[QFT]])

* [[sigma-model]]

  * **worldvolume**

    * worldsheet

      * [[Riemann surface]], [[elliptic curve]], [[algebraic curve]]

      * [[worldsheet parity operator]]

  * [[target space]]

  * [[background gauge field]]

  * [[genus]]

* [[worldsheet instanton]]

* [[world sheets for world sheets]]

[[!redirects worldline]]

[[!redirects worldsheet]]
[[!redirects worldsheets]]

[[!redirects worldvolumes]]

[[!redirects worldlines]]

[[!redirects world line]]
[[!redirects world volume]]
[[!redirects world sheet]]

[[!redirects world-sheet]]
[[!redirects world-sheets]]


worldvolume-target supersymmetry of brane sigma-models

**manifest [[supersymmetry]] for [[brane]] [[sigma-models]]**:

| manifest [[worldvolume]] supersymmetry |  manifest target+worldvolume supersymmetry | manifest [[target space]] supersymmetry |
|---|---|---|
| [[NSR action functional]] | [[superembedding approach]] | [[Green-Schwarz action functional]] |

> graphics grabbed from [[schreiber:Super-exceptional embedding construction of the M5-brane|FSS19c]]


wormhole

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## Related pages

* [[black hole]]

* [[ER = EPR]]

## References

The original article:

* {#EinsteinRosen35} [[Albert Einstein]], [[Nathan Rosen]], *The Particle Problem in the General Theory of Relativity*, Phys. Rev. **48** 73 (1935) ([doi:10.1103/PhysRev.48.73](https://doi.org/10.1103/PhysRev.48.73))

See also:

* Wikipedia, _[Wormhole](http://en.wikipedia.org/wiki/Wormhole)_

Argument that the image of *wormhole traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:

* [[Ping Gao]], [[Daniel Louis Jafferis]], [[Aron C. Wall]], *Traversable Wormholes via a Double Trace Deformation*, Journal of High Energy Physics **2017** 151 (2017) [[arXiv:1608.05687](https://arxiv.org/abs/1608.05687), doi:10.1007/JHEP12(2017)151]

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], *Diving into traversable wormholes*,Fortsch. Phys. **65** 5 (2017) 1700034 [[arXiv:1704.05333](https://arxiv.org/abs/1704.05333), [doi:10.1002/prop.201700034](https://doi.org/10.1002/prop.201700034)]

[[!redirects wormholes]]

[[!redirects Einstein-Rosen bridge]]
[[!redirects Einstein-Rosen bridges]]


Wout Merbis

## Selected writings

On [[3d gravity]] via [[Chern-Simons theory]]:

* [[Wout Merbis]], _Chern-Simons-like Theories of Gravity_ ([arXiv:1411.6888](https://arxiv.org/abs/1411.6888))

* [[Marc Henneaux]], [[Wout Merbis]], Arash Ranjbar, _Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions_ ([arXiv:1912.09465](https://arxiv.org/abs/1912.09465))

Discussion of [[holographic entanglement entropy]] in asymptotically flat [[3d quantum gravity]] via [[Wilson line observables]] in [[Chern-Simons theory]]:

* [[Wout Merbis]], [[Max Riegler]], _Geometric actions and flat space holography_ ([arXiv:1912.08207](https://arxiv.org/abs/1912.08207))

category: people


Wouter Swierstra

* [personal page](https://webspace.science.uu.nl/~swier004/)

* [institute page](https://www.uu.nl/staff/WSSwierstra)

## Selected writings

Introducing [[observational type theory]]:

* {#AltenkirchMcBrideSwierstra07} [[Thorsten Altenkirch]], [[Conor McBride]], [[Wouter Swierstra]], *Observational Equality, Now!*, PLPV '07: Proceedings of the 2007 workshop on Programming languages meets program verification (2007) 57-68 [ISBN:978-1-59593-677-6, [doi:10.1145/1292597.1292608](http://doi.org/10.1145/1292597.1292608), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/obseqnow.pdf)]

category: people


wrapped brane

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## Idea

A [[brane]] [[worldvolume]] $\phi \colon \Sigma \longrightarrow X$ is said to _wrap_ a [[cycle]] $c \in H_{\dim(\Sigma)}(X)$ in [[spacetime]] $X$ when the pushforward $\phi_\ast [\Sigma] \in H_\bullet(X)$ of the [[fundamental class]] of $\Sigma$ is the class, $[c]$, of the given cycle in $X$. If the pushforward is a multiple of $[c]$, then the brane is said to wrap $c$ multiple times.

## Examples

Via the identification of [[D-brane charge]] in [[K-theory]], the K-theoretic [[McKay correspondence]] formalizes how D-branes wrap the fundamental cycles in the [[blow-up]] [[resolution of singularities|resultion]] of an [[ADE-singularity]] ([Gonzalez-Sprinberg & Verdier 83](#GSV83))

* [[membrane instanton]]

* [[M5-brane instanton]]

* [[fractional M2-brane]]

* [[3d-3d correspondence]]

* [[giant graviton]]

* in the [[Witten-Sakai-Sugimoto model]] (see there) for [[non-perturbative field theory|non-perturbative]] [[quantum chromodynamics]] [[baryons]] appear as wrapped [[D4-branes]]

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

## Related concepts

* [[polarized brane]]

* [[brane intersection]]

* [[Dp-D(p+2)-brane bound state]]

* [[Dp-D(p+4)-brane bound state]]

## References

* {#GSV83} [[Gérard Gonzalez-Sprinberg]], [[Jean-Louis Verdier]], _Construction géométrique de la correspondance de McKay_, Ann. Sci. ́École Norm. Sup.16 (1983) 409–449. ([numdam](http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0))

* {#Sugimoto16} [[Shigeki Sugimoto]], _Skyrmion and String theory_, chapter 15 in [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

[[!redirects wrapped branes]]