stack semantics

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#### Type theory
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# Stack semantics
* table of contents
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## Idea

The **stack semantics** is an extension of the [[internal logic]] of a [[category]] (or [[higher category]]) $C$ which allows us to talk about, and [[quantifier|quantify over]], *objects* of $C$.  (Such quantification is sometimes called "unbounded" quantification.)

In the ordinary internal language, an internal "element" of an object $A$ is interpreted into a [[generalized element]] $X\to A$ for some object $X\in C$, which consists intuitively of one "actual" element of $A$ for every element of $X$.  Similarly, in the stack semantics, an internal "object" is interpreted as a "generalized object" defined at some stage $X$, which we define to mean an object of the [[slice category]] $C/X$.  This makes sense because an object of $C/X$ consists intuitively of one "actual" object of $C$ for each "element" of $X$ (namely, the [[fiber]] over that element).

The stack semantics is so-called because in most categories $C$ where we consider it (such as [[toposes]] or [[pretoposes]]), the [[self-indexing]] is a [[stack]] for some naturally defined [[Grothendieck topology]] on $C$, and the stack semantics can be regarded as the ordinary internal logic of a higher topos of stacks on $C$ (see below).  Analogously, the ordinary internal logic is often called *sheaf semantics*, because in a [[subcanonical topology]] all [[representable functors]] are [[sheaves]], and because it can often be regarded as part of the internal logic of the topos of sheaves on $C$.

## Definition

### Direct description

...

### As internal logic

While on the one hand, the stack semantics *generalizes* the internal logic of a category, it can also be regarded as a special case thereof.

The basic idea of this is that if we have a chosen morphism $\widetilde{U}\to U$ in $C$, then generalized elements of $U$ at stage $X$ can be regarded as "names" for generalized *objects* of $C$ at stage $X$.  Namely, the generalized element $u\colon X\to U$ names the [[pullback]] of $\widetilde{U}\to U$ along $u$.

There are two issues with this idea, which can prevent it from faithfully representing the *idea* of the stack semantics sketched above.

The first issue is that a given "generalized object" over $X$ may have multiple names, whereas the stack semantics ought to speak directly about objects rather than about names for them.  Moreover, since $C/X$ is actually a category (or a higher category, if $C$ is such), generalized objects [[evil|should]] be considered "the same" only if they are [[isomorphism|isomorphic]] or [[equivalence|equivalent]], whereas morphisms $X\to U$ may admit a stricter notion of sameness.  In particular, when $C$ is a [[1-category]] (the classical case), the morphisms $X\to U$ form a *set*, whereas $C/X$ should be regarded as a category, or at worst as a [[groupoid]] (its [[core]]).

More generally, if $C$ is an [[n-category]] for some finite $n$, then $C(X,U)$ is only an $(n-1)$-category, whereas $C/X$ is also an $n$-category or at worst an [[n-groupoid]].  Thus, there is no hope of solving this problem *inside* of $C$ unless $C$ is at least an [[(∞,1)-category]].  In that case, the notion of [[object classifier in an (∞,1)-category]] does solve this problem: the $\infty$-groupoid $C(X,U)$ is equivalent to the core of (a suitable subcategory of) $C/X$.  (The presence of noninvertible morphisms in $C/X$ is not nearly so big a problem; at least when $C$ is an $(n,1)$-category, these can be described already in the ordinary internal logic if $C$ is [[locally cartesian closed category]].)

If $C$ is only an $(n,1)$-category for some finite $n$, however, then in order to obtain its stack semantics as an internal logic, we must pass to some higher category containing it which is at least an $(n+1,1)$-category.  The obvious choice is a category of [[presheaves]] or [[sheaves]]/[[stacks]] on $C$ valued in $n$-categories or $n$-groupoids (or $m$-groupoids for some $m\ge n$).

The second issue is that not every "generalized object" over $X$ may have a name.  This is almost impossible to get around while remaining within $C$, due to issues with paradoxes.  Even [[object classifiers in an (∞,1)-category]] only classify the core of some [[subcategory]] of $C/X$.

One way to deal with this is to suppose some (possibly [[ordinal|transfinite]]) sequence of object classifiers, each contained in the next, such that every generalized object is eventually classified by some one of them.  In the internal logic, this corresponds roughly to the idea of [[universe polymorphism]] in [[type theory]].

Another way is, of course, to move outside of $C$.  In particular, if we work in a category of presheaves/sheaves/stacks on $C$ at one category-level up, then the [[self-indexing]] of $C$ (or its [[core]]) will be an object $U$ which classifies *exactly* the desired category $C/X$ of generalized objects for each $X\in C$.  (The exhaustive sequence of universes can also be regarded as moving outside of $C$, perhaps to a category of [[ind-objects]] in $C$.)  The fragment of the internal logic of this category of stacks which deals only with objects of $C$ itself can then be identified with the direct Kripke--Joyal style stack-semantics described above.

Of course, one can also embrace this second limitation and just live with a "partial" stack semantics that only allows us to talk about a particular subclass of generalized objects.  The [[Mitchell–Bénabou language]] of a [[topos]] can be regarded as a partial stack semantics in this sense, in which the only generalized objects we can classify over are the [[subterminal objects]] (i.e. the [[subobjects]] $A \hookrightarrow X$).  The object $U$ in this case is, of course, the [[subobject classifier]].

One can also embrace the first limitation and not worry about having multiple names for the same object.  This leads to notions such as a [[universe in a topos]] and the field of [[algebraic set theory]].

## Relationship to locally internal categories

The stack semantics for a topos $\mathcal{E}$ can be used to talk about [[locally internal category|locally internal categories]] of $\mathcal{E}$ (in the sense of ([Penon 1974](#Penon))) as if they're ordinary locally small categories over $\mathrm{Set}$.

Examples:

* A geometric morphism $f: \mathcal{F} \to \mathcal{E}$ looks like the canonical morphism $\mathcal{F} \to \mathrm{Set}$ when using the stack semantics of $\mathcal{E}$.

* The internal statement "$\mathcal{F}$ is a compact topos" is equivalent to $f$ being a [[proper geometric morphism|proper morphism]].

* The internal statement "$\mathcal{F}$ is an indiscrete topos" (i.e. "the canonical morphism $\mathrm{Sub}_{\mathcal{F}}(1) \to \mathrm{Sub}_{\mathrm{Set}}(1)$ is a bijection", where $\mathrm{Sub}$ refers to the corresponding [[poset of subobjects|posets of subobjects]]) is true iff $f$ is [[hyperconnected geometric morphism|hyperconnected]].

* Because the proof that indiscrete toposes are compact is constructive, it immediately follows that hyperconnected morphisms are proper.

## Related concepts

* [[semantics]]

* [[Kripke–Joyal semantics]]

## References

* [[Mike Shulman]], _[Stack semantics and the comparison of material and structural set theories](http://arxiv.org/abs/1004.3802)_ (2010)

* [[Mike Shulman]], _Large categories and quantifiers in topos theory_, [slides](http://home.sandiego.edu/~shulman/papers/cambridge-stacksem.pdf) (2021)

* J. Penon, _Categories localement internes_, C. R. Acad. Sci. Paris __278__ (1974) A1577--1580
{#Penon}

* Locally internal categories, Appendix in: [[Peter Johnstone]], _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977, xxiii+367 pp.
{#Johnstone}


stack* > history

< [[stack*]]
[[!redirects stack* -- history]]


stacked cover

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## Idea

A _stacked cover_ is a [[cover]] of a [[topological space]] which is indexed by a cover of another topological space, such that the product cover is a cover of the product space.

## Definition

+-- {: .un_defn}
###### Definition

Let $A,B$ be [[topological space]]s and $\mathcal{U}$ a [[numerable cover]] of $A$. Then a [[cover]] of the [[product]] space $A \times B$ is called a **stacked cover** of $A\times B$ over $\mathcal{U}$ -- denoted $\mathcal{U} \times \mathcal{S}$ -- , if there exists a [[function]] $\mathcal{S}$ -- called the **stacking function** -- which assignes to each set $U \in \mathcal{U}$ a cover $\mathcal{S}U$ of $B$, such that $\mathcal{U} \times\mathcal{S}$ consists of all the sets $U \times V$ with $V \in \mathcal{S}U$.

=--

## Properties

### General

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

A stacked cover is itself a [[numerable cover]].

=--

### Stacked covers of products with the interval {#OfProductsWithInterval}

In this section  we let $B = [0,1]$ the standard [[interval]] and consider properties of stacked covers of spaces of the form $A \times [0,1]$.

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

For $A$ a [[topological space]] and $\mathcal{W}$ a [[numerable cover]] of $A \times [0,1]$ there exists a refinement of $\mathcal{W}$ to a stacked cover $\mathcal{U} \times \mathcal{S}$ of $A \times [0,1]$ of the form

$$
  \{U_i \times [\frac{k-1}{r_i}, \frac{k+1}{r_i}] | r_i,k \in \mathbb{N}, 1 \leq k \leq r_i-1\}
  \,.
$$

=--

## References

Section A.2.17 of

* [[Albrecht Dold]], _Lectures on algebraic topology_ , Spring Verlag (1980)

[[!redirects stacked covers]]
[[!redirects stacked covering]]
[[!redirects stacked coverings]]


Stacy McGaugh

Stacy McGaugh is an astronomer who studies [[galaxies]], [[cosmology]], [[dark matter]], and [[MOND]].

* [website](http://astroweb.case.edu/ssm/index.html)

* [blog](https://tritonstation.com/)

## Selected writings

On [[superfluid dark matter]]:

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Galactic mass-to-light ratios with superfluid dark matter*, Astronomy and Astrophysics, Volume 664, Article 40, August 2022 ([doi:10.1051/0004-6361/202243216](https://doi.org/10.1051/0004-6361/202243216), [arXiv:2201.07282](https://arxiv.org/abs/2201.07282))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Aether Scalar Tensor theory confronted with weak lensing data at small accelerations*, ([arXiv:2301.03499](https://arxiv.org/abs/2301.03499))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Superfluid dark matter in tension with weak gravitational lensing data* ([arXiv:2303.08560](https://arxiv.org/abs/2303.08560))

That the [[Hubble tension]] might result from systematics in the [[cosmic microwave background]]:

* [[Stacy McGaugh]], *Early Galaxy Formation and the Hubble Tension* [[arXiv:2312.03127](https://arxiv.org/abs/2312.03127)]

## Related entries

* [[MOND]]

category: people


stalk

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### Context
#### Topos Theory
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#Contents#
* table of contents
{:toc}

## Definition

Recall that a [[point of a topos|point]] $x$ of a [[topos]] $E$ is a [[geometric morphism]]

$$
  x : Set \to E
  \,.
$$

The **stalk** at $x$ of an object $e \in E$ is the image of $e$ under the corresponding [[inverse image]] morphism

$$
  x^*
  \,\colon\,
  E \to Set
$$

i.e.

$$
  stalk_x(e)
    \,\coloneqq\,
  x^*(e)
  \,.
$$

### Special case of sheaf topoi

If $E$ is the [[category of sheaves]] on the [[category of open subsets]] $Op(X)$ of a [[topological space]] $X$

$$
  E
  \,=\,
  Sh(X)
  \,,
$$

then the [[point of a topos|topos points]] of $E$ come precisely from the ordinary points

$$
  (x \colon {*} \to X)   \in  Top
  \,,
$$

of the space $X$, where the [[direct image]] morphism

$$
  x_* \colon Set \to Sh(X)
$$

sends every set to the [[skyscraper sheaf]] at this point of $X$. By the general [[Kan extension]] formula for the [[inverse image]] (see there) one finds in this case for any [[sheaf]] $F \in Sh(X)$ the stalk

$$
  \begin{aligned}
    stalk_x(F)
    & =
    \underset{
      ({*} \to x^{-1}(V))
        \in
      (const_{*}, x^{-1})
    }{colim}
    F(V)
    \\
    &
    =
    \underset{
      { V \subset X }
      \atop
      { x \in V }
    }{
      colim
    }
    F(V)
  \end{aligned}
  \,.
$$

So for sheaves on (open subsets of) topological spaces the stalk at a given point is the colimit over all values of the sheaf on open subsets containing this point.

By the general definition of colimits in [[Set]] described at [[limits and colimits by example]], the elements in this colimit can in turn be described as [[equivalence classes]] represented pairs $(z, V)$ with $x \in V$ $z \in F(V)$, where the equivalence relation says that two such pairs $(z_1, V_1)$ and $(z_2, V_2)$ coincide if there is a third pair $(z,U)$ with $U \subset V_1$ and $U \subset V_2$ such that $z = z_1|_U = z_2|_U$.

for $F = C(-)$ a sheaf of functions on $X$, such an equivalence class, hence such an element in a stalk of $F$ is called a function **germ**.

#### Testing sheaf morphisms on stalks

For $E$ a topos with [[point of a topos|enough points]], the behaviour of morphisms $f : A \to B$ in $E$ can be tested on stalks:

+-- {: .num_theorem}
###### Theorem

A morphism $f : A \to B$ of sheaves on $X$ is a

* [[monomorphism]]

* resp. [[epimorphism]]

* resp. [[isomorphism]]

if and only if every induced map of stalk sets $stalk_x(f) : stalk_x(A) \to stalk_x(B)$ is, for all $x \in X$
=--

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

The statement for isomorphisms follows from the identification of sheaves with [[etale space]]s (e.g. section II, 6, corollary 3 in MacLane-Moerdijk, [[Sheaves in Geometry and Logic]]). The statement for epimorphisms/monomorphisms is proposition 6 there.

=--

#### Example

Let $X$ be a smooth manifold and let $\Omega^n(X)$ and $Z^{n+1}(X)$ be the sheaves of differential $n$-forms and that of _closed_ differential $(n+1)$-forms on $X$, respectively, for some $n \in \mathbb{N}$. Let

$$
  d : \Omega^n(X) \to Z^{n+1}
$$

be the morphism of sheaves that is given on each open subset by the deRham differential.

Then:

* for $U \subset X$ the map $d_U : \Omega^n(U) \to Z^{n+1}(U)$ need not be [[epimorphism|epi]], since not every closed form is exact;

* but by the [[Poincare lemma]] every closed form is _locally_ exact, so that for each $x \in X$ the map of stalks $d_x : stalk_x(\Omega^n(X)) \to stalk_x(Z^{n+1}(X))$ is an epimorphism.

Accordingly, the morphism $d : \Omega^n(X) \to Z^{n+1}(X)$ is an [[epimorphism]] of sheaves.

This kind of example plays a crucial role in the computation of [[abelian sheaf cohomology]], see the examples listed there.

## Examples

For a [[locally ringed topos]] with [[structure sheaf]] $\mathcal{O}$, the stalk of the [[multiplicative group]] $\mathbb{G}_m$ at a point $x$ is the multiplicative group $\mathcal{O}_x^\times$ in the stalk [[local ring]] of the structure sheaf.  (e.g. [Milne, example 6.13](#Milne))

## Related concepts

[[!include infinitesimal and local - table]]

## References

* [[James Milne]], section 6 of _[[Lectures on Étale Cohomology]]_
 {#Milne}
* [[Ofer Gabber]], Shane Kelly, _Points in algebraic geometry_, J. Pure App. Alg. __219__:10 (2015) 4667-4680 [doi](https://doi.org/10.1016/j.jpaa.2015.03.001)

[[!redirects stalks]]


Stam Nicolis

* [webpage](http://www.lmpt.univ-tours.fr/~nicolis/index_en.html)

category: people


standard Borel space

[[!redirects standard Borel]]
[[!redirects standard Borel]]

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#### Measure and probability theory
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# Contents
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{: toc}

## Idea

*Standard Borel spaces* are particular measurable spaces, well suited for a number of constructions in traditional probability theory, such as for forming [[conditional probability]].

The real line is in particular a standard Borel space.

From the point of view of [[categorical probability]] they form a particularly well-behaved [[Markov category]], [[Stoch#borelstoch|BorelStoch]].

## Definitions

A [[topological space]] is called **completely separably metrizable** if the [[topology]] can be metrized by a [[complete metric|complete]] and [[separable]] [[metric]].
These spaces are also called [[Polish spaces]], see there for more information.

A [[measurable space]] is called a **standard Borel space** if it can be written as a Polish space with its [[Borel sigma-algebra]].

By analogy, a [[measure space]] (for example a [[probability space]]) is called a **standard Borel measure space** if and only if its underlying measurable space is standard Borel as above.

## Examples

* The real line is a standard Borel space.

* Every finite or countable set, equipped with the discrete sigma-algebra, is a standard Borel spaces.

Up to isomorphism of measurable spaces, these are the only examples. Note in particular that all finite powers of the real line and all intervals, as measurable spaces, are isomorphic to the real line.

## Categories of standard Borel spaces

* The [[category]] *BorelMeas* (or *Pol*) has as [[objects]] standard Borel measurable spaces, and as [[morphisms]] [[measurable functions]].
(Sometimes, depending on the author, *Pol* denotes the category of completely metrizable topological spaces and *continuous* maps.)

* The category [[Stoch#borelstoch|BorelStoch]] has as objects standard Borel spaces, and as morphisms [[Markov kernels]] between them.

* The [[category of couplings]] has as objects standard Borel *measure* (not just *measurable*) spaces, and either equivalence classes of Markov kernels between them (under [[Markov kernel#almost_sure_equality|almost sure equality]]), or [[coupling (probability)|couplings]] between them.

## Properties

* Finite and countable [[products]] (in the category [[Meas]], i.e. with the tensor product [[sigma-algebra]]) of standard Borel spaces are again standard Borel.

* Measurable subsets of standard Borel spaces, with the induced sigma-algebra, are again standard Borel.

* Measurable retracts of a standard Borel space are again standard Borel.

* The [[sigma-algebra]] of a standard Borel space is countably generated and separating.

This has a number of consequences, for example:

* Every [[Markov kernel]] between standard Borel probability spaces admits a [[Markov kernel#bayesian_inversion|Bayesian inverse]].

* *Disintegration theorem:* For every sub-sigma-algebra of a standard Borel space, conditional expectation gives rise to a [[Markov kernel#regular_conditional_distributions|regular conditional distribution]].

* Standard Borel spaces admit countably [[infinite tensor products]].

* The [[Giry monad]] preserves standard Borel spaces: if $X$ is a standard Borel space, its functorial image $P X$ is standard Borel too. Therefore the Giry monad restricts to a monad on *Pol*, usually known under the same name.

## Related Concepts

* [[quasi-Borel space]]
* [[Polish space]]
* [[measurable space]]
* [[Stoch]]
* [[Markov kernel]]

## References

(...)

category: probability

[[!redirects standard Borel]]
[[!redirects standard Borel spaces]]
[[!redirects Pol]]


[[!redirects standard conjectures on algebraic cycles]]
[[!redirects standard conjectures]]

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* table of contents
{:toc}

## Idea

What are called the __standard conjectures on algebraic cycles__ are several [[conjectures]] brought up by [[Grothendieck]], concerned with the relation between [[algebraic cycles]] and [[Weil cohomology]] theories.

"The first (Lefschetz standard conjecture) is an existence assertion for algebraic cycles, the second (Hodge standard conjecture) is a statement of positivity, generalising Weil's well-known positivity theorem in the theory of [[abelian varieties]]" ([Grothendieck - 68](#Grothendieck68)).

"But in the category of [[pure motives]], from the start one dealt only with [[algebraic cycles]], represented by correspondences, and it was intuitively not at all clear how on earth they could convey information about transcendental cycles. Indeed, the main function of the "Standard Conjectures" was to serve as a convenient bridge from algebraic to transcendental. Everything that one could prove without them
was indeed "plus ou moins trivial" – until people started treating correspondences
themselves using sophisticated [[homological algebra]] (partly generated by the development of  ́étale cohomology and Grothendieck–Verdier's introduction of derived
and [[triangulated categories]]). ([Y. Manin - 2014](#YManin2014)).

They were also followed by the [[Beilinson conjectures]]".

## Conclusions

The proof of the two standard conjectures would yield results going considerably further than Weil's conjectures. They would form the basis of the so-called "theory of [[motive]]s" which is a systematic theory of "arithmetic properties" of [[algebraic variety]](ies), as embodied in their groups of classes of cycles for numerical equivalence. We have at present only a very small part of this theory in dimension one, as contained in the theory of [[abelian variety]](ies).
Alongside the problem of resolution of singularities, the proof of the standard conjectures seems to me to be the most urgent task in algebraic geometry.   ([Grothendieck - 68](#Grothendieck68))

## References

* {#Grothendieck68}[[A. Grothendieck]], _Standard Conjectures on Algebraic Cycles_, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford University Press, pp. 193–199, ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/StandardConjs.pdf))
* [[Steven Kleiman]], _Algebraic cycles and the Weil conjectures_, in: Dix exposés sur la cohomologie des schémas, North-Holland, pp. 359–386, 1968, [MR0292838](http://www.ams.org/mathscinet-getitem?mr=0292838); _The standard conjectures_, Motives (Seattle, WA, 1991), Proc. of Symposia in Pure Math. __55__, American Mathematical Society, pp. 3–20, 1994, [MR1265519](http://www.ams.org/mathscinet-getitem?mr=1265519)
* {#YManin2014}[[Yuri Manin]], _Forgotten Motives: the Varieties of Scientific Experience_ ([arXiv:1402.2155](http://arxiv.org/abs/1402.2155))
* [[Alexander Beilinson]], _Remarks on Grothendieck's standard conjectures_ ([arXiv:math/1006.116](http://arxiv.org/abs/1006.1116))
* [[James Milne]], _Polarizations and Grothendieck's standard conjectures_ ([ 	arXiv:math/0103175](http://arxiv.org/abs/math/0103175))
* [[Yves Andre]], _Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses 17, Paris: Société Mathématique de France (2204)_ ([pdf](http://tomlr.free.fr/Math%E9matiques/Andre,%20Y%20-%20Une%20Introduction%20aux%20Motifs%20%28SMF%202004%29.pdf))
* Mathoverflow, _Progress on the standard conjectures on algebraic cycles_ ([pdf](http://mathoverflow.net/questions/176122/progress-on-the-standard-conjectures-on-algebraic-cycles))
* [[Voevodsky]], _From motives to motivic homotopy types_, Visions in mathematics, Tel Aviv, August-September 1999 ([pdf](http://www.math.ias.edu/vladimir/files/from_motives_to_motivic_homotopy.pdf))
* [[K. Voelkel]], _Seminar on motives, Standard conjectures_ ([pdf](http://www.konradvoelkel.com/wp-content/uploads/talk7-conjectures.pdf))
* Secret blogging seminar, _The Weil conjectures: the approach via the standard conjectures_ ([pdf](https://sbseminar.wordpress.com/2010/05/03/the-weil-conjectures-the-approach-via-the-standard-conjectures/))
* wikipedia [standard conjectures on algebraic cycles](http://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles)
* [[Serre]], Analogues Kähleriens de certaines conjectures de Weil (extrait d'une lettre a A. Weil, 9 Nov. 1959). ([pdf](https://agrothendieck.github.io/divers/serre59kahscan.pdf))

For the [[Beilinson conjectures]], see the references there.

[[!redirects standard conjecture]]
[[!redirects standard conjectures on algebraic cycles]]


standard Courant algebroid

#Contents#
* table of contents
{:toc}

## Idea

A _standard Courant Lie algebroid_ of a [[manifold]] $X$ is a type of [[Courant algebroid]] constructed from the [[tangent bundle]] and [[cotangent bundle]] of $X$. This is the principal algebraic structure studied in [[generalized complex geometry]].

## Definition

Recall from the discussion at [[Courant algebroid]] that there are the following two equivalent definitions of Courant algebroids:

* as a [[vector bundle]] equipped with a bracket and a bilinear form on its space of sections, satisfying various identities;

* as a [[L-infinity algebroid|Lie 2-algebroid]] equivalently encoded in its [[Chevalley?Eilenberg algebra]], equivalently the function algebra on a certain type of [[dg-manifolds]].

### As a vector bundle with extra structure

In the first perspective a _standard Courant algebroid_ of a [[manifold]] $X$ is the vector bundle $E = T X\oplus T^* X$ -- the fiberwise [[direct sum]] of the [[tangent bundle]] and the cotangent bundle -- with

* bilinear form

  $$
    \langle X + \xi , Y +\eta \rangle = \eta(X) + \xi(Y)
  $$

  for $X,Y \in \Gamma(T X)$ and $\xi, \eta \in \Gamma(T^* X)$

* brackets

  $$
    [X + \xi, Y + \eta] = [X,Y] + \mathcal{L}_X \eta
    - \mathcal{L}_Y \xi + \frac{1}{2} d (\eta(X) - \xi(Y))
  $$

  where $\mathcal{L}_X \eta = \{d,\iota_X\} \eta$ denotes the [[Lie derivative]] of the 1-form $\eta$ by the vector field $X$.

### As a dg-manifold

As an [[dg-manifold]] a standard Courant algeebroid is is $\Pi T^* \Pi T X$, the shifted cotangent bundle of the [[shifted tangent bundle]],

where the differential (homological vector field) is on each local coordinate patch $\mathbb{R}^n \simeq U \subset X$ with coordinates

* $\{x^i\}$ in degree 0

* $\{d x^i\}$ and $\{\theta_i\}$ in degree 1

* and $\{p_i\}$ in degree 2

given by

  $$
    \begin{aligned}
      d_C &= d_{dR} + p_i \frac{\partial}{\partial \theta_i}
      \\
      &= d x^i \frac{\partial}{\partial x^i }
       + p_i \frac{\partial}{\partial \theta_i}
    \end{aligned}
    \,.
  $$

### As a Lie 2-algebroid

We may read the above [[dg-algebra]] as the [[Chevalley?Eilenberg algebra]] $CE(\mathfrak{c}(X))$ of the [[Lie ∞-algebroid|Lie 2-algebroid]] $\mathfrak{c}(X)$, the specification of which entirely specifies the Lie 2-algebroid itself.

More on this in the discussion below.

## Properties

### As the Atiyah Lie 2-algebroid of a $U(1)$-gerbe

A standard Courant algebroid may be understood as being related to $\mathbf{B}U(1)$ [[principal 2-bundle]]s ($U(1)$-[[gerbe]]s) as an [[Atiyah Lie algebroid]] is related to a $U(1)$-[[principal bundle]].

(...explain...)

### Connections and generalized Riemannian metrics

Write $\mathfrak{c}(X)$ for the standard Courant algebroid of the manifold $X$. It comes canonically equipped with a projection down to the [[tangent Lie algebroid]] $T X $ of $X$:

$$
  \pi : \mathfrak{c}(X) \to X
  \,.
$$

A [[section]]

$$
  \sigma : T X \to \mathfrak{c}(X)
$$

of this morphism of [[Lie ∞-algebroid]]s is often called a _connection_ on $\mathfrak{c}(X)$. One may regard it as being special flat [[schreiber:∞-Lie algebroid valued differential forms|∞-Lie algebroid valued differential form]] data on $X$.

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

On base manifolds of the form $X = \mathbb{R}^n$ sections of $\mathfrak{c}(X) \to T X$ in the 1-[[category]] of [[Lie ∞-algebroid]]s are in natural bijection with rank-2 tensor fields on $X$, i.e. with sections $q \in \Gamma(T X \oplus T X)$.

=--

The proof is straightforward and easy, but spelling it out in detail also serves to establish concepts and notation for the treatment of the Courant algebroid in terms of its [[Chevalley?Eilenberg algebra]].

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

The [[Chevalley?Eilenberg algebra]] of the Lie 2-algebroid $\mathfrak{c}(\mathbb{R}^n)$ is the [[semifree dga]] whose underlying algebra is the [[Grassmann algebra]]

$$
  CE(\mathfrak{c}(X))
  =
  \left(
  \wedge_{C^\infty(X)}^\bullet
  ( \langle \xi^i \rangle_{i=1}^n
    \oplus
    \langle \theta_i \rangle_{i=1}^n
    \oplus
    \langle p_i \rangle_{i=1}^n
  )
  \,,
  d_{\mathfrak{c}(X)}
  \right)
$$

where the generators $\xi_i$ and $\theta_i$ are in degree 1 and the $p_i$ in degree 2, equipped with the differential $d_{\mathfrak{c}(X)}$ that is defined on generators by

$$
  d_{\mathfrak{c}(X)} : x^i =  \xi^i
$$

$$
  d_{\mathfrak{c}(X)} : \xi^i = 0
$$

$$
  d_{\mathfrak{c}(X)} : \theta_i = p_i
$$

$$
  d_{\mathfrak{c}(X)} : p_i = 0
  \,
$$

where $\{x^i\}_{i=1}^n$ are the canonical coordinate functions on $\mathbb{R}^n$.

The [[Chevalley?Eilenberg algebra]] of the [[tangent Lie algebroid]] $T X$ is the [[deRham complex]]

$$
  CE(T X ) = (\Omega^\bullet(X), d_{dR})
  \,.
$$

The morphism $\mathfrak{c}(X) \to T X$ is given by the [[dg-algebra]] morphism

$$
  (\Omega^\bullet(X),d_{dR}) \hookrightarrow
  CE(\mathfrak{c}(X))
$$

that is the identity on $C^\infty(X)$ and identifies the $\xi^i$ with the deRham differentials of the standard coordinate functions

$$
  d x^i \mapsto \xi^i
  \,.
$$

A section $\sigma : \mathfrak{c}(X) \to T X$ is accordingly a dg-algebra morphism

$$
  \sigma^* : CE(\mathfrak{c}(X)) \to (\Omega^\bullet(X), d_{dR})
  \,.
$$

Being a section, it has to be the identity on $C^\infty(X)$ and send $\xi^i \mapsto d_{dR} x^i$.

The image of the generators $\theta_i$, being of degree 1, must be a linear combination over $C^\infty(X)$ of the degree-1 elements in $\Omega^\bullet(X)$, i.e. must be 1-forms on $X$. This defines the rank-2 tensor $q$ in question by

$$
  \hat{t}_i \mapsto q_{i j} \d x^i
  \,.
$$

For this assignment to qualify as part of a morphism of dg-algebras, it has in addition to be compatible with the differential. The condition is that for all $i$ we have the equality in the bottom right corner of

$$
  \array{
     \theta_i &\stackrel{d_{\mathfrak{c}(X)}}{\mapsto}&& p_i
     \\
     \downarrow^{\mathrlap{\sigma^*}}
     &&& \downarrow^{\mathrlap{\sigma^*}}
     \\
     q_{i j} d x^j &\stackrel{d_{dR}}{\mapsto}&
     (\partial_k q_{i j}) d x^k \wedge d x^j
     = &
     s^*(p_i)
  }
  \,.
$$

This uniquely fixes the image under $\sigma^*$ of the generators $p_i$ and the differential is respected. So, indeed, the section $\sigma^*$ is specified by the tensor $q \in \Gamma(T X \otimes T X)$ and every such tensor gives rise to a section.

=--

The rank-2 tensor $q$ appearing in the above may be uniquely writtes as sum of a symmetric and a skew-symmetric rank-2 tensor $g \Gamma(Sym^2(T X))$ and $b \in \Gamma(\wedge^2 T X)$

$$
  q = g + b
  \,.
$$

If the symmetric part happens to be non-degenerate, it may be regarded as a (possibly [[pseudo-Riemannian metric|pseudo]]-)[[Riemannian metric]]. In this case the combination $q = g + b$ is called a **generalized Riemannian metric** in [[generalized complex geometry]].

### Canonical $\infty$-Lie algebroid 3-cocycle

The standard Courant albebroid $\mathfrak{c}(X)$ is canonically equipped with the [[Lie ∞-algebroid cohomology|Lie ∞-algebroid 3-cocycle]] $\mu \in CE(\mathfrak{c}(X))$ that is on a local patch $\mathbb{R}^n \simeq U \to X$ given by

$$
  \mu|_U = \xi^i \wedge p_i
  \,.
$$

### Morphisms between standard Courant algebroids

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

In the 1-[[category]] of [[Lie ∞-algebroid]]s, [[automorphism]]s of the standard Courant algebroid of a cartesian space, $\mathfrak{c}(\mathbb{R}^n)$, that

* respect the projection $\mathfrak{c}(X) \to T X$

  $$
     \array{
       \mathfrak{c}(X) &&\stackrel{f}{\to}&& \mathfrak{c}(X)
       \\
       & \searrow && \swarrow
       \\
       && T X
     }
  $$

* fix the canonical 3-cocycle $\mu = \xi^i p_i$

come from (...say this more precisely...) rank-2 tensors $q = g + b$ such that the skew symmetric part $b$ is a closed 2-form, $d_{dR} b = 0$.

=--

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

With the same kind of reasoning as above, we find that the action on the generators $\theta_i$ and $p_i$ is of the form

$$
  \array{
     \theta_i &\stackrel{d_{\mathfrak{c}(X)}}{\mapsto}&& p_i
     \\
     \downarrow^{\mathrlap{f^*}}
     &&& \downarrow^{\mathrlap{f^*}}
     \\
     \theta_i + q_{i j} \theta^i  &
      \stackrel{d_{\mathfrak{c}(X)}}{\mapsto}&
     p_i
     +
     \partial_k q_{i j} \theta^k \wedge \theta^j
     = &
     f^*(p_i)
  }
  \,.
$$

For the 3-cocycle to be preserved, $f^*(\xi^i p_i) = \xi^i p_i$ we need that

$$
  0 =
  \partial_k q_{i j} \theta^i \wedge \theta^k \wedge \theta^j
 =
  \partial_k b_{i j} \theta^i \wedge \theta^k \wedge \theta^j
 =
  \pi^*(d_{dR} b)
  \,.
$$

=--

## Related concepts

* [[higher Atiyah groupoid]]

  * [[Atiyah groupoid]]

  * [[Courant 2-groupoid]]

## References

The description of the standard Courant algebroid in its incarnation as an [[dg-manifold]] is given for instance in section 5 of

* [[Dmitry Roytenberg]], _On the structure of graded symplectic supermanifolds and Courant algebroids_ ([arXiv:0203110](http://arxiv.org/abs/math/0203110))

[[!redirects standard Courant Lie 2-algebroid]]


standard model of cosmology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The standard [[model (in theoretical physics)]] for the [[observable universe]] on the largest length scales of [[cosmology]]:

it is an [[cosmic inflation|inflationary]] [[FRW spacetime]] with [[cosmological constant]] ("[[dark energy]]") and cold [[dark matter]]. The technical term for this is the _$\Lambda$ CDM concordance model_ (where "$\Lambda$" is the standard symbol for the [[cosmological constant]] and "CDM" is for "cold dark matter").

The current model assumes that the energy density of the [[observable universe]] consists of

* 4.9% [[baryon|baryonic]] (visible) [[matter]]

* 26.8% [[dark matter]]

* 68.3% [[dark energy]].

<img src="http://ncatlab.org/nlab/files/DarkEnergyConcordance.gif" width="400">

(e.g. [Einasto 09, fig 17](#Einasto09), [here](https://ned.ipac.caltech.edu/level5/Sept09/Einasto/Einasto6.html))

## Properties

### Structure formation

Computer simulation of [[cosmic structure formation]] on scales larger than that of [[galaxies]] had always shown very good agreement of the $\Lambda CDM$ [[standard model of cosmology]] and observation.

There used to be various discrepancies of [[cold dark matter]]-[[model (in theoretical physics)|models]] on the scale of [[galaxies]]

  * [core-cusp behaviour](dark+matter#CuspCoreProblem).

  * [[galaxy rotation curves]]

  * [[Tully-Fisher relation]]

But recent analysis seems to show that more fine-grained analysis shows that [[cold dark matter]]-[[model (in theoretical physics)|models]] match all of these observations well. See behind the above links for more.

### Vacuum energy and Cosmological constant

The [[renormalization]] freedom in [[perturbative QFT|perturbative]] [[quantization]] of [[gravity]] ([[perturbative quantum gravity]]) induces freedom in the choice of [[vacuum expectation value]] of the [[stress-energy tensor]] and hence in the [[cosmological constant]].

For review see [Hack 15, section 3.2.1](#Hack15)

For more see at [[cosmological constant]] [here](cosmological+constant#InPerturbativeQuantumGravity).

## Related concepts

* [[dark energy]], [[dark matter]], [[MOND]]

* [[structure formation]], [[inhomogeneous cosmology]]

* [[baryogenesis]]

* [[standard model of particle physics]]

* [[Randall-Sundrum model]]

* [[theory of everything]]

* [string theory FAQ -- Does string theory tell us anything about cosmology, such as the Big bang or cosmic inflation?](string+theory+FAQ#InsightsIntoCosmology)

[[!include standard model of fundamental physics - table]]

## References

### General
 {#ReferencesGeneral}

See also the references at *[[cosmology]]* ([here](cosmology#ReferencesGeneral)).

Lecture notes:

* [[Matthias Blau]], chapter 38 of: *Lecture notes on general relativity* ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

Review:

* Jorge L. Cervantes-Cota, George Smoot, *Cosmology today -- A brief review*, AIP Conf. Proc. **1396** (2011) 28-52 [[arXiv:1107.1789](http://arxiv.org/abs/1107.1789), [doi:10.1063/1.3647524](https://doi.org/10.1063/1.3647524)]

* {#Mukhanov16} [[Viatcheslav Mukhanov]], *Quantum Universe*, Phys.Usp. 59 (2016) no.10, 1021-1027 ([spire:1507528](http://inspirehep.net/record/1507528), [doi:10.3367/UFNe.2016.07.037857](https://iopscience.iop.org/article/10.3367/UFNe.2016.07.037857/meta), [video recording](https://youtu.be/xUIrj_97i0A))

> In March 2013, following an accurate processing of available measurement data, the Planck Scientific Collaboration published the highest-resolution photograph ever of the early Universe when it was only a few hundred thousand years old. The photograph showed galactic seeds in sufficient detail to test some nontrivial theoretical predictions made more than thirty years ago. Most amazing was that all predictions were confirmed to be remarkably accurate. With no exaggeration, we may consider it established experimentally that quantum physics, which is normally assumed to be relevant on the atomic and subatomic scale, also works on the scale of the entire Universe, determining its structure with all its galaxies, stars, and planets.

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

Discussion of open problems (see also at *[[inhomogeneous cosmology]]*):

* Benoit Famaey, Stacy McGaugh, _Challenges for Lambda-CDM and MOND_ ([arXiv:1301.0623](http://arxiv.org/abs/1301.0623))

* {#BCKRW15} [[Thomas Buchert]], Alan A. Coley, [[Hagen Kleinert]], Boudewijn F. Roukema, David L. Wiltshire, _Observational Challenges for the Standard FLRW Model_, Int. J. Mod. Phys. D 25, 1630007 (2016) ([arXiv:1512.03313](https://arxiv.org/abs/1512.03313))

See also:

* Wikipedia, _[Lambda-CDM model](http://en.wikipedia.org/wiki/Lambda-CDM_model)_

* {#Einasto09} Jaan Einasto, _Dark matter_ ([arXiv:0901.0632](https://arxiv.org/abs/0901.0632)) 2009

### Possible tension with observation

Possible tensions in the model, regarding determination of the [[Hubble constant]] in the early and late universe, apparently observed by various groups, since the late 2010s:

review:

* L. Verde, T. Treu, [[Adam G. Riess]], _Tensions between the Early and the Late Universe_, Nature Astronomy (2019) [[arXiv:1907.10625](https://arxiv.org/abs/1907.10625), [doi:10.1038/s41550-019-0902-0](https://doi.org/10.1038/s41550-019-0902-0)]

* {#SWP22} Snowmass White Paper: *Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies*, J. High En. Astrophys. **34** (2022) 49-211 [[arXiv:2203.06142](https://arxiv.org/abs/2203.06142), [doi:10.1016/j.jheap.2022.04.002](https://doi.org/10.1016/j.jheap.2022.04.002)]

Indication that the tension is not in the data but in systematic errors:

* Freedman et al. _The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch_ ([arxiv:1907.05922](https://arxiv.org/abs/1907.05922))

Indication from new data from the [[James Webb Space Telescope]] that the old data does not have systematic errors and the tension is in the data:

* [[Wenlong Yuan]], [[Adam G. Riess]], [[Stefano Casertano]], [[Lucas M. Macri]], *A First Look at Cepheids in a SN Ia Host with JWST* ([arXiv:2209.09101](https://arxiv.org/abs/2209.09101))

Review of further observed deviations (such as large-scale [[inhomogeneous cosmology|inhomogeneties]] like the [[KBC void]]):

* {#Peebles22} [[Phillip James E. Peebles]], *Anomalies in Physical Cosmology* [[arXiv:2208.05018](https://arxiv.org/abs/2208.05018)]

There are possible tensions with high redshift [[galaxies]] detected at the [[James Webb Space Telescope]]:

* Michael Boylan-Kolchin, *Stress Testing ΛCDM with High-redshift Galaxy Candidates*, ([arXiv:2208.01611](https://arxiv.org/abs/2208.01611))

For testing the standard model of cosmology in light of the [[James Webb Space Telescope]], see

* Nikita Lovyagin, Alexander Raikov, Vladimir Yershov, Yuri Lovyagin, *Cosmological Model Tests with JWST*. ([arXiv:2212.06575](https://arxiv.org/abs/2212.06575))

On the possibility that these tensions could, in part, by a sign/cause of [[inhomogeneous cosmology]] violating the widely assumed "cosmological principle":

* Pavan Kumar Aluri, et al., *Is the Observable Universe Consistent with the Cosmological Principle?*, Classical and Quantum Gravity **40** 9 (2023) [[doi:10.1088/1361-6382/acbefc](https://doi.org/10.1088/1361-6382/acbefc), [arXiv:2207.05765](https://arxiv.org/abs/2207.05765)]

### In AQFT on Curved spacetimes

Discussion in the rigorous context of[[AQFT on curved spacetimes]] includes

* [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], _Quantum field theory on curved spacetime and the standard cosmological model_ ([arXiv:1308.6773](http://arxiv.org/abs/1308.6773))

* {#Hack13} [[Thomas-Paul Hack]], _The Lambda CDM-model in quantum field theory on curved spacetime and Dark Radiation_ ([arXiv:1306.3074](https://arxiv.org/abs/1306.3074))

For review see

* {#Hack15} [[Thomas-Paul Hack]], section 3.2.1 _Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes_, Springer 2016 ([arXiv:1506.01869](https://arxiv.org/abs/1506.01869), [doi:10.1007/978-3-319-21894-6](https://doi.org/10.1007/978-3-319-21894-6))


standard model of fundamental physics - table

**[[standard model of particle physics]] [[standard model of cosmology|and cosmology]]**

| [[theory (physics)|theory]]: | [[Einstein-Hilbert action|Einstein]]- | [[Einstein-Yang-Mills theory|Yang-Mills]]- |  [[Einstein-Yang-Mills-Dirac theory|Dirac]]- | [[Einstein-Yang-Mills-Dirac-Higgs theory|Higgs]] |
|--|----------|------------|--------|-------|
| |  [[gravity]] | [[electroweak field|electroweak]] and [[strong nuclear force]] | [[fermion|fermionic]] [[matter]] | [[scalar field]] |
| [[field (physics)|field]] content: | [[vielbein field]] $e$ | [[principal connection]] $\nabla$  |  [[spinor]] $\psi$ |  [[scalar field]] $H$ |
| [[Lagrangian]]: | [[scalar curvature]] density | [[field strength]] squared | [[Dirac operator]] component density | [[field strength]] squared + potential density |
| $L = $ | $R(e) vol(e) + $ | $\langle F_\nabla \wedge \star_e F_\nabla\rangle + $ |  $ (\psi , D_{(e,\nabla)} \psi) vol(e) + $ |  $ \nabla \bar H \wedge \star_e \nabla H  + \left(\lambda  {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e) $ |


standard model of particle physics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **standard [[model (in particle physics)|model]] of [[particle physics]]** is a [[model (in particle physics)]]: a [[quantum field theory]] that describes the _[[fundamental particles]]_ currently experimentally known, containing

* **[[matter]]**: [[electrons]], [[neutrinos]], [[quarks]] and other [[fermions]],

as well as three of the four _fundamental [[forces]]_ as currently known, which, somewhat roughly, are

* **[[force]]**:  the [[electroweak field|electroweak force]] (including [[electromagnetism]]) and the [[strong nuclear force]].

It is defined as a [[local Lagrangian]] [[field (physics)|field]] [[theory (physics)|theory]] which is an  [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]].

The main ingredient missing from the standard model is the [[quantum physics|quantum]] version of the field of [[gravity]]. For decades, a large part of theoretical physics has been absorbed with attempts to understand how this last of the known fundamental forces might fit into the picture.

As a [[quantum field theory]], the standard model is in particular a [[Yang-Mills theory|Yang–Mills]] [[gauge theory]] with [[spinors in Yang-Mills theory|spinors in Yang–Mills theory]].

Although there are several approaches to formulate a mathematically precise definition of what a [[quantum field theory]] is, there is no rigorous formulation (yet) that comprises the whole standard model.

## Properties

### Gauge group
  {#GaugeGroup}

The _exact_ [[gauge group]] of the standard model is not quite the [[product group]]

$$
  U(1) \times SU(2) \times SU(3)
$$

of the [[circle group]] with [[special unitary groups]], but is the [[quotient group]]

\[
  \label{ExactGSM}
  G_{SM}
  \;=\;
  \big(
    U(1) \times SU(2) \times SU(3)
  \big)
  / \mathbb{Z}_6
\]

of that by a [[cyclic group]] $\mathbb{Z}_6 \subset U(1) \times SU(2) \times SU(3)$ which is the [[subgroup]] [[generators and relations|generated]] from an element of the form

$$
  (q_6, q_2 \mathbf{1}_2, q_3 \mathbf{1}_3) \;\in\;
  U(1) \times SU(2) \times SU(3)
  \,,
$$

where $q_n \in U(1)$ denotes an $n$th [[primitive root of unity]], i.e.

$$
  \left(
    e^{2 \pi i \tfrac{1}{6}}
    \;,\;
    e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2\;,\;
    e^{2 \pi i \tfrac{1}{3}} \mathbf{1}_3
  \right)
  \;\in\;
  U(1) \times SU(2) \times SU(3)
  \,.
$$

(See [Baez-Huerta 09, p. 33-36](#BaezHuerta09) for a fairly comprehensive discussion. See also e.g. [HMY 13, p. 2](#HMY13).)

Strikingly, this  exact gauge group (eq:ExactGSM) of the standard model of particle physics happens to coincide with...

1. ...the group

   $$
      G_{SM}
       \;\simeq\;
      S\big(U(2) \times U(3)\big)
       \;\subset\;
      SU(5)
   $$

   of [[determinant]]-1 elements in the [[direct product group]] $U(2) \times U(3)$, which makes manifest that the standard model gauge group is a [[subgroup]] of the [[simple Lie group]] [[special unitary group|SU(5)]].

   This is the basis of "[[grand unified theories]]" ([[GUT]]), speculative extensions of the standard model trying to understand its gauge group as being a [[spontaneously broken symmetry|spontaneously broken]] _[[simple Lie group]]_-[[symmetry]].

1. ...the [[subgroup]] of the [[Jordan algebra]] [[automorphism group]] of the [[octonions|octonionic]] [[Albert algebra]] that "[[stabilizer subgroup|stabilizes]] a 4d sub-[[Minkowski spacetime]]" (see [there](Albert+algebra#StabilizerOf4dMinkowskiInsideOctonionicAlbertAlgebra) for details).

   More concretely, it is identified with the [[subgroup]] of [[Spin(9)]] which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ ([Krasnov 19](Spin%289%29#Krasnov19), see also at [SO(10)-GUT](GUT#TheGUTKnownAsSO10))

  This is part of ongoing speculation that  [[octonion|octonionic]] [[exceptional structures]] might be behind the standard model.

### Fermion content

[[!include flavors of fundamental particles -- table]]

## Tension with experiment and Incompleteness
 {#TensionWithExperiment}

The standard model of particle physics is supported by a tremendous level of confirmation by [[experiment]], lately by the [[LHC]] experiment. But there are indictations of discrepancies, pointing to "new physics". See at

* [[flavour anomaly]]

* [[anomalous magnetic moment]] -- [anomalies](anomalous+magnetic+moment#Anomalies).

Not quite in tension with the standard model, but still possibly pointing to unaccounted effects is the

* [[Higgs field]]-[metastability](Higgs+field#MassAndVacuumInstability).

Then there are experimental phenomena which are known to be implied by the standard model by way of computer simulation ([[lattice QCD]]), but which are not conceptually understood:

* [[confinement]] ([[mass gap problem]])

Finally there is the evident and notorious fact that the standard model simply does not include any [[quantum field theory]] of the fourth known fundamental [[force]], [[gravity]]. See at

* [[quantum gravity]]

for more on this.

<br/>

## Variations and generalizations

There is a plethora of attempts and suggestions for variations and generalizations of the standard model into models that may resolve the tension with experiment mentioned [above](#TensionWithExperiment) and which may be conceptually more satisfying from the point of view of models in theoretical physics.

### Kaluza--Klein theory

Shortly after the conception of [[general relativity]], it was observed by Kaluza and Klein that the force of [[gravity]] alone may effectively appear -- if considered on a [[spacetime]] that is a [[bundle]] whose [[fiber]] has a tiny volume (as meaured by the [[Riemannian metric]]) -- as the field of gravity coupled to [[gauge field]]s on the base of the bundle. For details on this see [[Kaluza-Klein mechanism]].

The huge conceptual simplification that this observation suggested had excited theoreticians early on, but a problem of Kaluza--Klein models is that not only does the "compactified" theory of gravity as if by magic emulate [[gauge field]]s, but it also always contains further scalar fields that are _not_ experimentally observed.

For that reason interest in Kaluza--Klein theories had decreased in the middle of the last century. Physics departments saw a major revival of the idea when [[string theory]] (see below) gained interest, since that theory necessarily exhibits a Kaluza--Klein mechanism. Incidentally, the problem of the spurious fields -- the moduli -- was still present in this approach. For more on this see the entry [[landscape of string theory vacua]].

### GUTs

One of the oldest studies of variations of the standard model is the investigation of [[GUT|grand unified theories]] (GUTs), which are [[Yang-Mills theory|Yang–Mills theories]] that instead of the standard model gauge group have a bigger gauge group which is however a [[simple group]].

### Noncommutative geometry
 {#NCGeometry}

A widespread perception is that some of the conceptual problems with the standard model point to the fact that some basic assumption of 20th century physics on the nature of reality is oversimplified. According to the approach of [[noncommutative geometry]], modeling [[spacetime]] as a smooth [[manifold]] is an oversimplification that makes itself felt when the quantization of the force of [[gravity]] becomes relevant.

In a class of "noncommutative" generalizations of the standard model, spacetime is therefore replaced more generally by a [[spectral triple]] that models a possibly "noncommutative space". One of the more successful approaches in this direction is the [[Connes-Lott-Chamseddine model]]. This effectively is a [[Kaluza-Klein theory]] (see above), but with the crucial difference that the fiber in the KK-picture is a highly non-classical non-commutative space, whose classical dimension is that of a point, but whose intrinsic dimension is 6. (This is incidentally the same value of the internal dimension as suggest by [[string theory]].)

For more on this see

* [[Urs Schreiber]], _Connes on Spectral Geometry of the Standard Model_ ([part I](http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of.html), [part II](http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_1.html) [part III](http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_2.html), [part IV](http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_3.html))

### String theory

A more drastic theoretical modifications to the standard model is proposed in the context of [[string theory]], where the entire concept of [[quantum field theory]] is proposed to be refined by something else. As opposed to [[GUTs]], this approach at least suggests a way in which also the fourth remaining force field of [[gravity]] could be incorporated into the picture.

See

* [[intersecting D-brane models]]

* [[G2-MSSM]]

## Related concepts

* [[quantum field theory]]

  * [[perturbative quantum field theory]]

  * [[non-perturbative quantum field theory]]

* [[forces]]

  * [[electromagnetism]]

  * [[strong nuclear force]]

  * [[weak nuclear force]]

    * [[CKM matrix]]

* [[chiral fermions]]

* [[chiral perturbation theory]], [[quantum hadrodynamics]]

* [[flavor (particle physics)|flavoru]] [[generation of fundamental particles]]

* [[Yang-Mills theory]]

  * [[QCD]], [[electroweak field]], [[electromagnetism]]

* [[technicolor]], [[supersymmetry]]

* [[Grand Unified Theory]], [[theory of everything]]

* [[gauge coupling unification]]

* relevant [[experiments]]

  * [[LHC experiment]]

  * [[LHCb experiment]]

  * [[Belle collaboration]]

* anomalies

  * [anomalous magnetic moment of the muon](anomalous magnetic moment#Anomalies)

  * [[flavour anomaly]]

  * [[V_cb puzzle]]

* [[phenomenology]], [[hierarchy problem]]

* [[MSSM]]

  * [[G2-MSSM]]

* [[intersecting D-brane model]]

* [[standard model of cosmology]]

* [string theory FAQ -- Did string theory provide any insight relevant in experimental particle physics?](string+theory+FAQ#InsightInParticlePhysics)

[[!include standard model of fundamental physics - table]]

## References

### General

Textbook accounts:

* W. Noel Cottingham, ; Derek A. Greenwood, _An introduction to the standard model of particle physics_, Cambridge University Press 2012 ([doi:10.1017/CBO9780511791406 ](   https://doi.org/10.1017/CBO9780511791406 ), [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1126.81002&format=complete))

*  [[Paul Langacker]], _The Standard Model and Beyond_, CRC Press 2009, 2012, second edition 2017 ([spire:846915](http://inspirehep.net/record/846915/), [publisher webpage](https://www.crcpress.com/The-Standard-Model-and-Beyond/Langacker/p/book/9781498763219), [author webpage](http://www.sns.ias.edu/~pgl/SMB/))

*  J D Vergados, _The Standard Model and Beyond_, World Scientific 2017 ([doi:10.1142/10669](https://www.worldscientific.com/worldscibooks/10.1142/10669))

Review and introduction:

* [[Matthew Schwartz]], *Quantum Field Theory and the Standard Model*, Cambridge University Press (2014) [[ISBN:9781107034730](https://www.cambridge.org/de/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-field-theory-and-standard-model?format=HB&isbn=9781107034730), [doi:10.1017/9781139540940](https://doi.org/10.1017/9781139540940), [pdf](https://indico.ihep.ac.cn/event/11134/attachments/3783/4308/QFT-Schwartz.pdf)]

* [[Gerd Rudolph]], [[Matthias Schmidt]], Section 7.7 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

* Ben Gripaios, _Lectures: From quantum mechanics to the Standard Model_ ([arXiv:2005.06355](https://arxiv.org/abs/2005.06355))

Mathematical ([[representation theory|representation theoretic]]) review with an eye towards [[grand unified theory]]:

* [[Howard Georgi]], *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

* {#BaezHuerta09} [[John Baez]], [[John Huerta]], _The Algebra of Grand Unified Theories_, Bull. Am. Math. Soc. 47:483-552, 2010 ([arXiv:0904.1556](http://arxiv.org/abs/0904.1556))

and with an eye towards [[intersecting D-brane models]]L

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 1 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press (2012)

and with emphasis of the [[effective field theories]] involved ([[chiral perturbation theory]], [Fermi's theory](effective+quantum+field+theory#FermiTheoryOfWeakInteractions)):

* [[Luis Alvarez-Gaumé]], [[Miguel A. Vázquez-Mozo]], *Lectures on Field Theory and the Standard Model: A Symmetry-Oriented Approach* [[arXiv:2306.08097](https://arxiv.org/abs/2306.08097)]

On [[subgroups]] of Clifford algebras reminiscent of [[gauge groups]] of relevance in the [[standard model of particle physics]] and [[grand unified theories]] thereof:

* {#Wilson20} [[Robert A. Wilson]], *A group-theorist's perspective on symmetry groups in physics* ([arXiv:2009.14613](https://arxiv.org/abs/2009.14613))

* {#Wilson21} [[Robert A. Wilson]], *On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics*, Adv. Appl. Clifford Algebras 31, 59 (2021) ([doi:10.1007/s00006-021-01160-5]( https://doi.org/10.1007/s00006-021-01160-5))

See also:

* [Albert algebra -- possible relation to color gauge structure](Albert+algebra#ReferencesRelationStandardModel)

* [[Alain Connes]], [[Matilde Marcolli]], chapter I, section 12 of _[[Noncommutative Geometry, Quantum Fields and Motives]]_

Review with outlook "beyond the standard model":

* [[Paul Langacker]], _The standard model and beyond_, talk at TPFNP 2005 ([pdf](http://www.sns.ias.edu/~pgl/talks/neutron.pdf), [[Langacker05.pdf:file]])

* Dmitry Kazakov, _Beyond the Standard Model' 17_ ([arXiv:1807.00148](https://arxiv.org/abs/1807.00148))

* {#Wulzer19} Andrea Wulzer, _Behind the Standard Model_ ([arXiv:1901.01017](https://arxiv.org/abs/1901.01017))

* [[James Wells]], _The Once and Present Standard Model of Elementary Particle Physics_ ([arxiv:1911.04604](https://arxiv.org/abs/1911.04604))

### Anomaly cancellation
 {#ReferencesAnomalyCancellation}

Discussion of [[anomaly cancellation]] on the [[standard model of particle physics]]:

* {#HMY13} Takaaki Hashimoto, Mamoru Matsunaga, Kenta Yamamoto, _Quantization of hypercharge in gauge groups locally isomorphic but globally nonisomorphic to $SU(3)_c \times SU(2)_L \times U(1)_Y$_ ([arXiv:1302.0669](https://arxiv.org/abs/1302.0669))

* Nakarin Lohitsiri, [[David Tong]], _Hypercharge Quantisation and Fermat's Last Theorem_ ([arXiv:1907.00514](https://arxiv.org/abs/1907.00514))

  (relating to [[Fermat's last theorem]])

### History
 {#ReferencesHistory}

* {#Bjorken85} [[James Bjorken]], _The November Revolution: A Theorist Reminisces_, 1985 ([spire:214067](http://inspirehep.net/record/214067))

> The big international conference of $[$1974$]$ in
London was a turning point $[$...$]$ Ellis' catalog well reflected the state  of theoretical confusion and general disarray in trying to  interpret the $e^+ e^-$ data. But in the midst of all of this was a  talk by [[John Iliopoulos]] (I think I was there too). With  passionate zealotry, he laid out with great accuracy what we call  the [[standard model of particle physics|standard model]]. Everything was there: [[proton decay]], [[charm quark|charm]],  the GIM mechanism of course, [[QCD]], the $SU(2)\times U(1)$ [[electroweak  theory]], $SU(5)$ [[GUT|grand unification]], [[Higgs field|Higgs]], etc. It was all presented with absolute conviction and sounded at the time just a little mad, at least to me (I am a conservative).

* [[Tom Kibble]], _The Standard Model of Particle Physics_ ([arXiv:1412.4094](http://arxiv.org/abs/1412.4094))

* {#Iliopoulos16} [[John Iliopoulos]], _The making of the standard theory_, Adv.Ser.Direct.High Energy Phys. 26 (2016) 29-59 ([spire:1497884](http://inspirehep.net/record/1497884), [pdf](http://cds.cern.ch/record/2217096/files/9789814733519_0002.pdf))


standard site

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

# Contents
* table of contents
{:toc}

## Definition

A [[site]] is called a **standard site** if

1. the underlying category is a [[finitely complete category]];

2. its [[coverage]] is a [[subcanonical coverage]].

## Properties

Every [[sheaf topos]] has a standard site of definition. See [[site]] for more details.

## References

C2.1.11 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
  {#Johnstone}

[[!redirects standard site]]
[[!redirects standard sites]]


Stanislav Hronek

* [webpage](https://is.muni.cz/osoba/436691)

## related $n$Lab entries

* [[variational calculus]]

* [[jet bundle]]

* [[Lepage form]]

category: people


Stanisław Gołąb

[[!redirects Stanislav Golab]]

__Stanisław Gołąb__ was a Polish mathematician born 1902 in Travnik, Bosnia. He introduced one of the versions of the notion of a [[pseudogroup of transformations]], now sometimes referred to as the Gołąb pseudogroup. For this, see in particular

* W. Waliszewski, _Categories, groupoids, pseudogroups and analytical structures_, Rozprawy matematyczne XLV, Państwowe Wydawnictwo Naukowe, Warszawa 1965

He was working also in Netherlands with Schouten. He was particularly interested in tensor calculus and he authored the book _Rachunek tensorowy_ (Warszawa 1956) which has later been  translated from the Polish by Eugene Lepas as _Tensor calculus_ and published in 1974 by Elsevier.

* [biography page](http://www.matematycy.interklasa.pl/biografie/matematyk.php?str=golab) (in Polish)

category: people


Stanley Brodsky

* [Wikipedia page](https://en.m.wikipedia.org/wiki/Stanley_Brodsky)

## Selected writings

Introducing [[discretized light-cone quantization]]:

* [[Hans-Christian Pauli]], [[Stanley J. Brodsky]], *Solving field theory in one space and one time dimension*, Phys. Rev. D **32** (1985) 1993 [[doi:10.1103/PhysRevD.32.1993](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.32.1993)]

* {#PauliBrodsky85b} [[Hans-Christian Pauli]], [[Stanley J. Brodsky]], *Discretized light-cone quantization: Solution to a field theory in one space and one time dimension*, Phys. Rev. D **32** (1985) 2001 [[doi:10.1103/PhysRevD.32.2001](https://doi.org/10.1103/PhysRevD.32.2001)]

Applying [[discretized light-cone quantization]] to [[quantum electrodynamics]]:

* {#TangBrodskyPauli91} [[Andrew C. Tang]], [[Stanley J. Brodsky]], [[Hans-Christian Pauli]], *Discretized light-cone quantization: Formalism for quantum electrodynamics*, Phys. Rev. D **44** (1991) 1842 [[doi:10.1103/PhysRevD.44.1842](https://doi.org/10.1103/PhysRevD.44.1842)]

Applying [[light-front quantization]] to [[QCD]]:

* [[G. Peter Lepage]], [[Stanley J. Brodsky]], *Exclusive processes in perturbative quantum chromodynamics*, Phys. Rev. D **22** (1980) 2157 [[doi:10.1103/PhysRevD.22.2157](https://doi.org/10.1103/PhysRevD.22.2157)]

* [[Stanley J. Brodsky]], [[Gary McCartor]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *The challenge of light-cone quantization of gauge field theory*, Part. World **3** 3 (1993) 109-124  [[cds:240388](http://cds.cern.ch/record/240388), [inspire:335247](https://inspirehep.net/literature/335247)]

* [[Stanley Brodsky]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *Quantum Chromodynamics and Other Field Theories on the Light Cone*, Phys. Rept. **301** (1998) 299-486 [[arXiv:hep-ph/9705477](https://arxiv.org/abs/hep-ph/9705477), doi:10.1016/S0370-1573(97)00089-6]

* [[Stanley Brodsky]], [[Thomas Heinzl]] et al. *Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics*, White Paper of International Light Cone Advisory Committee, Nuclear Physics B - Proceedings Supplements **251**, **252** (2014) 165-174 [[doi:10.1016/j.nuclphysbps.2014.05.004](https://doi.org/10.1016/j.nuclphysbps.2014.05.004), [arXiv:1309.6333](https://arxiv.org/abs/1309.6333)]

and in relation to instant-time quantization:

* Philip D. Mannheim, Peter Lowdon, [[Stanley Brodsky]],
*Comparing light-front quantization with instant-time quantization*, Physics Reports **891** (2021) 1-65 [[arXiv:2005.00109](https://arxiv.org/abs/2005.00109), [doi:10.1016/j.physrep.2020.09.001](https://doi.org/10.1016/j.physrep.2020.09.001)]

Introducing [[holographic light front QCD]]:

* [[Stanley Brodsky]], [[Guy de Teramond]], _Light-Front Hadron Dynamics and AdS/CFT Correspondence_, Phys. Lett. B582:211-221, 2004 ([arXiv:hep-th/0310227](https://arxiv.org/abs/hep-th/0310227))

* [[Guy de Teramond]], [[Stanley Brodsky]], _Light-Front Holography: A First Approximation to QCD_, Phys. Rev. Lett. 102:081601, 2009 ([arXiv:0809.4899](https://arxiv.org/abs/0809.4899))

Review:

* [[Stanley Brodsky]], [[Guy de Teramond]],  [[Hans Günter Dosch]], [[Joshua Erlich]], _Light-Front Holographic QCD and Emerging Confinement_, Physics Reports Volume 584, 8 July 2015, Pages 1-105 ([arXiv:1407.8131](https://arxiv.org/abs/1407.8131))

* [[Stanley Brodsky]], [[Guy de Teramond]], [[Hans Günter Dosch]], _Light-Front Holography and Supersymmetric Conformal Algebra: A Novel Approach to Hadron Spectroscopy, Structure, and Dynamics_ ([arXiv:2004.07756](https://arxiv.org/abs/2004.07756))

On [[hadron supersymmetry]] via [[light cone gauge|light cone]] [[supersymmetric quantum mechanics]] in [[holographic light front QCD]];

* {#dTDB14}  [[Guy de Teramond]], [[Hans Günter Dosch]], [[Stanley Brodsky]], _Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding_, Phys. Rev. D 91, 045040 (2015) ([arXiv:1411.5243](https://arxiv.org/abs/1411.5243))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Superconformal Baryon-Meson Symmetry and Light-Front Holographic QCD_, Phys. Rev. D 91, 085016 (2015) ([arXiv:1501.00959](https://arxiv.org/abs/1501.00959))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum_, Phys. Rev. D 92, 074010 (2015) ([arXiv:1504.05112](https://arxiv.org/abs/1504.05112))

* {#BTDL16}  [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Cédric Lorcé, _Meson/Baryon/Tetraquark Supersymmetry from Superconformal Algebra and Light-Front Holography_, International Journal of Modern Physics AVol. 31, No. 19, 1630029 (2016) ([arXiv:1606.04638](https://arxiv.org/abs/1606.04638))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum II_, Phys. Rev. D 95, 034016 (2017) ([arXiv:1612.02370](https://arxiv.org/abs/1612.02370))

* {#NielsenBrodsky18} [[Marina Nielsen]], [[Stanley Brodsky]], _Hadronic Superpartners from Superconformal and Supersymmetric Algebra_, Phys. Rev. D 97, 114001 (2018) ([arXiv:1802.09652](https://arxiv.org/abs/1802.09652))

* [[Marina Nielsen]], [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Fernando S. Navarra, [[Liping Zou]], _Supersymmetry in the Double-Heavy Hadronic Spectrum_, Phys. Rev. D 98, 034002 (2018) ([arXiv:1805.11567](https://arxiv.org/abs/1805.11567))

Review:

* {#Brodsky18} [[Stanley Brodsky]], _Hadron Spectroscopy and Dynamics from Light-Front Holography and Superconformal Algebra_, Few-Body Syst 59, 83 (2018) ([arXiv:1802.08552](https://arxiv.org/abs/1802.08552))

* [[Stanley Brodsky]], *Supersymmetric and Other Novel Features of Hadron Physics from Light-Front Holography*,  Proceedings of the 24th Workshop, "What Comes Beyond the Standard Models" ([arXiv:2112.02453](https://arxiv.org/abs/2112.02453))

In view of [[tetraquarks]]:

* [[Hans Günter Dosch]], [[Stanley Brodsky]], [[Guy de Teramond]], [[Marina Nielsen]], [[Liping Zou]], _Exotic States in a Holographic Theory_ ([arXiv:2012.02496](https://arxiv.org/abs/2012.02496))

On the [[running coupling constant]] in [[QCD]]:

* [[Alexandre Deur]], [[Stanley Brodsky]], [[Guy de Teramond]], *The QCD Running Coupling*, Prog. Part. Nuc. Phys. 90 1 (2016) ([arXiv:1604.08082](https://arxiv.org/abs/1604.08082), [doi:10.1016/j.ppnp.2016.04.003](https://doi.org/10.1016/j.ppnp.2016.04.003))

category: people

[[!redirects Stanley J. Brodsky]]


Stanley Deser

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

## Selected writings

Making manifest the local [[worldsheet]] [[supersymmetry]] of the [[spinning string]] (via the supersymmetric [[Polyakov action]]):

* [[Stanley Deser]], [[Bruno Zumino]], *A complete action for the spinning string*, Physics Letters B **65** 4 (1976) 369-373 ([pdf](http://cdsweb.cern.ch/record/201648/files/CM-P00061857.pdf?version=1))

On [[3d gravity]] as a [[Chern-Simons theory]] and its [[holography|holographic]] relation to a [[2d CFT]] [[boundary field theory]] (well before [[AdS/CFT]] was conceived from [[string theory]]):

* [[Stanley Deser]], [[Roman Jackiw]], [[Gerard 't Hooft]], _Three-dimensional Einstein gravity: Dynamics of flat space_, Ann. Phys. 152 (1984) 220 (doi:10.1016/0003-4916(84)90085-X)

* [[Stanley Deser]], [[Roman Jackiw]], _Three-dimensional cosmological gravity: Dynamics of constant curvature_, Annals of Physics, Volume 153, Issue 2, 1 April 1984, Pages 405-416 (doi:10.1016/0003-4916(84)90025-3, [spire:192694](http://inspirehep.net/record/192694))

## Related $n$Lab entries

* [[positive energy theorem]]

* [[parent action functional]]

* [[quantum gravity]]

* [[3d quantum gravity]]

* [[spinning string]]

* [[11-dimensional supergravity]]

category: people


Stanley Kochman

[[!redirects Stanley Kochmann]]

* [webpage](http://www1.math.yorku.ca/new/node/55)

## Selected writings

On the [[symplectic cobordism ring]]:

* [[Stanley Kochman]], _The symplectic cobordism ring I_, Memoirs of the American Mathematical Society 1980, Volume 24, Number 271 ([doi:10.1090/memo/0271](http://dx.doi.org/10.1090/memo/0271))

* [[Stanley Kochman]], _The symplectic cobordism ring II_, Memoirs of the American Mathematical Society 1982
Volume 40, Number 271 ([doi:10.1090/memo/0271](http://dx.doi.org/10.1090/memo/0271))

On computation of [[stable homotopy groups of spheres]]:

*  _Stable Homotopy Groups of Spheres -- A Computer-Assisted Approach_, Lecture Notes in Mathematics, 1990

On [[cobordism theory]], [[stable homotopy theory]], [[complex oriented cohomology]], and the [[Adams spectral sequence]]:

* _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

category: people

[[!redirects Stan Kochmann]]


Stanley Mandelstam

* [webpage](http://www.physics.berkeley.edu/research/faculty/mandelstam.html)

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

* [[Charles B. Thorn]], _Scientific Biography of Stanley Mandelstam, Part I: 1955-1980_ ([arXiv:1612.01590](https://arxiv.org/abs/1612.01590))

## related entries

* [[S-matrix]]

* [[string scattering amplitude]]

category: people


star domain

#Contents#
* table of contents
{:toc}

## Idea

A star-shaped domain is a subset $U$ of an appropriate ambient space for which there exists a basepoint that can be connected to every other point by a "straight line in $U$".
Such a subset is a [[contractible space]], using the contraction that travels each point back to the basepoint along the given lines.

## Definition

For $X$ a [[real number|real]] [[affine space]], a **star domain** about a point $x\in X$ is an inhabited subset $U \subset X$ such that with $y \in X$, the straight line segment connecting $x$ with $y$ in $X$ is also contained in $U$.

For $V$ a [[vector space]], a star domain about the origin is an inhabited subset $U \subset V$ that is [[radial]].

These definitions can be modified in various obvious ways. For example, a **star shaped neighbourhood** of a point $x$ in an affine space $X$ is an open [[neighbourhood]] $U \subset X$ of $x$ that is a star domain about $x$. Or, a subset is a star domain if it is a star domain about one of its points.

A useful special case pertains to a [[simplicial complex]] $K$, where if $v$ is a vertex of $K$, then the **open star** of $v$ is the union of the interiors in ${|K|}$ of all the simplices containing $v$. Open stars of vertices provide a [[good open cover]] of a simplicial complex.

A [[convex set]] is the same as a set that is a star domain about each of its points.

## Related entries

* [[open ball]]

* [[good open cover]]

[[!redirects star-shaped]]
[[!redirects star shaped]]

[[!redirects star-shaped region]]
[[!redirects star shaped region]]

[[!redirects star-shaped neighborhood]]
[[!redirects star-shaped neighborhood]]
[[!redirects star shaped neighborhood]]
[[!redirects star-shaped neighbourhood]]
[[!redirects star shaped neighbourhood]]

[[!redirects star-shaped subset]]
[[!redirects star-shaped subsets]]

[[!redirects star domains]]
[[!redirects star-shaped domain]]
[[!redirects star-shaped domains]]


star exponential

* C. A. Alcalde, A. C. Cadavid, M. Nakashima, _The star exponential and $SU(2)$ path integrals_, Lett. Math. Phys. __22__, n. 1, 15-19, 1991 [doi](http://dx.doi.org/10.1007/BF00400373), [MR92m:81128](http://www.ams.org/mathscinet-getitem?mr=1121844)
* A. C. Cadavid, M. Nakashima, _The star exponential and path integrals on compact groups_, Lett. Math. Phys. __23__, n. 2, 111-115, 1991 [doi](http://dx.doi.org/10.1007/BF00703722), [MR93h:81074](http://www.ams.org/mathscinet-getitem?mr=1148501)
* M. Nakashima, _The star exponential and path integrals on compact groups. II. The Weyl character formula_, Lett. Math. Phys. __30__ (1994), no. 2, 119–123, [doi](http://dx.doi.org/10.1007/BF00939699)
* Carlos Alcalde, Daniel Sternheimer, _Analytic vectors, anomalies and star representations_, Lett. Math. Phys. __17__ (1989), no. 2, 117–127. [MR90h:22012](http://www.ams.org/mathscinet-getitem?mr=993017), [doi](http://dx.doi.org/10.1007/BF00402326)
* Jean-Marie Maillard, _Star exponentials for any ordering of the elements of the inhomogeneous symplectic Lie algebra_, J. Math. Phys. __45__ (2004), no. 2, 785–794, [MR2004m:81132](http://www.ams.org/mathscinet-getitem?mr=2029098), [doi](http://dx.doi.org/10.1063/1.1634352)
* H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, _Star exponential functions for quadratic forms and polar elements_, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), edited by T. Voronov, Contemp. Math. __315__ (A.M.S., Providence, RI, 2002), pp. 25–38, [MR2004b:53152](http://www.ams.org/mathscinet-getitem?mr=1958828)