Borel-Weil theorem

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## Idea

The _Borel-Weil-Bott theorem_ characterizes [[representations]] of suitable [[Lie groups]] $G$ as spaces of [[holomorphic function|holomorphic]] [[sections]] of [[complex line bundles]] over [[flag varieties]] $G/B$, for $B$ a [[Borel subgroup]].

With suitable qualifiers added this is Kirillov's _[[orbit method]]_, and the construction may be interpreted as sending a [[symplectic manifold]] equipped with $G$-[[Hamiltonian action]] to its [[geometric quantization]]. As discussed there, this is equivalently given by [[fiber integration]] in [[K-theory]] and accordingly the Borel-Weil-Bott theorem is naturally regarded in the context of [[generalized Schubert calculus]] (e.g. [Kumar 12](Kumar12)).

A common method of construction of [[representations]] of [[groups]] in [[representation theory]] is to consider the invariant subspaces of the [[induced representation]] (set-theoretic or $L^2$-version). The induced representation is too big and [[Frobenius reciprocity]] indicates that they are normally not [[irreducible representation|irreducible]]. Given a [[subgroup]] $B\subset G$ and a $B$-module $V$, with $\rho\colon B\to Aut(V)$ the induced module can be represented as a space of (set-theoretic or $L^2$‑) [[sections]] of the [[associated bundle]] $G\times_{G/B} V$ to the [[principal fiber bundle]] $G\to G/B$, at least when these words make sense. In [[geometric quantization]], the method to single out a sufficiently small space of sections is to look at sections which are horizontal in the sense of some [[polarization]], or equivalently horizontal for an appropriate choice of [[connection]] on the bundle.

The first instance is the theorem of Borel--Weil, (J-P. Serre, Bourbaki Seminar 100, 1953/54) which asserts that if $B$ is the [[Borel subgroup]] of the complex semisimple group $G^{\mathbb{C}}$ (which can be considered as the [[complexification]] of a [[compact Lie group]] $G$ with the [[maximal torus]] $T=G\cap B\subset G$), then all unitary irreducible representations can be obtained as the spaces of (anti)-holomorphic line bundles associated to the principal fibration $G\to G/B$ over the generalized [[flag variety]] $G^{\mathbb{C}}/B\cong G/T$ with the fiber $\mathbb{C}_\chi$, which is the $1$-dimensional representation corresponding to a dominant integral [[character]] $\chi$; and viceversa, all such spaces of sections are irreducible. The inner product is inherited from the hermitean structure on the line bundle.

There is an extension to higher cohomologies instead of spaces of sections, called the **Borel--Weil--Bott theorem** and numerous extensions, e.g. to Harish--Chandra sheaves to construct the infinite-dimensional representations. The original proof is by geometric and analytic methods; some of the modern extensions of the method use the algebraic [[D-module]] theory and are based on the [[Beilinson-Bernstein localization]] theorem. There are even extensions to [[quantum groups]].

## Related concepts

* [[orbit method]]

* [[Schubert calculus]]

## References

* Neil Chriss, [[Victor Ginzburg]], _Representation theory and complex geometry_, Birkhäuser 1994

* [[Jean-Pierre Serre]], _Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil)", Séminaire Bourbaki 100: 447–454, Paris: Soc. Math. France, 1953/54,  [numdam](http://www.numdam.org/item?id=SB_1951-1954__2__447_0)

* [[Bertram Kostant]], _Lie algebra cohomology and the generalized
Borel-Weil theorem_,

* [[Pierre Cartier]], _Remarks on "Lie algebra cohomology and the generalized Borel-Weil theorem" by B. Kostant_, Ann. of Math. _74_, 2,  1961 [pdf](http://www.math.tamu.edu/~jml/cartier61.pdf)

* [[Jacob Lurie]], _A proof of the Borel-Weil-Bott theorem_ ([pdf](http://www.math.harvard.edu/~lurie/papers/bwb.pdf))

Lecture notes include

* Wilfried Schmid (notes by Matvei Libine), _Geometric methods in representation theory_ ([pdf](http://www.math.harvard.edu/~schmid/articles/brussels_1_30_04.pdf))

* Shrawan Kumar, _Borel-Weil-Bott theorem and geometry of Schubert varieties_ ([pdf](http://www.unc.edu/math/Faculty/kumar/notes/kumar_notes_03.pdf))
 {#Kumar12}

* P. Woit, _Topics in representation theory: the Borel-Weil theorem_, ([pdf](http://www.math.columbia.edu/~woit/notes16.pdf)), _Quantum field theory and representation theory: A sketch_ ([pdf](http://www.math.columbia.edu/~woit/sketch.pdf))

* Lisa Jeffrey, _Remarks on geometric quantization and representation theory_ [pdf](http://www.math.toronto.edu/~jeffrey/mat1312/lec14.gq.pdf)

See also

* eom: [Bott-Borel-Weil theorem](http://eom.springer.de/b/b120400.htm)

* wikipedia: [Borel–Weil–Bott theorem](http://en.wikipedia.org/wiki/Borel–Weil–Bott_theorem), [Borel–Weil theorem](http://en.wikipedia.org/wiki/Borel–Weil_theorem)

[[!redirects Borel–Weil theorem]]
[[!redirects Borel--Weil theorem]]

[[!redirects Borel-Weil-Bott theorem]]


Borger's absolute geometry

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#Contents#
* table of contents
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## Idea

One proposal for a precise realization of the idea of "absolute" [[arithmetic geometry]] over [[F1|Spec(F1)]] is _Borger's absolute geometry_ ([Borger 09](#Borger09)). Here the structure of a [[Lambda-ring]] on a ring $R$, hence on its [[spectrum of a commutative ring|spectrum]] $Spec(R) \to Spec(\mathbb{Z})$,  is interpreted as a collection of lifts of all [[Frobenius morphisms]] and hence as [[descent]] data for descent to $Spec(\mathbb{F}_1)$ (which is defined thereby).  This definition yields an [[essential geometric morphism]] of [[gros topos|gros]] [[etale toposes]]

$$
  Et(Spec(\mathbb{Z}))
   \stackrel{\overset{}{\longrightarrow}}{\stackrel{\overset{}{\longleftarrow}}{\underset{}{\longrightarrow}}}
  Et(Spec(\mathbb{F}_1))
  \,,
$$

where on the right the notation is just suggestive, the [[topos]] is a suitable one over [[Lambda-rings]]. Here the middle [[inverse image]] is the [[forgetful functor]] which forgets the Lambda structure, and its [[right adjoint]] [[direct image]] is given by the [[ring of Witt vectors]] construction and may be thought of as producing [[arithmetic jet spaces]]. In this sense the [[adjoint triple]] here would be directly analogous to the [[base change]] along the [[unit of a monad|unit]] of an [[infinitesimal shape modality]] whose induced [[comonad]] is the [[jet comonad]].

This proposal seems to subsume many aspects of other existing proposals (see e.g. [Le Bruyn 13](#LeBruyn13)) and stands out as yielding an "absolute [[base topos]]" $Et(Spec(\mathbb{F}_1))$ which is rich and genuinely interesting in its own right.

## Motivation
 {#Motivation}

The following is an attempt to motivate or make intuitively clear why lifts of [[Frobenius morphisms]] may be related to "absolute geometry" over [[F1]].

First of all, the [[function field analogy]] says that $\mathbb{Z}$ is analogous to the [[polynomial ring]] $k[z]$ over a finite field $k$, as well as to the ring $\mathcal{O}_{\mathbb{C}}$ of ([[entire function|entire]]) [[holomorphic functions]] on the [[complex plane]].

To make this analogy more concrete, notice that one characteristic property of the rings $k[z]$ and $\mathcal{O}_{\mathbb{C}}$, witnessing
their affine-ness in one variable, is that they carry a canonical [[derivation]], namely $\frac{\partial}{\partial z}$, and that the [[ground field]] is recovered as the [[quotient]] by the [[ideal]] generated by $z$. More to the point, for each [[maximal ideal]] $(z-x)$ there is the
first order translation operator $\mathrm{id} + (z-x)\frac{\partial}{\partial (z-x)}$ and the quotient by its difference from the
identity is the [[residue field]] of functions at the point $x$.

Therefore if $\mathbb{Z}$ is analogous to $k[z]$ and to $\mathcal{O}_{\mathbb{C}}$, then it ought to
admit analogous operators, one for each of its [[maximal ideals]] $(p)$ given by a [[prime number]] $p \in \mathbb{Z}$.
Remarkably, such a collection of operations indeed exists on $\mathbb{Z}$: the $p$-power operations
$(-)^p : \mathbb{Z} \to \mathbb{Z}$ (acting on the underlying set of $\mathbb{Z}$) which by
[[Fermat's little theorem]] is indeed of the above form
$$
  (-)^p : n \mapsto n^p = n + p \cdot\partial_p n
  \,.
$$

Here the expression
$\partial_p n \in \mathbb{Z}$ is uniquely defined by this equation, it is given by the [[Fermat quotient]] operation

$$
  \partial_p : n \mapsto \partial_p n \coloneqq \frac{n^p - n}{p}
  \,.
$$

Hence by [[analogy]] it makes sense to think of $\partial_p : \mathbb{Z} \to \mathbb{Z}$ as being like a [[derivation]] on $\mathrm{Spec}(\mathbb{Z})$ -- it is called a _[[p-derivation]]_.
This is the beginning of the theory of _[[arithmetic differential equations]]_ ([Buium 05](#Buium05)).

More generally, a $k[z]$-algebra $A$ is to be thought of as a space $\mathrm{Spec}(A)$ over $\mathrm{Spec}(k[z])$.
The canonical [[derivation]] on the latter canonically lifts to the former, and is given by the same
formula: $\mathrm{id} + (z-x)\frac{\partial}{\partial (z-x)}$. This exhibits the fact that
$\mathrm{Spec}(A) \to \mathrm{Spec}(k[z])$ is simply a [[product]] of
$\mathrm{Spec}(A/(z)) \to \mathrm{Spec}(k)$ with the [[affine line]] $\mathbb{A}_k = \mathrm{Spec}(k[z])$ over the [[ground field]] $k$.
Analogously, any [[commutative ring]] $R$ is to be thought of as a space $\mathrm{Spec}(R)$ over $\mathrm{Spec}(\mathbb{Z})$ and the above arithmetic translation operator canonically
lifts to $\mathrm{Spec}(R)$, by the same formula: $a \mapsto a^p$.
However, inspecting this one finds that not only is the derivation-like part lifted non-trivially, but also the identity-part is lifted in general to some [[homomorphism]]: in particular if $R$ is an $\mathbb{F}_p$-[[associative algebra|algebra]], so that $\mathrm{Spec}(R)$
is [[support|supported]] over the point $\mathrm{Spec}(\mathbb{F}_p)$ in $\mathrm{Spec}(\mathbb{Z})$, then the standard fact that in [[positive characteristic|characteristic]] $p$ the
$p$-power operation is a ring homomorphism now means that
the derivation-like action vanishes here, as expected from the analogy, but that a pure homomorphism part covering the identity remains --  the [[Frobenius homomorphism]] $\mathrm{Frob}_p$.
If by some abuse of notation we allow ourselves to write [[Isbell duality|formally dual]] morphisms for maps between rings that are not necessarily homomorphisms, then this situation is, possibly, usefully visualized as follows.
$$
  \array{
    \mathrm{Spec}(R)
    &\stackrel{\mathrm{Frob}_p + 0}{\longrightarrow}
    &
    \mathrm{Spec}(R)
    \\
    \downarrow && \downarrow
    \\
    \mathrm{Spec}(\mathbb{F}_p)
    &\stackrel{\mathrm{id} + 0}{\longrightarrow}
    &
    \mathrm{Spec}(\mathbb{F}_p)
    \\
    \downarrow && \downarrow
    \\
    \mathrm{Spec}(\mathbb{Z})
    &\stackrel{\mathrm{id} + p \cdot \partial_p}{\longrightarrow}
    &
    \mathrm{Spec}(\mathbb{Z})
  }
$$

This suggests that we are to think of the operation $(-)^p = \mathrm{id} + p \cdot \partial_p$ on $\mathrm{Spec}(\mathbb{Z})$
not just as analogous to the identity transformation plus a derivation, but as analogous to the sum of a general finite transformation plus a derivation. In other words, a lift
of this operation to some $\mathrm{Spec}(R)$ is to be a choice of ring homomorphism
$\Phi : R \to R$ (a "finite translation")
in addition to a derivation-like operation $\partial_p : R \to R$ (the "infinitesimal translation"), such that
$$
  (-)^p = \Phi + p \cdot\partial_p
  \,.
$$
As before, when $p$ is invertible in $R$, hence away from the fiber over $\mathrm{Spec}(\mathbb{F}_p)$,
then $\partial_p$ is uniquely fixed by this equation once $\Phi$ is chosen, and hence
it is $\Phi$ alone which is to be chosen on $R$. A pair $(R,\Phi)$ satisfying the above
equation is equivalently a "$\Lambda_p$-ring" (see at _[[Lambda ring]]_). Or rather, by the analogy we are to lift
the whole collection of operators $\mathrm{id} + p \cdot\partial_p$ for all primes $p$ and hence are to ask that $\Phi$ satisfies the above equation for all $p$ and hence defines
for all primes $p$ the derivation-like operator (function on the underlying set of $R$)
$$
  \partial^\Phi_p  \coloneqq \frac{(-)^p - \Phi}{p} : R \to R
  \,.
$$
This is the general form of an "[[arithmetic jet space|arithmetic differential operator]]", see for instance the first pages of ([Buium 13](#Buium13)) for review.

In terms of the abuse of notation already employed previously, this situation is usefully visualized as follows.
$$
  \array{
    \mathrm{Spec}(R)
    &\stackrel{\Phi + p\cdot \partial^{\Phi}_p}{\longrightarrow}
    &
    \mathrm{Spec}(R)
    \\
    \downarrow && \downarrow
    \\
    \mathrm{Spec}(\mathbb{Z})
    &\stackrel{\mathrm{id} + p\cdot \partial_p}{\longrightarrow}
    &
    \mathrm{Spec}(\mathbb{Z})
  }
$$

A ring $R$ equipped with such an endomorphism $\Phi$ is equivalently a [[Lambda ring]] (an insight highlighted by [[James Borger]]).
In this way a $\Lambda$-ring structure
on a commutative ring $R$ is the [[arithmetic geometry]]-analog of exhibiting $\mathrm{Spec}(R) \to \mathrm{Spec}(\mathbb{Z})$ as being like a product by the [[affine line]] over the
non-existent field $\mathbb{F}_1$ of a map to "$\mathrm{Spec}(\mathbb{F}_1)$".
This should be one way to think of how $\Lambda$-rings
embody "geometry over $\mathbb{F}_1$" as proposed in ([Borger 09](#Borger09)).

Finally notice that the refinement of the $(-)^p$-operation from [[arithmetic geometry]] to [[E-infinity arithmetic geometry]] is given by the [[power operations]] in [[multiplicative cohomology theory]] ([Lurie, remark 2.2.9](#Lurie))

## Definition

> under construction

Write [[CRing]] for the [[category]] of (finitely generated) [[commutative rings]] and $\Lambda Ring$ for that of [[Lambda-rings]].

By the discussion at _[here](Lambda-ring#FreeAndCofreeLambdaRings)_
the [[forgetful functor]]
$U \;\colon\; \Lambda Ring \longrightarrow CRing$
from [[Lambda-rings]] to [[commutative rings]] has

* a [[left adjoint]], given by forming the ring $Symm$ of [[symmetric functions]];

* a [[right adjoint]] given by forming the [[ring of Witt vectors]] $W$.

$$
  (Symm \dashv U \dashv Witt)
  \;\colon\;
  \Lambda Ring
   \stackrel{\overset{Symm}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\overset{Witt}{\leftarrow}}}
  CRing
  \,.
$$

Hence

* [[rings of Witt vectors]] are the _[[co-free functors|co-free]] Lambda-rings;

* rings of [[symmetric functions]] are the [[free construction|free]] Lambda-rings.

Regarding $Ring^{op}$ as a [[site]] for [[arithmetic geometry]], the order of the adjoints is reversed by forming [[opposite categories]]

$$
  (Witt \dashv U \dashv Symm)
  \;\colon\;
  CRing^{op}
   \stackrel{\overset{Witt}{\longrightarrow}}{
     \stackrel{\overset{U}{\longleftarrow}}{
       \overset{Symm}{\longrightarrow}
     }
   }
  \Lambda Ring^{op}
  \,.
$$

Write $Spec(\mathbb{Z})_{et}$ for $CRing^{op}$ equipped with the [[etale topology]]. Hence

$$
  Et(Spec(\mathbb{Z}))
  \coloneqq
  Sh(Spec(\mathbb{Z})_{et})
$$

is the [[gros topos|gros]] [[etale topos]] of [[arithmetic geometry]].

Put a compatible [[Grothendieck topology]] on $\Lambda Ring^{op}$(...) and write the resulting [[site]] $Spec(\mathbb{F}_1)_{et}$.

In analogy we write

$$
  Et(Spec(\mathbb{F}_1))
  \coloneqq
  Sh(Spec(\mathbb{F}_1)_{et})
$$

and speak of the "[[etale topos]] over $Spec(\mathbb{F}_1)$", or the "absolute [[base topos]]" or something like this.

The above [[adjoint triple]] on sites then induces a sequence of adjoint functors on the [[categories of presheaves]] by left and right [[Kan extension]]

$$
  PSh(Spec(\mathbb{Z})_{et})
  \stackrel{\stackrel{Witt_!}{\longrightarrow}}{
    \stackrel{\stackrel{U_! \simeq Witt^\ast}{\longleftarrow}}{
      \stackrel{\stackrel{Symm_! \simeq U^\ast \simeq Witt_\ast}{\longrightarrow}}{
        \stackrel{\stackrel{Symm^\ast \simeq U_\ast}{\longleftarrow}}{
          \stackrel{Symm_\ast}{\longrightarrow}
        }
      }
    }
  }
  PSh(Spec(\mathbb{F}_1)_{et})
$$

The top three restrict to sheaves

$$
  Sh(Spec(\mathbb{Z})_{et})
  \stackrel{\stackrel{Witt_!}{\longrightarrow}}{
    \stackrel{\stackrel{U_! \simeq Witt^\ast}{\longleftarrow}}{
      \stackrel{Symm_! \simeq U^\ast \simeq Witt_\ast}{\longrightarrow}
    }
  }
  Sh(Spec(\mathbb{F}_1)_{et})
$$

The induced [[adjoint pair]] of [[monad]]/[[comonad]] is

$$
  (W^\ast \dashv W_\ast)
  \coloneqq
  ( U_! \circ Witt_!  \dashv U_! \circ Witt_\ast)
  \,.
$$

[[infinitesimal disk bundle]] $\dashv$ [[jet comonad]]

## References

The original article which proposes the [[topos]] over [[Lambda-rings]] as a realization of [[F1]]-geometry is

* {#Borger09} [[James Borger]], _Lambda-rings and the field with one element_ ([arXiv/0906.3146](http://arxiv.org/abs/0906.3146))

This is based on technical details laid out in

* {#Borger08} [[James Borger]], _The basic geometry of Witt vectors, I: The affine case_ ([arXiv:0801.1691](http://arxiv.org/abs/0801.1691))

* {#Borger10} [[James Borger]], _The basic geometry of Witt vectors, II: Spaces_ ([arXiv:1006.0092](http://arxiv.org/abs/1006.0092))

More discussion relating to this includes

* {#LeBruyn13} [[Lieven Le Bruyn]], _Absolute geometry and the Habiro topology_ ([arXiv:1304.6532](http://arxiv.org/abs/1304.6532))

Related discussion of [[arithmetic jet spaces]] is in

* {#Buium05} [[Alexandru Buium]], _Arithmetic differential equations_, volume 118 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005.

* {#Buium13} [[Alexandru Buium]], _Differential calculus with integers_ ([arXiv:1308.5194](http://arxiv.org/abs/1308.5194), [slighly differing pdf](http://www.math.unm.edu/~buium/statupdated.pdf))

Related discussion of [[power operations]] in [[E-infinity arithmetic geometry]] is around remark 2.2.9 of

* {#Lurie} [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

Further discussion of and speculation on an analogy between [[power operations]] and Borger's absolute geometry is in

* [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](http://arxiv.org/abs/math/0612327))

* {#MoravaSanthanam} [[Jack Morava]], Rakha Santhanam, _Power operations and Absolute geometry_ ([pdf](http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf))

[[!redirects Borger absolute geometry]]

[[!redirects Borger geometry]]
[[!redirects Borger geometries]]


Boris Apanasov

* [webpage](http://www2.math.ou.edu/~bapanasov/)

## Selected writings

On [[conformal geometry]]:

* [[Boris Apanasov]], _Doubles of Atoroidal Manifolds, Their Conformal Uniformization and Deformations_ In: J. Ławrynowicz (ed.) _Deformations of Mathematical Structures_ Springer, Dordrecht (1989) ([doi:10.1007/978-94-009-2643-1_9](https://doi.org/10.1007/978-94-009-2643-1_9))

* [[Boris Apanasov]], _Conformal Geometry of Discrete Groups and Manifolds_,  De Gruyter Expositions in Mathematics, 32, De Gruyter  (2000) ([doi:10.1515/9783110808056](https://doi.org/10.1515/9783110808056))

category: people


Boris Chorny

* [Homepage](http://math.haifa.ac.il/chorny/)

## Selected writings

On [[Goodwillie calculus]] via [[model structures on excisive functors]] and [[model structure for n-excisive functors]]:

* {#BiedermannChornyRondings06} [[Georg Biedermann]], [[Boris Chorny]], [[Oliver Röndigs]], _Calculus of functors and model categories_, Advances in Mathematics **214** (2007) 92-115 [[arXiv:math/0601221](http://arxiv.org/abs/math/0601221), [doi:10.1016/j.aim.2006.10.009](https://doi.org/10.1016/j.aim.2006.10.009)]

On [[class-locally presentable categories]]:

* [[Boris Chorny]], [[Jiří Rosický]], _Class-locally presentable and class-accessible categories_, Journal of Pure and Applied Algebra **216**, Issue 10 (2012) pp 2113–2125 ([doi:10.1016/j.jpaa.2012.01.015](https://doi.org/10.1016/j.jpaa.2012.01.015))

On [[Goodwillie calculus]], [[n-excisive functors]] and [[Whitehead products]]:

* {#SchererChorny11} [[Jerome Scherer]], [[Boris Chorny]], _Goodwillie calculus and Whitehead products_, Forum Math. 27 (2015), no. 1, 119 - 130 ([arXiv:1109.2691](https://arxiv.org/abs/1109.2691))

On enhancement of [[simplicial model categories|simplicial]] [[monoidal model categories]] up to [[strong monoidal functor|strong]] [[monoidal Quillen adjunction|monoidal]] [[Quillen equivalence]]:

* [[Haldun Özgür Bayındır]], [[Boris Chorny]], *Admissible replacements for simplicial monoidal model categories*, Algebr. Geom. Topol. **23** (2023) 43-73 [[arXiv:2008.00515](https://arxiv.org/abs/2008.00515), [doi:10.2140/agt.2023.23.43](https://doi.org/10.2140/agt.2023.23.43)]

## Related entries

* [[Goodwillie calculus]]

category:people


Boris Dubrovin

* Wikipedia entry

* [Homepage](https://people.sissa.it/~dubrovin/)

* [GoogleScholar page](https://scholar.google.it/citations?user=aWCzql8AAAAJ&hl=en)

## Selected writings

On 2d [[topological field theory]] ([[topological strings]]) and [[Frobenius manifolds]]:

* [[Boris Dubrovin]], *Geometry of 2d topological field theories*, in: *Integrable Systems and Quantum Groups*, Lecture Notes in Mathematics **1620** (1996) [[arXiv:hep-th/9407018](https://arxiv.org/abs/hep-th/9407018), [doi:10.1007/BFb0094793](https://doi.org/10.1007/BFb0094793)]

On [[Frobenius manifolds]]:

* [[Boris Dubrovin]], *Geometry and Analytic Theory of Frobenius Manifolds*, Extra Volume ICM II (1998) 315–326 [[arXiv:math/9807034](https://arxiv.org/abs/math/9807034)]

## Related entries

* [[quantum cohomology ring]]

category:people

[[!redirects B. A. Dubrovin]]


Boris Fedosov

* [webpage](http://www.math.tamu.edu/~comech/archive/fedosov/)

## Selected writings

Introducing what is now called *[[Fedosov deformation quantization]]*:

* {#Fedosov94} [[Boris Fedosov]], _A simple geometrical construction of deformation quantization_ Journal of Differential Geometry, 40(2):213–238, 1994 ([Euclid](https://projecteuclid.org/euclid.jdg/1214455536))

with a monograph account in

* [[Boris Fedosov]], *Deformation Quantization and Index Theory*, John Wiley & Sons (1995) [ISBN:9783055017162]

## Related entries

* [[formal deformation quantization|formal]] [[deformation quantization]]

category: people

[[!redirects Boris Vasilievich Fedosov]]


Boris Feigin

**Boris Feigin** is a Russian mathematician and mathematical physicist, former student of D. B. Fuks (Fuchs). His works include fundamental contributions to [[algebraic K-theory]], [[cyclic homology]], Lie theory, especially the cohomology of infinite-dimensional Lie algebras, characteristic classes of [[foliation]]s, representation theory of Virasoro and  Kac-Moody Lie algebras and loop groups, [[semiinfinite cohomology]], [[vertex operator algebras]] and [[CFT]], geometric Langlands program, [[quantum groups]], [[deformation theory]], [[integrable systems]] and [[algebraic geometry]]

* [dedications](http://www.ams.org/distribution/mmj/vol4-3-2004/dedication.html) to B.F. in a special volume in his honour

* [MathGenealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=52952)

* [German Wikipedia entry](http://de.wikipedia.org/wiki/Boris_Lwowitsch_Feigin)

## Selected writings

On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of [[conformal blocks]] for the [[sl(n)|$\mathfrak{sl}(n)$]] [[WZW model]]:

* {#FeiginSchechtmanVarchenko90} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by correlators in Wess-Zumino-Witten models*,  Lett Math Phys **20** (1990)  291–297  $[$[doi:10.1007/BF00626525](https://doi.org/10.1007/BF00626525)$]$

* {#FeiginSchechtmanVarchenko94} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. I*, Commun. Math. Phys. **163** (1994) 173–184 $[$[doi:10.1007/BF02101739](https://doi.org/10.1007/BF02101739)$]$

  > (for [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]])

* [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. II*, Comm. Math. Phys. **170** 1 (1995) 219-247  $[$[euclid:cmp/1104272957](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-170/issue-1/On-algebraic-equations-satisfied-by-hypergeometric-correlators-in-WZW-models/cmp/1104272957.full)$]$

On [[fractional level WZW models]] (non-integrable admissible [[affine Lie algebra]] [[irreps]]) for [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]]:

* [[Boris Feigin]], [[Feodor Malikov]], *Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level*, p. 357-405 in: Loday, Stasheff, Voronov (eds.) *Operads: Proceedings of Renaissance Conferences*,  Contemporary Mathematics **202**, AMS (1997) [[arXiv:q-alg/9511011](https://arxiv.org/abs/q-alg/9511011), [ams:conm-202](https://bookstore.ams.org/conm-202)]

  > (appearance of [[osp|$\mathfrak{osp}(1\vert2)$]]-contributions)

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:

* [[Boris Feigin]], [[Sergei Gukov]], $VOA[M_4]$, J. Math. Phys. **61** 012302 (2020) [[arXiv:1806.02470](https://arxiv.org/abs/1806.02470), [doi:10.1063/1.5100059](https://doi.org/10.1063/1.5100059)]

category: people

[[!redirects Boriw Feigin]]
[[!redirects Borya Feigin]]
[[!redirects B. Feigin]]


Boris Khesin

* [webpage](http://www.math.toronto.edu/khesin)

## Selected writings

On methods of [[topology]] and [[homotopy theory]] in [[hydrodynamics]]:

* [[Vladimir Arnold]], [[Boris Khesin]], _Topological methods in hydrodynamics_, Applied Mathematical Sciences __125__, Springer 1998 ([doi:10.1007/b97593](https://link.springer.com/book/10.1007/b97593))

category: people


Boris Kruglikov

* [webpage](https://uit.no/om/enhet/ansatte/person?p_document_id=40976&p_dimension_id=88140)

## related $n$Lab entries

* [[pseudoholomorphic vector bundle]]

category: people


Boris Pioline

* [webpage](http://arxiv.org/abs/1207.1109)

## Selected writings

On [[U-duality]], [[M-theory]] and [[M-branes]]:

* {#ObersPioline99} [[Niels Obers]], [[Boris Pioline]], *U-duality and M-Theory*, Phys. Rept. **318** (1999) 113-225 [[arXiv:hep-th/9809039](https://arxiv.org/abs/hep-th/9809039), doi:10.1016/S0370-1573(99)00004-6]

On [[2-loop]] [[string scattering amplitudes]] of the [[superstring]]:

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors_ ([arXiv:2006.05270](https://arxiv.org/abs/2006.05270))

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality_ ([doi:2008.08687](https://arxiv.org/abs/2008.08687))

## Related $n$Lab entries

* [[D3-brane instanton]]

category: people


Boris Podolsky

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

## Selected writings

Introducing the [[Einstein-Podolsky-Rosen paradox]] concerning [[quantum entanglement]]:

* {#EPR} [[Albert Einstein]], [[Boris Podolsky]], [[Nathan Rosen]], _Can the Quantum-Mechanical Description of Physical Reality be Considered Complete?_ Physical Review 47 (10): 777–780. (1935) ([doi:10.1103/PhysRev.47.777](https://doi.org/10.1103/PhysRev.47.777))

category: people


Boris S. Tsirelson

* [personal page](https://www.tau.ac.il/~tsirel/)

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

## Seelcted writings

Relating [[Grothendieck's inequality]] to [[Bell's inequality]] in [[quantum physics]]:

* [[Boris S. Tsirelson]], *Quantum analogues of the Bell inequalities. The case of two spatially separated domains*, Journal of Soviet Mathematics **36** (1987) 557–570 [[doi:10.1007/BF01663472](https://doi.org/10.1007/BF01663472)]

* [[Boris S. Tsirelson]], *Some results and problems on quantum Bell-type inequalities* Hadronic Journal Supplement **8** 4 (1993) 329-345 [[pdf](https://www.tau.ac.il/~tsirel/download/hadron.pdf), [[Tsirelson-QuantumBellType.pdf:file]] [web](https://www.tau.ac.il/~tsirel/download/hadron.html)]

  > (but see the erratum [here](https://www.tau.ac.il/~tsirel/Research/bellopalg/main.html))

category: people

[[!redirects Boris Tsirelson]]


Boris Tsygan

* [website](http://www.math.northwestern.edu/~tsygan/)

category: people

[[!redirects B. Tsygan]]
[[!redirects B. L. Tsygan]]


Boris Zilber

__Boris Zilber__ is a logician and [[model theory|model theorist]] at Oxford.

* [web](http://people.maths.ox.ac.uk/zilber), [some papers](http://people.maths.ox.ac.uk/zilber/publ.html)

* _On model theory, non-commutative geometry and physics_, (survey draft) [pdf](http://people.maths.ox.ac.uk/zilber/bul-survey.pdf)
* _Zariski geometries_, book, draft [pdf](http://people.maths.ox.ac.uk/zilber/s.pdf)
* On model theory, noncommutative geometry and physics, conference talk, [video](http://zomobo.net/play.php?id=wnhbpIQBpiE)

Related entries include [[Zariski geometry]]

[[!redirects B. Zilber]]


Born rule

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The *Born rule* is the core statement of [[coordination]] in the foundations of [[quantum physics]]:

In its version for [[pure states]] the Born rule says that

* for $A$ an [[observable]] in the form of a [[Hermitean operator]] on the given [[Hilbert space]] $\mathcal{H}$ of [[pure state|pure]] [[quantum states]] with [[Hermitian form|Hermitian]] [[inner product]] $\langle\text{-}\vert\text{-}\rangle$

* $P$ denoting the [[projection operator]] on a given [[eigenvector|eigenstate]] of this operator,

* $\pi \,\in\, \mathcal{H}$ a [[pure state]];

then the [[probability]] of observing the given eigenstate in a [[quantum system]] which is in state $\psi$ equals (in [[bra-ket notation]]):

$$
  \frac{
    \langle \psi \vert P \vert \psi \rangle
  }{
    \langle \psi \vert \psi \rangle
  }
  \,.
$$

In particular, if $W$ is an [[orthonormal basis]] for the [[Hilbert space]] [[space of quantum states|of states]] associated with a [[quantum measurement]]-procedure, then the probability of measuring the result $w$ on a system in state $\left\vert \psi \right\rangle$ is

$$
  \frac{
    \langle \psi \vert w \rangle \langle w \vert \psi \rangle
  }{
    \langle \psi \vert \psi \rangle
  }
  \;\;=\;\;
  \frac{
    \big\vert \langle w \vert \psi \rangle \big\vert^2
  }{
    \langle \psi \vert \psi \rangle
  }
$$

and so if the state is normalized to begin with ($\langle \psi \vert \psi \rangle$ = 1 ), then the probability is

$$
  P_\psi(w)
  \;=\;
  \big\vert \langle w \vert \psi \rangle \big\vert^2
  \,.
$$

Essentially in this form the rule was first formulated by [Born 1926b p 805](#Born26b).

## Related concepts

* [[quantum probability]]

[[!include states and observables -- content]]

## References
 {#References}

### Historical origins
  {#ReferencesHistoricalOrigins}

The Born rule is named in honor of

* {#Born26a} [[Max Born]], *Zur Quantenmechanik der Stoßvorgänge*, Zeitschrift für Physik **37** (1926) 863–867 [[doi:10.1007/BF01397477](https://doi.org/10.1007/BF01397477)]

where it appears (though disregarding the norm symbol) as a brief footnote-added-in-proof:

\begin{imagefromfile}
    "file_name": "Born_stating_the_BornRule-Footnote.jpg",
    "width": 560,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and its expanded version

* {#Born26b} [[Max Born]], *Quantenmechanik der Stoßvorgänge*, Zeitschrift für Physik **38** (1926) 803–827 [[doi:10.1007/BF01397184](https://doi.org/10.1007/BF01397184)]

which provides the mathematical details (p. 805):

\begin{imagefromfile}
    "file_name": "Born_stating_the_BornRule-Expanded.jpg",
    "width": 480,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and finally in the followup

* {#Born27} [[Max Born]], *Das Adiabatenprinzip in der Quantenmechanik*, Zeitschrift für Physik **40** (1927) 167–192 [[doi:10.1007/BF01400360](https://doi.org/10.1007/BF01400360)]

which makes the interpretation more explicit (p. 171):

\begin{imagefromfile}
    "file_name": "Born_stating_the_BornRule-ExpandedII.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

The generalization of this idea to probability *densities* for continuous [[observables]] is due to [[Wolfgang Pauli]], as first recounted in:

* {#Jordan27} [[Pascual Jordan]], *Über eine neue Begründung der Quantenmechanik*, Zeitschrift für Physik **40** (1927) 809–838 [[doi:10.1007/BF01390903](https://doi.org/10.1007/BF01390903)]

\begin{imagefromfile}
    "file_name": "Jordan-PauliBornRule.jpg",
    "width": 510,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and then by Pauli himself, again in a footnote:

* [[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)]

\begin{imagefromfile}
    "file_name": "PauliStatingBornPauliRule.jpg",
    "width": 540,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Early review of the Born-Pauli rule:

* {#HilbertvonNeumannNordheim} [[David Hilbert]], [[John von Neumann]], [[Lothar W. Nordheim]], *Über die Grundlagen der Quantenmechanik*,  Math. Ann. **98** (1928) 1–30 [[doi:10.1007/BF01451579](https://doi.org/10.1007/BF01451579)]

where it is once again a footnote:

\begin{imagefromfile}
    "file_name": "HilbertNeumannNordheim-BornRule.jpg",
    "width": 530,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

and in

*  [[Paul A. M. Dirac]], *The physical interpretation of the quantum dynamics*, Proceedings of the Royal Society of London **113** 765 (1927) [[doi:10.1098/rspa.1927.0012](https://doi.org/10.1098/rspa.1927.0012)]

\begin{imagefromfile}
    "file_name": "Dirac_OnBornPauliRule.jpg",
    "width": 530,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

The full recognition and amplification of the Born rule as a pillar of quantum physics making it a [[probability theory|probabilistic]] [[theory (physics)|theory]] (cf. *[[quantum probability]]*) is (maybe besides [Jordan 1927](#Jordan27)) due to:

* [[John von Neumann]], *Die quantenmechanische Statistik*, Part III in:

  *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

\begin{imagefromfile}
    "file_name": "vonNeumann-BornRule.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "vonNeumann-QMStatistics.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Historical survey:

* [[Jagdish Mehra]], Helmut Rechenberg, *The Probability Interpretation and the Statistical Transformation Theory, the Physical Interpretation, and the Empirical and Mathematical Foundations of Quantum Mechanics 1926-1932*, Part 1 in:  *The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics, 1926-1941*, Springer (2001)   [[ISBN:978-0-387-98971-6](https://link.springer.com/book/9780387989716)]

### Further discussion

Review:

* [[Klaas Landsman]], *The Born rule and its interpretation*, in: *Compendium of Quantum Physics*, Springer (2009) 64-70 [[doi:10.1007/978-3-540-70626-7_20](https://doi.org/10.1007/978-3-540-70626-7_20), [pdf](https://www.math.ru.nl/~landsman/Born.pdf), [[Landsman-BornRule.pdf:file]]]

See also:

* Wikipedia, *[Born rule](https://en.wikipedia.org/wiki/Born_rule)*

[[!redirects Born-Pauli rule]]


Born-Jordan quantization

* Maurice A. de Gosson, Fabio Nicola, _Born–Jordan pseudodifferential operators and the
Dirac correspondence: Beyond the Groenewold–van Hove Theorem_, [arxiv/1606.07796](http://arxiv.org/abs/1606.07796)
* Maurice A. de Gosson, _Symplectic covariance properties for Shubin and Born–Jordan pseudodifferential operators_, Trans. Amer. Math. Soc. 365(6) (2013) 3287–3307
* M. de Gosson, _Born-Jordan quantization: theory and applications_, Springer 2016
* M. de Gosson, F. Luef, _Preferred quantization rules: Born–Jordan vs. Weyl; applications to phase space quantization_, J. Pseudo-Differ. Oper. Appl. 2(1) (2011) 115–139


Born-Oppenheimer approximation

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[quantum mechanics]], the _Born--Oppenheimer approximation_ is a method for approximating the [[energy]] a [[quantum mechanical system]] by identifying "light" and "heavy" degrees of freedom and then treating the solving for the "light" dynamics as if depending only on the static configuration of the "heavy" degrees of freedom.

Historically this is motivated from, and still heavily used in practice, for the computation of energy spectra of [[molecule|molecules]], where the [[atomic nucleus|atomic nuclei]] are much heavier than the [[electron|electrons]], so that their dynamics can be split off to a good degree of approximation.

## References

The original reference is

* [[Max Born]], [[Robert Oppenheimer]], _Zur Quantentheorie der Molekeln_,  Annalen der Physik. 389, Nr. 20 (1927), p 457–484, doi:10.1002/andp.19273892002.

An early textbook-like account is

* J.C. Slater, _Quantum Theory of Molecules and Solids, Vol. 1: Electronic Structure of Molecules_  American Journal of Physics. 32, (1964), S. 65, doi:10.1119/1.1970097.

An review is for instance in section [2.2 The Born-Oppenheimer approximation](http://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node13.html) of

* Peter David Haynes, _Linear-scaling methods in ab initio quantum-mechanical calculations_ PhD thesis (1998) ([web](http://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/thesis.html))

[[!redirects Born-Oppenheimer approximation]]
[[!redirects Born-Oppenheimer approximations]]
[[!redirects Born–Oppenheimer approximation]]
[[!redirects Born–Oppenheimer approximations]]
[[!redirects Born--Oppenheimer approximation]]
[[!redirects Born--Oppenheimer approximations]]


bornological coarse space

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=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _bornological coarse space_ is a set $X$
equipped with a [[bornology]] $B$
and a [[coarse structure]] $C$
that are compatible: controlled thickenings of bounded subsets are bounded.

Here a U-controlled thickening ($U\subset X\times X$)
of a subset $B$ of $X$
is the projection of $X\times B\cap U$ onto its first factor.

A _morphism_ of bornological coarse spaces is a map $f$
of sets that is _controlled_ (for every $U\in C$ we
have $(f\times f)(U)\in C'$) and _proper_ (for every $B'\in B'$
we have $f^{-1}(B')\in B$).

Bornological coarse spaces can be used as a [[site with an interval]]
to construct [[motivic coarse spaces]].

## References

* {#BunkeEngel} [[Ulrich Bunke]], [[Alexander Engel]],
_Homotopy theory with bornological coarse spaces_,
[arXiv:1607.03657](http://arxiv.org/abs/1607.03657).

[[!redirects bornological coarse spaces]]


bornological group

+-- {: .rightHandSide}
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### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[group]] compatibly equipped with a [[bornology]].

## Related concepts

* [[bornological set]]

* [[bornological vector space]]

* [[normed group]]

## References

Discussion that bornological abelian groups form a [[quasi-abelian category]] is in

* {#Bambozzi14} [[Federico Bambozzi]], section 1 of _On a generalization of affinoid varieties_ ([arXiv:1401.5702](http://arxiv.org/abs/1401.5702))

[[!redirects bornological groups]]

[[!redirects bornological abelian group]]
[[!redirects bornological abelian groups]]


bornological set

# Contents
* table of contents
{: toc}

## Idea

A _bornological set_ is a notion of [[space]], where instead of considering [[open sets]] and [[continuous functions]] whose [[inverse images]] preserve open sets as one does for [[topological space]]s, one considers [[bounded sets]] (which constitute a _bornology_) and [[bounded maps]] whose [[direct images]] preserve bounded sets. Bornological [[topological vector spaces]], called [[bornological spaces]], are important in [[functional analysis]].

## Definitions

Let $X$ be a [[set]]. A **bornology** on $X$ is a collection $\mathcal{B} \subseteq P X$ of [[subsets]] of $X$ such that

* $\mathcal{B}$ covers $X$: $\bigcup_{B \in \mathcal{B}} B = X$,

* $\mathcal{B}$ is downward-closed: if $B \in \mathcal{B}$ and $A \subseteq B$, then $A \in \mathcal{B}$,

* $\mathcal{B}$ is closed under finite unions: if $B_1 \ldots, B_n \in \mathcal{B}$, then $\bigcup_{1 \leq \i \leq n} B_i \in \mathcal{B}$.

A **bornological set** is a set $X$ equipped with a bornology. The elements of $\mathcal{B}$ are called the **[[bounded sets]]** of a bornological set.

If $X$, $Y$ are bornological sets, a [[function]] $f\colon X \to Y$ is said to be **[[bounded function|bounded]]** if $f(B)$ is bounded in $Y$ for every bounded $B$ in $X$. One obtains a category of bornological sets and bounded maps.

## Examples

* If $X$ is any [[topological space]] such that every point is closed, then there is a bornology consisting of all [[precompact subsets]] of $X$ (subsets whose closure is [[compact set|compact]]).  Any [[continuous map]] is bounded with respect to this choice of bornology.

* If $X$ is any [[metric space]], there is a bornology where a set is bounded if it is contained in some open [[ball]]. Any [[Lipschitz map]] is bounded with respect to this choice of bornology.  A metric space is __bounded__ if it\'s a bounded subspace of itself.

* If $X$ is a [[measure space]], then the subsets of the sets of finite measure form a bornology .

* For [[linear operators]] between [[bornological spaces]], a map is continuous if and only if it is bounded.

## Properties

+-- {: .num_theorem}
###### Theorem
The category of bornological sets is a [[quasitopos]], in fact a topological universe.
=--

For a proof, see this [article](#AdamHerr) by Adamek and Herrlich.

+-- {: .num_theorem}
###### Theorem
Let $Alg_{\mathbb{C}}$ be the category of (noncommutative) finite-dimensional algebras over $\mathbb{C}$, the field of complex numbers. Let

$$U \colon Alg_{\mathbb{C}} \to Born$$

be the functor that takes an algebra $A$ to the set ${|A|}$ equipped with the bornology of precompact sets. Then there is a canonical identification of the monoid $Born^{Alg_\mathbb{C}}(U, U)$ with the monoid of entire holomorphic functions.
=--

This was proved by [Schanuel](#Schanuel).

## Related concepts

* [[bornological group]]

* [[bornological vector space]]

* [[bornological topos]]

## References

* {#AdamHerr} [[Jiří Adámek]] and [[Horst Herrlich]], _Cartesian closed categories, quasitopoi, and topological universes_, Comm. Math. Univ. Carol., Vol. 27, No. 2 (1986), 235-257. ([web](http://dml.cz/handle/10338.dmlcz/106447))

* {#Hogbe-Nlend70}H. Hogbe-Nlend, _Les racines historiques de la bornologie moderne_ , pp.1-6 in _Séminaire Choquet: Initiation à l'analyse_ 10.1 (1970-1971).  ([numdam](http://www.numdam.org/item?id=SC_1970-1971__10_1_A5_0))

* {#ProsmansSchneiders} [[Fabienne Prosmans]], [[Jean-Pierre Schneiders]], _A homological study of bornological spaces_, December 2000, Prepublications Mathematiques de l'Universite Paris 13, 46. ([pdf](http://www.analg.ulg.ac.be/jps/rec/hsbs.pdf))

* {#Schanuel} [[Stephen Schanuel]], _Continuous extrapolation to triangular matrices characterizes smooth functions_, J. Pure App. Alg. 24, Issue 1 (1982), 59–71. ([web](http://www.sciencedirect.com/science/journal/00224049/24/1))

Review includes

* {#Bambozzi14} [[Federico Bambozzi]], section 1 of _On a generalization of affinoid varieties_ ([arXiv:1401.5702](http://arxiv.org/abs/1401.5702))

[[!redirects bornology]]
[[!redirects bornologies]]
[[!redirects bornological]]
[[!redirects bornological set]]
[[!redirects bornological sets]]

[[!redirects bounded space]]
[[!redirects bounded spaces]]


bornological space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--

# Bornological spaces
* table of contents
{: toc}

## Idea

Much as a [[topological structure]] on a [[set]] is a notion of which [[subsets]] are 'open', so a bornological structure, or bornology, on a set is a notion of which subsets are 'bounded'.

## Definitions

So far, we only discuss [[bornological topological vector spaces]].  See [[bornological set]] for the general notion of bornological space.

However, we can tell that bornological spaces and certain [[morphisms]] between them form a [[category]] $Born$.

## Properties

### Relation to Banach spaces
 {#RelationToBanachSpaces}

Every [[inductive limit]] of [[Banach spaces]] is bornological. ([Alpay-Salomon 13, prop. 2.3](#AlpaySalomon13))

Conversely, every bornological space is an inductive limit of [[normed spaces]], and of [[Banach spaces]] if it is quasi-complete ([Schaefer-Wolff 99](#SchaeferWolff99)).

## References

* Wikipedia about [bornological spaces] (http://en.wikipedia.org/wiki/Bornological_space)

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

* {#AlpaySalomon13} Daniel Alpay, Guy Salomon, _On algebras which are inductive limits of Banach spaces_ ([arXiv:1302.3372](http://arxiv.org/abs/1302.3372))

* {#SchaeferWolff99} H. H. Schaefer with M. P. Wolff, _Topological vector spaces_, Springer 1999

[[!redirects bornological space]]
[[!redirects bornological spaces]]
[[!redirects bornalogical space]]

[[!redirects Bor]]
[[!redirects Born]]


bornological topological vector space

+-- {: .rightHandSide}
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###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

Linear operators on [[normed space|normed spaces]] are continuous precisely iff they are bounded. A bornological space retains this property by definition.

+-- {: .standout}
The discussion below is about bornological CVSes, but there is a more general notion of [[bornological space]].
=--

## Definition ##

A [[locally convex]] [[topological vector space]] $E$ is **bornological** if every circled, convex subset $A \subset E$ that absorbs every [[bounded set]] in $E$ is a neighbourhood of $0$ in $E$. Equivalently every [[seminorm]] that is bounded on bounded sets is continuous.

The **bornology** of a given [[TVS]] is the family of bounded subsets.

Given a locally convex [[TVS]] $E$ with initial topology $T_0$, there is a finest topology $T$ such that the family of bounded subsets of $T$ coincides with $T_0$. The space $E$ equipped with the topology $T$ is called the **bornologification** of $E$, or the **bornological space associated with** $(E, T_0)$

## Properties ##

### Maps on bornological spaces

+-- {: .num_theorem}
###### Theorem
Let $U$ be a linear map from a bornological space $E$ to any locally convex [[TVS]], then the following statements are equivalent:

* $U$ is continuous,

* $U$ is bounded on bounded sets,

* $U$ maps [[null sequence|null sequences]] to null sequences.
=--

### Relation to Banach spaces

Every [[inductive limit]] of [[Banach spaces]] is a bornological vector space. ([Alpay-Salomon 13, prop. 2.3](#AlpaySalomon13))

Conversely, every bornological vector space is an inductive limit of [[normed spaces]], and of [[Banach spaces]] if it is quasi-complete ([Schaefer-Wolff 99](#SchaeferWolff99))

## Examples ##

Every metrizible locally convex space is bornological, that is every [[Fréchet space]] and thus every [[Banach space]].

## Related concepts

* [[bornological set]]

* [[bornological group]]

* [[bornological space]]

## References ##

* Wikipedia about [bornological spaces] (http://en.wikipedia.org/wiki/Bornological_space)

* {#SchaeferWolff99} H. H. Schaefer with M. P. Wolff, section 8 of _Topological vector spaces_, Springer 1999

* {#AlpaySalomon13} Daniel Alpay, Guy Salomon, _On algebras which are inductive limits of Banach spaces_ ([arXiv:1302.3372](http://arxiv.org/abs/1302.3372))

Discussion of bornological vector spaces forming a [[quasi-abelian category]] is in

* {#ProsmansSchneiders} [[Fabienne Prosmans]], [[Jean-Pierre Schneiders]], _A homological study of bornological spaces_, December 2000, Prepublications Mathematiques de l'Universite Paris 13, 46 ([pdf](http://www.analg.ulg.ac.be/jps/rec/hsbs.pdf))

with review and generalization to [[bornological abelian groups]] in

* {#Bambozzi14} [[Federico Bambozzi]], section 1 of _On a generalization of affinoid varieties_ ([arXiv:1401.5702](http://arxiv.org/abs/1401.5702))

[[!redirects bornologification]]

[[!redirects bornological vector space]]
[[!redirects bornological vector spaces]]
[[!redirects bornological topological vector spaces]]
[[!redirects Bornological topological vector space]]


bornological topos

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

#Contents#
* table of contents
{:toc}

## Idea

The **bornological topos** is a [[Grothendieck topos]] based on the concept of a bounded sequence and was proposed by [[William Lawvere]] in unpublished lectures in 1983 as a convenient category to do [[functional analysis]] in.

## Definition

Let $\mathcal{C}$ be the category of [[countable set|countable sets]]. The _bornological topos_ $\mathcal{B}$ is the category of finite product preserving presheaves on $\mathcal{C}$.

## Properties

* Equivalently, $\mathcal{B}$ is the topos of sheaves on $\mathcal{C}$ for the _(finite) disjoint covering topology_ with coverings the finite families $(X_i\to Y)_{i\in I}$ , such that $\sum_{i\in I} X_i\to Y$ is an isomorphism.

* In the work of Español et al., $\mathcal{B}$ is accessed as a subtopos of the topos of actions of the monoid of endomorphisms of $N$.

* $\mathcal{B}$ contains the category of (Kolmogorov) [[bornological set|bornological spaces]] as a full reflective subcategory (cf. Español-Lambán [2002](#EspanolLamban02)).

* The object $R_D$ of [[Dedekind real numbers]] of $\mathcal{B}$ is the space $l^\infty$ of bounded sequences of real numbers (cf. Español-Mínguez [2001](#Espanol-Minguez01)).

* " _The category of modules over the Dedekind real number object includes all the inductive limits of Banach spaces as a full subcategory, but at the same time is itself a Grothendieck AB5 Abelian category with the accompanying exactness properties._ " (Lawvere [2008](#Lawvere08), p.15).
Thus " **functional analysis becomes linear algebra** " in the bornological topos (Lawvere [1994](#Lawvere94), p.10).

## Remark: bornology and cohesiveness

The bornological topos is the [[Gaeta topos]] on $\mathcal{C}$ and as such fits Lawvere's paradigm of doing abstract "algebraic geometry" (cf. Lawvere [1986](#Lawvere86), p.17). In particular, the geometry and cohesiveness of the objects $X$ in $\mathcal{B}$ arises _covariantly_ from the basic figure shape of a 'bounded' sequence $A$ via maps $A\to X$.

That bornology provides in the context of functional analysis "_a more basic notion of cohesiveness_" than the usual topological neighborhood concept and contravariant function algebra concept based on it is argued in Lawvere ([1997](#Lawvere97), pp.3f).

Although the bornological topos can be regarded as a cohesive category of "spaces" in a broad sense, it doesn't satisfy Lawvere's _axiomatic cohesion_ since it lacks the required left adjoint components functor $\Pi:\mathcal{B}\to Set$ (cf. Lawvere [2008](#Lawvere08)).

## Related entries

* [[bornology]]

* [[Johnstone's topological topos|topological topos]]

* [[topos of recursive sets|recursive topos]]

* [[cohesion]]

## References

* L. Español, _Extended real number object in the bornological topos_ , talk CT07 Carvoeiro 2007. ([pdf-slides](http://www.mat.uc.pt/~categ/ct2007/slides/espanol.pdf))

* L. Español, L. Lambán, _A tensor-hom adjunction in a topos related to vector topologies and bornologies_ , Journal of Pure and Applied Algebra **154** (2000) pp.143-158. (doi:[10.1016/S0022-4049(99)00188-7](http://dx.doi.org/10.1016/S0022-4049(99%2900188-7))

* {#Espanol-Lamban02}L. Español, L. Lambán, _On bornologies, locales and toposes of M-sets_ , Journal of Pure and Applied Algebra **176** (2002) pp.113–125. (doi:[10.1016/S0022-4049(02)00047-6](http://dx.doi.org/10.1016/S0022-4049(02%2900047-6))

* {#Espanol-Minguez01}L. Español, M. C. Mínguez, _Cortaduras Para $l^\infty$_ , pp.375-390 in Español, Varona (eds.), _Margarita Mathematica en Memoria de José Javier (Chicho) Guadalupe Hernández_ , Universidad de La Rioja 2001. ([pdf](http://www.emis.de/proceedings/Chicho2001/Espanol-Minguez.pdf))

* {#Lawvere86}[[F. William Lawvere]], _Taking Categories Seriously_, Revista Colombiana de Matemáticas **XX** (1986) pp.147-178. Reprinted as TAC Reprint no.8 (2005) pp.1-24. ([pdf](ftp://ftp.tac.mta.ca/tac/html/tac/reprints/articles/8/tr8.pdf))

* [[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_ , Contemporary Mathematics **92** (1989) pp.261-299. ([Google Books link](https://books.google.com.au/books?id=VxAcCAAAQBAJ&pg=PA261); doi:[10.1090/conm/092/1003203](http://dx.doi.org/10.1090/conm/092/1003203)).

* {#Lawvere94}[[F. William Lawvere]], _Cohesive Toposes and Cantor's 'lauter Einsen'_ , Phil. Math. **2** no.3 (1994) pp.5-15.

* {#Lawvere97}[[F. William Lawvere]], _Volterra's functionals and covariant cohesion of space_ , in _Proceedings of the May 1997 Meeting in Perugia_ , Perugia Studies in Mathematics 1997.

* {#Lawvere08}[[F. William Lawvere]], _[[Cohesive Toposes -- Combinatorial and Infinitesimal Cases]]_, Como Ms. 2008.

* [[F. William Lawvere]], Section 2 of _Toposes generated by codiscrete objects in combinatorial topology and functional analysis_, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, [pdf](http://tac.mta.ca/tac/reprints/articles/27/tr27abs.html).

[[!redirects Bornological topos]]


Borromean link

The __Borromean link__ is a famous [[link]] also known as the __Borromean rings__.  It is a [[Brunnian link]] with $3$ components.

[[!include Borromean link - SVG]]

+-- {: .un_remark}
###### Note
To include this picture in a page, write
<nowiki><code>[[!include Borromean link - SVG]]</code></nowiki>.
=--

[[!redirects Borromean link]]
[[!redirects Borromean rings]]
[[!redirects Borromean links]]
[[!redirects Borromean ring]]


Borromean link - SVG

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[[!redirects Borromean rings - SVG]]

category: svg


Borsuk's shape theory

#Contents#
* table of contents
{:toc}

##Reprise of the Idea##
The basic idea behind Borsuk's shape theory is explained in the entry on [[shape theory]], so will not be repeated here, except to say that it considers compact metric spaces embedded in the [[Hilbert cube]], then uses the open neighbourhoods of the space as a 'net' of approximations of the space.  The space is, of course, the intersection of all these open neighbourhoods.

Any compact metric space can be embedded in the [[Hilbert cube]], so it is sufficient to consider just compact subspaces of that space.

##Some details##
Let $s= \prod_{n=1}^\infty (-\frac{1}{n},\frac{1}{n})$ be the _pseudo-interior_ of the [[Hilbert cube]], $Q= \prod_{n=1}^\infty [-\frac{1}{n},\frac{1}{n}]$.

We will define (a category equivalent to) the *Borsuk Shape category*, $Shape_B$, to have compact subsets of $s$ as objects and some morphisms that need a bit of explaining.

If $X$ and $Y$ are compact subsets of $s$, then a _fundamental sequence_, $\underline{f} : X\to Y$, is defined to be a sequence of maps $f_n : Q\to Q$ such that for every neighbourhood $V$ of $Y$ in $Q$, there exists a neighbourhood $U$ of $X$ in $Q$ and an integer $n_0$ such that if $n, n^\prime \geq n_0$, the restrictions $f_n|_U$ and $f_{n^\prime}|_U$ are homotopic within $V$.

Note that the $f_n(X)$ do not have to be contained in $Y$, they only have to be 'near' $Y$.

Two fundamental sequences,  $\underline{f},\underline{f}^\prime : X\to Y$, are said to be _homotopic_, $\underline{f}\sim \underline{f}^\prime$ provided that for every neighbourhood $V$ of $Y$ in $Q$, there is a neighbourhood $U$ of $X$ in $Q$ and an integer $n_0$ such that if $n \geq n_0$, then $f_n|_U$ and $f^\prime_{n}|_U$ are homotopic within $V$.

The morphisms of $Shape_B$  and taken to be the homotopy classes of fundamental sequences between the corresponding spaces.

Two compacta contained in $s$ are said to have the _same shape_ if they are isomorphic in $Shape_B$.  As an example, the [[Warsaw circle]] has the same shape as the circle.

##Chapman complement theorem##
+--{.un_theorem}
######Theorem (Chapman, 1972)
If $X$ and $Y$ are compacta in $s$, then $X$ and $Y$ have the same shape if and only if their complements $Q\setminus X$ and $Q\setminus Y$ are homeomorphic.
=--

Chapman extended the association $X$ 'goes to' $Q\setminus X$ to a functor from the Borsuk shape category to the weak [[proper homotopy theory|proper homotopy]] category of complements in $Q$ of compacta.  This was the basis for Edwards-Hastings formulation of [[strong shape theory]], on replacing the weak form of proper homotopy by a strong form.

## Related concepts

* [[shape theory]]

  * [[strong shape theory]]

##References##

* K. Borsuk, _Concerning homotopy properties of compacta_, Fund Math. 62 (1968) 223-254
* K. Borsuk, _Theory of Shape_, Monografie Matematyczne Tom 59,Warszawa 1975.
* T. A.Chapman, _On Some Applications of Infinite Dimensional Manifolds to the Theory of Shape_, Fund. Math. 6 (1972), 181 - 193.
* D. A. Edwards and [[H. M. Hastings]], (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag.
[[!redirects Borsuk shape theory]]


Bose-Einstein condensate

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[condensate]] of [[bosons]].

## Related concepts

* [[boson star]]

* [[Bose-Einstein condensate dark matter]]

* [[superfluid]]

* [[superconductivity]]

* [[tachyon condensation]]

* [[quantum optics]]

## References

### General

Textbook accounts:

* [[Tim Byrnes]], [[Ebubechukwu Ilo-Okeke]], Section 2 of: *Quantum Atom Optics: Theory and Applications to Quantum Technology*, Cambridge University Press 2021 ([arXiv:2007.14601](https://arxiv.org/abs/2007.14601), [CUP](https://www.cambridge.org/core/books/quantum-atom-optics/2D867888B5C666D3A936F1C942C99568))

See also:

* Wikipedia, _[Bose-Einstein condensate](http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate)_

Proposal to use BEC for [[quantum computation]]:

* [[Tim Byrnes]], Kai Wen, Yoshihisa Yamamoto, *Macroscopic quantum computation using Bose-Einstein condensates*, Phys. Rev. A **85** (2012)  040306(R) [[doi:10.1103/PhysRevA.85.040306](https://doi.org/10.1103/PhysRevA.85.040306)]

Realization of [[quantum entanglement]]/[[EPR paradox]]:

* Paolo Colciaghi, Yifan Li, Philipp Treutlein, Tilman Zibold, *Einstein-Podolsky-Rosen experiment with two Bose-Einstein condensates* [[arXiv:2211.05101](https://arxiv.org/abs/2211.05101)]

Discussion of BECs via [[AdS-CFT in condensed matter physics]]:

* Yu-Kun Yan, Shanquan Lan, Yu Tian, Peng Yang, Shunhui Yao, Hongbao Zhang, *Towards an effective description of holographic vortex dynamics* [[arXiv:2207.02814](https://arxiv.org/abs/2207.02814)]

[[!include defect anyons -- references]]

[[!redirects Bose-Einstein condensates]]

[[!redirects BEC]]
[[!redirects BECs]]


boson

+-- {: .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

A __boson__ is a [[particle]]/[[quantum field]] that obeys [[Bose-Einstein statistics]].  By the [[spin-statistics theorem]], this is the same thing as a particle whose [[spin]] is an [[integer]].  Compare [[fermion]].

## Examples

* [[photon]]

* [[gluon]]

* [[graviton]]

* [[Higgs boson]]

* [[Z-boson]], [[Z'-boson]]

## Related concepts

* **boson**, [[fermion]], [[anyon]]

* [[spin-statistics theorem]]

* [[Bose-Einstein condensate]]

  * [[boson star]]

[[!redirects boson]]
[[!redirects bosons]]

[[!redirects boson field]]
[[!redirects boson fields]]

[[!redirects bosonic field]]
[[!redirects bosonic fields]]

[[!redirects Boson]]
[[!redirects Bosons]]

[[!redirects Boson field]]
[[!redirects Boson fields]]

[[!redirects Bosonic field]]
[[!redirects Bosonic fields]]


boson star

#Contents#
* table of contents
{:toc}

## Idea

A hypothetical [[Bose-Einstein condensate]] big enough to bind by its own [[gravity]] and form a star.

## Related concepts

* [[neutron star]]

## References

* Wikipedia, _[Exotic star -- Boson star](http://en.wikipedia.org/wiki/Boson_star#Boson_stars)_

[[!redirects boson stars]]


bosonic M-theory

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

#Contents#
* table of contents
{:toc}

## Idea

In analogy to how there is conjectured to be a [[non-perturbative effect|non-perturbative]] completion of [[superstring theory]] on 10-dimensional [[spacetimes]] to an 11-dimensional theory to be called "[[M-theory]]", there has been the speculation that the [[bosonic string]] with its 26-dimensional [[target spacetime]] might have a non-perturbative completion in a 27-dimensional theory of sorts: "bosonic M-theory" ([Horowitz-Susskind 00](#HorowitzSusskind00))

## Related concepts

* [[topological M-theory]]

## References

The term "bosonic M-theory" was coined in

* {#HorowitzSusskind00} [[Gary Horowitz]], [[Leonard Susskind]], _Bosonic M Theory_, Journal of Mathematical Physics, Volume 42, Issue 7  ([arXiv:hep-th/0012037](https://arxiv.org/abs/hep-th/0012037))

The suggestion that the 26 dimensions of the [[target spacetime]] of the [[bosonic string]] could somehow be identified with the $10_{even} + 16_{odd}$ dimensions of the [[heterotic string]] was previously voiced in

* {#BerkovitsVafa93} [[Nathan Berkovits]], [[Cumrun Vafa]], _On the Uniqueness of String Theory_, Mod.Phys.Lett. A9 (1994) 653-664 ([arXiv:hep-th/9310170](https://arxiv.org/abs/hep-th/9310170))

The possibility of a [[bosonic M-theory]] in 27-dimensions which is the lift of actual [[M-theory]] through [[KK-compactification]] on 16-dimensional [[Cayley plane]] fibers:

* [[Hisham Sati]], _$\mathbb{O}P^2$ bundles in M-theory_, Commun. Num. Theor. Phys 3:495-530,2009 ([arXiv:0807.4899](https://arxiv.org/abs/0807.4899))

* [[Hisham Sati]], _On the geometry of the supermultiplet in M-theory_, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 1-33 ([arXiv:0909.4737](https://arxiv.org/abs/0909.4737))

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D **99** (2019) 046004  [[arXiv:1811.06101](https://arxiv.org/abs/1811.06101), [doi:10.1103/PhysRevD.99.046004](https://doi.org/10.1103/PhysRevD.99.046004)]

  (relating also to [[12-dimensional supergravity]], [[D=14 supersymmetry]] et al.)

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], *Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B **808** (2020) 135674 [[arXiv:1906.10709](https://arxiv.org/abs/1906.10709), [doi:10.1016/j.physletb.2020.135674](https://doi.org/10.1016/j.physletb.2020.135674)]

reviewed in:

* [[Alessio Marrani]], *Exceptional super Yang-Mills in $27 + 3$ and worldvolume M-theory*, talk at *[M-Theory and mathematics 2023](https://ncatlab.org/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan, 2023) [[web](/nlab/show/M-Theory+and+Mathematics#Marrani2023)]

Relation to the [[Monster group]], [[Moonshine]] and the [[Monster vertex operator algebra]]:

* [[Alessio Marrani]], [[Michael Rios]], [[David Chester]], *Monstrous M-theory*, Symmetry **15** 2 (2023) 490; [[doi:10.3390/sym15020490](https://doi.org/10.3390/sym15020490), [arXiv:2008.06742](https://arxiv.org/abs/2008.06742)]


bosonic modality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On [[super smooth infinity-groupoids]] there is a [[modal operator]] $\stackrel{\rightsquigarrow}{(-)}$ which projects onto the [[boson|bosonic]] components of the [[supergeometry]]. On [[formal dual]] [[superalgebras]] this is given by passing to the _[[body]]_. In terms of [[physical fields]] this is the projection onto [[boson]] fields, which are hence the [[modal types]] of $\rightsquigarrow$, and so it makes sense to speak of the _bosonic modality_.

This has a [[left adjoint]] $\rightrightarrows$ (which on superalgebras passes to the even-graded sub-algebra) and hence together these form an [[adjoint modality]] which may be thought of as characterizing the [[supergeometry]]. See at _[super smooth infinity-groupoid -- Cohesion](http://ncatlab.org/nlab/show/super+smooth+infinity-groupoid#Cohesion)_. With $\rightrightarrows$ being opposite to $\rightsquigarrow$ thereby, it makes sense to call it the _fermionic modality_.

Notice that the fermionic [[currents]] in [[physics]] (e.g. the [[electron]] density current) are indeed fermionic bilinears, i.e. are indeed in the even subalgebras of the underlying superalgebra.

$$
  \array{
    fermions & \rightrightarrows &\stackrel{}{\dashv}& \rightsquigarrow & bosons
  }
$$

The [[modal objects]] for $\rightsquigarrow$ are the _bosonic objects_.

The [[right adjoint]] of the bosonic modality is the _[[rheonomy modality]]_.

## See also

[[!include cohesion - table]]

[[!redirects fermionic modality]]

[[!redirects bosonic]]
[[!redirects fermionic]]

[[!redirects bosonic object]]
[[!redirects bosonic objects]]


bosonic string

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## Idea

The incarnation of the concept of [[string]] without [[worldsheet]] [[supersymmetry]] is the _bosonic string_ of [[bosonic string theory]]. In contrast to the [[spinning string]]/[[superstring]].

## Definition

### In symplectic geometry / geometric quantization

See the reference [below](#ReferencesSymplecticGeometryAndGeometricQuantization)

## Properties

### Vacuum

* [[tachyon condensation]]

* [[Sen's conjecture]]

## Related concepts

* [[Polyakov action]]

* [[sigma model]], [[brane]]

  * [[particle]], [[spinning particle]], [[superparticle]]

  * [[string]], [[spinning string]], [[superstring]]

    * [[type II superstring]], [[F1-brane]], [[D1-brane]], [[(p,q)-string]]

    * [[heterotic string]]

  * [[membrane]]

* [[string theory]], [[string field theory]]

* [[Goddard-Thorn no-ghost theorem]]

* [[string scattering amplitude]]

* [[string field theory]]

* [[world sheets for world sheets]]

* [[D-brane]], [[Chan-Paton gauge field]], [[Freed-Witten-Kapustin anomaly cancellation]]

* [[string theory FAQ]]

## References

### General

The [[Polyakov action]] (but see there for more) for the bosonic string (and relation to [[Liouville theory]] in the non-critical case):

* [[Alexander Polyakov]], *Quantum geometry of bosonic strings*, Phys. Lett. B **103** (1981) 207-210 [doi:10.1016/0370-2693(81)90743-7, [pdf](http://qft.itp.ac.ru/polyakov-2.pdf)]

* [[Jürgen Jost]], *Bosonic Strings: A Mathematical Treatment*, AMS/IP Stud. Adv. Math. **21** (2001) [[ISBBN:978-0-8218-4336-9](https://bookstore.ams.org/view?ProductCode=AMSIP/21.S.B)[spire:1388134](https://inspirehep.net/literature/1388134)]

See the references at _[[string theory]]_ for more.

### Symplectic geometry and geometric quantization
 {#ReferencesSymplecticGeometryAndGeometricQuantization}

The ordinary [[symplectic geometry]] and ordinary [[geometric quantization]] of the bosonic string [[sigma-model]] is discussed in the following references.

The [[symplectic structure]] and [[Kähler geometry]] of [[loop space]] is discussed in

* M.J. Bowick, S.G. Rajeev, _String theory as the Kähler geometry of loop space_ Phys. Rev. Lett. 58, 535-538 (1987)

* M.J. Bowick, S.G. Rajeev, _The holomorphic geometry of closed bosonic string theory and $Diff(S^1)/S^1$_, Nucl. Phys. B293, 348-384 (1987)

* [[Jouko Mickelsson]], _String quantization on group manifolds and the holomorphic geometry of $Diff(S^1)/S^1$_ Commun. Math. Phys. 112, 653-661 (1987) ([EUCLID](http://projecteuclid.org/euclid.cmp/1104160059))

with further comments in

* Hendrik Grundlin, C. A. Hurst, _The operator quantization of the open bosonic string: field algebra_, Communications in mathematical physics 156 (1993) ([pdf](http://web.maths.unsw.edu.au/~hendrik/String-CMP.pdf))

A correction of some points in these articles is discussed in

* [[Sergei Merkulov]], _On the geometric quantization of bosonic string_,  Class. Quantum Grav. 9 2267 (1992) ([IOP](http://iopscience.iop.org/0264-9381/9/10/010))

See also

* Yue Yua, Han-Ying Guoa, _On the geometric quantization and BRST quantization for bosonic strings_, Physics Letters B Volume 216, Issues 1–2, (1989), Pages 68–74 ([web](http://inspirehep.net/record/279613))

* Yu-liang Liu, Su-qing Chen,Guang-jiong Ni, _Geometrical quantization of bosonic string with Wess-Zumino term on genus-g Riemann surface_, Phys. Rev. D 41, 472–477 (1990)

* A. D. Popov, _Geometric quantization of strings and reparametrization invariance_, Theoretical and Mathematical Physics, Volume 83, Number 3 (1990) ([journal](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=5817&option_lang=eng))

### In multisymplectic geometry and higher geometric quantization
 {#ReferencesMultisymplecticGeometry}

A discussion starting systematically with the correct [[symplectic form]]  obtained by transgression from an [[multisymplectic geometry|multisymplectic]] [[extended phase space]] and including the [[BRST-complex|BRST sector]] is in

* {#Yu} Yue Yu, _Symplectic geometry and geometric quantization for the open bosonic string in the BRST formalism_, Physics Letters B, Volume 216, Issue 1-2, (1989) p. 75-80.

A detailed exposition of the [[multisymplectic geometry]] of the bosonic string together with its interpretation in [[2-plectic geometry]] is in

* [[John Baez]], [[Alexander Hoffnung]], [[Chris Rogers]], _Categorified Symplectic Geometry and the Classical String_ ([arXiv:0808.0246](http://arxiv.org/abs/0808.0246/))

and the appearance of the [[string Lie 2-algebra]] as the [[Heisenberg Lie 2-algebra]] of the string [[WZW-model]] in this context is discussed in

* [[John Baez]], [[Chris Rogers]], _Categorified Symplectic Geometry and the String Lie 2-Algebra_. [arXiv:0901.4721](http://arxiv.org/abs/0901.4721).

### Vacuum

(...)

* [[Jan Ambjørn]], Y. Makeenko, _Stability of the nonperturbative bosonic string vacuum_ ([arXiv:1703.05382](https://arxiv.org/abs/1703.05382))

[[!redirects bosonic strings]]


bosonic string theory

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## Idea

The flavor of [[string theory]] for [[bosonic strings]] is called _bosonic string theory_.

For more see at _[[string theory]]_.

## Properties

### Leech lattice compactification

* [[Leech lattice]]

### D25 brane decay and true tachyon vacuum

See at _[[Sen's conjecture]]_ on the [[D25-brane]] decay.

## Related concepts

* [[type II superstring theory]]

* [[heterotic string theory]]

* [[M-theory]]


bosonization

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## Idea

In ([[2d CFT|2d]]) [[conformal field theories]], certain [[fermion]]-fields may be re-expressed as chiral [[boson]] fields -- this is called _bosonization_.

The converse of bosonization explains for instance why [[self-dual higher gauge theory|chiral bosons]] have [[partition functions]] like [[Pfaffian line bundles]] in 2d, namely [[theta functions]] (...) ([Alvares-Gaumé-, Bost, Moore, Nelson & Vafa 1987](AlvaresGaumeBostMooreNelsonVafa87)).

## Related concepts

* [[condensed matter physics]]

* [[string theory]]

## References

Lecture notes and surveys include

* D. Sénéchal, _An introduction to bosonization_ ([arXiv:cond-mat/9908262](http://arxiv.org/abs/cond-mat/9908262))

See also:

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

* {#AlvaresGaumeBostMooreNelsonVafa87} [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Gregory Moore]], Philip Nelson, [[Cumrun Vafa]], _Bosonization on higher genus Riemann surfaces_, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 ([Euclid](http://projecteuclid.org/euclid.cmp/1104159982))

Discussion for 3d [[CFT]]s (such as the [[ABJM model]]):

* [[Amit Sever]], _Line Operators in Chern-Simons-Matter Theories and Bosonization in Three Dimensions_, talk at *[[Strings 2022]]* [[indico:4940843](https://indico.cern.ch/event/1085701/contributions/4940843), [slides](https://indico.cern.ch/event/1085701/contributions/4940843/attachments/2482058/4261226/Slides_Sever.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=a9f7872c-1603-4954-9030-192a72e4f456)]

[[!redirects bosonizations]]


Bost-Connes system

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## Idea

The _Bost-Connes system_ ([Bost & Connes 1995](#BostConnes95)) is the [[semigroup]] [[crossed product C*-algebra]] $C^\ast(\mathbb{Q}/\mathbb{Z})\rtimes \mathbb{N}^\times$ equipped with a canonical 1-parameter flow $t\mapsto\exp(-t H)$, and as such thought of as a [[quantum mechanical system]].

The point is that the [[partition function]] of this system is the [[Riemann zeta function]] ([Bost-Connes 95, theorem 5 (c) (page 6)](#BostConnes95)).

[[Alain Connes]] has proposed that via this relation there might be a way to shed insight on the [[Riemann hypothesis]] using tools from [[quantum mechanics]] and [[statistical mechanics]] ([Connes-Marcolli 06](#ConnesMarcolli06)). See also at _[[Riemann hypothesis and physics]]_.

## References

The original article:

* {#BostConnes95} [[Jean-Benoit Bost]]; [[Alain Connes]], *Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory*, Selecta Mathematica, New Series **1** 3 (1995) 411-457 [[doi:10.1007/BF01589495](https://doi.org/10.1007/BF01589495), ISSN 1022-1824, [pdf](https://web.archive.org/web/20110806043925/https://alainconnes.org/docs/bostconnesscan.pdf)]

Review:

* {#ConnesMarcolli06} [[Alain Connes]], [[Matilde Marcolli]], section 22 of _A walk in the noncommutative garden_ ([arXiv:0601054](http://arxiv.org/abs/math/0601054))

Further developments:

* Marcelo Laca, Sergey Neshveyev, Mak Trifkovic, _Bost-Connes systems, Hecke algebras, and induction_ ([arXiv:1010.4766](http://arxiv.org/abs/1010.4766))

An abstract generalization:

* [[Matilde Marcolli]], [[Goncalo Tabuada]], _Bost-Connes systems, categorification, quantum statistical mechanics, and Weil numbers_ [[arxiv/1411.3223](http://arxiv.org/abs/1411.3223)]


Bott connection

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## Idea

In [[foliation]]-theory the _Bott connection_ is a certain characteristic [[connection on a bundle|connection]] on the [[conormal bundle]] of the [[foliation]].

## Definition

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###### Definition

For $X$ a [[smooth manifold]] and $\mathcal{P} \hookrightarrow T X$ a [[foliation]] of $X$, incarnated as a subbundle of the [[tangent bundle]], the corresponding [[conormal bundle]]

$$
  \mathcal{P}^\perp  \hookrightarrow T^* X
$$

is the annihilator of $\mathcal{P}$ under the pairing of covectors with vectors. The corresponding **Bott connection** is the [[covariant derivative]] of vectors $X \in \Gamma(\mathcal{P})$ on covectors $\xi \in \Gamma(\mathcal{P}^\perp)$ given by the [[Lie derivative]]

$$
  \nabla_X \;\colon\; \xi \mapsto \mathcal{L}_X \xi = \iota_X d_{dR} \xi
  \,.
$$

=--

Similarly there is a Bott connection along $\mathcal{P}$ along the [[normal bundle]] $T X / \mathcal{P}$. More generally, one speaks of _a Bott connection_ for any connection on the (co)normal bundle and defined on all vector fields on $X$ which restricts to the above along leaves.

## References

The notion originates in

* [[Raoul Bott]], _Lectures on characteristic classes and foliations, in Lectures on algebraic and differential topology_, Springer Lecture Notes in Mathematics, 279, (1972)

in the study of [[characteristic classes]] of [[foliations]]. Further developments are in

* Jonathan Bowden, _On foliated characteristic classes of transversally symplectic foliations_ ([arXiv:1108.1919](http://arxiv.org/abs/1108.1919))

[[!redirects Bott connections]]


Bott element

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## Idea

A _Bott element_ is an element in a [[periodic cohomology theory]], multiplication by which is [[invertible element|invertible]].

Specifically for complex [[topological K-theory]] there is the element of $1 - L \,\in\, K\big( \mathbb{C}P^1 \big)$ degree 2 (the [[virtual vector bundle|virtual difference]] of the [[trivial line bundle]] with the [[tautological line bundle]] on the [[Riemann sphere]]/[[complex projective space|complex projective 1-space]]) which induces [[Bott periodicity]]. This is where the term "Bott element" comes from. For details on this case see at

* _[[fundamental product theorem in topological K-theory]]_

* _[[equivariant K-theory of projective G-space]]_.

## Related entries

* [[Bott periodicity]]

## References

* [[A Survey of Elliptic Cohomology - cohomology theories]]

[[!redirects Bott class]]

[[!redirects Bott generator]]
[[!redirects Bott generators]]


Bott periodicity

#Contents#
* table of contents
{:toc}

## Idea

_Bott periodicity_ is the name of a periodicity phenomenon that appears throughout [[spin geometry]], [[supersymmetry]] and [[K-theory]]. Incarnations of it include the following:

### In topological K-theory

The [[complex numbers|complex]] [[reduced K-theory|reduced]] [[topological K-theory]] groups have a degree-2 periodicity:

$$
  \tilde K_{\mathbb{C}}^\bullet(X)
    \simeq
  \tilde K_{\mathbb{C}}^{\bullet + 2}(X)
  \,.
$$

This isomorphism is induced by [[external tensor product]] with the image of the [[basic line bundle on the 2-sphere]] in [[reduced K-theory]], called the _[[Bott element]]_.

For details see at _[[topological K-theory]]_ the section _[Bott periodicity](topological+K-theory#BottPeriodicity)_.

The periodicity lifts to the [[classifying spaces]] and makes the [[Brown representability theorem|representing]] [[spectrum]] [[KU]] of complex K-theory be an [[even periodic ring spectrum]].

In particular the 2-periodicity in the [[homotopy groups]] of the [[stable unitary group]] $U = \underset{\longrightarrow}{\lim}_n U(n)$ is thus a shadow of Bott periodicity.

$$
  \pi_i(U) = \pi_i(GL_{\mathbb{C}})
  =
  \left\lbrace
   \array{
     0 &\vert& i\, \text{even}
     \\
     \mathbb{Z} &\vert& i \, \text{odd}
   }
   \right.
$$

Similarly the [[real numbers|real]] [[reduced K-theory|reduced]] [[topological K-theory]] groups have an 8-periodicity

$$
 \tilde K^\bullet_{\mathbb{R}}(X)
   \simeq
  \tilde K^{\bullet + 8}_{\mathbb{R}}( X )
$$

a shadow of which is the 8-periodicity in the [[homotopy groups]] of the [[stable orthogonal group]]

$$
  \pi_i( O )
  =
  \pi_i(GL_{\mathbb{R}})
  =
  \left\{
    \array{
       \mathbb{Z}_2 &\vert& i = 0 \, \text{mod}\, 8
       \\
       \mathbb{Z}_2 &\vert& i = 1 \, \text{mod}\, 8
       \\
       0 &\vert& i = 2 \, \text{mod}\, 8
       \\
       \mathbb{Z} &\vert& i = 3 \, \text{mod}\, 8
       \\
       0 &\vert& i = 4 \, \text{mod}\, 8
       \\
       0 &\vert& i = 5 \, \text{mod}\, 8
       \\
       0 &\vert& i = 6 \, \text{mod}\, 8
       \\
       \mathbb{Z} &\vert& i = 7 \, \text{mod}\, 8
     }
   \right.
$$

### In Spin geometry

The complex [[Clifford algebras]] repeat -- up to _[[Morita equivalence]]_ -- with period 2, $Cl_{n}(\mathbb{C}) \simeq_{Morita} Cl_{n+2}(\mathbb{C})$.

The real Clifford algebras analogously have period 8, $Cl_n(\mathbb{R}) \simeq_{Morita} Cl_{n+8}(\mathbb{R})$.

Accordingly the basic properties of complex [[spinor representations]] are the same for $Spin(d-1,1)$ and $Spin(d+2-1,1)$. Those of the real spinor representations repeat with period 8.

## Related concepts

* [[ten-fold way]]

* [[Hopf fibration]]

* [[periodic cohomology theory]]

* [[supersymmetry and division algebras]].

## References

The original statement as periodicity of [[stable homotopy groups]] of [[classical Lie groups]]:

* [[Raoul Bott]], *The Stable Homotopy of the Classical Groups*, Proceedings of the National Academy of Sciences of the United States of America **43** 10 (1957) 933-935 [[jstor:89403](https://www.jstor.org/stable/89403)]

* [[Eldon Dyer]], [[Richard Lashof]], *A topological Proof of the Bott Periodicity Theorems*, Annali di Matematica Pura ed Applicata **54** (1961) 231–254 [[doi:10.1007/BF02415354](https://doi.org/10.1007/BF02415354)]

Proof of Bott periodicity for [[topological K-theory]], including [[equivariant K-theory]]:

* [[Michael Atiyah]], _Bott periodicity and the index of elliptic operators_, The Quarterly Journal of Mathematics, **19** 1 (1968) 113–140 [[doi:10.1093/qmath/19.1.113](https://doi.org/10.1093/qmath/19.1.113)]

Review:

* {#Segal68} [[Graeme Segal]], Prop. 3.2 in: _Equivariant K-theory_, Inst. Hautes Etudes Sci. Publ. Math.  No. 34 (1968) p. 129-151 ([numdam:PMIHES_1968__34__129_0](http://www.numdam.org/item/PMIHES_1968__34__129_0))

* {#Karoubi05} [[Max Karoubi]], _Bott Periodicity in Topological, Algebraic and Hermitian K-Theory_,  In: [[Eric Friedlander]], [[Daniel Grayson]] (eds.) _[[Handbook of K-Theory]]_, Springer 2005 ([doi:10.1007/978-3-540-27855-9_4](https://doi.org/10.1007/978-3-540-27855-9_4))

* [[Dai Tamaki]], [[Akira Kono]], Section 4.2 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#HJJS08} [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin  Schottenloher]], Section 15 of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))

For a list of proofs of Bott periodicity, see

* _Proofs of Bott periodicity_, [MO](https://mathoverflow.net/q/8800/447)

[[!redirects Bott periodicities]]

[[!redirects Bott periodicity theorem]]
[[!redirects Bott periodicity theorem]]


Bott periodicity theorem > history

> see _[[Bott periodicity]]_


Bott-Chern cohomology

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## Idea

In [[complex geometry]], Bott-Chern cohomolgy is a [[cohomology]]-theory of [[complex manifolds]] bridging between [[de Rham cohomology]] and [[Dolbeault cohomology]].

## References

General:

* [[Daniele Angella]], Adriano Tomassini, _On Bott-Chern cohomology and formality_, J. Geom. Phys. 93 (2015), 52-61 ([arXiv:1411.6037](https://arxiv.org/abs/1411.6037))

For [[orbifolds]]:

* [[Daniele Angella]], _Cohomologies of certain orbifolds_, J. Geom. Phys. 71 (2013), 117--126 ([arXiv:1211.2561](https://arxiv.org/abs/1211.2561))

* Xiaonan Ma, _Orbifolds and Analytic Torsions_, Transactions of the American Mathematical Society Vol. 357, No. 6 (Jun., 2005), pp. 2205-2233 ([jstor:3845255](https://www.jstor.org/stable/3845255))


bottom

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## Idea

In a [[partial order|poset]] $P$, a __bottom__ is a __least__ element: an element $\bot$ of $P$ such that $\bot \leq a$ for every element $a$.  Such a bottom may not exist; if it does, then it is unique.

In a [[preorder|proset]], a bottom may be defined similarly, but it need not be unique.  (However, it is still unique up the natural [[equivalence]] in the proset.)

A bottom of $P$ can also be understood as a [[join]] of zero elements in $P$.

A poset that has both [[top]] and bottom is called _bounded_.

As a poset is a special kind of [[category]], a bottom is simply an [[initial object]] in that category.

The bottom of the poset of [[subset]]s or [[subobject]]s of a given set or object $A$ is called the [[empty set|empty]] subset or subobject.  In a category (such as [[Set]]) with a strict [[initial object]] $\varnothing$, this will always serve as the bottom of any subobject poset.

## Related concepts

[[!include logic symbols -- table]]

[[!redirects bottom]]
[[!redirects bottoms]]
[[!redirects bottom element]]
[[!redirects bottom elements]]
[[!redirects least element]]
[[!redirects least elements]]


bottom quark

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## Idea

In the [[standard model of particle physics]] the _top quark_ is the down-type [[quark]] in the third [[generation of fermions]], a [[heavy quark]].

The [[bound states]] of two (anti-)bottom quarks are the _[[B mesons]]_.

Hints of [[flavour anomalies]] indicate that something peculiar may be going on with bottom quarks.

## Related concepts

* the [[mesons]] including [[bottom quarks]] are [[B-mesons]]

* the [[charmonium]] formed from [[bottom quarks]] are [[upsilon-mesons]]

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

## References

See also:

* Wikipedia, _[Bottom quark](https://en.wikipedia.org/wiki/Bottom_quark)_

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!redirects bottom quarks]]

[[!redirects beauty quark]]
[[!redirects beauty quarks]]


bottom-up and top-down model building

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## Idea

In [[particle physics]] [[phenomenology]] one distinguishes two strategies for constructing [[model (physics)|models]] of a [[theory (physics)|theory]] with desired properties:

* in a "top-down approach" one considers the full [[theory (physics)|theory]] right away and tries to narrow in, in its space of [[model (physics)|models]], to a subspace of models with the desired properties;

* in a "bottom-up approach" one first specifies exactly which properties a [[model (physics)|model]] is supposed to have, and then incrementally uses the remaining wiggle room to see if this specification may be completed to an actual model of the full theory.

This distinction was highlighted, and the terminology established, in ([Aldazabal-Ibáñez-Quevedo-Uranga 00](#AldazabalIbanezQuevedoUranga00)); there in the context of discussion of [[string phenomenology]] and specifically of [[intersecting D-brane models]] in [[type II string theory]]. In fact the article proposes and advertizes the bottom-up-strategy, felt to be in contrast with the, at that time, dominant top-down-strategy going back to ([Candelas-Horowitz-Strominger-Witten 85](#CandelasHorowitzStromingerWitten85)).

>  snippet grabbed from [Aldazabal-Ibáñez-Quevedo-Uranga 00](#AldazabalIbanezQuevedoUranga00)

For more on this see at [[string phenomenology]] the section _[Top-down models and Bottom-up models](string+phenomenology#TopDownAndBottomUpModels)_.

## Related entries

* [[string theory inspired model]]

* [[AdS/QCD correspondence]]

* [[G2-MSSM]]

## References

In [[string phenomenology]] the terminology as such was introduced and specifically the "bottom-up approach" was advertized in

* {#AldazabalIbanezQuevedoUranga00} G. Aldazabal, [[Luis Ibáñez]], [[Fernando Quevedo]], [[Angel Uranga]], _D-Branes at Singularities: A Bottom-Up Approach to the String Embedding of the Standard Model_, JHEP 0008:002 2000 ([arXiv:hep-th/0005067v2](https://arxiv.org/abs/hep-th/0005067))

while the "top-down approach" in [[string phenomenology]] originates with

* {#CandelasHorowitzStromingerWitten85} [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], [[Edward Witten]], _Vacuum configurations for superstrings_, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) (doi:10.1016/0550-3213(85)90602-9)

[[!redirects bottom-up model building]]
[[!redirects top-down model building]]

[[!redirects top-down and bottom-up model building]]

[[!redirects bottom-up model]]
[[!redirects bottom-up models]]

[[!redirects top-down model]]
[[!redirects top-down models]]


Boulatov model

This is a particular model in [[3d quantum gravity]]

* D. V. Boulatov, _A model of three-dimensional lattice gravity_,
Modern Phys. Lett. A 7 (1992), no. 18, 1629–1646, [MR94c:83039](http://www.ams.org/mathscinet-getitem?mr=1168831), [doi](http://dx.doi.org/10.1142/S0217732392001324)

There is a similar reasoning in 3+1 d in

* [[Hirosi Ooguri]], _Topological lattice models in four dimensions_, Modern Phys. Lett. A 7 (1992), no. 30, 2799--2810; [MR93h:57031](http://www.ams.org/mathscinet-getitem?mr=1184565), [doi](http://dx.doi.org/10.1142/S0217732392004171)

* Roberto De Pietri, Laurent Freidel, Kirill Krasnov, [[Carlo Rovelli]], _Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space_, Nuclear Phys. B __574__ (2000), no. 3, 785–806, [arxiv/hep-th/9907154](http://arxiv.org/abs/hep-th/9907154), [MR2001f:83050](http://www.ams.org/mathscinet-getitem?mr=1760223), <a href="http://dx.doi.org/10.1016/S0550-3213(00)00005-5">doi<a/>