Langlands program

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###Context###
#### Langlands correspondence
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[[!include Langlands correspondence -- contents]]
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#### Duality
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[[!include duality - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

What is called the _Langlands correspondence_ in [[number theory]] ([Langlands 67](#Langlands67)) is first of all a [[conjecture|conjectural]] correspondence (a [[bijection]] subject to various conditions) between

1. $n$-[[dimensional]] complex [[linear representations]] of the [[Galois group]] $Gal(\bar F/F)$ of a given [[number field]] $F$, and

1. certain [[representations]] -- called _[[automorphic representations]]_ -- of the $n$-dimensional [[general linear group]] $GL_n(\mathbb{A}_F)$ with [[coefficients]] in the [[ring of adeles]] of $F$, arising within the representations given by [[functions]] on the [[double coset space]] $GL_n(F) \backslash GL_n(\mathbb{A}_F)/GL_n(\mathcal{O})$ (where $\mathcal{O} = \prod_v \mathcal{O}_p$ is the [[ring of integers]] of all [[formal completions]] of $F$).

This is motivated from the abelian case ($n=1$), which is fully understood: For $n = 1$ then an $n$-dimensional representation of the [[Galois group]] factors through $GL_1$ and hence through an [[abelian group]]. Therefore, by [[adjunction]], it is equivalently a representation of the [[abelianization]] of the Galois group. The [[Kronecker-Weber theorem]] says that for $F = \mathbb{Q}$, then the abelianized Galois group is the [[idele class group]] $GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})$, and hence 1-dimensional representations of the Galois group are equivalently representations of this. Moreover, one finds that for any [[prime number]] $p$, then those representations which are "unramified at $p$" are invariant under the subring of [[p-adic integers]], hence are representations of the double quotient group $GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})/GL_1(\mathbb{Z}_p)$. More generally, the [[Artin reciprocity law]] says that for [[number fields]] there is an [[isomorphism]] between $GL_1(K) \backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O}_K)$ and the [[abelianization|abelianized]] [[Galois group]].

$\,$

Moreover:

{#Conjecture1} **Conjecture 1** To each such [[automorphic representation]] $\pi$ is associated an [[L-function]] -- the _[[automorphic L-function]]_ $L_\pi$ -- and in generalization of [[Artin reciprocity]] the conjecture of Langlands is that the [[Artin L-function]] $L_\sigma$ associated with the given [[Galois representation]] $\sigma$ is equal to this: $L_\sigma = L_\pi$ ([Gelbart 84, conjecture 1 (page 27 (204))](#Gelbart84)).

$\,$

More generally, analogous statements are supposed to hold for general [[reductive algebraic groups]] $G$ other than $GL_n$. Here now an [[L-function]] is assigned to data which in addition to the [[Galois representation]] consists of a [[linear representation]] ${}^L G \longrightarrow GL_n$ of the [[Langlands dual group]] of $G$.

First of all:

{#Conjecture2} **Conjecture 2** This more general L-function is conjectured to indeed behave like a decent L-function in that it has meromorphic [[analytic continuation]] to the [[complex plane]] and satisfies the "[[functional equation]]"-invariance under sending its parameter $s$ to $1-s$ ([Gelbart 84, conjecture 2' (page 29 (205))](#Gelbart84)).

Second:

{#Conjecture3} **Conjecture 3** This construction is supposed to behave well with respect to an analytic [[homomorphism]] ${}^L G \to {}^L G^{'}$ in that when changing the representation of ${}^L G^{'}$ by precomposition with this homomorphism one may find an accompanying change of Galois representation/automorphic representation from $G$ to $G^{'}$ such that the associated L-function remains invariant under these joint changes. This statement is what [[Robert Langlands]] calls _functoriality_ ([Gelbart 84, conjecture 3 (page 31 (207))](#Gelbart84))

In fact this last conjecture implies the previous two ([Gelbart 84, (page 32 (208))](#Gelbart84)).

$\,$

Various versions and refinements of this conjecture have since been considered, for some perspective see ([Taylor 02](#Taylor02), [Langlands 14](#Langlands14), [Harris 14](#Harris14)). On the one hand the "localization" of the program to [[local fields]] leads to the conjecture of _[[local Langlands correspondences]]_ ([Gelbart 84, (page 34 (210))](#Gelbart84)). On the other hand, the interpretation of the above story dually in [[arithmetic geometry]] in view of the [[function field analogy]] motivates the conjectural _[[geometric Langlands correspondence]]_, based on the following [[analogy]]:

* the [[Galois group]] is essentially the [[fundamental group]] of an [[algebraic curve]];

* hence a [[representation]] of it is a [[principal bundle|principal]]/[[vector bundle]] with [[flat connection]] on this curve;

* while that curious double quotient $GL_n(F) \backslash GL_n(\mathbb{A}_F)/GL_n(\mathcal{O}_F)$ is, in view of the [[Weil uniformization theorem]], analogous to the canonical [[Cech cohomology|Cech]]-realization of the [[moduli stack of bundles]] (see there for details) on that curve.

[[!include Langlands analogies -- table]]

From this [[arithmetic geometry]] point of view the Langlands conjecture seems to speak of a correspondence that sends [[Dirac distributions]] on the [[moduli space of flat connections]] over an [[algebraic curve]] to certain "automorphic" functions on the [[moduli stack of bundles]] on the same curve. This suggests that the Langlands correspondence should be understood as a nonabelian version of a [[Fourier-Mukai transform|Fourier-Mukai]]-type [[integral transform]]. This version of the conjecture is known as the _[[geometric Langlands correspondence]]_. See there for more details.

## Related concepts

* [[local Langlands correspondence]]

* [[automorphic form]]

* [[Galois representation]]

* [[Shimura variety]]

* [[modularity theorem]]

* [[Langlands functoriality]]

* [[relative Langlands program]]

## References
 {#References}

The original [[conjecture]] is due to

* {#Langlands67} [[Robert Langlands]], letter to [[Andre Weil]] 1967 ([IAS page](https://publications.ias.edu/rpl/paper/43) including scan and TeXed version, see also [this UBC page](http://sunsite.ubc.ca/DigitalMathArchive/Langlands/functoriality.html#weil1967) for a little more information)

Introductions and expository surveys include

* {#Gelbart84} [[Stephen Gelbart]], _An elementary introduction to the Langlands program_,  Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([euclid:1183551573](https://projecteuclid.org/euclid.bams/1183551573),  [doi:10.1090/S0273-0979-1984-15237-6](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))

* [[Edward Frenkel]], _Commentary on "An elementary introduction to the Langlands Program" by [[Stephen Gelbart]]_, Bull. Amer. Math. Soc. __48__ (2011), 513-515, ([pdf](http://www.ams.org/journals/bull/2011-48-04/S0273-0979-2011-01347-7/S0273-0979-2011-01347-7.pdf))

* Mark Goresky, _Langlands' conjectures for physicists_ ([pdf](http://www.math.ias.edu/~goresky/pdf/Physicists.pdf))

* {#Cogdell12} James W. Cogdell, _L-functions and non-abelian class field theory, from Artin to Langlands_, 2012 ([pdf](https://people.math.osu.edu/cogdell.1/Artin-www.pdf))

* {#MK14} [[Minhyong Kim]], _A superficial introduction to Langlands functoriality_, 2014 ([pdf](http://people.maths.ox.ac.uk/vonk/automorphic/functoriality.pdf))

*  {#BernsteinGelbart18} [[Joseph Bernstein]], [[Stephen Gelbart]] (eds.), _An Introduction to the Langlands Program_, Springer 2018 ([doi:10.1007/978-0-8176-8226-2](https://link.springer.com/book/10.1007/978-0-8176-8226-2))

Surveys of the state of the program include

* {#Taylor02} Richard Taylor, _Galois Representations_, Proceedings of the ICM 2002 ([long version pdf](http://www.math.ias.edu/~rtaylor/longicm02.pdf)).

* {#Langlands14} [[Robert Langlands]], _[[Problems in the theory of automorphic forms -- 45 years later]]_, Oxford 2014

* {#Harris14} Michael Harris, _Automorphic Galois representations and Shimura varieties_, Proceedings of the ICM 2014 ([pdf](https://www.imj-prg.fr/~michael.harris/2014.pdf)).

Discussion with an eye towards [[geometric class field theory]] and [[geometric Langlands duality]] and [[quantum field theory]] is in

* {#Kapranov95} [[Mikhail Kapranov]], _Analogies between the Langlands Correspondence and Topological Quantum Field Theory_, in _Functional Analysis on the Eve of the 21st Century_ Progress in Mathematics Volume 131/132, 1995, pp 119-151

* {#Toth11} Peter Toth, _Geometric abelian class field theory_, 2011 ([web](http://dspace.library.uu.nl/handle/1874/206061))

* {#Frenkel09} [[Edward Frenkel]], _Gauge Theory and Langlands Duality_ ([arXiv:0906.2747](http://arxiv.org/abs/0906.2747))

More resources are at

* the [digital archive](http://sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html) of [[Robert Langlands]]' articles at sunsite.ubc.ca, not currently being maintained, see:

* [The Work of Robert Langlands](https://publications.ias.edu/rpl) at the Institute of Advanced Study.

[[!redirects Langlands duality]]

[[!redirects Langlands correspondence]]
[[!redirects Langlands correspondences]]

[[!redirects Langlands conjecture]]
[[!redirects Langlands conjectures]]


language

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

## Idea

A _formal language_ is a collection of words over an [[alphabet]], i.e. strings of symbols from the alphabet, according to certain rules.

## Related concepts

* [[object language]]

* [[metalanguage]]

* [[syntax]], [[theory]]

* [[mathematical linguistics]]

* [[linguistics]]

[[!include mathematical statements --- contents]]

## References

* [[Joachim Lambek]] (with [[Michael Barr]]), *Programs, Grammars, Arguments*, lecture notes (2007) [[pdf](https://www.math.mcgill.ca/barr/papers/pga.pdf), [[LambekBarr-ProgramsGrammarArguments.pdf:file]]]

See also:

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

[[!redirects languages]]

[[!redirects formal language]]
[[!redirects formal languages]]

[[!redirects alphabet]]


language (in model theory)

## Idea

formal [[language]] as used in [[model theory]]

## Related concepts

* [[structures in model theory]]

[[!redirects language in model theory]]


language game

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

## Idea

Language games are a [[philosophy|philosophical concept]] introduced in [Wittgenstein (1953)](#Wittgenstein53).
They are only distantly related to [[game theory]], however they have recently been studied through the lens of [[game theory|compositional game theory]].

## References

* {#Wittgenstein53} [[Ludwig Wittgenstein]]. Philosophical investigations. 1953.
* [[Jules Hedges]] and [[Martha Lewis]]. Towards functorial language-games. CAPNS, 2018. ([arXiv:1807.07828](https://arxiv.org/abs/1807.07828))
* [[Giovanni de Felice]], [[Elena Di Lavore]], [[Mario Román]], [[Alexis Toumi]]. Functorial Language Games for Question Answering. ACT, 2020. ([arXiv:2005.09439](https://arxiv.org/abs/2005.09439))


Laplace operator

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###Context###
#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(..)

## Definition

+-- {: .num_defn }
###### Definition
**([[Laplace-Beltrami operator]])**

Given a [[Riemannian manifold]] $(X,g)$, the _[[Laplace-Beltrami operator]]_ $\Delta$ is the [[differential operator]]  on the space of [[smooth functions]]  $f \in C^\infty(X) = \Omega^0(X)$ given by the formula

\[
  \label{LaplaceOperatorViaHodgeDeRhamCalculus}
  \Delta f
  \;\coloneqq\;
  \star d \star d f
  \,,
\]

where

* $d \;\colon\; \Omega^\bullet(X) \to \Omega^{\bullet + 1}(X)$ is the [[de Rham differential]] (depending only on the [[smooth manifold]] $X$, not on the [[metric tensor]]);

* $\star \;\colon\; \Omega^\bullet(X) \to \Omega^{dim(X)-\bullet}(X)$ is the [[Hodge star operator]] of $(X,g)$ (this is where the dependence on the [[metric tensor]] enters).

The same formula makes sense more generally for [[pseudo-Riemannian manifolds]]. Even so, in the pseudo-Riemannian case one tends to speak of the _[[wave operator]]_ instead of the Laplace operator, and to use the symbol $\Box$ instead of $\Delta$ (at least for flat [[pseudo-Riemannian manifolds]]: [[Minkowski spacetime]]).

=--

+-- {: .num_prop #LaplaceOperatorInComponents}
###### Proposition
**([[coordinate]]-expression of [[Laplace operator]])**

If $U \subset X$ is a [[chart]] of $X$ with [[coordinate functions]] $\{x^i \colon U \to \mathbb{R}\}$, then the Laplace operator (eq:LaplaceOperatorViaHodgeDeRhamCalculus) is equivalently given by the following component-expression:

$$
  \Delta f_{\vert U}
  \;=\;
  \frac{
     sgn(g)
  }{
    \sqrt{
      \left\vert
        det\big( (g_{i j})  \big)
      \right\vert
    }
  }
  \partial_i
  \left(
    \sqrt{
      \left\vert
        det\big( (g_{i j})  \big)
      \right\vert
    }
    g^{i j}
    \partial_j
    f
  \right)
  \,,
$$

=--

Here:

* $sgn(g)$ is the [[signature of a quadratic form|signature]] of the [[metric tensor]], hence

  * $sgn(g) = +1$ for [[Riemannian manifolds]],

  * $sgn(g) = -1$ for [[pseudo-Riemannian manifolds]];

* $\partial_i \coloneqq \frac{\partial}{\partial x^i}$ denotes the [[partial derivative]] by the coordinate $x^i$;

* $(g_{i j})$ is the [[square matrix]] of components of the [[metric tensor]] $g$ in the given [[coordinate chart]], hence such that

  $$
    g
    \;=\;
    g_{i j}
    d x^i \otimes d x^j
  $$

* $det\big( (g_{i j}) \big)$ is the [[determinant]] of this [[matrix]],

  $\left\vert det\big( (g_{i j}) \big)\right\vert$ its [[absolute value]] (taking this is irrelevant for [[Riemannian manifolds]] but necessary for [[pseudo-Riemannian manifolds]], where the determinan of the metric is negative),

  $\sqrt{\left\vert det\big( (g_{i j}) \big)\right\vert}$ is the [[positive number|positive]] [[square root]] of that;

* $(g^{i j})$ is the corresponding [[inverse matrix]];

* the [[Einstein summation convention]] is used throughout, meaing that a [[sum]] over repeated indices is understood.

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

Using

1. the component formula for the [[de Rham differential]] $d = d x^i \partial_i$,

1. the component formula for the [[Hodge star operator]] (see [there](Hodge+star+operator#eq:ComponentFormulaForHodgeStar))

we compute as follows:

$$
  \begin{aligned}
    \star d \star d f
    & =
    \star d \star (\partial_j f) d x^j
    \\
    & =
    \star d
    \left(
      \tfrac{1}{ \color{green} (D-1)! }
      \sqrt{
        \left\vert
           det\big( (g_{i j}) \big)
        \right\vert
      }
      \,
      g^{ i  j} (\partial_j f)
      \,
      \epsilon_{
        i
        {\color{green} k_2 \cdots k_{D} }
      }
      d x^{
        \color{green} k_2
      }
      \wedge \cdots \wedge
      d x^{
        \color{green} k_{D}
      }
    \right)
    \\
    & =
    \star
    \partial_{ \color{magenta} k_1}
    \left(
      \tfrac{1}{ \color{green} (D-1)! }
      \sqrt{
        \left\vert
           det\big( (g_{i j}) \big)
        \right\vert
      }
      \,
      g^{i j} (\partial_j f)
      \,
      \epsilon_{
        i
        {\color{green} k_2 \cdots k_{D} }
      }
      d x^{ \color{magenta} k_1 }
      \wedge
      d x^{
        \color{green} k_2
      } \wedge \cdots \wedge
      d x^{
        \color{green} k_{D}
      }
    \right)
    \\
    & =
    \sqrt{
      \left\vert
        det\big( (g_{i j}) \big)
      \right\vert
    }
    \underset{
      =
      \det\big( (g_{i j})^{-1}  \big)
      \delta^{ \color{magenta} k_1 }_i
    }{
    \underbrace{
    \tfrac{1}{
       { \color{orange} D! }
       { \color{green} (D-1)! }
    }
    \epsilon_{
      \color{orange} l_1 l_2 \cdots l_D
    }
    g^{
      { \color{orange} l_1 }
      { \color{magenta} k_1  }
    }
    g^{
      { \color{orange} l_2 }
      { \color{green} k_2  }
    }
    \cdots
    g^{
      { \color{orange} l_D}
      { \color{green} k_D }
    }
    \epsilon_{
      i
      {\color{green} k_2 \cdots k_{D} }
    }
    }
    }
    \,
    \partial_{ \color{magenta} k_1 }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{i j} (\partial_j f)
    \right)
    \\
    & =
    \frac{1}{
       \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
    }
    \delta^{ \color{magenta} k_1 }_i
    \partial_{ \color{magenta} k_1 }
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{i j} (\partial_j f)
    \right)
    \\
    & =
    \frac{
      sgn(g)
    }{
       \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
    }
    \partial_{i}
    \left(
      \sqrt{
        \left\vert
          det\big( (g_{i j}) \big)
        \right\vert
      }
      g^{i j} (\partial_j f)
    \right)
  \end{aligned}
$$

=--

## Properties

### Functional determinant and Analytic torsion

The [[functional determinant]] of Laplace operator on a given space of [[differential p-forms]] appears as factor of the _[[analytic torsion]]_ of the given [[Riemannian manifold]].

## Related entries

* [[heat kernel]]

* [[harmonic form]], [[Hodge theory]]

* [[hearing the shape of a drum]]

* [[BV-Laplacian]]

## References

Textbook accounts:

* [[Nicole Berline]], [[Ezra Getzler]], [[Michele Vergne]], _Heat kernels and Dirac operators_, Grundlehren __298__, Springer 1992, "Text Edition" 2003.

See also:

* Wikipedia, _[Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator)_, _[Laplace-Beltrami operator](https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator)_

* MathOverflow: [why-is-the-laplacian-ubiquitous](http://mathoverflow.net/questions/54986/why-is-the-laplacian-ubiquitous)

[[!redirects Laplacian]]
[[!redirects Laplace operators]]

[[!redirects Laplace-Beltrami operator]]
[[!redirects Laplace-Beltrami operators]]

[[!redirects generalized Laplace operator]]
[[!redirects generalized Laplace operators]]


Laplace transform

#Contents#
* table of contents
{:toc}

## Idea

The _Laplace transform_ of a [[function]] $f$ on a suitable [[affine space]] is the result of an [[integration]] of the form

$$
  \hat f(a) \coloneqq \int f(x) e^{- x a} \, d x
  \,,
$$

when defined.

For [[imaginary number|imaginary]] numbers $a$, this is the _[[Fourier transform]]_. If $a$ is allowed to be a [[complex number]], this is also called the _[[Fourier-Laplace transform]]_.

## Related concepts

* [[Mellin transform]]

## References

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

[[!redirects Laplace transforms]]


Lara Anderson

Lara B. Anderson works on [[string phenomenology]], notably in [[heterotic string theory]].

* [publication list](http://arxiv.org/find/hep-th/1/au:+Anderson_Lara/0/1/0/all/0/1?skip=0&query_id=fc98e528f0b049a9)

## Selected writings

On [[heterotic string theory|heterotic]] [[string phenomenology]] with [[heterotic line bundle models]]:

* {#AndersonGrayLukasPalti11} [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_, Phys. Rev. D 84, 106005 (2011) ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804))

* {#AndersonGrayLukasPalti12} [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _Heterotic Line Bundle Standard Models_ JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757))

Scan of [[SU(5)]] [[GUT]] [[perturbative string theory vacuum|vacua]] among [[heterotic line bundle models]]:

* [[Lara Anderson]], [[Andrei Constantin]], [[James Gray]], [[Andre Lukas]], [[Eran Palti]], _A Comprehensive Scan for Heterotic $SU(5)$ GUT models_, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787))

On [[moduli stabilization]]:

* [[Lara Anderson]], [[James Gray]], [[Andre Lukas]], [[Burt Ovrut]], _Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories_ ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704))

category: people


large 1/N limit

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###Context###
#### Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#### String theory
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[[!include string theory - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of the [['t Hooft double line construction]] and the [[AdS/CFT correspondence]] with $N_c \in \mathbb{N}$ [[color charge|colors]], the _small $N$ region_ or _large $1/N_c$ limit_ is the situation _away from_ the [[large N limit]]. In this large $1/N_c$ limit [[perturbative string theory]]-corrections (for small [['t Hooft coupling]] $\lambda = g_{YM}^2 N_c$) and/or [[M-theory]]-corrections (for small [[color charge|$N_c$]]) to the [[supergravity]]-approximation of the [[AdS/CFT correspondence]] are relevant.

In application to [[phenomenology]], the small $N_c$ region (and hence [[M-theory]]) is what is ultimately relevant for the [[AdS/QCD correspondence]], given that [[quantum chromodynamics]] has a small number $N_c^{QCD} = 3$ of [[quark]] [[color charge|colors]]:

## References

### General

On the general relevance of [[M-theory]] at small $N_c$ in gauge/gravity duality:

* [[Leonard Susskind]], _Another Conjecture about M(atrix) Theory_ ([arXiv:hep-th/9704080](https://arxiv.org/abs/hep-th/9704080))

  (in the context of the [[BFSS matrix model]])

* [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 (1998) ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))

  ([[phase diagrams]] in [[color charge|$N_c$]] and [['t Hooft coupling]] for various [[D-brane|Dp-brane]] species)

On the [[logical equivalence]] between the [[four-colour theorem]] and a statement about transition from the [[small N limit]] to the [[large N limit]] for [[Lie algebra weight systems]] on [[Jacobi diagrams]] via the [['t Hooft double line construction]]:

* [[Dror Bar-Natan]], _Lie Algebras and the Four Color Theorem_, Combinatorica 17-1(1997) 43–52  ([arXiv:q-alg/9606016](https://arxiv.org/abs/q-alg/9606016), [doi:10.1007/BF01196130](https://doi.org/10.1007/BF01196130))

### In AdS/CFT

#### General

* B. Basso, _Cusp anomalous dimension in planar maximally supersymmetric Yang-Mills theory_ ([spire:858223](http://inspirehep.net/record/858223))

  > "The result $[$(29)$]$ coincides exactly with the recent two-loop stringy correction computed in [Alday-Maldacena 07](https://arxiv.org/abs/0708.0672), providing a striking confirmation of the AdS/CFT correspondence."

* David Jorrin, Nicolas Kovensky, Martin Schvellinger,  _Towards $1/N$ corrections to deep inelastic scattering from the gauge/gravity duality_, JHEP 04 (2016) 113 ([arXiv:1601.01627](https://arxiv.org/abs/1601.01627))

Discussion of [[small N corrections]] via a [[lattice QFT]]-Ansatz on the AdS side of [[AdS2/CFT1]]:

* Richard C. Brower, Cameron V. Cogburn, A. Liam Fitzpatrick, Dean Howarth, Chung-I Tan, _Lattice Setup for Quantum Field Theory in $AdS_2$_ ([arXiv:1912.07606](https://arxiv.org/abs/1912.07606))

#### Via conformal bootstrap

Using the [[conformal bootstrap]] for [[CFT]]s at small N to deduce [[M-theory]]-properties on the dual side:

* Nathan B. Agmon, [[Shai Chester]], Silviu S. Pufu, _Solving M-theory with the Conformal Bootstrap_, JHEP 06 (2018) 159 ([arXiv:1711.07343](https://arxiv.org/abs/1711.07343))

* [[Shai Chester]], _Bootstrapping M-theory_, 2018 ([[ChesterBootstrappingMTheory.pdf:file]])

Specifically for the [[D=6 N=(2,0) SCFT]] on the [[M5-brane]] via [[AdS7/CFT6]]:

* {#ChesterPerlmutter18} [[Shai Chester]], [[Eric Perlmutter]], _M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture_, J. High Energ. Phys. (2018) 2018: 116 ([arXiv:1805.00892](https://arxiv.org/abs/1805.00892))

* [[Luis Alday]], [[Shai Chester]], Himanshu Raj, _6d $(2,0)$ and M-theory at 1-loop_ ([arXiv:2005.07175](https://arxiv.org/abs/2005.07175))

Specifically for the [[D=3 SCFT]] ([[BLG-model]], [[ABJM model]]) on the [[M2-brane]] via [[AdS4/CFT3]]

* Nathan B. Agmon, [[Shai Chester]], Silviu S. Pufu, _The M-theory Archipelago_ ([arXiv:1907.13222](https://arxiv.org/abs/1907.13222))

Specifically in [[AdS5/CFT4]] via [[D3-brane]] contributions:

* Yosuke Imamura, *Finite-$N$ superconformal index via the AdS/CFT correspondence* ([arXiv:2108.12090](https://arxiv.org/abs/2108.12090))

### In AdS/QCD
 {#ReferencesInQAdSQCD}

Discussion of [[small N corrections]] specifically in [[holographic QCD]]:

* {#Basso08} B. Basso, _Cusp anomalous dimension in planar maximally supersymmetric Yang-Mills theory_, Continuous Advances in QCD 2008, pp. 317-328 (2008) ([spire:858223](http://inspirehep.net/record/858223), [doi:10.1142/9789812838667_0027](https://doi.org/10.1142/9789812838667_0027))

  > "The result $[$(29)$]$ coincides exactly with the recent two-loop stringy correction computed in [Alday-Maldacena 07](https://arxiv.org/abs/0708.0672), providing a striking confirmation of the AdS/CFT correspondence."

* H. Dorn, H.-J. Otto, _On Wilson loops and $Q\bar Q$-potentials from the AdS/CFT relation at $T\geq 0$_, In: A. Ceresole, C. Kounnas , [[Dieter Lüst]], [[Stefan Theisen]]  (eds.) _Quantum Aspects of Gauge Theories, Supersymmetry and Unification_ Lecture Notes in Physics, vol 525. Springer 2007 ([arXiv:hep-th/9812109](https://arxiv.org/abs/hep-th/9812109),
[doi:10.1007/BFb0104268](https://doi.org/10.1007/BFb0104268))

* Masayasu Harada, Shinya Matsuzaki, and Koichi Yamawaki, _Implications of holographic QCD in chiral perturbation theory with hidden local symmetry_, Phys. Rev. D 74, 076004 (2006) ([doi:10.1103/PhysRevD.74.076004](https://doi.org/10.1103/PhysRevD.74.076004))

  (with an eye towards [[hidden local symmetry]] in [[chiral perturbation theory]])

* Csaba Csaki, [[Matthew Reece]], John Terning, _The AdS/QCD Correspondence: Still Undelivered_, JHEP 0905:067, 2009 ([arXiv:0811.3001](https://arxiv.org/abs/0811.3001))

* Salvatore Baldino, [[Stefano Bolognesi]], Sven Bjarke Gudnason, Deniz Koksal, _A Solitonic Approach to Holographic Nuclear Physics_, Phys. Rev. D 96, 034008 (2017) ([arXiv:1703.08695](https://arxiv.org/abs/1703.08695))

* [[Edward Witten]], *Some Milestones in the Study of Confinement*, talk at *[[Prospects in Theoretical Physics 2023 -- Understanding Confinement]]*, IAS (2023) [[web](https://www.ias.edu/video/some-milestones-study-confinement), [YT](https://youtu.be/TvIz-6YOdKs)]

  > [26:26](https://youtu.be/TvIz-6YOdKs?t=1586): "by now it's clear [[Lucini & Teper 2001](large+N+limit#LuciniTeper01)] that [[lattice gauge theory]], at least for the [[glueball]] sector, has made it clear that the [[large N limit|$1/N$-expansion]] is a good approximation to the real world, especially if you include a leading correction to the [[large N limit|large $N$ limit]]. Now unfortunately this is best established in the [[glueball]] sector, which is not very accessible experimentally."

  > [28:56](https://youtu.be/TvIz-6YOdKs?t=1736): "but the $1/N$ expansion doesn't explain everything. In fact, it's not hard to find phenomena in meson physics where the $1/N$-expansion does *not* work well."

  > [36:15](https://youtu.be/TvIz-6YOdKs?t=2175): "I suspect the [[large N limit|$1/N$-expansion]] works reasonably well for many aspects of [[baryons]]. However, as for mesons, it is easy to point to things that won't work well for baryons. In particular, among other things, I don't think the [[large N limit|$1/N$ expansion]] will be successful for [[nuclei]] as opposed to individual [[nucleons]]."

  > [37:12](https://youtu.be/TvIz-6YOdKs?t=2232): "I don' t think the phenomenological models used by nuclear physicists would have any success at of if the [[large N limit]] was a good description of [[nuclei]]."

### In 2d QCD

On [[1/N corrections]] in [[2d QCD]]:

* [[Itzhak Bars]], _QCD and Strings in 2D_ ([arXiv:hep-th/9312018](https://arxiv.org/abs/hep-th/9312018))

[[!redirects small N limit]]

[[!redirects 1/N correction]]
[[!redirects 1/N corrections]]

[[!redirects small N correction]]
[[!redirects small N corrections]]


large cardinal

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## Idea

A **large cardinal** is a [[cardinal number]] that is larger than can be proven to exist in the ambient [[set theory]], usually [[ZF]] or ZFC.  Large cardinals arrange themselves naturally into a more or less linear order of size and consistency strength, and provide a convenient yardstick to measure the consistency strength of various other assertions that are unprovable from ZFC.

Set theorists often adopt the existence of certain large cardinals as [[axioms]] in the [[foundation of mathematics]].

## List of large cardinal conditions

* [[axiom of infinity]] -- a large cardinal axiom relative to finitist theories, but not relative to ZF
* [[regular cardinal]] - a large cardinal in strongly [[predicative mathematics]] where function sets and power sets do not exist.
* [[beth fixed point]] - a large cardinal in [[ETCS]] but small in ZF.
* [[axiom of replacement]], an axiom schemata that states that certain [[families]] of sets or [[diagrams]] in the [[category of sets]], which usually can be proven (for example, in [[Zermelo set theory]] or in a [[well-pointed topos]]) to be [[large sets|large]] or [[proper classes]], are instead [[small set|small]]. This implies that certain cardinals which previously were large are now small. Implies the existence of beth fixed points.
* [[inaccessible cardinal]] -- the smallest sort of large cardinal in ZF, equivalent to the existence of a [[Grothendieck universe]].
* [[Mahlo cardinal]]
* [[weakly compact cardinal]]
* [[measurable cardinal]] -- the boundary between "small" large cardinals and "large" large cardinals
* [[real-valued-measurable cardinal]], a “solution” to the Banach–Ulam problem.
* [[strongly compact cardinal]], whose existence controls properties images of [[accessible functors]]
* [[elementary embedding]] -- a tool used in the study of large large cardinals
* [[supercompact cardinal]]
* [[extendible cardinal]]
* [[C(n)-extendible cardinal]]
* [[Vopěnka's principle]] -- a large cardinal axiom with important implications for the behavior of [[locally presentable categories]] and [[accessible categories]].

Here is a diagram showing the relation between these:

<img src="http://websupport1.citytech.cuny.edu/faculty/vgitman/images/diagram.jpg"/>

In the context of [[ZFC]], certain axioms are inconsistent with large cardinal axioms:

[[ZFC-large-cardinals-consistency-strength.png:pic]]

## References

* Wikipedia has a [list of large cardinal properties](http://en.wikipedia.org/wiki/List_of_large_cardinal_properties).

A general axiomatic framework for large cardinal axioms is proposed in

* Arthur Apter, Carlos Diprisco, James Henle, William Swicker, _Filter spaces: towards a unified theory of large cardinal and embedding axioms_, Annals of Pure and Applied Logic Volume 41, Issue 2, 6 February 1989, Pages 93–106

* Arthur Apter, Carlos Diprisco, James Henle, William Swicker, _Filter spaces. II. Limit ultraproducts and iterated embeddings_, Acta Cient. Venezolana 40 (1989), no. 5-6, 311–318.

[[!redirects large cardinal]]
[[!redirects large cardinals]]

[[!redirects large cardinal axiom]]
[[!redirects large cardinal axioms]]

[[!redirects large set axiom]]
[[!redirects large set axioms]]


large category

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## Idea

A **large category** is a [[category]] which is not (necessarily) [[small category|small]].

There are some variations in usage depending on the [[foundations]] chosen.  Also, not all authors agree on whether a large category is *not* small, or merely not *necessarily* small (i.e., whether small categories are also large).

## Definitions

The precise meaning of the above definition depends on the [[foundations]] chosen.

* In [[ZF]], where a *small category* is one whose objects and morphisms form [[sets]], a *large category* is one whose objects and morphisms form (perhaps proper) [[classes]].

* In ZF with a specified [[Grothendieck universe]] $U$, where a *small category* is one whose objects and morphisms belong to $U$ (or have cardinalities belonging to $U$), a *large category* means a category whose objects and morphisms are still sets, but not (necessarily) belonging to $U$.  In this foundational system, one rarely considers categories whose objects and morphisms are proper classes.

* More generally, for any universe $U$, we may refer to categories whose objects and morphisms are elements of $U$ as **$U$-small**, and other categories as **$U$-large**.  In a context of multiple universes, it is perhaps more common to speak of "$V$-small categories" for some universe $V$ containing $U$, than "$U$-large categories."  If there are two fixed universes $U$ and $V$ with $U \in V$, then categories that are not even in $V$ may be called **very large** [[Shulman (2008), p. 18](#Shulman08)].

In all cases, it is somewhat ambiguous whether "large category" means "properly large," i.e., large and not small, or whether small categories should be considered as a subclass of large categories.  Usage may vary depending on need.

### Largeness and moderateness

A **[[moderate category]]** may be defined as one whose collections of objects and morphisms are no bigger than the size of the universe of small sets.  This is related to largeness in different ways depending on the foundations.

* In ZF, where all classes are subclasses of the class of all (small) sets, then *all* categories (small or large) are moderate.

* On the other hand, when smallness and largeness are defined with respect to a Grothendieck universe $U$, then a moderate category would be one whose objects and morphisms are (bijective to) subsets of $U$.  Thus, in this case, there are vastly more large categories than there are moderate ones.

* An in-between case, with some resulting ambiguity, is that of a class-set theory such as [[NBG]] or [[MK]], where *small category* means one whose objects and morphisms form sets.  Since such theories are closely related to ZF (indeed, NBG is conservative over ZF), it is natural to want to use "large category" to coincide with its ZF meaning: a category whose objects and morphisms form classes.

  However, in this case one can also consider categories which are "very large": their objects and morphisms are "superclasses" or "conglomerates" (collections of classes defined by first-order formulas, just as we do for classes in ZF).  Thus, in this case it may be more appropriate to say "moderate" for a category whose objects and morphisms are classes, and "large" for one which is not (necessarily) even moderate.

  This latter terminology also accords better with the usage for universes, since any Grothendieck universe $U$ gives rise to a model of NBG (and even MK) whose sets are the sets in $U$ and whose classes are the subsets of $U$.  One might even go backwards from this and start using "moderate" in the context of ZF for categories whose objects and morphisms are (not necessarily proper) classes.

  On the other hand, one could import terminology the other way, and use "large category" even in the context of a universe $U$ to mean one whose objects and morphisms are *subsets* of $U$ (i.e. what we have called above a "($U$-)moderate category").  This usage is that of [[Categories Work]]; it has the disadvantage that some categories are "too large to be large," although they are still "very large."  However, it does accord better with the ZF-usage, which is the original context in which "large category" was used.

## Properties

Many tools and results about small categories, in particular concerning [[limit]]s indexed by such a category and [[functor categories]], fail for large categories.  For instance:

* Many large categories have all *small* limits and/or colimits, but a large category with *large* limits or colimits must (at least in [[classical mathematics]]) be a [[preorder]].  Most large categories do have *some* large limits and colimits, however, although they are not always very useful; see for instance [[total category]].

* In a foundation of ZF, NBG, or MK, one cannot form the functor category $[C,D]$ unless $C$ is small.  In a foundation with universes, one can form $[C,D]$, but it will sometimes be oddly behaved.  In particular, it is not generally moderate, even if $C$ and $D$ are.

There are various notions and techniques to deal with this problem and reduce or relate large categories to small categories as much as possible:

* In the simplest case the large category is *essentially* small in that it is [[equivalence of categories|equivalent]] to a [[small category]].  Then (for all purposes adhering to the [[principle of equivalence]]) you can simply replace it with a small category.

* Many large categories that arise in practice are (even essentially) large but still *accessible*.  An [[accessible category]] is a large category which behaves like the category of [[ind-object]]s of a small category and is therefore, while itself large, entirely governed by a small category.

* Some problems can be avoided by simply assuming a [[Grothendieck universe]], so that large categories become small ones relative to a larger universe.

## Related concepts

* [[small category]], [[locally small category]], [[moderate category]]

* **large category**

* [[universe enlargement]]

* [[very large (∞,1)-sheaf (∞,1)-topos]]

## References

See most references on *[[category theory]]*, such as

* [[Saunders MacLane]], §I.7 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Horst Schubert]], §3.6 in: *Categories*, Springer (1972) [[doi:10.1007/978-3-642-65364-3](https://doi.org/10.1007/978-3-642-65364-3)]

See also:

* {#Shulman08} [[Mike Shulman]], *Set theory for category theory* [[arXiv:0810.1279](http://www.arxiv.org/abs/0810.1279)]

* {#Murfet06} [[Daniel Murfet]], _Foundations for category theory_ (2006) [[pdf](http://therisingsea.org/notes/FoundationsForCategoryTheory.pdf), [[Murfet-GrothendieckUniverses.pdf:file]]]

* [[Zhen Lin Low]], _Universes for category theory_ [[arxiv/1304.5227](http://arxiv.org/abs/1304.5227)]

* [[Paul Blain Levy]], _Formulating Categorical Concepts using Classes_ [[arXiv:1801.08528](https://arxiv.org/abs/1801.08528)]

[[!redirects large category]]
[[!redirects large categories]]
[[!redirects very large category]]
[[!redirects very large categories]]

[[!redirects moderate category]]
[[!redirects moderate categories]]

[[!redirects large]]


large cocompleteness

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## Idea

A [[locally small category]] cannot admit all large discrete [[colimits]] unless it is a [[preorder]] (by an argument of [[Peter Freyd|Freyd]] see *[[complete small category]]*). However, many nice locally small categories admit *some* large colimits.

This page is intended to act as a reference for classes of large colimits that are commonly encountered, and the class of categories that admit them.

On this page, categories will be assumed [[locally small]] unless stated otherwise.

## Colimits versus limits

While this page focuses on categories with large *colimit* properties, these often imply strong *[[complete category|completeness]]* properties. For instance, compact categories (below) are [[complete]], but not always [[cocomplete]].

## Cointersections of epimorphims

Many locally small categories admit small colimits and [[E-colimits]] for $E$ a class of epimorphisms, i.e. [[cointersections]] of morphisms in $E$.

Evidently, a small-[[cocomplete category]] that is [[well powered category|co-well-powered]] will admit cointersections of all [[epimorphisms]], though not all relevant examples are co-well-powered.

### Strong epimorphisms

When $E$ is the [[class]] of [[strong epimorphisms]], such a category is called **well-cocomplete**.

### Regular epimorphisms

The [[free cocompletion]] of a locally small category $A$ under small colimits and [[cointersections]] of [[regular epimorphisms]] is the [[full subcategory]] of the [[presheaf category]] $[A^{op}, Set]$ on the [[weak multilimit|weakly multirepresentable presheaves]] (also called **petty** presheaves). See Remark 4.39 of [Lack & Tendas 2024](#LackTendas2023) (such categories are called **well-cocomplete** in this paper, though conflicts with the earlier terminology).

Theorem 1 of [Kelly & Koubek 1981](#KellyKoubek1981) states that every [[functor]] $F \colon K \to A$, where $A$ has such colimits, admits a colimit if $F$ has a [[weakly terminal set]].

## Adjointness

### Total category

A [[total category]] is a category whose [[Yoneda embedding]] admits a [[left adjoint]]. Every total category is compact in the sense below.

### Compact categories

A category $A$ is **[[compact category|compact]]** in the sense of [Isbell (1968)](#Isbell68) if every (large-)[[cocontinuous functor]] from $A$ has a [[right adjoint]].

> Beware that this is *un-related* to the notion of [[compact closed category]].

Every compact category has small [[limits]] and [[intersections]] of [[monomorphisms]], but not necessarily small colimits. A [[counterexample]] is mentioned in §3.15 of [Börger et al.](#Borger1981).

A category $A$ is called **strongly compact** in [Brandenburg](#Brandenburg2021) if every small-[cocontinuous functor]] from $A$ has a [[right adjoint]]. Every strongly compact category is consequently compact.

## All large colimits

Despite its ill behaviour, it is possible to describe the cocompletion of a locally small category $A$ under large colimits (assuming the [[law of excluded middle]]): it is given by the [[functor category]] $[A^{op}, 2]$ where $2$ is the [[interval category]]. In particular, this gives rise to a [[lax-idempotent pseudomonad]] whose unit components are (unusually) not [[representably fully faithful]]. (One might argue this disqualifies it from being a "cocompletion" in the traditional sense.) See the discussion following Example 26 of [Walker](#Walker2019).

## Related pages

* [[total category]]

## References

* {#Isbell68} [[John Isbell]]. _Small subcategories and completeness_. Mathematical systems theory **2** 1 (1968) 27-50 [[doi:10.1007/BF01691344](https://doi.org/10.1007/BF01691344)]

* {#KellyKoubek1981} [[Max Kelly]] and V. Koubek. _The large limits that all good categories admit_. Journal of Pure and Applied Algebra 22.3 (1981): 253-263.

* {#Borger1981} R. Börger, [[Walter Tholen]], M. B. Wischnewsky & H. Wolff. _Compact and hypercomplete categories_. Journal of Pure and Applied Algebra 21.2 (1981): 129-144.

* {#Kelly1986} [[Max Kelly]], _A survey of totality for enriched and ordinary categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 2 (1986), p. 109-132, [numdam](http://www.numdam.org/item?id=CTGDC_1986__27_2_109_0)

* {#Walker2019} [[Charles Walker]]. "Distributive laws via admissibility." Applied Categorical Structures 27.6 (2019): 567-617.

* {#LackTendas2023} [[Stephen Lack]] and [[Giacomo Tendas]]. _Virtual concepts in the theory of accessible categories_. Journal of Pure and Applied Algebra 227.2 (2023): 107196.

* {#Brandenburg2021} [[Martin Brandenburg]]. _Large limit sketches and topological space objects_. [arXiv preprint 2106.11115](https://arxiv.org/abs/0810.1279) (2021).

[[!redirects compact category]]
[[!redirects compact categories]]


large gauge transformation

A [[gauge transformation]] is called **large** if it is not connected to the [[identity]] gauge transformation.

[[!redirects large gauge transformations]]


large N limit

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## Idea

For classes of [[gauge theories]], such as ([[super Yang-Mills theory|super]]) [[Yang-Mills theory]] or [[Chern-Simons theory]] or various [[matrix models]],  whose [[gauge groups]] may be $N \times N$ [[square matrices]] for any [[natural number]] $N$, notably in the [[special unitary group]] $SU(N)$, the [[special orthogonal group]] $SO(N)$ or the [[quaternionic unitary group]] ("symplectic group") $Sp(N)$, one may consider the [[limit of a sequence|limit]] of the theory's [[scattering amplitudes]] and other [[quantum observables]] as $N\to \infty$ ("large [[number of colours]]-limit"). In good cases the values close to but away from this _large $N$ limit_ scale with $1/N$ and allow a [[perturbation series]] around the large $N$ limit called the _$1/N$ expansion_.

This _large $N$ limit_ often has remarkable properties, often revealing an otherwise hidden relation to [[perturbative string theories]] with the parameter $1/N$ proportional to the [[string coupling constant]].

Notably for [[Yang-Mills theory]] and in particular for [[QCD]], the large $N$-behaviour is exhibited by rewriting the [[Feynman amplitudes]] in [['t Hooft double line notation]]. If the [['t Hooft coupling]] $g^2 N$  is held fixed as $N\to \infty$, this turns out to organize the gauge theory's [[Feynman perturbation series]] by the [[Euler characteristic]]/[[genus of a surface|genus]] of emerging [[string theory|string]] [[worldsheet]] [[surfaces]], with genus 0 ([[planar graphs]]) dominating in the large $N$ limit, whence also called the _[[planar limit]]_.

([[open/closed string duality|Open/closed string duality]] plays a subtle role in interpreting the [['t Hooft double line notation]] of [[gauge theory]] [[Feynman diagrams]] in the [[large N limit]] alternatively as [[open string]] or as [[closed string]] [[worldsheets]], see [Gopakumar-Vafa 98](#GopakumarVafa98), [Gaiotto-Rastelli 05](#GaiottiRastelli05), [Gopakumar 04](#Gopakumar04) and notably
[Marino 04, Section III, p. 14](#Marino04)).

At least for the case of [[super Yang-Mills theories]] the full statement of the relation of large-$N$ gauge theory to a [[perturbative string theory]] is the content of the _[[AdS/CFT correspondence]]_, which explains that the effective [[string]] [[worldsheets]] emerging from the [[gauge theory]] propagate in a higher-dimensional asymptotically [[anti-de Sitter spacetime]] (the [[near horizon geometry]] of a [[black brane]]) whose [[asymptotic boundary]] (the [[worldvolume]] of the [[black brane]] itself) is identified with the [[spacetime]] of the original [[gauge theory]].

An extreme case of this large $N$-limit is that of the [[BFSS matrix model]] in [[AdS2/CFT1 duality]] where _all_ spatial dependence of [[field (physics)|fields]] in the higher dimensional spacetime is supposedly encoded in the [[quantum mechanics]] of $N\times N$ [[matrix model|matrices]] as $N\to \infty$. And for the [[IKKT matrix model]] this includes also the temporal dependence.

For non-[[supersymmetry|supersymmetric]] [[gauge theories]] such as [[QCD]] this [[duality in string theory|duality]] still holds in suitably adjusted form such as in the _[[AdS/QCD correspondence]]_. Here the $1/N$-expansion serves to provide a computational tool for describing [[confinement|confined]] [[hadron]] states ([[mesons]] and [[baryons]], hence in particular [[nucleons]] and hence ordinary room-temperature [[matter]]) which are not seen by ordinary [[perturbative quantum field theory|perturbation theory]] in the gauge theory [[coupling constant]] (the [[confinement]]/[[mass gap problem]]).

## Related concepts

* [[planar limit]]

* [[large 1/N limit]]

* [['t Hooft coupling]]

* [[light-cone gauge quantization]]

* [[AdS-CFT duality]]

* [[BFSS matrix model]]

* [[volume conjecture]]

## References

### General

The original article observing the large $N$-behaviour and the [[planar limit]] of [[Yang-Mills theory]] in [['t Hooft double line notation]] is:

* {#tHooft74} [[Gerard 't Hooft]], _A Planar Diagram Theory for Strong Interactions_, Nucl. Phys. B72 (1974) 461 ([spire:80491](http://inspirehep.net/record/80491), doi:10.1016/0550-3213(74)90154-0)

First inkling of [[holographic QCD]] and [[matrix models]]:

* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of dynamical degrees of freedom in the large-$N$ gauge theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))

* A. Gonzalez-Arroyo, M. Okawa, _A twisted model for large $N$ lattice gauge theory_, Physics Letters B120:1--3 (1983) 174--178 (doi:10.1016/0370-2693(83)90647-0)

* A. Gonzalez-Arroyo, M. Okawa, _Twisted-Eguchi-Kawai model: A reduced model for large-
$N$ lattice gauge theory_, Phys. Rev. D 27, 2397 (1983) ([doi:10.1103/PhysRevD.27.2397](https://doi.org/10.1103/PhysRevD.27.2397))

First observation that various observables in [[QCD]] for $N=3$ are actually well-approximated by the large $N$-limit:

* {#LuciniTeper01} B. Lucini, M. Teper, *SU(N)* gauge theories in four dimensions: exploring the approach to $N = \infty$, JHEP 0106 050 (2001) [[doi:10.1088/1126-6708/2001/06/050](https://doi.org/10.1088/1126-6708/2001/06/050)]

See:

* [[Edward Witten]], *Some Milestones in the Study of Confinement*, talk at *[[Prospects in Theoretical Physics 2023 -- Understanding Confinement]]*, IAS (2023) [[web](https://www.ias.edu/video/some-milestones-study-confinement), [YT](https://youtu.be/TvIz-6YOdKs)]

  > [26:26](https://youtu.be/TvIz-6YOdKs?t=1586): "by now it's clear [[Lucini & Teper 2001](large+N+limit#LuciniTeper01)] that [[lattice gauge theory]], at least for the [[glueball]] sector, has made it clear that the [[large N limit|$1/N$-expansion]] is a good approximation to the real world, especially if you include a leading correction to the [[large N limit|large $N$ limit]]. Now unfortunately this is best established in the [[glueball]] sector, which is not very accessible experimentally."

  > [28:56](https://youtu.be/TvIz-6YOdKs?t=1736): "but the $1/N$ expansion doesn't explain everything. In fact, it's not hard to find phenomena in meson physics where the $1/N$-expansion does *not* work well."

  > [36:15](https://youtu.be/TvIz-6YOdKs?t=2175): "I suspect the [[large N limit|$1/N$-expansion]] works reasonably well for many aspects of [[baryons]]. However, as for mesons, it is easy to point to things that won't work well for baryons. In particular, among other things, I don't think the [[large N limit|$1/N$ expansion]] will be successful for [[nuclei]] as opposed to individual [[nucleons]]."

  > [37:12](https://youtu.be/TvIz-6YOdKs?t=2232): "I don' t think the phenomenological models used by nuclear physicists would have any success at of if the [[large N limit]] was a good description of [[nuclei]]."

Lecture notes:

* [[Sidney Coleman]], _$1/N$_,

  in: A. Zichichi  (ed.) _Pointlike Structures Inside and Outside Hadrons_. The Subnuclear Series, vol 17. Springer 1982 ([doi:10.1007/978-1-4684-1065-5_2](https://doi.org/10.1007/978-1-4684-1065-5_2))

  and Chapter 8 in: [[Sidney Coleman]], _Aspects of Symmetry_, Cambridge University Press 1985 ([doi:10.1017/CBO9780511565045.009](https://doi.org/10.1017/CBO9780511565045.009))

* [[Gerard 't Hooft]], _Large $N$_, workshop lecture ([hep-th/0204069](http://arxiv.org/abs/hep-th/0204069))

* A. V. Manohar, _Large $N$ QCD_, Les Houches Lecture 2004, (arXiv:[hep-ph/9802419](https://arxiv.org/abs/hep-ph/9802419), [pdf](http://einstein.ucsd.edu/manohar/pdf/LesHouches.pdf))

* Markus Gross, _Large $N$_, 2006 ([[GrossLargeN.pdf:file]])

* {#McGreevySwingle08} McGreevy, Swingle, _Large $N$ counting_, 2008 ([[GreevySwingle.pdf:file]])

* [[Paul Romatschke]], *Quantum Field Theory in Large $N$ Wonderland: Three Lectures* [[arXiv:2310.00048](https://arxiv.org/abs/2310.00048)]

See also:

* Wikipedia, _[1/N expansion](http://en.wikipedia.org/wiki/1/N_expansion)_

* E. Brézin, S.R. Wadia, eds. _The Large $N$ Expansion in Quantum Field Theory and Statistical Physics_, a book collection of reprinted historical articles, [gBooks](http://books.google.com/books?id=4TbOUqWBf-IC)

Application of [[WKB method]]:

* [[V. P. Maslov]], O. Yu. Shvedov, _Large-N expansion as a semiclassical approximation to the third-quantized theory_, Physical Review D60(10) 105012 [doi](https://doi.org/10.1103/PhysRevD.60.105012)

The refinement for [[super Yang-Mills theory]] to the [[AdS/CFT correspondence]] (see there for more) originates with

* [[Juan Maldacena]], _The large $N$ limit of superconformal field theories and supergravity_, Adv. Theor. Math. Phys.2:231-252, 1998 [hep-th/9711200](http://arxiv.org/abs/hep-th/9711200); _Wilson loops in large N field theories_, hep-th/9803002.

reviewed for instance in

* {#Maldacena97a} [[Juan Maldacena]], _The Large N limit of superconformal field theories and supergravity_, Adv. Theor. Math. Phys. 2:231, 1998 ([hep-th/9711200](http://arxiv.org/abs/hep-th/9711200))

But see at _[[AdS/CFT correspondence]]_ for a more comprehensive list of references.

Further discussion:

* {#Witten79} [[Edward Witten]], _Baryons in the $1/n$ Expansion_, Nucl. Phys. B160 (1979) 57--115 ([spire:140391](http://inspirehep.net/record/140391), doi:10.1016/0550-3213(79)90232-3)

  (on [[mesons]] and [[baryons]] in the [[large N limit]])

* A. Jevicki, _Instantons and the $1/N$ expansion in nonlinear $\sigma$ models_, Phys. Rev. D 20, 3331--3335 (1979) [pdf](http://prd.aps.org/pdf/PRD/v20/i12/p3331_1)

* Laurence G. Yaffe, _Large $N$ limits as classical mechanics_, Rev. Mod. Phys. __54__, 407--435 (1982) ([pdf](http://rmp.aps.org/pdf/RMP/v54/i2/p407_1))

* [[Alexander A. Migdal]], _Loop equations and $1/N$ expansion_, Physics Reports **102** (4) 199-290 (1983) [doi]

* M. Bershadsky, Z. Kakushadze, [[Cumrun Vafa]], _String expansion as large $N$ expansion of gauge theories_, Nucl.Phys. B523 (1998) 59--72 ([hep-th/9803076](http://arxiv.org/abs/hep-th/9803076), doi)

* [[Gary Horowitz]], [[Hirosi Ooguri]], _Spectrum of large $N$ gauge theory from supergravity_, hep-th/9802116

* S. Sinha, [[Cumrun Vafa]], _$SO$ and $Sp$ Chern-Simons at Large $N$_ ([arXiv:hep-th/0012136](https://arxiv.org/abs/hep-th/0012136))

  (for [[Chern-Simons theory as a topological string theory]])

* Hiroyuki Fuji, Yutaka Ookouchi, _Confining Phase Superpotentials for $SO/Sp$ Gauge Theories via Geometric Transition_, JHEP 0302:028, 2003 ([arXiv:hep-th/0205301](https://arxiv.org/abs/hep-th/0205301))

* {#OoguriVafa02} [[Hirosi Ooguri]], [[Cumrun Vafa]], _Worldsheet derivation of a large $N$ duality_, Nucl. Phys. B641:3--34, 2002 ([arXiv:hep-th/0205297](https://arxiv.org/abs/hep-th/0205297))

* Semyon Klevtsov, _Random normal matrices, Bergman kernel and projective embeddings_,  J. High Energ. Phys. 2014, 133 (2014) doi [arXiv:1309.7333](https://arxiv.org/abs/1309.7333)

* Jan Albert, Johan Henriksson, [[Leonardo Rastelli]], Alessandro Vichi, *Bootstrapping mesons at large $N$: Regge trajectory from spin-two maximization* [[arXiv:2312.15013](https://arxiv.org/abs/2312.15013)]

On a kind of [[BV-quantization]] of the [[Loday-Quillen-Tsygan theorem]] and relating to the [[large N limit|large $N$-limit]] of [[Chern-Simons theory]]:

* [[Grégory Ginot]], [[Owen Gwilliam]], [[Alastair Hamilton]], [[Mahmoud Zeinalian]], _Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem_, Adv. Math. __409A__ (2022) 108631 [[arXiv:2108.12109](https://arxiv.org/abs/2108.12109), [doi:10.1016/j.aim.2022.108631](https://doi.org/10.1016/j.aim.2022.108631)]
* [[Owen Gwilliam]], [[Alastair Hamilton]], [[Mahmoud Zeinalian]], _A homological approach to the Gaussian Unitary Ensemble_, [arXiv:2206.04256](https://arxiv.org/abs/2206.04256)

On the [[logical equivalence]] between the [[four-colour theorem]] and a statement about transition from the [[small N limit]] to the [[large N limit]] for [[Lie algebra weight systems]] on [[Jacobi diagrams]] via the [['t Hooft double line construction]]:

* [[Dror Bar-Natan]], _Lie Algebras and the Four Color Theorem_, Combinatorica 17-1(1997) 43--52  ([arXiv:q-alg/9606016](https://arxiv.org/abs/q-alg/9606016), [doi:10.1007/BF01196130](https://doi.org/10.1007/BF01196130))

On the [[large N limit]] in [[lattice gauge theory]]:

* Margarita Garcia Perez, _Prospects for large $N$ gauge theories on the lattice_ ([arXiv:2001.10859](https://arxiv.org/abs/2001.10859))

See also:

* Marco Bochicchio, Mauro Papinutto, Francesco Scardino, *On the structure of the large-$N$ expansion in $SU(N)$ Yang-Mills theory* [[arXiv:2401.09312](https://arxiv.org/abs/2401.09312)]

### Open/closed string duality

On the role of [[open/closed string duality]] in interpreting the [[large N limit]]:

* {#GopakumarVafa98} [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415--1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))

* {#GaiottiRastelli05} [[Davide Gaiotto]], [[Leonardo Rastelli]], _A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model_, JHEP 0507:053,2005 ([hep-th/0312196](https://arxiv.org/abs/hep-th/0312196))

> Nowadays we interpret $[$ the [['t Hooft double line notation]] $]$ quite literally as the perturbative expansion of an open string theory, either because the full open string theory is just equal to the gauge
theory (as e.g. for Chern-Simons theory [27]), or because we take an appropriate
low-energy limit (as e.g. for N = 4 SYM [31]).

> The general speculation [1] is that upon summing over the number of holes, (1.1) can be recast as the genus expansion for some closed string theory of coupling $g_s = g_{YM}^2$. This speculation is sometimes justified by appealing to the intuition that diagrams with
a larger and larger number of holes look more and more like smooth closed Riemann surfaces. This intuition is perfectly appropriate for the double-scaled matrix models, where the finite N theory is interpreted as a discretization of the closed Riemann surface; to recover the continuum limit, one must send $N\to \infty$ and tune $t$ to the
critical point $t_c$ where diagrams with a diverging number of holes dominate.

>  However, in AdS/CFT, or in the Gopakumar-Vafa duality [2], $t$ is a free parameter, corresponding on the closed string theory side to a geometric modulus. The intuition described above clearly goes wrong here.

> A much more fitting way in which the open/closed duality may come about in these cases is for each fatgraph of genus g and
with h holes to be replaced by a closed Riemann surface of the same genus g and with h punctures: each hole is filled and replaced by a single closed string insertion.

* {#Gopakumar04} [[Rajesh Gopakumar]], _Free Field Theory as a String Theory?_, Comptes Rendus Physique 5 (2004) 1111-1119 ([hep-th/0409233](https://arxiv.org/abs/hep-th/0409233))

* {#Marino04} [[Marcos Marino]], _Chern-Simons Theory and Topological Strings_, Rev. Mod. Phys. 77:675-720, 2005 ([arXiv:hep-th/0406005](https://arxiv.org/abs/hep-th/0406005))

In relation to [[quantum error correction]]:

* Alexey Milekhin, *Quantum error correction and large $N$* ([arXiv:2008.12869](https://arxiv.org/abs/2008.12869))

category: physics

[[!redirects large N limit]]
[[!redirects large N limits]]
[[!redirects large-N limit]]
[[!redirects large-N limits]]
[[!redirects large n limit]]
[[!redirects large n limits]]
[[!redirects large-n limit]]
[[!redirects large-n limits]]
[[!redirects large N expansion]]
[[!redirects large N expansions]]
[[!redirects large-N expansion]]
[[!redirects large-N expansions]]
[[!redirects large n expansion]]
[[!redirects large n expansions]]
[[!redirects large-n expansion]]
[[!redirects large-n expansions]]

[[!redirects 1/N expansion]]
[[!redirects 1/N expansions]]

[[!redirects 't Hooft limit]]
[[!redirects 't Hooft limits]]


large set

\tableofcontents

## Definition

Given a [[universe]] $U$, a **$U$-large set** is a set which is not a $U$-small set. $U$-large sets can be contrasted with $U$-[[small sets]].

## Relation to classes

Every [[proper class]] in [[set theory]] is a $U$-large set. However, in [[dependent type theory]], only material proper classes are always $U$-large [[h-sets]]; structural proper classes in dependent type theory are [[h-groupoids]] if the universe $U$ is a [[univalent universe]].

## See also

* [[universe]], [[Grothendieck universe]], [[well-pointed pretopos|well-pointed]] [[Heyting pretopos]], [[well-pointed category|well-pointed]] [[division allegory]], [[Tarski universe]], [[cumulative hierarchy]]

* [[class]]

* [[small set]]

* [[universe enlargement]]

[[!redirects large set]]
[[!redirects large sets]]

[[!redirects large h-set]]
[[!redirects large h-sets]]


large sheaf > history

< [[large sheaf]]
[[!redirects large sheaf -- history]]


large site

+-- {: .rightHandSide}
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#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Often [[sites]] are required to be [[small category|small categories]] ([[small sites]]).  If not, if one has a _large site_,  there are complications.

Many of the good properties of [[sheaf|sheaves]] depend on such smallness.  To begin with, even the category of sheaves may have to be [[very large category|extra-large]], but there are other issues, such as:

* the existence of [[sheafification]],
* the existence of [[colimit|colimits]] in the category of sheaves,
* [[cartesian closed category|cartesian closedness]] of the category of sheaves, and
* the category of sheaves being a [[topos]].

However, for many purposes it is desirable to consider the notion of [[sheaves]] on large sites.

## Small (dense) sub-sites

In some cases, sheaves on a large site can be identified with sheaves on some small full sub-site, for instance a [[dense sub-site]].
A large site with a small dense sub-site is called an [[essentially small site]].

For example, if $C$ is a [[Grothendieck topos]] with its [[subcanonical coverage|canonical coverage]], then every sheaf on $C$ is representable, so $C\simeq Sh(C)$; thus $Sh(C)$ is equivalent to the category of sheaves on some small site (a defining site for $C$ itself).

## Sheaves on a large site

On the other hand, one sometimes wants to consider sheaves on [[large category|large categories]] such as [[Top]] or [[Diff]], which are certainly not Grothendieck toposes.  One way to deal with this is to consider [[full subcategory|full subcategories]] of these large categories on objects whose size is bounded by some large (in the non-technical sense) [[cardinal number]] $\kappa$.  In an extreme case, $\kappa$ could be an [[inaccessible cardinal]].  The idea is that for sheaves and in particular for any [[homotopy theory]] of sheaves the choice of these cardinality bounds is "inessential."

See for instance [p. 2](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf#page=2) of Jardine, [Fields Lectuere: Simplicial presheaves](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf) where this issue arises in the study of [[simplicial presheaf|simplicial presheaves]] and the [[model structure on simplicial presheaves]].

+-- {: .query}
Can any of you size-issue experts help to clarify this?

_Mike_: I wish.  I added some stuff, but I still don't really understand this business.  In particular I don't really know what is meant by "inessential."  It certainly seems unlikely that you would  get _equivalent_ homotopy theories, but it does seem likely that you would get similar behavior no matter where you draw the line.  And if all you care about is, say, having a good category of sheaves in which you can embed any _particular_ space or manifold you happen to care about, then that may be good enough.  But I don't really know what the goal is of considering such large sites.
=--

[[!redirects large sheaf]]
[[!redirects sheaves on large sites]]

[[!redirects large site]]


large volume limit

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#Contents#
* table of contents
{:toc}

## Idea
In [[Riemannian geometry]] given any [[function]]/[[functional]] on the space of [[Riemannian metrics]] $g$ on some [[manifold]] $X$, then its _large volume limit_ is, if it exists, the [[limit of a sequence|limit]] of the functional evaluated on a sequence $t g$ of metrics as $t \to \infty$.

This plays a role in particular in the studies of [[sigma-model]] [[quantum field theory]] with [[target space]] $(X,g)$. Here the large volume limit may equivalently be thought of as the limit in which the extension of the [[brane]] described by the $\sigma$-model vanishes.

One example is the [[Witten genus]], which is the large volume limit of the [[partition function]] of the [[superstring]] $\sigma$-model ([Witten 87, p. 4](#Witten87))

## References

* {#Witten87} [[Edward Witten]], _Elliptic Genera And Quantum Field Theory_ , Commun.Math.Phys. 109 525 (1987) ([Euclid](http://projecteuclid.org/euclid.cmp/1104117076))

[[!redirects large volume limits]]


Larmore twisted cohomology

This entry is about particulars of the work of Larmore on [[local system|cohomology with local coefficients]] (Steenrod, compare also Reidemeister earlier), a special case of what is now called [[twisted cohomology]].

## Larmore's work ##

The phrase _twisted cohomology_ was  used by Larmore in

* Larmore, _Twisted cohomology theories and the single obstruction to lifting_,  Pac JM 41 (1972) 755-769

to describe [[cohomology]] $H'(-; E)$ with coefficients in a special kind of [[spectrum]] $E$  related to a [[fibration]] $p : E\to B$.

The result is what May and Sigurdsson (see references at [[twisted cohomology]]) call a _parameterized spectrum_, the "parameters" being the points of $B$, which might also be called, in the older topological terminology, an _ex-spectrum_.

For any map $f:K\to B$ and, for $L\subset K,$ a partial lift
$h:L\to E$ of $f$, he constructs a _single_ obstruction class $\Gamma(f)\in H'(K, L  ; E)$ to a full lift $g:K\to E.$

$$
  \array{
    && &\to& E
    \\
    &{}^h\nearrow& &{}^{g}\nearrow& \downarrow^p
    \\
    L &\hookrightarrow& K &\stackrel{f}{\to}& B
  }
$$

The vanishing of this obstruction is necessary for the existence of a lifting, but it is sufficient only in the usual stable range.

Notice that his cohomology with coefficients in a spectrum does _not_ mean the sequence of [[cohomology group]]s with coefficients in the sequence of spaces constituting the [[spectrum]], but rather a single group. He does explore the relation between his single obstruction and the classical obstructions.

[[!redirects Larmore]]


Larry Lambe

* [Link at MSSRC](http://lambe.mssrc.com)

* [ Notes on symbolic computation ]  ( being revised and extended )

category:people


Larry Siebenmann

Larry Siebenmann is a Canadian born mathematician who is based at Orsay (Paris sud). He is best known for his work in geometric topology.

* [English Wikipedia page](http://en.wikipedia.org/wiki/Laurent_C._Siebenmann)

* [Equipe de Topologie et Dynamique (Orsay)](http://www.math.u-psud.fr/topodyn/accueil.php)

## Selected writings

On the [[triangulation conjecture]] for [[topological manifolds]]:

* {#KirbySiebenmann69} [[Robion C. Kirby]], [[Laurent C. Siebenmann]], *On the triangulation of manifolds and the Hauptvermutung*, Bull. Amer. Math. Soc. 75(4): 742-749 (July 1969)  ([euclid:1183530633](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-75/issue-4/On-the-triangulation-of-manifolds-and-the-Hauptvermutung/bams/1183530633.full))

[[!redirects Siebenmann]]

[[!redirects L. C. Siebenmann]]

[[!redirects Laurent C. Siebenmann]]

category: people


Larry Smith

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=12023&fChrono=1)

## Selected writings

On the [[Eilenberg-Moore spectral sequence]]:

* [[Larry Smith]], _Homological Algebra and the Eilenberg-Moore Spectral Sequence_, Transactions of the American Mathematical Society, Vol. 129, No. 1, (1967) ([jstor:1994364](http://www.jstor.org/stable/1994364))

* [[Larry Smith]], _On the construction of the Eilenberg-Moore spectral sequence_, Bull. Amer. Math. Soc. 75 (1969), 873-878 ([euclid:bams/1183530666](https://projecteuclid.org/euclid.bams/1183530666))

On [[MU]]-[[bordism theory]]:

* {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], _On the complex bordism of finite complexes_, Publications Mathématiques de l'IHÉS, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0))

* [[Larry Smith]], _On Realizing Complex Bordism Modules: Applications to the Stable Homotopy of Spheres_, American Journal of Mathematics Vol. 92, No. 4 (Oct., 1970), pp. 793-856 ([doi:10.2307/2373397](https://doi.org/10.2307/2373397))

On [[MU]] and introducing the [[Todd character]]:

* [[Pierre Conner]], [[Larry Smith]],  _On the complex bordism of finite complexes. II_, J. Differential Geom. Volume 6, Number 2 (1971), 135-174 ([euclid:euclid.ijm/1256051760](https://projecteuclid.org/euclid.ijm/1256051760))

On [[MUFr]]:

* {#Smith71} [[Larry Smith]], _On characteristic numbers of almost complex manifolds with framed boundaries_, Topology Volume 10, Issue 3, August 1971, Pages 237-256 (doi:10.1016/0040-9383(71)90008-5)

More on the [[Todd character]]:

* [[Larry Smith]], _The Todd character and the integrality theorem for the Chern character_, Illinois J. Math. Volume 17, Issue 2 (1973), 301-310 ([euclid:ijm/1256051760](https://projecteuclid.org/euclid.ijm/1256051760))

* [[Larry Smith]], _The Todd character and cohomology operations_, Advances in Mathematics Volume 11, Issue 1, August 1973, Pages 72-92 (doi:10.1016/0001-8708(73)90003-0)

category:people


Lars Andersson

* [webpage](https://www.aei.mpg.de/~laan/)

## Selected writings

On a rigorous [[proof]] of [[moduli stabilization]] of vacuum [[spacetimes]], refuting [Penrose 03](moduli+stabilization#Penrose03) (for $\geq 10$-dimensions with a [[Killing spinor]] on the [[compact topological space|compact]] [[fiber]] and for [[Schwarzschild spacetime|Schwarzschild]]-asymptotics):

* {#AndersonBlueWyattYau20} [[Lars Andersson]], [[Pieter Blue]], [[Zoe Wyatt]], [[Shing-Tung Yau]], _Global stability of spacetimes with supersymmetric compactifications_ ([arXiv:2006.00824](https://arxiv.org/abs/2006.00824))

category: people


Lars Birkedal

* [website](https://cs.au.dk/~birke/)

## Selected writings

On [[modal type theory]]:

* [[Daniel Gratzer]], [[G. Alex Kavvos]], [[Andreas Nuyts]], [[Lars Birkedal]]: _Multimodal Dependent Type Theory_, Logical Methods in Computer Science **17** 3 (2021) lmcs:7713 [[arXiv:2011.15021](https://arxiv.org/abs/2011.15021), doi:10.46298/lmcs-17(3:11)2021]

category: people


Lars Brink

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

* [[Hermann Nicolai]]: *[Lars Brink 1943–2022](https://cerncourier.com/a/lars-brink-1943-2022/)*, CernCourier (1 Mar 2023):

## Selected writings

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

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], _A locally supersymmetric and reparametrization invariant action for the spinning string_ , Physics Letters B **65** Issue 5, 20 (1976) 471-474 [doi:10.1016/0370-2693(76)90445-7]

On [[spinning particles]] and their [[worldline]] [[supersymmetry]]:

* [[Lars Brink]], [[Paolo Di Vecchia]], [[Paul Howe]], *A Lagrangian formulation of the classical and quantum dynamics of spinning particles*, Nucl. Phys. B **118** (1977) 76 [doi:10.1016/0550-3213(77)90364-9]

Formulation of [[D=11 supergravity]] in [[superspace]] (via [[supergeometry]]):

* [[Lars Brink]], [[Paul Howe]], *Eleven-Dimensional Supergravity on the Mass-Shell in Superspace*, Phys.Lett. B **91** (1980) 384 [doi:10.1016/0370-2693(80)91002-3]

On the history of [[string theory]] as a theory of [[hadrons]] in [[QCD]]:

* [[Lars Brink]], *Hadronic Strings -- A Revisit in the Shade of Moonshine*,  J. Phys. A: Math. Theor. **53** (2020) 091001 [[arxiv:1911.06026](https://arxiv.org/abs/1911.06026), [doi:10.1088/1751-8121/ab6b93](https://iopscience.iop.org/article/10.1088/1751-8121/ab6b93)]

On historical work by [[Yoichiro Nambu]] on the [[Ising model]]:

* [[Lars Brink]], [[Pierre Ramond]], *Nambu and the Ising Model* [[arXiv:2209.01122](https://arxiv.org/abs/2209.01122)]

## Related entriees

* [[string theory]]

* [[11-dimensional supergravity]]

category: people


Lars Hesselholt

[website](http://www.math.nagoya-u.ac.jp/~larsh/)

Presently he is guest at the [centre for symmetry and deformation](http://sym.math.ku.dk/).


Lars Hörmander

**Lars Hörmander** was a Swedish analyst and Fields medalist. His work in [[harmonic analysis]], especially [[microlocal analysis]] and the analysis of linear [[partial differential equation|PDEs]] was a very central contribution to this theory in the 1960s and 1970s.

* [Wikipdia entry](https://en.wikipedia.org/wiki/Lars_H%C3%B6rmander)

## Writings

Hörmander wrote several books including  his magnus treatise

* _The analysis of linear partial differential operators_, in 4 vols.: I. Distribution theory and Fourier analysis, II. Differential operators with constant coefficients, III. Pseudo-differential operators, IV. Fourier integral operators.

## related $n$Lab entries

* [[distribution]], [[compactly supported distribution]]

* [[support of a distribution]]

* [[wave front set]], [[propagation of singularities theorem]]

* [[Hörmander's criterion]]

* [[product of distributions]]

* [[extension of distributions]]

* [[pullback of distributions]]

* [[tensor product of distributions]]

* [[Hörmander's condition]]

category: people

[[!redirects Lars Hormander]]


Lars Kjeseth

* [webpage](http://www.elcamino.edu/faculty/lkjeseth/)

## related $n$Lab entries

* [[Lie-Rinehart pair]],

* [[L-infinity algebras in physics]]

category: people


Lascar group

The Lascar group $Gal_L T$ is a variant of the [[Galois theory|Galois group]] in  [[model theory]] attached to a [[first order theory]].

##References

* D. Lascar, _On the category of models of a complete theory_, J. Symbolic Logic 47 (1982), no. 2, 249--266; [MR84g:03055](http://www.ams.org/mathscinet-getitem?mr=654786) [doi](http://dx.doi.org/10.2307/2273140)
* [[Martin Ziegler]], _Introduction to the Lascar group_, in: Tits buildings and the model theory of groups (Würzburg, 2000), 279–298,
London Math. Soc. Lecture Note Ser., 291, Cambridge Univ. Press, Cambridge, 2002, [MR2004j:03046](http://www.ams.org/mathscinet-getitem?mr=2018391), [doi](http://dx.doi.org/10.1017/CBO9780511549786.013), (Preprint version [here](http://home.mathematik.uni-freiburg.de/ziegler/preprints/lg10.pdf).)
* E. Casanovas, D. Lascar, A. Pillay, M. Ziegler, _Galois groups of first order theories_, J. Math. Log. 1 (2001), no. 2, 305–319, [MR2003e:03068](http://www.ams.org/mathscinet-getitem?mr=1864739) [doi](http://dx.doi.org/10.1142/S0219061301000119), (Preprint version [here](http://home.mathematik.uni-freiburg.de/ziegler/preprints/CaLaPiZi-Gal-1.6.pdf).)


laser

+-- {: .rightHandSide}
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#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[photon]], [[light]], [[electromagnetic wave]]

* [[topological laser]]

## References

See also:

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

[[!redirects lasers]]


Lasha Berezhiani

* [webpage](http://www.ccpp.nyu.edu/lasha_berezhiani.html)

## related $n$Lab entries

* [[MOND]], [[fuzzy dark matter]]

category: people


Lat

__$Lat$__ is the [[category]] whose [[objects]] are [[lattices]] and whose [[morphisms]] are lattice [[homomorphisms]], that is [[functions]] which preserve finitary [[meets]] and [[joins]] (equivalently, binary meets and joins and the [[top]] and [[bottom]] elements).  $Lat$ is a [[subcategory]] of [[Pos]].

$Lat$ is given by a finitary [[variety of algebras]], or equivalently by a [[Lawvere theory]], so has all the usual properties of such categories.  By general abstract nonsense, the __[[free object|free]] lattice__ on a [[set]] $X$ exists, but it is not easy to describe in general.

*  The free lattice on the [[empty set]] is the [[Boolean domain]] $\{\bot, \top\}$.
*  The free lattice on a [[singleton]] $\{x\}$ is the $3$-element [[totally ordered set]] $\{\bot, x, \top\}$.
*  The free lattice on $\{x,y\}$ is (if I haven\'t messed up) $\{\bot, x \wedge y, x, y, x \vee y, \top\}$.
*  The free lattice on $3$ generators, despite the simplicity of the previous examples, is [[infinite set|infinite]].
*  The free lattice on any [[countable set]] can be made into a [[subalgebra|sublattice]] of the free lattice on $3$ generators!
*  The free lattice on an uncountable set is uncountable, of course.

The results about the free lattice on $3$ generators may be found in Section 4.6 of _[[Stone Spaces]]_, which sites Skornjakov\'s _Elements of Lattice Theory_.

category: category

[[!redirects Lat]]

[[!redirects free lattice]]


latest changes > history


Latham Boyle

* [webpage](https://www.perimeterinstitute.ca/people/latham-boyle)

## Selected writings

Discussion of [[spectral triples]] over [[Jordan algebras]] in the [[Connes-Lott model]]:

* [[Latham Boyle]], [[Shane Farnsworth]],
_The standard model, the Pati-Salam model, and "Jordan geometry"_ ([arxiv:1910.11888](https://arxiv.org/abs/1910.11888))

On the [[Penrose tiling]] in relation to [[quantum error correcting codes]]:

* Zhi Li, [[Latham Boyle]], *The Penrose Tiling is a Quantum Error-Correcting Code* [[arXiv:2311.13040](https://arxiv.org/abs/2311.13040)]

## Related entries

* [[Connes-Lott-Chamseddine model]]

category: people


Latin square

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An $n \times n$ **Latin square** is a square array of numbers from $1$ to $n$ such that each row and each column contains every number from $1$ to $n$. For example, an ordinary sudoku square is a special type of $9 \times 9$ Latin square.

Latin squares can be regarded as precisely the multiplication tables for [[quasigroup]] structures on the set $\{1, 2, \ldots, n\}$, where $i j$ is the entry in the $i^{th}$ row and $j^{th}$ column. Indeed, the condition that no two entries in the $i^{th}$ row are the same says that left multiplication by $i$ is invertible, and that no two entries in the $j^{th}$ column are the same says that right multiplication by $j$ is invertible.

category: combinatorics


lattice

> This entry is about the notion in _[[order theory]]/[[logic]]_. For other notions of the same name, such as in [[bilinear form]]-theory, see at _[[lattice (disambiguation)]]_.

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## Definition

A **lattice** is a [[partial order|poset]] which admits all finite [[meets]] and finite [[joins]] (or all finite [[products]] and finite [[coproducts]], regarding a poset as a [[category]] (a [[(0,1)-category]])).

As [[(0,1)-limits]] are meets and [[(0,1)-colimits]] are joins, a lattice is a [[(0,1)-category]] that is [[finitely complete category|finitely complete]] and [[finitely cocomplete category|finitely cocomplete]].

A **lattice** can also be defined as an algebraic structure, with the binary operations $\wedge$ and $\vee$ and the constants $\top$ and $\bot$. (These correspond, respectively, to binary and nullary meets and joins in the poset-theoretic definition; accordingly, they are read 'meet', 'join', '[[top]]', and '[[bottom]]'.) Here are the axioms for these operations:

* $\wedge$ and $\vee$ are each [[idempotent]], [[commutative magma|commutative]], and [[associative magma|associative]];
* the _absorption laws_: $a \vee (a \wedge b) = a$, and $a \wedge (a \vee b) = a$;
* $\top$ and $\bot$ are the respective [[identity element|identities]] of $\wedge$ and $\vee$.

You can recover the original poset from either the meet or the join; $a \leq b$ iff $a \wedge b = a$, and $b \leq a$ iff $a \vee b = a$, and then prove that $a\wedge b$ is the greatest lower bound for $a$, $b$ and $a \vee b$ is the least upper bound for $a$, $b$. (Notice that the absorption laws guarantee that these two descriptions of $\leq$ agree.)  Indeed, we may say that a lattice is a _bisemilattice_ in that it has two semilattice structures that are compatible in that they define (but in dual ways) the same partial order.

Note that a poset with only finite meets _or_ finite joins is a (meet- or join-) [[semilattice]], while a lattice which has *all* joins and meets (not just finitary ones) is a [[complete lattice]].

## Bounded lattices and pseudolattices

Traditionally, a lattice need have only finite [[inhabited set|inhabited]] meets and joins; that is, it need not have a top or bottom element.  Algebraically, this means $\wedge$ and $\vee$ need not have identities.

Then one may call a lattice that *does* have a top and a bottom a __bounded lattice__; in general, a [[bounded poset]] is a poset that has top and bottom elements.

The other approach is to define a lattice, as above, to require a top and a bottom and then use the term __pseudolattice__ to allow for the possibility that it might not.

From an algebraic point of view, requiring top and bottom is quite natural, a special case of preferring [[monoids]] to more general [[semigroups]].  In any case, one can formally adjoin a top and a bottom if required.  On the other hand, many examples, especially from analysis, do not come with a top or a bottom, and adjoining them would break the other structure. For example, adjoining top ($\infty$) and bottom ($-\infty$) to the [[real line]] makes it no longer a [[field]] (addition is especially problematic); more generally, a [[Banach lattice]] need not (and, except in one degenerate case, cannot) have a top or a bottom. In fact, adjoining top and bottom to any [[total order|totally]] [[ordered ring]] makes it no longer a [[ring]].

## Lattice homomorphisms

A lattice homomorphism $f$ from a lattice $A$ to a lattice $B$ is a [[function]] from $A$ to $B$ (seen as sets) that preserves $\wedge$ and $\vee$ (and $\top$ and $\bot$, if these are required):
$$ f(x \wedge y) = f(x) \wedge f(y),\; f(\top) = \top,\; f(x \vee y) = f(x) \vee f(y),\; f(\bot) = \bot .$$
Note that such a homomorphism is necessarily a [[monotone function]], but the converse fails.

Thus, a lattice is a poset (or even a semilattice) with [[property-like structure]].

Lattices and lattice homomorphims form a [[concrete category]] [[Lat]].

## Related concepts

* [[sigma-complete lattice|$\sigma$-complete lattice]]

* [[complete lattice]]

* [[distributive lattice]]

* [[modular lattice]]

* [[orthomodular lattice]]

* [[geometric lattice]]

* [[continuous lattice]]

* [[complemented lattice]]

* [[semilattice]]

* [[suplattice]]

* [[Hilbert lattice]]

* [[frame]]

* [[pseudolattice ordered abelian group]]

* [[lattice object]]

## References

* [[Peter Johnstone]], chapter 1 of _[[Stone Spaces]]_

* [[Jacob Lurie]], section A.1.1 of _[[Spectral Algebraic Geometry]]_

[[!redirects lattices]]
[[!redirects bisemilattice]]
[[!redirects bounded lattice]]
[[!redirects pseudolattice]]


lattice (disambiguation)

* In [[order theory]]/[[logic]]: _[[lattice]]_ .

* In [[group theory]] / [[module]]-theory / [[quadratic form]]-theory: _[[ lattice (discrete subgroup)]]_ .

category: disambiguation


lattice (discrete subgroup)

> This entry is about the notion of _lattice_ in [[group theory]]/[[quadratic form]]-theory. For other notions see at _[[lattice (disambiguation)]]_.

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## Idea

Classically, a _lattice in the [[Cartesian space]]_ $\mathbb{R}^n$ is a [[discrete group|discrete]] [[subgroup]] (of the underlying [[topological group|topological]] [[abelian group]]) that [[spans]] $\mathbb{R}^n$ as a vector space over $\mathbb{R}$. This may be generalized, from $\mathbb{R}^n$ to a general locally compact abelian group.

## Definition

A *lattice* in a [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological abelian group|abelian group]] $A$ is a subgroup $L \hookrightarrow A$ that is [[discrete space|discrete]] and *cocompact*, meaning that the quotient group $A/L$ with the quotient topology is [[compact space|compact]].

Applying [[Pontryagin duality]], the dual of the quotient map $q: A \to A/L$ is in that case a discrete subgroup $\widehat{A/L} \hookrightarrow \widehat{A}$ that is also cocompact (its [[cokernel]] being the compact group $\widehat{L}$). This is called the *dual lattice* of $L$.

## Examples

Notable examples of classical lattices (in $\mathbb{R}^n$) include

* [[Leech lattice]], [[E8 lattice]]

* [[Niemeier lattice]]

The standard diagonal inclusion of a [[global field]] (such as a [[number field]]) $k$ into its [[ring of adeles]] $A_k$ is a lattice in the more general sense. Recalling that $A_k$ is Pontryagin dual to itself, the lattice $k$ is identified with its dual lattice.

## Related concepts

* [[integral lattice]]

* [[modular integral lattice]]

* [[unimodular integral lattice]]

* [[torus]], [[elliptic curve]]

* [[genus of a lattice]]

* [[crystallographic group]]

* [[Bravais lattice]]

* [[sphere packing]]

* [[lattice model]]

  * [[lattice gauge theory]]

## References

* [[Peter Engel]], [[Louis Michel]], [[Marjorie Senechal]], *Lattice Geometry*, IHES/P/04/45 (2004) $[$[pdf](https://cds.cern.ch/record/859509/files/cer-002542451.pdf), [cds:859509](https://cds.cern.ch/record/859509)$]$

* [[Andries E. Brouwer]], _Lattices_, Course notes (2002) $[$[pdf](http://www.win.tue.nl/~aeb/latt/lattices.pdf)$]$

* [[John Conway]], N. Sloane, _Low dimensional lattices I: Quadratic forms of small determinant_

* Wikipedia, _[Lattice (group)](http://en.wikipedia.org/wiki/Lattice_%28group%29)_

* Wikipedia, _[Lattice (module)](http://en.wikipedia.org/wiki/Lattice_%28module%29)_

Applications in [[heterotic string theory]] and [[F-theory]]:

* Dmitry Manning-Coe, *Lattices: From Roots to String Compactifications* [[arXiv:2304.05394](https://arxiv.org/abs/2304.05394)]

[[!redirects lattice in a vector space]]

[[!redirects lattices in a vector space]]

[[!redirects lattice (in a vector space, etc.)]]


lattice (in a vector space, etc.) > history

< [[lattice (discrete subgroup)]]