length

## Idea

Length is the [[volume]] of [[curves]].

## Examples

* [[circumference of a circle]]

## Related concepts

* [[area]]

* [[volume]]

* [[curve]]

* [[length scale]]

* [[wave length]]

* [[Planck length]]

* [[Compton wavelength]]

* [[string length]]

[[!redirects lengths]]


length of an object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _length_ of an [[object]] in an [[abelian category]] $\mathcal{C}$ is akin to the concept of [[dimension]] of [[vector spaces]], to which it reduces in the case that $\mathcal{C} = $ [[Vect]]. The 1-dimensional vector space is a [[simple object]] in [[Vect]], and the [[dimension]] of a vector space $V$, if it is finite, may be thought of as the number of times that one may split off such a simple object from $V$. The definition of _length_ generalizes this concept, notably to [[modules]] over some [[ring]].

## Definition

Let $\mathcal{C}$ be an [[abelian category]].

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

Given an [[object]] $X \in \mathcal{C}$, then a _Jordan-Hölder sequence_ or _[[composition series]]_ for $X$ is a finite [[filtration]], i.e. a finite sequence of [[subobject]] inclusions into $X$, starting with the [[zero objects]]

$$
  0 = X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n = X
$$

such that at each stage $i$ the [[quotient]] $X_i/X_{i-1}$ (i.e. the [[cokernel]] of the [[monomorphism]] $X_{i-1} \hookrightarrow X_i$) is a [[simple object]] of $\mathcal{C}$.

If a Jordan-Hölder sequence for $X$ exists at all, then $X$ is said to be of _finite length_.

=--

(e.g. [EGNO 15, def. 1.5.3](#EGNO15))

+-- {: .num_prop #JordanHolderSequenceHasDefiniteLength}
###### Proposition
**([[Jordan-Hölder theorem]])**

If $X \in \mathcal{C}$ has finite length according to def. \ref{FiniteLength}, then in fact all Jordan-Hölder sequences for $X$ have the same length $n \in \mathbb{N}$.

=--

(e.g. [EGNO 15, theorem 1.5.4](#EGNO15))

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

If an object $X \in \mathcal{C}$ has finite length according to def. \ref{FiniteLength}, then the length $n \in \mathbb{N}$ of any of its Jordan-Hölder sequences, which is uniquely defined according to prop. \ref{JordanHolderSequenceHasDefiniteLength}, is called the _length of the object_ $X$.

=--

(e.g. [EGNO 15, def. 1.5.5](#EGNO15))

## Properties

### Relation to Schur functors
 {#RelationToSchurFunctors}

In [[abelian categories]] that are also $k$-linear [[tensor categories]] over a [[field]] $k$ of [[characteristic zero]], then objects have finite length precisely if they are annihilated by some [[Schur functor]] for the [[symmetric group]].

This is a (considerable) generalization of the familiar fact that for every [[finite dimensional vector space]] $V$ there exists an [[symmetric algebra|exterior power]] that vanishes, $\wedge^n V = 0$ (namely for all $n \gt dim(V)$). Similarly, if $V$ is a [[super vector space]] of dimension $(d,p)$, then the combined $(d+1)$st skew-symmetric tensor power and $(p+1)$st symmetric tensor power annihilates it. In this way prop. \ref{LenghtOfObjectIsBounded} below goes in the direction of establishing that in a $k$-linear tensor category all objects of bounded length , in the sense of def. \ref{LenghtOfObjectIsBounded}, behave like having underlying [[super vector spaces]]. The completion of this statement is [[Deligne's theorem on tensor categories]], see there for more.

First we need to fix the precise meaning of "[[tensor category]]":

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

For $k$ an [[algebraically closed field]] of [[characteristic zero]], then a _$k$-[[tensor category]]_ $\mathcal{A}$ is an

1. [[abelian category|abelian]]

1. [[rigid monoidal category|rigid]]

1. [[symmetric monoidal category|symmetric]]

1. [[braided monoidal category|braided]]

1. [[monoidal category]]

1. [[enriched category|enriched]] over $k$[[Mod]] = $k$[[Vect]] (i.e. $k$-linear),

such that

1. the [[tensor product]] functor $\otimes \colon \mathcal{A} \times \mathcal{A} \longrightarrow \mathcal{A}$ is

   1. $k Mod$-[[enriched functor|enriched]] (i.e. $k$-linear);

   1. [[exact functor|exact]]

   in both arguments;

1. $End(1) \simeq k$ (the [[endomorphism ring]] of the [[tensor unit]] coincides with $k$).

Such a $k$-tensor category is called _finitely $\otimes$-generated_ if there exists an [[object]] $E \in \mathcal{A}$ such that every other object $X \in \mathcal{A}$ is a [[subquotient]] of a [[direct sum]] of [[tensor products]] $E^{\otimes^n}$, for some $n \in \mathbb{N}$:

$$
  \array{
    && \underset{i}{\oplus} E^{\otimes^{n_i}}
    \\
    && \downarrow
    \\
    X &\hookrightarrow& (\underset{i}{\oplus} E^{\otimes^{n_i}})/Q
  }
  \,.
$$

Such $E$ is called an _$\otimes$-generator_ for $\mathcal{A}$.

=--

([Deligne 02, 0.1](#Deligne02))

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

A [[tensor category]] $\mathcal{A}$ (def. \ref{TensorCategory}) is said to have _subexponential growth_ if for every [[object]] $X$ there exists a [[natural number]] $N$ such that $X$ is of length (def. \ref{LengthOfAnObject}) at most $N$, and that also all [[tensor product]] powers of $X$ are of length bounded by the corresponding powers of $N$:

$$
  \underset{X \in \mathcal{A}}{\forall}
  \underset{N \in \mathbb{N}}{\exists}
  \underset{n \in \mathbb{N}}{\forall}
    \;
   length(N^{\otimes^n}) \leq N^n
  \,.
$$

=--

(e.g. [EGNO 15, def. 9.11.1](#EGNO15))

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

For $(\mathcal{A},\otimes)$ a $k$-[[tensor category]] as in def.\ref{TensorCategory}, for $X \in \mathcal{A}$ an [[object]], for $n \in \mathbb{N}$ and $\lambda$ a [[partition]] of $n$, say that the value of the [[Schur functor]] $S_\lambda$ on $X$ is

$$
  S_{\lambda}(X) \coloneqq (V_\lambda \otimes X^{\otimes_n})^{S_n}
  \coloneqq
  \left(
    \frac{1}{n!}
    \underset{g\in S_n}{\sum}
    \rho(g)
  \right)
  \left(
    V_\lambda \otimes X^{\otimes_n}
  \right)
$$

where

* $S_n$ is the [[symmetric group]] on $n$ elements;

* $V_\lambda$ is the [[irreducible representation]] of $S_n$ corresponding to $\lambda$;

* $\rho$ is the diagional [[action]] of $S_n$ on $V_\lambda \otimes X^{\otimes_n}$, coming from the canonical [[permutation]] action on $X^{\otimes_n}$;

* $(-)^{S_n}$ denotes the subspace of [[invariants]] under the action $\rho$

* the last expression just rewrites this as a [[group averaging]].

=--

([Deligne 02, 1.4](#Deligne02))

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

For a [[tensor category]] $\mathcal{A}$ the following are equivalent:

1. the category has subexponential growth (def. \ref{SubexponentialGrowth}).

1. For every object $X \in \mathcal{A}$ there exists $n \in \mathbb{N}$ and a [[partition]] $\lambda$ of $n$ such that the corresponding value of the [[Schur functor]], def. \ref{SchurFunctor}, on $X$ vanishes: $S_\lambda(X) = 0$.

=--

([Deligne 02, prop. 05](#Deligne02))

## References

* {#EGNO15} [[Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[Victor Ostrik]], section 1.5 in _Tensor categories_, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 ([pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf
))

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

The relation to [[Schur functors]] is discussed in

* {#Deligne02} [[Pierre Deligne]], _Catégorie Tensorielle_, Moscow Math. Journal 2 (2002) no. 2, 227-248. ([pdf](https://www.math.ias.edu/files/deligne/Tensorielles.pdf))

For more on this see at _[[Deligne's theorem on tensor categories]]_.

[[!redirects lengths of objects]]

[[!redirects Jordan-Hölder series]]
[[!redirects Jordan-Holder series]]

[[!redirects Jordan-Hölder sequence]]
[[!redirects Jordan-Holder sequence]]

[[!redirects object of finite length]]
[[!redirects objects of finite length]]


length scale

> This entry is about scales in [[geometry]] and [[physics]]. For scales in [[algebra]] and [[linear logic]], see [[scale]].

#Contents#
* table of contents
{:toc}

## Idea

[[length]]$\,$[[scale]]

## In physics

[[energy]] and [[length scales]] in the [[observable universe]] (from [[cosmological constant|cosmic]] scales, over [[fundamental particle]]-[[masses]] around the [[electroweak symmetry breaking]] to [[GUT scale]] and [[Planck scale]]):

> graphics grabbed from [Zupan 19](flavour+in+particle+physics#Zupan19)

\begin{center}

\end{center}

> graphics grabbed from [here](http://sten.astronomycafe.net/the-particle-desert/)

\linebreak

[[!include fundamental scales -- contents]]

## Related concepts

* [[scaling transformation]]

## References

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

[[!redirects length scales]]


Lennart Meier

* [webpage](https://webspace.science.uu.nl/~meier007/)

## Selected writings

On [[equivariant elliptic cohomology]]/[[tmf]]:

* [[David Gepner]], [[Lennart Meier]], _On equivariant topological modular forms_, ([arXiv:2004.10254](https://arxiv.org/abs/2004.10254))

category: people


Lennart Schmidt

* [institute page](https://www.physics.nus.edu.sg/~strings/members.htm)

## Selected writings

Proposal for understanding [[D=6 N=(1,0) SCFTs]] as [[higher gauge theories]] for gauge-[[L-infinity algebras]] which are variants of the [[string Lie 2-algebra]]:

* [[Christian Saemann]], [[Lennart Schmidt]], _The Non-Abelian Self-Dual String and the $(2,0)$-Theory_ ([arXiv:1705.02353](https://arxiv.org/abs/1705.02353))

* [[Christian Saemann]], [[Lennart Schmidt]], _Towards an M5-Brane Model I: A 6d Superconformal Field Theory_, J. Math. Phys. 59 (2018) 043502 ([arxiv:1712.06623](https://arxiv.org/abs/1712.06623))

Discussion of [[adjusted Weil algebras]] for [[string Lie 2-algebra]] and the corresponding [[higher gauge theory]]:

* {#Schmidt19} [[Lennart Schmidt]], _Twisted Weil Algebras for the String Lie 2-Algebra_, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_ [Durham Symposium](http://www.maths.dur.ac.uk/lms/) 2018, Fortschritte der Physik 2019 [[arXiv:1903.02873](https://arxiv.org/abs/1903.02873), [doi:10.1002/prop.201910016](https://doi.org/10.1002/prop.201910016)]

* {#SaemannSchmidt20} [[Christian Saemann]], [[Lennart Schmidt]], *Towards an M5-Brane Model II: Metric String Structures*, Fortschr. Phys. **68** (2020) 2000051 [[arXiv:1908.08086](https://arxiv.org/abs/1908.08086), [doi:10.1002/prop.202000051](https://doi.org/10.1002/prop.202000051)]

On regularized [[quantization]] of [[Green-Schwarz sigma-models]] for [[super p-branes]], specifically of the [[super 4-brane in 9d]], in generalization of the [matrix model of the M2-brane](M2-brane#QuantizationViaBFSSMatrixModel):

* [[Meer Ashwinkumar]], [[Lennart Schmidt]], [[Meng-Chwan Tan]], Section 2 of: _Matrix Regularization of Classical Nambu Brackets and Super $p$-Branes_ ([arXiv:2103.06666](https://arxiv.org/abs/2103.06666))

category: people


lens (in computer science)

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[computer science]], originally in [[database theory]], a data [[structure]]-[[type]] now called (lawful) *lenses* [[Bohannon, Pierce & Vaughan (2006, §3)](#BohannonPierceVaughan06), [Foster, Greenwald, Moore, Pierce & Schmitt  (2007, §3)](#FosterGreenwaldMoorePierceSchmitt07), but considered under different names already by [Oles 1982](#Oles82), [1986](#Oles86) and [Hofmann & Pierce 1996](#HofmannPierce96)]
is used to encode [[data]]([[database|-base]]) types $S$ equipped with a read/write functionality (only) for a specified view-type $V$.

In particular, "very well-behaved" or "lawful" lenses (see Def. \ref{Lens} below) -- those for which updates of the "view" $V$ completely overwrite any previous such changes -- turn out to equivalently be data base types of product form $S = D \times V$ with "view" given by projecting onto the $V$-factor.  As such, lawful lenses are equivalent to (cf. *[[monads in computer science]]*):

* the [[module over a monad|modales]] over the $V$-[[possibility]]-monad

  ([Johnson, Rosebrugh & Wood 2010, Prop. 9](#JohnsonRosebrughWood10), see Prop. \ref{LawfulLensesArePossibilityModales} below)

* the [[comodule over a comonad|comodales]] over the [[costate comonad]],

  ([O’Connor (2010)](#O’Connor10), [(2011)](#O’Connor11), see Prop. \ref{LensesAreCostateCoalgebras} below).

Thus, lawful lenses may be thought of as encoding read/write functionality (only) at a certain address in a data base (compare the [[RAM]]-model given by the [[state monad]] which encodes read/write of the entire data base type at once).

More generally, there are two different approaches to lenses:

* Lawful lenses (as above) are an algebraic structure which axiomatize product projections. The laws govern the ways views and updates relate. These are generalized into [[delta lens|Delta lenses]], which are more flexible lawful lenses.

* Lawless lenses, and in particular bi-directional or polymorphic lenses, separate the two directions of view and update so that the laws no longer type-check. These lawless lenses are used to organize bi-directional flow of data, and are generalized into [[optics (in computer science)|optics]] in the [[functional programming]] literature.

An alternate generalization of lawless lenses is put forward in [Spivak 19](#Spivak19) as the [[Grothendieck construction]] of the fiberwise dual of an [[indexed category]]. This notion of lawless lens has been adopted in the context of [[categorical systems theory]] because they represent a *bidirectional stateful computation* which describes the way some systems expose and update their internal state. For instance, see [Myers](#Myers), [Spivak & Niu](#SpivakPoly) or [Hedges (2021)](#Hedges21).

In this sense, lawless lenses are applications of two mathematical frameworks which are interesting in their own right: [[fibration|fibrations]] and [[optics (in computer science)|optics]] (thus [[Tambara module|Tambara modules]]).

## Definition
 {#Definition}

\begin{definition}\label{Lens}
**(lenses)**
\linebreak
Given a [[category]] $(\mathbf{C}, 1, \times)$  with [[finite products]], a *lens* in $\mathbf{C}$ is:

1. a [[pair]] of [[objects]] of $\mathbf{C}$

   1. ("source")

      $$S \,\in\, \mathbf{C}$$

   1. ("view")

      $$V \,\in\, \mathbf{C}$$

1. a [[pair]] of [[morphisms]] of the form

   1. ("view", "read-out", "forward operation")

      $$
        \mathrm{get}
          \;\colon\;
        S \longrightarrow V
      $$

   1. ("update", "write", "backward operation")

      $$
        \mathrm{put}
          \;\colon\;
        V \times S \longrightarrow S
        \,.
      $$

The lens is called *well-behaved* if this data satisfies:

* (PutGet) "what has just been written is read out as such":

  $$
    v \colon V
    ,\,
    s \colon S
    \;\;\;\;\;\;
    \vdash
    \;\;\;\;\;\;
    get\big(
      put
     (v,\,s)
    \big)
    \;=\;
    v
    \,,
  $$

and

* (GetPut) "writing what has just been read out means no change":

  $$
    s \colon S
    \;\;\;\;\;\;
    \vdash
    \;\;\;\;\;\;
    put\big(
      get(s)
      ,\,
      s
    \big)
    \;=\;
    s
  $$

and it is called *very well-behaved* or *lawful* if in addition it satisfies:

* (PutPut) "writing means to over-write previous edits":

  $$
    v,\,v' \colon V
    ,\;
    s \colon S
    \;\;\;\;\;\;
    \vdash
    \;\;\;\;\;\;
    put\Big(
      v',\,
      put\big(
        v,\, s
      \big)
    \Big)
    \;=\;
    put(v',\, s)
  $$

\end{definition}

Definition \ref{Lens} appears, under no particular name, in [Hofmann & Pierce 1996, p. 12](#HofmannPierce96), where it is thought of as providing aspects of [[object-oriented programming]]: Here one thinks of $S$ as a "class" which inherits functionality from the class $V$. And indeed, for lawful lenses it follows (see [below](#PossibilityMonadicity)) that $S \simeq D \times V$ is a "record" which inherits all fields present in $V$ but may add some more).

The terminology "lens" for Def. \ref{Lens}
is due to [Bohannon, Pierce & Vaughan 2006, Def. 3.2](#BohannonPierceVaughan06).

## The category of lenses

Lenses (regardless of their lawfulness) organize in a category $\mathrm{Lens}(\mathbf C)$ whose objects are the same as $\mathbf C$ and whose morphisms $X \to Y$ are lenses with states $X$ and views $Y$. The identity lens is given by $(1_X, \pi_1) :X \to X$. Composition of $(\mathrm{get}_1, \mathrm{put}_{1}):X \to Y$ and $(\mathrm{get}_2, \mathrm{put}_{2}):Y \to Z$ is given by:
$$
  \mathrm{get}_{12} = \mathrm{get}_2 \circ \mathrm{get}_1
$$
$$
  \mathrm{put}_{12} = \mathrm{put}_1 \circ ((\mathrm{put}_2 \circ (1_Z \times \mathrm{get}_1)) \times 1_X) \circ (1_Z \times \Delta_X)
$$
The $\mathrm{put}_{12}$ morphism is probably easier to describe using generalized elements:
$$
  \mathrm{put}_{12} : (z,x) \mapsto \mathrm{put}_1(\mathrm{put}_2(z, \mathrm{get}_1(x)), x)
$$

\begin{remark}
Crucially, associativity of this composition relies on [[monoidal category with diagonals|naturality of the diagonals]], which is a given in cartesian categories but not in more general monoidal categories. [[optic (in computer science)|Optics]] are a sweeping generalization of lenses which overcomes this obstacle.
\end{remark}

Moreover, the [[cartesian product]] of $\mathbf C$ endows $\mathrm{Lens}(\mathbf{C})$ of a [[monoidal product]].

## Properties

### Monadicity over Possibility monad
 {#PossibilityMonadicity}

\begin{proposition}
\label{LawfulLensesArePossibilityModales}
**(lawful lenses are possibility modales)**
\linebreak
Lawful lenses in $\mathbf{C}$ with view $V$ (Def. \ref{Lens})
are equivalently the [[module of a monad|modules]] ([[modales]]) of the left [[base change]] [[monad]] on the [[slice category]] over $V$ (the $V$-[possibility](necessity+and+possibility#globally)-[[modality]] $\lozenge_V$):

\[
  \label{PossibilityMonadOverTheViewObject}
\]
\begin{tikzcd}
  \mathbf{C}
  \ar[
    r,
    shift right=8pt,
    "{
      \times V
    }"{swap}
  ]
  \ar[
    from=r,
    shift right=8pt,
    "{
      \coprod_{V}
    }"{swap}
  ]
  \ar[
    r,
    phantom,
    "{ \scalebox{.7}{$\bot$} }"
  ]
  &
  \mathbf{C}_{/V}
    \ar[out=40, in=-40,
      looseness=4.5,
      shorten=-3pt,
      "{
        \lozenge_{V}
      }"{description},
    ]
\end{tikzcd}

where

1. $view \,\colon\, S \to V$ is identified with the slicing morphism,

1. $get \,\colon\, V \times S \to S$ is identified with the module action $\lozenge_V(S) \to S$.

\end{proposition}
([Johnson, Rosebrugh & Wood 2010, Prop. 9](#JohnsonRosebrughWood10))
\begin{proof}
\label{ProofOfLawfulLensesArePossibilityModales}
By direct unwinding of the definitions:

\begin{imagefromfile}
    "file_name": "LawfulLensesAsPossibilityModales-230613.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 10,
        "right": 10,
        "left": 10
    }
\end{imagefromfile}

\end{proof}

\begin{proposition}
\label{LawfulLensesMonadicity}
  If the ambient category $\mathbf{C}$ is a [[topos]] (such as [[Sets]]) and the view $V$ is [[inhabited object|inhabited]],  then the adjunction (eq:PossibilityMonadOverTheViewObject) is [[monadic functor|monadic]].
\end{proposition}
For the case $\mathbf{C} = Sets$ this is [Johnson, Rosebrugh & Wood 2010, Thm. 12, Cor. 13](#JohnsonRosebrughWood10).
\begin{proof}
 The [[monadicity theorem]] ([here](monadicity+theorem#BeckMonadicityTheoremAlternative)) readily gives that the above adjunction (eq:PossibilityMonadOverTheViewObject) is [[monadic functor]]: $(\text{-})\times V$ [[preserved colimit|preserves]] all colimits (cf. [[universal colimits]]) hence in particular the relevant [[coequalizers]], and it is [[conservative functor|conservative]] by the assumption that $V$ is inhabited (this is clear by inspection in [[Sets]] -- which is what [JRW10](#JohnsonRosebrughWood10) observe on their p. 13 --  for the general case use [[Sketches of an Elephant|Johnstone 2002, vol 1, Lem. 1.3.2]]).
\end{proof}

\begin{remark}\label{MonadicityOfLawfulLensesAsMonadicDescent}
**(relation to [[monadic descent]])**
\linebreak
In ([[algebraic geometry|algebraic]]) [[geometry]], the statement of Prop. \ref{LawfulLensesMonadicity} is known as *[[monadic descent]]*. In this context one would say that:

* the [[epimorphism]] $V \to \ast$ (reflecting that $V$ is [[inhabited objects|inhabited]], necessarily an [[effective epimorphism]] given that $\mathbf{C}$ is assumed to be a [[topos]]) is the *morphism of effective descent*, representing a [[covering]],

* $\lozenge_V$-[[modale]]-[[structure]] is *[[descent data]]* on the [[underlying]] object in $\mathbf{C}_{/V}$,

* the [[Eilenberg-Moore category|EM-category]] $\big(\mathbf{C}_{/V}\big)^{\lozenge_V}$ is the *[[category of descent data]]* along $V \to \ast$,

and so the monadicity theorem $\mathbf{C} \,\simeq\, \big(\mathbf{C}_{/V}\big)^{\lozenge_V}$ in Prop. \ref{LawfulLensesMonadicity} states that objects over $V$ which are equipped with such descent data are equivalent to objects over $\ast$, hence "do descend" along $V \to \ast$.
\end{remark}

\begin{remark}
**(relation to [[epistemic logic]])**
\linebreak
Prop. \ref{LawfulLensesMonadicity} says that the category of $\lozenge_V$-[[modales]] among objects in $\mathbf{C}_{/V}$ is equivalent to $\mathbf{C}$. This makes good sense when viewing $\lozenge_V$ as the [[possibility]]-[[modality]], see the discussion of *potentiality* [there](necessity+and+possibility#RelationToPotentiality).
\end{remark}

### Monadicity over the CoState comonad
 {#MonadicityOverStateMonade}

Alternatively:

\begin{proposition}\label{LensesAreCostateCoalgebras}
**(lawful lenses are the [[costate comonad|costate]] [[coalgebra over a comonad|coalgebras]])**

For $\mathbf{C}$ a [[cartesian closed category]],  the lawful $V$-lenses in $\mathbf{C}$ (Def. \ref{Lens}) are equivalently the [[coalgebra over a comonad|coalgebras]] of the $V$-[[costate comonad|CoState]] [[comonad]],  i.e. that induced by the [[internal hom]]-[[adjoint functor|adjunction]]:

\begin{centre}

\begin{tikzcd}
  \mathbf{C}
  \arrow[r, shift right=7pt, "{[V, -]}"', "\bot"]
    \ar[out=135, in=-135,
      looseness=5.1,
      shorten=-3pt,
      "{
        V\mathrm{CoState}
      }"{description},
    ]
   &
  \mathbf{C}
  \arrow[l, shift right=7pt, "{(-) \times V}"']
\end{tikzcd}

\end{centre}

\end{proposition}
This is due to [O’Connor (2010)](#O’Connor10),  [O’Connor (2011)](#O’Connor11), for further discussion see  [Gibbons & Johnson (2012), Section 3.2](#GibbonsJohnson12).

A **proof** is spelled out [here](store+comonad#LabelStoreComodalesAreLenses) at *[[store comonad]]*.

## Generalizations

There are many generalizations of lenses which have been proposed, however they can be broadly classified into those which satisfy analogues of the _lens laws_, and those without any axioms or laws.

### Lenses without laws

* One generalization considers the lenses from the previous section as _monomorphic_ by contrast to _polymorphic_ lens which go between pairs of types, $\lambda: (S, T) \to (A, B)$, consisting of a view function, $v_{\lambda}: S \to A$, and an update function $u_{\lambda}: S \times B \to T$. Without further conditions, these are known as _bimorphic lenses_. To impose conditions comparable to the lens laws above requires that the types be related.

These sorts of lenses are generalized by [Spivak 19](#Spivak19). For a quick explanation of how these sorts of generalized lenses are of use in systems theory, see [Myers20](#Myers20); for a longer explanation, see Chapter 2 of [Myers](#Myers).

* An [[optic (in computer science)|optic]] generalizes the way lenses 'remember' state from the forward part to the backward part, avoiding the necessity of a cartesian structure by swapping it with a sufficiently rich actegorical context.

### Lenses with laws

* [[delta lens|Delta lenses]] are a generalization which does satisfy laws. Here we have categories $S$ and $V$ called the source and view, together with a *Get* functor $g : S \to V$ and a function $\varphi \colon S_{0} \times_{V_{0}} V_{1} \to S_{1}$ which takes a pair $(s \in S, u : g s \to v \in V)$ to a morphism
$\varphi(s, u) : s \to p(s, u)$ in $S$ where $p(s, u) = cod(\varphi(a, u))$ is the *Put* function. The function $\varphi$ must also satisfy three lens laws. When $S$ and $V$ are [[indiscrete category|codiscrete categories]], delta lenses are equivalent to a lens in [[Set]]; see ([Johnson-Rosebrugh 2016, Proposition 4](#JohnsonRosebrugh16)).

* A morphism between directed [[polynomial functor|containers]] is another kind of generalised lens satisfying laws called _update-update lenses_; see ([Ahman-Uustalu 2017, Section 5](#AhmanUustalu17)). These are equivalent to [[cofunctor|cofunctors]].

## Related entries

* [[optic (in computer science)]]

* [[polynomial functor]]

* [[delta lens]]

## References

### General

The definition of what are now called (lawful) lenses is already in

* {#Oles82} [[Frank J. Oles]], *A category-theoretic approach to the semantics of programming languages*, Syracuse University (1982) [[pdf](https://www.cs.cmu.edu/afs/cs.cmu.edu/project/fox-19/member/jcr/www/FrankOlesThesis.pdf)]

* {#Oles86} [[Frank J. Oles]], *Type algebras, functor categories and block structure*, Algebraic methods in semantics (1986) 543–573, Daimi Report No. 156 (1983): [[doi:10.7146/dpb.v12i156.7430](https://doi.org/10.7146/dpb.v12i156.7430), [pdf](https://tidsskrift.dk/daimipb/article/view/7430/6281)]

* {#HofmannPierce96} [[Martin Hofmann]], [[Benjamin Pierce]], p. 12 in: *Positive Subtyping*, Information and Computation **126** 1 (1996) 11-33 [[doi:10.1006/inco.1996.0031](https://doi.org/10.1006/inco.1996.0031)]

  > (motivated by [[object-oriented programming]])

but gained wider attention only with:

* {#BohannonPierceVaughan06} Aaron Bohannon, [[Benjamin C. Pierce]], Jeffrey A. Vaughan, *Relational lenses: a language for updatable views*, Proceedings of Principles of Database Systems (PODS) (2006) 338-347 [[doi:10.1145/1142351.1142399](https://doi.org/10.1145/1142351.1142399), [pdf](https://www.cis.upenn.edu/~bcpierce/papers/dblenses-pods.pdf)]

* {#FosterGreenwaldMoorePierceSchmitt07} J. N. Foster, M. B. Greenwald, J. T. Moore, [[Benjamin C. Pierce]], A. Schmitt, _Combinators for bidirectional tree transformations: A linguistic approach to the view-update problem_, ACM Transactions on Programming Languages and Systems **29** 3 (2007) 17-es [[doi:x10.1145/1232420.1232424](https://doi.org/10.1145/1232420.1232424)]

Further discussion:

* {#JohnsonRosebrughWood10} [[Michael Johnson]], [[Robert Rosebrugh]], [[Richard Wood]], _Algebras and Update Strategies_, Journal of Universal Computer Science **16** (2010) [[doi:10.3217/jucs-016-05-0729](http://dx.doi.org/10.3217/jucs-016-05-0729)]

* [[Michael Johnson]], [[Robert Rosebrugh]], and  [[Richard J. Wood]], *Lenses, fibrations and universal translations* Math. Structures Comput. Sci. **22** 1 (2012) 25–42 [[doi:10.1017/S0960129511000442](https://doi.org/10.1017/S0960129511000442)]

* {#JohnsonRosebrugh16} [[Michael Johnson]], [[Robert Rosebrugh]], *Unifying Set-Based, Delta-Based and Edit-Based Lenses*, CEUR Workshop Proceedings **1571** (2016) [[pdf](http://ceur-ws.org/Vol-1571/paper_13.pdf)]

* {#AhmanUustalu17} Danel Ahman, [[Tarmo Uustalu]], _Taking updates seriously_, CEUR Workshop Proceedings, 1827, 2017 ([pdf](http://ceur-ws.org/Vol-1827/paper11.pdf))

* {#Riley} [[Mitchell Riley]], *Categories of optics* [[arXiv:1809.00738](https://arxiv.org/abs/1809.00738), video:[YT](https://www.youtube.com/watch?v=Qy7Y4-mgwbw)]

* {#Spivak19} [[David Spivak]], _Generalized Lens Categories via functors $C^{op} \to Cat$_ ([arXiv:1908.02202](https://arxiv.org/abs/1908.02202))

* {#Myers20} [[David Jaz Myers]], _Double Categories of Dynamical Systems (Extended Abstract)_ ([arXiv:2005.05956](https://arxiv.org/abs/2005.05956))

* {#JohnsonRosebrugh20} [[Michael Johnson]], [[Robert Rosebrugh]], *The more legs the merrier: A new composition for symmetric (multi-)lenses*, EPTCS **333** (2021) 92-107 [[arXiv:2101.10482](https://arxiv.org/abs/2101.10482), [doi:10.4204/EPTCS.333.7](https://doi.org/10.4204/EPTCS.333.7)]

* {#Clarke20a} [[Bryce Clarke]], _Internal lenses as functors and cofunctors_, EPTCS, 323, 2020 ([doi:10.4204/EPTCS.323.13](http://dx.doi.org/10.4204/EPTCS.323.13))

* {#Clarke20b} [[Bryce Clarke]], _A diagrammatic approach to symmetric lenses_, EPTCS, 333, 2021 ([arXiv:2101.10481](https://arxiv.org/abs/2101.10481))

* {#Clarke20b} [[Bryce Clarke]], Derek Elkins, Jeremy Gibbons, [[Fosco Loregian]], [[Bartosz Milewski]], Emily Pillmore, Mario Román, _Profunctor optics, a categorical update_, ([arXiv:2001.07488](https://arxiv.org/abs/2001.07488))

* [[Matthew Di Meglio]], _The category of asymmetric lenses and its proxy pullbacks_, Master's Thesis, Macquarie University, (2021) [[doi:10.25949/20236449.v1](https://doi.org/10.25949/20236449.v1)]

* [[Matthew Di Meglio]], _Coequalisers under the Lens_, EPTCS **372**, (2022) 149-163. [[doi:10.4204/EPTCS.372.11](https://doi.org/10.4204/EPTCS.372.11)]

* [[Bryce Clarke]] and [[Matthew Di Meglio]], _An introduction to enriched cofunctors_, (2022) [[arXiv:2209.01144](https://arxiv.org/abs/2209.01144)]

* [[Matthew Di Meglio]], _Universal Properties of Lens Proxy Pullbacks_, EPTCS **380** (2023) 400-416 [[doi:10.4204/EPTCS.380.23](https://doi.org/10.4204/EPTCS.380.23)]

### Relation to the costate comonad
 {#ReferencesRelationToCostateComonad}

The observation that lenses are equivalently nothing but the [[coalgebra over a comonad|coalgebras]] of the [[costate comonad]] (cf.  *[[monads in computer science]]*) is due  to:

* {#O’Connor10} [[Russell O’Connor]], *Lenses Are Exactly the Coalgebras for the Store Comonad* (30th Nov 2010) [[r6research:23705](https://r6research.livejournal.com/23705.html)]

* {#O’Connor11} [[Russell O'Connor]], *Functor is to Lens as Applicative is to Biplate: Introducing Multiplate*, contribution to [ICFP '11: ACM SIGPLAN International Conference on Functional Programming](https://dl.acm.org/doi/proceedings/10.1145/2036918) (2011) [[arXiv:1103.2841](https://arxiv.org/abs/1103.2841)]

Early review:

* [[Jeremy Gibbons]], *[Lenses are the coalgebras for the costate comonad](https://patternsinfp.wordpress.com/2011/01/31/lenses-are-the-coalgebras-for-the-costate-comonad/)* (2011)

Further details:

* {#GibbonsJohnson12} [[Jeremy Gibbons]], [[Michael Johnson]], *Relating Algebraic and Coalgebraic Descriptions of Lenses*, Electronic Communications of the EASST **49** (2012) [[doi:10.14279/tuj.eceasst.49.726](http://dx.doi.org/10.14279/tuj.eceasst.49.726), [pdf](http://www.cs.ox.ac.uk/jeremy.gibbons/publications/colens.pdf)]

Further review:

* [[Bartosz Milewski]], *[Lenses, Stores, and Yoneda](https://bartoszmilewski.com/2013/10/08/lenses-stores-and-yoneda/)* (2013)

* [[Bartosz Milewski]], pp 240 in: *The Dao of Functional Programming* (2023) [[pdf](https://github.com/BartoszMilewski/Publications/blob/master/TheDaoOfFP/DaoFP.pdf), [github](https://github.com/BartoszMilewski/Publications/tree/master/TheDaoOfFP)]

### Blog posts, talk slides, and other material

* {#Hedges18} [[Jules Hedges]], _Lenses for philosophers_ (2018) [[blog post](https://julesh.com/2018/08/16/lenses-for-philosophers/)]

* {#SpivakACT19} [[David Spivak]], _Lenses:  applications and generalizations_, talk at [ACT 19](http://www.cs.ox.ac.uk/ACT2019/) ([slides](http://math.ucr.edu/home/baez/ACTUCR2019/ACTUCR2019_spivak.pdf), [[Spivak_Lenses.pdf:file]])

* {#Myers} [[David Jaz Myers]], *Categorical systems theory*, [[github](https://github.com/DavidJaz/DynamicalSystemsBook/tree/master/book), [pdf](http://davidjaz.com/Papers/DynamicalBook.pdf)]

* {#SpivakPoly} [[David Spivak]], [[Nelson Niu]], _Polynomial Functors: A General Theory of Interaction_ [[webpage](https://topos.site/poly-course/), [pdf](https://topos.site/poly-book.pdf)]

* {#Hedges21} [[Jules Hedges]], _Lenses as a foundation for cybernetics_, CyberCat seminar, [youtube](https://www.youtube.com/watch?v=WkZPH3Vb5ug&t=1927s)

[[!redirects lenses (in computer science)]]

[[!redirects lens in computer science]]
[[!redirects lenses in computer science]]

[[!redirects lawful lens]]
[[!redirects lawful lenses]]


lens space

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

#Contents#
* table of contents
{:toc}

## Idea

A _lens space_ is the [[quotient space]]

$$
  L\big(n,\{l_i\}_{i=0}^d\big)
  \;\coloneqq\;
  S^{2d+1} / \mathbb{Z}_n
  \;\coloneqq\;
  S
  \big(
    \mathbb{C}_{l_0}
    \oplus
    \cdots
    \oplus
    \mathbb{C}_{l_d}
  \big)
  / \mathbb{Z}_n
$$

of a [[n-sphere|(2d+1)-dimensional sphere]] by a [[free action]] of a [[finite group|finite]] [[cyclic group]] $\mathbb{Z}_n$; specifically --- under the identification of $S^{2d+1}$ with the [[unit sphere]] in the [[real vector space]] underlying $\mathbb{C}^{d+1}$ --- by the [[action]] induced from a [[linear representation]] of the form

$$
  \array{
    \mathbb{Z}_n
      \times
    \mathbb{C}^{d+1}
    &
      \overset{
      }{
        \longrightarrow
      }
    &
    \mathbb{C}^{d+1}
    \\
    \big(
      [k], (z_0, \cdots, z_{d})
    \big)
    &\mapsto&
    \Big(
      \exp( 2 \pi \mathrm{i} \tfrac{l_0}{n} ) \cdot z_0
      ,
      \cdots
      ,
      \exp( 2 \pi \mathrm{i} \tfrac{l_d}{n} ) \cdot z_d
    \Big)
  }
$$

for [[natural numbers]] $l_i$ [[coprime integer|coprime]] to $n$: $gcd(l_i,n) = 1$.

Often this is considered by default for $2d + 1 = 3$, in which case lens spaces are examples of [[3-manifolds]], and as such are in some sense the simplest after the [[3-sphere]] $S^{3}$ and the [[product manifold]] $S^{2} \times S^{1}$ of the [[2-sphere]] with the [[circle]].

## Properties

### Classification

\begin{theorem}

Two lens spaces $L\big(n,\{l_i\}_{i=0}^d\big)$ and $L\big(n,\{l'_i\}_{i=0}^d\big)$ are [[homotopy equivalent]] iff there exists a $k\in\mathbb{N}$ with
$$
  l_0\cdot\ldots\cdot l_d
  \;\equiv\;
  k^d l'_0 \cdot \ldots \cdot l'_d \mod n
  \,.
$$
\end{theorem}

([Olum 1953](#Olum53))

\begin{theorem}
Two lens spaces $L\big(n,\{l_i\}_{i=0}^d\big)$ and $L\big(n,\{l'_i\}_{i=0}^d\big)$ are [[homeomorphic]] iff there exists a $k\in\mathbb{N}$ and a [[permutation]] $\sigma\in Sym_d$ with
$$
  l_i
  \;\equiv\;
  \pm k l'_{\sigma(i)} \mod n
$$
for all $i=0,\ldots, d$.

\end{theorem}

([Brody 1960](#Brody60), [Milnor 1966](#Milnor66))

In both theorems, the same $n$ for both lens spaces is a necessary condition for them to be [[homeomorphic]] or [[homotopy equivalent]] as follows with their respective [[fundamental group]].

### Homotopy theory

Lens spaces are the only [[3-manifolds]] with [[finite group|finite]], [[cyclic group|cyclic]], [[trivial group|non-trivial]] [[fundamental group]]. The reverse direction of this statement (that a 3-manifold with such a fundamental group must be a lens space) is closely related to the [[Poincaré conjecture]].

Lens spaces are important in [[geometric topology]]. They offer the simplest examples of [[smooth manifolds]] which cannot be distinguished by [[homotopy theory]].

### Via Dehn surgery

The 3-dimensional lens space $L(p,q) \coloneqq L(p, \{1,q\})$, for [[coprime integers]] $p$ and $q$, can be constructed by [[Dehn surgery]] on an [[unknot]] with coefficient $-\frac{p}{q}$. One can instead construct $L(p,q)$ by an _integral_ Dehn surgery on the framed 'generalised Hopf link'

\begin{centre}
  \begin{tikzpicture}
     \draw[shorten >= 0.5em] (0,0) -- (2,0) -- (2,-1);
     \draw[shorten <= 0.5em] (2,-1) to (2,-2) to node[auto] {$a_{1}$} (0,-2) to (0,0);
     \draw[shorten >= 0.5em] (1,1) -- (1,0);
     \draw[shorten <= 0.5em, shorten >= 0.5em] (1,0) -- (1,-1) -- (2.25,-1);
     \draw[shorten <= 0.5em] (2.25, -1) to (3,-1) to (3,1) to node[auto, swap] {$a_{2}$} (1,1);
     \draw[shorten >= 0.5em] (4.25,0) -- (3,0);
     \draw[shorten <= 0.5em] (3,0) to (2.25,0) to (2.25,-2) to node[auto, swap] {$a_{3}$} (4.25,-2);
     \node at (5, -1) {$\cdots$};
     \draw[shorten >= 0.5em] (6,1) to node[auto] {$a_{n-1}$} (8,1) to (8,-1) to (7,-1);
     \draw[shorten <= 0.5em] (7,-1) -- (6,-1);
     \draw[shorten >= 0.5em] (9,0) -- (8,0);
     \draw[shorten <= 0.5em] (8,0) to (7,0) to (7,-2) to node[auto, swap] {$a_{n}$} (9,-2) to (9,0);
  \end{tikzpicture}
\end{centre}

where the unknots have framings $a_{1}$, $\ldots$, $a_{n}$ from left to right, for a [[continued fraction]] representation of $-\frac{p}{q}$ as follows.

$$
a_{1} - \frac{1}{a_{2} - \frac{1}{\cdots - \frac{1}{a_{n}}}}
$$

## Related concepts

* [[Seifert 3-manifold]]

## References

See also

* [[Manifold Atlas]], _[Lens spaces](http://www.map.mpim-bonn.mpg.de/Lens_spaces)_.

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

In the context of [[orbifolds]]:

* Siddhartha Gadgil, _Equivariant framings, lens spaces and contact structures_, Pacific Journal of Mathematics, Vol. 208 (2003), No. 1, 73–84 ([doi:10.2140/pjm.2003.208.73](http://dx.doi.org/10.2140/pjm.2003.208.73))

* Michel Boileau, Steven Boyer, Radu Cebanu, Genevieve S. Walsh, Section 3 of: _Knot commensurability and the Berge conjecture_, Geom. Topol. 16 (2012) 625-664 ([arXiv:1008.1034](https://arxiv.org/abs/1008.1034))

On the [[3d-3d correspondence]] for lens spaces:

* [[Du Pei]], Chapter III of: _3d-3d correspondence for Seifert manifolds_, 2016 ([spire:1469350](https://inspirehep.net/literature/1469350), [pdf](https://inspirehep.net/files/d509ff9e32448da3a5674f286b93224a))

On classification of lens spaces:

* {#Olum53} [[Paul Olum]], _Mappings of manifolds and the notion of degree_, Ann. of Math. (2) *58* (1953), p. 458–480. [JSTOR:1969748](https://www.jstor.org/stable/1969748)

* {#Brody60} [[E. J. Brody]], _The topological classification of the lens spaces_, Ann. of Math. (2) 71 (1960), 163–184. MR0116336 (22 #7125) Zbl 0119.18901

* {#Milnor66} [[John Milnor]], *Whitehead torsion*, Bull. Amer. Math. Soc. **72** (1966) 358–426 [[doi:10:1090/S0002-9904-1966-11484-2](https://www.ams.org/journals/bull/1966-72-03/S0002-9904-1966-11484-2), [maths.ed.ac.uk](https://web.archive.org/web/20160529051526/http://www.maths.ed.ac.uk/~s1122173/surgerygroup12/milnorwh.pdf), [jstor:bams/1183527946](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-72/issue-3/Whitehead-torsion/bams/1183527946.full)]

[[!redirects lens spaces]]


Leo Corry

* [webpage](http://www.tau.ac.il/~corry/)

## Selected writings

On the history of the notion of *[[mathematical structures]]* such as attributed to [[Bourbaki]]:

* [[Leo Corry]], *Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics*, in: *Changing Images of Mathematics in History. From the French Revolution to the new Millenium* Harwood Academic Publishers (2001) 167-186 [[ISBN:9780415868273](https://www.routledge.com/Changing-Images-in-Mathematics-From-the-French-Revolution-to-the-New-Millennium/Bottazini-Dalmedico/p/book/9780415868273), [pdf](https://www.leocorry.com/_files/ugd/e6fef7_15c7077b08ec4f249fb40eb961256ced.pdf)]

* [[Leo Corry]], *Modern Algebra and the Rise of Mathematical Structures*, Springer (2004) [[doi:10.1007/978-3-0348-7917-0](https://doi.org/10.1007/978-3-0348-7917-0)]

On the history of notions of *[[numbers]]* ([[natural numbers]], [[rational numbers]], [[real numbers]], [[complex numbers]], ...):

* [[Leo Corry]], *A Brief History of Numbers*, Oxford University Press (2015) [[ISBN:9780198702597](https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597)]

category: people


Leo P. Kadanoff

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

## Selected writings

Introducing [[para-fermions]]:

* [[Eduardo Fradkin]], [[Leo P. Kadanoff]], *Disorder variables and para-fermions in two-dimensional statistical mechanics*, Nuclear Physics B **170** 1 (1980) 1-15 $[$doi:10.1016/0550-3213(80)90472-1$]$

category: people

[[!redirects Leo Kadanoff]]


Leo Pochhammer

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

## Selected writings

Introducing the notion of [[Pochhammer contours]]:

* {#Pochhammer1890} [[Leo Pochhammer]], *Zur Theorie der Euler'schen Integrale*, Mathematische Annalen **35** (1890) 495–526 $[$[doi:10.1007/BF02122658](https://doi.org/10.1007/BF02122658)$]$

category: people


Leon Balents

* [personal page](https://www.kitp.ucsb.edu/balents)

## Selected writings

On [[topological insulators]]:

* [[Joel E. Moore]], [[Leon Balents]], *Topological invariants of time-reversal-invariant band structures*, Phys. Rev. B **75** (2007) 121306(R) $[$[doi:10.1103/PhysRevB.75.121306](https://doi.org/10.1103/PhysRevB.75.121306)$]$

On identifying [[topological order]] via [[topological entanglement entropy]] of the ground state:

* Hong-Chen Jiang, [[Zhenghan Wang]], [[Leon Balents]], *Identifying Topological Order by Entanglement Entropy*, Nature Physics 8, 902-905 (2012) ([arXiv:1205.4289](https://arxiv.org/abs/1205.4289))

category: people


Leon Menger

* [personal page](https://leonmenger.de/)

## Selected writings

On [[BF-theory]] in the [[BV-formalism]] and relating to the [[first-order formulation of gravity|first-order formulation]] of [[Einstein gravity]]:

* [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]

category: people


Leon Takhtajan

[[!redirects Leon A. Takhtajan]]
[[!redirects L. A. Takhtajan]]
[[!redirects L. Takhtajan]]
[[!redirects Lev Tahtajan]]

__Leon A. Takhtajan__ is a [[mathematical physicist]] at SUNY Stony Brook.

* [[Quantum mechanics for mathematicians]], Graduate Studies in Mathematics __95__, Amer. Math. Soc. 2008.

* [webpage](http://www.math.sunysb.edu/~leontak)

## Selected writings

On relations between [[quantum field theory]] and [[number theory]] and in particular about the analogy between the [[Weil reciprocity law]] for [[function fields]] and the [[Ward identities]] of quantum field theory:

* [[Leon Takhtajan]], _Quantum field theories on algebraic curves and A. Weil reciprocity law_, [arxiv/0812.0169](http://arxiv.org/abs/0812.0169); _Quantum field theories on an algebraic curve_, [pdf](http://www.math.sunysb.edu/~leontak/Takhtajan-00.pdf), 2000

On [[solitons]]:

* [[Ludwig D. Faddeev]], [[Leon Takhtajan]], *Hamiltonian methods in the theory of solitons*, Springer (2007) [[doi:10.1007/978-3-540-69969-9](https://doi.org/10.1007/978-3-540-69969-9)]

category: people


Leonard Dickson

Leonard Eugene Dickson (1874-1954) was an American mathematician best known for his work in [[algebra]] and [[number theory]].

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

## Selected writings

On [[finite group]] [[group theory|theory]]:

* {#MillerBlichfeldtDickson16} G. A. Miller, H. F. Blichfeldt, [[Leonard E. Dickson]], _Theory and applications of finite groups_, Dover, New York, 1916 ([ark:/13960/t0ht2kb4x](https://archive.org/details/theoryapplicatio00milluoft/))

On [[octonions]] and introducing the [[Cayley-Dickson construction]]:

* [[Leonard Dickson]], _On Quaternions and Their Generalization and the History of the Eight Square Theorem_,
Annals of Mathematics Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 ([jstor:1967865](https://www.jstor.org/stable/1967865))

## Related $n$Lab entries

* [[Cayley-Dickson construction]]

* [[Dickson invariant]]

* [[finite rotation group]]

category: people

[[!redirects Leonard Eugene Dickson]]

[[!redirects Leonard E. Dickson]]


Leonard Evens

* [webpage](https://www.math.northwestern.edu/people/emeriti/leonard-evens.html)

* [Mathematical genealogy page](http://www.genealogy.ams.org/id.php?id=12708)

## selected writings

* {#Evens63} [[Leonard Evens]], _A Generalization of the Transfer Map in the Cohomology of Groups_, Transactions of the American Mathematical Society Vol. 108, No. 1 (Jul., 1963), pp. 54-65 ([doi:10.1090/S0002-9947-1963-0153725-1](https://doi.org/10.1090/S0002-9947-1963-0153725-1), [jstor:1993825](https://www.jstor.org/stable/1993825))

* {#Evens65} [[Leonard Evens]], _On the Chern Classes of Representations of Finite Groups_, Transactions of the American Mathematical Society, Vol. 115 (Mar., 1965), pp. 180-193 ([doi:10.2307/1994264](https://www.jstor.org/stable/1994264))

  (on [[Chern classes of linear representations]])

## Related $n$Lab entries

* [[characteristic class of a linear representation]]

category: people


Leonard S. Charlap

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=9694)

## Selected writings

On [[Bieberbach's theorems]] about [[crystallographic groups]]:

* {#Charlap86} [[Leonard S. Charlap]], *Bieberbach Groups and Flat Manifolds*, Springer (1986) $[$[doi:10.1007/978-1-4613-8687-2](https://doi.org/10.1007/978-1-4613-8687-2)$]$

category: people

[[!redirects Leonard Charlap]]


Leonard Susskind

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

## Selected writings

Early suggestion that [[confinement|confined]] [[QCD]] is described by regarding the [[color charge|color]]-[[flux tubes]] as [[string]]-like dynamical degrees of freedoms (cf. [gauge/string duality](AdS-QCD+correspondence#PolyakovGaugeStringDualityReferences)):

* [[John Kogut]], [[Leonard Susskind]], *Vacuum polarization and the absence of free quarks in four dimensions*, Phys. Rev. D **9** (1974) 3501-3512 [[doi:10.1103/PhysRevD.9.3501](https://doi.org/10.1103/PhysRevD.9.3501)]

* [[John Kogut]], [[Leonard Susskind]], *Hamiltonian formulation of Wilson's lattice gauge theories*, Phys. Rev. D **11** (1975) 395 [[doi:10.1103/PhysRevD.11.395](https://doi.org/10.1103/PhysRevD.11.395)]

  > "The gauge-invariant configuration space consists of a collection of strings with quarks at their ends. The strings are lines of non-Abelian electric flux. In the strong coupling limit the dynamics is best described in terms of these strings. Quark confinement is a result of the inability to break a string without producing a pair. [...]"

  > "The confining mechanism is the appearance of one dimensional electric flux tubes which must link separated quarks. The appropriate description of the strongly coupled limit consists of a theory of interacting, propagating strings. [...]"

  > "This picture of the strongly coupled Yang-Mills theory in terms of a collection of stringlike flux lines is the central result of our analysis. It should be compared with the phenomenological use of stringlike degrees of freedom which has been widely used in describing hadrons."

Introducing the [[holographic principle]]:

* {#Susskind94} [[Leonard Susskind]], *The World as a hologram*, J. Math. Phys. **36** (1995) 6377-6396 [[hep-th/9409089](http://arxiv.org/abs/hep-th/9409089), [doi:10.1063/1.531249](https://doi.org/10.1063/1.531249)]

Introducing the [[BFSS matrix model]] ([[M-theory]] as [[D0-brane]] [[quantum mechanics]] in [[non-perturbative effect|non-perturbative]] [[type IIA string theory]]):

* {#BanksFischlerShenkerSusskind96} [[Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_  Phys. Rev. D55 (1997). ([arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043))

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

In view of [[flat space holography]]:

* [[Leonard Susskind]], *Holography in the flat space limit*, AIP Conf.Proc. 493 (1999) 1, 98-112,  ([spire](https://inspirehep.net/literature/482388), [arXiv:hep-th/9901079](https://arxiv.org/abs/hep-th/9901079), [doi:10.1063/1.1301570](https://doi.org/10.1063/1.1301570))

On [[black holes in string theory]] via the [[BFSS matrix model]]:

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]],  _Schwarzschild Black Holes from Matrix Theory_, Phys.Rev.Lett.80:226-229,1998 ([arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091))

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008,1998 ([arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005))

* [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzschild Black Holes in Various Dimensions from Matrix Theory_, Phys.Lett.B **416** 62-66 (1998) [[arXiv:hep-th/9709108](https://arxiv.org/abs/hep-th/9709108)]

Introducing the notion of [[giant gravitons]]:

* [[John McGreevy]], [[Leonard Susskind]], [[Nicolaos Toumbas]], _Invasion of the Giant Gravitons from Anti-de Sitter Space_, JHEP 0006:008, 2000 ([arXiv:hep-th/0003075](https://arxiv.org/abs/hep-th/0003075))

Proposing [[ER = EPR]]:

 * {#MaldacenaSusskind13} [[Juan Maldacena]], [[Leonard Susskind]], _Cool horizons for entangled black holes_, Fortsch. Phys. 61 (2013) 781-811, 2013 ([arXiv:1306.0533](http://arxiv.org/abs/1306.0533), [doi:10.1002/prop.201300020](https://doi.org/10.1002/prop.201300020))

## Interviews

* Y Combinator: *Leonard Susskind on Richard Feynman, the Holographic Principle, and Unanswered Questions in Physics* (Dec 2018) [[YT](https://youtu.be/CQAcLW6qdQY)]

  $\,$

  [49:00](https://www.youtube.com/watch?v=CQAcLW6qdQY&feature=youtu.be&t=49m)

  S:... rules of quantum mechanics which reflect the gravitational things..., but this is gonna get us into technical discussion

  Y: let's do it!

  S: No!

  Y: Yes!

  S grudgingly gives in

  ...

  [53:57](https://www.youtube.com/watch?v=CQAcLW6qdQY&t=3237s)

  Y: and what was the technical part you wanted to get to?

  S: uhm

* Kristiane Bernhard-Novotny: *[Lost in the landscape](https://cerncourier.com/a/lost-in-the-landscape/)*, interview with Susskind for CernCourier (3 Mar 2023)

category: people


Leonardo C. Mihalcea

* [personal page](https://personal.math.vt.edu/lmihalce/)

* [GoogleScholar page](https://scholar.google.com/citations?user=kJic2LsAAAAJ&hl=en)

## Selected writings

On [[quantum K-theory rings]]:

* [[Anders S. Buch]], [[Leonardo C. Mihalcea]], *Quantum K-theory of Grassmannians*, Duke Math. J. **156** 3 (2011) 501-538 [[arXiv:0810.0981](https://arxiv.org/abs/0810.0981), [doi:10.1215/00127094-2010-218](https://doi.org/10.1215/00127094-2010-218)]

* W. Gu, [[Leonardo C. Mihalcea]], [[Eric Sharpe]], H. Zou, *Quantum K theory of symplectic Grassmannians* [[arXiv:2008.04909](https://arxiv.org/abs/2008.04909)

On [[Pontryagin products]] not in [[ordinary homology]] but in [[topological K-theory|topological]] [[K-homology]]:

* [[Thomas Lam]], [[Changzheng Li]], [[Leonardo C. Mihalcea]], [[Mark Shimozono]], *A conjectural Peterson isomorphism in K-theory*, Journal of Algebra **513** (2018) 326-343 [[doi:10.1016/j.jalgebra.2018.07.029](https://doi.org/10.1016/j.jalgebra.2018.07.029), [arXiv:1705.03435](https://arxiv.org/abs/1705.03435)]

category: people

[[!redirects Leonardo Mihalcea]]


Leonardo Castellani

* [website]()

## Selected writings

On the [[D'Auria-Fré formulation of supergravity]]:

* {#CFGPN83} [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], *Gauging of $d = 11$ supergravity?*, Annals of Physics **146** 1 (1983) 35-77 [[spire:11998](https://inspirehep.net/literature/11998), doi:10.1016/0003-4916(83)90052-0]

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Leonardo Castellani]], *Supergravity in the group-geometric framework: a primer*, Fortschr. Phys. **66** 4 (2018) [[arXiv:1802.03407](https://arxiv.org/abs/1802.03407), [doi:10.1002/prop.201800014]( https://doi.org/10.1002/prop.201800014)]

* [[Leonardo Castellani]], *Group manifold approach to supergravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.04318](https://arxiv.org/abs/2211.04318)]

Discussion of the [[Hodge star operator]] on [[supermanifolds]] (in terms of [[picture changing operators]] and integral top-forms for [[integration over supermanifolds]]):

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _Hodge Dualities on Supermanifolds_, Nuclear Physics B Volume 899, October 2015, Pages 570-593 ([arXiv:1507.01421](https://arxiv.org/abs/1507.01421))

and generally in terms of [[Berezin integration]]:

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _The Hodge Operator Revisited_ ([arXiv:1511.05105](https://arxiv.org/abs/1511.05105))

On [[supersymmetric quantum mechanics]] in the perspective of [[supergeometry]] ([[integration over supermanifolds]], [[picture changing operators]]):

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Grassi]], _Super Quantum Mechanics in the Integral Form Formalism_, Ann. Henri Poincaré (2018) 19: 1385 ([arXiv:1706.04704](https://arxiv.org/abs/1706.04704))

On the work of [[Tullio Regge]] ([[Regge calculus]], [[D'Auria-Fre formulation of supergravity]], ...):

* [[Leonardo Castellani]],  [[Anna Ceresole]], [[Riccardo D'Auria]], [[Pietro Fré]] (eds.): _Tullio Regge: An Eclectic Genius_, World Scientific 2019 ([doi:10.1142/11643](https://doi.org/10.1142/11643))

category: people


Leonardo de Moura

* [webpage](http://leodemoura.github.io)

## related $n$Lab entries

* [[Lean]]

caegory: people


Leonardo Rastelli

* [webpage](http://www.physics.sunysb.edu/Physics/news/whatsnew_single.php?date=20130411)

## Selected writings

On [[multi-trace operators]] in the [[AdS/CFT correspondence]]:

* [[Eric D'Hoker]], [[Daniel Freedman]], [[Samir Mathur]], A. Matusis, [[Leonardo Rastelli]], _Extremal Correlators in the AdS/CFT Correspondence_, in: _[[The many faces of the superworld]]_ ([arXiv:hep-th/9908160](https://arxiv.org/abs/hep-th/9908160))

On [[open/closed string duality]]:

* [[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))

* [[Leonardo Rastelli]], _Lectures on open/closed string duality_, PiTP 2004 ([pdf](http://www.sns.ias.edu/pitp2/2004/IASrastelli.pdf), [[RastelliStringDuality.pdf:file]])

On [[double trace operators]] in [[AdS/CFT]]:

* Thomas Hartman, [[Leonardo Rastelli]], _Double-Trace Deformations, Mixed Boundary Conditions and Functional Determinants in AdS/CFT_, JHEP 0801:019, 2008 ([arXiv:hep-th/0602106](https://arxiv.org/abs/hep-th/0602106))

On [[conformal bootstrap]] for [[SCFT]]:

* [[Christopher Beem]], Madalena Lemos, Pedro Liendo, [[Leonardo Rastelli]], Balt C. van Rees, _The $N=2$ superconformal bootstrap_ ([arXiv:1412.7541](https://arxiv.org/abs/1412.7541))

* [[Christopher Beem]], Madalena Lemos, [[Leonardo Rastelli]], Balt C. van Rees, _The $(2,0)$ superconformal bootstrap_ ([arXiv:1507.05637](https://arxiv.org/abs/1507.05637))

On the [[Witten index]] of [[D=4 N=1 super Yang-Mills theory]]:

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

On using the [[conformal bootstrap]] to re-prove the solution of the [[sphere packing problem]] in [[dimensions]] 8 and 24, and interpretation of the result, via [[AdS/CFT]] in terms of the [[weak gravity conjecture]]:

* {#HMR19} [[Thomas Hartman]], [[Dalimil Mazáč]], [[Leonardo Rastelli]], _Sphere Packing and Quantum Gravity_ ([arXiv:1905.01319](https://arxiv.org/abs/1905.01319))

On [[mesons]] in the [[large N limit]] of [[QCD]]:

* 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)]

category: people


Leonhard Euler

Leonhard Euler was a major Swiss mathematician of 18th century, working most of his life in St. Petersburg, Russia and Berlin, Prussia. He prolifically worked in all areas of mathematics known to that period and introduced the first important theorems of topological content; he also worked on mathematical applications in physics (especially optics and hydrodynamics), astronomy and geodesy.

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

## Related entries

* [[Euler number]]

* [[Euler formula]]

* [[Euler product]]

* [[Euler characteristic]], [[Euler beta function]]

* [[Euler-Lagrange equation]]

category: people

[[!redirects Euler]]


Leonid Chekhov

* [webpage](http://www.mi.ras.ru/staff/chekhov_e.html)

## related $n$Lab entries

* [[topological recursion]]

* [[matrix model]]

category: people


Leonid Dickey

* [webpage](http://www.math.ou.edu/~ldickey/)

## related $n$Lab pages

* [[Dickey bracket]] on [[conserved currents]]

category: people


Leonid I. Glazman

* [webpage](https://physics.yale.edu/people/leonid-glazman)

## Selected writings

On [[quasiparticle]]-excitations among [superconductive qbits](superconductivity#SuperconductingQBitsReferences):

* [[Leonid I. Glazman]], [[Gianluigi Catelani]], *Bogoliubov Quasiparticles in Superconducting Qubits*, SciPost Phys. Lect. Notes **31** (2021) [[arXiv:2003.04366](https://arxiv.org/abs/2003.04366), [doi:10.21468/SciPostPhysLectNotes.31](https://doi.org/10.21468/SciPostPhysLectNotes.31)]

category: people

[[!redirects Leonid Glazman]]


Leonid Petrov

* [Institute page](https://math.virginia.edu/people/lap5r/)

## Selected writings

On [[random matrix theory]] (also [[Schur-Weyl measures]], ...):

* {#Petrov19} [[Leonid Petrov]], *Random Matrices*, lecture notes 2019 ([pdf slides](https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf), [[PetrovRandomMatrices.pdf:file]], [webpage](https://lpetrov.cc/2019/07/rmt-announce))

category: people


Leonid Positselski

* [webpage](http://iitp.ru/en/userpages/493/)

## related $n$Lab entries

* [[model structure on dg-comodules]]

category: people


Leonidas J. Guibas

* [institute page](https://geometry.stanford.edu/member/guibas/)

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

## Selected writings

On [[stability of persistence diagrams]]:

* [[Frédéric Chazal]], [[David Cohen-Steiner]], [[Marc Glisse]], [[Leonidas J. Guibas]], [[Steve Y. Oudot]], *Proximity of persistence modules and their diagrams*, SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 237–246 $[$[doi:10.1145/1542362.1542407](https://doi.org/10.1145/1542362.1542407)$]$

category: people

[[!redirects Leonidas Guibas]]


Leopold Infeld

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

## Selected writings

Introducing the [[DBI action]]:

* {#BornInfeld34} [[Max Born]], [[Leopold Infeld]], _Foundations of the New Field Theory_,  Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 144, No. 852 (Mar. 29, 1934), pp. 425-451 ([jstor:2935568](https://www.jstor.org/stable/2935568))

On [[gravity]]:

* [[Leopold Infeld]] (ed.), *Relativistic Theories of Gravitation*, Proceedings of a conference held in Warsaw and Jablonna 1962, Pergamon Press (1964) [[pdf](https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf)]

## Related entries

* [[Dirac-Born-Infeld action]]

* [[black brane]]

category: people


Leopold Kronecker

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

## related $n$Lab entries

* [[Grothendieck group]]

* [[Kronecker delta]]

* [[Kronecker pairing]]

category: people

[[!redirects Kronecker]]


Leopold Vietoris

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

* [English language Wikipedia entry](http://en.wikipedia.org/wiki/Leopold_Vietoris)

category: people


Leopold Zoller

* [personal page](https://www.mathematik.uni-marburg.de/~zoller/)

* [institute page](https://www.mathematik.uni-muenchen.de/~zoller/)

## Selected writings

On [[equivariant de Rham cohomology]]:

* [[Oliver Goertsches]], [[Leopold Zoller]], _Equivariant de Rham Cohomology: Theory and Applications_, São Paulo J. Math. Sci. (2019) ([arXiv:1812.09511](https://arxiv.org/abs/1812.09511), [doi:10.1007/s40863-019-00129-4](https://doi.org/10.1007/s40863-019-00129-4))

category: people


Leopoldo Nachbin

[[!redirects Leopold Nachbin]]

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

## related $n$Lab entries

* [[closed-projection characterization of compactness]]

category: people


Lepage form

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Lepage forms are certain [[differential forms]] on [[jet bundles]]. When of maximal horizontal degree then they may be identified with sums $\mathbf{L} + \theta$ of a [[local Lagrangian]] $\mathbf{L}$ with its induced presymplectic potential current $\theta$ (as discussed at [[covariant phase space]]). Lepage _equivalents_ associated with a local Lagrangian in particular allow to get the [[Euler-Lagrange form]] (which vanishes along the pullback of any section that satisfies the [[Euler-Lagrange equation]]) from its de Rham differential, thus avoiding the need to use the [[Euler operator]].

Lepage forms play a central role in the theory of [[variational sequences]].

## Definition

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

Given a [[smooth function|smooth]] [[bundle]] $E \to \Sigma$, then a [[differential n-form]] $\rho$ on its [[jet bundle]] $Jet(E)$ is called a _Lepage form_ if for every [[vector field]] $v$ on the jet bundle, the horizontal part $h(\iota_v \mathbf{d}\rho)$ (with respect to the [[variational bicomplex]]) of the contraction of $v$ into the [[de Rham differential]] of $\rho$ depends only on the projection of $v$ to a vector field on $E$ itself.

=--

e.g. [GMS 09, 2.1.2](#GMS09)

## In variational calculus

Let $\Sigma$ be of [[dimension]] $n$. Then a horizontal $n$-form $\mathbf{L} \in \Omega^{n,0}(Jet(E))$, may be regarded a [[local Lagrangian]]. Its de Rham differential has a unique decomposition

$$
  d \mathbf{L} = \mathbf{E} - d_H \theta
$$

where $E$ is a [[source form]], the [[Euler-Lagrange form]] of $\mathbf{L}$, and $\theta$, the induced presymplectic potential current, is defined up to addition of a horizontally exact form (e.g. [Zuckerman 87, section 2](#Zuckerman87)).

Then the combination

$$
  \rho \coloneqq \mathbf{L} + \theta
$$

is a Lepage form, and in fact such a form being Lepage is equivalent to $d \mathbf{L} + d_h \theta$ being a [[source form]].

The differential

$$
  d \rho = \mathbf{E} + \omega
$$

is the sum of the [[Euler-Lagrange form]] and the presymplectic current density $\omega = d_V \theta$.

To obtain the [[Euler-Lagrange form]] one needs to apply an operator, called the [[Euler operator]] to $\mathbf{L}$. In contrast, the same Euler-Lagrange form may be obtained from $\mathbf{L}+\theta$ by applying the usual de Rham operator. This serves as a motivation to replace the Lagrangian $\mathbf{L}$ with the form $\mathbf{L}+\theta$. Consistent replacements are known as _Lepage equivalents_.

###### Definition

Let $E \to \Sigma$ be a [[smooth function|smooth]] [[bundle]] with $\Sigma$ of [[dimension]] $n$. Given a horizontal $n$-form $\mathbf{L} \in \Omega^{n,0}(Jet(E))$, regarded as a [[local Lagrangian]] , a _Lepage equivalent_ $\rho_L$ is a $n$-form $\rho_L\in\Omega^n(Jet(E))$ (not necessarily horizontal) such that *a.* The horizontal component of $\rho_L$ is $mathbf{L}$, and *b.* The (n,1)-component of $d \rho_L$ is the [[Euler-Lagrange form]] $E$ of $\mathbf{L}$.

Lepage equivalents are rarely unique, and many times fail to be globally-defined even if the Lagrangian itself is globally defined. A notable exception is [[classical mechanics]] with $F$ a finite-dimensional smooth manifold and order-1 Lagrangian (meaning it depends on at most first-order derivatives fields), for which there is a unique Lepage equivalent, called the _Poincaré-Cartan form_.

Note that the definition of Lepage equivalent only involves restrictions on the horizontal and 1-vertical components. This means that given a Lepage equivalent $\rho_L$, one can shift by forms of bidegree $(n-k,k)$ for $k\geq 2$ and still obtain a Lepage equivalent even though this affects the resulting pre-$n$-plectic form. One way towards addressing this problem is to demand a further condition called _closure property_ , which removes the contributions from trivial Lagrangians.

###### Definition

A Lepage equivalent $\rho_L$ is said to satisfy the _closure property_ if whenever the [[Euler-Lagrange form]] $E$ of the Lagrangian $\mathbf{L}$ is _identically_ zero, then $d\rho_L=0$ vanishes identically too.

## Related concepts

* [[variational sequence]]

* [[multisymplectic geometry]]

* [[n-plectic geometry]]

## References

Lepage was a student of [[Théophile de Donder]] and did his work on the [[calculus of variations]] around the 1930s. Only much later, [[Demeter Krupka]] was responsible for naming Lepage forms around the 1970s.

See for instance

* {#GMS09} Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, _Advanced classical field theory_, World Scientific, 2009

* {#Zuckerman87} [[Gregg Zuckerman|G. J. Zuckerman]], _Action principles and global geometry_ , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259-284. ([[ZuckermanVariation.pdf:file]])

Exposition of [[variational calculus]] in terms of [[jet bundles]] and Lepage forms and aimed at examples from [[physics]] is in

* {#MusilovaHronek16} [[Jana Musilová]],  [[Stanislav Hronek]], _The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories_, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) ([doi.org/10.1515/cm-2016-0012]( https://doi.org/10.1515/cm-2016-0012))

* [[Demeter Krupka]]. _Introduction to global variational geometry_. Vol. 1. Amsterdam: Atlantis Press, (2015). ([doi:10.2991/978-94-6239-073-7] (https://doi.org/10.2991/978-94-6239-073-7}))

[[!redirects Lepage forms]]


leptogenesis

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[baryogenesis]]

* [[leptogenesis]]

## References

See also

* Wikipedia, _[Leptogenesis](https://en.wikipedia.org/wiki/Leptogenesis_(physics%29)_

Discussion in $Spin(10)$- ("$SO(10)$"-) [[grand unified theory]]:

* Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi, _Leptogenesis in $SO(10)$_ ([arXiv:1412.4776](https://arxiv.org/abs/1412.4776))


lepton

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]], a _lepton_ is a [[fermion]] that couples to the [[electromagnetic field]] or the [[weak nuclear force]] nut _not_ to the [[strong nuclear force]]. (Otherwise one speaks of _[[hadrons]]_.)

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

## Properties

* [[anomalous magnetic moment]]

* [[baryon-lepton symmetry]]

## Related concepts

* [[dilepton]]

* [[hadron]]

* [[lepton current]]

* [[leptonic decay]]

## References

* {#HalzenMartin84} [[Francis Halzen]], [[Alan Martin]], _Quarks and Leptons: An Introductory Course in Modern Particle Physics_, Wiley 1984 ([pdf](http://ajbell.web.cern.ch/ajbell/Documents/eBooks/Quarks%20&%20Leptons.pdf))

See also

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

[[!redirects leptons]]


lepton current

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

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum electrodynamics]], specifically when viewed in context with [[quantum chromodynamics]]/[[quantum hadrodynamics]], the _lepton current_ refers to the [[current]] of [[leptons]]. If $\psi$ denotes the given [[lepton]] [[spinor field]], this is

$$
  J^\mu
  \;\coloneqq\;
  \overline{\psi} \!\cdot\! \gamma^\mu \!\cdot\! \psi
  \,.
$$

This is in contrast, for instance, to the _[[hadron current]]_, given by the analogous expression, but with [[quarks]] instead of [[leptons]].

## Related concepts

* [[hadron current]]

* [[Dalitz decay]]

[[!redirects lepton currents]]


leptonic decay

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[quantum hadrodynamics]], the [[decay]]-mode of a [[meson]] via the [[weak interaction]] (i.e. by exchange of a [[W-boson]] or [[Z-boson]]) into a [[pair]] of [[leptons]] is called (shown are [[Feynman diagrams]] of example decays of [[kaons]]):

* a **purely leptonic decay** or **fully leptonic decay** (see also: _[[vector meson dominance]]_) if, besides the [[dilepton]], no further [[particle]] is among the decay products:

\begin{imagefromfile}
    "file_name": "LeptonicDecay.jpg",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -10,
        "right": 10,
        "bottom": -10,
        "left": 20
    }
\end{imagefromfile}

* a **semileptonic decay** if, in addition to the [[dilepton]], another [[meson]] is produced (see also: _[[semileptonic Dalitz decay]]_):

\begin{imagefromfile}
    "file_name": "SemileptonicDecay.jpg",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -10,
        "right": 10,
        "bottom": -10,
        "left": 20
    }
\end{imagefromfile}

* a **radiative leptonic decay** if, in addition to the [[dilepton]], a [[photon]] results (see also _[[radiative Dalitz decay]]_):

All this is in contrast to

* **non-leptonic decays** where other [[mesons]] but no [[leptons]] are produced:

\begin{imagefromfile}
    "file_name": "NonLeptonicDecay.jpg",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -10,
        "right": 10,
        "bottom": -10,
        "left": 20
    }
\end{imagefromfile}

## Related concepts

* [[Dalitz decay]]

* [[flavour anomaly]]

* leptonic conversion decay: [Landsberg 85, (3.3)](#Landsberg85)

## References

### General

* {#Landsberg85} L. G. Landsberg, _Electromagnetic decays of light mesons_, Physics Reports Volume 128, Issue 6, November 1985, Pages 301-376 (doi:10.1016/0370-1573(85)90129-2)

* R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, F. Feruglio, R. Gatto, G. Nardulli, _Phenomenology of Heavy Meson Chiral Lagrangians_, Phys. Rept. 281:145-238, 1997 ([arXiv:hep-ph/9605342](https://arxiv.org/abs/hep-ph/9605342))

#### Purely leptonic decays

* Tome Anticic, _Purely leptonic decays of heavy flavor mesons_ 1997 ([spire:454066](https://inspirehep.net/literature/454066))

### Leptonic decays of pions

Discussion of [[leptonic decays]] of [[pions]]:

#### General

* Dinko Pocanic, Emil Frlez, Andries van der Schaaf, _Experimental study of rare charged pion decays_, Journal of Physics G: Nuclear and Particle Physics, Volume 41, Number 11 ([arXiv:1407.2865](https://arxiv.org/abs/1407.2865))

#### Purely leptonic decays

See at _[[vector meson dominance]]_.

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

#### Raditive leptonic decays

Discussion of [[radiative leptonic decays]] of the [[pion]] (see also at _[[Dalitz decay]]_)

* Rene Unterdorfer, Hannes Pichl, _On the Radiative Pion Decay_, Eur. Phys. J. C55:273-283, 2008 ([arXiv:0801.2482](https://arxiv.org/abs/0801.2482))

* V. Bolotov, _The radiative pion decays_ ([[BolotovRadiativePionDecay.pdf:file]])

### Leptonic decays of $\omega$-mesons

Discussion of [[leptonic decays]] of [[omega-mesons]]:

* I. N. Mishustin, L. M. Satarov, H. Stoecker, W. Greiner, _Dilepton production by bremsstrahlung of meson fields in nuclear collisions_, Phys. Rev. C57:2552-2558, 1998 ([arXiv:hep-ph/9709384](https://arxiv.org/abs/hep-ph/9709384))

* Anastasiya Kuzmenko et al. , _The Study of the Conversion Decays of Omega Meson to $\pi^0$ Meson and $e^+ e^-$ Pair with the CMD-3 Detector_, CERN Proceedings Archives Vol 1 (2017) ([doi:10.23727/CERN-Proceedings-2017-001.91]( http://dx.doi.org/10.23727/CERN-Proceedings-2017-001.91)

### Leptonic decays of B-mesons

Discussion of [[leptonic decays]] of [[B-mesons]]:

#### General

* [[Patrick Koppenburg]], [[Zdenek Dolezal]], [[Maria Smizanska]], _Rare decays of b hadrons_, Scholarpedia, 11(6):32643 ([doi:10.4249/scholarpedia.32643](http://dx.doi.org/10.4249/scholarpedia.32643))

* {#Brambilla14} Brambilla et al., Section 4.2 of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* Jochen Dingfelder, Thomas Mannel, _Leptonic and semileptonic decays of $B$ mesons_, Rev. Mod. Phys. 88 (2016) 3, 035008 ([spire:1488176](https://inspirehep.net/literature/1488176), [doi:10.1103/RevModPhys.88.035008](https://doi.org/10.1103/RevModPhys.88.035008))

* Giovanni Banelli, Robert Fleischer, Ruben Jaarsma, Gilberto Tetlalmatzi-Xolocotzi, _Decoding (Pseudo)-Scalar Operators in Leptonic and Semileptonic B Decays_, The European Physical Journal C volume 78 ([arXiv:1809.09051](https://arxiv.org/abs/1809.09051))

#### Fully leptonic decays

* M. Lattery, _Fully Leptonic Decays of B Mesons_, 1996 ([spire](https://inspirehep.net/literature/438641))

#### Semileptonic decays

* J. P. Lees et al. (BABAR Collaboration), _Measurement of the $B^+ \to \omega \ell^+ \nu$ branching fraction with semileptonically tagged B mesons_, Phys. Rev. D 88, 072006 (2013) ([doi:10.1103/PhysRevD.88.072006](https://doi.org/10.1103/PhysRevD.88.072006))

* Xian-Wei Kang, Tao Luo, Yi Zhang, Ling-Yun Dai, Chao Wang, _Semileptonic $B$ and $B_s$ decays involving scalar and axial-vector mesons_, Eur. Phys. J. C 78, 909 (2018) ([doi:10.1140/epjc/s10052-018-6385-9](https://doi.org/10.1140/epjc/s10052-018-6385-9))

### In relation to flavor anomalies

Discussion of [[leptonic decays]] in view of observed [[flavour anomalies]]:

* N. Penalva, E. Hernández, J. Nieves, _Hadron and lepton tensors in semileptonic decays including new physics_ ([arXiv:2004.08253](https://arxiv.org/abs/2004.08253))

* N. Penalva, E. Hernández, J. Nieves, _$\bar B_c \to \eta_c$, $\bar B_c \to J/\psi$ and $\bar B \to D^{(\ast)}$ semileptonic decays including new physics_ ([arXiv:2007.12590](https://arxiv.org/abs/2007.12590))

[[!redirects leptonic decays]]

[[!redirects fully leptonic decay]]
[[!redirects fully leptonic decays]]

[[!redirects purely leptonic decay]]
[[!redirects purely leptonic decays]]

[[!redirects semileptonic decay]]
[[!redirects semileptonic decays]]

[[!redirects radiative leptonic decay]]
[[!redirects radiative leptonic decays]]


leptoquark

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

#Contents#
* table of contents
{:toc}

## Idea

_Leptoquarks_ is the name given to hypothetical [[fundamental particles]] which combine the properties of both [[quarks]] and [[leptons]] from the
[[standard model of particle physics]].

They naturally arise in [[grand unified theories]] (e.g. [Murayama-Yanagida 92](#MurayamaYanagida92), also [AFG 17, III.A](#AFG17))

## Properties

### Experimental hints
 {#ExperimentalHint}

The existence of leptoquarks could potentially explain _both_

1. the apparently observed [[flavour anomalies]]

   ([Bauer-Neubert 16](#BauerNeubert16), [CCDM 16](#CCDM16), [CCL 17](#CCL17), [Crivellin 17](#Crivellin17) [Falkowski17](#Falkowski17), [Müeller 18](#Mueller18), [Matsuzaki-Nishiwaki 18](#MatsuzakiNishiwaki18), [Monteux-Rajaraman 18](#MonteuxRajaraman18), [AMST 18](#AMST18), [BDFKFS 18](#BDFKFS18), [Crivellin 18](#Crivellin18), [MMR 18](#MMR18), [Blanke-Crivellin 18](#BlankeCrivellin18), [Kumar-London 19, section 2.2.1](#KumarLondon19), [MVT 19](#MVT19), [MVK 19](#MVK19), [Cata-Mannel 19](#CataMannel19))

1. the apparently observed [anomaly](anomalous+magnetic+moment#Anomalies) in the [[anomalous magnetic moments]] of the [[muon]] and also of the [[electron]] ([CCDM 16](#CCDM16), [Müller 18](#Mueller18))

From [Crivellin 18, p. 2](#Crivellin18):

> the global fit $[$ to [[flavour anomalies]] $]$ even shows compelling evidence for New Physics $[$... $]$ The vector leptoquark (LQ) $SU(2)_L$ singlet with hypercharge $-4/3$ arising in the famous [[Pati-Salam model]] is capable of explaining all the $[$[[flavour anomalies|flavour]] $]$ anomalies and therefore several attempts to construct a [[UV completion]] for this LQ to address the anomalies have been made. It can give a sizeable effect in $b \to c(u)\tau \nu$ data without violating bounds from $b \to s(d)\nu \bar \nu$ and/or direct searches, provides (at [[tree level]]) a $C_9 = - C_{10}$ solution to $b \to s \ell^+ \ell^-$ data and does not lead to [[proton decay]] at any order in [[perturbative quantum field theory|perturbation theory]].

### Experimental constraints

Constraints on the possible parameter space for leptoquarks from [[LHC]] data ([BGGSS 18](#BGGSS18), [BSS 18](#BSS18), [ATLAS 19](#ATLAS19))

### Theoretical conceptualization

Leptoquarks naturally arise in [[grand unified theories]] (..., [Murayama-Yanagida 92](#MurayamaYanagida92), [BDFKFS 18](#BDFKFS18))

## References

### General

* {#AFG17} Nima Assad, Bartosz Fornal, Benjamin Grinstein, _Baryon Number and Lepton Universality Violation in Leptoquark and Diquark Models_, Phys. Lett. B777 (2018) 324-331 ([arXiv:1708.06350](https://arxiv.org/abs/1708.06350))

* Achintya Rao, _[The hunt for leptoquarks is on](https://home.cern/news/news/physics/hunt-leptoquarks)_, 2018

Leptoquark [[effective field theory]]:

* Valerio Gherardi, David Marzocca, Elena Venturini, _Matching scalar leptoquarks to the SMEFT at one loop_ ([arXiv:2003.12525](https://arxiv.org/abs/2003.12525))

* [[Andreas Crivellin]], Christoph Greub, Dario Müller, Francesco Saturnino, _Scalar Leptoquarks in Leptonic Processes_ ([arXiv:2010.06593](https://arxiv.org/abs/2010.06593))

See also

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

### Experiment
 {#ReferencesExperiment}

Constraints from [[experiment]] ([[LHC]]):

* I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik, N. Košnik, _Physics of leptoquarks in precision experiments and at particle colliders_, Physics Reports Volume 641, 17 June 2016, Pages 1-68 ([arXiv:1603.04993](https://arxiv.org/abs/1603.04993))

* M. Tanabashi et al. (Particle Data Group), _115. Leptoquarks_  Phys. Rev. D 98, 030001 (2018) ([pdf](http://pdg.lbl.gov/2018/reviews/rpp2018-rev-leptoquark-quantum-numbers.pdf))

* CMS Collaboration, _Search for third-generation scalar leptoquarks decaying to a top quark and a $\tau$ lepton at $\sqrt{s} = 13$ TeV_, Eur. Phys. J. C 78 (2018) 707 ([arXiv:1803.02864](https://arxiv.org/abs/1803.02864))

* Andrei Angelescu, Damir Bečirević, Darius A. Faroughy, Olcyr Sumensari, _Closing the window on single leptoquark solutions to the $B$-physics anomalies_, JHEP10(2018)183 ([arXiv:1808.08179](https://arxiv.org/abs/1808.08179))

* {#BGGSS18} Aritra Biswas, Dilip Kumar Ghosh, Nivedita Ghosh, Avirup Shaw, Abhaya Kumar Swain, _Novel collider signature of $U_1$ Leptoquark and $B \to \pi$ observables_ ([arXiv:1808.04169](https://arxiv.org/abs/1808.04169))

* {#BSS18} Aritra Biswas, Avirup Shaw, Abhaya Kumar Swain, _Collider signature of $V_2$ Leptoquark with $b \to s$ flavour observables_ ([arXiv:1811.08887](https://arxiv.org/abs/1811.08887))

* {#Rapoccio18} [[Salvatore Rappoccio]], section 6 of _The experimental status of direct searches for exotic physics beyond the standard model at the Large Hadron Collider_, Reviews in Physics 4 (2019) 100027 ([arXiv:1810.10579](https://arxiv.org/abs/1810.10579))

* {#ATLAS19} [[ATLAS collaboration]], _Searches for scalar leptoquarks and differential cross-section measurements in dilepton-dijet events in proton-proton collisions at a centre-of-mass energy of s√ = 13 TeV with the ATLAS experiment _ ([arXiv:1902.00377](https://arxiv.org/abs/1902.00377))

* ZEUS Collaboration, _Limits on contact interactions and leptoquarks at HERA_ ([arXiv:1902.03048](https://arxiv.org/abs/1902.03048))

Alternative detection channel:

* Priyotosh Bandyopadhyay, Saunak Dutta, Anirban Karan, _Investigating the Production of Leptoquarks by Means of Zeros of Amplitude at Photon Electron Collider_ ([arXiv:2003.11751](https://arxiv.org/abs/2003.11751))

Claim that  $U(1)$-leptoquark at 1.5 TeV could both explain the [[flavor anomalies]] as well as fit all current experimental constraints:

* Arvind Bhaskar, Diganta Das, Tanumoy Mandal, Subhadip Mitra, Cyrin Neeraj, *Precise LHC limits on the U1 leptoquark parameter space* ([arXiv:2110.07638](https://arxiv.org/abs/2110.07638))

Claim of an apparent $3\sigma$-excess of possible leptoquark decay products:

* CMS Collaboration, *The search for a third-generation leptoquark coupling to a τ lepton and a b quark through single, pair and nonresonant production at $\sqrt{s} = 13 TeV$* (2022) [[cds:2815309](https://cds.cern.ch/record/2815309)]

  > Measurements of [[[B meson]] decay rates] collectively deviate from the SM predictions by about four standard deviations. New physics with larger couplings to the third-generation quarks and leptons could explain this anomaly. Leptoquark (LQ) models have recently received  onsiderable theoretical interest as a possible explanation.

  > $[\cdots]$

  > For a representative LQ mass of 2 TeV and a coupling strength of $2.5$, an excess with a significance of 3.4 standard deviations above the standard model expectation is observed in the data.

### Relation to flavour anomalies
 {#RelationToFloavourAnomalies}

As a potential solution of the apparent [[B meson]] [[flavour anomaly]]:

* {#BauerNeubert16} Martin Bauer, [[Matthias Neubert]], _One Leptoquark to Rule Them All: A Minimal Explanation for $R_{D^{(\ast)}}$, $R_K$ and $(g-2)_\mu$_, Phys. Rev. Lett. 116, 141802 (2016) ([arXiv:1511.01900](https://arxiv.org/abs/1511.01900))

* {#CCDM16} Estefania Coluccio Leskow, [[Andreas Crivellin]], Giancarlo D'Ambrosio, Dario Müller, _$(g-2)_\mu$, Lepton Flavour Violation and Z Decays with Leptoquarks: Correlations and Future Prospects_, Phys. Rev. D 95, 055018 (2017) ([arXiv:1612.06858](https://arxiv.org/abs/1612.06858))

* Yi Cai, John Gargalionis, Michael A. Schmidt, Raymond R. Volkas, _Reconsidering the One Leptoquark solution: flavor anomalies and neutrino mass_ ([arXiv:1704.05849](https://arxiv.org/abs/1704.05849))

* {#CCL17} Lorenzo Calibbi, [[Andreas Crivellin]], Tianjun Li, _A model of vector leptoquarks in view of the $B$-physics anomalies_, Phys. Rev. D 98, 115002 (2018) ([arXiv:1709.00692](https://arxiv.org/abs/1709.00692))

* {#Crivellin17} [[Andreas Crivellin]], _New Physics in Flavour Observables_ ([arXiv:1706.00929](https://arxiv.org/abs/1706.00929))

* {#Falkowski17} [[Adam Falkowski]], _[Leptoquarks strike back](http://resonaances.blogspot.com/2015/11/leptoquarks-strike-back.html)_, November 2017

* {#Mueller18} Dario Müller, _Leptoquarks in Flavour Physics_, EPJ Web of Conferences 179, 01015 (2018) ([arXiv:1801.03380](https://arxiv.org/abs/1801.03380))

* {#MatsuzakiNishiwaki18} Shinya Matsuzaki, Kenji Nishiwaki, Kei Yamamoto, _Simultaneous interpretation of K and B anomalies in terms of chiral-flavorful vectors_ ([arXiv:1806.02312](https://arxiv.org/abs/1806.02312))

* {#MonteuxRajaraman18} Angelo Monteux, [[Arvind Rajaraman]], _B Anomalies and Leptoquarks at the LHC: Beyond the Lepton-Quark Final State_, Phys. Rev. D 98, 115032 (2018) ([arXiv:1803.05962](https://arxiv.org/abs/1803.05962))

* {#AMST18} Ufuk Aydemir, Djordje Minic, Chen Sun, Tatsu Takeuchi, _$B$-decay anomalies and scalar leptoquarks in unified Pati-Salam models from noncommutative geometry_, JHEP 09 (2018) 117 ([arXiv:1804.05844](https://arxiv.org/abs/1804.05844))

* {#BDFKFS18} Damir Bečirević, Ilja Doršner, Svjetlana Fajfer, Nejc Košnik, Darius A. Faroughy, Olcyr Sumensari, _Scalar leptoquarks from GUT to accommodate the $B$-physics anomalies_, Phys. Rev. D 98, 055003 (2018) ([arXiv:1806.05689](https://arxiv.org/abs/1806.05689))

* {#MMR18} Tanumoy Mandal, Subhadip Mitra, Swapnil Raz, _$R_{D^{(\ast)}}$ in minimal leptoquark scenarios: impact of interference on the exclusion limits from LHC data_ ([arXiv:1811.03561](https://arxiv.org/abs/1811.03561))

* Jason Aebischer, [[Andreas Crivellin]], Christoph Greub, _QCD Improved Matching for Semi-Leptonic B Decays with Leptoquarks_ ([arXiv:1811.08907](https://arxiv.org/abs/1811.08907))

* {#KumarLondon19} Jacky Kumar, David London, _New physics in $b \to s e^+ e^-$?_ ([arXiv:1901.04516](https://arxiv.org/abs/1901.04516))

* {#MVT19} Ivo de Medeiros Varzielas, Jim Talbert, _Simplified Models of Flavourful Leptoquarks_ ([arXiv:1901.10484](https://arxiv.org/abs/1901.10484))

* Michael J. Baker, [[Javier Fuentes-Martin]], [[Gino Isidori]], [[Matthias König]], _High-pT Signatures in Vector-Leptoquark Models_ ([arXiv:1901.10480](https://arxiv.org/abs/1901.10480))

* Natascia Vignaroli, _Seeking leptoquarks in the $t \bar t$ plus missing energy channel at the high-luminosity LHC_ ([arXiv:1808.10309](https://arxiv.org/abs/1808.10309))

* {#AMM19} Ufuk Aydemir, Tanumoy Mandal, Subhadip Mitra, _A single TeV-scale scalar leptoquark in SO(10) grand unification and B-decay anomalies_ ([arXiv:1902.08108](https://arxiv.org/abs/1902.08108))

* {#MVK19} Ivo de Medeiros Varzielas, Stephen F. King, _Origin of Yukawa couplings for Higgs and leptoquarks_ ([arXiv:1902.09266](https://arxiv.org/abs/1902.09266))

* {#CataMannel19} Oscar Cata, Thomas Mannel, _Linking lepton number violation with $B$ anomalies_ ([arXiv:1903.01799](https://arxiv.org/abs/1903.01799))

* Bhubanjyoti Bhattacharya, Alakabha Datta, Saeed Kamali, David London, _CP Violation in $\bar B^0 \to D^{\ast +} \mu^- \bar \nu_\mu$_, ([arXiv:1903.02567](https://arxiv.org/abs/1903.02567))

* Han Yan, Ya-Dong Yang, Xing-Bo Yuan, _Phenomenology of $b \to c \tau \bar \nu$ decays in a scalar leptoquark model_ ([arXiv:1905.01795](https://arxiv.org/abs/1905.01795))

* Alakabha Datta, Divya Sachdeva, John Waite, _A unified explanation of $b \to s\mu^= \mu^-$ anomalies, neutrino masses and $B \to \pi K$ puzzle_ ([arXiv:1905.04046](https://arxiv.org/abs/1905.04046))

* Andrei Angelescu, _Single Leptoquark Solutions to the $B$-physics Anomalies_, contribution to the [2019 EW session](http://moriond.in2p3.fr/2019/EW/) of the [54th Rencontres de Moriond](http://moriond.in2p3.fr/2019/) ([arXiv:1905.06044](https://arxiv.org/abs/1905.06044))

* {#CrivellinSaturnino19} [[Andreas Crivellin]], Francesco Saturnino, _Explaining the Flavor Anomalies with a Vector Leptoquark (Moriond 2019 update)_ ([arXiv:1906.01222](https://arxiv.org/abs/1906.01222))

* {#BGV19} Innes Bigaran, John Gargalionis, Raymond R. Volkas, _A near-minimal leptoquark model for reconciling flavour anomalies and generating radiative neutrino masses_ ([arXiv:1906.01870](https://arxiv.org/abs/1906.01870))

* [[Javier Fuentes-Martin]], [[Gino Isidori]], [[Matthias König]], Nudzeim Selimovic, _Vector Leptoquarks Beyond Tree Level_ ([arxiv:1910.13474](https://arxiv.org/abs/1910.13474))

* {#HLN21} [[Gudrun Hiller]], Dennis Loose, Ivan Nišandžić, _Flavorful leptoquarks at the LHC and beyond: Spin 1_ ([arXiv:2103.12724](https://arxiv.org/abs/2103.12724))

* J. Kriewald, C. Hati, J. Orloff, A. M. Teixeira, _A combined explanation of the B-meson decay anomalies with a single vector leptoquark_ ([arXiv:2012.06315](https://arxiv.org/abs/2012.06315))

* Andrei Angelescu, Damir Bečirević, Darius A. Faroughy, Florentin Jaffredo, Olcyr Sumensari, _On the single leptoquark solutions to the B-physics anomalies_ ([arXiv:2103.12504](https://arxiv.org/abs/2103.12504))

* Jason Aebischer, [[Gino Isidori]], Marko Pesut, Ben A. Stefanek, Felix Wilsch, *Confronting the vector leptoquark hypothesis with new low- and high-energy data* [[arXiv:2210.13422](https://arxiv.org/abs/2210.13422)]

within [[GUT]]-models and specifically within [[Pati-Salam models]]:

* {#HeekTeresi18} Julian Heeck, Daniele Teresi, _Pati-Salam explanations of the B-meson anomalies_, JHEP 12 (2018) 103 ([arXiv:1808.07492](https://arxiv.org/abs/1808.07492))

* {#HeekTeresi19} Julian Heeck, Daniele Teresi, _Pati-Salam and lepton universality in B decays_ ([arXiv:1905.05211](https://arxiv.org/abs/1905.05211))

* Oleg Popov, Michael A. Schmidt, Graham White, _$R_2$ as a single leptoquark solution to $R_{D^{(\ast)}}$ and $R_{K^{(\ast)}}$_ ([arXiv:1905.06339](https://arxiv.org/abs/1905.06339))

* [[Andreas Crivellin]], Francesco Saturnino, _Correlating Tauonic B Decays to the Neutron EDM via a Scalar Leptoquark_ ([arXiv:1905.08257](https://arxiv.org/abs/1905.08257))

* {#Malinsky19} [[Michal Malinský]], _Lepton non-universality in $B$-decays in the minimal leptoquark gauge model_ ([arXiv:1906.09174](https://arxiv.org/abs/1906.09174))

* Jordan Bernigaud, Ivo de Medeiros Varzielas, Jim Talbert, _Finite Family Groups for Fermionic and Leptoquark Mixing Patterns_ ([arXiv:1906.11270](https://arxiv.org/abs/1906.11270))

* Junichiro Kawamura, Stuart Raby, Andreas Trautner, _Complete Vector-like Fourth Family and new $U(1)'$ for Muon Anomalies_ ([arXiv:1906.11297](https://arxiv.org/abs/1906.11297))

* Leandro Da Rold, Federico Lamagna, _A vector leptoquark for the B-physics anomalies from a composite GUT_ ([arXiv:1906.11666](https://arxiv.org/abs/1906.11666))

* C. Hati, J. Kriewald, J. Orloff, A.M. Teixeira, _A nonunitary interpretation for a single vector leptoquark combined explanation to the B-decay anomalies_ ([arXiv:1907.05511](https://arxiv.org/abs/1907.05511))

* Shyam Balaji, Michael A. Schmidt, _A unified $SU(4)$ theory for the $R_D^{(\ast)}$ and $R_K^{(\ast)}$ anomalies_ ([arXiv:1911.08873](https://arxiv.org/abs/1911.08873))

* Shaikh Saad, Anil Thapa, _A Common Origin of Neutrino Masses and $R_{D^\ast}$, $R_{K^\ast}$ Anomalies_ ([arXiv:2004.07880](https://arxiv.org/abs/2004.07880))

* P. S. Bhupal Dev, Rukmani Mohanta, Sudhanwa Patra, Suchismita Sahoo, _Unified explanation of flavor anomalies, radiative neutrino mass and ANITA anomalous events in a vector leptoquark model_ ([arXiv:2004.09464](https://arxiv.org/abs/2004.09464))

* Suman Kumbhakar, Rukmani Mohanta, _Investigating the effect of $U_1$ vector leptoquark on $b \to u \tau \bar\nu$ mediated B decays_ ([arXiv:2008.04016](https://arxiv.org/abs/2008.04016))

* Valerio Gherardi, David Marzocca, Elena Venturini, _Low-energy phenomenology of scalar leptoquarks at one-loop accuracy_ ([arXiv:2008.09548](https://arxiv.org/abs/2008.09548))

* [[Andreas Crivellin]], Dario Mueller, Francesco Saturnino, _Leptoquarks in Oblique Corrections and Higgs Signal Strength: Status and Prospects_ ([arXiv:2006.10758](https://arxiv.org/abs/2006.10758))

* Marco Ciuchini, Marco Fedele, Enrico Franco, Ayan Paul, Luca Silvestrini, Mauro Valli, _Lessons from the $B^{0,+} \to B^{\ast, 0,+} \mu^+ \mu^-$ angular analysis_ ([arXiv:2011.01212](https://arxiv.org/abs/2011.01212))

> In the Standard Model Effective Theory (SMEFT) [41, 42], the significance of NewPhysics from the global $b \to s \ell^+ \ell^-$ analysis increases with the inclusion of new data, reaching a maximum of almost 6σ for the simple hypothesis of a non-vanishing $[$[[leptoquark]] contribution$]$.

* [[Admir Greljo]], [[Peter Stangl]], Anders Eller Thomsen, _A Model of Muon Anomalies_ ([arXiv:2103.13991](https://arxiv.org/abs/2103.13991))

* [[Andreas Crivellin]], Luc Schnell, *Complete Set of Feynman Rules for Scalar Leptoquarks* ([arXiv:2105.04844](https://arxiv.org/abs/2105.04844))

* We-Fu Chang, *One colorful resolution to the neutrino mass generation, three lepton flavor universality anomalies, and the Cabibbo angle anomaly* ([arXiv:2105.06917](https://arxiv.org/abs/2105.06917))

* Mario Fernández Navarro, Stephen F. King, *$B$-anomalies in a twin Pati-Salam theory of flavour* [[arXiv:2209.00276](https://arxiv.org/abs/2209.00276)]

and [[intersecting D-brane model|on branes]]:

* [[Javier Fuentes-Martin]], [[Gino Isidori]], Julie Pagès, Ben A. Stefanek,
*Flavor Non-universal Pati-Salam Unification and Neutrino Masses*, Physics Letters B Volume 820, 10 September 2021, 136484 ([arXiv:2012.10492](https://arxiv.org/abs/2012.10492), [talk recording](https://www.youtube.com/watch?v=ko4R8S8XYV4&t=2s))

\linebreak

[[leptoquarks]] for explaining both the [[flavour anomalies]] and the [(g-2)-anomalies](anomalous+magnetic+moment#Anomalies):

* Hyun Min Lee, *Leptoquark Option for B-meson Anomalies and Leptonic Signatures* ([arXiv:2104.02982](https://arxiv.org/abs/2104.02982))

* Mingxuan Du, Jinhan Liang, Zuowei Liu, Van Que Tran, *A vector leptoquark interpretation of the muon $g-2$ and $B$ anomalies* ([arXiv:2104.05685](https://arxiv.org/abs/2104.05685))

* Kayoung Ban, Yongsoo Jho, Youngjoon Kwon, Seong Chan Park, Seokhee Park, Po-Yan Tseng, *A comprehensive study of vector leptoquark on the B-meson and Muon $g-2$ anomalies* ([arXiv:2104.06656](https://arxiv.org/abs/2104.06656))

* Alexandre Carvunis, [[Andreas Crivellin]], Diego Guadagnoli, Shireen Gangal, *The Forward-Backward Asymmetry in $B \to D^\ast \ell \nu$: One more hint for Scalar Leptoquarks?* ([arXiv:2106.09610](https://arxiv.org/abs/2106.09610))

* {#AliseEtAl2222} Alessandra D'Alise  et al.,  *Standard model anomalies: Lepton flavour non-universality, g-2 and W-mass* $[$[arXiv:2204.03686](https://arxiv.org/abs/2204.03686)$]$

\linebreak

A [[leptoquark]] model meant to address all of the [[flavour anomalies]], the [[(g-2)-anomaly]] and the [[Cabibbo anomaly]] at once:

* {#MarzoccaTrifinopoulos21} [[David Marzocca]], Sokratis Trifinopoulos, *A Minimal Explanation of Flavour Anomalies: B-Meson Decays, Muon Magnetic Moment, and the Cabbibo Angle* ([arXiv:2104.05730](https://arxiv.org/abs/2104.05730))

and within [[Randall-Sundrum models]]

* {#Crivellin18} [[Andreas Crivellin]], _Explaining the Flavour Anomalies with the Pati-Salam Vector Leptoquark_, PoS LHCP2018 (2018) 269 ([spire:1713260](http://inspirehep.net/record/1713260), [doi:10.22323/1.321.0269](https://doi.org/10.22323/1.321.0269))

* {#BlankeCrivellin18} Monika Blanke, [[Andreas Crivellin]], _$B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background_, Phys. Rev. Lett. 121, 011801 (2018) ([arXiv:1801.07256](https://arxiv.org/abs/1801.07256))

and as a possible explanation also of the anomalies seen at the [[ANITA experiment]]:

* Bhavesh Chauhan, Subhendra Mohanty, _A common leptoquark solution of flavor and ANITA anomalies_ ([arXiv:1812.00919](https://arxiv.org/abs/1812.00919))

### As dark matter candidates

Leptoquarks as candidates for [[dark matter]]:

* Francesco D'Eramo, Nejc Košnik, Federico Pobbe, Aleks Smolkovič, Olcyr Sumensari, _Leptoquarks and Real Singlets: A Richer Scalar Sector Behind the Origin of Dark Matter_ ([arXiv:2012.05743](https://arxiv.org/abs/2012.05743))

* Genevieve Bélanger et al. *Leptoquark Manoeuvres in the Dark: a simultaneous solution of the dark matter problem and the RD anomalies* ([arXiv:2111.08027](https://arxiv.org/abs/2111.08027))

### Appearance in GUTs

Appearance in [[GUT]]-theories:

* {#MurayamaYanagida92} H. Murayama, T. Yanagida, _A viable $SU(5)$ GUT with light leptoquark bosons_, Mod.Phys.Lett. A7 (1992) 147-152 ([spire:315898](http://inspirehep.net/record/315898), [doi:10.1142/S0217732392000070](https://doi.org/10.1142/S0217732392000070))

* [Heek-Teresi 18](#HeekTeresi18), [Heek-Teresi 19](#HeekTeresi19)

### In string phenomenology

Discussion of leptoquarks within [[string phenomenology]]:

* Ricardo Perez-Martinez, Saul Ramos-Sanchez, Patrick K.S. Vaudrevange , *The landscape of promising non-supersymmetric string models* ([arXiv:2105.03460](https://arxiv.org/abs/2105.03460))

[[!redirects leptoquarks]]


Leray spectral sequence

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#### Homological algebra
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[[!include homological algebra - contents]]
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#### Topos Theory
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[[!include topos theory - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

The _Leray spectral sequence_ is the special case of the [[Grothendieck spectral sequence]] for the case where the two [[functors]] being composed are a [[direct image|push-forward]] of [[sheaves of abelian groups]] along a [[continuous map]] $f : X \to Y$ between [[topological spaces]] or more generally the [[direct image]] of a [[morphism of sites]], followed by the push-forward $Y \to *$ to the [[point]] -- the [[global section]] functor. This yields a [[spectral sequence]] that computes the [[abelian sheaf cohomology]] on $X$ in terms of the abelian sheaf cohomology on $Y$.

## Properties

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###### Theorem

Let $X, Y$ be suitable [[sites]] let and $f : X \to Y$ be a [[morphism of sites]].  Let $\mathcal{C} = Ch_\bullet(Sh(X,Ab))$ and $\mathcal{D} = Ch_\bullet(Sh(Y,Ab))$ be the [[model structure on chain complexes|model categories of complexes of sheaves of abelian groups]]. The [[direct image]] $f_*$ and [[global section]] functor $\Gamma_Y$ compose to $\Gamma_X$:

$$
  \Gamma_X : \mathcal{C} \stackrel{f_*}{\to} \mathcal{D} \stackrel{\Gamma_Y}{\to} Ch_\bullet(Ab)
  \,.
$$

Then for $A \in Sh(X,Ab)$ a sheaf of abelian groups on $X$ there is a cohomology spectral sequence

$$
  E_2^{p,q} := H^p(Y, R^q f_* A)
$$

that converges as

$$
  E_2^{p,q} \Rightarrow H^{p+q}(X, A)
$$

and hence computes the [[abelian sheaf cohomology]] of $X$ with coefficients in $A$ in terms of the cohomology of $Y$ with coefficients in the derived [[direct image]] of $A$.

=--

## Related concepts

* [[Leray-Serre spectral sequence]]

## References

Lecture notes include

* Dan Petersen, _Leray spectral sequence_, November 2010  ([pdf](http://staff.science.uva.nl/~heinloth/StacksSeminar/DanLeray.pdf))

* [[Greg Friedman]], _Some extremely brief notes on the Leray spectral sequence_ ([pdf](http://faculty.tcu.edu/richardson/Seminars/Gregspecseq.pdf))

Textbook accounts with an eye specifically towards [[étale cohomology]]

* [[Günter Tamme]], section I 3.7 of _[[Introduction to Étale Cohomology]]_

* [[James Milne]], section 11 of _[[Lectures on Étale Cohomology]]_

[[!redirects Leray spectral sequences]]