loop (topology)

> This entry is about loops in [[topology]]. For loops in the sense of  [[algebra]] see at *[[loop (algebra)]]*, also at *[[quasigroup]]* and *[[Moufang loop]]*.

***

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# Loops
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## Definitions

In [[topology]], a (parametrised, oriented) __loop__ in a [[space]] $X$ is a map (a [[morphism]] in an appropriate [[category]] of spaces, such as a [[continuous function]] between [[topological spaces]]) to $X$ from the [[circle]] $S^1 = \mathbb{R}/\mathbb{Z}$. Hence a continuous [[path]] whose endpoints coincide.

A __loop at $a$__ is a loop $f$ such that $f(k) = a$ for any (hence every) [[integer]] $k$.  An __unparametrised loop__ is an [[equivalence class]] of loops, such that $f$ and $g$ are equivalent if there is an [[monotone function|increasing]] [[automorphism]] $\phi$ of $S^1$ such that $g = f \circ \phi$.  An __unoriented loop__ is an equivalence class of loops such that $f$ is equivalent to $(x \mapsto f(-x))$.  A __Moore loop__ has domain $\mathbb{R}/n \mathbb{Z}$ for some [[natural number]] (or possibly any [[real number]]) $n$.  All of these variations can be combined, of course.  (A Moore loop at $a$ has $f(k n) = a$ instead of $f(k) = a$.  Also, a Moore loop for $n = 0$ is simply a [[point]], so possibly there is a better way to define this to avoid making this exception.  Finally, there is not much difference between unparametrised loops and unparametrised Moore loops, since we may interpret $(t \mapsto n t)$ as a reparametrisation $\phi$.)

In [[graph theory]], a __loop__ is an edge whose endpoints are the same vertex.  Actually, this is a special case of the above, if we interpret $S^1$ as the graph with $1$ vertex and $1$ edge; in this way, the other variations become meaningful.  In this context, a Moore loop is called a __cycle__.  (However, as the only directed graph automorphism of $S^1$ is the [[identity morphism|identity]], parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs.)

Every loop may be interpreted as a [[path]].  Sometimes a loop, say at $a$, is *defined* to be a path from $a$ to $a$.  However, this is correct only in certain contexts.  In graph theory, it\'s incorrect, but only because of terminological conventions; the idea is sound.  In [[continuous spaces]], it is also correct.  However, in [[smooth spaces]], it is not correct, since the [[derivatives]] at the endpoints should also agree; the same holds in many other more structured contexts.

## Concatenation

Given two Moore loops $f$ and $g$ at $a$, the __concatenation__ of $f$ and $g$ is a Moore loop $f ; g$ or $g \circ f$ at $a$.  If the domain of $f$ is $\mathbb{R}/m \mathbb{Z}$ and the domain of $g$ is $\mathbb{R}/n \mathbb{Z}$, then the domain of $f ; g$ is $\mathbb{R}/(m+n) \mathbb{Z}$, and
$$ (f ; g)(x) \coloneqq \left \{ \array { f(x) & \quad x \leq m \\ g(m+x) & \quad x \geq m .} \right . $$
In this way, we get a [[monoid]] of Moore loops in $X$ at $a$, with concatenation as multiplication.  This monoid may called the __[[Moore loop monoid]]__.

Often we are more interested in a [[quotient]] monoid of the Moore loop monoid.  If we use unparametrised loops (in which case we may use loops with domain $S^1$ if we wish), then we get the __unparametrised [[loop monoid]]__.  If $X$ is a [[smooth space]], then we may additionally identify loops related through a [[thin homotopy]] to get the __[[loop group]]__.  Finally, if $X$ is a [[continuous space]] and we identify loops related through any (basepoint-preserving) [[homotopy]], then we get the __[[fundamental group]]__ of $X$.

See [[looping]] and [[delooping]] for more.

## Examples

* [[Pochhammer loop]]

## Related entries

* [[loop]]

* [[path]],

* [[loop space]], [[free loop space]]

  [[loop space object]], [[loop space type]]

* [[fundamental group]]

* [[loop group]]

* [[loop braid group]]

[[!redirects Moore loop]]
[[!redirects Moore loops]]


loop braid group

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## Idea

Where the [[braid group]] is the [[group]] of [[continuous function|continuous]] "motions" of sets of distinct [[points]] in the [[Euclidean plane]], the *loop braid group* ([Dahm 62](#Dahm62), [Goldsmith 81](#Goldsmith81)) is the group of continuous motions  of disjoint [[unknot|unknotted]] [[loop (topology)|loops]] in 3d [[Euclidean space]].

Accordingly, the role that the [[braid group]] plays in [[3d TQFTs]] such as [[Chern-Simons theory]]/[[Reshetikhin-Turaev theory]] with point-like defects ([[anyons]]), the loop braid group plays in [[4d TQFT]] with [[defect QFT|line defects]] (codimension-2 [[defect branes]]).

\begin{imagefromfile}
    "file_name": "LoopBraid.jpg",
    "width": 430,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": -10,
        "right": 0,
        "left": 10
    },
    "caption": "graphics from [Baez, Wise, Crans 06](#BaezWiseCrans06)"
\end{imagefromfile}

## Related concepts

* [[braid group]]

* [[defect brane]]

## References

Original articles on motion groups:

* {#Dahm62} D. Dahm, *A generalization of braid theory*, PhD thesis, Princeton University, 1962.

* {#Goldsmith81} Deborah L. Goldsmith, *The theory of motion groups*, Michigan Math. J. 28(1): 3-17 (1981) ([doi:10.1307/mmj/1029002454](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-28/issue-1/The-theory-of-motion-groups/10.1307/mmj/1029002454.full))

Specifically on the loop braid group:

* [[Xiao-Song Lin]], *The motion group of the unlink and its representations*, in:

  *Xiao-Song Lin's Unpublished Papers* ([doi:10.1142/9789812819116_others01](https://doi.org/10.1142/9789812819116_others01)), Part B of:

  [[Kevin Lin]], [[Zhenghan Wang]], [[Weiping Zhang]] (eds.) *Topology and Physics: Proceedings of the Nankai International Conference in Memory of Xiao-Song Lin*, Tianjin, China, 27-31 July 2007. World Scientific, 2008 ([doi:10.1142/6907](https://doi.org/10.1142/6907))

* Celeste Damiani, *A journey through loop braid groups*, Expositiones Mathematicae **35** 3 (2017) 252-285 [[doi:10.1016/j.exmath.2016.12.003](https://doi.org/10.1016/j.exmath.2016.12.003)]

* {#TorzewskaMartinsMartin23} [[Fiona Torzewska]], [[João Faria Martins]], [[Paul Purdon Martin]], *Motion groupoids and mapping class groupoids*, Comm. Math. Phys. **402** (2023) 1621-1705 [[arXiv:2103.10377](https://arxiv.org/abs/2103.10377), [doi:10.1007/s00220-023-04755-0](https://doi.org/10.1007/s00220-023-04755-0)]

See also:

* Wikipedia, *[Loop braid group](https://en.wikipedia.org/wiki/Loop_braid_group)*

Discussion via [[4d TQFT]] and [[higher gauge theory]]/[[higher parallel transport]]:

in 4d [[BF-theory]]:

* {#BaezWiseCrans06} [[John Baez]], [[Derek Wise]], [[Alissa Crans]], *Exotic Statistics for Strings in 4d BF Theory*, Adv. Theor. Math. Phys. 11:707-749, 2007 ([arXiv:gr-qc/0603085](https://arxiv.org/abs/gr-qc/0603085))

* Alex Bullivant, [[João Faria Martins]], Paul Martin, *Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory*, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 7 ([arXiv:1807.09551](https://arxiv.org/abs/1807.09551))

and in the [[3d toric code]]-[[4d TQFT]]:

* {#KongTianZhang20} [[Liang Kong]], Yin Tian, Zhi-Hao Zhang, Section 2.2 of: *Defects in the 3-dimensional toric code model form a braided fusion 2-category*, J. High Energ. Phys. 2020, 78 (2020) ([arXiv:2009.06564](https://arxiv.org/abs/2009.06564), doi:10.1007/JHEP12(2020)078)

* Zhen Bi, Yi-Zhuang You, Cenke Xu, *Anyon and loop braiding statistics in field theories with a topological $\Theta$ term*, Phys. Rev. B **90** (2014) 081110(R)  ([doi:10.1103/PhysRevB.90.081110](https://doi.org/10.1103/PhysRevB.90.081110))

and in 3d [[symmetry protected topological phases]]:

* Chao-Ming Jian, Xiao-Liang Qi, *Layer Construction of 3D Topological States and String Braiding Statistics*, Phys. Rev. X **4** (2014) 041043 $[$[doi:10.1103/PhysRevX.4.041043](https://doi.org/10.1103/PhysRevX.4.041043)$]$

* Shenghan Jiang, Andrej Mesaros, and Ying Ran, *Generalized Modular Transformations in (3+1)D Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding*, Phys. Rev. X **4** 031048 ([doi:10.1103/PhysRevX.4.031048](https://doi.org/10.1103/PhysRevX.4.031048))

and similar models in [[condensed matter theory]]:

* [[Chenjie Wang]], [[Michael Levin]], *Braiding Statistics of Loop Excitations in Three Dimensions*, Phys. Rev. Lett. **113** (2014) 080403 ([doi:10.1103/PhysRevLett.113.080403](https://doi.org/10.1103/PhysRevLett.113.080403))

* AtMa P. O. Chan, [[Peng Ye]], [[Shinsei Ryu]], *Braiding with Borromean Rings in (3+1)-Dimensional Spacetime*, Phys. Rev. Lett. 121, 061601 (2018) [[arXiv:1703.01926](https://arxiv.org/abs/1703.01926)]

* Zhi-Feng Zhang, Qing-Rui Wang, [[Peng Ye]], *Wilson Operators, Non-Abelian Fusion Rules, and Loop Shrinking Rules of Borromean Rings Topological Order* [[arXiv:2208.09228](https://arxiv.org/abs/2208.09228)]

Relation to [[braided monoidal category|braided]] [[tensor categories]]:

* Liang Chang,  *Representations of the loop braid groups from braided tensor categories*, Journal of Mathematical Physics 61, 051702 (2020) ([arXiv:2004.10896](https://arxiv.org/abs/2004.10896), [doi:10.1063/5.0005266](https://doi.org/10.1063/5.0005266))

Relation to [[integrable systems]]:

* Pramod Padmanabhan, Abhishek Chowdhury, *Loop braid groups and integrable models* [[arXiv:2210.12932](https://arxiv.org/abs/2210.12932)]

[[!redirects loop braid groups]]

[[!redirects motion group]]
[[!redirects motion groups]]

[[!redirects motion groupoid]]
[[!redirects motion groupoids]]


loop digraph object

[[!redirects directed loop graph object]]
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## Idea

A **loop digraph object** [[internalization|internal to]] a [[category]] is an [[object]] that behaves in that category like loop [[digraphs]] do in [[Set]].

## Definition

### In a category with finite products

A **loop digraph object** in a [[category]] $\mathcal{C}$ with [[finite products]] is

* a [[object]] $V$ in $\mathcal{C}$

* an object $E$ in $\mathcal{C}$

* a [[morphism]] $R:E \to V \times V$

such that $R$ is [[monic]]: for every object $A$ in $\mathcal{C}$ and morphisms $f: A \to E$ and $g: A \to E$, $R \circ f = R \circ g$ implies that $f = g$.

### In general categories

A **loop digraph object** in a category $\mathcal{C}$ is

* a [[object]] $V$ in $\mathcal{C}$

* an object $E$ in $\mathcal{C}$

* a [[morphism]] $s:E \to V$

* a morphism $t:E \to V$

such that $s$ and $t$ are [[jointly monic]]: for every object $A$ in $\mathcal{C}$ and morphisms $f: A \to E$ and $g: A \to E$, $s \circ f = s \circ g$ and $t \circ f = t \circ g$ imply that $f = g$.

## Properties

A loop digraph object is equivalently an object with an [[internal relation|internal binary endorelation]].

## Related concepts

* [[graph]]

* [[digraph]]

* [[relation]]

* [[loop graph object]]

* [[internal relation]]

[[!redirects loop digraph objects]]


loop graph object

[[!redirects directed loop graph object]]
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## Idea

A **loop graph object** [[internalization|internal to]] a [[category]] with [[finite products]] is an [[object]] that behaves in that category like loop [[graphs]] do in [[Set]].

## Definition

### In a category with finite products

A **loop graph object** in a category $\mathcal{C}$ with [[finite products]] is a [[loop digraph object]] $(V, E, R:E \to V \times V)$ with a morphism $i:E \to E$ such that

* $p_1 \circ R = p_2 \circ R \circ sym$
* $p_2 \circ R = p_1 \circ R \circ sym$
* $i \circ i = id_E$

where $p_1, p_2:V \times V \to V$ are the unique projection morphisms of the binary product.

### In a general category

A **loop graph object** in a category $\mathcal{C}$ is a [[loop digraph object]] $(V, E, s:E \to V, t:E \to V)$ with a morphism $i:E \to E$ such that

* $s \circ R = t \circ sym$
* $t \circ R = s \circ sym$
* $i \circ i = id_E$

## Properties

A loop graph object is equivalently an object with an [[internal relation|internal symmetric binary endorelation]].

## Related concepts

* [[graph]]

* [[symmetric relation]]

* [[loop digraph object]]

* [[internal relation]]

[[!redirects loop graph objects]]


loop group

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## Idea

The [[loop space]] of a [[topological group]] $G$ inherits the structure of a [[group]] under pointwise group multiplication of [[loops]]. This is called a _loop group_ of $G$.

(Notice that this is a group structure in addition to the [[infinity-group]]-structure of any [[loop space]] under _composition_ of loops.)

If $G$ is a [[Lie group]], then there is a smooth version of the loop group consisting of [[smooth functions]] $S^1 \to G$. By the discussion at [[manifold structure of mapping spaces]] the collection of such smooth maps is itself an [[infinite-dimensional smooth manifold]] and so the smooth loop group of a Lie group is an [[infinite-dimensional Lie group]].

Among all [[infinite-dimensional Lie groups]], loop groups are a most well behaved class. In particular their [[representation theory]] is similar to that of [[compact Lie groups]].

Some of these nice properties are solely due to the [[circle]] $S^1$ being a [[compact manifold]]. For $X$ any other compact manifold there is similarly an infinite-dimensional Lie group $[X,G]$ of smooth functions $X \to G$ under pointwise multiplication in $G$.

Such mapping groups appear in [[physics]] notably as groups of [[gauge transformations]] over a [[spacetime]]/[[worldvolume]] $X$. Accordingly, loop groups play a prominint role in 1- and 2-dimensional [[quantum field theory]], notably the [[WZW model]] describing the propagation of a [[string]] on $G$. The _[[current algebras]]_ ([[affine algebras]]) which arise as [[Lie algebras]] of ([[central extension|centrally extended]]) loop groups derive their name from this relation to physics. Accordingly, as for [[compact Lie groups]], the [[representation theory]] of loop groups is naturally understood in terms of their [[geometric quantization]] (by a loop variant of the [[orbit method]]).

On the other hand, for $X$ of [[dimension]] greater that 1 there are very few known results about the properties of the mapping group $[X,G]$.

## Properties

### Lie algebra

Let $G$ be a [[compact Lie group]]. Write $\mathfrak{g}$
for its [[Lie algebra]].

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

The [[Lie algebra]] of $L G$ is the [[loop Lie algebra]]

$$
  Lie(L G) \simeq L Lie(G) = L \mathfrak{g}
  \,.
$$

=--

### Complexification

Let $G$ be a [[compact Lie group]].

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

The [[complexification of a Lie group|complexification]] of $L G$ is the loop group of the complexification of $G$

$$
  (L G)_{\mathbb{C}}
  \simeq
  L (G_\mathbb{C})
  \,.
$$

=--

### Central extensions

Loop groups of [[compact space|compact]] [[Lie groups]] have canonical [[central extensions]], often called _Kac-Moody central extensions_ . A detailed discussion is in ([PressleySegal](#PressleySegal)). A review is in ([BCSS](#BCSS))

### Representations
 {#Representations}

#### Positive energy

Write

$$
  t_\theta \colon L G \to L G
$$

for the [[automorphism]] which rotates loops by an [[angle]] $\theta$.

The corresponding [[semidirect product group]] we write $L G \rtimes S^1$

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

Let $V$ be a [[topological vector space]]. A linear representation

$$
  S^1 \to Aut(V)
$$

of the [[circle group]] is called **positive** if $\exp(i \theta)$ acts by $\exp(i A \theta)$ where $A \in End(V)$ is a [[linear operator]] with positive [[spectrum of an operator|spectrum]].

A linear [[representation]]

$$
  \rho : L G \to Aut(V)
$$

is said to have **positive energy** or to be a **[[positive energy representation]]** if it extends to a representation of the [[semidirect product group]] $L G \rtimes S^1$ such that the restriction to $S^1$ is positive.

=--

#### By geometric quantization (looped orbit method)
 {#ByGeometricQuantization}

We discuss the [[quantization of loop groups]] in the sense of [[geometric quantization]] of their canonical [[prequantum bundle]].

Let $G$ be a [[compact Lie group]].
Let $T \hookrightarrow G$ be the inclusion of a [[maximal torus]]. There is a [[fiber sequence]]

$$
  \array{
    G/T &\to& L G / T
    \\
    && \downarrow
    \\
    && L G / G & \simeq \Omega G
  }
  \,.
$$

+-- {: .num_remark }
###### Remark

By the discussion at _[[orbit method]]_, if $G$ is a [[semisimple Lie group]], then $G/T$ is isomorphic to the [[coadjoint orbit]] of an element $\langle \lambda , -\rangle \in \mathfrak{g}^*$ for which $T \simeq G_\lambda$ is the [[stabilizer subgroup]].

If moreover $G$ is [[simply connected topological space|simply connected]], then the [[weight lattice]] $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is [[isomorphism|isomorphic]] to the group of [[group characters]]

$$
  \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1))
  \,.
$$

=--

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

The [[irreducible representation|irreducible]] [[projective representation|projective]] [[positive energy representations]] of $L G$ correspond precisley to the possible [[geometric quantizations]] of $L G / T$ (as in the [[orbit method]]).

More in detail:

The degree-2 [[integral cohomology]] of $L G / T$ is

$$
  H^2(L G / T) \simeq \mathbb{Z} \oplus H^2(G / T, \mathbb{Z}) \simeq \mathbb{Z} \oplus \hat T
  \,.
$$

Writing $L_{n,\lambda}$ for the corresponding [[complex line bundle]] with level $n \in \mathbb{Z}$ and weight $\lambda \in \hat T$ we have that

1. the space of [[holomorphic sections]] of $L_{n,\lambda}$ is either zero or is an irreducible positive energy representation;

1. every such arises this way;

1. and is non-zero precisely if $(n,\lambda)$ is positive  in the sense that for each positive [[coroot]] $h_\alpha$ of $G$

   $$
     0 \leq \lambda(h_\alpha) \leq n \langle h_\alpha, h_\alpha\rangle
    \,.
   $$

=--

This appears for instance as ([Segal, prop. 4.2](#Segal)).

### Relation to equivariant elliptic cohomology
 {#RelationToEquivariantEllipticCohomology}

Under mild conditions (but over the complex numbers) the representation ring of a loop group $L G$ is equivalent to the $G$-[[equivariant elliptic cohomology]] (see there for more) of the point ([Ando 00, theorem 10.10](#Ando00)).

This is a higher analog of how $G$-[[equivariant K-theory]] of the point gives the [[representation ring]] of $G$.

## Related concepts

* [[Kac-Moody group]]

* [[Verlinde ring]]

* [[string group]]

* [[caloron correspondence]]

* [[equivariant elliptic cohomology]]

* [[Kac character formula]]

* [[Kac-Weyl character]]

* [[double loop group]]

## References

The standard textbook on loop groups is

* {#PressleySegal} Andrew Pressley, [[Graeme Segal]], _Loop groups_ Oxford University Press (1988)

A review talk is

* {#Segal} [[Graeme Segal]], _Loop groups_ ([pdf](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084581/chapter08.pdf))

A review of some aspects with an eye towards loop groups as part of the [[crossed module]] of groups representing a [[string 2-group]] is in

* {#BCSS05} [[John Baez]], [[Alissa Crans]], [[Urs Schreiber]], [[Danny Stevenson]], _From loop groups to 2-groups_, Homology Homotopy Appl. Volume 9, Number 2 (2007), 101-135. ([arXiv:math/0504123](https://arxiv.org/abs/math/0504123))

The relation between [[representations]] of loop groups and [[twisted K-theory]] over the group is the topic of

* [[Dan Freed]], [[Mike Hopkins]], [[Constantin Teleman]], _[[Loop Groups and Twisted K-Theory]]

The relation between representations of loop groups an [[equivariant elliptic cohomology]] of the point is discussed in

* {#Ando00} [[Matthew Ando]], _Power operations in elliptic cohomology and representations of loop groups_ Transactions of the American
Mathematical Society 352, 2000, pp. 5619-5666. ([JSTOR](http://www.jstor.org/stable/221905), [pdf](http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf))

Discussion with respect to [[flag varieties]] is in

* {#Kumar02} [[Shrawan Kumar]], _Kac-Moody Groups, their Flag Varieties and Representation Theory_, Birkhäuser 2002

[[!redirects loop group]]
[[!redirects loop groups]]

[[!redirects loop group representation]]
[[!redirects loop group representations]]

[[!redirects Kac-Moody central extension]]
[[!redirects Kac-Moody central extensions]]

[[!redirects Kac-Moody loop group]]
[[!redirects Kac-Moody loop groups]]


loop groupoid

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## Definition

For $G$ a [[group]] and $\mathbf{B}G$ its one-object [[delooping]] [[groupoid]], the **loop groupoid** $\mathcal{L}\mathbf{B}G$ of $G$ is the [[free loop space object]] of $\mathbf{B}G$.

This is the groupoid whose objects are the elements of $G$, and whose [[morphism]]s are of the form

$$
  g \stackrel{h}{\to} h^{-1} g h
  \,.
$$

## Applications

This plays a role in [[Dijkgraaf-Witten theory]].


Loop Groups and Twisted K-Theory

This entry provides some hyperlinks for the articles

* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],

  1. _Loop Groups and Twisted K-Theory I_,

     J. Topology, 4 (2011), 737-789

     ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))

  1. {#PartII} _Loop Groups and Twisted K-Theory II_,

     J. Amer. Math. Soc. 26 (2013), 595-644

     ([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))

  1. _Loop Groups and Twisted K-Theory III_,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     ([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))

about the [[twisted ad-equivariant K-theory]] of [[compact Lie groups]], and the [[positive energy representations]] of their [[loop groups]] (forming the [[Verlinde ring]]). Part II also deals with [[Dirac induction]] and the [[orbit method]] for producing representations by the [[geometric quantization]] of [[coadjoint orbits]].

Exposition:

* [[Daniel S. Freed]], _The Verlinde algebra is twisted equivariant K-theory_, Turk. J. Math 25 (2001) 159-167 ([arXiv:math/0101038](https://arxiv.org/abs/math/0101038))

This identification enhanced to an [[equivalence of categories]]:

* [[Daniel S. Freed]], [[Constantin Teleman]],

  _Dirac families for loop groups as matrix factorizations_,

  Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

  ([arxiv/1409.6051](http://arxiv.org/abs/1409.6051), [doi:10.1016/j.crma.2015.02.011](https://doi.org/10.1016/j.crma.2015.02.011))

> We identify the category of integrable lowest-weight representations of the loop group $L G$ of a compact Lie group $G$ with the linear category of twisted, conjugation-equivariant curved Fredholm complexes on the group $G$: namely, the twisted, equivariant [[matrix factorizations]] of a super-potential built from the loop rotation action on $L G$. This lifts the isomorphism of K-groups of [FHT1,2, 3] to an equivalence of categories. The construction uses families of Dirac operators.

## Statement of the theorem

Every $\mathfrak{g}$-[[Lie algebra valued differential form|valued differential form]] $A \in \Omega^1(S^1, \mathfrak{g})$ on the [[circle]] defines an element in the free [[loop group]]. Also it can be understood as a [[connection on a bundle|connection]] on a $G$-[[principal bundle]] over the circle.

Given a [[Dirac operator]] $D$ on an [[associated bundle|associated]] [[spinor bundle]] over the circle, and using a [[loop group representation]] $V$ one can hence shift its defining connection by $A$ and form the associated Dirac operator $D_A$.

By remembering for each $A$ its [[holonomy]] $hol(A) \in G$, this construction defines a $G$-parameterized family of [[Fredholm operators]] over $G$, one for each loop group representation.

This construction is ([FHT, part II, cor. 3.39](#PartII)).

The FHT-theorem asserts that this construction gives an equivalence of the category of loop group representations at some level $\tau$ ([[Verlinde ring]]) and the $\tau$-[[twisted K-theory]], [[equivariant cohomology|equivariant]] with respect to $Ad_G$, on $G$.

This is ([FHT, part II, theorem. 3.43](#PartII)).

## Notes

> Some raw notes from a Teleman talk, to be polished

Group actions on categories and Langlands duality

I. Index formula on moduli of G_bundles and TQFT

* revolves around the [[Verlinde formula]]

* formulation in twisted K-theory

* generalization from [[line bundle]]s to general [[vector bundle]]s

* For line bundles, relation between [[loop groups]] reps on twisted K-theory

construction of twisted K-theory classes from families of Dirac operators

* puzzling appearance of Langlands dual group in relation to Lie group action on categories

start

$G$ a [[compact space|compact]] [[Lie group]] [[connected]] [[simply connected]]

$T \subset G$ [[maximal torus]], $T^\vee$ the dual torus, $W$ [[Weyl group]]

$k \in \mathbb{Z} \simeq H^4(B G, \mathbb{Z})$ called the "level"

$\Sigma$ a closed [[oriented]] [[Riemann surface]]

$K_G(\Sigma) = Flat G Bund(\Sigma) \simeq Hom(\pi_1 \Sigma, G)/G$ the [[moduli space]] of [[connection on a bundle|flat]] $G$-[[principal bundle]]s on $\Sigma$. This inherits a [[complex structure]] from $\Sigma$

$$
  \mathcal{O}(k) \to M_G(\Sigma)
$$

the [[determinant line bundle]] for $G = SU(n)$  $det^{-k} H^\bullet(\Sigma, standard vector bundle)$

Intense activity was generated in 1990s by the [[Verlinde formula]] suggested by [[CFT]]

$$
 d = dim H^0 ( M_G(\Sigma), \mathcal{O}(k))
$$

which is the [[partition function]] of a [[CFT]]
level-$k$ [[line bundle]]

Digression recall that [[2d TQFT]]s correspond to [[Frobenius algebra]]s.

If $A$ is a [[semi-simple algebra]] $\simeq \oplus \mathbb{C} P_i$, where

$\theta(P_i) = \theta_i \in \mathbb{C}^\times$ and $Z(\Sigma) = \sum \theta_i^{1-g(\Sigma)}$

In our case, the [[Verlinde ring]] is a Frobenius ring / $\mathbb{Z}$. It is a quotient of the ring of representations $R_G$ by the ideal of characters vanishing at the regular points  $F = ker(T \stackrel{k+c}{\to}T^\vee)$, where $c$ is the [[dual Coxeter number]]

the [[isogeny]] of $k$ being defined by the level

$$
  k \in H^4(B G, \mathbb{Z}) \to H^4(B T, \mathbb{Z}) = Sym^2 \pi_1(T)^\vee
$$

the projectors $P_f$ range over $f \in F^{reg}/W$ ($F$ a Weyl orbit)

The traces $\theta(P_f) = \frac{vol(c_f)}{|F|} = \frac{\Delta(f)^2}{|F|}$

Where $c_F$ is order of conjugacy class of $F$

Substantial work went into proving various casesy of the formula...

A distinction aroise between the [[moduli space]] [[topological space]] and the [[moduli stack]] of all algebraic $G_{\mathbb{C}}$-bundles;

turned out to be immaterial for holomorphic sections (and higher cohomology of line bundles, which vanishes in both cases)

But for more general vector bundles it became cleat that the Verline-style formulas apply to the [[moduli stack]]
 and not the space

Recall that Narasimhan-Sechadri identity the moduli space of semi-stable algebraic $G_{\mathbb{C}}$ bundles (modulo grade invariance)

The [[moduli stack]] of all holomorphic $G_{\mathbb{C}}$-bundles

Has an atractive complex analytic presentation due to [[Graeme Segal]]

choose a disk $\Delta \subset \Sigma$ with smooth parameterized boundary $\partial \Delta$. Then the stack $\mathcal{M}_G(\Sigma)$ is the double [[coset]] $  Hol(\Delta, G_{\mathbb{C}})\backslash L G_{\mathbb{C}} / Hol(\Sigma \backslash \Delta, G_{\mathbb{C}})$

where the holomorphic maps have smooth boundary values

The variety

$$
  X_{\Sigma \backslash \Delta} = L G / Hol(\Sigma \backslash \Delta, G_{\mathbb{C}})$$

is a complex Kähler homogeneous space for $L G_{\mathbb{C}}$ analogous in many ways to a [[flag variety]] of complex semsimple Lie groups

As a [[symplectic manifold]] it can be realized as

$$
  (flat G bundles in \Sigma \backslash \Delta) / {gauge equivalences trivial on \partial \Delta}
$$

The symplectic structure here is independent of the complex structure omn $\Sigma$

The $L G$ action is projective Hamiltonian, being a central extension lift to the prequantum line bundle

So holomorphic and cohomological questions about the stack of all $G_{\mathbb{C}}$-bundles on $\Sigma$ is equivalently

$Hol(\Delta, G)$-equivariant holomorphic and cohomological  on $X_{\Sigma \backslash \Delta}$

Example: What is $H^0(X_{\Sigma \backslash \Delta}, \mathcal{O}(k))$ as an $L G$-representation? Turns out the multiplicity of a certain "vacuum" representation inside is equal to $dim H^0(K_G(\Sigma), \mathcal{O}(k))$

While complex analysis on such infinite-dimensional manifolds is still out of reach, there exists an algebraic model for $X_{\Sigma \backslash \Delta}$ for which we can ask and answer analogous question

Example: $H^{\gt 0}(X^{ab}_{\Sigma \backslash \Delta}, \mathcal{O}(k)) = 0$
"Kodeira vanishing"

So we have an analytic index theorem for these varieties but without a topological side (this was "paradoxical", because usually the topological side is easier.)

But this requires finding a receptable fot the topological index ([[Riemann-Roch theorem]])

Whet $\mathcal{G}$ acts on R%X%, RR takes values in

$$
  Rep(\mathcal{G}) = K_{\mathcal{G}}(point) :
 K_{\mathcal{G}}(X) \stackrel{p_*}{\to} K_{\mathcal{G}}(point)
$$

[[fiber integration]] in [[generalized cohomology theory]]

Here $K_{L G}(point)$ is not "topological", not got equivariant K-theory for non-compact groups!, so that map $p_*$

But it turns out that both problems have a simultaneous solution.

Instead consider $K_{L G}(\mathcal{A}_{S^1})$ for the gauge action of $L G$ inb tge space of flat $G$-connections on the circle

Instead of projecting $X_{\Sigma \backslash \Delta}$ to a point, use the flat connection model and restrict to the boundary $\partial \Delta : X_{\Sigma \backslash \Delta} \o \mathcal{A}_{\partial \Delta}$

This is a [[proper map]]!

The based loop group $\Omega G$ acts freely and we can

which leads to a well-defined map

$$
  K_G(G^{2 g}) \stackrel{p_*}{\to} K_G(G)
$$

Amendments;

we wanted projective representations of $L G$ with cocycle

$$
  k \in H^2_{L G}(\mathcal{O}^\times)(\stackrel{\sim}{\to} H^3_G(G, \mathbb{Z}))
$$

So we should map to the [[twisted K-theory]] group

$$
  K_G^{k}(G)
$$

actually there is an extra shift by the dual Coxeter number $c$ ($= n $ for $SU(n)$) coming from the [[spinor]]s $\sqrt K$ on $X_{\Sigma \backslash \Delta}$, and the key theorem is

[[Freed-Hopkins-Teleman theorem]]:

$$
  {}^{k + c}K_G^{dim G}(G)
$$

is a free abelian group generated by the positive energey irreps of $L G$ at level $k$.

generalization by Teleman and Woodward:

analytical index = topological index for $X_{\Sigma \backslash \Delta}^alg$

for essentially all K-theory clases, at least after inverting $k+ c$

II. From loop group representations to K-theory

Preparation: Compact groups:

Recall the Kirillov correspondence between

irreducilble representations and co-adjoint orbits (+ line bundles)

In the connected case it can be summarized by saying that both of those correspond to the set of dominant integral regular wheights.

Example: $n$-dimensional rep of $SU(2)$: sphere of radius $n$ in $\mathfrak{su}(2) \simeq \mathbb{R}^3$

for $SO(3)$, odd radii,

but one can describe a canonical correspondence without directs reference to the classification of irreps

Each [[coadjoint orbit]] has a [[symplectic form]] (see at _[[orbit method]]_)

$$
  \omega_2(ad^*_\xi(\lambda), ad^*_\eta(\lambda))
  =
  \langle \lambda | [\xi,\eta]]\rangle
$$

For a vector bundle $V$ on the orbit $O_\lambda$ of $\lambda$, which is a sum of line bundles with curvature $\omega_\lambda$ and which carries a lifted $G$-action.

The Dirac index $D Ind(O_\lambda, V)$ is a representation of $G$ and this establishes the Kirillov correspondence

Remarks: For connected $G$, $V$ carries no information beyond its rank

Wehn $\pi_1(G) \neq 0$ the $G$-action on $V$ should be projective and cancel the spinor projective cocycle

The Dirac family consruction (Freed, Hopkins, Teleman) provides an inverse to this, assigning an orbit,...

Input: irreducible representation $V_\lambda$ of $G$

$\omega$ highest weight $\lambda$
invariant metric on $\mathfrak{g}$

Use Kostant's "cubic Dirac operator" on $G$, which is the [[Dirac operator]] on $G$ for the metric connection with canonical torsion coming from $H^3(G, \mathbb{Z}) \simeq \mathbb{Z}$.

Two key properties of this Dirac operator

$$
  [D, \psi(\xi)] = 2 T(\xi)
$$

(translation)

$$
  D^2 = - (\lambda + p)^2
$$

on $V_\lambda \otimes S^{\pm}$

The Dirac family is the $\mathbb{Z}/2$-graded vector bundle with fiber $V \otimes S^{\pm}$ over $\mathfrak{g}$ and odd operator

$$
  D_\xi^V := D^V + \psi(\xi)
$$

Theorem (FHT): The kernel of $D^V_\xi$ is supported on the orbit of $(\lambda + p)$ and equals $(Kirillov line bundle) \otimes (Spinors to normal bundle)$ In fact $D^V_\xi$ is a model for the Atiyah-Bott-Shapiro...

[[!redirects Freed-Hopkins-Teleman theorem]]
[[!redirects FHT theorem]]

category: reference


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
This page is to record links related to

* [[David Ben-Zvi]],

  _Loop Groups, Characters and Elliptic Curves_

  talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_, 17-19 December 2012, Oxford,

  ([pdf slides](https://people.maths.ox.ac.uk/tillmann/ASPECTSbenzvi.pdf))

on ([[geometric representation theory|geometric]]) [[representation theory]] reflected in and incarnated as low-dimensional [[extended TQFT|extended]] [[gauge field theory]], some relevant [[categorification]] in [[D-geometry]] (e. g. the [[Hecke category]]) and specifically on formalizations of [[geometric Langlands duality]] in terms of 2-dimensional [[QFT]] motivated from [[KK-compactification|compactification]] of [[N=4 D=4 super Yang-Mills theory]].

(Other talks at the same conference include [[Dan Freed]]: _[[4-3-2 8-7-6]]_.)

category: reference


loop order

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebraic Qunantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] the [[scattering amplitudes]] in the [[S-matrix]] are expressed as [[formal power series]] in (the [[coupling constant]] and) in [[Planck's constant]] $\hbar$. This formal power series may be expressed as a formal sum of contributions labeled by [[Feynman diagrams]]. The _loop order_ refers to something like the "number of loops" of [[edges]] in the [[Feynman diagram]] that contibutes to a given [[scattering amplitude]]. It turns out that the loop order corresponds to the order in $\hbar$ that is contributed by this diagram (see [below](#RelationToPowersInPlancksConstant)).

Therefore contributions of graphs at zero without loops (these are [[trees]], and hence these contributions are referred to as being at "tree level") correspond to the limit of [[classical field theory]] with $\hbar \to 0$. Indeed tree level Feynman diagrams yield [[perturbation theory|perturbative]] solutions of the [[classical field theory|classical]] [[equations of motion]] (see [Helling](#Helling)).

Most predictions of the [[standard model of particle physics]] have very good agreement with [[experiment]] already to very low loop order, first or second; inclusion of third loop order is used (at least in [[QCD]]) to constrain theoretical uncertainties of the result (see [Cacciari 05, slide 5](#Cacciari05), e.g. in [[Higgs field]] computation, see [ADDHM 15](#ADDHM15)).
In rare cases higher loop orders are used (for instance in the computation of the [[anomalous magnetic moments]] [AHKN 12](#AHKN12), but this is not a scattering experiment).

This usefulness of low loop order is fortunate because

1. the [[S-matrix]] [[formal power series]] for all [[theory (physics)|theories]] of interest has _vanishing_ [[radius of convergence]] ([Dyson 52](perturbation+theory#Dyson52)), hence is at best an [[asymptotic series]] for which the [[sum]] of more than some low order terms is meaningless;

1. the computational effort increases immensely with loop order.

## Relation to powers in Planck's constant
 {#RelationToPowersInPlancksConstant}

The following is taken from _[[geometry of physics -- A first idea of quantum field theory]]_. See there for more background.

+-- {: .num_prop #FeynmanDiagramLoopOrder}
###### Proposition
**([[loop order]] and [[tree level]] of [[Feynman perturbation series]])**

The [[effective action]] ([this def.](A+first+idea+of+quantum+field+theory#InPerturbationTheoryActionEffective)) contains no negative powers of [[Planck's constant]] $\hbar$, hence is indeed
a [[formal power series]] also in $\hbar$:

$$
  S_{eff}(g,j)
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]
  \,.
$$

and in particular

$$
  \left\langle
    S_{eff}(g,j)
  \right\rangle
  \;\in\;
  \mathbb{C}[ [ \hbar, g, j] ]
  \,.
$$

Moreover, the contribution to the effective action in the [[classical limit]] $\hbar \to 0$
is precisely that of [[Feynman amplitudes]] of those [[finite multigraphs]] ([this prop.](A+first+idea+of+quantum+field+theory#FeynmanPerturbationSeriesAwayFromCoincidingPoints)) which are [[trees]] ([this def.](A+first+idea+of+quantum+field+theory#GraphPlanar)); thus called the _[[tree level]]_-contribution:

$$
  S_{eff}(g,j)\vert_{\hbar = 0}
  \;=\;
  i \hbar
  \underset{\Gamma \in \mathcal{G}_{conn} \cap \mathcal{G}_{tree}}{\sum}
  \Gamma\left( (g S_{int} + j A)_{i = 1}^{\nu(\Gamma)} \right)
  \,.
$$

Finally, a [[finite multigraph]] $\Gamma$ ([this def.](A+first+idea+of+quantum+field+theory#Graphs)) which is [[planar graph|planar]] and [[connected graph|connected]] contributes to the effective action precisely at order

$$
  \hbar^{L(\Gamma)}
  \,,
$$

where $L(\Gamma) \in \mathbb{N}$ is the number of _[[faces]]_ of $\Gamma$, here called the _number of loops_ of the diagram;
here usually called the _[[loop order]]_ of $\Gamma$.

(Beware the terminology clash with [[graph theory]], see the discussion of [[tadpoles]] [here](A+first+idea+of+quantum+field+theory#Tadpoles))

=--

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

By [this def.](A+first+idea+of+quantum+field+theory#LagrangianFieldTheoryPerturbativeScattering) the explicit $\hbar$-dependence of the [[S-matrix]] is

$$
  \mathcal{S}
  \left(
    S_{int}
  \right)
  \;=\;
  \underset{k \in \mathbb{N}}{\sum}
    \frac{1}{k!}
    \frac{1}{(i \hbar)^k}
    T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots,  S_{int}}} )
$$

and by [this prop.](A+first+idea+of+quantum+field+theory#TimeOrderedProductAwayFromDiagonal) the further $\hbar$-dependence of the [[time-ordered product]] $T(\cdots)$ is

$$
  T(S_{int}, S_{int})
  \;=\;
  prod \circ
  \exp\left(
    \hbar
    \left\langle
      \Delta_F,
      \frac{\delta}{\delta \mathbf{\Phi}}
        \otimes
      \frac{\delta}{\delta \mathbf{\Phi}}
    \right\rangle
  \right)
 ( S_{int} \otimes S_{int} )
  \,,
$$

By the [[Feynman rules]] ([this prop.](A+first+idea+of+quantum+field+theory#FeynmanPerturbationSeriesAwayFromCoincidingPoints)) this means that

1. each [[vertex]] of a Feynman diagram contributes a power $\hbar^{-1}$ to its Feynman amplitude;

1. each [[edge]] of a Feynman diagram contributes a power $\hbar^{+1}$ to its Feynman amplitude.

If we write

$$
  E(\Gamma), V(\Gamma) \;\in\; \mathbb{N}
$$

for the total number of [[vertices]] and [[edges]], respectively, in $\Gamma$, this means that a Feynman amplitude
corresponding to some $\Gamma \in \mathcal{G}$ contributes precisely at order

$$
  \label{GeneralFeynmanDiagramhbarContribution}
  \hbar^{E(\Gamma) - V(\Gamma)}
  \,.
$$

So far this holds for arbitrary $\Gamma$. If however $\Gamma$ is [[connected graph|connected]] and [[planar graph|planar]], then _[[Euler's formula]]_ asserts that

$$
  \label{ConnectedPlanarGraphEulerCharacteristic}
  E(\Gamma) - V(\Gamma)
  \;=\;
  L(\Gamma) - 1
  \,.
$$

Hence $\hbar^{L(\Gamma)- 1}$ is the order of $\hbar$ at which $\Gamma$ contributes to the [[scattering matrix]] expressed as the [[Feynman perturbation series]].

But the [[effective action]], by definition ([this equation](A+first+idea+of+quantum+field+theory#eq:ExpansionEffectiveAction)), has the same contributions
of Feynman amplitudes, but multiplied by another power of $\hbar^1$, hence it contributes at order

$$
  \hbar^{E(\Gamma) - V(\Gamma) + 1} = \hbar^{L(\Gamma)}
  \,.
$$

This proves the second claim on [[loop order]].

The first claim, due to the extra factor of $\hbar$ in the definition of the effective action, is equivalent to saying
that the Feynman amplitude of every [[connected graph|connected]] [[finite multigraph]]
contributes powers in $\hbar$ of order $\geq -1$ and contributes at order $\hbar^{-1}$ precisely if the graph is a tree.

Observe that a [[connected graph|connected]] [[finite multigraph]] $\Gamma$ with $\nu \in \mathbb{N}$ vertices (necessarily $\nu \geq 1$) has at least $\nu-1$ edges and precisely $\nu - 1$ edges if it is a tree.

To see this, consecutively remove edges from $\Gamma$ as long as possible while retaining connectivity. When this process stops, the result must be a connected tree $\Gamma'$, hence a [[connected graph|connected]] [[planar graph]] with $L(\Gamma') = 0$. Therefore [[Euler's formula]] (eq:ConnectedPlanarGraphEulerCharacteristic) implies that that $E(\Gamma') = V(\Gamma') -1$.

This means that the connected multigraph $\Gamma$ in general has a Feynman amplitude of order

$$
  \hbar^{E(\Gamma) - V(\Gamma)}
  =
  \hbar^{ \overset{\geq 0}{\overbrace{E(\Gamma) - E(\Gamma')}} + \overset{= -1}{\overbrace{E(\Gamma') - V(\Gamma)}}  }
$$

and precisely if it is a tree its Feynman amplitude is of order $\hbar^{-1}$.

=--

## Related concepts

* [[radiative correction]]

## References

General discussion includes

* {#BrodskyHoyer10} Stanley J. Brodsky, Paul Hoyer, _The $\hbar$-Expansion in Quantum Field Theory_, Phys.Rev.D83:045026, 2011 ([arXiv:1009.2313](https://arxiv.org/abs/1009.2313))

Discussion of tree level Feynman diagrams as perturbative solutions in [[classical field theory]] is in

* {#Helling} Robert Helling, _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf))

Discussion of loop orders of relevance in comparison to [[experiment]] cited above includes for instance the following

* {#Cacciari05} Matteo Cacciari, _(Theoretical) review of heavy quark production_, BNL 14/12/2005 ([pdf](https://www.phenix.bnl.gov/WWW/publish/xiewei/RBRC_Workshop_Dec/heavyworkshop/cacciari.pdf))

* {#AHKN12} Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, Makiko Nio, _Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant_, 10.1103/PhysRevLett.109.111807 ([arXiv:1205.5368](#https://arxiv.org/abs/1205.5368))

* {#ADDHM15} Charalampos Anastasiou, Claude Duhr, Falko Dulat, Franz Herzog, Bernhard Mistlberger, _Higgs boson gluon-fusion production in N3LO QCD_, Phys. Rev. Lett. 114, 212001 (2015) ([arXiv:1503.06056](https://arxiv.org/abs/1503.06056))

[[!redirects loop orders]]

[[!redirects tree level]]
[[!redirects tree levels]]

[[!redirects tree-level]]
[[!redirects tree-levels]]

[[!redirects 1-loop]]
[[!redirects 2-loop]]
[[!redirects 3-loop]]
[[!redirects 4-loop]]

[[!redirects one-loop]]
[[!redirects two-loop]]
[[!redirects three-loop]]
[[!redirects four-loop]]


loop quantum gravity

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

#Contents#
* table of contents
{:toc}

## Idea

Loop quantum gravity is a [[theory (physics)|theory]] of [[quantum gravity]] in fundamental [[physics]].

## See also

* [[quantum gravity]]


loop space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Mapping space
+-- {: .hide}
[[!include mapping space - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In the strict sense of the word a _loop space_ in [[topology]] for a given [[pointed topological space]] $X$ is the [[mapping space]] (with its [[compact-open topology]], see the example [there](compact-open+topology#LoopSpace)) $Maps_\ast(S^1, X)$ of [[continuous functions]] from the [[circle]] to $X$, such that they take the given basepoint of the circle to the prescribed basepoint in $X$ (or if one drops this condition, then one speaks of the _[[free loop space]]_). One such continuous function may be thought of as a continuous loop in $X$, and hence the mapping space is the space of all these loops.

If here $X$ is equipped with further structure, such as [[smooth structure]] (e.g. a [[smooth manifold]]), then one may in good cases find such extra structure also on the loop space, for instance to form a _[[smooth loop space]]_, etc. See at _[[manifolds of mapping spaces]]_ for more on this.

If one regards this construction not in [[point-set topology]] but in [[classical model structure on topological spaces|classical homotopy theory of topological spaces]] ([[homotopy hypothesis|equivalently]] [[∞Grpd]]), then, up to [[weak homotopy equivalence]], the loop space is equivalent to the [[homotopy fiber product]] of the basepoint inclusion $\ast \to X$ along itself.

## Definition

Let [[Top]] be a [[nice category of spaces|nice category of topological spaces]], in particular one which is [[complete category|complete]], [[cocomplete category|cocomplete]], and [[cartesian closed category|cartesian closed]]. Let $(S^1, pt)$ be the [[circle]], i.e., 1-dimensional [[sphere]], with chosen basepoint, and let $(X, *)$ be a space with a chosen [[pointed set|basepoint]]. Then the **loop space** of $X$ (at $*$) is an [[internal hom]]

$$\Omega X = hom((S^1, pt), (X, *))$$

in the category $Top_*$ of based spaces. Explicitly, it is given by the [[pullback]] in $Top$

$$\array{
\Omega X & \to & 1\\
 \downarrow & & \downarrow *\\
X^{S^1} & \underset{X^{pt}}{\to} & X^1
}$$

(using [[exponential object|exponentials]] to denote internal homs in $Top$), in other words the [[function set|function space]] of basepoint-preserving maps $S^1 \to X$, whose basepoint is the constant map $S^1 \to X$ at the basepoint of $X$.

The category $Top_*$ is [[symmetric monoidal category|symmetric]] monoidal [[monoidal closed category|closed]]; its monoidal product is called the **[[smash product]]**, often denoted $\wedge$. In particular, the loop space functor

$$\Omega = hom((S^1, pt), -): Top_* \to Top_*$$

has a [[left adjoint]] obtained by taking smash product with $(S^1, pt)$. This left adjoint $S: Top_* \to Top_*$ is called the **[[suspension]] functor**. Explicitly, the suspension $S X$ is formed as the [[pushout]]

$$\array{
 & 1 \times X + S^1 \times 1& \to & 1\\
(pt \times X, S^1 \times *) & \downarrow & & \downarrow \\
 & S^1 \times X & \to & S X
}$$

with basepoint provided by the right vertical arrow.

## Properties

### Homotopy-associative structure
 {#AInfinityStructure}

A loop space is an example of a [[A-∞ space]], in particular it is an [[H-space]]. Loop spaces admit this rich algebraic structure which arises from the fact that the based space $S^1$ carries a correspondingly rich co-algebraic structure, starting from the fact that the based space $S^1$ is an H-[[cogroup]].

The description of this structure on loop spaces has been the very motivation for the introduction of the notion of [[operad]] and [[algebra over an operad]] in ([May](#May)).

An important theoretical consideration is when an H-space, and particularly one having the type of a [[CW-complex]], has the homotopy type of a loop space of another CW-complex: $X \simeq \Omega Y$. In this circumstance, one calls $Y$ a **[[delooping]]** of $X$; an important example is where $X$ carries a [[topological group]] structure $G$, and $Y$ is the [[classifying space]] of $G$.

The most basic fact about deloopings is the shift in [[homotopy group]]s:

* $\pi_n(\Omega Y) \cong \pi_{n+1}(Y)$

which follows straight from the [[adjunction]] $S \dashv \Omega$ plus the fact that the suspension of $S^n$ is $S^{n+1}$. (This isomorphism needs to be developed at greater length.)

The modern study of the question "when can an H-space be [[delooping|delooped]]?" was inaugurated by [[Jim Stasheff]]. The basic theorem is as follows (all spaces assumed to be CW-complexes):

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

An [[H-space]] $X$ admits a delooping if and only if the [[monoid]] $\pi_0(X)$ induced from the H-space structure is a [[group]], and the H-space $X$ structure can be extended to a structure of [[algebra over an operad]] over [[Jim Stasheff|Stasheff]]'s [[A-∞ operad]] $K$.
=--

This is due to ([Stasheff](#Stasheff)). The analogous statement holds true in every [[(∞,1)-topos]] other than [[Top]]. Details on this more general statement are at [[loop space object]] and at [[groupoid object in an (∞,1)-category]].

### Local homotopy properties ###

Let the space $X$ be [[locally connected space|locally 0-connected]] and [[semi-locally simply connected space|semi-locally 1-connected]] (i.e. it admits a [[universal covering space]]). The loop space $\Omega X$ for any basepoint is locally path connected, as is the free loop space $X^{S^1}$. If $X$ is locally 1-connected and admits a basis of open sets $U$ such that $\pi_2(U) \to \pi_2(X)$ is the zero map, then $\Omega X$ is locally 0-connected and semi-locally 1-connected, and so admits a universal covering space.

In general, if $X$ is locally $n$-[[n-connected space|connected]], $\Omega X$ is locally $(n-1)$-connected. This process can obviously be iterated up to $n$ times, so that $\Omega^n X$ is locally 0-connected. This can be weakened to locally $(n-1)$-connected and semi-locally $n$-connected: this is just like the $n=1$ case but replacing $\pi_1$ with $\pi_n$ (as was done in the previous paragraph with $\pi_2$). We will actually define a space to be semi-locally $n$-connected to include the condition that it is locally $(n-1)$-connected. This result was proved for more general mapping spaces $X^P$ and various subspaces when $X$ is Hausdorff and $P$ a finite [[polyhedron]] in ([Wada](#Wada)) but a much simpler and direct proof for general $X$ and $P = I$ or $P= S^1$ is possible.

+-- {: .un_theorem}
###### Conjecture

The fundamental $n$-groupoid of a space $X$ ([[Trimble n-category|Trimblean]] for choice) can be topologised to be an internal $n$-groupoid in $\Top$ when $X$ is semi-locally $n$-connected. Furthermore, the homotopy groups of the $n$-groupoid, _a priori_ topological groups, are discrete.
=--

For $n=2$, this is in [[David Roberts]]\'s [[Fundamental Bigroupoids and 2-Covering Spaces|thesis]].  For $n=1$, it has been known for ages and is in [[Ronnie Brown]]\'s topology textbook.

### Homology of loop spaces

See at _[[homology of loop spaces]]_.

## Models

There is a [[Quillen equivalence]]

$$
  (G \dashv \bar W)
   \;\colon\;
  sGrp
   \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_0
$$

between the [[model structure on simplicial groups]] and the [[model structure on reduced simplicial sets]], thus exhibiting both of these as models for [[infinity-groups]] ([Kan 58](#Kan58)). Its [[left adjoint]] $G$, the _[[simplicial loop space]] construction_, is a concrete model for the loop space construction with values in [[simplicial groups]].

See also [[simplicial group]] and [[groupoid object in an (∞,1)-category]] for more details.

## Related concepts

* [[path space]]

* [[loop space object]], [[free loop space object]]

  [[loop space type]]

  * [[Sullivan model of loop space]]

  * [[delooping]], [[looping and delooping]]

  * **loop space**, [[free loop space]],

  * [[relative loop space]]

  * [[iterated loop space]]

  * [[infinite loop space]]

  * [[formal loop space]]

  * [[derived loop space]]

  * [[smooth loop space]]

  * [[loop space of a wedge of circles]]

* [[caloron correspondence]]

* [[suspension object]]

  * [[suspension]]

[[!include k-monoidal table]]

* [[A-infinity space]], [[model structure for dendroidal Cartesian fibrations]]

## References

### General

* {#Kan58} [[Daniel Kan]], _On homotopy theory and c.s.s. groups_, Ann. of Math. 68 (1958), 38-53

* {#Stasheff} [[Jim Stasheff]], _Homotopy associative $H$-spaces I, II_, Trans. Amer. Math. Soc. 108, 1963, 275-312

* {#May} [[Peter May]], _The geometry of iterated loop spaces_ Lecture Notes in Mathematics 271 (1970) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

* H. Wada, _Local connectivity of mapping spaces_, Duke Mathematical Journal, vol ? (1955) pp 419-425

The simplicial loop group functor is discussed in chapter V, section 5 of

* [[Paul Goerss]], [[Rick Jardine]], _Simplicial homotopy theory_ ([web](http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html))

See also the references at [[looping and delooping]].

### Examples
 {#ReferencesExamples}

On loop spaces of [[configuration spaces of points]]:

* [[Edward Fadell]], [[Sufian Husseini]], _The space of loops on configuration spaces and the Majer-Terracini index_, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 ([euclid:tmna/1476842829](https://projecteuclid.org/euclid.tmna/1476842829))

* [[Fred Cohen]], [[Samuel Gitler]], _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Aguadé J., Broto C., [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))

* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))

* [[Fred Cohen]], [[Samuel Gitler]], _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013))

  (on [[ordinary homology]] of [[loop spaces]] of configuration spaces)

[[!redirects loop space]]
[[!redirects loop spaces]]

[[!redirects based loop space]]
[[!redirects based loop spaces]]


loop space object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
#### Stabe homotopy theory
+-- {: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Mapping space
+-- {: .hide}
[[!include mapping space - contents]]
=--
=--
=--

# Loop space objects
* tic
{: toc}

## Idea

In the [[(∞,1)-topos]] [[Top]] the construction of a [[loop space]] of a given [[topological space]] is familiar.

This construction may be generalized to any other [[(∞,1)-topos]] and in fact to any other [[(∞,1)-category]] with [[homotopy pullbacks]].

## Definition

Loop space objects are defined in any [[(∞,1)-category]] $\mathbf{C}$ with [[homotopy pullbacks]]: for $X$ any [[pointed object]] of $\mathbf{C}$ with point ${*} \to X$, its [[loop space object]] is [[generalized the|the]] [[homotopy pullback]] $\Omega X$ of this point along itself:

$$
  \array{
    \Omega X &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& X
  }
  \,.
$$

A ([[generalised element|generalised]]) [[global point|element]] of $\Omega X$ may be thought of as a [[loop]] in $X$ at the [[base point]] $*$.

When the point $x : {*} \to X$ is not clear from context, we can write $\Omega_x X$ or $\Omega(X,x)$ to indicate the point.

### Remarks

Since $\mathbf{C}(X,-)$ commutes with homotopy limits, one has a natural homotopy equivalence $\Omega_{\bar{y}}\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega_y Y)$, for any objects $X$ and pointed object $(Y,y)$ in $\mathbf{C}$, where $\bar{y}$ denotes the morphism $X \to * \to Y$.

See also

* [[fibration sequence]]

* [[delooping]]

* [[stable (∞,1)-category]]

## Explicit constructions

Usually the [[(∞,1)-category]] in question is [[presentable (infinity,1)-category|presented]] by concrete 1-categorical data, such as that of a [[model category]]. In that case the above [[homotopy pullback]] has various realizations as an ordinary [[pullback]].

Notably it may be expressed using [[path objects]] which may come from [[interval objects]]. Even if the context is not (or not manifestly) that of a [[homotopical category]], an [[interval object]] may still exist and may be used as indicated in the following to construct loop space objects.

### Free loop space objects
 {#FreeLoopSpaceObject}

In a category with [[interval object]] $ * \xrightarrow{0} I \xleftarrow{1} * $ the  **[[free loop space object]]** is the part of the [[path object]] $B^I = [I,B]$ which consists of closed paths, namely the [[pullback]]

\begin{tikzcd}
  \Lambda B
  \ar[r]
  \ar[d]
  &
  {}[I,B]
  \ar[d, "d_0 \times d_1"]
  \\
  B
    \ar[r, "\mathrm{Id} \times \mathrm{Id}"]
  &
  B \times B
\end{tikzcd}

where $ d_0 $ ($d_1$ resp.) is the composition of $ [0,B] $ ($[1,B]$ resp.) with the canonical identification of $[*, B]$ and $B$.

This is the same as the image of the [[co-span co-trace]] $cotr(I)$  of the interval object (which is the interval object closed to a loop!,  see the examples at [[co-span co-trace]]) in $B$:

$$
  \left[
  \array{
     && cotr(I)
     \\
     & \nearrow && \nwarrow
     \\
     pt &&&& I
     \\
     & {}_{Id \sqcup Id}\nwarrow && \nearrow_{in \sqcup out}
     \\
     && pt \sqcup pt
  }
  \;\;\;\;,\;\;\;\;
  B
  \right]
  \;,\;\;\;\;
  \simeq
  \;,\;\;\;\;
  \array{
     && \Lambda B
     \\
     & \swarrow && \searrow
     \\
     B &&&& [I,B]
     \\
     & {}_{Id \times Id}\searrow && \swarrow_{d_0 \times d_1}
     \\
     && B \times B
  }
$$

### Based loop space objects

If $B$ is a [[pointed object]] with point $pt \stackrel{pt_B}{\to} B$ then the **based loop space object** of $B$ is the pullback $\Omega_{pt} B$ in

$$
  \array{
    \Omega_{pt}B &\to& [I,B]
    \\
    \downarrow && \downarrow^{d_0 \times d_1}
    \\
    pt
    &\stackrel{pt_B \times pt_B}{\to}&
    B \times B
  }
  \,.
$$

#### Remarks

* $\Omega_{pt}B$ is the fiber of the [[generalized universal bundle]] $\mathbf{E}_{pt}B \to B$.

* the based loop space object $\Omega_{pt} B$ is the pullback of the free loop space object $\Lambda B$ to the point
$$
  \array{
    \Omega_{pt} B &\to& \Lambda B
    \\
    \downarrow && \downarrow
    \\
    pt &\stackrel{pt_B}{\to}& B
  }
  \,.
$$

## Remarks

* The loop space object $B$ can be regarded as the homotopy trace on the identity span on $B$, as described at [[span trace]].

* The free loop space object inherits the structure of an $A_\infty$-[[A-infinity-category|category]] from that of the [[path object]] $[I,B]$.

* In a [[generalized smooth space|suitable extension]] of $\operatorname{Diff}$, this construction does **not** give the usual _smooth_ loop space (free or based).  It gives the space of paths with coincident endpoints rather than the space of smooth maps from the circle.  Thus the [[smooth loop space]] is not a loop space object.

## Examples

* Let $C =$ [[Top]] with the standard [[interval object]]. Then for $B= X$ a topological space $\Lambda B = \Lambda X$ is the ordinary free [[loop space]] of $X$.

  The generalization of this to _smooth_ spaces is discussed at [[smooth loop space]].

* Let $C = $ [[Grpd]] with the standard interval object $I = \{a \stackrel{\simeq}{\to} b\}$ and let $\mathbf{B}G$ be the one-object groupoid corresponding to a group $G$, then
$$
  \Lambda \mathbf{B}G = G//_{Ad}G
$$
is the [[action groupoid]] of $G$ acting on itself by its adjoint action. Notice the example at [[co-span co-trace]] which says that the cotrace on $I$ is $cotr(I) = \mathbf{B}\mathbb{Z}$, and indeed
$$
  \Lambda \mathbf{B}G =  [\mathbf{B}\mathbb{Z}, \mathbf{B}G]
  \,.
$$
The role of this $\Lambda \mathbf{B}G$ as a loop object is amplified in particular in

  * Simon Willerton, _The twisted Drinfeld double of a finite group via gerbes and finite groupoids_ ([arXiv](http://arxiv.org/abs/math.QA/0503266))

  * Bruce Bartlett, _On unitary 2-representations of finite groups and topological quantum field theory_ ([arXiv](http://arxiv.org/abs/0901.3975))

* On the other hand, the _based_ loop object of $\mathbf{B}G$ is just $G$:
$$
  \Omega \mathbf{B}G = G
  \,.
$$

## Related concepts

* **loop space object**, [[free loop space object]]

  [[loop space type]]

  * [[delooping]], [[looping and delooping]]

  * [[loop space]], [[free loop space]], [[derived loop space]]

  * [[inertia orbifold]]

* [[suspension object]]

  * [[suspension]], [[reduced suspension]]

[[!redirects loop space object]]
[[!redirects loop space objects]]

[[!redirects loop object]]
[[!redirects loop objects]]
[[!redirects loop space functor]]
[[!redirects loop space functors]]
[[!redirects loop functor]]
[[!redirects loop functors]]

[[!redirects based loop space object]]
[[!redirects based loop space objects]]


loop space of a wedge of circles

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

This page considers the [[homotopy type]] of the [[loop space]] of a [[wedge sum]] of [[circles]].
For the moment, the following is entirely geared towards the language of [[homotopy type theory]].

Let $A$ be a [[set]], and consider the [[HIT]] $W_A$ with constructors

* $base:W_A$
* $loop : A \to (base = base)$.

This is the "wedge of $A$ circles". It can also be defined as the [[suspension]] of $A+1$.

## Known facts

* If $A$ has [[decidable equality]], then the [[loop space]] $\Omega W_A$ is the [[nlab:free group]] on $A$.

* For general $A$, it is probably still true that the [[fundamental group]] $\pi_1 W_A$ is the free group on $A$.

## An open problem

For a general set $A$ without decidable equality, is $W_A$ even a 1-type?  What can be said about $\Omega W_A$?

In the classical [[simplicial set model]], $W_A$ is a 1-type for all $A$, since classically all sets have decidable equality.  Moreover, since $W_A$ is definable as a colimit, and being a 1-type is a finite-limit property, this fact is inherited by all [[Grothendieck (infinity,1)-topos|Grothendieck (infinity,1)-topoi]]. Thus, no counterexample to $W_A$ being a 1-type can be found in such models, but it is also not clear how to prove that $W_A$ is a  1-type.

category: homotopy theory

[[!redirects loop spaces of a wedge of circles]]


loop space type

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

#Contents#
* table of contents
{:toc}

## Idea ##

In [[homotopy type theory]],
the [[loop space]] of a [[pointed type]] is simply the [[type]] of [[identifications]] from a [[term]] to itself i.e. [[loops]].

## Definition ##

Given a [[pointed type]] $(A,\star_A)$ the *loop space type* is the type $\Omega(A,\star_A)\equiv \star_{A} =_{A} \star_{A}$ and has basepoint $refl_{\star A}$.

The $n$-fold **iterated loop space** $\Omega^n(A,\star_A)$ can be defined by induction on $n$:
 * $\Omega^0(A,\star_A)=(A,\star_A)$
 * $\Omega^{n+1}(A,\star_A)=\Omega^n(\Omega(A,\star_A))$

[[the HoTT book|The HoTT book]] defines a 'loop space' in definitions 2.1.7 and 2.1.8:

> **Definition 2.1.7** A **pointed type** $(A,a)$ is a type $A\,:\,\mathcal{U}$ together with a point $a\,:\,A$, called its **basepoint**. We write $\mathcal{U}_\bullet :\equiv \sum_{(A:\mathcal{U})}A$ for the type of pointed types in the universe $\mathcal{U}$.

> **Definition 2.1.8** Given a pointed type $(A,a)$, we define the **loop space** of $(A,a)$ to be the following pointed type:

> $\Omega(A,a) = ((a =_A a),\,refl_a)$.

> An element of it will be called a **loop** at $a$. For $n\,:\,\mathbb{N}$, the _n-fold_ **iterated loop space** $\Omega^n(A,a)$ of a pointed type $(A,a)$ is defined recursively by:

> $\Omega^0(A,a) = (A,a)$
> $\Omega^{n+1}(A,a) = \Omega^n(\Omega(A,a))$.

> An element of it will be called an _n-loop_ or an _n-dimensional_ **loop** at $a$.

## Properties

The [[axiom K]] on a type states that for every element $a:A$, [[reflexivity]] $\mathrm{refl}_A(a)$ is the [[center of contraction]] of the [[loop space type]] of $a$.

### Pointed connected groupoids

Given a pointed connected groupoid $(A, a)$, the loop space type $\Omega(A, a)$ is a [[group]]. In fact, for every group $G$, one could form the [[delooping]] of $G$ as a [[higher inductive type]] generated by

* a type $B(G)$,
* an element $*:B(G)$,
* elements $a:G \vdash h(a):\Omega(B(G), *)$,
* identities $a:G, b:G \vdash \mathrm{isGrpHom}(h, a, b):h(a \cdot b) =_{\Omega(B(G), *)} \mathrm{comp}_{*, *, *}(h(a), h(b))$
* a witness $p:\mathrm{isGrpd}(B(G))$

## Related entries

* [[pointed type]]

* [[identity type]]

* [[autoequivalence type]]

## References

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

[[!redirects loop space types]]


loop type

\tableofcontents

## Disambiguation

Similar to [[path types]], there are a few different types which can be called *loop types* or *loop space types* in [[dependent type theory]].

* The [[identity type]] $x =_A x$ for type $A$ and element $x:A$, which is the type of self-identifications of $x:A$ or the based [[loop space type]] $\Omega_A(x)$ of $x:A$;

* The type of pointed type homomorphisms
$$S^1 \to_{*} A \equiv \sum_{f:S^1 \to A} f(\mathrm{base}) =_A x$$
from the [[circle type]] $S^1$ to a type $A$ with element $x:A$ - this parallels the topological definition of [[loop spaces]] as continuous functions out of the [[unit circle]] to a [[pointed topological space]];

The [[categorical semantics]] of loop types in any of the above forms is the [[loop space object]].

There are also other types which could be called the *free loop type* or *free loop space type*:

* The [[dependent sum type]] $\sum_{x:A} x =_A x$ for type $A$, which is the type of all self-identifications in $A$;

* The function type $S^1 \to A$ from the [[circle type]] $S^1$ to a type $A$ - this parallels the topological definition of [[free loop spaces]] as continuous functions out of the [[unit circle]] to a [[topological space]];

The [[categorical semantics]] of free loop types in any of the above forms is the [[free loop space object]].

## Related concepts

* [[path type]]

category: disambiguation

[[!redirects loop type]]
[[!redirects loop types]]

[[!redirects weak loop type]]
[[!redirects weak loop types]]
[[!redirects strict loop type]]
[[!redirects strict loop types]]

[[!redirects free loop type]]
[[!redirects free loop types]]

[[!redirects weak free loop type]]
[[!redirects weak free loop types]]
[[!redirects strict free loop type]]
[[!redirects strict free loop types]]

[[!redirects free loop space type]]
[[!redirects free loop space types]]

[[!redirects weak free loop space type]]
[[!redirects weak free loop space types]]
[[!redirects strict free loop space type]]
[[!redirects strict free loop space types]]


looping

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

For $X$ any kind of [[space]] (or possibly a [[directed space]], viewed as some sort of [[category]] or higher dimensional analogue of one), its [[loop space object]]s $\Omega_x X$ canonically inherit a [[monoidal category|monoidal structure]], coming from concatenation of [[loops]].

If $x \in X$ is essentially unique, then $\Omega_x X$ equipped with this monoidal structure remembers all of the structure of $X$: we say $X \simeq B \Omega_x X$ call $B A$ the _[[delooping]]_ of the monoidal object $A$.

What all these terms ("loops" $\Omega$, "delooping" $B$ etc.) mean in detail and how they are _presented_ concretely depends on the given setup. We discuss some of these below in the section [Examples](#Examples).

## Examples
 {#Examples}

### For plain groups delooping to groupoids
 {#ForPlainGroupsDeloopingToGroupoids}

Write

* [[Grpd]] for the [[(2,1)-category]] of [[groupoids]] ([[objects]] are [[groupoids]], [[1-morphisms]] are [[functors]] between these and [[2-morphisms]] are [[natural transformations]] between those, which are necessarily [[natural isomorphisms]]),

* [[Grp]] for the [[1-category]] of [[groups]] ([[discrete groups]]), also regarded as a [[(2,1)-category]];

* $Grpd^{\ast/}$ for the $(2,1)$-category of [[pointed objects]] in [[Grpd]],

* $Grpd_{\geq 1} \hookrightarrow Grpd$ for the [[full sub-(infinity,1)-category|full sub-(2,1)-category]] of [[connected object in an (infinity,1)-topos|connected]] groupoids, those for which $\pi_0 \simeq \ast$;

* $Grp^{\ast/}_{\geq 1}$ for the [[pointed objects]] in connected groupoids.

* $\pi_1(X,x) \in Grp$ for the [[fundamental group]] of a pointed groupoid $(\ast \stackrel{x}{\to} X) \in Grpd^{\ast/}$ at the given basepoint.

* $\mathbf{B}G \in Grpd$, given a group $G$, for the groupoid $(G\stackrel{\longrightarrow}{\longrightarrow} \ast)$, with composition given by the product in the group. There are two possible choices of conventions, we agree that

  $$
    \array{
      && \ast
      \\
      & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}}
      \\
      \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast
    }
    \,.
  $$

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

The [[(2,1)-category]] $Grp_{\geq 1}$ of connected groupoids is equivalent to its [[full sub-(infinity,1)-category|full sub-(2,1)-category]] on those objects of the form $\mathbf{B}G$, for $G$ a group.

=--

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

Given a connected groupoid $X$, pick any basepoint $x\in X$ and consider the canonical inclusion $\mathbf{B}\pi_1(X,x) \longrightarrow X$. By construction this is [[fully faithful functor|fully faithful]] and by assumption of connectedness it is [[essentially surjective functor|essentially surjective]], hence it is an [[equivalence of groupoids]].

=--

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

The [[hom-groupoids]] between connected groupoids with fundamental groups $G$ and $H$, respectively, are equivalent to the [[action groupoids]] of the set of group [[homomorphisms]] $G \to H$ [[action|acted]] on by [[conjugation]] with elements of $H$:

$$
  Grpd(\mathbf{B}G, \mathbf{B}H)
  \simeq
   Grp(G,H)//_{ad}H
$$

Given two group homomorphisms $\phi_1, \phi_2 \colon G \longrightarrow H$ then an [[isomorphism]] between them in this hom-groupoid is an element $h \in H$ such that

$$
  \phi_2 = Ad_h \circ \phi_1 \coloneqq h^{-1}\cdot \phi_1(-) \cdot h
  \,.
$$

=--

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

By direct inspection of the naturality square for the [[natural transformations]] which are the morphisms in $Grpd(\mathbf{B}G, \mathbf{B}H)$:

$$
  \array{
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
    \\
    \downarrow^{\mathrlap{g_1}} && && \downarrow^{\mathrlap{\phi_1(g_1)}} &&
    \downarrow^{\mathrlap{\phi_2(g_1)}}
    \\
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
    \\
    \downarrow^{\mathrlap{g_2}}
    && &&
    \downarrow^{\mathrlap{\phi_1(g_2)}}
    &&
    \downarrow^{\mathrlap{\phi_2(g_2)}}
    \\
    \ast && && \ast &\stackrel{h}{\longrightarrow}& \ast
  }
  \,.
$$

=--

+-- {: .num_remark #piAs2Functor}
###### Remark

The operation of forming $\pi_1$ is equivalently the operation of forming the [[homotopy fiber product]] of the point inclusion with itself, and hence extends to a [[(2,1)-functor]]

$$
  \pi_1 \colon Grpd^{\ast/} \longrightarrow Grp
  \,.
$$

=--

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

Restricted to [[connected object in an (infinity,1)-topos|connected groupoids]] among the pointed groupoids, the functor $\pi_1 \colon Grpd^{\ast/}_{\geq 1} \longrightarrow Grp$ of remark \ref{piAs2Functor} is an [[equivalence of (2,1)-categories]].

=--

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

It is clear that the functor is essentially surjective: for $G$ any [[group]] then  $\pi_1(\mathbf{B}G,\ast) \simeq G$.

The more interesting point to notice is that $\pi_1$ is indeed a fully faithful [[(2,1)-functor]], in that for any $(X,x), (Y,y) \in Grpd^{\ast/}_{\geq 1}$ then the functor

$$
  (\pi_1)_{X,Y}
  \colon
  Grpd^{\ast/}((X,y),(Y,y))
  \longrightarrow
  Grp(\pi_1(X,x), \pi_1(Y,y))
$$

is an [[equivalence of groupoids|equivalence]] of [[hom-groupoids]]. By prop. \ref{SkeletalRepresentativesForConnectedGroupoids} it is sufficient to check this for $X = \mathbf{B}G$ and $Y = \mathbf{B}H$ with their canonical basepoints, hence to check that for any two groups  $G,H$ the functor

$$
  (\pi_1)_{X,Y}
  \;\colon\;
  Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))
  \longrightarrow
  Grp(G,H)
$$

is an equivalence.

To see this, observe that, by definition of [[pointed objects]] via the [[undercategory]] under the point, a morphism in $Grpd^{\ast/}$ between groupoids of this form $\mathbf{B}(-)$ is a diagram in $Grp$ (unpointed) of the form

$$
  \array{
    && \ast
    \\
    & \swarrow &\swArrow_{h}& \searrow
    \\
    \mathbf{B}G
    &&
    \underset{\mathbf{B}\phi}{\longrightarrow}
    &&
    \mathbf{B}H
  }
$$

where the [[natural isomorphism]] is equivalently just the choice of an element $h \in H$. Hence these morphisms are pairs $(\phi,h)$ of a group homomorphism and an element of the domain.

We claim that the [[(2,1)-functor]] $\pi_1$ takes such $(\phi,h)$ to the homomorphism $Ad_{h^{-1}} \circ \phi \;\colon\; G \longrightarrow H$. To see this, consider via remark \ref{piAs2Functor} this functor as forming loops:

$$
  \pi_1(\mathbf{B}G,\ast)
  =
  \left\{
    \array{
      && \ast
      \\
      & \swarrow && \searrow
      \\
      \ast && \swArrow_{\mathrlap{g}} && \ast
      \\
      & \searrow && \swarrow
      \\
      && \mathbf{B}G
    }
  \right\}_{g\in G}
  \,.
$$

This shows that on a morphism as above this acts by forming the [[pasting]]

$$
  \array{
    && \ast
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{\mathrlap{g}} && \ast
    \\
    & \searrow && \swarrow &\swArrow_{\mathrlap{h}}& \searrow
    \\
    && \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H
  }
  \;\;\;\;
  =
  \:\;\;\;
  \array{
    && && \ast
    \\
    && & \swarrow && \searrow
    \\
    && \ast && \swArrow_{\mathrlap{h\phi(g)h^{-1}}} && \ast
    \\
    & \swarrow & \swArrow_{\mathrlap{h}} & \searrow && \swarrow
    \\
    \mathbf{B}G && \underset{\phi}{\longrightarrow} && \mathbf{B}H
  }
  \,.
$$

Unwinding the [[whiskering]] of [[natural transformations]] here, the claim follows, as indicated by the label of the upper 2-morphisms on the right.

One observes now that these extra labels $h$ are precisely the information that "trivializes" the conjugation action which in prop. \ref{HomsBetweenBGs} prevents the bare set of group homomorphism: a [[2-morphism]] $(\phi_1, h_1) \Rightarrow (\phi_2,h_2)$ in $Grp^{\ast/}$  is a natural isomorphism of groupoids

$$
  \array{
    \mathbf{B}G
    &\stackrel{\phi_1}{\longrightarrow}&
    \mathbf{B}H
    \\
    {}^{\mathllap{id}}\downarrow &\Downarrow^{\mathrlap{h}}& \downarrow^{\mathrlap{id}}
    \\
    \mathbf{B}G
    &\underset{\phi_2}{\longrightarrow}&
    \mathbf{B}H
  }
$$

(encoding a conjugation relation $\phi_2 = Ad_{h} \circ \phi_1$ as above) such that we have the [[pasting]] relation

$$
  \array{
    && \ast
    \\
    & \swarrow &\swArrow_{h_1}& \searrow
    \\
    \mathbf{B}G
    &&
    \stackrel{\phi_1}{\longrightarrow}
    &&
    \mathbf{B}H
    \\
    {}^{\mathllap{id}}\downarrow && \Downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{id}}
    \\
    \mathbf{B}G
    &&\underset{\phi_2}{\longrightarrow} &&
    \mathbf{B}H
  }
  \;\;\;\;\;
  =
  \;\;\;\;\;
  \array{
    && \ast
    \\
    & \swarrow &\swArrow_{h_2}& \searrow
    \\
    \mathbf{B}G
    &&
    \underset{\phi_2}{\longrightarrow}
    &&
    \mathbf{B}H
  }
  \,.
$$

But this says in components that $h_2 = h_1\cdot h$. Hence there is a _at most one_ morphism in $Grpd^{\ast/}((\mathbf{B}G,\ast),(\mathbf{B}H,\ast))$ from $(\phi_1,h_1)$ to $(\phi_2,h_2)$: it exists if $\phi_2 = Ad_h \circ \phi_1$ and $h_2 = h_1\cdot h$.

But since, by the previous argument, the functor $\pi_1$ takes $(\phi_1,h_1)$ to $Ad_{h_1^{-1}} \circ \phi_1$, this means that such a morphism exists precisely if both $(\phi_1,h_1)$ and $(\phi_2,h_2)$ are taken to the same group homomorphism by $\pi_1$

$$
  Ad_{h_2^{-1}} \circ \phi_2
  =
  Ad_{h^{-1}\cdot h_1^{-1}}\circ \phi_2
  =
  Ad_{h_1^{-1}} \circ \phi_1
  \,.
$$

This establishes that $\pi_1$ is alspo an equivalence on all [[hom-groupoids]].

=--

This proof also shows that $\mathbf{B}(-)$ is in fact the inverse equivalence:

+-- {: .num_cor}
###### Corollary

There is an [[equivalence of (2,1)-categories]] between pointed connected groupoids and plain groups

$$
  Grp
  \stackrel{\underoverset{\simeq}{\pi_1 = \Omega_\ast}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}}
  Grpd^{\ast/}_{\geq 1}
$$

given by forming [[loop space objects]] and by forming deloopings.

=--

### For topological spaces and $\infty$-groupoids

There is an [[equivalence of (∞,1)-categories]]

$$
  \infty Grpd^{\ast/}_{\geq 1}
    \stackrel{\overset{B}{\leftarrow}}{\underset{\Omega}{\to}}
  \infty Group
$$

between [[pointed object|pointed]] [[connected]] [[∞-groupoid]]s and [[∞-group]]s, where $\Omega$ forms [[loop space object]]s.

This is [[presentable (∞,1)-category|presented]] by a [[Quillen equivalence]] of [[model categories]]

$$
  sSet_*
    \stackrel{\overset{\bar W}{\leftarrow}}{\underset{G}{\to}}
  sGrp
$$

between the [[model structure on reduced simplicial sets]] and the [[transferred model structure]] on [[simplicial group]]s along the [[forgetful functor]] to the [[model structure on simplicial sets]].

(See [[groupoid object in an (infinity,1)-category]] for more details on this Quillen equivalence.)

### For parameterized $\infty$-groupoids ($\infty$-stacks / $(\infty,1)$-sheaves)

The following result makes precise for _parameterized [[∞-groupoid]]s_  -- for [[∞-stack]]s -- the general statement that $k$-fold [[delooping]] provides a correspondence between [[n-category|n-categories]] that have trivial [[k-morphism|r-morphism]]s for $r \lt k$ and  [[k-tuply monoidal n-category|k-tuply monoidal n-categories]].

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

An [[Ek-algebra]] object $A$ in an [[(∞,1)-topos]] $\mathbf{H}$ is called **groupal** if its [[homotopy groups in an (∞,1)-topos|connected components]] $\pi_0(A) \in \mathbf{H}_{\leq 0}$ is a [[group object]].

Write $Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H})$ for the [[(∞,1)-category]] of groupal $E_k$-algebra objects in $\mathbf{H}$.

A groupal $E_1$-algebra -- hence an groupal [[A-∞ algebra]] object in $\mathbf{H}$ -- we call an _[[∞-group]]_ in $\mathbf{H}$. Write $\infty Grp(\mathbf{H})$ for the [[(∞,1)-category]] of [[∞-group]]s in $\mathbf{H}$.

=--

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

Let $k \gt 0$, let $\mathbf{H}$ be an [[(∞,1)-category of (∞,1)-sheaves]] and let $\mathbf{H}_*^{\geq k}$ denote the [[full subcategory]] of the category $\mathbf{H}_{*}$ of [[pointed objects]], spanned by those pointed objects thar are $k-1$-[[connected]] (i.e. their first $k$ [[homotopy groups in an (∞,1)-topos|homotopy sheaves]]) vanish. Then there is a canonical equivalence of [[(∞,1)-category|(∞,1)-categories]]

$$
  \mathbf{H}^{\ast/}_{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathbf{H}) \,.
$$

between the pointed $(k-1)$-[[connected]] objects and the groupal [[Ek-algebra]] objects in $\mathbf{H}$.

=--

This is _[[Higher Algebra|Higher Algebra, theorem 5.2.6.10]]_

Specifically for $\mathbf{H} = $ [[Top]], this reduces to the following classical theorem due to [[Peter May]], known as the [[May recognition theorem]].

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

Let $Y$ be a [[topological space]] equipped with an action of the [[little cubes operad]] $\mathcal{C}_k$ and suppose that $Y$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space $\Omega^k X$ for some pointed topological space $X$.

=--

This is _[[Higher Algebra|Higher Algebra, theorem 5.2.6.15]]_

+-- {: .num_remark #LoopingDeloopingDegree1InTopos}
###### Remark

For $k = 1$ we have a looping/delooping equivalence

$$
  Grp(\mathbf{H})
  \stackrel{\overset{\Omega}{\longleftarrow}}{\underset{\mathbf{B}}{\longrightarrow}}
  \mathbf{H}_{\geq 1}^{\ast /}
$$

between pointed connected objects in $\mathbf{H}$ and grouplike [[A-∞ algebra]] objects in $\mathbf{H}$: [[∞-group]] objects in $\mathbf{H}$.

=--

+-- {: .num_remark}
###### Remark

If the ambient [[(∞,1)-topos]] has [[homotopy dimension]] 0 then every connected object $E$ admits a point $* \to E$. Still, the [[(∞,1)-category]] of pointed connected objects differs from that of unpointed connected objects (because in the latter the [[natural transformation]]s may have nontrivial components on the point, while in the former case they may not).

The connected objects $E$ which fail to be [[∞-group]]s by failing to admit a point are of interest: these are the _[[∞-gerbes]]_ in the [[(∞,1)-topos]].

=--

### For cohesive $\infty$-groupoids

A special case of the parameterized $\infty$-groupoids above are [[cohesive ∞-groupoid]]s. Looping and delooping for these is discussed at
[[cohesive (∞,1)-topos -- structures]] in the section Cohesive ∞-groups.

### For $(\infty,n)$-categories

See [[delooping hypothesis]].

## Relation to looping and suspension

For $A$ any monoidal [[space]], we may [[forgetful functor|forget]] its monoidal structure and just remember the underlying [[space]].  The formation of [[loop space object]]s composed with this [[forgetful functor]] has a [[left adjoint]] $\Sigma$ which forms _[[suspension objects]]_.

## Related concepts

* [[looping]], [[delooping]]

* [[May recognition theorem]]

* [[Quillen equivalence between simplicial groups and reduced simplicial sets]]

* [[pointed connected groupoid]]

* [[loop space object]], [[free loop space object]],

  * [[delooping]], **looping and delooping**

  * [[loop space]],

  * [[free loop space]], [[derived loop space]], [[free loop stack]]

* [[suspension object]]

  * [[suspension]], [[reduced suspension]]

[[!include k-monoidal table]]

## References

* [[Jacob Lurie]], Section 6.1.2 of: _[[Higher Topos Theory]]_

* {#LurieAlgebra} [[Jacob Lurie]],
Section 5.1.3 of: _[[Higher Algebra]]_

Discussion in [[homotopy type theory]]:

* [[David Wärn]], _Eilenberg-MacLane spaces and stabilisation in homotopy type theory_ [[arXiv:2301.03685](https://arxiv.org/abs/2301.03685)]

[[!redirects looping]]
[[!redirects looping and delooping]]


looping combinator

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

# Looping combinators
* table of contents
{: toc}

## Idea

In [[combinatory logic]], in the [[∞-calculus]], or more generally in [[type theory]], a *looping [[combinator]]* is closely akin to a [[fixed-point combinator]], but rather than producing a true [[fixed point]], it produces a sequence of points each of which is the [[image]] of the next.

## Definition

+-- {: .num_defn}
###### Definition
A term $Y$ is a **looping combinator** if for any [[function type|function]] [[term]] $f$ to which $Y$ can be applied, we have a [[beta reduction]]
$$ Y f \to_\beta f (Y' f) $$
where $Y'$ is a looping combinator.
=--

This is a [[coinductive definition]].  Unraveled explicitly, it means that a looping combinator $Y = Y_0$ comes with a sequence of combinators $Y_n$ for $n\in\mathbb{N}$ and reductions
$$ Y_n f \to_\beta f(Y_{n+1} f).$$

## Implementing general recursion

A looping combinator is essentially just as good as a [[fixed-point combinator]] for implementing general [[recursion]].  See the discussion there for details.

## Existence

[[Per Martin-Löf]]'s original [[dependent type theory]], which had the additional rule $\vdash Type:Type$, was shown to be [[inconsistency|inconsistent]] by [[Girard's paradox]].  In the 1980's, Meyer, Reinhold, and Howe (see references) showed that this paradox could be modified to construct a looping combinator.

## References

In the short paper

* Albert Meyer and Mark Reinhold, "'Type' is not a type", POPL 1986

it was claimed that from Girard's paradox one could actually construct a fixed-point combinator.  The proof turned out to be flawed, but it was sufficient to produce a looping combinator.  Details can be found in

* Mark Reinhold, "Typechecking is Undecidable When 'Type' is a Type", 1989, [citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.18.7073)

* Douglas Howe, "The Computational Behaviour of Girard's Paradox", Cornell University Technical Report, 1987, [link](http://hdl.handle.net/1813/6660).

Further analysis is in

* [[Herman Geuvers]], Joep Verkoelen, *On Fixed point and Looping Combinators in Type Theory* (2009) [[pdf](http://www.cs.ru.nl/~herman/PUBS/TLCApaper.pdf), [citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.158.1478)]

[[!redirects looping combinators]]


loopoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Just as a [[groupoid]] is the [[oidification]] of a [[group]] and a [[ringoid]] is the oidification of a [[ring]], a [[loopoid]] should be the oidification of a [[loop]].

## Definition

A __loopoid__ $Q$ is a [[magmoid]] where every object $a \in Ob(Q)$ has an [[identity morphism]] $id_a: a \to a$, such that for any morphism $f:a \to b$, $f \circ id_a = f$, and for any morphism $g:c \to a$, $id_a \circ g = g$, where every [[span]]

$$
  \array{
     && s
      \\
      & {}^{f}\swarrow
      && \searrow^{g}
     \\
     x
     &&&&
     y
  }
$$

in $Q$ has morphisms $i:x\to y$ and $j:y \to x$ such that $i \circ f = g$ and $j \circ g = f$, and where every [[cospan]]

$$
  \array{
     && a &&&& b
      \\
      &
      && {}_{f}\searrow
       &
       & \swarrow_g
      &&
     \\
     &&&&
     c
     &&&&
  }
$$

in $Q$ has morphisms $d:a\to b$ and $e:b \to a$ such that $g \circ d = f$ and $f \circ e = g$.

## Examples

* A [[groupoid]] is a [[loopoid]].

* A loopoid with only one object is called a [[loop]].

* A loopoid [[enriched magmoid|enriched]] on [[truth values]] is an [[equivalence relation]].

## Related concepts

[[!include oidification - table]]


LOOPS'22

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic quantum field theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Overview

*[LOOPS'22](https://indico.cern.ch/event/1100970/)* is a [[theoretical physics]] conference held in Lyon, France between 18 July and 22 July 2022. This conference consists of a variety of subjects at the frontier of theoretical physics, such as [[cosmology]], [[holography]], [[black hole]] physics, [[loop quantum gravity]], and other topics in [[quantum field theory]] and [[quantum gravity]]. The website for this conference could be found [here](https://indico.cern.ch/event/1100970/).

## Talks

### 18 July

#### Plenary talks

* [[Carlo Rovelli]], _Loop quantum Gravity: present and future_

* [[Ivan Agullo]], _Event horizons are tunable factories of quantum entanglement_

* [[Laurent Freidel]], _A new paradigm for quantum gravity based on symmetry_

* [[Daniele Oriti]], _Cosmology from full quantum gravity in the group field theory formalism_

#### Parallel session B: Black holes

* [[Jibril Ben Achour]], _Exploring black holes in modified gravity_

* [[Edward Wilson-Ewing]], _Shock waves from quantum black holes_

* [[Yuki Yokokura]], _Black hole as a bound state of semi-classical degrees of freedom_

* [[Farshid Soltani]], _Black to white hole transition_

* [[Rodrigo Eyheralde]], _Hawking radiation from a sandwich evaporating black hole_

* [[Jinzhao Wang]], _Demystifying the replica trick calculation of the entropy of Hawking radiation_

* [[Eugenia Colafranceschi]], _Modelling black hole horizons via random spin networks_

* [[Yili Wang]], _Spherically-symmetric geometries in a matter reference frame as quantum gravity condensates_

* [[Michał Bobula]], _Radiation in quantum gravitational collapse_

* [[Ian Bornhoeft]], _Residual diffeomorphisms in Kantowksi-Sachs spacetime_

* [[Rafael Guolo Dias]], _Diffeomorphism covariant dynamics in quantum Kantowski-Sachs_

#### Parallel session D: Quantum gravity amplitudes

* [[Alexander Jercher]], _Causal structure in the complete Barrett-Crane model_

* [[Tobias Bommer]], _Timelike faces and lightlike vectors within the CH-extended EPRL model asymptotics_

* [[Seth Asante]], _Gravity dynamics from effective spin foam models_

* [[Johanna Borissova]], _Towards a continuum effective action for spin foams_

* [[Athanasios Kogios]], _On the continuum limit of spin foams: graviton dynamics and area metric induced corrections_

* [[Hongguang Liu]], _Lefschetz thimbles, complex critical points and curved geometries in Lorentzian spinfoam model_

* [[Simone Speziale]], _Simplicity without simplicity constraints_

* [[Sebastian Steinhaus]], _Towards "matter matters" in spin foam quantum gravity_

* [[Andreas Pithis]], _Towards the phase structure of the complete Lorentzian Barrett-Crane model_

* [[Roukaya Dekhil]], _A new spin-foam model based on edge vectors_

* [[Steffen Gielen]], _Dirac quantisation and deparametrisation in group field theory_

* [[Gabriel Schmid]], _Transition amplitudes and holography in the coloured Boulatov-Ooguri model_

### 19 July

(...)

category: reference


Lord Kelvin

* [Wikipedia entry](http://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin)

## related $n$Lab entries

* _[[On Vortex Atoms]]_

category: people


Lorentz force

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

#Contents#
* table of contents
{:toc}

## Idea

The _Lorentz force_ is the [[force]] exerted on a [[particle]] that is charged under the [[electromagnetic field]].

See [[relativistic particle]] for details.

## Properties

### In Kaluza-Klein compactifications

If one models the [[electromagnetic field]] via the [[Kaluza-Klein mechanism]] as a field of [[gravity]] on a [[fiber bundle]], then [[trajectories]] of [[charged particles]] subject to the Lorentz force in the base space of that bundle are euivalently just [[geodesics]] on the total space, e.g. ([Bleecker 81](#Bleecker81)). An exposition of this fact is in ([Bartlett 13](#Bartlett13)).

## Related concepts

* [[strong nuclear force]]

* [[weak nuclear force]]

## References

Named after [[Hendrik Lorentz]].

Textbook accounts:

* V. C. A. Ferraro, §303 in: *Electromagnetic Theory*, The Athlone Press (1954, 1956)

Discussion in [[worldline]] [[Lagrangian mechanics]]:

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]], Exc. 7.2 (p. 179) in: *[[Gravitation]]*, W. H. Freeman, San Francisco (1973) [ISBN:9780716703440]

* {#Frankel97} [[Theodore Frankel]], §16.4b in: *[[The Geometry of Physics - An Introduction]]*, Cambridge University Press (1997, 2004, 2012) [[doi:10.1017/CBO9781139061377](https://doi.org/10.1017/CBO9781139061377)]

See also

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

On generalization to the [[spinning particle]]:

* [[Jan-Willem van Holten]], *Relativistic Dynamics of Spin in Strong External Fields* [[arXiv:hep-th/9303124](http://arxiv.org/abs/hep-th/9303124)]

Discussion of Lorentz force from the point of view of the [[Kaluza-Klein mechanism]]:

* {#Bleecker81} David Bleecker, _Gauge theory and variational principles_, Dover publications, 1981.

and exposition is in

* {#Bartlett13} [[Bruce Bartlett]], _The geometry of force_, Sept 2013 ([pdf](https://dl.dropboxusercontent.com/u/56141091/Lorentz.pdf)).

[[!redirects Lorentz forces]]

[[!redirects electromagnetic force]]
[[!redirects electromagnetic forces]]


Lorentz group

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

#Contents#
* table of contents
{:toc}

## Definition

The _Lorentz group_ is the [[orthogonal group]] for an invariant [[bilinear form]] of [[signature of a quadratic form|signature]] $(-+++\cdots)$, $O(d-1,1)$.

In [[physics]], in the [[theory (physics)|theory]] of _[[relativity]]_ the Lorentz group [[action|acts]] canonically as the group of linear [[isometries]] of [[Minkowski spacetime]] preserving a chosen basepoint. This is called the action by _Lorentz transformations_.

The elements in the Lorentz group in the image of the [[special orthogonal group]] $SO(d-1) \hookrightarrow O(d-1,1)$ are _[[rotations]]_ in space. The further elements in the special Lorentz group $SO(d-1,1)$, which mathematically are "hyperbolic rotations" in a space-time plane, are called _[[boost|boosts]]_ in [[physics]].

One distinguishes the following further [[subgroups]] of the [[Lorentz group]] $O(d-1,1)$:

* the _proper Lorentz group_ $SO(d-1,1)$ is the subgroup of elements which have [[determinant]] +1 (as elements  $SO(d-1,1)\hookrightarrow GL(d)$ of the [[general linear group]]);

* the _proper orthochronous_ (or _restricted_) Lorentz group $SO^+(d-1,1) \hookrightarrow SO(d-1,1)$ is the further [[subgroup]] of elements which do not act by [[reflection]] along the [[timelike]] axis.

## Properties

### Connected components
 {#ConnectedComponents}

As a [[smooth manifold]], the Lorentz group $O(3,1)$ has four [[connected components]]. The connected component of the identity is the the proper orthochronous group $SO^+(3,1)$. The other three components are

1. $SO^+(3,1)\cdot P$

1. $SO^+(3,1)\cdot T$

1. $SO^+(3,1)\cdot P T$,

where, as [[matrices]]

$$
  P \coloneqq diag(1,-1,-1, \cdots, -1)
$$

is the operation of point reflection at the origin in space,  where

$$
  T \coloneqq diag(-1,1,1, \cdots, 1)
$$

is the operation of reflection in time and hence where

$$
  P T = T P = diag(-1,-1, \cdots, -1)
$$

is point reflection in spacetime.

### Spin cover

While the proper orthochronous Lorentz group $SO^+(d-1,1)$ is [[connected topological space|connected]], it is not [[simply connected]]. Its [[universal covering space|universal]] [[double cover]] is the Lorentzian [[spin group]] $Spin(d-1,1)$

## Related concepts

* [[Lorentz Lie algebra]]

* [[quantum group]] version: [[quantum Lorentz group]]

* [[Lorentzian manifold]]

* [[rapidity]]

* [[Galilean group]]

* [[Minkowski spacetime]]

* [[de Sitter spacetime]]

* [[anti de Sitter spacetime]]

[[!include table of orthogonal groups and related]]

## References

Named after [[Hendrik Lorentz]]. (Not to be confused with [[Ludvik Lorenz]], whose name is attached to the [[Lorenz gauge]].)

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

[[!redirects Lorentz groups]]

[[!redirects Lorentz transformation]]
[[!redirects Lorentz transformations]]

[[!redirects proper Lorentz group]]
[[!redirects proper Lorentz groups]]

[[!redirects proper orthochronous Lorentz group]]
[[!redirects proper orthochronous Lorentz groups]]


Lorentz Lie algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Lorentz Lie algebra_ $\mathfrak{o}(d-1,1)$ is the [[Lie algebra]] of the [[Lorentz group]] $O(d-1,1)$ (the group of [[linear map|linear]] [[isometries]] of [[Minkowski spacetime]]). In particular, it is an [[orthogonal Lie algebra]].

It coincides with the Lie algebras $\mathfrak{so}(d-1,1)$ and $\mathfrak{so}^+(d-1,1)$ corresponding to the [[special orthogonal group]] $SO(d-1,1)$ and the proper orthochronous Lorentz group $SO^+(d-1,1)$, respectively. Indeed, the Lie algebra of any [[Lie group]] remains unchanged after passing to a subgroup of finite [[index of a subgroup|index]].

## Related concepts

* [[Poincare Lie algebra]]

* [[Lorentz group]]

* [[Minkowski spacetime]]

## References

See also

* Wikipedia, _[Lorentz group -- Lie algebra](https://en.wikipedia.org/wiki/Lorentz_group#Lie_algebra)_

[[!redirects Lorentz Lie algebras]]

[[!redirects Lorentz algebra]]
[[!redirects Lorentz algebras]]