string (computer science)

> This entry is about the notion in [[computer science]]. For the notion in [[particle physics]]/[[quantum gravity]] see at *[[string theory]]*.

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### Context
#### Computing
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[[!include constructivism - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[computer science]], "$string$" is a traditional name for the [[data type]] of [[lists]] of elements of a given [[alphabet]] $\mathscr{A}$.

From the point of view of [[categorical semantics]], the data type $String$ equipped with its evident [[concatenation]] functionality is the [[free monoid]] on $\mathscr{A}$.

Moreover, from the point of view of [[monads in computer science|monads]] in [[functional programming]], the writing of consecutive log messages of type $String$ is desribed by the [[writer monad]] $Writer\big((string, conc)\big)$

## Related concepts

* [[stream]]

* [[data type]]

* [[list]], [[free monoid]]

* [[language]], [[word]]

* [[syntax]], [[abstract syntax tree]]

## References

See also:

* Wikipedia, *String_(computer_science)*

[[!redirects strings (computer science)]]

[[!redirects string (in computer science)]]
[[!redirects strings (in computer science)]]

[[!redirects string in computer science]]
[[!redirects strings in computer science]]

[[!redirects String (computer science)]]
[[!redirects Strings (computer science)]]


string 2-group

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###Context###
#### Higher Lie theory
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[[!include infinity-Lie theory - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#### Higher spin geometry
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[[!include higher spin geometry - contents]]
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#### String theory
+-- {: .hide}
[[!include string theory - contents]]
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=--

#Contents#
* table of contents goes here
{:toc}

## Idea

The _string 2-group_ is a [[smooth ∞-groupoid|smooth 2-group]]-refinement of the [[topological group]] called the [[string group]]. It is the [[∞-group extension]] induced by the smooth/stacky version of the [[first fractional Pontryagin class]]/[[second Chern class]].

## Definition

A string 2-group extension $String(G)$ is defined for every [[simple Lie group|simple]] [[simply connected topological space|simply connected]] [[compact Lie group]] $G$, such as the [[spin group]] $G = Spin(n)$ or the [[special unitary group]] $G = SU(n)$ (for non-low $n$).

Since [[string structures]] arise predominantly as higher analogs of [[spin structures]], the default choice is $G = Spin$ and in that case one usually just writes $String = String(Spin)$, for short.

Recall first that the [[string group]] in [[Top]] is one step in the [[Whitehead tower]] of the [[orthogonal group]].

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

For $n \in \mathbb{N}$ let $Spin(n)$ denote the [[spin group]], regarded as a [[topological group]]. Write $B Spin(n) \in $ [[Top]] for its [[classifying space]] and

$$
  \frac{1}{2}p_1 : B Spin(n) \to B^4 \mathbb{Z}
$$

for a representative of the [[characteristic class]] called the first fractional [[Pontryagin class]]. Its [[homotopy fiber]] in the [[(∞,1)-topos]] [[Top]] $\simeq$ [[∞Grpd]] is denoted $B String(n) := B O(n)\langle 7 \rangle$

$$
  \array{
     B String(n) &\to& *
     \\
     \downarrow && \downarrow
     \\
     B Spin(n) &\stackrel{\frac{1}{2} p_1}{\to}& B^4 \mathbb{Z}
  }
  \,.
$$

The [[loop space]]

$$
  String(n) := \Omega B String(n)
$$

is the [[∞-group]]-object in [[Top]] called the **[[string group]]**.

=--

Write now

$$
  (\Pi \dashv Disc \dashv \Gamma \dashv coDisc)
   :
  Smooth\infty Grpd
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}
  \infty Grpd
    \simeq
  Top
$$

for the [[(∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s, regarded as a [[cohesive (∞,1)-topos]] over [[∞Grpd]].

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

There is a lift through $\Pi$ of $\frac{1}{2} p_1$ to the smooth first fractional Pontryagin class

$$
  \frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin(n) \to \mathbf{B}^3 U(1)
$$

in [[Smooth∞Grpd]], where

* $\mathbf{B}Spin$ is the [[delooping]] [[Lie groupoid]] of $Spin(n)$ regarded as a [[Lie group]] (see Smooth ∞-groupoids -- Cohesive ∞-groups -- Lie groups);

* $\mathbf{B}^3 U(1)$ is the three-fold delooping of the [[circle group]], regarded as a [[Lie group]] (see Smooth ∞-groupoids -- Cohesive ∞-groups -- Circle Lie n-group);

* $\frac{1}{2}\mathbf{p}_1$ is the image under [[Lie integration]] of the canonical [[Lie algebra cohomology|cocycle]]

  $$
    \mu = \langle -,[-,-]\rangle : \mathfrak{so}(n) \to b^2 \mathbb{R}
    \,.
  $$

  on the [[orthogonal Lie algebra]].

=--

This is shown in ([FSS](#FSS)).

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

Write $\mathbf{B}String(n)$ for the [[homotopy fiber]] of the smooth
first fractional Pontryagin class

$$
  \array{
    \mathbf{B}String &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \mathbf{B}^3 U(1)
  }
$$

in [[Smooth∞Grpd]]. Its [[loop space object]]

$$
  String(n) := \Omega \mathbf{B}String(n)
$$

is the [[∞-Lie group|smooth ∞-group]] called [[generalized the|the]] **smooth string 2-group**.

=--

## Properties
 {#Properties}

Write

$$
  \vert - \vert := \vert\Pi(-)\vert :
  Smooth \infty Grpd
    \stackrel{\Pi}{\to}
  \infty Grpd
    \stackrel{\vert - \vert}{\to}
  Top
$$

for the intrinsic geometric realization in [[Smooth∞Grpd]].

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

The smooth string 2-group, def. \ref{AbstractDefinitionSmooth},
indeed maps under $\vert-\vert$ to the topological string group:

$$
  \vert \mathbf{B}String(n) \vert
  \simeq
  B String(n)
  \,.
$$

=--

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

Since $\mathbf{B}^3 U(1)$ is presented by a simplicial presheaf that is degreewise presented by a paracompact smooth manifold (a finite product of the [[circle group]] with itself), it follows from the general properties of $\Pi$ discussed at [[Smooth∞Grpd]] that $\Pi$ preserves the [[homotopy fiber]] of $\frac{1}{2}\mathbf{p}_1$.

=--

## Presentations

Several explicit presentations of the string Lie 2-group are known.

### By Lie integration of the string Lie 2-algebra
 {#ByLieIntegrationOfStringLie2Algebra}

We discuss a presentation of the smooth string 2-group by [[Lie integration]] of the skeletal version of the [[string Lie 2-algebra]].

Recall the identification of [[L-∞ algebra]]s $\mathfrak{g}$ with their dual [[Chevalley-Eilenberg algebra]]s $CE(\mathfrak{g})$.

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

Write

$$
  \mu := \langle - ,[-,-]\rangle
  :
  \mathfrak{so}(n) \to b^2 \mathbb{R}
$$

for the canonical degree-3 [[cocycle]] in the [[Lie algebra cohomology]] of the [[special orthogonal group]], normalized such that the 3-form

$$
  \Omega^\bullet(Spin(n)) \hookleftarrow
  CE(\mathfrak{so}(n))
  \stackrel{\mu}{\leftarrow}
  CE(b^2 \mathbb{R})
$$

represents the image in [[de Rham cohomology]] of a generators of the [[integral cohomology]] group $H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$.

Define the **[[string Lie 2-algebra]]**

$$
  \mathfrak{string}(n) := \mathfrak{so}(n)_\mu
$$

to be given by the [[Chevalley-Eilenberg algebra]]

$$
  CE(\mathfrak{string}(n)) := \wedge^\bullet ( \mathfrak{so}(n)^* \oplus \langle b\rangle , d_{\mathfrak{string}})
$$

which is that of $\mathfrak{so}(n)$ with a single generator $b$ in degree 3 adjoined and the differential given by

$$
  d_{\mathfrak{string}}|_{\mathfrak{so}(n)^*} = d_{\mathfrak{so}(n)};
$$

$$
  d_{\mathfrak{string}} : b \mapsto \mu
  \,.
$$

=--

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

We have a [[pullback]] square in $L_\infty Alg$

$$
  \array{
    \mathfrak{string}(n) &\to& e b \mathbb{R}
    \\
    \downarrow && \downarrow
    \\
    \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R}
  }
  \,.
$$

=--

See [[string Lie 2-algebra]] for more discussion.

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

The [[Lie integration]] of $\mathfrak{string}(n)$ yields a presentation of the
smooth String 2-group, def. \ref{AbstractDefinitionSmooth}

$$
  \mathbf{cosk}_3 \exp(\mathfrak{string}(n))
  \simeq
  \mathbf{B} String(n)
  \,.
$$

=--

This is essentially the model considered in ([Henriques](#Henriques)), discussed here in the context of [[Smooth∞Grpd]] as described in ([FSS](#FSS)).

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

We observe the image under [[Lie integration]] of the $L_\infty$-algebra pullback diagram from prop. \ref{HomotopyFiberOfLInftyAlgebraCocycle} is a pullback diagram in $[CartSp_{smooth}^{op}, sSet]_{proj}$ that presents the defining [[homotopy fiber]]. Before applying the [[coskeleton]] operation we have immediately

$$
  \exp(-)
    \; :\;
  \left(
   \array{
     \mathfrak{string}(n) &\to& e b \mathbb{R}
     \\
     \downarrow && \downarrow
     \\
     \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R}
   }
  \right)
  \;\mapsto \;
  \left(
   \array{
     \exp(\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R})
     \\
     \downarrow && \downarrow
     \\
     \exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& \exp(b^2 \mathbb{R})
   }
  \right)
$$

such that on the right we still have a [[pullback]] diagram.

We discuss the descent o this pullback diagram along the projection $\exp(\mathfrak{so}(n)) \to \mathbf{cosk}_3 \exp(\mathfrak{so}(n))$.

Notice from [[Lie integration]] the weak equivalence

$$
  \int_{\Delta^\bullet} : \exp(b^n \mathbb{R}) \simeq \mathbf{B}^{n+1}\mathbb{R}_c
  \,.
$$

Let $I$ be the set of maps $\partial \Delta[4] \to \exp(b^2 \mathbb{R})$
that fit into a diagram

$$
  \array{
    \partial \Delta[4] &\to& \exp(b^2 \mathbb{R})
    \\
    \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}}
    \\
    && \mathbf{B}^3 \mathbb{R}_c
    \\
    \downarrow && \downarrow
    \\
    \Delta[4] &\to& \mathbf{B}^3 (\mathbb{Z} \to \mathbb{R})_c
  }
$$

(closed 3-forms on 3-balls whose integral is an integer).

Write

$$
  \exp(b^2 \mathbb{R}/\mathbb{Z})
    :=
  \mathbf{cosk}_3
   \left(
     (I \times \Delta[4])\coprod_{I \times \partial \Delta[4]} \mathbf{cosk_3} \exp(b^2 \mathbb{R})
   \right)
$$

for the result of filling all these by 4-cells. Similarly define
$\exp(e b \mathbb{R}/\mathbb{Z})$.

Then applying the [[coskeleton]] functor to the above pullback
diagram and using the projection ([FSS](#FSS))

$$
  \array{
    \exp(\mathfrak{so}(n)) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_3\exp(\mathfrak{so}(n))
      &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \exp(b^2 \mathbb{R}/\mathbb{Z})
  }
$$

we get the diagram

$$
  \array{
    \mathbf{cosk}_3 (\mathfrak{string}(n))
      &\to&
    \exp(e b \mathbb{R}/\mathbb{Z})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_3 \exp(\mathfrak{so})
    &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \exp(b^2 \mathbb{R}/\mathbb{Z})
  }
  \,.
$$

This is again a pullback diagram of a fibration resolution of the point inclusion, hence presents the homotopy fiber in question.

=--

### By strict Lie $2$-group
 {#PresentationByStrictTwoGroups}

A realization of the string 2-group as a [[strict 2-group]] [[internalization|internal]] to [[diffeological space]]s was given in ([BCSS](#BCSS)).

This is one of three different (there should be more), weakly equivalent such [[strict 2-group]] [[internalization|internal]] to [[diffeological space]] models that are discussed in the (to date unpublished)

* [nactwist, section 5.2.3](http://www.math.uni-hamburg.de/home/schreiber/nactwist.pdf#page=91)

(This particular section, and its results, are joint work of [[Urs Schreiber]] and [[Danny Stevenson]]).

We have the following pattern of routes through [[Lie integration]]:

$$
  \array{
    StrLie \omega Grpd
    &&&&
    StrLie \omega Grpd
    &\stackrel{\simeq}{\leftarrow}&
    LieCrsdCmplx
    \\
    \uparrow^{\Pi_n S CE}
    &&&&
    \uparrow
    &&
    \uparrow^{\exp(-)}
    \\
    L_\infty Algebras
    &&
    \leftarrow&&
    Str L_\infty Algebras
    &\to&
    DiffCrsdCmplx
  }
$$

Here $StrLie \omega Grpd$ is [[strict omega-groupoid]]s internal to [[diffeological space]]s, $LieCrsCmplx$ is accordingly smooth [[crossed complex]]es , $L_\infty Algebra$ is all [[L-infinity algebra]]s and $Str L_\infty Algebra$ is _strict_ $L_\infty$-algebras. The vertical morphism on the right is term-wise ordinary [[Lie integration]]. The other vertical morphisms take an [[L-infinity algebra]], form the [[sheaf]] on [[Diff]] of flat [[schreiber:∞-Lie algebroid valued differential forms|∞-Lie algebroid differential form]]s, and then take [[path n-groupoid]] $\Pi_n(-)$ of that.

For the String-case this yields

$$
  \array{
    \Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{so}_{\mu_3}))
    &\stackrel{\simeq}{\mapsto}&
    \mathbf{B} String_{Mick}
    &\stackrel{\simeq}{\mapsto}&
    \mathbf{B} String_{BCSS}
    &\leftarrow|&
    (\hat \Omega Spin \to P Spin)
    \\
    \uparrow &&&\nearrow& \uparrow  && \uparrow
    \\
    \mathfrak{so}_{\mu_3}
    &&\stackrel{\simeq}{\mapsto}&&
    \mathfrak{string}
    &\mapsto&
    (\hat \Omega \mathfrak{so} \to P \mathfrak{so})
  }
  \,,
$$

where

* $\mathfrak{so}_{\mu_3}$ denotes the weak, skeletal [[String Lie 2-algebra]]

* $\mathfrak{string}$ its equivalent strict version given by BCSS

* the diagonal morphism is the construction in BCSS.

* the strict 2-groupoid $\Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{g}_{\mu_3}))$ has, notice, as morphism smooth paths in $Spin(n)$ that are composed by concatenation

* the 2-groupoid $\mathbf{B}String_{Mick}$ is a version of the String Lie 2-group that manifestly uses the [[Mickelsson cocycle]] (morphism are paths in $Spin(n)$ that are composed using the group product)

* the 2-groupoid $\mathbf{B}String_{BCSS}$ is the version given in BCSS (morhisms again are paths in $Spin(n)$ that are composed using the group product).

### As a finite-dimensional weak Lie 2-group

([Schommer-Pries](#Schommer-Pries))

### As an automorphism 2-group of fermionic CFT

The string 2-group also appears as a certain [[automorphism 2-group]] inside the [[3-category of fermionic conformal nets]] ([Douglas-Henriques](#DouglasHenriques))

### As the automorphisms of the Wess-Zumino-Witten gerbe 2-connection

For $G$ a compact simply connected simple [[Lie group]], there is the "[[WZW gerbe]]", hence the [[circle n-bundle with connection|circle 2-bundle with connection]] on $G$ whose [[curvature]] 3-form is the [[left invariant differential form|left invariant]] extension $\langle \theta \wedge [\theta \wedge \theta]\rangle$ of the canonical Lie algebra 3-cocycle to the group

$$
  \mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2U(1)
  \,.
$$

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

The string 2-group is the [[smooth infinity-group|smooth 2-group]] of [[automorphism infinity-group|automorphism]] of $\mathcal{L}_{WZW}$ which cover the left [[action]] of $G$ on itself (hence the "[[Heisenberg 2-group]]" of $\mathcal{L}_{WZW}$ regarded as a [[prequantum 2-bundle]])

$$
  \mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G)
  \,,
$$

=--

This is due to ([Fiorenza-Rogers-Schreiber 13, section 2.6.1](#FiorenzaRogersSchreiber13)).

## Related concepts

* [[string Lie 2-algebra]]

* [[string Lie algebroid]]

* [[Platonic 2-group]]

[[fivebrane 6-group]] $\to$ **string 2-group** $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]] $\hookrightarrow$ [[general linear group]]

* [[spin group]], [[spin structure]], [[twisted spin structure]]

* [[spin^c]], [[spin^c structure]], [[twisted spin^c structure]]

* [[string group]], [[string 2-group]], [[string structure]], [[twisted string structure]]

  [[string cobordism]]

  [[stringor bundle]]

* [[T-duality 2-group]]

* [[string^c 2-group]]

* [[fivebrane group]], [[fivebrane 6-group]]

## References
 {#References}

A [[crossed module]] presentation of a topological realization of the string 2-group is implicit in

* {#StolzTeichner} [[Stephan Stolz]], [[Peter Teichner]], _What is an elliptic object?_ ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.5463&rep=rep1&type=pdf))

A realization of the string 2-group in [[∞-groupoid]]s [[internalization|internal to]] [[Banach space]]s by [[Lie integration]] of the skeletal version of the [[string Lie 2-algebra]] is in

* {#Henriques} [[Andre Henriques]], _Integrating $L_\infty$-algebras_, Compositio Mathematica, Volume 144, Issue 4 July 2008 , pp. 1017-1045 ([arXiv:math/0603563](http://arxiv.org/abs/math/0603563), [doi:10.1112/S0010437X07003405](https://doi.org/10.1112/S0010437X07003405))

A realization of the string 2-group in [[strict 2-group]]s internal to [[Frechet manifold]]s by  [[Lie integration]] of a [[strict Lie 2-algebra]] incarnation of the [[string Lie 2-algebra]] in in

* {#BCSS} [[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), [euclid:hha/1201127333](https://projecteuclid.org/euclid.hha/1201127333))

A realization of the string 2-group as a [[2-group]] in finite-dimensional [[smooth manifold]]s in in

* {#Schommer-Pries11} [[Chris Schommer-Pries]], *Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group* Geometry & Topology **15** (2011) 609-676 [[arXiv:0911.2483](http://arxiv.org/abs/0911.2483), [doi:10.2140/gt.2011.15.609](https://doi.org/10.2140/gt.2011.15.609)]

A discussion as an [[∞-group]] object in [[Smooth∞Grpd]] and the realization of the smooth first fractional Pontryagin class is in

* {#FSS} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _Cech cocycles for differential characteristic classes_ (web)

and in section 4.1 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

A 2-group model which has a smoothening of the _topological_ [[string group]] in lowest degree has been given in

* [[Thomas Nikolaus]], [[Christoph Sachse]], [[Christoph Wockel]], _A Smooth Model for the String Group_,  International Mathematics Research Notices, Volume: 2013 , Issue: 16 , 2013   ([arXiv:1104.4288](http://arxiv.org/abs/1104.4288), [doi:10.1093/imrn/rns154](https://doi.org/10.1093/imrn/rns154))

A construction explicitly in terms of the "[[basic bundle gerbe|basic]]" [[bundle gerbe]] on $G$:

* [[Konrad Waldorf]], *A Construction of String 2-Group Models using a Transgression-Regression Technique*, Contemp. Math. **584** (2012) 99-115 [[arXiv:1201.5052](http://arxiv.org/abs/1201.5052), [doi:10.1090/conm/584/11588](https://doi.org/10.1090/conm/584/11588)]

Via fermionic nets/[[2-Clifford algebra]]:

* {#DouglasHenriques} [[Chris Douglas]], [[André Henriques]], _Geometric string structures_ ([pdf](http://www.staff.science.uu.nl/~henri105/PDF/TringWP.pdf))

The realization of the string 2-group as the [[Heisenberg 2-group]] of the [[WZW gerbe]] is due to

* {#FiorenzaRogersSchreiber13} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_, 2013 ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

A model of the string 2-group using the smooth [[free loop space]] (instead of the based [[loop space]]) is dicussed in

* {#MurrayRobertsWockel17} [[Michael Murray]], [[David Roberts]], [[Christoph Wockel]], _Quasi-periodic paths and a string 2-group model from the free loop group_ ([arXiv:1702.01514](https://arxiv.org/abs/1702.01514))

Discussion in the context of [[matrix factorizations]] and [[equivariant K-theory]]:

* {#FreedTeleman14} [[Daniel S. Freed]], [[Constantin Teleman]], _Dirac families for loop groups as matrix factorizations_, [arxiv/1409.6051](http://arxiv.org/abs/1409.6051)

Further on [[2-group]]-[[higher central extensions|extensions]] by the [[circle 2-group]]:

of [[tori]]:

* [[Nora Ganter]], _Categorical Tori_, SIGMA 14 (2018), 014, 18 ([arXiv:1406.7046](https://arxiv.org/abs/1406.7046))

of [[finite subgroups of SU(2)]] (to [[Platonic 2-groups]]):

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192), [hs:30](https://journals.mq.edu.au/index.php/higher_structures/article/view/30))

Discussion of a general definition of smooth [[string group]] extensions $A\rightarrow \mathrm{String}(H)\rightarrow H$ of a [[compact]] [[simply connected]] [[Lie group]] $H$, with $A$ not necessarily chosen to be $\mathbf{B}U(1)$ but only of the same [[homotopy type]], in

* [[Severin Bunk]], _Principal $\infty$-Bundles and Smooth String Group Models_, [arXiv:2008.12263](http://arxiv.org/abs/2008.12263)

[[!redirects String 2-group]]
[[!redirects String Lie 2-group]]
[[!redirects string Lie 2-group]]

[[!redirects smooth string 2-group]]


string algebroid

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

#Contents#
* table of contents
{:toc}

## Idea

A *string Lie algebroid* is a kind of [[Courant Lie algebroid]] meant to play the role of [[higher structure|higher]] [[Atiyah Lie algebroids]] of [[principal 2-bundles]] for the [[string 2-group]].

## Related concepts

* [[string Lie 2-algebra]]

## References

* Yunhe Sheng, Xiaomeng Xu, [[Chenchang Zhu]], *String principal bundles and Courant algebroids*, International Mathematics Research Notices **2021** 7 (2021) 5290–5348 [[arXiv:1701.00959](https://arxiv.org/abs/1701.00959), [doi:10.1093/imrn/rnz017](https://doi.org/10.1093/imrn/rnz017)]

* [[Mario Garcia-Fernandez]], Roberto Rubio, Carl Tipler, *Holomorphic string algebroids*, Trans. Amer. Math. Soc. **373** 10 (2020) 7347-7382 [[arXiv:1807.10329](https://arxiv.org/abs/1807.10329), [doi:10.1090/tran/8149](https://doi.org/10.1090/tran/8149)]

* {#GarciaFernandezRubioTipler} [[Mario Garcia-Fernandez]], Roberto Rubio, Carl Tipler, *Gauge theory for string algebroids*, Journal of Differential Geometry [[arXiv:2004.11399](https://arxiv.org/abs/2004.11399), [hal:02557365](https://hal.science/hal-02557365)]

[[!redirects string algebroids]]

[[!redirects string Lie algebroid]]
[[!redirects string Lie algebroids]]


string bit model

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

#Contents#
* table of contents
{:toc}

## Idea

A kind of analogue of [[lattice gauge theory]] for [[worldsheet]] computations in [[perturbative string theory]]/[[string scattering amplitudes]].

## Properties

### Via single trace observables under AdS/CFT
 {#ViaSingleTraceObservablesUnderAdSCFT}

From the point of view of [[AdS-CFT|AdS/CFT-duality]], string bit observables are the incarnation of [[single trace observables]] on the [[CFT]]-side [Berenstein-Maldacena-Nastase 02, p. 13](single+trace+observable#BerensteinMaldacenaNastase02).

More in detail:

The [[single trace observables|single trace operators/observables]] in [[conformal field theories]] such as [[super Yang-Mills theories]] play a special role in the [[AdS-CFT correspondence]]: They correspond to single [[string]] excitations on the [[AdS spacetime|AdS]]-[[supergravity]] side of the correspondence, where, curiously, the "string of characters/letters" in the argument of the trace gets literally mapped to a [[superstring]] in [[spacetime]] (see the references [below](#ReferencesRelationToStringExcitations)).

From [Polyakov 02](single+trace+observable#Polyakov02), referring to gauge fields and their single trace operators as _letter_ and _words_, respectively:

> The picture which slowly arises from the above considerations is that of the space-time gradually disappearing in the regions of large curvature. The natural description in this case is provided by a gauge theory in which the basic objects are the texts formed from the gauge-invariant words. The theory provides us with the expectation values assigned to the various texts, words and sentences.

> These expectation values can be calculated either from the gauge theory or from the strongly coupled 2d sigma model. The coupling in this model is proportional to the target space curvature. This target space can be interpreted as a usual continuous space-time only when the curvature is small. As we increase the coupling, this interpretation becomes more and more fuzzy and finally completely meaningless.

From [Berenstein-Maldacena-Nastase 02](single+trace+observable#BerensteinMaldacenaNastase02), who write $Z$ for the elementary [[field observables]] ("letters") $\mathbf{\Phi}$ above:

> In summary, the "string of $Z$s" becomes the physical string and that each $Z$ carries one unit of $J$ which is one unit of $p_+$. Locality along the worldsheet of the string comes from the fact that planar diagrams allow only contractions of neighboring operators. So the Yang Mills theory gives a [[string bit model]] where each bit is a $Z$ operator.

On the [[CFT]] side these _[[BMN operators]]_ of fixed length (of "letters") are usefully identified as  [[spin chains]] which, with the [[dilatation operator]] regarded as their [[Hamiltonian]], are [[integrable systems]] ([Minahan-Zarembo 02](single+trace+observable#MinahanZarembo02), [Beisert-Staudacher 03](single+trace+observable#BeisertStaudacher03)).

This integrability allows a detailed matching between

* [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]]

* the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] $\times$ [[5-sphere|S5]]

under [[AdS/CFT duality]] ([Beisert-Frolov-Staudacher-Tseytlin 03](single+trace+observable#BeisertFrolovStaudacherTseytlin03), ...). For review see [BBGK 04](single+trace+observable#BBGK04), [Beisert et al. 10](single+trace+observable#BeisertEtAl10).

## Related concepts

* [[lattice gauge theory]]

## References
 {#References}

The idea originates with

* Roscoe Giles, [[Charles Thorn]], _Lattice approach to string theory_, Phys. Rev. D 16, 366 – Published 15 July 1977 ([doi:10.1103/PhysRevD.16.366](https://doi.org/10.1103/PhysRevD.16.366))

* [[Igor Klebanov]], [[Leonard Susskind]], _Continuum strings from discrete field theories_, Nuclear Physics B Volume 309, Issue 1, 31 October 1988, Pages 175-187 (doi:10.1016/0550-3213(88)90237-4)

* [[Charles Thorn]], _Reformulating String Theory with the $1/N$ Expansion_, in: L. V. Keldysh and V. Ya. Fainberg (eds.), Sakharov Memorial Lectures in Physics,  Nova Science Publishers Inc., Commack, New York,
1992 ([arXiv:hep-th/9405069](https://arxiv.org/abs/hep-th/9405069))

It is made explicit in

* Oren Bergman, [[Charles Thorn]], _String Bit Models for Superstring_, Phys.Rev.D52:5980-5996, 1995 ([arXiv:hep-th/9506125](https://arxiv.org/abs/hep-th/9506125))

Review includes

* [[Charles Thorn]], _Substructure of String_ ([arXiv:hep-th/9607204](https://arxiv.org/abs/hep-th/9607204))

Further developments include the following:

* [[Charles Thorn]], _Space from String Bits_, J. High Energ. Phys. (2014) 2014: 110 ([arXiv:1407.8144](https://arxiv.org/abs/1407.8144))

* [[Charles Thorn]], _String Bits at Finite Temperature and the Hagedorn Phase_, Phys. Rev. D 92, 066007 (2015) ([arXiv:1507.03036](https://arxiv.org/abs/1507.03036))

* [[Charles Thorn]], _$1/N$ Perturbations in Superstring Bit Models_, Phys. Rev. D 93, 066003 (2016) ([arXiv:1512.08439](https://arxiv.org/abs/1512.08439))

* Matteo Beccaria, _Thermal properties of a string bit model at large $N$_, J. High Energ. Phys. (2017) 2017: 200. ([arXiv:1709.01801](https://arxiv.org/abs/1709.01801))

[[!redirects string bit models]]

[[!redirects string bit]]
[[!redirects string bits]]


String bordism

+-- {: .rightHandSide}
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###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* tabe of contents
{:toc}

## Idea

String bordism theory, $MString$, is [[bordism homology theory]] for [[manifolds]] equipped with [[string structure]].

## Related concepts

* [[Witten genus]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References

* [Manifold Atlas](http://www.map.mpim-bonn.mpg.de/Main_Page), _[String bordism](http://www.map.mpim-bonn.mpg.de/String_bordism)_

Discussion of relation to the [[Witten genus]]:

* [[Carl McTague]], _The Cayley plane and String bordism_, Geom. Topol. 18 (2014) 2045-2078 ([arXiv:1111.4520](http://arxiv.org/abs/1111.4520))

Discussion of [[secondary characteristic classes]] in [[string cobordism cohomology theory]] and in [[tmf]]:

* [[Ulrich Bunke]], [[Niko Naumann]], _Secondary Invariants for String Bordism and tmf_, Bulletin des Sciences Mathématiques, Volume 138, Issue 8, December 2014, Pages 912-970 ([doi:10.1016/j.bulsci.2014.05.002](https://doi.org/10.1016/j.bulsci.2014.05.002))

String bordism of the [[classifying space]] of [[E8|$E_8$]]:

* [[Michael Hill]], *The $String$  bordism of $B E_8$ and $B E_8 \times B E_8$ through dimension 14*, Ill. J. Math. **53** 1 (2009) 183-196 [[doi:10.1215/ijm/1264170845](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-53/issue-1/The-string-bordism-of-BE_8-and-BE_8times-BE_8-through-dimension/10.1215/ijm/1264170845.full)]

Discussion of [[geometric string structure|geometric]] string bordism in degree 3 as a means to speak (via the [[Pontryagin-Thom theorem]]) about the [[third stable homotopy group of spheres]]:

* [[Domenico Fiorenza]], [[Eugenio Landi]], *Integrals detecting degree 3 string cobordism classes* [[arXiv:2209.12933](https://arxiv.org/abs/2209.12933)]

* [[Domenico Fiorenza]], *String bordism invariants in dimension 3 from $U(1)$-valued TQFTs*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#FiorenzaMar2023)]

[[!redirects string bordism]]

[[!redirects string cobordism]]
[[!redirects string cobordism cohomology]]

[[!redirects string bordism homology theory]]
[[!redirects string bordism homology theory]]

[[!redirects string cobordism cohomology theory]]
[[!redirects string cobordism cohomology theories]]

[[!redirects MString]]


string coupling constant

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

#Contents#
* table of contents
{:toc}

## Idea

The [[coupling constant]] in [[perturbative string theory]].

## Properties

### Behaviour under M/IIA duality

Under [[duality between type IIA string theory and M-theory]] the string coupling constant relates to the [[radius]] of the 11d [[circle]]-[[fiber]]:

Consider the [[M-theory]] [[scales]]

* $\ell_P$, the [[Planck length]] in 11-dimensions;

* $R_{10}$ the [[length]] (circumference) of the $S^1_{10}$ [[circle]] [[fiber]] for [[KK-compactification]] to 10 dimensions

and the [[string theory]] [[scales]]

* $\ell_s$, the [[string length scale]];

* $g_{s}$, the [[string coupling constant]] of [[perturbative string theory]].

Then under the [[duality between M-theory and type IIA string theory]] these scales are related as follows:

$$
  \ell_P \;=\; g_{st}^{1/3} \ell_s
  \,,
  \phantom{AAA}
  R_{10} \;=\; g_{st} \ell_s
$$

equivalently

$$
  \ell_s \;=\; R_{10}/ g_{st}
  \,,
  \phantom{AAA}
  \ell_P \;=\; g_{st}^{-2/3} R_{10}
$$

equivalently

$$
  g_{st} \;=\; (R_{10}/\ell_P)^{3/2}
  \,,
  \phantom{AAA}
  \ell_s \;=\; \ell_P (R_{10}/\ell_P)^{-1/2}
  \,.
$$

Hence a [[membrane instanton]], which on a 3-[[cycle]] $C_3$ gives a contribution

$$
  \exp\left(
    - \frac{ vol(C_3) }{ \ell^3_P }
  \right)
$$

becomes

1. if the cycle [[wrapped brane|wraps]], $C_3 = C_2 \cup S^1_{10}$, a [[worldsheet instanton]]

   $$
     \exp\left( - \frac{ vol(C_3) }{ \ell_P^3 }  \right)
     \;=\;
     \exp\left( - \frac{ R_{10} vol(C_2) }{ g_{st} \ell_s^3 }  \right)
     \;=\;
     \exp\left( - \frac{ vol(C_2) }{ \ell_s^2 }  \right)
   $$

1. the cycle does not [[wrapped brane|wrap]], a spacetime instanton contribution, specifically a [[D2-brane instanton]]

   $$
     \exp\left( - \frac{ vol(C_3) }{ \ell_P^3 }  \right)
     \;=\;
     \exp\left( - \frac{ vol(C_3)/\ell_s^3 }{ g_{st} }  \right)
   $$

(This unification of the two different [[non-perturbative effects]] in [[perturbative string theory]]
([[worldsheet instantons]] and spacetime [[instantons]]), to a single type of effect ([[membrane instanton]])
 in [[M-theory]] was maybe first made explicit in [Becker-Becker-Strominger 95](#BeckerBeckerStrominger95). Brief review includes [Marino 15, sections 1.2 and 1.3](#Marino15)).

## Related concepts

* [[axio-dilaton]]

* [[Yukawa coupling]]

[[!include fundamental scales -- contents]]

## References

The identification of [[non-perturbative effects]] in [[string theory]] with brane contributions is due to

* {#BeckerBeckerStrominger95} [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], _Five-branes, membranes and nonperturbative string theory_, Nucl. Phys. B 456, 130 (1995) ([hep-th/9507158](http://arxiv.org/abs/hep-th/9507158))

Review includes

* {#Marino15} [[Marcos Mariño]], _Non-perturbative effects in string theory and AdS/CFT_, [Spring School on Superstring Theory and Related Topics 2015](http://indico.ictp.it/event/a14251/)  ([[MarinoNonPerturbative.pdf:file]], [pdf](http://indico.ictp.it/event/a14251/session/89/contribution/401/material/0/0.pdf), [recording](https://youtu.be/5hw6l-JSpc4?list=PLzYFBWfshN8eqXZFYg2jBZSii7lHaeyXJ))

[[!redirects string coupling constants]]

[[!redirects string coupling]]
[[!redirects string couplings]]


string diagram

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

# String diagrams
* table of contents
{: toc}

## Idea
 {#Idea}

_String diagrams_ constitute a graphical calculus for expressing operations in [[monoidal categories]].

\begin{imagefromfile}
    "file_name": "HotzFigure2.jpg",
    "float": "right",
    "width": 440,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 20
    },
    "caption": "From [Hotz 65](#Hotz65)"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "PenroseDepictAContraction.jpg",
    "float": "right",
    "width": 440,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 20
    },
    "caption": "From [Penrose 71a](#Penrose71a)"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "PenroseRindlerFigureA1.jpg",
    "float": "right",
    "width": 440,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 20
    },
    "caption": "From [Penrose-Rindler 84, Append.](#PenroseRindler84)"
\end{imagefromfile}

In the archetypical cases of the [[Cartesian monoidal category]] of [[finite sets]] this is Hotz's notation ([Hotz 65](#Hotz65)) for [[automata]], while for  [[finite-dimensional vector spaces]] with their [[tensor product of vector spaces|usual tensor product]] this is Penrose's notation ([Penrose 71a](#Penrose71a), [Penrose-Rindler 84](#PenroseRindler84)) for [[tensor networks]]; but the same idea immediately applies more generally to any other [[monoidal category]] and yet more generally to [[bicategories]], etc.

The idea is roughly to think of [[objects]] in a monoidal category as "strings" and of [[morphisms]] from one [[tensor product]] to another as a node which the source strings enter and the target strings exit. Further structure on the monoidal category is encoded in geometrical properties on these strings.

For instance:

* putting strings next to each other denotes the [monoidal product](monoidal+category#eq:TensorProductFunctor), and having no string at all denotes the [[tensor unit]];

* braiding strings over each other corresponds to – yes, the [[braided monoidal category|monoidal braiding]] (if any);

* bending strings around corresponds to dualities on [[dualizable objects]] (if any).

Many operations in monoidal categories that look  unenlightening in symbols become obvious in string diagram calculus, such as the [[trace]]: an output wire gets bent around and connects to an input.

String diagrams may be seen as dual (in the sense of [[Poincaré duality]]) to [[commutative diagrams]]. For instance, in a [[2-category]], an example of a string diagram for a [[2-morphism]] (shown on the left) is shown on the right here:

\begin{centre}
\begin{tikzpicture}[every node/.style={node distance=1.5cm}]
\usetikzlibrary{calc,automata}

\node (a) {$A$};
\node[above right of=a] (b) {$B$};
\node[right of=b] (c) {$C$};
\node at ($(b) !.5! (c)$) (bc) {};
\node[below right of=c] (d) {$D$};
\node[below of=bc,node distance=1.875cm] (e) {$E$};

\draw[->] (a) edge node[above left] {$f$} (b);
\draw[->] (b) edge node[above] {$g$} (c);
\draw[->] (c) edge node[above right] {$h$} (d);
\draw[->] (a) edge node[below] {$k$} (e);
\draw[->] (e) edge node[below] {$l$} (d);

\node[below of=bc,node distance=0.9cm] (double) {\large $\Downarrow \alpha$};

\node[right of=d] (egal) {\large $\mapsto$};

\tikzstyle{point}=[circle,draw,fill,inner sep=1pt];
\node[right of=egal,node distance=3cm,point] (centre) {};
\draw ($(centre)-(1.5,1.5)$) rectangle ($(centre)+(1.5,1.5)$);

% Points
\tikzstyle{label}=[node distance=0.3cm]
\node[inner sep=0pt] at ($(centre)+(-1,1.52)$) (h) {};
\node[inner sep=0pt] at ($(centre)+(0,1.52)$) (g) {};
\node[inner sep=0pt] at ($(centre)+(1,1.52)$) (f) {};
\node[inner sep=0pt] at ($(centre)+(-0.5,-1.52)$) (l) {};
\node[inner sep=0pt] at ($(centre)+(0.5,-1.52)$) (k) {};
\node[above of=h,label] {$h$};
\node[above of=g,label] {$g$};
\node[above of=f,label] {$f$};
\node[below of=l,label] {$l$};
\node[below of=k,label] {$k$};
\node[right of=centre,label] {$\alpha$};

% Edges
\draw[thick] (f) edge[bend left] (centre);
\draw[thick] (h) edge[bend right] (centre);
\draw[thick] (l) edge[bend left] (centre);
\draw[thick] (k) edge[bend right] (centre);
\draw[thick] (g) -- (centre);

% Regions
\tikzstyle{region}=[node distance=0.9cm]
\node[below of=centre,region] {$E$};
\node[left of=centre,region] {$D$};
\node[right of=centre,region] {$A$};
\node[below left of=f,region] {$B$};
\node[below right of=h,region] {$C$};
\end{tikzpicture}

\end{centre}

String diagrams for monoidal categories can be obtained in the same way, by considering a monoidal category as a 2-category with a single object.

## Variants

There are many additional [[extra structures]] on monoidal categories, or similar structures, which can usually be represented by encoding further geometric properties of their string diagram calculus For instance:

* in monoidal categories which are _[[ribbon category|ribbon categories]]_ the strings from above behave as if they have a small transversal extension which makes them behave as ribbons. Accordingly, there is a _twist_ operation in the axioms of a ribbon category and graphically it corresponds to twisting the ribbons by 360 degrees.

* in a [[traced monoidal category]], the trace can be represented by bending an output string around to connect to an input, even though if the objects are not dualizable the individual "bends" do not represent anything.

* in monoidal categories which are _[[spherical category|spherical]]_ all strings behave as if drawn on a sphere.

* in a [[hypergraph category]], the string diagrams are labeled [[hypergraphs]].

* string diagrams can be extended to represent [[monoidal functors]] in several ways.  One nice way is described in [these slides](http://web.science.mq.edu.au/~mmccurdy/cms2010talk.pdf), and can also be done with "3D regions" as drawn [here](http://golem.ph.utexas.edu/category/2010/08/the_geometry_of_monoidal_fibra.html#c034428).

* there is also a string diagram calculus for [[bicategories]], which extends that for monoidal categories regarded as one-object bicategories.  Thus, the strings now represent 1-cells and the nodes 2-cells, leaving the two-dimensional planar regions cut out by the strings to represent the 0-cells.  This makes it manifest that in general, string diagram notation is _Poincaré dual_ to the [[globe|globular]] notation: where one uses $d$-dimensional symbols, the other uses $(2-d)$-dimensional symbols.

* string diagrams for bicategories can be generalized to string diagrams for [[double categories]] and [[proarrow equipments]] by distinguishing between "vertical" and "horizontal" strings.

* One can also [[categorification|categorify]] string diagrams to "[[surface diagram|surface diagrams]]", which are dual to commutative diagrams in [[3-category|3-categories]] (including [[monoidal bicategory|monoidal bicategories]], and [[braided monoidal category|braided monoidal categories]].

* Yet more generally, one can [[categorification|categorify]] string diagrams to diagrams that are dual to commutative diagrams in [[n-category|$n$-categories]], which yields the notion of [[manifold diagram|manifold $n$-diagrams]].

* As explained [here](http://sbseminar.wordpress.com/2007/07/12/the-operadic-periodic-table/), in the presence of certain levels of duality it may be better to work with diagrams on cylinders or spheres rather than in boxes. This relates to [[planar algebras]] and [[canopolis|canopolises]].

* A string diagram calculus for [[monoidal fibrations]] can be obtained as a generalization of C.S. Peirce's "existential graphs."  The ideas are essentially contained in ([Brady-Trimble 98](#BradyTrimble98)) and developed in ([Ponto-Shulman 12](#PontoShulman12)), and was discussed [here](http://golem.ph.utexas.edu/category/2010/08/the_geometry_of_monoidal_fibra.html).

* String diagrams for [[closed monoidal categories]] (see also at _[[Kelly-Mac Lane graph]]_) are similar to those for [[autonomous categories]], but a bit subtler, involving "boxes" to separate parts of the diagram.  They were used informally by Baez and Stay [here](#BaezQG06) and [here](#Rosetta), but can also be done in essentially the same way as the [[proof nets]] used in [[intuitionistic logic|intuitionistic]] [[linear logic]]; see [Lamarche](#Lamarche08).

* Proof nets for multiplicative [[linear logic]] with negation similarly give string diagrams for [[*-autonomous categories]], or more generally [[linearly distributive categories]]; see [Blute-Cockett-Seely-Trimble](#BCST).

* Proof surfaces for noncommutative multiplicative [[linear logic]] with negation, see [Dunn-Vicary](#DunnVicary17)

* _Sheet diagrams_, string diagrams drawn on a branching surface, may be used for [[rig categories]], see [Comfort-Delpeuch-Hedges](#CDH).

See also [Selinger 09](#Selinger09) for a review of different string diagram formalisms.

## Examples
 {#Examples}

### In linear algebra
 {#InLinearAlgebra}

String diagram calculus in [[linear algebra]]:

* {#PenroseNotation} String diagrams in [[finite-dimensional vector spaces]] with the [[tensor product of vector spaces|usual tensor product]] are *Penrose's notation* ([Penrose 71a](#Penrose71a), [71b](#Penrose71b), [72](#Penrose72), [Penrose-Rindler 84, Append.](#PenroseRindler84)) for working with [[tensors]].

* More recently, string diagrams in this category have come to be known as *[[tensor networks]]*, especially so in application to [[condensed matter physics]] and also in [[quantum computation]] and in particular in [[quantum error correction]].

### In quantum computation

* In [[quantum computation]], *[[quantum circuit diagrams]]* are a form of string diagrams in [[finite-dimensional vector spaces|finite-dimensional]] [[Hilbert spaces]]. See also at *[[finite quantum mechanics in terms of dagger-compact categories]]*.

### In representation theory
 {#InRepresentationTheory}

String diagram calculus in [[representation theory]]: [Mandula 81](#Mandula81), [Cvitanović 08](#Cvitanovic08)

*  String diagrams for the monoidal category of finite-dimensional [[linear representations]] of the group [[SU(2)]] on complex vector spaces are called [[spin networks]].

### In Lie theory
 {#ExamplesInLieTheory}

For string diagrams calculus in [[Lie theory]] see at:

* _[[Lie algebra object]]_

* _[[metric Lie algebra]]_

* _[[metric Lie representation]]_

* _[[Lie algebra weight system]]_.

* _[[M2-brane 3-algebra]]_

### In perturbative quantum field theory
 {#ExamplesInpQFT}

For applications of string diagram calculus in [[perturbative quantum field theory]], see at

* [['t Hooft double line notation]]

(...)

## Related concepts

* [[sharing graph]]

* [[surface diagram]]

* [[Kelly-Mac Lane diagram]]

* [[proof net]]

* [[Feynman diagram]]

* [[tensor network]],

* [[neural network]]

* [[Lie algebra weight system]]

* [[linguistics|natural language syntax]]

* [[Petri net]]

* [[string-net model]]

## References

### Introduction and survey

Discussion in [[representation theory]]:

* {#Mandula81} [[Jeffrey Ellis Mandula]], *Diagrammatic techniques in group theory*, Southampton Univ. Phys. Dept. (1981) ([cds:129911](https://cds.cern.ch/record/129911), [pdf](https://cds.cern.ch/record/129911/files/SHEP%2080-81-7.pdf))

* {#Cvitanovic08} [[Predrag Cvitanović]], _Group Theory: Birdtracks, Lie's, and Exceptional Groups_, Princeton University Press July 2008 ([PUP](https://press.princeton.edu/books/paperback/9780691202983/group-theory), [birdtracks.eu](http://birdtracks.eu/), [pdf](http://www.birdtracks.eu/version9.0/GroupTheory.pdf))

  > (aimed at [[Lie theory]] and [[gauge theory]])

* {#Peigné22} [[Stéphane Peigné]], *Introduction to color in QCD: Initiation to the birdtrack pictorial technique* [[arXiv:2302.07574](https://arxiv.org/abs/2302.07574)]

  > (further focus on [[QCD]])

Introductions to and surveys of string diagram calculus:

* [[Ross Street]], _Categorical structures_, in: M. Hazewinkel (ed.), _Handbook of algebra -- Volume 1_, Elsevier 1996 ([pdf](http://maths.mq.edu.au/~street/45.pdf), [978-0-444-82212-3](https://www.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-82212-3))

  > (in the generality of [[bicategories]])

* [[Ross Street]], _Low dimensional topology and higher-order categories_ ([ps](http://www.mta.ca/~cat-dist/CT95Docs/LowDim.ps))

  >   (on surface diagrams)

* {#Selinger09} [[Peter Selinger]], _A survey of graphical languages for monoidal categories_, in: [[Bob Coecke]] (ed.) _New Structures for Physics_, Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg (2010) ([arXiv:0908.334](http://arxiv.org/abs/0908.3347), [doi:10.1007/978-3-642-12821-9_4](https://doi.org/10.1007/978-3-642-12821-9_4))

* [[John Baez]], QG Seminar Fall 2000 ([web](http://math.ucr.edu/home/baez/qg-fall2000/)), Winter 2001 ([web](http://math.ucr.edu/home/baez/qg-winter2001/)), Fall 2006 ([web](http://math.ucr.edu/home/baez/qg-fall2006/index.html#computation)).

* [[John Baez]], [[Mike Stay]], _Physics, Topology, Logic and Computation: A Rosetta Stone_ ([arXiv:0903.0340](http://arxiv.org/abs/0903.0340))

* [[Aleks Kissinger]], _Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing_ ([arXiv:1203.0202](https://arxiv.org/abs/1203.0202))

* [[Simon Willerton]], _String diagrams_ ([YouTube](http://www.youtube.com/view_play_list?p=50ABC4792BD0A086))

* [[Robin Piedeleu]], [[Fabio Zanasi]], _An Introduction to String Diagrams for Computer Scientists_ [[arXiv:2305.08768](https://arxiv.org/abs/2305.08768)]

  > (in [[computer science]])

From the point of view of [[finite quantum mechanics in terms of dagger-compact categories]]:

* {#CoeckePicturalism} [[Bob Coecke]], _Quantum Picturalism_ ([arXiv:0908.1787](http://arxiv.org/abs/0908.1787))

* {#CoeckeIntroducing} [[Bob Coecke]], _Introducing categories to the practicing physicist_ ([arXiv:0808.1032](http://arxiv.org/abs/0808.1032))

* {#CoeckePractising} [[Bob Coecke]], _Categories for the practising physicist_ ([arXiv:0905.3010](http://arxiv.org/abs/0905.3010))

* [[Bob Coecke]], [[Ross Duncan]], _Interacting Quantum Observables: Categorical Algebra and Diagrammatics_, New J. Phys. 13 (2011) 043016 ([arXiv:0906.4725](http://arxiv.org/abs/0906.4725))

* {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], _Lectures on categorical quantum mechanics_, 2012 ([pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf))

From the point of view of [[tensor networks]] in [[solid state physics]]:

* [[Jacob Biamonte]], Ville Bergholm, _Tensor Networks in a Nutshell_, Contemporary Physics ([arxiv:1708.00006](https://arxiv.org/abs/1708.00006))

* [[Jacob Biamonte]], _Lectures on Quantum Tensor Networks_ ([arXiv:1912.10049](https://arxiv.org/abs/1912.10049))

Some philosophical discussion is given in

* [[David Corfield]], Section 10.4 of: _Towards a Philosophy of Real Mathematics_, CUP, 2003.

### Original articles
 {#OriginalArticles}

The development and use of string diagram calculus pre-dates its graphical appearance in print, due to the difficulty of printing non-text elements at the time.

> Many calculations in earlier works were quite clearly worked out with string diagrams, then painstakingly copied into equations. Sometimes, clearly graphical structures were described in some detail without actually being drawn: e.g. the construction of free compact closed categories in Kelly and Laplaza's 1980 "Coherence for compact closed categories".

> ([Pawel Sobocinski, 2 May 2017](http://angg.twu.net/categories-2017may02.html#.1))

> This idea that string diagrams are, due to technical issues, only useful for private calculation, is said explicitly by Penrose. Penrose and Rindler's book "Spinors and Spacetime" (CUP 1984) has an 11-page appendix full of all sorts of beautiful, carefully hand-drawn graphical notation for tensors and various operations on them (e.g. anti-symmetrization and covariant derivative). On the second page, he says the following:

> "The notation has been found very useful in practice as it grealy simplifies the appearance of complicated tensor or spinor equations, the various interrelations expressed being discernable at a glance. Unfortunately the notation seems to be of value mainly for private calculations because it cannot be printed in the normal way."

> ([Alex Kissinger, 2 May 2017](http://angg.twu.net/categories-2017may02.html#.2))

The first formal definition of string diagrams in the literature appears to be in

* {#Hotz65} [[Günter Hotz]], _Eine Algebraisierung des Syntheseproblems von Schaltkreisen_, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 ([part I](https://www.magentacloud.de/lnk/LiPMlYfh), [part II](https://www.magentacloud.de/lnk/YivslUWJ), [[HotzSchaltkreise.pdf:file]])

Application of string diagrams to [[tensor]]-calculus in [[mathematical physics]] (hence for the case that the ambient [[monoidal category]] is that of [[finite dimensional vector spaces]] equipped with the [[tensor product of vector spaces]]) was propagated by [[Roger Penrose]], whence [[physics|physicists]] know string diagrams as _Penrose notation for tensor calculus_:

* {#Penrose71a} [[Roger Penrose]], _Applications of negative dimensional tensors_, Combinatorial Mathematics and its Applications, Academic Press (1971) ([pdf](http://www2.math.uic.edu/~kauffman/Penrose.pdf), [[PenroseNegativeDimensionalTensors.pdf:file]])

* {#Penrose71b} [[Roger Penrose]], _Angular momentum: An approach to combinatorial spacetime_, in Ted Bastin (ed.) _Quantum Theory and Beyond_, Cambridge University Press (1971), pp.151-180 ([[PenroseAngularMomentum71.pdf:file]])

* {#Penrose72} [[Roger Penrose]], _On the nature of quantum geometry_, in: J. Klauder (ed.) _Magic Without Magic_, Freeman, San Francisco, 1972, pp. 333–354 ([spire:74082](http://inspirehep.net/record/74082), [[PenroseQuantumGeometry.pdf:file]])

* {#PenroseRindler84} [[Roger Penrose]], [[Wolfgang Rindler]], appendix (p. 424-434) of: _Spinors and space-time -- Volume 1: Two-spinor calculus and relativistic fields_, Cambridge University Press 1984 ([doi:10.1017/CBO9780511564048](https://doi.org/10.1017/CBO9780511564048))

See also

* Wikipedia, _[Penrose graphical notation](https://en.wikipedia.org/wiki/Penrose_graphical_notation)_

From the point of view of [[monoidal category|monoidal]] [[category theory]], an early description of string diagram calculus (without actually depicting any string diagrams, see the above comments) is in:

* {#KellyLaplaza80} [[Max Kelly]], [[M. L. Laplaza]], *Coherence for compact closed categories*, Journal of Pure and Applied Algebra, **19** 193-213 (1980) [doi:10.1016/0022-4049(80)90101-2, [pdf](https://core.ac.uk/download/pdf/82696829.pdf)]

  (proving the [[coherence theorem for monoidal categories|coherence for compact closed categories]])

following

* {#Kelly72} [[Max Kelly]], _Many-variable functorial calculus I_, in: [[Max Kelly]], M. Laplaza , [[L. Gaunce Lewis, Jr.]], [[Saunders Mac Lane]] (eds.) _Coherence in Categories_, Lecture Notes in Mathematics, vol 281. Springer, Berlin, Heidelberg 1972 ([doi:10.1007/BFb0059556](https://doi.org/10.1007/BFb0059556))

  (which does include the hand-drawn diagrams that are missing in [Kelly-Laplaza 80](#KellyLaplaza80)!)

and in

* [[André Joyal]], [[Ross Street]], _The geometry of tensor calculus I_, Advances in Math. 88 (1991) 55-112; MR92d:18011 ([pdf](https://core.ac.uk/download/pdf/82659437.pdf), doi:10.1016/0001-8708(91)90003-P)

* [[André Joyal]] and [[Ross Street]], _The geometry of tensor calculus II_  ([pdf](http://www.math.mq.edu.au/~street/GTCII.pdf))

String diagram calculus was apparently popularized by its use in

* [[Louis Kauffman]], _Knots and physics_, Series on _Knots and Everything_, Volume 1,  World Scientific, 1991 ([doi:10.1142/1116](https://doi.org/10.1142/1116))

  (in the context of [[knot theory]])

Probably [[David Yetter]] was the first (at least in public) to write string diagrams  with "coupons" (a term used by [[Nicolai Reshetikhin]] and [[Turaev]] a few months later) to represent maps which are not inherent in the (braided or symmetric compact closed) monoidal structure.

See also these:

* [[Peter Freyd]], [[David Yetter]], _Braided compact closed categories with applications to low dimensional topology_ Advances in Mathematics, 77:156–182, 1989.

* [[Peter Freyd]] and [[David Yetter]], _Coherence theorems via knot theory_. Journal of Pure and Applied Algebra, 78:49–76, 1992.

* [[David Yetter]], _Framed tangles and a theorem of Deligne on braided deformations of tannakian categories_ In M. Gerstenhaber and [[Jim Stasheff]] (eds.) _Deformation Theory and Quantum Groups with Applications to Mathematical Physics_, Contemporary Mathematics 134, pages 325–349. Americal Mathematical Society, 1992.

* {#Mellies06} [[Paul-André Melliès]], _Functorial boxes in string diagrams_, Procceding of _Computer Science Logic 2006_ in Szeged, Hungary. 2006 ([hal:00154243](https://dumas.ccsd.cnrs.fr/PPS/hal-00154243), [pdf](https://hal.archives-ouvertes.fr/hal-00154243/document), [[MelliesFunctorialBoxesInStringDiagrams.pdf:file]])

  > (see also *[[computational trilogy]]*)

For more on the history of the notion see the bibliography in ([Selinger 09](#Selinger09)).

### Details

String diagrams for [[monoidal categories]] are discussed in:

* [Hotz 65](#Hotz65)

* [[Andre Joyal]] and [[Ross Street]], _The geometry of tensor calculus I_, Advances in Math. 88 (1991) 55-112; MR92d:18011.  ([pdf](http://tqft.net/other-papers/Geometry%20of%20Tensor%20Calculus%20-%20Joyal%20&%20Street.pdf))

* [[Andre Joyal]] and [[Ross Street]], _The geometry of tensor calculus II_.  ([pdf](http://www.math.mq.edu.au/~street/GTCII.pdf).

* [[Andre Joyal]] and [[Ross Street]], _Planar diagrams and tensor algebra_, available [here](http://www.math.mq.edu.au/~street/PlanarDiags.pdf).

For [[1-categories]] in

* [[Dan Marsden]], _Category Theory Using String Diagrams_, ([arXiv:1401.7220](https://arxiv.org/abs/1401.7220)).

  (therein: many explicit calculations, colored illustrations, avoiding the common practice of indicating 0-cells by non-filled circles)

For [[traced monoidal categories]] in

* [[Andre Joyal]], [[Ross Street]] and [[Dominic Verity|Verity]], _Traced monoidal categories_.

*  David I. Spivak, Patrick Schultz, Dylan Rupel, _String diagrams for traced and compact categories are oriented 1-cobordisms_, [arxiv](https://arxiv.org/abs/1508.01069)

For [[closed monoidal categories]] in

* [[John Baez]], Quantum Gravity Seminar - Fall 2006.  <http://math.ucr.edu/home/baez/qg-fall2006/index.html#computation>
 {#BaezQG06}

*  {#Rosetta} [[John Baez]] and [[Mike Stay]], *Physics, Topology, Logic and Computation: A Rosetta Stone*, [arxiv](https://arxiv.org/abs/0903.0340)

* Francois Lamarche, *Proof Nets for Intuitionistic Linear Logic: Essential nets*, 2008 [pdf](http://hal.inria.fr/docs/00/34/73/36/PDF/prfnet1.pdf)
 {#Lamarche08}

* Ralf Hinze, *Kan Extensions for Program Optimisation, Or:
Art and Dan Explain an Old Trick*, [pdf](http://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf)

* [[Dan Ghica]], [[Fabio Zanasi]], _Hierarchical string diagrams and applications_ ([arXiv:2305.18945](https://arxiv.org/abs/2305.18945))

For [[biclosed monoidal categories]] in

* [[Bob Coecke]], Edward Grefenstette, and [[Mehrnoosh Sadrzadeh]], *Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus*, 2013 [link](https://www.sciencedirect.com/science/article/pii/S0168007213000626)

For [[linearly distributive categories]] in

* [[Richard Blute]] and [[Robin Cockett]] and [[R.A.G. Seely]] and [[Todd Trimble]], *Natural deduction and coherence for weakly distributive categories.
 {#BCST}

* {#DunnVicary17} Lawrence Dunn, [[Jamie Vicary]], _Surface Proofs for Nonsymmetric Linear Logic_ ([arXiv:1701.04917](http://arxiv.org/abs/1701.04917))

For [[indexed monoidal categories]] in

* {#BradyTrimble98} Geraldine Brady, [[Todd Trimble]], _[[A string diagram calculus for predicate logic]]_ (1998)

* {#PontoShulman12} [[Kate Ponto]], [[Michael Shulman]], _Duality and traces for indexed monoidal categories_, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 ([arXiv:1211.1555](http://arxiv.org/abs/1211.1555))

For [[symmetric monoidal category|symmetric]] [[traced monoidal categories]] in

* George Kaye, _The Graphical Language of Symmetric Traced Monoidal Categories_, ([arXiv:2010.06319](https://arxiv.org/abs/2010.06319))

The generalization of string diagrams to one dimension higher is discussed in

* [[Todd Trimble]], _[Surface diagrams](/toddtrimble/published/Surface+diagrams)_

* [[John Barrett]], [[Catherine Meusburger]], [[Gregor Schaumann]], _Gray categories with duals and their diagrams_, available [here](http://arxiv.org/abs/1211.0529).

The generalization to arbitrary dimension in terms of [[opetope|opetopic]] "zoom complexes" is due to

* {#KockJoyalBataninMascari07} [[Joachim Kock]], [[André Joyal]], [[Michael Batanin]], [[Jean-François Mascari]], _Polynomial functors and opetopes_ ([arXiv:0706.1033](http://arxiv.org/abs/0706.1033))

The generalization to arbitrary dimension in terms of [[manifold diagrams]] (generalizing, in particular, [[opetope|opetopic shapes]]) is due to

* {#DornDouglas22} [[Christoph Dorn]] and [[Christopher Douglas]], _Manifold diagrams and tame tangles_, 2022 ([arXiv](https://arxiv.org/abs/2208.13758), [latest](https://cxdorn.github.io/manifold-diagram-paper/))

Discussion of string diagram calculus for ([[virtual double category|virtual]]) [[double categories]] and ([[virtual equipment|virtual]]) [[2-category equipped with proarrows|pro-arrow equipments]]:

* {#Myers16} [[David Jaz Myers]], _String Diagrams For Double Categories and (Virtual) Equipments_ [[arXiv:1612.02762](https://arxiv.org/abs/1612.02762)]

* [[David Jaz Myers]], *String Diagrams for (Virtual) Proarrow Equipments* (2017) [slides: [pdf](http://www.mat.uc.pt/~ct2017/slides/myers_d.pdf), [[Myers-StringDiagrams2017.pdf:file]]]

See also at _[[opetopic type theory]]_.

Discussion of sheet diagrams for [[rig categories]] is in

* {#CDH} Cole Comfort, Antonin Delpeuch, [[Jules Hedges]], _Sheet diagrams for bimonoidal categories_, ([arXiv:2010.13361](https://arxiv.org/abs/2010.13361))

Discussion of the use of string diagrams to treat [[universal constructions]] such as [[limits]], [[Kan extensions]], and [[ends]]:

* Kenji Nakahira, _Diagrammatic category theory_ [[arXiv:2307.08891](https://arxiv.org/abs/2307.08891)]

### Software

The higher dimensional string diagrams ("zoom complexes" ([Kock-Joyal-Batanin-Mascari 07](#KockJoyalBataninMascari07))) used for presenting [[opetopes]] in the context of [[opetopic type theory]] are introduced in

* [[Eric Finster]], _Opetopic Diagrams 1 - Basics_ ([video](http://www.youtube.com/watch?v=OANwLohwJqk))

* [[Eric Finster]], _Opetopic Diagrams 2 - Geometry_ ([video](http://www.youtube.com/watch?v=E7OvuA1jRKM))

* [[Globular]] is a web-based proof assistant for finitely-presented semistrict globular higher categories. It allows one to formalize higher-categorical proofs in finitely-presented n-categories and visualize them as string diagrams.

[[!include quantum information theory via string diagrams -- references]]

[[!redirects string diagrams]]
[[!redirects circuit diagram]]
[[!redirects circuit diagrams]]

[[!redirects graphical calculus]]

[[!redirects Penrose notation]]
[[!redirects Penrose graphical notation]]