confluent category

# Confluent categories
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## Definition

A [[category]] is **confluent** if for any [[span]] $B \leftarrow A \to C$, there exists a [[cospan]] $B \to D \leftarrow C$.  Note that we do not require the resulting square to [[commutative diagram|commute]].

## Remarks

If the morphisms in a category represent (sequences of) "[[rewriting]]" operations, then confluence means that any two ways to rewrite the same thing can eventually be brought back together.

If $(X,\rightarrow)$ is an abstract rewriting system ie. a set equipped with a binary relation $\rightarrow$ and if $\rightarrow^{*}$ is the [[reflexive-transitive closure]] of $\rightarrow$, then we say that $\rightarrow$ is confluent iff for every $a,b,c \in X$ such that $a \rightarrow^{*} b$ and $a \rightarrow^{*} c$, there exists $d \in X$ such that $b \rightarrow^{*} d$ and $c \rightarrow^{*} d$.

This is a good property of rewriting in systems such as the [[lambda calculus]] (the [[Church-Rosser theorem]]), and as such it is a property one might expect for the hom-categories in a [[2-category|2-categorical]] model of lambda calculus.

Another good property one might want to assume is termination, i.e. the lack of infinite chains of nonidentity arrows.

As stated in the definition, the notion of confluence is perfectly sensible if we replace "category" by "[[directed graph]] (quiver)". However, it is also occasionally useful to strengthen the notion the notion of confluence so that commutativity of the resulting square holds in the category. An example is Mac Lane's proof of the coherence theorem for [[monoidal categories]] (as given in [[Categories Work]]), where the "rewrite" arrows are expanded (whiskered) instances of associativity morphisms $\alpha_{A B C}: (A \otimes B) \otimes C \to A \otimes (B \otimes C)$ (as opposed to their inverses) in a [[free object|free]] monoidal category.

## Related pages

* [[lambda-calculus]]

* [[diamond]]

* [[locally confluent]]

## References

* John Baez, [2-categories of computation](http://math.ucr.edu/home/baez/qg-winter2007/w07week04b.pdf).

* Barnaby P. Hilken, [Towards a proof theory of rewriting: the simply-typed 2λ-calculus](http://math.ucr.edu/home/baez/qg-winter2007/hilken_2lambda_calculus.ps),
*Theor. Comp. Sci.* **170** (1996), 407-444.

[[!redirects confluent category]]
[[!redirects confluent categories]]

[[!redirects confluence]]
[[!redirects confluent]]
[[!redirects confluences]]


conformal anomaly

#Contents#
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## Idea

In [[quantum field theory]] a _conformal anomaly_ is a [[quantum anomaly]] that breaks [[conformal invariance]].

## Examples

### Weyl anomaly of relativistic string

Discussion of the conformal anomaly (Weyl anomaly) of the [[relativistic string]] as an anomalous action functional is in ([Freed 86, 2.](#Freed86)).
The following summary of this is taken from [this MO answer](http://mathoverflow.net/a/99667/381) by Pavel Safranov.

Let $\Sigma$ be a compact surface (worldsheet) and $M$ a Riemannian manifold (spacetime). The string partition function looks like
$$Z_{string}=\int_{g\in Met(\Sigma)}dg\int_{\sigma\in Map(\Sigma,M)}d\sigma\exp(iS(g,\sigma)).$$
Here $Met(\Sigma)$ is the space of Riemannian metrics on $\Sigma$ and $S(g,\sigma)$ is the standard $\sigma$-model action $S(g,\sigma)=\int_{\Sigma} dvol_\Sigma \langle d\sigma,d\sigma\rangle$. In particular, $S$ is quadratic in $\sigma$, so the second integral $Z_{matter}$ does not pose any difficulty and one can write it in terms of the determinant of the Laplace operator on $\Sigma$. Note that the determinant of the Laplace operator is a section of the determinant line bundle $L_{det}\rightarrow Met(\Sigma)$. The measure $dg$ is a 'section' of the bundle of top forms $L_g\rightarrow Met(\Sigma)$. Both line bundles carry natural connections.

However, the space $Met(\Sigma)$ is enormous: for example, it has a free action by the group of rescalings $Weyl(\Sigma)$ ($g\mapsto \phi g$ for $\phi\in Weyl(\Sigma)$ a positive function). It also carries an action of the diffeomorphism group. The quotient $\mathcal{M}$ of $Met(\Sigma)$ by the action of both groups is finite-dimensional, it is the moduli space of conformal (or complex) structures, so you would like to rewrite $Z_{string}$ as an integral over $\mathcal{M}$.

Everything in sight is diffeomorphism-invariant, so the only question is how does the integrand change under $Weyl(\Sigma)$. To descend the integral from $Met(\Sigma)$ to $Met(\Sigma)/Weyl(\Sigma)$ you need to trivialize the bundle $L_{det}\otimes L_g$ along the orbits of $Weyl(\Sigma)$. This is where the critical dimension comes in: the curvature of the natural connection on $L_{det}\otimes L_g$ (local anomaly) vanishes precisely when $d=26$. After that one also needs to check that the connection is actually flat along the orbits, so that you can indeed trivialize it.

### QCD trace anomaly

In [[quantum chromodynamics]]: See at _[[QCD trace anomaly]]_.

## Related concepts

* [[gravitational anomaly]]

* [[Liouville cocycle]]

## References

### General

* [[Daniel Freed]], _Determinants, torsion, and strings_,  Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. ([Euclid](http://projecteuclid.org/euclid.cmp/1104116145))
  {#Freed86}

* Nicolas Boulanger, _Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions_, Phys. Rev. Lett.98:261302, 2007 ([arXiv:0706.0340](https://arxiv.org/abs/0706.0340))

See also:

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

### Via AdS/CFT

Discussion via [[AdS/CFT]]:

* Mans Henningson, Kostas Skenderis, _The Holographic Weyl anomaly_, JHEP 9807 (1998) 023 ([arXiv:hep-th/9806087](https://arxiv.org/abs/hep-th/9806087))

* Mozhgan Mir, _On Holographic Weyl Anomaly_, JHEP 1310:084, 2013 ([arXiv:1307.5514](https://arxiv.org/abs/1307.5514))

For more see at _[[QCD trace anomaly]]_ the references [there](QCD+trace+anomaly#ReferencesHolographicQCD).

[[!redirects conformal anomalies]]

[[!redirects Weyl anomaly]]
[[!redirects Weyl anomalies]]


conformal block

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## Idea

In a [[conformal field theory]] the conditions on [[correlators]] can be divided into two steps

1. for a fixed [[cobordism]] the correlators need to depend in a certain way on the choice of [[conformal structure]], they need to satisfy the [[Ward identities]] (e.g. [Gawedzki 99, around p. 30](#Gawedzki99));

1. the correlators need to glue correctly underly composition of cobordisms.

The spaces of functionals that satisfy the first of these conditions are called _conformal blocks_ . The second condition is called the [[sewing constraint]] on conformal blocks.

So conformal blocks are something like "precorrelators" or "potential correlators" of a [[CFT]].

The assignment of spaces of conformal blocks to surfaces and their [[isomorphisms]] under [[diffeomorphisms]] of these surfaces together constitutes the _[[modular functor]]_. Under [[AdS3-CFT2 and CS-WZW correspondence|CS/WZW holography]] this is essentially the data also given by the [[Hitchin connection]], see at _[[quantization of 3d Chern-Simons theory]]_ for more on this.

From a point of view closer to [[number theory]] and [[geometric Langlands correspondence]] elements of conformal blocks are naturally thought of ([Beauville-Laszlo 93](#BeauvilleLaszlo93)) as generalized [[theta functions]] (see there for more).

## Properties
 {#Properties}

### Relation to braiding of field insertions
 {#RelationToBraidingOfFieldInsertions}

In the case of [[2d CFT|2d]] [[rational CFT]] the [[representation category]] of the corresponding [[vertex operator algebra]] is a [[modular tensor category]] (see [there](modular+tensor+category#RepCategoriesOfVOAs)), hence a [[braided monoidal category]] with extra properties.

 In this case the [[monoidal category|monoidal]]- and the [[braided monoidal category|braided]] [[structure]] (hence the [[modular tensor category|modular tensor structure]]) on the underlying [[representation category]] is entirely fixed by the space of [[conformal blocks]] of the [[2d CFT]] on the [[Riemann sphere]] (the "genus zero conformal blocks").

This may be found highlighted in [EGNO 15, p. 266](#EGNO15), [Runkel, Sec. 4.3](#Runkel). The essentially equivalent fact that the genus=0 conformal blocks already determine the [[modular functor]] of the CFT is proven in [Andersen & Ueno 2012](modular+functor#AndersenUeno12).

### Holographic correspondence

The conformal blocks at least of the [[WZW model]] are by a [[holographic principle|holographic]] correspondence given by the [[space of quantum states]] of 3d [[Chern-Simons theory]]. See at _[[AdS3-CFT2 and CS-WZW correspondence]]_.

### Relation to equivariant elliptic cohomology

For the $G$-[[WZW model]] the assignment of spaces of conformal blocks, hence by the above equivalently modular functor for $G$-[[Chern-Simons theory]] restricted to [[genus of a surface|genus-1 surfaces]] ([[elliptic curves]]) is essentially what is encoded in the universal $G$-[[equivariant elliptic cohomology]] (equivariant [[tmf]]). In fact equivariant elliptic cohomology remembers also the [[local prequantum field theory|pre-quantum]] incarnation of the modular functor as a systems of [[prequantum line bundles]] over Chern-Simons [[phase spaces]] (which are [[moduli stacks of flat connections]]) and remembers the [[quantization]]-process from there to the actual [[space of quantum states]] by forming holomorphic [[sections]]. See at _[equivariant elliptic cohomology -- Idea -- Interpretation in Quantum field theory](equivariant+elliptic+cohomology#InterpretationInQuantumFieldTheory)_ for more on this.

## Related concepts

* [[FFRS-formalism]]

* [[Laughlin wavefunction]]

[[!include holographic principle -- table]]

## References

### For 2d CFT

A review is around p. 30 of

* {#Gawedzki99} [[Krzysztof Gawędzki]], _Conformal field theory: a case study_ ([arXiv:hep-th/9904145](http://arxiv.org/abs/hep-th/9904145))

On the [[Knizhnik-Zamolodchikov connection]] on [[configuration spaces of points]] and [[conformal blocks]]:

* {#Kohno14} [[Toshitake Kohno]], _Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks_,  Acta Math Vietnam 39, 575–598 (2014)  [[doi:10.1007/s40306-014-0088-6](https://doi.org/10.1007/s40306-014-0088-6), [pdf](https://www.ms.u-tokyo.ac.jp/~kohno/papers/kohno_config.pdf)]

* [[Toshitake Kohno]], §1.4 in: *Conformal field theory and topology*, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. (2002) [[AMS:mmono-210](https://bookstore.ams.org/mmono-210)]

and via [[Bohr-Sommerfeld leaves]] in [[geometric quantization]]:

* [[Andrei Tyurin]], *On Bohr-Sommerfeld bases*, Izvestiya: Mathematics **64** 5 (2000) 1033–1064 [[arXiv:math/9909084](https://arxiv.org/abs/math/9909084), [doi:10.1070/IM2000v064n05ABEH000308](https://iopscience.iop.org/article/10.1070/IM2000v064n05ABEH000308), [pdf](https://www.mathnet.ru/links/d8ad7540137cf2f33c44ab3e9455d392/im308_eng.pdf)]

Brief review:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], pp. 8 of: *Algebraic structures in two-dimensional conformal field theory*, Encyclopedia of Mathematical Physics [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

Discussion in terms of [[conformal nets]]:

* {#BartelsDouglasHenriques14} [[Arthur Bartels]], [[Christopher Douglas]], [[André Henriques]], _Conformal nets II: conformal blocks_ ([arXiv:1409.8672](http://arxiv.org/abs/1409.8672))

See also

* A. Tsuchiya, [[Kenji Ueno]], Y. Yamada, _Conformal field theory on universal family of stable curves with gauge symmetries_, Adv. Studies in Pure Math. __19__, 459--566, Academic Press (1989) [MR92a:81191](http://www.ams.org/mathscinet-getitem?mr=92a:81191)

* [[Kenji Ueno]], _Conformal field theory with gauge symmetry_, Fields Institute Monographs 2008 [book page](http://www.ams.org/bookstore-getitem/item=FIM-24)

* Ratul Mahanta, Tanmoy Sengupta, *Modular linear differential equations for four-point sphere conformal blocks* [[arXiv:2211.05158](https://arxiv.org/abs/2211.05158)]

* Mikhail Pavlov, *Global torus blocks in the necklace channel* [[arXiv:2302.10153](https://arxiv.org/abs/2302.10153)]

Conformal blocks for [[self-dual higher gauge theory]]:

* [[Kiyonori Gomi]], _An analogue of the space of conformal blocks in $(4k+2)$-dimensions_ ([pdf](http://www.math.kyoto-u.ac.jp/~kgomi/papers/acb.pdf))

In particular, argument that the higher conformal blocks of the [[D=6 N=(2,0) SCFT|$D=6$ $\mathcal{N}=(2,0)$ SCFT]] reduce to confomal blocks of the ordinary 2d [[WZW model]] after [[KK-compactification]] on a [[Riemann surface]]:

* {#Witten09} [[Edward Witten]], [p. 22](https://arxiv.org/pdf/0905.2720.pdf#page=22) of: *Geometric Langlands From Six Dimensions*, in Peter Kotiuga (ed.) _A Celebration of the Mathematical Legacy of Raoul Bott_, CRM Proceedings & Lecture Notes **50** AMS (2010) [[arXiv:0905.2720](http://arxiv.org/abs/0905.2720), [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/crmp-50)]

Conformal blocks for [[Liouville theory]]:

* [[Babak Haghighat]], [[Yihua Liu]], [[Nicolai Reshetikhin]], *Flat Connections from Irregular Conformal Blocks* [[arXiv:2311.07960](https://arxiv.org/abs/2311.07960)]

* [[Xia Gu]], [[Babak Haghighat]], [[Kevin Loo]], *Irregular Fibonacci Conformal Blocks* [[arXiv:2311.13358](https://arxiv.org/abs/2311.13358)]

### Relation to theta functions

Relation to [[theta functions]]:

* {#BeauvilleLaszlo93} [[Arnaud Beauville]], [[Yves Laszlo]], _Conformal blocks and generalized theta functions_, Comm. Math. Phys. __164__ (1994), 385 - 419, [euclid](http://projecteuclid.org/euclid.cmp/1104270837), [alg-geom/9309003](http://arxiv.org/abs/alg-geom/9309003), [MR1289330](http://www.ams.org/mathscinet-getitem?mr=1289330)

* [[Arnaud Beauville]], _Conformal blocks, fusion rings and the Verlinde formula_, Proc. of the Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc. __9__, 75-96 (1996) [pdf](http://math.unice.fr/~beauvill/pubs/Hirz65.pdf)

* [[Krzysztof Gawędzki]], _Lectures on CFT_ (from Park City, published in QFT and strings for mathematicians, Dijkgraaf at al editors, [site](http://www.math.ias.edu/qft), [source](http://www.math.ias.edu/QFT/fall/NewGaw.tex), [dvi](http://www.math.ias.edu/QFT/fall/NewGaw.dvi), [ps](http://www.math.ias.edu/QFT/fall/NewGaw.ps)

* [[Alexander Beilinson]], [[Yuri Manin]], [[Vadim Schechtman]], _Sheaves of Virasoro and Neveu-Schwarz algebras_, Lecture Notes in Math. __1289__, Springer 1987, 52–66

* A.Mironov, A.Morozov, Sh.Shakirov, _Conformal blocks as Dotsenko-Fateev integral discriminants_, [arxiv/1001.0563](http://arxiv.org/abs/1001.0563)

[[!include hypergeometric KZ-solutions -- references]]

[[!include Laughlin wavefunctions as conformal blocks -- references]]

[[!redirects conformal blocks]]


conformal bootstrap

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## Idea

The _conformal bootstrap program_ ([Belavin-Polyakov-Zamolofchikov 84](#BelavinPolyakovZamolofchikov84)) is an attempt to construct and classify [[conformal field theories]] [[non-perturbative field theory|non-perturbatively]] by [[axiom|axiomatizing]] the properties of their [[operator product expansion]]/[[correlation functions]].

The conformal bootstrap was proposed in the 1970s as a strategy for calculating the properties of second-order [[phase transitions]]. After spectacular success elucidating two-dimensional systems, little progress was made on systems in higher dimensions until a recent renaissance beginning in 2008 ([Poland-Simmons-Duffin 16](#PolandSimmonsDuffin16)).

The generalization of the conformal bootstrap to [[superconformal field theories]] has the potential to
provide, via [[AdS/CFT]], a precise and detailed construction of [[large-N limit|large-N]] and [[asymptotic boundary|asymptotically]] [[anti-de Sitter spacetime|AdS]] [[string theory|string]]/[[M-theory]].

## Related concepts

* [[Wightman axioms]], [[conformal net]]

* [[S-matrix]]

## References

### General

*  {#BelavinPolyakovZamolofchikov84} [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]],  (1984). _Infinite conformal symmetry in two-dimensional quantum field theory_. Nuclear Physics B. 241 (2): 333–380 (1984)

* {#PolandSimmonsDuffin16} [[David Poland]], [[David Simmons-Duffin]], _The conformal bootstrap_, Nature Physics 535–539 (2016) [doi:10.1038/nphys3761](http://www.nature.com.edgesuite.net/nphys/journal/v12/n6/full/nphys3761.html)

* [[David Simmons-Duffin]], _TASI Lectures on the Conformal Bootstrap_ ([arXiv:1602.07982](https://arxiv.org/abs/1602.07982))

* [[David Poland]], [[Slava Rychkov]], [[Alessandro Vichi]], _The Conformal Bootstrap: Numerical Techniques and Applications_, Rev. Mod. Phys. 91, 15002 (2019) ([arXiv:1805.04405](https://arxiv.org/abs/1805.04405))

* [[Shai Chester]], _Weizmann Lectures on the Numerical Conformal Bootstrap_ ([arXiv:1907.05147](https://arxiv.org/abs/1907.05147))

* Johan Henriksson, _Analytic Bootstrap for Perturbative Conformal Field Theories_ ([arXiv:2008.12600](https://arxiv.org/abs/2008.12600))

* Pedro Liendo, Zhengwen Liu, Junchen Rong, *The old conformal bootstrap revisited* ([arXiv:2108.07295](https://arxiv.org/abs/2108.07295))

See also:

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

* Zechuan Zheng, *Bootstrap Method in Theoretical Physics* [[arXiv:2401.00350](https://arxiv.org/abs/2401.00350)]

For [[gauge theories]]:

* Zhijin Li, [[David Poland]], _Searching for gauge theories with the conformal bootstrap_ ([arXiv:2005.01721](https://arxiv.org/abs/2005.01721))

For [[Liouville theory]]:

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ ([arxiv:2005.11530](https://arxiv.org/abs/2005.11530))

On [[manifolds with corners]]:

* António Antunes, _Conformal Bootstrap near the edge_ ([arXiv:2103.03132](https://arxiv.org/abs/2103.03132))

Comparison to [[experiment]] -- [[superfluid]]-transition:

* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))

  exposition in:

  [[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))

Solving the bootstrap constraints via [[machine learning]]:

* Gergely Kántor, Vasilis Niarchos, [[Constantinos Papageorgakis]], *Solving Conformal Field Theories with Artificial Intelligence* ([arXiv:2108.08859](https://arxiv.org/abs/2108.08859))

Relation to [[reflection positivity]]:

* Leandro Lanosa, Mauricio Leston, Mario Passaglia, *Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT* ([arXiv:2112.00232](https://arxiv.org/abs/2112.00232))

### Superconformal bootstrap
 {#ReferencesSuperconformal}

For [[superconformal field theory]], such as [[D=4 N=1 SYM]], [[D=4 N=2 SYM]], [[D=4 N=4 SYM]], [[D=6 N=(1,0) SCFT]], [[D=6 N=(2,0) SCFT]]:

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

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

* {#BRR29} [[Christopher Beem]], [[Leonardo Rastelli]], Balt C. van Rees, _More $N=4$ superconformal bootstrap_ ([arXiv:1612.02363](https://arxiv.org/abs/1612.02363))

* [[Luis F. Alday]], [[Shai M. Chester]], *Pure anti-de Sitter supergravity and the conformal bootstrap* [[arXiv:2207.05085](https://arxiv.org/abs/2207.05085)]

### In AdS/CFT

Discussion of [[superconformal bootstrap]] in view of [[AdS/CFT]], hence as a precise and detailed construction of [[large-N limit|large-N]] but also [[large 1/N limit|small N]]
and [[asymptotic boundary|asymptotically]] [[anti-de Sitter spacetime|AdS]] [[string theory]]/[[M-theory]]:

[[KK-compactification|extra dimensions]]:

* [[Luis Alday]], [[Eric Perlmutter]], _Growing Extra Dimensions in AdS/CFT_ ([arXiv1906.01477](https://arxiv.org/abs/1906.01477))

[[string scattering amplitudes]] :

* {#ABP18} [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], _Genus-One String Amplitudes from Conformal Field Theory_, JHEP06(2019) 010 ([arXiv:1809.10670](https://arxiv.org/abs/1809.10670))

Discussion of [[large 1/N limit|small N]] effects in [[M-theory]] using the [[conformal bootstrap]]:

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

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

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

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

  from p. 2:

  > On the other hand, given our utter lack of a complete description of M-theory, the bulk is  not  terribly  useful  for  determining  finite aspects of the dual CFT. However, we can turn this problem around using the modern perspective of the conformal bootstrap, which gives an a priori independent formulation of the (local sector of the) CFT. This provides an independent tool for constructing M-theory at the non-perturbative level, a philosophy that we will substantiate in this work.

* [[Luis Alday]], [[Shai Chester]], Himanshu Raj, _6d $(2,0)$ and M-theory at 1-loop_, Journal of High Energy Physics **2021** 133 (2021) [[arXiv:2005.07175](https://arxiv.org/abs/2005.07175)]

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

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

* Damon J. Binder, [[Shai Chester]], Max Jerdee, Silviu S. Pufu, _The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes_ ([arXiv:2011.05728](https://arxiv.org/abs/2011.05728))

[[!redirects conformal bootstrap program]]

[[!redirects superconformal bootstrap]]


conformal cobordism category

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## Idea

A [[cobordism category]] of [[cobordisms]] equipped with [[conformal structure]].

One way to axiomatize [[conformal field theory]] (see there) in [[FQFT]]-style is as a suitable [[monoidal functor]] on a category of conformal cobordisms ([Segal 88](#Segal88)).

## References

### General

* {#Segal88} [[Graeme Segal]],

  _The definition of conformal field theory_ , in _Differential geometrical methods in theoretical physics_ (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, (1988), 165-171

  _Two-dimensional conformal field theories and modular functors_ , in _Proceedings of the IXth International Congress on Mathematical Physics_ , Swansea, 1988, Hilger, Bristol (1989) 22-37.

  _The definition of conformal field theory_ , preprint, 1988; also in: _Topology, geometry and quantum field theory_ , ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577.

Review in the context of the [[Mumford conjecture]]:

* {#Madsen07} [[Ib Madsen]], section 2.2 of _Moduli spaces from a topological viewpoint_, Proceedings of the International Congress of Mathematics, Madrid 2006 (2007) ([pdf](http://www.icm2006.org/proceedings/Vol_I/20.pdf))

[[!include D=2 CFT a functorial field theory -- references]]

[[!redirects conformal cobordism categories]]

[[!redirects conformal cobordism]]
[[!redirects conformal cobordisms]]

[[!redirects category of conformal cobordisms]]
[[!redirects categories of conformal cobordisms]]


conformal compactification

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## Idea

A _conformal compactification_ is an [[embedding]] of a non-[[compact topological space|compact]] [[Lorentzian manifold]] into a [[compact]] Lorentzian manifold as a [[dense subspace|dense]] [[open subset|open]] [[subspace]], such that the embedding is a [[conformal map]] .

## Example

The (or rather, _any_) conformal compactification $\overline{M}$ of (flat) [[Minkowski space]] $M = \mathbb{R}^{n,1}$ is important in various treatments of behaviour at infinity of physics thereon (see e.g. _[[Penrose-Hawking theorem]]_).

Section 4.2 of ([Nikolov-Todorov](#NikolovTodorov)) gives a thorough geometric discussion of the construction and gives explicit expressions in coordinates, and shows that (the underlying manifold of) this $\overline{M}$ is in fact $(S^n\times S^1)/\{\pm 1\}$, for a specific embedding of $S^n\times S^1$ in $\mathbb{R}^{n+1,2}$, and the action by scalar multiplication.

Let $q(x,t)$ be the standard indefinite quadratic form of signature $(n,1)$ on $\mathbb{R}^{n,1}$ and define the following map $\mathbb{R}^{n,1} \to \mathbb{R}^{n+1,2} \simeq \mathbb{R}^{n,1}\times \mathbb{R}^{1,1}$:
$$
\tilde{C}\colon (x,t) \mapsto (x,t,\frac{1-q(x,t)}{2},\frac{-1-q(x,t)}{2})
$$
This is a diffeomorphism on its image, which can be described as the intersection of the hyperplane $v-w=1$ and the quadric $q(x,t) + q'(v,w)=0$, where $(v,w)$ are coordinates on $\mathbb{R}^{1,1}$ with the quadratic form $q'(v,w) = v^2-w^2$. In particular, the image of this map avoids the origin in $\mathbb{R}^{n+1,2}$, and rearranging the defining equation for the quadric we get $|x|^2 + v^2 = t^2 + w^2$ $=K$, say, where $K\neq 0$. We can then scale $\tilde{C}(x,t)$ to $C(x,t)$ so that $K=1$, which means $C(x,t) \in S^n\times S^1$. [TODO: calculate this normalisation] Then the final quotient by $\{\pm 1\}$ gives the desired dense embedding $\mathbb{R}^{n,1} \to (S^n\times S^1)/\{\pm 1\}$.

One could also skip the normalisation step if desired, and pass directly to the quotient by $\mathbb{R}^\times \simeq \mathbb{R}_{\gt0}^* \times \{\pm 1\}$, treating $\tilde{C}(x,t)$ as homogeneous coordinates.

The blog post ([Wong 2009](#Wong09))) gives a discussion of comformal compactification in general, with Minkowski space as an example. It describes the underlying manifold of (the 'usual' construction of) $\overline{M}$ is $S^n\times S^1$, but then clarifies that this is a 'double cover'.

A conformal compactification, of _complexified_ [[Minkowski spacetime]] $\mathbb{R}^{3,1}$, is given by the [[Klein quadric]]. (eg, [Fioresi-Lledo-Varadarajan 07, section 2](#FioresiLledoVaradarajan07)). This plays a key role in the _[[twistor correspondence]]_.

## Related entries

* [[conformal geometry]]

* [[compactification]]

* [[asymptotically flat spacetime]]

## References

* [[Roger Penrose]], _Relativistic Symmetry Groups_, in A.O.Barut (ed.), _Group Theory in Non-Linear Problems: Lectures Presented at the NATO Advanced Study Institute on Mathematical Physics_  (1974)

* Valentina, _Conformal compactifications_ ([pdf](http://www.fysik.su.se/~ingemar/relteori/valentinamaster.pdf) -- link appears to be dead)

* Jörg Frauendiener, _Conformal Infinity_ ([living reviews](https://doi.org/10.12942/lrr-2004-1))

* {#NikolovTodorov} Nikolay M. Nikolov, Ivan T. Todorov, _Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory_, arXiv:[math-ph/0412039](http://arxiv.org/abs/math-ph/0412039)
 ([inspirehep pdf](http://inspirehep.net/record/674206/files/page1.pdf)).

* {#FioresiLledoVaradarajan07} R. Fioresi, M. A. Lledo, [[Veeravalli Varadarajan]], _The Minkowski and conformal superspaces_, J.Math.Phys.48:113505,2007 ([arXiv:math/0609813](https://arxiv.org/abs/math/0609813))

* {#Wong09} Willie Wong, _Conformal compactification of space-time_ ([web](https://qnlw.info/post/conformal-compactification-spacetime/)) (2009).

[[!redirects conformal compactifications]]


conformal connection

#Contents#
* table of contents
{:toc}

## Idea

A _conformal connection_ is a $(H \hookrightarrow G)$-[[Cartan connection]] for $G= O^+(n,1)$ an orthochronous [[Lorentz group]] and $H$ the [[stabilizer subgroup]] of a [[lightlike]] line in [[Minkowski space]] $\mathbb{R}^{n,1}$. Here $G/H$ is the [[celestial sphere]] of [[dimension]] $n-1$.

An example of [[parabolic geometry|parabolic]] [[Cartan geometry]]-

## Related concepts

[[!include local and global geometry - table]]

## References

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

* [[Felipe Leitner]], _Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry_, 2007 ([pdf](http://elib.uni-stuttgart.de/opus/volltexte/2009/3922/pdf/HabilKopf.pdf))

* {#CapSlovak09} [[Andreas Cap]], [[Jan Slovák]], sections 1.1.5, 1.6 of _Parabolic Geometries I -- Background and General Theory_, AMS 2009

[[!redirects conformal connections]]


conformal field theory

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

#Contents#
* table of contents
{:toc}

## Idea

Recall that a [[TQFT]] is an [[FQFT]] defined on the $(\infty,n)$-[[(infinity,n)-category of cobordisms|category of cobordisms]] whose morphisms are plain cobordisms and diffeomorphisms between these.

In a _conformal_ quantum field theory the cobordisms are equipped with a [[conformal structure]] (a [[Riemannian metric]] structure modulo pointwise rescaling): _[[conformal cobordisms]]_.

A conformal field theory (CFT) is accordingly a functor on such a richer category of conformal [[cobordisms]]. See the discussion at [[FQFT]] for more details.

The conformally invariant quantum field theories have fields for whom the [[correlation functions]] have a specific behaviour accounting for the _conformal dimension_ of the [[field (physics)|fields]]. This kind of constraints coming from conformal invariance, leads to constraints on the possible behaviour of correlation functions; this has being formulated in 1971 by Polyakov as a **[[conformal bootstrap]]** program: the conformal invariance should be sufficient to classify consistent conformal QFTs directly from the analysis of symmetries, rather than computing the [[Feynman diagram]] [[perturbation series]] from some [[action functional]]; this avoids the usual problems with regularization and [[renormalization]].

Precisely in 2-dimensions is the representation theory of the conformal group exceptionally interesting. In $d \gt 2$, the global transformations, i.e. the elements of the [[conformal group]], are given by the [[Poincare group|Poincare algebra]], dilations and special conformal transformations. In $2$ dimensions, there are additionaly infinitely many local generators of conformal transformations whose commutation relations are given by the [[Virasoro algebra]]. This circumstance enables Belavin, Polyakov and Zamolodchikov to make a breakthrough in the bootstrap program in 1984 (what also helped a 1984-1985 revolution in string theory).
 There are well-developed tools for handling the theory locally ([[conformal net]]s, [[vertex operator algebra]]s) and at least in the [[rational conformal field theory]] case there are complete classification results for the full theories (defined on cobordisms of all genera).

For this reason often in the literature the term "CFT" often implicitly refers to 2d CFT.

$2$-dimensional conformal field theories have two major applications:

* they describe critical phenomena on surfaces in condensed matter physics;

* they are the building blocks used in [[string theory]]

In the former application it is mostly the _local_ behaviour of the CFT that is relevant. This is encoded in [[vertex operator algebra]]s.

In the [[string theory|string theoretic]] applications the extension of the local theory to a full representation of the 2d [[conformal cobordism category]] is crucial. This extension is called _solving the [[sewing constraints]]_ .

## Abstract ##

In the definition paragraph we will show how to define a conformal field theory using the axiomatic approach of Wightman resp. Osterwalder-Schrader. There are several approaches to axiomatically define conformal field theory, the said approach is not the most "popular" or "elegant" one. There are two reasons to consider the Wightman approach however: If one is already familiar with the Wightman approach, it helps to put conformal field theories into context. The second reason is that several notions often used by physicists can be easily and rigorously defined and explained using this approach. Therefore, it may serve as a bridge between mathematicians and physics literature.

## 2d CFT on the plane ##

### Definition in Terms of the Osterwalder-Schrader Axioms ###

A definition of conformal field theories can be formulated using the appropriate version of the [[Osterwalder--Schrader axioms]], that is by a system of axioms of the [[correlation functions]]. From the correlation functions it is then possible to deduce the existence of field operators in the sense of the [[Wightman axioms]], that is fields aka field operators are operator valued distributions. Both the Hilbert space and the field operators are therefore not defined in the axioms, but reconstructed from the correlations functions defined in the first three axioms.

Let
$$
M_n := \{(z_1, ..., z_n) \in \mathbb{C}^n : z_i \neq z_j \; \text{for} \; i \neq j  \}
$$
be the **space of configuration points**.
Let $B_0$ be a countable index set and $B := \bigcup_{n \in \mathbb{N}} B_0^{n}$.

The correlation functions are a family $(G_i)_{i \in B}$ of continuous and polynomially bound functions
$$
   G_{i_1, ..., i_n}: M_n \to \mathbb{C}
$$

+-- {: .num_defn}
###### Axiom 1
**locality**

For all indexes $(i_1, ..., i_n)$, configuration points $(z_1, ..., z_n)$ and permutations $\pi: \{1,..., n \} \to \{1,..., n \}$ one has
$$
G_{(i_1,..., i_n)}(z_1,..., z_n) = G_{(\pi(i_1),..., \pi(i_n))}(\pi(z_1),..., \pi(z_n))
$$
=--

In this context, covariance is meant with respect to the [[Euclidean group]] $E = E_2$.

+-- {: .num_defn}
###### Axiom 2
**covariance**

For every index $i \in B_0$ there are **conformal weights** $h_i, \overline{h_i} \in \mathbb{R}$ ($\overline{h_i}$ is completly independent from $h_i$, it is _not_ the complex conjugate, this notation is widly used in the physics literature so we use it here, too) such that for $n \ge 1$, $w \in E$ and $w_i := w(z_i)$, $h_j := h_{i_j}$ one has
$$
G_{(i_1,..., i_n)}(z_1, \overline{z_i}, ..., z_n, \overline{z_n}) = \prod (\frac{dw}{dz} (z_j))^{h_j} (\overline{\frac{dw}{dz}(z_j)})^{\overline{h_j}}   G_{(i_1,..., i_n)}(w_1, \overline{w_1}..., w_n, \overline{w_n})
$$

=--

The $s_i := h_i - \overline{h_i}$ are called **conformal spin** (for the index $i$) and $d_i := h_i + \overline{h_i}$ is the **scaling dimension**. As an axiom 2.2 we assume that all conformal spins and scaling dimensions are integers.

For the next axiom, which is about reflection positivity, we need some notation:

We write $z \in \mathbb{C}$ as $z = t + iy$ and identify $y$ as the space coordinate and $t$ as (imaginary) time.
So, time reflection $\theta$ is simply:
$$
\theta: \mathbb{C} \to \mathbb{C}
$$

$$
\theta:  z = t + iy \mapsto -t + iy = - \overline{z}
$$

Let $\mathcal{S}(\mathbb{C}^n)$ be the space of [[Schwartz functions]] and

$$
M_n^+ := \{ (z_1, ..., z_n) \in M_n: \operatorname{Re}(z_i) \gt 0 \}
$$

$$
\mathcal{S}_0^+ := \mathbb{C}
$$

$$
\mathcal{S}_n^+ := \{f \in \mathcal{S}(\mathbb{C}^n): \operatorname{Supp}(f) \subset M_n^+ \}
$$

Finally let $\underline{ \mathcal{S}^+ }$ the space of all sequences $\underline{f} = (f_i)_{i \in B}$ with $f_i \in \mathcal{S}_n^+$ fpr $i \in B_0^n$ with only finitely many entries $\ne 0$.

+-- {: .num_defn}
###### Axiom 3
**reflection positivity**

There is a map
$$
*: B_0 \to B_0
$$
with $*^2 = id_{B_0}$ that extends to a map

$$
*: B \to B
$$

$$
*: i \mapsto i^*
$$
so that

1. $G_i(z) = G_{i^*}(\theta(z)) = G_{i^*}(- \overline{z})$

2. $\langle \underline{f}, \underline{f} \rangle \ge 0$ for all $\underline{f} \in \underline{\mathcal{S}^+}$

The scalar product $\langle \underline{f}, \underline{f} \rangle$ is defined as

$$
\sum_{i, j \in B, m, n \in \mathbb{N} } \int_{M_{n+m}} G_{i^* j} (\theta(z_1), ...,\theta(z_n), w_1, ..., w_n) \overline{f_i(z)} f_j(w) d^n z d^m w
$$

=--

+-- {: .num_prop}
###### Proposition
Both a Hilbert space and the field operators of the theory can be (re-) constructed from axioms 1, 2 and 3.

TODO: Details
=--

+-- {: .num_defn}
###### Axiom 4
**scaling covariance**

The correlation functions satisfy the covariance condition also for dilatations $w(z) = e^t (t), t \in \mathbb{R}$.
=--

Explicitly, the last condition says that
$$
G_i(z_1, ..., z_n) = (e^t)^{h_1 + \overline{h_1} +...+ h_n + \overline{h_n}} G_i(e^t(z_1), ..., e^t(z_n))
$$

Given axioms 1-4, the 2-point functions can be fully classified, see below.

+-- {: .num_defn}
###### Axiom 5
**existence of the energy-momentum tensor**

There are four fields $T_{\mu, \nu}, \mu. \nu \in \{0, 1\}$ with the following properties:

Symmetry:
$$
T_{\mu, \nu} = T_{\nu, \mu}, \; T_{\mu, \nu}(z)^* = T_{\nu, \mu}(\theta (z))
$$

$$
\partial_t T_{\mu, 0} + \partial_y T_{\mu, 1} = 0
$$

$T_{\mu, \nu}$ has scaling dimension 2 and the conformal spin $s$ is restricted by
$$
s(T_{00} - T_{11} \pm 2i \; T_{01}) = \pm 2
$$
=--

The energy momentum tensor allows us to define two densly defined operators $L_{-n}, \overline{L_{-n}}$ that both define unitary representations of the [[Virasoro algebra]].

+-- {: .num_prop}
###### Proposition
**Lüscher-Mack**

T is holomorphic. Therefore, the operators
$$
L_{-n} := \frac{1}{2 \pi i} \oint_{| \zeta | = 1} \frac{T(\zeta)}{\zeta^{n+1}} d\zeta
$$
and
$$
\overline{L_{-n}} := \frac{1}{2 \pi i} \oint_{| \zeta | = 1} \frac{\overline{T}(\zeta)}{\zeta^{n+1}} d\zeta
$$
are well defined and satisfy the commutation relations of two commuting [[Virasoro algebra|Virasoro algebras]] with the same central charge.
=--

+-- {: .num_defn}
###### Definition
**primary field**

A conformal field $\psi_i, i \in B_0$, is called a primary field if
$$
[L_n, \psi_i] =  z^{n+1} \partial_z \psi_i(z) + h_i (n+1) z^n \psi_i(z)
$$
and likewise for $\overline{z}$.
=--

So, primary fields are the fields whose correlation functions are covariant "infinitesimally" with respect to all holomorphic functions, the "infinitesimal symmetry" expressed in the definition.

The fields are operator valued distributions and cannot be multiplied in general. The possibility of multiplication of fields evaluated at different points and some control of the singularities of this product are part of the axioms of conformal field theory:

+-- {: .num_defn}
###### Pseudodefinition
**operator product expansion**

An operator product expansion (**OPE**) for a family of fields means that there is for all fields and all $z_1 \neq \z_2$ a relation of the form
$$
\psi_i(z_1) \psi_j(z_2) \sim \sum_{k \in B_0} C_{ijk} (z_1 - z_2)^{h_k - h_i - h_j} \psi_k(z_2)
$$

Here $\sim$ means modulo regular functions.
=--

A rigorous interpretation of an OPE would interpret the given relation as a relation of e.g. matrix elements or vacuum expectation values.

An OPE is called **associative** if the expansion of a product of more than two fields does not depend on the order of the expansion of the products of two factors.
Since the OPE has no interpretation as defining products of operators, or more generally the product in a ring, the notion of associativity does not refer to the associativity of a product in a ring, as the term may suggest.

+-- {: .num_defn}
###### Axiom 6
**existence of associative operator product expansion**

The primary fields have an associative OPE.
=--

Many _formal_ calculations of physicists in CFT involving OPE can be justified by using [[vertex operator algebra]]s, so that [[vertex operator algebra]]s have become a standard way to formulate CFT.

### Properties ###

+-- {: .num_prop}
###### Proposition
Any 2-point function satisfying axioms 1-4 has the following form:
$$
G_{ij}(z_1, z_2) = C_{ij} (z_1 - z_2)^{-(h_i + h_j)} (\overline{z_1} - \overline{z_2})^{-(\overline{h_i} + \overline{h_j})}
$$
with some constant $C_{ij} \in \mathbb{C}$.
=--

## 2d CFT on surfaces or arbitrary genus

...

## Conformal anomaly

The [[Liouville cocycle]] appears when one moves from genuine [[representation]]s of 2-dimensional [[conformal cobordisms]] to [[projective representation|projective representations]]. The obstruction for such a projective representation to be a genuine representation is precisely given by the central charge $c$; when $c\neq 0$, one says that the conformal field theory has a _conformal [[quantum anomaly|anomaly]]_.

...

## CFT in AQFT language

In [[AQFT]] conformal field theory is modeled in terms of [[local net]]s that are [[conformal net]]s.

## Full versus chiral CFT
 {#FullAndChiral}

> some discussion of full vs. chiral CFT goes here... then:

The following result establishes which pairs of [[vertex operator algebras]] can appear as the left and right chiral parts of a _[[full field algebra]]_ in the sense of ([Huang-Kong 05](#HuangKong05)), etc. (see [[vertex operator algebra]]).

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

Two [[modular tensor categories]] $C$ and $D$ are said to have the same **Witt class** if there exist two [[spherical categories]] $S$ and $T$ such that we have an [[equivalence of categories|equivalence]] of [[ribbon categories]]

$$
  C \boxtimes \mathcal{Z}(S) \simeq D \boxtimes \mathcal{Z}(T)
$$

=--

Here $\mathcal{Z}(S)$ is the category whose objects are pairs $(Z, z)$ consisting of an
object $Z \in S$ and a [[natural isomorphism]]
$z_X : Z \otimes X \stackrel{\simeq}{\to} X \otimes Z$, such that for al
objects $X, Y \in S$ we have

$$
  \array{
    Z \otimes X \otimes Y &&\stackrel{z_{X \otimes Y}}{\to}&&
    X \otimes Y \otimes Z
    \\
    & {}_{\mathllap{z_x \otimes Id}}\searrow && \nearrow_{Id \otimes \mathrlap{z_y^{-1}}}
    \\
    && X \otimes Z \otimes Y
  }
$$

This becomes a monoidal category itself by setting

$$
  (Z,z) \otimes (W,w) := (Z \otimes W, z \otimes w)
  \,.
$$

+-- {: .num_theorem}
###### Theorem
**([[Michael Müger]])**

If $S$ is a [[spherical category]] then $\mathcal{Z}(S)$ is a [[modular tensor category]].

=--

+-- {: .num_theorem}
###### Theorem
**([[Michael Müger]])**

If $C$ is a [[modular tensor category]] then there is an equivalence of
[[ribbon categories]]

$$
  \mathcal{Z}
  \stackrel{\simeq}{\leftarrow}
  C \boxtimes \bar C
$$

=--

## Examples

* [[Ising model]]

* [[WZW model]]

* [[parafermion]]

* [[logarithmic CFT]]

* [[Brownian loop soup]]

* [[6d (2,0)-supersymmetric QFT]]

## Related concepts

* [[sewing constraints]]

  * [[modular invariance]]

  * [[Cardy condition]]

* [[conformal structure]], [[moduli space of conformal structures]], [[Teichmüller theory]]

* [[conformal transformation]], [[Möbius transformation]]

* [[vertex operator algebra]]

* [[conformal anomaly]]

* [[rational conformal field theory]]

* [[minimal model CFT]]

* [[Gepner model]], [[WZW model]]

* [[conformal net]]

* [[Liouville cocycle]]

* [[Yangian]]

* [[WZW model]]

  * [[current algebra]], [[affine Lie algebra]]

  * [[Knizhnik-Zamolodchikov connection]]

* [[Liouville theory]]

* [[SCFT]]

  * [[2d SCFT]]

* [[boundary conformal field theory]]

* [[automorphism of a 2d conformal field theory]]

* [[TT deformation]]

## References ##

### General

The mathematical axioms of CFT, as well as its relevance for surface phenomena goes back to

* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two–dimensional quantum field theory_, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (doi:10.1016/0550-3213(84)90052-X)

Textbook accounts:

* [[Philippe Di Francesco]], Pierre Mathieu, David Sénéchal: *Conformal field theory*, Springer (1997) [[doi:10.1007/978-1-4612-2256-9](https://doi.org/10.1007/978-1-4612-2256-9)]

* [[Martin Schottenloher]], *A Mathematical Introduction to Conformal Field Theory*, Lecture Notes in Physics **759**, Springer 2008 ([doi:10.1007/978-3-540-68628-6](https://link.springer.com/book/10.1007/978-3-540-68628-6), [web](https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/))

With emphases on [[braid group representations]] constituted by [[conformal blocks]] via the [[Knizhnik-Zamolodchikov equation]]:

* [[Toshitake Kohno]], *Conformal field theory and topology*, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs __210__. Iwanami Series in Modern Mathematics. Amer. Math. Soc. (2002) [[AMS:mmono-210](https://bookstore.ams.org/mmono-210)]

Introduction and surveys:

* [[Paul Ginsparg]], *Applied Conformal Field Theory*, lectures at: *[Fields, strings, critical phenomena, Les Houche Summer School 1988](https://inis.iaea.org/search/search.aspx?orig_q=RN:21047524)([arXiv:hep-th/9108028](https://arxiv.org/abs/hep-th/9108028))

* {#Gawedzki99} [[Krzysztof Gawedzki]], _Conformal field theory: a case study_ ([arXiv:hep-th/9904145](http://arxiv.org/abs/hep-th/9904145))

* [[Ingo Runkel]], _Boundary problems in conformal field theory_ ([pdf](http://www.math.uni-hamburg.de/home/runkel/PDF/phd.pdf))

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

* Yu Nakayama, _A lecture note on scale invariance vs conformal invariance_, [arXiv:1302.0884](http://arxiv.org/abs/1302.0884)

* [[Katrin Wendland]], _Snapshots of Conformal Field Theory_, in:
Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer 2015  ([arXiv:1404.3108](http://de.arxiv.org/abs/1404.3108), [doi:10.1007/978-3-319-09949-1_4](https://doi.org/10.1007/978-3-319-09949-1_4))

* [[Jörg Teschner]], _A guide to two-dimensional conformal field theory_ ([arXiv:1708.00680](http://arxiv.org/abs/1708.00680))

* Joaquin Liniado, *Two Dimensional Conformal Field Theory and a Primer to Chiral Algebras* ([arXiv:2110.15164](https://arxiv.org/abs/2110.15164))

* Marc Gillioz, *Conformal field theory for particle physicists* [[arXiv:2207.09474](https://arxiv.org/abs/2207.09474)]

* Satoshi Nawata, Runkai Tao, Daisuke Yokoyama, *Fudan lectures on 2d conformal field theory* [[arXiv2208.05180](https://arxiv.org/abs/2208.05180)]

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

* Andrew M. Evans, Alexandra Miller, Aaron Russell, *A Conformal Field Theory Primer in $D \geq 3$* [[arXiv:2309.10107](https://arxiv.org/abs/2309.10107)]

For a discussion of mathematical formalization ([[vertex operator algebras]], [[conformal nets]], [[functorial QFT]]) see

* {#Tener18} James E. Tener, _Representation theory in chiral conformal field theory: from fields to observables_ ([arXiv:1810.08168](https://arxiv.org/abs/1810.08168))

* {#HuangKong05} [[Yi-Zhi Huang]], [[Liang Kong]], _Full field algebras_, Commun.Math.Phys.272:345-396,2007 ([arXiv:0511328](http://arxiv.org/abs/math/0511328))

* [[Liang Kong]], _Full field algebras, operads and tensor categories_, Adv. Math.213:271-340, 2007 ([arXiv:0603065](http://arxiv.org/abs/math/0603065))

For a survey of perspectives in CFT with an eye towards [[string theory]] see various contributions in

* [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_

Discussion in relation to the [[AdS-CFT correspondence]]:

* Matteo Broccoli, *Aspects of Conformal Field Theory* [[arXiv:2212.11829](https://arxiv.org/abs/2212.11829)]

### CFT on complex curves/surfaces of arbitrary genus

For chiral 2d CFT:

* [[Edward Frenkel]], [[David Ben-Zvi]], _Vertex algebras and algebraic curves_, Math. Surveys and Monographs __88__, AMS 2001,
xii+348 pp. (Bull. AMS. [review](http://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00955-2/S0273-0979-02-00955-2.pdf), [ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1106.17035&format=complete))

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_, Colloqium Publications __51__,  Amer. Math. Soc. 2004, [gbooks](http://books.google.hr/books?id=yHZh3p-kFqQC&lpg=PP1&vq=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&dq=isbn%3A0817638296&pg=PP1#v=onepage&q=%22Two-dimensional%20conformal%20geometry%20and%20vertex%20operator%20algebras%22&f=false)

### Formulation by conformal nets

* [[Chris Douglas]], [[André Henriques]], _Topological modular forms and conformal nets_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_

For further references see [[conformal net]].

### Formulation in full AQFT

Formulation in full [[AQFT on curved spacetimes]]:

* [[Marco Benini]], [[Luca Giorgetti]], [[Alexander Schenkel]], *A skeletal model for 2d conformal AQFTs* ([arXiv:2111.01837](https://arxiv.org/abs/2111.01837))

[[!include D=2 CFT a functorial field theory -- references]]

### Formulation by algebra in modular tensor categories
 {#FRSFormalism}

For full CFT, The special case of _rational_ CFT has been essentially entirely formalized and classified. The classification result for full rational 2d CFT was given by Fjelstad--Fuchs--[[Ingo Runkel|Runkel]]--[[Christoph Schweigert|Schweigert]]

* [FRS reviews](http://golem.ph.utexas.edu/string/archives/000747.html)

* [The FRS theorem of RCFT](http://golem.ph.utexas.edu/string/archives/000813.html)

* {#KapustinSaulina} [[Anton Kapustin]], [[nLab:Natalia Saulina]] _Surface operators in 3d TFT and 2d Rational CFT_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ ([arXiv:1012.0911](http://arxiv.org/abs/1012.0911))

* {#Kong} [[Liang Kong]], _Conformal field theory and a new geometry_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ ([arXiv:1107.3649](http://arxiv.org/abs/1107.3649))

...

[[!redirects CFT]]
[[!redirects CFTs]]

[[!redirects conformal field theories]]

[[!redirects 2d CFT]]
[[!redirects 2d CFTs]]

[[!redirects 2-dimensional conformal field theory]]
[[!redirects 2-dimensional conformal field theories]]

[[!redirects 2d conformal field theory]]
[[!redirects 2d conformal field theories]]

[[!redirects D=2 conformal field theory]]
[[!redirects D=2 conformal field theories]]

[[!redirects D=2 CFT]]
[[!redirects D=2 CFT]]

[[!redirects D=2 CFT]]

[[!redirects conformal weight]]
[[!redirects conformal weights]]


The article

* [[Edward Witten]],

  _Conformal field theory in four and six dimensions_,

  in

  [[Ulrike Tillmann]] (ed.),

  _Topology, Geometry and Quantum Field Theory_,

  Cambridge University Press (2002)

  [arXiv:0712.0157](http://arxiv.org/abs/0712.0157)

  [doi:10.1017/CBO9780511526398.017](https://doi.org/10.1017/CBO9780511526398.017)

disucsses in a simplified setup the [[Kaluza-Klein mechanism|compactification]] of the [[6d (2,0)-superconformal QFT]] on a [[Riemann surface]] to 4-dimensional ([[super Yang-Mills theory|super]]) [[Yang-Mills theory]], and how this geometrically explains the [[S-duality]]/[[electric-magnetic duality]] of this theory.

For related references see at _[N=2 D=4 super-Yang-Mills theory -- References -- Construction from 5-branes](N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes)_

category: reference


conformal geometry

+-- {: .rightHandSide}
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###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _conformal structure_ on a [[manifold]] is the structure of a [[Riemannian metric]] modulo rescalings of the [[Riemannian metric|metric]] [[tensor]] by some real valued function on the manifold.  The [[homomorphisms]] of conformal structures are called _[[conformal transformations]]_.

In terms of [[Cartan geometry]] conformal structure is expressed by [[conformal connections]], conformal geometry is a special case of  [[parabolic geometry]] and hence of [[Cartan geometry]].

In the context of [[quantum field theory]] conformal structure underlies the formulation of [[conformal field theory]]. Due to the coincidence of the [[conformal group]] of [[Minkowski spacetime]] of [[dimension]] $d$ with the [[anti de Sitter group]] of [[anti de Sitter spacetime]] in dimension $d+1$ there is a close relation between certain [[conformal field theories]] and certain theories of [[gravity]]. This is the content of the _[[AdS-CFT correspondence]]_. This works most accurately in the context of [[supergeometry]], hence for [[superconformal groups]] acting on the [[asymptotic boundary]] of [[super anti de Sitter spacetimes]].

## Related concepts

* [[conformal map]], [[conformal transformation]], [[conformal group]]

* [[conformally flat manifold]]

* [[causal structure]]

* [[Fefferman-Graham ambient construction]]

* [[conformal compactification]]

* [[CFT]]

* [[twistor space]]

* [[conformal cobordism category]]

* [[conformal connection]]

* [[superconformal geometry]]

[[!include local and global geometry - table]]

## References

* David Gu, _Conformal structure_, slides ([pdf](http://www.cs.sunysb.edu/~gu/lectures/lecture_2_conformal_structure.pdf))

* [[eom]], _[Conformal structure](http://www.encyclopediaofmath.org/index.php/Conformal_structure)_

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

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

Discussion of [[conformal structure]] as  [[G-structure]]:

* Maks A. Akivis, Vladislav V. Goldberg, _Conformal and Grassmann structures_, Differential Geom. Appl. 8 (1998) no. 2 177-203 ([arXiv:math/9805107](https://arxiv.org/abs/math/9805107))

Discussion of conformal [[Cartan geometry]] ([[parabolic geometry]]) includes

* {#CapSlovak09} [[Andreas Čap]], [[Jan Slovák]], sections 1.1.5, 1.6 of _Parabolic Geometries I -- Background and General Theory_, AMS 2009

* [[Felipe Leitner]], part 1, section 6 of _Applications of Cartan and Tractor Calculus to Conformal and CR-Geometry_, 2007 ([pdf](http://elib.uni-stuttgart.de/opus/volltexte/2009/3922/pdf/HabilKopf.pdf))

* [[Hega Baum]], [[Andreas Juhl]], _Conformal Differential Geometry: Q-curvature and Conformal Holonomy_,  Oberwolfach Seminars, vol.  40,  Birkhäuser, 2010, 165pp.

* [[Andree Lischewski]], section 2 of _Conformal superalgebras via tractor calculus_, Class.Quant.Grav. 32 (2015) 015020 ([spire](https://inspirehep.net/record/1334198), [arXiv:1408.2238](http://arxiv.org/abs/1408.2238))

* Sean Curry, A. Rod Gover, _An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity_, 2014, ([ 	arXiv:1412.7559](http://arxiv.org/abs/1412.7559))

A survey of the field as of 2007 is in

* A. Rod Gover, [[Andreas Čap]], _Conformal and CR geometry: Spectral and nonlocal aspects_ [pdf](https://www.birs.ca/workshops/2007/07rit132/report07rit132.pdf)

Discussion with an eye towards combination with [[spin geometry]] is in

* [[Pierre Anglès]], _Conformal Groups in Geometry and Spin Structures_, Progress in Mathematical Physics 2008

[[!redirects conformal geometries]]

[[!redirects conformal structure]]
[[!redirects conformal structures]]

[[!redirects conformal manifold]]
[[!redirects conformal manifolds]]


conformal gravity

## Idea

[[gravity]] with local [[conformal transformation|conformal invariance]].

in [[first-order formulation of gravity]]: [[Cartan connection]] [[conformal connection]].

## related concepts

[[!include local and global geometry - table]]

## References

* Carlos Kozameh, Ezra T Newman, Pawel Nurowski, _Conformal Einstein equations and Cartan conformal connection_ ([arXiv:gr-qc/0302080](http://arxiv.org/abs/gr-qc/0302080))


conformal group

+-- {: .rightHandSide}
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#### Riemannian geometry
+--{: .hide}
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=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contemts#
* table of contents
{:toc}

## Idea

A _[[conformal transformation]]_ (conformal mapping) is a transformation of a space which preserves the angles between the curves. In other words, it preserves the angles infinitesimally. The __conformal group__ of a space which has well defined notion of angles between the curves is the group of space automorphisms which are also conformal transformations.

## Examples

### Of euclidean space

In euclidean $n$-space for $n\gt 2$ a general conformal transformation is some composition of a translation, dilation, rotation and possibly an inversion with respect to a $n-1$-sphere.

For $n=2$, i.e. in a complex plane, this still holds for (the group of) global conformal transformations but one also has nontrivial local automorphisms. One has in fact infinite-dimensional family of local conformal transformations, which can be described by an arbitrary holomorphic or an antiholomorphic automorphism (in fact one writes $z$ and $\bar{z}$ as independent coordinates in the complexification $\mathbb{C}^2$ and restricts to the real part $\mathbb{R}^2\cong \mathbb{C}$). This is important for CFT in 2d.

### Of $\mathbb{R}^{d,t}$

For $d,t \in \mathbb{N}$ write $\mathbb{R}^{d,t}$ for the [[pseudo-Riemannian manifold]] which is the [[Cartesian space]] $\mathbb{R}^{d+t}$ equipped with the constant metric of [[signature of a quadratic form|signature]]  $(d,t)$. I.e. for $t = 0$ this is [[Euclidean space]] and for $t=1 $ this is [[Minkowski spacetime]].

If $d+t \gt 2$ then the conformal group of $\mathbb{R}^{d,t}$ is the [[orthogonal group]] $P(d+1, t+1)/\{\pm 1\}$. The [[connected component]] of the [[neutral element]] is the [[special orthogonal group]] $SO(d+1,t+1)$. (e.g [Schottenloher 08, chapter 2, theorem 2.9](#Schottenloher08)).

Notice that for $t= 1$ this is also the [[anti de Sitter group]], the [[isometry group]] of [[anti de Sitter spacetime]] of dimension $d+1+t$. This equivalence is the basis of the [[AdS-CFT correspondence]].

### Cosets

(...) [[Möbius space]] (...)

([Arutyunov-Sokatchev 02](#ArutyunovSokatchev02))

(...)

## Related concepts

* [[conformal invariance and scale invariance]]

* [[Möbius group]]

* [[conformal geometry]]

* [[conformal connection]]

* [[conformal field theory]]

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

[[!include local and global geometry - table]]

## References

Textbook accounts include

* {#Schottenloher08} [[Martin Schottenloher]], _The conformal group_, chapter 2 of _A mathematical introduction to conformal field theory_, 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/02_978-3-540-68625-5_Ch02_23-08-08.pdf))

and (with an eye towards combination with [[spin geometry]])

* [[Pierre Anglès]], _Conformal Groups in Geometry and Spin Structures_, Progress in Mathematical Physics 2008

Details on the conformal Lie algebra of [[conformal Killing vectors]] acting on 3+1 dimensional [[Minkowski spacetime]] are spelled out for instance in

* [[Matthias Blau]], section 9.3 of _Lecture notes on general relativity_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

Discussion in [[conformal field theory]]

* {#ArutyunovSokatchev02} G. Arutyunov, E. Sokatchev, _Conformal fields in the pp-wave limit_, JHEP 0208 (2002) 014 ([arXiv:hep-th/0205270](https://arxiv.org/abs/hep-th/0205270))

See also

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

[[!redirects conformal groups]]

[[!redirects conformal symmetry]]
[[!redirects conformal invariance]]


conformal invariance and scale invariance

#Contents#
* table of contents
{:toc}

## Idea

An object is _scale invariant_ if it is preserved by global rescaling of length. An object is _conformally invariant_ if it is preserved by transformations that preserve angles, hence by _local_ rescaling of length.

## Related concepts

* [[conformal transformation]]

* [[fractal]]

## Referenes

* Yu Nakayama, _Scale invariance vs conformal invariance_ ([arXiv:1302.0884](http://arxiv.org/abs/1302.0884))

[[!redirects scale invariance and conformal invariance]]

[[!redirects scale invariance]]
[[!redirects conformal invariance]]


conformal map

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=--
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#Contents#
* table of contents
{:toc}

## Definition

A [[smooth function]] $f \;\colon\; X_1 \to X_2$ between two [[Riemannian manifold]] $(X_1,g_1)$, $(X_2,g_2)$ is called _conformal_ if it preserves the [[conformal geometry]] induced by the [[Riemannian metrics]] $g_1$ and $g_2$, hence if there is a [[smooth function]] $r\;\colon\; X_1 \to \mathbb{R}$ such that

$$
  f^\ast g_2 = r g_1
  \,.
$$

If $X_1 = X_2$ and $f$ is a [[diffeomorphism]], then this is also called a _[[conformal transformation]]_.

## Examples

* [[stereographic projection]]

* [[conformal compactification]]

## References

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

[[!redirects conformal maps]]

[[!redirects conformal mapping]]
[[!redirects conformal mappings]]

[[!redirects conformal function]]
[[!redirects conformal functions]]


conformal net

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

#Contents#
* table of contents
{:toc}

## Idea

A _conformal net_ is a [[local net of observables]] that describes ([[2d CFT|2-dimensional]]) [[conformal field theory]] in the framework of [[algebraic quantum field theory]].

## Definition

(...)

## Properties

### 3-Category of conformal nets

(...)

* [[3-category of fermionic conformal nets]]

### Relation to vertex operator algebras
 {#RelationToVertexOperatorAlgebras}

Subject to some conditions a [[vertex operator algebra]] may be turned into a conformal net and vice versa ([Carpi-Kawahigahshi-Longo-Weiner 15](#CarpiKawahigahshiLongoWeiner15), [Carpi 16](#Carpi16)) also ([Gui 18](#Gui18)).

### Relation to spectral triples

Superconformal nets encoding [[2d SCFTs]] may be related to [[local nets]] of [[spectral triples]] ([CHKL 09](#CHKL09)) (an incarnation of "[[2-spectral triples]]", see there for more).

## Examples

### Free fermion

([DH, section 4](#DouglasHenriquesGeometric))

$$
  Fer_n(I)
    \coloneqq
  Cl(L^2(I, \mathbb{R}^n \otimes S))
$$

## Related concepts

* [[conformal structure]], [[CFT]]

* [[conformal bootstrap]]

## References

### Fundamentals

Standard [[AQFT]] references on conformal nets include

* [[Roberto Longo]], _Conformal Subnets and Intermediate Subfactors_ ([arXiv:0102196](http://arxiv.org/abs/math/0102196))

and lots of others...

A somewhat novel take on conformal nets is presented and studied in

* [[Arthur Bartels]], [[Chris Douglas]], [[Andre Henriques]], _Conformal nets and local field theory_ ([arXiv:0912.5307](http://arxiv.org/abs/0912.5307))

* [[Arthur Bartels]], [[Chris Douglas]], [[Andre Henriques]], _Conformal nets I: coordinate-free nets_ ([arXiv:1302.2604](http://arxiv.org/abs/1302.2604))

where conformal nets are arranged into a [[tricategory]]  with [[symmetric monoidal category|symmetric monoidal structure]] that is a [[delooping]] of the [[bicategory]] of [[von Neumann algebra]]s and [[bimodule]]s between these. Among other things, this work connects the [[AQFT]] notion of conformal nets with the [[FQFT]] notion of [[cobordism]] representations.

More on this in

* [[Chris Douglas]], [[André Henriques]], _Topological modular forms and conformal nets_ , in [[Branislav Jurčo]], [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_

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

Relation to [[vertex operator algebras]] is discussed in

* {#CarpiKawahigahshiLongoWeiner15} [[Sebastiano Carpi]], [[Yasuyuki Kawahigashi]], [[Roberto Longo]], Mihály Weiner, _From vertex operator algebras to conformal nets and back_, Memoirs of the American Mathematical Society, Volume 254, Number 1213 2018;  ([doi:10.1090/memo/1213](https://doi.org/10.1090/memo/1213), [arXiv:1503.01260](http://arxiv.org/abs/1503.01260))

* {#Carpi16} [[Sebastiano Carpi]], _Operator algebras and vertex operator algebras_, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) ([arXiv:1603.06742](http://arxiv.org/abs/1603.06742))

and for [[super vertex operator algebras]]:

* [[Sebastiano Carpi]], [[Tiziano Gaudio]], [[Robin Hillier]], *From vertex operator superalgebras to graded-local conformal nets and back* [[arXiv:2304.14263](https://arxiv.org/abs/2304.14263)]

See also

* James E. Tener, _Representation theory in chiral conformal field theory: from fields to observables_ ([arXiv:1810.08168](https://arxiv.org/abs/1810.08168))

Relation of the corresponding [[ribbon categories]]:

* {#Gui18} Bin Gui, _Categorical extensions of conformal nets_ ([arXiv:1812.04470](https://arxiv.org/abs/1812.04470))

Relation to [[2-spectral triples]] is discussed in terms of [[vertex operator algebras]]

* [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ , in [[Branislav Jurčo]], [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_

and in terms of actual [[local nets]] in

* {#CHKL09} [[Sebastiano Carpi]], Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Spectral triples and the super-Virasoro algebra_, Commun.Math.Phys.295:71-97 (2010) ([arXiv:0811.4128](http://arxiv.org/abs/0811.4128))

### Classification
 {#ReferencesClassification}

* [[Yasuyuki Kawahigashi]], _Classification of operator algebraic conformal field theories in dimensions one and two_ ([arXiv:math-ph/0308029](http://arxiv.org/abs/math-ph/0308029))

* [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Classification of Two-dimensional Local Conformal Nets with $c \lt 1$ and 2-cohomology Vanishing for Tensor Categories_ ([arXiv](http://arxiv.org/abs/math-ph/0304022))

* [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Classification of Local Conformal Nets. Case $c \lt 1$ _ ([arXiv](http://arxiv.org/abs/math-ph/0201015))

### Representation theory

Articles that discuss the [[representation theory]] of conformal nets include

* [[Sebastiano Carpi]], [[Roberto Conti]], Robin Hillier, Mihaly Weiner, _Representations of Conformal Nets, Universal $C^\ast$-Algebras and K-Theory_ ([arXiv:1202.2543](http://arxiv.org/abs/1202.2543))

[[!redirects conformal nets]]

[[!redirects local conformal net]]
[[!redirects local conformal nets]]


conformal transformation

+-- {: .rightHandSide}
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###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A _conformal transformation_ is a [[diffeomorphism]] that preserves a [[conformal structure]] (a [[conformal map]]). The [[group]] of conformal transformations is the _[[conformal group]]_.

## Properties

* A bijective conformal transformation of the [[Riemann sphere]] (to itself) is also called a [[Möbius transformation]].

* For $p + q \geq 3$ then the [[group]] of conformal transformations of $\mathbb{R}^{p,q}$ is $\simeq O(p+1,q+1)$

## Related concepts

* [[conformal geometry]]

* [[conformal connection]]

* [[conformal field theory]]

## References

* Wolfgang Kühnel, Hans-Bert Rademacher, _Liouville's theorem in conformal geometry_ ([pdf](http://www.igt.uni-stuttgart.de/LstDiffgeo/Kuehnel/preprints/lv1.pdf))

* [[Isadore Singer]], [[Shlomo Sternberg]], _The infinite groups of Lie and Cartan. J. Anal. Math. 15, 1-114 (1965)

briefly reviewed in

* [[Shlomo Sternberg]], _Conformally flat geometry and supergeometry_, section 3 of  _On charge conjugation_, Comm. Math. Phys. Volume 109, Number 4 (1987), 649-679. ([Euclid](http://projecteuclid.org/euclid.cmp/1104117081))

[[!redirects conformal transformations]]

[[!redirects special conformal transformation]]
[[!redirects special conformal transformations]]


conformally flat manifold

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

#Contents#
* table of contents
{:toc}

## Idea

A [[Riemannian manifold]] $(X,g)$ is _conformally flat_ if it is locally taken to a flat manifold by a [[conformal transformation]]; more specifically, if there exists an [[open cover]] $\{U_i \to X\}_{i \in I}$ and on each $U_i$ a [[smooth function]] $f_i$, such that $e^{f_i} g_{\vert U_i}$ has vanishing [[Riemann curvature]]: $R\big( e^{f_i} g_{\vert U_i} \big) = 0$.

## Examples

* A [[hyperbolic manifold]] is conformally flat, see [there](hyperbolic+manifold#ConformalFlatness).

## Related concepts

* [[conformal compactification]]

* [[asymptotically flat spacetime]]

## References

See also

* Wikipedia, _[Conformally flat manifold](https://en.wikipedia.org/wiki/Conformally_flat_manifold)_

* Michael Kapovich, _Conformally flat metrics on 4-manifolds_,     J. Differential Geom. Volume 66, Number 2 (2004), 289-301 ([euclid:1102538612](https://projecteuclid.org/euclid.jdg/1102538612))

[[!redirects conformally flat manifolds]]


congruence

+-- {: .rightHandSide}
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### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Congruences
* table of contents
{: toc}

## Idea
 {#Idea}

In [[Euclidean geometry]], by *congruence* one means the [[equivalence relation]] on the collection of [[subsets]] of a [[Euclidean space]] which regards two of these as equivalent if one is carried into the other by an [[isometry]] (of the ambient Euclidean space).

Similarly, in [[algebra]], by a *congruence* one means certain [[equivalence relations]] on elements of algebraic [[structures]], such as [[groups]] or [[rings]] (cf. e.g. the [[multiplicative group of integers modulo n]]).

Therefore, in [[category theory]] the term *congruence* is used in the broad generality of
[[equivalence relations]] on (the [[generalized elements]] of) *any object* [[internalization|internal to]] *any* [[finitely complete category]].

## Definitions

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

In a [[category]] $C$ with [[pullbacks]], a **congruence** on an [[object]] $X$ is an [[internalization|internal]] [[equivalence relation]] on $X$ (i.e.: an [[internal groupoid]] --- hence an [[internal category]] with all [[morphisms]] being [[isomorphisms]] --- but with no non-[[identity morphism|identity]] [[automorphisms]]).

This means that it consists of a [[subobject]]

\[
  \label{TheSubobject}
  i
  \;\colon\;
  R\stackrel{(p_1,p_2)}\hookrightarrow X \times X
\]

of the [[Cartesian products]] of $X$ with itself, equipped with the following [[morphisms]]:

* internal [[reflexive relation|reflexivity]]: $r \colon X \to R$ which is a [[section]] both of $p_1$ and of $p_2$, i.e., $p_1 r = p_2 r = 1_X$;

* internal [[symmetric relation|symmetry]]: $s \colon R \to R$ which interchanges $p_1$ and $p_2$, i.e., $p_1\circ s = p_2$ and $p_2\circ s = p_1$;

* {#InternalTransitivity} internal [[transitive relation|transitivity]]:

  $t \,\colon\, R \times_X R \to R$

  (where on the left we have the [[fiber product]]
   of $R \hookrightarrow X \times X \overset{pr_2}{\to} X$
   with $R \hookrightarrow X \times X \overset{pr_1}{\to}$, i.e. the subobject of pairs of *composable* pairs in relation)

  which factors the left/right projection map $R \times_X R \to X \times X$ through $R$, i.e., the following [[commuting diagram|diagram commutes]]

  $$
    \array{
      && R
      \\
      &
        {}^{\mathllap{t}}\nearrow
      &
      \big\downarrow
      \\
      R \times_X R
      &
        \underset{(p_1 q_1,p_2 q_2)}{\longrightarrow}
      &
      X \times X
      \mathrlap{\,,}
    }
  $$
  where $q_1$ and $q_2$ are the [[projections]] defined by the [[pullback | pullback diagram]]
  $$\array{
    R \times_X R
    &
      \overset{q_2}\longrightarrow
    &
    R
    \\
    \big\downarrow{\mathrlap{{}^{q_1}}}
    &&
    \big\downarrow{\mathrlap{{}^{p_1}}}
    \\
    R
    &
      \underset{p_2}\longrightarrow
    &
    X
  }
  $$
=--

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

Since $i$ is a [[monomorphism]], the maps $r$, $s$, and $t$ are necessarily unique if they exist.

=--

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

Equivalently, a congruence on $X$ is an [[internal category]] with $X$ the object of [[objects]], such that the (source,target)-map is a [[monomorphism]] and such that if there is a morphism $x_1 \to x_2$ then there is also a morphism $x_2 \to x_1$ (internally).

=--

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

We can equivalently define a congruence $R$ as (a representing object of) a representable sub-presheaf of $\hom(-, X \times X)$ so that for each object $Y$, the composite of $R(Y) \hookrightarrow \hom(Y, X \times X) \cong \hom(Y, X) \times \hom(Y, X)$ exhibits $R(Y)$ as an equivalence relation on the set $\hom(Y, X)$. The upshot of this definition is that it makes sense even when $C$ is not finitely complete.

=--

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

A congruence which is the kernel pair of some morphism (example \ref{KernelPairIsCongruence}) is called **effective**.

=--

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

The [[coequalizer]] of a congruence is called a **[[quotient object]]**.

The quotient of an effective congruence is called an **effective quotient**.

=--

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

A [[regular category]] is called an [[exact category]] if every congruence is effective.

=--

## Properties

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

An effective congruence, def. \ref{EffectiveCongruence}, is always the [[kernel pair]] of its quotient, def. \ref{QuotientObject}, if that quotient exists.

=--

## Examples

+-- {: .num_example #DiagonalMorphismIsCongruence}
###### Example

Every [[diagonal morphism]] on an object is a congruence and has a [[quotient object]] [[isomorphic]] to the original object.

=--

+-- {: .num_example #KernelPairIsCongruence}
###### Example

Every [[kernel pair]] is a congruence.

=--

+-- {: .num_example}
###### Example

An [[equivalence relation]] is precisely a congruence in [[Set]].

=--

+-- {: .num_example}
###### Example

The eponymous example is congruence modulo $n$ (for a fixed [[natural number]] $n$), which can be considered a congruence on $\mathbb{N}$ in the category of [[rigs]], or on $\mathbb{Z}$ in the category of [[rings]].

=--

+-- {: .num_example}
###### Example

A [[quotient group]] by a  [[normal subgroup]] $K \hookrightarrow G$ is the quotient of the congruence $R = \{(x,y) : xy^{-1} \in K \}$.

Alternatively, a [[quotient group]] by a  [[normal subgroup]] $K \hookrightarrow G$ is the quotient of the congruence
 $G \times K \stackrel{(p_1,p_2)}{\hookrightarrow} G \times G$, where $p_1$ is projection on the first factor and $p_2$ is multiplication in $G$ (these are source and target maps in the [[action groupoid]] $G \sslash K$).

A special case of this is that of a _[[quotient module]]_.

=--

## Related pages

* The notions of [[regular category]] and [[exact category]] can naturally be formulated in terms of congruences.  A "higher arity" version, corresponding to [[coherent categories]] and [[pretoposes]] is discussed at [[familial regularity and exactness]].

* A [[Mal'cev category]] is a [[finitely complete category]] in which every internal relation satisfying reflexivity is thereby actually a congruence.

* [[higher category theory|Higher-categorical]] generalizations are that of a [[2-congruence]] and of a [[groupoid object in an (∞,1)-category]]. See also [[(n,r)-congruence]].

## See also

* [[preordered object]]

* [[internal pseudo-equivalence relation]]

* [[setoid object]]

## References

### In common mathematics

* Wikipedia, Congruence (geometry)

* Wikipedia, *[Congruence relation](https://en.wikipedia.org/wiki/Congruence_relation)*

### In category theory

References using terminology as above:

* [[Jiri Adamek]], [[Horst Herrlich]], [[George Strecker]], p. 195 of: *[[Abstract and Concrete Categories]]*, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories **17** (2006) 1-507 [[tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [book webpage](http://katmat.math.uni-bremen.de/acc/)]

* [[Francis Borceux]], Exp. 2.10.3.b in: *[[Handbook of Categorical Algebra]]* Vol. 1 *Basic Category Theory*,   Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

* [[Francis Borceux]], Exps. 2.5.6 in: *[[Handbook of Categorical Algebra]]* Vol. 2 *Categories and Structures*,   Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

* [[Marek A. Bednarczyk]], [[Andrzej M. Borzyszkowski]], [[Wieslaw Pawlowski]], *Generalized congruences -- epimorphisms in $\mathcal{C}at$*, Theory and Applications of Categories **5** 11 (1999) 266-280 [[tac:5-11](http://www.tac.mta.ca/tac/volumes/1999/n11/5-11abs.html), [dml:120226](https://eudml.org/doc/120226?lang=fr&limit=5)]

...

But, for what it's worth, a *different* use of the term "congruence" in category theory appears in Def. 3.5.1 on [p. 89](https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf#page=107) in:

* {#BarrWells95} [[Michael Barr]], [[Charles Wells]], *Category theory for computing science*, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories **22** (2012) 1-538 [[pdf](http://www.math.mcgill.ca/barr/papers/ctcs.pdf), [tac:tr22](http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html)]

[[!redirects congruence]]
[[!redirects congruences]]
[[!redirects congruence relation]]
[[!redirects congruence relations]]
[[!redirects internal equivalence relation]]
[[!redirects internal equivalence relations]]
[[!redirects internal congruence]]
[[!redirects internal congruences]]


congruence in geometry

Two triangles in Eucledean plane are said to be congruent if their corresponding sides are equal, $a' = a$, $b' = b$, $c' = c$, and the corresponding angles are also equal,
$\alpha'=\alpha$, $\beta' = \beta$ i $\gamma' = \gamma$.

It appears that two triangles are congruent iff there is an isometry of the plane which
sends one onto another bijectively (for this it is sufficient to send the vertices to vertices, as every isometry is a bijection and it sends convex hulls onto convex hulls).
Such an isometry is unique because if there are two, say $g$ and $h$ then $h^{-1}\circ g$ is an isometry fixing the first triangle and the only isometry of the plane fixing three noncolinear points is the identity.

This motivates how to define congruence of more complicated figures: two plane figures are congruent if there is an isometry sending one onto another bijectively; similarly for angles. For oriented angles and oriented figures we need to consider just isometries preserving orientation. In the plane these are isometries which can be represented as a composition of an even number of reflections.

Related pages: [[axioms of plane geometry]]

category: geometry


congruence rule

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

\tableofcontents

## Definition

### Judgmental congruence rules

In [[dependent type theory]] with [[judgmental equality]], a **(judgmental) congruence rule** is an [[inference rule]] which states that [[judgmental equality]] is respected.

For example, there is a congruence rule for element conversion, which states that given judgmentally equal types $A$ and $B$ and judgmentally equal elements $a$ and $b$ of $A$, $a$ and $b$ are judgmentally equal elements of $B$:

$$\frac{\Gamma \vdash A \equiv B \; \mathrm{type} \quad \Gamma \vdash a \equiv b:A}{\Gamma \vdash a \equiv b:B}$$

There are many other different judgmental congruence rules, one for every term or type that is formed in the [[natural deduction]] [[inference rules]] in the type theory:

#### Congruence rules for dependent function types

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \prod_{x:A} B(x) \equiv \prod_{x:A'} B'(x)\; \mathrm{type}}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\
	\Gamma, x:A \vdash b(x) \equiv b'(x):B(x)
\end{array}
}{\Gamma \vdash \lambda x:A.b(x) \equiv \lambda x:A.b'(x):\prod_{x:A}.B(x)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \quad f':\prod_{x:A} B(x) \\
	\Gamma \vdash f \equiv f':\prod_{x:A} B(x)
\end{array}
}{\Gamma, x:A \vdash f(x) \equiv f'(x):B(x)}$$

If the dependent function type is weak, then there are also the following congruence rules for the computation and uniqueness rules of dependent function types:

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\
	\Gamma, x:A \vdash b(x) \equiv b'(x):B(x)
\end{array}
}{\Gamma \vdash \beta_{\prod}^{A, B} x:A.b(x) \equiv \beta_{\prod}^{A, B} x:A.b'(x):\prod_{x:A} b(x) =_{B(x)} (\lambda x:A.b(x))(x)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash B'(x) \; \mathrm{type} \\
	\Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \eta_{\prod}^{A, B} \equiv \eta_{\prod}^{A, B'}:\prod_{f:\prod_{x:A} B(x)} f =_{\prod_{x:A} B(x)} \lambda x:A.f(x)}$$

#### Congruence rules for dependent pair types:

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \sum_{x:A} B(x) \equiv \sum_{x:A'} B'(x)\; \mathrm{type}}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}
\end{array}
}{\Gamma, x:A, y:B(x) \vdash \mathrm{pair}_{\sum}^{A, B} \equiv \mathrm{pair}_{\sum}^{A', B'}:\sum_{x:A} B(x)}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \mathrm{ind}_{\sum}^{A, B, C} \equiv \mathrm{ind}_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{z:\sum_{x:A} B(x)} C(z)}$$

If the dependent pair type is weak, then there is also the following congruence rule for the computation rule of dependent pair types:

$$\frac{
\begin{array}{c}
\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} B(x) \vdash C(z) \equiv C'(z) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \beta_{\sum}^{A, B, C} \equiv \beta_{\sum}^{A', B', C'}:\prod_{g:\prod_{x:A} \prod_{y:B(x)} C(\mathrm{pair}_{\sum}^{A, B}(x, y))} \prod_{x:A} \prod_{y:B(x)} \mathrm{ind}_{\sum}^{A, B, C}(g, \mathrm{pair}_{\sum}^{A, B}(x, y)) =_{C(\mathrm{pair}_{\sum}^{A, B}(x, y))} g(x, y)}$$

#### Congruence rules for identity types:

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma, x:A, y:A \vdash x =_A y \equiv x =_{A'} y}$$

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type}}{\Gamma \vdash \mathrm{refl}_A \equiv \mathrm{refl}_{A'}:\prod_{x:A} x =_A x}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \mathrm{ind}_{=}^{A, C} \equiv \mathrm{ind}_{=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \prod_{y:A} \prod_{p:x =_A y} C(x, y, p)}$$

If the identity types are weak, then there is also the following congruence rule for the computation rule of identity types:

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A, y:A, p:x =_A y \vdash C(x, y, p) \equiv C'(x, y, p) \; \mathrm{type}
\end{array}
}{\Gamma \vdash \beta_{ \mathrm{ind}_=}^{A, C} \equiv \beta_{\mathrm{ind}_=}^{A', C'}:\prod_{t:\prod_{x:A} C(x, x, \mathrm{refl}_A(x))} \prod_{x:A} \mathrm{ind}_{=}^{A, C}(t, x, x, \mathrm{refl}_A(x)) =_{C(x, x, \mathrm{refl}_A(x))} t(x)}$$

#### Congruence rules for the empty type:

$$\frac{\Gamma, x:\emptyset \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\emptyset^C \equiv \mathrm{ind}_\emptyset^{C'}:\prod_{x:\emptyset} C(x) \; \mathrm{type}}$$

#### Congruence rules for the type of booleans:

$$\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{2}^C \equiv \mathrm{ind}_\mathbb{2}^{C'}:\prod_{a:C(0)} \prod_{b:C(1)} \prod_{x:\mathbb{2}} C(x)}$$

If the type of booleans is weak, then there are also the following congruence rules for the computation rules of the type of booleans:

$$\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{0, C} \equiv \beta_\mathbb{2}^{0, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 0) =_{C(0)} a}$$

$$\frac{\Gamma, x:\mathbb{2} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{2}^{1, C} \equiv \beta_\mathbb{2}^{1, C'}:\prod_{a:C(0)} \prod_{b:C(1)} \mathrm{ind}_\mathbb{2}^C(a, b, 1) =_{C(1)} b}$$

#### Congruence rules for the natural numbers type:

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{ind}_\mathbb{N}^C \equiv \mathrm{ind}_\mathbb{N}^{C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N} C(x)}}$$

If the natural numbers type is weak, then there are also the following congruence rules for the computation rules of the natural numbers type:

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{0, C} \equiv \beta_\mathbb{N}^{0, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, 0) =_{C(0)} c_0}$$

$$\frac{\Gamma, x:\mathbb{N} \vdash C(x) \equiv C'(x) \; \mathrm{type}}{\Gamma \vdash \beta_\mathbb{N}^{s, C} \equiv \beta_\mathbb{N}^{s, C'}:\prod_{c_0:C(0)} \prod_{c_s:\prod_{x:\mathbb{N}} C(x) \to C(s(x))} \prod_{x:\mathbb{N}} \mathrm{ind}_\mathbb{N}^C(c_0, c_s, s(x)) =_{C(s(x))} c_s(x)(\mathrm{ind}_\mathbb{N}^C(c_0, c_s, x))}$$

#### Congruence rules for axioms

Similarly, we have congruence rules for every [[axiom]] in the dependent type theory, such as

* congruence rule for [[function extensionality]]:

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{funext}_{A, B} \equiv \mathrm{funext}_{A', B'}:\prod_{f;\prod_{x:A} B(x)} \prod_{g:\prod_{x:A} B(x)} (f =_{\prod_{x:A} B(x)} g) \simeq \prod_{x:A} f(x) =_{B(x)} g(x)}$$

* congruence rule for the [[axiom of choice]]:

$$\frac{\Gamma \vdash A \equiv A' \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \equiv B'(x) \; \mathrm{type} \quad \Gamma, x:A, y:B(x) \vdash C(x, y) \equiv C'(x, y) \; \mathrm{type}}{\Gamma \vdash \mathrm{choice}_{A, B, C} \equiv \mathrm{choice}_{A', B', C'}:\left(\mathrm{isSet}(A) \times \prod_{x:A} \mathrm{isSet}(B(x))\right) \to \forall x:A.\exists y:B(x).C(x, y) \to \exists g:\prod_{x:A} B(x).\forall x:A.C(x, g(x))}$$

### Typal congruence rules

In [[dependent type theory]], a **typal congruence rule** is an derivable [[inference rule]] which states that given [[identifications]] of the component terms and [[equivalences]] of the component types in the hypotheses of the [[natural deduction]] [[inference rules]], one could derive the corresponding identification and equivalences of the derived terms and types in the conclusion of the inference rules.

For example, the typal congruence rule for element conversion states that given equivalent types $A$ and $B$ with equivalence $e:A \simeq B$ and typally equal elements $a$ and $b$ of $A$ with identification $p:a =_A b$, $e(a)$ and $e(b)$ are typally elements of $B$, and is given by [[function application to identifications|application]] of $e$ to $p$:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash e:A \simeq B \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:a =_A b}{\Gamma \vdash \mathrm{ap}_e(a, b, p):e(a) =_B e(b)}$$

There are many other different typal congruence rules which are derivable, one for every term or type that is formed in the [[natural deduction]] [[inference rules]] in the type theory:

#### Congruence rules for dependent function types

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash A' \; \mathrm{type} \quad \Gamma \vdash e_A:A \simeq A' \\
        \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash B'(x) \; \mathrm{type} \quad \Gamma, x:A \vdash e_B(x):B(x) \simeq B'(x)
\end{array}
}{\Gamma \vdash \mathrm{congform}_{\prod_{x:A} B(x)}:\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A'} B'(x)\right)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma, x:A \vdash b'(x):B(x) \\
	\Gamma, x:A \vdash p(x):b(x) =_{B(x)} b'(x)
\end{array}
}{\Gamma \vdash \mathrm{congIntro}_{\prod_{x:A} B(x)}^{x:A.b(x), x:A.b'(x), x:A.p(x)}:\lambda x:A.b(x) =_{\prod_{x:A}.B(x)} \lambda x:A.b'(x)}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash f:\prod_{x:A} B(x) \quad f':\prod_{x:A} B(x) \\
	\Gamma \vdash p:f =_{\prod_{x:A} B(x)} f'
\end{array}
}{\Gamma, x:A \vdash \mathrm{congElim}_{\prod_{x:A} B(x)}(p, x):f(x) =_{B(x)} f'(x)}$$

#### Congruence rules for identity types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash A' \; \mathrm{type} \quad \Gamma \vdash e_A:A \simeq A'}{\Gamma, x:A, y:A \vdash \mathrm{congform}_{=_A}(e_A, x, y):(x =_A y) \simeq (e(x) =_{A'} e(y))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash A' \; \mathrm{type} \quad \Gamma \vdash e_A:A \simeq A'}{\Gamma \vdash \mathrm{congIntro}_{=_A}(e_A):\mathrm{congForm}_{\prod_{x:A} x =_A x}(\mathrm{refl}_A) =_{\prod_{x:A'} x =_A x} \mathrm{refl}_{A'}}$$

$$\frac{
\begin{array}{c}
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} \sum_{y:A} x =_A y \vdash C(z) \; \mathrm{type} \quad \Gamma, z:\sum_{x:A} \sum_{y:A} x =_A y \vdash C'(z) \; \mathrm{type} \\
	\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash e_C:\prod_{z:\sum_{x:A} \sum_{y:A} x =_A y} C(z) \simeq C'(z) \\
\end{array}
}{\Gamma \vdash \mathrm{congElim}_{=_A}(e_C):\mathrm{congForm}_{\prod_{t:\prod_{x:A} C'(\Delta_A(x))} \prod_{z:\sum_{x:A} \sum_{y:A} x =_A y} C'(z)}(\mathrm{congForm}_{\prod_{z:\sum_{x:A} \sum_{y:A} x =_A y} C'(z)}(\mathrm{ind}_{=}^{A, C})) =_{\prod_{t:\prod_{x:A} C'(\Delta_A(x))} \prod_{z:\sum_{x:A} \sum_{y:A} x =_A y} C'(z)} \mathrm{ind}_{=}^{A, C'}}$$

where $\Delta_A(x) \coloneqq (x, x, \mathrm{refl}_A(x)):\sum_{x:A} \sum_{y:A} x =_A y$ is the [[diagonal function]] of $A$.

## See also

* [[inference rule]]

* [[structural rule]]

## References

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

[[!redirects congruence rule]]
[[!redirects congruence rules]]

[[!redirects judgmental congruence rule]]
[[!redirects judgmental congruence rules]]

[[!redirects typal congruence rule]]
[[!redirects typal congruence rules]]


congruence subgroup

#Contents#
* table of contents
{:toc}

## Definition

### Of the modular group

Let $n \in \mathbb{N}$ be a [[natural number]]. Write

$$
  p_n \;\colon\; SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/n\mathbb{Z})
$$

for the [[projection]] from the [[special linear group]] induced by the [[quotient]] [[projection]] $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ to the [[integers modulo n]].

The _mod-$n$ congruence subgroups_ of the [[special linear group]] $SL_2(\mathbb{Z})$ (essentially the [[modular group]]) are the [[preimages]] under $p_n$ of [[subgroups]] of $SL_2(\mathbb{Z}/n\mathbb{Z})$.

Some of these have traditional names and symbols;

The **principal congruence subgroup** is the preimage of the trivial group:

$$
  \Gamma(n) \coloneqq ker(p_n) = p_n^{-1}\left(\left\{\array{ 1 & 0 \\ 0 & 1}\right\}\right)
  \,.
$$

This is the origin of the term: the elements of $\Gamma(n)$ are _congruent [[modular arithmetic|modulo]] $n$_ to the identity.

The other two congruence subgroups having special symbols are

$$
  \Gamma_0(n) \coloneqq p_n^{-1}\left(\left\{\array{ \ast & \ast \\ 0 & \ast}\right\}\right)
$$

$$
  \Gamma_1(n) \coloneqq p_n^{-1}\left(\left\{\array{ 1 & \ast \\ 0 & \ast}\right\}\right)
$$

## Properties

### Relation to spin structures

See at _[level structure -- relation to spin structure](level%20structure%20on%20an%20elliptic%20curve#RelationToSpinStructures)_.

## Examples

* [[level n subgroup]] of the [[modular group]]

## Related concepts

* [[modular curve]]

* [[modular equivariant elliptic cohomology]]

## References

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

A list of congruence subgroups is provided in

* [[Sebastian Pauli]], _[Congruence subgroups of PSL(2,Z)](http://www.uncg.edu/mat/faculty/pauli/congruence/)_

[[!redirects congruence subgroups]]

[[!redirects principal congruence subgroup]]
[[!redirects principal congruence subgroups]]