heterotic string theory

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#Contents#
* tale of contents
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## Idea

In [[string theory]] a [[spacetime]] [[landscape of string theory vacua|vacuum]] is encoded by a [[sigma-model]] 2-dimensional [[SCFT]]. In _heterotic string theory_ that SCFT is assumed to be the sum of a supersymmetric chiral piece and a  non-supersymmetric piece (therefore "heterotic").

## Properties

### Compactifications

An effective [[target space]] [[quantum field theory]] induced from a given heterotic 2d [[CFT]] [[sigma model]] that has a [[spacetime]] of the form $M^4 \times Y^6$ for $M^4$ the 4-dimensional [[Minkowski space]] that is experimentally observed locally (say on the scale of a particle accelerator) has $N= 1$ global [[supersymmetry]] precisely if the remaining 6-dimensional [[Riemannian manifold]] $Y^6$ is a [[Calabi-Yau manifold]]. See the [references below](#ReferencesNEqOne).

Since global $N=1$ supersymmetry for a long time has been considered a promising phenomenological model in high energy physics, this fact has induced a lot of interest in [[heterotic string theory on CY3-manifolds]].

### Enhanced supersymmetry

A priori the [[worldsheet]] [[2d SCFT]] describing the quantum heterotic string has $N=(1,0)$ [[supersymmetry]]. Precisely if the corresponding [[target space]] [[effective field theory]] has $N=1$ supersymmetry does the worldsheet theory enhance to $N=(2,0)$ supersymmetry. See at _[[2d (2,0)-superconformal QFT]]_ and at _[[Calabi-Yau manifolds and supersymmetry]]_ for more on this.

### Gauge groups

Precisely two ([[isomorphism classes]] of) [[gauge groups]] are consistent (give [[quantum anomaly cancellation]]) *while preserving [[supersymmetry]]*: one is the [[direct product group]] $E_8 \times E_8$ of the [[exceptional Lie group]] [[E8]] with itself, the other is in fact the [[semi-spin group]] [[SemiSpin(32)|$SemiSpin(32)$]].

If the supersymmetry requirement is dropped, then there is a third option, which [[Lie algebra|locally]] looks like the product of the [[special orthogonal group]] $SO(16)\times SO(16)$, as first described in ([Alvarez-Gaumé, Ginsparg, Moore, and Vafa (1986)](#AGMV86het)) and in ([Dixon, and Harvey (1986)](#DH86het)). It was suggested in ([Schellekens, and Warner (1987)](#SW87anom)) that this theory could be regarded as some sort of "difference" (what is better understood as a [[correspondence]], or duality) between the $E_8\times E_8$ and [[SemiSpin(32)|$\text{SemiSpin}(32)$]] theories, by looking at how the relevant representations of these two theories assemble into those appearing in the $SO(16)\times SO(16)$ theory.

The *global* character of the gauge group is not $SO(16)\times SO(16)$, however. In ([McInnes (1999)](#McIn99spin)) it is proposed that the actual gauge group is a quotient of [[spin group|spin groups]] $\text{Spin}(16) \times \text{Spin}(16)$ by a $\mathbb{Z}_2$-[[action]] (not corresponding to $\text{SemiSpin}(16)\times \text{SemiSpin}(16)$). This proposal allows to identify $\text{Spin} (16) \times \text{Spin}(16)/ \mathbb{Z}_2$ as a subgroup of both $E_8\times E_8$ and $\text{SemiSpin}(32)$, where the author's purpose for this is identifying the $\text{Spin} (16) \times \text{Spin} (16)/\mathbb{Z}_2$ theory as the realization of the [[T-duality]] between the supersymmetric heterotic strings.

### Dualities
 {#Dualities}

Some [[dualities in string theory]] involving the heterotic string:

#### Duality with type I string theory

* [[duality between heterotic and type I string theory]]

#### Duality with type II string theory

See _[[duality between heterotic and type II string theory]]_.

#### Duality with M-theory

See _[[duality between heterotic string theory and M-theory]]_

#### Duality with F-theory

See _[[duality between heterotic string theory and F-theory]]

and see [references below](#DualityWithFTheory).

### Partition function and Witten genus

[[!include genera and partition functions - table]]

### General gauge backgrounds and parameterized WZW models
 {#GeneralGaugeBackgroundsAndParameterizedWZWModels}

The traditional construction of the [[worldsheet]] theory of the heterotic string produces via the [[current algebra]] of the left-moving worldsheet [[fermions]] only those [[E8]]-[[background gauge fields]] which are reducible to $Spin(16)/\mathbb{Z}_2$-[[principal connections]] ([Distler-Sharpe 10, sections 2-4](#DistlerSharpe10)). But it is known that, for instance, the [[duality between F-theory and heterotic string theory]] produces more general gauge backgrounds ([Distler-Sharpe 10, section 5](#DistlerSharpe10)).

In ([Distler-Sharpe 10, section 7](#DistlerSharpe10)), following ([Gates-Siegel 88](#GatesSiegel88)), it is argued that the way to fix this is to consider [[parameterized WZW models]], parameterized over the [[E8]]-[[principal bundle]] over [[spacetime]]. This does allow the incorporation of all $E_8$-background gauge fields, and the [[Green-Schwarz anomaly]] (and its cancellation) of the heterotic string now comes out as being equivalently the [[obstruction]] (and its lifting) for such a parameterized WZW term to exist.

Moreover, where the traditional construction only produces level-1 [[current algebras]], this construction accommodates all levels, and it is argued ([Distler-Sharpe 10, section 8.5](#DistlerSharpe10)) that the [[elliptic genus]] of the resulting [[parameterized WZW models]] are the [[equivariant elliptic genera]] found by Liu and Ando ([Ando 07](#Ando07)).

However, presently questions remain concerning formulating a [[sigma-model]] for strings propagating on the total space of the bundle, as it is only the chiral part of the geometric WZW model that appears in the heterotic string. (...)

### Superspace formulation
 {#SuperspaceFormulation}

The [[gauge field]] [[field strength|strength]]:

$F_{\alpha \beta} = 0$ ([Witten 86](#Witten86), [Bonora-Bregola-Lechner-Pasti-Tonin 87, above (2.7)](#BonoraBregolaLechnerPastiTonin87), [Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.13)](#BonoraBregolaLechnerPastiTonin87)).

$F_{a \alpha} = \Gamma_{a \alpha \beta} \chi^\beta$ ([Witten 86 (8)](#Witten86), [Atick-Dhar-Ratra 86, (4.14)](#AtickDharRatra86), [Bonora-Pasti-Tonin 87, below (11)](#BonoraPastiTonin87), [Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.27)](#BonoraBregolaLechnerPastiTonin87)).

Here $\chi^\alpha$ is the [[gaugino]].

$F_{a b} = \tfrac{1}{4} (\Gamma_{a b})_\alpha{}^\beta D_\beta \chi^\alpha$ ([Bonora-Pasti-Tonin 87, below (11)](#BonoraPastiTonin87), [Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.28)](#BonoraBregolaLechnerPastiTonin87))

[[equations of motion]]:

$ (D^a \Gamma_a)_{\alpha\beta} \chi^\beta =0 $ ([Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.30)](#BonoraBregolaLechnerPastiTonin87))

$D^b F_{b a} + T_a{}^{b c} F_{b c} = - (\Gamma_a)_{\alpha \beta} \chi^\alpha \chi^\beta - \chi^\alpha L_{\alpha a}$ ([Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.31)](#BonoraBregolaLechnerPastiTonin87)) (where $L_{\alpha a}$ is defined by (2.20) there...)

$\,$

The [[B-field]] [[field strength|strength]]:

$H_{\alpha \beta \gamma} = 0$ ([Atick-Dhar-Ratra 86, (4.2)](#AtickDharRatra86), [Bonora-Pasti-Tonin 87, (15)](#BonoraPastiTonin87), [Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.14)](#BonoraBregolaLechnerPastiTonin87))

$H_{a \alpha \beta} = \phi \Gamma_{a \alpha \beta}$ ([Atick-Dhar-Ratra 86, (4.19)](#AtickDharRatra86), [Bonora-Pasti-Tonin 87, (15)](#BonoraPastiTonin87), [Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.15)](#BonoraBregolaLechnerPastiTonin87))

$\rho \coloneqq D_\alpha \phi$ ([Atick-Dhar-Ratra 86, (4.20)](#AtickDharRatra86))

$H_{a b \alpha} = -\tfrac{1}{2} \Gamma_{a b }_\alpha{}^\beta \rho_\beta$ ([Atick-Dhar-Ratra 86, (4.21)](#AtickDharRatra86))

$H_{a b c} = - \tfrac{3}{2} \phi T_{a b c} + \tfrac{c_1}{4} (\Gamma_{a b c})_{\alpha \beta} tr(\chi^\alpha \chi^\beta)$ ([Atick-Dhar-Ratra 86, (4.22)](#AtickDharRatra86))

According to ([Bonora-Bregola-Lechner-Pasti-Tonin 90](#BonoraBregolaLechnerPastiTonin90)) in fact all these constraints follow from just $T^a_{\alpha \beta}  \propto \Gamma^a_{\alpha \beta}$, up to field redefinition.

See also at _[[torsion constraints in supergravity]]_.

## Related concepts

* [[SemiSpin(16)]], [[SemiSpin(32)]]

* [[small instanton]]

* [[E-string]]

* [[string theory]]

  * **heterotic string theory**

    * [[2d SCFT]],

      * [[2-spectral triple]], [[Dirac-Ramond operator]]

    * [[2d (2,0)-superconformal QFT]]

    * [[Green-Schwarz mechanism]]

    * [[dual heterotic string theory]]

    * [[heterotic string theory on CY3-manifolds]]

    * [[Witten genus]], [[(2,1)-dimensional Euclidean field theories and tmf]]

  * [[heterotic line bundle]]

  * [[type II string theory]]

  * [[type 0 string theory]]

  * [[landscape of string theory vacua]]

  * [[supersymmetry and Calabi-Yau manifolds]]

* [[string field theory]]

* [[11-dimensional supergravity]]

  * [[Hořava-Witten theory]]

  * [[M-theory]]

* [[AdS-CFT correspondence]]

* [string theory FAQ -- Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)

## References

### General

Heterotic strings were introduced in

* [[David Gross]], [[Jeffrey Harvey]], [[Emil Martinec]], [[Ryan Rohm]]:

  _Heterotic string theory (I). The free heterotic string_  Nucl. Phys. B 256 (1985), 253 (doi:10.1016/0550-3213(85)90394-3)

  _Heterotic string theory (II). The interacting heterotic string_ , Nucl. Phys. B 267 (1986), 75 (doi:10.1016/0550-3213(86)90146-X)

* {#CHSW85} [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], [[Edward Witten]], _Vacuum configurations for superstrings_, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) (doi:10.1016/0550-3213(85)90602-9)

* {#Schellekens91} [[Bert Schellekens]], _Classification of Ten-Dimensional Heterotic Strings_, Phys.Lett. B277 (1992) 277-284 ([arXiv:hep-th/9112006](http://arxiv.org/abs/hep-th/9112006))

Relation to [[Niemeier lattices]]:

* [[Wolfgang Lerche]], [[Dieter Lüst]], [[Adrian Norbert Schellekens]], *Ten-dimensional heterotic strings from Niemeier lattices*, Physics Letters B **181** 1–2 (1986) 71-75 [doi:10.1016/0370-2693(86)91257-8, [inspire:233203](https://inspirehep.net/literature/233203)]

Textbook accounts:

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], vol 3 (which is part 6) of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Joseph Polchinski]], volume II, section 11 of _[[String theory]]_,

* [[Eric D'Hoker]], _String theory -- lecture 8: Heterotic strings_  in part 3 (p. 941 of volume II) of

  [[Pierre Deligne]], P. Etingof, [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds. . _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

See also:

* [[Eric Sharpe]], _Recent developments in heterotic compactifications_, AMS/IP Stud. Adv. Math. 44 (2008) 209-230 ([arXiv:0801.4080](https://arxiv.org/abs/0801.4080))

* Andrea Fontanella, Tomas Ortin, _On the supersymmetric solutions of the Heterotic Superstring effective action_ ([arxiv:1910.08496](https://arxiv.org/abs/1910.08496))

Relation to [[Donaldson-Thomas theory]] and [[quiver gauge theory]]:

* {#HeLee12} [[Yang-Hui He]], Seung-Joo Lee, _Quiver Structure of Heterotic Moduli_, J. High Energ. Phys. (2012) 2012: 119 ([arXiv:1208.3004](https://arxiv.org/abs/1208.3004))

Discussion of [[higher curvature corrections]]:

* Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, *Supersymmetry, T-duality and Heterotic $\alpha'$-corrections* ([arXiv:2104.09545](https://arxiv.org/abs/2104.09545))

* Hao-Yuan Chang, [[Ergin Sezgin]], Yoshiaki Tanii, *Dimensional reduction of higher derivative heterotic supergravity* ([arXiv:2110.13163](https://arxiv.org/abs/2110.13163))

### Orbifold and orientifold compactifications

Heterotic strings on [[orbifolds]]:

* {#DixonHarveyVafaWitten85} [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], _Strings on orbifolds_, Nuclear Physics B Volume 261, 1985, Pages 678-686 (doi:10.1016/0550-3213(85)90593-0)

* {#DixonHarveyVafaWitten86} [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], _Strings on orbifolds (II)_, Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 (doi:10.1016/0550-3213(86)90287-7)

* Joel Giedt, _Heterotic Orbifolds_ ([arXiv:hep-ph/0204315](https://arxiv.org/abs/hep-ph/0204315))

* Kang-Sin Choi, _Spectra of Heterotic Strings on Orbifolds_, Nucl. Phys. B708: 194-214, 2005 ([arXiv:hep-th/0405195](https://arxiv.org/abs/hep-th/0405195))

Specifically on [[ADE-singularities]]:

* [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_,  Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))

* {#Witten99} [[Edward Witten]], _Heterotic String Conformal Field Theory And A-D-E Singularities_, JHEP 0002:025, 2000 ([arXiv:hep-th/9909229](https://arxiv.org/abs/hep-th/9909229))

A kind of unusual boundary condition for heterotic strings, (analogous to open [[M5-branes]] ending in  [[Yang monopoles]] on [[M9-branes]]):

* [[Joseph Polchinski]], _Open Heterotic Strings_, JHEP 0609 (2006) 082 ([arXiv:hep-th/0510033](http://arxiv.org/abs/hep-th/0510033))

[[!include heterotic string phenomenology -- references]]

[[!include higher gauge theory of the Green-Schwarz mechanism -- references]]

### Superspace formulation of Heterotic supergravity

Discussion of heterotic supergravity in terms of [[superspace]] includes the following.

One solution of the heterotic superspace [[Bianchi identities]] is due to

* {#AtickDharRatra86} [[Joseph Atick]], Avinash Dhar, and Bharat Ratra, _Superspace formulation of ten-dimensional N=1 supergravity coupled to N=1 super Yang-Mills theory_, Phys. Rev. D 33, 2824, 1986 ([doi.org/10.1103/PhysRevD.33.2824](https://doi.org/10.1103/PhysRevD.33.2824))

* {#Witten86} [[Edward Witten]], _Twistor-like transform in ten dimensions_, Nuclear Physics B Volume 266, Issue 2, 17 March 1986

A second solution is due to [[Bengt Nilsson]], [[Renata Kallosh]] and others

* [[Bengt Nilsson]], _Simple 10-dimensional supergravity in superspace_, Nucl. Phys. B188 (1981) 176 (doi:10.1016/0550-3213(81)90111-5)

These two solutions are supposed to be equivalent under field redefinition.

See also at _[[torsion constraints in supergravity]]_.

Further references include these:

* {#BonoraPastiTonin87} [[Loriano Bonora]], [[Paolo Pasti]], [[Mario Tonin]], _Superspace formulation of 10D SUGRA+SYM theory a la Green-Schwarz_, Physics Letters B Volume 188, Issue 3, 16 April 1987, Pages 335–339 (doi:10.1016/0370-2693(87)91392-X)

* {#BonoraBregolaLechnerPastiTonin87} [[Loriano Bonora]], M. Bregola, [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], _Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions_, Nuclear Physics B
Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)

* {#BonoraBregolaLechnerPastiTonin90} [[Loriano Bonora]], M. Bregola; [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], _A discussion of the constraints in $N=1$ SUGRA-SYM in 10-D_,  International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 (doi:10.1142/S0217751X90000222)

* [Castellani-D'Auria-Fre 91, vol 3, part 6](#CastellaniDAuriaFre91)

* {#BBDFLPPRRTZ} L. Bonora, M. Bregola, R. D'Auria, P. Fré [[Kurt Lechner]], [[Paolo Pasti]], I. Pesando, M. Raciti, F. Riva, [[Mario Tonin]] and D. Zanon, _Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in $D = 10$ $N= 1$ supergravity theories_, Physics Letters B 277 (1992) ([[BonoraSuperGS.pdf:file]])

* {#LechnerTonin08} [[Kurt Lechner]], [[Mario Tonin]], _Superspace formulations of ten-dimensional supergravity_, JHEP 0806:021,2008 ([arXiv:0802.3869](https://arxiv.org/abs/0802.3869))

### In elliptic cohomology

For more mathematically precise discussion in the context of [[elliptic cohomology]] and the [[Witten genus]] see also the references at _[Witten genus -- Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models](Witten+genus#TwistedWittenGenus)_.

### General flux backgrounds and parameterized WZW models

Discussion of heterotic strings whoe [[current algebra]]-sector is parameterized by a [[principal bundle]] originates with

* {#GatesSiegel88} [[Jim Gates]], [[Warren Siegel]], _Leftons, Rightons, Nonlinear $\sigma$-Models, and Superstrings_, Phys.Lett. B206 (1988) 631 ([spire](https://inspirehep.net/record/251286/))

* [[Jim Gates]], _Strings, superstrings, and two-dimensional lagrangian field theory_, pp. 140-184 in Z. Haba, J. Sobczyk (eds.) _Functional integration, geometry, and strings_, proceedings of the XXV Winter School of Theoretical Physics, Karpacz, Poland (Feb. 1989), , Birkhäuser, 1989.

* {#GKKS91} [[Jim Gates]], S. Ketov, S. Kozenko, O. Solovev, _Lagrangian chiral coset construction of heterotic string theories in $(1,0)$ superspace_, Nucl.Phys. B362 (1991) 199-231 ([spire](http://inspirehep.net/record/314337/?ln=en))

and is further expanded on in

* {#DistlerSharpe10} [[Jacques Distler]], [[Eric Sharpe]], _Heterotic compactifications with principal bundles for general groups and general levels_, Adv. Theor. Math. Phys. 14:335-398, 2010 ([arXiv:hep-th/0701244](http://arxiv.org/abs/hep-th/0701244))

reviewed in

* {#Sharpe08} [[Eric Sharpe]], _Recent developments in heterotic compactifications_, in [[Eric Sharpe]], [[Arthur Greenspoon]], _[Advances in String Theory: The First Sowers Workshop in Theoretical Physics](http://www.ams.org/bookstore-getitem/item=AMSIP-44)_, AMS 2008 ([pdf slides (39-49)](http://www.phys.vt.edu/sowers/talks/sharpe-sowers.pdf))

The relation of this to [[equivariant elliptic cohomology]] is amplified in

* {#Ando07} [[Matthew Ando]], _Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe_, [talk 2007](http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php) ([lecture notes pdf](http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf))

### On elliptic fibrations

Compactified on an [[elliptic curve]] or, more generally, [[elliptic fibration]], heterotic string compactifictions are controled by a choice holomorphic [[stable bundle]] on the compact space. Dually this is an [[F-theory]] compactification on a [[K3]]-bundles.

The basis of this story is discussed in

* Robert Friedman, [[John Morgan]], [[Edward Witten]], _Vector Bundles And F-Theory_ ([arXiv:hep-th/9701162](http://arxiv.org/abs/hep-th/9701162))

A more formal discussion is in

* B. Andreas and D. Hernandez Ruiperez, Adv. Theor. Math. Phys. Volume 7, Number 5 (2003), 751-786 _Comments on N = 1 Heterotic String Vacua _ ([project Euclid](http://projecteuclid.org/euclid.atmp/1111510429))

### Dualities

#### With type I superstring theory

The original conjecture is due to

* {#Witten05} [[Edward Witten]], section 5 of _[[String Theory Dynamics In Various Dimensions]]_, Nucl.Phys.B443:85-126 (1995) ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

More details are then in

* {#PolchinskiWitten96} [[Joseph Polchinski]], [[Edward Witten]], _Evidence for Heterotic - Type I String Duality_, Nucl.Phys.B460:525-540,1996 ([arXiv:hep-th/9510169](http://arxiv.org/abs/hep-th/9510169))

#### With $F$-theory
 {#DualityWithFTheory}

The [[duality between F-theory and heterotic string theory]] originates in

* {#Sen96} [[Ashoke Sen]], _F-theory and Orientifolds_ ([arXiv:hep-th/9605150](http://arxiv.org/abs/hep-th/9605150))

* {#FriedmanMorganWitten97} Robert Friedman, [[John Morgan]], [[Edward Witten]], _Vector Bundles And F Theory_ ([arXiv:hep-th/9701162](http://arxiv.org/abs/hep-th/9701162))

Reviews include

* {#Donagi98} [[Ron Donagi]], _ICMP lecture on heterotic/F-theory duality_ ([arXiv:hep-th/9802093](http://arxiv.org/abs/hep-th/9802093))

* Björn Andreas, _$N=1$ Heterotic/F-theory duality_ PhD thesis ([pdf](http://edoc.hu-berlin.de/dissertationen/physik/andreas-bjoern/PDF/Andreas.pdf))

[[!include flavor on heterotic M5-branes -- references]]

[[!redirects heterotic superstring]]
[[!redirects heterotic superstrings]]

[[!redirects heterotic string]]
[[!redirects heterotic strings]]

[[!redirects heterotic supergravity]]

[[!redirects HET-theory]]
[[!redirects HET-theories]]


heterotic string theory on CY3-manifolds

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#Contents#
* table of contents
{:toc}

## Idea

The [[KK-compactification]] of [[heterotic string theory]] on [[Calabi-Yau manifolds]] of [[complex numbers|complex]] [[dimension]] 3, hence [[real number|real]] [[dimension]] 6. This choice of compactification means exactly that the resulting [[effective field theory]] on 4-dimension has $N =1$ [[supersymmetry]] (see at [[supersymmetry and Calabi-Yau manifolds]]).

## Related concepts

[[!include N=1 susy compactifications -- table]]

* [[supergravity]], [[supersymmetry]]

* [[Kaluza-Klein mechanism]]

* [[spontaneously broken symmetry]]

* [[hierarchy problem]]

* [string theory FAQ -- What does it mean to say that string theory has a "landscape" of solutions?](string%20theory%20FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)

## References
 {#References}

The idea originates in

* {#CandelasHorowitzStromingerWitten85} [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], and [[Edward Witten]], _Vacuum Configurations for Superstrings_ , Nucl. Phys. B 258 (1985), p. 46.

where in the introduction it says the following

> Recently, the discovery [6] of anomaly cancellation in a modified version of $d = 10$ supergravity and superstring theory with gauge group $O(32)$ or $E_8 \times E_8$ has opened the possibility that these theories might be phenomenologically realistic as well as mathematically consistent. A new string theory with  $E_8 \times E_8$ gauge group has recently been constructed [7] along with a second $O(32)$ theory.

> For these theories to be realistic, it is necessary that the vacuum state be of the form $M_4 \times K$, where $M_4$ is four-dimensional Minkowski space and K is some compact six-dimensional manifold. (Indeed, Kaluza-Klein theory -- with its now widely accepted interpretation that all dimensions are on the same logical footing -- was first proposed [8] in an effort to make sense out of higher-dimensional string theories). Quantum numbers of quarks and leptons are then determined by topological invariants of $K$ and of an $O(32)$ or $E_8 \times E_8$ gauge field defined on $K$ [9]. Such considerations, however, are far from uniquely determining $K$.

> In this paper, we will discuss some considerations, which, if valid, come very close to determining $K$ uniquely. We require

> (i) The geometry to be of the form $H_4 \times K$, where $H_4$ is a maximally symmetric spacetime.

> (ii) There should be an unbroken $N = 1$ supersymmetry in four dimensions. General arguments [10] and explicit demonstrations [11] have shown that supersymmetry may play an essential role in resolving the gauge hierarchy or Dirac large numbers problem. These arguments require that supersymmetry is unbroken at the Planck (or compactification) scale.

> (iii) The gauge group and fermion spectrum should be realistic.

> These requirements turn out to be extremely restrictive. In previous ten-dimensional supergravity theories, supersymmetric configurations have never given rise to chiral fermions -- let alone to a realistic spectrum. However, the modification introduced by Green and Schwarz to produce an anomaly-free field theory also makes it possible to satisfy these requirements. We will see that unbroken $N = 1$ supersymmetry requires that $K$ have, for perturbatively accessible configurations, $SU(3)$ holonomy and that the four-dimensional cosmological constant vanish. The existence of spaces with $SU(3)$ holonomy was conjectured by Calabi [12] and proved by Yau [13].

(Of course later it was understood that Calabi-Yau spaces, even those of complex dimension 3, are not "very close to unique".)

Lecture notes include

* {#Greene97} [[Brian Greene]], _String Theory on Calabi-Yau Manifolds_, lectures at TASI96 ([arXiv:hep-th/9702155](https://arxiv.org/abs/hep-th/9702155))

Further original references include

* {#BDFF88} [[Tom Banks]], [[Lance Dixon]], [[Dan Friedan]], [[Emil Martinec]], _Phenomenology and Conformal Field Theory or Can String Theory Predict the Weak Mixing Angle?_, Nucl. Phys. B299 (1988) 613.  ([pdf](http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4377.pdf))

* [[Jacques Distler]], [[Brian Greene]], _Aspects Of $(2,0)$ String Compactifications_, Nucl. Phys. B304 (1988)

* [[Andrew Strominger]], _Special Geometry_, Comm. Math. Phys. 133 (1990) 163.

* [[Philip Candelas]] and X. De la Ossa, _Moduli Space of Calabi-Yau Manifolds_, Nucl. Phys. B355 (1991) 455.

* [[Edward Witten]], _Phases of N=2 Theories in Two Dimensions_, Nucl. Phys. B403 (1993) 159 ([arXiv:hep-th/9301042](http://arxiv.org/abs/hep-th/9301042))

and chapters 12 - 16 of

* [[Michael Green]], [[John Schwarz]], [[Edward Witten]], _Superstring Theory_ , Vol. 2, Cambridge University Press, (1987)

A canonical textbook reference for the role of Calabi-Yau manifolds in compactifications of 10-dimensional [[supergravity]] is

* [[Andrew Strominger]] (notes by [[John Morgan]]), _Kaluza-Klein compactifications, Supersymmetry and Calabi-Yau spaces_ , volume II, starting on page 1091 in

  [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey,
[[David Kazhdan]],  [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds.  , _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

Lecure notes in a more general context of [[string phenomenology]] include

* {#Wijhnholt14} [[Martin Wijnholt]], _String compactification_, [PITP 2014](https://pitp2014.ias.edu) lecture notes ([[WijnholtCompactification14.pdf:file]], [slides for lecture 1](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P1_wijnholt.pdf), [slides for lecture 2](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P2_wijnholt.pdf), [slides for lecture 3](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P3_wijnholt.pdf))

Discussion of [[generalized Calabi-Yau manifold]] backgrounds is for instance in

* {#GMPT} [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], _Generalized structures of $N=1$ vacua_ ([arXiv:hep-th/0505212](http://arxiv.org/abs/hep-th/0505212))

Discussion of [[duality in string theory|duality]] with [[M-theory on G2-manifolds]]:

* {#BraunSchaeferNameki17} [[Andreas Braun]], [[Sakura Schafer-Nameki]], _Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality_, JHEP04(2018)126 ([arXiv:1708.07215](https://arxiv.org/abs/1708.07215))

[[!redirects hetrotic string theory on Calabi-Yau manifolds]]
[[!redirects hetrotic string theory on Calabi-Yau 3-folds]]


Heun equation

__Heun equation__ is a second order linear [[ordinary differential equation]] of [[Fuchsian equation|Fuchsian]] type with 4 regular singular points at $0,1,a,\infty$. Its normalized holomorphic solution is the Heun function. If some of the singular points coalesce we talk about the confluent Heun equation.

* wikipedia [Heun function](https://en.wikipedia.org/wiki/Heun_function)

An application of monodromy analysis of Heun equation (coming from separation of variables) to the physics of [[black hole]]s:

* F. Novaes, Bruno Carneiro da Cunha, _Isomonodromy, Painlevé transcendents and scattering off of black holes_, J. High Energ. Phys. 2014, 132 (2014) doi [arXiv:1404.5188](https://arxiv.org/abs/1404.5188)
* Bruno Carneiro da Cunha, João Paulo Cavalcante, _Confluent conformal blocks and the Teukolsky master equation_, Phys. Rev. D 102, 105013 (2020) [doi](https://doi.org/10.1103/PhysRevD.102.105013)  [arXiv:1906.10638](https://arxiv.org/abs/1906.10638); _Teukolsky master equation and Painlevé transcendents: numerics and extremal limit_, [arXiv:2105.08790](https://arxiv.org/abs/2105.08790)


heuristic introduction to sheaves, cohomology and higher stacks > history

< [[heuristic introduction to sheaves, cohomology and higher stacks]]
[[!redirects heuristic introduction to sheaves, cohomology and higher stacks -- history]]


hexagon

#Contents#
* table of contents
{:toc}

## Idea

The convex [[regular polygon]] with six sides.

## References

See also

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

[[!redirects hexagons]]


hexagonator

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

* [[associator]], [[unitor]], [[Jacobiator]]

* [[pentagonator]], **hexagonator**

***

#Contents#
* table of contents
{:toc}

## Idea

A _hexagonator_ in [[higher category theory]] is an [[isomorphism]] that relaxes the ordinary [[hexagon identity]].


HeytAlg

__$Heyt Alg$__ is the [[category]] whose [[objects]] are [[Heyting algebras]] and whose [[morphisms]] are Heyting algebra [[homomorphisms]], that is [[lattice]] homomorphisms which also preserve the [[Heyting implication]].  $Heyt Alg$ is a [[subcategory]] of [[Pos]].

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

*  The free Heyting algebra on the [[empty set]] is the [[Boolean domain]] $\{\bot, \top\}$.
*  The free Heyting algebra on a [[singleton]] $\{x\}$ is [[infinite set|infinite]], although it has a nice picture.

category: category

[[!redirects HeytAlg]]
[[!redirects Heyt Alg]]

[[!redirects free Heyting algebra]]
[[!redirects free heyting algebra]]


Heyting 2-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

+--{: .num_defn}
###### Definition
A 2-category is **Heyting** if it is [[coherent 2-category|coherent]]  and each pullback functor $f^*:Sub(Y)\to Sub(X)$ has a right adjoint $\forall_f$.
=--

As in the 1-categorical case, this also ensures that each $Sub(X)$ is a Heyting algebra and that each $f^*$ is a homomorphism of Heyting algebras.  Moreover, the right adjoints $\forall_f$ satisfy the [[Beck-Chevalley condition|Beck-Chevalley condition]] for pullback squares, because the left adjoints $\exists_f$ do.  This is just what is necessary for the [[michaelshulman:2-internal logic|internal]] interpretation of (finitary) first-order logic.

## Examples ##

* $Cat$ is Heyting.

* A 1-category is Heyting as a 2-category iff it is [[Heyting category|Heyting]] as a 1-category.

* A (0,1)-category is Heyting iff it is a [[Heyting algebra|Heyting algebra]].

* Any well-powered infinitary-coherent 2-category is Heyting, by the adjoint functor theorem for posets.  In particular, any [[michaelshulman:Grothendieck 2-topos]] is Heyting.

* Unlike in the 1-categorical case, there seems no reason why a coherent 2-category with [[michaelshulman:exponentials in a 2-category|exponentials]] must be Heyting.  It does, however have adjoints to pullback along fibrations and opfibrations, and in particular along product projections, which is enough for some uses of universal quantification in the internal logic.

## Preservation

Clearly if $K$ is Heyting, so are $K^{co}$, $gpd(K)$, $pos(K)$, $disc(K)$, and $Sub(1)$ (the Heyting algebra of subterminals).

Moreover, Heytingness is also preserved by [[michaelshulman:fibrational slice|fibrational slicing]].  First we show:

+--{: .num_theorem}
###### Theorem
If $K$ is Heyting, then for any $X$ and any $A\in Opf(X)$, $Sub_{Opf(X)}(A)$ is a coreflective sub-poset of $Sub_K(A)$.
=--
+--{: .proof}
###### Proof
As proven [[michaelshulman:fibrational slice|here]], $Sub_{Opf(X)}(A)$ is a full sub-poset of $Sub_K(A)$.  Moreover, it is easy to see that a ff $B\to A$ is in $Sub_{Opf(X)}(A)$ precisely when

1. $B\to X$ is an opfibration and
1. $B\to A$ is a morphism of opfibrations.

In general, the first condition does _not_ imply the second.  It is also easy to check that these two conditions together are equivalent to saying that the composite
$$B\times_X X ^{\mathbf{2}} \to A\times_X X ^{\mathbf{2}} \overset{a}{\to} A$$
factors through $B\to A$, where $a$ is the action making $A$ into a fibration.  Note that $B\times_X X ^{\mathbf{2}} $ is alternately written as $p^*B$, where $p:A\times_X X ^{\mathbf{2}} \to A$ is the projection onto the first factor.  Therefore, $B$ is in $Sub_{Opf(X)}(A)$ just when $\exists_p a^*B\le B$, or equivalently when $B\le \forall_a p^*B$.

We claim that $B\mapsto \forall_a p^* B$ coreflects $Sub(A)$ into $Sub_{Opf(X)}(A)$, and given the above, it suffices to verify that it is a comonad.  First, let $i:A\to A\times_X X ^{\mathbf{2}}$ be the inclusion of identities; then $a i\cong 1$ and $p i \cong 1$, so we have
$$
\begin{aligned}
  i^*a^*B =& B\\
  a^*B \le& \forall_i B\\
  \forall_p a^* B \le& \forall_p \forall_i B = B.
\end{aligned}
$$
This gives the counit of the comonad.  The comultiplication is a little more involved.  First note that we have a pullback square
$$\array{
  A\times_X X ^{\mathbf{2}} \times_X X ^{\mathbf{2}} &   \overset{q}{\to} & A\times_X X ^{\mathbf{2}} \\
  ^{a\times 1}\downarrow && \downarrow^a\\
  A\times_X X ^{\mathbf{2}}  & \underset{p}{\to} & A}$$
where $q$ denotes projection onto the first two factors.  Thus, by the Beck-Chevalley condition, $a^*\forall_p = \forall_q (a\times 1)^*$ and therefore
$$\forall_p a^*\forall_p a^* B =
\forall_p \forall_q (a\times 1)^* a^* B =
\forall_p \forall_q (1\times m)^* a^* B
$$
where $m:X ^{\mathbf{2}} \times_X X ^{\mathbf{2}} \to X^{\mathbf{2}} $ is the "composition" morphism.

Let us write $X^{\to\to}$ for $X ^{\mathbf{2}} \times_X X ^{\mathbf{2}}$, the object of composable arrows in $X$, and write $X^{\overset{\nearrow}{\to}\to}$ for the power of $X$ with the category
$$\array{&& \bullet\\ & \nearrow\\ \bullet &\to & \bullet & \to &\bullet}$$
Let $r:X^{\overset{\nearrow}{\to}\to} \to X^{\to\to}$ forget the "lonely" arrow, and let $c:X^{\to\to} \to X^{\overset{\nearrow}{\to}\to}$ make the lonely arrow into the composite of the other two.  Then $r c \cong 1$, and moreover $p' q' c\cong 1\times m$, where $p'$ and $q'$ are $p$ and $q$  applied to the non-lonely arrows.  Thus, we have
$$\begin{aligned}
  c^* (q')^* (p')^* a^* B =& (1\times m)^* a^* B\\
  (q')^* (p')^* a^* B \le& \forall_c (1\times m)^* a^* B\\
  \forall_r (q')^* (p')^* a^* B \le& \forall_r \forall_c (1\times     m)^* a^* B\\
  \forall_r (q')^* (p')^* a^* B \le& (1\times m)^* a^* B\\
  q^* p^* \forall_p a^* B \le& (1\times m)^* a^* B\\
  \forall_p a^* B \le& \forall_p \forall_q (1\times m)^* a^* B\\
  \forall_p a^* B \le& \forall_p a^*\forall_p a^* B.
\end{aligned}$$
as desired.  (Here we have used another Beck-Chevalley condition to move $r$ past $q'$ and $p'$, turning them into $p$ and $q$ and it into $p$.)
=--

+--{: .num_cor}
###### Corollary
If $K$ is Heyting, so is each [[fibrational slice]] $Opf(X)$ and $Fib(X)$.
=--
+--{: .proof}
###### Proof
This follows from the adjoint lifting theorem for posets: for any $f:A\to B$ in $Opf(X)$ we have a commutative square
$$\array{Sub_{Opf(X)}(B) & \overset{f^*}{\to} & Sub_{Opf(X)}(A)\\
  \downarrow && \downarrow\\
  Sub_K(B)& \underset{f^*}{\to} & Sub_K(A)}$$
where the vertical arrows are comonadic (inclusions of coreflective sub-posets) and the bottom horizontal arrow has a right adjoint; hence so does the top horizontal arrow.
=--

## Booleanness

We may naturally say that a coherent 2-category $K$ is **Boolean** if each $Sub(X)$ is a Boolean algebra.  As usual, this implies that $K$ is Heyting; we have $\forall_f = \neg \exists_f \neg$.

Note that a subobject and its complement in $Sub(X)$ need _not_ be "disjoint" in the sense introduced [[coherent 2-category|here]]; only their pullback, not their comma object, need be initial.  Note also that unlike in the 1-categorical case, Booleanness is _not_ stable under slicing; this fails already in the (1,2)-category $Pos$ (though it holds in any (2,1)-category).

## References

The above definitions and observations are originally due to

* [[Michael Shulman]], _[[michaelshulman:Heyting 2-category]]_

[[!redirects Heyting 2-categories]]

[[!redirects Heyting (2,1)-category]]
[[!redirects Heyting (2,1)-categories]]


Heyting algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea
 {#Idea}

Generally, the [[propositions]] in a [[formal logic]] may be thought of as having an [[algebra|algebraic]] structure with respect to logical operators like "[[logical conjunction|and]]" and "[[logical disjunction|or]]". Under this identification, different systems of logic correspond to different sets of operators and axioms.

For the [[intuitionistic logic|intuitionistic]] [[propositional calculus]], the operators (including nullary operators, i.e. constants) are "[[logical conjunction|and]]", "[[logical disjunction|or]]", "[[true]]", "[[false]]", and "[[implication|implies]]". The concept of a **Heyting algebra** captures these operators and their axioms, so that a Heyting algebra is precisely a [[model]] of the intuitionistic propositional calculus.

A Heyting algebra where [[excluded middle]] holds is a [[Boolean algebra]], a model of [[classical logic|classical]] propositional calculus.

To model [[quantifiers]] and variables, i.e. to extend from [[propositional calculus]] to [[first-order logic|first-order]] aka [[predicate logic]], one forms a [[hyperdoctrine]] on Heyting algebras, called a [[first-order hyperdoctrine]].

## Definition

\begin{definition}
A **Heyting algebra** is a [[lattice]] $L$ which as a [[poset]] admits an operation of [[implication]]
$\Rightarrow: L^{op} \times L \to L$
satisfying the condition (really a [[universal property]])

\[ \label{UniversalProperty}
(x \wedge a) \leq b \qquad \text{if and only if} \qquad x \leq (a \Rightarrow b)
\]

In other words, $a \Rightarrow {-}$ must be [[right adjoint]] to ${-} \wedge a$.

\end{definition}

This is equivalent to the following definition.

\begin{definition}
A **Heyting algebra** is a [[bicartesian closed category|bicartesian closed]] [[poset]], that is a poset which (when thought of as a [[thin category]]) is

* [[finitely complete category|finitely complete]],

* [[finitely cocomplete category|finitely cocomplete]],

* and [[cartesian closed category|cartesian closed]].

The implication $a\Rightarrow b$ is the [[exponential object]] $b^a$.

\end{definition}

Often (for instance, when doing forcing categorically) one is interested in *complete* Heyting algebras, which are Heyting algebras than admit *arbitrary* meets and joins (that is, not necessarily finite).

\begin{remark}
Insofar as all these properties of a poset are described by [[universal properties]], being a Heyting algebra is a [[property-like structure]] on a poset; a poset can be a Heyting algebra in at most one way.

\end{remark}

The definition of Heyting algebra may be recast into purely [[algebraic theory|equational form]]: add to the equational theory of lattices the inequalities $(x \Rightarrow y) \wedge x \leq y$ and $y \leq x \Rightarrow (y \wedge x)$, writing these inequalities in equational form via the equivalence $a \leq b$ iff $a = a \wedge b$. Hence we can speak of an __[[internalization|internal]] Heyting algebra__ in any [[cartesian monoidal category|category with products]].

\begin{definition}
A Heyting algebra [[homomorphism]] is a homomorphism of the underlying [[lattice]]s that preserve $\Rightarrow$.  Heyting algebras and their homomorphisms form a [[concrete category]] [[HeytAlg]].
\end{definition}

If one relaxes the requirement that $\leq$ be antisymmetric, so that instead of a [[poset]] $L$ is only a [[preorder]], the result is a [[Heyting prealgebra]].

## Properties

\begin{prop}\label{prop:distributive}
Any Heyting algebra is a [[distributive lattice]].
That is, finite meets distribute over finite joins and vice versa.
\end{prop}

\begin{proof}
As discussed at [[distributive lattice]], it suffices to prove the nontrivial direction of either of the two (dual) binary distributivity laws, e.g.
$ x \wedge (y \vee z) \leq (x \wedge y) \vee (x \wedge z) $.
This is a straightforward [[exercise]] applying the universal property \eqref{UniversalProperty}.

More abstractly: the distributivity law $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$ says precisely that $x \wedge {-}$ preserves binary joins, which it does because it has a [[right adjoint]] (namely $x \Rightarrow {-}$) and therefore [[left adjoints preserve colimits|preserves all colimits]].
\end{proof}

The relation of **modus ponens** is immediate from
\eqref{UniversalProperty}: for any $a, b$ in a Heyting algebra,
\[ \label{ModusPonens}
a \wedge (a \Rightarrow b) \leq b
\]

We list some further basic facts which are frequently useful,
including in the proofs below:

\begin{proposition} \label{BasicProperties}
The following hold for any $a, b, c$ in any Heyting algebra:

  1. Composition law:
     $(a \Rightarrow b) \wedge (b \Rightarrow c) \leq a \Rightarrow c$.

  1. Currying:
     $a \wedge b \Rightarrow c =
      a \Rightarrow (b \Rightarrow c)$.

  1. $a \Rightarrow {-}$ is monotone:
     $a \Rightarrow b \leq a \Rightarrow c$ if $b \leq c$.

  1. $a \Rightarrow {-}$ is increasing:
     $b \leq a \Rightarrow b$.

  1. $a \Rightarrow {-}$ is idempotent:
     $a \Rightarrow (a \Rightarrow b) = a \Rightarrow b$.

  1. ${-} \Rightarrow a$ is antimonotone:
     $c \Rightarrow a \leq b \Rightarrow a$ if $b \leq c$.

  1. ${-} \Rightarrow a$ is self-adjoint:
     ${-} \Rightarrow a: L \rightarrow L^{\mathrm{op}}$ is left adjoint
     to ${-} \Rightarrow a: L^{\mathrm{op}} \rightarrow L$.

     Explicitly, $b \leq (c \Rightarrow a)$ just if
     $(b \Rightarrow a) \leq^{\mathrm{op}} c$, i.e. just if
     $c \leq (b \Rightarrow a)$.

  1. $({-} \Rightarrow a) \Rightarrow a$ is increasing:
     $
     b \leq
     (b \Rightarrow a) \Rightarrow a
     $.

  1. $({-} \Rightarrow a) \Rightarrow a$ is idempotent:
     $
     (b \Rightarrow a) \Rightarrow a =
     (((b \Rightarrow a) \Rightarrow a) \Rightarrow a) \Rightarrow a
     $.

\end{proposition}
\begin{proof}
  Each of these is a short exercise using the universal property
  \eqref{UniversalProperty} and modus ponens \eqref{ModusPonens}.
\end{proof}

Most of these facts can be packaged up more abstractly like so:

\begin{proposition}\label{Monads}
  In any Heyting algebra $H$, for any element $a \in H$:

  * $a \Rightarrow {-}$ is a [[monad]].

  * $({-} \Rightarrow a) \Rightarrow a$ is a monad.

\end{proposition}
\begin{proof}
  A monad in a poset (or a [[preorder]], aka [[(0,1)-category]]) like
  $H$ is just a monotone, increasing, idempotent function.

  From Proposition \ref{BasicProperties} it is immediate that
  this describes both $a \Rightarrow {-}$
  and $({-} \Rightarrow a) \Rightarrow a$.
\end{proof}

### Negation

In any Heyting algebra $H$, we may define a [[negation]] operator:

\begin{definition}\label{Negation}
The **negation** operator
$\neg\colon H^{op} \to H$
is $\neg x = (x \Rightarrow 0)$, where $0$ is the bottom element of
the lattice.
\end{definition}

\begin{corollary}\label{DoubleNegationMonad}
$\neg\neg\colon H \to H$ is a [[monad]].
\end{corollary}
\begin{proof}
Immediate from \ref{Monads}.
\end{proof}

We collect some further frequently-useful facts:

\begin{proposition}\label{NegationFacts}
  In any Heyting algebra $H$, we have the following for all $x, y \in H$:

  1. Negation is antimonotone:
     $\neg y \leq \neg x$ if $x \leq y$.

  1. Currying:
     $\neg (x \wedge y) = (x \Rightarrow \neg y)$.

  1. Triple negation is just negation:
     $\neg \neg \neg x = \neg x$.

  1. Proof by contradiction:
     $x \leq \neg y$ just if $x \wedge y \leq 0$.

  1. [[De Morgan's law]] for negation over disjunction:
     $\neg (x \vee y) = \neg x \wedge \neg y$.

  1. One-way De Morgan's law for negation over conjunction:
     $\neg x \vee \neg y \leq \neg (x \wedge y)$.

  1. $x \wedge \neg x = 0$

  1. $\neg \neg (x \vee \neg x) = 1$
     (equivalently, $\neg (x \vee \neg x) = 0$)

  1. $\neg x \vee y \leq x \Rightarrow y$

  1. \[ \label{NegIfEq}
     \neg (x \Rightarrow y) = \neg\neg x \wedge \neg y
     \]

\end{proposition}
\begin{proof}
  Each of these is a short [[exercise]] using the universal property
  \eqref{UniversalProperty}, Proposition \ref{BasicProperties},
  and the preceding properties in the list.
\end{proof}

Several of the facts in \ref{NegationFacts} are weakenings of familiar propositional identities from [[classical logic]]:

* A Heyting algebra where the remaining [[De Morgan law]]
  $\neg (x \wedge y) = \neg x \vee \neg y$
  holds is precisely a [[De Morgan Heyting algebra]].

* A Heyting algebra satisfying any of the
 following (equivalent) conditions is precisely a [[Boolean algebra]]:

  * $\forall x.\; \neg\neg x = x$

  * $\forall x.\; x \vee \neg x = 1$

  * $\forall x, y.\; x \Rightarrow y = \neg x \vee y$

  These also imply the De Morgan law.

### Double negation

The double-negation map $\neg\neg \colon H \to H$ on a Heyting algebra
$H$ is often relevant, particularly in the relationship
[with Boolean algebras](#ToBooleanAlgebras).  We saw in
\ref{DoubleNegationMonad} that $\neg\neg$ is a [[monad]].
Further:

\begin{lem}\label{lem:NegnegMeets}
[[double negation|Double negation]] $\neg \neg\colon L \to L$ [[preserved limit|preserves]] [[finite limit|finite]] [[meets]].
\end{lem}

\begin{proof}
Nullary meets are trivial: $\neg \neg 1 = \neg 0 = 1$.
For binary meets, the direction
$\neg \neg (x \wedge y) \leq (\neg \neg x) \wedge (\neg \neg y)$
holds simply because $\neg \neg$ is monotone.

In the other direction, we show
$
(\neg \neg x) \wedge (\neg \neg y) \leq \neg \neg (x \wedge y)
$
by currying (per \ref{NegationFacts}) $\neg (x \wedge y)$ to calculate:
$$
\begin{aligned}
  (\neg \neg x) \wedge (\neg \neg y) \wedge \neg (x \wedge y)
  & =
  (\neg \neg x) \wedge (\neg y \Rightarrow 0) \wedge (x \Rightarrow \neg y)
  \\
  & \leq
  (\neg \neg x) \wedge (x \Rightarrow 0)
  \\
  & =
  (\neg x \Rightarrow 0) \wedge \neg x
  \\
  & \leq
  0
\end{aligned}
$$
where the two inequalities follow from composition
(Proposition \ref{BasicProperties}) and modus ponens \eqref{ModusPonens}.
\end{proof}

The following Lemma \ref{lem:NegnegImplication} is important for the [[double negation translation]].

\begin{lem}\label{lem:NegnegImplication}
[[double negation|Double negation]] $\neg \neg\colon L \to L$ preserves [[implication]]:
$ \neg\neg( a \Rightarrow b ) =
  (\neg\neg a \Rightarrow \neg\neg b) $.
\end{lem}
\begin{proof}
Applying \eqref{NegIfEq} and currying (by \ref{NegationFacts}), we have
$$
 \neg\neg (a \Rightarrow b) =
 \neg ((\neg\neg a) \wedge (\neg b)) =
 (\neg\neg a \Rightarrow (\neg b \Rightarrow 0)) =
 (\neg \neg a \Rightarrow \neg \neg b)
 \; .
$$
\end{proof}

## Relation to other concepts

### To toposes

An [[elementary topos]] is a [[vertical categorification]] of a Heyting algebra: the notion of Heyting algebra is essentially equivalent to that of [[(0,1)-topos]].  Note that a [[Grothendieck topos|Grothendieck]] $(0,1)$-topos is a [[frame]] or [[locale]].

In a [[Heyting category]], every [[subobject poset]] $Sub(A)$ is a Heyting algebra.  In particular, this holds for every [[topos]].  Furthermore, in a topos, the [[power object]] $\mathcal{P}(A)$ is an [[internalisation|internal]] Heyting algebra that corresponds to the external Heyting algebra $Sub(A)$.  In a [[boolean topos]], the internal Heyting algebras are all internal [[boolean algebras]].  In general, however, the [[internal logic]] of a topos (or other Heyting category) is intuitionistic.

The proof of Lemma \ref{lem:NegnegMeets} can be made purely equational, and is therefore internally valid in any category with products. Applied to the internal Heyting algebra $L = \Omega$ of a [[topos]], that is the [[subobject classifier]], this lemma says exactly that the double negation operator $\neg \neg\colon \Omega \to \Omega$ defines a [[Lawvere–Tierney topology]]. Similarly, we get the [[double negation sublocale]] of any [[locale]].

### To topology

One of the chief sources of Heyting algebras is given by [[topology|topologies]]. As a poset, the topology of a topological space $X$ is a [[complete lattice]] (it has arbitrary [[joins]] and [[meets]], although the infinitary meets are not in general given by [[intersection]]), and the implication operator is given by
$$(U \Rightarrow V) = int(U^c \vee V)$$
where $U, V$ are open sets, $U^c$ is the set-theoretic complement of $U$, and $int(S)$ denotes the interior of a subset $S \subseteq X$.

Somewhat more generally, a [[frame]] (a [[sup-lattice]] in which finite meets distribute over arbitrary sups) also carries a Heyting algebra structure. In a frame, we may define
$$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x$$
and the distributivity property guarantees that the universal property \eqref{UniversalProperty} holds. (The detailed proof is a "baby" application of an [[adjoint functor theorem]].)

Thus frames are extensionally the same thing as _[[complete lattice|complete]] Heyting algebras_. However, _intensionally_ they are quite different; that is, a morphism of frames is not usually a morphism of complete Heyting algebras: they do not preserve the implication operator.

A [[locale]] is the same thing as a frame, but again the morphisms are different; they are reversed.

Topologies that are [[Boolean algebras]] are the exception rather than the rule; basic examples include topologies of [[Stone spaces]]; see [[Stone duality]]. Another example is the topology of a [[discrete space]] $X$.

## Adjunctions with Boolean algebras
 {#ToBooleanAlgebras}

There are several ways of passing back and forth between Boolean algebras and Heyting algebras, having to do with the [[double negation]] operator.
By \ref{DoubleNegationMonad}, double negation $\neg \neg\colon L \to L$ is a [[monad]].
Further, by \ref{lem:NegnegMeets}, it [[preserved limit|preserves]] [[finite limit|finite]] [[meets]].

Now let $L_{\neg\neg}$ denote the poset of _[[regular element]]s_ of $L$, that is, those elements $x$ such that $\neg\neg x = x$. (When $L$ is the topology of a space, an open set $U$ is [[regular open subspace|regular]] if and only if it is the interior of its closure, that is if it is a regular element of the Heyting algebra of open sets described above.) With the help of Lemma \ref{lem:NegnegMeets}, we may prove

\begin{thm}\label{thm:NegnegAdjoint}
The poset $L_{\neg\neg}$ is a Boolean algebra. Moreover, the assignment $L \mapsto L_{\neg\neg}$ is the object part of a functor
$$F\colon Heyt \to Bool$$
called _Booleanization_, which is left adjoint to the full and faithful inclusion
$$i\colon Bool \hookrightarrow Heyt.$$
The unit of the [[adjoint functor|adjunction]], applied to a Heyting algebra $L$, is the map $L \to L_{\neg\neg}$ which maps each element $x$ to its _[[regular element|regularization]]_ $\neg\neg x$.
\end{thm}

\begin{proof}
To avoid confusion from two different maps called $\neg\neg$ that
differ only in their codomain, write $M: L \rightarrow L$ for the
monad and $U: L \rightarrow L_{\neg\neg}$ for the left adjoint.
Write $\iota: L_{\neg\neg} \rightarrow L$ for the right adjoint,
so that $M = \iota \circ U$.

We first show that $L_{\neg\neg}$ is a Heyting algebra and $U$ is a
Heyting-algebra map.  Because $U$ is surjective (and monotone), it
suffices to show that it preserves the Heyting-algebra operators:
finite meets, finite joins, and implication.

Because $\iota$ is full, it reflects meets.
Therefore because by Lemma \ref{lem:NegnegMeets} $M = \iota \circ U$
preserves finite meets, so too does $U$, and as a left adjoint
it preserves joins.

Finally, for $a, b \in L$, we show that $\neg\neg (a \Rightarrow b)$,
which by Lemma \ref{lem:NegnegImplication} is the $L$-implication
$\neg\neg a \Rightarrow \neg\neg b$, satisfies the universal property \eqref{UniversalProperty}
also in $L_{\neg\neg}$.
For any $x \in L_{\neg\neg}$,
$\iota(x) \leq \neg\neg a \Rightarrow_L \neg\neg b$
just if
$\iota(x) \wedge_L \neg\neg a \leq \neg\neg b$
by the universal property in $L$;
but because $\iota$ reflects meets this is equivalent to
$x \wedge_{L_{\neg\neg}} \neg\neg a \leq \neg\neg b$,
completing the universal property.

Now because $U$ preserves implication and (the empty join) $0$, it preserves
negation.  Therefore $\neg\neg$ in $L_{\neg\neg}$ is the identity,
so the latter is a Boolean algebra.

Therefore $U = \neg\neg \colon L \to L_{\neg\neg}$ is a Heyting algebra quotient which is the coequalizer of $1, \neg\neg \colon L \stackrel{\to}{\to} L$. It follows that a Heyting algebra map $L \to B$ to
any Boolean algebra $B$, i.e. any Heyting algebra where $1$ and $\neg\neg$ coincide,
factors uniquely through this coequalizer, and the induced map $L_{\neg \neg} \to B$ is a Boolean algebra map. In other words, $\neg\neg \colon L \to L_{\neg\neg}$ is the universal Heyting algebra map to a Boolean algebra, which establishes the adjunction.
\end{proof}

Thus $\neg\neg\colon L \to L_{\neg\neg}$ preserves finite joins and finite meets and implication. In the other direction, we have an inclusion $i\colon L_{\neg\neg} \to L$, and this preserves meets but not joins. It also preserves negations; more generally and perhaps surprisingly, it preserves implications as well.

Regular elements are not to be confused with _[[complemented element]]s_, i.e., elements $x$ in a Heyting algebra such that $x \vee \neg x = 1$, although it is true that every complemented element is regular. An example of a regular element which is not complemented is given by the unit interval $(0, 1)$ as an element of the topology of $\mathbb{R}$; a complemented element in a Heyting algebra given by a topology is the same as a [[clopen subset]].

Complemented elements furnish another universal relation between Boolean algebras and Heyting algebras: the set of complemented elements in a Heyting algebra $H$ is a Boolean algebra $Comp(H)$, and the inclusion $Comp(H) \to H$ is a Heyting algebra map which is universal among Heyting algebra maps $B \to H$ out of Boolean algebras $B$. In other words, we have the following result.

\begin{thm}\label{thm:CompAdjoint}
The assignment $H \mapsto Comp(H)$ is the object part of a right adjoint to the forgetful functor $Bool \to Heyt$.
\end{thm}

\begin{proof}
In a Heyting algebra $H$, the elements $0$ and $1$ are clearly
complemented. If $x$ and $y$ are complemented, then
so are $x \wedge y$, $x \vee y$, and $x \Rightarrow y$;
we leave meet and join as an [[exercise]] applying
\ref{NegationFacts}, and demonstrate implication
(using \eqref{NegIfEq}):
$$
\begin{aligned}
(x \Rightarrow y) \vee \neg(x \Rightarrow y)
&= (x \Rightarrow y) \vee (\neg\neg x \wedge \neg y)
\\
&= ((x \Rightarrow y) \vee \neg\neg x)
  \wedge ((x \Rightarrow y) \vee \neg y)
\\
&\geq (\neg x \vee x)
  \wedge (y \vee \neg y)
\\
&= 1
\; .
\end{aligned}
$$

Thus the complemented elements form a Heyting subalgebra $Comp(H) \hookrightarrow H$. Clearly $Comp(H)$ is a Boolean algebra, and clearly if $B$ is Boolean, then any Heyting algebra map $B \to H$ factors uniquely through $Comp(H) \hookrightarrow H$. This proves the theorem.
\end{proof}

## Examples

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

For $\mathcal{T}$ a [[topos]] and $X \in \mathcal{T}$ any [[object]], the poset $Sub(X)$ of [[subobject]]s of $X$ is a Heyting algebra.

In other words, every topos is a [[Heyting category]].
=--

In particular for $X = \Omega$ the [[subobject classifier]], $Sub(\Omega)$ is a Heyting algebra.

In $\mathcal{T} =$ [[Set]] for every set $S$ we have that $Sub(S)$ is the [[Boolean algebra]] of subset of $S$.

More details and examples are spelled out at [[internal logic#examples]].

+-- {: .num_prop}
###### Proposition
A [[frame]] $L$ is a Heyting algebra.
=--

+-- {: .proof}
###### Proof
By the [[adjoint functor theorem]], a right adjoint $x \Rightarrow -$ to the map $x \wedge -: L \to L$ exists since this map preserves arbitrary joins.
=--

## See also

* [[Heyting algebra object]]
* [[locally Heyting-algebraic 2-poset]]
* [[implicative structure]]

## References

The original reference:

* [[Arend Heyting]], *Die formalen Regeln der intuitionistischen Logik. I, II, III.* Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169

  abridged reprint in:

  Karel Berka, Lothar Kreiser (eds.), *Logik-Texte*, De Gruyter (1986) 188-192 [[doi:10.1515/9783112645826](https://doi.org/10.1515/9783112645826)]

A quick introduction can be found in §1.2 of

* [[Francis Borceux]],
_[[Handbook of Categorical Algebra]] 3.  Categories of sheaves_,
Cambridge University Press 1994.
ISBN: 0521441803.

[[!redirects Heyting algebra]]
[[!redirects Heyting algebras]]


Heyting algebra object

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#### Relations
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#### Category theory
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#### Limits and colimits
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#### $(0,1)$-Category theory
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# Contents
* table of contents
{: toc}

## Idea

The notion of a *Heyting algebra object* is the generalization of that of *[[Heyting algebra]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient  [[categories]] with suitable properties.

## Definition

In a [[finitely complete category]] $C$, a **Heyting algebra object** is a [[bicartesian closed preordered object]] that is also a [[partially ordered object]].

## See also

* [[partially ordered object]]

* [[bicartesian closed preordered object]]

* [[lattice object]]

* [[Boolean algebra object]]

* [[Heyting algebra]]

[[!redirects Heyting algebra objects]]


Heyting arithmetic

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# Contents #
* table of contents
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## Idea

**Heyting arithmetic** is the [[constructive]] counterpart of [[Peano arithmetic]]. It adopts all the axioms of Peano arithmetic, while using [[constructive logic]] as its rules of inference.

## References

Named after [[Arend Heyting]].


Heyting category

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#Contents#
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## Definition

A **Heyting category** (called a **[[logos]]** in [Freyd-Scedrov](#FreydScedrov)) is a [[coherent category]] in which each [[base change]] functor $f^*:Sub(Y)\to Sub(X)$ has a _[[right adjoint]]_ $\forall_f$ (the _[[universal quantifier]]_, in addition to the [[left adjoint]] [[existential quantifier]] that exists in any [[regular category]]). Thus, every Heyting category $\mathcal{C}$ is a [[first-order hyperdoctrine]] given by the [[subobject poset]] [[functor]] $\mathrm{Sub}:\mathcal{C}^{op} \to HeytAlg$.

It follows that each [[poset of subobjects]] $Sub(X)$ is a [[Heyting algebra]] and the base-change functors $f^*$ are Heyting algebra homomorphisms. The Heyting implication can be defined as follows: if $A$ and $B$ are subobjects of $X$, then we can define $(A\Rightarrow B) \in Sub(X)$ as $\forall_i(A\cap B) \hookrightarrow X$, where $i\colon A\hookrightarrow X$ is the subobject inclusion and we take $A \cap B\in Sub(A)$.  Any Heyting category has an [[internal logic]] which is full ([[type theory|typed]]) [[first order logic|first-order]] [[intuitionistic logic]]. The extra right adjoint $\forall_f$ provided by the above axiom gives the [[universal quantifier]] in this logic. On the other hand, any category defined in [[intuitionistic logic]] which satisfies [[bounded separation]] is a Heyting category.

A **Heyting functor** is a [[coherent functor]] between Heyting categories which additionally preserves the right adjoints $\forall_f$.  Such functors also preserve the [[internal logic|internal interpretation]] of [[first-order logic]].

## Examples

Any [[topos]], and in fact any [[well-powered category|well-powered]] [[infinitary coherent category]], is a Heyting category.  Any [[Boolean category]] is also a Heyting category.

Every [[locally cartesian closed category|locally cartesian closed]] [[coherent category]] is a [[Heyting category]], because the [[dependent product]] [[functors]] preserve [[monomorphisms]]. This implies that every [[Pi-pretopos]] is a [[Heyting pretopos]].

In particular, the [[h-sets]] in [[Martin-Löf type theory]] with [[propositional truncations]] form a Heyting category.

## Related concepts

* [[De Morgan Heyting category]]

* [[Heyting pretopos]]

* [[elementary topos]]

* [[Heyting algebra]]

[[!include categories and logic - table]]

## References

* [[Peter Freyd]], Andre Scedrov, _[[Categories, Allegories]]_
 {#FreydScedrov}

[[!redirects Heyting functor]]
[[!redirects Heyting categories]]
[[!redirects Heyting functors]]


Heyting dagger 2-poset

[[!redirects Heyting dagger 2-posets]]

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## Contents ##
* table of contents
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## Idea ##

A Heyting dagger 2-poset is a dagger 2-poset whose [[category of maps]] is a [[Heyting category]].

## Definition ##

A **Heyting dagger 2-poset** is a [[dagger 2-poset]] $C$ such that

* There is an object $0 \in Ob(C)$ such that for each object $A \in Ob(C)$, there is a [[monic map in a dagger 2-poset|monic map]] $i_{0,A} \in Hom(0,A)$ such that for each object $B \in Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, there is a monic map $i_{0,B} \in Hom(0,B)$ such that $i_{B,A} \circ i_{0,B} = i_{0,A}$.

* For each object $A \in Ob(C)$, $B \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{E,A} \in Hom(E,A)$, there is an object $B \cup E \in Ob(C)$ with monic maps $i_{B \cup E,A} \in Hom(B \cup E,A)$, $i_{B,B \cup E} \in Hom(B,B \cup E)$, $i_{E,B \cup E} \in Hom(E,B \cup E)$, such that $i_{B \cap E,A} \circ i_{B,B \cap E} = i_{B,A}$ and $i_{B \cap E,A} \circ i_{E,B \cap E} = i_{E,A}$,and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{B,D} \in Hom(B,D)$, $i_{E,D} \in Hom(E,D)$ such that $i_{D,A} \circ i_{B,D} = i_{B,A}$ and $i_{D,A} \circ i_{E,D} = i_{E,A}$, there is a monic map $i_{B \cup E,D} \in Hom(B \cup E,D)$ such that $i_{D,A} \circ i_{B \cup E,D} = i_{B \cup E,A}$

* For each object $A \in Ob(C)$, $B \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{E,A} \in Hom(E,A)$, there is an object $B \cap E \in Ob(C)$ with monic maps $i_{B \cap E,A} \in Hom(B \cap E,A)$, $i_{B \cap E,B} \in Hom(B \cap E,B)$, $i_{B \cap E,E} \in Hom(B \cap E,E)$, such that $i_{B,A} \circ i_{B \cap E,B} = i_{B \cap E,A}$ and $i_{E,A} \circ i_{B \cap E,E} = i_{B \cap E,A}$, and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{D,B} \in Hom(D,B)$, $i_{D,E} \in Hom(D,E)$ such that $i_{B,A} \circ i_{D,B} = i_{D,A}$ and $i_{E,A} \circ i_{D,E} = i_{D,A}$, there is a monic map $i_{D,B \cap E} \in Hom(D,B \cap E)$ such that $i_{B \cap E,A} \circ i_{D,B \cap E} = i_{D,A}$.

* For each object $A \in Ob(C)$, $B \in Ob(C)$, $D \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{D,A} \in Hom(D,A)$, there is an object $B \Rightarrow D \in Ob(C)$ with monic map $i_{D,A} \in Hom(D,A)$,
$$i_{(B \Rightarrow D) \cap D, B}:Hom((B \Rightarrow D) \cap D, B)$$
and
$$i_{D, B \Rightarrow (D \cap B)}:Hom(D, B \Rightarrow (D \cap B))$$

## Properties ##

* For each object $A \in Ob(C)$, the identity function $1_A \in Hom(A,A)$ is a monic map, and for each object $B:Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, $1_A \circ i_{B,A} = i_{B,A}$.

* The [[isomorphism classes]] of monic maps into every object $A$ is a [[Heyting algebra]]. Since every monic map is a map, the [[category of maps]] is a [[Heyting category]]. As a result, Heyting dagger 2-posets are the same as [[division allegories]].

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a Heyting dagger 2-poset.

## See also ##

* [[dagger 2-poset]]
* [[coherent dagger 2-poset]]
* [[geometric dagger 2-poset]]
* [[Heyting dagger 2-poset]]
* [[Boolean dagger 2-poset]]
* [[elementarily topical dagger 2-poset]]


Heyting dimension

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## Idea

A refinement of the notion of [[covering dimension]] for [[Heyting space]]s that takes [[ordinal]] values.

## Related concepts

[[!include notions of dimension -- contents]]

## References

Section 7.2.4 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_


Heyting field

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\tableofcontents

## Definition

In [[constructive mathematics]]:

\begin{definition}\label{TheDefinition}
A **Heyting field** is a [[local ring]] where every non-[[invertible]] [[element]] is [[equal]] to [[zero]]. More specifically, it is a [[commutative ring]] $R$ such that

1. $R$ is non-trivial ($0 \neq 1$);

2. given elements $a \in R$ and $b \in R$, if $a + b$ is invertible, then either $a$ is invertible or $b$ is invertible;

3. given an element $a \in R$, if $a$ is non-invertible, then $a = 0$.

\end{definition}

Equivalently, a Heyting field is:

* a [[local ring]] $R$ whose [[quotient ring]] by the [[ideal]] of non-invertible elements is [[isomorphic]] to $R$;

* a local ring $R$ such that the canonical [[apartness relation]] $\#$ --- defined as $a \# b$ if and only if $a - b$ is invertible --- is a [[tight apartness relation]].

* a [[local ring]] which is both [[reduced ring|reduced]] and [[Artinian ring|Artinian]].

* a [[local ring]] which is a [[semiprimitive ring]].

\begin{remark}\label{TrivialRingAsAField}
In [Lombardi & Quitté (2010)](#LombardiQuitté2010), the authors' definition of Heyting field do not include the non-equational axiom $1 \neq 0$. Instead, they define non-invertible to mean that if the element is invertible, then $1 = 0$. With such a definition, the [[trivial ring]] counts as a Heyting field and constitutes the [[terminal object]] in the [[category|categories]] of Heyting fields.
\end{remark}

## Properties

### Stable and decidable equality.

Every [[Heyting field]] $R$ has [[stable equality]], because it has a [[tight apartness relation]] defined as $a \# b$ if and only if $a - b$ is invertible. For all elements $a \in R$ and $b \in R$, $\neg \neg \neg (a \# b) \iff \neg (a \# b)$, and because $\neg (a \# b) \iff a = b$, $\neg \neg (a = b) \iff a = b$, and thus $R$ has stable equality.

A Heyting field with [[decidable equality]] is a [[discrete field]].

## Weak Heyting fields

A **weak Heyting field** (cf. [Richman (2020)](#Richman20)) or **Johnstone residue field** is a [[commutative ring]] which only satisfy [[axioms]] 1 and 3 in Def. \ref{TheDefinition}. [Johnstone (1977)](#Johnstone77) referred these as a *residue field*; however, this is not to be confused with the already existing notion of a [[residue field]] in [[algebraic geometry]] as the [[quotient field]] of a [[local ring]].

If every weak Heyting field with decidable equality is a Heyting field, then [[excluded middle]] follows. The following proof is due to Mark Saving:

\begin{definition}
Let $p$ be a proposition. We define the set $R_p$ to be the union of $\mathbb{Z}$ and $\{x \in \mathbb{Q} \vert p\}$
\end{definition}

$R_p$ is a [[subring]] of the [[rational numbers]] $\mathbb{Q}$.

Since $R_p$ is a [[subset]] of $\mathbb{Q}$ and $\mathbb{Q}$ has decidable equality, $R_p$ also has decidable equality. And of course $0\neq 1$ in $R_p$.

\begin{theorem}
$R_p$ is a weak Heyting field iff $\neg \neg p$.
\end{theorem}

\begin{proof}
Given a [[proposition]] $p$, suppose its [[double negation]] $\neg \neg p$, and consider some $x\in R_p$. Suppose $x$ does not have a multiplicative inverse. Now suppose $x\neq 0$. Then we see that $x^{-1}$ is not in $R_p$. If $p$ held, we would have $x{-1} \in R_p$. So we know $\neg p$ holds. But this is a [[contradiction]]. Therefore, $x$ must be zero (using decidable equality).

Conversely, suppose $R_p$ is a weak Heyting field. If $\neg p$ held, we would have $R_p = \mathbb{Z}$, which clearly is not a Johnstone residue field since $2$ is neither invertible nor zero. So we must have $\neg \neg p$.
\end{proof}

\begin{theorem}
$R_p$ is a Heyting field iff it is the case that $p$ iff $R_p$ is a discrete field.
\end{theorem}

\begin{proof}
Suppose $R_p$ is a Heyting field. Then either $2$ or $3$ has a multiplicative inverse, so either $2^{-1} \in R_p$ or $3^{-1} \in R_p$. In either case, we see that $p$ holds. If $p$ holds, then $R_p = \mathbb{Q}$, which is a discrete field. And if $R_p$ is a discrete field, it is clearly a Heyting field.
\end{proof}

\begin{theorem}
If every weak Heyting field with discrete equality is Heyting, then [[excluded middle]] is valid.
\end{theorem}

\begin{proof}
From the lemmas above, if every weak Heyting field with decidable equality is a Heyting field, then $p \iff \neg \neg p$ holds for all propositions $p$. So we have full excluded middle.
\end{proof}

## See also

* [[field]]

* [[local ring]]

* [[reduced local ring]]

* [[Artinian local ring]]

* [[ordered field]]

## References

* {#Johnstone77} [[Peter Johnstone]], *Rings, Fields, and Spectra*, Journal of Algebra **49** (1977) 238-260 [doi:[10.1016/0021-8693(77)90284-8](https://doi.org/10.1016/0021-8693%2877%2990284-8)]

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) [[doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf)]

* {#Richman20} [[Fred Richman]], *Laurent series over $\mathbb{R}$*. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [[doi:10.1080/00927872.2019.1710166](https://doi.org/10.1080/00927872.2019.1710166)]

[[!redirects Heyting field]]
[[!redirects Heyting fields]]
[[!redirects heyting field]]
[[!redirects heyting fields]]

[[!redirects weak Heyting field]]
[[!redirects weak Heyting fields]]
[[!redirects weak heyting field]]
[[!redirects weak heyting fields]]

[[!redirects reduced Artinian local ring]]
[[!redirects reduced local Artinian ring]]
[[!redirects Artinian reduced local ring]]

[[!redirects Johnstone residue field]]
[[!redirects Johnstone residue fields]]


Heyting prealgebra

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#### Relations
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#### Category theory
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#### Limits and colimits
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#### $(0,1)$-Category theory
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# Contents
* table of contents
{: toc}

## Definition

A __Heyting prealgebra__ is simply a [[preorder]] whose [[poset]]al reflection is a [[Heyting algebra]]; i.e., it is a preorder which is a [[bicartesian closed category]].

In most contexts, there is little need to maintain a difference between Heyting algebras whose carriers are preorders vs. those whose carriers are strictly posets; indeed, in most contexts, there is no need to consider such a distinction to even be meaningful, the only relevant notion of equality being that derived from the preorder structure.  See [[principle of equivalence]].

However, in contexts where there _is_ a separately provided relevant notion of equality on the carrier which is potentially finer-grained than that provided by the preorder structure alone (i.e., in situations where a Heyting algebra carries further structure $x = y$ not necessarily provided simply by $x \leq y \wedge y \leq x$), then it may be worthwhile to track such differences.

## Examples

* The [[propositions]] in a [[formal logic|logic]] naturally form a Heyting prealgebra under [[entailment]].

  One can take the quotient under bi-entailment to get a poset and a Heyting algebra, or (as discussed above) one can say that bi-entailment is the only relevant notion of equality so that the preorder is really already a poset. But if one wants to simultaneously talk about the syntactic equality of terms, then there will be unequal terms that entail each other, and so one has only a Heyting prealgebra.

## See also

* [[prelattice]]
* [[Heyting prealgebra object]]

[[!redirects Heyting prealgebras]]
[[!redirects bicartesian closed preorder]]
[[!redirects bicartesian closed preorders]]
[[!redirects bicartesian closed preordered set]]
[[!redirects bicartesian closed preordered sets]]
[[!redirects bicartesian closed proset]]
[[!redirects bicartesian closed prosets]]


Heyting scale

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#Contents#
* table of contents
{:toc}

## Idea

The idea of a Heyting scale comes from [[Peter Freyd]].

## Definition

### In terms of Heyting implication

A __Heyting scale__ or __chromatic scale__ is a [[scale]] $M$ with a [[Heyting implication]] operation $(-)\rightarrow(-):M \times M \to M$ such that

* $(\bot \rightarrow \bot)^\bullet = \bot$

* for all $a$ in $M$, $a \wedge (a \rightarrow \bot) = \bot$

* for all $a$ and $b$ in $M$, $((a \wedge b) \rightarrow \bot)^\bullet = (a \rightarrow \bot)^\bullet \wedge (b \rightarrow \bot)^\bullet$

### In terms of support

...

## Properties

Every Heyting scale with $\bot = \top$ is [[trivial object|trivial]].

## Relation to Girard's linear logic

...

## Examples

The set of truth values in Girard's [[linear logic]] is a Heyting scale.

## Related concepts

* [[scale]]

* [[linear logic]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

[[!redirects Heyting scales]]

[[!redirects chromatic scale]]
[[!redirects chromatic scales]]


Hida theory

# Contents
* table of contents
{: toc}

## Idea

_Hida theory_ is an approach to p-adic interpolation of [[modular forms]] in the ordinary case.

## Definitions

This section follows chapter 8 of [DarmonRotgerAWS](#DarmonRotgerAWS).

Let $\Lambda$ be the [[Iwasawa algebra]] $\mathbb{Z}_{p}[[\Gamma]]$ where $\Gamma=(1+p\mathbb{Z}_{p})^{\times}$. We have that $\Lambda\cong \mathbb{Z}_{p}[[T]]$. We can further identify $\Lambda$ as the ring of rigid analytic functions on the _weight space_

$$\mathcal{X}:=\Hom_{cts}(\Gamma,\mathbb{C}^{\times}).$$

A point $\nu\in \mathcal{X}$ is called an _arithmetic point_ if there exists and integer $k$ (called the _weight_) and a [[Dirichlet character]] $\chi$ of p-power order and values in $\mathbb{C}_{p}^{\times}$ such that

$$\nu(x)=\chi(x)x^{k}$$

for all $x\in (1+p\mathbb{Z}_{p})$.

More generally a finite extension $\widetilde{\Lambda}$ of $\Lambda$ can be identified with the ring of rigid analytic functions on an etale cover $\widetilde{\mathcal{X}}$ of $\mathcal{X}$.

A _$\Lambda$-adic modular form_ of tame level $N$ and tame character $\chi$ is a pair $(\widetilde{\Lambda},\underline{g})$ where $\widetilde{\Lambda}$ is a finite extension of $\Lambda$ and

$$\underline{g}:=\sum \underline{a}_{n}q^{n}$$

where $\underline{a}_{n}\in \widetilde{\Lambda}$, such that for almost all $\nu\in\widetilde{X}$ of weight $k\in\mathbb{Z}\geq 2$, the power series

$$\underline{g}:=\sum \underline{a}_{n}(\nu)q^{n}$$

is the q-expansion of a classical normalized ordinary eigenform of weight $k$ and level $N$.

## Related concepts

* [[eigenvariety]]

* [[p-adic modular form]]

## References

* Chris Williams, _An Introduction to Hida Theory_ ([pdf](http://www.im.ufrj.br/~aftab/An%20Introduction%20to%20Hida%20Theory.pdf))

* Cameron Franc, _Hida Theory_ ([pdf](http://virtualmath1.stanford.edu/~conrad/DarmonCM/2011Notes/hida_theory.pdf))

A discussion may also be found in chapter 8 of

* {#DarmonRotgerAWS} Henri Darmon and Victor Rotger, _Algebraic Cycles and Stark-Heegner Points_, ([pdf](http://virtualmath1.stanford.edu/~conrad/DarmonCM/2011Notes/hida_theory.pdf))


hidden local symmetry

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[effective field theory]] [[sigma-model|sigma-]][[model (in theoretical physics)|models]] for [[Goldstone bosons]] (such as [[mesons]] in [[chiral perturbation theory]] of [[QCD]]) the [[global symmetry]] (e.g. [[chiral symmetry]]) may typically be enhanced to a "hidden" [[local gauge symmetry]] that exhibits the source vector fields (notably: [[vector mesons]]) as [[gauge fields]] of a ([[flavour (particle physics)|flavour]]) [[gauge theory]].

In [[chiral perturbation theory]], the [[gauge bosons]] of the hidden local [[chiral symmetry]] are identified with the [[vector mesons]]. The [[minimal coupling]] of these two the [[baryons]] yields the _[[vector meson dominance]]_-model of [[quantum hadrodynamics]].

In [[intersecting D-brane models]], such as in [[holographic QCD]], this is [[geometric engineering of QFT|geometrically brought out]] by [[open strings]] ending on "[[flavor branes]]" instead of "[[color branes]]".

| | [[color charge]] | [[flavor physics|flavor charge]] |
|----|----|-----|
| **[[gauge bosons]]** | [[gluons]]
  ([[gauge group]]-[[local symmetry]])  | [[mesons]]
 ([[flavor physics|flavor]]-[[hidden local symmetry]]) |
| **[[fermions]]** | [[quarks]] | [[baryons]] |

## Properties

### Relation to Skyrmions

* [[Dmitri Diakonov]], [[Victor Petrov]], _Exotic baryon resonances in the Skyrme model_ ([arXiv:0812.1212](https://arxiv.org/abs/0812.1212), [doi:10.1142/9789814704410_0004](https://doi.org/10.1142/9789814704410_0004)), Chapter 3 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> It is astounding that [[Tony Skyrme|Skyrme]] had suggested [[Skyrmion|his model]] as early as in 1961 before it has been generally accepted that [[pions]] are (pseudo) [[Goldstone bosons]] associated with the [[spontaneous symmetry breaking|spontaneous breaking]] of [[chiral symmetry]], and of course long before [[QCD|Quantum Chromodynamics (QCD)]] has been put forward as the microscopic theory of [[strong nuclear force|strong interactions]].

> The revival of the Skyrme idea in 1983 is [due to Witten](Skyrmion#Witten83) who explained the _raison d'ˆetre_ of the [[Skyrmion|Skyrme model]] from the viewpoint of [[QCD]]. In the [[chiral perturbation theory|chiral limit]] when the light [[quark]] masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons [[pion|π]], [[kaon|K]] , η become nearly massless (pseudo) [[Goldstone bosons]], they are the lightest degrees of freedom of [[QCD]]. The [[effective field theory|effective]] chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative [[WZW-term|Wess–Zumino term]] to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark [[color charge|colours]] $N_c$.

> $[...]$

> Soon after Witten's work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$[6](#DiakonovEides83), [7a](#DharWadia84), [7b](#DarShankarWadia85), [8](#DiakonovPetrov86)$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,

> $$\mathcal{L} = \overline{Q} \left( \partial\!\!\!\!/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi}  }  \right) Q$$

> induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including
automatically the Wess–Zumino term, with the correct coefficient!

> $[...]$

> The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$[9](#DiakonovPetrovPobylitsa88)$]$ The Skyrmion is, thus, nothing but the **mean chiral field binding quarks in a baryon**.

\linebreak

* {#DiakonovEides83} [[Dmitri Diakonov]], Michael I. Eides, _Chiral Lagrangian from a functional integral over quarks_, JETP Letters 38.7 (1983): 433-436 ([pdf](http://jetpletters.ac.ru/ps/1483/article_22635.pdf), [[DiakonovEides83.pdf:file]])

* {#DharWadia84} A. Dhar, Spenta R. Wadia, _Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales_, Phys. Rev. Lett. 52, 959 (1984) ([doi:10.1103/PhysRevLett.52.959](https://doi.org/10.1103/PhysRevLett.52.959))

* {#DarShankarWadia85} Avinash Dhar, R. Shankar, Spenta R. Wadia, _Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD_, Phys. Rev. D 31, 3256 (1985) ([doi:10.1103/PhysRevD.31.3256](https://doi.org/10.1103/PhysRevD.31.3256))

* {#DiakonovPetrov86} [[Dmitri Diakonov]], [[Victor Petrov]], _A theory of light quarks in the instanton vacuum_, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (doi:10.1016/0550-3213(86)90011-8)

* {#DiakonovPetrovPobylitsa88} [[Dmitri Diakonov]], [[Victor Petrov]], P.V. Pobylitsa, _A Chiral Theory of Nucleons_, Nucl. Phys. B306 (1988) 809 ([spire:247700](http://inspirehep.net/record/247700), doi:10.1016/0550-3213(88)90443-9)

## Related concepts

[[!include effective field theories of nuclear physics -- contents]]

## References

### General

Origin of [[effective field theory]] of [[mesons]] in [[nuclear physics]], via [[hidden local gauge symmetry]]:

* [[Jun John Sakurai]], _Theory of strong interactions_, Annals Phys. 11 (1960) 1-48 ([spire:3218](http://inspirehep.net/record/3218), doi:10.1016/0003-4916(60)90126-3)

Review:

* [[Masako Bando]], [[Taichiro Kugo]], Koichi Yamawaki, _Nonlinear realization and hidden local symmetries_, Physics Reports Volume 164, Issues 4–5, July 1988, Pages 217-314 (doi:10.1016/0370-1573(88)90019-1)

* Yong-Liang Ma, Masayasu Harada, Section 6 of: _Lecture notes on the Skyrme model_ ([arXiv:1604.04850](https://arxiv.org/abs/1604.04850), [spire:1448311](https://inspirehep.net/literature/1448311))

  (in relation to [[Skyrmions]])

Further original articles:

* [[Masako Bando]], [[Taichiro Kugo]], S. Uehara, Koichi Yamawaki, and T. Yanagida, _Is the $\rho$ Meson a Dynamical Gauge Boson of Hidden Local Symmetry?_, Phys. Rev. Lett. 54, 1215 (1985) ([doi:10.1103/PhysRevLett.54.1215](https://doi.org/10.1103/PhysRevLett.54.1215))

* [[Masako Bando]], Takanori Fujiwara, Koichi Yamawaki, _Generalized Hidden Local Symmetry and the $A_1$ Meson_, Progress of Theoretical Physics, Volume 79, Issue 5, May 1988, Pages 1140–1166 ([doi:10.1143/PTP.79.1140](https://doi.org/10.1143/PTP.79.1140))

* [[Howard Georgi]], _Vector realization of chiral symmetry_, Nuclear Physics B Volume 331, Issue 2, 12 February 1990, Pages 311-330 (doi:10.1016/0550-3213(90)90210-5)

Equivalence between [[hidden local symmetry]]- and [[massive Yang-Mills theory]]-description of [[Skyrmion]] [[quantum hadrodynamics]]:

* [[Atsushi Hosaka]], H. Toki, W. Weise, _Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 ([spire:24079](http://inspirehep.net/record/24079))

In view of the [[chiral anomaly]] [[WZW term]]:

* Michio Hashimoto, _Hidden Local Symmetry for Anomalous Processes with Isospin/$SU(3)$ Breaking Effects_, Phys. Rev. D54 (1996) 5611-5619 ([arXiv:hep-ph/9605422](https://arxiv.org/abs/hep-ph/9605422))

With emphasis on [[Skyrmions]] involving [[vector mesons]]:

* Yong-Liang Ma, Ghil-Seok Yang, Yongseok Oh, Masayasu Harada, _Skyrmions with vector mesons in the hidden local symmetry approach_, Phys. Rev. D87:034023, 2013 ([arXiv:1209.3554](https://arxiv.org/abs/1209.3554))

* [[Mannque Rho]], *Skyrmions and Fractional Quantum Hall Droplets Unified by Hidden Symmetries in Dense Matter* ([arXiv:2109.10059](https://arxiv.org/abs/2109.10059))

[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]

### In D-brane models

Further interpretation/derivation of [[hidden local symmetry]] in [[holographic QCD]]:

* D. T. Son, M. A. Stephanov, _QCD and dimensional deconstruction_, Phys. Rev. D69 (2004) 065020 ([arXiv:hep-ph/0304182](https://arxiv.org/abs/hep-ph/0304182), [doi:10.1103/PhysRevD.69.065020](https://doi.org/10.1103/PhysRevD.69.065020))

  (also derivation of [[vector meson dominance]])

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

  (on [[small N corrections]] to [[holographic QCD]])

* Hyun Kyu Lee, [[Mannque Rho]], Section 6.2.1 of: _Half-Skyrmion Hadronic Matter at High Density_ ([doi:10.1142/9789814280709_0006](https://doi.org/10.1142/9789814280709_0006)), Chapter 6 in: [[Mannque Rho]] et al., _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

[[!redirects hidden local symmetries]]

[[!redirects hidden local gauge symmetry]]


hidden sector

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

#Contents#
* table of contents
{:toc}

## Idea

In ([[perturbative quantum field theory|perturbative]]) [[quantum field theory]] a _hidden sector_ is a subspace of the space of [[field (physics)|fields]] whose only [[interaction]] with the remaining fields is via [[gravity]].

If one interprets the remaining fields as constituting a [[model (physics)|model]] for [[phenomenology]], then in the world this describes the fields in the subspace are literally hidden, they can only bee "seen" via their gravitational pull (or via [[gravitational waves]]).

Hidden sectors as a model for the real world remain hypothetical. They have been postulated notably as a means to explain [[supersymmetry breaking]] (see [below](#SupersymmetryBreaking)). They appear generically in [[perturbative string theory vacua]] (further [below](#InStringTheory)) after [[KK-compactification]].

## Definition

Specifically in [[Lagrangian field theory]] this means that the [[field bundle]] $E \overset{fb}{\to} \Sigma$ over [[spacetime]] $\Sigma$ is a [[fiber product]]

$$
  E \underoverset{\simeq}{(p_1,P_2)}{\longrightarrow} E_{1} \times_\Sigma E_2
$$

such that the [[interaction]] [[Lagrangian density]] $\mathbf{L}_{int}$ is of the form

$$
  \mathbf{L}_{int}
  =
  p_1^\ast \mathbf{L}_{int,1}
  +
  p_2^\ast \mathbf{L}_{int,2}
  +
  \mathbf{L}_{grav}
$$

where the first two summands are [[pullback of differential forms|pullbacks]] of interactions of the fields in $E_1$ and $E_2$ separately, while the last summand is the interaction term of [[gravity]] coupled to the respective fields jointly.

In this case $E_2$ is a _sector hidden from_ $E_1$, and conversely.

## Examples

### In supersymmetry breaking
 {#SupersymmetryBreaking}

The concept of [[gravity-mediated supersymmetry breaking]] resulted from study of [[gaugino condensation]] in a hidden sector. ([Nilles 01](#Nilles01))

### In string theory
  {#InStringTheory}

In [[string theory]] hidden sectors appear naturally in various ways.

[[Hořava-Witten theory]] at finite interval length gives rise to a copy of [[heterotic supergravity]] at each of the "endpoints", which, as [[gauge theories]], interact with each others only via [[bulk]] gravity.

In [[model (physics)|models]] of [[M-theory on G2-manifolds]] such as the [[G2-MSSM]], the 4d gauge-field content arises under [[KK-compactification]] from 3-cycles in the [[G2-manifold]] [[fiber bundle|fiber]], and if two such do not intersect, they give rise to hidden sectors in the 4d physics. (e.g. [Kane 17, section 4.4](#Kane17))

## Related concepts

* [[soft supersymmetry breaking term]]

## References

Discussion in the context of [[supersymmetry breaking]] is in

* {#Nilles01} [[Hans-Peter Nilles]], _Hidden Sector Supergravity Breakdown_, Nucl. Phys. Proc. Suppl. 101 (2001) 237-250 ([arXiv:hep-ph/0106063](https://arxiv.org/abs/hep-ph/0106063))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 2.6.5 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

Discussion specifically in the context of the [[G2-MSSM]] is in

* {#Kane17} [[Gordon Kane]], section 4.4 of _String theory and the real world_, Morgan & Claypool, 2017 (doi:0.1088/978-1-6817-4489-6)

[[!redirects hidden sectors]]


hidden smoothness principle

The [[hidden smoothness principle]] refers to the conjectural picture envisioned in 1980s by [[Deligne]], [[Drinfeld]], [[Beilinson]], [[Kontsevich]] that [[moduli space]]s in algebraic geometry which are often singular, should be just truncations of a moduli spaces in some derived sense (precursors seen often at the level of [[derived categories of coherent sheaves]] and conjecturally these moduli spaces would locally involve [[commutative differential graded algebras]]); furthermore these moduli spaces should be smooth, and this property is lost due truncation. The derived moduli spaces were indeed realized in [[derived algebraic geometry]] and often indeed smoothness fails for truncation reasons. However, the smoothness is, even for the derived moduli spaces, not always present, hence the conjecture resulted in a heuristic rather than in a general rule.

One of the principal concrete motivations was the idea to give the intrinsic geometric presentation and proof of the [[Serre intersection formula]].

* [[Maxim Kontsevich]], _Enumeration of rational curves via torus actions_, in: The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math. __129__, Birkhäuser 1995. MR1363062 (97d:14077), [hep-th/9405035](http://arxiv.org/abs/hep-th/9405035)
* [[Clark Barwick]], _Why derived algebraic geometry_, 2007, [pdf](http://web.archive.org/web/20110125061347/http://www.math.ias.edu/~clarkbar/docs/research200711.pdf)


hidden variable theory

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

# Contents
* table of contents
{: toc}

## Idea

In its standard formulation, [[quantum physics]] is a probabilistic theory of nature, since it provides only [[probabilities]] (in fact [[probability amplitudes]]) for values of [[observable]] in [[experiment]]. This is in contrast to [[classical mechanics]], which (on classical [[pure states]]) predicts events with certainty.

However, on very small scales classical mechanics predicts the _wrong_ events, while [[quantum mechanics]] and [[quantum field theory]] predicts the right [[probabilities]].

On the other hand, it is well known that upon "coarse graining", which means after averaging over certain details, also classical mechanics induces a probabilistic theory of nature, namely [[statistical mechanics]]/[[thermodynamics]].

Therefore it is natural to speculate that maybe also [[quantum physics]] is the [[statistical mechanics]] of a more refined theory of as yet unseen "degrees of freedom of nature" on small scales, which does predict events with certainty on very small scales, and which reduces to quantum mechanics on larger scales as one averages over these unseen new degrees of freedom.

These "unseen degrees of freedom" are usually called _hidden variables_, and a [[theory (physics)|theory]] which is non-probabilistic but designed to reproduce [[quantum mechanics]] as its statistical coarse grained theory are called _hidden variable theories_. (These are hence one potential [[interpretation of quantum mechanics]].)

There have been various attempts to construct such hidden variable theories. However, there are also [[theorems]] about the characteristic properties of [[quantum mechanics]] which assert (under some (natural) assumptions, of course) that there cannot be a hidden variable theory.

These theorems are

* [[Bell's inequalities]] ([Bell 64](#Bell64));

* the [[Kochen-Specker theorem]] ([Kochen-Specker 68](#KochenSpecker68)).

One well-developed attempt to construct a hidden variable theory is [[Bohmian mechanics]]; this makes hidden variables out of the entire [[wavefunction]] and violates the assumption of [[locality]].

## Related concepts

* [[quantum contextuality]]

* [[interpretation of quantum mechanics]]

* [[Einstein-Podolsky-Rosen paradox]]

* [[Kochen-Specker theorem]]

* [[Bell's theorem]]

* [[Hardy's paradox]]

## References

Hidden variables (and their non-existence) are already mentioned in:

* {#vonNeumann32} [[John von Neumann]],

  p. 172 in: *Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  p. 198 in: *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

Textbook account:

* {#Moretti16} [[Valter Moretti]], §5.1 of: *Mathematical Foundations of Quantum Mechanics: An Advanced Short Course*, Int. J. Geom. Methods Mod. Phys. **13** Supp. 1 (2016) 1630011 [[arXiv:1508.06951](http://arxiv.org/abs/1508.06951), [doi:10.1142/S0219887816300117](https://doi.org/10.1142/S0219887816300117)]

See also:

* Wikipedia, _[Hidden variable theory](http://en.wikipedia.org/wiki/Hidden_variable_theory)_

The original article on [[Bell's theorem]]:

* {#Bell64} [[John Bell]], _On the Einstein Podolsky Rosen paradox_, Physics **1** (1964) 195-200 [[pdf](http://www.drchinese.com/David/Bell_Compact.pdf)]

The original article on the [[Kochen-Specker theorem]] is

* [[Simon Kochen]], [[Ernst Specker]], _The problem of hidden variables in quantum mechanics_ 1968 , Journal of Mathematics and Mechanics, [pdf](http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1968/17/17004/pdf).
  {#KochenSpecker68}

Discussion of hidden variable theories in terms of [[quantum logic]] is in section 5 of

* Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto
Giuntini and Francesco Paoli, _Quantum Logic and Nonclassical Logics_, p. 127 in  Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) _Handbook of Quantum Logic and Quantum Structures: Quantum Logic_, 2009 North Holland
  {#CCGP09}

[[!redirects hidden variable theory]]
[[!redirects hidden variable theories]]

[[!redirects hidden variable]]
[[!redirects hidden variables]]


Hidetoshi Awata

* [personal page](http://www.math.nagoya-u.ac.jp/~awata/)

## Selected writings

On the [[sl(2)|$\mathfrak{sl}(2)$]]-[[WZW model]] [[2d CFT]] at fractional [[level (Chern-Simons theory)|level]] (cf. also [as logarithmic CFTs](logarithmic+CFT#FractionalLevelWZWModelReferences)):

* [[Hidetoshi Awata]], [[Yasuhiko Yamada]], *Fusion rules for the fractional level $\widehat{\mathfrak{sl}(2)}$ algebra*,     Mod. Phys. Lett. A **7** (1992) 1185-1196 $[$[spire:332974](https://inspirehep.net/literature/332974), [doi:10.1142/S0217732392003645](https://doi.org/10.1142/S0217732392003645)$]$

category: people