lattice gauge theory

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## Idea

_Lattice gauge theory_ (introduced in [Wegner 71](#Wegner71), [Wilson 74](#Wilson74)) is [[gauge theory]] ([[Yang-Mills theory]], such as [[quantum chromodynamics]]) where [[continuum]] [[spacetime]] is replaced by a [[discrete group|discrete]] [[lattice (in a vector space, etc.)|lattice]], hence a [[lattice model]] for [[gauge field theory]].

Usually this is considered after [[Wick rotation]] from [[Minkowski spacetime]] $\mathbb{R}^{3,1}$ to [[Euclidean field theory]] on a [[lattice (in a vector space, etc.)|lattice]] inside $\mathbb{R}^3 \times S^1$, and typically one further identifies the spatial directions periodically to arrive at [[Euclidean]] [[gauge field theory]] on a [[lattice (in a vector space, etc.)|lattice]] inside the [[4-torus]] $T^4$.

This discretization and further [[KK-compactification|compactification]] has the effect that the would-be [[path integral]] of the theory becomes an ordinary [[finite number|finite]]- (albeit high-)[[dimension|dimensional]] [[integral]], hence well defined and in principle amenable to explicit computation.

This allows to consider ([[Wick rotation|Wick-rotated]]) [[path integral quantization]] at fixed lattice spacing, this being, in principle, a [[non-perturbative field theory|non-perturbative]] [[quantization]], in contrast to [[perturbative quantum field theory]] in terms of a [[Feynman perturbation series]]. On the other hand, much of the subtlety of the latter now appears in issues of taking the continuum limit where the the lattice spacing is sent to zero. In particular, different choices of discretizing the [[path integral]] over the lattice correspond to the [[renormalization]]-freedom seen in [[perturbative quantum field theory]].

Hence lattice gauge theory lends itself to brute-force simulation of [[quantum field theory]] on electronic computers, and the term is often understood by default in this sense. See [Fodor-Hoelbling 12](#FodorHoelbling12) for a good account.

Since the explicit [[non-perturbative quantum field theory|non-perturbative]] formulation of [[Yang-Mills theories]] such as [[QCD]] is presently wide open (see the references at [[mass gap]] and at _[[quantization of Yang-Mills theory]]_) these numerical simulations provide, besides actual [[experiment]], key insights into the non-perturbative nature of the theory, such as its [[instanton sea]] ([Gruber 13](#Gruber13)) and notably the phenomenonon of [[confinement]]/[[mass gap]] and explicit computation of [[hadron]] [[masses]] ([Durr et al. 09](#Durr09), see [Fodor-Hoelbling 12, section V](#FodorHoelbling12))

Despite the word "theory", lattice gauge theory is more like "computer-simulated [[experiment]]". While it allows to see phenomena of QCD, it usually cannot provide a conceptual explanation, and of course not a mathematical derivation of problems such as [[confinement]]/[[mass gap]]. Lattice gauge theory is to the [[confinement]]/[[mass gap]]-problems as explicit computation of zeros of the [[Riemann zeta-function]] is to the [[Riemann hypothesis]] (see [there](#Riemann+hypothesis#ReferencesComputerChecks))).

## Properties

### Sign problem
 {#SignProblem}

See at *[[sign problem in lattice QCD]]*.

## Related concepts

* [[Symanzik effective field theory]]

* [[lattice model]]

* [[non-perturbative effect]]

* [[lattice renormalization]]

* Discussion of [[QCD instantons]] in LGT includes ([Moore 03, section 7](#Moore03), [Gruber 13](#Gruber13))

* [[QCD cosmology]]

* [[AdS/QCD correspondence]]

* [[Euclidean field theory]]

* [[string bit model]]

## References
 {#References}

### General

The concept was introduced in

* {#Wegner71} [[Franz Wegner]], _Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters_, Journal of Mathematical Physics 12, 2259 (1971) ([doi:10.1063/1.1665530](https://doi.org/10.1063/1.1665530))

* {#Wilson74} [[Kenneth Wilson]],  _Confinement of quarks, Phys. Rev. D10, 2445, 1974 ([doi:10.1103/PhysRevD.10.2445](https://doi.org/10.1103/PhysRevD.10.2445))

([Wilson 1974](#Wilson74) envisioned to take the continuum limit by way of [[block spin transformations]], but [Creutz 1979](#Creutz79) successfully applied [[Monte Carlo methods]], and that has been the method of choice ever since.)

Introduction and review:

* S. Gupta, _Introduction to lattice field theory_, March 2011, ([pdf](http://theory.tifr.res.in/~sgupta/talks/11aslft.pdf))

* G. Münster, M. Walzl, _Lattice Gauge Theory - A short Primer_ ([arXiv:hep-lat/0012005](http://arxiv.org/abs/hep-lat/0012005))

* [[Kenneth Wilson]], _The Origins of Lattice Gauge Theory_, ([arXiv:hep-lat/0412043](http://arxiv.org/abs/hep-lat/0412043))

* {#Moore03} Guy Moore, _Informal lectures on lattice gauge theory_, 2003 ([pdf](https://theorie.ikp.physik.tu-darmstadt.de/qcd/moore/latt_lectures.pdf))

* {#PeetersZamaklar09} [[Kasper Peeters]], [[Marija Zamaklar]], section 5 of _Euclidean Field Theory_, Lecture notes 2009-2011 ([web](http://maths.dur.ac.uk/users/kasper.peeters/eft.html), [pdf](http://maths.dur.ac.uk/users/kasper.peeters/pdf/eft.pdf))

* {#Brambilla14} Brambilla et al., Section 3.3.1 of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* Mattia Dalla Brida, _Past, present, and future of precision determinations of the QCD parameters from lattice QCD_ ([arXiv:2012.01232](https://arxiv.org/abs/2012.01232))

* [[Franz Gross]], [[Eberhard Klempt]] et al., Chapter 4 of: *50 Years of Quantum Chromodynamics*, EJPC [[arXiv:2212.11107](https://arxiv.org/abs/2212.11107)]

Rigorous discussion in view of the [[mass gap problem]]:

* {#Chatterjee20} [[Sourav Chatterjee]], _A probabilistic mechanism for quark confinement_, Comm. Math. Phys. 2020 ([arXiv:2006.16229](https://arxiv.org/abs/2006.16229))

Visualization:

* James Biddle et al. _Publicising Lattice Field Theory through Visualisation_ ([arXiv:1903.08308](https://arxiv.org/abs/1903.08308))

Relation to [[string theory]]/[[M-theory]] (such as via [[BFSS matrix model]]) in view [[AdS-CFT duality]]:

* {#Hanada16} [[Masanori Hanada]], _What lattice theorists can do for superstring/M-theory_,  International Journal of Modern Physics A Vol. 31, No. 22, 1643006 (2016)  ([arXiv:1604.05421](https://arxiv.org/abs/1604.05421))

See also

* Wikipedia, _[Lattice gauge theory](https://en.wikipedia.org/wiki/Lattice_gauge_theory)_

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

For [[topological phases of matter]] via [[higher gauge theory|higher]] lattice gauge theory:

* [[Joe Huxford]], *The higher lattice gauge theory model for topological phases of matter*, PhD thesis, Oxford (2022) [[uuid:e789bf29-179b-4b79-9168-605bf8c035ba](https://ora.ox.ac.uk/objects/uuid:e789bf29-179b-4b79-9168-605bf8c035ba), [pdf](https://ora.ox.ac.uk/objects/uuid:e789bf29-179b-4b79-9168-605bf8c035ba/files/ddj52w5202)]

* [[Joe Huxford]], [[Steven H. Simon]], *Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview* [[arXiv:2202.08294](https://arxiv.org/abs/2202.08294)]

Further on [[higher gauge theory|higher]] lattice gauge theory:

* [[Juan Orendain]], [[José A. Zapata]], *Higher homotopy and lattice gauge fields* [[arXiv:2311.02363](https://arxiv.org/abs/2311.02363)]

### Computer simulations
 {#ReferencesMontoCarloSimulations}

Introduction:

* [[Masanori Hanada]], _Markov Chain Monte Carlo for Dummies_ ([arXiv:1808.08490](https://arxiv.org/abs/1808.08490))

* Anosh Joseph, _Markov Chain Monte Carlo Methods in Quantum Field Theories: A Modern Primer_, Springer Briefs in Physics 2020 ([arXiv:1912.10997](https://arxiv.org/abs/1912.10997))

Account of computer simulation results in lattice [[QCD]]:

* {#FodorHoelbling12} Zoltan Fodor, Christian Hoelbling, sections II-IV of _Light Hadron Masses from Lattice QCD_, Rev. Mod. Phys. 84, 449 (2012) ([arXiv:1203.4789](https://arxiv.org/abs/1203.4789))

The idea of using [[Monte Carlo method]] for lattice gauge theory originates with:

* {#Creutz79} [[Michael Creutz]], *Confinement and the Critical Dimensionality of Space-Time*, Phys. Rev. Lett. **43**  (1979) 553 [[doi:10.1103/PhysRevLett.43.553](https://doi.org/10.1103/PhysRevLett.43.553)]

* [[Michael Creutz]], _Monte Carlo study of quantized $SU(2)$ gauge theory_, Phys. Rev. D **21** (1980) 2308-2315 [[doi:10.1103/PhysRevD.21.2308](https://doi.org/10.1103/PhysRevD.21.2308), [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub037.pdf)]

* [[Michael Creutz]], *Monte Carlo study of renormalization in lattice gauge theory* Phys.Rev. D **23** (1981) 1815  [[doi:10.1103/PhysRevD.23.1815](https://doi.org/10.1103/PhysRevD.23.1815), [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub045.pdf)&rbrack

* [[Michael Creutz]], Laurence Jacobs, Claudio Rebbi, *Monte Carlo computations in lattice gauge theories*, Physics Reports **95** 4 (1983) 201-282 [doi:10.1016/0370-1573(83)90016-9, [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub068.pdf)]

Specifically computation of [[hadron]]-[[masses]] (see [[mass gap problem]]) in lattice QCD is reported here:

* {#Durr09} S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S.D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K.K. Szabo, G. Vulvert,

  _Ab-initio Determination of Light Hadron Masses_,

  Science 322:1224-1227, 2008 ([arXiv:0906.3599](https://arxiv.org/abs/0906.3599))

reviewed in

* [Fodor-Hoelbling 12, section V of ](#FodorHoelbling12)

Discussion specifically of numerical computation of [[form factors]]:

* B.B. Brandt, S. Capitani, M. Della Morte, D. Djukanovic, J. Gegelia, G. von Hippel, A. Juttner, B. Knippschild, H.B. Meyer, H. Wittig, _Form factors in lattice QCD_, Eur. Phys. J. ST 198:79-94, 2011 ([arXiv:1106.1554](https://arxiv.org/abs/1106.1554))

Relation to [[tensor networks]]:

* [[Luca Tagliacozzo]], Alessio Celi, Maciej Lewenstein, _Tensor Networks for Lattice Gauge Theories with continuous groups_, Phys. Rev. X 4, 041024 (2014) ([arXiv:1405.4811](https://arxiv.org/abs/1405.4811))

* {#BBCCCDFJLMMRRTVV19} M.C. Bañuls, R. Blatt, J. Catani, A. Celi, J.I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C.A. Muschik, B. Reznik, E. Rico, [[Luca Tagliacozzo]], K. Van Acoleyen, [[Frank Verstraete]], U.-J. Wiese, M. Wingate, J. Zakrzewski, P. Zoller:

  _Simulating Lattice Gauge Theories within Quantum Technologies_

  ([arXiv:1911.00003](https://arxiv.org/abs/1911.00003))

### Renormalization

A proposal for a rigorous formulation of [[renormalization]] in lattice gauge theory is due to

* [[Tadeusz Balaban]], _Renormalization group approach to lattice gauge field theories: I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions_, Communications in Mathematical Physics, Volume 109, Issue 2, pp.249-301 ([web](https://link.springer.com/article/10.1007%2FBF01215223))

* ...

reviewed in

* [[Jonathan Dimock]], _The renormalization group according to Balaban, I. Small fields_, Rev. Math. Phys., 25, 1330010 (2013) ([doi:10.1142/S0129055X13300100](https://doi.org/10.1142/S0129055X13300100))

* ...

### Topological effects and instantons

Discussion of [[instantons]] in lattice QCd

* {#Gruber13} Florian Gruber, _Topology in dynamical Lattice QCD simulations_, 2013 ([web](http://epub.uni-regensburg.de/27631/), [pdf](http://epub.uni-regensburg.de/27631/1/dissertation.pdf))

### For super Yang-Mills theories
  {#ForSuperYangMills}

Lattice simulation of [[torus]]-[[KK-compactifications]] of [[10d super Yang-Mills theory]] and numerical test of [[AdS/CFT]]:

#### General

* {#Joseph15} Anosh Joseph, _Review of Lattice Supersymmetry and Gauge-Gravity Duality_ ([arXiv:1509.01440](https://arxiv.org/abs/1509.01440))

* {#Hanada16} [[Masanori Hanada]], _What lattice theorists can do for superstring/M-theory_,  International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016)  ([arXiv:1604.05421](https://arxiv.org/abs/1604.05421))

#### Compactification to $D = 1$
 {#ReferencesBFSS}

The [[BFSS matrix model]]:

* Veselin G. Filev, Denjoe O'Connor, _The BFSS model on the lattice_, JHEP 1605 (2016) 167 ([arXiv:1506.01366](https://arxiv.org/abs/1506.01366))

* [[Masanori Hanada]], Paul Romatschke, _Lattice Simulations of 10d Yang-Mills toroidally compactified to 1d, 2d and 4d_ ([arXiv:1612.06395](https://arxiv.org/abs/1612.06395))

Including the [[BMN matrix model]]:

* Hrant Gharibyan, [[Masanori Hanada]], Masazumi Honda, Junyu Liu, _Toward simulating Superstring/M-theory on a quantum computer_ ([arXiv:2011.06573](https://arxiv.org/abs/2011.06573))

* Georg Bergner, Norbert Bodendorfer, Masanori Hanada, Stratos Pateloudis, Enrico Rinaldi, Andreas Schäfer, Pavlos Vranas, [[Hiromasa Watanabe]], *Confinement/deconfinement transition in the D0-brane matrix model -- A signature of M-theory?*, JHEP 05 (2022) 096 [[arXiv:2110.01312](https://arxiv.org/abs/2110.01312)]

#### Compactification to $D= 0$
 {#ReferencesIKKT}

The [[IKKT matrix model]] and claims that it predicts spontaneous [[KK-compactification]] of the $D = 10$ [[M-theory|non-perturbative]] [[type IIB string theory]]/[[F-theory]] to $D = 3+1$ macrocopic [[spacetime]] [[dimensions]]:

* {#KimNishimuraTsuchiya12} S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions_, Phys. Rev. Lett. 108, 011601 (2012), ([arXiv:1108.1540](https://arxiv.org/abs/1108.1540)).

* S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Late time behaviors of the expanding universe in the IIB matrix model_, JHEP 10, 147 (2012), ([arXiv:1208.0711](https://arxiv.org/abs/1208.0711)).

* Yuta Ito, Jun Nishimura, Asato Tsuchiya, _Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model_ ([arXiv:1512.01923](https://arxiv.org/abs/1512.01923))

* {#AHINT19} Toshihiro Aoki, Mitsuaki Hirasawa, Yuta Ito, Jun Nishimura, Asato Tsuchiya, _On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model_ ([arXiv:1904.05914](https://arxiv.org/abs/1904.05914))

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lattice group

A lattice group is a [[set]] with compatible [[lattice]] and [[group]] structures.

This should not be confused with a lattice in group theory (which is a [[discrete group|discrete]] [[subgroup]] of the additive group of a [[topological vector space]]).

Somebody should write down the compatibility requirements!

[[!redirects lattice group]]
[[!redirects lattice groups]]


lattice model

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## Idea

A [[model (in theoretical physics)]] whose underlying [[spacetime]] is a [[lattice in a vector space|lattice]], hence a [[discrete space]], or rather a [[graph]].

## Examples

* [[Ising model]]

* [[lattice gauge theory]]

* [[string-net model]]

## Related concepts

* [[semi-holomorphic 4d Chern-Simons theory]]

## References

### General

See also

* Wikipedia, _Lattice model (physics)_

### For 2d CFT

For [[2d CFTs]]:

* [[Philippe Di Francesco]], *Integrable lattice models, graphs and modular invariant conformal field theories*, International Journal of Modern Physics A **7** 03 (1992) 407-500 $[$[doi:10.1142/S0217751X92000223](https://doi.org/10.1142/S0217751X92000223)$]$

Further development

* Roger E Behrend, David E Evans, *Integrable lattice models for conjugate $A^{(1)}_n$*, J. Phys. A: Math. Gen. **37** (2014) 2937 $[$[doi:10.1088/0305-4470/37/8/006](https://iopscience.iop.org/article/10.1088/0305-4470/37/8/006), [arXiv:hep-th/0309068](https://arxiv.org/abs/hep-th/0309068)$]$

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lattice object

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## Idea

The notion of a *lattice object* is the generalization of that of *[[lattice]]* 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 **lattice object** is a [[bicartesian preordered object]] that is also a [[partially ordered object]].

## See also

* [[partially ordered object]]

* [[bicartesian preordered object]]

* [[meet-semilattice object]]

* [[join-semilattice object]]

* [[Heyting algebra object]]

* [[lattice]]

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lattice of subgroups

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## Idea

The [[lattice]] of [[subgroups]] of a given [[group]].

## Related concepts

* [[maximal subgroup]]

* [[lattice of subobjects]]

* [[conjugacy class of subgroups]]

## References

See also

* Wikipedia, _[Lattice of subgroups](https://en.wikipedia.org/wiki/Lattice_of_subgroups)_

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lattice of TVS properties

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<a xlink:href="http://ncatlab.org/nlab/show/reflexive topological vector space" xlink:title="Reflexive">
<polygon style="fill:#ffffbb;stroke:black;" points="381,-396 327,-396 327,-360 381,-360 381,-396"/>
<text fill="#008B26" text-anchor="middle" x="354" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Re</text>
</a>
</g>
<g id="edge14" class="edge"><title>Hi->Re</title>
<path style="fill:none;stroke:black;" d="M339,-576C342,-538 349,-451 352,-406"/>
<polygon style="fill:black;stroke:black;" points="355.488,-406.299 353,-396 348.522,-405.602 355.488,-406.299"/>
</g>
<g id="node23" class="node"><title>AP</title>
<a xlink:href="http://ncatlab.org/nlab/show/approximation property" xlink:title="Approximation Property">
<polygon style="fill:#ffffbb;stroke:black;" points="420,-540 366,-540 366,-504 420,-504 420,-540"/>
<text fill="#008B26" text-anchor="middle" x="393" y="-517.9" style="font-family:Times New Roman;font-size:14.00;">AP</text>
</a>
</g>
<g id="edge88" class="edge"><title>Hi->AP</title>
<path style="fill:none;stroke:black;" d="M352,-576C358,-568 366,-557 373,-548"/>
<polygon style="fill:black;stroke:black;" points="375.8,-550.1 379,-540 370.2,-545.9 375.8,-550.1"/>
</g>
<g id="node8" class="node"><title>Sc</title>
<a xlink:href="http://ncatlab.org/nlab/show/Schwartz space" xlink:title="Schwartz">
<polygon style="fill:#ffffbb;stroke:black;" points="492,-540 438,-540 438,-504 492,-504 492,-540"/>
<text fill="#008B26" text-anchor="middle" x="465" y="-517.9" style="font-family:Times New Roman;font-size:14.00;">Sc</text>
</a>
</g>
<g id="edge24" class="edge"><title>Nu->Sc</title>
<path style="fill:none;stroke:black;" d="M465,-576C465,-568 465,-559 465,-550"/>
<polygon style="fill:black;stroke:black;" points="468.5,-550 465,-540 461.5,-550 468.5,-550"/>
</g>
<g id="edge90" class="edge"><title>Nu->AP</title>
<path style="fill:none;stroke:black;" d="M447,-576C438,-567 428,-557 418,-547"/>
<polygon style="fill:black;stroke:black;" points="420.546,-544.596 411,-540 415.596,-549.546 420.546,-544.596"/>
</g>
<g id="node9" class="node"><title>UB</title>
<a xlink:href="http://ncatlab.org/nlab/show/ultrabornological topological vector space" xlink:title="Ultrabornological">
<polygon style="fill:#ffffbb;stroke:black;" points="448,-468 394,-468 394,-432 448,-432 448,-468"/>
<text fill="#008B26" text-anchor="middle" x="421" y="-445.9" style="font-family:Times New Roman;font-size:14.00;">UB</text>
</a>
</g>
<g id="edge26" class="edge"><title>Ba->UB</title>
<path style="fill:none;stroke:black;" d="M274,-511C304,-499 351,-479 385,-465"/>
<polygon style="fill:black;stroke:black;" points="386.283,-468.26 394,-461 383.44,-461.863 386.283,-468.26"/>
</g>
<g id="node10" class="node"><title>Fr</title>
<a xlink:href="http://ncatlab.org/nlab/show/Fréchet space" xlink:title="Fréchet">
<polygon style="fill:#ffffbb;stroke:black;" points="238,-468 184,-468 184,-432 238,-432 238,-468"/>
<text fill="#008B26" text-anchor="middle" x="211" y="-445.9" style="font-family:Times New Roman;font-size:14.00;">Fr</text>
</a>
</g>
<g id="edge28" class="edge"><title>Ba->Fr</title>
<path style="fill:none;stroke:black;" d="M238,-504C234,-496 229,-486 224,-477"/>
<polygon style="fill:black;stroke:black;" points="227.26,-475.717 220,-468 220.863,-478.56 227.26,-475.717"/>
</g>
<g id="node11" class="node"><title>DF</title>
<a xlink:href="http://ncatlab.org/nlab/show/DF topological vector space" xlink:title="DF">
<polygon style="fill:#ffffbb;stroke:black;" points="310,-468 256,-468 256,-432 310,-432 310,-468"/>
<text fill="#008B26" text-anchor="middle" x="283" y="-445.9" style="font-family:Times New Roman;font-size:14.00;">DF</text>
</a>
</g>
<g id="edge30" class="edge"><title>Ba->DF</title>
<path style="fill:none;stroke:black;" d="M256,-504C260,-496 265,-486 270,-477"/>
<polygon style="fill:black;stroke:black;" points="273.137,-478.56 274,-468 266.74,-475.717 273.137,-478.56"/>
</g>
<g id="node12" class="node"><title>No</title>
<a xlink:href="http://ncatlab.org/nlab/show/normed vector space" xlink:title="Normable space">
<polygon style="fill:#ffffbb;stroke:black;" points="94,-468 40,-468 40,-432 94,-432 94,-468"/>
<text fill="#008B26" text-anchor="middle" x="67" y="-445.9" style="font-family:Times New Roman;font-size:14.00;">No</text>
</a>
</g>
<g id="edge32" class="edge"><title>Ba->No</title>
<path style="fill:none;stroke:black;" d="M220,-513C193,-503 149,-487 103,-468"/>
<polygon style="fill:black;stroke:black;" points="104.56,-464.863 94,-464 101.717,-471.26 104.56,-464.863"/>
</g>
<g id="node22" class="node"><title>LB</title>
<a xlink:href="http://ncatlab.org/nlab/show/LB space" xlink:title="strict inductive sequence of Banach spaces">
<polygon style="fill:#ffffbb;stroke:black;" points="166,-468 112,-468 112,-432 166,-432 166,-468"/>
<text fill="#008B26" text-anchor="middle" x="139" y="-445.9" style="font-family:Times New Roman;font-size:14.00;">LB</text>
</a>
</g>
<g id="edge104" class="edge"><title>Ba->LB</title>
<path style="fill:none;stroke:black;" d="M220,-504C206,-495 189,-484 175,-474"/>
<polygon style="fill:black;stroke:black;" points="176.262,-470.635 166,-468 172.379,-476.459 176.262,-470.635"/>
</g>
<g id="edge34" class="edge"><title>IP->No</title>
<path style="fill:none;stroke:black;" d="M94,-504C90,-496 85,-486 80,-477"/>
<polygon style="fill:black;stroke:black;" points="83.2598,-475.717 76,-468 76.8631,-478.56 83.2598,-475.717"/>
</g>
<g id="edge20" class="edge"><title>Mo->Re</title>
<path style="fill:none;stroke:black;" d="M493,-439C468,-430 431,-414 390,-396"/>
<polygon style="fill:black;stroke:black;" points="391.56,-392.863 381,-392 388.717,-399.26 391.56,-392.863"/>
</g>
<g id="node29" class="node"><title>Pc</title>
<a xlink:href="http://ncatlab.org/nlab/show/paracompact space" xlink:title="Paracompact">
<polygon style="fill:#bbffff;stroke:black;" points="595,-324 541,-324 541,-288 595,-288 595,-324"/>
<text fill="#008B26" text-anchor="middle" x="568" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">Pc</text>
</a>
</g>
<g id="edge22" class="edge"><title>Mo->Pc</title>
<path style="fill:none;stroke:black;" d="M547,-440C567,-432 592,-417 604,-396 615,-375 603,-351 590,-332"/>
<polygon style="fill:black;stroke:black;" points="592.8,-329.9 584,-324 587.2,-334.1 592.8,-329.9"/>
</g>
<g id="node16" class="node"><title>Bo</title>
<a xlink:href="http://ncatlab.org/nlab/show/bornological topological vector space" xlink:title="Bornological">
<polygon style="fill:#ffffbb;stroke:black;" points="480,-324 426,-324 426,-288 480,-288 480,-324"/>
<text fill="#008B26" text-anchor="middle" x="453" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">Bo</text>
</a>
</g>
<g id="edge40" class="edge"><title>UB->Bo</title>
<path style="fill:none;stroke:black;" d="M448,-438C465,-429 484,-415 494,-396 504,-374 492,-350 477,-332"/>
<polygon style="fill:black;stroke:black;" points="479.8,-329.9 471,-324 474.2,-334.1 479.8,-329.9"/>
</g>
<g id="node14" class="node"><title>Cn</title>
<a xlink:href="http://ncatlab.org/nlab/show/convenient space" xlink:title="Convenient">
<polygon style="fill:#ffffbb;stroke:black;" points="485,-396 431,-396 431,-360 485,-360 485,-396"/>
<text fill="#008B26" text-anchor="middle" x="458" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Cn</text>
</a>
</g>
<g id="edge42" class="edge"><title>Fr->Cn</title>
<path style="fill:none;stroke:black;" d="M238,-435C241,-434 244,-433 247,-432 319,-407 344,-420 421,-395"/>
<polygon style="fill:black;stroke:black;" points="422.427,-398.226 431,-392 420.416,-391.521 422.427,-398.226"/>
</g>
<g id="node21" class="node"><title>LF</title>
<a xlink:href="http://ncatlab.org/nlab/show/LF space" xlink:title="strict inductive sequence of Fréchet spaces">
<polygon style="fill:#ffffbb;stroke:black;" points="95,-396 41,-396 41,-360 95,-360 95,-396"/>
<text fill="#008B26" text-anchor="middle" x="68" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">LF</text>
</a>
</g>
<g id="edge102" class="edge"><title>Fr->LF</title>
<path style="fill:none;stroke:black;" d="M184,-436C161,-425 129,-408 104,-396"/>
<polygon style="fill:black;stroke:black;" points="105.56,-392.863 95,-392 102.717,-399.26 105.56,-392.863"/>
</g>
<g id="node24" class="node"><title>Pt</title>
<a xlink:href="http://ncatlab.org/nlab/show/Ptak space" xlink:title="Ptak Space">
<polygon style="fill:#ffffbb;stroke:black;" points="167,-396 113,-396 113,-360 167,-360 167,-396"/>
<text fill="#008B26" text-anchor="middle" x="140" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Pt</text>
</a>
</g>
<g id="edge100" class="edge"><title>Fr->Pt</title>
<path style="fill:none;stroke:black;" d="M193,-432C184,-423 174,-413 165,-404"/>
<polygon style="fill:black;stroke:black;" points="167.219,-401.221 158,-396 161.951,-405.831 167.219,-401.221"/>
</g>
<g id="node33" class="node"><title>Cp</title>
<a xlink:href="http://ncatlab.org/nlab/show/complete topological vector space" xlink:title="Complete">
<polygon style="fill:#bbffff;stroke:black;" points="189,-252 135,-252 135,-216 189,-216 189,-252"/>
<text fill="#008B26" text-anchor="middle" x="162" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">Cp</text>
</a>
</g>
<g id="edge44" class="edge"><title>Fr->Cp</title>
<path style="fill:none;stroke:black;" d="M208,-432C202,-402 189,-340 176,-288 175,-280 172,-270 170,-262"/>
<polygon style="fill:black;stroke:black;" points="173.226,-260.573 167,-252 166.521,-262.584 173.226,-260.573"/>
</g>
<g id="node34" class="node"><title>Br</title>
<a xlink:href="http://ncatlab.org/nlab/show/Baire topological vector space" xlink:title="Baire">
<polygon style="fill:#bbffff;stroke:black;" points="309,-396 255,-396 255,-360 309,-360 309,-396"/>
<text fill="#008B26" text-anchor="middle" x="282" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Br</text>
</a>
</g>
<g id="edge46" class="edge"><title>Fr->Br</title>
<path style="fill:none;stroke:black;" d="M229,-432C238,-423 248,-413 257,-404"/>
<polygon style="fill:black;stroke:black;" points="260.049,-405.831 264,-396 254.781,-401.221 260.049,-405.831"/>
</g>
<g id="node35" class="node"><title>Me</title>
<a xlink:href="http://ncatlab.org/nlab/show/metrisable topological vector space" xlink:title="Metrisable">
<polygon style="fill:#bbffff;stroke:black;" points="595,-396 541,-396 541,-360 595,-360 595,-396"/>
<text fill="#008B26" text-anchor="middle" x="568" y="-373.9" style="font-family:Times New Roman;font-size:14.00;">Me</text>
</a>
</g>
<g id="edge48" class="edge"><title>Fr->Me</title>
<path style="fill:none;stroke:black;" d="M238,-435C241,-434 244,-433 247,-432 352,-398 385,-417 494,-396 506,-394 519,-390 531,-387"/>
<polygon style="fill:black;stroke:black;" points="531.881,-390.393 541,-385 530.508,-383.529 531.881,-390.393"/>
</g>
<g id="node15" class="node"><title>SR</title>
<a xlink:href="http://ncatlab.org/nlab/show/semi-reflexive topological vector space" xlink:title="Semi-Reflexive">
<polygon style="fill:#ffffbb;stroke:black;" points="370,-252 316,-252 316,-216 370,-216 370,-252"/>
<text fill="#008B26" text-anchor="middle" x="343" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">SR</text>
</a>
</g>
<g id="edge38" class="edge"><title>Re->SR</title>
<path style="fill:none;stroke:black;" d="M353,-360C351,-335 348,-291 345,-262"/>
<polygon style="fill:black;stroke:black;" points="348.478,-261.602 344,-252 341.512,-262.299 348.478,-261.602"/>
</g>
<g id="node27" class="node"><title>Bl</title>
<a xlink:href="http://ncatlab.org/nlab/show/barrelled topological vector space" xlink:title="Barrelled">
<polygon style="fill:#ffbbff;stroke:black;" points="326,-324 272,-324 272,-288 326,-288 326,-324"/>
<text fill="#008B26" text-anchor="middle" x="299" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">Bl</text>
</a>
</g>
<g id="edge36" class="edge"><title>Re->Bl</title>
<path style="fill:none;stroke:black;" d="M340,-360C334,-352 326,-341 319,-332"/>
<polygon style="fill:black;stroke:black;" points="321.8,-329.9 313,-324 316.2,-334.1 321.8,-329.9"/>
</g>
<g id="edge50" class="edge"><title>Cn->Bo</title>
<path style="fill:none;stroke:black;" d="M457,-360C456,-352 456,-343 455,-334"/>
<polygon style="fill:black;stroke:black;" points="458.478,-333.602 454,-324 451.512,-334.299 458.478,-333.602"/>
</g>
<g id="node17" class="node"><title>LC</title>
<a xlink:href="http://ncatlab.org/nlab/show/locally complete topological vector space" xlink:title="Locally Complete">
<polygon style="fill:#ffffbb;stroke:black;" points="480,-36 426,-36 426,-1.06581e-14 480,-0 480,-36"/>
<text fill="#008B26" text-anchor="middle" x="453" y="-13.9" style="font-family:Times New Roman;font-size:14.00;">LC</text>
</a>
</g>
<g id="edge52" class="edge"><title>Cn->LC</title>
<path style="fill:none;stroke:black;" d="M470,-360C476,-350 484,-336 489,-324 503,-285 508,-274 508,-234 508,-234 508,-234 508,-162 508,-119 486,-73 470,-45"/>
<polygon style="fill:black;stroke:black;" points="472.916,-43.0418 465,-36 466.797,-46.4414 472.916,-43.0418"/>
</g>
<g id="node18" class="node"><title>QC</title>
<a xlink:href="http://ncatlab.org/nlab/show/quasi-complete topological vector space" xlink:title="Quasi-Complete">
<polygon style="fill:#ffffbb;stroke:black;" points="370,-180 316,-180 316,-144 370,-144 370,-180"/>
<text fill="#008B26" text-anchor="middle" x="343" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">QC</text>
</a>
</g>
<g id="edge62" class="edge"><title>SR->QC</title>
<path style="fill:none;stroke:black;" d="M343,-216C343,-208 343,-199 343,-190"/>
<polygon style="fill:black;stroke:black;" points="346.5,-190 343,-180 339.5,-190 346.5,-190"/>
</g>
<g id="node19" class="node"><title>QB</title>
<a xlink:href="http://ncatlab.org/nlab/show/quasi-barrelled topological vector space" xlink:title="Quasi-Barrelled">
<polygon style="fill:#ffffbb;stroke:black;" points="480,-252 426,-252 426,-216 480,-216 480,-252"/>
<text fill="#008B26" text-anchor="middle" x="453" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">QB</text>
</a>
</g>
<g id="edge64" class="edge"><title>Bo->QB</title>
<path style="fill:none;stroke:black;" d="M453,-288C453,-280 453,-271 453,-262"/>
<polygon style="fill:black;stroke:black;" points="456.5,-262 453,-252 449.5,-262 456.5,-262"/>
</g>
<g id="node31" class="node"><title>Sq</title>
<a xlink:href="http://ncatlab.org/nlab/show/sequentially complete topological vector space" xlink:title="Sequentially Complete">
<polygon style="fill:#bbffff;stroke:black;" points="425,-108 371,-108 371,-72 425,-72 425,-108"/>
<text fill="#008B26" text-anchor="middle" x="398" y="-85.9" style="font-family:Times New Roman;font-size:14.00;">Sq</text>
</a>
</g>
<g id="edge68" class="edge"><title>QC->Sq</title>
<path style="fill:none;stroke:black;" d="M357,-144C363,-136 371,-125 378,-116"/>
<polygon style="fill:black;stroke:black;" points="380.8,-118.1 384,-108 375.2,-113.9 380.8,-118.1"/>
</g>
<g id="node20" class="node"><title>Mk</title>
<a xlink:href="http://ncatlab.org/nlab/show/Mackey topological vector space" xlink:title="Mackey">
<polygon style="fill:#ffffbb;stroke:black;" points="480,-180 426,-180 426,-144 480,-144 480,-180"/>
<text fill="#008B26" text-anchor="middle" x="453" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">Mk</text>
</a>
</g>
<g id="edge72" class="edge"><title>QB->Mk</title>
<path style="fill:none;stroke:black;" d="M453,-216C453,-208 453,-199 453,-190"/>
<polygon style="fill:black;stroke:black;" points="456.5,-190 453,-180 449.5,-190 456.5,-190"/>
</g>
<g id="edge80" class="edge"><title>LF->Cp</title>
<path style="fill:none;stroke:black;" d="M73,-360C78,-341 89,-310 104,-288 111,-277 121,-267 130,-259"/>
<polygon style="fill:black;stroke:black;" points="132.779,-261.219 138,-252 128.169,-255.951 132.779,-261.219"/>
</g>
<g id="edge78" class="edge"><title>LB->LF</title>
<path style="fill:none;stroke:black;" d="M121,-432C112,-423 102,-413 93,-404"/>
<polygon style="fill:black;stroke:black;" points="95.2191,-401.221 86,-396 89.9511,-405.831 95.2191,-401.221"/>
</g>
<g id="node25" class="node"><title>BC</title>
<a xlink:href="http://ncatlab.org/nlab/show/Ptak space" xlink:title="Br Space">
<polygon style="fill:#ffffbb;stroke:black;" points="167,-324 113,-324 113,-288 167,-288 167,-324"/>
<text fill="#008B26" text-anchor="middle" x="140" y="-301.9" style="font-family:Times New Roman;font-size:14.00;">BC</text>
</a>
</g>
<g id="edge96" class="edge"><title>Pt->BC</title>
<path style="fill:none;stroke:black;" d="M140,-360C140,-352 140,-343 140,-334"/>
<polygon style="fill:black;stroke:black;" points="143.5,-334 140,-324 136.5,-334 143.5,-334"/>
</g>
<g id="edge98" class="edge"><title>BC->Cp</title>
<path style="fill:none;stroke:black;" d="M146,-288C148,-280 151,-270 154,-262"/>
<polygon style="fill:black;stroke:black;" points="157.479,-262.584 157,-252 150.774,-260.573 157.479,-262.584"/>
</g>
<g id="node28" class="node"><title>TS</title>
<a xlink:href="http://ncatlab.org/nlab/show/topological space" xlink:title="Topological Space">
<polygon style="fill:#bbffff;stroke:black;" points="58,-540 4,-540 4,-504 58,-504 58,-540"/>
<text fill="#008B26" text-anchor="middle" x="31" y="-517.9" style="font-family:Times New Roman;font-size:14.00;">TS</text>
</a>
</g>
<g id="edge107" class="edge"><title>TVS->TS</title>
<path style="fill:none;stroke:black;" d="M31,-576C31,-568 31,-559 31,-550"/>
<polygon style="fill:black;stroke:black;" points="34.5001,-550 31,-540 27.5001,-550 34.5001,-550"/>
</g>
<g id="edge66" class="edge"><title>Bl->QB</title>
<path style="fill:none;stroke:black;" d="M326,-293C351,-281 389,-264 417,-251"/>
<polygon style="fill:black;stroke:black;" points="418.283,-254.26 426,-247 415.44,-247.863 418.283,-254.26"/>
</g>
<g id="node32" class="node"><title>Nm</title>
<a xlink:href="http://ncatlab.org/nlab/show/normal topological space" xlink:title="Normal">
<polygon style="fill:#bbffff;stroke:black;" points="595,-252 541,-252 541,-216 595,-216 595,-252"/>
<text fill="#008B26" text-anchor="middle" x="568" y="-229.9" style="font-family:Times New Roman;font-size:14.00;">Nm</text>
</a>
</g>
<g id="edge70" class="edge"><title>Pc->Nm</title>
<path style="fill:none;stroke:black;" d="M568,-288C568,-280 568,-271 568,-262"/>
<polygon style="fill:black;stroke:black;" points="571.5,-262 568,-252 564.5,-262 571.5,-262"/>
</g>
<g id="node30" class="node"><title>CP</title>
<a xlink:href="http://ncatlab.org/nlab/show/countably paracompact space" xlink:title="Countably Paracompact">
<polygon style="fill:#bbffff;stroke:black;" points="595,-180 541,-180 541,-144 595,-144 595,-180"/>
<text fill="#008B26" text-anchor="middle" x="568" y="-157.9" style="font-family:Times New Roman;font-size:14.00;">CP</text>
</a>
</g>
<g id="edge76" class="edge"><title>Sq->LC</title>
<path style="fill:none;stroke:black;" d="M412,-72C418,-64 426,-53 433,-44"/>
<polygon style="fill:black;stroke:black;" points="435.8,-46.1 439,-36 430.2,-41.9 435.8,-46.1"/>
</g>
<g id="edge74" class="edge"><title>Nm->CP</title>
<path style="fill:none;stroke:black;" d="M568,-216C568,-208 568,-199 568,-190"/>
<polygon style="fill:black;stroke:black;" points="571.5,-190 568,-180 564.5,-190 571.5,-190"/>
</g>
<g id="edge54" class="edge"><title>Cp->QC</title>
<path style="fill:none;stroke:black;" d="M189,-223C220,-210 272,-190 306,-177"/>
<polygon style="fill:black;stroke:black;" points="308.015,-179.964 316,-173 305.415,-173.464 308.015,-179.964"/>
</g>
<g id="edge56" class="edge"><title>Br->Bl</title>
<path style="fill:none;stroke:black;" d="M286,-360C288,-352 290,-343 292,-334"/>
<polygon style="fill:black;stroke:black;" points="295.479,-334.584 295,-324 288.774,-332.573 295.479,-334.584"/>
</g>
<g id="edge58" class="edge"><title>Me->Bo</title>
<path style="fill:none;stroke:black;" d="M541,-361C526,-352 506,-339 489,-329"/>
<polygon style="fill:black;stroke:black;" points="490.262,-325.635 480,-323 486.379,-331.459 490.262,-325.635"/>
</g>
<g id="edge60" class="edge"><title>Me->Pc</title>
<path style="fill:none;stroke:black;" d="M568,-360C568,-352 568,-343 568,-334"/>
<polygon style="fill:black;stroke:black;" points="571.5,-334 568,-324 564.5,-334 571.5,-334"/>
</g>
<g id="node36" class="node"><title>Se</title>
<a xlink:href="http://ncatlab.org/nlab/show/separable space" xlink:title="Separable">
<polygon style="fill:#bbffff;stroke:black;" points="602,-540 548,-540 548,-504 602,-504 602,-540"/>
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<g id="node38" class="node"><title>QCBo</title>
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lattice ordered group

##Idea

A lattice ordered group is a group which is also a lattice in a compatible way.

##Partially ordered groups.

 First we introduce _partially ordered groups_.

A group $G$ is said to be _partially ordered_ if it is equipped with a partial order, $\le$, which is compatible with the group multiplication, $\cdot$, so, if $g\leq h$ then $g\cdot k\leq h\cdot k$ and also $k\cdot g\leq k\cdot h$ for any $g,h,k \in G$.

##Positive elements

The set, $G^+=\{g\in G\mid g\geq 1\}$ is called the _positive cone_ of $(G,\leq)$. It completely determines the partial order since $g\leq h$ if and only if $hg^{-1}\in G^*\cup \{0\}$.

##Lattice ordered groups.

If $G,\leq)$ is a partially ordered group, and the partial order is a lattice, then we say $G$ is a _lattice ordered group_, so for each pair , $a,b$ in $G$ has a [[join]], $a \vee b$ and a [[meet]], $a\wedge b$.

Many sources abbreviate 'lattice ordered group' to '$\ell$-group'.

##Properties

* The group operation distibutes over both $\vee$ and $\wedge$.

* Inversion reverses the order so it $g\le h$, then $h^{-1}\leq g^{-1}$. (It is worth noting that this property is the opposite of that encountered with [[ordered groupoid]]s.)

* A sort of de Morgan's law holds, so $(a\vee b)^{-1}= a^{-1}\wedge b^{-1}$. This is easily proved once one notes that inversion reverses the order.

* The compatibility of the group multiplication with the lattice structure interpretas as saying that left and right multiplication by elements of the group give automorphisms of the lattice structure.

## Lattice ordered groups and residuated lattices

We note that given any $x, y \in G$, we can form $x / y = x\cdot y^{-1}$, and $y\backslash x = y^{-1}\cdot x$. In this case, if $x\cdot y \leq z$, then, of course, $y\leq x\backslash z$ and $x\leq z / y$.  We then also have $x^{-1}= 1 / x$ - which is neat!

This gives that any lattice ordered group gives a [[residuated lattice]].

The lattice ordered groups form a variet ub the category of residuated lattices that is axiomatised by the identity $x\cdot (1/x) = x$.

##References
The original article that considered ordered groups in the non-commutative case is by Garrett Birkhoff:

* [[Garrett Birkhoff]]. _Lattice-ordered groups,_ Annals of Math. 43, (1942) pp. 298 - 331.

He later considered Lattice-ordered Lie groups.

There are several monographs or books on the subject of lattice ordered groups.

* Marlow Anderson and Todd Feil, _Lattice Ordered Groups: An Introduction,_ Reidel Texts in the Mathematical Sciences, vol 4, (1988), (doi:10.1007/978-94-009-2871-8).

* V. M. Kopytov, N. Ya. Medvedev, _The Theory of Lattice-Ordered Groups_,  Mathematics and Its Applications, volume 307, 1994), Springer, [doi](https://doi.org/10.1007/978-94-015-8304-6).


lattice renormalization

#Contents#
* table of contents
{:toc}

## Idea

[[renormalization]] in [[lattice gauge field theory]].

## References

*  Barry M. Mc Coy, Tai Tsun Wu, _Lattice Renormalization of Non-Perturbative Quantum Field Theory_, Bifurcation Phenomena in Mathematical Physics and Related Topics NATO Advanced Study Institutes Series Volume 54, 1980, pp 69-90

* G.C. Rossi, _Non-perturbative renormalization in lattice QCD_, Nucl.Phys.Proc.Suppl. 53 (1997) 3-15 ([arXiv:hep-lat/9609038](http://xxx.lanl.gov/abs/hep-lat/9609038))

* Yasumichi Aoki, _Non-perturbative renormalization in lattice QCD_ ([arXiv:1005.2339](http://arxiv.org/abs/1005.2339))

[[!redirects lattice renormalizations]]


Laughlin wavefunction

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

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum physics]] a  *Laughlin wavefunction* is a certain Ansatz for an [[quantum many-body physics|n-particle]] [[wavefunction]] which is meant to capture at least aspects of [[ground states]] with [[anyon|anyonic]] properties, such as for the levels in the [[quantum Hall effect]].

(...)

## Related concepts

* [[quantum Hall effect]]

* [[conformal block]]

## References

### General

The original article:

* [[Robert B. Laughlin]], *Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations*, Phys. Rev. Lett. **50** (1983) 1395 $[$[doi:10.1103/PhysRevLett.50.1395](https://doi.org/10.1103/PhysRevLett.50.1395)$]$

Review:

* [[Steven M. Girvin]], Section 2.1 of: *Introduction to the Fractional Quantum Hall Effect*, Séminaire Poincaré **2** (2004) 53–74, reprinted in *The Quantum Hall Effect*, Progress in Mathematical Physics **45**, Birkhäuser (2005)  $[$[pdf](http://www.bourbaphy.fr/girvin.pdf), [doi:10.1007/3-7643-7393-8_4](https://doi.org/10.1007/3-7643-7393-8_4)$]$

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], Section III.D.2.c (pp. 1125) of: *Non-Abelian Anyons and Topological Quantum Computation*, Rev. Mod. Phys. **80** 1083 (2008) $[$[arXiv:0707.1888] (http://arxiv.org/abs/0707.1889)$]$

* {#Tong16} [[David Tong]], Section 3.1 of: *The Quantum Hall Effect* (2016) $[$[course webpage](https://www.damtp.cam.ac.uk/user/tong/qhe.html), [pdf](http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf), [[Tong-QuantumHallEffect.pdf:file]]$]$

* Roman Remme, *The Laughlin Wavefunction*, talk notes (2017) $[$[pdf](https://www.mathi.uni-heidelberg.de/~walcher/teaching/sose17/atqft/writeups/Roman.pdf), [[Remme-LaughlinWavefunction.pdf:file]]$]$

See also:

* Wikipedia, *[Laughlin wavefunciton](https://en.wikipedia.org/wiki/Laughlin_wavefunction)*

Characterization as a [[braid representation]]:

* [[Janusz E. Jacak]], *Formal derivation of the Laughlin function and its generalization for other topological phases of FQHE*, Sci Rep **12** 616 (2022) [[doi:10.1038/s41598-021-04672-z](https://doi.org/10.1038/s41598-021-04672-z)]

A "hierarchy" of Laughlin-like states:

* [[Bertrand I. Halperin]], *Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States*, Phys. Rev. Lett. **52** (1984) 1583 $[$[doi:10.1103/PhysRevLett.52.1583](https://doi.org/10.1103/PhysRevLett.52.1583)$]$

* {#Lan19} [[Tian Lan]], *Matrix formulation for non-Abelian families*, Phys. Rev. B **100** 241102 (2019) $[$[arXiv:1908.02599](https://arxiv.org/abs/1908.02599), [doi:10.1103/PhysRevB.100.241102](https://doi.org/10.1103/PhysRevB.100.241102)$]$

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

[[!redirects Laughlin wavefunctions]]

[[!redirects Laughlin state]]
[[!redirects Laughlin states]]


Laughlin wavefunctions as conformal blocks -- references

### Laughlin wavefunctions as conformal blocks
 {#LaughlinWavefunctionAsConformalBlocksReferences}

Relating [[anyon|anyonic]] [[topological order|topologically ordered]] [[Laughlin wavefunctions]] to [[conformal blocks]]:

* {#MooreRead91} [[Gregory Moore]], [[Nicholas Read]], Section 2.2 of: *Nonabelions in the fractional quantum hall effect*, Nuclear Physics B **360** 2–3 (1991) 362-396 $[$doi:10.1016/0550-3213(91)90407-O, [pdf](https://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf)$]$

* {#Wen91} [[Xiao-Gang Wen]], *Non-Abelian statistics in the fractional quantum Hall states*, Phys. Rev. Lett. **66** (1991) 802 $[$[doi:10.1103/PhysRevLett.66.802](https://doi.org/10.1103/PhysRevLett.66.802), [pdf](https://xgwen.mit.edu/sites/default/files/documents/nab.pdf)$]$

* B. Blok, [[Xiao-Gang Wen]], *Many-body systems with non-abelian statistics*, Nuclear Physics B **374** 3 (1992) 615-646 $[$doi:10.1016/0550-3213(92)90402-W$]$

* [[Xiao-Gang Wen]], Yong-Shi Wu, *Chiral operator product algebra hidden in certain fractional quantum Hall wave functions*, Nucl. Phys. B **419** (1994) 455-479 $[$doi:10.1016/0550-3213(94)90340-9$]$

Review in the broader context of the [[CS-WZW correspondence]]:

* [[Daniel C. Cabra]], [[Gerardo L. Rossini]], *Explicit connection between conformal field theory and 2+1 Chern-Simons theory*, Mod. Phys. Lett. A **12** (1997) 1687-1697 $[$[arXiv:hep-th/9506054](https://arxiv.org/abs/hep-th/9506054), [doi:10.1142/S0217732397001722](https://doi.org/10.1142/S0217732397001722)$]$

Specifically for [[logarithmic CFT]]:

* {#GurarieFlohrNayak97} [[Victor Gurarie]], [[Michael Flohr]], [[Chetan Nayak]], *The Haldane-Rezayi Quantum Hall State and Conformal Field Theory*, Nucl. Phys. B **498** (1997) 513-538 $[$[arXiv:cond-mat/9701212](https://arxiv.org/abs/cond-mat/9701212), [doi:10.1016/S0550-3213%2897%2900351-9](https://doi.org/10.1016/S0550-3213%2897%2900351-9)$]$

* {#Flohr03} [[Michael Flohr]], §5.4 in: *Bits and pieces in logarithmic conformal field theory*, International Journal of Modern Physics A, **18** 25 (2003) 4497-4591 $[$[doi:10.1142/S0217751X03016859](https://doi.org/10.1142/S0217751X03016859), [arXiv:hep-th/0111228](https://arxiv.org/abs/hep-th/0111228)$]$

Specifically for [[su(2)-anyons]]:

* Kazusumi Ino, *Modular Invariants in the Fractional Quantum Hall Effect*, Nucl. Phys. B **532** (1998) 783-806 $[$doi:10.1016/S0550-3213(98)00598-7, [arXiv:cond-mat/9804198](https://arxiv.org/abs/cond-mat/9804198)$]$

* {#ReadRezayi99} [[Nicholas Read]], [[Edward Rezayi]], *Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level*, Phys. Rev. B **59** (1999) 8084 $[$[doi:10.1103/PhysRevB.59.8084](https://doi.org/10.1103/PhysRevB.59.8084)$]$


Laura Andrianopoli

* [webpage](https://didattica.polito.it/portal/pls/portal/sviluppo.scheda_pers_swas.show?m=003711)

## Selected writings

Discussion of [[multi-trace operators]] in [[super Yang-Mills theory]] and of their [[AdS-CFT duality|AdS-CFT dual]] [[gravity]]/[[string theory]] incarnation:

* [[Laura Andrianopoli]], [[Sergio Ferrara]], _On short and long $SU(2,2/4)$ multiplets in the AdS/CFT correspondence_, Lett. Math.Phys. 48 (1999) 145-161 ([arXiv:hep-th/9812067](https://arxiv.org/abs/hep-th/9812067))

On the [[supergravity Lie 3-algebra]] (see also at *[[M-theory supersymmetry algebra]]*):

* {#AndrianopoliDAuriaRavera16} [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _Hidden Gauge Structure of Supersymmetric Free Differential Algebras_, JHEP 1608 (2016) 095 ([arXiv:1606.07328](https://arxiv.org/abs/1606.07328))

* [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _More on the Hidden Symmetries of 11D Supergravity_ ([arXiv:1705.06251](https://arxiv.org/abs/1705.06251))

Usage of full [[supergravity]] (retaining the [[gravitino]]) for [[AdS-CMT duality]], with application to [[graphene]]-like systems:

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], [[Mario Trigiante]], *Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity*,  J. High Energ. Phys. **2018** 7 (2018) [[arXiv:1801.08081](https://arxiv.org/abs/1801.08081), doi:10.1007/JHEP04(2018)007]

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], A. Gallerati, R. Noris, [[Mario Trigiante]], [[Jorge Zanelli]], *$\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene*, JHEP 01 (2020) 084 [[arXiv:1910.03508](https://arxiv.org/abs/1910.03508), doi:10.1007/JHEP01(2020)084]

On the [[self-dual higher gauge field]] on the [[M5-brane]] lifted to [[superspace]] in the [[D'Auria-Fré formulation of supergravity]]:

* {#ACDGMNRT22} [[Laura Andrianopoli]], [[C. A. Cremonini]], [[Riccardo D'Auria]], [[Pietro A. Grassi]], R. Matrecano, R. Noris, [[Lucrezia Ravera]], [[Mario Trigiante]], *M5-brane in the superspace approach*, Phys. Rev. D **106** 2 (2022) 026010 [[arXiv:2206.06388](https://arxiv.org/abs/2206.06388), [doi:10.1103/PhysRevD.106.026010](https://doi.org/10.1103/PhysRevD.106.026010)]

## Related entries

* [[D'Auria-Fré formulation of supergravity]]

* [[M-theory super Lie algebra]]

* [[7d supergravity]]

* [[4d supergravity]]

category: people


Laura Donnay

* [personal page](https://lauradonnay.wixsite.com/webpage/about-me)

* [GoogleScholar page](https://scholar.google.com/citations?user=cFzJ3vEAAAAJ&hl=en)

## Selected writings

On [[3d gravity]]:

* [[Laura Donnay]], _Asymptotic dynamics of three-dimensional gravity_, Proceedings of *Eleventh Modave Summer School in Mathematical Physics*, POS **271** (2016)  $[$[arXiv:1602.09021](http://arxiv.org/abs/1602.09021), [doi:10.22323/1.271.0001 ](https://doi.org/10.22323/1.271.0001 )$]$

On [[celestial holography]]:

* [[Laura Donnay]], *Celestial holography: An asymptotic symmetry perspective* [[arXiv:2310.12922](https://arxiv.org/abs/2310.12922)]

category: people


Laura Fontanella

* [webpage](https://lacl.fr/~lfontanella/)

## Selected publications

* [[Laura Fontanella]], *How to Choose New Axioms for Set Theory?*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_2](https://doi.org/10.1007/978-3-030-15655-8_2), [pdf](https://lacl.fr/~lfontanella/papers/paper%20philo%202016.pdf)]

category: people


Laura Fredrickson

* [webpage](https://web.stanford.edu/~ljfred4/)

## Selected writings

On the [[Yang-Mills equations]] in [[Yang-Mills theory]] ([[gauge theory]]):

* {#Uhlenbeck12} [[Karen Uhlenbeck]], notes by [[Laura Fredrickson]], _Equations of Gauge Theory_, lecture at Temple University, 2012 ([pdf](https://web.stanford.edu/~ljfred4/Attachments/TempleLectures.pdf), [[UhlenbeckGaugeTheory.pdf:file]])

category: people


Laura Scull

* [webpage](http://faculty.fortlewis.edu/scull_l/)

## Selected writings

On [[equivariant rational homotopy theory]]:

* {#Scull01} [[Laura Scull]], _Rational $S^1$-equivariant homotopy theory_, Transactions of the AMS, Volume 354, Number 1, Pages 1-45 2001 ([pdf](http://www.ams.org/journals/tran/2002-354-01/S0002-9947-01-02790-8/S0002-9947-01-02790-8.pdf), [doi:10.1090/S0002-9947-01-02790-8](https://doi.org/10.1090/S0002-9947-01-02790-8))

Introducing the [[model structure on equivariant dgc-algebras]], generalizing the [[projective model structure on dgc-algebras]] to [[equivariant rational homotopy theory]], such that equivariant [[minimal Sullivan models]] are [[cofibrant objects]], and proving the [[fundamental theorem of dg-algebraic equivariant rational homotopy theory]]:

* [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

On certain [[pushouts]] ([[amalgamations]]) in the [[1-category]] [[Cat]] of [[small categories]]:

* {#MacdonalScull09} [[John MacDonald]], [[Laura Scull]], _Amalgamations of categories_, Canad. Math. Bull. **52** 2 (2009) 273–284 [[pdf](http://faculty.fortlewis.edu/Scull_L/pushouts.pdf), journal:[pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A414F5D3FA9FB08ACBBB72143C70FE28)]

On [[orbifold cohomology]] (such as [[orbifold K-theory]]):

* {#PronkScull07} [[Dorette Pronk]], [[Laura Scull]], _Translation Groupoids and Orbifold Bredon Cohomology_, Canad. J. Math. 62 (2010), 614-645 ([arXiv:0705.3249](https://arxiv.org/abs/0705.3249), [doi:10.4153/CJM-2010-024-1](https://doi.org/10.4153/CJM-2010-024-1))

## Related $n$Lab entries

* [[equivariant rational homotopy theory]]

category: people


Laurence Taylor

* [webpage](https://math.nd.edu/people/faculty/laurence-taylor/)

## related $n$Lab entries

* [[cohomotopy]]

category: people


Laurens Molenkamp

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

* [institute page](https://www.physik.uni-wuerzburg.de/ep3/staff/professors/prof-dr-laurens-w-molenkamp/)

* [GoogleScholar page](https://scholar.google.com/citations?user=ptmy2MsAAAAJ&hl=en)

## Selected writings

On [[topological insulators]]:

*  [[Marcel Franz]], [[Laurens Molenkamp]] (eds.), *Topological Insulators*, Contemporary Concepts of Condensed Matter Science **6** (2013) $[$[ISBN:978-0-444-63314-9](https://www.sciencedirect.com/bookseries/contemporary-concepts-of-condensed-matter-science/vol/6/suppl/C)$]$

category: people


Laurent Bartholdi

* [webpage](http://www.uni-math.gwdg.de/laurent/)

category: people


Laurent Fargues

* [website](https://webusers.imj-prg.fr/~laurent.fargues/)

## Selected writings

On the [[local Langlands conjecture]]:

* [[Laurent Fargues]], [[Peter Scholze]], _Geometrization of the local Langlands correspondence_ ([pdf](http://www.math.uni-bonn.de/people/scholze/Geometrization.pdf))

## Related entries

* [[local Langlands correspondence]]

* [[Fargues-Fontaine curve]]

category: people


Laurent Freidel

__Laurent Freidel__ is a French theoretical physicist. Main works are related to [[quantum gravity]] and mathematical formalisms related to attempts to study quantum gravity.

* [wikipedia](https://en.wikipedia.org/wiki/Laurent_Freidel)
* [gScholar](https://scholar.google.com/citations?user=ADpHAh0AAAAJ)

## Selected publications

On ([[universal Vassiliev invariant|universal]]) [[Vassiliev invariants]] and relation to [perturbative](quantization+of+3d+Chern-Simons+theory#ReferencesPerturbativeCSTheory) [[Chern-Simons theory]]

* [[Daniel Altschuler]], [[Laurent Freidel]], _On Universal Vassiliev Invariants_, Commun. Math. Phys. **170** (1995) 41-62 [[arXiv:hep-th/9403053](https://arxiv.org/abs/hep-th/9403053), [doi:10.1007/BF02099438](https://doi.org/10.1007/BF02099438)]

* {#AF96} [[Daniel Altschuler]], [[Laurent Freidel]], _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. **187** (1997) 261-287 [[arXiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010), [doi:10.1007/s002200050136](https://doi.org/10.1007/s002200050136)]

See also:

* [[Laurent Freidel]], [[Shahn Majid]], _Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity_, Classical Quantum Gravity __25__ (2008), no. 4, 045006 [MR2009f:83058](https://mathscinet.ams.org/mathscinet-getitem?mr=2388191), [doi](https://doi.org/10.1088/0264-9381/25/4/045006)
* [[Laurent Freidel]], _Group Field Theory: An overview_, Int.J.Theor.Phys.44:1769-1783 (2005) arXiv:[hep-th/0505016](http://arxiv.org/abs/hep-th/0505016), [doi](http://dx.doi.org/10.1007/s10773-005-8894-1)

* Laurent Freidel, Etera R. Livine, _The fine structure of $SU(2)$ intertwiners from $U(N)$ representations_, J. Math. Phys. __51__ (2010), no. 8, 082502, [MR2011j:81140](http://mathscinet.ams.org/mathscinet-getitem?mr=2683530), [doi](https://doi.org/10.1063/1.3473786)
* [[Laurent Freidel]], Marc Geiller, Wolfgang Wieland,  _Corner symmetry and quantum geometry_, in [[Handbook of Quantum Gravity]], Springer (2023) [arXiv:2302.12799](https://arxiv.org/abs/2302.12799)
* [[John Baez|John C. Baez]], Aristide Baratin, [[Laurent Freidel]], Derek K. Wise, _Infinite-dimensional representations of 2-groups_, [arXiv](https://arxiv.org/abs/0812.4969)
* Laurent Freidel, Robert G. Leigh, Djordje Minic, _Quantum spaces are modular_, Phys. Rev. D __94__, 104052 (2016) [arXiv:1606.01829](https://arxiv.org/abs/1606.01829) [doi](https://doi.org/10.1103/PhysRevD.94.104052)
* Laurent Freidel, Artem Starodubtsev, _Quantum gravity in terms of topological observables_, arXiv:[hep-th/0501191](https://arxiv.org/abs/hep-th/0501191)

category: people


Laurent Houart

* [inspire page](https://inspirehep.net/authors/1005509)

## Selected writings

On [[M-branes]]:

* [[Laurent Houart]], [[Yolanda Lozano]], _Brane Descent Relations in M-theory_, Phys.Lett. B **479** (2000) 299-307 [[arXiv:hep-th/0001170](https://arxiv.org/abs/hep-th/0001170), doi:10.1016/S0370-2693(00)00317-8]

On [[E9]] [[BPS-states]] in [[D=11 supergravity]]:

* {#EHKNT07} [[François Englert]], [[Laurent Houart]], [[Axel Kleinschmidt]], [[Hermann Nicolai]], Nassiba Tabti, _An E9 multiplet of BPS states_, JHEP 0705:065 (2007) [[arXiv:hep-th/0703285](http://arxiv.org/abs/hep-th/0703285)]

category: people


Laurent Lafforgue

Laurent Lafforgue is a French mathematician. He has made important contributions to [[Langlands program]] in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism group of a function field.

He has emphasised the importance of investigating the technique 'toposes as bridges' in connection to the Langlands program. Explicitly mentioning the work of [[Olivia Caramello]].

In 2002, he received the Fields Medal together with [[Vladimir Voevodsky]].  He is a permanent professor at the [[math institutions|IHÉS]].

###Publications, preprints, Seminar notes etc.

* _La théorie de Caramello : un cadre en
construction pour des correspondances du type
de celle de Langlands ?_, Notes 2013, [IHES](http://www.ihes.fr/~lafforgue/math/TheorieCaramello.pdf)

* [Webpage on seminar](https://sites.google.com/site/logiquecategorique/Contenus/20130227_Lafforgue).

###References

* [Homepage](http://www.ihes.fr/~lafforgue/)

* [Wikipedia (English)](http://en.wikipedia.org/wiki/Laurent_Lafforgue)
category:people


Laurent Moret-Bailly

* [institute page](https://perso.univ-rennes1.fr/laurent.moret-bailly/)

## Selected writings

On [[algebraic stacks]]:

* {#LaumontMoret-Bailly} [[Gérard Laumon]], [[Laurent Moret-Bailly]], _Champs algébriques_, Ergebn. der Mathematik und ihrer Grenzgebiete **39**, Springer (2000) [[doi:10.1007/978-3-540-24899-6](https://doi.org/10.1007/978-3-540-24899-6)]

On [[principal bundles]] ([[pseudo torsors]]):

* {#MoretBailly13} [[Laurent Moret-Bailly]], slide 5 of: *Principal Bundles over Valued Fields*, Oberwolfach 2013 ([[MoretBaillyPrincipalBundlesValuedFields.pdf:file]])

category: people


Laurent phenomenon

While the composition of birational maps given by [[Laurent polynomial]]s is in general not a Laurent polynomial there is a number of systematic situations when this is true. In particular, in a [[cluster algebra]], any cluster variable is expressed in terms of any given cluster as a Laurent polynomial with coefficients in the integral group ring $\mathbf{Z}\mathbf{P}$ where $\mathbf{P}$ is the coefficient group. An explanation in the quantum case, from a Hodge-theoretic point of view, has recently being offered by A. Efimov.

* Sergey Fomin, [[Andrei Zelevinsky]], _The Laurent phenomenon_, [math.CO/0104241](http://arxiv.org/abs/math/0104241)
* Sergey Fomin, [[Andrei Zelevinsky]], _Cluster algebras I: Foundations_, [arxiv/0104151](http://arxiv.org/abs/math/0104151)
* [[Alexander I. Efimov]], _Quantum cluster variables via vanishing cycles_,  [arxiv/1112.3601](http://arxiv.org/abs/1112.3601)


Laurent polynomial

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given indeterminates $t_1,\ldots, t_k$ a _Laurent polynomial_ over a [[field]] or [[ring]] $F$ is a [[polynomial]] in $t_1,\ldots,t_k,t_1^{-1},\ldots,t_k^{-1}$ over $F$.

## Properties

### Ring of Laurent polynomials

Let $R$ be a [[commutative ring]], and let $S$ be a set. The ring of Laurent polynomials generated by $S$ is the [[localization of a commutative ring|localization]] of the [[polynomial ring]] $R[S]$, with canonical [[ring homomorphism]] $h:R \to R[S]$ and canonical function $i:S \to R[S]$, away from the image of $i$, $R[S, S^{-1}]$.

Now, assume that $S$ is a [[singleton]] set $\{x\}$. The underlying $R$-[[module]] of the ring of Laurent polynomials $R[x, x^{-1}]$ is the [[integers object]] in the category $R$[[Mod]] of $R$-modules and $R$-[[linear maps]], with $c:R \to R[x, x^{-1}]$ taking [[scalars]] to constant Laurent polynomials and the [[linear isomorphism]] $P \mapsto P x:R[x, x^{-1}] \to R[x, x^{-1}]$ multiplying Laurent polynomials by $x$, with [[inverse function|inverse]] [[isomorphism]] $P \mapsto P x^{-1}:R[x, x^{-1}] \to R[x, x^{-1}]$.

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[Laurent series]]

* [[integers object]]

## References

* Wikipedia, _[Laurent polynomial](http://en.wikipedia.org/wiki/Laurent_polynomial)_

[[!redirects Laurent polynomial]]
[[!redirects Laurent polynomials]]

[[!redirects ring of Laurent polynomial]]
[[!redirects ring of Laurent polynomials]]


Laurent Regnier

Laurent Regnier is a professor in the _département de mathématiques de l'UFR Sciences de l'Université d'Aix-Marseille_ and a researcher in the _[Groupe scientifique Arithmétique, Géométrie, Logique et Représentations](https://www.i2m.univ-amu.fr/la-recherche/groupes-scientifiques/groupe-arithmetique-geometrie-logique-et-representations-aglr/)_ of the _[Institut de Mathématiques de Marseille](http://www.i2m.univ-amu.fr/en/home/)_.

* [Webpage](https://www.i2m.univ-amu.fr/perso/laurent.regnier/)

## Selected writings

On [[differential linear logic]]:

* {#EhrhardtRegnier09} [[Thomas Ehrhard]], [[Laurent Regnier]]: _The differential lambda-calculus_, Theor. Comput. Sci. **309** 1 (2003) 1–41  (doi:10.1016/S0304-3975(03)00392-X)

* [[Thomas Ehrhard]], [[Laurent Regnier]]: _Differential interaction nets,_ Theor. Comput. Sci. **364** 2 (2006) 166–195 ([doi:10.1016/j.tcs.2006.08.003](https://doi.org/10.1016/j.tcs.2006.08.003))

## Related $n$Lab entries

* [[differential linear logic]]

* [[differential category]]

[[!redirects L.Regnier]]
[[!redirects L. Regnier]]

category:people


Laurent Schwartz

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

## Related entries

* [[distribution]], [[tempered distribution]], [[compactly supported distribution]]

* [[Schwartz space]]

category: people

[[!redirects L. Schwartz]]


Laurent series

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Laurent series_ generalize [[power series]] by allowing both positive and negative powers. In particular, _Laurent series_ with complex coefficients generalize [[Taylor series]] of analytic functions to [[meromorphic functions]]. A Laurent series for a meromorphic function $f(z)$ at finite $z\in\mathbb{C}$ has the form
$$ f(z) = \sum_{n=k}^{\infty}f_n z^n, $$
where $k$ is merely constrained to be finite and is often negative.

Or, in some contexts one wants to take $k = -\infty$. Here such a formal sum (with powers extending infinitely in both directions) is suggestive notation for an element belonging to the dual $\prod_{n \in \mathbb{Z}} R \cdot z^n$
of a ring $\oplus_{n \in \mathbb{Z}} R \cdot z^n$ (see Laurent polynomials below). In such contexts, Laurent series can be likened to distributions, i.e., functionals on the algebra of functions $\mathbb{Z} \to R$ with compact support.

## Definition

+-- {: .num_defn}
###### Definition
A _Laurent series_ in one variable $z$ over a commutative unital ring $k$ is a [[doubly infinite series]]
$$ f(z) = \sum_{n=-\infty}^{\infty} f_n z^n ,$$
where $f_n\in k$. Equivalently: a Laurent series is a function $\mathbb{Z} \to k: n \mapsto f_n$. The $k$-module of Laurent series is denoted $k[ [z, z^{-1}] ]$.
=--

A _Laurent polynomial_ is a Laurent series for which all but finitely many $f_n$ are zero. Laurent polynomials form a ring which may be described as $k[z, z^{-1}]$ or abstractly as $k[x, y]/(x y - 1)$. Observe that Laurent series in the generality discussed here do not analogously form a ring: the obvious definition of the coefficients of the product $h(z) = f(z)g(z)$ of two Laurent series $f(z), g(z)$,

$$h_n = \sum_{k \in \mathbb{Z}} f_k g_{n-k},$$

doesn't make sense in general (although it could sometimes make sense in topological contexts where some such infinite sums can converge, as in the case $k = \mathbb{C}$).

+-- {: .num_remark}
###### Remark
The $k$-vector space of Laurent series does however form a module over the ring of Laurent polynomials, i.e., if $f(z) \in k[z, z^{-1}]$ and $g(z)$ is a Laurent series, then the product $h(z) = f(z)g(z)$ as defined above always makes sense.
=--

In general, questions of convergence are treated as separate issues. In complex analysis, the Laurent series $\sum_{n \in \mathbb{Z}} a_n z^n$ describes a meromorphic function in a neighborhood around the point $z = 0$ (possibly with a pole there) if all but finitely many negatively indexed terms are zero. Similarly, series of the form $\sum_{n = -N}^{\infty} a_n (z-a)^n$ describe meromorphic functions in a neighborhood of $z=a$ with poles of order at most $N$.

On the other hand, in algebra one often hears of the ring of _formal Laurent series_. Here, the presence of the word "ring" signifies that we are restricting the coefficients so that multiplication makes sense. Thus,

+-- {: .num_defn #ring}
###### Definition
The _ring of formal Laurent series_ over a [[commutative ring]] $A$ in an indeterminate $x$ consists of Laurent series $\sum_{n \in \mathbb{Z}} f_n z^n$, with $f_n \in A$ but where all but finitely many $f_n$ for $n \lt 0$ vanish.
=--

Multiplication defined as above clearly makes sense. If $A$ is a field $k$, then this ring is usually denoted $k((x))$ and is in fact a [[field]]; indeed it is the [[field of fractions]] of the ring $k[ [x] ]$ of [[formal power series]], where $k[ [x] ]$ is often viewed as a discrete [[valuation ring]].

+-- {: .num_remark #restrict}
###### Remark
It would perhaps be clearer if we used the term "restricted Laurent series" to cover the Laurent series considered in Definition \ref{ring}, and let "Laurent series" be the term that covers doubly infinite series.
=--

## Properties

### Laurent series as distributions

Another point of view on Laurent series is given in the following alternative definition.

+-- {: .num_defn}
###### Definition
A Laurent series over $k$ is a $k$-linear functional

$$\phi: k[z, z^{-1}] \to k$$

on the algebra of Laurent polynomials.
=--

The associated series is $\sum_{n \in \mathbb{Z}} \phi(z^n)z^n$. But here we emphasize the point of view that Laurent series have a distribution-like character, with Laurent polynomials being considered a space of functions $\mathbb{Z} \to k$ with compact (finite) support, via the evident inclusion

$$k[z, z^{-1}] \cong \oplus_{n \in \mathbb{Z}} k \cdot z^n \hookrightarrow \prod_{n \in \mathbb{Z}} k \cdot z^n = k^{\mathbb{Z}}.$$

Of particular interest as a Laurent series is the formal Dirac distribution,

$$\delta(z) = \sum_{n \in \mathbb{Z}} z^n,$$

which intuitively is like the Fourier transform $\sum_{n = -\infty}^\infty e^{i n x}$ of the pointwise multiplicative identity $\mathbf{1}$ given by $\mathbf{1}(n) = 1$ for all $n$. It has the following property:

* For $f(z)$ a Laurent polynomial, $f(z)\delta(z) = f(1)\delta(z)$.

Indeed, notice that $\delta(z)$ is the distribution $p \mapsto p(1)$ which evaluates a Laurent polynomial at the multiplicative identity $z=1$.

More generally, for $a \in k^\ast$ an invertible unit, the series $\delta(a z) = \sum_{n \in \mathbb{Z}} a^n z^z$ satisfies

* $f(z)\delta(a z) = f(a^{-1})\delta(a z)$.

Just as it doesn't make sense in general to multiply distributions (at least not without heavy qualifications, as in the theory of [Colombeau](#Col)), we cannot make sense of expressions like $\delta(z)^2$. However, one can meaningfully work with derivatives of distributions, so we have for example

$$\delta'(z) = \sum_{n \in \mathbb{Z}} n z^n$$

which has the property

* $f(z)\delta'(z) = f(1)\delta'(z) - f'(1)\delta(z)$.

Indeed, there is a formal calculus of the Dirac distribution that plays an important role in the theory of vertex operator algebras. See for example the treatment in [Frenkel, Lepowsky, Meurman](#FLM).

One can also work with _convolution products_ $h(z) = (f \ast g)(z)$, defined by multiplying coefficients degree-wise:

$$h(z) = \sum_{n \in \mathbb{Z}} f_n g_n z^n.$$

### Algebraic closure

+-- {: .num_theorem #Puiseux}
###### Theorem
If $K$ is [[algebraically closed field|algebraically closed]] and has [[characteristic]] 0, then the [[algebraic closure]] of the field of (restricted) Laurent series $K((x))$ over $K$ is the field of [[Puiseux series]] over $K$.

=--

Here "restricted" refers to Remark \ref{restrict}. See _[[Puiseux series]]_ for more details on this result.

### Function field analogy

[[!include function field analogy -- table]]

## References

* Igor Frenkel, James Lepowsky, and Arne Meurman, _Vertex Operator Algebras and the Monster_, Volume 134 in Pure and Applied Mathematics, Academic Press 1988.
{#FLM}
* [Doubly Infinite Laurent Series](http://math.rutgers.edu/~sdurst/DILS.html), lectures for formal Laurent series in [[vertex operator algebra]] context
* Alexander Zheglov, _Wild division algebras over Laurent series fields_, [math.NT/0503637](http://arxiv.org/abs/math/0503637)

* [[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)]

For discussion of products of distributions, see

* J.F. Colombeau, _Multiplication of distributions_, Bull. Amer. Math. Soc. (N.S.) Volume 23, Number 2 (1990), 251-268. ([web](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183555881))
{#Col}


Laurie M. Brown

* Wikipedia entry

## Selected writings

On the early history of [Feynman 1942](path+integral#Feynman42) discovering the [[path integral]]-formulation of [[quantum mechanics]], following [Dirac 1933](path+integral#Dirac33):

* [[Laurie M. Brown]], *Feynman's Thesis — A New Approach to Quantum Theory*, World Scientific (2005) [[doi:10.1142/5852](https://doi.org/10.1142/5852)]

[[!redirects Laurie Brown]]


law of a random variable

+-- {: .rightHandSide}
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### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[probability space]] $(S,\mathcal{F},\mu)$ and a [[random variable]] or [[random element]] $f \colon S \to X$, the **law** or **distribution** of $f$ (or sometimes, of $X$) is the [[probability measure]] on $X$ which to a [[measurable subset]] $A\subseteq X$ assigns the [[probability]] that $f$ takes values in $A$.

In particular, the law of $f$ is the [[pushforward measure|pushforward]] of $\mu$ along $f$.

## Examples

\begin{example}
  The law of a [[deterministic random variable]] is a [[Dirac measure]].
\end{example}

## See also

* [[random variable]]
* [[probability space]]
* [[probability monad]]
* [[joint and marginal probability]]

category: probability

[[!redirects laws of random variables]]


Law of Cosines


law of double negation

\tableofcontents

## Definition

The **law of double negation** is the statement that the [[double negation]] of a [[proposition]] [[implication|implies]] that proposition

$$
  \not \not A \Rightarrow A
  \,.
$$

equivalently, that the double negation of a [[proposition]] is [[logically equivalent]] to that proposition:

$$
  \not \not A \iff A
  \,.
$$

In [[classical logic]], this is simply true.  In [[constructive logic]], it is equivalent to the law of [[excluded middle]] (because $\not \not (P \vee \not P)$ is a constructive theorem), and is not assertable in general.

## In dependent type theory

In [[dependent type theory]], we define the [[negation]] of a type $A$ as the function type from $A$ to the empty type $\mathbb{0}$; $\neg A \coloneqq A \to \mathbb{0}$. The law of double negation states that given a type $A$, one could derive that one could get an element of $A$ in the context of a witness that $A$ is an [[h-proposition]] and a witness of the double negation of $A$:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:\mathrm{isProp}(A), c:\neg \neg A \vdash \mathrm{doubleneg}_A(p, c):A}$$

Given [[function types]], this could be wrapped up as

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{doubleneg}_A:\mathrm{isProp}(A) \to (\neg \neg A \to A)}$$

The requirement that $A$ be an [[h-proposition]] is necessary; the law of double negation for [[h-sets]] is the [[set-theoretic choice operator]], which implies the [[set-theoretic axiom of choice]] in addition to [[excluded middle]], and the law of double negation for general [[types]] is the [[type-theoretic choice operator]], which implies [[UIP]] in addition to the set-theoretic axiom of choice and excluded middle.

If one additionally has a [[Russell type of all propositions]] $\Omega$ or a [[Tarski type of all propositions]] $(\Omega, \mathrm{El}_\Omega)$, the law of double negation could be expressed as an axiom:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} (\neg \neg P) \to P}\mathrm{Russell} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} (\neg \neg \mathrm{El}_\Omega(P)) \to \mathrm{El}_\Omega(P)}\mathrm{Tarski}$$

There is an alternative presentation of the law of double negation, which uses an [[equivalence type]] for [[logical equivalence]] rather than a [[function type]] for [[implication]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{doubleneg}_A:\mathrm{isProp}(A) \to ((\neg \neg A) \simeq A)}$$

In [[dependent type theory]] as [[foundations of mathematics]], the requirement that $A$ be an [[h-proposition]] is also necessary. This alternate law of double negation for general [[types]] implies that all types are [[h-propositions]] a la [[propositional logic as a dependent type theory]], since the double negation of a type is always an [[h-proposition]]:

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{doubleneg}_A:(\neg \neg A) \simeq A}$$

Similarly, if one additionally has a [[Russell type of all propositions]] $\Omega$ or a [[Tarski type of all propositions]] $(\Omega, \mathrm{El}_\Omega)$, this alternate law of double negation could be expressed as an axiom:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} (\neg \neg P) \simeq P}\mathrm{Russell} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{doubleneg}:\prod_{P:\Omega} (\neg \neg \mathrm{El}_\Omega(P)) \simeq \mathrm{El}_\Omega(P)}\mathrm{Tarski}$$

## Related concepts

* [[excluded middle]]

* [[ex falso quodlibet]]

* [[closed subspace]]

* [[type-theoretic law of double negation]]

## References

* [[eom]], _[Double negation, law of](https://www.encyclopediaofmath.org/index.php/Double_negation,_law_of)_

[[!redirects law of double negation]]
[[!redirects double negation law]]
[[!redirects principle of double negation]]
[[!redirects double negation principle]]
[[!redirects axiom of double negation]]
[[!redirects double negation axiom]]

[[!redirects choice operator for propositions]]
[[!redirects choice operator for h-propositions]]
[[!redirects choice operator for mere propositions]]


Lawrence Breen

* [website](http://www.math.univ-paris13.fr/~breen/)

#related $n$Lab entries#

* [[nonabelian cohomology]]

* [[gerbe]]

* [[braided monoidal 2-category]]

* [[differential forms in synthetic differential geometry]]

* [[display of a p-divisible group]]

* [[Alexander Grothendieck]], letter to [[Larry Breen]] 17/2/1975 [pdf](https://agrothendieck.github.io/divers/LGB17275en.pdf), [scan](https://agrothendieck.github.io/divers/LGB17275scanen.pdf)

## Selected works

* _Extensions du groupe additif_, Publ. Math. I.H.E.S. 48 (1978) 39-125.
* L. Breen, T. Ekedahl, _Construction et propriétés de la sphère schématique_, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985) 665-668.
* _On the classification of 2-gerbes and 2-stacks_, Astérisque 225 SMF, Paris (1994) [numdam](http://www.numdam.org/item/?id=AST_1994__225__1_0)
* [[Lawrence Breen]], _Notes on 1- and 2-gerbes_, in [[John Baez]], [[Peter May]] (eds.) _[[Towards Higher Categories]]_, [arXiv:math/0611317](https://arxiv.org/abs/math/0611317) [doi](https://doi.org/10.1007/978-1-4419-1524-5_5)
* _Tannakian categories_, in _Motives_, Proceedings of Symposia in Pure Mathematics, 55, Providence, R.I.: American Mathematical Society, 1994
* [[Lawrence Breen]], Bitorseurs et cohomologie non abélienne_, in [[Grothendieck Festschrift]], vol. I, Progress in Math. __86__ (1990) 401--476

category: people

[[!redirects Larry Breen]]
[[!redirects L. Breen]]


Lawrence Sklar

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

## related $n$Lab entries

* [[philosophy of physics]]

category: people


Lawvere distribution

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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Lawvere distribution_ is a [[cosheaf]] valued in some [[topos]], such as the [[category of sets]] [[Set]].
The concept of _Lawvere distribution_ is a kind of _[[categorification]]_ of the concept of _[[distribution]]_ in [[functional analysis]].

To some extent one may think of a [[sheaf]] $F$ on a [[topological space]] as being a [[Set]]-valued [[function]] on that space: to each point $x \in X$ it assigns the [[stalk]] $x^* F \in Set$. In this analogy a _Lawvere distribution_ is the analog of a [[distribution]] in the sense of [[functional analysis]]: where the latter is a [[continuous linear functional]], the former is a [[colimit]]-preserving [[functor]].
(Here we think of a [[coproduct]] of sets as the [[categorification]] (under set [[cardinality]]) of the [[sum]] of [[numbers]] and hence read preservation of colimits as _linearity_ .)

Better yet, under [[∞-groupoid cardinality]] we may think of tame [[∞-groupoid]]s as [[real numbers]] and hence of [[(∞,1)-sheaves]] as a [[higher category theory|higher]]/[[homotopy theory|homotopical]] [[categorification]] of real-number valued functions. This yields a more general notion of Lawvere distributions on [[(∞,1)-toposes]] given by [[(∞,1)-colimit]] preserving [[(∞,1)-functors]].

Still more generally one may allow to generalize $(\infty,1)$-toposes to general [[locally presentable (∞,1)-categories]]. Viewed this way, Lawvere distributions are the morphism in $Pr(\infty,1)Cat$, the [[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]].

## Definition

Throughout $\mathcal{S}$ is some base [[topos]] or [[(∞,1)-topos]] and all notions are to be understood as indexed over this base.

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

(Lawvere, see Definition 1.3.4 in [Bunge and Funk](#SCT).)

Let $\mathcal{E}$ and $\mathcal{K}$ be [[(∞,1)-toposes]] over [[Set]] (more generally, over an [[elementary topos]] $S$ with a [[natural numbers object]]). A **distribution** on $\mathcal{E}$ with values in $\mathcal{K}$ is
an [[(∞,1)-cosheaf]] on $E$ with values in $K$, i.e.,
an [[(∞,1)-functor]]
$$
  \mu : \mathcal{E} \to \mathcal{K}
$$
that preserves $S$-small [[(∞,1)-colimit]]s.
We write
$$
  Dist_S(\mathcal{E}, \mathcal{K}) \subset (\infty,1)Func(\mathcal{E}, \mathcal{K})
$$
for the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that are [[(∞,1)-cosheaves]], i.e., preserve $S$-small colimits.

=--

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

By the [[adjoint (∞,1)-functor theorem]] this is equivalently a pair
$$
  (\mu \dashv \mu^*) : \mathcal{E} \to \mathcal{K}
$$
of ($S$-indexed) [[adjoint (∞,1)-functor]]s.

=--

+-- {: .un_remark}
###### Notation

To amplify the interpretation in analogy with [[distribution]]s in [[functional analysis]] one sometimes writes
$$
  \int_{\mathcal{E}} (-) d\mu : \mathcal{E} \to \mathcal{K}
$$
for a Lawvere distribution $\mu$.

Notably in the case that $\mathcal{K} = $ [[∞Grpd]] and $F$ is an [[(∞,1)-sheaf]] such that $\mu(F)$ is tame, we may use

$$
   \int_{\mathcal{E}} F d \mu \in \mathbb{R}
$$

for the corresponding [[∞-groupoid cardinality]].

=--

## Examples

### Dirac $\delta$-distributions {#DiracDeltaDistribution}

A [[point of a topos]] is a [[geometric morphism]] of the form
$$
  (p^* \dashv p_*) : \mathcal{S} \stackrel{\leftarrow}{\to} \mathcal{E}
  \,.
$$

The [[left adjoint]] $p^*$ is therefore a Lawvere distribution. This sends any [[(∞,1)-sheaf]] to its [[stalk]] at the point $p$. So this behaves like the [[Dirac distribution]] on functions.

### The canonical distribution on a locally $\infty$-connected topos

If $\mathcal{E}$ is a [[locally ∞-connected (∞,1)-topos]] then its terminal [[global section]] [[(∞,1)-geometric morphism]] by definition has a further left adjoint

$$
  (\Pi \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \mathcal{S}
  \,.
$$

This left adjoint $\Pi$ (the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]) is therefore a canonical $\mathcal{S}$-valued distribution on $\mathcal{E}$. It is also written
$$
  \int_{\mathcal{E}}(-) d x : \mathcal{E} \to \infty Grpd
  \,.
$$

This construction also works in the relative setting:
if $e\colon E\to K$ [[locally ∞-connected geometric morphism]] of [[(∞,1)-toposes]],
then $e_!\colon E\to K$ is a $K$-valued Lawvere distribution on $E$.

### Local toposes

If $E$ is a [[local (∞,1)-topos]],
more generally, $e\colon E\to K$ is a [[local (∞,1)-geometric morphism]] of [[(∞,1)-toposes]],
then the right adjoint $e_*$ is a (left exact) Lawvere distribution on $E$.

### Multiplication of distributions by functions

For $F \in \mathcal{E}$ an $(\infty,1)$-sheaf and $\mu : \mathcal{E} \to \mathcal{S}$ a distribution, there is a new distribution

$$
  F \cdot \mu :
  G \mapsto
  \mu(F \times G)
  \,.
$$

In the functional notation this is the formula

$$
  \int_{\mathcal{E}} G d(F \times \mu)
  =
  \int_{\mathcal{E}} G \times F d \mu
  \,.
$$

### Distributions on the point

The [[∞Grpd]]-valued distributions on $\infty Grpd \simeq Sh_{(\infty,1)}(*)$ itself coincide with the value at the single point

$$
  Dist(\infty Grpd, \infty Grpd) \simeq \infty Grpd.
$$

## Related entries

* [[symmetric topos]]
* [[bagdomain topos]]
* [[cosheaf]]

For the $(\infty,1)$-category theory generalization and related references:

* [[Pr(∞,1)Cat]]

## References

The 1-categorical notion has been described by [[Bill Lawvere]] in a series of talks and expositions. For instance in the context of [[cohesive topos]]es in

* [[Bill Lawvere]], _Axiomatic cohesion_ , Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

A comprehensive discussion is in

* {#SCT} [[Marta Bunge]] and [[Jonathon Funk]], _Singular coverings of toposes_ , Lecture Notes in Mathematics vol. 1890 Springer Heidelberg (2006). (chap. 1)

See also:

* [[Marta Bunge]], _Cosheaves and Distributions on Toposes_ , Alg. Univ. **34** (1995) pp.469-484.

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos_ , JPAA **113** (1996) pp.1-38.

* [[Marta Bunge]], [[Jonathon Funk]], _Spreads and the Symmetric Topos II_ , JPAA **130** (1998) pp.49-84.

* [[Anders Kock]], [[Gonzalo E. Reyes]], _A Note on Frame Distributions_ , Cah. Top. Géom. Diff. Cat.**40** (1999) pp.127-140.

* [[Andrew Pitts]], _On Product and Change of Base for Toposes_ , Cah. Top. Géom. Diff. Cat.**26** (1985) pp.43-61.

[[!redirects topos distribution]]
[[!redirects Lawvere distributions]]


Lawvere interval

[[!redirects lawvere interval]]
[[!redirects lawvere interval]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $A$ be a [[small category]], and let $Psh(A)=Set^{A^{op}}$ be the [[category of presheaves]] on $A$. Since $Psh(A)$ is a [[Grothendieck topos]], it has a unique [[subobject classifier]], $L$.

Let $\mathbf{0}$ and $\mathbf{1}$ denote the [[initial object]] and [[terminal object]], respectively, of $Psh(A)$.  The terminal presheaf $\mathbf{1}$ has two distinguished [[subobjects]] $\mathbf{0}\hookrightarrow \mathbf{1}$ and $\mathbf{1}\hookrightarrow \mathbf{1}$, which correspond to global points $\lambda^0,\lambda^1\in L(\mathbf{1})=Hom(\mathbf{1},L)$ of the subobject classifier.

\begin{definition}\label{LawvereInterval}
The triple $\mathfrak{L}=(L,\lambda^0,\lambda^1)$ is called the **Lawvere interval** for the [[topos]] $Psh(A)$. This object determines a [[cylinder functor]] given by taking the [[cartesian product]] with $L$, called the **Lawvere cylinder**.
\end{definition}
This is due to [Cisinski (2006), §1.3.9](#Cisinski06).

By Prop. \ref{LawvereIntervalIsFibrant} below, the Lawvere interval $\mathfrak{L}$ may be regarded as the universal [[cylinder object]] for [[Cisinski model structures]] on presheaf toposes.

## Properties

\begin{proposition}
\label{LawvereIntervalIsFibrant}
The [[subobject classifier]] in any [[topos]] is an [[injective object]], whence the Lawvere interval $L$ (Def. \ref{LawvereInterval}) is a [[fibrant resolution]] of the [[terminal object]] in any [[Cisinski model structure]] on $Psh(A)$.
\end{proposition}
This is highlighted at the end of [Cisinski (2006), §1.3.9](#Cisinski06) with reference (up to a typo) to [[Sheaves in Geometry and Logic|MacLane & Moerdijk (1992), IV §10.1]]:
\begin{proof}
Recall that for $L$ to be an [[injective object]] means that every solid [[span]] as below, where the vertical [[map]] is a [[monomorphism]], admits a dashed [[lifting]] as shown:

\begin{tikzcd}
  B
    \ar[d, hook]
    \ar[r, "{ f }"]
  &
  L
  \ar[d, gray]
  \\
  A
  \ar[ur, dashed, "{ \underline{f} }"{swap}]
  \ar[r, gray]
  &
  \color{gray}\ast
\end{tikzcd}

Since in a [[Cisinski model structure]], by definition, the monomorphisms are precisely the [[cofibrations]], injective objects here are equivalently those for which the [[terminal object|terminal]] [[map]] $L \to \ast$ is an [[acyclic cofibration]].

Hence assuming the solid diagram above, we show the existence of $\underline{f}$:

Here $f \colon B \to L$ classifies a subobject of $B$, which we denote $C \hookrightarrow B$. The point now is that, with $B \hookrightarrow A$ being a monomorphism, the composite
$$
  C \hookrightarrow B \hookrightarrow A
$$
exhibits $C$ also as a subobject of $A$, which as such is classified by some map $\underline{f} \colon A \to L$, and we claim that this serves as the desired extension.
To see that indeed this $\underline{f}$ makes the above triangle commute, consider the following [[commuting diagram]]:

\begin{tikzcd}
  C
  \ar[d, hook]
  \ar[r, equals]
  \ar[dr, phantom, "{ \scalebox{.6}{(pb)} }"]
  &
  C
    \ar[d, hook]
    \ar[r]
    \ar[ddr, phantom, "{ \scalebox{.6}{(pb)} }"]
  &
  \ast
    \ar[dd]
  \\
  B
  \ar[r, equals]
  \ar[d, equals]
  \ar[dr, phantom, "{ \scalebox{.6}{(pb)} }"]
  &
  B \ar[d, hook]
  &
  \\
  B \ar[r, hook]
  &
  A \ar[r, "{ \underline{f} }"{swap}] & L
\end{tikzcd}

Here

* the right rectangle is the [[pullback]] square that witnesses $\underline{f}$ as the classifying map of $C \hookrightarrow A$, by definition,

* the left rectangle is the [[fiber product]] $B \cap_A C$ computed via the [[pasting law for pullbacks]] to be isomorphic to $C$, using that $C$ is already a subobject of $A$ (which gives the factorization in the middle) and then using (for the two squares on the left) that:

  1. the [[fiber product]] of a monomorphism with itself is its [[domain]] ([this Prop.](monomorphism#BasicCharacterizationOfMonomorphisms)),

  1. [[isomorphisms]] are preserved by pullback ([this Prop.](pullback#PullbackPreservesMonomorphisms)).

This implies, again by the [[pasting law]], that the total square is a pullback, hence that the total bottom map classifies $C$. But $C$ was defined to be classified by $f$,
and so the uniqueness of subobject classifying maps implies that the total bottom map equals $f$, which was to be shown.
\end{proof}

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

Given any [[small set]] of monomorphisms in $Psh(A)$, there exists the smallest [[Cisinski model structure]] for which those monomorphisms are [[acyclic cofibrations]].

=--

By applying a theorem of [[Denis-Charles Cisinski]]. (...)

## Examples

Suppose $A=\Delta$ is the [[simplex category]], and  let $S$ consist only of the inclusion $\{1\}:\Delta^0\to\Delta^1$. Applying Cisinski's anodyne completion of $S$ by Lawvere's cylinder $\mathbf{\Lambda}_\mathfrak{L}(S,M)$, we get exactly the [[model structure for right fibrations|contravariant model structure]] on the category of simplicial sets.

## References

* {#Cisinski06} [[Denis-Charles Cisinski]], *Les préfaisceaux comme types d'homotopie*, Astérisque **308** Soc. Math. France (2006), 392 pages [[numdam:AST_2006__308__R1_0](http://www.numdam.org/item/?id=AST_2006__308__R1_0) [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf)]

[[!redirects Lawvere intervals]]
[[!redirects Lawvere cylinder]]