AGT correspondence

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#### Algebraic Quantum Field Theory
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#Contents#
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## Idea

The _AGT correspondence_ ([AGT 09](#AGT09)) is a relation between

1. the [[instanton]]-[[partition function]] of $SU(2)^{n+3g-3}$-[[N=2 D=4 super Yang-Mills theory]] (Nekrasov's partition function, e.g. [Szabo 15 (2.1)](#Szabo15))

1. the [[conformal blocks]] of [[Liouville theory]] on an $n$-punctured [[Riemann surface]] $C_{g,n}$ of [[genus]] $g$

Here the idea is that $C_{g,n}$  is the [[super Yang-Mills theory]] obtained by [[Kaluza-Klein mechanism|compactifying]] the worldvolume [[6d (2,0)-supersymmetric QFT]] of two [[M5-branes]], see at [[N=2 D=4 super Yang-Mills theory]], the section [Construction by compactification](N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes)). This method generalizes a bit beyond super Yang-Mills, to [[class S]].

In particular, the [[N=2 D=4 super Yang-Mills theory]] is a [[quiver gauge theory]] and the correspondence matches the shape of its [[quiver]]-diagram to the [[genus of a surface|genus]] and punctures of the [[Riemann surface]]:

More generally, this construction yields something like a decomposition of the [[6d (2,0)-superconformal QFT]] into a [[2d SCFT]] "with values in [[super Yang-Mills theory|4d SYM field theory]]" (e.g. [Tachikawa 10, slide 25 (33 of 54)](#Tachikawa10)). Hence composition with any kind of suitable invariant of the 4d field theories yields an actual [[2d SCFT]], for instance taking the superconformal index in 4d yields a [[2d TQFT]] ([GPRR 10](#GPRR10)). In this picture of "4d-SYM field theory-valued [[2d SCFT]]" one has the following correspondences:

* the [[complex structure]] in 2d is the [[coupling constants]] and [[theta angles]] etc in the 4d [[super Yang-Mills theory]];

* the [[mapping class group]] (large [[conformal transformations]]) in 2d is the (generalized) [[S-duality]] of the 4d theory.

## Related concepts

* [[M5-brane elliptic genus]]

* [[duality in physics]], [[duality in string theory]]

* [[Nekrasov function]]

* [[McKay correspondence]]

* [[3d-3d correspondence]]

* [[knots-quivers correspondence]]

* [[class S]]

Wrapping the [[M5-brane]] on a [[3-manifold]] instead yields: [[3d-3d correspondence]].

## References

A [[2d SCFT]] argued to describe the  [[KK-compactification]] of the [[M5-brane]] on a [[4-manifold]] (specifically: a [[complex surface]]) originates with

* [[Juan Maldacena]], [[Andrew Strominger]], [[Edward Witten]], _Black Hole Entropy in M-Theory_, JHEP 9712:002, 1997 ([arXiv:hep-th/9711053](https://arxiv.org/abs/hep-th/9711053))

The origin of the AGT correspondence is:

* {#Gaiotto09} [[Davide Gaiotto]], _$N=2$ dualities_, JHEP 08 (2012) 034 [[arXiv:0904.2715](https://arxiv.org/abs/0904.2715)]

* {#AGT09} [[Luis Alday]], [[Davide Gaiotto]], [[Yuji Tachikawa]], _Liouville Correlation Functions from Four-dimensional Gauge Theories_, Lett.Math.Phys.91:167-197, 2010 ([arXiv:0906.3219](https://arxiv.org/abs/0906.3219))

* [[Davide Gaiotto]], [[Gregory Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics __234__ (2013) 239--403 ([arXiv:0907.3987](https://arxiv.org/abs/0907.3987) [doi](https://doi.org/10.1016/j.aim.2012.09.027))

The [[2d TQFT]] obtained from this by forming the 4d index is discussed in

* {#GPRR10} Abhijit Gadde, Elli Pomoni, [[Leonardo Rastelli]], Shlomo S. Razamat, _S-duality and 2d Topological QFT_, JHEP 1003:032, 2010 ([arXiv:0910.2225](https://arxiv.org/abs/0910.2225))

Relation of the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]]

* Benjamin Assel, [[Sakura Schafer-Nameki]], Jin-Mann Wong, _M5-branes on $S^2 \times M_4$: Nahm's Equations and 4d Topological Sigma-models_, J. High Energ. Phys. (2016) 2016: 120 ([arxiv:1604.03606](https://arxiv.org/abs/1604.03606))

  (relating to the [[moduli space of monopoles]])

and to the [[3d-3d correspondence]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Toda Theory From Six Dimensions_, J. High Energ. Phys. (2017) 2017: 106 ([arxiv:1605.03997](https://arxiv.org/abs/1605.03997))

* Sam van Leuven, Gerben Oling, _Generalized Toda Theory from Six Dimensions and the Conifold_, J. High Energ. Phys. (2017) 2017: 50 ([arxiv:1708.07840](https://arxiv.org/abs/1708.07840))

Brief surveys include

* {#Tachikawa10} [[Yuji Tachikawa]], _M5-branes, 4d gauge theory and 2d CFT_, 2010 ([pdf](http://member.ipmu.jp/yuji.tachikawa/transp/4d-2d-caltech.pdf))

* Abhijit Gadde, _$\mathcal{N}= 2$ Dualities and 2d TQFT_ 2012 ([[Gadde2dTQFT.pdf:file]])

* Nikolay Bovev, _New SCFTs from wrapped branes_, 2013 ([pdf](http://ipht.cea.fr/Meetings/Itzykson2013/Talks/bobev-Itzykson-july2013.pdf))

* Giulio Bonelli, _Variations on AGT Correspondence_, 2013 ([pdf](https://indico.cern.ch/event/217384/attachments/348854/486363/Bonelli.pdf))

* {#Taki16} Masato Taki, _String Theory as an Attempt of PolyMathematics_, talk at 2016.4/28 iTHES-AIMR-IIS  ([pdf](https://indico2.riken.jp/event/2215/attachments/4106/4775/Taki_invited.pdf))

More detailed review is in

* Rober Rodger, _A pedagogical introduction to the AGT conjecture_, Master Thesis Utrecht (2013) ([pdf](http://testweb.science.uu.nl/ITF/teaching/2013/R.J.Rodger.pdf))

* {#Szabo15} [[Richard Szabo]], _$N=2$ gauge theories, instanton moduli spaces and geometric representation theory_, Journal of Geometry and Physics Volume 109, November 2016, Pages 83-121 ([arXiv:1507.00685](https://arxiv.org/abs/1507.00685))

* Bruno Le Floch, _A slow review of the AGT correspondence_, J. Phys. A: Math. Theor. __55__ (2022) 353002 ([doi](https://doi.org/10.1088/1751-8121/ac5945)[arXiv:2006.14025](https://arxiv.org/abs/2006.14025))

See also

* [[Alexander Belavin]], M. A. Bershtein, B. L. Feigin, A. V. Litvinov, G. M. Tarnopolsky, _Instanton moduli spaces and bases in coset conformal field theory_ ([arxiv/1111.2803](http://arxiv.org/abs/1111.2803))

* [[Volker Schomerus]], Paulina Suchanek, _Liouville's imaginary shadow_ ([arxiv/1210.1856](http://arxiv.org/abs/1210.1856))

* A.Mironov, A.Morozov, _The power of Nekrasov functions_ ([arxiv/0908.2190](http://arxiv.org/abs/0908.2190))

* D. Galakhov, A. Mironov, A. Morozov, _S-duality as a beta-deformed Fourier transform_ ([arxiv/1205.4998](http://arxiv.org/abs/1205.4998))

* A. Mironov, _Spectral duality in integrable systems from AGT conjecture_ ([arxiv/1204.0913](http://arxiv.org/abs/1204.0913))

* A. Belavin, V. Belavin, _AGT conjecture and integrable structure of conformal field theory for $c=1$_, Nucl.Phys.B850:199-213 (2011) ([arxiv/1102.0343](http://arxiv.org/abs/1102.0343))

* A. Belavin, V. Belavin, M. Bershtein, _Instantons and 2d Superconformal field theory_ ([arxiv/1106.4001](http://arxiv.org/abs/1106.4001))

* Kazunobu Maruyoshi, _Quantum integrable systems, matrix models, and AGT correspondence_, seminar ([slides pdf](http://db.ipmu.jp/seminar/sysimg/seminar/428.pdf))

* Giulio Bonelli, Alessandro Tanzini, _Hitchin systems, N=2 gauge theories and W-gravity_ ([arxiv/0909.4031](http://arxiv.org/abs/0909.4031))

* Giulio Bonelli, Kazunobu Maruyoshi, Alessandro Tanzini, _Quantum Hitchin systems via beta-deformed matrix models_ ([arxiv/1104.4016](http://arxiv.org/abs/1104.4016))

* {#ChacaltanaDistler10} [[Oscar Chacaltana]], [[Jacques Distler]], _Tinkertoys for Gaiotto Duality_, JHEP 1011:099,2010, ([arXiv:1008.5203](http://arxiv.org/abs/arXiv:1008.5203))

* Satoshi Nawata, _Givental J-functions, Quantum integrable systems, AGT relation with surface operator_ ([arXiv/1408.4132](http://arxiv.org/abs/1408.4132))

The AGT correspondence is treated with the help of a [[Riemann-Hilbert problem]] in

* G. Vartanov, [[Jörg Teschner]], _Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory_ ([arxiv/1302.3778](http://arxiv.org/abs/1302.3778))

* [[Andrei Mironov]], A. Morozov, *Superintegrability as the hidden origin of Nekrasov calculus* [[arXiv:2207.08242](https://arxiv.org/abs/2207.08242)]

Proof of the AGT conjecture in special cases:

* [[Andrei Mironov]], [[Andrey Morozov]], [[Sh. Shakirov]], *A direct proof of AGT conjecture at $\beta = 1$*, JHEP 1102:067 (2011) [[arXiv:1012.3137](https://arxiv.org/abs/1012.3137), doi:10.1007/JHEP02(2011)067]

* Qing-Jie Yuan, Shao-Ping Hu, Zi-Hao Huang, Kilar Zhang, *A proof of An AGT conjecture at $\beta = 1$* [[arXiv:2305.11839](https://arxiv.org/abs/2305.11839)]

See also:

* Leszek Hadasz, Błażej Ruba, *Decomposition of $\widehat{sl}_{2,k} \otimes \widehat{sl}_{2,1}$
highest weight representations for generic level $k$ and equivalence between two dimensional CFT models* [[arXiv:2312.14695](https://arxiv.org/abs/2312.14695)]

category: physics

[[!redirects AGT-correspondence]]

[[!redirects AGT conjecture]]
[[!redirects Class S]]


Agustí Roig

[[!redirects Agusti Roig]]

* [webpage](https://mat.upc.edu/en/people/agustin.roig)

* [FUTUR website](https://futur.upc.edu/AgustinRoigMarti)

## related $n$Lab entries

* [[Grothendieck construction for model categories]]

* [[minimal fibration]]

* [[KS-model]]

* [[circle action]]

* [[4-sphere]]

category: people


Aharon Casher

* [InSpire page](https://inspirehep.net/authors/1014341)

## Selected writings

Early discussion of [[discretized light-cone quantization]]:

* {#Casher76} [[Aharon Casher]], *Gauge fields on the null plane*, Phys. Rev. D **14** (1976) 452 [[doi:10.1103/PhysRevD.14.452](https://doi.org/10.1103/PhysRevD.14.452)]

category: people


Aharonov-Bohm effect

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## Idea

The _Aharonov-Bohm effect_ is a configuration of the [[electromagnetic field]] which has vanishing electric/magnetic [[field strength]] (vanishing [[Faraday tensor]] $F = 0$) and but is nevertheless non-trivial, in that the [[vector potential]] $A$ is non-trivial. Since the vector potential affects the [[phase|quantum mechanical phase]] on the [[wavefunction]] of [[electrons]] moving in an electromagnetic field, in such a configuration classical physics sees no effect, but the phase of quantum particles, which may be observed as a [[quantum interference|interference]] pattern on some screen, does.

More technically, a configuration of the [[electromagnetic field]] is generally given by a [[circle]]-[[principal connection]] and an Aharonov-Bohm configuration is one coming from a [[flat connection]], whose [[curvature]]/[[field strength]] hence vanishes, but which is itself globally non-trivial. This is only possible on [[spaces]] ([[spacetimes]]) which have a non-trivial [[fundamental group]], hence for instance it doesn't happen on [[Minkowski spacetime]].

In practice one imagines an idealized [[electric current]]-carrying [[solenoid]] in [[Euclidean space]]. Away from the solenoid itself the [[magnetic field]] produced by it gives such a configuration.

## Details

Let $\mathbb{R}^2 - \{0\}$ be the [[plane]] with the origin removed, and consider the space $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}$ (thought of as 3d [[Cartesian space]] with the z-axis removed) and [[spacetime]] $(\mathbb{R}^2 - \{0\}) \times \mathbb{R}^2$ (thought of as the previous configuration statically moving in time).

For the following argument only the topological structure of the space matters, and nothing needs to explicitly depend on the $z$-[[coordinate]] and the time-coordinate, so for notational simplicity we may suppress these and consider just $\mathbb{R}^2 - \{0\}$.

On this space minus the x-axis consider the [[polar coordinates]]  $(\phi,r)$ with

$$
  x = r cos(\phi)\,,\;\;\; y = r sin(\phi)
  \,.
$$

Accordingly we have the [[differential 1-forms]]

$$
  \mathbf{d}x = cos(\phi)\mathbf{d}r - r sin(\phi) \mathbf{d}\phi
$$

$$
  \mathbf{d}y = sin(\phi)\mathbf{d}r + r cos(\phi) \mathbf{d}\phi
$$

hence

$$
  \begin{aligned}
    \mathbf{d}\phi
     & =
     \frac{1}{r}cos(\phi)\mathbf{d}y - \frac{1}{r}sin(\phi) \mathbf{d}x
     \\
     & =
     \frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x
  \end{aligned}
  \,.
$$

Here the expression on the right extends smoothly also to the $x$-axis and this extension we call

$$
  \theta \coloneqq
  \frac{1}{r^2} x \mathbf{d}y - \frac{1}{r^2} y \mathbf{d}x
  \;\;
  \in \Omega^1(\mathbb{R}^2 - \{0\})
  \,.
$$

From the way this is constructed it is clear that $\theta$ is a closed differential form

$$
  \mathbf{d}\theta = 0
  \,.
$$

However, on $\mathbb{R}^2 - \{0\}$ this is not an exact form. In other words, if one regards $\theta$ as the [[vector potential]] being the configuration of an  [[electromagnetic field]]

$$
  A \coloneqq \theta
$$

then:

1. the [[field strength]] vanishes $F = \mathbf{d}A = 0$;

1. but there is no [[gauge transformation]] relating $A$ to the trivial field configuration.

This is possible because $\mathbb{R}^2 - \{0\}$ is not [[simply connected topological space|simply connected]] and hence the [[Poincaré lemma]] does not apply.

## Related concepts

* [[fiber bundles in physics]]

* [[Dirac charge quantization]], [[magnetic monopole]]

* anyonic [[braid group statistics]] as Aharanov-Bohm effect for a *[[fictitious gauge field]]*

## References

The effect was first predicted by

* W. Ehrenberg, R. E. Siday, *The Refractive Index in Electron Optics and the Principles of Dynamics*,  Proceedings of the Physical Society. Section B, **62** 8 (1949) 1 [[doi:10.1088/0370-1301/62/1/303](https://iopscience.iop.org/article/10.1088/0370-1301/62/1/303)]

It is named after:

* [[Yakir Aharonov]], [[David Bohm]], *Significance of Electromagnetic Potentials in the Quantum Theory*, Phys. Rev. **115** (1959) 485 [[doi:10.1103/PhysRev.115.485](https://doi.org/10.1103/PhysRev.115.485), [pdf](https://journals.aps.org/pr/pdf/10.1103/PhysRev.115.485)]

Early discussion with emphasis of the role of [[connection on a bundle|connections]] on [[fiber bundles in physics]] and generalization to non-abelian [[Yang-Mills theory]]:

* [[Tai Tsun Wu]], [[Chen Ning Yang]], *Concept of nonintegrable phase factors and global formulation of gauge fields*, Phys. Rev. D **12** (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

See also:

* L. Mangiarotti, [[Gennadi Sardanashvily]], section 6.6 of _Connections in Classical and Quantum Field Theory_, World Scientific, 2000

* [[Mikio Nakahara]], Section 10.5.3 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

* Wikipedia, _[Aharonov-Bohm effect](https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect)_


Ahmadnagar


Ahmed Almheiri

* [webpage](https://www.ias.edu/scholars/ahmed-almheiri)

## Selected writings

On the interpretation of [[tensor networks]] encoding [[holographic entanglement entropy]] as [[quantum error correcting codes]]:

* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))

* [[Ahmed Almheiri]], _Holographic Quantum Error Correction and the Projected Black Hole Interior_ ([arXiv:1810.02055](https://arxiv.org/abs/1810.02055))

On the [[black hole firewall problem]]:

* [[Ahmed Almheiri]], [[Donald Marolf]], [[Joseph Polchinski]], James Sully, _Black holes: complementarity or firewalls?_ ([arXiv:1207.3123](http://arxiv.org/abs/arXiv:1207.3123))

On [[AdS/CFT]]-duality for [[AdS2-CFT1|$AdS_2/CFT_1$]] via [[Jackiw-Teitelboim gravity]]:

* {#AlmheiriPolchinski14} [[Ahmed Almheiri]], [[Joseph Polchinski]], _Models of $AdS_2$ Backreaction and Holography_, J. High Energ. Phys. (2015) 2015: 14. ([arXiv:1402.6334](https://arxiv.org/abs/1402.6334))

Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. **2020** 13 (2020) [[arXiv:1911.12333](https://arxiv.org/abs/1911.12333)]

Review:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. **93** 35002 (2021) [[arXiv:2006.06872](https://arxiv.org/abs/2006.06872), [doi:10.1103/RevModPhys.93.035002](https://doi.org/10.1103/RevModPhys.93.035002)]

## Activity

The case for a Kavli-like theoretical physics institute in Abu Dhabi, UAE:

* with [[David Gross]]: *Majlis Mohamed bin Zayed lecture: “The Future of Scientific Research”*, Abu Dhabi (Jan 2023)  [video: [TW](https://twitter.com/i/broadcasts/1mnxeRYyomQKX), [YT](https://www.youtube.com/watch?v=PpMWrajguzg)]

category: people


Aidan Herderschee

* [Institute page](https://www.ias.edu/scholars/aidan-herderschee)

* [InSpire page](https://inspirehep.net/authors/1869780)

* [GoogleScholar page](https://scholar.google.com/citations?user=FI0WW9kAAAAJ&hl=en)

## Selected writings

On [[graviton]] [[scattering amplitudes]] in the [[BFSS matrix model]] and [[soft graviton theorems]]:

* [[Aidan Herderschee]], [[Juan Maldacena]], *Three Point Amplitudes in Matrix Theory* [[arXiv:2312.12592](https://arxiv.org/abs/2312.12592)]

* [[Aidan Herderschee]], [[Juan Maldacena]], *Soft Theorems in Matrix Theory* [[arXiv:2312.15111](https://arxiv.org/abs/2312.15111)]

category: people


Airy function

#Contents#
* table of contents
{:toc}

## Idea

An **Airy function** is a special function satisfying the [[ordinary differential equation]]

$$
 y''(x) - x \cdot y(x) = 0
$$

and with the Airy integral representation

$$
Ai(x) = \frac{1}{\pi}\int_0^\infty cos\left(\frac{t^3}{3}+tx\right) dt
= \frac{1}{2\pi}\int_{-\infty}^\infty exp\left(\frac{i t^3}{3}+i t x\right) dt
$$

## Properties

This function appears often in the study of oscillating ([[path integral|path]]) [[integrals]], e.g. in [[semiclassical approximation]] to [[quantum mechanics]] and in the geometric approximation to [[wave mechanics]]/[[optics]]. Its asymptotics is important in the study of the singular behaviour of [[light]] in the vicinity of [[caustics]].

The asymptotic expansions for the Airy function have sharp changes at certain lines, observed by G. G. Stokes, and present often in the stationary phase method (cf. [[semiclassical approximation]]). This is called the _[[Stokes phenomenon]]_ and is a special case of the [[wall crossing]].

Airy function appears in the subject of integrable models, related to Painleve transcendents and also in the study of Hermitean [[random matrices]] (work of Tracy and Widom). Airy function has also a remarkable role in the [[Kontsevich]]'s solution to the [[Witten conjecture]].

## References

* [wikipedia](http://en.wikipedia.org/wiki/Airy_function)

* G. B. Airy,  _On the intensity of light in the neighbourhood of a caustic_, Trans. Camb. Phil. Soc., 6 (1838), 379-403.

* [[Maxim Kontsevich]], _Intersection theory on the moduli space of curves and the matrix Airy function_, Comm. Math. Phys. 147 (1992), no. 1, 1--23, [euclid](http://projecteuclid.org/euclid.cmp/1104250524)
* C. A. Tracy, H. Widom, _Level-spacing distributions and the Airy kernel_, Physics Letters B 305 (1-2): 115–118 (1993) [hep-th/9210074](http://arxiv.org/abs/hep-th/9210074), doi; _Level-spacing distributions and the Airy kernel_, Commun. in Math. Physics 159 (1): 151–174 (1994) [euclid](http://projecteuclid.org/euclid.cmp/1104254495) [doi](http://dx.doi.org/10.1007/BF02100489), [MR1257246](http://www.ams.org/mathscinet-getitem?mr=1257246); _On orthogonal and symplectic matrix ensembles_, Commun. in Math. Phys. __177__ (3): 727–754 (1996) [doi](http://dx.doi.org/10.1007/BF02099545), [MR1385083](http://www.ams.org/mathscinet-getitem?mr=1385083)

See also sec. 7.2 in

* Alain Connes, Caterina Consani, _The universal thickening of the field of real numbers_, [arxiv/1202.4377](https://arxiv.org/abs/1202.4377)

For generalizations see the references

* R. N. Fernandez, V. S. Varadarajan, _Matrix Airy functions for compact Lie groups_, Internat. J. Math. __20__ (2009), no. 8, 945–977, [doi](http://dx.doi.org/10.1142/S0129167X09005595), [MR2554728](http://www.ams.org/mathscinet-getitem?mr=2554728)
* R. N. Fernandez, V. S. Varadarajan, D. Weisbart, _Airy functions over local fields_, Lett. Math. Phys. __88__ (2009), no. 1-3, 187–206,  [MR2010d:11138](http://www.ams.org/mathscinet-getitem?mr=2512146), [doi](http://dx.doi.org/10.1007/s11005-009-0311-x)
* Marco Bertola, Boris Dubrovin, Di Yang, _Simple Lie algebras and topological ODEs_, [arxiv/1508.03750](http://arxiv.org/abs/1508.03750)

category: analysis
[[!redirects Airy functions]]
[[!redirects Airy integral]]


Aise Johan de Jong

* [website](http://www.math.columbia.edu/~dejong/)

de Jong is running a wiki on [[algebraic stack]]-theory:

* [[The Stacks Project]]

category: people

[[!redirects Johan de Jong]]


AJ Tolland

* [website](http://www.math.sunysb.edu/~ajt/)

[[!redirects A. J. Tolland]]

category: people


Akash Singh

* [spire page](https://inspirehep.net/authors/1919675?ui-citation-summary=true)

* [institute page](https://web.iisermohali.ac.in/Faculty/anoshjoseph/hep_group/members.html)

## Selected writings

On the [[AdS-QCD correspondence]] (embedding the "hard-wall model" into [[type IIB string theory]]):

* [[Akash Singh]], [[K. P. Yogendran]], *Phases of a 10-D Holographic hard wall model* [[arXiv:2208.09387](https://arxiv.org/abs/2208.09387)]

category: people


Akhil Mathew

* [Webpage](http://math.berkeley.edu/~amathew/)

## Selected writings

On [[stable homotopy theory]]:

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

* [[Akhil Mathew]], *The Galois group of a stable homotopy theory* ([arXiv](http://arxiv.org/abs/1404.2156))

On [[Hurewicz cofibrations]]:

* {#Mathew10a} [[Akhil Mathew]], *Cofibrations*, 2010 ([web](https://amathew.wordpress.com/2010/10/07/cofibrations/))

* {#Mathew10b} [[Akhil Mathew]], *Examples of cofibrations*, 2010 ([web](https://amathew.wordpress.com/2010/10/08/examples-of-cofibrations/))

## Related $n$Lab entries

* [[Galois group]]

category: people


Akihiko Sonoda

## Selected writings

Interpretation in [[holographic QCD]] of the [[Schwinger effect]] as exhibited by the [[DBI-action]] on [[flavor branes]]:

* {#HashimotoOkaSonoda14a} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Magnetic instability in AdS/CFT : Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD_, J. High Energ. Phys. 2014, 85 (2014) ([arXiv:1403.6336](https://arxiv.org/abs/1403.6336))

* {#HashimotoOkaSonoda14b} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Electromagnetic instability in holographic QCD_,  J. High Energ. Phys. 2015, 1 (2015) ([arXiv:1412.4254](https://arxiv.org/abs/1412.4254))

Review:

* [[Akihiko Sonoda]], _Electromagnetic instability in AdS/CFT_, 2016 ([spire:1633963](http://inspirehep.net/record/1633963), [[Sonoda16.pdf:file]])

category: people


Akikazu Hashimoto

* [webpage](https://www.physics.wisc.edu/people/akihashimoto)

## Selected writings

On the [[non-abelian DBI action]] for [[intersecting branes]] with non-abelian [[gauge enhancement]] on their worldvolume:

* [[Akikazu Hashimoto]], [[Washington Taylor]], _Fluctuation Spectra of Tilted and Intersecting D-branes from the Born-Infeld Action_, Nucl. Phys. B503: 193-219, 1997 ([arXiv:hep-th/9703217](https://arxiv.org/abs/hep-th/9703217))

category: people

[[!redirects Aki Hashimoto]]


Akira Furusaki

* [Research group page](https://cems.riken.jp/en/laboratory/qmtrt)

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

## Selected writings

Precursor to the [[K-theory classification of topological phases of matter]]:

* [[Andreas P. Schnyder]], [[Shinsei Ryu]], [[Akira Furusaki]], [[Andreas W. W. Ludwig]], *Classification of topological insulators and superconductors in three spatial dimensions*, Phys. Rev. B **78** 195125 (2008) $[$[doi:10.1103/PhysRevB.78.195125](https://doi.org/10.1103/PhysRevB.78.195125), [arXiv:0803.2786](https://arxiv.org/abs/0803.2786)$]$

category: people


Akira Kono

* [ResearchGate page](https://www.researchgate.net/scientific-contributions/Akira-Kono-21367148)

## Selected writings

On [[unstable topological K-theory]]:

* {#HK03} [[Hiroaki Hamanaka]], [[Akira Kono]]: *On $[X, U(n)]$ when $\text{dim}(X)$ is $2n$*, Journal of Mathematics of Kyoto University **43** 2 (2003) 333-348 [[doi:10.1215/kjm/1250283730](https://doi.org/10.1215/kjm/1250283730)[

* [[Hiroaki Hamanaka]], [[Akira Kono]]: *An application of unstable K-theory*, Journal of Mathematics of Kyoto University **44** 2 (2004) 451-456 [[doi:10.1215/kjm/1250283560](https://doi.org/10.1215/kjm/1250283560)]

* [[Hiroaki Hamanaka]]: *Adams $ e $-invariant, Toda bracket and $[X, U (n)]$*, Journal of Mathematics of Kyoto University **43** 4 (2003) 815-827 [[doi:10.1215/kjm/1250281737](https://doi.org/10.1215/kjm/1250281737)]

On [[algebraic topology]] with focus on [[complex oriented cohomology|complex oriented]] [[Whitehead-generalized cohomology]]:

* [[Dai Tamaki]], [[Akira Kono]], _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

## Related entries

* [[generalized (Eilenberg-Steenrod) cohomology]]

  * [[Brown representability theorem]]

  * [[multiplicative cohomology theory]]

category: people


Akshay Venkatesh

[Website](https://www.ias.edu/math/people/faculty/venkatesh)

## Selected writings

On the [[Mordell conjecture]]:

* {#LawrenceVenkatesh18} [[Brian Lawrence]], [[Akshay Venkatesh]]: _Diophantine problems and $p$-adic period mappings_,  Invent. math. **221** (2020) 893–999  [[arxiv:1807.02721](https://arxiv.org/abs/1807.02721), [doi:10.1007/s00222-020-00966-7](https://doi.org/10.1007/s00222-020-00966-7)]

On the [[relative Langlands program]]

* [[David Ben-Zvi]], [[Yiannis Sakellaridis]], [[Akshay Venkatesh]], _Relative Langlands duality_ ([pdf](https://www.math.ias.edu/~akshay/research/BZSVpaperV1.pdf)).

category: people


AKSZ sigma-model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

What is called the _AKSZ formalism_ -- after the initials of its four authors -- Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], [[Oleg Zaboronsky]] -- is a technique for constructing [[action functional]]s in [[BV-BRST formalism]]  for [[sigma model]] [[quantum field theories]] whose
[[target space]] is an [[symplectic Lie n-algebroid]] $(\mathfrak{P}, \omega)$.

The [[action functional]] of AKSZ theory is that of [[∞-Chern-Simons theory]] induced from the [[Chern-Simons element]] that correspondonds to the [[invariant polynomial]] $\omega$. Details on this are at [∞-Chern-Simons theory -- Examples -- AKSZ theory](http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#ASKZTheory).

## Examples

* to a [[Poisson Lie algebroid]] corresponds the [[Poisson sigma-model]];

* the a [[Courant algebroid]] corresponds the [[Courant sigma-model]];

  in particular to a [[semisimple Lie algebra]] corresponds [[Chern-Simons theory]].

* [[BF-theory]]+[[topological Yang-Mills theory]],

Also, the [[A-model]] and the [[B-model]] topological 2d [[sigma-models]] are examples.

## Definition

A [[sigma-model]] [[quantum field theory]] is, roughly, one

* whose fields are maps $\phi : \Sigma \to X$  to some space $X$;

* whose [[action functional]] is, apart from a
  [[kinetic action|kinetic term]], the [[transgression]]
  of some kind of [[cocycle]]  on $X$
  to the [[mapping space]] $\mathrm{Map}(\Sigma,X)$.

Here the terms "space", "maps" and "cocycles"
are to be made precise in a suitable context.
One says that $\Sigma$ is the _[[worldvolume]]_,
$X$ is the _[[target space]]_ and the cocycle is
the _[[background gauge field]]_ .

For instance the ordinary charged [[particle]]
(for instance an electron) is described by a $\sigma$-model
where $\Sigma = (0,t) \subset \mathbb{R}$ is the abstract
_[[worldline]]_, where $X$ is a smooth
([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]]
(for instance our [[spacetime]]) and where the background cocycle
is a [[circle bundle]] with [[connection on a bundle|connection]]
on $X$ (a degree-2 cocycle in [[ordinary differential cohomology]] of $X$, representing a background _[[electromagnetic field]]_ :
up to a kinetic term the action functional is the [[holonomy]] of the connection over  a given [[curve]] $\phi : \Sigma \to X$.

The $\sigma$-models to be considered here are
_higher_ generalizations of this example,
where the background gauge field is a cocycle of higher degree
(a [[connection on an infinity-bundle|higher bundle with connection]])
and where the worldvolume is
accordingly higher dimensional -- and where $X$ is allowed to be
not just a manifold but an approximation to a
_higher [[orbifold]] (a [[smooth ∞-groupoid]]).

More precisely, here we take the [[category]] of [[space]]s
to be [[dg-geometry|smooth dg-manifolds]].
One may imagine that we can equip this
with an [[internal hom]]
$\mathrm{Maps}(\Sigma,X)$ given by $\mathbb{Z}$-graded objects.
Given [[dg-geometry|dg-manifolds]] $\Sigma$ and $X$
their canonical degree-1 vector fields
$v_\Sigma$ and $v_X$ acting on the mapping space from the
left and right. In this sense their linear combination
$v_\Sigma + k \, v_X$ for some $k \in \mathbb{R}$
equips also $\mathrm{Maps}(\Sigma,X)$ with the structure of
a differential graded smooth manifold.

Moreover, we take the "cocycle" on $X$ to be a
graded [[symplectic structure]] $\omega$,
and assume that there is a kind of Riemannian structure on $\Sigma$
that allows to form the [[transgression]]

$$
  \int_\Sigma \mathrm{ev}^* \omega
  :=
  p_! \mathrm{ev}^* \omega
$$

by [[integral transform|pull-push]] through the canonical
[[correspondence]]

$$
  \mathrm{Maps}(\Sigma,X)
  \stackrel{p}{\leftarrow}
  \mathrm{Maps}(\Sigma,X) \times \Sigma
  \stackrel{ev}{\to}
   X
  \,,
$$

where on the right we have the [[evaluation map]].

Assuming that one succeeds in making precise sense of all this
one expects to
find that $\int_\Sigma \mathrm{ev}^* \omega$ is in turn a symplectic
structure on the mapping space. This implies that the vector field
$v_\Sigma + k\, v_X$ on mapping space has a
[[Hamiltonian]]
$\mathbf{S} \in C^\infty(\mathrm{Maps}(\Sigma,X))$.
The grade-0 components $S_{\mathrm{AKSZ}}$ of $\mathbf{S}$
then constitute a functional on the space
of maps of graded manifolds $\Sigma \to X$. This is
the **AKSZ action functional** defining the AKSZ $\sigma$-model
with target space $X$ and background field/cocycle $\omega$.

In ([AKSZ](#AKSZ)) this procedure is indicated only somewhat vaguely.
The focus of attention there is a discussion, from this perspective,
of the action functionals
of the 2-dimensional $\sigma$-models called the
_[[A-model]]_ and the _[[B-model]]_ .

In ([Roytenberg](#Roytenberg)), a more detailed discussion of the general construction is given, including an explicit

and general formula for $\mathbf{S}$ and hence for $S_{\mathrm{AKSZ}}$ .
For $\{x^a\}$ a coordinate chart on $X$ that formula is
the following.

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

For $(X,\omega)$ a [[symplectic Lie n-algebroid|symplectic dg-manifold]] of grade $n$, $\Sigma$ a smooth compact manifold of dimension $(n+1)$
and $k \in \mathbb{R}$, the
**AKSZ action functional**

$$
  S_{\mathrm{AKSZ},k} :
   \mathrm{SmoothGrMfd}(\mathfrak{T}\Sigma, X)
  \to
  \mathbb{R}
$$

(where $\mathfrak{T}\Sigma$ is the shifted tangent bundle)

is

$$
  S_{\mathrm{AKSZ},k}
   :
  \phi
  \mapsto
  \int_\Sigma
  \left(
    \frac{1}{2}\omega_{ab} \phi^a \wedge d_{\mathrm{dR}}\phi^b
    +
    k
    \,
    \phi^* \pi
  \right)
  \,,
$$

where $\pi$ is the [[Hamiltonian]] for $v_X$ with respect to $\omega$
and where on the right we are interpreting fields as forms on $\Sigma$.

=--

This formula hence defines an infinite class of $\sigma$-models
depending on the target space structure $(X, \omega)$, and on the
relative factor $k \in \mathbb{R}$.  In ([AKSZ](#AKSZ)) it was
already noticed that ordinary [[Chern-Simons theory]] is a special
case of this for $\omega$ of grade 2,  as is the
[[Poisson sigma-model]]
for $\omega$ of grade 1 (and hence, as shown there, also the [[A-model]]
and the [[B-model]]).
The main example in ([Roytenberg](#Roytenberg)) is spelling out the
general case for $\omega$ of grade 2, which is called the
_[[Courant sigma-model]]_ there.

One nice aspect of this construction is that it follows immediately
that the full Hamiltonian $\mathbf{S}$ on mapping space satisfies
$\{\mathbf{S}, \mathbf{S}\} = 0$. Moreover, using the standard formula
for the internal hom of chain complexes one finds that the cohomology
of $(\mathrm{Maps}(\Sigma,X), v_\Sigma + k v_X)$ in degree 0
is the space of functions on those fields that satisfy the
[[Euler-Lagrange equations]] of $S_{\mathrm{AKSZ}}$.
Taken together this implies that
$\mathbf{S}$ is a solution of the "master equation"
of a [[BV-BRST complex]] for the quantum field theory
defined by $S_{\mathrm{AKSZ}}$. This is a crucial ingredient for the
quantization of the model, and this is what the AKSZ construction
is mostly used for in the literature.

## Related concepts

* [[sigma-model]]

* [[schreiber:infinity-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[11d Chern-Simons theory]]

  * [[string field theory]]

  * [[infinite-dimensional Chern-Simons theory]]

  * **AKSZ $\sigma$-model**

    * [[Poisson sigma-model]]

      * [[A-model]], [[B-model]]

    * [[Courant sigma-model]]

      * [[Chern-Simons theory]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

The original reference is

* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], [[Oleg Zaboronsky]], _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))

Dmitry Roytenberg wrote a useful exposition of the central idea of the original work and studied the case of the [[Courant sigma-model]] in

* {#Roytenberg} [[Dmitry Roytenberg]], _AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories_ Lett.Math.Phys.79:143-159,2007 ([arXiv:hep-th/0608150](http://arxiv.org/abs/hep-th/0608150)).

Other reviews include

* [[Noriaki Ikeda]], _Deformation of graded (Batalin-Volkvisky) Structures_ in Dito, Lu, Maeda, Weinstein (eds.) _Poisson geometry in mathematics and physics_ Contemp. Math. 450, AMS (2008)

* [[Noriaki Ikeda]], _Lectures on AKSZ Topological Field Theories for Physicists_ ([arXiv:1204.3714](http://arxiv.org/abs/1204.3714))

A cohomological reduction of the formalism is described in

* F. Bonechi, P. Mnëv, [[Maxim Zabzine]], _Finite dimensional AKSZ-BV-theories_ ([arXiv](http://arxiv.org/abs/0903.0995))

That the AKSZ action on bounding manifolds $\partial \hat \Sigma$ is the integral of the graded symplectic form over $\hat \Sigma$ is theorem 4.4 in

* A. Kotov, T. Strobl, _Characteristic classes associated to Q-bundles_ ([arXiv:0711.4106v1](http://arxiv.org/abs/0711.4106v1))

The discussion of the AKSZ action functional as the [[nLab:∞-Chern-Simons theory]]-functional induced from a [[symplectic Lie n-algebroid]] in [[∞-Chern-Weil theory]] is due discussed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]_, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 ([arXiv:1108.4378](http://arxiv.org/abs/1108.4378))

In the broader context of smooth [[higher geometry]] this is discussed in section 4.3 of

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

Discussion of [[boundary conditions]] for the AKSZ sigma model includes

* [[Peter Bouwknegt]], [[Branislav Jurco]], _AKSZ construction of topological open $p$-brane action and Nambu brackets_, [arxiv/1110.0134](http://arxiv.org/abs/1110.0134)

* {#IkedaXu13} [[Noriaki Ikeda]], Xiaomeng Xu, _Canonical functions and differential graded symplectic pairs in supergeometry and AKSZ sigma models with boundary_ ([arXiv:1301.4805](http://arxiv.org/abs/1301.4805))

The AKSZ model is extended to coisotropic boundary conditions in

* [[Theo Johnson-Freyd]], _Exact triangles, Koszul duality, and coisotropic boundary conditions_ ([arxiv/1608.08598](https://arxiv.org/abs/1608.08598))

An example in [[higher spin gauge theory]] is discussed in

* {#AlkalevGrigorievSkvortsov14} K.B. Alkalaev, Maxim Grigoriev, E.D. Skvortsov, _Uniformizing higher-spin equations_ ([arXiv:1409.6507](http://arxiv.org/abs/1409.6507))

See also

* Theodore Th. Voronov, _Vector fields on mapping spaces and a converse to the AKSZ construction_, [arxiv/1211.6319](http://arxiv.org/abs/1211.6319)

[[!redirects AKSZ sigma-models]]
[[!redirects AKSZ sigma model]]
[[!redirects AKSZ functional]]
[[!redirects AKSZ model]]
[[!redirects AKSZ formalism]]
[[!redirects AKSZ theory]]
[[!redirects AKSZ]]

[[!redirects AKSZ-sigma models]]


Alain Aspect

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

## Selected writings

On [[experiment|experimental]] tests of [[Bell's inequality]] in [[quantum physics]]:

* {#Aspect15} [[Alain Aspect]], *[Closing the Door on Einstein and Bohr’s Quantum Debate](https://physics.aps.org/articles/v8/123)*, Physics **8** 123 (2015)

category: people


Alain Bruguières

* [personal page](https://imag.umontpellier.fr/~bruguieres/)

## Selected writings

On [[Hopf monads]] and [[Hopf adjunctions]]:

* [[Alain Bruguières]], [[Alexis Virelizier]], *Hopf monads*, Advances in Mathematics **215** 2 (2007) 679-733 [[doi:10.1016/j.aim.2007.04.011](https://doi.org/10.1016/j.aim.2007.04.011), [arXiv:math/0604180](https://arxiv.org/abs/math/0604180)]

* [[Alain Bruguières]], [[Steve Lack]], [[Alexis Virelizier]], *Hopf monads on monoidal categories*,  Adv. Math. __227__ 2 (2011) 745-800 [[arXiv:1003.1920](https://arxiv.org/abs/1003.1920), [doi:10.1016/j.aim.2011.02.008](https://doi.org/10.1016/j.aim.2011.02.008)]

category: people

[[!redirects Alain Bruguieres]]


Alain Connes

Alain Connes (born on April 1, 1947) is a French mathematician, Fields medalist (1982), Crafoord prize winner (2001), Professor at [IHÉS](http://www.ihes.fr), Professor at Collège de France and part-time working as a Professor at Vanderbilt University. His interests include [[geometry]], [[topology]], especially K-theory and index theory, [[operator algebras]], the connections between noncommutative geometry and number theory, and more recently also the absolute geometry over a [[field with one element]].

#Contents#
* table of contents
{:toc}

## Research

Most recently Connes is studying number theory including connections to the "thermodynamic aspect" via the [[KMS state|KMS states]], to the absolute geometry via the [[field with one element]] and a related approach to the [[Riemann hypothesis]].

Alain Connes uses intuition from [[mathematical physics]] like the notion of KMS state and has introduced  the noncommutative extensions of [[standard model of particle physics]] predicting a value for the mass of [[Higgs particle]]. He proposed elements of a  unification of [[gravity]] and [[gauge theories]] via [[spectral action]] functionals on spaces of [[spectral triple|spectral triples]].

Connes is most well known for introducing a dominant direction to  [[noncommutative geometry]] where his contributions include, most remarkably, the introduction of [[cyclic homology]] and its connections to the [[K-theory]] of spaces and of [[operator algebra|operator algebras]], study of fundamental examples of noncomutative spaces like
the space of leaves of a foliation, noncommutative tori, groupoid operator algebras, spaces of Penrose tilings, introducing noncommutative motives into operator algebras (with [[Matilde Marcolli]]), Baum-Connes hypothesis, the [[local index formula]], combinatorial approach to Feynman diagrams (Connes-Kreimer Hopf algebra of renormalization) and analytic aspects connecting them to the Birkhoff decomposition, introducing Hopf-cyclic homology etc.

As an inspirative and energetic lecturer Connes also directly contibuted to the popularization of noncommutative geometry and its connections to physics.

## Selected writings

Introducing the [[cyclic category]] and [[cyclic objects]] ([[cyclic sets]], [[cyclic spaces]]) for [[cyclic homology]]:

* [[Alain Connes]], _Cohomologie cyclique et foncteurs $Ext^n$_, C.R.A.S. **269** (1983), Série I, 953-958

On the [[equivariant Chern character]]:

* [[Alain Connes]], [[Paul Baum]], _Chern character for discrete groups_, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 ([doi:10.1016/B978-0-12-480440-1.50015-0](https://doi.org/10.1016/B978-0-12-480440-1.50015-0))

On [[noncommutative geometry]]:

* Alain Connes, _Noncommutative geometry_, Academic Press 1994, 661 p. [PDF](http://www.alainconnes.org/docs/book94bigpdf.pdf)

* Alain Connes, [[Matilde Marcolli]], _Noncommutative geometry, quantum fields and motives_, draft [pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf)

Introducing the [[Bost-Connes system]]:

* {#BostConnes95} [[Jean-Benoit Bost]]; [[Alain Connes]], *Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory*, Selecta Mathematica, New Series **1** 3 (1995) 411-457 [[doi:10.1007/BF01589495](https://doi.org/10.1007/BF01589495), ISSN 1022-1824, [pdf](https://web.archive.org/web/20110806043925/https://alainconnes.org/docs/bostconnesscan.pdf)]

For more see Connes' official [website](http://www.alainconnes.org/en).

## Related entries

* [[noncommutative geometry]], [[operator algebra]], [[C-star algebra]]

* [[spectral triple]]
* [[Hochschild-Kostant-Rosenberg theorem]]
* [[cyclic object]], [[cycle category]], [[cyclic cohomology]]
* [[renormalization]], [[field with one element]]
* [[Bost-Connes system]]

category: people

[[!redirects A. Connes]]

[[!redirects Connes]]


Alain Joye

* [institute page](https://www-fourier.ujf-grenoble.fr/~joye/)

* [GoogleScholar page](https://scholar.google.fr/citations?user=4NUK5AgAAAAJ&hl=en)

## Selected writings

On [[open quantum systems|open]] [[quantum systems]] ([[quantum probability]], [[quantum noise]], [[quantum channels]], ...):

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems I -- The Hamiltonian Approach*, Lecture Notes in Mathematics **1880**, Springer (2006) [[doi:10.1007/b128449](https://doi.org/10.1007/b128449)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems II -- The Markovian approach*,  Lecture Notes in Mathematics **1881**, Springer (2006) [[doi:10.1007/b128451](https://doi.org/10.1007/b128451)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems III -- Recent Developments*, Lecture Notes in Mathematics **1882**, Springer (2006) [[doi:10.1007/b128453](https://doi.org/10.1007/b128453)]

category: people


Alain Prouté

* [Home page](http://www.logique.jussieu.fr/~alp/)

## Related entries

* [[A-infinity-algebra]]

##References:

* Alain Prouté, _Introduction à la Logique Catégorique_, [Lecture Notes](http://www.logique.jussieu.fr/~alp/cours_2010.pdf)

* Alain Prouté, _Modèles minimaux de Baues-Lemaire et Kadeishvili et homologie des fibrations_, [TAC reprints](http://www.tac.mta.ca/tac/reprints/articles/21/tr21abs.html). This is a reprint of his thesis.

category:people


Alain Valette

* [webpage](http://www2.unine.ch/alain.valette)

## Selected writings

On [[equivariant K-theory]] and the [[Baum-Connes conjecture]]:

* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))

category: people


Alain Verschoren

__Alain Verschoren__ was an [[algebraist]] at Antwerp. With [[Fred van Oystaeyen]] he coauthored the first monograph on noncommutative algebraic geometry

_Non-commutative algebraic geometry_ (Springer LNM 887, 1981).

* [webpage](http://www.ua.ac.be/main.aspx?c=alain.verschoren)

Verschoren died suddenly in 2020 at the age of 66, [obituary](https://www.uantwerpen.be/en/about-uantwerp/organisation/in-the-spotlight/alain-verschoren)

* [[Alain Verschoren]], _Compatibility and stability_,
Notas de Matemática [Mathematical Notes], 3. Universidad de Murcia, Secretariado de Publicaciones e Intercambio Científico, Murcia, 1990. xii+81 pp. ISBN: 84-7684-934-6

* J. Mulet, A. Verschoren, _On compatibility. II._, Comm. Algebra 20 (1992), no. 7, 1897--1905. [doi](http://dx.doi.org/10.1080/00927879208824438)

* M. I. Segura, D. Tarazona, A. Verschoren, _On compatibility_,  Comm. Algebra __17__ (1989),  no. 3, 677--690.

category: people

[[!redirects A. Verschoren]]


Alan Carey

* [website](http://www.cirs-tm.org/researchers/researchers.php?id=251)

## Selected writings

On [[twisted K-theory]]:

* [[Alan Carey]], [[Bai-Ling Wang]], p. 5 of: *Thom isomorphism and Push-forward map in twisted K-theory*, Journal of K-Theory **1** 2 (2008) 357-393  ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011))

On [[twisted differential K-theory]]:

* [[Alan Carey]], [[Jouko Mickelsson]], [[Bai-Ling Wang]], _Differential Twisted K-theory and Applications_, Journal of Geometry and Physics, Volume 59, Issue 5, May 2009, Pages 632-653 ([arXiv:0708.3114](https://arxiv.org/abs/0708.3114), [doi:10.1016/j.geomphys.2009.02.002](https://doi.org/10.1016/j.geomphys.2009.02.002))

## Related $n$Lab entries

* [[bundle gerbe]]

* [[Chern-Simons 2-gerbe]]

* [[noncommutative geometry]]

* [[quantum anomaly]]

category: people


Alan Coley

* [webpage](https://www.mscs.dal.ca/~aac/)

## related $n$Lab entries

* [[gravity]], [[general relativity]]

* [[mathematical physics]]

* [[inhomogeneous cosmology]]

* [[Penrose-Hawking singularity theorem]]

category: people


Alan Guth

* [webpage](http://web.mit.edu/physics/people/faculty/guth_alan.html)

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

## Selected writings

On [[false vacuum eternal inflation]]:

* {#GarrigaGuthVilenkin06} [[Jaume Garriga]], [[Alan Guth]], [[Alexander Vilenkin]], _Eternal inflation, bubble collisions, and the persistence of memory_, Phys. Rev. D76:123512, 2007 ([arXiv:hep-th/0612242](https://arxiv.org/abs/hep-th/0612242), [doi:10.1103/PhysRevD.76.123512](https://doi.org/10.1103/PhysRevD.76.123512))

## Related entries

* [[cosmic inflation]]

category: people


Alan Hatcher > history

> see [[Allen Hatcher]]


Alan Hatcher > history 2

[[!redirects Alan Hatcher > history 2]]
* [Home page](https://www.math.cornell.edu/~hatcher/)

category:people


Alan J. Deschner

## Selected writings

On [[Sullivan models]] in [[rational homotopy theory]]

* [[Alan J. Deschner]], *Sullivan's theory of minimal models*, MSc thesis, Univ. British Columbia (1976) [[doi:10.14288/1.0080132](http://hdl.handle.net/2429/20052), [pdf](https://open.library.ubc.ca/media/stream/pdf/831/1.0080132/2)]

## Related entries

* [[Sullivan model of n-spheres]]

* [[Sullivan model of complex projective space]]

[[!redirects Alan Deschner]]
[[!redirects Alan Joseph Deschner]]


Alan M. Turing

A British mathematician, logician and computer scientist (before computers were invented).

## References

* [Wikipedia page](https://en.wikipedia.org/wiki/Alan_Turing)

category: people

[[!redirects Alan Turing]]
[[!redirects A. M. Turing]]


Alan Martin

* Wikipedia entry

## selected writings

* {#HalzenMartin84} [[Francis Halzen]], [[Alan Martin]], _Quarks and Leptons: An Introductory Course in Modern Particle Physics_, Wiley 1984 ([pdf](http://ajbell.web.cern.ch/ajbell/Documents/eBooks/Quarks%20&%20Leptons.pdf))

## related $n$Lab entries

* [[quark]]

* [[lepton]]

* [[QCD]]

* [[standard model of particle physics]]

category: people


Alan Mycroft

* [personal page](https://www.cl.cam.ac.uk/~am21/)

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

* [GoogleScholar page](https://scholar.google.com/citations?user=0ycSEz8AAAAJ&hl=en)

## Selected writings

On [[comonads]] and [[graded modalities]] [[comonads in computer science|in computer science]];

* {#POM13} [[Tomas Petricek]], [[Dominic Orchard]], [[Alan Mycroft]], *Coeffects: Unified Static Analysis of Context-Dependence*, in: *Automata, Languages, and Programming. ICALP 2013*, Lecture Notes in Computer Science **7966** Springer (2013) [[doi:10.1007/978-3-642-39212-2_35](https://doi.org/10.1007/978-3-642-39212-2_35)]

* [[Dominic Orchard]], [[Alan Mycroft]], *A Notation for Comonads*, in: *Implementation and Application of Functional Languages. IFL 2012*, Lecture Notes in Computer Science **8241** [[doi:10.1007/978-3-642-41582-1_1](https://doi.org/10.1007/978-3-642-41582-1_1)]

  > (on [[codo-notation]])

category: people


Alan Robinson

* [personal page](http://homepages.warwick.ac.uk/~maspas/)

* [institute page](https://warwick.ac.uk/fac/sci/maths/people/staff/alan_robinson/)

## Selected writings

Introducing the "[[stable Dold-Kan correspondence]]" (equivalence between [[Eilenberg-MacLane spectrum|$H R$-]][[module spectra]] and [[chain complexes]]) on the level of [[stable homotopy category|stable]] [[homotopy categories]]:

* {#Robinson87} [[Alan Robinson]], *The extraordinary derived category*, Math. Z. **196** 2 (1987) 231-238 [[doi:10.1007/BF01163657](https://doi.org/10.1007/BF01163657)]

category: people


Alan S. Cigoli

* [institute page](https://www.matematica.unito.it/do/docenti.pl/Alias?alanstefano.cigoli#tab-profilo)

## Selected writings

On [[final functors|finality]] of [[internal functors]] between [[internal groupoids]] in [[exact categories]]:

* [[Alan S. Cigoli]], *A characterization of final functors between internal groupoids in exact categories*, Theory and Applications of Categories **33** 11 (2018) 265-275.  [[arXiv:1711.10747](https://arxiv.org/abs/1711.10747), [tac:33-11](http://www.tac.mta.ca/tac/volumes/33/11/33-11abs.html)]

category: people

[[!redirects Alan Cigoli]]


Alan Weinstein

* [website](http://math.berkeley.edu/~alanw/)

## Selected writings

On [[Lagrangian submanifolds]] in [[symplectic geometry]]:

* [[Alan Weinstein]], _Symplectic Manifolds and Their Lagrangian Submanifolds_, Advances in Math. **6** (1971) 329346 \[doi:10.1016/0001-8708(71)90020-X\]

On [[symplectic geometry]]:

* [[Alan Weinstein]], _Symplectic geometry_, Bulletin Amer. Math. Soc. **5** (1981) 1-13 [[doi:10.1090/S0273-0979-1981-14911-9](http://dx.doi.org/10.1090/S0273-0979-1981-14911-9)]

On [[Poisson manifolds]]:

* [[Alan Weinstein]], *The local structure of Poisson manifolds*,  J. Differential Geom. **18** 3 (1983) 523-557 [[doi:10.4310/jdg/1214437787](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-18/issue-3/The-local-structure-of-Poisson-manifolds/10.4310/jdg/1214437787.full)]

On [[geometric quantization]] via [[symplectic groupoids]]:

* {#Weinstein87} [[Alan Weinstein]], _Symplectic groupoids and Poisson manifolds_, Bull. Amer. Math. Soc. (N.S.) **16** (1987) 101-104 [[euclid:bams/1183553676](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-16/issue-1/Symplectic-groupoids-and-Poisson-manifolds/bams/1183553676.full)]

* [[Alan Weinstein]], _Symplectic groupoids, geometric quantization, and irrational rotation algebras_, in:
_Symplectic geometry, groupoids, and integrable systems_ (Berkeley, CA, 1989), Springer (1991) 281-290 [[doi:10.1007/978-1-4613-9719-9_19](https://doi.org/10.1007/978-1-4613-9719-9_19), MR1104934]

* [[Alan Weinstein]], _Tangential deformation quantization and polarized symplectic groupoids_, in: _Deformation theory and symplectic geometry_ (Ascona, 1996), 301-314, Kluwer (1997) [[ISBN:9780792345251](https://link.springer.com/book/9780792345251), [MR1480730](http://www.ams.org/mathscinet-getitem?mr=1480730)]

On [[formal deformation quantization]]:

* [[Alan Weinstein]], *Deformation quantization*, Séminaire Bourbaki volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Talk no. 789 [[numdam:SB_1993-1994__36__389_0](http://www.numdam.org/item/?id=SB_1993-1994__36__389_0)]

On [[geometric quantization]]:

* Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_, AMS (1997) [[pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)]

On [[groupoids]]:

* [[Alan Weinstein]], _Groupoids: Unifying Internal and External Symmetry -- A Tour through some Examples_, Notices of the AMS **43** 7 (1996) [[pdf](http://www.ams.org/notices/199607/weinstein.pdf), [[Weinstein_Groupoids.pdf:file]]]

## Related entries

* [[symplectic geometry]]

  [[symplectic manifold]], [[symplectic groupoid]]

* [[Lagrangian correspondence]], [[canonical transformation]]

* [[symplectic category]]

* [[geometric quantization]]

* [[geometric quantization of symplectic groupoids]]

* [[n-symplectic manifold]]

* [[volume of a Lie groupoid]]

category: people


Alastair Craw

Alastair Craw is a mathematician at University of Bath.

* [webpage](http://people.bath.ac.uk/ac886/)

## related $n$Lab entries

* [[mirror symmetry]]

* [[McKay correspondence]]

category: people


Alastair Grant-Stuart

* [InSpire page](https://inspirehep.net/authors/2016419)

## Selected writings

Formulation of the [[CS/WZW correspondence]] in [[homotopical AQFT]]:

* [[Marco Benini]], [[Alastair Grant-Stuart]], [[Alexander Schenkel]], *The linear CS/WZW bulk/boundary system in AQFT*, Annales Henri Poincaré (2023) [[arXiv:2302.06990](https://arxiv.org/abs/2302.06990)]

category: people


Alastair Hamilton

__Alastair Hamilton__ is a mathematician at the University of Connecticut.

* [webpage](http://www.math.uconn.edu/~hamilton)

## Selected writings

* On the extension of a TCFT to the boundary of the moduli space. [arXiv/1002.2670](http://arxiv.org/abs/1002.2670).

* Cohomology theories for homotopy algebras and noncommutative geometry (with A. Lazarev).  Algebr. Geom. Topol. 9 (2009), 1503--1583, [arxiv/0707.3937](http://arxiv.org/abs/0707.3937)

* Classes on compactifications of the moduli space of curves through solutions to the quantum master equation. Lett. Math. Phys. 89 (2009), no. 2, 115--130.

* Noncommutative geometry and compactifications of the moduli space of curves. Journal of Noncommutative Geometry __4__, 2, pp. 157–188, 2010, [arXiv/0710.4603](http://arxiv.org/abs/0710.4603)

* Graph cohomology classes in the Batalin-Vilkovisky formalism (with A. Lazarev).  J. Geom. Phys. __59__ (2009), no. 5, 555--575, [arxiv/0701825](http://arxiv.org/abs/math/0701825)

* Characteristic classes of A-infinity algebras (with A. Lazarev). J. Homotopy Relat. Struct. __3__ (2008), no. 1, 65--111, [math.QA/0608395](http://arxiv.org/abs/math/0608395)

* Symplectic C-infinity algebras (with A. Lazarev). Mosc. Math. J. __8__ (2008), no. 3, 443--475, 615, [arxiv/0707.3951](http://arxiv.org/abs/0707.3951)

* Symplectic A-infinity algebras and string topology operations (with A. Lazarev). Amer. Math. Soc. Transl. (2), Vol. 224, 2008, 147--157, [arxiv/0707.4003](http://arxiv.org/abs/0707.4003)

* A super-analogue of Kontsevich's theorem on graph homology. Lett. Math. Phys. 76 (2006), no. 1, 37--55, [math.QA/0510390](http://arxiv.org/abs/math/0510390)

* On the classification of Moore algebras and their deformations. Homology, Homotopy Appl. __6__ (2004), no. 1, 87--107, [math.QA/0304314](http://arxiv.org/abs/math/0304314)

* Homotopy algebras and noncommutative geometry (with A. Lazarev). [math.QA/0410621](http://arxiv.org/abs/math/0410621)

On a kind of [[BV-quantization]] of the [[Loday-Quillen-Tsygan theorem]] and relating to the [[large N limit|large $N$-limit]] of [[Chern-Simons theory]]:

* [[Grégory Ginot]], [[Owen Gwilliam]], [[Alastair Hamilton]], [[Mahmoud Zeinalian]], _Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem_, Adv. Math. __409A__ (2022) 108631 [[arXiv:2108.12109](https://arxiv.org/abs/2108.12109), [doi:10.1016/j.aim.2022.108631](https://doi.org/10.1016/j.aim.2022.108631)]

category: people

[[!redirects A. Hamilton]]


Alastair King

* [webpage](http://people.bath.ac.uk/masadk/)

## related $n$Lab entries

* [[geometric invariant theory]]

* [[McKay correspondence]]

* [[Bridgeland stability condition]]

category: people


Alastair Wilson

* [personal page](https://alastairwilson.org/)

* [institute page](https://www.birmingham.ac.uk/staff/profiles/philosophy/wilson-alastair.aspx)

## Selected writings

Relating the "[[possible worlds]]" of [[modal logic]] to the "[[many-worlds interpretation of quantum mechanics|many worlds]]" of [[interpretations of quantum mechanics|quantum philosophy]]:

* [[Alastair Wilson]], *Modal Metaphysics and the Everett Interpretation* (2006) [[philsci:2635](http://philsci-archive.pitt.edu/2635), [pdf](http://philsci-archive.pitt.edu/2635/1/modalmetaphysicsandeverett.pdf)]

* [[Alastair Wilson]], *The Nature of Contingency: Quantum Physics as Modal Realism*, Oxford University Press (2020) [[ISBN:9780198846215](https://global.oup.com/academic/product/the-nature-of-contingency-9780198846215)]

category: people


Albanese variety

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The Albanese variety $Alb(X)$ of a [[projective variety|projective]] [[algebraic variety]] $X$ with a chosen basepoint is the universal way of turning this pointed variety into an [[abelian variety]].  Moreover, the Albanese variety of the Albanese variety is the Albanese variety.  Thus, taking the Albanese variety defines an [[idempotent monad]] on the category of pointed projective algebraic varieties.

## Definition

By 'variety' let us mean a connected complete algebraic variety over an algebraically closed field.  Given any variety $X$ with a chosen basepoint there is an abelian variety called the **Albanese variety** $Alb(X)$.  This is defined by the following universal property: there is a map of pointed varieties called the **Albanese map**

$$i_X \colon X \to A(X)$$

such that any map of pointed varieties $f: X \to A$ where $A$ is abelian factors uniquely as $i_X$ followed by a map of abelian varieties (in particular, a group homomorphism):

$$\overline{f} \colon Alb(X) \to A.$$

That is:

$$  f = \overline{f} \circ i_X $$

This process defines a functor

$$ Alb: Var_* \to AbVar $$

from pointed varieties to abelian varieties which has a right adjoint

$$ U: AbVar \to Var_* $$

sending any abelian variety to its underlying pointed variety.  The right adjoint $U$ is [[faithful functor|faithful]], but more remarkably it is also [[full functor|full]]: any basepoint-preserving map of varieties between abelian varieties is automatically a group homomorphism.  (A proof of this fact is outlined in the article [[abelian variety]].)   Moreover, $U$ is [[monadic functor|monadic]].   As a consequence the composite functor

$$ T = U \circ Alb $$

is an [[idempotent monad]] on $Var_*$, and its algebras are the abelian varieties.

It follows that the Albanese map $i_X \colon X \to A(X)$ is the unit of the monad $T$, and $Alb(Alb(X)) \cong Alb(X)$.   For more details, see the nCafé discussion [Two miracles in algebraic geometry](https://golem.ph.utexas.edu/category/2016/08/the_magic_of_algebraic_geometr.html).

## Properties

The Albanese variety of $X$ is dual, as an abelian variety, to its [[Picard scheme| Picard variety]].

For $X$ a suitably well behaved ([[smooth variety|smooth]] [[complex variety|complex]], [[projective variety|projective]]) [[algebraic variety]] of [[dimension]] $dim(X)$, its Albanese variety is the [[intermediate Jacobian]] in degree $2 dim(X)-1$:

$$
  Alb(X) \coloneqq J^{2 dim(X)-1}(X)
 \,.
$$

## Related concepts

* [[Picard scheme]], [[Jacobian variety]]

## References

* [Is forming the Albanese variety a monad?](http://mathoverflow.net/questions/247104/is-forming-the-albanese-variety-a-monad), MathOverflow.

* [[Patrick Walls]], _Intermediate Jacobians and Abel-Jacobi maps_, 2012 ([[WallsJacobian.pdf:file]])

[[!redirects Albanese varieties]]

---
<http://mathoverflow.net/questions/2548/albanese-schemes-when-does-an-initial-abelian-scheme-exist-under-a-given-sch>

nLab page on [[nlab:Albanese variety]]


Albert algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The _octonionic Albert algebra_ is the [[Jordan algebra]] of $3$-by-$3$ [[hermitian matrices]] over the [[octonions]] $\mathbb{O}$

$$
  \label{3Times3HermitianMatrix}
  \mathfrak{h}_3(\mathbb{O})
  \;\coloneqq\;
  \left\{
  \left(
    \array{
      (x_0 + x_1) & y & \psi_1
      \\
      y^\ast & (x_0 - x_1) & \psi_2
      \\
      \psi_1^\ast & \psi_2^\ast & \phi
    }
  \right)
  \;|\;
  \array{
    x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O}
    \\
    y, \psi_1, \psi_2 \in \mathbb{O}
  }
  \right\}
$$

Similarly the _split-octonionic Albert algebra_ is the algebra of $3$-by-$3$ [[hermitian matrices]] over the [[split-octonions]].

The construction is due to ([Albert 1934](#Albert)), originating in an algebraic approach to [[quantum mechanics]].

## Properties

### Uniqueness

The octonionic and split-octonionic Albert algebras are (up to [[isomorphism]]) the only [[simple algebra|simple]] [[finite-dimensional space|finite-dimensional]] [[formally real algebra|formally real]] [[Jordan algebras]] over the [[real numbers]] that are not [[special Jordan algebra|special]], together comprising the _real Albert algebras_.

Their [[complexifications]] are [[isomorphism|isomorphic]], the _complex-octonionic Albert algebra_, or simply the _complex Albert algebra_.  Analogues exist over any [[field]].

An _exceptional Jordan algebra_ (over any [[field]]) is any [[Jordan algebra]] in which an Albert algebra appears as a [[direct summand]].  Every formally real Jordan algebra over the real numbers is either special or exceptional (so they all have excellent self-esteem).  The exceptional Jordan algebras are related to the [exceptional Lie algebras](exceptional+Lie+group#lie_algebras_2).

### Relation to 10d super-Spacetime
  {#RelationTo10dSuperSpacetime}

The form of the $3 \times 3$-hermitian matrix in (eq:3Times3HermitianMatrix) makes it manifest that the exceptional Jordan algebra is naturally a [[linear map|linear]] [[direct sum]] of the form

$$
  \mathfrak{h}_3(\mathbb{O})
  \;\simeq_{\mathbb{R}}\;
  \mathfrak{h}_2(\mathbb{O})
  \oplus
  \mathbb{O}^2
  \oplus
  \mathbb{R}
$$

via

$$
  \underset{
    \mathfrak{h}_3(\mathbb{O})
  }{
  \underbrace{
  \left\{
  \left(
    \array{
      (x_0 + x_1) & y & \psi_1
      \\
      y^\ast & (x_0 - x_1) & \psi_2
      \\
      \psi_1^\ast & \psi_2^\ast & \phi
    }
  \right)
  \right\}
  }}
  \;\simeq\;
  \underset{
    \mathfrak{h}_2(\mathbb{O})
  }{
  \underbrace{
  \left\{
  \left(
    \array{
      (x_0 + x_1) & y & 0
      \\
      y^\ast & (x_0 - x_1) & 0
      \\
      0  & 0 & 0
    }
  \right)
  \right\}
  }
  }
  \oplus
  \underset{
    \mathbb{O}^2
  }{
  \underbrace{
  \left\{
  \left(
    \array{
      0 & 0 & \psi_1
      \\
      0 & 0 & \psi_2
      \\
      \psi_1^\ast  & \psi_2^\ast & 0
    }
  \right)
  \right\}
  }
  }
  \oplus
  \underset{
    \mathbb{R}
  }{
  \underbrace{
  \left\{
  \left(
    \array{
      0 & 0 & 0
      \\
      0 & 0 & 0
      \\
      0  & 0 & \phi
    }
  \right)
  \right\}
  }
  }
$$

with

$$
\array{
    x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O}
    \\
    y, \psi_1, \psi_2 \in \mathbb{O}
}$$

By the discussion at _[[geometry of physics -- supersymmetry]]_ in the section _[Real spinors in dimension 3,4,6,10](geometry+of+physics+--+supersymmetry#InTermsOfNormedDivisionAlgebraInDimension3To10)_ these summands may be further identified as follows:

* $\mathfrak{h}_2(\mathbb{O}) \simeq \mathbb{R}^{9,1}$ is the incarnation of 10-dimensional [[super-Minkowski spacetime]] via octonionic [[Pauli matrices]];

* under this identification $\mathbb{O}^2 \simeq \mathbf{16}$ is the [[Majorana-Weyl spinor]] [[real spin representation]] of the [[spin group]] $Spin(9,1)$.

$$
  \mathfrak{h}_3(\mathbb{O})
  \;
  \simeq_{\mathbb{R}}
  \;
  \underset{
    dim_{\mathbb{R}} = 26
  }{
  \underbrace{
    \mathbb{R}^{9,1}
      \oplus
    \mathbf{16}
  }}
  \oplus
  \mathbb{R}
  \,.
$$

Under these identifications, $\phi \in \mathbb{R}$ looks like the size of $S^1/(\mathbb{Z}_2)$ in [[Horava-Witten theory]].

This decomposition hence induces an [[action]] of the [[spin group]] $Spin(9,1)$ on the exceptional Jordan algebra. While only the subgroup $Spin(9) \hookrightarrow Spin(9,1)$ of that is an [[isomorphism]] of the [[Jordan algebra]]-[[structure]] itself, the full $Spin(9,1)$-[[action]] does preserve the [[determinant]] on $\mathfrak{h}_3(\mathbb{O})$.

### Automorphisms and exceptional Lie groups
 {#Automorphisms}

+-- {: .num_prop}
###### Proposition
**([[general linear group]] of $Mat_{3\times 3}^{herm}(\mathbb{O})$ preserving [[determinant]] is [[E6]])**

The [[group]] of [[determinant]]-preserving [[linear map|linear]] [[isomorphisms]] of the vector space underlying the octonionic Albert algebra is  the [[exceptional Lie group]] [[E6]]${}_{(-26)}$.

=--

(see e.g. ([Manogue-Dray 09](#ManogueDray09))).

+-- {: .num_prop #JordanAutomorphisms}
###### Proposition
**([[Jordan algebra]] [[automorphism group]] of $Mat_{3\times 3}^{herm}(\mathbb{O})$ is [[F4]])**

The [[group]] of [[automorphism]] with respect to the [[Jordan algebra]] structure $\circ$ on the octonionic Albert algebra is the [[exceptional Lie group]] [[F4]]:

$$
  Aut\left(
    Mat_{3\times 3}^{herm}(\mathbb{O}), \circ
  \right)
  \;\simeq\;
  F_4
  \,.
$$

=--

(e.g. [Yokota 09, section 2.2](#Yokota09))

+-- {: .num_prop #JordanAlgebraAutomorphismsFixingAnImaginaryOctonion}
###### Proposition
**([[Jordan algebra]] [[automorphism group]] of $Mat_{3 \times 3}^{herm}(\mathbb{O})$ fixing an [[imaginary number|imaginary]] [[octonion]])**

Fix an [[imaginary number|imaginary]] [[octonion]] $i \in \mathbb{O}$, hence a $\mathbb{R}$-[[linear map|linear]] [[direct sum]] decomposition

$$
  \mathbb{O}
  \;\simeq_{\mathbb{R}}\;
  \mathbb{C} \oplus V
  \phantom{AA}\text{with}\phantom{AA}
  V \simeq_{\mathbb{R}} \mathbb{C}^3
  \,,
$$

and let

\[
  \label{ComponentwiseiFixingAutomorphism}
  \array{
    Mat_{3 \times 3}^{herm}(\mathbb{O})
     &\overset{w}{\longrightarrow}&
    Mat_{3 \times 3}^{herm}(\mathbb{O})
  }
\]

be given componentwise by the identity on $\mathbb{C}$ and by multiplication with some fixed non-vanishing number on $V$.

Then the [[subgroup]] of the [[Jordan algebra]] [[automorphism]] group $Aut\left(Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4$ (Prop. \ref{JordanAutomorphisms}) of elements that commute with $w$ (eq:ComponentwiseiFixingAutomorphism)

$$
  F_4^w
  \;\coloneqq\;
  \left\{
    \alpha \in F_4 \;\vert\;  w \alpha = \alpha w
  \right\}
$$

is

$$
  F_4^w
  \;\simeq\;
  \big(
    SU(3) \times SU(3)
  \big)/ \mathbb{Z}_3
  \,,
$$

where every element in the [[direct product group]] of [[SU(3)]] with itself

$$
  (A, B) \in SU(3) \times SU(3)
$$

[[action|acts]] on an element

$$
  \underset{
    \in Mat_{3\times 3}^{herm}(\mathbb{O})
  }{\underbrace{\;X\;}}
  \;\simeq\;
  \underset{
    \in Mat_{3\times 3}^{herm}(\mathbb{C})
  }{\underbrace{\;X_{\mathbb{C}}\;}}
  \;+\;
  \underset{
    \in Mat_{3 \times 3}(\mathbb{C})
  }{\underbrace{X_{V}}}
$$

via [[matrix multiplication]] as

\[
  \label{MatrixMultiplicationRepresentationOfiFixingJordanAutomorphism}
  X_{\mathbb{C}}
  +
  X_{\mathbb{V}}
  \;\mapsto\;
  A  X_{\mathbb{C}} A^\dagger
  \;+\;
  B X_{V} A^\dagger
\]

(with $(-)^\dagger$ being the [[conjugate transpose matrix]], hence the [[inverse matrix]] for the [[unitary matrices]] under consideration) and where the [[quotient group|quotient]] is by the [[cyclic group|cyclic]] [[subgroup]]

$$
  \mathbb{Z}_3
  \;\subset\;
  SU(3) \times SU(3)
$$

which is [[generators and relations|generated]] by the [[pair]] of [[diagonal matrices]]

\[
  \label{Z3Generator}
  \left(
    e^{2\pi i \tfrac{1}{3}}   \mathbf{1}_3
    \;,\;
    e^{2\pi i \tfrac{1}{3}}  \mathbf{1}_3
  \right)/
  \;\in \;
  SU(3) \times SU(3)
  \,.
\]

=--

([Yokota 09, theorem 2.12.2](#Yokota09))

\begin{prpn} \label{StabilizerOf4dMinkowskiInsideOctonionicAlbertAlgebra} The further [[subgroup]] of
$
  F_4^w
  \simeq
  \big(
    SU(3) \times SU(3)
  \big) / \mathbb{Z}_3
  \;\subset\;
  F_4
$
(Prop. \ref{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion}) which fixes a subspace

$$
  Mat_{2 \times 2}^{herm}(\mathbb{C})
  \;\subset\;
  \underset{
    Mat_{3 \times 3}^{herm}( \mathbb{O} )
  }{
  \underbrace{
    Mat_{3 \times 3}^{herm}(\mathbb{C})
    \;\oplus\;
    Mat_{3 \times 3}(V)
  }}
$$

(hence, by the [above](#RelationTo10dSuperSpacetime), a 4d [[Minkowski spacetime]] (incarnated via its [[Pauli matrices]]) inside the 10d [[Minkowski spacetime]] inside the octonionic Albert algebra)

is

$$
  \big(
    U(1)
    \times
    SU(2)
    \times
    SU(3)
  \big) /  \mathbb{Z}_6
  \,,
$$

where the [[quotient group|quotient]] is by the [[cyclic group|cyclic]] [[subgroup]] which is [[generators and relations|generated]] by the element

\[
  \left(
    \exp\left(2 \pi i \tfrac{1}{6}\right)\;,\;
    \exp\left(2 \pi i \tfrac{1}{2}\right) \mathbf{1}_2\;,\;
    \exp\left(2 \pi i \tfrac{1}{3}\right) \mathbf{1}_3
  \right)
  \;\in\;
  U(1) \times SU(2) \times SU(3)
  \,.
\]

(Hence this group happens to coincide with the _exact_ [[gauge group]] of the [[standard model of particle physics]], see [there](standard+model+of+particle+physics#GaugeGroup)).

\end{prpn}

This was claimed without proof in [Dubois-Violette & Todorov 18](#DuboisVioletteTodorov18). See also [Krasnov 19](#Krasnov19).

\begin{proof} By Prop. \ref{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion} (eq:MatrixMultiplicationRepresentationOfiFixingJordanAutomorphism) it is clear that the subgroup in question is that represented by those [[pairs]] $(A,B) \in SU(3) \times SU(3)$ for which $A$ is $(1 + 2)$-block diagonal. Such matrices $A$ form the [[subgroup]] of [[SU(3)]] of [[matrices]] that may be written in the form

$$
  diag\left(
    c^2, c^{-1} \mathbf{\sigma}
  \right)
$$

for $c \in U(1)$ and $\mathbf{\sigma} \in $ [[SU(2)]]. The [[kernel]] of the [[group homomorphism]]

\[
  \label{IdentifyingU1SU2inSU3}
  \array{
    U(1) \times SU(2) &\longrightarrow& SU(3)
    \\
    (c,\mathbf{\sigma}) &\mapsto& diag\left(
    c^{2}, c^{-1} \mathbf{\sigma}
  \right)
  }
\]

is clearly the [[cyclic group]]

\[
  \label{Z2Generator}
  \left\{
    (1,\mathbf{1}_2)\;,\;
    \left(
      e^{2\pi i \tfrac{1}{2}},e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2
    \right)
  \right\}
  \;\simeq\;
  \mathbb{Z}_2
  \,.
\]

Hence the subgroup in question is

$$
  \begin{aligned}
    \Big(
      \big( U(1) \times SU(2) \big)/ \mathbb{Z}_2
      \;\times\;
      SU(3)
    \Big)/ \mathbb{Z}_3
    & \simeq
    \Big(
    \Big(
      \big( U(1) \times SU(2) \big)
      \;\times\;
      SU(3)
    \Big) / \mathbb{Z}_2
    \Big) / \mathbb{Z}_3
    \\
    &\simeq
    \Big(
      \big( U(1) \times SU(2) \big)
      \;\times\;
      SU(3)
    \Big) / \mathbb{Z}_6
    \,,
  \end{aligned}
$$

where in the first step we extended the $\mathbb{Z}_2$-[[action]] as the [[trivial action]] on the $SU(3)$-factor, and in the second step we used the evident [[isomorphism]] $\mathbb{Z}_2 \times \mathbb{Z}_3 \simeq \mathbb{Z}_6$ (an application of the "[[fundamental theorem of cyclic groups]]", if you wish).

It remains to see that the [[action]] of $\mathbb{Z}_6$ is as claimed. By the above identification $\mathbb{Z}_6 \simeq \mathbb{Z}_2 \times \mathbb{Z}_3$, it is generated by the _joint_ action of that of the generators of $\mathbb{Z}_3$ and of $\mathbb{Z}_2$, which, by (eq:Z3Generator) and (eq:Z2Generator), is

$$
  \underset{
    \text{generator of}\, \mathbb{Z}_3
  }{
  \underbrace{
  \Big(
    e^{2\pi i \tfrac{1}{3}}  \;,\;
    e^{2\pi i \tfrac{1}{3}}  \mathbf{1}_2\;,\;
    e^{2\pi i \tfrac{1}{3}}  \mathbf{1}_3
  \Big)
  }
  }
  \underset{
    \text{generator of} \, \mathbb{Z}_2
  }{
  \underbrace{
    \Big(
      1 , (e^{2 \pi i \tfrac{1}{2}}) (e^{2 \pi i \tfrac{1}{2}}\mathbf{1}_2), \mathbf{1}_3
    \Big)
  }
  }
  \;=\;
  \left(
    e^{2\pi i \tfrac{1}{3}} \;,\;
    \underset{
      = e^{2\pi i \tfrac{-1}{6}}
    }{
    \underbrace{
      e^{2\pi i \tfrac{1}{2}} e^{2\pi i \tfrac{1}{3}}
    }}
   \;
   ( e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2)
   \;,\;
    e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3
  \right)
$$

as an element in $SU(3) \times SU(3)$, hence is

$$
  \Big(
    e^{2\pi i \tfrac{1}{6}} \;,\;
    e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2 \;,\;
    e^{2\pi i \tfrac{1}{3}}\mathbf{1}_3 \;,\;
  \Big)
  \;\in\;
  U(1) \times SU(2) \times SU(3)
$$

under the lift through (eq:IdentifyingU1SU2inSU3).

\end{proof}

## Related concepts

* [[Kantor-Koecher-Tits construction]] $\rightarrow$ [[E7]]

## References

### General

The original article is

* {#Albert} [[Abraham Adrian Albert]], _On a Certain Algebra of Quantum Mechanics_, Annals of Mathematics, Second Series 35 (1): 65–73, (1934)(doi:[10.2307/1968118](http://dx.doi.org/10.2307/1968118), [JSTOR](https://www.jstor.org/stable/1968118)).

A textbook account is in

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 5 of _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000

Further discussion:

* {#Baez02} [[John Baez]], [section 3.4](http://math.ucr.edu/home/baez/octonions/node12.html) _$\mathbb{O}P^2$ and the Exceptional Jordan Algebra_ of _The Octonions_,  Bull. Amer. Math. Soc. 39 (2002), 145-205. ([web](http://math.ucr.edu/home/baez/octonions/octonions.html))

* {#Yokota09} Ichiro Yokota, _Exceptional Lie groups_ ([arXiv:0902.0431](https://arxiv.org/abs/0902.0431))

See also

* Wikipedia, _[Albert algebra](https://en.wikipedia.org/wiki/Albert_algebra)_

### Possible relation to color gauge structure
{#ReferencesRelationStandardModel}

Attempts to identify aspects of the [[color charge|color]] [gauge group](standard+model+of+particle+physics#GaugeGroup) of the [[standard model of particle physics]] within the exceptional Jordan algebra:

* {#ManogueDray09} [[Corinne Manogue]], [[Tevian Dray]], _Octonions, $E_6$, and Particle Physics_, J. Phys. Conf.Ser.254:012005,2010 ([arXiv:0911.2253](http://arxiv.org/abs/0911.2253))

* {#DuboisVioletteTodorov18} Michel Dubois-Violette, Ivan Todorov, _Exceptional quantum geometry and particle physics II_ ([arXiv:1808.08110](https://arxiv.org/abs/1808.08110))

* Ivan Todorov, _Exceptional quantum algebra for the standard model of particle physics_ ([arXiv:1911.13124](https://arxiv.org/abs/1911.13124))

* {#Krasnov19} [[Kirill Krasnov]], _$SO(9)$ characterisation of the Standard Model gauge group_ ([arXiv:1912.11282](https://arxiv.org/abs/1912.11282))

* Latham Boyle, _The Standard Model, The Exceptional Jordan Algebra, and Triality_ ([arXiv:2006.16265](https://arxiv.org/abs/2006.16265))

[[!redirects exceptional Jordan algebra]]
[[!redirects exceptional Jordan algebras]]

[[!redirects Albert algebra]]
[[!redirects Albert algebras]]

[[!redirects real Albert algebra]]
[[!redirects real Albert algebras]]

[[!redirects octonionic Albert algebra]]
[[!redirects octonionic Albert algebras]]

[[!redirects split-octonionic Albert algebra]]
[[!redirects split-octonionic Albert algebras]]
[[!redirects split octonionic Albert algebra]]
[[!redirects split octonionic Albert algebras]]

[[!redirects complex Albert algebra]]
[[!redirects complex Albert algebras]]
[[!redirects complexified Albert algebra]]
[[!redirects complexified Albert algebras]]
[[!redirects complex-octonionic Albert algebra]]
[[!redirects complex-octonionic Albert algebras]]
[[!redirects complex octonionic Albert algebra]]
[[!redirects complex octonionic Albert algebras]]


Albert Blakers

* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=24490)

## related $n$Lab entries

* [[Blakers-Massey theorem]]

category: people

[[!redirects Albert L. Blakers]]


Albert Burroni

Albert Burroni is a French mathematician based in the University of Paris 7. He has been very influential in applying categorical methods to certain classes of logical structures.

* [Home page](http://www.pps.univ-paris-diderot.fr/~burroni/)

* Article by [[Pierre Ageron]] on [Albert Burroni dans l'école d'Ehresmann. Constructivisme et structuralisme](http://www.math.unicaen.fr/~ageron/histoire-philo.html)in : Actes de la journée mathématique en l'honnneur d'Albert Burroni : catégories, théories algébriques et informatique, Institut de mathématiques de Jussieu (2004) 11-24.

## Selected writings
 {#SelectedWritings}

On (among other things) [[virtual double categories]]:

* [[Albert Burroni]], _$T$-catégories (catégories dans un triple)_, Cahiers de topologie et géométrie différentielle catégoriques **12**.3 (1971) 215-321 [[dml:91097](https://eudml.org/doc/91097), [pdf](http://www.numdam.org/article/CTGDC_1971__12_3_215_0.pdf)]

category:people

[[!redirects Burroni]]


Albert Einstein

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

* Jeroen van Dongen, *Einstein's Unification*, Cambridge University Press (2013) [[doi:10.1017/CBO9780511781377](https://doi.org/10.1017/CBO9780511781377)]

## Writings and References
  {#References}

### The 1919 eclipse

On the (eventual) [[experiment|experimental]] confirmation of [[general relativity]]:

* {#Coles01} Peter Coles, _Einstein, Eddington, and the 1919 Eclipse_ ([arXiv:astro-ph/0102462](https://arxiv.org/abs/astro-ph/0102462))

* Gerard Gilmore, Gudrun Tausch-Pebody, _The 1919 eclipse results which verified General Relativity and their later detractors: a story re-told_ ([arXiv:2010.13744](https://arxiv.org/abs/2010.13744))

### Gravitational waves

The first article that _correctly_ derived [[gravitational waves]] from the [[Einstein equations]] is

* {#Einstein18} [[Albert Einstein]], _Über Gravitationswellen_, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften Berlin (1918), 154-167

In particular this correctly stated that gravitational waves require a [[quadrupole moment]] as a [[source]] (e.g. a rotating binary star system) and not just a [[dipole moment]] (e.g. an oscillating charge) as for [[electromagnetic waves]] (the [[graviton]] has [[spin]] 2, the [[photon]] has spin 1...), thereby correcting a mistake to this effect in the earlier article

* {#Einstein16} [[Albert Einstein]], _Näherungsweise Integration der Feldgleichungen der Gravitation_, Sitzung der physikalisch-mathematischen Klasse vom 22. Juni 1916 ([web](http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/BGG54UCY/index.meta&ww=0.7142&wh=0.7142&wx=0.2315))

The reality of gravitational wave solutions however kept being a cause of concern for many years (Einstein himself was concerned that the linearization approximation used in their derivation might have been too coarse), for a brief account of the early history see

* {#Steinicke05} [[Wolfgang Steinicke]], _Einstein and the Gravitational waves_, Astron. Nachr. / AN 326 (2005), No. 7 – Short Contributions AG 2005 Köln ([pdf](http://www.hs.uni-hamburg.de/DE/GNT/events/pdf/steinicke05.pdf))

* [[Sean Carroll]], _[Einstein vs. Physical Review](http://www.preposterousuniverse.com/blog/2005/09/16/einstein-vs-physical-review/)_, 2005

### Wormholes

Introducing the idea of [[spacetime]] [[wormholes]] in [[gravity]] ([[Einstein-Rosen bridges]]):

* [[Albert Einstein]], [[Nathan Rosen]], *The Particle Problem in the General Theory of Relativity*, Phys. Rev. **48** 73 (1935) ([doi:10.1103/PhysRev.48.73](https://doi.org/10.1103/PhysRev.48.73))

### Entanglement

Introducing the [[Einstein-Podolsky-Rosen paradox]] concerning [[quantum entanglement]]:

* {#EPR} [[Albert Einstein]], [[Boris Podolsky]], [[Nathan Rosen]], _Can the Quantum-Mechanical Description of Physical Reality be Considered Complete?_ Physical Review 47 (10): 777–780. (1935) ([doi:10.1103/PhysRev.47.777](https://doi.org/10.1103/PhysRev.47.777))

### Philosophy of science

* [[Albert Einstein]], *Physik und Realität*, Journal of The Franklin Institute **221** 3 (1936) [[pdf](https://informationphilosopher.com/solutions/scientists/einstein/Physik_und_Realitat.pdf), [[Einstein-PhysikRealitate.pdf:file]]]

On the assumption of a [[spacetime]] [[continuum]] in view of [[quantum physics]] ([[quantum gravity]]):

> Es ist allerdings darauf hingewiesen worden, dass bereits
die Einführung eines raum-zeitlichen Kontinuums angesichts der molekularen Struktur allen Geschehens im Kleinen möglicherweise als naturwidrig anzusehen sei. Vielleicht weise der Erfolg von Heisenbergs Methode auf eine rein algebraische Methode der Naturbeschreibung, auf die Ausschaltung kontinuierlicher Funktionen aus der Physik hin. Dann aber muss auch auf die Verwendung des Raum-Zeit Kontinuums prinzipiell verzichtet werden. Es ist nicht undenkbar, dass der menschliche Scharfsinn einst Methoden finden wird, welche die Beschreitung dieses Weges möglich machen. Einstweilen aber erscheint dieses Projekt ähnlich wie der Versuch, in einem luftleeren Raum zu atmen.

> To be sure, it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuous functions from physics. Then, however, we must also give up, by principle, the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an attempt to breathe in empty space.

## Further selected writings

* [[Albert Einstein]], *Why Socialism?*, Monthly Review (1949) [[web version](https://monthlyreview.org/2009/05/01/why-socialism/), [Wikipedia entry](https://en.wikipedia.org/wiki/Why_Socialism%3F)]

## Quotes

In his Spencer lecture, delivered at Oxford in 1933, Einstein stressed the importance to be accorded to formal beauty:

> Experience can of course guide us in our choice of serviceable mathematical concepts; it cannot possibly be the source from which they are derived; experience of course remains the sole criterion of the serviceability of a mathematical construction for physics, but the truly creative principle resides in mathematics.

Concerning what are now called [[Einstein equations]] ([[equations of motion]] for [[Einstein gravity]]):

> I have learned something else from the theory of gravitation: No ever so inclusive collection of empirical facts can ever lead to the setting up of such complicated equations. A theory can be tested by experience, but there is no way from experience to the setting up of a theory. Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition which determines the equations completely or [at least] almost completely.

(From A. Einstein, *Autobiographical Notes*, as translated in P. A. Schilpp: *Albert Einstein -- Philosopher-Scientist*, MJF books, New York (1949, 1951, 1969, 1970), p. 89, [scan](https://ncatlab.org/nlab/files/Einstein-AutobioNotes-from-Schilpp-p88.jpg))

category: people

[[!redirects Einstein]]


Albert Georg Passegger

category: people


Albert Lautman

__Albert Lautman__ was a French [[philosophy of mathematics|mathematical philosopher]].

He was interested in the structure of advanced mathematics and its creativity and critiziced the [[analytic philosophy|analytic philosophers]] like [[Bertrand Russell|Russell]] and [[Gottlob Frege|Frege]] from the early 20th century who dealt mainly with the issues of a particular logical [[foundation]] and formal aspects and not much on the nature of doing mathematics and its meaning.

As a member of the Resistance, a former prisoner of war and also of Jewish origin, he was killed by a German squad on August 1, 1944.

He influenced the French philosophers [[Gilles Deleuze]] and [[Alain Badiou]], the mathematician and historian of culture and mathematical philosophy [[Fernando Zalamea]], the mathematician, semiolinguist, and philosopher (of science) [[Jean Petitot]], and the philosopher [[David Corfield]].

According to

* Jean Petitot, _Refaire le « Timée » : Introduction à la philosophie mathématique d'Albert Lautman_, 1987

>Although studied very little, Albert Lautman has already been labelled a neo-platonist. Regarded as too speculative despite his exceptional mathematical erudition and his close relationship to Hilbert's axiomatic structuralism, his philosophy of mathematics has not been a subject of specific attention until now. And yet it is, in our opinion, of noteworthy importance. In separating from mathematical theories an additional level of reality lying above, level made up of dialectic- problematic ideas whose understanding is equivalent to the genesis of real theories in which they are determined and achieved, this philosophy of mathematics allows a (transcendental) doctrine of relationships between mathematics and reality to be developed, which goes beyond the dogmatism of logical empiricism without ending up in post-positivistic skepticisms for all that and which articulates the indefinite evolution of the autonomisation and unification of mathematics toward the indefinite production of scientific ontogenèses.

>...

>We believe Albert Lautman is, unemphatically, one of the most inspired philosophers of the century. His theses are of real importance and if we would devote to him only a fraction of the thoughts which we have devoted to another philosopher, who is comparable in stature and opposed in ideas, namely Wittgenstein, he would undoubtedly become one of the most glorious figures of our modernity.

### Lautman on effective and advanced mathematics

According to

* Fernando Zalamea, _Albert Lautman and the Creative
Dialectic of Modern Mathematics_

> With the term 'effective mathematics', Lautman tackles the theories, structures and constructions conceived in the very activity of the mathematician. The term refers to the structure of mathematical knowledge, and what is effective refers to the concrete _action_ of the mathematician to gradually build the mathematical edifice, that such action is constructivist or existential. The mathematical – beyond its ideal set theoretical reconstruction – develops along a hierarchy of real configurations of rather diverse complexity, in which the concepts and examples are connected through structural processes of liberation and saturation, resulting in mathematical creations like mixes between opposite polarities. Lautman detects some _specific_ features of advanced mathematics that are not given in elementary mathematics:

> a) the _complex hierarchisation_ of various theories, irreducible to systems of _intermediate_ deduction;

> b) the _richness_ of the models, irreducible to linguistic manipulation;

> c) the _unity_ of structural methods and of conceptual polarities, beyond the effective multiplicity of models;

> d) the _dynamics_ of the creative activity, in a permanent back-and-forth between freedom and saturation, open to the Platonic division and the Platonic dialectic;

> e) the _mathematically demonstrable relation_ between what is multiple on a given level and what is singular on another, through a sophisticated lattice of mixed ascents and descents.

###Lautman on the abstract world of mathematics and the sensible world of physics as both participating in the same dialectical structure

According to

* Lautman, _Mathematics, Ideas, and the Physical Real_, p231:

>In seeing the sensible thus defined by a mixture of symmetry and dissymmetry, of identity and difference, it is impossible not to recall Plato's [[Timaeus]] (1997). The existence of bodies is based there on the existence of this receptacle that Plato calls the place and whose function consists, as Rivaud has shown in the preface to his edition of the Timaeus (Plato 1932), in making possible the multiplicity of bodies and their alternation in a sin- gle place in the sensible world, just as the role of the Idea of the Other in the intelligible world is to ensure, by its mixture with the Same, both the connection and the separation of types. This reference to Plato enables the understanding that the materials of which the universe is formed are not so much the atoms and molecules of the physical theory as these great pairs of ideal opposites such as the Same and the Other, the Symmetrical and Dissymmetrical, related to one another according to the laws of a harmonious mixture. Plato also suggests more. The properties of place and matter, according to him, are not purely sensible, they are, as Rivaud goes on to say, the geometric and physical transposition of a dialectical theory. It is also possible that the distinction between left and right, as observed in the sensible world, is only the transposition on the plane of experience of a dissymmetrical symmetry which is equally constitutive of the abstract reality of mathematics. A common participation in the same dialectical structure would thus bring to the fore an analogy between the structure of the sensible world and that of mathematics, and would allow a better understanding of how these two realities accord with one another.

### Literature

* wikipedia: [en](http://en.wikipedia.org/wiki/Albert_Lautman)

His collected works have appeared now also in English:

* Albert Lautman, _Mathematics, ideas and the physical real_, collected works, 2011

Jean Petitot wrote an introduction to his philosophy:

* Jean Petitot, [Refaire le « Timée » : Introduction à la philosophie mathématique d'Albert Lautman](http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1987_num_40_1_4488), 1987

An edition of the French-language journal _Philosophiques_ was dedicated to Lautman ([2010, vol. 37 no. 1](http://www.er.uqam.ca/nobel/philuqam/philosophiques/index.php?section=sommaire_par_numeros&vol=37&no=1)). For an English version of one these articles see

* David Corfield, [Lautman and the Reality of Mathematics](http://philsci-archive.pitt.edu/9210/)

category: people, philosophy

[[!redirects Albert Lautman]]
[[!redirects A. Lautman]]


Albert Nijenhuis

Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a mathematician at the University of Pennsylvania.

He got his PhD degree in 1952 from the University of Amsterdam, advised by [[Jan Schouten]].

He introduced the notion of a [[natural bundles]] and [[natural operation]] in [[differential geometry]], which was subsequently developed by others.

## Selected writings

On the [[Schouten-Nijenhuis bracket]]:

* [[Albert Nijenhuis]], _Jacobi-type identities for bilinear differential concomitants of certain tensor fields I_, Indagationes Mathematicae **17** (1955) 390–403 [[doi:10.1016/S1385-7258(55)50054-0](https://doi.org/10.1016/S1385-7258(55)50054-0)]

Introducing the notion of [[natural operations]] in [[differential geometry]]:

* [[Albert Nijenhuis]], _Geometric aspects of formal differential operations on tensor fields_, Proceedings of the International Congress of Mathematicians 1958, 463–469.

* [[Albert Nijenhuis]], _Natural bundles and their general properties_, in: _Differential Geometry (in honor of Kentaro Yano)_, Kinokuniya, Tokyo, 1972, pp. 317–334.

Introducing the [[Frölicher-Nijenhuis bracket]]:

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part I.  Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) **59** (1956) 338–350 and 351–359 [doi:10.1016/s1385-7258(56)50046-7, doI;10.1016/s1385-7258(56)50047-9]

and its refinement for [[almost complex structures]]:

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part II.  Almost-complex structures_, Indagationes Mathematicae (Proceedings) **61** (1958) 414–421 and 422-429 [doi:10.1016/S1385-7258(58)50058-4, doi:10.1016/S1385-7258(58)50057-2]

Giving an explicit formula for what came to be known as the [[Nijenhuis–Richardson bracket]]:

* {#NijenhuisRichardson64} [[Albert Nijenhuis]], [[Roger W. Richardson]], *Cohomology and deformations of algebraic structures*,  Bulletin of the American Mathematical Society **70** 3 (1964) 406–412  [[doi:10.1090/s0002-9904-1964-11117-4](http://dx.doi.org/10.1090/s0002-9904-1964-11117-4)]

* [[Albert Nijenhuis]], [[R. W. Richardson]], *Deformations of Lie Algebra Structures*, Journal of Mathematics and Mechanics **17** 1 (1967) 89-105 [[jstor:24902154](https://www.jstor.org/stable/24902154)]

Beware that this other article has a similar title to [Nijenhuis & Richardson 1964](#NijenhuisRichardson64), but does not seem to mention the [[Nijenhuis–Richardson bracket]]:

* [[Albert Nijenhuis]], [[R. W. Richardson]], *Cohomology and deformations in graded Lie algebras*, Bull. Amer. Math. Soc. **72** (1966) 1-29 [[doi:10.1090/S0002-9904-1966-11401-5](https://doi.org/10.1090/S0002-9904-1966-11401-5)]

## Related entries

* [[Nijenhuis tensor]]

* [[Schouten–Nijenhuis bracket]]

* [[Frölicher–Nijenhuis bracket]]

* [[Nijenhuis–Richardson bracket]]

category: people


Albert R. Hibbs

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

## Selected writings

Introducing [[path integral quantization]]:

* [[Richard Feynman]], [[Albert R. Hibbs]], _Quantum Mechanics and Path Integrals_, New York: McGraw-Hill (1965)

* [[Daniel F. Styer]], [[Richard Feynman]], [[Albert R. Hibbs]], *Quantum Mechanics and Path Integrals: Emended Edition*, Dover (2010) [[ISBN:0486477223](https://store.doverpublications.com/0486477223.html), [pdf](http://www-f1.ijs.si/~ramsak/km1/FeynmanHibbs.pdf)]

category: people

[[!redirects Albert Hibbs]]


Albert Schwartz > history

> see _[[Albert Schwarz]]_