Bernoulli number

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## Motivation

### In Lie theory

In [[Lie theory]], _Bernoulli numbers_ appear as [[coefficients]]
in the linear part of the [[Hausdorff series]] (and the [[recursion|recursive]]
relation for the Dynkin Lie polynomials appearing in the Hausdorff series); this has consequences in [[deformation theory]].
The (determinant of the square root of the) inverse of its generating function appears (for variables in an [[adjoint representation]]) in an expression for the [[Duflo isomorphism]] in Lie theory and in its generalizations in [[knot theory]] etc.

### In algebraic topology

In [[algebraic topology]]/[[cohomology]] Bernoulli numbers appear as the [[coefficients]] of the [characteristic series](genus#LogarithmAndCharacteristicSeries) of the [[A-hat genus]] (see there), and they (or equivalently, their [[generating functions]]) also appear  in the expression for the [[Todd class]].

The Bernoulli numbers are also proportional to the constant terms of the [[Eisenstein series]] and as such appear in the exponential form of the characteristic series of the [[Witten genus]].

Finally they appear as the [[order of a group|order]] of some groups in the [[image of the J-homomorphism]]  (cf. [Adams 65, section 2](#Adams65)).

Of course, all of these cases are related to [[formal group laws]]. Formal groups bear also some other connections to Bernoulli numbers and generalizations like [[Bernoulli polynomial]]s.

### Elsewhere

The [[Riemann zeta-function]] $\zeta$ at negative integral values is proportional to the Bernoulli numbers as

$$
  \zeta(-n) = - \frac{B_{n+1}}{n+1}
  \,.
$$

Bernoulli numbers appear also in [[umbral calculus]]. There are generalizations, for example, [[Bernoulli polynomials]].

They also have applications in analysis ([[Euler-MacLaurin formula]], with applications in numerical analysis).

## Definition

The Bernoulli numbers $B_k$ are [[rational numbers]] given by their [[generating function]], i.e. by the [[equation]] of [[functions]]/[[power series]] $x \mapsto f(x)$

$$
  \frac{x}{
    e^x -1
  }
  =
  \sum_{k = 0}^{\infty} \frac{\beta_k}{k !}x^k
  \,.
$$

The $k$th Bernoulli number $B_k \in \mathbb{Q}$ is, depending on convention, either equal to $\beta_k$ (or sporadically, in older literature, to $(-1)^{k-1} \beta_{2k}$).
If we take generating function
$x+\frac{x}{e^x-1}=\frac{x e^x}{e^x -1}=\frac{-x}{e^{-x}-1}$ this only changes the sign of $B_1$ as all other odd Bernoulli numbers (in standard convention) vanish.

## Properties

The __von Staudt-Clausen theorem__ states that

$$
B_{2n} + \sum_{p-1|2n} \frac{1}{p} \in\mathbb{Z}.
$$

## Related concepts

* [[modified Bernoulli number]]

* [[Bernoulli polynomial]]

* [[umbral calculus]]

## References

### General

Textbook accounts;

* Tom Apostol, Section 12.12 of: _Introduction to Analytic Number Theory_, Springer 1976 ([springer:book/9780387901633](https://www.springer.com/de/book/9780387901633))

Exposition:

* [[Pierre Cartier]], chapter 3 in: _Mathemagics_, Séminaire Lotharingien de Combinatoire 44 (2000), Article B44d ([pdf](http://www.mat.univie.ac.at/~slc/wpapers/s44cartier1.pdf), [[CartierMathemagics.pdf:file]])

* [[John Baez]], _The Bernoulli numbers_, 2003 expository notes ([pdf](http://math.ucr.edu/home/baez/qg-winter2004/bernoulli.pdf))

See also:

* Wikipedia, _[Bernoulli number](http://en.wikipedia.org/wiki/Bernoulli_number)_

* Wolfram MathWorld:  [Bernoulli number](http://mathworld.wolfram.com/BernoulliNumber.html)

* Bernoulli numbers page [bernoulli.org](http://bernoulli.org)

### Lie theory, deformation theory, knot theory, geometry

* MathOverflow [Todd class and Baker-Campbell-Hausdorff, or the curious number 12](http://mathoverflow.net/questions/31972/todd-class-and-baker-campbell-hausdorff-or-the-curious-number-12)

* [[Nikolai Durov|N. Durov]], S. Meljanac, A. Samsarov, [[Z. Škoda]], _A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra_, Journal of Algebra __309__:1, pp.318-359 (2007) [math.RT/0604096](http://arxiv.org/abs/math.RT/0604096), MPIM2006-62

* Vinay Kathotia, _Kontsevich's universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, I_, [math.QA/9811174](http://arxiv.org/abs/math/9811174)

* Emanuela Petracci, _Functional equations and Lie algebras_, PhD thesis, [pdf](http://www1.mat.uniroma1.it/didattica/dottorato/TESI/ARCHIVIO/petracciemanuela.pdf)
* E. Meinrenken, _Clifford algebras and Lie theory_, Springer

* [[Anton Alekseev]], _Bernoulli numbers, Drinfeld associators, and the Kashiwara–Vergne problem_, slides, [pdf](http://www.6ecm.pl/docs/Alekseev.pdf)

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))

  ([[modified Bernoulli numbers]] in the [[universal Vassiliev invariant]] of the [[unknot]])

### In QFT and string theory

Appearance of [[Bernoulli numbers]] in [[perturbative quantum field theory]] and [[string theory]]:

* [[Gerald Dunne]], [[Christian Schubert]], _Bernoulli Number Identities from Quantum Field Theory and Topological String Theory_, Communications in Number Theory and Physics, Volume 7 (2013) Number 2, 225 - 249 ([arXiv:math/0406610](https://arxiv.org/abs/math/0406610))

### In algebraic topology

In [[algebraic topology]]:

* {#Adams65} [[John Adams]], _On the groups $J(X)$ II_, Topology 3 (2) (1965) ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/J-II.pdf))

In the context of the [[A-hat genus]] the Bernoulli numbers are discussed in

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], Section 10.2 of: _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

### Formal groups and Bernoulli polynomials

* Piergiulio Tempesta, _Formal groups, Bernoulli-type polynomials and L-series_, Comptes Rendus de l Académie des Sciences - Series I - Mathematics 07/2007; 345
[doi](http://dx.doi.org/10.1016/j.crma.2007.05.016)

* Stefano Marmi, Piergiulio Tempesta, _Hyperfunctions, formal groups and generalized Lipschitz summation formulas_, Nonlinear Analysis 03/2012; 75:1768-1777 [doi](http://dx.doi.org/10.1016/j.na.2011.09.013)

### Analysis

* [[Terence Tao]]'s blog: [The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation](http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation)

[[!redirects Bernoulli numbers]]


Bernoulli polynomial

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## Idea

Bernoulli polynomials are a family of [[rational number|rational]] [[polynomials]] which generalize [[Bernoulli numbers]].

## Related concepts

* [[Bernoulli number]]

* [[modified Bernoulli number]]

## References

* Piergiulio Tempesta, _Formal groups, Bernoulli-type polynomials and L-series_, Comptes Rendus de l Académie des Sciences - Series I - Mathematics 07/2007; 345
[doi](http://dx.doi.org/10.1016/j.crma.2007.05.016)

* Stefano Marmi, Piergiulio Tempesta, _Hyperfunctions, formal groups and generalized Lipschitz summation formulas_, Nonlinear Analysis 03/2012; 75:1768-1777 [doi](http://dx.doi.org/10.1016/j.na.2011.09.013)

* Francis Clarke, _The universal von Staudt theorems_, Trans. Amer. Math. Soc. 315 (1989), 591-603 [doi](http://dx.doi.org/10.1090/S0002-9947-1989-0986687-3)

[[!redirects Bernoulli polynomials]]


Bernstein-Sato polynomial

Bernstein-Sato polynomial or $b$-function (in [[D-module]] theory)

* I.N. Bernšte ̆ın, The analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen. 6 (1972) 26–40 = Funct. Anal. Appl. 6 (1972) 273–285
* Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello, _Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians_, [arxiv/1105.6270](http://arxiv.org/abs/1105.6270)


Berry connection

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## Idea

Generally, a *Berry phase* is a non-trivial phase picked up by a [[quantum state]] of definite [[energy]] under [[quantum adiabatic theorem|adiabatic]] changes of the [[quantum system]]'s [[Hamiltonian]] around a [[loop]] in its parameter space.

Specifically, in [[condensed matter theory]] and  for a [[crystal|crystalline]] material with a set of isolated (gapped) [[electronic band structure|electronic bands]], the [[partial derivatives]] along [[momentum]]-[[vectors]] of the corresponding [[Bloch states]], projected back onto these states, turn out to canonically define a [[connection on a vector bundle|connection]] [[differential 1-form|1-form]] on the [[vector bundles]] (over the [[Brillouin torus]]) which is spanned by these states. This is called a *Berry connection*, due to ([Berry 84](#Berry84), [Simon 83](#Simon83)). The [[parallel transport]] along this connection describes the change of [[Bloch states]] under the [[quantum adiabatic theorem|adiabatic]] change of their [[momentum]]/[[wave vector]].  The [[curvature 2-form]] of a Berry connection is accordingly called the *Berry curvature*.

The [[holonomy]] of such a Berry connection is called a *[[Berry phase]]*, in general, and a *[[Zak phase]]* ([Zak 89](#Zak89)) when evaluated along one of the non-trivial 1-[[cycles]] of the [[Brillouin torus]].

By default, this is understood to apply to the [[valence bundle]] of a [[crystal|crystalline]] material, but the construction works more generally.

{#TopologicalOrder} If in the case of a gapped [[valence bundle]], hence a [[topological insulator]]-phase, the [[holonomy]] of the Berry connection is [[non-abelian group|non-abelian]] (which may happen, as originally highlighted in [Wilczek & Zee 84](#WilczekZee1984)) then one also says that the topological phase exhibits *[[topological order]]*.

For [[semimetals]] the Berry phases of the [[valence bundle]] around their nodal loci of [[codimension]] 2 are a measure for the [[obstruction]] to adiabatically deforming the semimetal such as to open its gap closures, hence to become a ([[topological insulator|topological]]) [[insulator]] (eg. [Vanderbilt 18, 5.5.2](#Vanderbilt18)).

## Related concepts

* [[Berry's phase]] (needs to be merged)

* [[quantum adiabatic theorem]]

## References

### General

The original article:

* {#Berry84} [[Michael V. Berry]], *Quantal phase factors accompanying adiabatic changes*, Proc. R. Soc. Lond. A **392** (1984) 45–57 ([doi:10.1098/rspa.1984.0023](https://doi.org/10.1098/rspa.1984.0023), [jstor:2397741](https://www.jstor.org/stable/2397741))

The formulation in terms of [[connections]] on [[fiber bundles]] and their [[holonomy]]:

* {#Simon83} [[Barry Simon]], *Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase*, Phys. Rev. Lett. **51** (1983) 2167 ([doi:10.1103/PhysRevLett.51.2167](https://doi.org/10.1103/PhysRevLett.51.2167))

Generalization to (connections with) [[non-abelian group|non-abelian]] [[holonomies]]:

* {#WilczekZee1984} [[Frank Wilczek]],  [[Anthony Zee]], *Appearance of gauge structure in simple dynamical systems*, Physical Review Letters **52** 24 (1984) 2111 $[$[doi:10.1103/PhysRevLett.52.2111](https://doi.org/10.1103/PhysRevLett.52.2111)$]$

The special case of Berry phases around the 1-cycles of a [[Brillouin torus]] -- [[Zak phases]]:

* {#Zak89} [[Joshua Zak]], *Berry’s phase for energy bands in solids*, Phys. Rev. Lett. **62** (1989) 2747 ([doi:10.1103/PhysRevLett.62.2747](https://doi.org/10.1103/PhysRevLett.62.2747))

Review and further discussion:

* [[Michael Berry]], *The quantum phase, five years after*, in: *Geometric phases in physics*,  Adv. Ser. Math. Phys. **5**, World Scientific (1989) 7--28 ([[Berry-FiveYearsAfter.pdf:file]], [doi:10.1142/0613](https://doi.org/10.1142/0613))

* Ming-Che Chang, Qian Niu, *Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields*, J. Phys.: Condens. Matter **20** (2008) 193202 ([doi:10.1088/0953-8984/20/19/193202](https://iopscience.iop.org/article/10.1088/0953-8984/20/19/193202))

* Di Xiao, Ming-Che Chang, Qian Niu, *Berry Phase Effects on Electronic Properties*, Rev. Mod. Phys. **82** (2010) 1959-2007 ([arXiv:0907.2021](https://arxiv.org/abs/0907.2021), [doi:10.1103/RevModPhys.82.1959](https://doi.org/10.1103/RevModPhys.82.1959))

With focus on [[topological phases of matter]] ([[topological insulators]], [[semimetals]], etc.):

* {#Vanderbilt18} [[David Vanderbilt]],  *Berry Phases in Electronic Structure Theory -- Electric Polarization, Orbital Magnetization and Topological Insulators*, Cambridge University Press (2018) ([doi:10.1017/9781316662205](https://doi.org/10.1017/9781316662205))

* [[Jérôme Cayssol]], [[Jean-Noël Fuchs]], Section IV.C of: *Topological and geometrical aspects of band theory*, J. Phys. Mater. **4** (2021) 034007 ([arXiv:2012.11941](https://arxiv.org/abs/2012.11941), [doi:10.1088/2515-7639/abf0b5](https://doi.org/10.1088/2515-7639/abf0b5))

* [[Tudor D. Stanescu]], Chapter 2 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

See also:

* Wikipedia, *[Berry connection and curvature](https://en.wikipedia.org/wiki/Berry_connection_and_curvature)*

### For anyon statistics

On [[anyon statistics|anyon phases]] (specifically in the [[quantum Hall effect]]) as [[Berry phases]] of a [[quantum adiabatic theorem|adiabatic]] transport of anyon positions:

* {#AvrosSchriefferWilczek84} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$

### Experimental observation

Experimental observation of [[Zak phases]]:

* Marcos Atala, Monika Aidelsburger, Julio T. Barreiro, Dmitry Abanin, Takuya Kitagawa, Eugene Demler, Immanuel Bloch: *Direct measurement of the Zak phase in topological Bloch bands*, Nature Physics **9** (2013) 795–800 ([doi:10.1038/nphys2790](https://doi.org/10.1038/nphys2790))

Proposal for experimental realization of [[Berry phases]] around [[codimension]]=2 nodal loci of (quantum simulations of time+space inversion symmetric) [[semi-metals]]:

* Dan-Wei Zhang, Y. X. Zhao, Rui-Bin Liu, Zheng-Yuan Xue, Shi-Liang Zhu, Z. D. Wang, *Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice*, Phys. Rev. A **93** (2016) 043617  ([arXiv:1601.00371](https://arxiv.org/abs/1601.00371), [doi:10.1103/PhysRevA.93.043617](https://doi.org/10.1103/PhysRevA.93.043617))

  > (see sec II.A, these authors stand out as mentioning the relevant [[KO-theory]])

[[!redirects Berry connections]]

[[!redirects Berry curvature]]
[[!redirects Berry curvatures]]

[[!redirects Berry phase]]
[[!redirects Berry phases]]

[[!redirects Zak phase]]
[[!redirects Zak phases]]


Berry's phase

#Contents#
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{:toc}

## Idea

__Berry's geometric phase__ is a correction to the [[wave function]] arising in the study of adiabatic quantum systems; it has been discovered by M. V. Berry. There are analogous effects for other [[wave]] phenomena; there has been also much earlier work of Pancharatnam on the related phenomenon in optics. The origin of the Berry's phase is in nonflatness of a [[parallel transport]] which appears in the corresponding phase factors.

## Related concepts

* [[Berry connection]]

* [[quantum adiabatic theorem]]

* [[adiabatic quantum computation]]

## References

* [[Michael Berry]], _Quantal phase factors accompanying adiabatic changes_, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45--57, [doi](http://dx.doi.org/10.1098/rspa.1984.0023)

* [[Michael Berry]], _The quantum phase, five years after_, in: Geometric phases in physics,  7--28, Adv. Ser. Math. Phys., 5, World Sci. Publ., Teaneck, NJ, 1989.

* Barry Simon, _Holonomy, the quantum adiabatic theorem, and Berry's phase_, Phys. Rev. Lett. __51__ (1983), no. 24, 2167--2170, [MR85e:81024](http://www.ams.org/mathscinet-getitem?mr85e:81024)

* J. M. Robbins, [[Michael Berry]], _The geometric phase for chaotic systems_,  Proc. Roy. Soc. London Ser. A  436  (1992),  no. 1898, 631--661, [doi](http://dx.doi.org/10.1098/rspa.1992.0039), [94a:81036](http://www.ams.org/mathscinet-getitem?mr=94a:81036)

* [[V. I. Arnold]], _Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect_,
Selecta Mathematica __1__:1, 1--19 (1995) [doi](http://dx.doi.org/10.1007/BF01614072)

* Dariusz Chruściński, Andrzej Jamioƚkowski, _Geometric phases in classical and quantum mechanics_, Progress in Math. Physics __36__, Birkhäuser 2004. xiv+333 pp. ISBN: 0-8176-4282-X

* [[Mikio Nakahara]], Chapter 10.6 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

A relation to Chern-Bott connection is explained in 4.1 of lecture notes

* [[Mauro Spera]], _Geometric methods of quantum mechanics_, J.  Geometry and Symmetry in Physics __24__ 1-44 (2011) [euclid](https://projecteuclid.org/euclid.jgsp/1495677645)

* D. Rohrlich, _Berry's phase_, entry in _Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy_, ed. F. Weinert, K. Hentschel, D. Greenberger and B. Falkenburg (Springer), to appear; [arxiv/0708.3749](http://arxiv.org/abs/0708.3749)

* wikipedia: [geometric phase](http://en.wikipedia.org/wiki/Geometric_phase)

* <http://www.mi.infm.it/manini/berryphase.html>

* [[Frank Wilczek]], Alfred Shapere, _Geometric phases in physics_, World Scientific, 1989

* M. O. Katanaev, _On geometric interpretation of the Berry phase_, Rus. Phys. J. 54(2012)1082--1092, Izv. VUZov. Fizika 10(2011) 26--35 [arxiv/1212.1782](http://arxiv.org/abs/1212.1782)

* Maxim Braverman, _The Berry phase and the phase of the determinant_, [arxiv/1310.6332](http://arxiv.org/abs/1310.6332)

Review in a context of ([[adiabatic quantum computation|adiabatic]]) [[quantum computation]]:

* Jiang Zhang, Thi Ha Kyaw, Stefan Filipp, Leong-Chuan Kwek, Erik Sjöqvist, Dianmin Tong, *Geometric and holonomic quantum computation* [[arXiv:2110.03602](https://arxiv.org/abs/2110.03602)]

category: physics

[[!redirects Berry phase]]
[[!redirects Berry's geometric phase]]


Bert Guillou

* [webpage](http://www.ms.uky.edu/~guillou/)

## Selected writings

On [[enriched model categories]]:

* [[Bertrand Guillou]], [[Peter May]], _Enriched model categories and presheaf categories_, New York J. Math. 26 (2020) 37–9 ([arXiv:1110.3567](http://arxiv.org/abs/1110.3567), [](https://www.emis.de/journals/NYJM/j/2020/26-3.html), [NYJM:2020/26-3](https://www.emis.de/journals/NYJM/j/2020/26-3.html))

On [[model category]]-presentations for [[Elmendorf's theorem]] in [[equivariant homotopy theory]]:

* {#Guillou06} [[Bert Guillou]], _A short note on models for equivariant homotopy theory_, 2006 ([pdf](http://www.math.uiuc.edu/~bertg/EquivModels.pdf), [[GuillouEquivariantHomotopy.pdf:file]])

On [[classifying spaces]]/[[universal principal bundles]] for [[equivariant principal bundles]]:

* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]] _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))

On [[equivariant homotopy theory]] and [[Elmendorf's theorem]] via [[enriched model categories]]:

* {#GuillouMayRubin13} [[Bertrand Guillou]], [[Peter May]], [[Jonathan Rubin]], _Enriched model categories in equivariant contexts_, Homology, Homotopy and Applications 21 (1), 2019 ([arXiv:1307.4488](https://arxiv.org/abs/1307.4488), [arXiv:10.4310/HHA.2019.v21.n1.a10](https://dx.doi.org/10.4310/HHA.2019.v21.n1.a10))

On [[equivariant homotopy theory]] and [[equivariant cohomology]]:

* {#Guillou06} [[Bert Guillou]], *Equivariant Homotopy and Cohomology*, lecture notes, 2020 ([pdf](http://www.ms.uky.edu/~guillou/F20/751Notes.pdf), [[GuillouEquivariantHomotopyAndCohomology.pdf:file]])

## Related $n$Lab entries

* [[equivariant homotopy theory]], [[equivariant stable homotopy theory]]

* [[tom Dieck splitting]]

* [[Mackey functor]]

* [[Borel model structure]]

category: people

[[!redirects Bertrand Guillou]]


Bert Janssen

* [webpage](http://www.ugr.es/~bjanssen/)

## Selected writings

On [[brane intersection]] laws of [[M-branes]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter van der Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** (1997) 2757 [[doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]

On [[D8-branes]] as [[black branes]] in [[massive type IIA string theory]]:

* {#JanssenMeessenOrtin99} [[Bert Janssen]], [[Patrick Meessen]], [[Tomas Ortin]], _The D8-brane tied up: String and brane solutions in massive type IIA supergravity_, Phys. Lett. B453 (1999) 229-236 ([spire:494174](http://inspirehep.net/record/494174), doi:10.1016/S0370-2693(99)00315-9)

## Related $n$Lab entries

* [[M-brane]]

* [[brane intersection]]

category: people


Bert Lindenhovius

* [institute page](https://www.jku.at/institut-fuer-mathematische-methoden-in-medizin-und-datenbasierter-modellierung/ueber-uns/team/bert-lindenhovius/)

## Selected writings

Review of the [[ADHM construction]] for [[Yang-Mills instantons]]:

* [[Bert Lindenhovius]], *Instantons and the ADHM construction*, Amsterdam (2011) [[[Lindenhovius-Instantons.pdf:file]]]

Introducing a notion of [[quantum CPOs]] for [[quantum computation]] (via [[quantum sets]] carrying [[quantum relations]]):

* [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], *Quantum CPOs*, EPTCS **340** (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)]

category: people


Bert Schroer

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

## Selected writings

On [[anyons]]:

* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- I: General theory_,   Comm. Math. Phys. Volume 125, Number 2 (1989), 201-226.  ([euclid:cmp/1104179464](http://projecteuclid.org/euclid.cmp/1104179464))

* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- II: Geometric aspects and conformal covariance_,  Reviews in Mathematical PhysicsVol. 04, No. spec01, pp. 113-157 (1992)  ([doi:10.1142/S0129055X92000170](https://doi.org/10.1142/S0129055X92000170) [pdf](ftp://ftp.theorie.physik.uni-goettingen.de/pub/papers/rehren/92/superselection_sectors_II_RMP.pdf))

category: people


Bertfried Fauser

* [webpage](http://www.cs.bham.ac.uk/~fauserb/)

category: people


Bertram Kostant

* [website](http://www-math.mit.edu/~kostant/)

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

## Selected writings

On the [[Jacobson-Morozov theorem]]:

* {#Kostant59} [[Bertram Kostant]], _The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group_, Amer. J. Math. 81 (1959), 973–1032 ([jstor:2372999](https://www.jstor.org/stable/2372999))

category: people

[[!redirects B. Kostant]]


Bertrand Eynard

* [webpage](http://ipht.cea.fr/en/Pisp/bertrand.eynard/)

## related $n$Lab entries

* [[topological recursion]]

* [[spectral curve]]

* [[integrable system]]

category: people


Bertrand Halperin

* [institute page](https://www.physics.harvard.edu/people/facpages/halperin)

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

## Selected writings

On a hierarchy of [[Laughlin wavefunctions]]:

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

On the [[fictitious gauge field]]-[[model (in theoretical physics)|model]] for [[anyon statistics]] and the resulting [[superfluidity]]/[[superconductivity]] *of* [[anyons]]:

* [[Yi-Hong Chen]], [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*,  International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990](#Wilczek90))  $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$

category: people

[[!redirects Bertrand I. Halperin]]


Bertrand Russel > history


Bertrand Russell

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

* [St. Andrews History entry](http://www-history.mcs.st-and.ac.uk/Biographies/Russell.html)

* [Stanford  Encyclopedia of Philosophy](http://plato.stanford.edu/entries/russell/)
*  The Bertrand Russell society, Russell's texts online, [html](http://users.drew.edu/~jlenz/brtexts.html)

## Selected writings

* _[[An Essay on the Foundations of Geometry]]_ (1897)

* _The Principles of Mathematics_ (1903) ([pdf](https://bertrandrussellsocietylibrary.org/br-pom/br-pom.pdf))

* _[[Principia Mathematica]]_  (1910) with [[Alfred Whitehead]]

* _[[Logic as the Essence of Philosophy]]_ (1914)

* _Introduction to Mathematical Philosophy_ (1919) - Russell’s informal account of the mathematics and logic of _Principia Mathematica_ - ([web](https://people.umass.edu/klement/imp/imp.html))

* _[[A History of Western Philosophy]]_ (1945)

## Quotes
 {#Quotes}

{#OnEducation} On education:

From *[Free Thought and Official Propaganda](https://www.gutenberg.org/files/44932/44932-h/44932-h.htm)* (1922):

> "Meanwhile the whole machinery of the State, in all the different countries, is turned on to making defenceless children believe absurd propositions the effect of which is to make them willing to die in defence of sinister interests under the impression that they are fighting for truth and right. This is only one of countless ways in which education is designed, not to give true knowledge, but to make the people pliable to the will of their masters. Without an elaborate system of deceit in the elementary schools it would be impossible to preserve the camouflage of democracy."

> "[...]"

> "It must not be supposed that the officials in charge of education desire the young to become educated. On the contrary, their problem is to impart information without imparting intelligence. Education should have two objects: first, to give definite knowledge—reading and writing, languages and mathematics, and so on; secondly, to create those mental habits which will enable people to acquire knowledge and form sound judgments for themselves. The first of these we may call information, the second intelligence. The utility of information is admitted practically as well as theoretically; without a literate population a modern State is impossible. But the utility of intelligence is admitted only theoretically, not practically; it is not desired that ordinary people should think for themselves, because it is felt that people who think for themselves are awkward to manage and cause administrative difficulties. Only the guardians, in Plato’s language, are to think; the rest are to obey, or to follow leaders like a herd of sheep. This doctrine, often unconsciously, has survived the introduction of political democracy, and has radically vitiated all national systems of education."

## Related entries

* [[Russell's paradox]]

* [[analytic philosophy]]

* [[type theory]]

* [[assertion]]

category: people


Bertrand Souères

* [spire page](https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=a%20B.Soueres.1)

## Selected writings

On [[higher curvature corrections]] to [[D=11 supergravity]] (particularly the [[I8]]-term):

* {#SoueresTsimpis17} [[Bertrand Souères]], [[Dimitrios Tsimpis]], _The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity_, Phys. Rev. D 95, 026013 (2017) ([arXiv:1612.02021](https://arxiv.org/abs/1612.02021))

* [[Bertrand Souères]], _Supergravities in Superspace_, Lyon 2018 ([tel:01998725](https://tel.archives-ouvertes.fr/tel-01998725), [pdf](https://tel.archives-ouvertes.fr/tel-01998725/document))

category: people

[[!redirects Betrand Soueres]]


Bertrand Toen > history

< [[Bertrand Toen]]
[[!redirects Bertrand Toen -- history]]


Bertrand Toën

Bertrand Toën is a mathematician at Université Paul Sabatier in Toulouse.

Together with [[Gabriele Vezzosi]], Bertrand Toën
has laid foundations of what is now called [[derived geometry]].

(The picture [shows](http://owpdb.mfo.de/detail?photo_id=7420) Vezzosi on the left and Toën on the right during a research in pairs stay at [Oberwolfach](http://www.mfo.de/) in 2002).

* [web page](http://www.math.univ-toulouse.fr/~btoen/)

## Selected writings

On [[algebraic stacks]], [[derived algebraic geometry]] and [[rational homotopy theory]]:

* {#Toen} [[Bertrand Toën]], _Champs affines_, Selecta Math. new series **12** (2006), no. 1, 39-135 ([arXiv:math/0012219](https://arxiv.org/abs/math/0012219), [doi:10.1007/s00029-006-0019-z](https://doi.org/10.1007/s00029-006-0019-z))

On [[stacks]] and [[non-abelian cohomology]]:

* {#Toen02} [[Bertrand Toën]], _Stacks and Non-abelian cohomology_, lecture at _[Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory](https://www.msri.org/realvideo/index04.html)_, MSRI 2002 ([slides](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html), [ps](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/meta/aux/toen.ps), [pdf](https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf))

On [[infinity-stacks]] and [[derived stacks]]:

* [[Bertrand Toën]], _Higher and derived stacks: a global overview_, In: _Algebraic Geometry Seattle 2005_, Proceedings of Symposia in Pure Mathematics, Vol. 80.1, AMS 2009 ([arXiv:math/0604504](http://arxiv.org/abs/math/0604504), [doi:10.1090/pspum/080.1](https://doi.org/10.1090/pspum/080.1))

On [[derived algebraic geometry]]:

* {#ToenVezzosi04} [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry II: geometric stacks and applications_, Memoirs of the AMS **193** (2008) [[arXiv:math/0404373](http://arxiv.org/abs/math/0404373), [ams:memo-193-902](https://bookstore.ams.org/memo-193-902)]

See also

* _Vers une interprétation Galoisienne de la théorie de l’homotopie_. Cahiers de Top. et de Geo. Diff. Cat. 43, No. 4 (2002), 257-312. [pdf](https://perso.math.univ-toulouse.fr/btoen/files/2012/04/Galhom.pdf)

Discussion of [[opposite categories]] of [[commutative monoids in a symmetric monoidal category]] regarded as categories of generalized [[affine schemes]] (and [[Spec(Z)]]):

* [[Bertrand Toën]], [[Michel Vaquié]], *Au-dessous de $Spec \mathbb{Z}$*, Journal of K-Theory **3** 3 (2009) 437-500 [[doi:10.1017/is008004027jkt048](https://doi.org/10.1017/is008004027jkt048)]

## Related entries

* [[∞-stack]], [[derived stack]], [[locally constant ∞-stack]]

* [[model structure on simplicial presheaves]]

* [[model structure on cosimplicial abelian groups]]

* [[simplicial ring]], [[derived algebraic geometry]], [[complicial algebraic geometry]]

* [[nonabelian cohomology]]

* [[higher topos theory]]

* [[dg-categories]], [[derived noncommutative algebraic geometry]]

* [[Hall algebra]]

* [[Northwestern TFT Conference 2009]]

category: people

[[!redirects Bertrand Toen]]
[[!redirects Toen]]
[[!redirects Toën]]
[[!redirects B. Toën]]
[[!redirects B. Toen]]


Bertrand's postulate

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\tableofcontents

\section{Introduction}

_Bertrand's postulate_, first proven by analytic methods by Chebyshev in 1852,  is one of the few simple facts providing precise bounds on the occurrence of [[prime numbers]].

\section{Bertrand's postulate}

Given an [[integer]] $i \geq 1$, let $p_{i}$ denote the $i^{th}$ [[prime number|prime]]. _Bertrand's postulate_ is the following.

\begin{thm} \label{TheoremBertrandsPostulate} For any integer $n \geq 1$, we have that $p_{n+1} \lt 2p_{n}$. \end{thm}

\begin{rmk} \label{RemarkAlternativeFormulations} There are a number of equivalent formulations of Theorem \ref{TheoremBertrandsPostulate}. One is that, for any integer $n \gt 3$, there is a prime $p$ such that $n \lt p \lt 2n$. \end{rmk}

\section{Proof given Goldbach's conjecture}
 {#ProofFromGoldbach}

Around 2005, it was noticed by Henry J. Ricardo and Yoshihiro Tanaka ([Ricardo 05](#Ricardo05)) that Bertrand's postulate follows from the [[Goldbach conjecture]]. We record the [[proof]] (of the formulation in Remark \ref{RemarkAlternativeFormulations}).

\begin{proof} If the [[Goldbach conjecture]] holds, there are prime numbers $p_{1}$ and $p_{2}$ such that $2n = p_{1} + p_{2}$. We must have that at least one of $p_{1}$ and $p_{2}$ is greater than or equal to $n$. Without loss of generality, suppose that $p_{1}$ has this property.

If $n$ is not prime, then it is immediate that $n \lt p_{1} \lt 2n$, as required. Suppose instead that $n$ is prime. Then $n+1$ is composite, since it must be divisible by $2$.

By Goldbach's conjecture once more, there are prime numbers $p_{1}'$ and $p_{2}'$ such that $2(n+1) = p_{1}' + p_{2}'$. As above, at least one of $p_{1}'$ and $p_{2}'$ must be greater than or equal to $n+1$. Without loss of generality, suppose that $p_{1}'$ has this property.

Then since $n+1$ is not prime, we have that $n+1 \lt p_{1}' \lt 2(n+1)$, and thus that $n \lt p_{1}' \lt 2(n+1)$.

Now, $p_{1}'$ is not equal to $2n+1$, for this would imply that $p_{2}'=1$, which is impossible. Moreover, $p_{1}'$ is not equal to $2n$, since $2n$ is not prime. We deduce that $p_{1}' \lt 2n$, as required.

\end{proof}

\section{References}

* {#Ricardo05} Henry J. Ricardo, _Goldbach's conjecture implies Bertrand's postulate_, Amer. Math. Monthly, 112, pg. 492, 2005


Bertrand's theorem

Bertrand's theorem in [[classical mechanics]]: the only central forces for which all orbits are closed are $f(r) = a r$ and $f(r) = - b r^{-2}$ where  $a,b\gt 0$. In both cases the orbits are ellipses, centered at origin in the first case and with the focus in the center in the second case.

* [wikipedia](http://en.wikipedia.org/wiki/Bertrand%27s_theorem)


Bessel function

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Properties

### Integral representations
 {#IntegralRepresentations}

$$
  \label{J0AsIntSinOfxCoshtdt}
  J_0(x)
  \;=\;
  \frac{2}{\pi}
  \int_0^\infty
  \sin\left(
    x \, \cosh(t)
  \right)
  \,
  dt
  \phantom{AAAA}
  \text{if}
  \,\,
  x \gt 0
$$

([DLMF](#DLMF) [10.9.9](http://dlmf.nist.gov/10.9#E9))

$$
  \label{N0AsIntSinOfxCoshtdt}
  N_0(x)
  \coloneqq
  Y_0(x)
  \;=\;
  -\frac{2}{\pi}
  \int_0^\infty
  \cos\left(
    x \,\cosh(t)
  \right)
  \,
  dt
  \phantom{AAAA}
  \text{if}
  \,\,
  x \gt 0
$$

([DLMF](#DLMF) [10.9.9](http://dlmf.nist.gov/10.9#E9))

$$
  \label{K0AsIntSinOfxCoshtdt}
  K_0(x)
  \;=\;
  \int_0^\infty
  \cos\left(
    x \,\sinh(t)
  \right)
  \,
  dt
  \phantom{AAAA}
  \text{if}
  \,\,
  x \gt 0
$$

([DLMF](#DLMF) [10.32.6](http://dlmf.nist.gov/10.32#E6))

## Related concepts

* [[special function]]

* [[causal propagator]], [[Hadamard propagator]]

## References

* Milton Abramowitz, Irene Stegun, sections 9, 10, 11 of_Handbook of mathematical functions_, 1964 ([pdf](http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf))

  chapter 9: F. W. J. Olver, _Bessel functions of integer order_ ([pdf](http://www.dam.brown.edu/people/mariom/AM282-01/HANDOUTS/Abramowitz_Stegun_Chapter_9.pdf))

* G. N. Watson, _A treatise on the theory of Bessel functions_, Cambridge University Press 1966 ([web](https://archive.org/details/ATreatiseOnTheTheoryOfBesselFunctions))

* {#DLMF} Digital Library of Mathematical Functions, chapter 10 _[Bessel functions](http://dlmf.nist.gov/10)_

See also

* Wikipedia, _[Bessel function](https://en.wikipedia.org/wiki/Bessel_function)_

[[!redirects Bessel functions]]

[[!redirects Neumann function]]
[[!redirects Neumann functions]]

[[!redirects modified Bessel function]]
[[!redirects modified Bessel functions]]


beta decay

#Contents#
* table of contents
{:toc}

## Idea

A process caused by the [[weak nuclear force]] by which a [[neutron]] decays into a [[proton]], an [[electron]] and a [[neutrino]].

$$
  n^0 \mapsto p^+ + e^- + \overline{\nu}_e
$$

## References

* Wikipedia, _[Beta decay](http://en.wikipedia.org/wiki/Beta_decay)_

[[!redirects weak beta decay]]


beta function

> This entry is about the concept if [[quantum field theory]]. For the [[Euler beta function]], related to the [[Gamma function]], see there.

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* table of contents
{:toc}

## Idea

In statistical field theory and in [[perturbative quantum field theory]], what is called the _beta function_ is the [[logarithm|logarithmic]] [[derivative]] of the [[running of the coupling constants]] under [[renormalization group flow]]. See there for more.

## Examples

### For Yang-Mills theory

In ([Metsaev-Tseytlin 88](#MetsaevTseytlin88)) the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] was obtained as the point-particle limit of the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in ([Nepomechie 83](#Nepomechie83)) that when computed via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].

For more on this see at _[[worldline formalism]]_

## References

The original informal discussion of beta functions for [[scaling transformations]] is due to

* {#GellMannLow54} [[Murray Gell-Mann]] and F. E. Low, _Quantum Electrodynamics at Small Distances_, Phys. Rev. 95 (5) (1954), 1300–1312 ([pdf](http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf))

there denoted "$\psi$". The notation "$\beta$" is due to

* {#Callan70} [[Curtis Callan]], _Broken Scale Invariance in Scalar Field Theory_, Phys. Rev. D 2, 1541, 1970 ([doi:10.1103/PhysRevD.2.1541](https://doi.org/10.1103/PhysRevD.2.1541))

* {#Symanzik70} [[Kurt Symanzik]],  _Small distance behaviour in field theory and power counting_, Communications in Mathematical Physics. 18 (3): 227–246 ([doi:10.1007/BF01649434](https://doi.org/10.1007/BF01649434))

Formulation in the rigorous context of [[causal perturbation theory]]/[[pAQFT]], via the [[main theorem of perturbative renormalization]], is due to

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

reviewed in

* {#Duetsch18} [[Michael Dütsch]], section 3.5.3 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

Discussion for [[Yang-Mills theory]] includes

* {#Nepomechie83} R.I. Nepomechie, _Remarks on quantized Yang-Mills theory in 26 dimensions_, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 (doi:10.1016/0370-2693(83)90385-4)

* {#MetsaevTseytlin88} [[Ruslan Metsaev]], [[Arkady Tseytlin]], _On loop corrections to string theory effective actions_, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (doi:10.1016/0550-3213(88)90306-9)

[[!redirects beta functions]]

[[!redirects beta-function]]
[[!redirects beta-functions]]


beta-gamma system

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#Contents#
* table of contents
{:toc}

## Idea

What is called the _$\beta$-$\gamma$ system_ is a 2-dimensional [[quantum field theory]] defined on [[Riemann surfaces]] $X$ whose [[field (physics)|fields]] are pairs consisting of a $(0,0)$-form and a $(1,0)$-form and whose [[equations of motion]] demand these fields to be [[holomorphic differential forms]].

The name results from the traditional symbols for these fields, which are

$$
  (\gamma,\beta) \in \Omega^{0,\bullet}(X) \oplus \Omega^{1, \bullet}(X)
  \,.
$$

## Definition

We state the definition of the $\beta$-$\gamma$-system as a [[free field theory]] (see there) in [[BV-BRST formalism]], following ([Gwilliam, section 6.1](#Gwilliam)).

We first give the standard variant of the theory, the

* [Abelian massless theory](#AbelianMasslessTheory).

Then we consider the

* [Abelian massive theory](#AbelianMassiveTheory)

* [Holomorphic Chern-Simons theory](#HolomorphicChernSimonsTheory)

### Abelian massless theory
 {#AbelianMasslessTheory}

Let $X$ be a [[Riemann surface]].

**[[kinematics]]**

* the [[field bundle]] $E \to X$ is

  $$
    E \coloneqq \wedge^{0,\bullet}\Gamma(T X) \oplus \wedge^{1,\bullet} \Gamma(T X)
  $$

* hence the ([[abelian sheaf|abelian]]) [[sheaf]] of [[local sections]] is

  $$
    \mathcal{E} = \Omega_X^{0,\bullet} \oplus \Omega_X^{1, \bullet}
    \,,
  $$

  we write

  $\mathcal{E}_c \hookrightarrow \Gamma_X(E)$

  for the sections of [[compact support]]

* the local pairing

  $$
    \langle -,-\rangle_{loc} \colon E \otimes E \to Dens_X
  $$

  with values in the [[density bundle]] is given by [[wedge product]] followed by projection on the $(1,1)$-forms

  $$
    \langle \gamma_1 + \beta_1, \gamma_2, \beta_2\rangle_{loc}
    \coloneqq
    (\gamma_1 \wedge \beta_2 + \gamma_2 \wedge \beta_1)_{|(1,1)}
  $$

* hence the global pairing

  $$
    \langle -,-\rangle \colon \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C}
  $$

  is given by

  $$
    \langle \gamma_1 + \beta_1, \gamma_2, \beta_2\rangle_{loc}
    \coloneqq
    \int_{X}\left(\gamma_1 \wedge \beta_2 + \gamma_2 \wedge \beta_1\right)
  $$

**[[dynamics]]**

* the [[differential operator]]

  $$
    Q \colon \mathcal{E} \to \mathcal{E}
  $$

  is the [[Dolbeault differential]] $\bar \partial$

* hence the [[elliptic complex]] of fields is

  $$
    (\mathcal{E}, Q) = (\Omega_X^{0,\bullet}\oplus \Omega_X^{1,\bullet}, \bar \partial)
  $$

  is the [[Dolbeault complex]];

* and hence the [[action functional]]

  $$
    S \colon \mathcal{E}_c \to \mathcal{C}
  $$

  is

  $$
    \begin{aligned}
      (\gamma + \beta) & \mapsto
     \frac{1}{2}\int_X \langle \gamma+ \beta, \; \bar \partial (\gamma + \beta)\rangle
     \\
     & =  \int_X \beta \wedge \bar \partial \gamma
    \end{aligned}
  $$

### Abelian massive theory
 {#AbelianMassiveTheory}

(...)

### Holomorphic Chern-Simons theory
 {#HolomorphicChernSimonsTheory}

...[[holomorphic Chern-Simons theory]]...

## Properties

### Euler-Lagrange equations of motion

The [[equations of motion]] are

$$
  \bar \partial \gamma = 0
  \;\;,
  \;\;
  \bar \partial\beta = 0
  \,.
$$

## Relation with $\sigma$-models

Consider a [[sigma-model]] $X\hookrightarrow \mathbb{R}^{1,1}$ to the [[target space]] $\mathbb{R}^{1,1}$

$$ S[q,\gamma] = \int_X \partial q \wedge \bar{\partial}\gamma, $$

which has an abelian right-moving Kac-Moody symmetry $q\mapsto q+\lambda$ with $\partial\lambda=0$. We can can consider a theory where this symmetry is promoted to a [[gauge symmetry]], i.e.

$$ S_{\mathrm{gauged}}[q,\gamma] = \int_X \partial_\beta q \wedge \bar{\partial}\gamma, $$

where $\partial_\beta q := \partial q + \beta$ where $\beta\in\Omega^{1,\bullet}(X)$ is the [[connection]].
If we choose the gauge with $q=0$, we obtain the $\beta$-$\gamma$ system with action

$$ S_{\beta\gamma}[\beta,\gamma] = \int_X \beta \wedge \bar{\partial}\gamma. $$

Thus, a $\beta$-$\gamma$ system can be interpreted as a chiral (or Kac-Moody) quotient along a null killing vector of a [[sigma-model]] with target space $\mathbb{R}^{1,1}$ ([LinRoc20](#LinRoc20)).

## Related concepts

* [[pure spinor formalism]]

## References

### General

* [[Nikita Nekrasov]], _Lectures on curved beta-gamma systems, pure spinors, and anomalies_, ([hep-th/0511008](https://arxiv.org/abs/hep-th/0511008))

* [[Anton Zeitlin]], _Beta-gamma systems and the deformations of the BRST operator_, J.Phys. A42:355401 (2009) ([doi](https://doi.org/10.1088/1751-8113/42/35/355401) [arXiv/0904.2234](https://arxiv.org/abs/0904.2234))

Discussion in the context of [[BV-quantization]] and [[factorization algebras]] is in chapter 6 of

* [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](https://people.math.umass.edu/~gwilliam/thesis.pdf))
 {#Gwilliam}

A construction of [[chiral differential operator]]s via quantization of $\beta\gamma$ system in [[BV formalism]] with an intermediate step using factorization algebras:

* [[Vassily Gorbounov]], [[Owen Gwilliam]], Brian Williams, _Chiral differential operators via Batalin-Vilkovisky quantization_, [pdf](http://people.mpim-bonn.mpg.de/gwilliam/cdo.pdf)

* {#LinRoc20} [[Ulf Lindstrom]], [[Martin Rocek]], _$\beta$-$\gamma$-systems interacting with sigma-models_, ([arXiv:2004.06544](https://arxiv.org/abs/2004.06544v3))

### As a fractional level or logarithmic CFT

On the beta-gamma system as the [[su(2)|$\mathfrak{su}(2)$]] [[WZW model]] at fractional [[level (Chern-Simons theory)|level]] $-1/2$:

* {#LesageMathieuRasmussenSaleur02} F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, *The $\mathfrak{su}(2)_{-1/2}$ WZW model and the beta-gamma system*, Nucl. Phys. B **647** (2002) 363-403  $[$[arXiv:hep-th/0207201](https://arxiv.org/abs/hep-th/0207201), doi:10.1016/S0550-3213(02)00905-7a$]$

and its lift to a [[logarithmic CFT]]:

* {#LesageMathieuRasmussenSaleur04} F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, *Logarithmic lift of the $\mathfrak{su}(2)_{-1/2}$ model*, Nuclear Physics B **686** 3 (2004) 313-346 [[doi:10.1016/j.nuclphysb.2004.02.039](https://doi.org/10.1016/j.nuclphysb.2004.02.039)]


beta-reduction

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# $\beta$-reduction
* table of contents
{: toc}

## Idea

In [[type theory]], $\beta$-reduction is a [[conversion rule]] of "computation", which generally replaces more complicated terms with simpler ones.  It was originally identified in the [[lambda-calculus]], where it contrasts with $\alpha$-[[alpha-equivalence|equivalence]] and $\eta$-[[eta expansion|expansion]]; this is the version described below for [[function types]].  The analogous reduction for [[inductive types]] may also be known as $\iota$-reduction.

## "Definition"

In its most general form, $\beta$-reduction consists of rules which specify, for any given [[type]] $T$, if we apply an "eliminator" for $T$ to the result of a "constructor" for $T$, how to "evaluate" the result.  We write

$$s \to_\beta t$$

if the term $s$ beta-reduces to the term $t$.  Sometimes we write $s \to_\beta^* t$ if this reduction takes $n$ steps (leaving off the $*$ to denote $n=1$).  The relation "reduces to" generates an [[equivalence relation]] on the set of terms called **beta equivalence** and often denoted $s =_\beta t$ or $s \equiv_\beta t$.

### Function types

The most common (and original) example is when $T$ is a [[function type]] $A \to B$.

In this case, the constructor of $A \to B$ is a *$\lambda$-expression*: given a term $b$ of type $B$ containing a free variable $x$ of type $A$, then $\lambda x.\, b$ is a term of type $A \to B$.

The eliminator of $A \to B$ says that given a term $f$ of type $A \to B$ and a term $a$ of type $A$, we can [[function application|apply]] $f$ to $a$ to obtain a term $f(a)$ of type $B$.

Now if we first construct a term $\lambda x.\, b\colon A \to B$, and then apply *this term* to $a\colon A$, we obtain a term $(\lambda x.\, b)(a)\colon B$.  The rule of $\beta$-reduction then tells us that this term *evaluates* or *computes* or *reduces* to $b[a/x]$, the result of [[substitution|substituting]] the term $a$ for the variable $x$ in the term $b$.

See [[lambda calculus]] for more.

### Product types

Although function types are the most publicized notion of $\beta$-reduction, basically all types in type theory have a form of it.  For instance, in the negative presentation of a [[product type]] $A \times B$, the constructor is an ordered pair $(a,b)\colon A \times B$, while the eliminators are projections $\pi_1$ and $\pi_2$ which yield elements of $A$ or $B$.

The beta reduction rules then say that if we first apply a constructor $(a,b)$, then apply an eliminator to this, the resulting terms $\pi_1(a,b)$ and $\pi_2(a,b)$ compute to $a$ and $b$ respectively.

## Informal usage

Informally, one sometimes speaks of a "$\beta$-reduction" of a definition or a proof to mean the elimination of levels of abstraction.  For instance, if Theorem A is proven by invoking the existence of a green widget, which is proven by Lemma B, then a $\beta$-reduced proof of Theorem A would proceed instead by using the specific green widget constructed in the proof of Lemma B.

It makes some sense to call this $\beta$-reduction because under [[propositions as types]], the proof of Lemma B would be a [[term]] $\lemmab:B$, whereas the proof of Theorem A would be an application $(\lambda x.\theorema)(b)$, where $\theorema$ is the proof of Theorem A using an unspecified green widget $x$.  This application $(\lambda x.\theorema)(lemmab)$ can then be literally $\beta$-reduced, in the above sense, to $\theorem[lemmab/x]$, in which the specific green widget constructed in the proof of Lemma B is used instead of the unspecified one $x$.

## Related concepts

* [[lambda-calculus]]

* [[eta-reduction]]

* [[confluent category]] / [[diamond]] / [[Church-Rosser theorem]]

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[[!redirects iota-reduction]]
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beta-ring

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## Idea

A **$\beta$-ring** is a [[commutative ring]], $R$, equipped with a set of operations, $\beta_H: R \to R$, indexed by subgroups of [[symmetric groups]], $S_n$, satisfying a number of conditions. They may be seen as collections of integral linear combinations of generalized symmetric powers defined on [[Burnside rings]]. The [[cohomotopy]] of a space, $\pi^0(X)$, is a $\beta$-ring ([Guillot 06, Thrm 4.5](#Guillot06)).

They are completely unrelated to [[relational beta-modules]].

## Related concepts

* [[lambda-ring]]

## References

Note that there are variations in the literature as to the definition of $\beta$-rings. For a close comparison with [[λ-rings]], see

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](http://arxiv.org/abs/math/0612327))

Other references

* Ernesto Vallejo, _The free $\beta$-ring on one generator_, Journal of Pure and Applied Algebra 86(1), 1993, pp. 95-108, ([doi](https://doi.org/10.1016/0022-4049%2893%2990156-N))

* N.W. Rymer, Power operations on the Burnside ring, J. London Math. Sot. (2) 15 (1977) 75-80.

* E. Vallejo, Polynomial operations from Burnside rings to representation functors, J. Pure Appl. Algebra 65 (1990) 163-190.

* G. Ochoa, _Outer plethysm, Burnside rings and $\beta$-rings_, J. Pure Appl. Algebra 55 (1988), 173-195.

* I. Morris and C.D. Wensley, _Adams operations and λ-operations in β-rings_, Discrete Mathematics Volume 50, 1984, Pages 253-270, ([doi](https://doi.org/10.1016/0012-365X%2884%2990053-0))

[[!redirects beta-rings]]
[[!redirects β-rings]]
[[!redirects β-ring]]


Beth definability theorem

##Idea

The _Beth definability theorem_ is a classical result in the [[model theory]] of [[first-order logic]].

It is probably the oldest 'result' in model theory, as it goes back to _Alessandro Padoa_ in 1900. The first proof for the case of predicate logic appeared in Beth (1953) whereas [[Alfred Tarski]] had treated the type-logical case in 1935. Modern proofs rely on the [[Robinson consistency theorem]] or the [[Craig interpolation theorem]].

##Related Concepts

* [[definability]]
* [[first-order logic]]
* [[conceptual completeness]]
* [[Robinson consistency theorem]]
* [[Craig interpolation theorem]]
* [[descent theory]]

##References

*  Evert Willem Beth, _On Padoa's method in the theory of definition_ , Indagationes Mathemathicae **15** (1953) pp.30-39.

For an English translation of Padoa's contribution and historical background information consult:

* J. van Heijenoort (ed.), _From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931_ , Harvard UP 1976.

* M. Aiguier, F. Barbier, _An institution-independent Proof of the Beth Definability Theorem_ , Studia Logica **85** no. 3 (2007) pp.333-359. ([pdf](http://perso.ecp.fr/~aiguierm/publications/communications/sl07.pdf))

A categorical generalization that uses methods of [[descent theory]] appears in:

* [[Michael Makkai]], _Duality and Definability in First Order Logic_ , Memoirs AMS no. 503, Providence 1993.


Betti number

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## Definition

For $n \in \mathbb{Z}$,  the $n$-**Betti number** of a [[chain complex]] $V$ (of [[module]]s over a [[ring]] $R$) is the [[rank]]

$$
  b_n(V) := rk_R H_n(V)
$$

of its $n$th [[homology group]], regarded as an $R$-[[module]].

For $X$ a [[topological space]], its $n$th Betti number is that of its [[singular homology]]-complex

$$
  b_n(V) = rk_R H_n(X, R)
  \,.
$$

For $X$ moreover a [[smooth manifold]] then by the [[de Rham theorem]] this is equivalently the dimension of the [[de Rham cohomology]] groups.

## Properties

### Euler characteristic

The alternating sum of all the Betti numbers is -- if it exists -- the [[Euler characteristic]].

## Related concepts

* [[Hodge number]]

* [[etale topos]]

## References

Named after [[Enrico Betti]].

(...)

[[!redirects Betti numbers]]


BF-theory

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## Idea

What is called **BF-theory** is a [[topological quantum field theory]] defined by an [[action functional]] $S_{BF}$  on a space of certain [[connection on a bundle|connections]] and [[differential forms]] over a [[4-manifold]] $X$, such that locally on $X$ the [[space of field histories]] is given by

1. [[Lie algebra-valued 1-forms]] $\;A$ with values in some [[Lie algebra]] $\mathfrak{g}_1$ and with [[field strength]]/[[curvature]] 2-form $F_A$;

1. [[differential 2-forms]] $\;B$ with values in some [[Lie algebra]] $\mathfrak{g}_2$,

1. together with

   1. a homomorphism $\partial \colon \mathfrak{g}_2 \to \mathfrak{g}_1$

   1. an [[invariant polynomial]] $\langle -,- \rangle$

by

$$
  S_{BF}
  \;\colon\;
  (A,B) \;\mapsto\;
  \int_X \langle F_A \wedge \partial B\rangle
  \,.
$$

There is not much of a proposal in the literature for how to make sense of this expression globally. It has been observed that it looks like the action functional is one on [[∞-Lie algebra-valued forms]] with values in a [[strict Lie 2-algebra]] $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g}_1)$.

This would suggest that the BF-action functional is to be regarded as a functional on the [[space]] ([[2-groupoid]]) of $G$-[[principal 2-bundles]]
with [[connection on a 2-bundle]], where $G = (G_2 \to G_1)$ is a [[Lie 2-group]] integrating $\mathfrak{g}$.

If one couples to the above action functional that for [[topological Yang-Mills theory]] and a [[cosmological constant]] with coefficients as in

$$
  \int_X( \langle F_A \wedge B\rangle - \frac{1}{2} \langle F_A \wedge F_A\rangle - \frac{1}{2}\langle \partial B \wedge \partial B\rangle)
$$

then this is the [[generalized Chern-Simons theory]] action functional induced from the canonical [[Chern-Simons element]] on the [[strict Lie 2-algebra]] $\mathfrak{g}$. See [[Chern-Simons element]] for details.

## Applications

Much of the interest in BF-theory results from the fact that on a 4-dimensional manifold, to some extent the [[Einstein-Hilbert action]] for [[gravity]] may be encoded in BF-theory form. See [[gravity as a BF-theory]].

## References

BF theory was maybe first considered in

* [[Gary Horowitz]], _Exactly soluable diffeomorphism invariant theories_ Commun. Math. Phys. 125, 417-437 (1989)

The observation that the BF-theory action functional looks like it should be read as a functional on a space of  [[∞-Lie algebra valued forms]] with values in a [[strict Lie 2-algebra]] possibly appears in print first in section 3.9 of

* [[Florian Girelli]], Hendryk Pfeiffer, _Higher gauge theory -- differential versus integral formulation_, [arXiv:hep-th/0309173](http://arxiv.org/abs/hep-th/0309173)

The observation that coupled to [[topological Yang-Mills theory]] it can be read as the [[schreiber:∞-Chern-Simons theory]] action functional on [[connections on 2-bundles]] is in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-conections_ (web)
{#SSSI}

and a more comprehensive discussion is in section 4.3 of

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

See also

* Aristide Baratin, Florian Girelli, Daniele Oriti,  _Diffeomorphisms in group field theories_, Physical Review D, vol. 83, Issue 10, id. 104051, [doi](http://dx.doi.org/10.1103/PhysRevD.83.104051), [arxiv/1101.0590](http://arxiv.org/abs/1101.0590)

There is a more general BFCG action introduced by Girelli, Pfeiffer and Popescu
which has been shown to be a special case of the categorified BF-theory
in

* João Martins, Aleksandar Miković, _Lie crossed modules and gauge-invariant actions for 2-BF theories_, Adv. Theor. Math. Phys. __15__:4 (2011), 913-1199 [euclid](http://projecteuclid.org/euclid.atmp/1339438351)

Relation to [[Einstein gravity]] (in [[first-order formulation of gravity|first-order formulation]]):

* Mariano Celada, Diego González, Merced Montesinos, _BF gravity_ [[arxiv/1610.02020](https://arxiv.org/abs/1610.02020)]

* [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]

On a version of BF-theory in [[arithmetic]] related to non-[[orientation|orientability]] of [[arithmetic schemes]]:

* [[Magnus Carlson]], [[Minhyong Kim]], _A note on abelian arithmetic BF-theory_, ([arXiv:1911.02236](https://arxiv.org/abs/1911.02236))

[[!redirects BF theory]]


BFSS matrix model

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## Idea

The _BFSS matrix model_ ([Banks-Fischler-Shenker-Susskind 96](#BanksFischlerShenkerSusskind96), [Seiberg 97](#Seiberg97)) is the description of the [[worldline]] dynamics of interacting  [[D0-branes]]. In the [[large N limit]] of a large number of [[D0-branes]] this is  supposed to encode the [[non-perturbative quantum field theory|strong coupling limit]] of [[type IIA string theory]] known as _[[M-theory]]_  at least in certain corners of its [[moduli space]].

The BFSS model is a limiting case of the _[[BMN matrix model]]_, which improves on some of its shortcomings (see the [Open problems](#OpenProblems) below).

The BFSS matrix model was argued to arise in several seemingly rather different (but apparently secretly equivalent) ways:

1. as the [[worldline]] theory of a large number of [[D0-branes]] in [[type IIA string theory]],

1. as the [[Kaluza-Klein compactification]] of [[10d super Yang-Mills theory]] to 1+0 space dimensions,

1. {#AsM2Regularization} as a certain non-commutative regularization  of the [[light-cone gauge quantization]] of the [[Green-Schwarz sigma-model]] for the [[M2-brane]] ([Nicolai-Helling 98](#NicolaiHelling98), [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)).

   In this picture matrix blocks around the diagonal correspond to blobs of [[membrane]], while off-diagonal matrix elements correspond to thin tubes of membrane connecting these blobs.

> graphics grabbed from [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)

In any case, the BFSS matrix model ends up being a [[quantum mechanics|quantum mechanical]] system whose bosonic degrees of freedom are a set of 9+1 large [[matrices]]. These play the role of would-be [[coordinate functions]] and their [[eigenvalues]] may be interpreted as points in a [[non-commutative geometry|non-commutative]] [[spacetime]] thus defined.

There is also the [[IKKT matrix model]], which takes this one step further by reducing one dimension further down to [[D(-1)-branes]] in [[type IIB string theory]].

See also at _[[membrane matrix model]]_.

## Open problems
 {#OpenProblems}

### General
 {#OpenProblemsGeneral}

In the 1990s there was much excitement about the BFSS model (and then its cousin, the [[IKKT matrix model]]), as people hoped it might provide a definition of [[M-theory]], whose formulation remains [elusive](M-theory#TheOpenProblem). It is from these times that [[Edward Witten]] changed the original suggestion that "M" is for "magic, mystery and membrane" to the suggestion that it is for "magic, mystery and matrix". (See [Witten's 2014 Kyoto prize speech](M-theory#Witten14), last paragraph.)

However, while the BFSS matrix model clearly sees something M-theoretic, just as clearly it is not the full answer. Notably it needs for its definition an ambient [[asymptotic boundary|asymptotic]] [[Minkowski spacetime]] [[background field|background]], a light cone limit and a peculiar scaling of [[string coupling]] over [[string length]], all of which means that it pertains to a particular corner of a full theory.

From [Nicolai-Helling 1998, p. 2](#NicolaiHelling98):

> Despite the recent excitement, however, we do not think that M(atrix) theory and the $d= 11$ supermembrane in their present incarnation are already the final answer in the search for M-Theory, even though they probably are important pieces of the puzzle. There are still too many ingredients missing that we would expect the final theory to possess. For one thing, we would expect a true theory of quantum gravity to exhibit certain pregeometrical features corresponding to a “dissolution” of space-time and the emergence of some kind of non-commutative geometry at short distances; although the matrix model does achieve that to some extent by replacing commuting coordinates by non-commuting matrices, it seems to us that a still more radical departure from conventional ideas about space and time may be required in order to arrive at a truly background independent formulation (the matrix model “lives” in nine _flat_ transverse dimensions only). Furthermore, there should exist some huge and so far completely hidden symmetries generalizing not only the duality symmetries of extended supergravity and string theory, but also the principles underlying general relativity.

From [Mohammed-Murugan-Nastase 2010, p. 6](#MohammedMuruganNastase10):

>  If Matrix theory is to correctly describe M-theory (and its dimensional reduction to type IIA string theory) then it should be able to describe all D−branes in the theory and not just D2−branes. For example, a D4−brane wrapping an $S^4$ was found in [9], following the earlier works of [10, 11], but the solution is not without several unresolved subtleties. In general, finding the complete spectrum of D−branes from Matrix
theory remains a very difficult problem. The D2− and D4−branes already found are reductions to
ten dimensions of M2− and M5−branes, and while they are a minimum necessary for the spectrum of M-theory, they are by no means sufficient. Indeed, we would also need to find a D6−brane, coming from an eleven dimensional KK monopole, and a D8−brane.

Then, even assuming that all the crucial [[generalized (Eilenberg-Steenrod) cohomology|cohomological]] aspects of [[D-brane]] and [[M-brane]] charges (in [[twisted differential K-theory]], [[twisted cohomotopy]] etc.) are secretly encoded in the matrix model, somehow, none of this is manifest, making the matrix model spit out numbers about a conceptually elusive theory in close analogy to how [[lattice QCD]] produces numbers without informing us about the actual conceptual nature of [[confinement|confined]] [[hadron]] physics.

A similar assessment has been given by [[Greg Moore]], from pages 43-44 of his _[[Physical Mathematics and the Future]]_ (2014, [here](Physical+Mathematics+and+the+Future#AGoodStartWasGivenByTheMatrixTheory)):

> A good start $[$with defining M-theory$]$ was given by the Matrix theory approach of Banks, Fischler, Shenker and Susskind. We have every reason to expect that this theory produces the correct scattering amplitudes of modes in the 11-dimensional supergravity multiplet in 11-dimensional Minkowski space - even at energies sufficiently large that black holes should be created. (This latter phenomenon has never been explicitly demonstrated). But Matrix theory is only a beginning and does not give us the whole picture of M-theory. The program ran into increasing technical difficulties when more complicated compactifications were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[$...$]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics.

> If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue.

### Ground state
 {#ProblemsGroundState}

There are furcher technical open issues, such as the open question whether the theory has a decent ground state the way it needs to have to make sense (see the references [below](#ReferencesGroundStateProblem)).

## Related concepts

[[!include brane matrix models -- content]]

* [[SYK model]]

* [[Myers effect]]

## References
 {#References}

### General

First inkling of [[matrix models]] from the [[large N limit]] of [[QCD]]:

* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))

* A. Gonzalez-Arroyo, M. Okawa, _A twisted model for large $N$ lattice gauge theory_, Physics Letters B Volume 120, Issues 1–3, 6 January 1983, Pages 174-178 (doi:10.1016/0370-2693(83)90647-0)

* A. Gonzalez-Arroyo, M. Okawa, _Twisted-Eguchi-Kawai model: A reduced model for large-
$N$ lattice gauge theory_, Phys. Rev. D 27, 2397 (1983) ([doi:10.1103/PhysRevD.27.2397](https://doi.org/10.1103/PhysRevD.27.2397))

The original articles on the BFSS matrix model:

* {#BanksFischlerShenkerSusskind96} [[Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_,  Phys. Rev. D **55** (1997) [[doi:10.1103/PhysRevD.55.5112](https://doi.org/10.1103/PhysRevD.55.5112), [arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043)]

* [[Leonard Susskind]], _Another Conjecture about M(atrix) Theory_ ([arXiv:hep-th/9704080](https://arxiv.org/abs/hep-th/9704080))

  > (argument for [[small N limit|small N]]-validity)

and with more details on the [[discrete light front quantization]] involved:

* {#Sen97} [[Ashoke Sen]], *D0 Branes on $T^n$ and Matrix Theory*, Adv. Theor. Math. Phys. **2** (1998) 51-59 [[arXiv:hep-th/9709220](https://arxiv.org/abs/hep-th/9709220)]

* {#Seiberg97} [[Nathan Seiberg]], *Why is the Matrix Model Correct?*, Phys. Rev. Lett. **79** (1997) 3577-3580 [[arXiv:hep-th/9710009](https://arxiv.org/abs/hep-th/9710009), [doi:10.1103/PhysRevLett.79.3577](https://doi.org/10.1103/PhysRevLett.79.3577)]

* [[Adel Bilal]], *DLCQ of M-Theory as the Light-Like Limit*, Phys. Lett. B **435** (1998) 312-318 [[arXiv:hep-th/9805070](https://arxiv.org/abs/hep-th/9805070), doi:10.1016/S0370-2693(98)00811-9]

In view of [[flat space holography]]:

* [[Leonard Susskind]], *Holography in the flat space limit*, AIP Conf.Proc. 493 (1999) 1, 98-112,  ([spire](https://inspirehep.net/literature/482388), [arXiv:hep-th/9901079](https://arxiv.org/abs/hep-th/9901079), [doi:10.1063/1.1301570](https://doi.org/10.1063/1.1301570))

Review:

* [[Adel Bilal]], *M(atrix) Theory : a Pedagogical Introduction*, Fortsch. Phys. **47** (1999) 5-28 [[arXiv:hep-th/9710136](https://arxiv.org/abs/hep-th/9710136), doi:10.1002/(SICI)1521-3978(199901)47]

* [[Tom Banks]], _Matrix Theory_, Nucl. Phys. Proc. Suppl. 67 (1998) 180-224 [[arXiv:hep-th/9710231](https://arxiv.org/abs/hep-th/9710231), doi:10.1016/S0920-5632(98)00130-3]

* [[Washington Taylor]], _M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory_, Rev. Mod. Phys. **73** (2001) 419-462 [[arXiv:hep-th/0101126](https://arxiv.org/abs/hep-th/0101126)]

* {#Ydri18} [[Badis Ydri]], _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ [[arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9)]

Review in the context of the [[holographic principle]]:

* [[Juan Maldacena]], *A simple quantum system that describes a black hole* [[arXiv:2303.11534](https://arxiv.org/abs/2303.11534), inspire:2644529]

A review of further developments:

* [[David Berenstein]], _Classical dynamics and thermalization in holographic matrix models_, talk at Leiden, October (2012) [[pdf](http://www.lorentzcenter.nl/lc/web/2012/514/presentations/Berenstein.pdf)]

See also

* [[Paul Townsend]], _M(embrane) theory on $T^0$_, Nucl. Phys. Proc. Suppl. **68** (1998) 11-16 [[arXiv:hep-th/9708034](http://arxiv.org/abs/hep-th/9708034)]

* [[Chris D. A. Blair]], Johannes Lahnsteiner, [[Niels A. Obers]], Ziqi Yan. *Unification of Decoupling Limits in String and M-theory*. (2023). ([arXiv:2311.10564](https://arxiv.org/abs/2311.10564))

Discussion as a solution to the open problem of defining [[M-theory]] is in

* {#Moore14} [[Gregory Moore]], section 12, p. 43-44 in: _[[Physical Mathematics and the Future]]_, talk at *[Strings 2014](http://physics.princeton.edu/strings2014/)*  [[talk slides](http://physics.princeton.edu/strings2014/slides/Moore.pdf), [companion text pdf](http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf), [[MooreVisionTalk2014.pdf:file]]]

where it says:

> {#AGoodStartWasGivenByTheMatrixTheory} A good start was given by the [[BFSS matrix model|Matrix theory]] approach of [[Tom Banks|Banks]], [[Willy Fischler|Fischler]], Shenker and [[Leonard Susskind|Susskind]]. We have every reason to expect that this theory produces the correct [[scattering amplitudes]] of modes in the [[11-dimensional supergravity]] multiplet in 11-dimensional [[Minkowski space]] - even at energies sufficiently large that [[black holes]] should be created. (This latter phenomenon has never been explicitly demonstrated). But Matrix theory is only a beginning and does not give us the whole picture of [[M-theory]]. The program ran into increasing technical difficulties when more complicated compactifications were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[...]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics.

> If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of [[M-theory]] is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue.

Derivation from [[open string field theory]] is discussed in

* [[Taejin Lee]], _Covariant Open String Field Theory on Multiple D$p$-Branes_ ([arXiv:1703.06402](https://arxiv.org/abs/1703.06402))

Relation to the [[6d (2,0)-supersymmetric QFT]]:

* [[Micha Berkooz]], [[Moshe Rozali]], [[Nathan Seiberg]], _Matrix Description of M-theory on $T^3$ and $T^5$_ ([arXiv:hep-th/9704089](http://arxiv.org/abs/hep-th/9704089))

[[!include quantization of M2-brane on Minkowski spacetime to BFSS matrix model -- references]]

See also in relation to the [[ABJM model]]:

* {#MohammedMuruganNastase10} Asadig Mohammed, Jeff Murugan, [[Horatiu Nastase]], _Looking for a Matrix model of ABJM_, Phys. Rev. D82:086004, 2010 ([arXiv:1003.2599](https://arxiv.org/abs/1003.2599))

### Relation to M5-branes

Discussion of [[light cone longitudal]] [[M5-branes]] in the [[BFSS matrix model]] (for [[light cone transversal]] M5-s see at _[[BMN matrix model]]_):

* [[Tom Banks]], [[Nathan Seiberg]], [[Stephen Shenker]], _Branes from Matrices_, Nucl. Phys. B490:91-106, 1997 ([arXiv:hep-th/9612157](https://arxiv.org/abs/hep-th/9612157))

* Judith Castelino, Sangmin Lee, [[Washington Taylor]], _Longitudinal 5-branes as 4-spheres in Matrix theory_, Nucl. Phys. B526:334-350, 1998 ([arXiv:hep-th/9712105](https://arxiv.org/abs/hep-th/9712105))

  (introducing the [[fuzzy 4-sphere]])

Discussion of a [[BFSS matrix model|BFSS-like]] [[matrix model]] for [[MK6-branes]]:

* [[Amihay Hanany]], Gilad Lifschytz, _M(atrix) Theory on $T^6$ and a m(atrix) Theory Description of KK Monopoles_, Nucl. Phys. B519:195-213, 1998 ([arXiv:hep-th/9708037](https://arxiv.org/abs/hep-th/9708037))

### Ground state problem
  {#ReferencesGroundStateProblem}

There remains the problem of existence of a sensible ground state of the BFSS model.

* [[Bernard de Wit]], M. Luscher, [[Hermann Nicolai]], _The Supermembrane Is Unstable_, Nucl.Phys. B320 (1989) 135-159 ([spire:266584](http://inspirehep.net/record/266584), doi:10.1016/0550-3213(89)90214-9)

For a new attempt at solving this problem, and for pointers to previous attempts see

* L. Boulton, M.P. Garcia del Moral, A. Restuccia, _The ground state of the D=11 supermembrane and matrix models on compact regions_, Nuclear Physics B
Volume 910, September 2016, Pages 665-684 ([arXiv:1504.04071](https://arxiv.org/abs/1504.04071))

* L. Boulton, M.P. Garcia del Moral, A. Restuccia, _Measure of the potential valleys of the supermembrane theory_, Physics Letters B
Volume 797, 2019, 134873 ([arXiv:1811.05758](https://arxiv.org/abs/1811.05758))

### Graviton scattering
 {#ReferencesGravitonScattering}

Computation of [[graviton]] [[scattering amplitudes]] with the BFSS matrix model:

* [[Katrin Becker]], [[Melanie Becker]], _A Two-Loop Test of M(atrix) Theory_, Nucl.Phys. B506 (1997) 48-60 ([arXiv:hep-th/9705091](https://arxiv.org/abs/hep-th/9705091))

* [[Katrin Becker]], [[Melanie Becker]], [[Joseph Polchinski]], [[Arkady Tseytlin]], _Higher Order Graviton Scattering in M(atrix) Theory_, Phys. Rev. D **56** (1997) 3174-3178 [[arXiv:hep-th/9706072](https://arxiv.org/abs/hep-th/9706072), [doi:10.1103/PhysRevD.56.R3174](https://doi.org/10.1103/PhysRevD.56.R3174)]

* also [Kabat-Taylor 97](#KabatTaylor97)

* M. Fabbrichesi, _Graviton scattering in matrix theory and supergravity_, in: Ceresole A., Kounnas C., [[Dieter Lüst]], [[Stefan Theisen]] (eds.) _Quantum Aspects of Gauge Theories, Supersymmetry and Unification_, Lecture Notes in Physics, vol 525. Springer, Berlin, Heidelberg ([arXiv:hep-th/9811204](https://arxiv.org/abs/hep-th/9811204))

* [[Robert Helling]], [[Jan Plefka]], Marco Serone, Andrew Waldron, *Three-graviton scattering in M-theory*, Nuclear Physics B **559** 1–2 (1999) 184-204 [[arXiv:hep-th/9905183](https://arxiv.org/abs/hep-th/9905183), doi:10.1016/S0550-3213(99)00451-4]

* Robert Echols, _M-theory, supergravity and the matrix model: Graviton scattering and non-renormalization theorems_, PhD thesis, 1999 [[pdf](https://web.calpoly.edu/~rechols/phys403/mythesis.pdf)]

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

In relation to the [[soft graviton theorem]]:

* Noah Miller, [[Andrew Strominger]], Adam Tropper, Tianli Wang, *Soft Gravitons in the BFSS Matrix Model* [[arXiv:2208.14547](https://arxiv.org/abs/2208.14547)]

* Adam Tropper, Tianli Wang, *Lorentz Symmetry and IR Structure of The BFSS Matrix Model* [[arXiv:2303.14200](https://arxiv.org/abs/2303.14200)]

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

### Black holes

Relation to [[black holes in string theory]]:

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]],  _Schwarzschild Black Holes from Matrix Theory_, Phys.Rev.Lett.80:226-229,1998 ([arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091))

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008,1998 ([arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005))

* [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzschild Black Holes in Various Dimensions from Matrix Theory_, Phys.Lett.B416:62-66,1998 ([arXiv:hep-th/9709108](https://arxiv.org/abs/hep-th/9709108))

* Edi Halyo, _Six Dimensional Schwarzschild Black Holes in M(atrix) Theory_ ([arXiv:hep-th/9709225](https://arxiv.org/abs/hep-th/9709225))

* [[Gary Horowitz]], [[Emil Martinec]], _Comments on Black Holes in Matrix Theory_, Phys. Rev. D 57, 4935 (1998) ([arXiv:hep-th/9710217](https://arxiv.org/abs/hep-th/9710217))

* {#KabatTaylor97} Daniel Kabat, [[Washington Taylor]], _Spherical membranes in Matrix theory_, Adv.Theor.Math.Phys.2:181-206,1998 ([arXiv:hep-th/9711078](https://arxiv.org/abs/hep-th/9711078))

* Yoshifumi Hyakutake, _Black Hole and Fuzzy Objects in BFSS Matrix Model_
([arXiv:1801.07869](https://arxiv.org/abs/1801.07869))

* {#DuSahakian18} Haoxing Du, Vatche Sahakian, _Emergent geometry from stochastic dynamics, or Hawking evaporation in M(atrix) theory_ ([arXiv:1812.05020](https://arxiv.org/abs/1812.05020))

  (combination with [[random matrix theory]])

### Relation to lattice gauge theory

Relation to [[lattice gauge theory]] and numerical tests of [[AdS/CFT]]:

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

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

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

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

### Holography

On [[AdS/CFT]] in the form of [[AdS2/CFT1]] with the [[BFSS matrix model]] on the CFT side and [[black hole in string theory|black hole-like solutions]] in [[type IIA supergravity]] on the AdS side:

* [[Juan Maldacena]], Alexey Milekhin, _To gauge or not to gauge?_, JHEP 04 (2018) 084 ([arxiv:1802.00428](https://arxiv.org/abs/1802.00428))

and concerning the analog of its [[holographic entanglement entropy]]:

* Tarek Anous, Joanna L. Karczmarek, Eric Mintun, [[Mark Van Raamsdonk]], Benson Way, _Areas and entropies in BFSS/gravity duality_ ([arXiv:1911.11145](https://arxiv.org/abs/1911.11145))

[[!redirects BFSS model]]


BGG resolution

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

#Contents#
* table of contents
{:toc}

## Idea

A _BGG resolution_ is a certain [[projective resolution]] of certain [[representations]].

More in detail, give a finite dimensional semisimple complex [[Lie algebra]] $\mathfrak{g}$ with [[Cartan subalgebra]] $\mathfrak{h}$, and a positive [[root system]], and let $P^+\subset \mathfrak{h}^*$ be the set of dominant integral [[weights]]. Then for every $\lambda\in P^+$ consider the corresponding finite dimensional left $U(\mathfrak{g})$-[[module]] $L(\lambda)$. Certain [[resolutions]] of $L(\lambda)$ are defined in a series of papers of Bernstein, Gel'fand and Gel'fand ([BBG 7x](#BBG7x)) and are now called _BGG resolutions_. There are also generalizations, e.g. for Kac-Moody algebras.

These resolutions have a natural incarnation in terms of [[complexes]] of sections of [[tractor bundles]] over [[flag varieties]] or more generally over [[homogeneous space|homogeneous]] [[parabolic geometry|parabolic]] [[Klein geometries]]. As such, there are generalizations of the construction to more general [[parabolic geometry|parabolic]] [[Cartan geometries]], called _curved BGG sequences_. See for instance ([Calderbank-Diemer 00, theorem 3.6](#CalderbankDiemer00)).

BGG resolutions may be used to construct [[resolutions]] of sheaves of constant functions on [[Klein geometries]]/[[coset space]] $G/H$ that are more efficient (smaller) that the general resolution given by the [[de Rham complex]] (the [[Poincare lemma]]). In this way BGG resolutions are used notably for computation in [[Leray spectral sequences]] as they appear in [[Penrose transforms]] ([Baston-Eastwood 89, chapter 8](#BastonEastwood89)).

## Properties

### Curved $A_\infty$-Structure

Under a [[cup product]] the BGG sequence becomes a [[curved A-infinity algebra]]. ([Calderbank-Diemer 00, section 6](#CalderbankDiemer00))

## Related concepts

* [[Verma module]]

* [[parabolic geometry]]

## References

### Original case over flag variety

* [[eom]]: Alvany Rocha, [BGG resolution](http://www.encyclopediaofmath.org/index.php?title=BGG_resolution&oldid=14028); wikipedia, [category O](http://en.wikipedia.org/wiki/Category_O)

* {#BBG7x} I.N. Bernstein, [[I. M. Gelfand]], S. I. Gelfand, _Structure of representations generated by vectors of highest weight_, Funkts. Anal. Prilozh. __5__: 1 (1971) pp. 1–9; _A certain category of -modules_, Funkts. Anal. Prilozh. __10__: 2 (1976) pp. 1–8; _Differential operators on the base affine space and a study of $\mathfrak{g}$-modules), I.M. Gelfand (ed.), Lie groups and their representations, Proc. Summer School on Group Representations, Janos Bolyai Math. Soc.& Wiley (1975) pp. 39–64

* James E. Humphreys, _Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$_, 2008, [pdf](http://www.math.umass.edu/~jeh/bgg/main.pdf)

* M. Falk, V. Schechtman, A. Varchenko, _BGG resolutions via configuration spaces_, [arxiv/1309.7811](http://arxiv.org/abs/1309.7811)

> We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik-Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the $sl_2$ Bernstein - Gelfand - Gelfand resolution as an Aomoto complex.

* Sergey Arkhipov, _A new construction of the semi-infinite BGG resolution_, [q-alg/9605043](http://arxiv.org/abs/q-alg/9605043)

Discussion in the context of the [[Penrose transform]] includes

* Michael Eastwood, _Variations on the de Rham complex_, Notices Amer. Math. Soc. __46__ (1999), no. 11, 1368–1376 [pdf](http://www.ams.org/notices/199911/fea-eastwood.pdf)

* {#BastonEastwood89} R. J. Baston, M. G. Eastwood, _The Penrose transform_, Oxford Univ. Press, New York, 1989; [MR92j:32112](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1038279)

### Curved generalization to parabolic Cartan geometries
 {#ReferencesCurved}

The generalization from [[coset spaces]] to [[parabolic geometry|parabolic]] [[Cartan geometries]] (curved BGG sequences) is discussed in

* {#Baston90} R. J. Baston, _Verma modules and differential conformal invariants_, J. Diff. Geom. 32 (1990) 851–898.

* [[Andreas Čap]], [[Jan Slovák]], [[Vladimír Souček]], _Bernstein-Gelfand-Gelfand sequences_, ESI preprint 722 (1999).

* [[Andreas Čap]], [[Jan Slovák]], [[Vladimír Souček]], _Bernstein-Gelfand-Gelfand sequences_, Ann. Math. (II) __154__:1 (2001) pp. 97-113 ([jstor](http://www.jstor.org/stable/3062111))

* {#CalderbankDiemer00} [[David Calderbank]], Tammo Diemer, _Differential invariants and curved Bernstein-Gelfand-Gelfand sequences_, J.Reine Angew.Math. 537 (2001) 67-103 ([arXiv:math/0001158](http://arxiv.org/abs/math/0001158))

* {#CapSoucek07} [[Andreas Čap]], [[Vladimír Souček]], _Curved Casimir operators and the BGG machinery_, SIGMA __3__, 2007, 111 ([arxiv:0708.3180](http://arxiv.org/abs/0708.3180) [doi](http://dx.doi.org/10.3842/SIGMA.2007.111))

category: representation theory

[[!redirects BGG resolutions]]
[[!redirects Bernstein-Gelfand-Gelfand resolution]]

[[!redirects BGG sequence]]
[[!redirects BGG sequences]]


Bharath Sriperumbudur

* [webpage](http://www.personal.psu.edu/bks18/)

* [webpage](https://science.psu.edu/stat/people/bks18)

## Selected writings

On [[Bochner's theorem]] generalized to [[non-abelian groups]] in the context of [[kernel methods]]:

* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))

category: people


Bhargav Bhatt

* [webpage](http://www.math.ias.edu/~bhatt/)

* [wikipedia](https://en.wikipedia.org/wiki/Bhargav_Bhatt_%28mathematician%29)

## Selected writings

On the [[Mordell conjecture]]:

* {#BhattSnowden} [[Bhargav Bhatt]], [[Andrew Snowden]] (org.): _Faltings' proof of the Mordell conjecture_ (seminar notes, 2016) [[pdf](https://web.math.princeton.edu/~takumim/Mordell.pdf), [[BhattSnowden-FaltingsProofofMordell.pdf:file]]]

On [[crystalline cohomology]] and [[de Rham cohomology]]:

* Bhargav Bhatt, [[Aise Johan de Jong]], _Crystalline cohomology and de Rham cohomology_ ([pdf](http://www.math.columbia.edu/~dejong/papers/crystalline-comparison.pdf))

On [[Witt vectors]]:

* Bhargav Bhatt, [[Peter Scholze]], _Projectivity of the Witt vector affine Grassmannian_, Invent. math. 209, 329-423 (2017) [doi](https://doi.org/10.1007/s00222-016-0710-4) [arXiv:1507.06490](https://arxiv.org/abs/1507.06490)

On [[p-adic Hodge theory|$p$-adic Hodge theory]]:

* Bhargav Bhatt, [[Matthew Morrow]], [[Peter Scholze]] *Integral $p$-adic Hodge Theory* [[arXiv:1602.03148](https://arxiv.org/abs/1602.03148)]

and on [[topological Hochschild homology]]:

* Bhargav Bhatt, [[Matthew Morrow]], [[Peter Scholze]], *Topological Hochschild homology and integral $p$-adic Hodge theory* [[arXiv:1802.03261](https://arxiv.org/abs/1802.03261)]

On [[prismatic cohomology]]:

* Bhargav Bhatt, [[Peter Scholze]], _Prisms and prismatic cohomology_, Ann. of Math. (2) 196(3): 1135-1275 [doi](https://10.4007/annals.2022.196.3.5) arXiv:[1905.08229](https://arxiv.org/abs/1905.08229)

* Bhargav Bhatt, [[Jacob Lurie]], _Absolute prismatic cohomology_, arXiv:[2201.06120](https://arxiv.org/abs/2201.06120)

## Related entries:

* [[pro-étale site]]

* [[p-adic Hodge theory]]

category: people