Hadamard lemma

> This page is about the lemma on [[Taylor series|Taylor polynomials]] of [[smooth functions]]. For [[Hadamard's formula]] in [[Lie theory]] see there.

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#### Differential geometry
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# Contents
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## Idea

The _Hadamard lemma_ says that the [[Taylor series]] of a [[smooth function]] on the [[real line]] around the origin has a remainder at order $n$ which is the product of $x^{n+1}$ (for $x$ the canonical [[coordinate]] function) with another [[smooth function]].

Simple as this may sound, it has profound consequences, as it means that [[smooth functions]] behave more like [[polynomials]] than the classical definition might suggest. For instance the Hadamard lemma directly implies that:

1. [[derivations of smooth functions are vector fields]]: For $X$ a [[smooth manifold]] and $C^\infty(X)$ its $\mathbb{R}$-[[associative algebra|algebra]] of [[smooth functions]] (under pointwise multiplication), then there is a [[natural bijection]] between the smooth [[tangent vector fields]] on $X$ and the purely algebraic [[derivations]] of the algebra $C^\infty(X)$;

1. the smooth [[infinitesimally thickened point]] $\mathbb{D}$ is the same as in [[algebraic geometry]]: The [[quotient]] of the algebra of [[smooth functions]] $C^\infty(\mathbb{R}^1)$ on the [[real line]] by the [[ideal] generated by the square $x^2$ of the canonical [[coordinate]] function $x$ is the [[ring of dual numbers]]:

   $$
     C^\infty(\mathbb{R})/(x^2) \simeq (\mathbb{R} \oplus \epsilon \mathbb{R})/(\epsilon^2)
     \,.
   $$

1. together this implies that a [[tangent vector]] in a smooth manifold $X$ is equivalently a morphism of the form

   $$
     \mathbb{D} \longrightarrow X
   $$

   of [[formal duals]] of $\mathbb{R}$-alghebras, from the [[infinitesimally thickened point]] $\mathbb{D}$.

This means that [[differential geometry]] has more in common with [[algebraic geometry]] than is manifest from the traditional definitions. In _[[synthetic differential geometry]]_ one makes use of these facts to find a useful unified perspective. For exposition of this point see at _[[geometry of physics -- supergeometry]]_.

More generally one may ask for other types of function algebras which satisfy the conclusion of the Hadamard lemma. These turn out to be the [[algebras over a Lawvere theory|algebras over]] those [[algebraic theories]] which are called _[[Fermat theories]]_. These are hence a crucial ingredient for [[Models for Smooth Infinitesimal Analysis|well-adapted models]] of [[synthetic differential geometry]].

## Statement

+-- {: .num_prop #HadamardLemma}
###### Proposition
**(Hadamard lemma)**

For every [[smooth function]] $f \in C^\infty(\mathbb{R})$ on the [[real line]] there is a smooth function $g$ such that

\[
  \label{TheDifferenceEquation}
  f(x) = f(0) + x \cdot g(x)
  \,.
\]

=--

This function $g$ is also called a **Hadamard quotient**.

+-- {: .num_cor #AndItsCorollary}
###### Corollary

It follows that

$$
  g(0) \,=\, f'(0)
$$

is the [[derivative]] of $f$ at 0. By applying this repeatedly the lemma says that $f$ has a partial [[Taylor series]] expansion whose remainder $h$ is a smooth function:

$$
  f(x) = f(0) + x f'(0) + \frac1{2} x^2 f''(0) + \cdots + \frac1{n!} x^n h(x)
  \,.
$$

More generally, for smooth functions on any [[Cartesian space]] $\mathbb{R}^n$ the lemma says that there are for each $f \in C^\infty(X)$ $n$ smooth functions $g_i$ such that

$$
 f(\vec x) = f(0) + \sum_i x_i g_i(x)
 \,.
$$

So at the origin these smooth functions compute the partial derivatives of $f$

$$
  g_i(0) = \frac{\partial f}{\partial x_i}(0)
  \,.
$$

=--

+-- {: .proof}
###### Proof

Holding $x$ fixed, put $h(t) = f(t x)$. Then

$$
  f(x) - f(0) = \int_{0}^{1} h'(t) d t = \int_{0}^{1} \sum_{i=1}^n \frac{\partial f}{\partial x_i} (t x) \cdot x_i d t
  $$

where the second equality uses the [[chain rule]].
The lemma follows by putting

$$
  g_i(x) = \int_{0}^{1} \frac{\partial f}{\partial x_i}(t x) d t
  \,.
$$

=--

## Applications

The Hadamard lemma implies in particular that

* [[derivations of smooth functions are vector fields]] (see there for the proof);

* the [[Taylor series]] of a [[smooth function]] is an [[asymptotic series]] ([this example](asymptotic+series#TaylorSeriesOfSmoothFunctionIsAsymptoticSeries)).

## Fermat theory

The notion of a [[Fermat theory]]
makes Hadamard's lemma into an axiom.
See there for more information.

## Related theorems
 {#RelatedTheorems}

* [[Borel's theorem]]

* the [[Tietze extension theorem]]

* the [[Whitney extension theorem]]

* the [[Steenrod-Wockel approximation theorem]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[smooth Serre-Swan theorem]]

* [[derivations of smooth functions are vector fields]]

## References

The Hadamard lemma is due to [[Jacques Hadamard]].

Review includes

* [[Jet Nestruev]], _Smooth manifolds and observables_, Graduate texts in mathematics **218** (2003) [[doi:10.1007/978-3-030-45650-4](https://doi.org/10.1007/978-3-030-45650-4)]

The Hadamard lemma is what makes the standard _convenient models_ for [[synthetic differential geometry]] tick. Its role in this respect can be seen from proposition 1.2 on in

* [[Ieke Moerdijk]], [[Gonzalo Reyes|Gonzalo E. Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_ Springer (1991)

Mathema-philosophical discussion of the ingredients of the Hadamard lemma as analogous to an [[adjoint cylinder]] resembling a [[duality of opposites]]:

* [[William Lawvere]], *Unity and Identity of Opposites in Calculus and Physics*, Proceedings of ECCT 1994 Tours Conference, Applied Categorical Structures **4** 167-174 (1996) Kluwer Academic Publishers [[doi:10.1007/BF00122250](https://doi.org/10.1007/BF00122250), [github](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1996-unity-and-identity-of-opposites-in-calculus-and-physics.pdf)]

[[!redirects Hadamard\'s lemma]]
[[!redirects Hadamard's lemma]]
[[!redirects Hadamard quotient]]
[[!redirects Hadamard quotients]]


Hadamard product

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\tableofcontents

\section{Definition}

\subsection{Of matrices}

The **Hadamard product** or **Schur product** or **element-wise product** of two matrices $A$ and $B$ in a commutative ring $R$ is a matrix $A \odot B$ whose elements $c_{i, j}$ are the product $a_{i, j} \cdot b_{i, j}$ of the respective elements $a_{i, j}$ in the matrix $A$ and $b_{i, j}$ in the matrix $B$.

\subsection{Of power series}

The **Hadamard product** of two [[power series]] $p = \sum_{i = 0}^\infty a_i x^i$ and $q = \sum_{i = 0}^\infty b_i x^i$ with [[coefficients]] from a [[commutative ring]] $R$ is the [[power series]] defined by

$$p \odot q \coloneqq \sum_{i = 0}^\infty a_i b_i x^i$$

\subsection{More generally}

Let $R$ be a [[commutative ring]], and let $[n]$ denote the finite set with $n$ elements. The $m$ by $n$ [[matrix algebra]] $M_{m, n}(R)$ is [[isomorphic]] to the [[function algebra]] $[m n] \to R$, where the Hadamard product in $M_{m, n}(R)$ corresponds to pointwise multiplication in $[m n] \to R$. Similarly, the [[power series ring]] $R[[x]]$ is [[isomorphic]] to the [[function algebra]] $\mathbb{N} \to R$, where the Hadamard product in $R[[x]]$ corresponds to pointwise multiplication in $\mathbb{N} \to R$.

Thus, there is a generalization of the definition of Hadamard product from [[matrix algebras]] and [[power series rings]] to any [[commutative algebra]] with a [[isomorphism]] of [[commutative algebras]] to a [[function algebra]]:

Let $S$ be a set, let $R$ be a commutative ring, and let $A$ be a commutative $R$-algebra with an isomorphism of commutative $R$-algebras $i:A \cong (S \to R)$. Then the **Hadamard product** in $A$ is a [[binary operation]] $(X, Y) \mapsto X \odot Y:A \times A \to A$ defined as

$$X \odot Y \coloneqq i^{-1}(x \mapsto i(X)(x) \cdot i(Y)(x))$$

\section{Related concept}

* [[matrix]]

* [[power series]]

\section{References}

* Wikipedia, Hadamard product (matrices)

* Wikipeida, Generating function transformation#Hadamard products and diagonal generating functions

[[!redirects Hadamard product]]
[[!redirects Schur product]]
[[!redirects element-wise product]]


Hadamard propagator

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## Definition

In [[perturbative quantum field theory]] the term "Hadamard propagator" refers to the [[vacuum expectation value]] of the [[anticommutator]] of [[field observables]]:

$$
  \left\langle
    \mathbf{\Phi}^a(x)\mathbf{\Phi}^b(y)
    +
    \mathbf{\Phi}^b(y) \mathbf{\Phi}^a(x)
  \right\rangle
$$

(e.g. [Calzetta 99, III.6](#Calzetta99), [Millington 14 (7.29)](#Millington14))

Beware that this is _different_ from the concept of _[[Hadamard 2-point function]]_  ("[[Wightman propagator]]").

## References

* {#Calzetta99} Esteban Calzetta, _Stochastic dynamics of correlations in quantum field theory: From Schwinger-Dyson to Boltzmann-Langevin equation_ ([arXiv:hep-ph/9903291](https://arxiv.org/abs/hep-ph/9903291))

* {#Millington14} Peter Millington, _Thermal Quantum Field Theory and Perturbative Non-Equilibrium Dynamics_, Springer 2014

[[!redirects Hadamard propagators]]


Hadamard's formula

Sometimes called Hadamard lemma, however, in $n$Lab, [[Hadamard lemma]] is another result of [[Jacques Hadamard]], in [[analysis]].

## Geometrical motivation

__Hadamard's formula__ relates the Ad action of a Lie group on its [[tangent Lie algebra]], with the ad action of the Lie algebra on itself.

## Statement

$$
e^X Y e^{-X} = e^{ad X} (Y)
$$

## References

* wikipedia [Baker-Campbell-Hausdorff formula](http://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula), section *Hadarmard lemma*.

[[!redirects Hadamard formula]]


hadron

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## Idea

In [[QCD]] a [[bound state]] of [[quarks]] via the [[strong nuclear force]].

Binding two quarks: [[meson]],

Binding three quarks: [[baryon]],

Binding four quarks: [[tetraquark]].

## Properties

### Computation via lattice QCD

Comparison of light hadron [[masses]] as seen in accelertator [[experiment]] and in [[lattice QCD]]-computer simulation (from [Fodor-Hoelbling 12](#FodorHoelbling12))

### Conceptualization and computation in AdS/QCD
 {#ConceptualizationAndComputationInAdSQCD}

In the [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|strongly coupled]] [[QCD]] via an [[intersecting D-brane model]], the [[hadrons]] in [[QCD]] correspond to [[string theory|string-theoretic]]-phenomena in an ambient [[bulk field theory]] on an approximately [[anti de Sitter spacetime]]:

1. the [[mesons]] ([[bound states]] of 2 [[quarks]]) correspond to [[open strings]] in the bulk, whose two endpoints on the [[asymptotic boundary]] correspond to the two [[quarks]];

1. [[baryons]] ([[bound states]] of $N_c$ [[quarks]]) appear in two different but equivalent ([Sugimoto 16, 15.4.1](#Sugimoto16)) guises:

   1. as [[wrapped brane|wrapped]] [[D4-branes]] with $N_c$ [[open strings]] connecting them to the [[D8-brane]]

      ([Witten 98b](#Witten98b), [Gross-Ooguri 98](#GrossOoguri98))

   1. as [[skyrmions]]

      ([Sakai-Sugimoto 04, section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, section 3.3](#SakaiSugimoto05), see [Bartolini 17](#Bartolini17)).

For review see [Sugimoto 16](#Sugimoto16), also [Rebhan 14, around (18)](#Rebhan14).

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

This produces [[baryon]] [[mass]] spectra with moderate quantitative agreement with [[experiment]] ([HSSY 07](#HSSY07)):

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

For more see at _[[hadron Kaluza-Klein theory]]_.

## Related concepts

* [[hadron current]]

* [[confinement]]

  * [[chiral perturbation theory]]

  * [[quark bag model]], [[Cheshire cat principle]]

  * [[vector meson dominance]]

* [[quark-hadron duality]]

* [[proton spin crisis]]

* [[flavour anomaly]]

* [[hadron supersymmetry]]

* [[D-term]]

## References

### General

* Claude Amsler, _The Quark Structure of Hadrons: An Introduction to the Phenomenology and Spectroscopy_, Lecture Notes in Physics 949 ([doi:10.1007/978-3-319-98527-5](https://link.springer.com/book/10.1007/978-3-319-98527-5))

See also:

* Wikipedia, _[Hadron](http://en.wikipedia.org/wiki/Meson)_

Discussion in a general context of [[bound states]]:

* [[Paul Hoyer]], _Bound states -- from QED to QCD_, Lectures at "Mini-school on theoretical methods in particle physics" at the Higgs Centre for Theoretical Physics, University of Edinburgh on 30 September to 4 October 2013 ([arXiv:1402.5005](https://arxiv.org/abs/1402.5005))

* [[Paul Hoyer]], _Journey to the Bound States_ ([arXiv:2101.06721](https://arxiv.org/abs/2101.06721))

On the issue of hadron [[mass]] ([[confinement]]):

* Minghui Ding, Craig D. Roberts, Sebastian M. Schmidt, *Emergence of Hadron Mass and Structure* *lbrack;[arXiv:2211.07763](https://arxiv.org/abs/2211.07763)]

List of hadrons discovered at the [[LHC experiment]]:

* {#Koppenburg21} [[Patrick Koppenburg]] for the [[LHCb collaboration]]: *List of hadrons observed at the LHC* (2021) [[pdf](https://cds.cern.ch/record/2749030/files/FIGURE-2021-001.pdf), [webpage](https://www.nikhef.nl/~pkoppenb/particles.html), [cds:2749030](https://cds.cern.ch/record/2749030)]

[[!include quantum hadrodynamics -- references]]

[[!include Skyrme hadrodynamics with vector mesons -- references ]]

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

### Experiment

Observation of hadrons at [[LHC]]:

* Maria Vasileiou on behalf of the ALICE Collaboration, _Measurement of the Hadronic Resonance Production with ALICE at the CERN LHC_, Universe 2019, 5(1), 6 ([doi:10.3390/universe5010006](https://doi.org/10.3390/universe5010006))

### In lattice QCD

Computation of the [[mass]] of hadrons using [[lattice QCD]] (see also _[[confinement]]_ and _[[mass gap problem]]_) is discussed in

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

  _Ab-initio Determination of Light Hadron Masses_,

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

  conclusion on p. 4:

> our study strongly suggests that [[QCD]] is the theory of the strong interaction, at low energies as well

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

* S. Aoki et. al. _Review of lattice results concerning low-energy particle physics_ ([arXiv:1607.00299](https://arxiv.org/abs/1607.00299))

### In Witten-Sakai-Sugimoto model for AdS-QCD
 {#ReferencesInWSSModel}

QCD hadrons realized in the [[Witten-Sakai-Sugimoto model]] of [[holographic QCD]]:

* {#Witten98b} [[Edward Witten]], _Baryons And Branes In Anti de Sitter Space_, JHEP 9807:006, 1998 ([arXiv:hep-th/9805112](https://arxiv.org/abs/hep-th/9805112))

* {#GrossOoguri98} [[David Gross]], [[Hirosi Ooguri]], _Aspects of Large N Gauge Theory Dynamics as Seen by String Theory_, Phys.Rev.D58:106002,1998 ([arXiv:hep-th/9805129](https://arxiv.org/abs/hep-th/9805129))

* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Low energy hadron physics in holographic QCD_, Prog.Theor.Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))

* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _More on a holographic dual of QCD_, Prog.Theor.Phys.114:1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))

* {#HSSY07} Hiroyuki Hata, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Shinichiro Yamato, _Baryons from instantons in holographic QCD_, Prog.Theor.Phys.117:1157, 2007 ([arXiv:hep-th/0701280](https://arxiv.org/abs/hep-th/0701280))

Review

* {#Rebhan14} [[Anton Rebhan]], _The Witten-Sakai-Sugimoto model: A brief review and some recent results_, 3rd International Conference on New Frontiers in Physics, Kolymbari, Crete, 2014 ([arXiv:1410.8858](https://arxiv.org/abs/1410.8858))

* {#Sugimoto16} [[Shigeki Sugimoto]], _Skyrmion and String theory_, chapter 15 in [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

* Sophia K Domokos, Robert Bell, Trinh La, Patrick Mazza, *A Pedagogical Introduction to Holographic Hadrons* ([arXiv:2106.13136](https://arxiv.org/abs/2106.13136))

[[!redirects hadrons]]


hadron current

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## Idea

In [[quantum hadrodynamics]], the _hadron current_ is the [[current]] of [[hadrons]].

In models with fundamental [[quark]][[field (physics)|fields]] $q$ ([[quantum chromodynamics]] or [[quark-meson coupling models]]) the (electromagnetic) hadron current is given by

$$
  J^\mu
  \;=\;
  \overline{q} Q \gamma^\mu q
$$

where $Q$ denotes the quark charge matrix (e.g. [Piller-Weise 90](#PillerWeise90)).

(This is in contrast, for instance, to the _[[lepton current]]_, given by the analogous expression, but with [[leptons]] instead of [[quarks]].)

In [[confinement|confined]] [[chiral perturbation theory]] the hadron current is given by [[vector mesons]] ([[vector meson dominance]]), and including the $A$-variation of the [[chiral anomaly]] [[WZW term]] (see e.g. [Wakamatsu 11](#Wakamatsu11)).

## Related concepts

* [[lepton current]]

* [[vector meson dominance]]

## References

General:

* Oleg Boyarkin, Section 5.2 of:
_Advanced Particle Physics: The Standard Model and Beyond_ Volume II, CRC press, 2011

Related to [[vector meson dominance]]:

* {#PillerWeise90} G. Piller, [[Wolfram Weise]], _Vector meson dominance: Selected topics_ 1990 ([spire310958](https://inspirehep.net/literature/310958), [[PillerWeiseVMD.pdf:file]])

In terms of [[chiral perturbation theory]]:

* {#Wakamatsu11} Masashi Wakamatsu, _On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action_,  ([arXiv:1108.1236](https://arxiv.org/abs/1108.1236), [spire:922302](https://inspirehep.net/literature/922302))

[[!redirects hadron currents]]

[[!redirects hadronic current]]
[[!redirects hadronic currents]]


hadron Kaluza-Klein theory

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## Idea

The idea of [[Kaluza-Klein theory]] has traditionally been applied mostly to "[[color charge|color physics]]", such as in attempts to realize the [[color charges]] of [[quantum chromodynamics]] ([[quarks]] and [[gluons]]) as a Kaluza-Klein compactification of [[heterotic supergravity]] (see for instance at _[[string phenomenology]] -- [heterotic models](string+phenomenology#ReferencesHeteroticModels)_). The success of this approach remains somewhat elusive (see also at _[[landscape of string theory vacua]]_).

Alternatively, Kaluza-Klein theory may be considered for "[[flavor physics]]" to produce the charges of [[flavor physics|flavor]]-"[[hidden local symmetries]]", namely the [[baryons]] and [[mesons]], respectively, hence the [[hadrons]] of [[quantum hadrodynamics]]. In terms of [[geometric engineering of QFT]] via [[intersecting D-brane models]] this means to consider [[gauge theory]] on _[[flavor branes]]_ (instead of on _[[color branes]]_), such as in the [[Witten-Sakai-Sugimoto model]] of [[holographic QCD]].

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

Indeed, the [[experiment|experimentally]] observed [[mesons]] appear in towers of increasing mass ("higher resonances"), which may usefully be identified as a Kaluza-Klein tower of the single [[gauge boson]] of an [[SU(2)]]-[[D=5 Yang-Mills theory]] ([Son-Stephanov 03](#SonStephanov03)).

Moreover, the [[pion field]] appears as the gauge 0-mode of this tower, right away in its [[soliton|solitonic]] incarnation as the [[Skyrmion]]-excitation in 4d, hence reflecting [[baryons]]. (This phenomenon is secretly the old theorem of [Atiyah-Manton 89](#AtiyahManton89), as explained from the modern perspective of [[holographic QCD]] in [Sutcliffe 10](#Sutcliffe10), [Sutcliffe 15](#Sutcliffe15)).

Various qualitative phenomena of the [[phenomenology]] of [[quantum hadrodynamics]] find a natural explanation in hadron Kaluza-Klein theory this way, notably:

1. [[hidden local symmetry]] itself (by the very KK-reduction of a gauge theory)

1. [[vector meson dominance]] (as discussed there)

1. QCD sum rules (...)

1. (...)

In terms of [[string phenomenology]], the [[flavor brane]]-[[D=5 Yang-Mills theory]] which gives [[quantum hadrodynamics]] this way naturally arises on [[D4/D8-brane intersections]] in the [[Witten-Sakai-Sugimoto model]] ([Sakai-Sugimoto 04](#SakaiSugimoto04), [Sakai-Sugimoto 05](#SakaiSugimoto05)) or else on [[M5-branes]] [[wrapped brane|wrapped]] on a [[closed interval]] ([Ivanova-Lechtenfeld-Popov 18](#IvanovaLechtenfeldPopov18))

Already to first approximation, this produces for instance [[baryon]] [[mass]] spectra with moderate quantitative agreement with [[experiment]] ([HSSY 07](#HSSY07)):

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

An extensive review of hadron Kaluza-Klein theory may be found in [Rho et al 16](#RhoEtAl16).

Strikingly, the [[experiment|experimentally]] observed [[hadron]]-spectrum also exhibits [[supersymmetry]]: see at _[[hadron supersymmetry]]_.

## Related concepts

* [[hadron supersymmetry]]

* [[Witten-Sakai-Sugimoto model]]

* [[quantum hadrodynamics]]

## References

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

### History

The late [[Michael Atiyah]], following up on his visionary early work in [Atiyah-Manton 89](#AtiyahManton89), saw the relevance of further develop [[hadron Kaluza-Klein theory]], and suggested using advanced tools of [[complex geometry]] for this purpose; for a reminiscence see

* [[Bernd Schroers]], p. 2-3 of:  _Michael Atiyah and Physics: the Later Years_,  ([arXiv:1910.10630](https://arxiv.org/abs/1910.10630)) in: Notices of the AMS, _Memories of Sir Michael Atiyah_ ([pdf](https://www.ams.org/journals/notices/201911/rnoti-p1834.pdf))

This led to a sequence of visionary but speculative articles, including the following:

* [[Michael Atiyah]], [[Nicholas Manton]], [[Bernd Schroers]], _Geometric Models of Matter_, Proceedings of the Royal Society A  ([arXiv:1108.5151](https://arxiv.org/abs/1108.5151), [doi:10.1098/rspa.2011.0616](https://doi.org/10.1098/rspa.2011.0616))

* [[Michael Atiyah]], [[Nicholas Manton]], _Complex Geometry of Nuclei and Atoms_,  International Journal of Modern Physics AVol. 33, No. 24, 1830022 (2018) ([arXiv:1609.02816](https://arxiv.org/abs/1609.02816), [doi:10.1142/S0217751X18300223](https://doi.org/10.1142/S0217751X18300223))

* [[Michael Atiyah]], _Geometric Models of Helium_,  Modern Physics Letters AVol. 32, No. 14, 1750079 (2017) ([arXiv:1703.02532](https://arxiv.org/abs/1703.02532), [doi:10.1142/S0217732317500791](https://doi.org/10.1142/S0217732317500791))

The idea here is to try to match patterns in the [[characteristic classes]] ([[Chern classes]]) of [[complex surfaces]] to properties of [[nuclei]].


hadron supersymmetry

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## Idea

Hadron supersymmetry
([Miyazawa 66](#Miyazawa66),
[Miyazawa 68](#Miyazawa68)) is an approximate ([[supersymmetry breaking|dynamically broken]]) [[supersymmetry]] among the [[experiment|experimentally]] observed spectra of [[hadrons]], hence an approximate [[symmetry]] of [[quantum hadrodynamics]], relating the [[masses]] of [[mesons]] (which are [[bosons]]) to [[baryons]] (which are [[fermions]]) and further to [[tetraquarks]] (which are again [[bosons]]).

This experimentally observed (see [below](#Phenomenology)) hadron [[flavor physics|flavor]] supersymmetry is in contrast to the more popular but also more speculative idea of [[color charge]] [[supersymmetry]] ([[super QCD]], [[MSSM]]), which hypothesizes that [[quarks]] and [[gluons]] have [[superpartners]]. While these ([[squarks]] and [[gluinos]]) have not been observed, even single [[quarks]] have never been observed, due to their [[confinement]] inside [[hadrons]].

A model of [[confinement|confined]] [[QCD]] which brings out the hadron supersymmetry is [[holographic light front QCD]].

In terms of [[geometric engineering of QFT]] via [[intersecting D-brane models]], [[hadron supersymmetry]] means that the [[flavor brane]] is [[supersymmetry|supersymmetric]], while the [[color brane]] may not be.

## Theory

From the [[constituent quark model]] of [[hadrons]] in [[quantum chromodynamics]], the phenomenon of [[hadron supersymmetry]] follows ([Catto-Gürsey 85](#CattoGuersey85), [Catto-Gürsey 88](#CattoGuersey88)) from the fact that

1. both single [[antiparticle|anti]][[quarks]] $\bar q$ as well as [[diquarks]] $q\!q$ carry the same [[color charge]] $\overline{\mathbf{3}} \in Rep_\mathbb{C}(SU(3)_c)$,

1. [[color charge]] predominantly determines the properties of [[bound states]] in [[QCD]],

1. the exchange of anti-quarks with diquarks inside the [[bound state]] with yet another [[quark]] exchanges [[mesons]] with [[baryons]],

this gives an approximate  [[symmetry]] of [[confinement|confined]] [[QCD]] as shown on the left here:

$$
  \array{
     \text{antiquark} & & \text{diquark}
     \\
     \phantom{q} \bar q & \leftrightarrow & \phantom{q} q\!q
     \\
     q \bar q & \leftrightarrow & q q\!q & \leftrightarrow & \bar q\!\!\bar{q} q\!q
     \\
     \text{meson} &  &\text{baryon} & & \text{tetraquark}
   }
$$

Dually, both [[quarks]] as well as [[antiparticle|anti]]-[[diquarks]] carry [[color charge]] $\mathbf{3} \in Rep_\mathbb{C}(SU(3)_c)$, which leads to a further such transformation exchanging [[baryons]] with [[tetraquarks]], as shown on the right.

These transformations organize into a [[supersymmetry]] operation, successively relating [[mesons]], [[baryons]] and [[tetraquarks]]:

\begin{imagefromfile}
    "file_name": "HadronSupersymmetry.jpg",
    "width": 600,
    "unit": "px",
    "caption": "from [Nielsen-Brodsky 18](#NielsenBrodsky18)"
\end{imagefromfile}

In the formulation of [[confinement|confined]] [[QCD]] as [[holographic light front QCD]], this [[hadron supersymmetry]] is brought out explicitly by the fact that the [[light cone gauge|light cone quantum mechanics]] of the system refines to [[supersymmetric quantum mechanics]] ([dTDB 14](#dTDB14), see e.g.
[Nielsen-Brodsky 18](#NielsenBrodsky18)).

This [[supersymmetry]] would be an exact [[symmetry]] of [[quantum hadrodynamics]] if [[anti-particle|anti]][[quarks]] and [[diquarks]] had not only the same [[color charge]] but also the same [[mass]] (and other properties), for instance if the quarks involved all had vanishing mass, as assumed, to lowest order, in [[chiral perturbation theory]]. Any discrepancy causes [[symmetry breaking|breaking of the symmetry]], hence here: [[supersymmetry breaking]].

\linebreak

## Phenomenology
 {#Phenomenology}

A direct consequence of [[hadron supersymmetry]] is the equality of [[slopes]] of the [[Regge trajectories]] of [[mesons]] and [[baryons]]. This is indeed observed in [[experiment]], with high accuracy:

\begin{imagefromfile}
    "file_name": "MesonBaryonReggeTrajectoriesKlemptMetsch.jpg",
    "width": 420,
    "unit": "px",
    "caption": "from [Klempt-Metsch 12](Regge+trajectory#KlemptMetsch12)"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "MesonBaryonReggeTrajectories.jpg",
    "width": 600,
    "unit": "px",
    "caption": "from [BTDL 16](#BTDL16)"
\end{imagefromfile}

## Related concepts

* [[hadron Kaluza-Klein theory]]

* [[baryon-lepton symmetry]]

## References

### Original articles

The phenomenon of [[hadron supersymmetry]] was first noted and formalized (together with early classification of [[supersymmetry]] [[super Lie algebras]]) in:

* {#Miyazawa66} [[Hironari Miyazawa]], _Baryon Number Changing Currents_, Prog. Theor. Phys. 36 (1966) 6, 1266-1276 ([spire:1235194](https://inspirehep.net/literature/1235194), [doi:10.1143/PTP.36.1266](https://doi.org/10.1143/PTP.36.1266))

* {#Miyazawa68} [[Hironari Miyazawa]], _Spinor Currents and Symmetries of Baryons and Mesons_, Phys. Rev. 170, 1586 (1968) ([doi:10.1103/PhysRev.170.1586](https://doi.org/10.1103/PhysRev.170.1586))

In view of [[diquark]] models:

* [[Don B. Lichtenberg]], *Baryon Supermultiplets of $SU(6) \times O(3)$ in a Quark-Diquark Model*, Phys. Rev. **178** (1969) 2197 ([doi:10.1103/PhysRev.178.2197](https://doi.org/10.1103/PhysRev.178.2197))

An argument that [[hadron supersymmetry]] is in fact implied by the [[constituent quark model]] in [[quantum chromodynamics]]:

* {#CattoGuersey85} [[Sultan Catto]], [[Feza Gürsey]], _Algebraic treatment of effective supersymmetry_, Nuov Cim A 86, 201–218 (1985) ([doi:10.1007/BF02902548](https://doi.org/10.1007/BF02902548))

* {#CattoGuersey88} [[Sultan Catto]], [[Feza Gürsey]], _New realizations of hadronic supersymmetry_, Nuov Cim A 99, 685–699 (1988) ([doi:10.1007/BF02730633](https://doi.org/10.1007/BF02730633))

* [[Sultan Catto]], _Algebraic Realization of Quark-Diquark Supersymmetry_ ([arXiv:hep-th/9811069](https://arxiv.org/abs/hep-th/9811069), [spire:479005](https://inspirehep.net/literature/479005))

* [[Sultan Catto]], Yasemin Gürcan, A. Khalfan, L. Kurt, B. Nicolescu, E. Yu, _Hadronic Supersymmetry from QCD_, Ukrainian Journal of Physics, Vol. 64 No. 8 (2019) ([doi:10.15407/ujpe64.8.672](https://doi.org/10.15407/ujpe64.8.672))

* {#CGNY19} [[Sultan Catto]], Yasemin Gürcan, B. Nicolescu, E. Yu, _Effective Hadronic Supersymmetry from Quantum Chromodynamics_, Ukrainian Journal of Physics, Vol 64 No 12 (2019)  ([doi:10.15407/ujpe64.12.1096](https://doi.org/10.15407/ujpe64.12.1096))

Formulation via [[octonions]]:

* [[Sultan Catto]], Yasemin Gürcan, Amish Khalfan and Levent Kurt, _Algebraic formulation of hadronic supersymmetry based on octonions: new mass formulas and further applications_, Journal of Physics: Conference Series, Volume 563, XXII International Conference on Integrable Systems and Quantum Symmetries (ISQS-22) 23–29 June 2014, Prague, Czech Republic ([doi:10.1088/1742-6596/563/1/012006/meta](https://iopscience.iop.org/article/10.1088/1742-6596/563/1/012006/meta))

See also:

* Čestmir Burdik, [[Sultan Catto]], Yasemin Giircan, Amish Khalfan, V. Kato La and Enxi Yu, _Generalized Four-Dimensional Effective Hadronic Supersymmetry based on QCD: New Results_, Journal of Physics: Conference Series, Volume 1416, XXVI International Conference on Integrable Systems and Quantum symmetries 8–12 July 2019, Prague, Czech Republic ([doi:10.1088/1742-6596/1416/1/012008](https://iopscience.iop.org/article/10.1088/1742-6596/1416/1/012008))

### Review

Review of the original work [Miyazawa 66](#Miyazawa66), [Miyazawa 68](#Miyazawa68), [Catto-Gürsey 85](#CattoGuersey85), [Catto-Gürsey 88](#CattoGuersey88):

* Mauro Anselmino, Enrico Predazzi, Svante Ekelin, Sverker Fredriksson, [[Don B. Lichtenberg]], Chapter IX of: *Diquarks*, Rev. Mod. Phys. **65** (1993) 1199 ([doi:10.1103/RevModPhys.65.1199](https://doi.org/10.1103/RevModPhys.65.1199))

* [[Don Lichtenberg]], _Hadron Supersymmetry and Relations between Meson and Baryon Masses_, In: Kursunoglu B.N., Mintz S.L., Perlmutter A. (eds.) _Neutrino Mass, Dark Matter, Gravitational Waves, Monopole Condensation, and Light Cone Quantization_, Springer 1996 ([doi:10.1007/978-1-4899-1564-1_29](https://doi.org/10.1007/978-1-4899-1564-1_29))

* {#Lichtenberg99} [[Don Lichtenberg]], _Whither Hadron Supersymmetry?_, Contribution to: _[International Conference on Orbis Scientiae 1999: Quantum Gravity, Generalized Theory of Gravitation and Superstring Theory Based Unification_ (28th Conference on High-Energy Physics and Cosmology Since 1964)](https://inspirehep.net/conferences/972660) (1999) 203-208
([arXiv:hep-ph/9912280](https://arxiv.org/abs/hep-ph/9912280), [spire:511339](https://inspirehep.net/literature/511339))

Review with emphasis on the formulation via [[holographic light front QCD]]:

* [[Guy de Téramond]], _Superconformal and Supersymmetric Constraints to Hadron Masses in Light-Front Holographic QCD_, talk at _[The Proton Mass: At the Heart of Most Visible Matter](https://www.ectstar.eu/node/2218)_, Trento 2017 ([pdf](https://www.ectstar.eu/sites/www.ectstar.eu/files/talks/de-teramond-pm.pdf), [[TeramondHadronSupersymmetry2017.pdf:file]])

* {#Brodsky18} [[Stanley Brodsky]], _Hadron Spectroscopy and Dynamics from Light-Front Holography and Superconformal Algebra_, Few-Body Syst 59, 83 (2018) ([arXiv:1802.08552](https://arxiv.org/abs/1802.08552))

* [[Stanley Brodsky]], _Color Confinement and Supersymmetric Properties of Hadron Physics from Light-Front Holography_, International Conference on Beauty, Charm and Hyperon Hadrons (BEACH 2018) 17–23 June 2018, Peniche, Portugal ([arXiv:1912.12578](https://arxiv.org/abs/1912.12578))

* [[Guy de Téramond]], *Emergent phenomena in QCD: The holographic perspective* [[arXiv:2212.14028](https://arxiv.org/abs/2212.14028)]

### Exotic hadron phenomenology

Discussion of [[exotic hadron]]-[[phenomenology]] using broken [[hadron supersymmetry]]:

* [[Don Lichtenberg]],  Renato Roncaglia, Enrico Predazzi, *Predicting exotic hadron masses from supersymmetry and a quark -- diquark model*, J. Phys. G: Nucl. Part. Phys. **23** (1997) 865 ([doi:10.1088/0954-3899/23/8/001](https://iopscience.iop.org/article/10.1088/0954-3899/23/8/001))

* [[Harry J. Lipkin]], *Exotic Hadrons in the Constituent Quark Model*, Prog. Theor. Phys. Suppl. **168** (2007) 15-22 ([arXiv:hep-ph/0703190](https://arxiv.org/abs/hep-ph/0703190), [doi:10.1143/PTPS.168.15](https://doi.org/10.1143/PTPS.168.15))

Specifically regarding [[tetraquarks]]:

* [[Hans Günter Dosch]], [[Stanley Brodsky]], [[Guy de Teramond]], [[Marina Nielsen]], [[Liping Zou]], _Exotic States in a Holographic Theory_ ([arXiv:2012.02496](https://arxiv.org/abs/2012.02496))

  > (via [[holographic light front QCD]])

* M. A. Bedolla, J. Ferretti, C. D. Roberts, E. Santopinto, *Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective* ([arXiv:1911.00960](https://arxiv.org/abs/1911.00960), [doi:10.1140/epjc/s10052-020-08579-3](https://doi.org/10.1140/epjc/s10052-020-08579-3))

  > (in view of [experimental results in 2021](tetraquark#ExperimentalDetection))

### Via holographic light front QCD

Discussion of [[hadron supersymmetry]] via [[light cone gauge|light cone]] [[supersymmetric quantum mechanics]] in [[holographic light front QCD]]:

* {#dTDB14}  [[Guy de Teramond]], [[Hans Günter Dosch]], [[Stanley Brodsky]], _Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding_, Phys. Rev. D 91, 045040 (2015) ([arXiv:1411.5243](https://arxiv.org/abs/1411.5243))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Superconformal Baryon-Meson Symmetry and Light-Front Holographic QCD_, Phys. Rev. D 91, 085016 (2015) ([arXiv:1501.00959](https://arxiv.org/abs/1501.00959))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum_, Phys. Rev. D 92, 074010 (2015) ([arXiv:1504.05112](https://arxiv.org/abs/1504.05112))

* [[Alexandre Deur]], [[Stanley Brodsky]], [[Guy de Teramond]], Section 4.2 of: *The QCD Running Coupling*, Prog. Part. Nuc. Phys. 90 1 (2016) ([arXiv:1604.08082](https://arxiv.org/abs/1604.08082), [doi:10.1016/j.ppnp.2016.04.003](https://doi.org/10.1016/j.ppnp.2016.04.003))

* {#BTDL16}  [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Cédric Lorcé, _Meson/Baryon/Tetraquark Supersymmetry from Superconformal Algebra and Light-Front Holography_, International Journal of Modern Physics AVol. 31, No. 19, 1630029 (2016) ([arXiv:1606.04638](https://arxiv.org/abs/1606.04638))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum II_, Phys. Rev. D 95, 034016 (2017) ([arXiv:1612.02370](https://arxiv.org/abs/1612.02370))

* {#NielsenBrodsky18} [[Marina Nielsen]], [[Stanley Brodsky]], _Hadronic Superpartners from Superconformal and Supersymmetric Algebra_, Phys. Rev. D 97, 114001 (2018) ([arXiv:1802.09652](https://arxiv.org/abs/1802.09652), [doi:10.1103/PhysRevD.97.114001](https://journals.aps.org/prd/references/10.1103/PhysRevD.97.114001))

* [[Marina Nielsen]], [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Fernando S. Navarra, [[Liping Zou]], _Supersymmetry in the Double-Heavy Hadronic Spectrum_, Phys. Rev. D 98, 034002 (2018) ([arXiv:1805.11567](https://arxiv.org/abs/1805.11567))

* [[Stanley Brodsky]], *Supersymmetric and Other Novel Features of Hadron Physics from Light-Front Holography*,  Proceedings of the 24th Workshop, "What Comes Beyond the Standard Models" ([arXiv:2112.02453](https://arxiv.org/abs/2112.02453))


hadrons as KK-modes of 5d Yang-Mills theory -- references

### Hadrons as KK-modes of 5d Yang-Mills theory
 {#ReferencesHadronsAsKKModesOf5dYangMillsTheory}

The suggestion that the tower of observed [[vector mesons]] -- when regarded as [[gauge fields]] of [[hidden local symmetry|hidden local symmetries]] of [[chiral perturbation theory]] -- is reasonably modeled as a [[Kaluza-Klein theory|Kaluza-Klein tower]] of [[D=5 Yang-Mills theory]]:

* {#SonStephanov03} D. T. Son, M. A. Stephanov, _QCD and dimensional deconstruction_, Phys. Rev. D69 (2004) 065020 ([arXiv:hep-ph/0304182](https://arxiv.org/abs/hep-ph/0304182), [doi:10.1103/PhysRevD.69.065020](https://doi.org/10.1103/PhysRevD.69.065020))

That the pure [[pion]]-[[Skyrmion]]-model of [[baryons]] is approximately the [[Kaluza-Klein theory|KK-reduction]] of [[instantons]] in [[D=5 Yang-Mills theory]] is already due to:

* {#AtiyahManton89} [[Michael Atiyah]], [[Nicholas Manton]], _Skyrmions from instantons_, Phys.  Lett.  B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

with a [[hyperbolic space]]-variant in:

* [[Michael Atiyah]], [[Paul Sutcliffe]], _Skyrmions, instantons, mass and curvature_,  	Phys. Lett. B605 (2005) 106-114 ([arXiv:hep-th/0411052](https://arxiv.org/abs/hep-th/0411052))

Further discussion of this approximation:

* [[Josh Cork]], [[Chris Halcrow]], *ADHM skyrmions* ([arXiv:2110.15190](https://arxiv.org/abs/2110.15190))

The observation that the result of [Atiyah-Manton 89](#AtiyahManton89) becomes an exact [[Kaluza-Klein theory|Kaluza-Klein construction]] of [[Skyrmions]]/[[baryons]] from [[D=5 Yang-Mills theory|D=5]] [[instantons]] when the full [[Kaluza-Klein theory|KK-tower]] of [[vector mesons]] as in [Son-Stephanov 03](#SonStephanov03) is included into the Skyrmion model (see also [there](skyrmion#ReferencesSkyrmeHadrodynamicsWithVectorMesons)) is due to:

* {#Sutcliffe10} [[Paul Sutcliffe]], _Skyrmions, instantons and holography_, JHEP 1008:019, 2010 ([arXiv:1003.0023](https://arxiv.org/abs/1003.0023))

* {#Sutcliffe15} [[Paul Sutcliffe]], _Holographic Skyrmions_, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 ([spire:1383608](http://inspirehep.net/record/1383608), [doi:10.1142/S0217984915400515](https://doi.org/10.1142/S0217984915400515))

In the [[Sakai-Sugimoto model]] of [[holographic QCD]] the [[D=5 Yang-Mills theory]] of this [[hadron Kaluza-Klein theory]] is identified with the [[worldvolume]]-theory of [[D8-brane|D8]]-[[flavour branes]] [[D4/D8-brane intersection|intersected with D4-branes]] in an [[intersecting D-brane model]]:

* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], section 5.2 of _Low energy hadron physics in holographic QCD_, Prog.Theor.Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))

* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], section 3.3. of _More on a holographic dual of QCD_, Prog.Theor.Phys.114:1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))

* {#HSSY07} Hiroyuki Hata, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Shinichiro Yamato, _Baryons from instantons in holographic QCD_, Prog.Theor.Phys.117:1157, 2007 ([arXiv:hep-th/0701280](https://arxiv.org/abs/hep-th/0701280))

* {#BolognesiSutcliffe13} [[Stefano Bolognesi]], [[Paul Sutcliffe]], _The Sakai-Sugimoto soliton_, JHEP 1401:078, 2014 ([arXiv:1309.1396](https://arxiv.org/abs/1309.1396))

* {#Bartolini17} [[Lorenzo Bartolini]], [[Stefano Bolognesi]], Andrea Proto, _From the Sakai-Sugimoto Model to the Generalized Skyrme Model_, Phys. Rev. D **97** 014024 (2018) \[doi:10.1103/PhysRevD.97.014024, arXiv:1711.03873\]

* [[Lorenzo Bartolini]], [[Stefano Bolognesi]], [[Sven Bjarke Gudnason]], Tommaso Rainaldi, *Mass and Isospin Breaking Effects in the Skyrme Model and in Holographic QCD* \[arXiv:2312.15404\]

Extensive review of this [[holographic QCD|holographic]]/[[Kaluza-Klein theory|KK-theoretic]]-realization of [[quantum hadrodynamics]] from [[D=5 Yang-Mills theory]] is in:

* {#RhoEtAl16} [[Mannque Rho]], [[Ismail Zahed]] (eds.)  _[[The Multifaceted Skyrmion]]_,  World Scientific, Second edition, 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

  * {#Sugimoto16} [[Shigeki Sugimoto]], _Skyrmion and String theory_, chapter 15 in [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

Via the realization of [[D4/D8 brane bound states]] as [[instantons]] in the [[D8-brane]] [[worldvolume]]-theory (see [there](DpDpPlus4BraneBoundState#AsYangMillsInstantons) and [there](D4-D8+brane+bound+state#AsInstantonsOnTheD8Brane)), this relates also to the model of [[baryons]] as [[wrapped brane|wrapped]] [[D4-branes]], originally due to

* {#Witten98b} [[Edward Witten]], _Baryons And Branes In Anti de Sitter Space_, JHEP 9807:006, 1998 ([arXiv:hep-th/9805112](https://arxiv.org/abs/hep-th/9805112))

* {#GrossOoguri98} [[David Gross]], [[Hirosi Ooguri]], _Aspects of Large $N$ Gauge Theory Dynamics as Seen by String Theory_, Phys. Rev. D58:106002, 1998 ([arXiv:hep-th/9805129](https://arxiv.org/abs/hep-th/9805129))

and further developed in the [[nuclear matrix model]]:

* {#HashimotoIizukaYi10} [[Koji Hashimoto]], Norihiro Iizuka, Piljin Yi, _A Matrix Model for Baryons and Nuclear Forces_, JHEP 1010:003, 2010 ([arXiv:1003.4988](https://arxiv.org/abs/1003.4988))

* Si-wen Li, Tuo Jia, _Matrix model and Holographic Baryons in the D0-D4 background_, Phys. Rev. D 92, 046007 (2015) ([arXiv:1506.00068](https://arxiv.org/abs/1506.00068))

* {#HashimotoMatsuoMorita19} [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))

* Yasuhiro Hayashi, Takahiro Ogino, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Stringy excited baryons in holographic QCD_, Prog Theor Exp Phys (2020) ([arXiv:2001.01461](https://arxiv.org/abs/2001.01461))

In relation to [[Yang-Mills monopoles]]:

* [[Stefano Bolognesi]], *Solitons, Large $N$ and Holography*, 2015 ([[Bolognesi_SolitonsHolography.pdf:file]])

Discussion, in this context, of [[D-term]] effects affecting [[hadron]] stability:

* Mitsutoshi Fujita, Yoshitaka Hatta, [[Shigeki Sugimoto]], Takahiro Ueda, *Nucleon D-term in holographic QCD* $[$[arXiv:2206.06578](https://arxiv.org/abs/2206.06578)$]$

More on [[baryons]] in the [[Sakai-Sugimoto model]] of [[holographic QCD]]:

* Salvatore Baldino, [[Lorenzo Bartolini]], [[Stefano Bolognesi]], [[Sven Bjarke Gudnason]], _Holographic Nuclear Physics with Massive Quarks_,  Phys. Rev. D **103** 126015 (2021) \[doi:10.1103/PhysRevD.103.126015, arXiv:2102.00680\]

More on [[mesons]] in [[holographic QCD]]:

* [[Johanna Erdmenger]], [[Nick Evans]], [[Ingo Kirsch]], [[Ed Threlfall]], *Mesons in Gauge/Gravity Duals - A Review*, Eur. Phys. J. A **35** (2008) 81-133 [[arXiv:0711.4467](https://arxiv.org/abs/0711.4467), [doi:10.1140/epja/i2007-10540-1](https://doi.org/10.1140/epja/i2007-10540-1)]

* Daniel Ávila, Leonardo Patiño, _Melting holographic mesons by cooling a magnetized quark gluon plasma_ ([arXiv:2002.02470](https://arxiv.org/abs/2002.02470))

* Xuanmin Cao, Hui Liu, Danning Li, _Pion quasiparticles and QCD phase transitions at finite temperature and isospin density from holography_, Phys. Rev. D 102, 126014 (2020) ([arXiv:2009.00289](https://arxiv.org/abs/2009.00289))

* Xuanmin Cao, Songyu Qiu, Hui Liu, Danning Li, _Thermal properties of light mesons from holography_ ([arXiv:2102.10946](https://arxiv.org/abs/2102.10946))

* Artur Amorim, Miguel S. Costa, Matti Järvinen, _Regge theory in a Holographic dual of QCD in the Veneziano Limit_ ([arXiv:2102.11296](https://arxiv.org/abs/2102.11296))

* {#daRocha21} [[Roldão da Rocha]], *Information in AdS/QCD: mass spectroscopy of isovector mesons*, Phys. Rev. D **103** 106027 (2021) [[arXiv:2103.03924](https://arxiv.org/abs/2103.03924), [doi:10.1103/PhysRevD.103.106027](https://doi.org/10.1103/PhysRevD.103.106027)]

* Shahin Mamedov, Narmin Nasibova, _Temperature dependence of $\rho$ meson-nucleon coupling constant from the  soft-wall model_ ([arXiv:2103.10494](https://arxiv.org/abs/2103.10494))

An alternative scenario of derivation of 4d [[Skyrmions]] by [[KK-compactification]] of [[D=5 Yang-Mills theory]], now on a [[closed interval]], motivated by [[M5-branes]] instead of by [[D4/D8-brane intersections]] as in the [[Sakai-Sugimoto model]], is discussed in:

* {#IvanovaLechtenfeldPopov18} [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Skyrme model from 6d $\mathcal{N}= (2,0)$ theory_, Physics Letters B Volume 783, 10 August 2018, Pages 222-226 ([arXiv:1805.07241](https://arxiv.org/abs/1805.07241))

following

* [[Tatiana Ivanova]], [[Olaf Lechtenfeld]], [[Alexander Popov]], _Non-Abelian sigma models from Yang-Mills theory compactified on a circle_, Physics Letters B Volume 781, 10 June 2018, Pages 322-326 ([arXiv:1803.07322](https://arxiv.org/abs/1803.07322))

See also:

* Y. H. Ahn, Sin Kyu Kang, Hyun Min Lee, *Towards a Model of Quarks and Leptons* ([arXiv:2112.13392](https://arxiv.org/abs/2112.13392))


Hadwiger's theorem

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

#Contents#
* table of contents
{:toc}

## Idea

A theorem in [[Euclidean geometry]] about uniqueness of measures of subsets of [[Cartesian space]].

## References

* [[Tom Leinster]],

  _Hadwiger's theorem, Part 1_ ([blog post](http://golem.ph.utexas.edu/category/2011/06/hadwigers_theorem_part_1.html)),

  _Hadwiger's theorem, Part 2_ ([blog post](http://golem.ph.utexas.edu/category/2011/08/hadwigers_theorem_part_2.html)),


haecceity

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

\tableofcontents

## Idea

In [[philosophy]], **haecceity** or **primitive thisness** is the irreducible determination of a thing that makes it the particular thing that it is.

## See also

* [[philosophy of physics]]

## References

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

* French, Steven (2019), Zalta, Edward N. (ed.), *Identity and Individuality in Quantum Theory*, The Stanford Encyclopedia of Philosophy (Winter 2019 ed.), Metaphysics Research Lab, Stanford University. ([web](https://plato.stanford.edu/archives/win2019/entries/qt-idind/))

[[!redirects haecceity]]
[[!redirects haecceities]]

[[!redirects primitive thisness]]


Haefliger groupoid

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

#Contents#
* table of contents
{:toc}

## Definition

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

For $n \in \mathbb{N}$, the **Haefliger groupoid** $\Gamma^n$ is the [[groupoid]] whose set of [[objects]] is the [[Cartesian space]] $\mathbb{R}^n$ and for which a [[morphism]] $x \to y$ is a [[germ]] of a [[diffeomorphism]] $(\mathbb{R}^n ,x) \to (\mathbb{R}^n ,y)$.

This is regarded as a [[topological groupoid|topological]] or [[Lie groupoid|Lie]] [[étale groupoid]] via the canonical [[topological space|topology]]/[[smooth structure]] on $(\Gamma^n)_0 = \mathbb{R}^n$ and taking $s \colon (\Gamma^n)_1 \to (\Gamma^n)_0$ to be the [[étale space]] [associated](etale+space#RelationToSheaves) to the [[sheaf]] on $\mathbb{R}^n$ (with its canonical [[open cover]] [[Grothendieck topology]]) which is the [[sheafification]] of the presheaf that sends $U \subset \mathbb{R}^n$ to the set of all open [[embedding of differentiable manifolds|embeddings]] of $U$ into $\mathbb{R}^n$.

The [[smooth stack]] represented by the smooth Haefliger groupoid is also called the _Haefliger stack_.

=--

## Variants and Generalizations
 {#Variants}

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

There is also the full smooth structure on the space of germs of diffeomorphisms. This gives a Lie groupoid whose underlying bare groupoid is the same as that of the Haefliger groupoid, but whose smooth structure is different.

=--

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

More generally for a given [[integrable G-stucture]] there is a corresponding Haefliger groupoid, for instance for [[symplectic structures]].

=--

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

Instead of considering [[germs]] of local diffeomorphisms one may consider (just) order-$k$ [[jets]] of these. The resulting Lie groupoids are known as [[jet groupoids]] (see [Lorenz 09](#Lorenz09))

=--

## Properties

### Classification of foliations

The Haefliger groupoid classifies [[foliations]]. See at _[[Haefliger theorem]]_.

### Universal characterization

Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.

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

The Haefliger stack is a [[terminal object]] in the [[2-category]] of [[étale stacks]] on the site of [[smooth manifolds]] with [[étale morphisms]] between them.

=--

([Carchedi 12, theorem 3.3.](#Carchedi12))

This implies ([Carchedi 12, 3,2](#Carchedi12))

+-- {: .num_theorem #StacksOnEtaleSiteAndEtaleStacks}
###### Theorem

There is an equivalence

$$
  \Theta \colon St(SmthMfd^{et}) \simeq EtSt(SmthMfd)^{et}
$$

between [[stacks]] on the [[site]] of smooth manifolds with [[local diffeomorphisms]] between them and [[étale stacks]] with [[étale morphisms]] between them inside all [[smooth stacks]].

=--

([Carchedi 12, theorem 1.3](#Carchedi12))

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

This in turn implies for instance that the Haefliger groupoid for [[complex structures]] ([Carchedi 12, p. 38](#Carchedi12)) is simply the image under the equivalence $\Theta$ in theorem \ref{StacksOnEtaleSiteAndEtaleStacks} of the sheaf on $SmthMfd^{et}$ which sends each smooth manifold to its set of [[complex structures]]. (...)

=--

### Sheaves and stacks on the Haefliger groupoid.

Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.

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

The [[category of sheaves]] over $\mathcal{H}$ is equivalently the category of sheaves on the [[site]] of [[smooth manifolds]] with [[local diffeomorphism]] between them.

=--

([Carchedi 12, theorem 3.1](#Carchedi12)).

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

The [[2-topos]] over the Haefliger stack is equivalent to the 2-topos over the site $SmthMfd^{et}$ of smooth manifolds with local diffeomorphisms between them:

$$
  St(\mathcal{H}) \simeq St(SmthMfd^{et})
$$

=--

([Carchedi 12, 3.2](#Carchedi12)).

## References

Original articles:

* [[André Haefliger]], _Homotopy and integrability_,  In: N.H. Kuiper (ed.) _Manifolds_ — Amsterdam 1970. Lecture Notes in Mathematics, vol 197. Springer 1971 ([doi:10.1007/BFb0068615](https://doi.org/10.1007/BFb0068615))

* [[André Haefliger]], _Groupoïdes d'holonomie et espaces classiants_ , Astérisque 116 (1984), 70-97 ([numdam:AST_1984__116__70_0](http://www.numdam.org/item/?id=AST_1984__116__70_0))

* [[Raoul Bott]], _Lectures on characteristic classes and foliations_, Springer LNM 279, 1-94 ([doi:10.1007/BFb0058509](https://doi.org/10.1007/BFb0058509))

A textbook account is in

* [[Ieke Moerdijk]], [[Janez Mrčun]], _[[Introduction to foliations and Lie groupoids]]_, Cambridge Studies in Advanced Mathematics __91__, 2003. x+173 pp. ISBN: 0-521-83197-0

See also

* Wikipedia, _[Haefliger structure](http://en.wikipedia.org/wiki/Haefliger_structure)_

Discussion in a broader context of [[étale stacks]] and [[étale ∞-stacks]]:

* {#Carchedi12} [[David Carchedi]], section 2.2, section 3 of: _Étale Stacks as Prolongations_, Advances in Mathematics Volume 352, 20 August 2019, Pages 56-132 ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))

* [[David Carchedi]], p. 104-105 of: _Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi_, Memoirs of the American Mathematical Society 2020; 120 ([arXiv:1312.2204](https://arxiv.org/abs/1312.2204), [ISBN:978-1-4704-5810-2](https://bookstore.ams.org/memo-264-1282))

* {#Carchedi15} [[David Carchedi]], _On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces_, Advances of Mathematics, Volume 294, 2016, Pages 756-818 ([arXiv:1504.02394](http://arxiv.org/abs/1504.02394))

Discussion of [[jet groupoids]] includes

* {#Lorenz09} Arne Lorenz, _Jet Groupoids, Natural Bundles
and the Vessiot Equivalence Method_, Thesis ([pdf](https://web.archive.org/web/20120619105223/http://wwwb.math.rwth-aachen.de/~arne/Dissertation_Lorenz_Arne.pdf), [pdf](https://publications.rwth-aachen.de/record/50704)) 2009

The [[geometric realization]]/[[shape modality]] for Haefliger-type groupoids is discussed in

* {#Carchedi15} [[David Carchedi]], _On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces_ ([arXiv:1504.02394](http://arxiv.org/abs/1504.02394))

[[!redirects Haefliger groupoids]]

[[!redirects Haefliger stack]]
[[!redirects Haefliger stacks]]


Haefliger structure

## Idea

The notion of _Haefliger structure_ is a slight generalization of that of _[[foliation]]_. A _Haefliger structure_ on a [[manifold]] is a [[cocycle]] with coefficients in the [[Haefliger groupoid]], a _[[Haefliger cocycle]]_.

## References

* [[Ieke Moerdijk]], [[Janez Mrčun]], _[[Introduction to foliations and Lie groupoids]]_, Cambridge Studies in Advanced Mathematics __91__, 2003. x+173 pp. ISBN: 0-521-83197-0

[[!redirects Haefliger structures]]

[[!redirects Haefliger cocycle]]
[[!redirects Haefliger cocycles]]


Haefliger theorem

#Contents#
* table of contents
{:toc}

## Idea

On the classification of [[foliations]]:

## Statement

+-- {: .num_theorem}
###### Theorem

If $X$ is an open [[manifold]], then there is a [[bijection]] between

1. [[codimension]] q [[foliations]] on $X$ up to integral [[homotopy]];

1. [[homotopy classes]] of fiberwise surjective [[vector bundle]] maps $T X \to N \Gamma_q$,

where $\Gamma_q$ is the [[topological groupoid]] of [[germs]] of local [[diffeomorphisms]] of $\mathbb{R}^q$.

=--

## Related concepts

* [[Haefliger groupoid]]

## References

* [[John Francis]], notes by M. Hoyois, _Haefliger's theorem classifying foliations over open manifold_ ([pdf](http://math.northwestern.edu/~jnkf/classes/hprin/14haefligerthm.pdf))

[[!redirects Haefliger's theorem]]


hafnian

Hafnian is a special determinant-like expression evaluated for the pair $(F,I)$ of a matrix corresponding to an element $F$ of symplectic Lie algebra $sp(2n)$ and a subsequence $\mathbf{i} = (i_1,\ldots,i_{2k})$ of the ascending sequence $(-n,\ldots,-1, 1,\ldots,n)$. Let $\Sigma_n$ be the symmetric group on $n$ letters. Then
$$
Hf F^{\mathbf{i}} := \frac{1}{2^k k!}\sum_{\sigma\in\Sigma_n}
(-1)^{\sum_{l = 0}^{k-1} i_{\sigma(2 l + 1)}} F_{i_{\sigma(1)}i_{-\sigma(2)}}\cdots F_{i_{\sigma(2k-1)}i_{-\sigma(2k)}}
$$

Related entries include [[determinant]], [[Pfaffian]], [[Pfaffian line bundle]]

* J.-G. Luque, J.-Y. Thibon, _Pfaffian and hafnian identities in shuffle algebras_, [math.CO/0204026](http://arxiv.org/abs/math.CO/0204026)

* sec 4.4. in: A. I. Molev, _Yangians and their applications_, in "Handbook of Algebra", Vol. 3, (M. Hazewinkel, Ed.), Elsevier, 2003, pp. 907-959 [http://arxiv.org/abs/math.QA/0211288](http://arxiv.org/abs/math.QA/0211288)


HAG context

A **HAG context** is a setting for [[homotopical algebraic geometry]].


Hagen Kleinert

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

## Selected writings

On [[path integrals]]:

* [[Hagen Kleinert]], *Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets*, World Scientific (1990, 1994, 2003, 2006, 2009) [[doi:10.1142/7305](https://doi.org/10.1142/7305), [pdf](https://hagenkleinert.de/documents/pi/HagenKleinert_PathIntegrals.pdf)]

## Related entries

* [[path integral quantization]]

category: people


Hagen Triendl

* [spire page](https://inspirehep.net/authors/1070922)

## Selected writings

On [[BPS-states]] in [[string theory]]/[[supergravity]] using [[G-structures]]:

* [[Ruben Minasian]], [[Daniël Prins]], [[Hagen Triendl]], _Supersymmetric branes and instantons on curved spaces_, JHEP 10 (2017) 159 ([arXiv:1707.07002](https://arxiv.org/abs/1707.07002))

category: people


Hahn series

# Hahn series

* table of contents
{: toc}

## Idea

A Hahn series is a generalization of a [[formal power series]] which allows non-integer exponents and transfinite sums, as long as the exponents are well-ordered.

## Definition

Let $G$ be a [[linear order|linearly]] [[ordered abelian group]], and $k$ a field.

+-- {: .un_defn}
###### Definition
The ring of **Hahn series** with [[value group]] $G$, denoted $k[t^G]$ (or sometimes $k[[t^G]]$ or $k((t^G))$), is the ring of functions $f\colon G \to k$ such that $\{x \in G : f(x) \neq 0\}$ is [[well-ordered]] as as a subset of $G$ (or, sometimes, as a subset of the opposite poset $G^{op}$). Addition is defined pointwise, and multiplication is defined by the [[convolution]] product:
$$(f \cdot g)(x) = \sum_{y+z = x \in G} f(y)g(z)$$
=--

Notationally, we may write a Hahn series $f\colon G \to k$ as $\sum_{x\in G} f(x) t^x$.

## Properties

+-- {: .un_theorem}
###### Theorem
The ring $k[t^G]$ is a field. If $k$ is [[algebraically closed field|algebraically closed]], then $k[t^G]$ is algebraically closed provided that $G$ is [[divisible group|divisible]].
=--

As a corollary, if $G$ is divisible, $k[t^G]$ is [[real closed field|real closed]] if $k$ is real closed. This is because the adjunction of a square root of $-1$ would make $k[t^G]$ algebraically closed, since this gives the same result as constructing the Hahn series over the algebraically closed field $k[\sqrt{-1}]$.

The [[valuation ring|multiplicative valuation]] $v(f)$ of an element $f\in k[t^G]$ is the least $x \in G$ for which $f(x) \neq 0$.  This yields a [[valuation ring]], such a valuation ring determines and is determined by a valuation on a [[field]].  The field $k[t^G]$ is a [[complete metric space|complete]] [[ultrametric space]] (and indeed a [[spherically complete field]]) with respect to this valuation, *but* unlike for [[formal power series]] the formal series $\sum_{x\in G} f(x) t^x$ do not actually "converge" in this metric.

## Examples

* If $G=\mathbb{R}$, then we have Hahn series such as
  $$ t^1 + t^{3/2} + t^{5/3} + t^{7/4} + \cdots $$

* Via Conway normal forms, the ring of Hahn series with $k=\mathbb{R}$ and with value group the [[surreal numbers]] is isomorphic to the surreal numbers themselves.  In other words, the surreals are a [[fixed point]] of the "Hahn series with real coefficients" [[functor]] on the [[category]] of abelian groups.  However, the surreals are not the [[inductive type|initial fixed point]] of this functor, since there are surreals that appear as exponents in their own Conway normal form --- for instance, the [[∞-numbers]].

* Well-based [[transseries]] can be constructed by iterating the Hahn series construction.

* If $G=\mathbb{Z}$, then a Hahn series is precisely a [[formal Laurent series]], i.e. $k[t^{\mathbb{Z}}] = k((t))$.

* If $G=\mathbb{Q}$, then $k[t^{\mathbb{Q}}]$ properly contains the [[Levi-Civita field]], as the subclass of those Hahn series whose support is finite below any fixed rational.

* More generally, any Hahn series field $k[t^G]$ contains the [[Novikov field]] of $k$ with value group $G$.

## Related pages

* [[formal power series]]
* [[formal Laurent series]]
* [[Puiseux series]]
* [[Novikov series]]
* [[Ribenboim power series]]
* [[finiteness space]]


Hahn-Banach separation theorem

## Idea

The **Hahn-Banach Separation Theorem** is a variant of the [[Hahn-Banach Theorem]] for general [[topological vector spaces]]. A version also holds for [[topological affine spaces]].

#### References ####

* Wikipedia article on [Hahn-Banach Separation Theorem](https://en.wikipedia.org/wiki/Hahn_banach_theorem#Hahn.E2.80.93Banach_separation_theorem).

[[!redirects hahn banach separation theorem]]
[[!redirects hahn-banach separation theorem]]


Hahn-Banach theorem

# Contents
* the following line creates the automatic table of contents
{:toc}

## Idea

The Hahn--Banach theorem explains why the concept of [[locally convex spaces]] is of interest in the analysis of [[topological vector spaces]]: It ensures that such a space will have enough [[continuous function|continuous]] [[linear functionals]] such that the [[dual vector space|topological dual space]] is interesting.

## Statement

+-- {: .num_theorem}
###### Theorem
Let $V$ be a [[vector space]] over a [[local field]] $K$ (usually $\mathbb{R}$ or $\mathbb{C}$), equipped with a [[seminorm]] $p: V \to [0, \infty)$. Let $W \subseteq V$ be a [[linear subspace]], and $f: W \to K$ a [[linear functional]] such that ${|f(x)|} \leq p(x)$ for all $x \in W$, then there exists an [[extension]] of $f$ to a linear functional $g: V \to K$ such that ${|g(x)|} \leq p(x)$ for all $x \in V$.
=--

## Foundational issues

The full Hahn--Banach theorem may be seen as a weak form of the [[axiom of choice]]; this is the perspective taken in, for example, _[[HAF]]_.  It fails in [[dream mathematics]] and is generally not accepted in [[constructive mathematics]].  (Does it actually imply [[excluded middle]]? No, but the [[ultrafilter principle]] frequently used to prove the Hahn-Banach theorem implies [[De Morgan's law]] which is usually not accepted in constructive mathematics.)

The Hahn-Banach theorem can be proven in [[set theory]] with the axiom of choice, or more weakly in set theory assuming the [[ultrafilter theorem]], itself a weak form of choice. (To be continued...)

However, the Hahn--Banach theorem for [[separable spaces]] is much weaker.  It may be proved constructively using only [[dependent choice]].

There is also a version of the theorem for [[locales]] proven
in [Pelletier 1991](#Pelletier91). This constructs a locale of functionals (whose points are the unit ball of the dual space) and proves that it is compact and completely regular.

## Consequences

(...)

## References ##

See also:

* Wikipedia, *[Hahn-Banach theorem] (http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem)*

Discussion in the generality of [[locales]]:

* {#Pelletier91} Joan Wick Pelletier, *Locales in Functional Analysis*, Journal of Pure and Applied Algebra Volume 70, Issues 1–2, 15 March 1991, Pages 133-145 (doi:10.1016/0022-4049(91)90013-R)

[[!redirects Hahn-Banach theorem]]
[[!redirects Hahn-Banach theorems]]
[[!redirects Hahn–Banach theorem]]
[[!redirects Hahn–Banach theorems]]
[[!redirects Hahn--Banach theorem]]
[[!redirects Hahn--Banach theorems]]


Hai Lin

## Selected writings

On the [[BMN matrix model]] in relation to the [[Myers effect]] and [[D0-D2 brane bound states]]:

* {#Lin04} [[Hai Lin]], _The Supergravity Dual of the BMN Matrix Model_, JHEP 0412:001, 2004 ([arXiv:hep-th/0407250](https://arxiv.org/abs/hep-th/0407250))

On [[M5-branes]] in the [[BMN matrix model]] and a limit where aspects of [[little string theory]] become visible:

* [[Hai Lin]], [[Juan Maldacena]], _Fivebranes from gauge theory_, Phys. Rev. D74:084014, 2006 ([arXiv:hep-th/0509235](https://arxiv.org/abs/hep-th/0509235))

category: people


Hajime Ishihara

* [Institute page](https://fp.jaist.ac.jp/public/Default2.aspx?id=48&l=1)

## Selected writings

On [[constructive mathematics|constructive]] [[reverse mathematics]]:

* [[Hajime Ishihara]], _Reverse Mathematics in Bishop's Constructive Mathematics_, Philosophia Scientiæ, CS 6 (2006) ([doi:10.4000/philosophiascientiae.406](https://doi.org/10.4000/philosophiascientiae.406),  [pdf](https://philosophiascientiae.revues.org/pdf/406))

category: people


Haken manifold

## References

* Wikipedia, _[Haken manifold](http://en.wikipedia.org/wiki/Haken_manifold)_

[[!redirects Haken manifolds]]

[[!redirects Haken 3-manifold]]
[[!redirects Haken 3-manifolds]]


Hal Sadofsky

* [webpage](http://pages.uoregon.edu/sadofsky/)

category: people


Hal Schenck

* [website](http://www.math.uiuc.edu/~schenck/)

category: people


Haldane model

> For the *[[Haldane phase]]* see instead at *[[Heisenberg model]]*.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], the *Haldane model* is a [[model (in theoretical physics)|model]] for effectively 2-[[dimension of a manifold|dimensional]] [[crystalline topological insulator|crystalline]] [[Chern insulators]] (ie. [[topological phases of matter]] "without [[symmetry protected topological phase|symmetry protection]]") obtained by starting with a simple [[model (in theoretical physics)|model]] for a [[graphene]]-like 2d [[semi-metal]] and then adding a [[mass term]] and [[interactions]] (similar to [[spin-orbit coupling]]) which [[symmetry breaking|break]] the [[time-reversal symmetry]] and the [[crystallographic point group|spatial inversion symmetry]].

## Definition

Consider the [[honeycomb lattice|honeycomb crystal lattice]] and consider the following three lattice sites $a_i \in \mathbb{R}^2$:

\begin{imagefromfile}
    "file_name": "HoneycombCrystalLattice.jpg",
    "width": 310,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [DTC](#DTC))"
\end{imagefromfile}

### The graphene-like semi-metal model

The [[Hilbert space]] [[space of quantum states|of quantum states]] of the Haldane model is the [[direct sum]] of copies of $\mathbb{C}^2$ (regarded as the defining/[[fundamental representation]] of [[SU(2)]]) indexed by the sites in a Brillouin zone of this hexagonal lattice.

The un-deformed [[Bloch theory|Bloch]]-[[Hamiltonian]] at [[momentum]]/[[wave vector]] $k$ in the [[Brillouin torus]] $\widehat{\mathbb{T}}^2$ is

\[
  \label{GrapheneLikeHamiltonian}
  H_0(k)
  \;\coloneqq\;
  t_1
  \cdot
  \underoverset{i = 1}{3}{\sum}
  \;
  \big(
    cos(k \cdot a_i)
    \sigma_x
    -
    \sigma_y
    sin(k \cdot a_i)
  \big)
  \,,
\]

where

1. $t_1 \in \mathbb{R}$ is some real parameter,

1. the [[sum]] ranges over the three unit cell sites shown above,

1. $k \cdot a_i$ denotes the canonical [[inner product]] (evaluation pairing) of the [[wave vector]] with a position space lattice vector,

1. $\sigma_x, \sigma_y, \sigma_z$ are the [[Pauli matrices]] acting as [[linear operators]] of the on-site copies of the [[SU(2)]]-[[linear representation|representation]] $\mathbb{C}^2$.

This undeformed Hamiltonian (eq:GrapheneLikeHamiltonian) is a simple but good model for the [[electronic band structure]] of a [[graphene]]-like 2-dimensional [[semi-metal]].

### Deformation by the Haldane mass term

Consider now also the following next-to-mearest neighbour site vectors $b_i \in \mathbb{R}^2$ in the above honeycomb lattice:

\begin{imagefromfile}
    "file_name": "NextNearNeighbSitesInHoneycLattice-220518.jpg",
    "width": 170,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [DTC](#DTC))"
\end{imagefromfile}

The full [[Bloch theory|Bloch]]-[[Hamiltonian]] of the Haldane model is the deformation of $H_0(k)$ (eq:GrapheneLikeHamiltonian) by a [[mass term]] of the following form:

\[
  \label{HaldaneHamiltonian}
  H(k)
  \;\coloneqq\;
  H_0(k)
  \;+\;
  \underset{
    \mathrm{mass}\;\mathrm{term}
  }{
  \underbrace{
    \left(
       M
       \;+\;
       2 t_2
       \underoverset{j = 1}{3}{\sum}
       sin(k \cdot b_i)
    \right)
    \cdot
    \sigma_z
  }
  }
  \,,
\]

where

1. $t_2 \in \mathbb{R}$ is a second [[real number|real]] parameter

1. the sum is now over the above three next-to-nearest neighbour site vectors $b_i$.

Notice that (only) for $t_2 = 0$ this reduces to the deformation by a *constant* mass term (which is often understood as the default meaning of "[[mass term]]"):

\[
  \label{HaldaneHamiltonianForConstantMassTerm}
  H(k)_{t_2 = 0}
  \;=\;
  t_1
  \cdot
  \underoverset{i = 1}{3}{\sum}
  \;
  \big(
    cos(k \cdot a_i)
    \sigma_x
    -
    \sigma_y
    sin(k \cdot a_i)
  \big)
  \;+\;
  \underset{
    \mathclap{
      \mathrm{constant}\;\mathrm{mass}\;\mathrm{term}
    }
  }{
  \underbrace{
    M
    \sigma_z
  }
  }
  \,,
\]

## Properties

### Phase diagram
 {#PhaseDiagram}

First of all, the Haldane model

* at *M = 0* constitutes a non-trivial [[graphene]]-like [[topological semi-metal]]-phase with *two* Dirac points.

\begin{imagefromfile}
    "file_name": "HaldanePhaseDiagram-220518.jpg",
    "float": "right",
    "width": 320,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": -10,
        "right": 0,
        "left": 30
    },
    "caption": "(graphics from [SS 22](#SS22))"
\end{imagefromfile}

For $M \neq 0$ and as the parameter $t_2$  (eq:HaldaneHamiltonian) increases, the Haldane model passes, consecutively,  through

1. at $0 \leq t_2/M \lt \frac{1}{3 \sqrt{3}}$ -- in particular where the *mass term is constant* (eq:HaldaneHamiltonianForConstantMassTerm) at $t_2 = 0$ --, a *topologically trivial* [[insulator]]-phase with [[Berry curvature]] concentrated (see [below](#BerryCurvature)) around the would-be Dirac points of the [[graphene]]-like [[semi-metal]] phase which has been gapped out by the constant [[mass term]];

1. at $t_2/M = \frac{1}{3 \sqrt{3}} \sim 0.19$ a non-trivial [[topological semi-metal]]-phase with a *single* band node;

1. at $\frac{1}{3\sqrt{3}} \lt t_2/M$ a *non-trivial* [[topological insulator]]-phase.

   This is the non-trivial *[[Chern insulator]]*-phase of the Haldane model.

### Berry curvature
 {#BerryCurvature}

A curious property of the Haldane model, possibly not shared by all 2d [[Chern insulators]], is that its [[Berry curvature]] is strongly localized around the (would-be) nodal [[Dirac points]], hence that the [[Berry connection]] is essentially a [[flat connection]] on the [[complement]] of a small [[neighbourhood retract]] of the (would-be) nodal points.

(eg. [Atteia 16, Sec. 2-3.4, p. 4](#Atteia16), [DTC](#DTC) [here](https://topocondmat.org/w4_haldane/haldane_model.html#gap-closings-are-sources-of-berry-curvature))

\begin{imagefromfile}
    "file_name": "BerryCurvatureInHaldaneModelFromAtteia-220518.jpg",
    "width": 620,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [Atteia 16, Fig. 2.7](#Atteia16))"
\end{imagefromfile}

Concretely, the above graphics shows the Berry curvature in the Haldane model for constant mass term, hence in the case that the interaction paramater $t_2$ (eq:HaldaneHamiltonian) vanishes:

\begin{imagefromfile}
    "file_name": "BerryCurvatureInHaldaneModelFromDTC-220518.jpg",
    "width": 340,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [DTC](#DTC))"
\end{imagefromfile}

Since blue and red coloring denotes [[Berry curvature]] of opposite [[sign]], the figure makes it plausibly manifest that the integrated [[Berry curvature]] -- and thus the [[first Chern class|first Chern number]] of the [[valence bundle]] -- *vanishes* for $M \neq 0$, $t_2 = 0$. This is the statement that the [[Chern insulator]]-phase of the Haldane model at $t_2 = 0$ is topologically trivial.

In constract, as $t_2/M \gt \frac{1}{3 \sqrt{3}}$, the [[Berry curvature]] is still concentrated around the would-be Dirac points, but now it has the same sign everywhere:

\begin{imagefromfile}
    "file_name": "PosBerryCurvInHaldaneModelFromDTC-220518.jpg",
    "width": 340,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [DTC](#DTC))"
\end{imagefromfile}

This makes it clear that the [[integration|integral]] of the [[Berry curvature]] over the [[Brillouin torus]] -- hence the [[first Chern class|first Chern number]] of the [[valence bundle]] -- is [[positive number|positive]] in this phase. This is the statement that the [[Chern insulator]]-phase of the Haldane model at $t_2/M \gt \frac{1}{3\sqrt{3}}$ is non-trivial.

### K-Theory classification?
 {#KTheoryClassification}

The established [[K-theory classification of topological phases of matter]] asserts that ([[twisted equivariant K-theory|twisted equivariant]]) [[topological K-theory]] of gapped [[valence bundles]] over a full [[Brillouin torus]] classifies [[crystalline topological insulator|crystalline]] [[topological insulator]]-phases. For the non-[[symmetry protected topological phases|symmetry protected]] phases relavant for the Haldane model, this is the [[KU-theory]]-classification of [[Chern insulators]].

There has not been much systematic discussion of how to generalize this statement to a classification of [[topological semi-metal]]-phases in some flavor of [[topological K-theory]]. But the [above](#PhaseDiagram) phase behavior of of the Haldane model under deformation by *constant* [[mass terms]] (i.e. at $t_2 = 0$) supports the conjecture that topological semi-metal phases are classified by the (twisted, equivariant) *flat* [[differential K-theory]] of the [[complement]] $\widehat{T}^2 \setminus \{k_1, \cdots, k_N\}$ of the $N$ nodal points $\vec k := k_1, k_2, \cdots, k_N$ inside the [[Brillouin torus]].

Namely, under this assumption the relevant K-theory charges form the following [[double complex]] of [[exact sequences]]:

\begin{imagefromfile}
    "file_name": "SemimetalDoubleSESinKTheory-220519.jpg",
    "width": 750,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": -10,
        "right": 0,
        "left": 30
    },
    "caption": "(graphics from [SS 22](#SS22))"
\end{imagefromfile}

(To see that the boundary of this diagram is as shown, use the [[stable homotopy type]] of the punctured [[torus]] as shown [there](torus#AsAHomotopyType).)

Here the squiggly dashed arrow indicates the mathematical lifting problem which accurately encodes the above statement that topological charges associated to accidental/spurious band nodes (those which actually make for a trivial semi-metal phase, in that there exists a constant [[mass term]] that gaps out the node crossings), give rise to local Berry curvatures whose global topological charge cancels out and thus implies that also the globally gapped phase is topologically trivial:

\begin{imagefromfile}
    "file_name": "LiftingSemimetalDoubleSESinKTheory-220519.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": -10,
        "right": 0,
        "left": 30
    },
    "caption": "(graphics from [SS 22](#SS22))"
\end{imagefromfile}

This accurately reflects what is seen [above](#PhaseDiagram) in the phase diagram of the Haldane mode, for constant mass terms $M$ (which are those that correspond to passage to $\mathrm{K}^{-1}$, by [[Karoubi K-theory]]) but in the absence of interactions ($t_2 = 0$).

\linebreak

## Related concepts

* [[topological phase of matter]]

* [[K-theory classification of topological phases of matter]]

* [[Kane-Mele model]]

## References

### Theory

The original article:

* [[Duncan Haldane]], *Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly"*, Phys. Rev. Lett. **61** (1988) 2015-2018 $[$[doi:10.1103/PhysRevLett.61.2015](https://doi.org/10.1103/PhysRevLett.61.2015)$]$

see also

* Doru Sticlet, Frederic Piéchon, [[Jean-Noël Fuchs]], Pavel Kalugin, Pascal Simon, Section 2.B.1 (pp. 3) of: *Geometrical engineering of a two-bands Chern insulator in two dimensions with arbitrary topological index*, Phys. Rev. B **85** 165456 (2012) $[$[arXiv:1201.6613](https://arxiv.org/abs/1201.6613), [doi:10.1103/PhysRevB.85.165456](https://doi.org/10.1103/PhysRevB.85.165456)$]$

Review:

* Online course on topology in condensed matter*, *Haldane model, Berry curvature, and Chern number* (2015-) $[$[topocondmat.org/w4_haldane/haldane_model.html](https://topocondmat.org/w4_haldane/haldane_model.html)$]$

* {#Atteia16} [[Jonathan Atteia]], *Topology and electronic transport in Dirac systems under irradiation* (2016) $[$[tel:02426217](https://tel.archives-ouvertes.fr/tel-02426217), [pdf](https://tel.archives-ouvertes.fr/tel-02426217/document)$]$

* {#Vanderbilt18} [[David Vanderbilt]],  Section 5.1.1 of: *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))

* [[Tudor D. Stanescu]], Section 7.2.1 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))

* [[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))

Modified Haldene model for [[Chern insulators]]:

* Doru Sticlet, [[Frédéric Piéchon]], [[Jean-Noël Fuchs]], Pavel Kalugin, Pascal Simon, *Geometrical engineering of a two-bands Chern insulator in two dimensions with arbitrary topological index*, Phys. Rev. B **85** 165456 (2012) [[arXiv:1201.6613](https://arxiv.org/abs/1201.6613), [doi:10.1103/PhysRevB.85.165456](https://doi.org/10.1103/PhysRevB.85.165456)]

* [[Marwa Mannaï]], [[Jean-Noël Fuchs]], [[Frédéric Piéchon]], [[Sonia Haddad]], *Stacking-induced Chern insulator* [[arXiv:2208.02491](https://arxiv.org/abs/2208.02491)]

Much of the above material follows:

* {#DTC} Delft Topology Course team, *[Haldane model, Berry curvature, and Chern number](https://topocondmat.org/w4_haldane/haldane_model.html#first-try)*

Some of the above material is taken from:

* {#SS22} [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Anyonic topological order in TED K-theory]]*

### Experiment

Realization in [[experiment]]:

* {#JMDLUGE14} Gregor Jotzu, Michael Messer, Rémi Desbuquois, Martin Lebrat, Thomas Uehlinger, Daniel Greif,  Tilman Esslinger:

  *Experimental realization of the topological Haldane model with ultracold fermions*, Nature **515** (2014) 237–240  ([doi:10.1038/nature13915](https://doi.org/10.1038/nature13915))

[[!redirects Haldane models]]


Haldun Özgür Bayındır

* [personal page](https://sites.google.com/view/ozgurbayindir/home)

## Selected writings

On enhancement of [[simplicial model categories|simplicial]] [[monoidal model categories]] up to [[strong monoidal functor|strong]] [[monoidal Quillen adjunction|monoidal]] [[Quillen equivalence]]:

* [[Haldun Özgür Bayındır]], [[Boris Chorny]], *Admissible replacements for simplicial monoidal model categories*, Algebr. Geom. Topol. **23** (2023) 43-73 [[arXiv:2008.00515](https://arxiv.org/abs/2008.00515), [doi:10.2140/agt.2023.23.43](https://doi.org/10.2140/agt.2023.23.43)]

category: people

[[!redirects Haldun Bayındır]]

[[!redirects Haldun Özgür Bayindir]]
[[!redirects Haldun Bayindir]]


half-twisted model

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=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]] the term _half-twisted model_ refers to a [[topological twist]] of the [[2d (2,0)-superconformal QFT]] (see there for more) which is "in-between" the [[A-model]] and the [[B-model]].

## References

The half-twisted model was introduced in

* {#Witten91} [[Edward Witten]], _Mirror Manifolds And Topological Field Theory_, AMS IP studies in Advanced mathematics 9 (1998): 121-160. ([arXiv:hep-th/9112056](http://arxiv.org/abs/hep-th/9112056))

as a variant of the [[A-model]]/[[B-model]],

For more references see at _[[2d (2,0)-superconformal QFT]]_.

[[!redirects half-twisted models]]


Hall algebra

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

#Contents#
* table of contents
{:toc}

## Idea

### In terms of 2-Segal spaces

Given a [[2-Segal space]] $X_\bullet$ such that the [[spans]]

$$
  X_1 \times X_1 \stackrel{(\partial_2, \partial_0)}{\leftarrow} X_2 \stackrel{\partial_1}{\rightarrow} X_1
$$

and

$$
  pt \leftarrow X_0 \stackrel{s_0}{\to} X_1
$$

admit pull-push [[integral transforms]] in some given [[cohomology theory]] $h$. Then the **Hall algebra** of $X$ with [[coefficients]] in $H$ is the [[associative algebra]] structure on $h(X_1)$ induced by these pull-push operations.

This is the perspective of [Dyckerhoff-Kapranov 12, def. 8.1.8](#DyckerhoffKapranov12).

### Motivic Hall algebra
 {#MotivicHallAlgebra}

Specifically for a given [[algebraic stack]] $X$ and with

$$
  X \leftarrow X^{(2)} \rightarrow X\times X
$$

denoting the moduli stack of 2-flags of coherent sheaves on $X$, then the corresponding pull-push multiplication on the motivic Grothendieck ring $K(X)$ is called the _motivic Hall algebra_ of $X$ (due to [[Dominic Joyce]] reviewed e.g. in [Bridgeland 10, 4.2](#Bridgeland10)). Discussion of motivic Hall algebras of [[Calabi-Yau 3-folds]] is in ([Kontsevich-Soibelman 08](#KontsevichSoibelman08)).

### In terms of constructible sheaves

The _Hall algebra_ of an [[abelian category]] is the [[Grothendieck group]] of [[constructible sheaves]]/[[perverse sheaves]] on the [[moduli stack]] of [[object]]s in the category. The Hall algebra is an algebra because the constructible derived category of the moduli stack of objects in an [[abelian category]] is [[monoidal category|monoidal]] in a canonical way.

This perspective is taken from ([Webster11](#Webster)). See there for more details.

## References

Survey:

* [[Claus M. Ringel]], *The Hall multiplication]*, lecture notes  (2006) [[web](https://www.math.uni-bielefeld.de/~ringel/lectures/green/Welcome.html)]

* {#Webster} [[Ben Webster]], _Hall algebras are Grothendieck groups_ (2011) [[webpage](http://sbseminar.wordpress.com/2011/04/18/hall-algebras-are-grothendieck-groups/#more-3988)]

* [[Olivier Schiffmann]], *Lectures on Hall algebras* [[arXiv:math/0611617](https://arXiv.org/abs/math/0611617)]

* [[Matthew B. Young]], *Self-Dual Hall modules*, PhD thesis, Stony Brook (2013) [[pdf](https://www.math.stonybrook.edu/alumni/2013-Matthew-Young.pdf), [[Young-HallModules.pdf:file]]]

* Wikipedia: [Hall algebra](https://en.wikipedia.org/wiki/Hall_algebra), [Ringel-Hall algebra](https://en.wikipedia.org/wiki/Ringel–Hall_algebra)

The characterization via [[2-Segal spaces]]/decomposition spaces is independently due to

* [[Tobias Dyckerhoff]], [[Mikhail Kapranov]], _Higher Segal spaces I_, ([arxiv:1212.3563](http://arxiv.org/abs/1212.3563))
 {#DyckerhoffKapranov12} (now part of their book _Higher Segal spaces_, Springer Lec. Notes in Math. 2244  [doi](https://doi.org/10.1007/978-3-030-27124-4))
* [[Imma Gálvez-Carrillo]], [[Joachim Kock]], [[Andrew Tonks]], _Decomposition spaces, incidence algebras and Möbius inversion_, [arXiv:1404.3202](https://arxiv.org/abs/1404.3202)

Anew light on the interplay between 2-Segal and Hall product:

* [[Matthew B. Young]], _Relative 2-Segal spaces_,
Algebraic & Geometric Topology **18&& (2018) 975--1039 [[doi:10.2140/agt.2018.18.975](https://doi.org/)]

  > We introduce a relative version of the 2–Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative 2–Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the $\mathcal{R}_\bullet$-construction from Grothendieck–Witt theory. We show that a relative 2–Segal space defines a categorical representation of the Hall algebra associated to the base 2–Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative 2–Segal spaces.

Further references:

* [[Mikhail Kapranov]], _Eisenstein series and quantum affine algebras_, Journal Math. Sciences __84__ (1997), 1311–1360.

* [[Mikhail Kapranov]], [[Eric Vasserot]], _Kleinian singularities, derived categories and Hall algebras_,  Math. Ann.  __316__  (2000) 565-576, [arXiv/9812016](https://arxiv.org/abs/math/9812016)

* [[Bernhard Keller]], Dong Yang, Guodong Zhou, _The Hall algebra of a spherical object_, J. London Math Soc. (2) __80__ (2009) 771--784, [doi](https://doi.org/10.1112/jlms/jdp054), [pdf](www.math.jussieu.fr/~keller/publ/HallAlgSphericalObject.pdf)

* [[Claus M. Ringel]], _Hall algebras and quantum groups_, Invent. Math. __101__ (1990), no. 3, 583--591

Green has introduced a coproduct on Hall algebras of a quiver which are related to quantum groups; Green's theorem states that the Hall algebra of the category of representations of a quiver over a finite field is a twisted bialgebra. The coproduct than agrees with the one on quantum groups:

* J. A. Green, _Hall algebras, hereditary algebras and quantum groups_, Invent. Math. __120__ (2), 361--377
(1995)
* J.  Xiao, F. Xu, M. Zhao, _Ringel-Hall algebras beyond their quantum groups I: Restriction functor and Green formula. Algebras & Represent. Theory __22__ (2019) , 1299--1329 [doi](https://doi.org/10.1007/s10468-018-9821-5)
* J. Xiao, _Drinfeld double and Ringel-Green theory of Hall algebras_, J Algebra __190__ (1997) 100--144 [Zbl0874.16026](https://zbmath.org/0874.16026)
* Matthew B. Young, _Degenerate versions of Green's theorem for Hall modules_, J. Pure & Applied Algebra
__225__:4 (2021) 106557 [doi](https://doi.org/10.1016/j.jpaa.2020.106557)

Elliptic Hall algebras

* [[Olivier Schiffmann]], [[Eric Vasserot]], _The elliptic Hall algebra_, Cherednik Hecke algebras and Macdonald polynomials, Compositio Mathematica __147__:1 (2011) 188-234 [doi](https://doi.org/10.1112/S0010437X10004872) [arXiv:0802.4001](https://arxiv.org/abs/0802.4001), (2008); _The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbf{A}^2$_,  Duke Math. J. 162(2): 279-366 (2013) [doi](https://doi.org/10.1215/00127094-1961849) [arXiv:0905.2555](http://arXiv.org/abs/0905.2555)
* O. Schiffman, _Drinfeld realization of the elliptic Hall algebra_, J. Algebr. Comb. 35 (2012) 237--262  [doi](https://doi.org/10.1007/s10801-011-0302-8) [arxiv/1004.2575](https://arxiv.org/abs/1004.2575)
* O. Schiffmann, [[Eric Vasserot]], _Hall algebras of curves, commuting varieties and Langlands duality_, Math. Ann. 353 (2012) 1399-1451 [doi](https://doi.org/10.1007/s00208-011-0720-x) [arXiv:1009.0678](https://arxiv.org/abs/1009.0678)

Toen's derived Hall algebras

* [[Bertrand Toen]], _Derived Hall algebras_,  Duke Math. J. 135(3): 587-615 (2006) [doi](https://doi.org/10.1215/S0012-7094-06-13536-6) [arXiv:0501343](https://arxiv.org/abs/math/0501343)

* [[Alexander I. Efimov]], _Cohomological Hall algebra of a symmetric quiver_, Compositio Mathematica __148__:4 (2012) 1133-1146 [doi](https://doi.org/10.1112/S0010437X12000152) [arxiv/1103.2736](https://arxiv.org/abs/1103.2736)

* Description of seminar on stability conditions, Hall algebras and [[Stokes phenomenon|Stokes factors]] in Bonn 2009 ([[Daniel Huybrechts|D. Huybrechts]]), [pdf](http://www.math.uni-bonn.de/people/compgeo/Hall.pdf)

* sbseminar blog: [Hall algebras and Donaldson-Thomas invariants-i](http://sbseminar.wordpress.com/2009/03/25/hall-algebras-and-donaldson-thomas-invariants-i)

* Bangming Deng, Jie Du, [[Brian Parshall]], Jianpan Wang, _Finite dimensional algebras and quantum groups_, Mathematical Surveys and Monographs __150__, Amer. Math. Soc. 2008. xxvi+759 pp. (chap. 10: Ringel-Hall algebras) [MR2009i:17023)](https://mathscinet.ams.org/mathscinet-getitem?mr=2457938)

* [[David Hernandez]], [[Bernard Leclerc]], _Quantum Grothendieck rings and derived Hall algebras_, J. Reine Angew. Math. 701 (2015) 77--126 [doi](https://doi.org/10.1515/crelle-2013-0020) [arXiv/1109.0862](https://arxiv.org/abs/1109.0862)

* Parker E. Lowrey, _The moduli stack and motivic Hall algebra for the bounded derived category_, [arxiv/1110.5117](http://arxiv.org/abs/1110.5117)

* [[Tobias Dyckerhoff]], _Higher categorical aspects of Hall Algebras_, In: Herbera D., Pitsch W., Zarzuela S. (eds) Building Bridges Between Algebra and Topology. Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser 2015 [arXiv:1505.06940](https://arxiv.org/abs/1505.06940) [doi](https://doi.org/10.1007/978-3-319-70157-8_1)

* [[Mauro Porta]], Francesco Salla, _Two-dimensional categorified Hall algebras_, J. Eur. Math. Soc. __25__:3 (2023) 1113--1205 [doi](https://doi.org/10.4171/jems/1303) (review in [[zoranskoda:Zbl076835085]])

* J. Xiao, F. Xu, _Hall algebras associated to triangulated categories_, Duke Math. J. 143 (2) (2008) 357--373

A realization of (Drinfeld-Jimbo) [[quantum group]]s via a "Bridgeland" version of Hall algebra,

* [[Tom Bridgeland]], _Quantum groups via Hall algebras of complexes_, Ann of Math __177__ (2) (2013) 739--759

Motivic Hall algebras:

* {#Bridgeland10} [[Tom Bridgeland]], _An introduction to motivic Hall algebra_ ([arXiv:1002.4372](https://arxiv.org/abs/1002.4372))

* {#KontsevichSoibelman08} [[Maxim Kontsevich]], [[Yan Soibelman]], _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_ ([arXiv:0811.2435](https://arxiv.org/abs/0811.2435))

* [[Kontsevich|M. Kontsevich]], [[Yan Soibelman|Y. Soibelman]], _Motivic Donaldson-Thomas invariants: summary of results_, [arXiv/0910.4315](https://arxiv.org/abs/0910.4315)

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants_, [arXiv/1006.2706](https://arxiv.org/abs/1006.2706)

* Richard Rimányi, _Motivic characteristic classes in cohomological Hall algebras_, Advances in Mathematics __360__ (2020) 106888 [doi](https://doi.org/10.1016/j.aim.2019.106888) [arXiv:1808.05654](https://arxiv.org/pdf/1808.05654)

* Alexei Latyntsev, _Vertex algebras, moduli stacks,
cohomological Hall algebras and quantum groups_, PhD thesis, Oxford 2022 [pdf](https://people.maths.ox.ac.uk/joyce/theses/LatyntsevDPhil.pdf)

[[!redirects Hall algebras]]

[[!redirects motivic Hall algebra]]
[[!redirects motivic Hall algebras]]


Hall coherent state

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[coherent state]]

* [[geometric quantization]]

## Literature

* [[Brian Charles Hall]], _The Segal-Bargmann coherent state transform for Lie groups_. J. Funct. Anal. __122__:103–151, 1994 ([doi:10.1006/jfan.1994.1064](http://dx.doi.org/10.1006/jfan.1994.1064))

* [[Brian Charles Hall]], _Holomorphic methods in analysis and mathematical physics_, First Summer School in Analysis and Mathematical Physics (Cuernavaca Morelos, 1998). Contemp. Math., __260__:1–59, 2000;

* [[Brian Charles Hall]], _Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type_, Comm. Math. Phys., __226__:233–268, 2002 ([doi:10.1007/s002200200607](http://dx.doi.org/10.1007/s002200200607))

* B. Driver, [[Brian Hall]], _Yang-Mills theory and the Segal-Bargmann transform_, Commun. Math. Phys. 201, 249–290 (1999) ([MR2000c:58064](http://www.ams.org/mathscinet-getitem?mr=1682238))

* William D. Kirwin, José M. Mourão, João P. Nunes, _Coherent state transforms and the Mackey-Stone-Von Neumann theorem_, [arxiv/1211.2145](http://arxiv.org/abs/1211.2145); _Complex time evolution in geometric quantization and generalized coherent state transforms_, [arxiv/1203.4767](http://arxiv.org/abs/1203.4767)

* Carlos Florentino, Pedro Matias, Jose Mourao, Joao P. Nunes, _Geometric quantization, complex structures and the coherent state transform_, [math.DG/0402313](http://arxiv.org/abs/math/0402313)

category: geometry, physics

[[!redirects Hall coherent states]]


Hall effect

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## Idea

In [[electromagnetism]] the _Hall effect_ ([Hall 1879](#Hall1879)) is essentially the manifestation of the [[Lorentz force]] in [[conductors]]:

It is the phenomenon that an [[electric current]] flowing [[perpendicular]] to a [[magnetic field]] induces an [[electric field]] perpendicular to both of these, sourced from the separation of positive and magnetic charge carriers in the current, which are driven apart by the [[Lorentz force]].

## Details

Write

* $B$ for the [[absolute value]] of the ambient [[magnetic field]];

* $E_{Hall}$ for the [[absolute value]] of the [[electric field]] induced by charge separation;

* $e$ for the [[electron]]'s [[electric charge]];

* $v$ for the [[absolute value]] of the drift [[velocity]] of the electrons in the material;

* $\rho$ for the [[number]] [[density]] of electrons;

* $j = \rho e v$ for the [[absolute value]] of the electric [[current]] [[density]].

Now, assuming, for simplicity and as usual, that the magnetic field and the electric current are both (as [[vector fields]]) constant in space and time as well as [[perpendicular]] to each other:

* the [[absolute value]] of the [[Lorentz force]] on the [[electrons]] is

  $$
    F_L = e v B
  $$

* the [[absolute value]] of the [[electrostatic force]] on the [[electrons]] is

  $$
    F_S = e E_{Hall}
  $$

* and hence in equilibrium $F_S = F_L$ we have

  $$
    E_{Hall} \;=\; v B
    \,.
  $$

  This is called the _Hall electric field_.

In a [[conductor]] of extension $d$ perpendicular to both the current and the magnetic field (hence: along the electric field), this produces a [[voltage]]

$$
  V_{Hall} \;=\; d E_{Hall} \;=\; d v  B
  \,,
$$

called the _Hall voltage_.

In practice one typically refers to the _Hall resistance_ $R_{Hall}$, defined to be the ratio of Hall voltage over the electron current $J = A j$ through the cross-section [[area]] $A$ of the [[conductor]]:

$$
  R_{Hall}
  \;\coloneqq\;
  \frac{
    V_{Hall}
  }{
    J
  }
  \;=\;
  \frac{
    d
  }{
    A
  }
  \cdot
  \frac{
    v
  }{
    \rho
  }
  \cdot
  \frac{
    B
  }{
    e
  }
  \,.
$$

Here

* the first factor on the right encodes the geometry of the [[conductor]];

* the second factor encodes the charge transport properties of its material;

* the third factor encodes the external magnetic field.

Since the first term is typicall known by construction,  knowledge of either of the remaining two terms makes the Hall effect applicable for _Hall sensors_ measuring the remaining term:

* when the material properties (second term) are known, the Hall effect serves in magnetometers (measuring the last term);

* when the external magentic field is known (the third term) the Hall effect provides information about the current and/or its conducting material (the second term).

## Related concepts

* [[quantum Hall effect]]

* [[spin Hall effect]], [[quantum spin Hall effect]]

## References

The original article:

* {#Hall1879} [[Edwin Hall]], _On a New Action of the Magnet on Electric Currents_, American Journal of Mathematics Vol. 2, No. 3 (Sep., 1879), pp. 287-292 ([doi:10.2307/2369245](https://doi.org/10.2307/2369245))

See also:

* Wikipedia, _[Hall effect](https://en.wikipedia.org/wiki/Hall_effect)_

[[!redirects Hall effects]]

[[!redirects Hall field]]
[[!redirects Hall fields]]

[[!redirects Hall voltage]]
[[!redirects Hall voltages]]

[[!redirects Hall resistance]]


halting theorem

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### Context
#### Constructivism, Realizability, Computability
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## Related concepts

* [[Lawvere's fixed point theorem]]

## References

* Wikipedia _[Halting problem](http://en.wikipedia.org/wiki/Halting_problem)_

[[!redirects halting problem]]


Halvard Fausk

* [webpage](http://www.fausk.com)

## Selected writings

On [[pro-homotopy theory]] via [[model categories]] of [[pro-objects]]:

* {#FauskIsaksen07} [[Halvard Fausk]], [[D. C. Isaksen]], _Model structures on pro–categories_, Homology, Homotopy and Applications, Vol. 9 (2007), 367–398 ([arXiv:math/0511057](https://arxiv.org/abs/math/0511057))

On [[six operations]] and [[Wirthmüller contexts]]:

* [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]

On [[equivariant stable homotopy theory]] for [[pro-objects|prop-]][[spectra]] (and the [[fine model structure on topological G-spaces]]):

* {#Fausk08} [[Halvard Fausk]], *Equivariant homotopy theory for pro-spectra*,  Geom. Topol. 12 (2008) 103-176 ([doi:10.2140/gt.2008.12.103](http://dx.doi.org/10.2140/gt.2008.12.103))

On [[RO(G)-grading]] and the [[Picard ∞-group]] of [[equivariant stable homotopy theory]]:

* {#FauskLewisMay01} [[Halvard Fausk]], [[L. Gaunce Lewis]], [[Peter May]],  _The Picard group of equivariant stable  homotopy theory_, Advances in Mathematics Volume 163, Issue 1, 15 October 2001, Pages 17–33  ([pdf](http://www.math.uchicago.edu/~may/PAPERS/FLMJan01.pdf))

category: people