hierarchy of universes

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## Idea

In [[dependent type theory]], having a [[type of all types]] results in various paradoxes, such as [[Russell's paradox]] and [[Girard's paradox]]. There are two ways to resolve this issue. One way is to simply add a [[universe of small types]] and accept that not all types are small. On the other hand, one can use a **hierarchy of universes** or **universe hierarchy** and postulate that every type is an element of universes in this universe hierarchy: Given an universe level $\kappa$, each [[type universe]] $U_\kappa$ consists of only the $U_\kappa$-small types, and $U_\kappa$ itself, while not an element of $U_\kappa$, is an element of the successor universe $U_{\kappa^+}$, where $\kappa^+$ is the successor universe level. Most commonly used for the universe levels are the [[natural numbers]], but other options are available as well, such as the [[integers]] or the [[ordinal numbers]].

There are a few advantages of using a hierarchy of universes where every type is an element of the hierarchy, instead of a single separate type judgment, in the formulation of dependent type theory:

1. Having a hierarchy of universes where every type in the type theory is an element of the hierarchy of universes avoids having to use long annotations everywhere. For example, if there is a single separate type judgment where not all types are elements of types, then given the type family $x:A \vdash B(x)$, if it is not small relative to the hierarchy of universes, then the [[heterogeneous identity type]], elements $a:A$, $b:A$, $p:\mathrm{Id}_A(a, b)$, $y:B(a)$, and $z:B(b)$ is represented by $\mathrm{hId}_{x:A.B(x)}(a, b, p, y, z)$, where the subscript $x:A.B(x)$ is a long annotation used to represent the type family $x:A \vdash B(x)$. With a hierarchy of universes where every type in the type theory is an element of the hierarchy of universes, the type family $x:A \vdash B(x):U$ is represented by the function $B:A \to U$, and the same heterogeneous identity type is then represented by $\mathrm{hId}_{B}(a, b, p, y, z)$, which is more concise when written out, and in addition, the subscript $B$ is now represents that the type depends upon the universe, rather than merely being an annotation, and can be written as the dependent type $\mathrm{hId}(B, a, b, p, y, z)$.

1. In the [[split context]] formalization of [[spatial type theory]] and [[cohesive type theory]] with an additional judgment for crisp terms of types, with a hierarchy of universes, one can define crisp types by simply postulating the type to crisply be an element of a type universe. However, without [[type universes]], one has to add to the theory a separate judgment for crisp types, and all the requisite [[inference rules]], [[structural rules]], and [[congruence rules]] for crisp type judgments.

In addition, in [[dependent type theory]] with a hierarchy of universes,

* the [[congruence rules]] for [[substitution]] implies the [[congruence rules]] for every type in the type theory

* one can define a symbol $B$ as a type $A:U$ using the [[propositional equality]] $B =_U A$ or [[judgmental equality of terms]] $A \equiv B:U$.

However, with a single separate judgment for types, where not all types are elements of the hierarchy of universes, not all types can be compared using judgmental equality of terms, so neither of these are possible. There are two solutions to this, both of which are unwieldy:

* One can add [[judgmental equality]] of types, and the associated [[structural rules]] for strict judgmental equality of types. However, the [[congruence rules]] for [[substitution]] no longer implies the [[congruence rules]] for every type former in the type theory, because not all types are terms of universes. Thus, the [[congruence rules]] for every type in the type theory have to be added separately.

* Alternatively, one could add [[strict transport]] across [[judgmental equality of terms]] as [[strict equivalences of types]], whether using the [[strict equivalence type]] or using the [[natural deduction]] [[inference rules]] of [[strict negative copies]]. This results in a simpler formal theory, since one doesn't need any of the structural and congruence rules for judgmental equality of types.

* If one doesn't even have [[judgmental equality of terms]], one would need to use [[transport]] across [[identity types]] or [[path types]]. However, one needs to define [[weak equivalence of types]] and prove the congruence rules for weak equivalences of types, before weak equivalence of types could be used for definitions. Weak equivalences are significantly more complex to define, since the symbol $B \simeq A$ usually representing the [[weak equivalence type]] hasn't been formally defined in the theory yet, and any definition of [[weak equivalence type]] or [[isEquiv]] for functions is itself a very complex expression when only using [[dependent function types]], [[function types]], [[dependent pair types]], [[pair types]], and [[identity types]] in the expression. Alternatively, one can add [[natural deduction]] [[inference rules]] for $B$ as a [[positive copy]] or [[weak negative copy]] of $A$, from which one can prove that the function in the introduction rule carries the structure of a weak equivalence of types. The proofs of the various congruence rules of type formers are also very complex; see [dependent product type § typal congruence rules § using weak equivalences of types](https://ncatlab.org/nlab/show/dependent+product+type#using_weak_equivalences_of_types) for a proof of the associated congruence rule for the formation rule of dependent product types and section 11.1.6 of [Rijke22](#Rijke22) for a proof of the associated congruence rule for the formation rule of dependent sum types.

However, this all comes at the cost of having to formalize the theory of universe levels of the hierarchy of universes before formalizing the type theory.

## Definition

There are two main ways to define a hierarchy of universes:

* The first way is done in dependent type theories with only a single separate type judgment, by defining a type of universe levels $\mathrm{Level}$ inside the type theory and then defining a family of [[type universes]] indexed by $\mathrm{Level}$. Not all types are elements of universes in the universe hierarchy.

* The second way is done in dependent type theories with either no separate type judgment [[à la Russell]], or a separate type judgment for every type universe in the type theory [[à la Coquand]], where every type in the theory is an element of the hierarchy of universes. Instead of having an internal type of universe levels, one has either a separate judgment $\kappa \mathrm{level}$ which is in an untyped [[first-order theory]] or a meta-theoretic [[sort]] $\mathrm{Level}$ which is in a typed [[first-order theory]] or a [[dependent type theory]], with an operation which takes level $\kappa$ to the successor level $\kappa^+$. Then for type theories à la Russell one has for each level $\kappa$ a universe $U_\kappa:U_{\kappa^+}$, and for type theories à la Coquand one has for each level $\kappa$ a type judgment $A \; \mathrm{type}_\kappa$ and a universe $U_\kappa \; \mathrm{type}_{\kappa^+}$.

### Cumulativity

Types of a universe $U_\kappa$ in an hierarchy of universes are also types of the successor universe $U_{\kappa^+}$; i.e. $U_\kappa$ is a [[subtype]] of $U_{\kappa^+}$. There are two different ways of representing this, reminiscent of the distinction between [[Russell universes]] and [[Tarski universes]].

* Given a universe level $\kappa$, the universe $U_\kappa$ is **cumulative** if given an element $A:U_\kappa$, one can derive that $A:U_{\kappa^+}$. This is similar to Russell universes in that elements of a cumulative universe $U_\kappa$ are literally elements of $U_{\kappa^+}$, similar to how elements of a Russell universe are literally types.

* Given a universe level $\kappa$, the universe $U_\kappa$ is **non-cumulative** if given a $U_\kappa$-small type $A:U_\kappa$, one can derive that $\mathrm{Lift}_\kappa(A):U_{\kappa^+}$. This is similar to Tarski universes in that elements of a cumulative universe $U_\kappa$ are codes for elements of $U_{\kappa^+}$, represented by the family of elements $A:U_\kappa \vdash \mathrm{Lift}_\kappa(A):U_{\kappa^+}$, similar to how elements of a Tarski universe are codes for types, represented by the family of types $A:U_\kappa \vdash \mathrm{Lift}_\kappa(A) \; \mathrm{type}$.

## Examples

Some examples of type theories with a hierarchy of universes are as follows:

* [[Coq]] uses a hierarchy of Russell universes. For practical purposes, it also has cumulativity, although there is some question (perhaps mainly semantic) of whether this is true internally or whether it uses casts that are simply hidden from the user.

* [[Agda]] uses a hierarchy of non-cumulative Russell universes.

* [UFP13](#UFP13) (first edition) uses a hierarchy of cumulative Russell universes.

* [Rijke 22](#Rijke22) uses a hierarchy of non-cumulative Tarski universes.

## Related concepts

* [[type universe]]

* [[coercion]]

* [[reflection principle]]

## References

The hierarchy of universes is discussed in section 1.3 of:

* {#UFP13} *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

and in section 6.2 of:

* {#Rijke22} [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

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hierarchy problem

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## Idea

What is called the _hierarchy problem_ is an issue in [[phenomenology]] and [[model (in theoretical physics)|model building]], notably in the [[standard model of particle physics]], concerning discrepancies between the [[electroweak]] and [[Planck scale]]. For instance, the masses of the [[W-boson|W]] and [[Z-boson|Z]] particles, the force carriers of the [[weak nuclear force]], are about 10,000 trillion times smaller than the [[Planck mass]]. Thus there is a huge hierarchy in the mass scales of [[weak nuclear forces]] and [[gravity]]. See [Strassler](#Strassler) for review.

## Proposed solutions

### Just an artifact

In 2021, Mooij and Shaposhnikov suggested that the hierarchy problem might not have any physical meaning, rather being a mere artifact of a certain formulation of the quantum field theory and/or model of the Standard model, and that if a different formulation was chosen the hierarchy problem can be avoided entirely. See [Mooij and Shaposhnikov](#Mooij-Shaposhnikov) for more details.

### In M-theory on $G_2$-manifolds
  {#InMTheoryOnG2Manifols}

In the context of [[M-theory on G2-manifolds]] (the [[G2-MSSM]]) a generic solution to the [[hierarchy problem]] is proposed in ([Acharya-Kane-Kumar 12, section  V.A.2 (p. 10-11)](#AcharyaKaneKumar12)) due to control over [[non-perturbative effects]] from [[membrane instantons]], which due to their exponential nature, provide the exponential hierarchy.

## Related concepts

* [[naturalness]]

* [[multiverse]]

## References

* {#Strassler} [[Matthew Strassler]], _The Hierarchy Problem_ ([web](http://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-hierarchy-problem/))

* Seth Koren, _The Hierarchy Problem: From the Fundamentals to the Frontiers_ ([arXiv:2009.11870](https://arxiv.org/abs/2009.11870))

* {#Mooij-Shaposhnikov} [[Sander Mooij]], [[Mikhail Shaposhnikov]], _QFT without infinities and hierarchy problem_ [[arXiv:2110.05175](https://arxiv.org/abs/2110.05175)]

  > The aim of this work has been to stress that there is no physical sense in the [[naturalness]] criterion.

In the context of [[M-theory on G2-manifolds]]/[[G2-MSSM]] a solution is proposed in

* {#AcharyaKaneKumar12} [[Bobby Acharya]], [[Gordon Kane]], Piyush Kumar, section V.A.2 (p. 10-11) of _Compactified String Theories -- Generic Predictions for Particle Physics_,  Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 ([arXiv:1204.2795](http://arxiv.org/abs/1204.2795))

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Higgs boson

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## Idea

The _Higgs particle_ is a [[boson]] hypothesized to be the quantum of the [[Higgs field]]. See there for more.

## Related concepts

* [[Higgs mechanism]], [[electroweak symmetry breaking]]

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## References

See the references at _[[Higgs field]]_.

* [[Antonio Pich]], _Electroweak Symmetry Breaking and the Higgs Boson_, [Acta Phys. Pol. B 47, 151](https://www.actaphys.uj.edu.pl/index_n.php?I=R&V=47&N=1#151) (2016) ([arXiv:1512.08749](https://arxiv.org/abs/1512.08749))

* [[Particle Data Group]] [Review of particle physics](http://pdg.lbl.gov/), _Status of Higgs Boson Physics_, 2018 ([pdf](http://pdg.lbl.gov/2018/reviews/rpp2018-rev-higgs-boson.pdf), [[StatusOfHiggsBosonPhysics2018.pdf:file]])

* Johannes Haller, Andreas Hoecker, Roman Kogler, Klaus Mönig, Thomas Peiffer, Jörg Stelzer, _Update of the global electroweak fit and constraints on two-Higgs-doublet models_,  [10.1140/epjc/s10052-018-6131-3](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-018-6131-3) ([arXiv:1803.01853](https://arxiv.org/abs/1803.01853))

See also

* Wikipedia, _[Higgs boson](https://en.wikipedia.org/wiki/Higgs_boson)_

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Higgs branch

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## Idea

For [[N=2 D=4 super Yang-Mills theory]] the [[moduli space]] of [[vacuum expectation values]] (VEVs) of the theory is meant to locally be a [[Cartesian product]] of spaces of

1. [[moduli]] of the [[vector multiplet]] (the [[gauge field]] sector)

   this is called the *[[Coulomb branch]]*

1. moduli of the [[hypermultiplet]] (the scalar matter field sector)

   this is called the *Higgs branch*.

These are thought to be [[duality in physics|dual]] to each other to it under a version of [[mirror symmetry]]. This is largely the topic of [[Seiberg-Witten theory]].

Definitions of the Coulomb and Higgs branches have been extended to [[N=4 D=3 super Yang-Mills theory]].

## Definition

### D=4

### D=3

## Related concepts

* [[symplectic duality]]

* [[fractional D-brane]], [[McKay correspondence]]

## References

The terminology "Coulomb branch" and "Higgs branch" first appears in

* {#SeibergWitten94} [[Nathan Seiberg]], [[Edward Witten]], _Monopoles, Duality and Chiral Symmetry Breaking in $N=2$ Supersymmetric QCD_ ([arXiv:hep-th/9408099](http://arxiv.org/abs/hep-th/9408099))

Review and exposition:

* {#Albertsson03} Cecilia Albertsson, around p. 31 of _Superconformal D-branes and moduli spaces_ ([arXiv:hep-th/0305188](http://arxiv.org/abs/hep-th/0305188))

* [[Yuji Tachikawa]], Section 7 of: *$\mathcal{N}=2$ supersymmetric dynamics for pedestrians*, Lecture Notes in Physics, vol. 890, 2014 ([arXiv:1312.2684](https://arxiv.org/abs/1312.2684), [doi:10.1007/978-3-319-08822-8](https://link.springer.com/book/10.1007%2F978-3-319-08822-8), [web version](http://www-hep.phys.s.u-tokyo.ac.jp/~yujitach/tmp/dummies/dummies.html#dummiesse7.html))

On [[mirror symmetry]] between [[Higgs branches]]/[[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]] (with emphasis of [[Hilbert schemes of points]]):

* [[Jan de Boer]], [[Kentaro Hori]], [[Hirosi Ooguri]], [[Yaron Oz]], _Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes_, Nucl. Phys. B493:101-147, 1997 ([arXiv:hep-th/9611063](https://arxiv.org/abs/hep-th/9611063))

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Higgs bundle

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## Idea

A _Higgs bundle_ is a [[holomorphic vector bundle]] $E$ together with a [[differential form|1-form]] $\Phi$ with values in the [[endomorphisms]] of (the [[fibers]] of) $E$, such that $\Phi \wedge \Phi = 0$.

Higgs bundles play a central role in [[nonabelian Hodge theory]].

## Terminology
 {#Terminology}

The term was introduced by [[Nigel Hitchin]] as a reference to roughly analogous structures in the [[standard model of particle physics]] related to the _[[Higgs field]]_.

> ([Witten 08, remark 2.1](#Witten08)): As an aside, one may ask how closely related $\phi$, known in the present context as the Higgs field, is to the Higgs fields of [[standard model of particle physics|particle physics]]. Thus, to what extent is the terminology that was introduced in Hitchin (1987a) actually justified? The main difference is that [[Higgs fields]] in particle physics are [[scalar fields]], while $\phi$ is a [[differential 1-form|one-form]] on $C$ (valued in each case in some [[representation]] of the [[gauge group]]). However, although Hitchin's equations were first written down and studied directly, they can be obtained from [[N=4 D=4 super Yang-Mills theory|N = 4 supersymmetric gauge theory]] via a sort of twisting procedure (similar to the procedure that leads from N = 2 supersymmetric gauge theory to Donaldson theory). In this twisting procedure, some of the Higgs-like scalar fields of $N = 4$ super Yang-Mills theory are indeed converted into the Higgs field that enters in Hitchin's equations. $[$ [Kapustin-Witten 06](#KapustinWitten06) $]$ This gives a reasonable justification for the terminology.

## Definition

### In components

Let $\mathcal{E}$ be a [[sheaf]] of [[section]]s of a [[holomorphic vector bundle]] $E$ on [[complex manifold]] $M$ with [[structure sheaf]] $\mathcal{O}_M$ and [[module]] of [[Kähler differential]]s $\Omega^1_M$.

A **Higgs field** on $\mathcal{E}$ is an $\mathcal{O}_M$-linear map

$$

  \Phi : \mathcal{E}\to \Omega^1_M\otimes_{\mathcal{O}_M}\mathcal{E}
$$

satisfying the integrability condition $\Phi\wedge\Phi = 0$. The pair of data $(E,\Phi)$ is then called a **Higgs bundle**.

(Notice that this is similar to but crucially different the definition of a [[flat connection]] on a [[vector bundle]]. For that the map $\Phi$ is just $\mathbb{C}$-linear and the integrability condition is $\mathbf{d}\phi + \Phi\wedge\Phi = 0$.)

Higgs bundles can be considered as a limiting case of a (holomorphic) [[curvature|flat]] [[connection on a bundle|connection]] in the limit in which its exterior differential $d\Phi$ is small in comparison to the exterior square $\Phi\wedge\Phi$, what can be obtained by rescaling the connection form. So the equation $d u/dz = A(z)u$ where $A(z)$ is a matrix of connection can be rescaled by putting a small parameter in front of $d u/dz$, or equivalently a large parameter in front of $A(z)$. Then the Maurer-Cartan equation has a term quadratic in the large parameter and the exterior derivative term is linear in large parameter and can be neglected, so the flatness reduces to the Higgs bundle condition  $\Phi\wedge\Phi = 0$.

In fact, there is a notion interpolating between flat connections and Higgs bundles due Deligne, the Deligne's $\lambda$-[[lambda-connection|connection]].

### Formulation in D-geometry

Analogous to how the [[de Rham stack]] $\int_{inf} X = X_{dR}$ of $X$ is the ([[homotopy quotient|homotopy]]) [[quotient]] of $X$ by the first order [[infinitesimal neighbourhood]] of the [[diagonal]] in $X \times X$, so there is a space ([[stack]]) $X_{Dol}$ which is the [[formal completion]] of the 0-section of the [[tangent bundle]] of $X$ ([Simpson 96](#Simpson96)).

Now a [[flat vector bundle]] on $X$ is essentially just a [[vector bundle]] on the [[de Rham stack]] $X_{dR}$, and a [[Higgs bundle]] is essentially just a [[vector bundle]] on $X_{Dol}$. Therefore in this language the nonabelian Hodge theorem reads (for $G$ a linear [[algebraic group]] over $\mathbb{C}$)

$$
  \mathbf{H}(X_{dR}, \mathbf{B}G)
  \simeq
  \mathbf{H}(X_{Dol}, \mathbf{B}G)^{ss,0}
  \,,
$$

where the superscript on the right denotes restriction to semistable Higgs bundles with vanishing [[first Chern class]] (see [Raboso 14, theorem 4.2](#Raboso14)).

## Properties

### Stability

For a Higgs bundle to admit a harmonic metric (...) it needs to be _stable_ (...).

Stability is defined similarly to stability for holomorphic vector bundles except that instead of quantifying over all proper non-zero sub-bundles, one only considers proper $\Phi$ invariant sub-bundles. So in particular for $\Phi=0$, this is stability of the underlying vector bundle.

### In nonabelian Hodge theory

In [[nonabelian Hodge theory]] the [[moduli space]] of stable Higgs bundles over a [[Riemann surface]] $X$ is identified with that of  [[special linear group]] $SL(n,\mathbb{C})$ [[irreducible representations]] of its [[fundamental group]] $\pi_1(X)$.

## Examples

### Rank 1

In the special case that $E$ has [[rank]] 1, hence is a [[line bundle]], the form $\Phi$ is simply any holomorphic 1-form. This case is also called that of an **abelian Higgs bundle**.

### Bundles of holomorphic forms

Let $X$ be a [[complex manifold]] and $\omega \in \Omega^{k,0}(X)$ for odd $k$. Then $\Omega^{\bullet,0}(X)$ becomes a Higgs bundle when equipped with the endomorphis-valued 1-form which sends a holomorphic vector $v$ to the wedge product operation with the contraction of $\omega$ with $v$.

This is discussed in ([Seaman 98](#Seaman98))

## Related concepts

* [[Hitchin system]]

* [[nonabelian Hodge theory]]

* [[topological recursion]]

## References

### General

The [[moduli space]] of Higgs bundles over an [[algebraic curve]] is one of the principal topics in works of [[Nigel Hitchin]] and [[Carlos Simpson]] in late 1980-s and 1990-s (and later Ron Donagi, Tony Pantev, etc....), hence also known as the *Hitchin moduli space*.

* [[Nigel Hitchin|N.J. Hitchin]], _Stable bundles and integrable systems_,  Duke Math. J. , 54  (1987)  pp. 91–114 [MR89a:32021](http://www.ams.org/mathscinet-getitem?mr=89a:32021) [doi](http://dx.doi.org/10.1215/S0012-7094-87-05408-1) [euclid](http://projecteuclid.org/euclid.dmj/1077305506); _The self-duality equations on a Riemann surface_, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, [MR887284](http://www.ams.org/mathscinet-getitem?mr=887284) [doi](http://dx.doi.org/10.1112/plms/s3-55.1.59); _Flat connections and geometric quantization_, Comm. Math. Phys. __131__, n 2 (1990), 347-380, [euclid](http://projecteuclid.org/euclid.cmp/1104200841)

* [[Carlos Simpson]], _Higgs bundles and local systems_, Publ. Mathématiques de l'IHÉS __75__ (1992), p. 5-95, [numdam](http://www.numdam.org/item?id=PMIHES_1992__75__5_0), [MR94d:32027](http://www.ams.org/mathscinet-getitem?mr=94d:32027)

* [[Ludmil Katzarkov]], [[Dmitri Orlov]], [[Tony Pantev]], _Notes on Higgs bundles and D-branes_, (delivered as a lecture by Tony Pantev at winter school at Guanajuato 2013 [link](http://www.cimat.mx/Eventos/moduli_spaces2013)) draft [pdf](http://www.cimat.mx/Eventos/moduli_spaces2013/img/tony_pantev.pdf)

* S. B. Bradlow, O. García-Prada, P. B. Gothen, _WHAT IS...a Higgs Bundle?_, Notices AMS, [pdf](http://www.ams.org/notices/200708/tx070800980p.pdf)

* [[Michael Murray]], [[Danny Stevenson]], _Higgs fields, bundle gerbes and string structures_, arxiv:math.DG/0106179](https://arxiv.org/abs/math.DG/0106179)

* [[David Baraglia]], _Cyclic Higgs bundles and the affine Toda equations_, [arxiv:1011.6421](https://arxiv.org/abs/1011.6421)

* [[Kevin Costello]], around lemma 6.4.1 in: *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

Review:

* [[Jan Swoboda]], *Moduli Spaces of Higgs Bundles -- Old and New*, Dtsch. Math. Ver. **123** (2021) 65–130 [[doi:10.1365/s13291-021-00229-1](https://doi.org/10.1365/s13291-021-00229-1)]

See also:

* Donu Arapura, _[MO comment on Higgs bundles](http://mathoverflow.net/questions/101326/what-is-the-current-state-of-the-mathematics-of-higgs-fields/101347#101347)_

* math.stackexchange:[difference-between-higgs-bundle-and-vector-bundle-with-connection](https://math.stackexchange.com/questions/2498255/difference-between-higgs-bundle-and-vector-bundle-with-connection)

Discussion of the topology of the moduli space of Higgs bundles is in

* {#Rayan18} [[Steven Rayan]], _Aspects of the topology and combinatorics of Higgs bundle moduli spaces_, SIGMA 14 (2018) 129, 18 pages ([arXiv:1809.05732](https://arxiv.org/abs/1809.05732))

Discussion in terms of $X_{Dol}$ is in

* {#Simpson96} [[Carlos Simpson]], _The Hodge filtration on nonabelian cohomology_, Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217{281. MR
1492538 (99g:14028) ([arXiv:9604005](http://arxiv.org/abs/alg-geom/9604005))

* {#Raboso14} [[Alberto García Raboso]], _A twisted nonabelian Hodge correspondence_, PhD thesis 2014 ([pdf slides](http://www.math.toronto.edu/agraboso/files/TwistedNAHT_Talk_Handout.pdf))

Discussion of the example of homolorphic forms is in

* {#Seaman98} Walter Seaman, _Higgs Bundles and Holomorphic Forms_ ([arXiv:9811097](http://arxiv.org/abs/math/9811097))

Discussion in the context of [[geometric Langlands duality]] includes

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_, Communications in Number Theory and Physics Volume 1 (2007) Number 1 ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

* {#Witten08} [[Edward Witten]], _Mirror Symmetry, Hitchin's Equations, And Langlands Duality_ ([arXiv:0802.0999](http://arxiv.org/abs/0802.0999))

The following paper starts with a survey of [[nonabelian Hodge correspondence]],

* Peter B. Gothen, _Higgs bundles and the real symplectic group_, [arXiv:1102.4175](https://arxiv.org/abs/1102.4175)

### Relation to $Spin(7)$-manifolds

Relating [[M-theory on Spin(7)-manifolds]] with [[F-theory on Spin(7)-manifolds]] via [[Higgs bundles]]:

* {#CHRTZ20} [[Mirjam Cvetic]], [[Jonathan Heckman]], Thomas B. Rochais, Ethan Torres, [[Gianluca Zoccarato]], _Geometric Unification of Higgs Bundle Vacua_ ([arXiv:2003.13682](https://arxiv.org/abs/2003.13682))

[[!redirects Higgs bundles]]

[[!redirects Hitchin moduli space]]
[[!redirects Hitchin moduli spaces]]


Higgs field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Higgs field_ or _[[Higgs boson]]_ is a [[scalar field|scalar]] [[physical field]]/[[fundamental particle]]  in a [[gauge theory]] such as the [[standard model of particle physics]] supposedly responsible for the [[spontaneously broken symmetry]] of the [[electroweak field]] ([[electroweak symmetry breaking]]) and for giving elementary particles their [[mass|masses]] by the _[[Higgs mechanism]]_.

## Properties

### Mass and vacuum (in-)stability
 {#MassAndVacuumInstability}

The [[rest mass]] of the Higgs particle observed at the [[LHC]] [[experiment]] is about $125$ [[GeV]] ([ATLAS Collaboration 12](#ATLAS12), [CMS Collaboration 12](#CMS12), [ATLAS Collaboration 15](#ATLAS15), see [Gibbs 11a](#Gibbs11a), [HLL 11](#HLL11) for early discussion).

This is determined by a [[local minimum]] of the Higgs [[potential energy|potential]] (see [Kusenko 15](#Kusenko15) for exposition):

\begin{imagefromfile}
  "file_name": "HiggsPotentials.png",
  "width": 500
\end{imagefromfile}

{#Curiously} Curiously, the Higgs [[potential energy|potential]] is such that the Higgs field at this mass is at least close to being at the border between [[vacuum stability]] and [[false vacuum]]. This was highlighted before the actual measurement ([EEGHR 09](#EEGHR09), [Gibbs 11b](#Gibbs11b)):

\begin{imagefromfile}
  "file_name": "HiggsVacuumStabilityIII.png",
  "width": 700
\end{imagefromfile}

Then it was amplified again after the detection of the Higgs particle at the [[LHC]] ([DVEEGI 12](#DVEEGI12)):

\begin{imagefromfile}
  "file_name": "HiggsVacuumStability.png",
  "width": 700
\end{imagefromfile}

More detailed computation at [[2-loop]] confirmed this result, showing that the observed Higgs vacuum is indeed very close to the boundary between the [[vacuum stability|stable]] and the meta-stable region ([BKPV 15](#BKPV15)):

\begin{imagefromfile}
  "file_name": "HiggsVacuumStabilityII.png",
  "width": 400
\end{imagefromfile}

{#SeeAlsoAFS18} See also [AFS 18](#AFS18):

\begin{imagefromfile}
  "file_name": "HiggsVacuumStability4.png",
  "width": 800
\end{imagefromfile}

In summary ([Kusenko 15](#Kusenko15)):

> $[$The$]$ conclusion is that the best theoretical fit to measured parameters, including the Higgs and top-quark masses, points to a metastable Universe. However, their analysis also concludes that values of parameters are closer to a region of absolute stability than suggested by previous studies: it is possible for the Universe to be fully stable (and for the standard model to work all the way up to the Planck scale), if the true values of measured parameters are only 1.3 standard deviations away from the current best estimates.

And more recently ([PDG 18, p. 9](#PDG18)):

> {#PDG18OnVacuumMetastability} However, for the value of Higgs mass experimentally measured, the EW vacuum of the Higgs potential is most likely metastable. Indeed, the high energy evolution of $\lambda$ shows that it becomes [[negative number|negative]] at [[energies]] $\Lambda = \mathcal{O}(10^{10}-10^{12})$ [[GeV]], with a broader range if the [[top quark]] [[mass]] exceeds its current measured value by [[statistical significance|3σ]]. When this occurs, the [[standard model of particle physics|SM]] Higgs potential develops an instability and the long term existence of the [[electroweak symmetry breaking|EW]] [[vacuum]] is challenged. This behavior may call for new physics at an intermediate scale before the instability develops, i.e., below [[Planck mass|M_Planck]], otherwise, the electroweak vacuum remains metastable.

>

However [[quantum tunneling]]/[[vacuum decay]] is an intrinsically [[non-perturbative effect]] which needs careful treatment beyond [[perturbative quantum field theory]] (e.g. [AFFS 17](#AFFS17))

\begin{centre}
  \begin{imagefromfile}
    "file_name": "NonPerturbativeTunneling.jpg",
    "width": 600
  \end{imagefromfile}
\end{centre}

> graphics grabbed form [Schwartz 15, slide 44](#Schwartz15)

\linebreak

### Possible causes of the near criticality
 {#PossibleCausesOfNEarCriticality}

Given, by the [above](#MassAndVacuumInstability) discussion, that the parameters of the Higgs field are observed to be inside a special reason of their potential parameter space, it is natural to speculate that there is some mechanism that enforces or at least prefers this position.

In [BDGGSSS 13, Section 5.2](#BDGGSSS13) is speculation that for dynamics over a [[landscape of string theory vacua|landscape of vacua]] critical points may generally serve as attractors.

More specifically, [Isidori-Pattori 17](#IsidoriPattori17) claims that, under reasonable assumptions (including [[gauge coupling unification]], but excluding "[[naturalness]]") [[supersymmetry|supersymmetric]]-extensions of the [[standard model of particle physics]] ([[MSSM]]) predict a parameter range for the [[top quark]] and Higgs field [[mass]] that is close to coinciding with the corresponding parameter space for Higgs field near criticality:

\begin{center}
\begin{imagefromfile}
  "file_name": "IsidoriPattoriViable.jpg",
  "width": 600
\end{imagefromfile}
\end{center}

> graphics grabbed from [Isidori-Pattori 17](#IsidoriPattori17)

That [[supersymmetry]], possibly in a [[G2-MSSM]], would be a natural mechanism behind the near criticality of the Higgs field was also claimed in [Kane 18, "Clue 4"](#Kane18).

### Cosmological instability?
 {#CosmologicalInstability}

However, it has been argued that an actual [[false vacuum]] of the Higgs is incompatible with [[cosmology]], as due to [[vacuum fluctuations]] during [[inflation]] the [[vacuum decay]] would not have been avoided ([EGR 07](#EGR07), [HKSZ 14](#HKSZ14), [EGMRSST 15](#EGMRSST15), [EKSYZ 16](#EKSYZ16)). In ([BDGGSSS 13, section 7](#BDGGSSS13), [Kane 18 , "Clue 4"](#Kane18)) it was argued that this suggests a further principle which prevents the vacuum instability and that a natural such principle is [[supersymmetry]]. (This argument has a long history, see [Gibbs 11b](#Gibbs11b)).

### Asymptotic safety?
 {#AsymtoticSafetyOrNot}

The near criticality of the Higgs field [[vacuum]] discussed [above](#MassAndVacuumInstability) implies that the [[coefficient]] $\lambda$ of the quartic part of the Higgs potential is close to zero after [[renormalization group flow]] ("RGE") to around the [[Planck scale]] of about $10^{19}$ [[GeV]] (e.g. [BDGGSSS 13, p. 17-18](#BDGGSSS13)):

\begin{imagefromfile}
  "file_name": "HiggsQuarticCoupling.png",
  "width": 400
\end{imagefromfile}

In fact also the [[beta function]] $\beta_\lambda$ of the quartic coupling $\lambda$ (i.e. its logarithmic [[derivative]] with respect to [[scale]]) is close to zero around the [[Planck scale]] of about $10^{19}$ [[GeV]] ([BDGGSSS 13, p. 18](#BDGGSSS13)):

\begin{imagefromfile}
  "file_name": "HiggsQuarticBetaFunctionRelative.png",
  "width": 400
\end{imagefromfile}

Earlier it has been suggested that this reflects the principle of [[asymptotic safety]] ([Shaposhnikov-Wetterich 09](#ShaposhnikovWetterich09)). But this would mean that not only $\lambda$ and its [[beta function|RGE-derivative]] $\beta_\lambda$ vanish around the [[Planck scale]], but that in fact all higher derivatives do, too (see e.g. [Niedermaier 06, equation (1.5)](asymptotic+safety#Niedermaier06)) hence that $\beta_\lambda$ asymptotes to zero. But this does not seem to be the case; in ([BDGGSSS 13, p. 17-18](#BDGGSSS13)) it says:

>  As shown in fig. 2 (upper right), the corresponding Higgs quartic [[beta-function]] vanishes at a [[scale]] of about $10^{17}$-$10^{18}$ [[GeV]]. In order to quantify the degree of cancellation in the β-function, we plot in fig. 2 (lower right) $\beta_\lambda$ in units of its pure [[top quark]] contribution.  The vanishing of $\beta_\lambda$ looks  more  like  an  accidental  cancellation  between  various  large  contributions, rather than an asymptotic approach to zero.

\begin{imagefromfile}
  "file_name": "HiggsQuarticBetaFunction.png",
  "width": 800
\end{imagefromfile}

### Higgs inflation?
 {#HiggsInflation}

Starting with [Jegerlehner 13](cosmic+inflation#Jegerlehner13) has been argued that the near-criticality of the Higgs potential is in fact consistent or even necessary for [[Higgs inflation]] to be a viable model. See [Jegerlehner 18](cosmic+inflation#Jegerlehner18).

## Models

There is no lack of proposals for realizing the Higgs field in various big schemes of mathematical structures modelling physics.

### Gauge-Higgs unification

A mechanis, of _gauge-Higgs unification_ ([Manton 79](#Manton79), [Hosotani 83](#Hosotani83)) also known as the _Hosotani mechanism_ (see [Hosotani 12](#Hosotani12)) refers to hypothetical [[model (in theoretical physics)|models]] of [[particle physics]] where the [[Higgs field]] on 4d [[spacetimes]] arises as a component of a [[gauge field]] in higher dimensions, either after [[Kaluza-Klein compactification]] or by passage to a 4d [[asymptotic boundary]] in a [[Randall-Sundrum model]] or similar.

> graphics grabbed from [Lim 09](#Lim09)

Gauge-Higgs unification is naturally combined with [[GUT|grand unification]] of gauge groups, whence one then also speaks of _gauge-Higgs grand unification_ (e.g. [Hosotani-Yamatsu 15](#HosotaniYamatsu15))

Also the [[Connes-Lott models]] realize gauge-Higgs unification, in a [[non-commutative geometry|non-commutative geometric]] [[KK-compactification]].

### In string theory

In [[string theory]] (see _[[string phenomenology]]_) a Higgs mechanism can arise in a variety of ways.

#### In intersecting D-brane models

Notably in [[intersecting D-brane models]] it arises naturally from brane recombination at intersections: ([Cremades-Ibanez-Marchesano 02, section 7](#CremadesIbanezMarchesano02)):

\begin{imagefromfile}
  "file_name": "BraneRecombination.png",
  "width": 600
\end{imagefromfile}

Further developments in [Ibanez-Marchesano-Rabadan 01](#IbanezMarchesanoRabadan01), [Hebecker-Knochel-Weigand 13](#HebeckerKnochelWeigand13) and specifically via [[string field theory]] in [HINSS 18](#HINSS18).

For review see [Ibanez-Uranga 12, fig 10.2](intersecting+D-brane+model#IbanezUranga12):

#### In $G_2$-MSSM

Discussion of the Higgs mechanism in the [[G2-MSSM]]: [Kane 11](#Kane11), [Kane-Kumar-Lu-Zheng 11](#KaneKumarLuZheng11), for review see [Kane 17](#Kane17), [Kane 18](#Kane18)

### Other

* in the [[technicolor]] model the Higgs field is not a fundamental particle but a compound of [[fermions]]. This realizes the Higgs effect entirely in ordinary [[gauge theory]];

* in [[Connes-Lott-Chamseddine-Barrett models]] it has been [shown](standard+model+of+particle+physics#NCGeometry) that the Higgs may be modeled as a component of the gauge bosons assuming that the [[Kaluza-Klein mechanism|KK-reduction]] is over a certain non-commutative space of classical dimension 0.

## History

The Higgs mechanism was proposed in 1963-1964 by a fair number of authors essentially simultaneously, see the [References](#References) below. The explicit prediction of the Higgs boson implied by this mechanism though seems to be solely due to ([Higgs 64](#Higgs64)).

The Higgs boson (or at least something very much like it) was finally detected in 2013 at the [[LHC]] [[experiment]].

So for the Higgs particle prediction and experimental detection lie apart by about 50 years. Compare maybe to the [[neutrino]], which was predicted in 1930 and detected in 1956, about 26 years later.

## Related concepts

* [[gauge-Higgs unification]]

* [[massive Yang-Mills theory]]

* [['t Hooft-Polyakov monopole]]

* [[phi^n interaction]]

* [[electroweak field]]

* [[inflaton field]]

* [[composite Higgs model]]

* [[higgsino]]

* [[Higgs bundle]]

[[!include standard model of fundamental physics - table]]

## References
 {#References}

### General

The original articles explaining what is now called the Higgs mechanism by [[spontaneous symmetry breaking]] were

* P. Anderson, _Plasmons, gauge invariance and mass_, Physical Review 130: 439. (1963)

* [[François Englert]], [[Robert Brout]],  _Broken Symmetry and the Mass of Gauge Vector Mesons_, Physical Review Letters 13 (9): 321–23. (1964)

*  Gerald Guralnik, C. R. Hagen, ; T. W. B. Kibble, _Global Conservation Laws and Massless Particles_ Physical Review  (1964)

* {#Higgs64} [[Peter Higgs]], _Broken Symmetries and the Masses of Gauge Bosons_, Physical Review Letters 13 (16): 508–509. (1964)

While all these articles essentially describe the Higgs mechanism, apparently only the one by Peter Higgs explicitly points out that this mechanism predicts the existence of a new, then unobserved, boson, the one therefore now called the _Higgs boson_.

Review includes

* {#PDG18} [[Particle Data Group]], [Review of particle physics](http://pdg.lbl.gov/), _Status of Higgs Boson Physics_, 2018 ([pdf](http://pdg.lbl.gov/2018/reviews/rpp2018-rev-higgs-boson.pdf), [[StatusOfHiggsBosonPhysics2018.pdf:file]])

* [[Antonio Pich]], _Electroweak Symmetry Breaking and the Higgs Boson_, [Acta Phys. Pol. B 47, 151](https://www.actaphys.uj.edu.pl/index_n.php?I=R&V=47&N=1#151) (2016) ([arXiv:1512.08749](https://arxiv.org/abs/1512.08749))

* Sally Dawson, Christoph Englert, Tilman Plehn, _Higgs Physics: It ain't over till it's over_, Physics Reports ([arXiv:1808.01324](https://arxiv.org/abs/1808.01324))

The general theory of [[spontaneous symmetry breaking]] is reviewed in

* Jeremy Bernstein, _Spontaneous symmetry breaking, gauge theories, the Higgs mechanism and all that_, Rev. Mod. Phys. 46, 7–48 (1974)  ([pdf](http://www.calstatela.edu/faculty/kaniol/p544/rmp46_p7_higgs_goldstone.pdf))

The [[phenomenology]] of Higgs [[model (in particle physics)|models]] is discussed in

* Marcela Carena, Howard E. Haber, _Higgs Boson Theory and Phenomenology_, Prog.Part.Nucl.Phys.50:63-152,2003 ([arXiv:hep-ph/0208209](http://arxiv.org/abs/hep-ph/0208209))

Discussion in "string localized" [[AQFT]] and claim of derivation of the quartic form of the Higgs potential:

* Jens Mund, Karl-Henning Rehren, [[Bert Schroer]], *How the Higgs potential got its shape* [[arXiv:2209.06133](https://arxiv.org/abs/2209.06133)]

### Detection

Early discussion of the detection of a Higgs field of 125 [[GeV]] at [[LHC]] is in

* {#Gibbs11a} [[Philip Gibbs]], _Seminar Watch (Higgs Special), Rumoured Higgs at 125 GeV and What Would a Higgs at 125 GeV Tell Us?_, Prespacetime Journal, December 2011, Vol. 2 Issue 12 pp. 1899-1905 ([web](http://prespacetime.com/index.php/pst/article/view/308/298))

* {#HLL11} Martin Holthausen, Kher Sham Lim, Manfred Lindner, _Planck Scale Boundary Conditions and the Higgs Mass_, JHEP 1202 (2012) 037 ([arXiv:1112.2415](https://arxiv.org/abs/1112.2415))

The official announcement of the detection at [[LHC]] is due to

* {#ATLAS12} G. Aad et al. (ATLAS Collaboration), _Observation of a New Particle in the Search for the Standard Model Higgs Boson with the ATLAS Detector at the LHC,_ Phys. Lett. B 716, 1 (2012) ([arXiv:1207.7214](https://arxiv.org/abs/1207.7214))

* {#CMS12} S. Chatrchyan et al. (CMS Collaboration), _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_, 716, 30 (2012) ([arXiv:1207.7235](https://arxiv.org/abs/1207.7235))

* {#ATLAS15} G. Aad et al. (ATLAS Collaboration, CMS Collaboration), _Combined Measurement of the Higgs Boson Mass in $p p$ Collisions at $\sqrt{s} = 7$ and 8 TeV with the ATLAS and CMS Experiments_, Phys. Rev. Lett. 114, 191803 (2015) ([arXiv:1503.07589](https://arxiv.org/abs/1503.07589))

### Vacuum (in-)stability
 {#ReferencesVacuumInstability}

Before experimental observation of the Higgs mass at the [[LHC]], the three possible outcomes for [[vacuum stability]] were analyzed in

* {#EEGHR09} [[John Ellis]], J.R. Espinosa, [[Gian Giudice]], A. Hoecker, A. Riotto, _The Probable Fate of the Standard Model_, Phys.Lett.B679:369-375,2009 ([arXiv:0906.0954](https://arxiv.org/abs/0906.0954))

* {#Gibbs11b} [[Philip Gibbs]], _[What would a Higgs at 125 GeV tell us?](https://vixra.wordpress.com/2011/12/04/what-would-a-higgs-at-125-gev-tell-us/)_, in _Seminar Watch (Higgs Special), Rumoured Higgs at 125 GeV and What Would a Higgs at 125 GeV Tell Us?_, Prespacetime Journal, December 2011, Vol. 2 Issue 12 pp. 1899-1905 ([web](http://prespacetime.com/index.php/pst/article/view/308/298))

More discussion of the near-criticality of the [[vacuum stability]] after the observation of the Higgs mass is due to

* {#DVEEGI12} Giuseppe Degrassi, Stefano Di Vita, Joan Elias-Miró, José R. Espinosa, [[Gian Giudice]], Gino Isidori, _Higgs mass and vacuum stability in the Standard Model at NNLO_, J. High Energ. Phys. (2012) 2012: 98 ([arrXiv:1205.6497](https://arxiv.org/abs/1205.6497), [doi:10.1007/JHEP(2012)098](https://doi.org/10.1007/JHEP08%282012%29098))

with a more precise analysis due to

* {#BKPV15} A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, _Stability of the Electroweak Vacuum: Gauge Independence and Advanced Precision_, Phys. Rev. Lett. 115, 201802 ([arXiv:1507.08833](https://arxiv.org/abs/1507.08833), [doi:10.1103/PhysRevLett.115.201802](https://doi.org/10.1103/PhysRevLett.115.201802))

Careful discussion of the stability issue under [[renormalization]] is in

* Anders Andreassen, William Frost, Matthew D. Schwartz, _Consistent Use of the Standard Model Effective Potential_, Phys. Rev. Lett. 113, 241801 (2014) ([arXiv:1408.0292](https://arxiv.org/abs/1408.0292))

and with emphasis on [[non-perturbative effects]] in [[quantum tunneling]]/[[vacuum decay]]

* {#AFFS17} Anders Andreassen, David Farhi, William Frost, Matthew D. Schwartz, _Precision decay rate calculations in quantum field theory_, Phys. Rev. D 95, 085011 (2017) ([arXiv:1604.06090](https://arxiv.org/abs/1604.06090))

Review includes

* {#Kusenko15} Alexander Kusenko, _[Are We on the Brink of the Higgs Abyss?](https://physics.aps.org/articles/v8/108)_, Physics 8, 108 (2015)

* Sebastian Baum, _On the metastability of the Standard Model_, 2015 ([pdf](https://uu.diva-portal.org/smash/get/diva2:819933/FULLTEXT01.pdf))

* Holger Gies, _Vacuum stability and the mass of the Higgs boson_, 2015 ([pdf](http://physik.uni-graz.at/~dk-user/talks/Gies_20150828.pdf))

* {#Schwartz15} Matthew Schwartz, _Do we know if our universe is stable?_, Rutgers Seminar 2015 ([pdf](https://www.physics.rutgers.edu/het/video/schwartz15a.pdf))

* {#Strumia17} [[Alessandro Strumia]], _Higgs and Vacuum (In)Stability_, talk at GGI 2017 ([pdf](https://indico.cern.ch/event/660870/contributions/2746200/attachments/1538374/2411246/2017-HiggsDecay.pdf))

* {#AFS18} Anders Andreassen, William Frost, Matthew D. Schwartz, _Scale Invariant Instantons and the Complete Lifetime of the Standard Model_, Phys. Rev. D 97, 056006 (2018) ([arXiv:1707.08124](https://arxiv.org/abs/1707.08124))

Speculation about what this near-criticality of the Higgs vacuum could be pointing to:

attractor mechanism in a [[landscape of string theory vacua|landscape of vacua]]:

* {#BDGGSSS13} Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, [[Gian Giudice]], Filippo Sala, Alberto Salvio, [[Alessandro Strumia]], section 7 of _Investigating the near-criticality of the Higgs boson_, JHEP12(2013)089 ([arXiv:1307.3536](https://arxiv.org/abs/1307.3536))

an [[supersymmetry|supersymmetric]]-extension of the [[standard model of particle physics]] (e.g. [[MSSM]], [[G2-MSSM]], ...)

* {#IsidoriPattori17} Gino Isidori, Andrea Pattori, _On the tuning in the $(m_h, m_t)$ plane: Standard Model criticality vs. High-scale SUSY_, Physics Letters B Volume 782, 10 July 2018, Pages 551-558 ([arXiv:1710.11060](https://arxiv.org/abs/1710.11060))

as well as [Kane 18, "Clue 4"](#Kane18)

and more generally in view of [[split supersymmetry]]:

* {#GiudiceStrumia11} [[Gian Giudice]], [[Alessandro Strumia]], _Probing High-Scale and Split Supersymmetry with Higgs Mass Measurements_, Nuclear Physics B Volume 858, Issue 1, 1 May 2012, Pages 63-83 ([arXiv:1108.6077](https://arxiv.org/abs/1108.6077))

Arguments that a false Higgs vacuum is incompatible with [[cosmology|cosmological]] evolution ([[inflation]]) include the following:

* {#EGR07} J.R. Espinosa, [[Gian Giudice]], A. Riotto, _Cosmological implications of the Higgs mass measurement_, JCAP 0805:002, 2008 ([arXiv:0710.2484](https://arxiv.org/abs/0710.2484))

* {#HKSZ14} Anson Hook, John Kearney, Bibhushan Shakya, Kathryn M. Zurek, _Probable or Improbable Universe? Correlating Electroweak Vacuum Instability with the Scale of Inflation_, J. High Energ. Phys. (2015) 2015: 61 ([arXiv:1404.5953](https://arxiv.org/abs/1404.5953))

* {#EGMRSST15} Jose R. Espinosa, [[Gian Giudice]], Enrico Morgante, Antonio Riotto, Leonardo Senatore, [[Alessandro Strumia]], Nikolaos Tetradis, _The cosmological Higgstory of the vacuum instability_ ([arXiv:1505.04825](https://arxiv.org/abs/1505.04825))

* {#EKSYZ16} William E. East, John Kearney, Bibhushan Shakya, Hojin Yoo, Kathryn M. Zurek, _Spacetime Dynamics of a Higgs Vacuum Instability During Inflation_, Phys. Rev. D 95, 023526 (2017) ([arXiv:1607.00381](https://arxiv.org/abs/1607.00381))

But see

* Mark P. Hertzberg, Mudit Jain, _Explanation for why the Early Universe was Stable and Dominated by the Standard Model_, JCAP 2020 ([arXiv:1911.04648](https://arxiv.org/abs/1911.04648))

for a possible solution by strong [[interaction]] between the Higgs field and the [[inflaton]].

An interpretation in terms of [[asymptotic safety]]:

* {#ShaposhnikovWetterich09} [[Mikhail Shaposhnikov]], [[Christof Wetterich]], _Asymptotic safety of gravity and the Higgs boson mass_, Phys. Lett. B 683 (2010) 196 [[arXiv:0912.0208](https://arxiv.org/abs/0912.0208)]

Proposals for further [[experiments]]:

* Roberto Franceschini, [[Alessandro Strumia]], Andrea Wulzer, *The collider landscape: which collider for establishing the SM instability?*, Journal of High Energy Physics **2022** 229 (2022) [[arXiv:2203.17197](https://arxiv.org/abs/2203.17197), doi:10.1007/JHEP08(2022)229]

See also:

* Thomas Steingasser, David I. Kaiser, *Higgs Criticality beyond the Standard Model* [[arXiv:2307.10361](https://arxiv.org/abs/2307.10361)]

* [[Gudrun Hiller]], Tim Höhne, Daniel F. Litim, Tom Steudtner, *Vacuum Stability in the Standard Model and Beyond* [[arXiv:2401.08811](https://arxiv.org/abs/2401.08811)]

### Diphoton decay

On [[renormalization]] of the [[Higgs field]]'s diphoton decay using [[causal perturbation theory]]:

* [[Pawel Duch]], [[Michael Duetsch]], [[Jose Gracia-Bondia]], _Diphoton decay of the higgs from the Epstein--Glaser viewpoint_ ([arXiv:2011.12675](https://arxiv.org/abs/2011.12675))

### Gauge-Higgs unification

Discussion of [[gauge-Higgs unification]]:

The mechanism is due to

* {#Manton79} [[Nicholas Manton]], _A new six-dimensional approach to the Weinberg-Salam model_, Nuclear Physics B Volume 158, Issue 1, 8 October 1979, Pages 141-153 (doi:10.1016/0550-3213(79)90192-5)

* {#Hosotani83} [[Yutaka Hosotani]], _Dynamical Mass Generation by Compact Extra Dimensions_, Phys. Lett. 126B (1983) 309-313 ([spire:188768](http://inspirehep.net/record/188768), doi:10.1016/0370-2693(83)90170-3)

* {#Hosotani89} [[Yutaka Hosotani]], _Dynamics of non-integrable phases and gauge symmetry breaking_, Annals of Physics Volume 190, Issue 2, March 1989, Pages 233-253 (doi:10.1016/0003-4916(89)90015-8)

and was revived with:

* H. Hatanaka, T. Inami, C.S. Lim, _The Gauge Hierarchy Problem and Higher Dimensional Gauge Theories_, Mod. Phys. Lett. A13 (1998) 2601-2612 ([arXiv:hep-th/9805067](https://arxiv.org/abs/hep-th/9805067))

Review:

* {#Hosotani12} [[Yutaka Hosotani]], _Gauge-Higgs Unification Approach_, AIP Conference Proceedings 1467, 208 (2012) ([arXiv:1206.0552](https://arxiv.org/abs/1206.0552))

See also:

* Naoyuki Haba, Kazunori Takenaga, Toshifumi Yamashita, _Higgs mass in the gauge-Higgs unification_, Phys. Lett. B615 (2005) 247-256 ([arXiv:hep-ph/0411250](https://arxiv.org/abs/hep-ph/0411250))

* {#Lim09} C. S. Lim, _Precision Tests and CP Violation in Gauge-Higgs Unification_, 2009 ([slides pdf](https://www.iop.vast.ac.vn/theor/conferences/icfp09/files/Lim))

Discussion of [[gauge-Higgs unification]] in [[Spin(11)]]-[[GUT]] ("SO(11)-GUT"):

* {#HosotaniYamatsu15} [[Yutaka Hosotani]], Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([arXiv:1504.03817](https://arxiv.org/abs/1504.03817), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))

* {#FuruiHosotaniYamatsu16} Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))

* {#Hosotoni16} [[Yutaka Hosotani]], _Gauge-Higgs EW and Grand Unification_, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016)  ([arXiv:1606.08108](https://arxiv.org/abs/1606.08108))

* {#Hosotani17} [[Yutaka Hosotani]], _New dimensions from gauge-Higgs unification_ ([arXiv:1702.08161](https://arxiv.org/abs/1702.08161))

* {#HosotaniYamatsu17} [[Yutaka Hosotani]], Naoki Yamatsu, _Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification_ ([arXiv:1710.04811](https://arxiv.org/abs/1710.04811))

Discussion of [[gauge-Higgs unification]] in [[Spin(11)]]-[[GUT]] ("SO(11)-GUT"):

* S. Rajpoot and P. Sithikong, _Implications of the $SO(12)$ gauge symmetry for grand unification_, Phys. Rev. D 23, 1649 (1981) ([doi:10.1103/PhysRevD.23.1649](https://doi.org/10.1103/PhysRevD.23.1649))

* {#NomuraSato08} Takaaki Nomura, Joe Sato, _Standard(-like) Model from an $SO(12)$ Grand Unified Theory in six-dimensions with $S^2$ extra-space_, Nucl.Phys.B811:109-122, 2009 ([arXiv:0810.0898](https://arxiv.org/abs/0810.0898))

* {#Nomura09} Takaaki Nomura, _Physics beyond the standard model with $S^2$ extra-space_, 2009 ([pdf](http://krishna.th.phy.saitama-u.ac.jp/joe/doctor/nomura-doctor.pdf), [[NomuraSO12GUT.pdf:file]])

* {#ChiangNomuraSato11} Cheng-Wei Chiang, Takaaki Nomura, Joe Sato, _Gauge-Higgs unification models in six dimensions with $S^2/\mathbb{Z}_2$ extra space and GUT gauge symmetry_ ([arXiv:1109.5835](https://arxiv.org/abs/1109.5835))

### In string theory

Discussion of the Higgs field from [[intersecting D-brane models]] is due to

* {#IbanezMarchesanoRabadan01} [[Luis Ibáñez]], [[Fernando Marchesano]], R. Rabadan, section 7 of _Getting just the Standard Model at Intersecting Branes_, JHEP 0111:002, 2001 ([arXiv:hep-th/0105155](https://arxiv.org/abs/hep-th/0105155))

* {#CremadesIbanezMarchesano02} D. Cremades, [[Luis Ibáñez]], [[Fernando Marchesano]], _Intersecting brane models of particle physics and the Higgs mechanism_, JHEP, 0207, 022 2002 ([arXiv:hep-th/0203160](https://arxiv.org/abs/hep-th/0203160))

* {#HebeckerKnochelWeigand13} Arthur Hebecker, Alexander K. Knochel, [[Timo Weigand]], _The Higgs mass from a String-Theoretic Perspective_,  Nucl.Phys. B874 (2013) 1-35 ([arXiv:1304.2767](https://arxiv.org/abs/1304.2767))

* {#HINSS18} Manami Noumi Hashi, Hiroshi Isono, Toshifumi Noumi, [[Gary Shiu]], Pablo Soler, _Higgs Mechanism in Nonlocal Field Theories_, JHEP08 (2018) 064 ([arXiv:1805.02676](https://arxiv.org/abs/1805.02676))

Discussion of the Higgs mechanism in the [[G2-MSSM]] and related models is due to

* {#Kane11} [[Gordon Kane]], _String theory and generic predictions for our world – superpartner masses, LHC signatures, dark matter, EWSB, cosmological history of universe, etc_, talk at String phenomenology 2011, August 2011 ([pdf](http://conferencing.uwex.edu/conferences/stringpheno2011/documents/kane.pdf))

* {#KaneKumarLuZheng11} [[Gordon Kane]], Piyush Kumar, Ran Lu, Bob Zheng, _Higgs Mass Prediction for Realistic String/M Theory Vacua_, Phys. Rev. D 85, 075026 ([arXiv:1112.1059](http://arxiv.org/abs/1112.1059))

  (a useful informed comment is [here](http://www.math.columbia.edu/~woit/wordpress/?p=4262&cpage=1#comment-101525))

and related to the issue of the [[vacuum stability]] in

* {#Kane18} [[Gordon Kane]], _Exciting Implications of LHC Higgs Boson Data_ ([arXiv:1802.05199](https://arxiv.org/abs/1802.05199))

which is based on

* {#Kane17} [[Gordon Kane]], _String theory and the real world_, Morgan & Claypool, 2017 ([doi:0.1088/978-1-6817-4489-6](http://iopscience.iop.org/book/978-1-6817-4489-6">doi:0.1088/978-1-6817-4489-6))

See also

* Lawrence J. Hall, Yasunori Nomura, _Grand Unification and Intermediate Scale Supersymmetry_, JHEP02 (2014) 129 ([arXiv:1312.6695](https://arxiv.org/abs/1312.6695))

* Lawrence J. Hall, Yasunori Nomura, Satoshi Shirai, _Grand Unification, Axion, and Inflation in Intermediate Scale Supersymmetry_ ([arXiv:1403.8138](https://arxiv.org/abs/1403.8138))

* David Dunsky, Lawrence J. Hall, Keisuke Harigaya, _Dark Matter Detection, Standard Model Parameters, and Intermediate Scale Supersymmetry_ ([arXiv:2011.12302](https://arxiv.org/abs/2011.12302))

Discussion in [[holographic QCD]]:

* {#EspiruKatanaeva18} Domenec Espriu, Alisa Katanaeva, _Composite Higgs Models: a new holographic approach_ ([arXiv:1812.01523](https://arxiv.org/abs/1812.01523))

* [[Johanna Erdmenger]], [[Nick Evans]], [[Werner Porod]], [[Konstantinos Rigatos]], *Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models*, JHEP **58** (2021) [[arXiv:2010.10279](https://arxiv.org/abs/2010.10279), doi:10.1007/JHEP02(2021)058]

Discussion in [[heterotic string theory]]:

* Steven Abel, [[Keith Dienes]],
*Calculating the Higgs Mass in String Theory* ([arXiv:2106.04622](https://arxiv.org/abs/2106.04622))

[[!redirects Higgs fields]]

[[!redirects Higgs mechanism]]

[[!redirects Higgs effect]]
[[!redirects Higgs effects]]

[[!redirects Higgs particle]]
[[!redirects Higgs particles]]


higgsino

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

In the [[MSSM]] the [[superpartner]] of the [[Higgs field]], or rather its quanta, are calle the _higgsino_.

[[!redirects higgsinos]]
[[!redirects Higgsino]]
[[!redirects Higgsinos]]


high energy physics

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

#Contents#
* table of contents
{:toc}

## Idea

[[physics]] at very higher [[energy]] [[density]], hence small [[distance]]:

[[quantum field theory]] and [[particle physics]]

## Related concepts

* [[mathematical physics]]

* [[solid state physics]]

* [[mesoscopic physics]]

* [[nuclear physics]]

* [[quantum gravity]]

## References

See the references at *[[particle physics]]*.

See also:

* Wikipedia, *[Particle physics](https://en.wikipedia.org/wiki/Particle_physics)*


high performance distributed ledger

Bitcoin with only about 8 transaction per second (TPS) and also big latency times (for final confirmations) is not very suitable for most [[blockchain]] applications. Similarly with most early generation distributed ledgers. Important things for performance include also the efficiency of [[virtual machine]] employed (trend is that the newest projects go toward versions of [[WebAssembly]] VM).

Faster and even scalable solutions are now in works.
We first describe improvements on Bitcoin and Ethernet
networks in the form of lightning and plasma networks on top of those, and then various projects which claim to be scalable, "3rd generation" etc. Essential improvements are various algorithms trying to go beyond linearity of classical blockchain projects, especially those relying on DAGs (directed acyclic graphs) and on so called sharding. As far as consensus algorithms there are scalability improvements with many new algorithms, while PoW is in a way most secure (for public blockchains).

While mining is computationally intensive it is considered that the optimizations with use of GPU and ASIC hardware are bad forces of centralization (not to mention unhealthy and energy consuming chase for success with single purpose hardware). It is considered better to use computational resources for more useful work like validation of smart contracts, security of computation, cryptographic tools for privacy protection and so on. With FPGA the situation is a little better, and some FPGA are claimed to be useful for smart contract processing and alike, see [here](https://www.embedded.com/electronics-blogs/say-what-/4461708/FPGA-based-frameworks-speed-blockchain-processing) there are also Chaincode [[virtual machine]]-specific ASICs by [accelor.io](https://accelor.io).

### Lightning network

Bitcoin solves some of the problems with   so called lightning network on top of it, which makes temporary transactions with lots of intermediate transfers only off chain and only small footprint on the main chain, ensuring small price of transactions.

* [lightning.network](https://lightning.network)
* Joseph Poon, Tadge Dryja, _Lightning Network_, [pdf](https://lightning.network/
lightning-network-paper.pdf) Mar 2015.
* A van Wirdum _Understanding the lightning network [1](https://bitcoinmagazine.com/articles/understanding-the-lightning-network-part-building-a-bidirectional-payment-channel-1464710791) [2](https://bitcoinmagazine.com/articles/understanding-the-lightning-network-part-creating-the-network-1465326903)
* Amazon using lightning network for e-commerce, April 2019
[news](https://www.coindesk.com/you-can-now-shop-with-bitcoin-on-amazon-using-lightning)

### Ethernet improvements and plasma

Ethernet recognizes the same problem as Bitcoin, but has significant plans of substantial upgrades of its Mainnet, namely the new Casper protocol, plans for deployment of [[WebAssembly]] virtual machine and sharding which, according to Buterin, could together bring million TPS when all employed as planned.

In addition, there is an analogue algorithm to lightning network targeted to Ethereum, with substantial new ideas in the algorithm, called plasma network.

* Joseph Poon, Vitalik Buterin, _Plasma: scalable autonomous smart contracts_
[pdf](https://www.plasma.io/plasma.pdf) draft whitepaper

There are several realizations of plasma, mostly used being the Loom network.

### Projects based on DAGs

There are several major algorithms. Several projects are based on "hashgraph" paper

* Leemon Baird, _The Swirlds hashgraph consensus algorithm: fair, fast, byzantine fault tolerance_, [pdf](https://www.swirlds.com/downloads/SWIRLDS-TR-2016-01.pdf) (May 2016); _Hashgraph consensus: detailed examples_ [pdf](https://www.swirlds.com/downloads/SWIRLDS-TR-2016-02.pdf); _Overview of Swirlds Hashgraph: An Introduction To The Hashgraph (SDK Available Now)_ [online](https://steemit.com/technology/@swirlds/overview-of-swirlds-hashgraph-an-introduction-to-the-hashgraph-sdk-available-now)

including [Hedera](https://www.hedera.com/whitepaper) which claims 50000 to 500000 TPS tested,

* Hedera Hashgraph whitepaper [pdf](https://www.hedera.com/hh-whitepaper-v1.5-190219.pdf); intro to whitepaper [web](https://www.hedera.com/whitepaper);

then so called Consensus project etc.

Somewhat similar but with a new innovative algorithm is [Tolar Hashnet](https://www.tolar.io), see whitepaper [pdf](https://www.tolar.io/static/media/Whitepaper-Tolar.2f13a40e.pdf) with around 150000 TPS achieved on testnet and low power consumption.

Another kind of scalable algorithms using DAGs is the tangle blockchain (of the [iota](https://www.iota.org) project)

* Serguei Popov, _The tangle_ [pdf](https://www.iotatoken.com/IOTA_Whitepaper.pdf), 2016

### Telegram Open Network

[Telegram](https://telegram.org) (see [FAQ](https://telegram.org/faq) and [wikipedia](https://en.wikipedia.org/wiki/Telegram_(messaging_service)) is developing Telegram Open Network (TON) centered around a new generation blockchain project. Besides the blockchain and user interfaces, including on Telegram, it will contain analogues of other parts of the internet structure including TON proxy, TON DNS and TON storage. The blockchain uses delegated proof of stake with a number of innovations including infinite sharding paradigm and instant hypercube rooting. The whitepaper predicts that the platform will be able to support millions of TPS (private access to testnet allegedly given in April 2019). For more details see [[Telegram Open Network]] in this wiki and original

* TON whitepaper, 23 page [pdf](https://cdn.crowdfundinsider.com/wp-content/uploads/2018/03/TON-White-Paper-Telegram-ICO.pdf), mirrors [pdf](https://drive.google.com/file/d/1ucUeKg_NiR8RxNAonb8Q55jZha03WC0O/view) [pdf](https://icorating.com/upload/whitepaper/gNQ7e9z3lCGi519Wz8mmC0Kg8aA0goeZKAQ802vo.pdf)
* [[nlab:Nikolai Durov]]'s 132-page technical overview _Telegram Open Network_, Dec 2017 [pdf](https://toncoin.tech/TON-official-whitepaper.pdf), mirror [pdf](https://www.kriptovaluta.hr/wp-content/uploads/2018/03/TON-Technology.pdf); _Telegram Open Network Blockchain_, Sep 2018, 121 pp. [pdf](https://www.docdroid.net/qY4sQEv/telegram-open-network-blockchain-september-5-2018.pdf); _Telegram Open Network Virtual Machine_, Sep. 2018, 148 pp. [pdf](https://www.docdroid.net/R3vEKBY/telegram-open-network-virtual-machine-september-5-2018.pdf)

### RedBelly

Australian academic [RedBelly](https://redbellyblockchain.io) blockchain (also referred as CSIRO) has confirmed 40000 TPS with Amazon platform worldwide testing. Since mid 2017 it had local tests which could achieve up to 400000 but with all nodes in the same location; see the paper about various performance experiments (the largest performance experiment, with measuring various parameters, with up to 1000 nodes in 10 countries!),

* Concurrent Systems Research Group, University of Sydney, Data61-CSIRO, _The Red Belly Blockchain Experiments_, [pdf](https://redbellyblockchain.io/papers/redbellyblockchain-experiments.pdf)

and selection of [news](https://redbellyblockchain.io/Press.html). The academic group behind the projects has also a number of interesting [research papers](https://redbellyblockchain.io/Research.html).

### MIT Vault

* Derek Leung, Adam Suhl, Yossi Gilad, Nickolai Zeldovich (MIT CSAIL), _Vault: fast bootstrapping for cryptocurrencies_, Jan 2019, preprint [pdf](http://people.csail.mit.edu/nickolai/papers/leung-vault-eprint.pdf)
* [news.mit.edu:faster-more-efficient-cryptocurrency](https://news.mit.edu/2019/vault-faster-more-efficient-cryptocurrency-0124)

### Algorand

[Algorand](https://www.algorand.com) project, founded by a group at MIT lead by S. Micali, claims 1000 tps and latency of only 5 seconds in a setup with 10000 participants and 500 nodes on testnet world wide. It got 62 M dollars equity funding in 2018 ([news](https://www.the-blockchain.com/2018/10/24/algorand-announces-62m-in-funding-and-appoints-new-executive-team)) and is issuing Algos coins in auction in June 2019, [news](https://www.the-blockchain.com/2019/06/06/mit-professor-silvio-micalis-algorand-foundation-announces-date-for-first-auction). Undergone security audits from Trail of Bits and NCC. While the first prototype in 2016 has been coded in C++, the full implementation is in Golang, see [github](https://github.com/algorand/go-algorand). The underlying algorithms are described in

* Yossi Gilad, Rotem Hemo, Silvio Micali, Georgios Vlachos, Nickolai Zeldovich (MIT CSAIL), _Algorand: scaling Byzantine agreements for cryptocurrencies_, [pdf](https://people.csail.mit.edu/nickolai/papers/gilad-algorand-eprint.pdf)
* Jing Chen, Silvio Micali, _Algorand_, [arxiv/1607.01341](https://arxiv.org/abs/1607.01341)
* github/[algobet/kirin](https://github.com/algobet/kirin) - Official javascript framework of the Digital Asset Platform (DAP) protocol atop the POS/POR consensus mechanism. Kirin is the Quarkonium's first generation constitutional protocols deeply built with the Algorand blockchain
* S. Gorbounov, _Algorand releases first open-source code: verifiable random function_, [medium](https://medium.com/algorand/algorand-releases-first-open-source-code-of-verifiable-random-function-93c2960abd61)
* _Announcing full open source availability of the Algorand blockchain_ june 12, 2019 [medium](https://medium.com/algorand/announcing-full-open-source-availability-of-the-algorand-blockchain-4585f1329348)

### Other projects which claim to be very fast or scalable

* [bitconch](https://bitconch.io) claims 120000 TPS
* [Futurepia](https://futurepia.io) uses Double Delegated Proof of Stake. "KOLAS (Korea Laboratory Accreditation Scheme) has assessed and certified FUTUREPIA’s Social Media Mainnet transaction speed as 300,000 TPS".
* [[EOS]] is a high performance blockchain using [[WebAssembly]] [[virtual machine]] for [[smart contract]]s. [eos.io](https://eos.io) claims speeds up to 300000 TPS.  [eos.io](https://eos.io) claims speeds up to 300000 TPS. EOS technical white paper v2 (March 2018) is at github, [md](https://github.com/EOSIO/Documentation/blob/master/TechnicalWhitePaper.md).
* NEO "smart economy" [neo.org](http://neo.org) claims 1000-10000 TPS with plans to go to 100000 in 2020
* Ripple [XRP](https://ripple.com/xrp) blockchain has over 1700 TPS in regular usage. They claim to test speeds up to 50000
* [devv](https://devv.io/pages/tech) algorithm of [devv.io](https://devv.io) uses massive sharding and the claims are to test-support up to 8 million TPS (Jan 2019) [news](https://currencyjournals.com/blockchain/start-up-devvio-claims-its-blockchain-can-handle-8m-transactions-a-second-computerworld), [whitepaper](https://cdn.shopify.com/s/files/1/0009/0079/2378/files/2018-09-12_Devv_Whitepaper.pdf?1984954902434967058) "Devvio’s approach emphasizes maintaining only representations of value on the blockchain, with non-essential processing occurring off-chain"
* [mimblewimble](https://mimblewimble.support) cryptocurrency protocol is focused on privacy and also created to be pretty scalable and very secure (even in comparison to bitcoin) with less computer power for PoW, while more suitable for GPUs than ASICs. There are two leading implementations, [Grin](https://grin-tech.org) and [Beam](https://www.beam.mw). Grin is coded in [[Rust]], see [github](https://github.com/mimblewimble/grin).
* [thundercore](https://www.thundercore.com) (see also description at [medium](https://medium.com/boxmining/thundercore-explained-breakthrough-scaling-for-ethereum-dapps-8cfb942331ce)) is EVM-compatible blockchain with some claims on scalability

There is a theoretical proposal to scale up by 3
orders of magnitude by the BlockReduce algorithm "which only segments consistency". The paper shortly reflects on other approaches

* K. J. Kreder III, _BlockReduce: scaling blockchain to human commerce_, [arxiv/1811.00125](https://arxiv.org/abs/1811.00125)

[[!redirects high performance DL]]
[[!redirects high performance blockchain]]
[[!redirects high performance DLs]]


Higher Algebra

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#### Higher algebra
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[[!include higher algebra - contents]]
=--
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=--

This entry collects links related to the book

* [[Jacob Lurie]]:

  \linebreak

  **Higher Algebra**

  \linebreak

  ($\sim$2017)

  [pdf](https://www.math.ias.edu/~lurie/papers/HA.pdf)

on [[higher algebra]].

This book lays the basis of [[categorical algebra]] -- the study of the trinity [[Lawvere theory|algebra]], [[monads]], [[operads]] -- in the context of [[(∞,1)-category theory]]: [[(∞,1)-algebraic theory|(∞,1)-algebras]], [[(∞,1)-monad]]s, [[(∞,1)-operad]]s.

In particular it discusses the analogs of [[abelian groups]] and [[commutative rings]], namely [[abelian ∞-groups]], [[spectra]] and [[E-∞ rings]].

#Contents#
* table of contents
{:toc}

The following is (or will eventually be) a linked list of keywords.

## 1. Stable $\infty$-categories

### 1.1 Foundations

#### 1.1.1 Stability

* [[stable (∞,1)-category]]

  * [[stabilization]]

#### 1.1.2 The homotopy category of a stable $\infty$-category

* [[triangulated category]]

#### 1.1.3 Closure properties of stable $\infty$-categories

#### 1.1.4 Exact functors

* [[exact (∞,1)-functor]]

### 1.2 Stable $\infty$-categories and homological algebra

* [[homological algebra]]

#### 1.2.1 t-Structures on stable $\infty$-categories

* [[t-structure]]

* [[t-structure on a stable (∞,1)-category]]

* [[heart of a stable (∞,1)-category]]

#### 1.2.2 Filtered objects and spectral sequences

* [[spectral sequence]]

* [[spectral sequence of a filtered stable homotopy type]]

  * [[Adams spectral sequence]]

#### 1.2.3 The Dold-Kan correspondence

* [[Dold-Kan correspondence]]

#### 1.2.4 The $\infty$-categorical Dold-Kan correspondence

* [[infinity-Dold-Kan correspondence]]

* [[spectral sequence of a simplicial stable homotopy type]]

### 1.3 Homological algebra and derived categories

#### 1.3.1 Nerves of differential graded categories

* [[dg-nerve]]

#### 1.3.2 Derived $\infty$-categories

* [[derived category]]

#### 1.3.3 The universal property of $\mathcal{D}^-(\mathcal{A})$

#### 1.3.4 Inverting Quasi-isomorphisms

#### 1.3.5 Grothendieck abelian categories

* [[Grothendieck category]]

#### 1.3.6 Complexes of injective objects

* [[injective object]]

* [[chain complex]]

### 1.4 Spectra and stabilization

#### 1.4.1 The Brown representability theorem

* [[Brown representability theorem]]

#### 1.4.2 Spectrum objects

* [[excisive (∞,1)-functor]]

* [[spectrum object]]

#### 1.4.3 The $\infty$-category of spectra

* [[(∞,1)-category of spectra]]

#### 1.4.4 Presentable stable $\infty$-categories

* [[presentable (∞,1)-category]]

* [[stable (∞,1)-category]]

## 2. $\infty$-Operads

### 2.1 Foundations

#### 2.1.1 From colored operads to $\infty$-operads

* [[Segal Gamma-category]]

* [[inert morphism]]

* [[category of operators]]

* [[(∞,1)-operad]]

#### 2.1.2 Maps of $\infty$-operads

* [[morphism of (∞,1)-operads]]

* [[fibration of (∞,1)-operads]]

#### 2.1.3 Algebra objects

* [[algebra over an (∞,1)-operad]]

* $\mathcal{O}$-[[monoidal (∞,1)-category]]

#### 2.1.4 $\infty$-Preoperads

### 2.2 Constructions of $\infty$-Operads

### 2.3 Disintegration and assembly

#### 2.3.1 Disintegration and assembly

#### 2.3.2 Generalized $\infty$-operads

* [[generalized (∞,1)-operad]]

* [[family of (∞,1)-operads]]

#### 2.3.3 Approximations to $\infty$-operads

#### 2.3.4 Disintegration of $\infty$-operads

### 2.4 Products and coproducts

#### 2.4.1 Cartesian symmetric monoidal structure

* [[cartesian monoidal (∞,1)-category]]

#### 2.4.2 Monoid objects

* [[monoid]]

* [[Gamma space]]

#### 2.4.3 CoCartesian Symmetric Monoidal Structure

* [[cocartesian monoidal (∞,1)-category]]

#### 2.4.4 Wreath Products

## 3. Algebras and modules over $\infty$-operads

### 3.1 Free algebras

### 3.2 Limits and colimits of algebras

### 3.3 Modules over $\infty$-operads

* [[module over an algebra over an (∞,1)-operad]]

#### 3.3.1 Coherent $\infty$-operads

* [[coherent (∞,1)-operad]]

### 3.4 General features of module $\infty$-categories

## 4. Associative algebras and their modules

### 4.1 Associative algebras

#### 4.1.1 The $\infty$-Operad $\mathcal{Ass}^\otimes$

* [[associative operad]]

* [[A-∞ operad]]

* [[planar (∞,1)-operad]]

#### 4.1.2 Simplicial models for associative algebras

* [[simplicial algebra]]

* [[model structure on simplicial T-algebras]]

#### 4.1.3 Monoidal Model Categories

* [[monoidal model category]]

#### 4.1.4 Rectification of Associative Algebras

* [[model structure on algebras over an operad]]

* [[homotopy T-algebras]]/[[model structure on simplicial T-algebras]]

### 4.2 Left and Right Modules

#### 4.2.1 The $\infty$-Operad $\mathcal{L M}^{\otimes}$

* [[(∞,1)-operad for (∞,1)-modules over an A-∞ algebra]]

* [[(∞,1)-module]]

#### 4.2.2 Simplicial models for algebras and modules

#### 4.2.3 Limits and colimits of algebras

#### 4.2.4 Free modules

#### 4.2.5 Duality in monoidal $\infty$-categories

### 4.3 Bimodules

#### 4.3.1 The $\infty$-Operad $\mathcal{BM}^{\otimes}$

* [[operad for bimodules over algebras]]

* [[∞-bimodule]]

* [[bitensoring]]

#### 4.3.2 Simplicial models for algebras and modules

* [[model structure on modules over an algebra over an operad]]

#### 4.3.3 Limits, Colimits, and Free Bimodules

(...)

#### 4.3.4 Multilinear maps

* [[multilinear map]]

* [[bilinear map in an (∞,1)-category]]

#### 4.3.5 Tensor Products and the Bar Construction

* [[tensor product of ∞-modules]]

* [[bar construction]]

#### 4.3.6 Associtivity of the Tensor Product

* [[associativity]]

#### 4.3.7 Duality of Bimodules

(...)

### 4.4 Modules over commutative algebras

## 5. Little cubes and factorizable sheaves

* [[little cubes operad]]

### 5.1 Definitions and basic properties

### 5.2 Little cubes and manifold topology

#### 5.2.1 Embeddings of topological manifolds

#### 5.2.2 Variations on the little cubes operad

#### 5.2.3 Nonunital $\mathbb{E}_k$-Algebras

* [[nonunital ring]]

* [[nonunital Ek-algebra]]

* [[unitalization]]

* [[augmented A-∞ algebra]]

#### 5.2.4 Little cubes in a manifold

### 5.3 Topological chiral homology

* [[topological chiral homology]]

## 6. Algebraic structures on $\infty$-categories

### 6.1 Endomorphism objects

### 6.2 Monads and Barr-Beck theorem

* [[(∞,1)-monad]]

* [[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]]

#### 6.2.1 Split simplicial objects

#### 6.2.2 The Barr-Beck theorem

* [[Barr-Beck theorem]]

#### 6.2.3 BiCartesian Fibrations

#### 6.2.4 Descent and the Beck-Chevalley condition

* [[higher monadic descent]]

## 7. The calculus of functors

* [[Goodwillie calculus]]

### 7.1 The calculus of functors

#### 7.1.1 $n$-Excisive Functors

* [[differentiable (∞,1)-category]]

* [[n-excisive (∞,1)-functor]]

* [[n-reduced (∞,1)-functor]]

* [[n-homogeneous (∞,1)-functor]]

#### 7.1.2 Taylor Tower

* [[Goodwillie-Taylor tower]]

#### 7.1.6 Norm maps

* [[norm map]]

* [[Tate construction]]

## 8. Algebra in stable homotopy theory

* [[stable homotopy category]]

* [[stable homotopy theory]]

### 8.1 Structured ring spectra

#### 8.1.1 $\mathbb{E}_1$-rings and their modules

* [[A-∞ ring]], [[ring spectrum]], [[algebra spectrum]]

* [[∞-module]]. [[module spectrum]]

#### 8.1.2 Recognition principle

#### 8.1.3 Change of ring

* [[connective cover]]

#### 8.1.4 Algebras over Commutative Rings

* [[A-∞ algebra]]

* [[stable Dold-Kan correspondence]]

### 8.2 Properties of rings and modules

#### 8.2.1 Free resolutions and Spectral Sequences

#### 8.2.2 Flat and projective modules

#### 8.2.3 Injective objects of stable $\infty$-categories

#### 8.2.4 Localization and Ore conditions

#### 8.2.5 Finiteness  properties of rings and modules

* [[perfect module]]

* [[compactly generated (∞,1)-category|compactly generated]] [[stable (∞,1)-category]]

* [[compactly generated triangulated category]]

### 8.3 The cotangent complex formalism

* [[tangent (∞,1)-category]]

  * [[cotangent complex]]

  * [[tangent complex]]

### 8.4 Deformation theory

* [[deformation theory]]

### 8.5 Étale morphisms

* [[etale morphism]]

## A Constructible sheaves and exit paths

* [[higher homotopy van Kampen theorem]]

## B Categorical patterns

## References

The book is based on the series of articles

* [[Jacob Lurie]],

  _[[Stable ∞-Categories]]_ ([arxiv](http://www.arxiv.org/abs/math.CT/0608228))

  _[[Noncommutative Algebra]]_ ([arXiv](http://arxiv.org/abs/math/0702299))

  _[[Commutative Algebra]]_ ([arXiv](http://arxiv.org/abs/math/0703204))

* _[[Deformation Theory]]_ ([arXiv](http://arxiv.org/abs/0709.3091))

* _$\mathbb{E}[k]$-[[Ek-Algebras|Algebras]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf))

category: reference


higher algebra

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

[[higher geometry]] $\leftarrow$ [[Isbell duality]] $\to$ **higher algebra**

***

#Contents#
* table of contents
{:toc}

## Idea

The notion of _higher algebra_ or _[[homotopical algebra]]_ refers to generalizations of [[algebra]] in the context of [[homotopy theory]] and more general of [[higher category theory]].

### General

Ordinary [[algebra]] concerns itself in particular with structures such as [[associative algebra]]s, which are [[monoid]]s [[internalization|internal to]] [[monoidal category|monoidal categories]]:

* a [[monoid]] [[internalization|internal to]] [[Set]] is just an ordinary [[monoid]];

* a [[monoid]] [[internalization|internal to]] [[Ab]], the category of abelian groups, is a [[ring]];

* a [[monoid]] [[internalization|internal to]] [[Vect]] is an ordinary algebra: a vector space equipped with a linear binary associative product with unit;

* a [[monoid]] in a [[category of chain complexes]] is a [[differential graded algebra]];

etc.

Of course, there are other aspects to algebra such as those resulting from non-associative theories such as Lie algebras and there are many aspects such as questions within Galois theory, and representation theory for which the above is too limited a view, but for the moment let it stand.

**Higher algebra** (or **[[homotopical algebra]]**) is similarly, but  in particular, the study of monoids internal to [[infinity-category|higher categories]].

A central motivating example for - or special case of the study of higher algebra was

* monoids internal to the [[stable (infinity,1)-category of spectra]] -- called [[commutative ring spectrum|commutative ring spectra]]

The "higher algebra" embodied by [[commutative ring spectrum|commutative ring spectra]] has been called **[[brave new algebra]]** by F. Waldhausen.

More generally, [[algebra]] is partially about [[algebraic theories]], about [[monads]] and about [[operads]]. All these have higher analogs in higher algebra.

###  Other aspects of higher algebra?

The parts of algebra that we set aside at the end of the *idea* are not outside the  possible range of higher algebra, they just have not yet been that developed and it is not always clear in what directions they most naturally 'should' be developed. To take an example, [[Lie infinity-algebroid]] is clearly a higher algebraic analogue of a Lie algebra, and is a 'multi-object' one as well. Questions in representation theory are often phrased in terms of monoidal categories, and their higher algebraic analogues have new structural facets that look very interesting and useful. Finally Galois theory naturally falls into the context of Grothendieck's extensive work both on higher stacks but also the Grothendieck-Teichmuller theory. Here the theory is awaiting clear indications what higher Galois theory might mean.

## Concepts

### Monads, algebraic theories, operads

* [[algebraic theory]] / [[2-algebraic theory]] / [[(∞,1)-algebraic theory]]

* [[monad]] / [[2-monad]]/[[doctrine]] / [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

### Algebras and modules

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

* [[algebra over an operad]]

   [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

### Monoidal $(\infty,1)$-Categories

* [[monoidal category]]

  * [[monoid]]

  * [[algebra]]

* [[monoidal (∞,1)-category]]

* [[algebra in an (∞,1)-category]]

### The monoidal structure on stable homotopy theory

* [[stable homotopy theory]]

  * [[stable (infinity,1)-category of spectra]]

  * [[stable homotopy category]]

* [[smash product of spectra]]

  * [[symmetric monoidal smash product of spectra]]

### Symmetric monoidal $(\infty,1)$-categories and commutative algebra

* [[symmetric monoidal (infinity,1)-category]]

* [[commutative algebra in an (infinity,1)-category]]

* examples

  * [[stable (infinity,1)-category of spectra]]

  * [[symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories]]

### Commutative ring spectra

* [[stable (infinity,1)-category of spectra]]

  * [[spectrum]]

  * [[spectrum object]]

* [[associative ring spectrum]]

  * [[A-infinity ring]]

* [[commutative ring spectrum]]

  * [[E-infinity ring]]

### Symmetric monoidal model categories

* [[model category]]

* [[monoidal model category]]

## Related concepts

* [[higher structures]]

* [[higher linear algebra]]

[[!include Isbell duality - table]]

## References

A comprehensive development of the theory is in

* [[Jacob Lurie]], _[[Higher Algebra]]_

  * _[[Commutative Algebra]]_

  * _[[Noncommutative Algebra]]_

  * _[[Deformation Theory]]_

See also

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], _Rings, modules and algebras in stable homotopy theory_, Mathematical surveys and monographs 47, American Mathematical Society, 1997


higher algebra - contents

**[[higher algebra]]**

[[universal algebra]]

## Algebraic theories

* [[algebraic theory]] / [[2-algebraic theory]] / [[(∞,1)-algebraic theory]]

* [[monad]] / [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

## Algebras and modules

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

* [[algebra over an operad]]

  [[∞-algebra over an (∞,1)-operad]]

* [[action]], [[∞-action]]

* [[representation]], [[∞-representation]]

* [[module]], [[∞-module]]

* [[associated bundle]], [[associated ∞-bundle]]

## Higher algebras

* [[monoidal (∞,1)-category]]

  * [[symmetric monoidal (∞,1)-category]]

  * [[monoid in an (∞,1)-category]]

  * [[commutative monoid in an (∞,1)-category]]

* symmetric monoidal (∞,1)-category of spectra

  * [[smash product of spectra]]

  * [[symmetric monoidal smash product of spectra]]

  * [[ring spectrum]], [[module spectrum]], [[algebra spectrum]]

* [[A-∞ algebra]]

  * [[A-∞ ring]], [[A-∞ space]]

* [[C-∞ algebra]]

* [[E-∞ ring]], [[E-∞ algebra]]

  * [[∞-module]], [[(∞,1)-module bundle]]

  * [[multiplicative cohomology theory]]

* [[L-∞ algebra]]

  * [[deformation theory]]

## Model category presentations

* [[model structure on simplicial T-algebras]] / [[homotopy T-algebra]]

* [[model structure on operads]]

  [[model structure on algebras over an operad]]

## Geometry on formal duals of algebras

* [[Isbell duality]]

* [[derived geometry]]

## Theorems

* [[Deligne conjecture]]

* [[delooping hypothesis]]

* [[monoidal Dold-Kan correspondence]]

[Edit this sidebar](/nlab/edit/higher+algebra+-+contents)


Higher Algebra > history

< [[Higher Algebra]]
[[!redirects Higher Algebra -- history]]


The article

* [[Daniel Freed]], _Higher Algebraic Structures and Quantization_, 	Commun.Math.Phys. 159 (1994) 343-398 ([arXiv:hep-th/9212115](http://arxiv.org/abs/hep-th/9212115))

and the companion

* [[Daniel Freed]], _Quantum groups from path integrals_ ([arXiv:q-alg/9501025](http://arxiv.org/abs/q-alg/9501025))

contain an early proposal that ([[topological field theory|topological]]) [[quantum field theory]] should have an "[[extended quantum field theory|extended]]" or "multi-tiered" formulation which involves not just [[vector spaces]] of phases/staes in [[codimension]] 1, but also [[n-vector space|(n-k)-vector spaces]] in [[codimension]] $k$.

This is a first indication of the structure that was later made precise by [[Jacob Lurie]] in _[[On the Classification of Topological Field Theories]]_ by [[extended field theory]] in terms of [[(infinity,n)-functors]] on the [[(infinity,n)-category of cobordisms]].

See at _[[higher category theory and physics]]_ for more comments.

category: reference