anti-dual linear space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

If the [[ground ring]] ([[ground field]]) is equipped with a [[star-ring|star structure]] (an [[anti-involution]], such as [[complex conjugation]] in the complex numbers), then the operation of forming [[dual linear spaces]] may be "twisted" by this involution to yield a notion of "anti-dual" spaces. These relate to [[Hermitian forms]] as ordinary dual spaces relate to ordinary [[inner products]].

## Definition

Let $\mathbb{K}$ be a [[star-ring]], hence a [[unital ring]] (often a [[field]], whence our notation, and here often the [[complex numbers]]) equipped with an [[anti-involution]] (for instance [[complex conjugation]]):

$$
  \overline{(-)} \,\colon\, \mathbb{K} \to \mathbb{K}
$$
$$
  \overline{x \cdot y}
    \,=\,
  \overline{y} \cdot \overline{x}
  \,.
$$

Consider a $\mathbb{K}$-[[module]] $\mathscr{V} \,\in\, \mathbb{K}Mod$ (say *right* modules for possibly non-commutative $\mathbb{K}$, but being just $\mathbb{K}$-[[vector spaces]] if $\mathbb{K}$ is a [[field]]).

\begin{definition}
The *anti-dual* space $\overline{\mathscr{V}^\ast}$ of $\mathscr{V}$ is the space of [[anti-linear maps]] to the [[ground ring]] ([[ground field]]):

$$
  \overline{\mathscr{V}^\ast}
  \;\coloneqq\;
  \Big\{
    \phi \,\colon\,
    \mathscr{V} \to \mathbb{K}
    \,\Big\vert\,
    \underset{v, v' \in \mathscr{V}}{\forall}
    \,
    \phi(v + v') = \phi(v) + \phi(v')
    ,\;
    \underset{
      { v \in \mathscr{V} }
      \atop
      { \lambda \in \mathbb{K} }
    }{\forall}
    \,
    \phi(v \cdot \lambda) \,=\, \overline{\lambda} \cdot v
  \Big\}
$$

equipped itself with the structure of a (right) module by

$$
  \phi \in \overline{\mathscr{V}^\ast}
  ,\,
  \,\lambda \in \mathbb{K}
  \;\;\;
    \vdash
  \;\;\;
  (\phi \cdot \lambda)(c)
    \;=\;
  \phi(v) \cdot \lambda
  \,.
$$

\end{definition}
(eg. [Karoubi & Villamayor 1973, Ex. 1](KaroubiVillamayor73); [Karoubi 2010, §1](#Karoubi10))

As the notation already indicates, an equivalent definition is:
\begin{definition}
  The anti-dual $\overline{\mathscr{V}^\ast}$ has as [[underlying]] [[abelian group]] the ordinary [[linear dual ]] $\mathscr{V}^\ast$ but equipped with its anti-linear structure, namely with the (right) module action twisted by the involution as:
$$
  \phi \,\in\, Hom_{\mathbb{K}}(\mathscr{V}, \mathbb{K})
  \;\;\;
    \vdash
  \;\;\;
  \underset{v \in \mathscr{V}}{\forall}
  \,
  (\phi \cdot \lambda)(v)
  \;=\;
  \overline{\lambda} \cdot \phi(v)
  \,.
$$
\end{definition}
(eg. [Mishchenko 1976 §1.1](#Mishchenko76)).

An [[isomorphism]] between the two definitions
$$
  \array{
    \mathscr{V}^\ast
    &\longrightarrow&
    \mathscr{V}^\ast
    \\
    \phi &\mapsto& \overline{\phi}
  }
$$
is established by postcomposing linear forms with the star-involution
$$
  \overline{\phi}
    \,\colon\,
  v \,\mapsto\, \overline{\phi(v)}
$$
which respects the above (right) $\mathbb{K}$-actions due to
$$
  \array{
    \big(
      v \,\mapsto\,
      \phi(v)\cdot \lambda
    \big)
    &\mapsto&
    \big(
      v
      \,\mapsto\,
      \overline{\phi(v) \cdot \lambda}
    \big)
    \\
    & = &
    \big(
      v
      \,\mapsto\,
      \overline{\lambda} \cdot \overline{\phi(v)}
    \big)
    \mathrlap{\,.}
  }
$$

## Related concepts

* [[hyperbolic functor]]

## References

Discussion in the context of [[Hermitian K-theory]]:

* {#KaroubiVillamayor73} [[Max Karoubi]], [[Orlando Villamayor]], Ex. 1 in: *K-théorie algébrique et K-théorie topologique II*, Math. Scand. **32** (1973) 57-86 [[jstor:24490565](https://www.jstor.org/stable/24490565)]

* {#Mishchenko76} [[Alexandr S. Mishchenko]], §1.1 (pp. 76) in: *Hermitian K-Theory. The Theory of characteristic classes and methods of functional analysis*, Uspeki Mat. Nauk **31** 2 (1976) 69-134,  Russ. Math. Surv. **31** 71 (1976) [[doi:10.1070/RM1976v031n02ABEH001478](https://iopscience.iop.org/article/10.1070/RM1976v031n02ABEH001478), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mishch5.pdf)]

* {#Karoubi10} [[Max Karoubi]], §1 in: *Le théorème de périodicité en K-théorie hermitienne*, Quanta of Maths **1**, AMS and Clay Math Institute Publications (2010) [[arXiv:0810.4707](https://arxiv.org/abs/0810.4707)]

See also:

* Wikipedia, *[Anti-dual space](https://en.wikipedia.org/wiki/Antilinear_map#Anti-dual_space)*

[[!redirects anti-dual linear spaces]]

[[!redirects anti-dual vector space]]
[[!redirects anti-dual vector spaces]]


anti-ghost field

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

#Contents#
* table of contents
{:toc}

## Idea

In [[BV-BRST formalism]], for [[gauge fixing]] [[Yang-Mills theory]] (to [[Lorenz gauge]] or similar) a [[contractible chain complex]] of [[auxiliary field|auxiliary]] [[field bundles]] is introduced for two [[Lie algebra]]-valued [[field (physics)|fields]], one in degree zero, called the _[[Nakanishi-Lautrup field]]_, usually denoted "$B$" and one in degree -1, called the _antighost field_, usually denoted $\overline{C}$.

Beware that there are also the [[antifields]] of the [[ghost fields]], which technically are hence "anti-ghostfields" as opposed to the Nakanishi-Lautrup "antighost-fields".

Whoever is responsible for this bad terminology should be blamed.

## Related concepts

* [[field (physics)]]

  * [[ghost]],

  * [[antifield]], **antighost**

## References

Review for the case of [[electromagnetism]] and with [[path integral]] terminology is in

* {#Henneaux90} [[Marc Henneaux]], section 9.1 of _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))

while discussion for general [[Yang-Mills theory]] in the context of [[causal perturbation theory]]/[[perturbative algebraic quantum field theory]] is in

* {#Rejzner16} [[Katarzyna Rejzner]], section 7.2 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

[[!redirects antighost]]
[[!redirects antighosts]]

[[!redirects anti-ghost fields]]

[[!redirects antighost field]]
[[!redirects antighost fields]]


anti-ideal predicate

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

\tableofcontents

## Idea

In [[material set theory]], an [[anti-ideal]] of a [[commutative ring]] $R$ with an [[apartness relation]] $\#$ is an $\#$-[[open subset]] $I \subseteq R$ which satisfies

* $\neg(0 \in I)$
* for all $x \in R$ and $y \in R$, $x + y \in I$ implies that $x \in I$ or $y \in I$
* for all $x \in R$ and $y \in R$, $x \in I$ and $y \in I$ implies that $x \cdot y \in I$

In [[dependently sorted set theory]], where membership $x \in S$ is not a [[relation]], the above statement that $x \in I$ for every element $x \in R$ in material set theory is equivalently a predicate in the logic $x \in R \vdash P_I(x)$. The given [[anti-ideal]] $I$ is then defined by [[restricted separation]] as the set $I \coloneqq \{x \in R \vert P_I(x)\}$, which in structural set theory automatically comes with an [[injection]]
$$i:\{x \in R \vert P_I(x)\} \hookrightarrow R$$
such that
$$\exists y \in \{x \in R \vert P_I(x)\}.x = i(y) \iff P(x)$$
Hence, the notion of [[anti-ideal]] predicate, a formulation of the notion of anti-ideal as a predicate rather than a subset.

## Definition

Given a commutative ring $R$ with an [[apartness relation]] $\#$, an **anti-ideal predicate** on $R$ is an $\#$-[[apartness-open predicate|open predicate]] $x \in R \vdash P_I(x)$ which satisfies

* $\neg P_I(0)$
* for all $x \in R$ and $y \in R$, $P_I(x + y)$ implies that $P_I(x)$ or $P_I(y)$
* for all $x \in R$ and $y \in R$, $P_I(x)$ and $P_I(y)$ implies that $P_I(x \cdot y)$

The **[[anti-ideal]]** $I$ is then defined by [[restricted separation]] as
$I \coloneqq \{x \in R \vert P_I(x)\}$

## Examples

The various definitions of anti-ideals translate over from material set theory to anti-ideal predicates in dependently sorted structural set theory by replacing $x \in I$ with $P_I(x)$ throughout the definition:

* A **[[proper anti-ideal]] predicate** on a commutative ring $R$ with apartness relation $\#$ is an anti-ideal predicate $P_I$ where $P_I(1)$ is true.
* A **[[anti-prime anti-ideal]] predicate** on a commutative ring $R$ with apartness relation $\#$ is a proper anti-ideal predicate $P_I$ where for all $x \in R$ and $y \in R$, $P_I(x)$ and $P_I(y)$ implies that $P_I(x \cdot y)$.
$$\forall x \in R.\forall y \in R.P_I(x) \wedge P_I(y) \implies P_I(x \cdot y)$$
* A **[[principal anti-ideal]] predicate** on a commutative ring $R$ with apartness relation $\#$ anti-generated by an element $a \in R$ is an ideal predicate $P_I$ where for all $x \in R$ and for all $y \in R$, $P_I(x)$ implies that $x \# a \cdot y$.
$$\forall x \in R.\forall y \in R.P_I(x) \implies x \# a \cdot y$$

## See also

* [[anti-ideal]]
* [[predicate]]
* [[ideal predicate]]
* [[restricted separation]]
* [[antisubalgebra]]

[[!redirects anti-ideal predicate]]
[[!redirects anti-ideal predicates]]


anti-reduced type

[[!redirects anti-reduced object]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesion
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _anti-reduced object_ or simple _infinitesimal type_ is one whose [[reduced object|reduction]] is the point, hence one consisting entirely of "[[infinitesimal object|infinitesimal]] extension", i.e. an [[infinitesimally thickened point]].

## Definition

In the context of [[differential cohesion]], an anti-reduced obect is an [[comodal type]] $X$ for the [[infinitesimal shape modality]] $\Im$

$$
  \Im(X) \simeq \ast
  \,.
$$

## Examples

### Formal moduli problems

In [[homotopy type theory]]/[[higher topos theory]] anti-reduced types are essentially what is also called "[[formal moduli problems]]" (these are typically required to satisfy one more condition besides being anti-reduced, namely being [[cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive]] in the sense of Lurie).

## Related concepts

[[!include cohesion - table]]

[[!redirects anti-reduced obects]]

[[!redirects infinitesimal type]]
[[!redirects infinitesimal types]]


antibracket

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[BV-BRST formalism]] the _antibracket_ is a canonical product operation on an [[associative algebra]] generated by [[field (physics)|fields]] and [[antifields]]. If antifields are regarded as [[vector fields]] on the space of fields, then the antibracket is just the (graded) _[[Schouten bracket]]_.

There are different incarnations of the antibracket associated with different incarnations of the algebra of fields/antifields:

Applied to [[horizontal differential forms]] on the [[jet bundle]] of the [[field bundle]] this refines to the _[[local antibracket]]_.

By [[transgression of variational differential forms]] this yields the ("global") antibracket on [[polynomial observables]] of a [[Lagrangian field theory]].

For details see at _[[A first idea of quantum field theory]]_ the chapter _[Reduced phase space](A+first+idea+of+quantum+field+theory#ReducedPhaseSpace)_ for the antibracket before [[quantization]], and the chapter _[Free quantum fields](A+first+idea+of+quantum+field+theory#FreeQuantumFields)_ for the ([[time-ordered product|time-ordered]]) antibracket after quantization.

The global [[antibracket]] is closely related to the _[[BV-operator]]_. See there fore more.

## Related concepts

*  [[BV-operator]]

## References

Review includes

* {#Henneaux90} [[Marc Henneaux]], section 7.3 of _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))

* {#HenneauxTeitelboim92} [[Marc Henneaux]], [[Claudio Teitelboim]], section 15.5.2 of _[[Quantization of Gauge Systems]]_, Princeton University Press 1992.

* {#GomisParisSamuel94} [[Joaquim Gomis]], Jordi Paris, Stuart Samuel, section 4.2 of _Antibracket, Antifields and Gauge-Theory Quantization_, Phys. Rept. 259 (1995) 1-145 ([arXiv:hep-th/9412228](https://arxiv.org/abs/hep-th/9412228))

* {#CattaneoRossi01} [[Alberto Cattaneo]], Carlo Rossi, section 4.2 of _Higher-dimensional BF theories in the Batalin-Vilkovisky formalism: The BV action and generalized Wilson loops_, Commun.Math.Phys. 221 (2001) 591-657 ([arXiv:0010172](https://arxiv.org/abs/math/0010172))

[[!redirects antibrackets]]


antichain

#Contents#
* table of contents
{:toc}

In a [[preorder]] or [[poset]] $P$, an **antichain** is a subset $S \subseteq P$ such that no two distinct elements of $S$ are comparable.

Assuming $P$ has a bottom element $0$, a *strong antichain* is a subset $S \subseteq P$ such that for distinct $a, b \in S$, the only lower bound of $\{a, b\}$ is $0$. This definition may be extended to posets $P$ *without* a bottom element, by declaring $A \subseteq P$ to be a strong antichain if $A$ is a strong antichain in $P^+$, the poset formed by freely adjoining a bottom element to $P$.

In the context of [[set theory]], for example in discussions of [[forcing]] and [[countable chain conditions]], "strong antichain" is often abbreviated to just "antichain".

[[!redirects antichains]]


[[!redirects Anticommutative graded codifferential category]]
## Idea

This is a notion in development which is meant to link [[differential categories]] with [[homological algebra]].

For example, the category of modules over a [[commutative ring]], equipped with the [[exterior powers]] is an anticommutative graded codifferential category.

It is hoped for example that the concept of [[differential algebra]] can be studied in this context.

## Related concepts

* [[differential category]]
* [[Koszul complex]]


antifield

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

#Content#
* table of contents
{:toc}

## Idea

In a [[BV-complex]] the dual of an element under the [[antibracket]] is called its  _antifield_.

(There is _no_ relation to [[antiparticle]].)

## Example

In the standard example of the [[BV-complex]] of [[multivector fields]] over a [[smooth manifold]] with [[volume form]], the antifields of the functions on the manifolds are the [[vector fields]].

## Related concepts

[[BV-BRST formalism]]

* [[antibracket]]

* [[field (physics)]]

  * [[ghost]],

  * **antifield**, [[antighost]]

[[!redirects antifields]]


antihomomorphism

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

# Anti-Homomorphisms
* table of contents
{: toc}

## Idea

While a [[homomorphism]] of [[magmas]] (including [[groups]], [[rings]], etc) must preserve multiplication, an _antihomomorphism_ must instead _reverse_ multiplication.

## Definitions

Let $A$ and $B$ be [[magmas]], or more generally [[magma objects]] [[internalisation|in]] any [[symmetric monoidal category]] $C$.  (Examples include [[groups]], which are magmas with extra properties; [[rings]], which are magma objects in [[Ab]] with extra proprties; etc.)

An __antihomomorphism__ from $A$ to $B$ is a [[homomorphism]] $f\colon A \to B^\op$ where $B^\op$ is the [[opposite magma]] of $B$, or equivalently, it is a [[function]] (or $C$-morphism) $f\colon A \to B$ such that:

*  for every two ([[generalised element|generalised]]) elements $x, y$ of $A$, $f(x y) = f(y) f(x)$.

Note that for magma objects in $C$, the left-hand side of this equation is a generalised element of $B$ whose source is ${|x|} \otimes {|y|}$ (where ${|x|}$ and ${|y|}$ are the sources of the generalised elements $x$ and $y$ and $\otimes$ is the [[tensor product]] in $C$), while the right-hand side is a generalised element of $B$ whose source is ${|y|} \otimes {|x|}$.  Therefore, this definition only makes unambiguous sense because $C$ is symmetric monoidal, using the unique [[natural isomorphism]] ${|x|} \otimes {|y|} \cong {|y|} \otimes {|x|}$.

An __antiautomorphism__ is an antihomomorphism whose underlying $C$-morphism is an [[automorphism]].

## Examples

* The [[inverse function]] of a [[group]] is a [[monoid]] anti-homomorphism, and in fact an anti-automorphism, hence an [[anti-involution]], since $(x y) (x y)^{-1} = 1 = x x^{-1} = x y y^{-1} x^{-1}$, which means that $(x y)^{-1} = y^{-1} x^{-1}$.

* The [[antipode]] in a [[Hopf algebra]] is an anti-homomorphism (by [this Prop.](Hopf+algebra#AntipodeIsAnAntihomomorphism)). The same is separately required for antipodes on associative [[bialgebroid]]s.

* In a [[star-algebra]] the star-operation is an anti-homomorphism, in fact an anti-automorphism, hence an anti-[[involution]].

* Combining these two examples, in an [[involutive Hopf algebra]] the [[antipode]] is an anti-automorphism.

## Related concepts

* [[opposite magma]]

[[!redirects antihomomorphism]]
[[!redirects antihomomorphisms]]
[[!redirects anti-homomorphism]]
[[!redirects anti-homomorphisms]]

[[!redirects antiautomorphism]]
[[!redirects antiautomorphisms]]
[[!redirects anti-automorphism]]
[[!redirects anti-automorphisms]]


antilinear map

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

#Contents#
* table of contents
{:toc}

## Idea

An *antilinear map* or *conjugate linear map* is much like a [[linear map]], but instead of commuting with "[[scalar]] [[multiplication]]" it "anti-commutes" with it, in that multiplication by a scalar $c$ is mapped to multiplication by the scalar's [[star-conjugation|conjugate]] $\overline{c}$.

In order to make sense of this notion, the [[ground ring]] of the [[modules]] (or [[ground field]] of the [[vector spaces]]) that serve as the map's [[source|domain]] and [[codomain]] must have the additional structure of an [[involution]], to serve as the [[star-conjugation|conjugation]] map $c \mapsto \overline{c}$.

An antilinear map has a central role in the concept of [[star-algebra]]. Conversely, an antilinear map can be seen as built on a star-algebra, in that the [[involution]] makes the [[ground ring]] into a [[star-algebra]] over itself.

## Definition

Given a [[commutative ring]] (often a [[field]], or possibly just a [[rig]]) $K$, equipped with an [[involution]] $x \mapsto \overline{x}$, meaning an [[endomorphism]] with $\overline{\overline{x}} = x$ for all $x \in K$.

Then for $K$-[[modules]] (or $K$-[[linear spaces]]) $V, W$, a __$K$-antilinear map__ is a function $T \colon V \to W$ such that for all $x, y \in V$ and $r \in K$,

$$
  T(r \cdot x + y)
  \;=\;
  \overline{r} \cdot T(x) + T(y)
  \,.
$$

This differs from the definition of a [[linear map]] in the appearance of $\overline{(-)}$ on the right-hand side.

## Examples

### Simple general examples

Every $K$-linear map is also a $K$-antilinear map, for $K$ regarded as equipped with the [[identity function|identity]] [[involution]].

Any involution $\overline{(-)} \colon K \to K$ is itself an antilinear map.

### Complex vector spaces

A motivating class of examples is when $K = \mathbb{C}$ is the [[complex numbers]], and $\overline{(-)}$ is [[complex conjugation]].

In particular, the [[Hermitian adjoint]] is an antilinear map from a space of $\mathbb{C}$-linear operators to itself.

### Further examples

A [[star-algebra|$*$-algebra]] requires by definition its [[anti-involution]] to be antilinear.

## Related concepts

* [[real structure]]

* [[anti-dual linear space]]

* [[antiunitary operator]]

* [[star-algebra]]

## References

See also

* Wikipedia, *[Antilinear map](https://en.wikipedia.org/wiki/Antilinear_map)*

and see also the references at *[[Wigner's theorem]]*.

[[!redirects antilinear maps]]

[[!redirects anti-linear map]]
[[!redirects anti-linear maps]]

[[!redirects antilinear function]]
[[!redirects antilinear functions]]

[[!redirects conjugate linear map]]
[[!redirects conjugate linear maps]]

[[!redirects conjugate linear function]]
[[!redirects conjugate linear function]]


antiparticle

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

#Contents#
* table of contents
{:toc}

## Idea

If [[particles]] species are identified [[simple objects]] of a [[DHR category]], then the elements of the corresponding [[dual object]] are called **antiparticles**.

Similarly, in the [[topological quantum field theory | (2+1)-TQFT]] setting isomorphism classes of quasiparticles will correspond to simple objects of a [[modular tensor category]]. The anti-quasiparticle of a quasiparticle is its dual, under the [[rigid category | rigidity]] structure of your category.

## Examples

In the [[standard model of particle physics]], every [[fundamental particle | particle]] has an antiparticle. For example, the antiparticle of the [[electron]] is the [[positron]].

In the realm of (2+1)-TQFTs, the most simple example is the [[quantum double anyon model]] applied to an abelian group. For every finite abelian group $G$ such a model can be created, and quasiparticles can be explicitly described. They correspond to pairs $(g,\chi)$ where $g\in G$ is a group element and $\chi\in \widehat{G}$ is a [[character]]. The antiparticle is $(g^{-1},\chi^{-1})$, where both the inverses are taken with respect to the group operation.

Just like inverses play a key role in the theory of finite groups, antiparticles play a key role in the theory of (2+1)-TQFTs and modular tensor categories.

## Related concepts

* [[dilepton]]

* [[CPT theorem]]

* [[anti-D-brane]]

* [[topological quantum field theory]]

* [[rigid category]]

## References

A discussion of traditional and of formalized discussions of antimatter, with an eye towards [[AQFT]], is in

* David Baker, [[Hans Halvorson]], _Antimatter_  ([pdf](http://philsci-archive.pitt.edu/4467/1/Antimatter.pdf))

In

* [[Hans Halvorson]], _Algebraic quantum field theory_ ([pdf](http://www.princeton.edu/~hhalvors/aqft.pdf))

the topic appears around remark 8.79.

See also

* R. Ascoli, G. Teppati und S. Termini, _Some remarks about particle-antiparticle superselection rules_, Lettere Al Nuovo Cimento (1969 - 1970)
Volume 1, Number 4 (1969), 223-227, DOI: 10.

[[!redirects antiparticles]]

[[!redirects anti-particle]]
[[!redirects anti-particles]]

[[!redirects antimatter]]


antipode

* **In [[topology]]:**

  Regarding the [[n-sphere]] as the [[unit sphere]] inside [[Cartesian space]], $S^n \,\simeq\, S(\mathbb{R}^{n+1})$, the *antipode* of any point $p \,\in\, S^n \hookrightarrow \mathbb{R}^{n=1}$ is the point $-p \,\in\, S^n \hookrightarrow \mathbb{R}^{n+1}$ obtained by sending the [[coordinates]] of $p$ to their negatives.

  The [[quotient space]] of the [[n-sphere]] by the [[cyclic group of order two|$\mathbb{Z}/2$]]-[[group action|action]] given by switching antipodes is the [[real projective space]] $\mathbb{R}P^{n-1}$.

  See also *[[equator]]*, *[[hemisphere]]*, *[[meridian]]*.

* **In [[algebra]]:**

  The *antipode* of a *[[Hopf algebra]]* is the [[formal duality|formal dual]] of the operation of passing to [[inverse elements]] in a [[group]].

  If the Hopf algebra is the [[Pontrjagin ring]]-structure on the [[homology of loop spaces|homology of loop spaces]], then this antipode-operation corresponds to reversal of orientation of loops.

## Literature

See also

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

category: disambiguation

[[!redirects antipodes]]


antisubalgebra

# Antisubalgebras
* table of contents
{: toc}

## Idea

In [[constructive mathematics]], we often do [[algebra]] by equipping an [[gebra|algebra]] with a [[tight apartness]] (and requiring the algebraic operations to be [[strongly extensional function|strongly extensional]]).  In this context, it is convenient to replace [[subalgebras]] with _anti_-subalgebras, which [[classical mathematics|classically]] are simply the [[complements]] of subalgebras.

## Definitions

Let us work in the context of [[universal algebra]], so an __algebra__ is a [[set]] $X$ equipped with a family of [[functions]] $f_i\colon X^{n_i} \to X$ (where each [[arity]] $n_i$ is a [[cardinal number]]) that satisfy certain equational identities (which are irrelevant here).  As usual, a __subalgebra__ of $X$ is a [[subset]] $S$ such that $f_i(p_1,\ldots,p_{n_i}) \in S$ whenever each $p_k \in S$.  There is no need, in general, to require that any arity $n_i$ be finite or that there be finitely many $f_i$; however, for a few results, we will need a special case of these that we will call having __well-behaved constants__:

* each arity is either $0$ or at least $1$ (so each operation either is a constant or has at least one operand), and
* the number of constants is [[Kuratowski-finite]] (so there is an exhaustive list $c_1, c_2, c_3, \ldots c_n$ of constants for some [[natural number]] $n$, where it remains possible that $c_i = c_j$ might be an equational law).

The first item is true of all derived operations in the theory as long as it is true of the fundamental operations in the signature; but in this last item, we\'re counting all derived constants, not just the fundamental ones.  For example, the theory of (unital) [[rings]] does *not* have well-behaved constants, because there are infinitely many constants (one for each [[integer]]).

Now we require $S$ to have a [[tight apartness]] $\ne$, which induces a tight apartness on each $X^{n_i}$ (via [[existential quantification]]), and we require the operations $f_i$ to be [[strongly extensional function|strongly extensional]].  An algebra $X$ with these properties is called an **inequality algebra**.  (For much of the theory we don't need the apartness to be tight, but for some purposes it is necessary.)

A [[subset]] $A$ of $X$ is __[[open subset|open]]__ (or $\ne$-open) if, whenever $p \in A$, $q \in A$ or $p \ne q$.  An __antisubalgebra__ of $X$ is an open subset $A$ such that $p_j \in A$ for some $j$ whenever $f_i(p_1,\ldots,p_{n_i}) \in A$ for any $i$.  By taking the [[contrapositive]], we see that the [[complement]] of $A$ is a subalgebra $S$, but we cannot (in general) start with a subalgebra $S$ and get an antisubalgebra $A$.  (Impredicatively, we can take the antisubalgebra generated, as described below, by the $\ne$-complement of $S$, that is the set of those elements of $X$ that $\ne$ every element of $S$, but its complement will generally only be a superset of $S$.)

## Examples

Unless otherwise noted, all of the constructions in these examples should be [[predicative mathematics|predicative]].

The [[empty subset]] of any algebra is an antisubalgebra, the __empty antisubalgebra__ or __improper antisubalgebra__, whose complement is the [[improper subset|improper]] subalgebra (which is all of $X$).  An antisubalgebra is __[[proper subset|proper]]__ if it is [[inhabited subset|inhabited]]; the ability to have a positive definition of when an antisubalgebra is proper is a significant motivation for the concept.

If $A$ is an antisubalgebra and $c$ is a constant (given by an operation $X^0 \to X$ or a composite of same with other operations), then $p \ne c$ whenever $p \in A$.  If the theory has well-behaved constants, then we can define the __trivial antisubalgebra__ to be the subset of those elements $p$ such that $p \ne c$ for each constant $c$ (the $\ne$-complement of the [[trivial subalgebra]]).  In general, we may also take the __trivial antisubalgebra__ to be the [[union]] of all antisubalgebras (but this is not predicative).

Instead of [[subgroups]], use antisubgroups.  In this case the definition can be simplified a bit: a subset $A$ of an inequality group $X$ is an __antisubgroup__ if $p \ne 1 $ whenever $p \in A$, $p \in A$ or $q \in A$ whenever $p q \in A$, and $p \in A$ whenever $p^{-1} \in A$.  We need not assume that $A$ is open; this can be proved from strong extensionality of the group operations on $X$ and the stronger form of the nullary anticlosure condition ("$p \ne 1 $ whenever $p \in A$" is a strengthening of the condition $\neg (1\in A)$ that would be the literal nullary case of the general definition.)  An antisubgroup $A$ is __[[normal subgroup|normal]]__ if $p q \in A$ whenever $q p \in A$.  The __[[trivial subgroup|trivial]] antisubgroup__ is the $\ne$-complement of $\{1\}$.

Instead of [[ideals]] (of [[rings]]), use antiideals.  (Technically, these are antisubalgebras of the ring as a module over itself.)  Again we can omit $\ne$-openness by strengthening the nullary condition.  In detail, a subset $A$ of $X$ is a __two-sided antiideal__ (or simply an __antiideal__ in the commutative case) if $p \ne 0 $ whenever $p \in A$, $p \in A$ or $q \in A$ whenever $p + q \in A$, and $p \in A$ and $q \in A$ whenever $p q \in A$.  $A$ is a __left antiideal__ if instead the last condition requires only that $p \in A$, and $A$ is a __right antiideal__ if instead the last condition requires only that $q \in A$.  It follows that an antiideal $A$ is proper iff $1 \in A$.  $A$ is __[[prime ideal|prime]]__ (or _antiprime_) if it is proper and $p r q \in A$ for some $r$ whenever $p \in A$ and $q \in A$; in the commutative case, we can say that $p q \in A$ whenever $p \in A$ and $q \in a$.  $A$ is __[[maximal ideal|minimal]]__ (or _antimaximal_) if it is proper and, for each $p \in A$, for some $q$, for each $r \in A$, $p q + r \ne 1$ and $q p + r \ne 1$ (which is constructively stronger than being prime and [[minimal element|minimal]] among proper ideals); of course, we only need one of these two inequalities in the commutative case.  The __[[trivial ideal|trivial]] antiideal__ is the $\ne$-complement of $\{0\}$.

Note that a [[union]] of antisubalgebras is again an antisubalgebra.  Given any subset $B$ of $X$, the antisubalgebra __generated__ by $B$ is the union of all antisubalgebras contained in $B$.  (This construction is not predicative, although it may still be true predicatively that the generated subalgebra exists in some situations.)

In some cases, we may prefer to anti-generate: given any subset $B$ of $X$, the antisubalgebra __antigenerated__ by $B$ is the union of all antisubalgebras whose elements are all distinct from ($\ne$) each element of $B$, in other words the antisubalgebra generated by the $\ne$-complement of $B$.  For example, the trivial antisubgroup of a group is antigenerated by $\{1\}$, and the trivial antiideal of a ring is antigenerated by $\{0\}$.  More generally than these examples, we may talk of the __[[cyclic subgroup|cyclic antisubgroup]]__ or __[[principal ideal|principal antiideal]]__ antigenerated by a given element of the group or ring.

## Quotient algebras

To form a [[quotient group]] or a [[quotient ring]], it\'s enough to have a [[normal subgroup]] or a [[two-sided ideal]].  However, if we want the quotient algebra to inherit an apartness from the original algebra, then we need antisubgroups and antiideals.

In general, instead of [[congruence relations]], use anticongruence relations.  An __anticongruence relation__ $K$ on $X$ is an [[apartness relation]] on $X$ that is also an antisubalgebra of $X \times X$.  Given this, let $R$ be the [[negation]] of $K$; then $R$ is a congruence relation, giving a quotient algebra $X/R$.  Furthermore, $K$ becomes a [[tight relation|tight]] apartness on $X/R$, relative to which the algebra operations on $X/R$ are strongly extensional.  We denote the resulting algebra-with-apartness by $X/K$.  (This notation should cause no confusion; if an apartness relation on a set $X$ is also an equivalence relation, then $X$ must be the [[empty set]], which has a unique apartness and at most one algebra structure, and the only [[quotient set]] of the empty set is itself.)  The quotient map $X \twoheadrightarrow X/K$ is also strongly extensional.

Conversely, any strongly extensional map $f\colon X \to Y$ between algebras with apartness gives rise to an anticongruence $\aker f$ on $X$ (the __antikernel__ of $f$), where $(p, q) \in \aker f$ iff $f(p) \ne f(q)$.  The complement of the antikernel is (because the apartness of $Y$ is tight) the [[kernel]] in the usual sense of universal algebra.  Thus, the quotient algebra $X/(\aker f)$ is [[naturally isomorphic]] to a [[subalgebra]] $im f$ of $Y$; the maps $X \twoheadrightarrow X/(\aker f) \cong \im f \hookrightarrow Y$ are strongly extensional.  Similarly, a sequence $X \overset{f}\to Y \overset{g} \to Z$ is [[exact sequence|exact]] iff $\im f$ is the complement of $\aker g$.

(We would like to say that there is an antisubalgebra $\aim f$ of $Y$ whose complement is $\im f$; then we could, for example, define a stronger notion of exactness requiring that $\aker g$ equal the antiimage of $f$.  In principle, $\aim f$ should be the $\ne$-complement of $\im f$.  If $X$ is Kuratowski-finite, then this works, but in general, we can prove neither that this is open nor that its complement is all of $\im f$.)

Given a group-with-apartness and a normal antisubgroup $A$, we define an anticongruence $K$, where $(p, q) \in K$ iff $p q^{-1} \in A$.  Similarly, given a ring-with-apartness and a two-sided antiideal $A$, we define an anticongruence $K$, where $(p, q) \in K$ iff $p - q \in A$.  This allows us to form quotient groups or quotient rings by modding out by normal antisubgroups or two-sided antiideals.  Conversely, we can interpret the antikernel as a normal antisubgroup or two-sided antiideal: $p \in \aker f$ iff $f(p) \ne 1$, $p \in \aker f$ iff $f(p) \ne 0$, etc.  In general, this works for any [[Omega-group]] structure.

## Localic point of view

As noted at [[apartness relation]], an apartness relation on a set $X$ is equivalent to a (strongly) closed [[equivalence relation]] on the corresponding [[discrete locale]], and the $\ne$-open subsets are those whose complementary closed sublocales are stable under this equivalence relation, and the $\ne$-topology itself is the corresponding quotient locale.  From this point of view, an algebra structure is strongly extensional if it respects the equivalence relation, hence passes to the quotient; and an antisubalgebra is an $\ne$-open set whose complementary closed sublocale is additionally a localic subalgebra, since the operation $\mathsf{C}$ from open sublocales to closed ones takes arbitrary (not only finite) unions to intersections.

In other words, antisubalgebras of an inequality algebra are equivalent to closed subalgebras of a localic algebra, in the case when the latter is the quotient of a discrete algebra by a closed localic congruence.

## References

According to [Troelstra and van Dalen](#TvD):

> The study of algebraic structures in an intuitionistic setting was undertaken by Heyting ([1941](#Heyting1941))... in full generality, equipped with an apartness relation.  The notion of an antisubstructure, implicit in Heyting's treatment of ideals in polynomial rings, was formulated explicitly by D.S. Scott ([1979](#Scott1979)) (N.B. the first draft of this paper contains a good deal more than the published version).  Ruitenburg ([1982](#Ruitenberg1982Thesis), [1982A](#Ruitenberg1982)) deals with intuitionistic algebra in the spirit of Heyting and Scott.

* [[Arend Heyting]], _Untersuchungen über intuitionistische Algebra_, 1941
  {#Heyting1941}

* [[Dana Scott]], _Identity and existence in intuitionistic logic_, 1979
  {#Scott1979}

* [[Wim Ruitenberg]], _Intuitionistic Algebra_, Ph.D. Thesis, Rijksuniversiteit Utrecht, 1982
  {#Ruitenberg1982Thesis}

* [[Wim Ruitenberg]], _Primality and invertibility of polynomials_, 1982
  {#Ruitenberg1982}

* Surprisingly, antisubalgebras make hardly any appearence in [[Ray Mines]], [[Fred Richman]], [[Wim Ruitenburg]].  _A Course in Constructive Algebra_.  Springer, 1987.

* More can be found in [[Anne Troelstra]] and [[Dirk van Dalen]], _Constructivism in Mathematics_ (volume 2).

[[!redirects antisubalgebra]]
[[!redirects antisubalgebras]]
[[!redirects anti-subalgebra]]
[[!redirects anti-subalgebras]]

[[!redirects empty antisubalgebra]]
[[!redirects empty antisubalgebras]]
[[!redirects empty anti-subalgebra]]
[[!redirects empty anti-subalgebras]]

[[!redirects improper antisubalgebra]]
[[!redirects improper antisubalgebras]]
[[!redirects improper anti-subalgebra]]
[[!redirects improper anti-subalgebras]]

[[!redirects proper antisubalgebra]]
[[!redirects proper antisubalgebras]]
[[!redirects proper anti-subalgebra]]
[[!redirects proper anti-subalgebras]]

[[!redirects antisubgroup]]
[[!redirects antisubgroups]]
[[!redirects anti-subgroup]]
[[!redirects anti-subgroups]]

[[!redirects normal antisubgroup]]
[[!redirects normal antisubgroups]]
[[!redirects normal anti-subgroup]]
[[!redirects normal anti-subgroups]]

[[!redirects trivial antisubgroup]]
[[!redirects trivial antisubgroups]]
[[!redirects trivial anti-subgroup]]
[[!redirects trivial anti-subgroups]]

[[!redirects antiideal]]
[[!redirects antiideals]]
[[!redirects anti-ideal]]
[[!redirects anti-ideals]]

[[!redirects left antiideal]]
[[!redirects left antiideals]]
[[!redirects left anti-ideal]]
[[!redirects left anti-ideals]]

[[!redirects right antiideal]]
[[!redirects right antiideals]]
[[!redirects right anti-ideal]]
[[!redirects right anti-ideals]]

[[!redirects two-sided antiideal]]
[[!redirects two-sided antiideals]]
[[!redirects two-sided anti-ideal]]
[[!redirects two-sided anti-ideals]]

[[!redirects prime antiideal]]
[[!redirects prime antiideals]]
[[!redirects prime anti-ideal]]
[[!redirects prime anti-ideals]]

[[!redirects minimal antiideal]]
[[!redirects minimal antiideals]]
[[!redirects minimal anti-ideal]]
[[!redirects minimal anti-ideals]]

[[!redirects trivial antiideal]]
[[!redirects trivial antiideals]]
[[!redirects trivial anti-ideal]]
[[!redirects trivial anti-ideals]]

[[!redirects generated antisubalgebra]]
[[!redirects generated antisubalgebras]]
[[!redirects antisubalgebra generated]]
[[!redirects antisubalgebras generated]]
[[!redirects generated anti-subalgebra]]
[[!redirects generated anti-subalgebras]]
[[!redirects anti-subalgebra generated]]
[[!redirects anti-subalgebras generated]]

[[!redirects generated antisubgroup]]
[[!redirects generated antisubgroups]]
[[!redirects antisubgroup generated]]
[[!redirects antisubgroups generated]]
[[!redirects generated anti-subgroup]]
[[!redirects generated anti-subgroups]]
[[!redirects anti-subgroup generated]]
[[!redirects anti-subgroups generated]]

[[!redirects generated antiideal]]
[[!redirects generated antiideals]]
[[!redirects antiideal generated]]
[[!redirects antiideals generated]]
[[!redirects generated anti-ideal]]
[[!redirects generated anti-ideals]]
[[!redirects anti-ideal generated]]
[[!redirects anti-ideals generated]]

[[!redirects antigenerated antisubalgebra]]
[[!redirects antigenerated antisubalgebras]]
[[!redirects antisubalgebra antigenerated]]
[[!redirects antisubalgebras antigenerated]]
[[!redirects anti-generated anti-subalgebra]]
[[!redirects anti-generated anti-subalgebras]]
[[!redirects anti-subalgebra anti-generated]]
[[!redirects anti-subalgebras anti-generated]]

[[!redirects antigenerated antisubgroup]]
[[!redirects antigenerated antisubgroups]]
[[!redirects antisubgroup antigenerated]]
[[!redirects antisubgroups antigenerated]]
[[!redirects anti-generated anti-subgroup]]
[[!redirects anti-generated anti-subgroups]]
[[!redirects anti-subgroup anti-generated]]
[[!redirects anti-subgroups anti-generated]]

[[!redirects antigenerated antiideal]]
[[!redirects antigenerated antiideals]]
[[!redirects antiideal antigenerated]]
[[!redirects antiideals antigenerated]]
[[!redirects anti-generated anti-ideal]]
[[!redirects anti-generated anti-ideals]]
[[!redirects anti-ideal anti-generated]]
[[!redirects anti-ideals anti-generated]]

[[!redirects cyclic antisubgroup]]
[[!redirects cyclic antisubgroups]]
[[!redirects cyclic anti-subgroup]]
[[!redirects cyclic anti-subgroups]]

[[!redirects principal antiideal]]
[[!redirects principal antiideals]]
[[!redirects principal anti-ideal]]
[[!redirects principal anti-ideals]]

[[!redirects anticongruence]]
[[!redirects anticongruences]]
[[!redirects anti-congruence]]
[[!redirects anti-congruences]]
[[!redirects anticongruence relation]]
[[!redirects anticongruence relations]]
[[!redirects anti-congruence relation]]
[[!redirects anti-congruence relations]]

[[!redirects antikernel]]
[[!redirects antikernels]]
[[!redirects anti-kernel]]
[[!redirects anti-kernels]]

[[!redirects strongly exact sequence]]
[[!redirects strongly exact sequences]]


antisymmetric relation

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

A (binary) [[relation]] $\sim$ on a set $A$ is __antisymmetric__ if any two elements that are related in both orders are [[equality|equal]]:
$$\forall (x, y: A),\; x \sim y \;\wedge\; y \sim x \;\Rightarrow\; x = y$$

In the language of the $2$-poset-with-duals [[Rel]] of sets and relations, a relation $R: A \to A$ is __antisymmetric__ if its intersection with its reverse is contained in the identity relation on $A$:
$$R \cap R^{op} \subseteq \id_A$$
If an antisymmetric relation is also [[reflexive relation|reflexive]] (as most are in practice), then this containment becomes an equality.

## See also ##

* [[internal antisymmetric relation]]

[[!redirects antisymmetry]]
[[!redirects antisymmetric]]
[[!redirects antisymmetric relations]]


antiunitary operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An *anti-unitary operator* on a [[Hilbert space]] is an [[anti-linear map]] which preserves the [[inner product]]/[[norm]] up to [[complex conjugation]].

## Related concepts

* [[antilinear map]], [[anti-dual linear space]]

* [[time-reversal symmetry]]

* [[Wigner's theorem]]

## References

See also:

* Wikipedia, *[Antiunitary operator](https://en.wikipedia.org/wiki/Antiunitary_operator)*

and see the references at *[[Wigner's theorem]]*.

[[!redirects antiunitary operators]]

[[!redirects anti-unitary operator]]
[[!redirects anti-unitary operators]]


Antoine Allioux

## Selected writings

On combining [[homotopy type theory]] with [[opetopic type theory]]:

* {#AlliouxFinsterSozeau21} [[Antoine Allioux]], [[Eric Finster]], [[Matthieu Sozeau]], _Types are internal infinity-groupoids_, 2021 ([hal:03133144](https://hal.inria.fr/hal-03133144), [pdf](https://hal.inria.fr/hal-03133144/document))

category: people


Antoine Van Proeyen

* [webpage](http://itf.fys.kuleuven.be/~toine/home.htm)

## Selected writings:

Formulation of ([[Lagrangian densities]] for) [[type II supergravity]] with "democratic"/"[[pregeometric RR-field|pregeometric]]" [[RR-fields]] subject to [[self-dual higher gauge theory|self-duality]]:

* [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 [[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)]

On [[supergravity]]:

* [[Antoine Van Proeyen]], [[Daniel Freedman]], _Supergravity_, Cambridge University Press (2012) [[doi:10.1017/CBO9781139026833]( https://doi.org/10.1017/CBO9781139026833)]

On [[D=4 supergravity]], [[D=5 supergravity]], [[D=6 supergravity]]:

* Edoardo Lauria, [[Antoine Van Proeyen]], _$\mathcal{N}=2$ Supergravity in $D=4,5,6$ Dimensions_ ([arXiv:2004.11433](https://arxiv.org/abs/2004.11433))

## Rekated entries

* [[supersymmetry]]

category: people


Anton Alekseev

__Anton Alekseev__ is a mathematician at the [Université de Genève](http://www.unige.ch), working mainly on the interface of mathematical physics, differential geometry and algebra.

* [web](http://www.unige.ch/math/people/alekseev.html)
* [arxiv preprints](http://arxiv.org/find/math/1/au:+Alekseev_A/0/1/0/all/0/1)

category: people


Anton Deitmar

* [webpage](http://www.math.uni-tuebingen.de/user/deitmar/)

## related $n$Lab entries

* [[field with one element]]

* [[algebraic K-theory]], [[K-theory of a permutative category]]

category: people


Anton Geraschenko

[[!redirects Anton Greschenko]]
[[!redirects Anton Greschenko]]

* [webpage](http://stacky.net/wiki/index.php?title=Main_Page)

category: people


Anton Gerasimov

* [webpage](http://www.maths.tcd.ie/people/AntonGerasimov.php)

category: people


Anton Hilado

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

* [blog](https://ahilado.wordpress.com/author/ahilado/)

* [institute page](https://www.uvm.edu/cems/mathstat/profiles/anton-hilado)

## Selected writings

On [[Mochizuki's corollary 3.12]] and [[Szpiro's conjecture]]:

* [[Taylor Dupuy]], [[Anton Hilado]], _Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12_ [[arxiv:2004.13108](https://arxiv.org/abs/2004.13108)]

* [[Taylor Dupuy]], [[Anton Hilado]], _The Statement of Mochizuki's Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies_ [[arxiv:2004.13228](https://arxiv.org/abs/2004.13228)]

## Related pages

* [[inter-universal Teichmüller theory]]

* [[initial Θ-data]]

category: people


Anton Kapustin

__Anton Kapustin__ is a Russian mathematical physicist, now a Professor at Caltech University in USA. His works include the study of noncommutative analogues of ADHM construction, [[homological mirror symmetry]], derived categories of [[coherent sheaves]] on algebraic varieties, Landau-Ginzburg models, TQFT-s and Langlands duality.

* [homepage](http://www.theory.caltech.edu/~kapustin) at Caltech

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

## Selected writings

A popular exposition of formal deformation quantization with emphasis of the [[string theory|stringy]] [[Poisson sigma-model]]-construction:

* [[Anton Kapustin]], _Quantum geometry: The reunion of math and physics_ [[[Kapustin-QuantumGeometry.pdf:file]]]

On [[topological quantum field theory]]:

* [[Sergei Gukov]], [[Anton Kapustin]], _[[Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories]]_ ([arXiv:1307.4793](http://arxiv.org/abs/1307.4793))

On [[Rozansky-Witten theory]] as a [[boundary field theory]]:

* [[Anton Kapustin]], [[Lev Rozansky]], [[Natalia Saulina]],
_Three-dimensional topological field theory and symplectic algebraic geometry I_ ([arXiv:0810.5415](https://arxiv.org/abs/0810.5415))

On the relation between [[Rozansky-Witten theory]] and [[Chern-Simons theory]]:

* [[Anton Kapustin]], [[Natalia Saulina]], _Chern-Simons-Rozansky-Witten topological field theory_, Nucl. Phys. B823 (2009) 403-427 ([arXiv:0904.1447](https://arxiv.org/abs/0904.1447), [spire:817599/](http://inspirehep.net/record/817599/))

On [[generalized global symmetries]]:

* {#GKSW14} [[Davide Gaiotto]], [[Anton Kapustin]], [[Nathan Seiberg]], [[Brian Willett]], *Generalized Global Symmetries*  J. High Energ. Phys. **2015** 172 (2015) [[arXiv:1412.5148](https://arxiv.org/abs/1412.5148), doi:10.1007/JHEP02(2015)172]

On the [[theta angle]] in [[QCD]]:

* [[Davide Gaiotto]], [[Anton Kapustin]], [[Zohar Komargodski]], [[Nathan Seiberg]], _Theta, Time Reversal, and Temperature_, JHEP05(2017)091 ([arXiv:1703.00501](https://arxiv.org/abs/1703.00501))

category: people


Anton Petrunin

* [webpage](https://www.math.psu.edu/petrunin/)

## related $n$Lab entries

* [[Euclidean geometry]]

* [[synthetic geometry]]

category: people


Anton Rebhan

* [personal page](http://www.itp.tuwien.ac.at/Homepage_Anton_Rebhan)

* [Wikipedia entry](https://de.wikipedia.org/wiki/Anton_Rebhan) (Deutsch)

## Selected writings

On the [[quark-gluon plasma]]:

* {#BlaizotIancuRebhan03} Jean-Paul Blaizot, Edmond Iancu, [[Anton Rebhan]], _Thermodynamics of the high temperature quark gluon plasma_, Quark–Gluon Plasma 3, pp. 60-122 (2004) ([arXiv:hep-ph/0303185](https://arxiv.org/abs/hep-ph/0303185), [spire:615570](http://inspirehep.net/record/615570))

On the [[Witten-Sakai-Sugimoto model]] for [[holographic QCD]]:

* {#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))

On the [[anomalous magnetic moment]] of the [[muon]] via [[holographic QCD]]:

* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the anomalous magnetic moment of the muon_ ([arXiv:1912.01596](https://arxiv.org/abs/1912.01596))

* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the muon $g-2$_ ([arXiv:2012.06514](https://arxiv.org/abs/2012.06514))

* [[Josef Leutgeb]], [[Anton Rebhan]], *Hadronic light-by-light contribution to the muon $g-2$ from holographic QCD with massive pions* ([arXiv:2108.12345](https://arxiv.org/abs/2108.12345))

* [[Josef Leutgeb]], Jonas Mager, [[Anton Rebhan]], *Holographic QCD and the muon anomalous magnetic moment* ([arXiv:2110.07458](https://arxiv.org/abs/2110.07458))

category: people


Anton Setzer

* [webpage](http://www.cs.swan.ac.uk/~csetzer/)

## Selected writings

Introducing the notion of [[inductive-recursive types]]:

* [[Peter Dybjer]], [[Anton Setzer]], *Indexed induction-recursion*, in *Proof Theory in Computer Science* PTCS 2001. Lecture Notes in Computer Science**2183** Springer (2001) [[doi:10.1007/3-540-45504-3_7](https://doi.org/10.1007/3-540-45504-3_7), [pdf](http://www.cse.chalmers.se/~peterd/papers/InductionRecursionInitialAlgebras.pdf)]

## Related entries

* [[inductive-inductive type]]

category: people


Anton Suschkewitsch

Anton Suschkewitsch (Антон Казимирович Сушкевич) was a Russian mathematician working on [[semigroup]] theory.

[Math-Net.Ru page](http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=42597).

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

## Selected writings

* [[Anton Suschkewitsch]], _On a generalization of the associative law_.  Transactions of the American Mathematical Society 31:1 (1929), 204–204.  [doi](http://dx.doi.org/10.1090/s0002-9947-1929-1501476-0).

* [[А. К. Сушкевич]], _Теория обобщенных групп_ (Theory of generalized groups), Государственное научно-техническое издательство Украины, 1937.

[[!redirects Антон Казимирович Сушкевич]]
[[!redirects А. К. Сушкевич]]
[[!redirects А. Сушкевич]]
[[!redirects Anton Sushkevich]]
[[!redirects A. Sushkevich]]
[[!redirects A. K. Sushkevich]]


Anton Zabrodin

* [webpage](https://www.hse.ru/en/org/persons/35436251)

## Selected writings

Suggestion that the [[disk]] [[worldsheet]] of the [[open string|open]] [[p-adic string]] is to be identified with the [[Bruhat-Tits tree]] $T_p$:

* [[Anton Zabrodin]], _Non-Archimedean strings and Bruhat-Tits trees_, Comm. Math. Phys.
Volume 123, Number 3 (1989), 463-483 ([euclid.cmp/1104178891](https://projecteuclid.org/euclid.cmp/1104178891))

category: people


Anton Zeilinger

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

* [research group](https://www.iqoqi-vienna.at/people/staff/anton-zeilinger)

## Selected writings

On [[experiment|experimental]] [[quantum physics]] and its [[epistemology|epistemological]] [[interpretation of quantum mechanics]]:

* [[Daniel Greenberger]], [[Wolfgang L. Reiter]], [[Anton Zeilinger]], *Epistemological and Experimental Perspectives on Quantum Physics*, Vienna Circle Institute Yearbook (VCIY) **7** (1999) [[doi:10.1007/978-94-017-1454-9](https://doi.org/10.1007/978-94-017-1454-9)]

On [[quantum teleportation]] realized in [[experiment]]:

* [[Dirk Bouwmeester]], Jian-Wei Pan, Klaus Mattle, Manfred Eibl, [[Harald Weinfurter]], [[Anton Zeilinger]], *Experimental quantum teleportation*, Nature **390** 575–579 (1997) [[doi:10.1038/37539](https://doi.org/10.1038/37539)]

On the [[quantum physics]] of [[quantum information]]:

* [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]

## Related entries

* [[quantum teleportation]]

category: people


Anton Zeitlin

* [webpage](http://users.math.yale.edu/~az84/)

## Selected writings

On [[Maurer-Cartan equations]] as [[equations of motion]] of [[Yang-Mills theory]] and [[gravity]] (by truncation of [[string field theory]]):

* [[Anton M. Zeitlin]], *Formal Maurer-Cartan Structures: from CFT to Classical Field Equations*, JHEP 0712:098, 2007 ([arXiv:0708.0955](https://arxiv.org/abs/0708.0955))

On [[super Riemann surfaces]] and [[fat graphs]]:

* [[Albert S. Schwarz]], [[Anton M. Zeitlin]], *Super Riemann surfaces and fatgraphs* [[arXiv:2307.02706](https://arxiv.org/abs/2307.02706)]

## Related $n$Lab pages

* [[beta-gamma system]]

* [[vertex operator algebra]]

* [[super Riemann surface]]

category: people

[[!redirects Anton M. Zeitlin]]


Antonella Grassi

* [webpage](https://www.math.upenn.edu/~grassi/)

## related $n$Lab entries

* [[M-theory on G2-manifolds]], [[ADE singularity]]

category: people


Antoni Kosinski

* [Institute page](https://www.math.rutgers.edu/component/comprofiler/userprofile/kosinski?Itemid=774)

* [Mathematics Genealogy Page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=5936)

## Selected writings

On [[differential geometry]] ([[differentiable manifolds]], [[smooth manifolds]]) with applications to [[cobordism theory]] ([[Cohomotopy]], [[Pontryagin-Thom theorem]]):

* {#Kosinski93} [[Antoni Kosinski]], _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf), [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C))

## Related $n$Lab entries

* [[differentiable manifold]], [[smooth manifold]]

* [[cobordism ring]],

* [[cohomotopy]]

* [[Pontrjagin-Thom construction]]

category: people


Antonin Delpeuch

* [personal page](https://antonin.delpeuch.eu/index.html)

* [institute page](https://www.cs.ox.ac.uk/people/antonin.delpeuch/)

## Selected writings

On the [[free construction]] of [[autonomous categories]] (aka [[rigid monoidal categories]]) from plain [[monoidal categories]]:

* {#Delpeuch2020} [[Antonin Delpeuch]], *Autonomization of monoidal categories*, EPTCS 323 (2020) 24-43 [[arXiv:1411.3827](https://arxiv.org/abs/1411.3827), [doi:10.4204/EPTCS.323.3](https://dx.doi.org/10.4204/EPTCS.323.3)]

category: people


Antonio Alarcón

* [webpage](http://www.ugr.es/~alarcon/invest.html)

## Selected writings

On the [[Oka principle]] applied to discussion of [[minimal surfaces]]:

* [[Antonio Alarcon]], [[Franc Forstnerič]], *New complex analytic methods in the theory of minimal surfaces: a survey*,  Journal of the Australian Mathematical Society, 106(3), 287-341 ([arXiv:1711.08024](https://arxiv.org/abs/1711.08024), [doi:10.1017/S1446788718000125](https://doi.org/10.1017/S1446788718000125))

category: people

[[!redirects Antonio Alarcon]]


Antonio Cegarra

Antonio Martínez Cegarra is a Spanish mathematician, who is in the [Departamento de Álgebra](http://www.ugr.es/~algebra/) of the Universidad de Granada, Spain.  He has published extensively in homotopical and homological algebra, and non-abelian cohomology.  A (partial) list of his publications is [here](https://www.ugr.es/~acegarra/Publicaciones.html).

##Related entries

* [[hypercrossed complex]]

* [[Moore complex]]

## Selected writings

On [[cat-n-groups]] and [[homotopy types]]:

* [[Manuel Bullejos]], [[Antonio M. Cegarra]], [[John W. Duskin]], *On cat$^n$ -groups and homotopy types*, J. Pure
Appl. Alg. **86** (1993) 135-154 \[doi:10.1016/0022-4049(93)90099-F]

* [[Pilar Carrasco]] and [[Antonio Cegarra|A. M. Cegarra]], _Group-theoretic Algebraic Models for Homotopy Types_, J. Pure Appl. Alg., 75, (1991), 195 – 235.

On [[non-abelian cohomology]] with [[coefficients]] in [[braided 2-groups]] and [[symmetric 2-groups]]:

* [[Manuel Bullejos]], [[Pilar Carrasco]], [[Antonio M. Cegarra]], *Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem*, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 ([doi:10.1017/S0305004100071498](https://doi.org/10.1017/S0305004100071498))

On "classifying spaces" (ie: [[topological realization]] of [[simplicial nerves]]) of [[2-categories]]:

* [[Manuel Bullejos]], [[Antonio M. Cegarra]],  *On the geometry of 2-categories and their classifying spaces*, K-Theory **29** 3 (2003) 211-229  [[doi:10.1023/B:KTHE.0000006921.50151.00](http://dx.doi.org/10.1023/B:KTHE.0000006921.50151.00), [pdf](http://www.ugr.es/\%7Ebullejos/geometryampl.pdf)]

On the [[diagonal of a bisimplicial set|diagonal]] of [[bisimplicial sets]]:

* [[Antonio Cegarra]], [[Josué Remedios]], _The relationship between the diagonal and the bar constructions on a bisimplicial set_, Topology and its applications, volume 153 (1)  (2005) ([pdf](http://www.ugr.es/~acegarra/Paperspdfs/TRBDWC.pdf), [doi:10.1016/j.topol.2004.12.003](https://doi.org/10.1016/j.topol.2004.12.003))

On [[double groupoids]] and [[homotopy 2-types]]:

* [[Antonio Martínez Cegarra]], Benjamın A. Heredia, [[Josué Remedios]],  _Double groupoids and homotopy 2-types_, Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also [arXiv:1003.3820](http://arxiv.org/abs/1003.3820).

category:people

[[!redirects Antonio M. Cegarra]]
[[!redirects Antonio Martínez Cegarra]]
[[!redirects A. M. Cegarra]]
[[!redirects Cegarra]]


Antonio Gómez-Tato

* [institute page](https://www.usc.gal/en/department/mathematics/directory/antonio-gomez-tato-1899)

## Selected writings

On [[rational homotopy theory]] of non-[[nilpotent spaces]] via [[deck transformation|deck]]-[[Borel-equivariant rational homotopy theory]] of their [[universal cover|universal]] [[covering spaces]]:

* [[Antonio Gómez-Tato]], [[Stephen Halperin]], [[Daniel Tanré]], *Rational homotopy theory for non-simply connected spaces*, Trans. Amer. Math. Soc. 352 (2000), 1493-1525 ([doi:10.1090/S0002-9947-99-02463-0](https://doi.org/10.1090/S0002-9947-99-02463-0), [jstor:118074](https://www.jstor.org/stable/118074))

category: people

[[!redirects Antonio Gomez-Tato]]


Antonio N. Bernal

* [GoogleScholar page](https://scholar.google.com/citations?user=m72DV5EAAAAJ)

* [InSpire page](https://inspirehep.net/authors/2276636)

## Selected writings

On [[spacetime]] [[causality]], [[globally hyperbolic spacetimes]] and [[Cauchy surfaces]]:

* [[Antonio N. Bernal]], [[Miguel Sánchez]], _On smooth Cauchy hypersurfaces and Geroch's splitting theorem_, Commun. Math. Phys. **243** (2003) 461-470 [[arXiv:gr-qc/0306108v2](http://arxiv.org/abs/gr-qc/0306108), [doi:10.1007/s00220-003-0982-6](https://doi.org/10.1007/s00220-003-0982-6)]

* {#BernalSánchez05} [[Antonio N. Bernal]], [[Miguel Sánchez]], *Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes*, Commun. Math. Phys. **257** (2005) 43-50 [[arXiv:gr-qc/0401112](https://arxiv.org/abs/gr-qc/0401112), [doi:10.1007/s00220-005-1346-1](https://doi.org/10.1007/s00220-005-1346-1)]

* [[Antonio N. Bernal]], [[Miguel Sánchez]], *Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions*, Lett. Math. Phys. **77** (2006) 183-197 [[arXiv:gr-qc/0512095](https://arxiv.org/abs/gr-qc/0512095), [doi:10.1007/s11005-006-0091-5](https://doi.org/10.1007/s11005-006-0091-5)]

category: people

[[!redirects Antonio Bernal]]


Antonio Pich

* [webpage](https://webific.ific.uv.es/web/en/content/pich-zardoya-antonio)

## Selected writings

On [[chiral perturbation theory]]:

* [[Gerhard Ecker]], [[Jürg Gasser]], [[Antonio Pich]], E. DeRafael, Sec 3 and Appendix A in: _The role of resonances in chiral perturbation theory_, Nuclear Physics B
Volume 321, Issue 2, 24 July 1989, Pages 311-342 (doi:10.1016/0550-3213(89)90346-5)

On [[kaon]] [[decay]] in relation to the [[chiral anomaly]]:

* [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], _Non-leptonic kaon decays and the chiral anomaly_, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (doi:10.1016/0550-3213(94)90623-8)

On [[electroweak symmetry breaking]] and the [[Higgs boson]]/[[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))

On [[flavour physics]], [[CP violation]] and [[flavour anomalies]]:

* [[Antonio Pich]], _Flavour Dynamics and Violations of the CP Symmetry_, Lectures at the 2017 and 2019 CERN - Latin-American Schools of High-Energy Physics ([arXiv:1805.08597](https://arxiv.org/abs/1805.08597))

On [[flavour anomalies]]:

* [[Antonio Pich]], _Flavour Anomalies_, PoS LHCP2019 (2019) 078 ([arxiv:1911.06211](https://arxiv.org/abs/1911.06211), [spire:1765034](https://inspirehep.net/literature/1765034))

On [[right-handed neutrinos]] in relation to [[flavor anomalies]]:

* Rusa Mandal, Clara Murgui, Ana Peñuelas, [[Antonio Pich]], _The role of right-handed neutrinos in $b \to c \tau \bar \nu$ anomalies_ ([arXiv:2004.06726](https://arxiv.org/abs/2004.06726))

category: people


Antonio R. Garzón

Antonio Garzón is a Spanish mathematician, who is in the [Departamento de Álgebra](http://www.ugr.es/~algebra/) of the Universidad de Granada, Spain.  He has published extensively in [[homotopical algebra|homotopical]] and [[homological algebra]], and [[non-abelian cohomology]].

## Selected writings

On [[homotopy theory]] of [[2-groups]] and [[braided 2-groups]]:

* [[Antonio R. Garzón]], Jesus G Miranda, Sec. 1 of: *Homotopy theory for (braided) cat-groups*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 38 (1997) no. 2, pp. 99-139 ([numdam:CTGDC_1997__38_2_99_0](http://www.numdam.org/item/?id=CTGDC_1997__38_2_99_0))

On [[k-tuply groupal n-groupoids]]:

* [[Antonio R. Garzón]], Jesus G. Miranda, Section 6 of: _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

category:people

[[!redirects A.R. Garzón]]
[[!redirects Antonio Garzon]]
[[!redirects Antonio Garzón]]
[[!redirects A. Garzon]]


Antony Wassermann

* [course notes](https://www.dpmms.cam.ac.uk/~ajw/)

category: people


Antti J. Niemi

* [institute page](https://www.su.se/english/profiles/aniem-1.306055)

* [research group](https://www.su.se/english/research/research-groups/quantum-frontiers?open-collapse-boxes=research-group-members) ([[Frank Wilczek]])

## Selected writings

On [[anyon|anyonic]] [[vortices]] in [[Bose-Einstein condensates]]:

* Julien Garaud, Jin Dai, [[Antti J. Niemi]], *Vortex precession and exchange in a Bose-Einstein condensate*, J. High Energ. Phys. **2021** 157 (2021)
$[$[arXiv:2010.04549](https://arxiv.org/abs/2010.04549)$]$

category: people

[[!redirects Antti Niemi]]


Antti Kupiainen

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

## Selected writings

On the rigorous construction of [[Liouville theory]]

* [[François David]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Liouville Quantum Gravity on the Riemann sphere_, Communications in Mathematical Physics volume 342, pages869–907 (2016) ([arxiv:1410.7318](https://arxiv.org/abs/1410.7318))

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Integrability of Liouville theory: proof of the DOZZ Formula_, Annals of Mathematics, Pages 81-166 from Volume 191 (2020), Issue 1, ([arxiv:1707.08785](https://arxiv.org/abs/1707.08785), [doi:10.4007/annals.2020.191.1.2](https://doi.org/10.4007/annals.2020.191.1.2))

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *The DOZZ formula from the path integral*, Journal of High Energy Physics volume 2018, Article number: 94 (2018) ([arXiv:1803.05418](https://arxiv.org/abs/1803.05418) doi:10.1007/JHEP05(2018)094)

and via the [[conformal bootstrap]]:

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ ([arxiv:2005.11530](https://arxiv.org/abs/2005.11530))

and as a [[functorial field theory]] following [Segal 1988](conformal+field+theory#Segal88):

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *Segal's axioms and bootstrap for Liouville Theory* [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)]

category: people


anyonic braiding in momentum space -- references

### Anyons in momentum-space
 {#ReferencesAnyonicBraidingInMomentumSpace}

On non-trivial [[braid group statistics|braiding]] of nodal points in the [[Brillouin torus]] of [[topological semi-metals]] ("braiding in momentum space"):

* {#AhnParkYang19} Junyeong Ahn, Sungjoon Park, Bohm-Jung Yang, *Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle*, Phys. Rev. X **9** (2019) 021013 $[$[doi:10.1103/PhysRevX.9.021013](https://doi.org/10.1103/PhysRevX.9.021013), [arXiv:1808.05375](https://arxiv.org/abs/1808.05375)$]$

  > "here are band crossing points, henceforth called [[vortices]]"

* QuanSheng Wu, Alexey A. Soluyanov, [[Tomáš Bzdušek]], *Non-Abelian band topology in noninteracting metals*, Science **365** (2019) 1273-1277 $[$[arXiv:1808.07469](https://arxiv.org/abs/1808.07469), [doi:10.1126/science.aau8740](https://doi.org/10.1126/science.aau8740)$]$

  > [[fundamental group]] of *[[complement]]* of nodal points/lines considered above (3)

* {#TiwariBzdusek20} Apoorv Tiwari, [[Tomáš Bzdušek]], *Non-Abelian topology of nodal-line rings in PT-symmetric systems*, Phys. Rev. B **101** (2020) 195130 $[$[doi:10.1103/PhysRevB.101.195130](https://doi.org/10.1103/PhysRevB.101.195130)$]$

> "a new type non-Abelian 'braiding' of nodal-line rings inside the momentum space"

* {#BWSWYB20} [[Adrien Bouhon]], QuanSheng Wu, [[Robert-Jan Slager]], Hongming Weng, Oleg V. Yazyev, [[Tomáš Bzdušek]], *Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe*, Nature Physics **16** (2020)  1137-1143 $[$[arXiv:1907.10611](https://arxiv.org/abs/1907.10611), [doi:10.1038/s41567-020-0967-9](https://doi.org/10.1038/s41567-020-0967-9)$]$

> "Here we report that Weyl points in three-dimensional (3D) systems with $\mathcal{C}_2\mathcal{T}$ symmetry carry non-Abelian topological charges. These charges are transformed via non-trivial phase factors that arise upon braiding the nodes inside the reciprocal momentum space."

{#InTwistedBilayerGraphene} Braiding of Dirac points in twisted bilayer [[graphene]]:

* Jian Kang, Oskar Vafek, *Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene*, Phys. Rev. B **102** (2020) 035161 $[$[arXiv:2002.10360](https://arxiv.org/abs/2002.10360), [doi:10.1103/PhysRevB.102.035161](https://doi.org/10.1103/PhysRevB.102.035161)$]$

* {#JBLZHLSJ21} Bin Jiang, [[Adrien Bouhon]], Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou, Feng Li, [[Robert-Jan Slager]], Jian-Hua Jiang *Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions*, Nature Physics **17** (2021) 1239-1246 $[$[arXiv:2104.13397](https://arxiv.org/abs/2104.13397), [doi:10.1038/s41567-021-01340-x](https://doi.org/10.1038/s41567-021-01340-x)$]$

  (analog realization in [[phononic crystals]])

> Here, we consider an exotic type of topological phases beyond the above paradigms that, instead, depend on topological charge conversion processes when band nodes are braided with respect to each other in momentum space or recombined over the Brillouin zone. The braiding of band nodes is in some sense the reciprocal space analog of the non-Abelian braiding of particles in real space.

> $[$...$]$

> we experimentally observe non-Abelian topological semimetals and their evolutions using acoustic Bloch bands in kagome acoustic metamaterials. By tuning the geometry of the metamaterials, we experimentally confirm the creation, annihilation, moving, merging and splitting of the topological band nodes in multiple bandgaps and the associated non-Abelian topological phase transitions

* {#ParkWongZhangOh21} Haedong Park, Stephan Wong, Xiao Zhang, and Sang Soon Oh, *Non-Abelian Charged Nodal Links in a Dielectric Photonic Crystal*, ACS Photonics **8**
(2021) 2746–2754 [[doi:10.1021/acsphotonics.1c00876](https://doi.org/10.1021/acsphotonics.1c00876)]

* {#CBSM22} Siyu Chen, [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions* (formerly: *Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions*), Phys. Rev. B **105** (2022) L081117 $[$[arXiv:2108.10330](https://arxiv.org/abs/2108.10330), [doi:10.1103/PhysRevB.105.L081117](https://doi.org/10.1103/PhysRevB.105.L081117)$]$

  > "Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding."

* {#PBSM22} [[Bo Peng]], [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Multi-gap topology and non-Abelian braiding of phonons from first principles*, Phys. Rev. B **105** (2022) 085115 ([arXiv:2111.05872](https://arxiv.org/abs/2111.05872), [doi:10.1103/PhysRevB.105.085115](https://doi.org/10.1103/PhysRevB.105.085115))

  (analog realization in [[phononic crystals]])

> new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies.

> Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding.

* {#PBMS22} [[Bo Peng]], [[Adrien Bouhon]], [[Bartomeu Monserrat]], [[Robert-Jan Slager]], *Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates*, Nature Communications **13** 423 (2022) $[$[doi:10.1038/s41467-022-28046-9](https://doi.org/10.1038/s41467-022-28046-9)$]$

  (analog realization in [[phononic crystals]])

  > it is possible to controllably braid Kagome band nodes in monolayer $\mathrm{Si}_2 \mathrm{O}_3$ using strain and/or an external electric field.

* {#ParkGaoZhangOh22} Haedong Park, Wenlong Gao, Xiao Zhang, Sang Soon Oh, *Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems*, Nanophotonics **11** 11  (2022) 2779–2801 $[$[doi:10.1515/nanoph-2021-0692](https://doi.org/10.1515/nanoph-2021-0692)$]$

  (analog realization in [[photonic crystals]])

* [[Adrien Bouhon]], [[Robert-Jan Slager]], *Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs* $[$[arXiv:2203.16741](https://arxiv.org/abs/2203.16741)$]$

See also:

* [[Robert-Jan Slager]], [[Adrien Bouhon]], [[Fatma Nur Ünal]], *Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase* $[$[arXiv:2208.12824](https://arxiv.org/abs/2208.12824)$]$

* Huahui Qiu et al., *Minimal non-abelian nodal braiding in ideal metamaterials*, Nature Communications **14** 1261 (2023) $[$[doi:10.1038/s41467-023-36952-9](https://doi.org/10.1038/s41467-023-36952-9)$]$

* Wojciech J. Jankowski, Mohammedreza Noormandipour, [[Adrien Bouhon]], [[Robert-Jan Slager]], *Disorder-induced topological quantum phase transitions in Euler semimetals* $[$[arXiv:2306.13084](https://arxiv.org/abs/2306.13084)$]$

Incidentally, references indicating that the required toroidal (or yet higher genus) geometry for anyonic topological order in position space is dubious (as opposed to the evident toroidal geometry of the momentum-space [[Brillouin torus]]): [Lan 19, p. 1](Laughlin+state#Lan19), ....

**Knotted nodal lines in 3d semimetals**

Beware that various authors consider [[braids]]/[[knots]] formed by nodal *lines* in *3d* semimetals, i.e. knotted nodal lines in 3 spatial dimensions, as opposed to [[worldlines]] (in 2+1 spacetime dimensions) of nodal points in effectively 2d semimetals needed for the [[anyon]]-[[braiding]] considered above.

An argument that these nodal lines in 3d space, nevertheless, may be  controlled by [[Chern-Simons theory]]:

* {#LianVafaVafaZhang17} [[Biao Lian]], [[Cumrun Vafa]], [[Farzan Vafa]], [[Shou-Cheng Zhang]], *Chern-Simons theory and Wilson loops in the Brillouin zone*, Phys. Rev. B **95** (2017) 094512 [[doi:10.1103/PhysRevB.95.094512](https://doi.org/10.1103/PhysRevB.95.094512)]


anyonic topological order via braided fusion categories -- references

### Anyonic topological order in terms of braided fusion categories
 {#AnyonicTopologicalOrderInTermsOfBraidedFusionCategoriesReferences}

#### Claim and status

In [[condensed matter theory]] it is [[folklore]] that species of [[anyon|anyonic]] [[topological order]] correspond to [[braided monoidal categories|braided]] [[unitary fusion category|unitary]] [[fusion categories]]/[[modular tensor categories]].

The origin of the claim is:

* {#Kitaev06} [[Alexei Kitaev]], Section 8 and Appendix E of: *Anyons in an exactly solved model and beyond*, Annals of Physics **321** 1 (2006) 2-111 $[$[doi:10.1016/j.aop.2005.10.005](https://doi.org/10.1016/j.aop.2005.10.005)$]$

Early accounts re-stating this claim (without attribution):

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], pp. 28 of: *Non-Abelian Anyons and Topological Quantum Computation*, Rev. Mod. Phys. **80** 1083 (2008) $[$[arXiv:0707.1888] (http://arxiv.org/abs/0707.1889), [doi:10.1103/RevModPhys.80.1083](https://doi.org/10.1103/RevModPhys.80.1083)$]$

* [[Zhenghan Wang]], Section 6.3 of: *Topological Quantum Computation*, CBMS Regional Conference Series in Mathematics **112**, AMS (2010) $[$[ISBN-13: 978-0-8218-4930-9](http://www.ams.org/publications/authors/books/postpub/cbms-112), [pdf](http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf)$]$

Further discussion (mostly review and mostly without attribution):

* [[Simon Burton]], *A Short Guide to Anyons and Modular Functors* $[$[arXiv:1610.05384](https://arxiv.org/abs/1610.05384)$]$

  > (this one stands out as still attributing the claim to [Kitaev (2006), Appendix E](#Kitaev06))

* {#RowellWang18} [[Eric C. Rowell]], [[Zhenghan Wang]], _Mathematics of Topological Quantum Computing_, Bull. Amer. Math. Soc. 55 (2018), 183-238 ([arXiv:1705.06206](https://arxiv.org/abs/1705.06206), [doi:10.1090/bull/1605](https://doi.org/10.1090/bull/1605))

* [[Tian Lan]], *A Classification of (2+1)D Topological Phases with Symmetries* $[$[arXiv:1801.01210](https://arxiv.org/abs/1801.01210)$]$

* *From categories to anyons: a travelogue* $[$[arXiv:1811.06670](https://arxiv.org/abs/1811.06670)$]$

* [[Colleen Delaney]], *A categorical perspective on symmetry, topological order, and quantum information* (2019) $[$[pdf](https://crdelane.pages.iu.edu/dissertationch1-5.pdf), [uc:5z384290](https://escholarship.org/uc/item/5z384290)$]$

* [[Colleen Delaney]], *Lecture notes on modular tensor categories and braid group representations* (2019) $[$[pdf](http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf), [[DelaneyModularTensorCategories.pdf:file]]$]$

* Liang Wang, [[Zhenghan Wang]], *In and around Abelian anyon models*, J. Phys. A: Math. Theor. **53** 505203 (2020) $[$[doi:10.1088/1751-8121/abc6c0](https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0)$]$

* [[Parsa Bonderson]], *Measuring Topological Order*, Phys. Rev. Research **3**, 033110 (2021) $[$[arXiv:2102.05677](https://arxiv.org/abs/2102.05677), [doi:10.1103/PhysRevResearch.3.033110](https://doi.org/10.1103/PhysRevResearch.3.033110)$]$

* Zhuan Li, Roger S.K. Mong, *Detecting topological order from modular transformations of ground states on the torus* $[$[arXiv:2203.04329](https://arxiv.org/abs/2203.04329)$]$

* [[Eric C. Rowell]], *Braids, Motions and Topological Quantum Computing* $[$[arXiv:2208.11762](https://arxiv.org/abs/2208.11762)$]$

* [[Sachin Valera]], *A Quick Introduction to the Algebraic Theory of Anyons*, talk at *[[CQTS]] Initial Researcher Meeting* (Sep 2022) $[$[[CQTS-InitialResearcherMeeting-Valera-220914.pdf:file]]$]$

* Willie Aboumrad, *Quantum computing with anyons: an F-matrix and braid calculator* $[$[arXiv:2212.00831](https://arxiv.org/abs/2212.00831)$]$

Emphasis that the expected description of [[anyons]] by [[braided fusion categories]] had remained [[folklore]], together with a list of minimal assumptions that would need to be shown:

* {#Valera21} [[Sachin J. Valera]], *Fusion Structure from Exchange Symmetry in (2+1)-Dimensions*, Annals of Physics **429** (2021) $[$
[arXiv:2004.06282](https://arxiv.org/abs/2004.06282), [doi:10.1016/j.aop.2021.168471](https://doi.org/10.1016/j.aop.2021.168471)$]$

An argument that the statement at least for [[SU(2)-anyons]] does follow from an enhancement of the [[K-theory classification of topological phases of matter]] to interacting [[topological order]]:

* [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Anyonic topological order in TED K-theory]]*, Rev. Math. Phys. (20223) $[$[arXiv:2206.13563](https://arxiv.org/abs/2206.13563), [doi:10.1142/S0129055X23500010](https://doi.org/10.1142/S0129055X23500010)$]$

#### Further discussion

Relation to [[ZX-calculus]]:

* [[Fatimah Rita Ahmadi]], [[Aleks Kissinger]], *Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics* $[$[arXiv:2211.03855](https://arxiv.org/abs/2211.03855)$]$

On detection of [[topological order]] by observing [[modular transformations]] on the [[ground state]]:

* [[Zhuan Li]], [[Roger S. K. Mong]], *Detecting topological order from modular transformations of ground states on the torus*, Phys. Rev. B **106** (2022) 235115 $[$[doi:10.1103/PhysRevB.106.235115](https://doi.org/10.1103/PhysRevB.106.235115), [arXiv:2203.04329](https://arxiv.org/abs/2203.04329)$]$

See also:

* [[Liang Kong]], *Topological Wick Rotation and Holographic Dualities*, [talk at](CQTS#KongOct2022) [[CQTS]] (Oct 2022) $[$[[Kong-TalkAtCQTS-20221019.pdf:file]]$]$


anyons in the quantum Hall effect -- references

### Anyons in the quantum Hall liquids
 {#AnyonsInTheQuantumHallEffectReferences}

References on [[anyon]]-excitations (satisfying [[braid group statistics]]) in the [[quantum Hall effect]] (for more on the application to [[topological quantum computation]] see the references [there](topological+quantum}computation#TopologicalQuantumComputationWithAnyons)):

The prediction of [[abelian group|abelian]] [[anyon]]-excitations in the [[quantum Hall effect]] (i.e. satisfying [[braid group statistics]] in 1-dimensional [[linear representations]] of the [[braid group]]):

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

  Erratum Phys. Rev. Lett. 52, 2390 (1984) ([doi:10.1103/PhysRevLett.52.2390.4](https://doi.org/10.1103/PhysRevLett.52.2390.4))

* Daniel Arovas, J. R. Schrieffer, [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$

The original discussion of [[non-abelian group|non-abelian]] [[anyon]]-excitations in the [[quantum Hall effect]] (i.e. satisfying [[braid group statistics]] in higher dimensional [[linear representations]] of the [[braid group]], related to [[modular tensor categories]]):

* [[Gregory Moore]], [[Nicholas Read]], _Nonabelions in the fractional quantum Hall effect_, Nucl. Phys. 360B(1991)362 ([pdf](http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf), doi:10.1016/0550-3213(91)90407-O)

Review:

* [[Ady Stern]], _Anyons and the quantum Hall effect -- A pedagogical review_, Annals of Physics Volume 323, Issue 1, January 2008, Pages 204-249 ([doi:10.1016/j.aop.2007.10.008](https://doi.org/10.1016/j.aop.2007.10.008), [arXiv:0711.4697](https://arxiv.org/abs/0711.4697))


anyons in topological superconductors -- references

### Anyons in topological superconductors
 {#AnyonsInTopologicalSuperconductorsReferences}

On [[anyon]]-excitations in [[topological superconductors]].

via [[Majorana zero modes]]:

Original proposal:

* [[Nicholas Read]], Dmitry Green, _Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect_,  Phys. Rev. B61:10267, 2000 ([arXiv:cond-mat/9906453](https://arxiv.org/abs/cond-mat/9906453))

Review:

* {#DasSarmaFreedmanNayak15} [[Sankar Das Sarma]], [[Michael Freedman]], [[Chetan Nayak]], _Majorana Zero Modes and Topological Quantum Computation_, npj Quantum Information 1, 15001 (2015) ([nature:npjqi20151](https://www.nature.com/articles/npjqi20151))

* Nur R. Ayukaryana, Mohammad H. Fauzi, Eddwi H. Hasdeo, _The quest and hope of Majorana zero modes in topological superconductor for fault-tolerant quantum computing: an introductory overview_ ([arXiv:2009.07764](https://arxiv.org/abs/2009.07764))

* Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, *Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors* [[arXiv:2301.11614](https://arxiv.org/abs/2301.11614)]

Further developments:

* [[Meng Cheng]], Victor Galitski, [[Sankar Das Sarma]], _Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors_, Phys. Rev. B 84, 104529 (2011) ([arXiv:1106.2549](https://arxiv.org/abs/1106.2549))

* {#ShabaniEtAl15} [[Javad Shabani]] et al., *Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconducting networks*, Phys. Rev. B **93** 155402 (2016) \[doi:10.1103/PhysRevB.93.155402, [arXiv:1511.01127](https://arxiv.org/abs/1511.01127)\]

* {#ShabaniEtAl17} [[Javad Shabani]] et al., *Zero-Energy Modes from Coalescing Andreev States in a Two-Dimensional Semiconductor-Superconductor Hybrid Platform*, Phys. Rev. Lett. **119** (2017) 176805 \[doi:10.1103/PhysRevLett.119.176805, [arXiv:1703.03699](https://arxiv.org/abs/1703.03699)\]

* {#ShabaniEtAl22} [[Javad Shabani]] et al., *Fusion of Majorana Bound States with Mini-Gate Control in Two-Dimensional Systems*,  	Nature Communications **13** (2022) 1738-1747 \[doi:10.1038/s41467-022-29463-6, [arXiv:2101.09272](https://arxiv.org/abs/2101.09272)\]

* {#ShabaniEtAl23} [[Javad Shabani]] et al., *Quasiparticle dynamics in epitaxial Al-InAs planar Josephson junctions*, PRX Quantum **4** 030339 (2023) \[doi:10.1103/PRXQuantum.4.030339, [arXiv:2303.04784](https://arxiv.org/abs/2303.04784)\]

via [[Majorana zero modes]] restricted to edges of [[topological insulators]]:

* Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, Xiao-Liang Qi, and Shou-Cheng Zhang, _Topological quantum computation based on chiral Majorana fermions_, PNAS October 23, 2018 115 (43) 10938-10942; first published October 8, 2018 ([doi:10.1073/pnas.1810003115](https://doi.org/10.1073/pnas.1810003115))

See also:

* Yusuke Masaki, Takeshi Mizushima, [[Muneto Nitta]], *Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors* $[$[arXiv:2301.11614](https://arxiv.org/abs/2301.11614)$]$