Dror Bar-Natan

* [website](http://www.math.toronto.edu/drorbn/)

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

* [Algebraic Knot Theory - A Call for Action](http://drorbn.net/index.php?title=Algebraic_Knot_Theory_-_A_Call_for_Action)

## Selected writings

On [[noncommutative probability theory]]:

* [[Dror Bar-Natan]], *Two examples in noncommutative probability*, Foundations of Physics, **19** (1989) 97–104  ([doi:10.1007/BF00737769](https://doi.org/10.1007/BF00737769))

On [[Vassiliev invariant|Vassiliev]] [[knot invariants]]  from [[perturbative quantum field theory|perturbative]] [[Chern-Simons theory]] [[Feynman amplitudes]] (see also at [[graph complex]]):

* {#BarNatan91} [[Dror Bar-Natan]], _Perturbative aspects of the Chern-Simons topological quantum field theory_, thesis 1991 ([spire:323500](http://inspirehep.net/record/323500), [proquest:303979053](https://search.proquest.com/docview/303979053), [[BarNatanPerturbativeCS91.pdf:file]])

On [[Vassiliev invariants]] for [[knots]], the proof that [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]] etc.:

* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_, in: *Geometry and Physics*, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996 ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009), [ISBN:0-8247-9791-4](https://users-math.au.dk/swann/Aarhus-proceedings-1995/index.html))

* [[Dror Bar-Natan]], _Finite Type Invariants_, in:  J.-P. Francoise, G.L. Naber and Tsou S.T. (eds.) Encyclopedia of Mathematical Physics, Oxford: Elsevier, 2006, volume 2 page 340
([arXiv:math/0408182](https://arxiv.org/abs/math/0408182))

On [[Vassiliev invariants of braids]] via [[horizontal chord diagrams]]:

* [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

Relation of the [[colored Jones polynomial]] to [[Lie algebra weight systems]] on [[chord diagrams]]:

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], _On the Melvin–Morton–Rozansky conjecture_, Invent math (1996) 125: 103 ([doi:10.1007/s002220050070](https://doi.org/10.1007/s002220050070), [[BarNatanGaroufaldis96.pdf:file]])

On the [[logical equivalence]] between the [[four-colour theorem]] and a statement about transition from the [[small N limit]] to the [[large N limit]] for [[Lie algebra weight systems]] on [[Jacobi diagrams]] via the [['t Hooft double line construction]]:

* [[Dror Bar-Natan]], _Lie Algebras and the Four Color Theorem_, Combinatorica 17-1(1997) 43–52  ([arXiv:q-alg/9606016](https://arxiv.org/abs/q-alg/9606016), [doi:10.1007/BF01196130](https://doi.org/10.1007/BF01196130))

Computation of the perturbative [[Wilson loop observable]] ([[universal Vassiliev invariant]] of the [[unknot]] ("[[Wheels theorem]]"):

* [[Dror Bar-Natan]], [[Le Tu Quoc Thang]], [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

following

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

List of literature on [[Vassiliev invariants]]:

* [[Dror Bar-Natan]], [[Sergei Duzhin]], _[Bibliography of Vassiliev Invariants](http://www.pdmi.ras.ru/~duzhin/VasBib/Long)_

On the [[cochain cohomology]] of [[graph complexes]]:

* [[Dror Bar-Natan]], [[Brendan McKay]], _Graph cohomology -- An overview and some computations_ ([[BarNatanMcKayGraphCohomology.pdf:file]])

On the [[Cayley distance kernel]]:

* {#BarNatan21} [[Dror Bar-Natan]], *[AcademicPensieve](http://drorbn.net/AcademicPensieve/)* [2021-05](http://drorbn.net/AcademicPensieve/2021-05/), *The Cayley distance kernel following arXiv://2105.0287 by Corfield, Sati, and Schreiber* ([nb](http://drorbn.net/AcademicPensieve/2021-05/CayleyDistanceKernel.nb), [pdf](http://drorbn.net/AcademicPensieve/2021-05/nb/CayleyDistanceKernel.pdf), [[BarNatanCayleyDistanceKernelnb.pdf:file]])

## Related $n$Lab entries

* [[knot theory]]

* [[Khovanov homology]]

category: people


dSet

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

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] of [[dendroidal sets]] is often denoted $dSet$ or similar, following the common denotation _[[sSet]]_ for the [[category]] of [[simplicial sets]].

## Properties

Some [[mathematical structure|structure]] carried by the [[category]] [[dSet]] of [[dendroidal sets]]:

### $sSet$-enriched structure

Using the fact that [[dSet]] is a [[closed monoidal category]] with [[internal hom]] dendroidal sets $[C,D]$ for dendroidal sets $C$ and $D$, and using the functor $i^* : dSet \to SSet$ we obtain canonically the structure of an [[simplicially enriched category]] / [[sSet]]-[[enriched category]] on $dSet$ with the hom-simplicial set between $C$ and $D$ being $i^*[C,D]$.

### Model category structure

The category $dSet$ of [[dendroidal sets]] carries a [[monoidal model category|monoidal]] [[model category]]-[[structure]] -- the [[model structure on dendroidal sets]] -- which serves to [[presentable (infinity,1)-category|present]] the [[(∞,1)-category]] of [[(∞,1)-operads]]:

Together with the fact that $i^*: dSet \to sSet$ is a [[right Quillen functor]] (with respect to the [[model structure for quasi-categories]]) this imples that [[dSet]] is an $sSet_{Joyal}$-[[enriched model category]] (but not, without further work, an $sSet_{Quillen}$-enriched model category!).

[[!include table - models for (infinity,1)-operads]]

## Related entries

* [[model structure on dendroidal sets]]

* [[sSet]]

## References

See the [list of references](dendroidal+set#References) at _[[dendroidal set]]_.

For instance:

* [[Ieke Moerdijk]] and [[Ittay Weiss]], _Dendroidal Sets_, Algebraic and Geometric Topology, Volume 7, Number 3 (2007), 1441-1470. ([doi:10.2140/agt.2007.7.1441](https://msp.org/agt/2007/7-3/p13.xhtml), [arXiv:math/0701293](https://arxiv.org/abs/math/0701293))

category: category


Du Pei

* [webpage](https://sites.google.com/site/dupeiphysics/)

category: people

## Selected writings

On the [[3d-3d correspondence]]:

* [[Du Pei]], Ke Ye, _A 3d-3d appetizer_, JHEP 11 (2016) 008 ([arXiv:1503.04809](https://arxiv.org/abs/1503.04809))

and specifically for [[Seifert 3-manifolds]] (such as [[lens spaces]]):

* [[Du Pei]], _3d-3d correspondence for Seifert manifolds_, 2016 ([spire:1469350](https://inspirehep.net/literature/1469350), [pdf](https://inspirehep.net/files/d509ff9e32448da3a5674f286b93224a))

On the [[M5-brane elliptic genus]]:

* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*,  J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), doi:10.1007/JHEP05(2021)084, [spire:1704312](https://inspirehep.net/literature/1704312)]

category: people


dual

# Duals

There are various kinds of duals.

Specific:

* [[dual vector space]],
* [[Pontrjagin dual]],
* [[dual object]]
* [[dual object in a closed category]]
* etc.

General:

* [[category with duals]],
* [[(infinity,n)-category with duals]],
* etc.

See also [[duality]].

[[!redirects dual]]
[[!redirects duals]]


dual abelian group scheme

#Contents#
* table of contents
{:toc}

## Idea

Given an [[abelian group scheme]] $A$, its _dual_ $A^\vee$ is essentially the [[moduli space]] of [[line bundles]] over $A$, the [[Picard scheme]] of $A$. Accordingly, there is a canonical line bundle over the [[product]] $A \times A^\vee$ (which over each point $(a,P)$ is the [[fiber bundle|fiber]] over $a$ of the line bundle $P$): the _[[Poincaré line bundle]]_.

## Related concepts

* [[Picard scheme]]

* [[Cartier duality]]

## References

A standard textbook is

* Mumford, _Abelian varieties_

A review of some basics is in

* [[Akhil Mathew]], _[Duality for abelian varieties](http://amathew.wordpress.com/2013/03/21/duality-for-abelian-varieties/)_

[[!redirects dual abelian group schemes]]

[[!redirects dual abelian variety]]
[[!redirects dual abelian varieties]]


dual adjunction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* tableofcontents
{:toc}

## Idea

A _dual adjunction_ between [[category|categories]] $C$ and $D$ is an [[adjoint functor|adjunction]] between the [[opposite category]] $C^{op}$ of $C$ and $D$.

The concept arises in the context of [[duality]].

Dual adjunctions between [[concrete category|concrete categories]] are frequently represented by [[dualizing objects]].

Dual adjunctions between [[posets]] are also called _[[Galois connections]]_.

## Definition

A _dual adjunction_ consists of [[contravariant functors]] $F: C \to D$, $G: D \to C$ together with natural transformations $\eta: 1_C \to G F$ and $\theta: 1_D \to F G$ such that $F \eta \circ \theta F = 1_F$ and $G \theta \circ \eta G = 1_G$. In diagrams, the following must commute.
\begin{tikzcd}
F \ar{dr}[swap]{1} \ar{r}{\theta F} & FGF \ar{d}{F\eta}  & & G \ar{dr}[swap]{1} \ar{r}{\eta G} & GFG \ar{d}{G\theta} \\
& F & & & G
\end{tikzcd}

Reformulated in terms of covariant functors, a dual adjunction can be viewed as an ordinary adjunction $F \dashv G$ with $F: C \to D^{op}$ and $G: D^{op} \to C$, or as $G \dashv F$ with $G: D \to C^{op}$ and $F: C^{op} \to D$. However, it is often useful not to break the symmetry of the contravariant formulation.

A _self-dual_ adjunction is a dual adjunction for which $F = G: C \to C$ and $\eta = \theta: 1 \to F F$. An example is where $C$ is a [[symmetric monoidal closed category]] and $F = [-, d]$ is an internal hom into an object $d$, where the unit is the usual double-dual embedding $\delta_c: c \to [[c, d], d]$.

## Related concepts

* [[adjunction]]

  * [[strong adjoint functor]]

  * [[adjoint triple]], [[adjoint quadruple]], [[recollement]]

  * [[ambidextrous adjunction]]

  * [[adjoint monad]]

  * [[fixed point of an adjunction]]

  * [[enriched adjunction]]

  * [[two-variable adjunction]]

[[!redirects dual adjunctions]]


dual basis

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of a _dual basis_ is a way of characterizing [[projective modules]], alternative to their characterization as [[direct sum|direct summands]] of [[free modules]]. The terminology derives from a similarity with a situation involving [[dual vector spaces]], see [below](#RelationToDualVectorSpaces).

## Details

Let $R$ be a [[ring]] or, more generally, an [[associative algebra]] over a unital [[commutative ring]] $k$.

Assuming the [[axiom of choice]], one of the standard characterizations of the [[projective modules]] $N$ (say left) over $R$ is that there is an [[epimorphism]] $F\to N$ from a [[free module]] over $R$  which is [[split epimorphism|split]] ("$N$ is a [[direct summand]] of a [[free module]]", [this prop.](projective%20module#ProjectiveIsPreciselyDirectSummandOfFreeModule)).

Equivalently, an $R$-module $M$ is projective iff it has a __dual basis__ $\{(x_i,x_i^*)\in M\times Hom_k(M,R)\}_{i\in I}$ in the following sense. There is a free $R$-module $F = \oplus_{i\in I} R$, an epimorphism of $R$-modules $F\to M$, $F=\oplus_i R \ni \sum_i r_i\mapsto \sum_i r_i x_i$ which is split, i.e. has a right inverse, where this inverse, by the universal property of the direct sum, must be of the form $x\mapsto \sum_i x_i^*(x)\in \oplus_i R$ where $x_i^*\in Hom_k(M,R)$. The right inverse condition translates to  $\forall x \in M : x=\sum_i x_i^*(x)x_i$. In particular for every $x\in M$ $x_i^*(x)\neq 0$ for only finitely many $i\in I$.

## Relation to dual vector spaces
 {#RelationToDualVectorSpaces}

This terminology is related to but a bit different than in the case of $k$-vector spaces (cf. at [[dual vector space]]).
If $V$ has a vector space
basis $(x_j)_{j\in I}$ is a [[linear basis]] of $V$ then $x_i^*$ _defined by_ $x_i^*(x_j) = \delta_i^j$ is not necessarily a basis of $V^* = Hom_k(V,k)$; it is
if $V$ is [[finite dimensional vector space|finite dimensional]].  In the case of projective module $M$,
$x_i$ do not form a basis in a free sense,
but only a set of generators and with the property that there exist another set $x_i^*$ in $M^*$
such that they together form a "dual basis". (Still, one sometimes says
that $x_i^*$ form a basis dual to $x_i$.)

category: algebra

[[!redirects dual basis]]
[[!redirects dual basises]]
[[!redirects dual bases]]


dual bialgebra

See also [[dual gebra]] for a more general entry.

## Definition

Given a field $k$, a $k$-vector space pairing between $k$-bialgebras $H$ and $K$  such that

$$
\langle h\otimes h', \Delta_K k\rangle = \langle h h', k\rangle
$$

$$
\langle \Delta_H h, k\otimes k' \rangle = \langle h, k k'\rangle
$$

$$
\langle h, 1_K\rangle = \epsilon_H(h),
\,\,\,\,\,
\langle 1_H, k\rangle = \epsilon_K(k).
$$

(where on the left hand side $\langle,\rangle$ denotes the map $H\otimes H\otimes K\otimes K\to k$ given by
$\langle h\otimes h', k\otimes k'\rangle = \langle h,k\rangle \langle h',k' \rangle$), is called the __bialgebra pairing__.

The bialgebra pairing which is __perfect__ (nondegenerate in each variable) as a $k$-vector space pairing (i. e. if $\langle h, K\rangle = 0$ implies that $h$ is $0$ and $\langle H,k\rangle$ implies that $k$ is $0$) is called the bialgebra duality.

## Finite duals

As stated in [[dual gebra]], for an algebra $A = (A,m)$, a finite (or restricted or Hopf) dual is the subspace
$$ A^\circ = \{f\in A^* | \exists ideal I\subset Ker(f)\subset A, dim(A^*/I)\lt\infty \}\subset A^*.$$
Then $f\in A^\circ$ iff $m^*(f)\subset A^*\otimes A^*$ which is iff $m^*(f)\subset A^\circ\otimes A^\circ$, see [Sweedler1969](Sweedler1969). This subspace is consequently a coalgebra, via the corestriction of $m^*:A^*\to (A\otimes A)^*$. If $A$ is a bialgebra (Hopf algebra) then $A^\circ$ is such as well.
We can say this differently: If $H$ is a bialgebra (resp. Hopf algebra) then its finite dual $H^\circ$ has a unique structure of a bialgebra (resp. Hopf algebra) such that
the evaluation is a bialgebra pairing.

A general pairing $\langle -,-\rangle : H\otimes K\to K$
of vector spaces underlying Hopf algebras $H$ and $K$ is
a bialgebra pairing iff the induced map $H\to K^\circ$, $h\mapsto \langle h, -\rangle$
is a bialgebra map.

## Hopf algebra case

If $H$ and $K$ are Hopf algebras then the bialgebra pairing is a Hopf pairing if $\langle h, S k \rangle = \langle S h, k\rangle$. This is however, automatic.

## Literature

* {#Sweedler1969} [[Moss E. Sweedler]], $()^\circ$, Chapter VI in _Hopf algebras_, Benjamin, N.Y. 1969

* MO: [is-a-bialgebra-pairing-of-hopf-algebras-automatically-a-hopf-pairing](https://mathoverflow.net/questions/20666/is-a-bialgebra-pairing-of-hopf-algebras-automatically-a-hopf-pairing), [dual-of-a-hopf-algebra](https://math.stackexchange.com/questions/3324098/dual-of-a-hopf-algebra), [hopf-dual-of-the-hopf-dual](https://mathoverflow.net/questions/308374/hopf-dual-of-the-hopf-dual), [on-reflexive-bialgebras](https://mathoverflow.net/questions/357738/on-reflexive-bialgebras)
* Stephen Donkin, _On the Hopf algebra dual of an enveloping algebra_, Math. Proc. Camb. Phil. Soc. __91__ (1982) 215-224
* R. G. Heyneman, D. E. Radford, _Reflexivity and coalgebras of finite type_,J. Alg. __28__ (1974) 215-246 [pdf](https://core.ac.uk/download/pdf/82125553.pdf)

category: algebra

[[!redirects dual Hopf algebra]]
[[!redirects Hopf algebra dual]]
[[!redirects bialgebra dual]]
[[!redirects bialgebra pairing]]
[[!redirects Hopf algebra pairing]]
[[!redirects Hopf pairing]]


dual equivalence

A __dual equivalence__, or __antiequivalance__, between [[categories]] (or [[higher category theory|higher categories]]) $C$ and $D$ is simply an [[equivalence of categories|equivalence]] between one and the [[opposite category]] of the other.  It is usually most convenient to think of the equivalence as going from $C^{op}$ to $D$, as with a [[contravariant functor]] in general.

Many examples of abstract [[duality]] involve dual equivalences.

[[!redirects antiequivalence]]


dual fibration

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

#Contents#
* table of contents
{:toc}

## Idea

The **dual fibration** of a (Grothendieck) [[Grothendieck fibration|fibration]] is a fibration over the same base but with all of the fibers dualized.

## Definition

Let $p : E \to B$ be a fibration, and let $F : B^{op} \to Cat$  be the corresponding [[indexed category]]. The dual fibration $p^d : E^d \to B$ can be defined [(Borceux)](#Borceux) as the fibration associated (via the [[Grothendieck construction]]) to the composite

$$
 B^{op} \stackrel{F}{\longrightarrow} Cat \stackrel{(-)^{op}}{\longrightarrow} Cat
$$

where $(-)^{op} : Cat \to Cat$ is the operation sending any category to its opposite (note this operation preserves the direction of the 1-cells, although it reverses the direction of the 2-cells in Cat). (See [Borceux](#Borceux) reference.) Alternatively, the dual fibration may be defined in more elementary terms as a category with the same objects as $E$, and whose morphisms are equivalence classes of [[spans]] $L \stackrel{v}{\longleftarrow} X \stackrel{h}{\longrightarrow} R$ where $v$ is _vertical_ (i.e., $p(v) = id$) and $h$ is _horizontal_ (i.e., $p$-[[cartesian morphism|cartesian]]). (See [Pavlovic](#Pavlovic90) and [Kock](#AKock15) references.)

## Related concepts

* [[hyperdoctrine]]

* [[trifibration]]

## References

* {#Pavlovic90} [[Dusko Pavlovic|Duško Pavlović]], _Categorical Interpolation: Descent and the Beck-Chevalley Condition without Direct Images_ , pp.306-325  in _Category theory Como 1990_, LNM **1488** Springer Heidelberg 1991. ([pdf](http://www.isg.rhul.ac.uk/dusko/papers/1990-BCDE-Como.pdf))

* {#Borceux} [[Francis Borceux]], [[Handbook of Categorical Algebra]] (see Volume 2, 8.3).

* {#AKock15} [[Anders Kock]], The dual fibration in elementary terms, [arXiv:1501.01947](https://arxiv.org/abs/1501.01947).

[[!redirects opposite fibration]]


dual gebra

# Dual gebras
* table of contents
{: toc}

## What is this entry about

Following Serre, [[gebra]] is a common term for [[associative algebra]]s and coassociative [[coalgebra]]s (also called cogebras), and sometimes more involved variants and combinations, like [[bialgebra]]s (also, more properly, called bigebras) and (co)rings.

When working over a field, finite dimensional algebras are duals to finite dimensional cogebras. When the dimension is infinite, even for algebraic duals, the situation is more complicated. This entry should eventually sort out these issues (for now only the simplest
cases are discussed).

## Algebraic dual

For a commutative ring $k$, a coassociative
$k$-coalgebra $(C,\Delta)$ and an associative $k$-algebra $(A,m)$
the $k$-module $Hom_k(C,A)$ is equipped with an associative __convolution product__ $\star$ given by
$(f\star g)(c) = m(f\otimes g)(\Delta(c))$.
In particular, for $k$ a field,
the algebraic dual $C^*:= Hom_k(C,k)$ of
a $k$-coalgebra $C$ is an associative algebra, called its __dual coalgebra__ whose product is also often referred to as convolution.
Correspondence $C\mapsto C^*$ extends to a contravariant functor
$Cog_k\to Alg_k$, where for $f:C\to D$, $f^*:D^*\to C^*$ is simply the transpose, hence $f^*(d^*)(c) = d^*(f(c))$.

Now for a $k$-algebra $(A,m)$, its algebraic dual $A^*$ is _not_ necessarily a coalgebra; namely the natural candidate for the
comultiplication $\Delta$ is the transpose operator $m^*: A^*\to (A\otimes A)^*$ of the multiplication $m: A\otimes A\to A$.
There is a canonical injection
$A^*\otimes A^*\to (A\otimes A)^*$; over a field $k$ it is an isomorphism (hence taken as an identification) iff $A$ is finite dimensional over $k$. In topological cases (e.g., if $A$ is filtered with filtered pieces finite-dimensional), one can replace the tensor product with some completed tensor product $\hat\otimes$ and define a topological comultiplication $m^*:A^*\to A^*\hat\otimes A^*$. In algebraic situation, one usually employs so called finite dual which is the maximal subspace $A^\circ$ for which $m^*$ factors through
$A^\circ\otimes A^\circ$.

## Finite dual

If $k$ is a field, the finite dual functor $()^\circ:Alg_k\to Cog_k$ is the [[left adjoint functor]] to the algebraic dual as a functor $()^*:Cog_k\to Alg_k$.  $A^\circ\subset A^*=Hom_k(A,k)$ as a vector spaces (actually as functors $Alg_k\to Vec_k$, $()^\circ$ is a [[subfunctor]] of $()^*:Alg_k\to Vec_k$). For a concrete construction
below, the statement of adjointness is Theorem 1.5.22 in Dascalescu et al.

We say that a subspace $W$ of a vector space $V$ is of finite codimension if $V/W$ is of finite dimension. As a vector subspace of
$A^*$,

$$
A^\circ = \{ f\in A^* \, | \, Ker(f)\, \text{contains an ideal of finite codimension in}\, A\}
$$

There are several other characterizations of the finite dual. Alternative terminologies are restricted dual and Hopf dual.

## Actions on duals

We define here left actions ⇀ (or in LaTeX $\rightharpoonup$), ⇁ (LaTeX $\rightharpoondown$) and right actions ↽ (LaTeX $\leftharpoondown$), ↼ (LaTeX $\leftharpoonup$).

(Montgomery 1.6.5) If $C$ is a coalgebra, and $C^*$ the dual algebra, then $C^*$ acts from the left on $C$ by the transpose to the left multiplication

$$
\langle g, c ↼ f\rangle = \langle f g, c\rangle
$$

or equivalently by the formula

$$
f ↼ c \coloneqq \langle f, c_{(1)}\rangle c_{(2)},
$$

where the [[Sweedler notation]] has been used.

Similarly for the right-hand action:

$$
f ⇀ c \coloneqq \langle f, c_{(2)}\rangle c_{(1)},
$$

or

$$
\langle g, f ⇀ c\rangle = \langle f, c_{(2)}\rangle
\langle g, c_{(1)}\rangle = \langle g f, c\rangle
$$

According to the suggestion of Nichols, one reads $f$⇀$c$
as "$f$ hits $c$" and $f$↼$c$ as "$f$ is hit by $c$".

Similarly (cf. Montgomery 1.6.6), if $A$ is an algebra and $A^*$ its algebraic dual, one also defines harpoon actions as transposes to left and right multiplications, for example for right multiplication

$$
\langle h ↼ f, k\rangle = \langle f, k h\rangle.
$$

Now, if $f\in A^\circ$ is in finite dual, then $\Delta(f)$ makes sense, hence, in Sweedler notation,
$h$⇀$f=\langle f_{(2)}, h\rangle f_{(1)}$.

We also define here left and right coadjoint actions and coactions, cf. Majid.

One should also treat rationality: a module is rational if it corresponds to a comodule of the finite dual coalgebra.

## Paired bialgebras

For [[bigebra]]s (and [[Hopf algebra]]s n particular)
one may consider the duality pairings which are
compatible with their structure.

Two $k$-bigebras $B$ and $H$ are __paired__
if there is a bilinear map
$\langle,\rangle: B\otimes H\to k$ such that for all $a,b\in B$ and $h,g\in H$ the equations

$$
\langle a b, h\rangle = \langle a\otimes b,\Delta h\rangle,
\,\,\,\,\,\,\,\,\,\,\langle 1_B,h\rangle = \epsilon(h),
$$
$$
\langle \Delta a, h\otimes g\rangle = \langle a, h g\rangle,
\,\,\,\,\,\,\,\,\,\,\epsilon(a) = \langle a, 1_H\rangle
$$

They are a __strictly dual pair of bigebras__ if the pairing is in also nondegenerate. If $B$ and $H$ are paired then one can quotient out [[biideal]]s $J_B\subset B$, $J_H\subset H$ of all those elements in each of them which pair as zero with all elements in the other bigebra; the quotients $B'$ and $H'$ will then be strictly paired bigebras.

See also [[dual bialgebra]].

## Literature

Related $n$Lab entries: [[dual]], [[Heisenberg double]], [[gebra]]

Quite detailed treatment of duality of gebras is in

* Sorin Dăscălescu, Constantin Năstăsescu, Serban Raianu, _Hopf algebras: an introduction_, Marcel & Dekker 2000

and the entire Chapter VI (titled $()^\circ$) of

* [[Moss E. Sweedler]], _Hopf algebras_, Benjamin, N.Y. 1969

Other sources are

* Susan Montgomery, _Hopf algebras and their actions on rings_, AMS 1994, 240 pp.
* Shahn Majid, _Foundations of quantum group theory_, Cambridge Univ. Press
* Eiichi Abe, _Hopf algebras_, Cambridge Tracts in Mathematics 74, 1977

and for gebras with involution

* A. van Daele, _Dual pairs of Hopf $\ast$-algebras_,  Bull. London Math. Soc. (1993) 25 (3): 209-230 [doi](http://dx.doi.org/10.1112/blms/25.3.209)

Hit-actions are recently studied in

* M. Cohen, S. Westreich, _Hit-actions and commutators for Hopf algebras_, Bull. Math. Soc. Sci. Math. Roumanie __56__(104) No. 3, 2013, 299--313 [pdf](http://ssmr.ro/bulletin/pdf/56-3/articol_4.pdf)

[[Cartier duality]] and related earlier issues on [[linearly compact topological spaces]] due Dieudonné are in the first chapter of

* [[Jean Dieudonné]], _Introduction to the theory of formal groups_, Marcel Dekker, New York 1973.

Some newer applications are in

* Lowell Abrams, [[Charles Weibel]], _Cotensor products of modules_, [math.RA/9912211](https://arxiv.org/abs/math/9912211)

Duality of dg-algebras vs. dg-coalgebras is studied recently in great detail in

* [[Mathieu Anel]], [[André Joyal]], _Sweedler Theory for (co)algebras and the bar-cobar constructions_, 260 pp. [arxiv/1309.6952](https://arxiv.org/abs/1309.6952); cf. also Boston 2012 [slides](http://thales.math.uqam.ca/~anelm/mat/doc/boston.pdf)

Some special cases of finite duals are treated in

* Stephen Donkin, _On the Hopf algebra dual of an enveloping algebra_, Math. Proc. Camb. Phil. Soc. (1982), 91, 215-224, [doi](http://dx.doi.org/10.1017/S0305004100059260)
* Jahn Astrid, _The finite dual of crossed products_, thesis, [pdf](http://theses.gla.ac.uk/6158/1/2015jahnphd.pdf)
* MathOverflow: [Hopf algebra duality and algebraic groups](http://mathoverflow.net/questions/31237/hopf-algebra-duality-and-algebraic-groups)

category: algebra

[[!redirects dual gebras]]
[[!redirects Hopf pairing]]
[[!redirects dual bialgebra]]
[[!redirects dual Hopf algebra]]
[[!redirects finite dual]]
[[!redirects restricted dual]]
[[!redirects Hopf dual]]


dual graviton

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

#Contents#
* table of contents
{:toc}

## Idea

By _dual graviton_ one refers to the dual of the [[graviton]] under [[electric-magnetic duality]] (e.g. [Curtright 85](#Curtright85), [Hull 01](#Hull01), [Bekaert-Boulanger-Henneaux 02](#BekaertBoulangerHenneaux02), [Godazger-Godazger-Nicolai 13, section 2.2](#GodazgerGodazgerNicolai13)).

Dual gravitons have been considered in particular for manifestly EM-duality-symmetric formulations of [[supergravity]] [[Lagrangians]]. See for instance ([de Wit-Nicolai 13, section 6](#deWitNicolai13)) and see at _[3-d supergravity -- possible gaugings](3-dimensional+supergravity#PossibleGaugings)_.

The spin-2 dual graviton is described by the [Curtright field](https://en.wikipedia.org/wiki/Curtright_field) in five spacetime dimensions. Dual graviton can be coupled to topological [[BF-theory]] in _D_ = 5 ([Bizdadea 09](#Bizdadea09)), so it should be also coupled with [[gravity as a BF theory]] in extra dimensions _D_ > 4.

## References

The dual graviton was maybe first discussed in

* {#Curtright85} [[Thomas Curtright]], _Generalised Gauge Fields_,  Phys.Lett. B 165:304, 1985 (doi:[10.1016/0370-2693(85)91235-3](https://doi.org/10.1016%2F0370-2693%2885%2991235-3))

then further in

* {#Hull01} [[Chris Hull]], _Duality in Gravity and Higher Spin Gauge Fields_, JHEP 0109:027, 2001 ([arXiv:hep-th/0107149](http://arxiv.org/abs/hep-th/0107149))

* {#BekaertBoulangerHenneaux02} [[Xavier Bekaert]], Nicolas Boulanger, [[Marc Henneaux]], _Consistent deformations of dual formulations of linearized gravity: A no-go result_, Phys. Rev. D67 (2003) 044010 ([arXiv:hep-th/0210278](http://arxiv.org/abs/hep-th/0210278))

* {#deWitNicolai13} [[Bernard de Wit]], [[Hermann Nicolai]], _Deformations of gauged SO(8) supergravity and supergravity in eleven dimensions_ ([arXiv:1302.6219](http://arxiv.org/abs/1302.6219))

Discussion in the context of [[E7]]-[[exceptional generalized geometry]] includes

* {#GodazgerGodazgerNicolai13} Hadi Godazgar, Mahdi Godazgar, [[Hermann Nicolai]], _Generalised geometry from the ground up_, Journal of High Energy Physics February 2014, 2014:75 ([arXiv:1307.8295](http://arxiv.org/abs/1307.8295))

Double-dual graviton:

* [[Marc Henneaux]], Victor Lekeu, Amaury Leonard, _A note on the double dual graviton_ ([arxiv:1909.12706](https://arxiv.org/abs/1909.12706))

Discussion in terms of [[E11|$E_{11}$]] [[U-duality]]:

* {#Wess11} [[Peter West]], section N (pp. 19) of _Generalised geometry, eleven dimensions and $E_{11}$_ ([arXiv:1111.1642](http://arxiv.org/abs/1111.1642))

* Keith Glennon, [[Peter West]],
_Gravity, Dual Gravity and $A_1^{+++}$_ ([arXiv:2004.03363](https://arxiv.org/abs/2004.03363))

* Nicolas Boulanger, [[Paul P. Cook]], Josh A. O'Connor, [[Peter West]], *Higher dualisations of linearised gravity and the $A_1^{+++}$ algebra* [[arXiv:2208.11501](https://arxiv.org/abs/2208.11501)&rbrack

Coupling with topological [[BF-theory]] in

* {#Bizdadea09} C. Bizdadea, E. M. Cioroianu, A. Danehkar, M. Iordache, S. O. Saliu, S. C. Sararu, _Consistent interactions of dual linearized gravity in D=5: couplings with a topological BF model_, The European Physical Journal C (Particles and Fields) October 2009, 63:491 ([arXiv:0908.2169](https://arxiv.org/abs/0908.2169))

[[!redirects dual gravitons]]


dual heterotic string theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

Ordinary [[heterotic string theory]] is the study of the perturbation series of correlators of a certain 2-dimensional [[SCFT]] over surfaces. The resulting [[S-matrix]] is interpreted as encoding the scattering amplitudes of strings propagating in some target space.

Among the [[gauge field]]s that these strings are charged under is the [[Kalb-Ramond field]], which is a [[circle n-bundle with connection|circle 2-bundle with connection]].

By general [[electric-magnetic duality]] (see [Freed 00](#Freed00) for a formal treatment in [[differential cohomology]]) one expects there to be a dual theory theory where the $(p=1)$-dimensional strings are replaced by their magnetic duals, which are $10-(p+2)-2 = 5$-branes.

The study of the corresponding perturbation series over correlators of the 6-dimensional fivebrane worldvolume [[SCFT]] over 6-volumes is far from tractable,  but a handful of consistency checks exist, that the corresponding _dual heterotic string theory_ makes sense.

Where the [[quantum anomaly]]-cancellation for the heterotic string involves the demand for (twisted) [[string structure]]s, that for the dual theory involves [[fivebrane structure]]s (which gives these their name).

The dual degree-8 anomaly polynomial in [[Green-Schwarz mechanism]] for dual heterotic string theory is (e.g. [Freed 00 (3.48)](#Freed00))

$$
  d H_7
  \;=\;
  ch_4(A)
  -
  \tfrac{1}{48}
  p_1(\omega) \wedge ch_2(A)
  +
  \tfrac{1}{64}
  p_1(\omega) \wedge p_1(\omega)
  -
  \tfrac{1}{48} p_2(\omega)
  \,,
$$

where $A$ denotes the [[gauge field]] [[connection on a bundle|connection]], $\omega$ the [[gravity]] [[spin connection]], $ch_n$ the degree-$n$ component of the [[Chern character]] and $p;_i$ the [[Pontryagin classes]].

## Related concepts

* [[string theory]]

* [[heterotic string theory]]

  * [[Green-Schwarz mechanism]]

* **dual heterotic string theory**

## References

The heterotic [[NS5-brane]] (see also at _[[half M5-brane]]_) was found as a [[black brane]] in [[heterotic string theory]] in

* {#Strominger91} [[Andrew Strominger]],

  _Heterotic solitons_, Nucl. Phys. B343 (1990) 167-184 (doi:10.1016/0550-3213(90)90599-9)

  Erratum: Nucl. Phys. B353 (1991) 565-565 (doi:10.1016/0550-3213(91)90349-3, [spire:27900](http://inspirehep.net/record/27900))

Discussion of the heterotic [[NS5-brane]] as a [[Green-Schwarz sigma-model]] and its fermion [[worldvolume]] [[quantum anomaly]]:

* [[Kurt Lechner]], [[Mario Tonin]], _Worldvolume and target space anomalies in the D=10 super--fivebrane sigma--model_ ([arXiv:hep-th/9603094](http://arxiv.org/abs/hep-th/9603094))

* [[Kurt Lechner]], [[Mario Tonin]], _World-manifold and target space anomalies in heterotic Green-Schwarz strings and five-branes_, in: _Gauge Theories, Applied Supersymmetry and Quantum Gravity II_, pp. 311-318 (1997) ([arXiv:hep-th/9610110](https://arxiv.org/abs/hep-th/9610110), [doi:10.1142/9781848160927_0022](https://doi.org/10.1142/9781848160927_0022))

* J. Mourad, _Anomalies of the $SO(32)$ five-brane and their cancellation_, Nucl.Phys. B512 (1998) 199-208 ([arxiv:hep-th/9709012](https://arxiv.org/abs/hep-th/9709012))

* {#Lechner10} [[Kurt Lechner]], _Quantum properties of the heterotic five-brane_, Phys. Lett. B693:323-329, 2010 ([arxiv:1005.5719](https://arxiv.org/abs/1005.5719))

* J. A. Dixon, [[Mike Duff]], [[Jan Plefka]], _Putting the string/fivebrane duality to the test_ ([arXiv:hep-th/9208055](http://arxiv.org/abs/hep-th/9208055))

Review of the history of the dual heterotic string conjecture is in

* [[Mike Duff]], chapter 6 of _[[The World in Eleven Dimensions]]: Supgergravity, Supermembranes and M-theory_ IoP 1999

On p. 3 and 4 the latter reviews the worldsheet description of the relevant terms for the string and then look at the corresponding situation for the 5-brane on p. 6.

So this is the EM-dual analog of the Killingback-Witten-computation of the fermionic  anomaly that leads to [[string structure]]s in the heterotic string (see there). Where the former involves cancelling the first fractional [[Pontryagin class]], this involves the second.

For a description of the general mechanism in [[differential cohomology]] at work here, see

* {#Freed00} [[Daniel Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_, 2000

[[!redirects dual heterotic string theory]]


dual linear map

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

#Contents#
* table of contents
{:toc}

## Definition

The operation $V \mapsto V^*$ of forming [[dual vector spaces]] extends to a [[contravariant functor]].

+-- {: .num_defn}
###### Definition (transpose map)
The **dual linear map** or __transpose map__ of a [[linear map]] $A\colon V\to W$, is the linear map $A^* = A^T\colon W^*\to V^*$, given by
$$ \langle{A^*(w), v}\rangle = \langle{w, A(v)}\rangle $$
for all $w$ in $W^*$ and $v$ in $V$.
=--

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

This functor is, of course, the [[representable functor]] represented by $K$ as a vector space over itself (a [[line]]).

=--

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

This construction is the notion of [[dual morphism]] applied in the [[monoidal category]] [[Vect]] with its [[tensor product]] monoidal structure.

=--

## Related concepts

* [[Chevalley-Eilenberg algebra]]

* [[duality]]

[[!redirects dual linear maps]]

[[!redirects transpose linear map]]
[[!redirects transpose linear maps]]

[[!redirects linear dual map]]
[[!redirects linear dual maps]]


dual morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Monoidal categories
+-- {: .hide}
[[!include monoidal categories - contents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of dual morphism is the generalization to arbitrary [[monoidal categories]] of the notion of [[dual linear map]] in the [[category]] [[Vect]] of [[vector spaces]].

## Definition
 {#Definition}

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

Given a [[morphism]] $f \colon X \to Y$ between two [[dualizable objects]] in a [[monoidal category]] $(\mathcal{C}, \otimes)$, the corresponding **dual morphism**

$$
  f^\ast \colon Y^\ast \to X^\ast
$$

is the one obtained by $f$ by using the duality [[unit of an adjunction|unit]] of $X$ (the [[coevaluation map]]) and the duality [[counit of an adjunction|counit]] of $Y$ (the [[evaluation map]]) as follows:

$$
  Y^*
  \to Y^*\otimes I
  \to Y^*\otimes X\otimes X^*
  \to Y^*\otimes Y\otimes X^*
  \to I\otimes X^*
  \to X^*
$$

=--

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

This notion is a special case of the the notion of _[[mate]]_ in a [[2-category]].

Namely if $K \coloneqq \mathbf{B}_\otimes \mathcal{C}$ is the [[delooping]] [[2-category]] of the [[monoidal category]] $(\mathcal{C}, \otimes)$, then [[objects]] of $\mathcal{C}$ correspond to [[morphisms]] of $K$, [[dual objects]] correspond to [[adjunctions]] and [[morphisms]] in $\mathcal{C}$ correspond to [[2-morphisms]] in $K$. Under this identification a morphism $f \colon X \to Y$ in $\mathcal{C}$ may be depicted as a [[2-morphism]] of the form

$$
  \array{
     \ast &\stackrel{\mathbb{1}}{\to}& \ast
     \\
     {}^{\mathllap{Y}}\downarrow &\swArrow_{\mathrlap{f}}& \downarrow^{\mathrlap{X}}
     \\
     \ast &\underset{\mathbb{1}}{\to}& \ast
  }
$$

and duality on morphisms is then given by the [[mate]] [[bijection]]

$$
  \array{
    \ast & \overset{\mathbb{1}}{\to} & \ast
    \\
    {}^{\mathllap{Y}} \downarrow & \swArrow_{\mathrlap{f}} & \downarrow^{\mathrlap{X}}
    \\
    \ast & \underset{\mathbb{1}}{\to} & \ast
  }
  \;\;\;\;\;
    \mapsto
  \;\;\;\;\;
  \array{
    \ast & \overset{\mathbb{1}}{\to} & \ast
    \\
    {}^{\mathllap{X^\ast}} \downarrow & \swArrow_{\mathrlap{f^\ast}} & \downarrow^{\mathrlap{Y^\ast}}
    \\
    \ast & \underset{\mathbb{1}}{\to} & \ast
  }
  \;\;\;\;
  \coloneqq
  \;\;\;\;
  \array{
    \ast & \overset{Y^\ast}{\to} & \ast & \overset{\mathbb{1}}{\to}
    & \ast &   \overset{\mathbb{1}}{\to} &    \ast
   \\
    {}^\mathllap{\mathbb{1}}\downarrow &
   \swArrow_{\epsilon_Y} & {}^{\mathllap{Y}} \downarrow &
    \swArrow_{f}     & \downarrow^{\mathrlap{X}} &
    \swArrow_{\eta_X}
    & \downarrow \mathrlap{1}
    \\
    \ast & \underset{\mathbb{1}}{\to} & \ast & \underset{\mathbb{1}}{\to}
    & \ast & \underset{X^\ast}{\to}    & \ast
  }
  \,.
$$

=--

## Examples

+-- {: .num_example}
###### Example

In $\mathcal{C} = $ [[Vect]] with its standard [[tensor product]] monoidal structure, a [[dual object]] is a [[dual vector space]] and a dual morphism is a [[dual linear map]].

=--

+-- {: .num_example}
###### Example

If $A$, $B$ are [[C*-algebras]] which are [[Poincaré duality algebras]], hence [[dualizable objects]] in the [[KK-theory]]-category, then for $f \colon A \to B$ a morphism it is [[K-orientation|K-oriented]], the corresponding [[Umkehr map]] is (postcomposition) with the dual morphism of its [[opposite algebra]] version:

$$
  f! \coloneqq (f^op)^\ast
  \,.
$$

=--

See at _[KK-theory -- Push-forward in KK-theory](KK-theory#UmkehrMap)_.

+-- {: .num_example}
###### Example

More generally, [[twisted Umkehr maps]] in [[generalized cohomology theory]] are given by dual morphisms in [[(∞,1)-category of (∞,1)-modules]]. See at _[[twisted Umkehr map]]_ for more.

=--

## Related concepts

* [[trace]]

[[!redirects dual morphisms]]


dual net of von Neumann algebras

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

#Contents#
* table of contents
{:toc}

## Idea ##

Duality for [[nets]] of [[von Neumann algebras]] is a concept that was introduced for [[local nets]] in the [[Haag-Kastler approach]] to [[quantum field theory]]. Many results of this approach need some sort of duality in the sense described here or close to it as a precondition.

## Definition ##
Let $(M_k)$ be a [[net]] of [[von Neumann algebras]] with a [[causal index set]] $I$ that is a [[causal net of algebras]]. Let $R$ be the global algebra of the net, that is
$$
R = (\bigcup_k M_k){''}
$$
An index $i \in I$ satisfies **duality** if
$$
M_i = (\bigcup_{j \perp i} M_j)' \bigcap R = (\bigcap_{j \perp i} M_j^') \bigcap R = \bigcap_{j \perp i} M^c_j
$$
Here $M^c_j$ is the [[relative commutant]] of $M_j$ with respect to $R$.

The net is called **dual** if every index is dual i.e. satisfies duality.

A weaker concept is that of **essentially dual**:

Define an extension $(\hat M_k)$ of the net $(M_k)$ via

$$
\hat M_i = (\bigcup_{j \perp i} M_j)'
$$
This extension is not necessarily a [[causal net]] anymore. If it is, then it is dual by definition. The net $(M_k)$ is **essentially dual**, if the extended net $(\hat M_k)$ is dual, which is true iff $(\hat M_k)$ is causal.

The net $(M_k)$ is called **maximal** if there is no proper extension which satisfies the [[causality condition]].

## Examples ##
A [[Haag-Kastler vacuum representation]] satisfies **Haag duality** if every double cone aka diamond is a dual index. The reason for this relaxation is that full duality of every index is often too restrictive, so that the less restrictive Haag duality plays an important role in the theory.

Let $J_0$ be the index set of diamonds, a [[Haag-Kastler vacuum representation]] is **essentially Haag dual** if the net $M(J_0)$ (that is the original net restricted to diamonds as indices) is essentially dual.

[[!redirects Haag duality]]
[[!redirects Haag dual]]
[[!redirects essentially Haag dual]]
[[!redirects essential Haag duality]]
[[!redirects maximal net]]
[[!redirects maximal nets]]
[[!redirects dual net]]
[[!redirects dual nets]]
[[!redirects essentially dual net]]
[[!redirects essentially dual nets]]


dual number

# Contents
* automatic table of contents goes here
{: toc}

## Idea

A __dual number__ is given by an expression of the form $a + \epsilon b$, where $a$ and $b$ are [[real numbers]] and $\epsilon^2 = 0$ (but $\epsilon \ne 0$).  The set of dual numbers is a [[topological vector space]] and a [[commutative algebra]] over the real numbers.

We can generalise (at least the algebraic aspects) from $\mathbb{R}$ to any [[commutative ring]] $R$.

## Interpretations

This can be thought of as:

*  the vector space $\mathbb{R}^2$ made into an [[associative algebra|algebra]] by the rule
   $$ (a, b) \cdot (c, d) = (a c, a d + b c) ;$$
*  the subalgebra of those $2$-by-$2$ real [[matrix|matrices]] of the form
   $$ \left(\array { a & b \\ 0 & a } \right);$$
*  the [[polynomial]] ring $\mathbb{R}[\mathrm{x}]$ modulo $\mathrm{x}^2$;
*  the parabolic $2$-dimensional algebra of [[hypercomplex number]]s;
*  the algebra of functions on the [[infinitesimal object|infinitesimal interval]] (the smallest of the [[infinitesimally thickened points]]) in [[synthetic differential geometry]].
*  if $\epsilon$ is regarded as being of degree 1 and $\mathbb{R} \oplus \epsilon \mathbb{R}$ is regarded accordingly as a [[superalgebra]] then this is the algebra of functions on the [[odd line]] $\mathbb{R}^{0|1}$.
*  the square-0-extension corresponding to the $\mathbb{R}$-[[module]] (see there) given by $\mathbb{R}$ itself.

We think of $\mathbb{R}$ as a [[subset]] of $\mathbb{D}$ by identifying $a$ with $a + 0 \epsilon$.

## Properties

$\mathbb{D}$ is equipped with an [[involution]] that maps $\epsilon$ to $\bar{\epsilon} = -\epsilon$:
$$ \overline{a + \epsilon b} = a - \epsilon b .$$
$\mathbb{D}$ also has an [[absolute value]]:
$$ {|a + \epsilon b|} = {|a|} ;$$
notice that the absolute value of a dual number is a non-negative real number, with
$$ {|z|^2} = z \bar{z}. $$
But this absolute value is degenerate, in that ${|z|} = 0$ need not imply that $z = 0$.

Some concepts in analysis can be extended from $\mathbb{R}$ to $\mathbb{D}$, but not as many as work for the [[complex numbers]].  Even algebraically, the dual numbers are not as nice as the real or complex numbers, as they do not form a [[field]].

[[!redirects parabolic complex number]]
[[!redirects parabolic complex numbers]]
[[!redirects dual number]]
[[!redirects dual numbers]]
[[!redirects dual number system]]
[[!redirects dual number systems]]
[[!redirects ring of dual numbers]]
[[!redirects rings of dual numbers]]

[[!redirects algebra of dual numbers]]


dual object in a closed category

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

# Contents
* table of contents
{: toc}

## Conventional meaning

Let $V$ be a [[closed category]] (for example a [[closed monoidal category]]), with the [[internal hom]] denoted as $A \multimap B$ and the [[unit object]] as $I$,

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

The ("weak") **dual** $\mathbb{D}A$ of an object $A \in V$ is the [[internal hom]] out of $A$ into the [[unit object]]

$$
  \mathbb{D}A \coloneqq A \multimap I
$$

=--

This applies for example to categories of [[modules]] over a [[commutative ring]], to the category of [[Banach spaces]], etc., with their usual closed monoidal structures.

+-- {: .num_remark}
###### Warning
A dual in the sense above might not be a [[dual object]] in other established senses of the word. For example,

* The dual $X^\ast$ of a [[Banach space]] $X$ is generally not the dual in any sense that guarantees $X^\ast \otimes -$ is [[adjoint functor|left or right adjoint]] to $X \otimes -$, unless $X$ is finite-dimensional. (Here the monoidal product is the [[projective tensor product]], which makes the closed category of Banach spaces into a closed monoidal category.) In particular, there is generally no appropriate unit of the form $I \to X^\ast \otimes X$ (where $I$ is the [[ground field]]).

=--

For a [[compact closed category]], a dual object (in one or another monoidal category sense; see [[category with duals]]) _is_ the same as a dual in the closed category sense. But in general, monoidal duals are a stronger notion than duals in the closed category sense.

Of course, where the monoidal product is not [[braided monoidal category|braided or symmetric monoidal]], due care must be exercised in distinguishing between between different types of monoidal dual (left or right dual).

* If $V$ is a [[star-autonomous category]], "dual object" has a different meaning: it usually is taken to mean $A^\ast \cong A \multimap \bot$ where $\bot$ is the [[dualizing object]]. In a $\ast$-autonomous category where $\bot$ and $I = \top$ are not isomorphic (e.g., a [[Boolean algebra]]), this gives something genuinely different from "dual" in the sense of this page. But again, these two meanings, $A \multimap \top$ and $A \multimap \bot$, coincide in the case where the $\ast$-autonomous category is compact closed.

## Applications

Closed duals play a central role in [[six operations]] yoga, notably in
[[Verdier duality]] and in the [[Wirthmüller isomorphism]].

## Related concepts

* [[dualizing object in a closed category]]

* [[dual object]]

* [[duality]]

[[!redirects dual object in a closed monoidal category]]
[[!redirects dual objects in a closed monoidal category]]
[[!redirects dual objects in closed monoidal categories]]
[[!redirects dual in a closed monoidal category]]
[[!redirects duals in a closed monoidal category]]
[[!redirects duals in closed monoidal categories]]
[[!redirects dual object in a closed category]]
[[!redirects dual objects in a closed category]]
[[!redirects dual objects in closed categories]]
[[!redirects dual in a closed category]]
[[!redirects duals in a closed category]]
[[!redirects duals in closed categories]]

[[!redirects algebraic dual]]
[[!redirects algebraic duals]]


dual photon

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

#Contents#
* table of contents
{:toc}

## Idea

A dual of the [[photon]] under [[electric-magnetic duality]] (e.g. [Bakas 01](#Bakas01), [Bliokh 13](#Bliokh13)).

## Related concepts

* [[dual graviton]]
* [[Seiberg duality]]
* [[S-duality]]

## References

* {#Bakas01} [[Ioannis Bakas]], _Dual photons and gravitons_,  Publ. Astron. Obs. Belgrade 88:113, 2010 (arXiv:[0910.1739 ](https://inspirehep.net/record/833555))

* {#Bliokh13} Konstantin Y. Bliokh, Aleksandr Y. Bekshaev, Franco Nori, _Dual electromagnetism: helicity, spin, momentum and angular momentum_,  New J. Phys. 15:033026, 2013 (doi:[10.1088/1367-2630/15/3/033026](https://doi.org/10.1088/1367-2630/15/3/033026))

[[!redirects dual photons]]


dual space

There are many general and specific notions of "[[space]]" and of "[[duality]]", and accordingly "dual space" can mean many things.

Perhaps the most common usage is as in **[[dual vector space]]** in [[linear algebra]] and [[functional analysis]]. For others see at _[[duality]]_.

category: disambiguation

[[!redirects dual space]]
[[!redirects dual spaces]]


dual space > history

< [[dual space]]


dual vector bundle

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[vector bundle]], its _dual_ is the vector bundle obtained by passing [fiber-wise](topological+vector+bundle#FiberwiseOperations) to the [[dual vector space]].

## Examples

* Given a [[smooth manifold]] $X$ with [[tangent bundle]] $T X$, the dual vector bundle is the [[cotangent bundle]] $T^\ast X$.

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

Let $X$ be a [[topological space]] and let $E_i \overset{p_i}{\to} X$ be two [[topological vector bundles]] over $X$, of [[finite number|finite]] [[rank of a vector bundle]]. Then a [[homomorphism]] of vector bundles

$$
  f \;\colon\; E_1 \rightarrow E_2
$$

is equivalently a [[section]] of the [[tensor product of vector bundles]] of $E_2$ with the dual vector bundle of $E_1$.

$$
  Hom_{Vect(X)}(E_1, E_2)
   \;\simeq\;
  \Gamma_X( E_1^\ast \otimes_X E_2 )
  \,.
$$

Moreover, this section is a trivializing section ([this example](topological+vector+bundle#FiberwiseLinearlyIndependentSectionsTrivialize)) precisely if the corresponding morphism is an [[isomorphism]].

=--

## Related concepts

* [[direct sum of vector bundles]]

* [[tensor product of vector bundles]], [[external tensor product of vector bundles]]

* [[tensor category]]

## Related concepts

* [[Max Karoubi]], §4.8 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften **226** Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3)]

[[!redirects dual vector bundes]]


dual vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _dual vector space_ is a [[dual in a closed category]] of [[vector spaces]] (or similar [[algebraic structures]]).

Of course, this is a very restricted notion of [[space]]; but for spaces in [[geometry]], one usually uses the [[duality between space and quantity]] and speaks of the [[spectrum]] (not 'dual space') of an [[algebra]].  In [[homotopy theory]], there are also [[Spanier-Whitehead dual]]s; and there are other notions of dual space in [[physics]].

## Definitions

Let $K$ be a [[field]] (or any [[commutative ring|commutative]] [[rig]]), and let $V$ be a [[vector space]] (or [[module]]) over $K$.

+-- {: .num_defn}
###### Definition
The __dual space__ or __dual module__ of $V$ is the vector space $V^*$ of [[linear functionals]] on $V$.  That is, $V^*$ is the [[internal hom]] $[V,K]$ (thinking of $K$ as a vector space over itself: a [[line]]).
=--

More generally, let $K$ and $L$ be [[rings]] (or [[rigs]]) *not* assumed commutative, and let $V$ be a $K$-$L$-[[bimodule]].

+-- {: .num_defn}
###### Definitions
The __left dual module__ of $V$ is the right $K$-module $^*V$ of left $K$-module [[homomorphisms]] from $V$ to $K$.  The __right dual module__ of $V$ is the left $L$-module $V^*$ of right $L$-module homomorphisms from $V$ to $L$.
=--

Now let $V$ be a [[topological vector space]] over the [[ground field]] $K$.

+-- {: .num_defn #LinearDualOfATopologicalVectorSpace}
###### Definitions
**(linear dual of a topological vector space)**

Let $V$ be a [[topological vector space]] over the [[ground field]] $K$.

The __dual space__ of $V$ is the topological vector space $V^*$ of [[continuous linear functionals]] on $V$, equipped with the [[weak-star topology|weak-* topology]] (meaning the [[initial topology]] generated by the elements of $V$, viewed as themselves linear functionals on $V^*$).

=--

In principle, there is no conflict among these definitions, the most general case (so far) being a topological bimodule over two topological rigs; the non-topological cases simply involve [[discrete spaces]].  In practice, however, some complications are possible:

*  If the rig $K$ is an [[associative algebra|algebra]] over another rig $L$, then any $K$-module $V$ is also an $L$-module, but the dual as a $K$-module is different from the dual as an $L$-module.  So one may speak of the __$K$-dual__ or the __dual over $K$__.

*  A topological vector space $V$ has an [[underlying]] discrete vector space, and these have different duals.  So one speaks of the __topological dual__ and the __algebraic dual__ (respectively).  If $V$ is considered with several different topologies (say called 'weak' and 'strong'), then one may speak of the __weak dual__ and the __strong dual__ (etc).

Logically, these duals take place in different [[categories]], which are related by various [[functors]]; the objects whose duals are being taken should not be conflated.  In practice, however, these objects are identified, so the duals must be distinguished.

## Transpose maps

The operation $V \mapsto V^*$ extends to a [[contravariant functor]].

+-- {: .num_defn}
###### Definition (transpose map)
The [[dual linear map]] or __transpose map__ of a [[linear map]] $A\colon V\to W$, is the linear map $A^* = A^T\colon W^*\to V^*$, given by
$$ \langle{A^*(w), v}\rangle = \langle{w, A(v)}\rangle $$
for all $w$ in $W^*$ and $v$ in $V$.
=--

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

This functor is, of course, the [[representable functor]] represented by $K$ as a vector space over itself (a [[line]]).

=--

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

This construction is the notion of [[dual morphism]] applied in the [[monoidal category]] [[Vect]] with its [[tensor product]] monoidal structure.

=--

## Dual bases

If $B$ is any [[basis]] of $V$, then we can sometimes turn $B$ into a basis $B^*$ of the dual space $V^*$.

We will begin with the definition of what *might* be the dual basis, cautioning that it is not always a basis:

+-- {: .num_defn}
###### Definition (dual basis)
Treating the basis $B$ as a [[family]] $(b_i)$ with index set $I$, the __dual basis__ $B^*$ is the family $(b^i)$ (with the same index set) such that
$$ b^i(b_j) = \delta^i_j $$
(the [[Kronecker delta]]).
=--

Since $B$ is a basis of $V$, this formula defines $b^i$ for each index $I$, so $B^*$ exists; but in general there is no reason why $B^*$ should be a basis of $V^*$.  However, if $V$ has [[finite set|finite]] [[dimension]], then $B^*$ is a basis of $V^*$. If $V$ is a [[Hilbert space]], and $B$ is a basis of $V$ in the Hilbert space sense (i.e., $B$ is a linearly independent set whose span is topologically dense in $V$), then also $B^*$ is a basis of the dual Hilbert space $V^*$.

This is related to but different from the sort of [[dual basis]] applicable generally to [[projective modules]].

## Double duals
 {#DoubleDualVectorSpaces}

The dual $\big(V^{\ast}\big)^\ast$ of the dual $V^\ast$ of a vector space $V$ is also called its *[[double dual]]*.

There is a [[natural transformation]] from $V$ to its double dual:
\begin{tikzcd}[sep=0pt]
  V \ar[rr] && \big(V^\ast\big)^\ast
  \\
  v
    &\mapsto&
  \big(
    \omega(-) \mapsto \omega(v)
  \big)
\end{tikzcd}

> (In fact, historically this was the motivating example for the notion of natural transformations in the first place, see [there](natural+transformation#UnitOfDoubleDualization))

The space $V$ is called __[[reflexive Banach space|reflexive]]__ if this natural transformation is an [[isomorphism]].  The reflexive spaces include all [[finite-dimensional vector spaces]] (or more generally [[modules]]) over [[fields]] (or more generall [[division rings]]), as well as all [[Hilbert spaces]], the [[Lebesgue spaces]] $L^p$ over a [[localizable measure|localisable measure space]] for $1 \lt p \lt \infty$, and others.

## Dual spaces as dual objects

A dual vector space is a [[dual object]] in the [[monoidal category]] [[Vect]] equipped with its [[tensor product]] monoidal structure.

In general, the duality between $V$ and $V^*$ does *not* make $Vect$ into a [[monoidal category with duals]].  However, if we restrict to spaces of [[F-finite|finite]] [[dimension]], then we get a [[compact category]]; finite-dimensional [[Hilbert spaces]] form a $\dagger$-[[dagger-compact category|compact category]], which is very nice indeed.

## Examples

+-- {: .num_example #DualityBetweenVectorsAndCovectors}
###### Example
**(duality between [[vectors]] and [[covectors]])**

* [[tensor|Tensors]] in finite-dimensional [[differential geometry]] make heavy use of the duality between [[tangent spaces]] and [[cotangent spaces]].  That [[infinite-dimensional manifold|infinite-dimensional differential geometry]] is much harder is largely because this duality is no longer perfect.

=--

+-- {: .num_example #VonNeumannAlgebras}
###### Example
**([[von Neumann algebras]])

A [[von Neumann algebra]] (abstractly) is precisely a $C^*$-[[C-star-algebra|algebra]] whose underlying [[Banach space]] is the dual space of some (other) Banach space.  One may equivalently *define* a von Neumann algebra as a Banach space together with a $C^*$-algebra structure on its dual space (except that the [[morphisms]] go the other way, so one is more directly defining a noncommutative [[measurable space]], along the lines of [[noncommutative geometry]]).

=--

+-- {: .num_example #SpacesOFDistributionsAsDualVectorSpaces}
###### Example
**(spaces of [[distributions]])**

The standard [[topological space|topology]] on the spaces $\mathcal{D}'$ of [[distributions]] is the dual space topology according to def. \ref{LinearDualOfATopologicalVectorSpace}.

(e.g. [Hörmander 90, p. 38](#Hoermander90))

=--

See _[[Riesz representation theorem]]_ for more examples from [[functional analysis]].

## Related concepts

* [[dual space]]

* [[anti-dual linear space]]

* [[dual vector bundle]]

* [[bra-ket]]

* [[hyperbolic functor]]

## References

Discussion in the context of [[distributions]]:

* {#Hoermander90} [[Lars Hörmander]], section 2.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

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

[[!redirects dual linear space]]
[[!redirects dual linear spaces]]

[[!redirects dual module]]
[[!redirects dual modules]]

[[!redirects left dual module]]
[[!redirects left dual modules]]
[[!redirects right dual module]]
[[!redirects right dual modules]]

[[!redirects dual topological vector space]]
[[!redirects dual topological vector spaces]]
[[!redirects dual Banach space]]
[[!redirects dual Banach spaces]]
[[!redirects dual Hilbert space]]
[[!redirects dual Hilbert spaces]]

[[!redirects alebraic dual space]]
[[!redirects alebraic dual spaces]]
[[!redirects alebraic dual]]
[[!redirects alebraic duals]]

[[!redirects topological dual space]]
[[!redirects topological dual spaces]]
[[!redirects topological dual]]
[[!redirects topological duals]]

[[!redirects linear dual]]
[[!redirects linear duals]]

[[!redirects linear dual space]]
[[!redirects linear dual spaces]]

[[!redirects linear form]]
[[!redirects linear forms]]

[[!redirects double dual]]
[[!redirects double duals]]


duality

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

# Duality
* table of contents
{: toc}

## Idea

Instances of "dualities" relating two different, maybe opposing, but to some extent equivalent concepts or phenomena are ubiquitous in [[mathematics]] (and in [[mathematical physics]], see at _[[dualities in physics]]_). The term "duality" is widespread and doesn't have a single crisp meaning, but a rough guiding intuition is of pairs of concepts that are mirror images of one another.

## General notions

In terms of general abstract concepts in [[logic]] and [[category theory]], instances of dualities might be (and have been) organized as follows.

### Abstract/formal/axiomatic duality

Instances here include projective duality, duality in lattice theory, and duality in category theory. In each case, one has a [[theory]] whose [[signature]] admits a nontrivial [[involution]], in such a way that "dualizing" or applying the involution to any axiom of the theory (but otherwise preserving the logical structure of formulae) results in another theorem of the theory. For example, for the theory of projective planes, the involution swaps points and lines, meets and joins, etc., and for each theorem there is a dual theorem. Similarly, in [[category]] theory, the involution swaps the domain and codomain and order of composition, etc., and for any theorem of formal category theory, the corresponding dual statement is also a theorem (because the set of axioms of category theory are closed under taking formal duals).

Such formal duality can also be expressed at the level of [[models]] of the theory $T$: for each model $M$ there is a "dual" or "opposite" model $M^{op}$ obtained by re-interpreting each formal sort/function/relation of the theory as the dual sort/function/relation. This further induces an involution $Mod(T) \to Mod(T)$ on the category of models. For example, in the case of category theory, this operation $C \mapsto C^{op}$, mapping a category $C$ to its [[opposite category]], gives an involution $(-)^{op}: Cat \to Cat$ on the category [[Cat]] of (small) categories, viewing $Cat$ here as a 1-category. In fact this is the only non-trivial automorphism of [[Cat]], see [here](%28infinity%2Cn%29Cat#automorphisms)).

This involution $(-)^{op}: Cat \to Cat$ is also known as _abstract duality_. While the construction is a priori tautologous, any given [[opposite category]] often is equivalent to a [[category]] known by other means, which makes abstract duality interesting (particularly so in cases of concrete duality, which we discuss next).

### Concrete dualities

Instances here include linear duality, Stone duality, Pontryagin duality, and projective inversions with respect to a conic hypersurface. In each such case there is some [[contravariant functor|contravariant]] process of "homming into" a suitable structure $V$ called a [[dualizing object]], which in the classical cases of what we will call "perfect duality", induces an equivalence of categories $C^{op} \stackrel{\sim}{\to} D$ where typically $C$ is the category of models of one type of concrete structure, and the equivalence maps the formal categorical dual $C^{op}$ to a category of models $D$ consisting of "dual" concrete structures.

In other cases one might not obtain an equivalence or perfect duality, but in any case a contravariant [[adjoint pair]] of functors $S: C \to D$, $T: D \to C$ between categories which can be termed a duality of sorts, in that concepts developed in $C$ are mapped to dual concepts in $D$ and vice-versa. Quoting ([Lawvere-Rosebrugh, chapter 7](#LawvereRosebrugh)):

> Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, and indeed a large part of the study of mathematics

>>space vs. quantity

> and of logic

>>theory vs. example

>may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond. (p. 122)

Some examples follow.

* In linear duality, say for vector spaces over a field $k$, the dual of a space $W$ is $W^\ast = \hom(W, k)$, the hom of $k$-linear maps into $k$. This induces a contravariant functor $(-)^\ast: Vect_k \to Vect_k$ that is adjoint to itself, in that there is a double-dual embedding $\delta_W: W \to W^{\ast\ast}$ such that
$$1_{W^\ast} = (W^\ast \stackrel{\delta_{W^\ast}}{\to} W^{\ast\ast\ast} \stackrel{(\delta_W)^\ast}{\to} W^\ast).$$
This becomes a perfect duality if we restrict to finite-dimensional vector spaces, i.e., the contravariant functor $(-)^\ast: Vect_{fd} \to Vect_{fd}$ is adjoint-equivalent to itself (i.e., the covariant functor $((-)^\ast)^{op}: Vect_{fd} \to Vect_{fd}^{op}$ is left adjoint to $(-)^\ast: Vect_{fd}^{op} \to Vect_{fd}$ and the adjunction is an [[adjoint equivalence]]).

* More generally, given a (usually [[symmetric monoidal category|symmetric]]) [[monoidal closed category]] $\mathcal{C}$, any object $D$ induces an operation $[-,D] : \mathcal{C} \to \mathcal{C}^{op}$ obtained by forming the [[internal hom]] into $D$, sending each object to what may be termed its $D$-dual object. There is a corresponding double-dual embedding $\delta_C: C \to [ [C, D], D]$ obtained as the composite
$$C \to [ [C, D], C \otimes [C, D]] \to [ [C, D], [C, D] \otimes C] \to [ [C, D], D]$$
where the first arrow uses the unit of an adjunction $(- \otimes B) \dashv [B, -]$ where $B = [C, D]$, the second uses a symmetry isomorphism $C \otimes B \cong B \otimes C$, and the third uses the counit of an adjunction $(- \otimes C) \dashv [C, -]$, aka an evaluation map. It may be shown that the contravariant functor $(-)^\ast \coloneqq [-, D]: \mathcal{C} \to \mathcal{C}$ is again dual to itself, exactly as in the case of linear duality above, where we have a triangular equation
$$1_{C^\ast} = (\delta_C)^\ast \circ \delta_{C^\ast}$$
for an adjunction $[-, D]^{op} \dashv [-, D]$. Under certain circumstances, we have perfect duality, i.e., double dualization $[ [-, D], D]: \mathcal{C} \to \mathcal{C}$ is an equivalence; see [[dualizing object in a closed category]] and [[star-autonomous category]]. Particular special cases of this may obtain when $D = I$, the monoidal unit, or even more particularly when every object has a [[dual object]] in the sense of [[monoidal categories]]; see also [[compact closed category]]. On the other hand, there is also a mild generalization of this type of example where we deal with a [[biclosed monoidal category]]; here the double dualization will involve both the left and right internal hom.

* More general still is a concrete duality induced by a [[dualizing object]]. In this case one is given a pair of categories together with underlying-set functors
$$U: \mathcal{C} \to Set, \qquad V: \mathcal{D} \to Set$$
(and often when one says "concrete", one intends that these functors be faithful as well, so that $\mathcal{C}, \mathcal{D}$ can be viewed as "sets with structure"; see [[stuff, structure, property]]). The concrete duality consists of a pair of objects $C \in \mathcal{C}, D \in \mathcal{D}$ together with an isomorphism $\omega: U C \cong V D$, such that the contravariant homs $\hom(-, C): C^{op} \to Set$ and $\hom(-, D): D^{op} \to Set$ lift to a contravariant adjunction between $C$ and $D$, in the sense described [here](/nlab/show/dualizing+object#ambi). Frequently in practice, such concrete dualities are "naturally represented" in the sense described [here](/nlab/show/dualizing+object#natural), involving certain lifts adapted from the theory of [[topological concrete categories]].

Again, in all of these examples, one can consider the further condition of "perfect duality" where the units and counits of the (lifted) adjunctions are isomorphisms.

### Adjunctions
 {#Adjunctions}

Perhaps the loosest general notion of "duality" is that of [[adjunction]], as in pairs of [[adjoint functors]] ([Lambek 81](#Lambek81)). Here one may omit any concretizations via functors to $Set$, or even for that matter any explicit mention of opposite categories, and just work at the level of abstract categories themselves.

Nevertheless, many adjunctions come packaged in "dual pairs". A famous slogan from [[Categories for the Working Mathematician]] is that "all concepts are Kan extensions", and in that light the dual pairs are instances of the general dual pair "(right Kan extension, left Kan extension)" which are formal duals in the axiomatic sense described earlier. Via the many incarnations of [[universal constructions]] in [[category theory]], we have for example

   * [[limit]] and [[colimit]]

   * [[end]] and [[coend]]

   * [[dependent sum]] and [[dependent product]]

   * [[existential quantification]] and [[universal quantification]]

When the adjoint functors are [[monads]] and hence [[modalities]], then adjointness between them has been argued to specifically express the concept of [[duality of opposites]].

Again, adjunctions and specifically [[dual adjunctions]] ("[[Galois connections]]") may be thought of as generalized dualities, more general than "perfect duality" which involves equivalences between categories ("Galois correspondences"). However, it should also be noted that any such adjunction (or [[dual adjunction]]) restricts to a _maximal (dual) equivalence_ between [[subcategories]], by considering objects where the appropriate units and counits are isomorphisms. This generalizes the manner by which any Galois connection induces a Galois correspondence (where in this special case, one need only take the images of the poset maps which constitute the connection).

## Examples

* [[opposite category]]

* [[dual object]], [[dualizing object]], [[dualizing object in a closed category]]

* [[dual space]], [[dual vector space]]
* the duality between [[space and quantity]]
* [[Poincaré duality]] for finite dimensional (oriented) closed manifolds

* [[Spanier-Whitehead duality]], [[Brown-Comenetz duality]], [[Anderson duality]]

* [[Pontryagin duality]] for commutative ([[Hausdorff space|Hausdorff]]) [[topological group|topological groups]]
* [[Cartier duality]] of a finite flat commutative [[group scheme]]
* [[Serre duality]] on nonsingular projective algebraic varieties which has as a special case the statement of the [[Riemann-Roch theorem]]
* [[Grothendieck duality]], [[coherent duality]] for [[coherentsheave|coherent sheaves]]
* [[Verdier duality]] for abelian categories of sheaves; e.g. for a category of sheaves of abelian groups.
* [[Artin-Verdier duality]] generalizing [[Tate duality]] for constructible sheaves over the spectrum of a ring of [[algebraic number|algebraic numbers]]
* [[Koszul duality]]
* [[Stone duality]]
* [[syntax - semantics duality]]
* [[Eckmann-Hilton duality]]
* [[Gabriel-Roos-Oberst duality]]
* [[Tannaka duality]]
* [[Hochster duality]]

## Dualizing objects

Of particular interest are concrete dualities between [[concrete categories]] $C, D$, i.e. categories equipped with faithful functors
$$ f : C \to Set$, $\hat f : D \to Set $$
to [[Set]], which are represented by objects $a \in C$, $\hat a \in D$ with the same underlying set $f(a) = \hat f(\hat a)$. Such objects are known as [[dualizing objects]].

## Related concepts

* [[de Morgan duality]]

* [[Isbell duality]]

* [[unity of opposites]]

* [[duality in physics]]

* [[fixed point of an adjunction]]

* [[mysterious duality]]

## References

* {#Lambek81} [[Joachim Lambek]], _The Influence of Heraclitus on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1981)

* {#PorstTholen91} [[Hans-E. Porst]], [[Walter Tholen]], _Concrete Dualities_ in H. Herrlich, [[Hans-E. Porst]] (eds.) _Category Theory at Work_, Heldermann Verlag 1991 ([pdf](http://www.heldermann.de/R&E/RAE18/ctw07.pdf))

* [[Saunders MacLane]], §II.1 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 1.10 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

* {#LawvereRosebrugh} [[William Lawvere]], [[Bob Rosebrugh]], chaper 7 of: _[[Sets for Mathematics]]_ (2004) [[web](http://www.mta.ca/~rrosebru/setsformath/)]

* {#Corfield17} [[David Corfield]], _Duality as a category-theoretic concept_, Science Part B: Studies in History and Philosophy of Modern Physics Volume 59, August 2017, Pages 55-61 ([doi:10.1016/j.shpsb.2015.07.004](https://doi.org/10.1016/j.shpsb.2015.07.004))

Discussion of duality specifically in [[homological algebra]] and [[stable homotopy theory]] with emphasis on the concept of [[dualizing object in a closed category]] (and the induced [[Umkehr maps]] etc.) is in

* [[William Dwyer]], [[John Greenlees]], S. Iyengar, _Duality in algebra and topology_, [Hopf archive](http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Greenlees-Iyengar/duality)

See also

* [[David Corfield]]; _[More on duality](http://golem.ph.utexas.edu/category/2007/01/more_on_duality.html)_ (blog)

* wikipedia [duality (mathematics)](http://en.wikipedia.org/wiki/Duality_%28mathematics%29)

* MathOverflow: [the-concept-of-duality](http://mathoverflow.net/questions/73711/the-concept-of-duality)

[[!redirects duality]]
[[!redirects dualities]]

[[!redirects dualization]]

[[!redirects formal duality]]
[[!redirects formal dualities]]

[[!redirects abstract duality]]
[[!redirects abstract dualities]]

[[!redirects formal dual]]
[[!redirects formal duals]]

[[!redirects formally dual]]

[[!redirects concrete duality]]
[[!redirects concrete dualities]]


duality - contents

**[[duality]]**

* abstract duality: [[opposite category]],

  [[Eckmann-Hilton duality]]

* concrete duality: [[dual object]], [[dualizable object]], [[fully dualizable object]], [[dualizing object]]

  * [[dual vector space]]

**Examples**

* between [[higher geometry]]/[[higher algebra]]

  * [[Isbell duality]]

  * [[Stone duality]]

  * [[Gelfand duality]]

* [[Tannaka duality]]

* [[Langlands duality]], [[geometric Langlands duality]], [[quantum geometric Langlands duality]]

* [[Pontryagin duality]]

  * [[Cartier duality]]

* [[Poincaré duality]] for [[manifolds]]

* [[Koszul duality]]

* [[Spanier-Whitehead duality]]

* [[Grothendieck duality]]

**In QFT and String theory**

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

* [[Seiberg duality]], [[AGT conjecture]]

* [[S-duality]]

  * [[electro-magnetic duality]], [[Montonen-Olive duality]], [[geometric Langlands duality]]

* [[T-duality]]

  * [[mirror symmetry]]

* [[U-duality]]

* [[open/closed string duality]]

  * [[AdS/CFT|AdS/CFT duality]]

  * [[KLT relations]]


duality between algebra and geometry

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

#Contents#
* table of contents
{:toc}

## Idea

In [[mathematics]], one often has an [[equivalence of categories]]
between [[algebra]]-like objects and [[space]]-like objects.
Such an idea has many incarnations: [[Stone duality]], [[Gelfand duality]], etc.,
and in this article we make some observations that are common to these dualities.

Given an [[algebra]]-like object $A$, we assign to it its [[poset]] of [[ideals]]
(typically defined as [[kernels]] of [[homomorphisms]] $A\to B$),
which is interpreted as the [[poset]] of opens of some [[space]] $S$.

The technical term for such [[posets]] is [[locale]],
which is a notion very closely related to [[topological spaces]].
In particular, from any [[locale]] one can canonically [extract a topological space](https://ncatlab.org/nlab/show/locale#RelationToTopologicalSpaces),
and this is the [[topological space]] $S$ produced in many classical Stone-type dualities.
The points of $S$ are [[ideals]] corresponding to morphisms $A\to k$, where $k$ is often a particularly simple algebra.
These often turn out to be [[maximal ideals]] in $A$.

Conversely, given a [[space]]-like object $S$, we assign to it the [[algebra]] of morphisms $S\to k$,
where $k$ is often the “same” [[algebra]] $k$ as above,
only this time its underlying object is a [[space]], not just a [[set]].

Some examples from [[general topology]], [[measure theory]], [[differential geometry]], [[algebraic geometry]], and [[complex geometry]]
(the list is very much incomplete):

|algebra|homomorphism|$k$|ideal|space|maps|duality|
|-|-|-|-|-|-|-|
|[[Boolean algebra]]|homomorphism|$\mathbf{Z}/2$|ideal|compact totally disconnected Hausdorff space ([[Stone space]])|continuous map|[[Stone duality]]|
|[[complete Boolean algebra]]|complete homomorphism|$\mathbf{Z}/2$|closed ideal|compact extremally disconnected Hausdorff space ([[Stonean space]])|open continuous map|[[Stonean duality]]|
|[[localizable Boolean algebra]]|complete homomorphism|$\mathbf{Z}/2$|closed ideal|[[hyperstonean space]]|open continuous map||
|[[localizable Boolean algebra]]|complete homomorphism|$\mathbf{Z}/2$|closed ideal|[compact strictly localizable enhanced measurable space](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical/20820#20820)|measurable map||
|[[commutative von Neumann algebra]]|normal \*-homomorphism|$\mathbf{C}$|ultraweakly closed \*-ideal|compact strictly localizable enhanced measurable space|measurable map||
|commutative unital [[C*-algebra]]|\*-homomorphism|$\mathbf{C}$|norm-closed \*-ideal|compact Hausdorff space|continuous map|[[Gelfand duality]]|
|[[commutative algebra]] over $k$|homomorphism|$k$|ideal|[[coherent space]] / [[affine scheme]]|[[continuous map]] / [[morphism of schemes]]|[[Zariski duality]]|
|finitely generated germ-determined [[C^∞-ring]]|C$^\infty$-homomorphism|$\mathbf{R}$|germ-determined ideal|[[smooth locus]] (e.g., [[smooth manifold]])|[[smooth map]]|[[Milnor duality]]|
|finitely presented complex [[EFC-algebra]]|EFC-homomorphism|$\mathbf{C}$|ideal|globally finitely presented [[Stein space]]|holomorphic map|[[Stein duality]]|

and:

[[!include Isbell duality - table]]

The duality relevant to the [[spectral theory]] is the duality between [[commutative von Neumann algebras]]
and [compact strictly localizable enhanced measurable spaces](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical/20820#20820).

Given a [[normal operator]] $T$ on a [[Hilbert space]] $H$,
$T$ generates a [[commutative von Neumann algebra]] $A$ inside $B(H)$, i.e., [[bounded operators]] on $H$.
(This is precisely the point where normality is crucial; without the relation $T^*T=TT^*$
the algebra generated by $T$ will be noncommutative.)

By the cited duality, the [[commutative von Neumann algebra]] $A$
is dual to a compact strictly localizable enhanced measurable space $Spec A$.
This is indeed the spectrum of $T$ in the usual sense.
Under this equivalence, the element $T\in A$
corresponds to the measurable map $Spec A\to \mathbf{C}$ given by inclusion of $Spec A$ into $\mathbf{C}$.

One may ask whether we can recover the full [[spectral theorem]] for a [[normal operator]] in this manner.
This is possible once [[Stone duality]] is upgraded to Serre–Swan-type duality
between [[modules]] and [[vector bundle]]-like objects (including, e.g., [[sheaves of modules]] etc.).

Given a [[vector bundle]]-like object $V\to S$, we assign to it its [[module]] of [[sections]],
which is a [[module]] over the [[algebra]] of maps $S\to k$.
Conversely, given a [[module]] $M$ over $A$,
the corresponding [[vector bundle]]-like object $V\to S$ over $S=Spec A$
has as its [[fiber]] over some point $s\in S$ the [[vector space]] $M/IM$,
where $I$ is the [[ideal]] corresponding to $s$.
(Many details are necessarily omitted in this brief sketch.)

Typically, genuine [[vector bundles]] correspond to [[dualizable modules]]
(dualizable with respect to the [[tensor product]] over $A$).
Non-dualizable module tend to correspond to [[sheaves of modules]] that are not [[vector bundles]],
e.g., [[skyscraper sheaves]] etc.

|[[module]]|[[vector bundle]]-like object|
|-|-|
|module over a Boolean algebra|sheaf of $\mathbf{Z}/2$-vector spaces|
|[[Hilbert W*-module]] over a commutative von Neumann algebra|measurable field of Hilbert spaces|
|[[W*-representations]] of a [[commutative von Neumann algebra]] on a [[Hilbert space]]|[[measurable field of Hilbert spaces]]|
|[[Hilbert C*-module]] over a commutative unital C\*-algebra|continuous field of Hilbert spaces|
|[[module]] over a commutative ring|[[quasicoherent sheaf]] of $\mathcal{O}$-modules over an affine scheme|
|[[dualizable module]] over a commutative algebra over $k$|algebraic vector bundle|
|dualizable module over a finitely generated germ-determined C$^\infty$-ring|smooth vector bundle|
|dualizable module over finitely presented complex EFC-algebra|holomorphic vector bundle|

The duality relevant to the [[spectral theory]] is the duality between representations of a commutative von Neumann algebras on a [[Hilbert space]]
and [[measurable fields]] of Hilbert spaces.

Given a [[normal operator]] $T$ on a [[Hilbert space]] $H$,
$T$ generates a [[commutative von Neumann algebra]] $A$ inside $B(H)$,
whose [[spectrum]] $Spec A$ is a compact strictly localizable enhanced measurable space.

Furthermore, the inclusion of $A$ into $B(H)$ is a representation of $A$ on $H$.
As such, it corresponds under the Serre–Swan-type duality to a [[measurable field]] of Hilbert spaces over $A$.
This is precisely the [[measurable field]] produced by the classical [[spectral theorem]].

Under the duality, the operator $T$ corresponds to the operator that
multiplies a given [[section]] of this [[measurable field]] of Hilbert spaces
by the complex-valued function $Spec A\to \mathbf{C}$ produced above.
Thus, we recovered the entire content of the classical [[spectral theorem]].

In fact, the above considerations work equally well to establish the [[spectral theorem]]
for an arbitrary family (not necessarily finite) of commuting normal operators.

## Related entries

* [[Isbell duality]]

* [[Gelfand duality]]

* [[Stone duality]]

* [[Zariski spectrum]]

* [[modules over a ring are equivalent to quasicoherent sheaves over its spectrum]]

* [[Serre-Swan theorem]], [[smooth Serre-Swan theorem]]

* [[noncommutative geometry]]

* [[reconstruction theorem]], [[Gabriel-Rosenberg theorem]]

[[!redirects dualities between algebra and geometry]]
[[!redirects duality between geometry and algebra]]
[[!redirects dualities between geometry and algebra]]


duality between heterotic and type II string theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[duality in string theory]] relating [[heterotic string theory]] to [[type II string theory]] via [[KK-compactification]] on [[K3-surfaces]]. This is the archetypical example of [[string-string duality]].

The [[non-perturbative effect|non-perturbative]] enhancement of this is [[duality between heterotic string theory and M/F-theory]].

## References
 {#References}

Precursor articles:

* [[Paul Townsend]], _A New Anomaly Free Chiral Supergravity Theory From Compactification on K3_, Phys.Lett. 139B (1984) 283-287 ([spire:14293](http://inspirehep.net/record/14293))

* [[Michael Green]], [[John Schwarz]] [[Peter West]], _Anomaly-free chiral theories in six dimensions_, Nuclear Physics B Volume 254, 1985, Pages 327-348 (doi:10.1016/0550-3213(85)90222-6)

Original articles

* [[Michael Duff]], [[Ruben Minasian]], _Putting string/string duality to the test_, Nucl.Phys. B436 (1995) 507-528 ([arXiv:hep-th/9406198](https://arxiv.org/abs/hep-th/9406198))

* [[Chris Hull]], [[Paul Townsend]], Section 6 of _Unity of Superstring Dualities_, Nucl.Phys.B438:109-137, 1995 ([arXiv:hep-th/9410167](https://arxiv.org/abs/hep-th/9410167))

* [[Ashoke Sen]], _String string duality conjecture in six dimensions and charged solitonic strings_, Nucl.Phys.B450:103-114, 1995 ([arXiv:hep-th/9504027](https://arxiv.org/abs/hep-th/9504027))

* K. Davis, _M-Theory and String-String Duality_ ([arXiv:hep-th/9601102](https://arxiv.org/abs/hep-th/9601102))

Survey:

* [[Paul Aspinwall]], _K3 Surfaces and String Duality_ ([arXiv:hep-th/9611137](https://arxiv.org/abs/hep-th/9611137))

Specifically in view of [[moonshine]]:

* S. Chaudhuri, D.A. Lowe, _Monstrous String-String Duality_, Nucl.Phys.B469:21-36, 1996 ([arXiv:hep-th/9512226](https://arxiv.org/abs/hep-th/9512226))

* {#ChengHarrisonVolpatoZimet16} [[Miranda Cheng]], Sarah M. Harrison, [[Roberto Volpato]], Max Zimet, _K3 String Theory, Lattices and Moonshine_, Research in the Mathematical Sciences 2018 5:32 ([arXiv:1612.04404](https://arxiv.org/abs/1612.04404), [doi:10.1007/s40687-018-0150-4](https://doi.org/10.1007/s40687-018-0150-4))

Relation to [[little string theory]]:

* [[Ofer Aharony]], Bartomeu Fiol, David Kutasov, David A. Sahakyan, _Little String Theory and Heterotic/Type II Duality_, Nucl. Phys. B679:3-65, 2004 ([arXiv:hep-th/0310197](https://arxiv.org/abs/hep-th/0310197))

Further developments:

* I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, _N=2 Type II- Heterotic duality and Higher derivative F-terms_ ([arXiv:hep-th/9507115](https://arxiv.org/abs/hep-th/9507115))

* [[Oren Bergman]], [[Matthias Gaberdiel]], _Non-BPS States in Heterotic - Type IIA Duality_ ([arXiv:hep-th/9901014](https://arxiv.org/abs/hep-th/9901014))

* [[Jan Louis]], Roberto Valandro, _Heterotic-Type II duality in the hypermultiplet sector_, JHEP05 (2012) 016 ([arXiv:1112.3566](https://arxiv.org/abs/1112.3566))

* Sergei Alexandrov, Boris Pioline, _Heterotic-type II duality in twistor space_ ([arXiv:1210.3037](https://arxiv.org/abs/1210.3037))

* [[Eric Bergshoeff]], C. Condeescu, G. Pradisi, F. Riccioni, _Heterotic-Type II duality and wrapping rules_, JHEP12(2013) 057 ([arXiv:1311.3578](https://arxiv.org/abs/1311.3578))

* Sergei Alexandrov, Jan Louis, Boris Pioline, Roberto Valandro, _N=2 Heterotic-Type II duality and bundle moduli_, JHEP08 (2014) 092 ([arXiv:1405.4792](https://arxiv.org/abs/1405.4792))

* [[Andreas Braun]], Taizan Watari, _Heterotic-Type IIA Duality and Degenerations of K3 Surfaces_, JHEP08 (2016) 034 ([arXiv:1604.06437](https://arxiv.org/abs/1604.06437))

[[!redirects duality between type II and heterotic string theory]]

[[!redirects II - HET]]
[[!redirects II - HET duality]]

[[!redirects HET - II]]
[[!redirects HET - II duality]]


duality between M-theory and F-theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[duality in string theory]] between [[M-theory]] and [[F-theory]]:

The following line of argument shows why first compactifying M-theory on a torus $S_1^A \times S_1^B$ to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape $\frac{R_A}{R_B}$ of the torus, but not on its other geometry.

By the [[dualities in string theory]], 10-dimensional [[type II string theory]] is supposed to be obtained from the [[UV-completion]] of [[11-dimensional supergravity]] by first [[Kaluza-Klein mechanism|dimensionally reducing]] over a circle $S^1_A$ -- to obtain [[type IIA supergravity]] -- and then applying [[T-duality]] along another circle $S^1_B$ to obtain [[type IIB supergravity]].

To obtain type IIB sugra in noncompact 10 dimensions this way, also $S^1_B$ is to be compactified (since [[T-duality]] sends the radius $r_A$ of $S^1_A$ to the inverse radius $r_B = \ell_s^2 / R_A$ of $S^1_B$).
Therefore type IIB sugra in $d = 10$ is obtained from 11d sugra compactified on the [[torus]] $S^1_A \times S^1_B$. More generally, this torus may be taken to be an [[elliptic curve]] and this may vary over the 9d base space as an [[elliptic fibration]].

Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity [[vacuum]] is the input data that F-theory is concerned with.

A schematic depiction of this story is the following:

|  |  |  |
|--|--|--|
| [[11d supergravity|M-theory]] in $d = 11$ | | F-theory in $d = 12$ |
| $\downarrow$ [[Kaluza-Klein mechanism|KK-reduction]] along [[elliptic fibration]] | | $\downarrow$ [[axio-dilaton]] is modulus of [[elliptic fibration]] |
| [[type IIA string theory|IIA string theory]] in $d = 9$ | $\leftarrow$[[T-duality]]$\rightarrow$ | [[type IIB string theory|IIB string theory]] in $d = 10$ |

In the simple case where the elliptic fiber is indeed just $S^1_A \times S^1_B$, the [[imaginary part]] of its complex modulus is

$$
  Im(\tau) = \frac{R_A}{R_B}
  \,.
$$

By following through the above diagram, one finds how this determines the [[coupling constant]] in the [[type II string theory]]:

First, the KK-reduction of M-theory on $S^1_A$ yields a type IIA string coupling

$$
  g_{IIA} = \frac{R_A}{\ell_s}
  \,.
$$

Then the T-duality operation along $S^1_B$ divides this by $R_B$:

$$
  \begin{aligned}
    g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B}
    \\
    & = \frac{R_A}{R_B}
    \\
    & = Im(\tau)
   \end{aligned}
   \,.
$$

## References

See at _[[F-theory]]_ for more


duality between M-theory and type IIA string theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[conjecture|conjectured]] _[[duality in string theory|duality]]_ between [[type IIA string theory]] and the theory with working title _[[M-theory]]_ is essentially one half of what defines [[M-theory]] in the first place, namely the statement that [[M-theory]] is the [[non-perturbative effect|non-perturbative]] [[theory (physics)|theory]] of which the [[string perturbation series]] of [[type IIA string theory]] is the [[perturbation series]].

The second defining characteristic of [[M-theory]] is that at low [[energy]], hence at large [[wavelength]], it is approximated by [[11-dimensional supergravity]]. On the other hand, the low energy limit of [[type IIA string theory]] is [[10d type IIA supergravity]].

In view of this, the conjectured relation between M-theory and [[type IIA string theory]] says, more in detail, that in the low energy limit the relation becomes that exhibited exhibited by the [[Kaluza-Klein compactification]] of [[11-dimensional supergravity]] to [[10d type IIA supergravity]] on a [[circle]] [[fiber]], such that the [[radius]] of the circle fiber is proportional to the [[coupling constant]] of type II A string theory.

$$
  \array{
      M-Theory(?) &\stackrel{low\;energy\;limit}{\to}& \text{11d Supergravity}
      \\
      {}^{\mathllap{ \array{ small \\ coupling \\ limit}}}\bigg\downarrow && \bigg\downarrow{}^{\mathrlap{ \array{KK\;reduction \\ on S^1 }}}
      \\
      \text{type IIA string theory}
        &\stackrel{low\;energy\;effective\;QFT}{\to}&
      \text{10d Supergravity}
  }
$$

(e.g. [Obers-Pioline 98, p. 12](#ObersPioline98))

One indication for this conjecture is that the [[double dimensional reduction]] of the [[Green-Schwarz sigma-model]] that describes the [[super membrane]] (later called the _[[M2-brane]]_) propagating on an [[11-dimensional supergravity]] [[background field|background]] [[spacetime]] yields the [[Green-Schwarz sigma-model]] for the [[type IIA superstring]] propagating on a [[10d type IIA supergravity]] [[background field|background]] ([Duff-Howe-Inami-Stelle 87](#DuffHoweInamiStelle87)).

This observation is the origin of the name "[[M-theory]]", in the first place: As a [non-commital shorthand](M-theory#NonCommittal) for "membrane theory", where the caveat is that, there is at least no _direct_ analog of the [[string perturbation series]] with [[strings]] replaced by [[membranes]].

What convinced the community that [[11-dimensional supergravity]] really is the low energy limit of some non-perturbative version of [[perturbative string theory]] was the arguments given in ([Witten 95](#Witten95)).

## Related concepts

* [[graviphoton]]

* [[duality in string theory]]

* [[duality between F-theory and heterotic string theory]]

* [[duality between type I and heterotic string theory]]

[[KK-compactifications of M-theory -- table]]

## References

The original insight is due to

* {#DuffHoweInamiStelle87} [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], _Superstrings in $D=10$ from Supermembranes in $D=11$_, Phys. Lett. B **191** (1987) 70 [doi:10.1016/0370-2693(87)91323-2] and in: [[Michael Duff]] (ed.) _[[The World in Eleven Dimensions]]_ 205-206 (1987) [[spire](http://inspirehep.net/record/245249)]

* [[Paul K. Townsend]], *The eleven-dimensional supermembrane revisited*, Phys. Lett. B **350** (1995) 184-187 [[arXiv:hep-th/9501068](https://arxiv.org/abs/hep-th/9501068), doi:10.1016/0370-2693(95)00397-4]

but it gained popularity only with:

* {#Witten95} [[Edward Witten]], §2.2, §2.3 in: _[[String Theory Dynamics In Various Dimensions]]_, Nucl. Phys. B **443** (1995) 85-126 [[arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124), doi:10.1016/0550-3213(95)00158-O]

Review:

* {#Duff96} [[Michael Duff]], §2(ii) in: _M-Theory (the Theory Formerly Known as Strings)_,  Int. J. Mod. Phys. A **11** (1996) 5623-5642 [[arXiv:hep-th/9608117](http://arxiv.org/abs/hep-th/9608117), [doi:10.1142/S0217751X96002583](https://doi.org/10.1142/S0217751X96002583)]

* {#Duff99} [[Mike Duff]], _[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and M-theory_, IoP 1999 ([publisher](https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720))

* {#ObersPioline98} [[Niels A. Obers]], [[Boris Pioline]], §2.1 in: *U-duality and M-Theory*, Phys. Rept. **318** (1999) 113-225 [[arXiv:hep-th/9809039](http://arxiv.org/abs/hep-th/9809039), doi:10.1016/S0370-1573(99)00004-6]

The actual [[double dimensional reduction]] over a non-trivial [[circle group|circle]]-[[principal bundle]] of the [[supergravity C-field]] [[flux densities]] $G_4$ and $G_7$ to the [[B-field]] and (most of) the [[RR-field]] [[flux densities]] ois worked out (only?) in:

* {#MathaiSati04} [[Varghese Mathai]], [[Hisham Sati]], §4 in: *Some Relations between Twisted K-theory and $E_8$ Gauge Theory*, J. High Energ. Phys. **2004** 03 (2004) 016 [[arXiv:hep-th/0312033](https://arxiv.org/abs/hep-th/0312033), [doi:10.1088/1126-6708/2004/03/016](https://doi.org/10.1088/1126-6708/2004/03/016)]

[[!redirects duality between M-theory and type IIA superstring theory]]

[[!redirects duality between type IIA string theory and M-theory]]
[[!redirects duality between type IIA superstring theory and M-theory]]

[[!redirects M-theory on a circle]]


duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One of the conjectured [[dualities in string theory]]:

[[M-theory]] in a configuration whose low-energy [[super spacetime]] is an [[orbifold]] locally of the form $\mathbb{R}^{5,1} \times (\mathbb{R}^5\sslash_{refl} \mathbb{Z}_2)$ is argued to be equivalent to [[KK-compactifications]] of [[type IIB string theory]] on [[K3-surfaces]].

(Either this refers to the [[perturbation series]]-limit of M-theory, or else, more generally, to [[non-perturbative effect|non-perturbative]] type IIB theory, now also known as [[F-theory]].)

This was originally argued in ([Dasgupta-Mukhil 96, section 2](#DasguptaMukhil96)) and ([Witten 95b](#Witten95b), [Sen 96, top. of p .5](#Sen96)).

## Properties

### Geometric engineering of the 6d $(2,0)$-superconformal QFT
  {#GeometricEngineeringOfThe6d2SuperconformalQFT}

On either side of the [[duality in string theory|duality]], this is the setup that exhibits [[geometric engineering of quantum field theory|geometric engineering]] of the [[6d (2,0)-superconformal QFT]] in the [[ADE-classification]]:

On the type IIB-side this is argued in ([Witten95a, section 1](#Witten95a)), for [[ADE singularities]] in the [[K3-surface]].

On the M-theory side, ([Witten 95b, p. 9-10](#Witten95b)) identifies the $\mathbb{Z}_2$-[[orbifold]] singularity with the worldvolume of [[M5-branes]]. Placing this furthermore inside the [[ADE singularity]] locus of a [[MK6-brane]] should be the corresponding setup (any reference saying this explicitly, i.e. speaking about M5 $\cap$ (MK6 at ADE) ?)

## References

* {#DasguptaMukhil96} [[Keshav Dasgupta]], [[Sunil Mukhi]], _Orbifolds of M-theory_, Nucl.Phys. B465 (1996) 399-412 ([arXiv:hep-th/9512196](https://arxiv.org/abs/hep-th/9512196))

* {#Witten95b} [[Edward Witten]], _Five-branes And M-Theory On An Orbifold_, Nucl.Phys.B463:383-397, 1996 ([arXiv:hep-th/9512219](https://arxiv.org/abs/hep-th/9512219))

* {#Sen96} [[Ashoke Sen]], _Orbifolds of M-Theory and String Theory_, Mod.Phys.Lett.A11:1339-1348, 1996 ([arXiv:hep-th/9603113](https://arxiv.org/abs/hep-th/9603113))

The expected relation to the [[6d (2,0)-superconformal QFT]] is discussed in

* {#Witten95a} [[Edward Witten]], _Some Comments On String Dynamics_ ([arXiv:hep-th/9507121](https://arxiv.org/abs/hep-th/9507121))