Contents

duality

# Contents

## Idea

In fundamental physics, notably in quantum field theory and string theory one often says that a non-trivial equivalence of quantum field theories between two models (in theoretical physics) is a “duality”.

## Examples

duality in string theory

general mechanisms

string-fivebrane duality

string-string dualities

M-theory

F-theory

string-QFT duality

QFT-QFT duality:

## Formalization and Relation to mathematical duality

One should beware that the use of the word “duality” in physics is in general different from concepts called “duality” in mathematics.

For instance in T-duality only simple cases exhibit such obviously “dual” behaviour and in general cases such as U-duality really only the notion of equivalence remains. This more closely resembles the mathematical concept of Morita equivalence, see Relation to Morita equivalence. However, in some cases such as Montonen-Olive duality/S-duality the equivalence involves some actual duality in the mathematical sense, as in replacing the gauge group by its Langlands dual group.

One way to pseudo-formalize accurately most of what is usually meant by “duality” in physics might instead be the following.

Write $LagrangianData$ for a moduli stack of prequantum field theory data consisting of species of fields and of Lagrangians/action functionals defined on these.

###### Example

For the well-understood case of mirror symmetry this would be the usual moduli space of Calabi-Yau manifolds regarded as the Lagrangian data for the 2d (2,0)-superconformal QFT.

One imagines that quantization gives a map from such prequantum data to a moduli stack $QFTs$ of actual quantum field theories

$quantization \;\colon\; LagrangianData \longrightarrow QFTs \,.$
###### Example

Continuing example in the case of mirror symmetry this would be the TCFT-construction that takes a Calabi-Yau manifold to its Calabi-Yau A-∞ category (“of branes”) which defines the corresponding 2d TQFT via the noncompact version of the cobordism hypothesis.

The 1-image of this map would be the moduli space of Lagrangian quantum field theories

$quantization \;\colon\; LagrangianData \longrightarrow LagrangianQFTs \hookrightarrow QFTs \,.$

By assumption this is now a 1-epimorphism and hence an atlas of moduli stacks.

The physical concept of duality, such as in mirror symmetry, says that two points $L_1, L_2 \colon \ast \to LagrangianData$ in the space of Lagrangian data are “dual” to each other, if they become equivalent as quantum field theories after quantization.

Mathematically this means that the space of such “dualities” is the homotopy fiber product

$\array{ && Dualities \\ & \swarrow && \searrow \\ LagrangianData && \swArrow_{\simeq} && LagrangianData \\ & {}_{\mathllap{quantization}}\searrow && \swarrow_{\mathrlap{quantization}} \\ && LagrangianQFTs }$

By definition, an element of $Dualities$ is two Lagrangians and a choice of equivalence of their associated quantum field theories:

$quantization(L_1) \simeq quantization(L_2) \,.$

This construction is the first step in associating the groupoid object in an (∞,1)-category which is induced by the atlas “quantization” via Giraud's theorem of Higher Topos Theory.

It continues in the way that Cech covers do (whence one speaks of the Cech nerve construction of the quantization map $LagrangianData \to LagrangianQFTs$): above “$Dualities$” there is the space of triples of Lagrangian data that all have the same quantization, equipped with dualities between any two of them, and equipped with an equivalence of dualities (hence a “duality of dualities”) between the composite of two of these and the third:

$\array{ && \vdots \\ DualitiesOfdualities &\simeq& LagrangianData \underset{LagrangianQFT}{\times} LagrangianData \underset{LagrangianQFT}{\times} LagrangianDat \\ && \downarrow \downarrow \downarrow \\ Dualities &\simeq& LagrangianData \underset{LagrangianQFT}{\times} LagrangianData \\ && \downarrow \downarrow \\ && LagrangianData }$

It continues this way through all $n$-fold dualities of dualities. The resulting $\infty$-groupoid object has as moduli stack of objects $LagrangianData$ and as moduli stack of 1-morphisms $Dualities$. Its corresponding stack realization is $LagrangianQFTs$ and so the corresponding augmented simplicial object looks as

$\array{ \vdots \\ \downarrow\downarrow\downarrow\downarrow \\ DualitiesOfDualities \\ \downarrow\downarrow\downarrow \\ Dualities \\ \downarrow \downarrow \\ LagrangianData \\ \downarrow^{\mathrlap{quantization}} \\ LagrangianQFTs } \,.$

Such towers are to be thought of as the incarnation of equivalence relations as we pass to (∞,1)-category theory: A plain equivalence relation is just the first stage of such a tower

$\array{ Dualities \\ \downarrow \downarrow \\ Lagrangians }$

The conditions on an equivalence relation – reflexivity, transitivity, symmetry – may be read as those on a groupoid object – identity, composition, inverses. So now in homotopy logic this is boosted to an groupoid object in an (∞,1)-category by relaxing all three to hold only up to higher coherent homotopies.

The bottom-most arrow

$\array{ LagrangianData \\ \downarrow^{\mathrlap{quantization}} \\ LagrangianQFTs }$

is the quotient projection of the equivalence relation. In 1-logic this would be its cokernel, here in homotopy logic it is the homotopy colimit over the full simplicial diagram.

So the perspective of the full diagram gives the usual way of speaking in QFT also a reverse:

a) that two Lagrangians are dual if there is an equivalence between the QFTs which they induce under quantization,

we may turn this around and say that therefore

b) quantization is the result of forming the homotopy quotient of the space of Lagrangian data by these duality relations.

It is one of the clauses of the Giraud theorem in (∞,1)-topos theory that these two perspectives are equivalent.

## Relation to Morita equivalence

According to Albert Schwarz,

I am convinced that the mathematical notion of Morita equivalence of associative algebras and its generalization for differential associative algebras should be regarded as the mathematical foundation of dualities in string/M-theory. (My life in science)

Schwarz showed that compactifications on Morita equivalent noncommutative tori are physically equivalent (Schwarz98). This work is followed up in relation to T-duality in Pioline99 and CNS11.

BMRS08 discusses an axiomatic definition of topological T-duality generalizing and refining T-duality between noncommutative spaces in terms of Morita equivalence to a special type of KK-equivalence, which defines a T-duality action that is of order two up to Morita equivalence.

In Okada09 a variant of mirror symmetry is shown to be a form of derived Morita equivalence.

Morita theoretic ideas are also involved in factorization homology, the blob complex and premodular TQFTS, see MW10, BZBJ15, and Scheimbauer for the Morita $(\infty, n)$-category of $E_n$-algebras.

There is also a duality in the description of physics:

Isbell duality between algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

## References

Last revised on September 16, 2019 at 03:07:51. See the history of this page for a list of all contributions to it.