duality in physics



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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”.


While most of these dualities refer to equivalences between quantum field theories, they find their conceptual explanation in string theory. See at

for more.

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


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 QFTsQFTs of actual quantum field theories

quantization:LagrangianDataQFTs. quantization \;\colon\; LagrangianData \longrightarrow QFTs \,.

Continuing example 1 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:LagrangianDataLagrangianQFTsQFTs. 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:*LagrangianDataL_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

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

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

quantization(L 1)quantization(L 2). 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 LagrangianDataLagrangianQFTsLagrangianData \to LagrangianQFTs): above “DualitiesDualities” 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:

DualitiesOfdualities LagrangianData×LagrangianQFTLagrangianData×LagrangianQFTLagrangianDat Dualities LagrangianData×LagrangianQFTLagrangianData LagrangianData \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 nn-fold dualities of dualities. The resulting \infty-groupoid object has as moduli stack of objects LagrangianDataLagrangianData and as moduli stack of 1-morphisms DualitiesDualities. Its corresponding stack realization is LagrangianQFTsLagrangianQFTs and so the corresponding augmented simplicial object looks as

DualitiesOfDualities Dualities LagrangianData quantization LagrangianQFTs. \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

Dualities Lagrangians \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

LagrangianData quantization LagrangianQFTs \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:

instead of saying

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.

There is also a duality in the description of physics:

duality between algebra and geometry in physics:

Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation


Revised on December 20, 2014 13:12:13 by Urs Schreiber (