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 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.

Relation to mathematical duality

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. 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.


One way to pseudo-formalize accurately most of what is usually meant by “duality” in physics might 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 instance for well-understood 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 \,.

For instance, in the case of mirror symmetry this would be the TCFT-construction that takes a Calabi-Yau manifold to the corresponding Calabi-Yau category.

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 (infinity,1)-category which is induced by the atlas “quantization” via Giraud's theorem of Higher Topos Theory.

This groupoid has as moduli stack of objects LagrangianDataLagrangianData and as moduli stack of 1-morphisms DualitiesDualities. Its corresponding stack realization is LagrangianQFTsLagrangianQFTs

Dualities LagrangianData LagrangianQFTs. \array{ \vdots \\ \downarrow\downarrow\downarrow \\ Dualities \\ \downarrow \downarrow \\ LagrangianData \\ \downarrow \\ LagrangianQFTs } \,.

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 November 18, 2014 10:09:29 by Urs Schreiber (