Types of quantum field thories
abstract duality: opposite category,
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”.
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.
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 physical concept of duality, such as in mirror symmetry, says that two points 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
By definition, an element of is two Lagrangians and a choice of equivalence of their associated quantum field theories:
It continues in the way that Cech covers do (whence one speaks of the Cech nerve construction of the quantization map ): above “” 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:
It continues this way through all -fold dualities of dualities. The resulting -groupoid object has as moduli stack of objects and as moduli stack of 1-morphisms . Its corresponding stack realization is and so the corresponding augmented simplicial object looks as
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
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.
There is also a duality in the description of physics:
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|