Types of quantum field thories
This is the generalization of the notion of symplectic manifold to higher symplectic geometry. It is also the image under Lie integration of the notion of symplectic L-∞ algebroid, which is also a higher analog of symplectic manifolds, but in an infinitesimal way.
Notice that every symplectic manifold is in particular a Poisson manifold and that the structure of a Poisson manifold is equivalently encoded in the corresponding Poisson Lie algebroid. A symplectic groupoid is the Lie integration of such a Poisson Lie algebroid. Therefore, strictly speaking, already “ordinary” symplectic geometry secretly involves Lie groupoids. This insight is exploited in the refinement of geometric quantization of symplectic groupoids.
representing a class .
We say an object on is a symplectic smooth -groupoid.
from the Lie integration of to the de Rham coefficient object: this is an -form on a smooth ∞-groupoid (as discussed at smooth ∞-groupoid – structures – de Rham cohomology) and hence equips with the structure of a symplectic -groupoid.
We spell this out in some special cases.
We discuss the Lie integration of Poisson Lie algebroids to symplectic groupoids. For more details and applications of this see at extended geometric quantization of 2d Chern-Simons theory.
The symplectic groupoid associated to this is (by the discussion there) supposed to be the fundamental groupoid of equipped on its space of morphisms with the differential form , where are the two endpoint projections from paths in to .
We demonstrate in the following how this is indeed the result of applying the ∞-Chern-Weil homomorphism to this situation.
For simplicity we shall start with the simple situation where has a global Darboux coordinate chart . Write for the components of the symplectic form in these coordinates, and for the components of the inverse.
Then the Chevalley-Eilenberg algebra is generated from in degree 0 and in degree 1, with differential given by
The differential in the corresponding Weil algebra is hence
Clearly it is useful to introduce a new basis of generators with
In this new basis we have a manifest isomorphism
It remains to show that the symplectic form on makes this a symplectic groupoid.
Notice that in the new basis the invariant polynomial reads
and that we may regard this as a morphism of -algebroids
Over a test space in degree 1 an element in is a pair
subject to the verticality constraint, which says that along we have
The vertical morphism has in fact a section whose image is given by those pairs for which has no leg along . We therefore find the desired form on by evaluating the top morphism on pairs of this form.
Such a pair is taken by the top morphism to
Using the above verticality constraint and the condition that has no leg along , this becomes
By the Stokes theorem the integration over yields
This completes the proof.
The notion of symplectic manifold formalizes in physics the concept of a classical mechanical system . The notion of geometric quantization of a symplectic manifold is one formalization of the general concept in physics of quantization of such a system to a quantum mechanical system .
Or rather, the notion of symplectic manifold does not quite capture the most general systems of classical mechanics. One generalization requires passage to Poisson manifolds . The original methods of geometric quantization become meaningless on a Poisson manifold that is not symplectic.
However, a Poisson structure on a manifold is equivalent to the structure of a Poisson Lie algebroid over . This is noteworthy, because the latter is again symplectic, as a Lie algebroid, even if the underlying Poisson manifold is not symplectic: it is a symplectic Lie algebroid .
Based on related observations it was suggested that the notion of symplectic groupoid (see the references there) should naturally replace that of symplectic manifold for the purposes of geometric quantization to yield a notion of geometric quantization of symplectic groupoids .
More or less implicitly, there has been strong evidence that this shift in perspective is substantial: the deformation quantization (see there for references) of a Poisson manifold turns out to be constructible in terms of correlators of the 2-dimensional TQFT called the Poisson sigma-model associated with the corresponding Poisson Lie algebroid. The fact that this is 2-dimensional and not 1-dimensional, as the quantum mechanical system that it thus encodes, is a direct reflection of this categorification shift of degree – see holographic principle for more on this.
On general abstract grounds this already suggests that it makes sense to pass via higher categorification further to symplectic Lie 2-algebroids, and generally symplectic Lie n-algebroids, as well as to symplectic 2-groupoids, symplectic 3-groupoids, etc. up to symplectic -groupoids.
Formal hints for such a generalization had been noted in (Ševera), in particular in its concluding table. More indirect – but all the more noteworthy – hints came from quantum field theory, where it was observed that a generalization of symplectic geometry to multisymplectic geometry of degree more naturally captures the description of -dimensional QFT (notice that quantum mechanics may be understood as -dimensional QFT). For, observe that the symplectic form on a symplectic Lie n-algebroid is, while always “binary”, nevertheless a representative of de Rham cohomology in degree .
There is a natural formalization of these higher symplectic structures in the context of any cohesive (∞,1)-topos. Moreover, with (FRS) we may observe that symplectic forms on L-∞ algebroids have a natural interpretation in ∞-Lie theory: they are -invariant polynomials. This means that the ∞-Chern-Weil homomorphism applies to them.
We shall show below that all notions of geometric quantization of symplectic -groupoids have a natural interpretation in terms of these canonical structures. For instance the higher “prequantum line bundle” is nothing but the circle n-bundle with connection that the ∞-Chern-Weil homomorphism assigns to the symplectic form, regarded as an -invariant polynomial, and the corresponding “holographic” TQFT – the AKSZ sigma-model – is that given by the induced ∞-Chern-Simons functional.
What is called (geometric) prequantization is a refinement of symplectic 2-forms to curvature 2-forms on a line bundle with connection. This is called a choice of prequantum line bundle for the given symplectic form.
This has an evident generalization to closed forms of degree . If integral, these may be refined to a curvature -form on a circle n-bundle with connection . Since in the context of smooth ∞-groupoids we can have circle -bundles over other smooth -groupoids, this means that we canonically have the notion of prequantum circle -bundles on a symplectic -groupoid.
the passage to the prequantum -bundle with connection corresponds to passing to the refined ∞-Chern-Weil homomorphism
(as discussed there).
Let be a symplectic -groupoid. Then represents a class
We say this form is integral if it is in the image of the curvature-projection
from the ordinary differential cohomology of .
In this case we say a prequantum circle (n+1)-bundle with connection for is a lift of to .
classified by :
This definition generalizes verbatim to n-plectic geometry.
We observe below that this condition is equivalent to the fact that the flow of preserves the connection on any prequantum line bundle, up to homotopy (up to gauge transformation). In this form the definition has an immediate generalization to symplectic -groupoids.
Let be a symplectic -groupoid and let
Regard it as an object in the over-(∞,1)-topos .
Consider the internal automorphism ∞-group
An Hamiltonian diffeomorphism is given by a diagram
where is an ordinary diffeomorphism. To compute the Lie algebra of this, we need to consider smooth 1-parameter families of such and differentiate them.
Assume first that the connection 1-form in is globally defined with . Then the above diagram is equivalent to
where . Differentiating this at 0 yields the Lie derivative
By Cartan calculus this is
In the general case that the prequantum circle n-bundle with connection is not trivial, we can present it by a Cech cocycle on the Cech nerve of the based path space surjective submersion (regarding as a diffeological space and choosing one base point per connected component, or else assuming without restriction that is connected).
Any diffeomorphism lifts to a diffeomorphism by setting .
So we get a diagram
of simplicial presheaves. Now the same argument as above applies on .
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||discussed in:|
|0||symplectic manifold||symplectic manifold||symplectic geometry||Lagrangian submanifold||–||ordinary space of states (in geometric quantization)||geometric quantization|
|1||Poisson Lie algebroid||symplectic groupoid||2-plectic geometry||Poisson sigma-model||coisotropic submanifold (of underlying Poisson manifold)||brane of Poisson sigma-model||2-module = category of modules over strict deformation quantiized algebra of observables||extended geometric quantization of 2d Chern-Simons theory|
|2||Courant Lie 2-algebroid||symplectic 2-groupoid||3-plectic geometry||Courant sigma-model||Dirac structure||D-brane in type II geometry|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
Some ideas pointing to higher symplectic groupoids were indicated in
Aspects of the relation to multisymplectic geometry are in
A discussion of higher symplectic geometry in a general context is in
See also section 4.3 of
Some ingredients for the geometric quantization of symplectic Lie -algebroids are constructed in