These are expanded notes for a talk that I gave at the workshop
on a topic in the context of differential cohomology in a cohesive topos.
The phase space of a mechanical system is essentially a symplectic manifold, and geometric quantization is a process that sends this to the corresponding quantum mechanical system. On the other hand, the extended phase space of an -dimensional classical field theory is an n-plectic manifold, and we expext that some form of higher geometric quantization takes this to the corresponding extended quantum field theory.
So consider a symplectic manifold with symplectic form . For the following we want to view this in terms of sheaves: closed differential 2-forms naturally form a sheaf on the site SmthMfd of smooth manifolds, as does of course itself, via .
in the (2,1)-topos of (2,1)-sheaves is equivalently a circle bundle with connection (and such that a homotopy between two such morphisms is a smooth gauge transformation of these bundles). This comes with a morphism, of (2,1)-sheaves , sending a connection to its curvature 2-form , expressed the the commutativity of the diagram
If on the other hand is given (and symplectic, hence non-degenerate), then the lift is called a prequantum circle bundle for . In geometric quantization one tries to obtain the quantum mechanics of regarded as a phase space by constructions on such a prequantum bundles.
But ordinary geometric quantization is fully well suited only for quantum mechanics, hence for 1-dimensional quantum field theory. More generally, the extended phase space of an -dimensional quantum field theory naturally carries a closed differential form of degree . A natural question to ask is therefore if there is a higher-degree analog of geometric quantization usefully adapted to these higher degree forms on extended phase spaces. In other words: What is higher geometric quantization?
This question reaches into largely unexplored territory with currently only a few hints giving orientation. Accordingly, different people have made different proposals for how the theory should proceed.
For instance in the context of multisymplectic geometry, Kanatchikov observed that an n-plectic manifold carries a graded Leibniz algebra generalizing the Poisson bracket Lie algebra of an ordinary symplectic maniifold, and Hrabak made proposals (see here) for how to extend BV-BRST formalism to such a situation, crucial for the quantization of gauge theories. There is however no analog of the prequantum circle bundle in this work so far and hence the nature of geometric quantization is not retained.
It is noteworthy that Kanachikov’s construction involves dividing out certain Hamiltonian forms: the straightforward generalization of the definition of the Poisson bracket from symplectic forms for -plectic forms fails to satisfy the Jacobi identity by an exact form. Quotienting out these Jacobiators Kanatchikov enforces a binary bracket structure.
But the appearance of nontrivial Jacobiators is of course the hallmark of Lie theory generalized to homotopy theory: of L-∞ algebras. Indeed, Chris L. Rogers observed that if not dividing out anything, then the evident Poisson bracket of an -plectic form on a smooth manifold extends to a Lie n-algebra. He also considers circle n-bundles with connection as the natural higher generalization of prequantum circle bundles. What has however not been clarified yet is how the BV-BRST formalism would generalize to this approach.
In summary these two proposals for how to start the theory of higher geometric quantization are:
At the moment it is not clear how these two approached would be usefully related, and therefore one is left wondering what the right direction to proceed would be. Evidently it is difficult and dangerous to proceed just by guesswork and intuition. One needs more guidance from
The purpose of the present note is to consider guidance from abstract generals. The theorem presented below states that a notion of higher Poisson bracket Lie algebras – and in fact even of the higher Heisenberg group/quantomorphism group that Lie integrates it – is automatically induced by a formalization of the foundations of differential geometry/differential cohomology in the formal language of cohesive homotopy type theory (Schreiber-Shulman). And when interpreted in the relevants models, we show that it reproduces Chris Rogers’ proposal for Poisson Lie -brackets, and thereby generalizes this from smooth manifolds to more general geometric objects.
Here we indicate this natural general abstract formalization of the notion of higher quantomorphism groups. The actual proof of the main theorem below, that this subsumes Rogers’ construction, comes down to a lengthy computation in differential geometry, which the reader may find spelled out in (Schreiber, section 4.4.17). This is joint work with Chris Rogers. A writeup should appear later this year.
Since our goal is to see how deep in the grand scheme of things we can root the notion of higher geometric quantization, we consider here a formalism that reaches deep to the foundations of mathematics and at the same time provides a natural foundation for many aspects of quantum field theory (namely cohesive homotopy type theory, the relevant aspects of which we briefly introduce now).
So: consider an∞-topos_ . That means:
One concrete way to think of this is as follows: if is a site with enough points, then is obtained from the category of simplicial presheaves by simplicial localization, by universally sending stalkwise weak homotopy equivalences to homotopy equivalence:
But it is noteworthy that apart from this sheaf-theoretic description, ∞-toposes also have an entirely abstract characterization by just a handful of general properties that do not mention any notion of stack explicitly. Our theorem below can be and will be stated entirely in terms of these abstract properties only. This way of speaking about ∞-toposes – or rather: internal to ∞-toposes – has come to be known as homotopy type theory, a system of formal logic that captures modern homotopy theory. Since we are after a fundamental theoretical justification of notions of higher geometric quantization, this seems to be a suitable framework to base our discussion on, and hence we spend here a few introductory words on the basics that we need below.
The abstract properties of an ∞-topos that we need are
it has an object classifier;
which we turn to now.
The corresponding equivalence of ∞-groupoids
sends an -family of objects to its “name”:
defined by fitting into an ∞-pullback diagram of the form
Since this way the -topos “reflects on itself” via , it is useful to consistently speak of all objects in a slice entirely in terms of their “names” this way (the “internal language” of ). This equivalent point of view is highlighted with a slight change of notation, adopted from formal logic:
instead of drawing the symbols for a morphism
we will draw symbols like this:
called a sequent, but meaning precisely the same as the morphism-notation before.
Similarly, instead of the symbols
to express the same state of affairs (behind all this is the relation between type theory and category theory).
In summary we have the following dictionary
|homotopy type theory|
While nothing changes about our statements whether we use symbols as on the top or on the bottom of this disctionary, the top row is more closely related to the higher geometry that we want to describe, whereas the bottom row is more closely related to the type theoretic foundations of mathematics. It turns out that the general definition of the quantomorphism n-group is simplest actually in the latter formulation, and so we amplify that one here as support for our claim that we give a deep theoretical justification for Chris Rogers’ definition of n-plectic geometry.
Since this is by definition internal to the slice, if follows that an internal hom is again given by a morphism over . This means by the above that there should be notation of the form :
Even if this is just notation, it is in fact more suggestive and more illuminating than the standard category theoretic notation: it expresses manifestly that the internal hom in a slice ∞-topos is given by forming the collection of morphisms homotopy fiber-wise over generalized elements .
In particular, this means that the expression
We need one last notational ingredient to be able to formulate quantomorphism n-groups in the foundational internal language of homotopy type theory, namely for base change geometric morphisms. (Their existene is actually equivalent to the internal homs discussed before, see at locally cartesian closed category).
For instance if
is a morphism in over , then we obtain from this the object , which in homotopy type theory notation is just
This is the object of sections of .
The extra axiom needed for all matters of differential geometry/differential cohomology on those of plain homotopy type theory is cohesion, and expression of the fact that in a context of higher geometry the points of a space “hand togther” or “cohere” in a geometric way, as the points in an open ball do in topology (continuously) and in differential geometry (smoothly).
The details of this extra axiom are described
Here we just need that this axiom implies in particular that for every object there is an object
is a flat -principal -connection on . With comes a canonical morphism
which sends such a flat connection to its underlying principal ∞-bundle
Using the formal properties of in cohesive homotopy type theory, one finds that for the case that is an abelian (E-∞) ∞-group and using a given notion of manifolds, there is canonically a refinement of to an object
(for each ) which is the moduli for non-flat -connections.
(Details on this are at structures in a cohesive ∞-topos – Differential cohomology).
the circle group, or
is the corresponding circle n-group,
Hence in this case a circle n-bundle with connection is
This has a curvature -form
For this we now conside aspects of higher geometric quantization.
(This construction is a differential refinement of that of -action homomorphisms, see the comments in General notion in representation theory below for more background on this.)
Let Smooth∞Grpd, and .
This is spelled out in (Schreiber, section 4.4.17).
The expected statement is true for all , but the fully general proof is not written up yet. The full statement should appear in (RogersSchreiber).
The relevance of the ordinary quantomorphism group is that it acts on the space of prequantum states by prequantum operators. In particular if itself has abelian group-structure, then those quantomorphisms covering the -action? on itself form the Heisenberg group.
The following additions to theorem 1 say that all these structures are similarly reproduced form fundamental constructions in cohesive homotopy type theory. However, to appreciate the following statements the reader will need a little bit more background on ∞-actions of ∞-groups than we have mentioned so far. The relevant notions are briefly discussed below in Background on higher group actions.
With the assumptions as in theorem 1 and for equipped with the canonical -action, the interpretation of def. 2 in Smooth∞Grpd reproduces the ordinary notion of space of states (in geometric quantization).
This is spelled out in (Schreiber, section 22.214.171.124).
Under this equivalence the object that is being acted on is the homotopy fiber
From this immeditatel derive various standard notions in representation theory.
The conjugation action on maps between two representations and is
In particular, the type of -action homomorphisms is the type of invariants of the conjugation action
An atlas is
is circle bundle on .
The defining representation of
is space of smooth sections of the associated complex line bundle
is canonical action of , then for
(generally covariant field of gravity)
The computations going into the proof of theorem 1 are spelled out in section 4.4.17 of
A dedicated discussion of the result discussed here should appear as