higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
under construction
Traditional prequantum geometry is the differential geometry of smooth manifolds which are equipped with a twist in the form of a circle group-principal bundle and a circle-principal connection. In the context of geometric quantization of symplectic manifolds these arise as prequantum bundles. Equivalently, prequantum geometry is the contact geometry of the total spaces of these bundles, equipped with their Ehresmann connection differential 1-form and thought of as regular contact manifolds. Prequantum geometry notably studies the automorphisms of prequantum bundles covering diffeomorphisms of the base – the prequantum operators or contactomorphisms – and the action of these on the space of sections of the associated line bundle – the prequantum states. This is an intermediate step in the genuine geometric quantization of the curvature differential 2-form of these bundles, which is obtained by “dividing the above data in half” (polarization), important for instance in the the orbit method.
But prequantum geometry is of interest in its own right. For instance the above automorphism group naturally provides the Lie integration of the Poisson bracket Lie algebra of the underlying symplectic manifold, together with the canonical injection into the group of bisections of the Lie integration of the Atiyah Lie algebroid which is associated with the given circle bundle, all of which are fundamental objects of interest in the study of line bundles over manifolds.
For a plethora of applications in differential geometry, one wants to generalize this to higher differential geometry (see at motivation for higher differential geometry) and accordingly study higher prequantum geometry.
A sequence of time-honored traditional concepts in geometric quantization/prequantum geometry is
Lie groups: | Heisenberg group | $\hookrightarrow$ | quantomorphism group | $\hookrightarrow$ | gauge group |
---|---|---|---|---|---|
Lie algebras: | Heisenberg Lie algebra | $\hookrightarrow$ | Poisson Lie algebra | $\hookrightarrow$ | twisted vector fields |
For instance in the geometric quantization of the electrically charged particle sigma-model we have a prequantum circle bundle $P$ with connection on a bundle $\nabla$ on a cotangent bundle $X = T^* Y$ which is essentially the pullback of the electromagnetic field-bundle on target spacetime $Y$. Its quantomorphism group is the group of diffeomorphisms $P \stackrel{\simeq}{\to} P$ of the total space of the prequantum bundle which preserve the connection (also called the contactomorphism of $(P,\nabla)$ regarded as a regular contact manifold). For the following it is convenient to say this using the language of moduli stacks: we may regard $X$ as a representable sheaf on the site of smooth manifolds (a “smooth space”) and then moreover as a representable stack on this site (a “smooth groupoid”) and make use of the tautological existence of the moduli stack of $U(1)$-principal connections, which we write $\mathbf{B}U(1)_{conn}$ (we don’t need further details right now, but they can be found for instance at circle n-bundle with connection for details). By definition this is such that for any $X$ a map $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ is equivalently a $U(1)$-principal connection and such that a homotopy $\eta \colon \nabla_1 \to \nabla_2$ between two such maps is equivalently a gauge transformation between two such connections. With this formulation a quantomorphism of the prequantum bundle $\nabla$ is equivalently a diagram of the form as on the right of
in the (2,1)-category of stacks, namely a diffeomorphism $\phi \colon X \stackrel{\simeq}{\to} X$ of the base space of the bundle together with a gauge transformation of $U(1)$-principal connections $\eta \colon \phi^* \nabla \stackrel{\simeq}{\to} \nabla$.
The quantomorphism group is naturally an (infinite dimensional) Lie group. Its Lie algebra is the Poisson bracket Lie algebra. If $X$ is equipped with the structure of a Lie group itself (notably if it is a vector space), then the sub-Lie algebra of that on the invariant vectors is the Heisenberg Lie algebra and the Lie group corresponding to that is the Heisenberg group.
One also says that a triangular diagram as above is an autoequivalence of the “modulating” map $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ in the slice (2,1)-category of stacks/smooth groupoids over $\mathbf{B}U(1)_{conn}$.
Such autoequivalences in slices are familiar from basic concepts of Lie groupoid theory. For $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ a Lie groupoid, we may regard the inclusion of its manifold of objects as an atlas being a map $p_\mathcal{G} \colon\mathcal{G}_0 \to \mathcal{G}$. Regarding this atlas as an object in the slice (2,1)-category of stacks/smooth groupoids over $\mathcal{G}$, its autoequivalences are diagrams as on the right of
This is a diffeomorphism $\phi \colon \mathcal{G}_0 \stackrel{\simeq}{\to} \mathcal{G}_0$ of the smooth manifold of objects equipped with a natural transformation $\eta$ whose component map is a smooth function that assigns to each point q \in \mathcal{G_0}
a morphism in $q \in \mathcal{G_\mathcal{G}$ of the form $\eta_q \colon q \to \phi(q)$. This collection of data is known as a bisection of a Lie groupoid. Bisections naturally form a group $\mathbf{BiSect}(p_{\mathcal{G}})$ , which is all the more manifest if we understand them as autoequivalences of the atlas in the slice, called the group of bisections.
This perspective of regarding maps of smooth groupoids as objects in the slice over their codomain (an elementary step in higher category theory/higher topos theory, but not common in traditional differential geometry) turns out to be useful and drives all of the refinements, generalizations and theorems that we discuss in the following: we will see that higher prequantum geometry is essentially the geometry insice higher slice categories of higher stacks over higher moduli stacks of higher principal connections.
Before we get there, notice the following…
The tools of geometric quantization mainly apply to quantum mechanics and only partially to quantum field theory. In particular in the context of extended prequantum field theory in dimension $n$ a prequantum bundle over the (phase-)space of fields is to be refined (de-transgressed) to a prequantum n-bundle over the moduli ∞-stack of fields. Therefore in order to apply geometric quantization to extended prequantum field theory to obtain extended quantum field theory we first need extended/higher prequantum geometry.
For instance the prequantum 3-bundle for standard 3d Spin group Chern-Simons theory is modulated by the differential smooth first fractional Pontryagin class
modulating/clsasifying the universal Chern-Simons circle 3-bundle with connection (also known as a bundle 2-gerbe) over the moduli stack of fields of $G$-Chern-Simons theory, which is the moduli stack $\mathbf{B}G_{conn}$ of $G$-principal connection.
Similarly the prequantum 7-bundle for 7d Chern-Simons theory on string 2-group principal 2-connections is given by the differential smooth second fractional Pontryagin class
modulating/classifying the universal Chern-Simons circle 7-bundle with connection over the moduli 2-stack $\mathbf{B}String_{conn}$ of string 2-group principal 2-connections.
Therefore we want to lift the above table of traditional notions to higher geometry…
In order to say this, clearly we need some basics of higher geometry…
Important construction principle for (∞,1)-categories: simplicial localization. For $\mathcal{C}$ a category with some subset of morphisms $W \hookrightarrow Mor(\mathcal{C})$ declared to be “weak equivalences”, the simplicial localization
is the universal $(\infty,1)$-category obtained from $\mathcal{C}$ by universally turning each weak equivalence into an actual homotopy equivalence in the sense of homotopy theory.
In particular let $C$ be a site, assumed for simplicity to have enough points. Declare then that in the functor category $Func(C^{op}, KanCplx)$, hence in Kan complex-valued presheaves, the weak equivalences are the stalkwise homotopy equivalences of Kan complexes. Then
is called the (∞,1)-topos of (∞,1)-sheaves/∞-stacks on $C$.
An A-∞ algebra-object $G$ in such an $(\infty,1)$-topos such that $\pi_0(G)$ is a group is called an ∞-group “with geometric structure as encoded by the test spaces $C$”. The canonical source of $\infty$-groups are the homotopy fiber products of point inclusions $* \to X$ of any object X, the loop space object
In fact this are all the ∞-groups that there are, up to equivalence: forimg loop space objects is an equivalence of (∞,1)-categories
between ∞-groups and pointed connected objects. The inverse equivalence $\mathbf{B}$ is the delooping operation.
We say that such an $(\infty,1)$-topos $\mathbf{H}$ is cohesive if it is equipped with an adjoint triple of idempotent (co)/(∞,1)-monads
shape modality | flat modality | sharp modality | ||
---|---|---|---|---|
idemp. monad | idemp. comonad | idemp. monad | ||
$\Pi$ | $\dashv$ | $\flat$ | $\dashv$ | $\sharp$ |
This implies (strictly speaking we need differential cohesion for that, coming from another adjoint triple of (co)monads) that for every braided ∞-group $\mathbb{G} \in Grp(\mathbf{H})$ there is a canonical object $\mathbf{B}\mathbb{G}_{conn}$ which modulats $\mathbb{G}$-principal ∞-connections.
Looking at the above table and noticing the above need for higher prequantum bundles, we should try to find an analogous table of concepts in higher geometry, something like this:
slice-automorphism ∞-groups in higher prequantum geometry
cohesive ∞-groups: | Heisenberg ∞-group | $\hookrightarrow$ | quantomorphism ∞-group | $\hookrightarrow$ | ∞-bisections of higher Courant groupoid | $\hookrightarrow$ | ∞-bisections of higher Atiyah groupoid |
---|---|---|---|---|---|---|---|
L-∞ algebras: | Heisenberg L-∞ algebra | $\hookrightarrow$ | Poisson L-∞ algebra | $\hookrightarrow$ | Courant L-∞ algebra | $\hookrightarrow$ | twisted vector fields |
(…)
The way all these notions and theorems work is by considering automorphism ∞-groups of the classifying (or rather: modulating) maps $\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}$ of a prequantum ∞-bundle in the slice (∞,1)-topos over the domain. For instance
The others are obtained by succesively forgetting connection data. For instance
and
The extension sequence is then schematically simply the following
in this generality this now includes various other notions, too:
higher Atiyah groupoid: | standard higher Atiyah groupoid | higher Courant groupoid | groupoid version of quantomorphism n-group |
---|---|---|---|
coefficient for cohomology: | $\mathbf{B}\mathbb{G}$ | $\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$ | $\mathbf{B} \mathbb{G}_{conn}$ |
type of fiber ∞-bundle: | principal ∞-bundle | principal ∞-connection without top-degree connection form | principal ∞-connection |
For $\mathbb{G}$ a braided ∞-group and $\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}$ a higher prequantum geometry with respect to $\mathbb{G}$ there is a long homotopy fiber sequence
Similarly there is the Heisenberg infinity-group extension
The Lie differentiation of the ∞-group extension sequence of prop. \ref{QuantomorphismExtension} is a homotopy fiber sequence of L-∞ algebras
where
$\mathfrak{Poisson}(X,\omega)$ is the Poisson Lie n-algebra as defined in (Rogers 11).
$\mathcal{X}_{Ham}$ is the Lie algebra of vector fields restricted to the Hamiltonian vector fields;
$\mathbf{H}(X, \flat (\mathbf{B}^{n-1})\mathbb{R})$ is the chain complex for flat de Rham cohomology in the given degree, regarded as an abelian L-∞ algebra.
The following table shows what this sequence reduces to when one chooses $\mathbb{G} = \mathbf{B}^{n-1}U(1)$.
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)
(extension are listed for sufficiently connected $X$)
For $G$ a simply connected semisimple compact Lie group such as the spin group, let
be the canonical circle 2-bundle with connection over it. Then the Heisenberg 2-group extension
is the string 2-group extension
(by classification of extensions by cohomology… by Lie 2-algebra computation…)
(and analogously for fivebrane 6-group…)
slice-automorphism ∞-groups in higher prequantum geometry
cohesive ∞-groups: | Heisenberg ∞-group | $\hookrightarrow$ | quantomorphism ∞-group | $\hookrightarrow$ | ∞-bisections of higher Courant groupoid | $\hookrightarrow$ | ∞-bisections of higher Atiyah groupoid |
---|---|---|---|---|---|---|---|
L-∞ algebras: | Heisenberg L-∞ algebra | $\hookrightarrow$ | Poisson L-∞ algebra | $\hookrightarrow$ | Courant L-∞ algebra | $\hookrightarrow$ | twisted vector fields |
higher Atiyah groupoid: | standard higher Atiyah groupoid | higher Courant groupoid | groupoid version of quantomorphism n-group |
---|---|---|---|
coefficient for cohomology: | $\mathbf{B}\mathbb{G}$ | $\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$ | $\mathbf{B} \mathbb{G}_{conn}$ |
type of fiber ∞-bundle: | principal ∞-bundle | principal ∞-connection without top-degree connection form | principal ∞-connection |
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)
(extension are listed for sufficiently connected $X$)
See also the references at n-plectic geometry and at higher geometric quantization
Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher $U(1)$-gerbe connections in geometric prequantization, Rev. Math. Phys., Vol. 28, Issue 06, 1650012 (2016) (arXiv:1304.0236)
Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)