Schreiber ∞-geometric prequantization

This is a project on higher geometric quantization that I am working on with Domenico Fiorenza and Chris Rogers; in the context of differential cohomology in a cohesive topos. This page contains indications of some contents and pointers to further details.

• Domenico Fiorenza, Chris Rogers, Urs Schreiber,

  Geometric prequantization over cohesive $\infty$-stacks

  ( talk handout pdf (4 pages))

Overview

There are two main formalizations of the notion of quantization: algebraic deformation quantization and geometric quantization. They are closely related where they both apply, but, as the names suggest, are in nature roughly dual to each other in a way that reflects the general duality between algebra and geometry. Moreover, deformation quantization is more natural with respect to the Heisenberg picture of quantum physics, hence the algebras of observables, whereas geometric quantization is more natural with respect to the Schrödinger picture and its emphasis on spaces of states. Indeed, geometric quantization is first and foremost a theory concerned with the construction and investigation of spaces of sections of line bundles equipped with certain extra structure.

We discuss here the generalization of this to the context of higher geometry and notably to higher differential geometry, where line bundles associated to circle bundles with connection are generalized to associated ∞-bundles of circle n-bundles with connection. As discussed at Principal ∞-bundles – models and general theory, sections of such associated ∞-bundles are equivalently cocycles in some flavor of twisted cohomology, geometrically represented by twisted ∞-bundles, and so higher geometric prequantization may also be thought of as studying aspects of twisted differential cohomology. For instance in degree 2 a section of the canonical 2-bundle associated to a circle 2-bundle is a twisted vector bundle and 2-geometric quantization overlaps to a large degree with the study of differential refinements of twisted K-theory.

These higher structures in prequantization are supposed to match corresponding higher structures known in extended quantum field theory. One expects that a notion of higher geometric quantization completes the following table, whose entries we further explain below:

classical mechanics	–quantization$\to$	quantum mechanics
symplectic geometry	–geometric quantization$\to$	quantum field theory
higher symplectic geometry	–higher geometric quantization$\to$	extended quantum field theory

Infinitesimally, ordinary geometric quantization is of course based on symplectic geometry: the curvature 2-form of the circle bundle with connection mentioned before is a symplectic form, and, conversely, the circle bundle with connection is the very prequantization of this symplectic structure – the prequantum line bundle – that gives geometric quantization its name. As we generalize to higher circle n-bundles with connection, the underlying symplectic geometry is generalized to what is called multisymplectic geometry or, of more direct relevance for us, n-plectic geometry. But not only may the degree of the symplectic structure increase, but also the base space itself may generalize from a symplectic manifold to a symplectic ∞-groupoid/∞-stack. For instance there is a canonical 3-plectic structure on the moduli stack $\mathbf{B}G$ of $G$-principal bundles for $G$ any simply connected compact simple Lie group, encoded by the prequantum circle 3-bundle known as the Chern-Simons circle 3-bundle. Or for instance there is a canonical 7-plectic structure on the moduli 2-stack $\mathbf{B}String$ of string 2-group-Chern-Simons circle 7-bundle.

To capture this, write $\mathbf{B}^n \mathbb{G}_{conn} \in \mathbf{H}$ for the cohesive moduli ∞-stack of circle n-bundles with connection, in the ambient cohesive (∞,1)-topos $\mathbf{H}$. Then for $X \in \mathbf{H}$ any object to be thought of as the moduli ∞-stack of fields or as the target space for a sigma-model, a morphism

$$\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}$$

modulates a circle n-bundle with connection on $X$. We regard this as a extended action functional in that for $\Sigma_{k} \in \mathbf{H}$ of cohomological dimension $k \leq n$ and sufficiently compact so that fiber integration in ordinary differential cohomology $\exp(2 \pi i \int_{\Sigma}_k(-))$ applies, the transgression of $\mathbf{c}_{conn}$ to low codimension reproduces the traditional ingredients

$k =$	transgression of $\mathbf{c}_{conn}$ to $[\Sigma_{n-1},X]$	meaning in geometric quantization
$n$	$\exp(2 \pi i S(-)) : [\Sigma_n, X] \stackrel{[\Sigma_n, \mathbf{c}_{conn}]}{\to} [\Sigma_n, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_n}(-))}{\to} \mathbb{G}$	action functional
$n-1$	$\exp(2 \pi i S(-)) : [\Sigma_{n-1}, X] \stackrel{[\Sigma_{n-1}, \mathbf{c}_{conn}]}{\to} [\Sigma_{n-1}, \mathbf{B}^n \mathbb{G}_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_{n-1}}(-))}{\to} \mathbf{B}\mathbb{G}_{conn} \,$	ordinary (off-shell) prequantum circle bundle

The idea is to consider the higher geometric quantization not just of the low codimension transgressions, but of all transgressions of $\mathbf{c}_{conn}$.

While a large class of different structures and phenomena are examples or ingredients of $\infty$-geometric prequantization, we find that the basic principle which controls all of the theory is a single concise general abstract construction in (∞,1)-topos theory:

any ∞-representation $\rho$ of the circle n-group $\mathbf{B}^{n-1}U(1)$ on some object $V$ is given by a fiber sequence in Smooth∞Grpd

$$\array{ V &\to& V//\mathbf{B}^{n-1}U(1) \\ && \downarrow^{\mathrlap{\rho}} \\ && \mathbf{B}^n U(1) } \,,$$

which at the same time is identified as the universal $V$-associated ∞-bundle corresponding to the universal principal ∞-bundle over $\mathbf{B}^n U(1)$. Then for $(P \to X,\; \nabla)$ a circle n-bundle with connection classified (or rather: modulated) by a morphism $g :X \to \mathbf{B}^n U(1)_{conn}$, the ∞-groupoid of sections of the associated $V$-$\infty$-bundle $P \times_{\mathbf{B}^{n-1}U(1)} V$ is canonically identified with the $g$-twisted $\Omega V$-cohomology of $X$:

$$\Gamma_X(P \times_{\mathbf{B}^{n-1}U(1)}V) \simeq \mathbf{H}_{/\mathbf{B}^n U(1)}(g, \rho) \,.$$

Since the ∞-groupoid on the right is nothing but an (∞,1)-categorical hom space in the slice (∞,1)-topos, there is a canonical ∞-action of the smooth ∞-group $Aut_{/\mathbf{B}^n U(1)_{conn}}(\nabla)$ on it.

This ∞-group, it turns out, is the higher generalization of the group that integrates the Poisson bracket in ordinary symplectic geometry: the quantomorphism group. In particular, over linear base spaces it contains the higher analog of the Heisenberg group and its action on the space of states.

These structures are summarized in the following table:

higher geometric quantization	cohesive homotopy type theory	twisted cohomology
n-plectic ∞-groupoid	$X \stackrel{\omega}{\to} \Omega^{n+1}_{cl}(-,\mathbb{G})$	twisting cocycle in de Rham cohomology
symplectomorphism group	$\mathbf{Aut}_{/\Omega^{n+1}(-,\mathbb{G})}(\omega) = \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\omega}}\searrow && \swarrow_{\mathrlap{\omega}} \\ && \Omega^{n+1}_{cl}(-,\mathbb{G}) } \right\}$
prequantum circle n-bundle	$\array{ && \mathbf{B}^n \mathbb{G}_{conn} \\ & {}^{\mathllap{\mathbf{c}_{conn}}}\nearrow & \downarrow^{\mathrlap{curv}} \\ X &\stackrel{\omega}{\to}& \Omega^{n+1}(-,\mathbb{G})}$	twisting cocycle in differential cohomology
Planck's constant $\hbar$	$\tfrac{1}{\hbar}\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}$	divisibility of twisting class
quantomorphism group $\supset$ Heisenberg group	$\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) = \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\mathbf{c}_{conn}}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{c}_{conn}}} \\ && \mathbf{B}^n \mathbb{G}_{conn} } \right\}$	twist automorphism ∞-group
Hamiltonian quantum observables with Poisson bracket	$Lie(\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}))$	infinitesimal twist automorphisms
Hamiltonian actions of a smooth ∞-group $G$ / dual moment maps	$\mu : \mathbf{B}G \to \mathbf{B}\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn})$	$G$-∞-action on the twisting
gauge reduction	$\mathbf{c}_{conn}//G \,:\, X//G \to \mathbf{B}^n \mathbb{G}_{conn}$	$G$-∞-quotient of the twisting
Hamiltonian symplectomorphisms	∞-image of $\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{Aut}_{/\Omega^{n+1}_{cl}(-,\mathbb{G})}(\omega)$	twists in de Rham cohomology that lift to differential cohomology
∞-representation of n-group $\mathbf{B}^{n-1}\mathbb{G}$ on $V_n$	$\array{ V_n &\to& V_n//\mathbf{B}^{n-1}\mathbb{G} \\ && \downarrow^{\mathbf{p}} \\ && \mathbf{B}^n \mathbb{G} }$	local coefficient bundle
prequantum space of states	$\mathbf{\Gamma}_X(E) := [\mathbf{c},\mathbf{p}]_{/\mathbf{B}^n \mathbb{G}} = \left\{ \array{ X &&\stackrel{\sigma}{\to}&& V//\mathbf{B}^{n-1}\mathbb{G} \\ & {}_{\mathllap{\mathbf{c}}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{p}}} \\ && \mathbf{B}^n \mathbb{G} } \right\}$	cocycles in $[\mathbf{c}]$-twisted V-cohomology
prequantum operator	$\widehat{(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E)$	∞-action of twist automorphisms on twisted cocycles
trace to higher dimension	$\array{ [S^1, V_n//\mathbf{B}^{n-1}\mathbb{G}_{conn}] &\stackrel{tr\,hol_{S^1}}{\to}& V_{n-1}//\mathbf{B}^{n-2}\mathbb{G}_{conn} \\ \downarrow^{\mathrlap{\mathbf{p}^{V_n}_{conn}}} && \downarrow^{\mathrlap{\mathbf{p}^{V_{n-1}}_{conn}}} \\ \mathbf{B}^n \mathbb{G}_{conn} &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}^{n-1} \mathbb{G}_{conn} }$	fiber integration in ordinary differential cohomology adjoined with one in nonabelian differential cohomology

Therefore $\infty$-geometric prequantization, understood this way, is in itself a fundamental topic in higher twisted cohomology and nonabelian cohomology. But of course, as the name suggests, we are interested in it as a theory of quantum physics. We expect that $\infty$-geometric prequantization is to ordinary geometric quantization as extended functorial quantum field theory is to 1-functorial FQFT: it refines all structures to full codimension and reproduces them by transgression back to mapping spaces.

Notably, $\infty$-geometric prequantization applies seamlessly to the “extended action functionals” of ∞-Chern-Simons theory: as discussed there, these are precisely defined to be differential refinements $\mathbf{c}_{conn} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{\mathrm{conn}}$ of universal characteristic classes to moduli ∞-stacks of connections on G-principal ∞-bundles, for $G$ some smooth ∞-group. Hence these systems – which secretly contain many known quantum field theories that are traditionally not identified as being of ∞-Chern-Simons type – canonically come to us as prequantum circle $n$-bundles with connection on moduli $\infty$-stacks of field configurations.

Applying $\infty$-geometric prequantization to these systems is clearly not only possible, but seems to be the compelling step to take. This is what we are investigating in this project.

General theory

Let $\mathbf{H}$ be a choice of ambient cohesive ∞-topos, that models the kind of higher geometry – the kind of cohesion – in which we consider geometric prequantization. For instance for differential geometry we take $\mathbf{H} \coloneqq$ Smooth∞Grpd or for supergeometry we take $\mathbf{H} \coloneqq$ SmoothSuper∞Grpd.

We call an object $X \in \mathbf{H}$ equivalently

• a cohesive ∞-stack;
• a cohesive ∞-groupoid;
• a cohesive homotopy type,

depending on taste and situation.

(The first term has the advantage that it enjoys a certain tradition in the literature (for instance: moduli stacks), while it has the disadvantage that there is no good reason to introduce a new term for what should be called an ∞-sheaf. The second term has the advantage that it is true to the perspective of (∞,1)-topos theory, but the disadvantage that it may invoke in the reader associations with structures much more restricted than ∞-stacks. The third term has the advantage that is quite possibly the one with the most promising future, the main disadvantage being that the future is hard to predict.)

Let furthermore

• $\mathbb{A} \in \mathbf{H}$ be a line object;

• $\mathbb{G} \in Grp(\mathbf{H})$ the corresponding multiplicative group.

A default setup for $\infty$-geometric prequantization in the context of differential geometry would be to choose

• $\mathbf{H} =$ Smooth∞Grpd;

• $\mathbb{A} = \mathbb{C}$ the field of complex numbers

• $\mathbb{G} = \mathbb{C}^\times$ the Lie group, under multiplication, of invertible complex numbers.

Given this ambient data, we conceptualize geometric prequantization as follows, in four items.

1. Prequantum n-bundles;

2. Symplectomorphisms, Heisenberg- and Quantomorphism group

3. Spaces of states

4. Prequantum operators

1) Prequantum $n$-bundles over $n$-plectic moduli $\infty$-stacks

In the following objects denoted $\Sigma_k \in \mathbf{H}$ for $k \in \mathb{N}$ generically denote objects of cohomology dimension $k$.

Definition

For $n \in \mathbb{N}$ an extended ∞-model? Lagrangian is a morphism $\mathbf{c}$ in $\mathbf{H}$ of the form

$$\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn} \,.$$

Here we say that $X$ is, equivalently;

• the target space of the sigma-model;

• the moduli ∞-stack of gauge fields of a (higher) gauge theory.

And we say equivalently that $\mathbf{c}$ itself is the

• background gauge field;

• off-shell prequantum circle n-bundle.

Its curvature we call the n-plectic structure on $X$.

2) Symplectomorphism-, Quantomorphism- and Heisenberg-$\infty$-groups

Consider $\mathbf{c}$ as an object $\mathbf{H}_{/\mathbf{B}\mathbb{G}_{conn}}$ of the slice ∞-topos.

Definition

The quantomorphism ∞-group of $\mathbf{c}$ is the internal automorphism ∞-group

$$\mathbf{Q} := \mathbf{Aut}(\mathbf{c}_{conn}) \,.$$

3) Spaces of states

By the discussion at ∞-action, defining an $\infty$-action of the circle n-group $\mathbf{B}^{n-1} \mathbb{G}$ on an object $V \in \mathbf{B}$

$$\rho : V \times \mathbf{B}^{n-1} \mathbb{G} \to V$$

is equivalently giving a fiber sequence of the form

$$\array{ V &\to& V//\mathbf{B}^n \mathbb{G} \\ && \downarrow^{\mathbf{p}_\rho} \\ && \mathbf{B}^n \mathbb{G} } \,.$$

This is interpreted as the universal $\rho$-associated ∞-bundle.

Definition

For $\mathbf{c} : X \stackrel{\mathbf{c}_{conn}}{\to} \mathbf{B}^n \mathbb{G}_{conn} \to \mathbf{B}^n \mathbb{G}$ the moduli of the given prequantum circle n-bundle $P \to X$, the space of sections of the $\rho$-associated $\infty$-bundle is

$$\mathbf{\Gamma}_X( P \times_{\mathbf{B}^{n-1}}\mathbb{G} V ) \simeq [\mathbf{c}, \mathbf{p}_\rho]_{/\mathbf{B}^{n-1}\mathbb{G}} \,.$$

This is the space of prequantum states.

Remark

If we regard $\mathbf{p}_\rho$ as a local coefficient bundle, then this is equivalently the space of $[\mathbf{c}]$-twisted cohomology with coefficients in $V$.

4) Prequantum operators

There is a canonical action of the quantomorphism group on the space of sections

$$\widehat{(-)} : \mathbf{Aut}(\mathbf{c}_{conn} V) \times \mathbf{\Gamma}_X( P \times_{\mathbf{B}^{n-1} \mathbb{G}} V) \to \mathbf{\Gamma}_X( P \times_{\mathbf{B}^{n-1} \mathbb{G}} V) \,.$$

For $\phi \in \mathbf{Aut}(\mathbf{c}_{conn})$, we call

$$\widehat \phi : \mathbf{\Gamma}_X( P \times_{\mathbf{B}^{n-1} \mathbb{G}}) \to \mathbf{\Gamma}_X( P \times_{\mathbf{B}^{n-1} \mathbb{G}})$$

the corresponding prequantum operator.

(…)

Examples and applications

Ordinary geometric prequantization

This is discussed in (DCT, 4.4.17.1).

2-Plectic prequantization

This is discussed in (DCT, 4.4.17.2).

Extended $(4k+3)d$ abelian Chern-Simons theory

higher dimensional Chern-Simons theory

prequantum circle (4k+3)-bundle

from Beilinson-Deligne cup product

$$\mathbf{B}^{2k+1}U(1)_{conn} \stackrel{(-)\cup (-)}{\to} \mathbf{B}^{4k+3}U(1)_{conn}$$

The quantomorphism $\infty$-group of this should be

$$\mathbb{Z}_2 \simeq Aut(U(1)) \,.$$

For there is, up to equivalence, a unique autoequivalence

$$\mathbf{B}^{2k+1}U(1)_{conn} \stackrel{\simeq}{\to} \mathbf{B}^{2k+1}U(1)_{conn} \,,$$

the one induced by the nontrivial automorphism of $U(1)$. Since the cup-product is strictly invariant under this, this extends to

$$\array{ \mathbf{B}^{2k+1}U(1)_{conn} &&\stackrel{\simeq}{\to}&& \mathbf{B}^{2k+1}U(1)_{conn} \\ & {}_{\mathllap{(-)\cup(-)}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{(-)\cup(-)}} \\ && \mathbf{B}^{4k+3}U(1)_\conn } \,.$$

But for any further nontrivial such autoequivalence in the slice we would need in particular a gauge transformation parameterized by $(2k+1)$-forms over test manifolds from $C \wedge d C$ to itself. But the only closed $2k$-forms that we can produce naturally from $C$ are multiples of $C \wedge C$. But these all vanish since $C$ is of odd degree $2k+1$.

Quantomorphism 3-group of 3d $Spin$-Chern-Simons theory

• quantomorphism 3-group of 3d Spin-Chern-Simons theory

• prequantum circle 3-bundle:

  differential first fractional Pontryagin class

  $\tfrac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn}$

The total space of the prequantum 3-bundle is

$$\array{ \mathbf{B}String_{conn'} &\to& \Omega^{1 \leq \bullet \leq 3} &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} &\to& \mathbf{B}^3 U(1) }$$

as it appears in The moduli 3-stack of the C-field.

But the quantomorphism group of this will be small, as the Chern-Simons form is far from being gauge invariant.

Extended 3d $G \times G$-Chern-Simons theory

However, when we consider $G \times G$ CS theory given by

$$\mathbf{B}(G \times G)_{conn} \stackrel{\mathbf{c}^1_{conn}- \mathbf{c}^2_{conn}}{\to} \mathbf{B}^3 U(1)_{conn}$$

then diagonal gauge transformations $\mathbf{B}(G \times G)_{conn} \to \mathbf{B}(G \times G)_{conn}$ have interesting extensions to quantomorphisms, because for $g : U \to G$ the given gauge transformation at stage of definition $U$, the Chern-Simons form transforms by an exact term

$$CS(A_1^g,A_2^g) = CS(A_1,A_2) + d \langle A_1 - A_2, g^* \theta\rangle \,.$$

Quantomorphism 7-group of 7d $String$-Chern-Simons theory

• quantomorphism 7-group of 7d String-Chern-Simons theory?

• prequantum circle 7-bundle:

  differential second fractional Pontryagin class

  $\tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^z U(1)_{conn}$

The total space of the prequantum 7-bundle is

$$\array{ \mathbf{B}Fivebrane_{conn'} &\to& \Omega^{1 \leq \bullet \leq 7} &\to& * \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}String_{conn} &\stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} &\to& \mathbf{B}^7 U(1) }$$

References

For literature on traditional geometric quantization, see there.

Geometric prequantization of n-plectic geometry over smooth manifolds is discussed in

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)

Formulations in (∞,1)-topos theory are in sections 2.9.11 and 3.4.17 of

• Urs Schreiber, differential cohomology in a cohesive topos

For the broader context see also

• Thomas Nikolaus, Danny Stevenson, Danny Stevenson, Principal ∞-bundles – theory, presentations and applications

and the applications discussed there.

Related projects

• differential cohomology in a cohesive topos

  • Quantum gauge field theory in Cohesive homotopy type theory

  • cohomology

    • Principal ∞-bundles – theory, presentations and applications

    • Fivebrane structures

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • nonabelian 7d Chern-Simons theory and the 5-brane

    • Extended higher cup-product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

  • ∞-geometric prequantization