nLab higher geometric quantization

under construction, for a more coherent account see (hpqg).

Idea

Higher geometric quantization is meant to complete this table:

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

Being a concept in higher geometry, higher geometric quantization is formulated naturally in (∞,1)-topos theory. More precisely, since it involves not just cohomology but differential cohomology, it is formulated in cohesive (∞,1)-topos theory (cohesive homotopy type theory).

In this context, 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 $\mathbf{c}_{conn}$. The basic constructions that higher geometric quantization is concerned with are indicated in the following table. All of them have also a fundamental interpretation in twisted cohomology (independent of any interpretation in the context of quantization) this is indicated in the right column of the 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

Examples

Ordinary symplectic manifolds

• prequantum circle bundle

  $X \to \mathbf{B} U(1)_{conn}$

• local coefficient bundle

  $$\array{ \mathbb{C} &\to& \mathbb{C}//U(1) \\ && \downarrow \\ && \mathbf{B} U(1) }$$

Of non-integral 2-forms

While only integral presymplectic forms have a prequantization to a prequantum circle bundle with connection, hence to a $(\mathbb{Z} \to \mathbb{R})$-principal 2-bundle, a general 2-form has a higher prequantization given by a connection on a 2-bundle on a principal 2-bundle with structure-2-group that coming from the crossed module $(\Gamma \hookrightarrow \mathbb{R})$, where $\Gamma$ is the discrete group of periods of the 2-form.

This is discussed further at prequantization of non-integral 2-forms.

Of 2-plectic $\infty$-groupoids

In codimension 2

• prequantum circle 2-bundle

• local coefficient bundle

  $$\array{ \mathbf{B}U(n) &\to& \mathbf{B}PU(n) \\ && \downarrow^{\mathrlap{\mathbf{dd}_n}} \\ && \mathbf{B}^2 U(1) }$$

• space of states: twisted bundles

In codimension 1

Proposition There is a lift of coefficient bundles to loop space

$$\array{ [S^1,(\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] &\stackrel{tr hol_{S^1}}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow^{\mathrlap{\mathbf{p}^{\mathbf{B}U}}} && \downarrow^{\mathrlap{\mathbf{p}^{\mathbb{C}}}} \\ [S^1,\mathbf{B}^2 U(1)_{conn}] &\stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to}& \mathbf{B}U(1)_{conn} }$$

where on the left we have loop space objects formed in $\mathbf{H}$ and on the bottom we have fiber integration in ordinary differential cohomology.

Forming the pasting composite with this sends 2-states and 2-operators in codimension 2 to ordinary states and operators in codimension 1.

In particular it sends twisted bundles to sections of a line bundle.

For $X$ a D-brane and $\mathbf{c}_{conn}$ the B-field, this reproduces Freed-Witten anomaly cancellation mechanism.

$\infty$-Chern-Simons theory

∞-Chern-Simons theory

$G$ a smooth ∞-group,

$$\mathbf{c}_{conn} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}$$

a universal differential characteristic map.

The following examples are of this form.

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 $2k+1$.

For $k = 1$ the total space of the prequantum circle 3-bundle of $U(1)$-Chern-Simons theory over the point is the smooth moduli 2-stack of differential T-duality structures.

Extended 3d $\mathrm{Spin}$-Chern-Simons theory

• prequantum circle 3-bundle:

  differential first fractional Pontryagin class

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

So Planck's constant here is $\hbar = 2$ (relative to the naive multiple).

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.

See the discussion at Chern-Simons theory – Geometric quantization – In higher codimension.

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

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

Extended 7d $\mathrm{String}$-Chern-Simons theory

• prequantum circle 7-bundle:

  differential second fractional Pontryagin class

  $\tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn}$

So Planck's constant here is $\hbar = 6$ (relative to the naive multiple).

The total space of the prequantum 7-bundle is

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

$\infty$ Wess-Zumino-Witten theory

• ∞-Wess-Zumino-Witten theory

Differentially twisted looping of $\infty$-Chern-Simons theory

$$\Omega \mathbf{c} : G \to \mathbf{B}^{n-1}\mathbb{G}$$

Ordinary $G$-WZW model

• WZW model

$$\tilde\Omega \tfrac{1}{2}\mathbf{p}_1 : G \to \mathbf{B}^2 U(1)_{conn}$$

studied in (Rogers PhD, section 4.2).

$String$-WZW model

(…)

$Fivebrane$-WZW model

(…)

Related concepts

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

References

2-geometric quantization over smooth manifolds is discussed in section 6 and section 7 of

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

with further indications in

• Chris Rogers, Higher geometric quantization, at Higher Structures 2011 (pdf)

The special case of geometric quantization over infinitesimal action groupoids can be described in terms of BRST complexes. For references on this see Geometric quantization – References – Geometric BRST quantization.

Higher geometric quantization in a cohesive (∞,1)-topos over smooth ∞-groupoids is discussed in

• Domenico Fiorenza, Urs Schreiber, Chris Rogers, infinity-geometric prequantization

and the examples of higher Chern-Simons theories in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber Higher geometric prequantum theory

Review:

• Luigi Alfonsi, §3 in: Higher geometry in physics, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.07308]