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.

Contents

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 mechanicsquantization\toquantum mechanics
symplectic geometrygeometric quantization\toquantum field theory
higher symplectic geometry–higher geometric quantization\toextended 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 BG\mathbf{B}G of GG-principal bundles for GG 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 BString\mathbf{B}String of string 2-group-Chern-Simons circle 7-bundle.

To capture this, write B n𝔾 connH\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 H\mathbf{H}. Then for XHX \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

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

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

k=k = transgression of c conn\mathbf{c}_{conn} to [Σ n1,X][\Sigma_{n-1},X]meaning in geometric quantization
nnexp(2πiS()):[Σ n,X][Σ n,c conn][Σ n,B n𝔾 conn]exp(2πi Σ n())𝔾\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
n1n-1exp(2πiS()):[Σ n1,X][Σ n1,c conn][Σ n1,B n𝔾 conn]exp(2πi Σ n1())B𝔾 conn\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 c conn\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 ρ\rho of the circle n-group B n1U(1)\mathbf{B}^{n-1}U(1) on some object VV is given by a fiber sequence in Smooth∞Grpd

V V//B n1U(1) ρ B nU(1), \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 VV-associated ∞-bundle corresponding to the universal principal ∞-bundle over B nU(1)\mathbf{B}^n U(1). Then for (PX,)(P \to X,\; \nabla) a circle n-bundle with connection classified (or rather: modulated) by a morphism g:XB nU(1) conng :X \to \mathbf{B}^n U(1)_{conn}, the ∞-groupoid of sections of the associated VV-\infty-bundle P× B n1U(1)VP \times_{\mathbf{B}^{n-1}U(1)} V is canonically identified with the gg-twisted ΩV\Omega V-cohomology of XX:

Γ X(P× B n1U(1)V)H /B nU(1)(g,ρ). \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 /B nU(1) conn()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 quantizationcohesive homotopy type theorytwisted cohomology
n-plectic ∞-groupoidXωΩ cl n+1(,𝔾)X \stackrel{\omega}{\to} \Omega^{n+1}_{cl}(-,\mathbb{G})twisting cocycle in de Rham cohomology
symplectomorphism groupAut /Ω n+1(,𝔾)(ω)={X X ω ω Ω cl n+1(,𝔾)}\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 B n𝔾 conn c conn curv X ω Ω n+1(,𝔾)\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 \hbar1c conn:XB n𝔾 conn\tfrac{1}{\hbar}\mathbf{c}_{conn} : X \to \mathbf{B}^n \mathbb{G}_{conn}divisibility of twisting class
quantomorphism group \supset Heisenberg groupAut /B n𝔾 conn(c conn)={X X c conn c conn B n𝔾 conn}\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 bracketLie(Aut /B n𝔾 conn(c conn))Lie(\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn}))infinitesimal twist automorphisms
Hamiltonian actions of a smooth ∞-group GG / dual moment mapsμ:BGBAut /B n𝔾 conn(c conn) \mu : \mathbf{B}G \to \mathbf{B}\mathbf{Aut}_{/\mathbf{B}^n \mathbb{G}_{conn}}(\mathbf{c}_{conn})GG-∞-action on the twisting
gauge reductionc conn//G:X//GB n𝔾 conn\mathbf{c}_{conn}//G \,:\, X//G \to \mathbf{B}^n \mathbb{G}_{conn}GG-∞-quotient of the twisting
Hamiltonian symplectomorphisms∞-image of Aut /B n𝔾 conn(c conn)Aut /Ω cl n+1(,𝔾)(ω)\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 B n1𝔾\mathbf{B}^{n-1}\mathbb{G} on V nV_nV n V n//B n1𝔾 p B n𝔾\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 statesΓ X(E):=[c,p] /B n𝔾={X σ V//B n1𝔾 c p B n𝔾}\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 [c][\mathbf{c}]-twisted V-cohomology
prequantum operator()^:Γ X(E)×Aut /B n𝔾 conn(c conn)Γ X(E)\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[S 1,V n//B n1𝔾 conn] trhol S 1 V n1//B n2𝔾 conn p conn V n p conn V n1 B n𝔾 conn exp(2πi S 1()) B n1𝔾 conn\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 c conn:BG connB nU(1) conn \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 GG 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 nn-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 H\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 H\mathbf{H} \coloneqq Smooth∞Grpd or for supergeometry we take H\mathbf{H} \coloneqq SmoothSuper∞Grpd.

We call an object XHX \in \mathbf{H} equivalently

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

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

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 nn-bundles over nn-plectic moduli \infty-stacks

In the following objects denoted Σ kH\Sigma_k \in \mathbf{H} for kmathbNk \in \mathb{N} generically denote objects of cohomology dimension kk.

Definition

For nn \in \mathbb{N} an extended σ-model Lagrangian is a morphism c\mathbf{c} in H\mathbf{H} of the form

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

Here we say that XX is, equivalently;

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

Its curvature we call the n-plectic structure on XX.

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

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

Definition

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

Q:=Aut(c conn). \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 B n1𝔾\mathbf{B}^{n-1} \mathbb{G} on an object VBV \in \mathbf{B}

ρ:V×B n1𝔾V \rho : V \times \mathbf{B}^{n-1} \mathbb{G} \to V

is equivalently giving a fiber sequence of the form

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

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

Definition

For c:Xc connB n𝔾 connB n𝔾\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 PXP \to X, the space of sections of the ρ\rho-associated \infty-bundle is

Γ X(P× B n1𝔾V)[c,p ρ] /B n1𝔾. \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 p ρ\mathbf{p}_\rho as a local coefficient bundle, then this is equivalently the space of [c][\mathbf{c}]-twisted cohomology with coefficients in VV.

4) Prequantum operators

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

()^:Aut(c connV)×Γ X(P× B n1𝔾V)Γ X(P× B n1𝔾V). \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 ϕAut(c conn)\phi \in \mathbf{Aut}(\mathbf{c}_{conn}), we call

ϕ^:Γ X(P× B n1𝔾)Γ X(P× B n1𝔾) \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(4k+3)d abelian Chern-Simons theory

higher dimensional Chern-Simons theory

prequantum circle (4k+3)-bundle from Beilinson-Deligne cup product

B 2k+1U(1) conn()()B 4k+3U(1) conn \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

2Aut(U(1)). \mathbb{Z}_2 \simeq Aut(U(1)) \,.

For there is, up to equivalence, a unique autoequivalence

B 2k+1U(1) connB 2k+1U(1) conn, \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)U(1). Since the cup-product is strictly invariant under this, this extends to

B 2k+1U(1) conn B 2k+1U(1) conn ()() ()() B 4k+3U(1) conn. \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)(2k+1)-forms over test manifolds from CdCC \wedge d C to itself. But the only closed 2k2k-forms that we can produce naturally from CC are multiples of CCC \wedge C. But these all vanish since CC is of odd degree 2k+12k+1.

Quantomorphism 3-group of 3d SpinSpin-Chern-Simons theory

The total space of the prequantum 3-bundle is

BString conn Ω 13 * BSpin conn 12p^ 1 B 3U(1) conn B 3U(1) \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×GG \times G-Chern-Simons theory

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

B(G×G) connc conn 1c conn 2B 3U(1) conn \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 B(G×G) connB(G×G) conn\mathbf{B}(G \times G)_{conn} \to \mathbf{B}(G \times G)_{conn} have interesting extensions to quantomorphisms, because for g:UGg : U \to G the given gauge transformation at stage of definition UU, the Chern-Simons form transforms by an exact term

CS(A 1 g,A 2 g)=CS(A 1,A 2)+dA 1A 2,g *θ. 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 StringString-Chern-Simons theory

The total space of the prequantum 7-bundle is

BFivebrane conn Ω 17 * BString conn 16p^ 2 B 7U(1) conn B 7U(1) \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

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

For the broader context see also

and the applications discussed there.

Revised on October 17, 2012 12:51:41 by David Corfield (129.12.18.29)