nLab
higher prequantum geometry

Context

Higher geometry

Geometric quantization

under construction

Contents

Idea

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.

Motivation and survey of results

Ordinary prequantum geometry in terms of automorphisms in slices

A sequence of time-honored traditional concepts in geometric quantization/prequantum geometry is

Lie groups:Heisenberg group\hookrightarrowquantomorphism group\hookrightarrowgauge group
Lie algebras:Heisenberg Lie algebra\hookrightarrowPoisson Lie algebra\hookrightarrowtwisted vector fields

For instance in the geometric quantization of the electrically charged particle sigma-model we have a prequantum circle bundle PP with connection on a bundle \nabla on a cotangent bundle X=T *YX = T^* Y which is essentially the pullback of the electromagnetic field-bundle on target spacetime YY. Its quantomorphism group is the group of diffeomorphisms PPP \stackrel{\simeq}{\to} P of the total space of the prequantum bundle which preserve the connection (also called the contactomorphism of (P,)(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 XX 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)U(1)-principal connections, which we write BU(1) conn\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 XX a map :XBU(1) conn\nabla \colon X \to \mathbf{B}U(1)_{conn} is equivalently a U(1)U(1)-principal connection and such that a homotopy η: 1 2\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

QuantMorph()={X ϕ X η BU(1) conn} \mathbf{QuantMorph}(\nabla) = \left\{ \array{ X &&\underoverset{\simeq}{\phi}{\to}&& X \\ & \searrow &\swArrow_{\eta}& \swarrow \\ && \mathbf{B}U(1)_{conn} } \right\}

in the (2,1)-category of stacks, namely a diffeomorphism ϕ:XX\phi \colon X \stackrel{\simeq}{\to} X of the base space of the bundle together with a gauge transformation of U(1)U(1)-principal connections η:ϕ *\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 XX 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 :XBU(1) conn\nabla \colon X \to \mathbf{B}U(1)_{conn} in the slice (2,1)-category of stacks/smooth groupoids over BU(1) conn\mathbf{B}U(1)_{conn}.

Such autoequivalences in slices are familiar from basic concepts of Lie groupoid theory. For 𝒢=(𝒢 1𝒢 0)\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 𝒢:𝒢 0𝒢p_\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 𝒢\mathcal{G}, its autoequivalences are diagrams as on the right of

BiSect(p 𝒢)={𝒢 0 ϕ 𝒢 0 η 𝒢}. \mathbf{BiSect}(p_{\mathcal{G}}) = \left\{ \array{ \mathcal{G}_0 &&\stackrel{\phi}{\to}&& \mathcal{G}_0 \\ & \searrow &\swArrow_\eta & \swarrow \\ && \mathcal{G} } \right\} \,.

This is a diffeomorphism ϕ:𝒢 0𝒢 0\phi \colon \mathcal{G}_0 \stackrel{\simeq}{\to} \mathcal{G}_0 of the smooth manifold of objects equipped with a natural transformation η\eta whose component map is a smooth function that assigns to each point q \in \mathcal{G_0} a morphism in 𝒢q \in \mathcal{G_\mathcal{G} of the form η q:qϕ(q)\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 BiSect(p 𝒢)\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 need for higher prequantum bundles

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 nn 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

BSpin conn 12p^ 1 B 3U(1) conn forgetconnections BSpin 12p 1 B 3U(1) geometricrealization BSpin 12p 1 K(,4), \array{ \mathbf{B}Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} \\ \downarrow && \downarrow & forget \; connections \\ \mathbf{B}Spin &\stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) \\ \downarrow && \downarrow & geometric\;realization \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& K(\mathbb{Z},4) } \,,

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 GG-Chern-Simons theory, which is the moduli stack BG conn\mathbf{B}G_{conn} of GG-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

BString conn 16p^ 2 B 7U(1) conn forgetconnections BString 16p 2 B 7U(1) geometricrealization BString 16p 2 K(,8), \array{ \mathbf{B}String_{conn} &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} \\ \downarrow && \downarrow & forget\; connections \\ \mathbf{B}String &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 U(1) \\ \downarrow && \downarrow & geometric\; realization \\ B String &\stackrel{\frac{1}{6}p_2}{\to}& K(\mathbb{Z},8) } \,,

modulating/classifying the universal Chern-Simons circle 7-bundle with connection over the moduli 2-stack BString conn\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

Brief recollection: Higher geometry

In order to say this, clearly we need some basics of higher geometry

Groupoids nerve Categories Kancomplexes (,1)Categories. \array{ && Groupoids \\ & \swarrow && \searrow^{\mathrlap{nerve}} \\ Categories && && Kan complexes \\ & \searrow && \swarrow \\ && (\infty,1)-Categories } \,.

Important construction principle for (∞,1)-categories: simplicial localization. For 𝒞\mathcal{C} a category with some subset of morphisms WMor(𝒞)W \hookrightarrow Mor(\mathcal{C}) declared to be “weak equivalences”, the simplicial localization

L W𝒞(,1)Cat L_W \mathcal{C} \in (\infty,1)Cat

is the universal (,1)(\infty,1)-category obtained from 𝒞\mathcal{C} by universally turning each weak equivalence into an actual homotopy equivalence in the sense of homotopy theory.

In particular let CC be a site, assumed for simplicity to have enough points. Declare then that in the functor category Func(C op,KanCplx)Func(C^{op}, KanCplx), hence in Kan complex-valued presheaves, the weak equivalences are the stalkwise homotopy equivalences of Kan complexes. Then

HSh (C)L WFunc(C op,KanCplx) \mathbf{H} \coloneqq Sh_{\infty}(C) \coloneqq L_{W} Func(C^{op}, KanCplx)

is called the (∞,1)-topos of (∞,1)-sheaves/∞-stacks on CC.

An A-∞ algebra-object GG in such an (,1)(\infty,1)-topos such that π 0(G)\pi_0(G) is a group is called an ∞-group “with geometric structure as encoded by the test spaces CC”. The canonical source of \infty-groups are the homotopy fiber products of point inclusions *X* \to X of any object X, the loop space object

ΩX*×X*. \Omega X \coloneqq {*} \underset{X}{\times} {*} \,.

In fact this are all the ∞-groups that there are, up to equivalence: forimg loop space objects is an equivalence of (∞,1)-categories

Grp(H)BΩH 1 */ Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\leftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\to}} \mathbf{H}^{*/}_{\geq 1}

between ∞-groups and pointed connected objects. The inverse equivalence B\mathbf{B} is the delooping operation.

We say that such an (,1)(\infty,1)-topos H\mathbf{H} is cohesive if it is equipped with an adjoint triple of idempotent (co)/(∞,1)-monads

shape modalityflat modalitysharp modality
idemp. monadidemp. comonadidemp. monad
Π\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 𝔾Grp(H)\mathbb{G} \in Grp(\mathbf{H}) there is a canonical object B𝔾 conn\mathbf{B}\mathbb{G}_{conn} which modulats 𝔾\mathbb{G}-principal ∞-connections.

Higher Atiyah groupoids

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\hookrightarrowquantomorphism ∞-group\hookrightarrow∞-bisections of higher Courant groupoid\hookrightarrow∞-bisections of higher Atiyah groupoid
L-∞ algebras:Heisenberg L-∞ algebra\hookrightarrowPoisson L-∞ algebra\hookrightarrowCourant L-∞ algebra\hookrightarrowtwisted vector fields

(…)

The way all these notions and theorems work is by considering automorphism ∞-groups of the classifying (or rather: modulating) maps :XB𝔾 conn\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn} of a prequantum ∞-bundle in the slice (∞,1)-topos over the domain. For instance

QuantMorph()={X ϕ X B𝔾 conn}. \mathbf{QuantMorph}(\nabla) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \,.

The others are obtained by succesively forgetting connection data. For instance

BiSect(Cou())={X ϕ X 1 1 B(B𝔾 conn)}. \BiSect(Cou(\nabla)) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla_1}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_1}} \\ && \mathbf{B}(\mathbf{B}\mathbb{G}_{conn}) } \right\} \,.

and

BiSect(At())={X ϕ X 0 0 B𝔾}. \BiSect(At(\nabla)) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla_0}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_0}} \\ && \mathbf{B}\mathbb{G} } \right\} \,.

The extension sequence is then schematically simply the following

{ X B𝔾 conn}{X X B𝔾 conn}{X X} \left\{ \array{ && X \\ & \swarrow & & \searrow \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \; \to \; \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \; \to \; \left\{ \array{ X && \stackrel{\simeq}{\to} && X } \right\}

in this generality this now includes various other notions, too:

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid version of quantomorphism n-group
coefficient for cohomology:B𝔾\mathbf{B}\mathbb{G}B(B𝔾 conn)\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})B𝔾 conn\mathbf{B} \mathbb{G}_{conn}
type of fiber ∞-bundle:principal ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

The central theorem: Quantomorphism \infty-group extensions

Theorem

For 𝔾\mathbb{G} a braided ∞-group and :XB𝔾 conn\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn} a higher prequantum geometry with respect to 𝔾\mathbb{G} there is a long homotopy fiber sequence

(Ω𝔾)FlatConn()QuantMorph()HamSympl()()B((Ω𝔾)FlatConn()). \left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right) \to \mathbf{QuantMorph}(\nabla) \to \mathbf{HamSympl}(\nabla) \stackrel{\nabla \circ (-)}{\to} \mathbf{B}\left(\left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right) \right) \,.

Similarly there is the Heisenberg infinity-group extension

(Ω𝔾)FlatConn(X)Heis()G (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{Heis}(\nabla) \to G
Theorem

The Lie differentiation of the ∞-group extension sequence of prop. \ref{QuantomorphismExtension} is a homotopy fiber sequence of L-∞ algebras

H(X,B n1)𝔓𝔬𝔦𝔰𝔰𝔬𝔫(X,ω)𝒳 Ham(X,ω)ι ()ωBH(X,B n1), \mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \to \mathfrak{Poisson}(X,\omega) \to \mathcal{X}_{Ham}(X,\omega) \stackrel{\iota_{(-)\omega}}{\to} \mathbf{B}\mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \,,

where

The following table shows what this sequence reduces to when one chooses 𝔾=B n1U(1)\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 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

Examples: StringString and FivebraneFivebrane as Heisenberg \infty-groups

Example

For GG a simply connected semisimple compact Lie group such as the spin group, let

exp(2πi S 1[S 1,12p^ 1]):GB 2U(1) conn \nabla \coloneqq \exp\left(2 \pi i \int_{S^1} [S^1, \tfrac{1}{2}\hat \mathbf{p}_1]\right) \;\colon\; G \to \mathbf{B}^2 U(1)_{conn}

be the canonical circle 2-bundle with connection over it. Then the Heisenberg 2-group extension

U(1)FlatConn(G)Heis()G U(1)\mathbf{FlatConn}(G) \to \mathbf{Heis}(\nabla) \to G

is the string 2-group extension

BU(1)String(G)G. \mathbf{B}U(1) \to String(G) \to G \,.

(by classification of extensions by cohomology… by Lie 2-algebra computation…)

(and analogously for fivebrane 6-group…)

B 6U(1)Heis(exp(2πi S 1[S 1,12p^ 2]))String \mathbf{B}^6 U\left(1\right) \to \mathbf{Heis}\left(\exp\left(2 \pi i \int_{S^1} \left[S^1, \tfrac{1}{2}\hat \mathbf{p}_2\right] \right)\right) \to String

Constructions in higher prequantum geometry

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:Heisenberg ∞-group\hookrightarrowquantomorphism ∞-group\hookrightarrow∞-bisections of higher Courant groupoid\hookrightarrow∞-bisections of higher Atiyah groupoid
L-∞ algebras:Heisenberg L-∞ algebra\hookrightarrowPoisson L-∞ algebra\hookrightarrowCourant L-∞ algebra\hookrightarrowtwisted vector fields

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid version of quantomorphism n-group
coefficient for cohomology:B𝔾\mathbf{B}\mathbb{G}B(B𝔾 conn)\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})B𝔾 conn\mathbf{B} \mathbb{G}_{conn}
type of fiber ∞-bundle:principal ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

higher and integrated Kostant-Souriau extensions:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

References

See also the references at n-plectic geometry and at higher geometric quantization

Revised on September 21, 2017 16:25:37 by Urs Schreiber (77.56.177.247)