Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Idea

We discuss a refinement of the traditional notion of Atiyah Lie groupoids (the Lie groupoids which are the Lie integration of Atiyah Lie algebroids of $G$-principal bundles) from differential geometry to higher differential geometry and generally to higher geometry.

### General idea

Briefly, for $G$ an ∞-group in an (∞,1)-topos and $P \to X$ a $G$-principal ∞-bundle, its higher Atiyah groupoid is the groupoid object $At(P)$ such that the

• object of objects is $X$;

• object of morphisms is the collection of all $G$-equivariant maps between all pairs of fibers of $P$.

In these vague words this is precisely the same description as for the traditional Atiyah groupoid. Definition below makes precise what this means in higher geometry.

Besides generalizing the traditional definition to homotopy theory, the notion of higher Atiyah groupoids also generalizes from concrete objects such as Lie groups to general objects in an (∞,1)-topos (general ∞-stacks, not necessarily “supported on points”), notably to moduli ∞-stacks for cocycles in differential cohomology. For instance if we assume that the ambient (∞,1)-topos $\mathbf{H}$ is cohesive and consider $\mathbb{G} \in \mathrm{Grp}(\mathbf{H})$ a sylleptic ∞-group, then there is the moduli ∞-stack $\mathbf{B}\mathbb{G}_{\mathrm{conn}}$ of $\mathbb{G}$-principal ∞-connections and this is itself again a group object. A $(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$-principal ∞-bundle is equivalently a $(\mathbf{B}^2\mathbb{G})$-principal ∞-connection “without curving”. For instance if $\mathbb{G} = U(1)$ is the circle group in smooth ∞-groupoids, then $\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$ classifies circle 2-bundle with connection without 2-form part: in parts of the literature this is known as “bundle gerbes with connective structure but without curving”.

So the general definition considered here assigns a higher Atiyah groupoid to a “bundle gerbe with connective structure but no curving”. It turns out that this is the Courant 2-groupoid which Lie integrates the standard Courant Lie 2-algebroid traditionally induced by this data.

The notion of higher Atiyah groupoids is more general still: the definition does not really require that the object fed into the construction is a plain principal ∞-bundle. It may notably also be a genuine principal ∞-connection (hence withcurving”). We show below that the corresponding higher Atiyah groupoid is that groupoid object whose ∞-group of bisections is the quantomorphism n-group of the principal $\infty$-connection regarded as a prequantum n-bundle.

In summary, the higher geometric generalization of the notion of Atiyah groupoids unifies all three of the traditional notions of Atiyah groupoid, of Courant 2-groupoid and of quantomorphism group and refines each of these to higher geometry:

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid 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 ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

At the same time the definition of higher Atiyah groupoids in (∞,1)-topos theory, def. below, is very simple, almost tautological, identifying it as a very fundamental notion in (∞,1)-topos theory/homotopy type theory.

### In higher prequantum geometry: motivation and survey

Higher Atiyah groupoids play a central role in higher prequantum geometry.

under construction

#### 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$\hookrightarrow$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 charted 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

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

in the (2,1)-category of stacks, namely a diffeomorphism $\phi \colon X \stackrel{\simeq}{\to} X$ 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

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

This is a diffeomorphism $\phi \colon \mathcal{G}_0 \stackrel{\simeq}{\to} \mathcal{G}_0$ 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 $\mathcal{G}$ 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 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 $n$ 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

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

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 $G$-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

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

modulating/classifying the universal Chern-Simons circle 7-bundle with connection over the moduli 2-stack $\mathbf{B}String_{conn}$ 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

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

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

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

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

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

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

$\Omega X \coloneqq {*} \underset{X}{\times} {*} \,.$

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

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

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 modalityflat modalitysharp modality
$\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.

#### 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$\hookrightarrow$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

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

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

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

and

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

The extension sequence is then schematically simply the following

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

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:$\mathbf{B}\mathbb{G}$$\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$$\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 $\mathbb{G}$ 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

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

Similarly there is the Heisenberg infinity-group extension

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

The Lie differentiation of the ∞-group extension sequence of prop. is a homotopy fiber sequence of L-∞ algebras

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

where

The following table shows what this sequence reduces to when one chooses $\mathbb{G} = \mathbf{B}^{n-1}U(1)$.

higher and integrated Kostant-Souriau extensions:

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

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$
$n$geometrystructureunextended structureextension byquantum extension
$\infty$higher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$quantomorphism ∞-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
$n$n-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
$n$smooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected $X$)

#### Examples: $String$ and $Fivebrane$ as Heisenberg $\infty$-groups

###### Example

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

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

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

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

is the string 2-group extension

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

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

(and analogously for fivebrane 6-group…)

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

## Definition

We now turn to the formal definition of higher Atiyah groupoids and the basic constructions on them.

Let $\mathbf{H}$ be an (∞,1)-topos. Let $G \in Grp(\mathbf{H})$ be a group object in $\mathbf{H}$, an ∞-group.

We define now for every $G$-principal ∞-bundle $P \to X$ in $\mathbf{H}$ a groupoid object $At(P) \in Grpd(\mathbf{H})$ in $\mathbf{H}$. In order to do so we invoke two basic facts.

###### Proposition

The construction of forming the Cech nerve of a morphism consitutes an equivalence of (∞,1)-categories from that of 1-epimorphisms to that of groupoid objects in $\mathbf{H}$:

$(\mathbf{H}^{\Delta^1})_{1epi} \stackrel{\simeq}{\to} Grpd(\mathbf{H}) \,.$

This is a refined version of one of the Giraud-Rezk-Lurie axioms characterizing (∞,1)-topos, discussed at groupoid object in an (∞,1)-category.

###### Remark

In terms of traditional terminology in the literature on topological stacks/differentiable stacks etc, this says that a groupoid object in $\mathbf{H}$ is equivalently an object $X \in \mathbf{H}$ which is equipped with an atlas $X_0 \to X$.

Write $\mathbf{B}G \in \mathbf{H}$ for the delooping of $G$ in $\mathbf{H}$ (the moduli ∞-stack of $G$-principal ∞-bundles, as the following proposition asserts:

###### Proposition

The operation of forming (∞,1)-fibers (homotopy fibers) constitutes an equivalence of ∞-groupoids

$fib \;\colon\; \mathbf{H}(X, \mathbf{B}G) \to G Bund(X) \,.$

This is discussed at principal ∞-bundle.

Using these two facts we now set:

###### Definition

For $P \to X$ a $G$-principal ∞-bundle in $\mathbf{H}$, its Atiyah groupoid is the groupoid object $At \in \mathrm{Grpd}(\mathbf{H}) \simeq (\mathbf{H}^{\Delta^1})_{1epi}$ which is the 1-image projection of the classifying map $g \colon X \to \mathbf{B}G$:

$g \;\colon\; X \stackrel{}{\to} At(P) \coloneqq im_1(g) \hookrightarrow \mathbf{B}G \,.$
###### Remark

By the discussion at 1-image, the 1-image projection of any morphism $f \colon X \to Y$ in an (∞,1)-topos is equivalently given as the canonical map given by the (∞,1)-colimit over the Cech nerve

$X \to im_1(f) \simeq \underset{\rightarrow}{\lim} (X^{\times^{\bullet+1}_Y}) \,.$

This means that regarded as an object of $Grpd(\mathbf{H})$, the Atiyah groupoid $At(P)$ is simply the Cech nerve of the classifying map. This means that the definition of Atiyah groupoids in higher geometry is much more fundamental than in traditional geometry.

## Properties

### Equivalence of Atiyah-groupoid bisections to slice automorphisms

We discuss how the ∞-group of bisections of a higher Atiyah groupoid is canonically equivalent to the $\mathbf{H}$-valued automorphism ∞-group of the modulating map that gave rise to it, regarded as an object in the slice (∞,1)-topos over its codomain.

To this end we need the following two definitions

###### Definition

For $f \colon X \to Y$ a morphism in an (∞,1)-topos $\mathbf{H}$, its $\mathbf{H}$-valued automorphism ∞-group $\mathbf{Aut}_{\mathbf{H}}(f)$ is the dependent product over $Y$ over the automorphism ∞-group of $f$ regarded as an object in the slice (∞,1)-topos $\mathbf{H}_{/Y}$:

$\mathbf{Aut}_{\mathbf{H}}(f) \coloneqq \underset{Y}{\prod} \mathbf{Aut}_{/Y}(f) \,.$
###### Remark

For non-concrete codomains $Y$ one is usually interested in the concretification of this group. To be discussed… For an example see at The quantomorphism $n$-group below.

###### Proposition

For $f \colon X \to Y$ a morphism in $\mathbf{H}$, its $\mathbf{H}$-valued slice automorphism $\infty$-group according to prop. sits in an (∞,1)-pullback diagram

$\array{ \mathbf{Aut}_{\mathbf{H}}(f) &\to& \mathbf{Aut}(X) \\ \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{f \circ (-)}} \\ {*} &\stackrel{\vdash f}{\to}& [X,Y] } \,,$

where $\mathbf{Aut}(X) \hookrightarrow [X,X]$ is the ordinary automorphism ∞-group of $X$ in $\mathbf{H}$.

###### Definition

For $\mathcal{G} \in Grpd(\mathbf{H})$ a groupoid object in an (∞,1)-topos $\mathbf{H}$, its ∞-group of bisections $\mathbf{BiSect}(\mathcal{G}) \in Grpd(\mathbf{H})$ is the $\mathbf{H}$-valued automorphism $\infty$-group, def. , of the atlas $\phi_{\mathcal{G}} \colon \mathcal{G}_0 \to \mathcal{G}$ of $\mathcal{G}$ under prop. :

$\mathbf{BiSect}(\mathcal{G}) \coloneqq \mathbf{Aut}_{\mathbf{H}}(\phi_{\mathcal{G}}) \,.$

With this the following proposition is immediate, but important for the interpretation of higher Atiyah groupoids:

###### Propostition

For $c \;\colon\; X \to \mathbf{F}$ a morphism in an (∞,1)-topos $\mathbf{H}$, modulating an fiber ∞-bundle $P \to X$, there is an canonical equivalence of ∞-groups

$\mathbf{BiSect}(At(P)) \simeq \mathbf{Aut}_{\mathbf{H}}(c) \,.$
###### Remark

We may read this as saying that the higher Atiyah groupoid of an fiber ∞-bundle is the universal solution to the problem of finding a groupoid object whose ∞-group of bisections reproduces a given slice automorphism ∞-group. In many applications, this is indeed the crucial property that drives the interest in higher Atiyah groupoids, see the Examples below.

### Sequences of inclusions of Atiyah-bisection $\infty$-groups

Let $\mathbf{H}$ be an (∞,1)-topos which is cohesive. As discussed there, this implies that there is an internal notion of differential cohomology and in particular of principal ∞-connections in $\mathbf{H}$. We note here how the canonical forgetful maps between moduli ∞-stacks of principal ∞-bundles equipped with differing degree of differential refinement induce canonical inclusions of the corresponding higher Atiyah groupoids.

Let $\mathbb{G} \in Grp(\mathbf{H}) be a$braided ∞-group. Then there exists, by cohesion, a canonical notion of $\mathbb{G}$-principal ∞-connections, whose moduli ∞-stack we denote $\mathbf{B}\mathbb{G}_{\mathrm{conn}}$. This is equipped with a canonical map

$\mathbf{B}\mathbb{G}_{conn} \to \mathbf{B}\mathbb{G}$

which “forgets the connection”. Then for $\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}$ a $\mathbb{G}$-principal ∞-connection we write

$\nabla_0 \;\colon\; X \stackrel{\nabla}{\to} \mathbf{B}\mathbb{G}_{conn} \to \mathbf{B}\mathbb{G}$

for the corresponding underlying map.

###### Proposition

The dependent sum along this map induces a canonical map of ∞-groups

$\mathbf{BiSect}(At(\nabla)) \to \mathbf{BiSect}(At(\nabla_0)) \,.$

If we regard $\nabla$ as a prequantum n-bundle then this is a canonical inclusion of the quantomorphism n-group into the $\infty$-group of “$\nabla_0$-twisted diffeomorphisms”.

If $\mathbb{G}$ is even a sylleptic ∞-group, then the above moduli $\infty$-stacks have a further delooping and we obtain a 2-step sequence of forgetful maps

$\mathbf{B}^2 \mathbb{G}_{conn} \to \mathbf{B}(\mathbf{B}\mathbb{G}_{conn}) \to \mathbf{B}^2 \mathbb{G} \,.$

Accordingly:

###### Proposition

The dependent sum along these maps induces inclusions of $\infty$-groups

$\mathbf{BiSect}(At(\nabla)) \to \mathbf{BiSect}(At(\nabla_1)) \to \mathbf{BiSect}(At(\nabla_0)) \,.$
###### Remark

This now interprets as the inclusion

1. of the quantomorphism n-group into the $\infty$-group of bisections of the higher Courant groupoid;

2. $\infty$-group of bisections of the higher Courant groupoid into that of “$\nabla_0$-twisted diffeomorphisms”.

In summary we have the following table of inclusions

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

See below at Examples– The traditional Courant Lie 2-algebroid for more on this.

## Examples

We first show how the general notion of higher Atiyah groupoid reproduces various traditonal structures.

### The traditional Atiyah Lie groupoid

We discuss how the traditional notion of Atiyah Lie groupoids in traditional differential geometry is a special case of higher Atiyah groupoids of def. .

To set this up we take the ambient (∞,1)-topos to be

$\mathbf{H} \coloneqq$ Smooth∞Grpd and make use of the canonical embeddings

SmthMfd$\hookrightarrow$ DiffeologicalSpace $\hookrightarrow$ SmoothSpace $\hookrightarrow$ Smooth∞Grpd,

and

LieGrpd$\simeq$ differentiable stack $\hookrightarrow$ Smooth∞Grpd

which are understood in the following.

###### Proposition

For $G$ a Lie group, $X$ a smooth manifold and $P \to X$ a $G$-principal bundle, the traditional Atiyah Lie groupoid of $P$ is equivalent to that of def. .

###### Proof

Write $g \colon X \to \mathbf{B}G$ for the classifying map of $P \to X$, by prop. .

By remark the higher Atiyah groupoid $At(P)$ is simply the Cech nerve of this map. Since $G$ and $X$ and hence $P$ are all 0-truncated objects, hence $\mathbf{B}G$ a 1-truncated object, this Cech nerve is 2-coskeletal and hence is sufficient to consider the first three degrees. By definition these are

$At(P) \simeq \left( X \underset{\mathbf{B}G}{\times}X \underset{\mathbf{B}G}{\times}X \stackrel{\to}{\stackrel{\to}{\to}} X \underset{\mathbf{B}G}{\times}X \stackrel{\to}{\to} X \right) \,,$

where $X \underset{\mathbf{B}G}{\times}X$ denotes the homotopy fiber product of $g$ with itself, and so forth. To see what this object is, pick any $U \in$ CartSp, and observe that

$\mathbf{H}(U, X\underset{\mathbf{B}G}{\times}X ) \simeq \mathbf{H}(U,X) \underset{\mathbf{H}(U,\mathbf{B}G)}{\times} \mathbf{H}(U,X)$

(using that the (∞,1)-categorical hom-(∞,1)-functor $\mathbf{H}(U,-)$ preserves (∞,1)-limits) is equivalently the set of triples consisting of two smooth functions $\phi_1, \phi_2 \colon X$ and a gauge transformation between the pulled-back bundles $\eta \colon \phi_1^* P \to \phi_2^* P$ on $U$.

Since $U$ is topologically contractible, and hence every $G$-principal bundle over $U$ admits a section, every such triple induces a function, in fact a bijection, from the set of lifts $\hat \phi_1 \colon U \to P$ of $\phi_1$ to the set of lifts $\hat \phi_2 \colon U \to P$ which are $C^\infty(U,G)$-equivariant. By $G$-equivariant every pair consisting of a single lift $\hat \phi_1$ and its image $\eta(\hat \phi_1)$ already uniquely determes $\eta$. Therefore the above set of triples is naturally isomorphic to the set of smooth functions $U \to P \times_G P \coloneqq (P \times P)/_{diag} G$. This is precisely the smooth manifold of morphisms of the traditional Atiyah Lie groupoid. Since this is true for all $U \in$ CartSp and naturally so, and since CartSp is a site of definition of Smooth∞Grpd it follows by the (∞,1)-Yoneda lemma (which in the present cases reduces to the ordinary Yoneda lemma), we have a natural equivalence

$X \underset{\mathbf{B}G}{\times} X \simeq P \times_G P \,.$

In this manner it is immediate to check that this identification respects all the structure maps, and hence the above Cech nerve is indeed identified as the simplicial manifold which is the nerve of the traditional Atiyah Lie groupoid $(P \times_G P \stackrel{\to}{\to} X)$.

### The traditional Courant Lie 2-algebroid

There is a traditional construction which assigns to a bundle gerbe “with connective structure but without curving” a Courant Lie 2-algebroid. We discuss here how this is the Lie differentiation of the corresponding higher Atiyah groupoid.

In order to do so, we pick again, as above, as ambient context $\mathbf{H} =$ Smooth∞Grpd.

###### Proposition

For $\mathbb{G} \coloneqq U(1) \in LieGrp \hookrightarrow Grp(\mathbf{H})$ the circle Lie group (which is in particular a sylleptic ∞-group), the sequence of maps of moduli ∞-stacks

$\mathbf{B}^2 U(1)_{conn} \to \mathbf{B}(\mathbf{B}U(1)_{conn}) \to \mathbf{B}^2 U(1)$

of prop. is presented under the canonical equivalence Smooth∞Grpd $\simeq _{L_{lwhe}} Func(CartSp^{op}_{smooth}, sSet)$ by the image under the Dold-Kan correspondence of the evident sequence of chain maps

$\array{ U(1) &\stackrel{id}{\to}& U(1) &\stackrel{id}{\to}& U(1) \\ \downarrow^{\mathrlap{d log}} && \downarrow^{\mathrlap{d log}} && \downarrow^{\mathrlap{0}} \\ \Omega^1 &\stackrel{id}{\to}& \Omega^1 &\stackrel{\mathrlap{0}}{\to}& 0 \\ \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{0}} && \downarrow^{\mathrlap{0}} \\ \Omega^2 &\to& 0 &\to& 0 } \,,$

where on the left we have the Deligne complex for degree-3-ordinary differential cohomology.

###### Proof

This is a direct consequence of the discussion at circle n-bundle with connection.

###### Remark

This makes precise how

Let

$\nabla_1 \colon X \to \mathbf{B}(\mathbf{B}U(1)_{conn})$

be the map modulating circle 2-bundle with connection but “without curving”. Then then higher Atiyah groupoid of th $(\mathbf{B}U(1)_{conn})$-principal 2-bundle classified by this map has as higher Atiyah groupoid the corresponding Courant Lie 2-groupoid: the object which is the Lie integration of the traditional Courant Lie 2-algebroid associated with $\nabla_1$.

To see we observe that the corresponding 2-group of bisections is

$\mathbf{Aut}_{\mathbf{H}}(\nabla_1) \coloneqq \underset{\mathbf{B}(\mathbf{B}U(1)_{conn})}{\prod} \mathbf{Aut}_{/\mathbf{B}(\mathbf{B}U(1)_{conn})}(\nabla_1) \,.$

This has as objects diagrams in $\mathbf{H}$ of the form

$\array{ X &&\underoverset{\simeq}{\phi}{\to}&& X \\ & {}_{\mathllap{\nabla_1}}\searrow &\swArrow_\eta& \swarrow_{\mathrlap{\nabla_1}} \\ && \mathbf{B}(\mathbf{B}U(1)_{conn}) } \,,$

hece equivalently pairs consisting of a diffeomorphism $\phi \colon X \to X$ and a gauge transformation (of 2-connections without curving)

$\eta \;\colon \; \phi^* \nabla_1 \to \nabla_1 \,.$

The morphisms are accordingly the suitable natural transformations of these diagrams.

This is precisely the 2-group of “bundle gerbe symmetries” of $\nabla_1$ which is studient in (Collier). With this identification the main result there is the above claim.

Moreover, the canonical inclusions of smooth 2-groups of prop. reproduces, under Lie differentiation, the inclusion of the Poisson Lie 2-algebra into that Lie 2-algebra of sections of the corresponding Courant Lie 2-algebroid observed in (Rogers 10).

For $\nabla \colon X \to \mathbf{B}U(1)_{conn}$ the group of bisections of the corresponding Atiyah groupoid is the quantomorphism group of $\nabla$ regarded as a prequantum bundle.

(…)

### The quantomorphism $n$-groups

In (Rogers 11) is a proposal for the generalization of the notion of Poisson bracket Lie algebra of a symplectic manifold to a notion of Poisson Lie n-algebra induced by an n-plectic manifold. Since the Lie integration of the Poisson bracket is traditionally known as the quantomorphism group, the Lie integration of these Poisson Lie n-algebras should be called an quantomorphism n-group.

We here discuss a general abstract theory of quantomorphism n-groups as ∞-groups of bisections of a higher Atiyah groupoid associated with a principal ∞-connections. Then we show that under Lie differentiation this reproduces the construction in (Rogers 11).

higher and integrated Kostant-Souriau extensions:

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

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$
$n$geometrystructureunextended structureextension byquantum extension
$\infty$higher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$quantomorphism ∞-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
$n$n-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
$n$smooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected $X$)

For all of the following, let $\mathbf{H}$ be a cohesive (∞,1)-topos equipped with differential cohesion. Let $\mathbb{G} \on Grpd(\mathbf{H})$ be equipped with the structure of a braided ∞-group. Then there is a canonical object $\mathbf{B}\mathbb{G}_{conn} \in \mathbf{H}$ which is the moduli ∞-stack of $\mathbb{G}$-principal ∞-connections.

Fox such a principal ∞-connection given by a map

$\nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn} \,.$
###### Remark

By prop. the $\mathbf{H}$-valued automorphism $\infty$-group $\mathbf{Aut}_{\mathbf{H}}(\nabla)$ according to def. sits in an (∞,1)-pullback diagram of the form

$\array{ \mathbf{Aut}_{\mathbf{H}}(\nabla) &\to& \mathbf{Aut}(X) \\ \downarrow && \downarrow^{\mathrlap{\nabla \circ (-)}} \\ {*} &\stackrel{\vdash \nabla}{\to}& [X, \mathbf{B}\mathbb{G}_{conn}] } \,.$

By remark we want to pass to its concretification. Indeed, in the above diagram the mapping ∞-stack $[X, \mathbf{B}\mathbb{G}_{conn}]$ is not quite yet the correct moduli ∞-stack for $\mathbb{G}$-principal ∞-connections on $X$, but instead its differential concretification $\mathbb{G}\mathbf{Conn}(X)$ is, as defined at concretification - Examples - Of differential moduli. Therefore the following definition states the above pullback diagram with that replacement.

###### Definition

Let $\mathbb{G}$ be a braided ∞-group as above and let $\nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn}$ be a $\mathbb{G}$-principal ∞-connection.

The quantomorphism ∞-group $QuantMorph(\nabla) \in \mathrm{Grp}(\mathbf{H})$ of a $\nabla$ is the object fitting into the (∞,1)-pullback

$\array{ \mathbf{QuantMorph}(\nabla) &\to& \mathbf{Aut}(X) \\ \downarrow && \downarrow^{\mathrlap{\nabla \circ (-)}} \\ {*} &\stackrel{\vdash \nabla}{\to}& \mathbb{G}\mathbf{Conn}(X) } \,.$
###### Definition

The Hamiltonian symplectomorphism ∞-group $\mathbf{HamSympl}(\nabla)$ is the 1-image of the canonical map $\mathbf{QuantMorph}(\nabla) \to \mathbf{Aut}(X)$.

###### Proposition

The quantomorphism ∞-group $\mathbf{QuantMorph}(\nabla)$ in an ∞-group extension of the Hamiltonian symplectomorphism $\infty$-group of def. by the ∞-group $(\Omega\mathbb{G})\mathbf{FlatConn}(X)$ of concretified flat ∞-connections on $X$: we have a homotopy fiber sequence

$(\Omega\mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(\nabla) \to \mathbf{HamSympl}(\nabla) \,.$

Moreover, at least at the level of the underlying objects, this extension is classified by the cocycle $\mathbf{HamSympl}(\nabla) \stackrel{\nabla \circ (-)}{\to} \mathbf{B}((\Omega\mathbb{G})\mathbf{FlatConn}(X))$ in that we have a long homotopy fiber sequence

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

We now restrict this to a special case and describe it more in detail:

Let $\mathbf{H} =$ Smooth∞Grpd, let $X \in$ SmthMfd $\hookrightarrow$ Smooth∞Grpd and let $\mathbb{G} \coloneqq \mathbf{B}^{n-1}U(1) \in \mathrm{Grp}(\mathbf{H})$ be the circle n-group. Finally let $\omega \colon X \to \Omega^{n+1}$ be an n-plectic form and $\nabla \;\colon\; X \to \mathbf{B}^n U(1)_{conn}$ a prequantization by a prequantum circle n-bundle.

###### Proposition

The Lie differentiation of the ∞-group extension sequence of prop. is a homotopy fiber sequence of L-∞ algebras

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

where

(…) Heisenberg group (…)

### The Heisenberg $n$-group

If $X = G \in \mathbf{H}$ has itself ∞-group structure, then it is natural to restrict the quantomorphism ∞-group to that subgroup of the Hamiltonian symplectomorphism ∞-group whose elements come from the $G$-∞-action on itself. This is the corresponding Heisenberg ∞-group.

###### Definition

With all assumtions as above, let $G \in Grp(\mathbf{H})$ be an ∞-group and let

$G \hookrightarrow \mathbf{Aut}(G)$

(where on the right we have the automorphism ∞-group of the underlying object $G \in \mathbf{H}$) the inclusion that exhibits the left $G$-∞-action on itself.

The the Heisenberg ∞-group $\mathbf{Heis}(\nabla)$ is the (∞,1)-pullback in the diagram

$\array{ \mathbf{Heis}(\nabla) &\to& \mathbf{QuantMorph}(\nabla) \\ \downarrow && \downarrow \\ G &\to& \mathbf{Aut}(G) } \,.$

The following is an immediate consequence of the definition

###### Proposition

The Heisenberg ∞-group $\mathbf{Heis}(\nabla)$ is an ∞-group extension of $G$ by $(\Omega \mathbb{G})\mathbf{FlatConn}(G)$: we have a homotopy fiber sequence of ∞-groups

$(\Omega \mathbb{G})\mathbf{FlatConn}(G) \to \mathbf{QuantMorph}(\nabla) \to G \,.$
###### Example

In $\mathbf{H} =$ Smooth∞Grpd consider $G \in LieGrp \hookrightarrow Grp(\mathbf{H})$ a connected, simply connected compact semisimple Lie group, say the Spin group $G = Spin$. Then the Killing form invariant polynomial is a pre-3-plectic form on the moduli stack of $G$-principal connections:

$\langle -,-\rangle \;\colon\; \mathbf{B}G_{conn} \to \mathbf{\Omega}_{cl}^4 \,.$

This has a higher geometric prequantization by the smooth first fractional Pontryagin class, a map

$\tfrac{1}{2}\hat\mathbf{p}_1 \;\colon\; \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} \,.$

The transgression of this to maps oout of the circle yields a circle 2-bundle with connection

$\nabla \;\colon\; G \to [S^1, G] \stackrel{[S^1, \frac{1}{2}\hat\mathbf{p}_1]}{\to} [S^1, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to} \mathbf{B}^2 U(1) \,.$

This is a prequantum circle 2-bundle which prequantizes the canonical differential 3-form on $G$, the one which is left invariant and at the neutral element is $\langle -,[-,-]\rangle$.

Consider now the higher prequantum geometry of this 2-connection. So now $\mathbb{G} = \mathbf{B}U(1)$.

Observe that

\begin{aligned} (\Omega \mathbb{G})\mathbf{FlatConn}(G) & \simeq U(1) \mathbf{FlatConn}(G) \\ & = \mathbf{B}U(1) \end{aligned}

because $G$ is assumed to be simply connected. (Notice that $\mathbf{B}U(1)$ does appear here with its canonical smooth structure: while a gauge transformation from the trivial $U(1)$-principal connection to itself is a constant function along $X$, the smooth structure in $U(1)\mathbf{FlatConn}(G)$ comes from how this may vary in parameterized collections ).

Therefore by prop. we have an ∞-group extension

$\mathbf{B}U(1) \to \mathbf{Heis}(\nabla) \to G \,.$

This exhibits the Heisenberg 2-group here as the string 2-group $String(G)$:

$\mathbf{Heis}(\nabla) \simeq String(G) \,.$

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid 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 ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

## References

The above identification of higher Atiyah groupoids of “bundle gerbes with connective structure but without curving” with those Lie integrating the corresponding standard Courant Lie 2-algebroid is directly implied (under the above translations) by the main result in

The corresponding inclusion of the Poisson Lie 2-algebra into the Lie 2-algebra of bisections of the Courant Lie 2-algebroid was first observed in

in the context of 2-plectic geometry over smooth manifolds.

The Poisson Lie n-algebra over an n-plectic manifold, which by prop. is the Lie differentiation of the quantomorphism n-group of any prequantum circle n-bundle prequantizing the $n$-plectic form, has been proposed in

Most further statements here will appear in

Last revised on February 24, 2021 at 00:08:10. See the history of this page for a list of all contributions to it.