nLab symplectic infinity-groupoid

under construction

Contents

Idea

A symplectic $\infty$-groupoid is a smooth ∞-groupoid equipped with a symplectic form, or, more generally, with an n-plectic form.

This is the generalization of the notion of symplectic manifold to higher symplectic geometry. It is also the image under Lie integration of the notion of symplectic L-∞ algebroid, which is also a higher analog of symplectic manifolds, but in an infinitesimal way.

Notice that every symplectic manifold is in particular a Poisson manifold and that the structure of a Poisson manifold is equivalently encoded in the corresponding Poisson Lie algebroid. A symplectic groupoid is the Lie integration of such a Poisson Lie algebroid. Therefore, strictly speaking, already “ordinary” symplectic geometry secretly involves Lie groupoids. This insight is exploited in the refinement of geometric quantization of symplectic groupoids.

Definition

For any $n \in \mathbb{N}$, a symplectic Lie n-algebroid $(\mathfrak{P}, \omega)$ is an L-∞ algebroid $\mathfrak{P}$ that is equipped with a quadratic and non-degenerate $L_\infty$-invariant polynomial.

Under Lie integration $\mathfrak{P}$ integrates to a smooth n-groupoid $\tau_n \exp(\mathfrak{P})$. Under the ∞-Chern-Weil homomorphism the invariant polynomial induces an differential form on an ∞-groupoid

$\omega : \tau_n \exp(\mathfrak{P}) \to \flat_{dR} \mathbf{B}^{n+2} \mathbb{R}$

representing a class $[\omega] \in H^{n+2}_{dR}(\tau_n \exp(\mathfrak{P}))$.

Let

$SymplSmooth\infty Grpd \hookrightarrow Smooth\infty Grpd/(\coprod_{n}\mathbf{\flat}_{dR}\mathbf{B}^{n+2}\mathbb{R})$

be the full sub-(∞,1)-category of the over-(∞,1)-topos of Smooth∞Grpd over the de Rham coefficient objects on those objects in the image of this construction.

We say an object on $SymplSmooth \infty Grpd$ is a symplectic smooth $\infty$-groupoid.

(There are evident variations of this for the ambient Smooth∞Grpd replaced by some variant, such as SynthDiff∞Grpd or SmoothSuper∞Grpd.)

Examples

The symplectic form $\omega$ on a symplectic Lie n-algebroid $\mathfrak{a}$ is Lie theoretically an invariant polynomial. Therefore by infinity-Chern-Weil theory it induces a moprhism

$\exp(\omega) : \tau_n\exp(\mathfrak{a}) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+2} \mathbb{R}$

from the Lie integration of $\mathfrak{a}$ to the de Rham coefficient object: this is an $(n+2)$-form on a smooth ∞-groupoid (as discussed at smooth ∞-groupoid – structures – de Rham cohomology) and hence equips $\exp(\mathfrak{a})$ with the structure of a symplectic $\infty$-groupoid.

We spell this out in some special cases.

$n = 0$ – Symplectic manifolds

A symplectic Lie 0-algebroid is simply a symplectic manifold, and so is its Lie integration.

$n = 1$ – Symplectic groupoids from Poisson Lie algebroids

We discuss the Lie integration of Poisson Lie algebroids to symplectic groupoids. For more details and applications of this see at extended geometric quantization of 2d Chern-Simons theory.

Let $\mathfrak{P}$ be the Poisson Lie algebroid corresponding to a Poisson manifold that comes from a symplectic manifold $(X,\omega)$.

The symplectic groupoid associated to this is (by the discussion there) supposed to be the fundamental groupoid $\Pi_1(X)$ of $X$ equipped on its space of morphisms with the differential form $p_1^* \omega - p_2^* \omega$, where $p_1,p_2$ are the two endpoint projections from paths in $X$ to $X$.

We demonstrate in the following how this is indeed the result of applying the ∞-Chern-Weil homomorphism to this situation.

For simplicity we shall start with the simple situation where $(X,\omega)$ has a global Darboux coordinate chart $\{x^i\}$. Write $\{\omega_{i j}\}$ for the components of the symplectic form in these coordinates, and $\{\omega^{i j}\}$ for the components of the inverse.

Then the Chevalley-Eilenberg algebra $CE(\mathfrak{P})$ is generated from $\{x^i\}$ in degree 0 and $\{\partial_i\}$ in degree 1, with differential given by

$d_{CE} x^i = - \omega^{i j} \partial_j$
$d_{CE} \partial_i = \frac{\partial \pi^{j k}}{\partial x^i} \partial_j \wedge \partial_k = 0 \,.$

The differential in the corresponding Weil algebra is hence

$d_{W} x^i = - \omega^{i j} \partial_j + \mathbf{d}x^i$
$d_{W} \partial_i = \mathbf{d} \partial_i \,.$

By the discussion at Poisson Lie algebroid, the symplectic invariant polynomial is

$\mathbf{\omega} = \mathbf{d} x^i \wedge \mathbf{d} \partial_i \in W(\mathfrak{P}) \,.$

Clearly it is useful to introduce a new basis of generators with

$\partial^i := -\omega^{i j} \partial_j \,.$

In this new basis we have a manifest isomorphism

$CE(\mathfrak{P}) = CE(\mathfrak{T}X)$

with the Chevalley-Eilenberg algebra of the tangent Lie algebroid of $X$.

Therefore the Lie integration of $\mathfrak{P}$ is the fundamental groupoid of $X$, which, since we have assumed global Darboux oordinates and hence contractible $X$, is just the pair groupoid:

$\tau_1 \exp(\mathfrak{P}) = \Pi_1(X) = (X \times X \stackrel{\overset{p_2}{\to}}{\underset{p_1}{\to}} X) \,.$

It remains to show that the symplectic form on $\mathfrak{P}$ makes this a symplectic groupoid.

Notice that in the new basis the invariant polynomial reads

\begin{aligned} \mathbf{\omega} &= - \omega_{i j} \mathbf{d}x^i \wedge \mathbf{d} \partial^j \\ & = \mathbf{d}( \omega_{i j} \partial^i \wedge \mathbf{d}x^j) \end{aligned}

and that we may regard this as a morphism of $L_\infty$-algebroids

$\mathbf{\omega} : \mathfrak{T}\mathfrak{P} \to \mathfrak{T}b^3 \mathbb{R}$

The corresponding infinity-Chern-Weil homomorphism that we need to compute is given by the ∞-anafunctor

$\array{ \exp(\mathfrak{P})_{diff} &\stackrel{\exp(\mathbf{\omega})}{\to}& \exp(b \mathbb{R})_{dR} &\stackrel{\int_{\Delta^\bullet}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^3 \mathbb{R} \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{P}) } \,.$

Over a test space $U$ in degree 1 an element in $\exp(\mathfrak{P})_{diff}$ is a pair $(X^i, \eta^i)$

$X^i \in C^\infty(U \times \Delta^1)$
$\eta^i \in \Omega^1_{vert}(U \times \Delta^1)$

subject to the verticality constraint, which says that along $\Delta^1$ we have

$d_{\Delta^1} X^i + \eta^i_{\Delta^1} = 0 \,.$

The vertical morphism $\exp(\mathfrak{P})_{diff} \to \exp(\mathfrak{P})$ has in fact a section whose image is given by those pairs for which $\eta^i$ has no leg along $U$. We therefore find the desired form on $\exp(\mathfrak{P})$ by evaluating the top morphism on pairs of this form.

Such a pair is taken by the top morphism to

\begin{aligned} (X^i, \eta^j) & \mapsto \int_{\Delta^1} \omega_{i j} F_{X^i} \wedge F_{\partial^j} \\ & = \int_{\Delta^1} \omega_{i j} (d_{dR} X^i + \eta^i) \wedge d_{dR} \eta^j \in \Omega^3(U) \end{aligned} \,.

Using the above verticality constraint and the condition that $\eta^i$ has no leg along $U$, this becomes

$\cdots = \int_{\Delta^1} \omega_{i j} d_U X^i \wedge d_U d_{\Delta^1} X^j \,.$

By the Stokes theorem the integration over $\Delta^1$ yields

$\cdots = \omega_{i j} d_{dR} x^i \wedge \eta^j|_{0} - \omega_{i j} d_{dR} x^i \wedge \eta^j|_{1} \,.$

This completes the proof.

Geometric quantization of symplectic $\infty$-groupoids

Idea

The notion of symplectic manifold formalizes in physics the concept of a classical mechanical system . The notion of geometric quantization of a symplectic manifold is one formalization of the general concept in physics of quantization of such a system to a quantum mechanical system .

Or rather, the notion of symplectic manifold does not quite capture the most general systems of classical mechanics. One generalization requires passage to Poisson manifolds . The original methods of geometric quantization become meaningless on a Poisson manifold that is not symplectic.

However, a Poisson structure on a manifold $X$ is equivalent to the structure of a Poisson Lie algebroid $\mathfrak{P}$ over $X$. This is noteworthy, because the latter is again symplectic, as a Lie algebroid, even if the underlying Poisson manifold is not symplectic: it is a symplectic Lie algebroid .

Based on related observations it was suggested that the notion of symplectic groupoid (see the references there) should naturally replace that of symplectic manifold for the purposes of geometric quantization to yield a notion of geometric quantization of symplectic groupoids .

Since a symplectic manifold can be regarded as a symplectic Lie 0-algebroid and also as a symplectic smooth 0-groupoid, this step amounts to a kind of categorification of symplectic geometry.

More or less implicitly, there has been strong evidence that this shift in perspective is substantial: the deformation quantization (see there for references) of a Poisson manifold turns out to be constructible in terms of correlators of the 2-dimensional TQFT called the Poisson sigma-model associated with the corresponding Poisson Lie algebroid. The fact that this is 2-dimensional and not 1-dimensional, as the quantum mechanical system that it thus encodes, is a direct reflection of this categorification shift of degree – see holographic principle for more on this.

On general abstract grounds this already suggests that it makes sense to pass via higher categorification further to symplectic Lie 2-algebroids, and generally symplectic Lie n-algebroids, as well as to symplectic 2-groupoids, symplectic 3-groupoids, etc. up to symplectic $\infty$-groupoids.

Formal hints for such a generalization had been noted in (Ševera), in particular in its concluding table. More indirect – but all the more noteworthy – hints came from quantum field theory, where it was observed that a generalization of symplectic geometry to multisymplectic geometry of degree $n$ more naturally captures the description of $n$-dimensional QFT (notice that quantum mechanics may be understood as $(0+1)$-dimensional QFT). For, observe that the symplectic form on a symplectic Lie n-algebroid is, while always “binary”, nevertheless a representative of de Rham cohomology in degree $(n+2)$.

There is a natural formalization of these higher symplectic structures in the context of any cohesive (∞,1)-topos. Moreover, with (FRS) we may observe that symplectic forms on L-∞ algebroids have a natural interpretation in ∞-Lie theory: they are $L_\infty$-invariant polynomials. This means that the ∞-Chern-Weil homomorphism applies to them.

We shall show below that all notions of geometric quantization of symplectic $\infty$-groupoids have a natural interpretation in terms of these canonical structures. For instance the higher “prequantum line bundle” is nothing but the circle n-bundle with connection that the ∞-Chern-Weil homomorphism assigns to the symplectic form, regarded as an $L_\infty$-invariant polynomial, and the corresponding “holographicTQFT – the AKSZ sigma-model – is that given by the induced ∞-Chern-Simons functional.

Prequantum circle $(n+1)$-bundle

Idea

What is called (geometric) prequantization is a refinement of symplectic 2-forms to curvature 2-forms on a line bundle with connection. This is called a choice of prequantum line bundle for the given symplectic form.

This has an evident generalization to closed forms of degree $(n+2)$. If integral, these may be refined to a curvature $(n+2)$-form on a circle n-bundle with connection . Since in the context of smooth ∞-groupoids we can have circle $n$-bundles over other smooth $\infty$-groupoids, this means that we canonically have the notion of prequantum circle $(n+1)$-bundles on a symplectic $n$-groupoid.

Moreover, since, as discussed above, the symplectic form on a symplectic $n$-groupoid may be regarded as the image of an invariant polynomial under the unrefined ∞-Chern-Weil homomorphism

$\omega : X \to \mathbf{\flat}_{dR} \mathbf{B}^{n+2}\mathbb{R} \,,$

the passage to the prequantum $(n+1)$-bundle with connection corresponds to passing to the refined ∞-Chern-Weil homomorphism

$\hat \omega : X \to \mathbf{B}^{n+1}U(1)_{conn}$

(as discussed there).

Definition

Let $(X, \omega)$ be a symplectic $\infty$-groupoid. Then $\omega$ represents a class

$[\omega] \in H^{n+2}_{dR}(X) \,.$

We say this form is integral if it is in the image of the curvature-projection

$curv : H^{n+1}_{diff}(X,U(1)) \to H^{n+2}_{dR}(X)$

from the ordinary differential cohomology of $X$.

In this case we say a prequantum circle (n+1)-bundle with connection for $(X,\omega)$ is a lift of $\omega$ to $\mathbf{H}_{diff}(X, \mathbf{B}^{n+1}U(1))$.

Write $\hat X \to X$ for the underlying circle (n+1)-group-principal ∞-bundle.

Proposition

If $(X, \omega)$ indeed comes from the Lie integration of a symplectic Lie n-algebroid $(\mathfrak{P}, \omega)$ such that the periods of the L-∞ cocycle $\pi$ that $\omega$ transgresses to are integral, then $\hat X$ is the Lie integration of the L-∞ extension

$b^{n}\mathbb{R} \to \hat \mathfrak{P} \to \mathfrak{P}$

classified by $\pi$:

$\hat X \simeq \tau_{n+1} \exp(\hat \mathfrak{P}) \,.$

Examples

$n = 2$ – String Lie 2-algebra

For $\mathfrak{g}$ a semisimple Lie algebra with quadratic invariant polynomial $\omega$, the pair $(b \mathfrak{g}, \omega)$ is a symplectic Lie 2-algebroid (Courant Lie 2-algebroid) over the point.

In this case the infinitesimal prequantum line 2-bundle is the delooping of the string Lie 2-algebra

$\widehat b \mathfrak{g} \simeq b \mathfrak{string}$

and the prequantum circle 2-group principal 2-bundle is the delooping of the smooth string 2-group

$(\hat X \to X) = (\mathbf{B}String \to \mathbf{B}G) \,.$

Poisson $L_\infty$-algebras

Idea

A Hamiltonian vector field on an ordinary symplectic manifold is a vector field $v$ whose contraction with the symplectic form yields an exact form

$\iota_v \omega = d \alpha \,.$

This definition generalizes verbatim to n-plectic geometry.

We observe below that this condition is equivalent to the fact that the flow $\exp(v) : X \to X$ of $v$ preserves the connection on any prequantum line bundle, up to homotopy (up to gauge transformation). In this form the definition has an immediate generalization to symplectic $n$-groupoids.

Definition

Definition

Let $\omega : X \to \mathbf{\flat}_{dR} \mathbf{B}^{n+2} U(1)$ be a symplectic $(n-1)$-groupoid and let

$\hat \omega : X \to \mathbf{B}^{n+2} U(1)_{conn}$

Regard it as an object in the over-(∞,1)-topos $\mathbf{H}/\mathbf{B}^{n+2}U(1)_{conn}$.

Consider the internal automorphism ∞-group

$\underline{Aut}_{\mathbf{H}/\mathbf{B}^{n+1}U(1)_{conn}}(X) \in \mathbf{H}$

of auto-equivalences that respect the ∞-connection that refines $\omega$.

Examples

Ordinary Hamiltonian vector fields

Proposition

For $\omega : X \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)$ an ordinary symplectic manifold, regarded as a symplectic 0-groupoid, the general definition 1 reproduces the standard notion of Hamiltonian vector fields.

Proof

An Hamiltonian diffeomorphism is given by a diagram

$\array{ X &&\stackrel{\phi}{\to} && X \\ & {}_{\mathllap{\hat \omega}}\searrow &\swArrow_{\alpha}& \swarrow_{\mathrlap{\hat \omega}} \\ && \mathbf{B} U(1)_{conn} } \,,$

where $\phi$ is an ordinary diffeomorphism. To compute the Lie algebra of this, we need to consider smooth 1-parameter families of such and differentiate them.

Assume first that the connection 1-form in $\hat \omega$ is globally defined $A \in \Omega^1(X)$ with $d A = \omega$. Then the above diagram is equivalent to

$(\phi(t)^* A - A) = d \alpha(t) \,,$

where $\alpha(t) \in C^\infty(X)$. Differentiating this at 0 yields the Lie derivative

$\mathcal{L}_v A = d \alpha' \,,$

where $v$ is the vector field of which $t \mapsto \phi(t)$ is the flow.

By Cartan calculus this is

$d \iota_v A + \iota_v d_{dR} A = d \alpha'$

hence

$\iota_v \omega = d (\alpha' - \iota_v A) \,.$

This says that for $v$ to be Hamiltonian, its contraction with $\omega$ must be exact. This is precisely the definition of Hamiltonian vector fields. The corresponding Hamiltonian here is $\alpha'-\iota_v A$.

In the general case that the prequantum circle n-bundle with connection is not trivial, we can present it by a Cech cocycle on the Cech nerve $C(P_* X \to X)$ of the based path space surjective submersion (regarding $P_* X$ as a diffeological space and choosing one base point per connected component, or else assuming without restriction that $X$ is connected).

Any diffeomorphism $\phi = \exp(v) : X \to X$ lifts to a diffeomorphism $P_*\phi : P_* X \to P_* X$ by setting $P_* \phi(\gamma) : (t \in [0,1]) \mapsto \exp(t v)(\gamma(t))$.

So we get a diagram

$\array{ C(P_* \to X) &&\stackrel{P_*\phi}{\to} && C(P_* \to X) \\ & {}_{\mathllap{\hat \omega}}\searrow &\swArrow_{\alpha}& \swarrow_{\mathrlap{\hat \omega}} \\ && \mathbf{B} U(1)_{conn} }$

of simplicial presheaves. Now the same argument as above applies on $P_* X$.

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-modelhigher symplectic geometry$(n+1)$d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension $(n+1)$discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
$n$symplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometry$d = n+1$ AKSZ sigma-model

References

Some ideas pointing to higher symplectic groupoids were indicated in

Aspects of the relation to multisymplectic geometry are in

A discussion of higher symplectic geometry in a general context is in

Some ingredients for the geometric quantization of symplectic Lie $n$-algebroids are constructed in