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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.
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
representing a class $[\omega] \in H^{n+2}_{dR}(\tau_n \exp(\mathfrak{P}))$.
Let
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.)
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
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.
A symplectic Lie 0-algebroid is simply a symplectic manifold, and so is its Lie integration.
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
The differential in the corresponding Weil algebra is hence
By the discussion at Poisson Lie algebroid, the symplectic invariant polynomial is
Clearly it is useful to introduce a new basis of generators with
In this new basis we have a manifest isomorphism
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:
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
and that we may regard this as a morphism of $L_\infty$-algebroids
The corresponding infinity-Chern-Weil homomorphism that we need to compute is given by the ∞-anafunctor
Over a test space $U$ in degree 1 an element in $\exp(\mathfrak{P})_{diff}$ is a pair $(X^i, \eta^i)$
subject to the verticality constraint, which says that along $\Delta^1$ we have
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
Using the above verticality constraint and the condition that $\eta^i$ has no leg along $U$, this becomes
By the Stokes theorem the integration over $\Delta^1$ yields
This completes the proof.
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 “holographic” TQFT – the AKSZ sigma-model – is that given by the induced ∞-Chern-Simons functional.
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
the passage to the prequantum $(n+1)$-bundle with connection corresponds to passing to the refined ∞-Chern-Weil homomorphism
(as discussed there).
Let $(X, \omega)$ be a symplectic $\infty$-groupoid. Then $\omega$ represents a class
We say this form is integral if it is in the image of the curvature-projection
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.
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
classified by $\pi$:
See geometric quantization of symplectic groupoids.
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
and the prequantum circle 2-group principal 2-bundle is the delooping of the smooth string 2-group
A Hamiltonian vector field on an ordinary symplectic manifold is a vector field $v$ whose contraction with the symplectic form yields an exact form
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.
Let $\omega : X \to \mathbf{\flat}_{dR} \mathbf{B}^{n+2} U(1)$ be a symplectic $(n-1)$-groupoid and let
be a prequantization circle n-bundle with connection.
Regard it as an object in the over-(∞,1)-topos $\mathbf{H}/\mathbf{B}^{n+2}U(1)_{conn}$.
Consider the internal automorphism ∞-group
of auto-equivalences that respect the ∞-connection that refines $\omega$.
Its image under $p_! : \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \to \mathbf{H}$ we call the Hamiltonian symplectomorphism $\infty$-group.
Its ∞-Lie algebra we call the Poisson ∞-Lie algebra of $(X, \omega)$.
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.
An Hamiltonian diffeomorphism is given by a diagram
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
where $\alpha(t) \in C^\infty(X)$. Differentiating this at 0 yields the Lie derivative
where $v$ is the vector field of which $t \mapsto \phi(t)$ is the flow.
By Cartan calculus this is
hence
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
of simplicial presheaves. Now the same argument as above applies on $P_* X$.
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
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
See also section 4.3 of
Some ingredients for the geometric quantization of symplectic Lie $n$-algebroids are constructed in