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There are two dual kinds of mathematical formalization of quantization, one is algebraic, usually known as deformation quantization, the other is geometric, known as geometric quantization. The latter involves an intermediate stage called prequantization. This is the refinement of symplectic geometry obtained by lifting closed differential 2-forms to being curvature forms of circle bundles with connection. Since circle bundles and circle bundles with connection, as opposed to plain differential forms, have non-trivial automorphisms (they form a groupoid not just a set) the group of symmetries of a prequantized symplectic manifold differs from the group of symmetries of the underlying symplectic manifold (the symplectomorphisms): the freedom in identifying the prequantum $U(1)$-bundle with itself makes it is a central extension, specifically a $U(1)$-central extension if the base manifold is connected. This circle extension with the integrality that it relates to (e.g. via its Chern class with coefficients in $\mathbb{Z} = \pi_1(U(1))$) is behind much of what is quanta-like about quantum physics. Hence while prequantization is not yet full quantization, much of the hallmark structure of quantization is present already in these “quantomorphism” $U(1)$-extensions of the symmetries of prequantum bundles, notably its Lie algebra subsumes the Heisenberg algebra of symplectic vector spaces, this we survey below in The geometric idea of quantization.
Hence it makes good sense to study prequantized smooth manifolds as geometries in their own right, and a sensible name for this is prequantum geometry.
As one passes from quantum mechanics to $(p+1)$-dimensional quantum field theory such as to preserve the locality of local quantum field theory, then the process of quantization involves not just symplectic manifolds equipped with symplectic differential 2-forms, but involves “symplectic currents” which are $(p+2)$-forms. Such higher degree analogs of symplectic forms are known historically already from the de Donder-Weyl-Hamilton equation of local classical field theory and approaches to promote their theory to a higher analog of symplectic geometry go by the names multisymplectic geometry and (p+1)-plectic geometry. The higher prequantization of such a higher degree form is naturally a circle (p+1)-bundle with connection (a bundle gerbe with connection for $p = 1$, generally a cocycle in Deligne cohomology of degree $p+2$), and hence higher prequantum geometry is the study of spaces that are equipped with such abelian principal infinity-connections.
This is what we discuss here. More abstractly, if we we write $\mathbf{B}\mathbb{G}_{conn}$ for the higher moduli stack of $\mathbb{G}$-principal infinity-connections, for instance $\mathbf{B}^{p+1}U(1)_{conn}$ the moduli stack for cocycles in Deligne cohomology in degree $(p+2)$, then a prequantum $\mathbb{G}$-bundle over some smooth homotopy type $X$ is equivalently just a morphism $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$, and a symmetry of such a higher prequantum geometry $(X,\nabla)$ is a diagram of the form
This means that for $\mathbf{H}$ a cohesive (∞,1)-topos of geometric spaces of sorts, then the corresponding $\mathbb{G}$-prequantum geometry is that encoded by the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}\mathbb{G}_{conn}}$. Hence prequantum geometry is the geometry in slices over higher moduli stacks for differential cohomology.
Quantization of course was and is motivated by experiment, hence by observation of the observable universe: it just so happens that quantum mechanics and quantum field theory correctly account for experimental observations where classical mechanics and classical field theory gives no answer or incorrect answers. A historically important example is the phenomenon called the “ultraviolet catastrophe”, a paradox predicted by classical statistical mechanics which is not observed in nature, and which is corrected by quantum mechanics.
But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from classical mechanics to quantum mechanics. Could one have been led to quantum mechanics by just pondering the mathematical formalism of classical mechanics? (Hence more precisely: is there a natural Synthetic Quantum Field Theory?)
The following spells out an argument to this effect. It will work for readers with a background in modern mathematics, notably in Lie theory, and with an understanding of the formalization of classical/prequantum mechanics in terms of symplectic geometry.
So to briefly recall, a system of classical mechanics/prequantum mechanics is a phase space, formalized as a symplectic manifold $(X, \omega)$. A symplectic manifold is in particular a Poisson manifold, which means that the algebra of functions on phase space $X$, hence the algebra of classical observables, is canonically equipped with a compatible Lie bracket: the Poisson bracket. This Lie bracket is what controls dynamics in classical mechanics. For instance if $H \in C^\infty(X)$ is the function on phase space which is interpreted as assigning to each configuration of the system its energy – the Hamiltonian function – then the Poisson bracket with $H$ yields the infinitesimal time evolution of the system: the differential equation famous as Hamilton's equations.
Something to take notice of here is the infinitesimal nature of the Poisson bracket. Generally, whenever one has a Lie algebra $\mathfrak{g}$, then it is to be regarded as the infinitesimal approximation to a globally defined object, the corresponding Lie group (or generally smooth group) $G$. One also says that $G$ is a Lie integration of $\mathfrak{g}$ and that $\mathfrak{g}$ is the Lie differentiation of $G$.
Therefore a natural question to ask is: Since the observables in classical mechanics form a Lie algebra under Poisson bracket, what then is the corresponding Lie group?
The answer to this is of course “well known” in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that has found its way into the basic educational textbooks. The answer is that this Lie group which integrates the Poisson bracket is the “quantomorphism group”, an object that seamlessly leads to the quantum mechanics of the system.
Before we spell this out in more detail, we need a brief technical aside: of course Lie integration is not quite unique. There may be different global Lie group objects with the same Lie algebra.
The simplest example of this is already one of central importance for the issue of quantization, namely, the Lie integration of the abelian line Lie algebra $\mathbb{R}$. This has essentially two different Lie groups associated with it: the simply connected translation group, which is just $\mathbb{R}$ itself again, equipped with its canonical additive abelian group structure, and the discrete quotient of this by the group of integers, which is the circle group
Notice that it is the discrete and hence “quantized” nature of the integers that makes the real line become a circle here. This is not entirely a coincidence of terminology, but can be traced back to the heart of what is “quantized” about quantum mechanics.
Namely, one finds that the Poisson bracket Lie algebra $\mathfrak{poiss}(X,\omega)$ of the classical observables on phase space is (for $X$ a connected manifold) a Lie algebra extension of the Lie algebra $\mathfrak{ham}(X)$ of Hamiltonian vector fields on $X$ by the line Lie algebra:
This means that under Lie integration the Poisson bracket turns into an central extension of the group of Hamiltonian symplectomorphisms of $(X,\omega)$. And either it is the fairly trivial non-compact extension by $\mathbb{R}$, or it is the interesting central extension by the circle group $U(1)$. For this non-trivial Lie integration to exist, $(X,\omega)$ needs to satisfy a quantization condition which says that it admits a prequantum line bundle. If so, then this $U(1)$-central extension of the group $Ham(X,\omega)$ of Hamiltonian symplectomorphisms exists and is called… the quantomorphism group $QuantMorph(X,\omega)$:
While important, for some reason this group is not very well known, which is striking because it contains a small subgroup which is famous in quantum mechanics: the Heisenberg group.
More precisely, whenever $(X,\omega)$ itself has a compatible group structure, notably if $(X,\omega)$ is just a symplectic vector space (regarded as a group under addition of vectors), then we may ask for the subgroup of the quantomorphism group which covers the (left) action of phase space $(X,\omega)$ on itself. This is the corresponding Heisenberg group $Heis(X,\omega)$, which in turn is a $U(1)$-central extension of the group $X$ itself:
At this point it is worth pausing for a second to note how the hallmark of quantum mechanics has appeared as if out of nowhere simply by applying Lie integration to the Lie algebraic structures in classical mechanics:
if we think of Lie integrating $\mathbb{R}$ to the interesting circle group $U(1)$ instead of to the uninteresting translation group $\mathbb{R}$, then the name of its canonical basis element $1 \in \mathbb{R}$ is canonically “$i$”, the imaginary unit. Therefore one often writes the above central extension instead as follows:
in order to amplify this. But now consider the simple special case where $(X,\omega) = (\mathbb{R}^2, d p \wedge d q)$ is the 2-dimensional symplectic vector space which is for instance the phase space of the particle propagating on the line. Then a canonical set of generators for the corresponding Poisson bracket Lie algebra consists of the linear functions $p$ and $q$ of classical mechanics textbook fame, together with the constant function. Under the above Lie theoretic identification, this constant function is the canonical basis element of $i \mathbb{R}$, hence purely Lie theoretically it is to be called “$i$”.
With this notation then the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads
Since the choice of basis element of $i \mathbb{R}$ is arbitrary, we may rescale here the $i$ by any non-vanishing real number without changing this statement. If we write “$\hbar$” for this element, then the Poisson bracket instead reads
This is of course the hallmark equation for quantum physics, if we interpret $\hbar$ here indeed as Planck's constant. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) Lie integration of the Poisson bracket.
This is only the beginning of the story of quantization, naturally understood and indeed “derived” from applying Lie theory to classical mechanics. From here the story continues. It is called the story of geometric quantization. We close this motivation section here by some brief outlook.
The quantomorphism group which is the non-trivial Lie integration of the Poisson bracket is naturally constructed as follows: given the symplectic form $\omega$, it is natural to ask if it is the curvature 2-form of a $U(1)$-principal connection $\nabla$ on complex line bundle $L$ over $X$ (this is directly analogous to Dirac charge quantization when instead of a symplectic form on phase space we consider the the field strength 2-form of electromagnetism on spacetime). If so, such a connection $(L, \nabla)$ is called a prequantum line bundle of the phase space $(X,\omega)$. The quantomorphism group is simply the automorphism group of the prequantum line bundle, covering diffeomorphisms of the phase space (the Hamiltonian symplectomorphisms mentioned above).
As such, the quantomorphism group naturally acts on the space of sections of $L$. Such a section is like a wavefunction, except that it depends on all of phase space, instead of just on the “canonical coordinates”. For purely abstract mathematical reasons (which we won’t discuss here, but see at motivic quantization for more) it is indeed natural to choose a “polarization” of phase space into canonical coordinates and canonical momenta and consider only those sections of the prequantum line bundle which depend only on the former. These are the actual wavefunctions of quantum mechanics, hence the quantum states. And the subgroup of the quantomorphism group which preserves these polarized sections is the group of exponentiated quantum observables. For instance in the simple case mentioned before where $(X,\omega)$ is the 2-dimensional symplectic vector space, this is the Heisenberg group with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.
We discuss here the infinitesimal symmetries of prequantum geometries, exhibited by the Poisson bracket Lie algebra and its higher analogs, the Poisson bracket Lie n-algebra.
We discuss the traditional definition of the Poisson bracket of a (pre-)symplectic manifold, highlighting how conceptually it may be understood as the algebra of infinitesimal symmetries of any of its prequantizations.
Let $X$ be a smooth manifold. A closed differential 2-form $\omega \in \Omega_{cl}^2(X)$ is a symplectic form if it is non-degenerate in that the kernel of the operation of contracting with vector fields
is trivial: $ker(\iota_{(-)}\omega) = 0$.
If $\omega$ is just closed with possibly non-trivial kernel, we call it a presymplectic form. (We do not require here the dimension of the kernel restricted to each tangent space to be constant.)
Given a presymplectic manifold $(X, \omega)$, then a Hamiltonian for a vector field $v \in Vect(X)$ is a smooth function $H \in C^\infty(X)$ such that
If $v \in Vect(X)$ is such that there exists at least one Hamiltonian for it then it is called a Hamiltonian vector field. Write
for the $\mathbb{R}$-linear subspace of Hamiltonian vector fields among all vector fields
When $\omega$ is symplectic then, evidently, there is a unique Hamiltonian vector field, def. 2, associated with every Hamiltonian, i.e. every smooth function is then the Hamiltonian of precisely one Hamiltonian vector field (but two different Hamiltonians may still have the same Hamiltonian vector field uniquely associated with them). As far as prequantum geometry is concerned, this is all that the non-degeneracy condition that makes a closed 2-form be symplectic is for. But we will see that the definitions of Poisson brackets and of quantomorphism groups directly generalize also to the presymplectic situation, simply by considering not just Hamiltonian fuctions but pairs of a Hamiltonian vector field and a compatible Hamiltonian.
Let $(X,\omega)$ be a presymplectic manifold. Write
for the linear subspace of the direct sum of Hamiltonian vector fields, def. 2, and smooth functions on those pairs $(v,H)$ for which $H$ is a Hamiltonian for $v$
Define a bilinear map
by
called the Poisson bracket, where $[v_1,v_2]$ is the standard Lie bracket on vector fields. Write
for the resulting Lie algebra. In the case that $\omega$ is symplectic, then $Ham(X,\omega) \simeq C^\infty(X)$ and hence in this case
Let $X = \mathbb{R}^{2n}$ and let $\omega = \sum_{i = 1}^n d p_i \wedge d q^i$ for $\{q^i\}_{i = 1}^n$ the canonical coordinates on one copy of $\mathbb{R}^n$ and $\{p_i\}_{i = 1}^n$ that on the other (“canonical momenta”). Hence let $(X,\omega)$ be a symplectic vector space of dimension $2n$, regarded as a symplectic manifold.
Then $Vect(X)$ is spanned over $C^\infty(X)$ by the canonical bases vector fields $\{\partial_{q^i}, \partial_{p^i}\}$. These basis vector fields are manifestly Hamiltonian vector fields via
Moreover, since $X$ is connected, these Hamiltonians are unique up to a choice of constant function. Write $\mathbf{i} \in C^\infty(X)$ for the unit constant function, then the nontrivial Poisson brackets between the basis vector fields are
This is called the Heisenberg algebra.
More generally, the Hamiltonian vector fields corresponding to quadratic Hamiltonians, i.e. degree-2 polynomials in the $\{q^i\}$ and $\{p_i\}$, generate the affine symplectic group of $(X,\omega)$. The freedom to add constant terms to Hamiltonians gives the extended affine symplectic group.
Example 1 serves to motivate a more conceptual origin of the definition of the Poisson bracket in def. 3.
Write
for the canonical choice of differential 1-form satisfying
If we regard $\mathbb{R}^{2n} \simeq T^\ast \mathbb{R}^n$ as the cotangent bundle of the Cartesian space $\mathbb{R}^n$, then this is what is known as the Liouville-Poincaré 1-form.
Since $\mathbb{R}^{2n}$ is contractible as a topological space, every circle bundle over it is necessarily trivial, and hence any choice of 1-form such as $\theta$ may canonically be thought of as being a connection on the trivial $U(1)$-principal bundle. As such this $\theta$ is a prequantization of $(\mathbb{R}^{2n}, \sum_{i=1}^n d p_i \wedge d q^i)$.
Being thus a circle bundle with connection, $\theta$ has more symmetry than its curvature $\omega$ has: for $\alpha \in C^\infty(\mathbb{R}^{2n}, U(1))$ any smooth function, then
is the gauge transformation of $\theta$, leading to a different but equivalent prequantization of $\omega$.
Hence while a vector field $v$ is said to preserve $\omega$ (is a symplectic vector field) if the Lie derivative of $\omega$ along $v$ vanishes
in the presence of a choice for $\theta$ the right condition to ask for is that there is $\alpha$ such that
For more on this see also at prequantized Lagrangian correspondence.
Notice then the following basic but important fact.
For $(X,\omega)$ a presymplectic manifold and $\theta \in \Omega^1(X)$ a 1-form such that $d \theta = \omega$ then for $(v,\alpha) \in Vect(X)\oplus C^\infty(X)$ the condition $\mathcal{L}_v \theta = d \alpha$ is equivalent to the condition that makes
a Hamiltonian for $v$ according to def. 2:
Moreover, the Poisson bracket, def. 3, between two such Hamiltonian pairs $(v_i, \alpha_i -\iota_v \theta)$ is equivalently given by the skew-symmetric Lie derivative of the corresponding vector fields on the $\alpha_i$:
Using Cartan's magic formula and by the prequantization condition $d \theta = \omega$ the we have
This gives the first statement. For the second we first use the formula for the de Rham differential and then again the definition of the $\alpha_i$
For $(X,\omega)$ a presymplectic manifold with $\theta \in \Omega^1(X)$ such that $d \theta = \omega$, consider the Lie algebra
with Lie bracket
Then by (1) the linear map
is an isomorphism of Lie algebras
from the Poisson bracket Lie algebra, def. 3.
This shows that for exact pre-symplectic forms the Poisson bracket Lie algebra is secretly the Lie algebra of infinitesimal symmetries of any of its prequantizations. In fact this holds true also when the pre-symplectic form is not exact:
For $(X,\omega)$ a presymplectic manifold, a Cech-Deligne cocycle
for a prequantization of $(X,\omega)$ is
an open cover $\{U_i \to X\}_i$;
1-forms $\{\theta_i \in \Omega^1(U_i)\}$;
smooth function $\{g_{i j} \in C^\infty(U_{i j}, U(1))\}$
such that
$d \theta_i = \omega|_{U_i}$ on all $U_i$;
$\theta_j = \theta_i + d log g_{ij}$ on all $U_{i j}$;
$g_{i j} g_{j k} = g_{i k}$ on all $U_{i j k}$.
The quantomorphism Lie algebra of $\overline{\theta}$ is
with bracket
For $(X,\omega)$ a presymplectic manifold and $(X,\{U_i\},\{g_{i j}, \theta_i\})$ a prequantization, def. 4, the linear map
constitutes an isomorphism of Lie algebras
between the Poisson bracket algebra of def. 3 and that of infinitesimal quantomorphisms, def. 4.
The condition $\mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i$ on the infinitesimal quantomorphisms, together with the Cech-Deligne cocycle condition $d log g_{i j} = \theta_j - \theta_i$ says that on $U_{i j}$
and hence that there is a globally defined function $H \in C^\infty(X)$ such that $\iota_v \theta_i - \alpha_i = H|_{U_i}$. This shows that the map is an isomrophism of vector spaces.
Now over each $U_i$ the the situation for the brackets is just that of corollary 1 implied by (1), hence the morphism is a Lie homomorphism.
The following fact is immediate, but important.
Given a presymplectic manifold $(X,\omega)$, then the Poisson bracket Lie algebra $\mathfrak{poiss}(X,\omega)$, def. 3, is a central Lie algebra extension of the algebra of Hamiltonian vector fields, def. 2, by the degree-0 de Rham cohomology group of $X$: there is a short exact sequence of Lie algebras
Hence when $X$ is connected, then $\mathfrak{poiss}(X,\omega)$ is an $\mathbb{R}$-extension of the Hamiltonian vector fields:
Moreover, given any choice of splitting of the underlying short exact sequence of vector spaces as $\mathfrak{pois}(X,\omega) \simeq_{vs} HamVect(X,\omega)\oplus H^0(X)$, which is equivalently a choice of Hamitlonian $H_v$ for each Hamiltonian vector field $v$, the Lie algebra cohomology 2-cocycle which classifies this extension is
The morphism $\mathfrak{poiss}(X,\omega) \to HamVect(X,\omega)$ is on elements given just by projection onto the direct summand of vector fields, taking a Hamiltonian pair $(v,H)$ to $v$. This is surjective by the very definition of $HamVect(X,\omega)$, in fact $HamVect(X,\omega)$ is the image of this map regarded as a morphism $\mathfrak{poiss}(X,\omega) \longrightarrow Vect(X)$. Moreover, the kernel of this projection is manifestly the space of Hamiltonian pairs of the form $(v = 0,H)$. By the defining constraint $\iota_v \omega = d H$ these are precisely the pairs for which $d H = 0$. This gives the short exact sequence as stated.
Generally, given a Lie algebra $\mathfrak{g}$ and an $\mathbb{R}$-valued 2-cocycle $\mu_2$ in Lie algebra cohomology, then the Lie algebra extension that it classifies is $\hat \mathfrak{g} =_{vs} \mathfrak{g}\oplus \mathbb{R}$ with bracket
Applied to the case at hand, given a choice of splitting $v\mapsto (v,H_v)$ this yields
Consider again example 1 where $(X,\omega) = (\mathbb{R}^{2n}, d p_i \wedge d q^i)$ is a symplectic vector space and where we restrict along the inclusion of the translation vector fields to get the Heisenberg algebra. Then the KS-extension of prop. 3 also pulls back:
The Lie algebra cohomology 2-cocycle which classifiesthe Kostant-Souriau extension, $\iota_{(-)}\iota_{(-)}\omega$ manifestly restricts to the Heisenberg cocycle $(q^i, p_j) = \delta^i_j$.
Recall the definition of L-∞ algebras.
For $V$ a graded vector space, for $v_i \in V_{\vert v_i\vert}$ homogenously graded elements, and for $\sigma$ a permutation of $n$ elements, write $\chi(\sigma, v_1, \cdots, v_n)\in \{-1,+1\}$ for the product of the signature of the permutation with a factor of $(-1)^{\vert v_i \vert \vert v_j \vert}$ for each interchange of neighbours $(\cdots v_i,v_j, \cdots )$ to $(\cdots v_j,v_i, \cdots )$ involved in the permutation.
An $L_\infty$-algebra is
a graded vector space $V$;
for each $n \in \mathbb{N}$ a multilinear map called the $n$-ary bracket
$l_n(\cdots) \coloneqq [-,-, \cdots, -] \colon V^{\wedge n} \to V$
of degree $n-2$
such that
each $l_n$ is graded antisymmetric, in that for every permutation $\sigma$ and homogeneously graded elements $v_i \in V_{\vert v_i \vert}$ then
the generalized Jacobi identity holds:
for all $n$, all and homogeneously graded elements $v_i \in V_i$ (here the inner sum runs over all $(i,j)$-unshuffles $\sigma$).
There are various different conventions on the gradings possible, which lead to similar formulas with different signs.
In lowest degrees the generalized Jacobi identity says that
for $n = 1$: the unary map $\partial \coloneqq l_1$ squares to 0:
1: for $n = 2$: the unary map $\partial$ is a graded derivation of the binary map
hence
When all higher brackets vanish, $l_{k \gt 2}= 0$ then for $n = 3$:
this is the graded Jacobi identity. So in this case the $L_\infty$-algebra is equivalently a dg-Lie algebra.
When $l_3$ is possibly non-vanishing, then on elements $x_i$ on which $\partial = l_1$ vanishes then the generalized Jacobi identity for $n = 3$ gives
This shows that the Jacobi identity holds up to an “exact” term, hence up to homotopy.
On connective $L_\infty$-algebras (those whose underlying chain complex is concentrated in non-negative degrees), passage to degree-0 chain homology constitutes a functor (“0-truncation”) to plain Lie algebras
In the discussion above we amplified that the definition of the Poisson bracket of a symplectic form has an immediate generalization to presymplectic forms, hence to any closed differential 2-form. This naturally suggests to ask for higher analogs of this bracket for the case of of closed differential (p+2)-forms $\omega \in \Omega^{p+2}(X)$ for $p \gt 0$.
Indeed, the natural algebraic form of definition 2 of Hamiltonian vector fields makes immediate sense for higher $p$, with the Hamiltonians $H$ now being $p$-forms, and the natural algebraic form of the binary Poisson bracket of def. 3 makes immediate sense as a bilinear pairing for any $p$:
However, one finds that for $p \gt 0$ then this bracket does not satisfy the Jacobi identity. On the other hand, the failure of the Jacobi identity turns out to be an exact form, and hence in the spirit of regarding the shift of a differential form by a de Rham differential as a homotopy or gauge transformation this suggests that the bracket might still give a Lie algebra upto higher coherent homotopy, called a strong homotopy Lie algebra or L-∞ algebra. This turns out to indeed be the case (Rogers 10).
For $p \in \mathbb{N}$, we say that a pre-(p+1)-plectic manifold is a smooth manifold $X$ equipped with a closed degree-$(p+2)$ differential form $\omega \in \Omega^{p+2}(X)$.
This is called an (p+1)-plectic manifold if the kernel of the contraction map
is trivial.
Given a pre-$(p+1)$-plectic manifold $(X,\omega)$, def. 6, write
for the subspace of the direct sum of vector fields $v$ on $X$ and differential p-forms $J$ on $X$ satisfying
We call these the pairs of Hamiltonian forms with their Hamiltonian vector fields.
Given a pre-$(p+1)$-plectic manifold $(X,\omega)$, def. 6, define an L-∞ algebra $\mathfrak{poiss}(X,\omega)$, to be called the Poisson bracket Lie (p+1)-algebra as follows.
The underlying chain complex is the truncated de Rham complex ending in Hamiltonian forms as in def. 7:
with the Hamiltonian pairs, def. 7, in degree 0 and with the 0-forms (smooth functions) in degree $p$.
The non-vanishing $L_\infty$-brackets are defined to be the following
$l_1(J) = d J$
$l_{k \geq 2}(v_1 + J_1, \cdots, v_k + J_k) \coloneqq - (-1)^{\left(k+1 \atop 2\right)} \iota_{v_k}\cdots \iota_{v_1}\omega$.
Definition 8 indeed gives an L-∞ algebra in that the higher Jacobi identity is satisfied.
For the special case of $(p+1)$-plectic $\omega$ this is due to (Rogers 10, lemma 3.7), for the general pre-$(p+1)$-plectic case this is (FRS 13b, prop. 3.1.2).
Repeatedly apply Cartan's magic formula $\mathcal{L}_v = \iota_v \circ d + d \circ \iota_v$ as well as the consequence $\mathcal{L}_{v_1} \circ \iota_{v_2} - \iota_{v_2} \circ \mathcal{L}_{v_1} = \iota_{[v_1,v_2]}$ to find that for all vector fields $v_i$ and differential forms $\beta$ (of any degree, not necessarily closed) one has
With this, the statement follows straightforwardly.
Recall the Cech complex.
(Čech complex)
Let $X$ be a smooth manifold and let $A_\bullet \in Ch_+(Sh(X))$ be a sheaf of chain complexes on $X$. Let $\{U_i \to X\}$ be a good open cover of $X$, i.e. an open cover such that each finite non-empty intersection $U_{i_0, \cdots, i_k}$ is diffeomorphic to an open ball/Cartesian space.
The Čech cochain complex $C^\bullet((X,\{U_i\}),A_\bullet)$ of $X$ with respect to the cover $\{U_i \to X\}$ and with coefficients in $A_\bullet$ is in degree $k \in \mathbb{N}$ given by the abelian group
which is the direct sum of the values of $A_\bullet$ on the given intersections as indicated; and whose differential
is defined componentwise (see at matrix calculus for conventions on maps between direct sums) by
where on the right the sum is over all components of $a$ obtained via the canonical restrictions obtained by discarding one of the original $(k+1)$ subscripts.
The Cech cohomology groups of $X$ with coefficients in $A_\bullet$ relative to the given cover are the chain homology groups of the Cech complex
The Cech cohomology groups as such are the colimit (“direct limit”) of these groups over refinements of covers
Recall the Deligne complex.
The Deligne complex for Deligne cohomology of degree $(p+2)$ is the chain complex of abelian sheaves given by
So for $\{U_i\to X\}$ an open cover, then we have the Cech-Deligne double complex
where vertically we have the de Rham differential and horizontally the Cech differential given by alternating sums of pullback of differential forms.
The corresponding total complex has in degree $n$ the direct sum of the entries in this double complex which are on the $n$th nw-se off-diagonal and has the total differential
with $deg$ denoting form degree.
A Cech-Deligne cocycle in degree $3$ (a “bundle gerbe with connection”) is data $(\{B_{i}\}, \{A_{i j}\}, \{g_{i j k}\})$ such that
The curvature of a Cech-Deligne cocycle
is the uniquely defined $(p+2)$-form $F_{\overline{\theta}} \in \Omega^{p+2}(X)$ such that
There is an evident generalization of the prequantization, def. 4, of closed 2-forms by circle bundles with connection, hence by degree-2 cocycles in Deligne cohomology, to the prequantization of closed $(p+2)$-forms by degree-$(p+2)$-cocycles in Deligne cohomology.
Given a pre-(p+1)-plectic manifold $(X,\omega)$, then a prequantization is a Cech-Deligne cocycle $\overline{\theta}$, the prequantum (p+1)-bundle, whose curvature, def. 10, equals $\omega$:
In terms of diagrams in the homotopy theory $\mathbf{H}$ of smooth homotopy types, def. 11 describes lifts of the form
This way there is an immediate generalization of def. 4 to forms and cocycles of higher degree:
Let $\overline{\theta}$ be any Cech-Deligne-cocycle relative to an open cover $\mathcal{U}$ of $X$, which gives a prequantum n-bundle for $\omega$. The L-∞ algebra $\mathfrak{quantmorph}(X,\overline{\theta})$ is the dg-Lie algebra (regarded as an $L_\infty$-algebra) whose underlying chain complex is the Cech total complex made to end in Hamiltonian Cech cocycles
$\mathfrak{quantmorph}(X,\overline{\theta})^0 \coloneqq \{v+ \overline{\alpha} \in Vect(X)\oplus Tot^{n-1}(\mathcal{U}, \Omega^\bullet) \;\vert\; \mathcal{L}_v \overline{\theta} = \mathbf{d}_{Tot}\overline{\alpha}\}$;
$\mathfrak{quantmorph}(X,\overline{\theta})^{i \gt 0} \coloneqq C^{n-1-i}(\mathcal{U},\Omega^\bullet)$
with differential given by $d_{tot} = d + (-1)^{deg} \delta$.
The non-vanishing dg-Lie brackets on this complex are given by the evident action of vector fields on all the components of the Cech cochains by Lie derivative:
$[v_1 + \overline{\alpha}_1, v_2 + \overline{\alpha}_2] \coloneqq [v_1, v_2] + \mathcal{L}_{v_1}\overline{\alpha}_2 - \mathcal{L}_{v_2}\overline{\alpha}_1$
$[v+ \overline{\alpha}, \overline{\eta}] = - [\overline{\eta}, v + \overline{\alpha}] = \mathcal{L}_v \overline{\eta}$.
One then finds a direct higher analog of corollary 1 (its proof however is requires a bit more work):
There is an equivalence in the homotopy theory of L-∞ algebras
between the $L_\infty$-algebras of def. 8 and def. 12 (in particular def. 12 does not depend on the choice of $\overline{A}$) whose underlying chain map satisfies
Proposition 7 says that all the higher Poisson $L_\infty$-algebras are $L_\infty$-algebras of symmetries of Deligen cocycles prequantizing the give pre-$(p+1)$-plectic form, higher “quantomorphisms”.
In fact the dg-algebra $\mathfrak{quantmorph}(X,\overline{\theta})$ makes yet another equivalent interpretation of $\mathfrak{poiss}(X,\omega)$ manifest: it is also a resolution of the Dickey bracket of conserved currents for WZW sigma-models. This we come to below.
The higher Poisson brackets come with a higher analog of the Kostant-Souriau extension, prop. 3.
Write
for the truncated de Rham complex regarded as an abelian L-∞ algebra.
Given a pre-(p+1)-plectic manifold $(X,\omega)$, the Poisson bracket Lie (p+1)-algebra $\mathfrak{poiss}(X,\omega)$, def. 8, is an L-∞ extension of the Hamiltonian vector fields by the truncated de Rham complex, def. 13, there is a homotopy fiber sequence of $L_\infty$-algebras of the form
To better see what this means, we may truncate this down to a statement about ordinary Lie algebras.
Given a pre-(p+1)-plectic manifold $(X,\omega)$, the 0-truncation, prop. 4, of the higher Kostant-Souriau extension of prop. 7 is a Lie algebra extension of the Hamiltonian vector fields by the de Rham cohomology group $H^p(X)$.
By the discussion at geometry of physics -- WZW terms, prequantization $\mathbf{L} \colon X \to \mathbf{B}^{p+1}U(1)_{conn}$ of $\omega \in \Omega^{p+2}(X)$ may be thought of as a parameterized WZW term for a sigma model field theory decribing the propagation of a p-brane on $X$.
Under this perspective, a Hamiltonian vector field $v$ on $X$ is a point symmetry of this sigma-model field theory and a Hamiltonian form $J$ for $v$ is is the conserved current corresponding to this via Noether's theorem. Inspection then shows that the bracket in def. 12 is the Dickey bracket of conserved currents, while the differential in def. 12 expresses the shift of currents by trivial currents (KS).
Hence under this perspective, def. 12 gives a dg-Lie algebra resolution of the Dickey bracket Lie algebra of conserved Noether currents for point symmetries of higher WZW sigma-models from gauge equivalence classes of conrved currents to the currents themselves.
For the case that $X$ is a super spacetimes and $\omega$ is a definite form on a super cocycle in the brane scan then this is known as the algebra of supergravity BPS charges of $X$. Therefore we also write
From this perspective the higher Kostant-Souriau extesion as in prop. 8 says the following:
For $\omega \in \Omega^{p+2}(X)$ the curvature of a WZW term on the smooth manifold $X$, then the Lie algebra of conserved currents covering target space point symmetries of the corresponding $p$-brane sigma-model is a Lie algebra extension of the taget space symmetry by the $p$th de Rham cohomology group
For $X$ a superspacetime and $\omega$ a definite form, definite on a super-cocycle in the brane scan, then corollary 2 is folklore in the string theory literature, due to (AGIT 89).
This is discussed further in geometry of physics -- BPS charges.
We here discuss the full finite version of quantomorphism n-groups.
Throughout, let $\mathbb{G} \in Grp(\mathbf{H})$ be a braided ∞-group equipped with a Hodge filtration. Write $\mathbf{B}\mathbb{G}_{conn}\in$ for the corresponding moduli stack of differential cohomology.
For $\mathbf{H} =$ Smooth∞Grpd we have $\mathbb{G} = \mathbf{B}^p (\mathbb{R}/\Gamma)$ for $\Gamma = \mathbb{Z}$ is the circle (p+1)-group. Equipped with its standard Hodge filtration this gives $\mathbf{B}\mathbb{G}_{conn} = \mathbf{B}^p U(1)_{conn}$ presented via the Dold-Kan correspondence by the Deligne complex in degree $(p+2)$.
For $X \in \mathbf{H}$, for write
for the differential concretification of the internal hom.
This is the proper moduli stack of $\mathbb{G}$-principal ∞-connections on $X$ in that a family $U \longrightarrow \mathbb{G}\mathbf{Conn}(X)$ is a vertical $\mathbb{G}$-principal $\infty$-connection on $U \times X\to U$.
For $\mathbf{H} =$Smooth∞Grpd or =FormalSmooth∞Grpd, for $\mathbb{G} = \mathbf{B}^p U(1)$ the circle (p+1)-group with its standard Hodge filtration as in example 8, then for $X$ any smooth manifold or formal smooth manifold, $(\mathbf{B}^p U(1))\mathbf{Conn}(X)$ is presented via the Dold-Kan correspondence by the sheaf $U \mapsto Ch_\bullet$ of vertical Deligne complexes on $U \times X$ over $U$.
For $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then the loop space object of the moduli stack of $\mathbb{G}$-principal $\infty$-connections on $X$ is the moduli stack of flat ∞-connections with gauge group $\Omega \mathbb{G}$
The canonical precomposition ∞-action of the automorphism ∞-group $\mathbf{Aut}(X)$ on $[X,\mathbf{B}\mathbb{G}_{conn}]$ passes along $conc$ to an ∞-action on $\mathbb{G}\mathbf{Conn}(X)$.
Given a $\mathbb{G}$-principal ∞-connection $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ there are the following concepts in higher geometric prequantum theory.
The quantomorphism ∞-group is the stabilizer ∞-group of $\nabla \in \mathbb{G}\mathbf{Conn}(X)$, def. 14, under the $\mathbf{Aut}(X)$-action of 9;
The Hamiltonian symplectomorphism ∞-group
is the 1-image of the canonical morphism $\mathbf{QuantMorph}(X,\nabla) \longrightarrow \mathbf{Aut}(X)$.
A Hamiltonian action of an ∞-group $G$ on $(X,\nabla)$ is an ∞-group homomorphism
An ∞-moment map is an $\infty$-group homomorphism
The Heisenberg ∞-group for a given Hamiltonian $G$-action $\rho$ is the homotopy pullback
For $\mathbf{H} =$ Smooth∞Grpd, for $X \in SmoothMfd \hookrightarrow \mathbf{H}$ a smooth manifold and for $\nabla$ a prequantum line bundle on $X$, then $\mathbf{QuantMorph}(X,\nabla)$ is Souriau’s quantomorphism group covering the Hamiltonian diffeomorphism group. In the case that $(X, F_\nabla)$ is a symplectic vector space $X = V$ regarded as a linear symplectic manifold with Hamiltonian action on itself by translation, then $\mathbf{Heis}_{V}(X,\nabla)$ is the traditional Heisenberg group.
Since $\mathbf{HamSymp}(X,\nabla)\hookrightarrow \mathbf{Aut}(X)$ is by construction a 1-monomorphism, given any $G$-action $\rho \colon G \longrightarrow \mathbf{Aut}(X)$ on $X$, not necessarily Hamiltonian, then the homotopy pullback $\rho^\ast \mathbf{QuantMorph}(X,\nabla)$ is the Heisenberg ∞-group of the maximal sub-$\infty$-group of $G$ which does act via Hamiltonian symplectomorphisms. Therefore we will also write $\mathbf{Heis}_G(X,\nabla)$ in this case.
The following is the refinement of the Kostant-Souriau extension to higher differential geometry
Given a $\mathbb{G}$-principal ∞-connection $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$, there is a homotopy fiber sequence of the form
if $\mathbb{G}$ is 0-truncated then
if $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then
exhibiting the quantomorphism ∞-group as an ∞-group extension of the Hamiltonian symplectomorphism ∞-group by the moduli stack of $\Omega \mathbb{G}-$flat ∞-connections, classified by a cocycle $\mathbf{KS}$.
(FRS 13a)
In $\mathbf{H} =$ Smooth∞Grpd, let $\mathbb{G} = \mathbf{B}^p U(1)$ be the circle (p+1)-group and let $X \in SmoothMfd \hookrightarrow Smooth \infty Grpd$ be p-connected, then $(\Omega\mathbf{B}^p U(1))\mathbf{FlatConn}(X)\simeq \mathbf{B}^{p}U(1)$. Hence here prop. 10 gives
Given a $\mathbb{G}$-principal ∞-connection $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$, and for $\rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla)$ a $G$-Hamiltonian action, then there is a homotopy fiber sequence
if $\mathbb{G}$ is 0-truncated then
if $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then
exhibiting the Heisenberg ∞-group as an ∞-group extension of the $G$ by the moduli stack of $\Omega \mathbb{G}-$flat ∞-connections, classified by a cocycle $\mathbf{KS}(\rho)$.
The class of the cocycle $\mathbf{KS}(\rho)$ is the obstruction to prequantizing $\rho$ to a moment map (the classical anomaly of $\rho$); and the the Heisenberg ∞-group extension of $G$ is the universal cancellation of this anomaly.
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