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geometry of physics -- prequantum geometry

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Prequantum geometry

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)U(1)-bundle with itself makes it is a central extension, specifically a U(1)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 =π 1(U(1))\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 “quantomorphismU(1)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)(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)(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=1p = 1, generally a cocycle in Deligne cohomology of degree p+2p+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 B𝔾 conn\mathbf{B}\mathbb{G}_{conn} for the higher moduli stack of 𝔾\mathbb{G}-principal infinity-connections, for instance B p+1U(1) conn\mathbf{B}^{p+1}U(1)_{conn} the moduli stack for cocycles in Deligne cohomology in degree (p+2)(p+2), then a prequantum 𝔾\mathbb{G}-bundle over some smooth homotopy type XX is equivalently just a morphism :XB𝔾 conn\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}, and a symmetry of such a higher prequantum geometry (X,)(X,\nabla) is a diagram of the form

X X B𝔾 conn. \array{ X && \stackrel{\simeq}{\longrightarrow} && X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}\mathbb{G}_{conn} } \,.

This means that for H\mathbf{H} a cohesive (∞,1)-topos of geometric spaces of sorts, then the corresponding 𝔾\mathbb{G}-prequantum geometry is that encoded by the slice (∞,1)-topos H /B𝔾 conn\mathbf{H}_{/\mathbf{B}\mathbb{G}_{conn}}. Hence prequantum geometry is the geometry in slices over higher moduli stacks for differential cohomology.

The geometric idea of quantization

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,ω)(X, \omega). A symplectic manifold is in particular a Poisson manifold, which means that the algebra of functions on phase space XX, 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 HC (X)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 HH 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 𝔤\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) GG. One also says that GG is a Lie integration of 𝔤\mathfrak{g} and that 𝔤\mathfrak{g} is the Lie differentiation of GG.

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

U(1)=/. U(1) = \mathbb{R}/\mathbb{Z} \,.

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 𝔭𝔬𝔦𝔰𝔰(X,ω)\mathfrak{poiss}(X,\omega) of the classical observables on phase space is (for XX a connected manifold) a Lie algebra extension of the Lie algebra 𝔥𝔞𝔪(X)\mathfrak{ham}(X) of Hamiltonian vector fields on XX by the line Lie algebra:

𝔭𝔬𝔦𝔰𝔰(X,ω)𝔥𝔞𝔪(X). \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X) \,.

This means that under Lie integration the Poisson bracket turns into an central extension of the group of Hamiltonian symplectomorphisms of (X,ω)(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)U(1). For this non-trivial Lie integration to exist, (X,ω)(X,\omega) needs to satisfy a quantization condition which says that it admits a prequantum line bundle. If so, then this U(1)U(1)-central extension of the group Ham(X,ω)Ham(X,\omega) of Hamiltonian symplectomorphisms exists and is called… the quantomorphism group QuantMorph(X,ω)QuantMorph(X,\omega):

U(1)QuantMorph(X,ω)Ham(X,ω). U(1) \longrightarrow QuantMorph(X,\omega) \longrightarrow Ham(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,ω)(X,\omega) itself has a compatible group structure, notably if (X,ω)(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,ω)(X,\omega) on itself. This is the corresponding Heisenberg group Heis(X,ω)Heis(X,\omega), which in turn is a U(1)U(1)-central extension of the group XX itself:

U(1)Heis(X,ω)X. U(1) \longrightarrow Heis(X,\omega) \longrightarrow X \,.

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)U(1) instead of to the uninteresting translation group \mathbb{R}, then the name of its canonical basis element 11 \in \mathbb{R} is canonically “ii”, the imaginary unit. Therefore one often writes the above central extension instead as follows:

i𝔭𝔬𝔦𝔰𝔰(X,ω)𝔥𝔞𝔪(X,ω) i \mathbb{R} \longrightarrow \mathfrak{poiss}(X,\omega) \longrightarrow \mathfrak{ham}(X,\omega)

in order to amplify this. But now consider the simple special case where (X,ω)=( 2,dpdq)(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 pp and qq 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 ii \mathbb{R}, hence purely Lie theoretically it is to be called “ii”.

With this notation then the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads

[q,p]=i. [q,p] = i \,.

Since the choice of basis element of ii \mathbb{R} is arbitrary, we may rescale here the ii by any non-vanishing real number without changing this statement. If we write “\hbar” for this element, then the Poisson bracket instead reads

[q,p]=i. [q,p] = i \hbar \,.

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 ω\omega, it is natural to ask if it is the curvature 2-form of a U(1)U(1)-principal connection \nabla on complex line bundle LL over XX (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,)(L, \nabla) is called a prequantum line bundle of the phase space (X,ω)(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 LL. 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,ω)(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.

Infinitesimal symmetries

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.

Poisson brackets and Heisenberg 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.

Definition

Let XX be a smooth manifold. A closed differential 2-form ωΩ cl 2(X)\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

ι ()ω:Vect(X)Ω 1(X) \iota_{(-)}\omega \colon Vect(X) \longrightarrow \Omega^1(X)

is trivial: ker(ι ()ω)=0ker(\iota_{(-)}\omega) = 0.

If ω\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.)

Definition

Given a presymplectic manifold (X,ω)(X, \omega), then a Hamiltonian for a vector field vVect(X)v \in Vect(X) is a smooth function HC (X)H \in C^\infty(X) such that

ι vω+dH=0. \iota_{v} \omega + d H = 0 \,.

If vVect(X)v \in Vect(X) is such that there exists at least one Hamiltonian for it then it is called a Hamiltonian vector field. Write

HamVect(X,ω)Vect(X) HamVect(X,\omega) \hookrightarrow Vect(X)

for the \mathbb{R}-linear subspace of Hamiltonian vector fields among all vector fields

Remark

When ω\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.

Definition

Let (X,ω)(X,\omega) be a presymplectic manifold. Write

Ham(X,ω)HamVect(X,ω)C (X) Ham(X,\omega) \hookrightarrow HamVect(X,\omega) \oplus C^\infty(X)

for the linear subspace of the direct sum of Hamiltonian vector fields, def. 2, and smooth functions on those pairs (v,H)(v,H) for which HH is a Hamiltonian for vv

Ham(X,ω){(v,H)|ι vω+dH=0}. Ham(X,\omega) \coloneqq \left\{ (v,H) | \iota_v \omega + d H = 0 \right\} \,.

Define a bilinear map

[,]:Ham(X,ω)Ham(X,ω)Ham(X,ω) [-,-] \;\colon\; Ham(X,\omega) \otimes Ham(X,\omega) \longrightarrow Ham(X,\omega)

by

[(v 1,H 1),(v 2,H 2)]([v 1,v 2],ι v 2ι v 1ω), [(v_1,H_1), (v_2,H_2)] \coloneqq ([v_1,v_2], \iota_{v_2}\iota_{v_1} \omega) \,,

called the Poisson bracket, where [v 1,v 2][v_1,v_2] is the standard Lie bracket on vector fields. Write

𝔭𝔬𝔦𝔰𝔰(X,ω)(Ham(X,ω),[,]) \mathfrak{poiss}(X,\omega) \coloneqq (Ham(X,\omega),[-,-])

for the resulting Lie algebra. In the case that ω\omega is symplectic, then Ham(X,ω)C (X)Ham(X,\omega) \simeq C^\infty(X) and hence in this case

𝔭𝔬𝔦𝔰𝔰(X,ω)(C (X),[,]). \mathfrak{poiss}(X,\omega) \simeq (C^\infty(X),[-,-]) \,.
Example

Let X= 2nX = \mathbb{R}^{2n} and let ω= i=1 ndp idq i\omega = \sum_{i = 1}^n d p_i \wedge d q^i for {q i} i=1 n\{q^i\}_{i = 1}^n the canonical coordinates on one copy of n\mathbb{R}^n and {p i} i=1 n\{p_i\}_{i = 1}^n that on the other (“canonical momenta”). Hence let (X,ω)(X,\omega) be a symplectic vector space of dimension 2n2n, regarded as a symplectic manifold.

Then Vect(X)Vect(X) is spanned over C (X)C^\infty(X) by the canonical bases vector fields { q i, p i}\{\partial_{q^i}, \partial_{p^i}\}. These basis vector fields are manifestly Hamiltonian vector fields via

ι q iω=dp i \iota_{\partial_{q^i}} \omega = - d p_i
ι p iω=+dq i. \iota_{\partial_{p_i}} \omega = + d q^i \,.

Moreover, since XX is connected, these Hamiltonians are unique up to a choice of constant function. Write iC (X)\mathbf{i} \in C^\infty(X) for the unit constant function, then the nontrivial Poisson brackets between the basis vector fields are

[q i,p j][( p i,q i),( q j,p j)]=δ j i(0,i)=δ j ii. [q^i, p_j] \coloneqq [(-\partial_{p_i}, q^i), (\partial_{q^j}, p_j)] = - \delta_j^i (0, \mathbf{i}) = - \delta_j^i \mathbf{i} \,.

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}\{q^i\} and {p i}\{p_i\}, generate the affine symplectic group of (X,ω)(X,\omega). The freedom to add constant terms to Hamiltonians gives the extended affine symplectic group.

Infinitesimal quantomorphisms

Example 1 serves to motivate a more conceptual origin of the definition of the Poisson bracket in def. 3.

Example

Write

θ i=1 np idq iΩ 1( 2n) \theta \coloneqq \sum_{i = 1}^n p_i d q_i \in \Omega^1(\mathbb{R}^{2n})

for the canonical choice of differential 1-form satisfying

dθ=ω. d \theta = \omega \,.

If we regard 2nT * n\mathbb{R}^{2n} \simeq T^\ast \mathbb{R}^n as the cotangent bundle of the Cartesian space n\mathbb{R}^n, then this is what is known as the Liouville-Poincaré 1-form.

Since 2n\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 θ\theta may canonically be thought of as being a connection on the trivial U(1)U(1)-principal bundle. As such this θ\theta is a prequantization of ( 2n, i=1 ndp idq i)(\mathbb{R}^{2n}, \sum_{i=1}^n d p_i \wedge d q^i).

Being thus a circle bundle with connection, θ\theta has more symmetry than its curvature ω\omega has: for αC ( 2n,U(1))\alpha \in C^\infty(\mathbb{R}^{2n}, U(1)) any smooth function, then

θθ+dα \theta \mapsto \theta + d\alpha

is the gauge transformation of θ\theta, leading to a different but equivalent prequantization of ω\omega.

Hence while a vector field vv is said to preserve ω\omega (is a symplectic vector field) if the Lie derivative of ω\omega along vv vanishes

vω=0 \mathcal{L}_v \omega = 0

in the presence of a choice for θ\theta the right condition to ask for is that there is α\alpha such that

vθ=dα. \mathcal{L}_v \theta = d \alpha \,.

For more on this see also at prequantized Lagrangian correspondence.

Notice then the following basic but important fact.

Proposition

For (X,ω)(X,\omega) a presymplectic manifold and θΩ 1(X)\theta \in \Omega^1(X) a 1-form such that dθ=ωd \theta = \omega then for (v,α)Vect(X)C (X)(v,\alpha) \in Vect(X)\oplus C^\infty(X) the condition vθ=dα\mathcal{L}_v \theta = d \alpha is equivalent to the condition that makes

Hι vθα H \coloneqq \iota_v \theta - \alpha

a Hamiltonian for vv according to def. 2:

vθ=dαι vω+d(ι vθαH)=0. \mathcal{L}_v \theta = d \alpha \;\;\;\Leftrightarrow\;\;\; \iota_v \omega + d (\underset{H}{\underbrace{\iota_v \theta - \alpha}}) = 0 \,.

Moreover, the Poisson bracket, def. 3, between two such Hamiltonian pairs (v i,α iι vθ)(v_i, \alpha_i -\iota_v \theta) is equivalently given by the skew-symmetric Lie derivative of the corresponding vector fields on the α i\alpha_i:

(1)ι [v 1,v 2]θι v 2ι v 1ω= v 1α 2 v 2α 1 \iota_{[v_1,v_2]} \theta - \iota_{v_2}\iota_{v_1}\omega = \mathcal{L}_{v_1} \alpha_2 - \mathcal{L}_{v_2} \alpha_1
Proof

Using Cartan's magic formula and by the prequantization condition dθ=ωd \theta = \omega the we have

vθ =ι vdθ+dι vθ =ι vω+dι vθ. \begin{aligned} \mathcal{L}_v \theta &= \iota_v d\theta + d \iota_v \theta \\ & = \iota_v\omega + d \iota_v \theta \end{aligned} \,.

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 α i\alpha_i

ι v 2ι v 1ω =ι v 2ι v 1dθ =ι v 1dι v 2θι v 2dι v 1θι [v 1,v 2]θ =ι v 1dα 2ι v 1ι v 2ωι v 2dα 1+ι v 2ι v 1ωι [v 1,v 2]θ =2ι v 2ι v 1ω+ v 1α 2 v 2α 1ι [v 1,v 2]θ. \begin{aligned} \iota_{v_2}\iota_{v_1} \omega & = \iota_{v_2}\iota_{v_1} d\theta \\ & = \iota_{v_1} d \iota_{v_2} \theta - \iota_{v_2} d \iota_{v_1} \theta - \iota_{[v_1,v_2]} \theta \\ & = \iota_{v_1} d \alpha_2 - \iota_{v_1} \iota_{v_2}\omega - \iota_{v_2} d \alpha_1 + \iota_{v_2} \iota_{v_1}\omega - \iota_{[v_1,v_2]} \theta \\ & = 2 \iota_{v_2} \iota_{v_1}\omega + \mathcal{L}_{v_1} \alpha_2 -\mathcal{L}_{v_2} \alpha_1 - \iota_{[v_1,v_2]} \theta \end{aligned} \,.
Corollary

For (X,ω)(X,\omega) a presymplectic manifold with θΩ 1(X)\theta \in \Omega^1(X) such that dθ=ωd \theta = \omega, consider the Lie algebra

𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ)={(v,α)| vθ=dα}Vect(X)C (X) \mathfrak{quantmorph}(X,\theta) = \left\{ (v,\alpha) | \mathcal{L}_v \theta = d \alpha \right\} \subset Vect(X) \oplus C^\infty(X)

with Lie bracket

[(v 1,α 1),(v 2,α 2)]=([v 1,v 2], v 1α 2 v 2α 1). [(v_1,\alpha_1), (v_2,\alpha_2)] = ([v_1,v_2], \mathcal{L}_{v_1}\alpha_2 - \mathcal{L}_{v_2}\alpha_1) \,.

Then by (1) the linear map

(v,H)(v,ι vθH) (v,H) \mapsto (v, \iota_v \theta - H)

is an isomorphism of Lie algebras

𝔭𝔬𝔦𝔰𝔰(X,ω)𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ) \mathfrak{poiss}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{quantmorph}(X,\theta)

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:

Definition

For (X,ω)(X,\omega) a presymplectic manifold, a Cech-Deligne cocycle

(X,θ¯)(X,{U i},{g ij,θ i}) (X,\overline{\theta}) \coloneqq (X,\{U_i\},\{g_{i j}, \theta_i\})

for a prequantization of (X,ω)(X,\omega) is

  1. an open cover {U iX} i\{U_i \to X\}_i;

  2. 1-forms {θ iΩ 1(U i)}\{\theta_i \in \Omega^1(U_i)\};

  3. smooth function {g ijC (U ij,U(1))}\{g_{i j} \in C^\infty(U_{i j}, U(1))\}

such that

  1. dθ i=ω| U id \theta_i = \omega|_{U_i} on all U iU_i;

  2. θ j=θ i+dlogg ij\theta_j = \theta_i + d log g_{ij} on all U ijU_{i j};

  3. g ijg jk=g ikg_{i j} g_{j k} = g_{i k} on all U ijkU_{i j k}.

The quantomorphism Lie algebra of θ¯\overline{\theta} is

𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ¯)={(v,{α i})| vlogg ij=α jα i, vθ i=dα i}Vect(X)(iC (U i)) \mathfrak{quantmorph}(X,\overline{\theta}) = \left\{ (v, \{\alpha_i\}) | \mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i \,, \mathcal{L}_v \theta_i = d \alpha_i \right\} \subset Vect(X) \oplus \left(\underset{i}{\bigoplus} C^\infty(U_i)\right)

with bracket

[(v 1,{(α 1) i}),(v 2,{(α 2) i})]([v 1,v 2],{ v 1(α 2) i v 2(α 1) i}). [(v_1, \{(\alpha_1)_i\}), (v_2, \{(\alpha_2)_i\})] \coloneqq ([v_1,v_2], \{\mathcal{L}_{v_1}(\alpha_2)_i - \mathcal{L}_{v_2} (\alpha_1)_i\}) \,.
Proposition

For (X,ω)(X,\omega) a presymplectic manifold and (X,{U i},{g ij,θ i})(X,\{U_i\},\{g_{i j}, \theta_i\}) a prequantization, def. 4, the linear map

(v,H)(v,{ι vθ iH| U i}) (v,H) \mapsto (v, \{\iota_v \theta_i - H|_{U_i}\})

constitutes an isomorphism of Lie algebras

𝔭𝔬𝔦𝔰𝔰(X,ω)𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ¯) \mathfrak{poiss}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{quantmorph}(X,\overline{\theta})

between the Poisson bracket algebra of def. 3 and that of infinitesimal quantomorphisms, def. 4.

Proof

The condition vlogg ij=α jα i\mathcal{L}_v log g_{i j} = \alpha_j - \alpha_i on the infinitesimal quantomorphisms, together with the Cech-Deligne cocycle condition dlogg ij=θ jθ id log g_{i j} = \theta_j - \theta_i says that on U ijU_{i j}

ι vθ jα j=ι vθ iα i \iota_v \theta_j - \alpha_j = \iota_v \theta_i - \alpha_i

and hence that there is a globally defined function HC (X)H \in C^\infty(X) such that ι vθ iα i=H| U i\iota_v \theta_i - \alpha_i = H|_{U_i}. This shows that the map is an isomrophism of vector spaces.

Now over each U iU_i the the situation for the brackets is just that of corollary 1 implied by (1), hence the morphism is a Lie homomorphism.

The Kostant-Souriau extension

The following fact is immediate, but important.

Proposition

Given a presymplectic manifold (X,ω)(X,\omega), then the Poisson bracket Lie algebra 𝔭𝔬𝔦𝔰𝔰(X,ω)\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 XX: there is a short exact sequence of Lie algebras

0H 0(X)𝔭𝔬𝔦𝔰𝔰(X,ω)HamVect(X,ω)0. 0 \to H^0(X) \stackrel{}{\longrightarrow} \mathfrak{poiss}(X,\omega) \stackrel{}{\longrightarrow} HamVect(X,\omega) \to 0 \,.

Hence when XX is connected, then 𝔭𝔬𝔦𝔰𝔰(X,ω)\mathfrak{poiss}(X,\omega) is an \mathbb{R}-extension of the Hamiltonian vector fields:

0𝔭𝔬𝔦𝔰𝔰(X,ω)HamVect(X,ω)0. 0 \to \mathbb{R} \stackrel{}{\longrightarrow} \mathfrak{poiss}(X,\omega) \stackrel{}{\longrightarrow} HamVect(X,\omega) \to 0 \,.

Moreover, given any choice of splitting of the underlying short exact sequence of vector spaces as 𝔭𝔬𝔦𝔰(X,ω) vsHamVect(X,ω)H 0(X)\mathfrak{pois}(X,\omega) \simeq_{vs} HamVect(X,\omega)\oplus H^0(X), which is equivalently a choice of Hamitlonian H vH_v for each Hamiltonian vector field vv, the Lie algebra cohomology 2-cocycle which classifies this extension is

(v 1,v 2)=ι v 2ι v 1ωH [v 1,v 2]. (v_1, v_2) = \iota_{v_2}\iota_{v_1}\omega - H_{[v_1,v_2]} \,.
Proof

The morphism 𝔭𝔬𝔦𝔰𝔰(X,ω)HamVect(X,ω)\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)(v,H) to vv. This is surjective by the very definition of HamVect(X,ω)HamVect(X,\omega), in fact HamVect(X,ω)HamVect(X,\omega) is the image of this map regarded as a morphism 𝔭𝔬𝔦𝔰𝔰(X,ω)Vect(X)\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)(v = 0,H). By the defining constraint ι vω=dH\iota_v \omega = d H these are precisely the pairs for which dH=0d H = 0. This gives the short exact sequence as stated.

Generally, given a Lie algebra 𝔤\mathfrak{g} and an \mathbb{R}-valued 2-cocycle μ 2\mu_2 in Lie algebra cohomology, then the Lie algebra extension that it classifies is 𝔤^= vs𝔤\hat \mathfrak{g} =_{vs} \mathfrak{g}\oplus \mathbb{R} with bracket

[(x 1,a 1),(x 2,x 2)]=([x 1,x 2],μ 2(a 1,a 2)). [(x_1,a_1), (x_2,x_2)] = ([x_1,x_2], \mu_2(a_1,a_2)) \,.

Applied to the case at hand, given a choice of splitting v(v,H v)v\mapsto (v,H_v) this yields

[(v 1,H v 1+a 1),(v 2,H v 2+a 2)]=([v 1,v 2],H [v 1,v 2]+ι v 2ι v 1ωH [v 1,v 2])=([v 1,v 2],ι v 2ι v 1ω). [(v_1,H_{v_1} + a_1), (v_2, H_{v_2} + a_2) ] = ([v_1,v_2], H_{[v_1,v_2]} + \iota_{v_2}\iota_{v_1}\omega - H_{[v_1,v_2]}) = ([v_1,v_2], \iota_{v_2}\iota_{v_1}\omega ) \,.
Example

Consider again example 1 where (X,ω)=( 2n,dp idq i)(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:

H dR 0(X) H dR 0(X) 𝔥𝔢𝔦𝔰(X,ω) 𝔭𝔬𝔦𝔰𝔰(X,ω) 2n HamVect(X,ω). \array{ H_{dR}^0(X) &\longrightarrow& H_{dR}^0(X) \\ \downarrow && \downarrow \\ \mathfrak{heis}(X,\omega) &\longrightarrow& \mathfrak{poiss}(X,\omega) \\ \downarrow && \downarrow \\ \mathbb{R}^{2 n} &\longrightarrow& HamVect(X,\omega) } \,.

The Lie algebra cohomology 2-cocycle which classifiesthe Kostant-Souriau extension, ι ()ι ()ω\iota_{(-)}\iota_{(-)}\omega manifestly restricts to the Heisenberg cocycle (q i,p j)=δ j i(q^i, p_j) = \delta^i_j.

Interlude: L L_\infty-algebras

Recall the definition of L-∞ algebras.

For VV a graded vector space, for v iV |v i|v_i \in V_{\vert v_i\vert} homogenously graded elements, and for σ\sigma a permutation of nn elements, write χ(σ,v 1,,v n){1,+1}\chi(\sigma, v_1, \cdots, v_n)\in \{-1,+1\} for the product of the signature of the permutation with a factor of (1) |v i||v j|(-1)^{\vert v_i \vert \vert v_j \vert} for each interchange of neighbours (v i,v j,)(\cdots v_i,v_j, \cdots ) to (v j,v i,)(\cdots v_j,v_i, \cdots ) involved in the permutation.

Definition

An L L_\infty-algebra is

  1. a graded vector space VV;

  2. for each nn \in \mathbb{N} a multilinear map called the nn-ary bracket

    l n()[,,,]:V nVl_n(\cdots) \coloneqq [-,-, \cdots, -] \colon V^{\wedge n} \to V

    of degree n2n-2

such that

  1. each l nl_n is graded antisymmetric, in that for every permutation σ\sigma and homogeneously graded elements v iV |v i|v_i \in V_{\vert v_i \vert} then

    l n(v σ(1),v σ(2),,v σ(n))=χ(σ,v 1,,v n)l n(v 1,v 2,v n) l_n(v_{\sigma(1)}, v_{\sigma(2)},\cdots ,v_{\sigma(n)}) = \chi(\sigma,v_1,\cdots, v_n) \cdot l_n(v_1, v_2, \cdots v_n)
  2. the generalized Jacobi identity holds:

    (2) i+j=n+1 σUnShuff(i,ni)χ(σ,v 1,,v m)(1) i(j1)l j(l i(v σ(1),,v σ(i)),v σ(i+1),,v σ(n))=0, \sum_{i+j = n+1} \sum_{\sigma \in UnShuff(i,n-i)} \chi(\sigma,v_1, \cdots, v_m) (-1)^{i(j-1)} l_{j} \left( l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right), v_{\sigma(i+1)} , \cdots , v_{\sigma(n)} \right) = 0 \,,

    for all nn, all and homogeneously graded elements v iV iv_i \in V_i (here the inner sum runs over all (i,j)(i,j)-unshuffles σ\sigma).

There are various different conventions on the gradings possible, which lead to similar formulas with different signs.

Example

In lowest degrees the generalized Jacobi identity says that

  1. for n=1n = 1: the unary map l 1\partial \coloneqq l_1 squares to 0:

    ((v 1))=0 \partial (\partial(v_1)) = 0

1: for n=2n = 2: the unary map \partial is a graded derivation of the binary map

[v 1,v 2](1) |v 1||v 2|[v 2,v 1]+[v 1,v 2]=0 - [\partial v_1, v_2] - (-1)^{\vert v_1 \vert \vert v_2 \vert} [\partial v_2, v_1] + \partial [v_1, v_2] = 0

hence

[v 1,v 2]=[v 1,v 2]+(1) |v 1|[v 1,v 2]. \partial [v_1, v_2] = [\partial v_1, v_2] + (-1)^{\vert v_1 \vert}[v_1, \partial v_2] \,.
Example

When all higher brackets vanish, l k>2=0l_{k \gt 2}= 0 then for n=3n = 3:

[[v 1,v 2],v 3]+(1) |v 1|(|v 2|+|v 3|)[[v 2,v 3],v 1]+(1) |v 2|(|v 1|+|v 3|)[[v 1,v 3],v 2]=0 [[v_1,v_2],v_3] + (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)} [[v_2,v_3],v_1] + (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)} [[v_1,v_3],v_2] = 0

this is the graded Jacobi identity. So in this case the L L_\infty-algebra is equivalently a dg-Lie algebra.

Example

When l 3l_3 is possibly non-vanishing, then on elements x ix_i on which =l 1\partial = l_1 vanishes then the generalized Jacobi identity for n=3n = 3 gives

[[v 1,v 2],v 3]+(1) |v 1|(|v 2|+|v 3|)[[v 2,v 3],v 1]+(1) |v 2|(|v 1|+|v 3|)[[v 1,v 3],v 2]=[v 1,v 2,v 3]. [[v_1,v_2],v_3] + (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)} [[v_2,v_3],v_1] + (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)} [[v_1,v_3],v_2] = - \partial [v_1, v_2, v_3] \,.

This shows that the Jacobi identity holds up to an “exact” term, hence up to homotopy.

Proposition

On connective L 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

τ 0H 0:L Alg 0LieAlg. \tau_0 \coloneqq H^0 \colon L_\infty Alg_{\geq 0} \longrightarrow LieAlg \,.

Higher Poisson brackets and higher Heisenberg algebra

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 ωΩ p+2(X)\omega \in \Omega^{p+2}(X) for p>0p \gt 0.

Indeed, the natural algebraic form of definition 2 of Hamiltonian vector fields makes immediate sense for higher pp, with the Hamiltonians HH now being pp-forms, and the natural algebraic form of the binary Poisson bracket of def. 3 makes immediate sense as a bilinear pairing for any pp:

[(v 1,H 1),(v 2,H 2)]([v 1,v 2],ι v 2ι v 1ω). [(v_1, H_1), (v_2, H_2)] \coloneqq ([v_1,v_2], \iota_{v_2} \iota_{v_1} \omega) \,.

However, one finds that for p>0p \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).

Definition

For pp \in \mathbb{N}, we say that a pre-(p+1)-plectic manifold is a smooth manifold XX equipped with a closed degree-(p+2)(p+2) differential form ωΩ p+2(X)\omega \in \Omega^{p+2}(X).

This is called an (p+1)-plectic manifold if the kernel of the contraction map

ι ():Vect(X)Ω p+1(X) \iota_{(-)} \colon Vect(X) \longrightarrow \Omega^{p+1}(X)

is trivial.

Definition

Given a pre-(p+1)(p+1)-plectic manifold (X,ω)(X,\omega), def. 6, write

Ham p(X)Vect(X)Ω p(X) Ham^{p}(X) \subset Vect(X) \oplus \Omega^{p}(X)

for the subspace of the direct sum of vector fields vv on XX and differential p-forms JJ on XX satisfying

ι vω+dJ=0. \iota_v \omega + d J = 0 \,.

We call these the pairs of Hamiltonian forms with their Hamiltonian vector fields.

Definition

Given a pre-(p+1)(p+1)-plectic manifold (X,ω)(X,\omega), def. 6, define an L-∞ algebra 𝔭𝔬𝔦𝔰𝔰(X,ω)\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:

Ω 0(X)dΩ 1(X)ddΩ p1(X)(0,d)Ham p(X) \Omega^0(X) \stackrel{d}{\to} \Omega^1(X) \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^{p-1}(X) \stackrel{(0,d)}{\longrightarrow} Ham^{p}(X)

with the Hamiltonian pairs, def. 7, in degree 0 and with the 0-forms (smooth functions) in degree pp.

The non-vanishing L L_\infty-brackets are defined to be the following

  • l 1(J)=dJl_1(J) = d J

  • l k2(v 1+J 1,,v k+J k)(1) (k+12)ι v kι v 1ωl_{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.

Proposition

Definition 8 indeed gives an L-∞ algebra in that the higher Jacobi identity is satisfied.

(3) i+j=n+1 σUnShuff(i,j)(1) sgn(σ)l i(l j(x σ(1),,x σ(j)),x σ(j+1),,x σ(n))=0, \sum_{i+j = n+1} \sum_{\sigma \in UnShuff(i,j)} (-1)^{sgn(\sigma)} l_i \left( l_j \left( x_{\sigma(1)}, \cdots, x_{\sigma(j)} \right), x_{\sigma(j+1)} , \cdots , x_{\sigma(n)} \right) = 0 \,,

For the special case of (p+1)(p+1)-plectic ω\omega this is due to (Rogers 10, lemma 3.7), for the general pre-(p+1)(p+1)-plectic case this is (FRS 13b, prop. 3.1.2).

Proof

Repeatedly apply Cartan's magic formula v=ι vd+dι v\mathcal{L}_v = \iota_v \circ d + d \circ \iota_v as well as the consequence v 1ι v 2ι v 2 v 1=ι [v 1,v 2]\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 iv_i and differential forms β\beta (of any degree, not necessarily closed) one has

(1) kdι v kι v 1β= 1i<jk(1) i+jι v kι v j^ι v i^ι [v i,v j] +i=1k(1) iι v kι v i^ι v 1 v iβ +ι v kι v 1dβ. \begin{aligned} (-1)^k d \iota_{v_k} \cdots \iota_{v_1} \beta = & \underset{1 \leq i \lt j \leq k}{\sum} (-1)^{i+j} \iota_{v_k} \cdots \widehat{\iota_{v_j}} \cdots \widehat{\iota_{v_i}} \cdots \iota_{[v_i,v_j]} \\ & + \underoverset{i=1}{k}{\sum} (-1)^i \iota_{v_k} \cdots \widehat{\iota_{v_i}} \cdots \iota_{v_1} \mathcal{L}_{v_i} \beta \\ & + \iota_{v_k} \cdots \iota_{v_1} d \beta \end{aligned} \,.

With this, the statement follows straightforwardly.

Interlude: Cech-Deligne complexes

Recall the Cech complex.

Definition

(Čech complex)

Let XX be a smooth manifold and let A Ch +(Sh(X))A_\bullet \in Ch_+(Sh(X)) be a sheaf of chain complexes on XX. Let {U iX}\{U_i \to X\} be a good open cover of XX, i.e. an open cover such that each finite non-empty intersection U i 0,,i kU_{i_0, \cdots, i_k} is diffeomorphic to an open ball/Cartesian space.

The Čech cochain complex C ((X,{U i}),A )C^\bullet((X,\{U_i\}),A_\bullet) of XX with respect to the cover {U iX}\{U_i \to X\} and with coefficients in A A_\bullet is in degree kk \in \mathbb{N} given by the abelian group

C k((X,{U i}),A ) l,nk=nl i 0,i 1,,i nA l(U i 0,,i n) C^k((X,\{U_i\}),A_\bullet) \coloneqq \oplus_{{l,n} \atop {k = n-l}} \oplus_{i_0, i_1, \cdots, i_n} A_l(U_{i_0, \cdots, i_n})

which is the direct sum of the values of A A_\bullet on the given intersections as indicated; and whose differential

d:C k((X,{U i}),A )C k+1((X,{U i}),A ) d \colon C^{k}((X,\{U_i\}),A_\bullet) \longrightarrow C^{k+1}((X,\{U_i\}),A_\bullet)

is defined componentwise (see at matrix calculus for conventions on maps between direct sums) by

(da) i 0,,i k+1 ( Aa+(1) kδa) i 0,,i k+1 Aa i 0,,i k+1+(1) k 0jk+1(1) ja i 0,,i j1,i j+1,,i k+1| U i 0,,i k+1 \begin{aligned} (d a)_{i_0, \cdots, i_{k+1}} & \coloneqq (\partial_A a + (-1)^k \delta a)_{i_0, \cdots, i_{k+1}} \\ & \coloneqq \partial_A a_{i_0, \cdots, i_{k+1}} + (-1)^k \sum_{0 \leq j \leq k+1} (-1)^{j} a_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}} |_{U_{i_0, \cdots, i_{k+1}}} \end{aligned}

where on the right the sum is over all components of aa obtained via the canonical restrictions obtained by discarding one of the original (k+1)(k+1) subscripts.

The Cech cohomology groups of XX with coefficients in A A_\bullet relative to the given cover are the chain homology groups of the Cech complex

H Cech k((X,{U i}),A )H k(C ((X,{U i}),A )). H_{Cech}^k((X,\{U_i\}), A_\bullet) \coloneqq H^k(C^\bullet((X,\{U_i\}),A_\bullet)) \,.

The Cech cohomology groups as such are the colimit (“direct limit”) of these groups over refinements of covers

H Cech k(X,A )lim {U iX}H Cech k((X,{U i}),A ). H^k_{Cech}(X, A_\bullet) \coloneqq \underset{\longrightarrow}{\lim}_{\{U_i \to X\}} H_{Cech}^k((X,\{U_i\}), A_\bullet) \,.

Recall the Deligne complex.

Definition

The Deligne complex for Deligne cohomology of degree (p+2)(p+2) is the chain complex of abelian sheaves given by

(B p+1U(1) conn) =[C (,U(1))dlogΩ 1()dΩ 2()ddΩ p+1()]Ch 0(Sh(X)). (\mathbf{B}^{p+1}U(1)_{conn})_{\bullet} = \left[ C^\infty(-,U(1)) \stackrel{d log}{\longrightarrow} \Omega^1(-) \stackrel{d}{\to} \Omega^2(-) \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^{p+1}(-) \right] \in Ch_{\bullet \geq 0}(Sh(X)) \,.
Remark

So for {U iX}\{U_i\to X\} an open cover, then we have the Cech-Deligne double complex

0 0 0 Ω p+1( iU i) δ Ω p+1( i,jU ij) δ Ω p+1( i,j,kU ijk) δ d d d d d d Ω 2( iU i) δ Ω 2( i,jU ij) δ Ω 2( i,j,kU ijk) δ d d d Ω 1( iU i) δ Ω 1( i,jU ij) δ Ω 1( i,j,kU ijk) δ dlog dlog dlog C ( iU i,U(1)) δ C ( i,jU ij,U(1)) δ C ( i,j,kU ijk,U(1)) δ \array{ 0 &\longrightarrow& 0 &\longrightarrow& 0 &\longrightarrow & \cdots \\ \uparrow && \uparrow && \uparrow \\ \Omega^{p+1}(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^{p+1}(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^{p+1}(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \vdots && \vdots && \vdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \Omega^2(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^2(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^2(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \Omega^1(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^1(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^1(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} && \\ C^\infty(\coprod_i U_i, U(1)) &\stackrel{\delta}{\longrightarrow}& C^\infty(\coprod_{i,j} U_{i j}, U(1)) &\stackrel{\delta}{\longrightarrow}& C^\infty(\coprod_{i,j,k} U_{i j k}, U(1)) &\stackrel{\delta}{\longrightarrow}& \cdots }

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 nn the direct sum of the entries in this double complex which are on the nnth nw-se off-diagonal and has the total differential

d tot=d+(1) degδ d_{tot} = d + (-1)^{deg} \delta

with degdeg denoting form degree.

Example

A Cech-Deligne cocycle in degree 33 (a “bundle gerbe with connection”) is data ({B i},{A ij},{g ijk})(\{B_{i}\}, \{A_{i j}\}, \{g_{i j k}\}) such that

{B i} δ {B jB i}=dA ij d {A ij} δ {A jk+A ikA ij}={dlogg ijk} dlog {g ijk} δ {g jklg ikl 1g ijlg ijk 1}=1 \array{ \{B_i\} &\stackrel{\delta}{\longrightarrow}& {{\{B_j - B_i\}} = {d A_{i j}}} && && \\ && \uparrow^{\mathrlap{d}} && && \\ && \{A_{i j}\} &\stackrel{\delta}{\longrightarrow}& \{-A_{ j k} + A_{i k} - A_{i j}\} = \{d log g_{i j k}\} && \\ && && \uparrow^{\mathrlap{d log}} && \\ && && \{g_{i j k}\} &\stackrel{\delta}{\longrightarrow}& \{g_{j k l} g_{i k l}^{-1} g_{i j l} g_{i j k}^{-1} \} = 1 }
Definition

The curvature of a Cech-Deligne cocycle

θ¯=(B i,,g i 1i p+2) \overline{\theta} = (B_{i}, \cdots, g_{i_1 \cdots i_{p+2}})

is the uniquely defined (p+2)(p+2)-form F θ¯Ω p+2(X)F_{\overline{\theta}} \in \Omega^{p+2}(X) such that

F θ¯| U i=dB i. F_{\overline{\theta}}|_{U_i} = d B_i \,.

Higher infinitesimal quantomorphisms and conserved currents

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)(p+2)-forms by degree-(p+2)(p+2)-cocycles in Deligne cohomology.

Definition

Given a pre-(p+1)-plectic manifold (X,ω)(X,\omega), then a prequantization is a Cech-Deligne cocycle θ¯\overline{\theta}, the prequantum (p+1)-bundle, whose curvature, def. 10, equals ω\omega:

F θ¯=ω. F_{\overline{\theta}} = \omega \,.
Remark

In terms of diagrams in the homotopy theory H\mathbf{H} of smooth homotopy types, def. 11 describes lifts of the form

B p+1U(1) conn θ¯ F () X ω Ω cl p+2. \array{ && \mathbf{B}^{p+1}U(1)_{conn} \\ & {}^{\mathllap{\overline{\theta}}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\longrightarrow} & \mathbf{\Omega}^{p+2}_{cl} } \,.

This way there is an immediate generalization of def. 4 to forms and cocycles of higher degree:

Definition

Let θ¯\overline{\theta} be any Cech-Deligne-cocycle relative to an open cover 𝒰\mathcal{U} of XX, which gives a prequantum n-bundle for ω\omega. The L-∞ algebra 𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ¯)\mathfrak{quantmorph}(X,\overline{\theta}) is the dg-Lie algebra (regarded as an L L_\infty-algebra) whose underlying chain complex is the Cech total complex made to end in Hamiltonian Cech cocycles

  • 𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ¯) 0{v+α¯Vect(X)Tot n1(𝒰,Ω )| vθ¯=d Totα¯}\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}\};

  • 𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ¯) i>0C n1i(𝒰,Ω )\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) degδd_{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+α¯ 1,v 2+α¯ 2][v 1,v 2]+ v 1α¯ 2 v 2α¯ 1[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+α¯,η¯]=[η¯,v+α¯]= vη¯[v+ \overline{\alpha}, \overline{\eta}] = - [\overline{\eta}, v + \overline{\alpha}] = \mathcal{L}_v \overline{\eta}.

(FRS 13b, def./prop. 4.2.1)

One then finds a direct higher analog of corollary 1 (its proof however is requires a bit more work):

Proposition

There is an equivalence in the homotopy theory of L-∞ algebras

f:𝔭𝔬𝔦𝔰𝔰(X,ω)𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ¯) f \colon \mathfrak{poiss}(X,\omega) \stackrel{\simeq}{\longrightarrow} \mathfrak{quantmorph}(X,\overline{\theta})

between the L L_\infty-algebras of def. 8 and def. 12 (in particular def. 12 does not depend on the choice of A¯\overline{A}) whose underlying chain map satisfies

  • f(v+J)=(v, i=0 n(1) iι vθ niJ| 𝒰)f(v + J) = (v,\; \sum_{i = 0}^n (-1)^i \iota_v \theta^{n-i} - J|_{\mathcal{U}}).

(FRS 13b, theorem 4.2.2)

Proposition

Proposition 7 says that all the higher Poisson L L_\infty-algebras are L L_\infty-algebras of symmetries of Deligen cocycles prequantizing the give pre-(p+1)(p+1)-plectic form, higher “quantomorphisms”.

In fact the dg-algebra 𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,θ¯)\mathfrak{quantmorph}(X,\overline{\theta}) makes yet another equivalent interpretation of 𝔭𝔬𝔦𝔰𝔰(X,ω)\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.

Higher Kostant-Souriau extension

The higher Poisson brackets come with a higher analog of the Kostant-Souriau extension, prop. 3.

Definition

Write

H(X,B p)(Ω 0(X)dΩ 1(X)dΩ (p1)(X)dΩ cl p(X)) \mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R}) \coloneqq (\Omega^0(X) \stackrel{d}{\to} \Omega^1(X) \stackrel{d}{\to} \cdots \Omega^{(p-1)}(X) \stackrel{d}{\to}\Omega^p_{cl}(X))

for the truncated de Rham complex regarded as an abelian L-∞ algebra.

Proposition

Given a pre-(p+1)-plectic manifold (X,ω)(X,\omega), the Poisson bracket Lie (p+1)-algebra 𝔭𝔬𝔦𝔰𝔰(X,ω)\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 L_\infty-algebras of the form

H(X,B p) 𝔭𝔬𝔦𝔰𝔰(X,ω) HamVect(X,ω) ι ω BH(X,B p). \array{ \mathbf{H}(X,\flat\mathbf{B}^p \mathbb{R}) &\longrightarrow& \mathfrak{poiss}(X,\omega) &\longrightarrow& HamVect(X,\omega) \\ && && \downarrow^{\mathrlap{\iota_{\cdots}\omega} } \\ && && \mathbf{B} \mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R}) } \,.

(FRS 13b, theorem 3.3.1)

To better see what this means, we may truncate this down to a statement about ordinary Lie algebras.

Proposition

Given a pre-(p+1)-plectic manifold (X,ω)(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)H^p(X).

0H p(X)τ 0𝔭𝔬𝔦𝔰𝔰(X,ω)HamVect(X,ω)0. 0 \to H^p(X) \longrightarrow \tau_0 \mathfrak{poiss}(X,\omega) \longrightarrow HamVect(X,\omega) \to 0 \,.

Applications

Higher conserved current algebras and BPS charge extensions

By the discussion at geometry of physics -- WZW terms, prequantization L:XB p+1U(1) conn\mathbf{L} \colon X \to \mathbf{B}^{p+1}U(1)_{conn} of ωΩ p+2(X)\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 XX.

Under this perspective, a Hamiltonian vector field vv on XX is a point symmetry of this sigma-model field theory and a Hamiltonian form JJ for vv 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 XX is a super spacetimes and ω\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 XX. Therefore we also write

𝔟𝔭𝔰(X,L)𝔮𝔲𝔞𝔫𝔱𝔪𝔬𝔯𝔭𝔥(X,L). \mathfrak{bps}(X,\mathbf{L}) \coloneqq \mathfrak{quantmorph}(X,\mathbf{L}) \,.

From this perspective the higher Kostant-Souriau extesion as in prop. 8 says the following:

Corollary

For ωΩ p+2(X)\omega \in \Omega^{p+2}(X) the curvature of a WZW term on the smooth manifold XX, then the Lie algebra of conserved currents covering target space point symmetries of the corresponding pp-brane sigma-model is a Lie algebra extension of the taget space symmetry by the ppth de Rham cohomology group

0H p(X)𝔟𝔭𝔰(X,ω)HamVect(X,ω)0. 0 \to H^p(X) \longrightarrow \mathfrak{bps}(X,\omega) \longrightarrow HamVect(X,\omega) \to 0 \,.
Remark

For XX a superspacetime and ω\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.

Finite symmetries

We here discuss the full finite version of quantomorphism n-groups.

Differential coefficients

Throughout, let 𝔾Grp(H)\mathbb{G} \in Grp(\mathbf{H}) be a braided ∞-group equipped with a Hodge filtration. Write B𝔾 conn\mathbf{B}\mathbb{G}_{conn}\in for the corresponding moduli stack of differential cohomology.

Example

For H=\mathbf{H} = Smooth∞Grpd we have 𝔾=B p(/Γ)\mathbb{G} = \mathbf{B}^p (\mathbb{R}/\Gamma) for Γ=\Gamma = \mathbb{Z} is the circle (p+1)-group. Equipped with its standard Hodge filtration this gives B𝔾 conn=B pU(1) conn\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)(p+2).

Definition

For XHX \in \mathbf{H}, for write

conc:[X,B𝔾 conn]𝔾Conn(X) conc \colon [X,\mathbf{B}\mathbb{G}_{conn}] \longrightarrow \mathbb{G}\mathbf{Conn}(X)

for the differential concretification of the internal hom.

This is the proper moduli stack of 𝔾\mathbb{G}-principal ∞-connections on XX in that a family U𝔾Conn(X)U \longrightarrow \mathbb{G}\mathbf{Conn}(X) is a vertical 𝔾\mathbb{G}-principal \infty-connection on U×XUU \times X\to U.

Example

For H=\mathbf{H} = Smooth∞Grpd or =FormalSmooth∞Grpd, for 𝔾=B pU(1)\mathbb{G} = \mathbf{B}^p U(1) the circle (p+1)-group with its standard Hodge filtration as in example 8, then for XX any smooth manifold or formal smooth manifold, (B pU(1))Conn(X)(\mathbf{B}^p U(1))\mathbf{Conn}(X) is presented via the Dold-Kan correspondence by the sheaf UCh U \mapsto Ch_\bullet of vertical Deligne complexes on U×XU \times X over UU.

Proposition

For 𝔾B𝔾\mathbb{G} \simeq \mathbf{B}\mathbb{G}' then the loop space object of the moduli stack of 𝔾\mathbb{G}-principal \infty-connections on XX is the moduli stack of flat ∞-connections with gauge group Ω𝔾\Omega \mathbb{G}

Ω 0(𝔾Conn(X))(Ω𝔾)FlatConn(X). \Omega_0 (\mathbb{G}\mathbf{Conn}(X)) \simeq (\Omega\mathbb{G})\mathbf{FlatConn}(X) \,.
Proposition

The canonical precomposition ∞-action of the automorphism ∞-group Aut(X)\mathbf{Aut}(X) on [X,B𝔾 conn][X,\mathbf{B}\mathbb{G}_{conn}] passes along concconc to an ∞-action on 𝔾Conn(X)\mathbb{G}\mathbf{Conn}(X).

Higher quantomorphism groups and Heisenberg groups

Definition

Given a 𝔾\mathbb{G}-principal ∞-connection :XB𝔾 conn\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn} there are the following concepts in higher geometric prequantum theory.

  1. The quantomorphism ∞-group is the stabilizer ∞-group of 𝔾Conn(X)\nabla \in \mathbb{G}\mathbf{Conn}(X), def. 14, under the Aut(X)\mathbf{Aut}(X)-action of 9;

    QuantMorph(X,)Stab Aut(X)(conc()). \mathbf{QuantMorph}(X,\nabla) \coloneqq \mathbf{Stab}_{\mathbf{Aut}(X)}(conc(\nabla)) \,.
  2. The Hamiltonian symplectomorphism ∞-group

    HamSymp(X,)Aut(X) \mathbf{HamSymp}(X,\nabla) \longrightarrow \mathbf{Aut}(X)

    is the 1-image of the canonical morphism QuantMorph(X,)Aut(X)\mathbf{QuantMorph}(X,\nabla) \longrightarrow \mathbf{Aut}(X).

  3. A Hamiltonian action of an ∞-group GG on (X,)(X,\nabla) is an ∞-group homomorphism

    ρ:GHamSymp(X,) \rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla) \;
  4. An ∞-moment map is an \infty-group homomorphism

    GQuantMorph(X,) G \longrightarrow \mathbf{QuantMorph}(X,\nabla)
  5. The Heisenberg ∞-group for a given Hamiltonian GG-action ρ\rho is the homotopy pullback

    Heis G(X,)ρ *QuantMorph(X,). \mathbf{Heis}_G(X,\nabla) \coloneqq \rho^\ast \mathbf{QuantMorph}(X,\nabla) \,.
Example

For H=\mathbf{H} = Smooth∞Grpd, for XSmoothMfdHX \in SmoothMfd \hookrightarrow \mathbf{H} a smooth manifold and for \nabla a prequantum line bundle on XX, then QuantMorph(X,)\mathbf{QuantMorph}(X,\nabla) is Souriau’s quantomorphism group covering the Hamiltonian diffeomorphism group. In the case that (X,F )(X, F_\nabla) is a symplectic vector space X=VX = V regarded as a linear symplectic manifold with Hamiltonian action on itself by translation, then Heis V(X,)\mathbf{Heis}_{V}(X,\nabla) is the traditional Heisenberg group.

Remark

Since HamSymp(X,)Aut(X)\mathbf{HamSymp}(X,\nabla)\hookrightarrow \mathbf{Aut}(X) is by construction a 1-monomorphism, given any GG-action ρ:GAut(X)\rho \colon G \longrightarrow \mathbf{Aut}(X) on XX, not necessarily Hamiltonian, then the homotopy pullback ρ *QuantMorph(X,)\rho^\ast \mathbf{QuantMorph}(X,\nabla) is the Heisenberg ∞-group of the maximal sub-\infty-group of GG which does act via Hamiltonian symplectomorphisms. Therefore we will also write Heis G(X,)\mathbf{Heis}_G(X,\nabla) in this case.

The higher Kostant-Souriau extension

The following is the refinement of the Kostant-Souriau extension to higher differential geometry

Proposition

Given a 𝔾\mathbb{G}-principal ∞-connection :XB𝔾 conn\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}, there is a homotopy fiber sequence of the form

  1. if 𝔾\mathbb{G} is 0-truncated then

    𝔾ConstFunct(X) QuantMorph(X,) HamSymp(X,) KS B(𝔾ConstFunct(X)) \array{ \mathbb{G}\mathbf{ConstFunct}(X) &\longrightarrow& \mathbf{QuantMorph}(X,\nabla) \\ && \downarrow \\ && \mathbf{HamSymp}(X,\nabla) &\stackrel{\mathbf{KS}}{\longrightarrow}& \mathbf{B} (\mathbb{G}\mathbf{ConstFunct}(X)) }
  2. if 𝔾B𝔾\mathbb{G} \simeq \mathbf{B}\mathbb{G}' then

    (Ω𝔾)FlatConn(X) QuantMorph(X,) HamSymp(X,) KS B((Ω𝔾)FlatConn(X)) \array{ (\Omega \mathbb{G})\mathbf{FlatConn}(X) &\longrightarrow& \mathbf{QuantMorph}(X,\nabla) \\ && \downarrow \\ && \mathbf{HamSymp}(X,\nabla) &\stackrel{\mathbf{KS}}{\longrightarrow}& \mathbf{B} ((\Omega \mathbb{G})\mathbf{FlatConn}(X)) }

exhibiting the quantomorphism ∞-group as an ∞-group extension of the Hamiltonian symplectomorphism ∞-group by the moduli stack of Ω𝔾\Omega \mathbb{G}-flat ∞-connections, classified by a cocycle KS\mathbf{KS}.

(FRS 13a)

Example

In H=\mathbf{H} = Smooth∞Grpd, let 𝔾=B pU(1)\mathbb{G} = \mathbf{B}^p U(1) be the circle (p+1)-group and let XSmoothMfdSmoothGrpdX \in SmoothMfd \hookrightarrow Smooth \infty Grpd be p-connected, then (ΩB pU(1))FlatConn(X)B pU(1)(\Omega\mathbf{B}^p U(1))\mathbf{FlatConn}(X)\simeq \mathbf{B}^{p}U(1). Hence here prop. 10 gives

B pU(1) QuantMorph(X,) HamSymp(X,) KS B p+1U(1) \array{ \mathbf{B}^{p}U(1) &\longrightarrow& \mathbf{QuantMorph}(X,\nabla) \\ && \downarrow \\ && \mathbf{HamSymp}(X,\nabla) &\stackrel{\mathbf{KS}}{\longrightarrow}& \mathbf{B}^{p+1}U(1) }
Corollary

Given a 𝔾\mathbb{G}-principal ∞-connection :XB𝔾 conn\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}, and for ρ:GHamSymp(X,)\rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla) a GG-Hamiltonian action, then there is a homotopy fiber sequence

  1. if 𝔾\mathbb{G} is 0-truncated then

    𝔾ConstFunct(X) Heis G(X,) G KS(ρ) B(𝔾ConstFunct(X)) \array{ \mathbb{G}\mathbf{ConstFunct}(X) &\longrightarrow& \mathbf{Heis}_G(X,\nabla) \\ && \downarrow \\ && G &\stackrel{\mathbf{KS}(\rho)}{\longrightarrow}& \mathbf{B} (\mathbb{G}\mathbf{ConstFunct}(X)) }
  2. if 𝔾B𝔾\mathbb{G} \simeq \mathbf{B}\mathbb{G}' then

    (Ω𝔾)FlatConn(X) Heis G(X,) G KS(ρ) B((Ω𝔾)FlatConn(X)) \array{ (\Omega \mathbb{G})\mathbf{FlatConn}(X) &\longrightarrow& \mathbf{Heis}_G(X,\nabla) \\ && \downarrow \\ && G &\stackrel{\mathbf{KS}(\rho)}{\longrightarrow}& \mathbf{B} ((\Omega \mathbb{G})\mathbf{FlatConn}(X)) }

exhibiting the Heisenberg ∞-group as an ∞-group extension of the GG by the moduli stack of Ω𝔾\Omega \mathbb{G}-flat ∞-connections, classified by a cocycle KS(ρ)\mathbf{KS}(\rho).

The class of the cocycle KS(ρ)\mathbf{KS}(\rho) is the obstruction to prequantizing ρ\rho to a moment map (the classical anomaly of ρ\rho); and the the Heisenberg ∞-group extension of GG is the universal cancellation of this anomaly.

References

Revised on June 5, 2015 13:02:08 by Urs Schreiber (50.207.161.2)