nLab
Poisson bracket Lie n-algebra

Contents

Idea

The Lie n-algebra that generalizes the Poisson bracket from symplectic geometry to n-plectic geometry: the Poisson bracket L L_\infty-algebra of local observables in higher prequantum geometry.

More discussion is here at n-plectic geometry.

Applied to the symplectic current (in the sense of covariant phase space theory, de Donder-Weyl field theory) this is the higher current algebra (see there) of conserved currents of a prequantum field theory.

Definition

Throughout, Let XX be a smooth manifold, let n1n \geq 1 a natural number and ωΩ cl n+1(X)\omega \in \Omega^{n+1}_{cl}(X) a closed differential (n+1)-form on XX. The pair (X,ω)(X,\omega) we may regard as a pre-n-plectic manifold.

We define two L-∞ algebras defined from this data and discuss their equivalence. Either of the two or any further one equivalent to the two is the Poisson bracket Lie nn-albebra of (X,ω)(X,\omega). The first definition is due to (Rogers 10), the second due to (FRS 13b). Here in notation we follow (FRS 13b).

Definition

Write

Ham n1(X)Vect(X)Ω n1(X) Ham^{n-1}(X) \subset Vect(X) \oplus \Omega^{n-1}(X)

for the subspace of the direct sum of vector fields vv on XX and differential (n-1)-forms JJ on XX satisfying

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

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

(FRS 13b, def. 2.1.3)

Definition

The L-∞ algebra L (X,ω)L_\infty(X,\omega) has as underlying chain complex the truncated and modified de Rham complex

Ω 0(X)dΩ 1(X)ddΩ n2(X)(0,d)Ham n1(X) \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^{n-2}(X) \stackrel{(0,\mathbf{d})}{\longrightarrow} Ham^{n-1}(X)

with the Hamiltonian pairs, def. 1, in degree 0 and with the 0-forms (smooth functions) in degree n1n-1, and its non-vanishing L L_\infty-brackets are as follows:

  • l 1(J)=dJl_1(J) = \mathbf{d}J

  • l k2(v 1+J 1,,v k+J k)=(1) (k+12)ι v 1v kωl_{k \geq 2}(v_1 + J_1, \cdots, v_k + J_k) = - (-1)^{\left(k+1 \atop 2\right)} \iota_{v_1 \wedge \cdots \wedge v_k}\omega.

(FRS 13b, prop. 3.1.2)

Definition

Let A¯\overline{A} 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 dgLie Qu(X,A¯)dgLie_{Qu}(X,\overline{A}) is the dg-Lie algebra (regarded as an L L_\infty-algebra) whose underlying chain complex is

dgLie Qu(X,A¯) 0={v+θ¯Vect(X)Tot n1(𝒰,Ω )| vA¯=d Totθ¯}dgLie_{Qu}(X,\overline{A})^0 = \{v+ \overline{\theta} \in Vect(X)\oplus Tot^{n-1}(\mathcal{U}, \Omega^\bullet) \;\vert\; \mathcal{L}_v \overline{A} = \mathbf{d}_{Tot}\overline{\theta}\};

dgLie Qu(X,A¯) i>0=Tot n1i(𝒰,Ω )dgLie_{Qu}(X,\overline{A})^{i \gt 0} = Tot^{n-1-i}(\mathcal{U},\Omega^\bullet)

with differential given by d Tot\mathbf{d}_{Tot} (where TotTot refers to total complex of the Cech-de Rham double complex).

The non-vanishing dg-Lie bracket on this complex are defined to be

  • [v 1+θ¯ 1,v 2+θ¯ 2]=[v 1,v 2]+ v 1θ¯ 2 v 2θ¯ 1[v_1 + \overline{\theta}_1, v_2 + \overline{\theta}_2] = [v_1, v_2] + \mathcal{L}_{v_1}\overline{\theta}_2 - \mathcal{L}_{v_2}\overline{\theta}_1;

  • [v+θ¯,η¯]=[η,v+θ¯]= vη¯[v+ \overline{\theta}, \overline{\eta}] = - [\eta, v + \overline{\theta}] = \mathcal{L}_v \overline{\eta}.

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

Proposition

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

f:L (X,ω)dgLie Qu(X,A¯) f \colon L_\infty(X,\omega) \stackrel{\simeq}{\longrightarrow} dgLie_{Qu}(X,\overline{A})

between the L L_\infty-algebras of def. 2 and def. 3 (in particular def. 3 does not depend on the choice of A¯\overline{A}) whose underlying chain map satisfies

  • f(v+J)=vJ| 𝒰+ i=0 n(1) iι vA nif(v + J) = v - J|_{\mathcal{U}} + \sum_{i = 0}^n (-1)^i \iota_v A^{n-i}.

(FRS 13b, theorem 4.2.2)

Properties

The extension theorem

Proposition

Given a pre n-plectic manifold (X,ω n+1)(X,\omega_{n+1}), then the Poisson bracket Lie nn-algebra 𝔓𝔬𝔦𝔰(X,ω)\mathfrak{Pois}(X,\omega) from above is an extension of the Lie algebra of Hamiltonian vector fields Vect Ham(X)Vect_{Ham}(X), def. 1 by the cocycle infinity-groupoid H(X,B n1)\mathbf{H}(X,\flat \mathbf{B}^{n-1} \mathbb{R}) for ordinary cohomology with real number coefficients in that there is a homotopy fiber sequence in the homotopy theory of L-infinity algebras of the form

H(X,B d1) 𝔓𝔬𝔦𝔰(X,ω) Vect Ham(X,ω) ω[] BH(X,B d1), \array{ \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) &\longrightarrow& \mathfrak{Pois}(X,\omega) \\ && \downarrow \\ && Vect_{Ham}(X,\omega) &\stackrel{\omega[\bullet]}{\longrightarrow}& \mathbf{B} \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) } \,,

where the cocycle ω[]\omega[\bullet], when realized on the model of def. 2, is degreewise given by by contraction with ω\omega.

This is FRS13b, theorem 3.3.1.

As a corollary this means that the 0-truncation τ 0𝔓𝔬𝔦𝔰(X,ω)\tau_0 \mathfrak{Pois}(X,\omega) is a Lie algebra extension by de Rham cohomology, fitting into a short exact sequence of Lie algebras

0H dR d1(X)τ 0𝔓𝔬𝔦𝔰(X,ω)Vect Ham(X)0. 0 \to H^{d-1}_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Vect_{Ham}(X) \to 0 \,.
Remark

These kinds of extensions are known traditionally form current algebras.

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:Heisenberg ∞-group\hookrightarrowquantomorphism ∞-group\hookrightarrow∞-bisections of higher Courant groupoid\hookrightarrow∞-bisections of higher Atiyah groupoid
L-∞ algebras:Heisenberg L-∞ algebra\hookrightarrowPoisson L-∞ algebra\hookrightarrowCourant L-∞ algebra\hookrightarrowtwisted vector fields

higher and integrated Kostant-Souriau extensions:

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

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

References

The Poisson bracket L L_\infty-algebra L (X,ω)L_\infty(X,\omega) was introduced in

Discussion in the broader context of higher differential geometry and higher prequantum geometry is in

See also

Revised on July 6, 2015 03:40:39 by Urs Schreiber (89.204.138.26)