-plectic geometry is a generalization of symplectic geometry to higher category theory.
It is closely related to multisymplectic geometry and n-symplectic manifolds.
For , an n-plectic vector space is a vector space (over the real numbers) equipped with an -linear skew function
such that regarded as a function
is has trivial kernel.
Let be a smooth manifold, a differential form.
We say is a -plectic manifold if
is closed: ;
for all the map
given by contraction of vectors with forms
is injective.
For a -plectic manifold and a submanifold, we say is a -isotropic, -Lagrangian, and -coisotropic submanifold is for every , the tangent space is a -isotropic, -Lagrangian, and -coisotropic subspace, respectively (see n-plectic vector space for the definition).
Given a diffeomorphism for a -plectic manifold, its graph is a -Lagrangian submanifold of the -plectic manifold if and only if is an -plectomorphism, meaning
See e.g. Proposition 3.7 in de León & Vilariño 2012.
See also the definition at multisymplectic geometry.
– this is the case of an ordinary symplectic manifold this appears in Hamiltonian mechanics;
, this appears naturally in 1+1 dimensional quantum field theory.
For orientable, take the volume form. This is -plectic.
let be the canonical Lie algebra 3-cocycle and extend it left-invariantly to a 3-form on . Then is 2-plectic.
To an ordinary symplectic manifold is associated its Poisson algebra. Underlying this is a Lie algebra, whose Lie bracket is the Poisson bracket.
We discuss here how to an -plectic manifold for there is correspondingly assoociated not a Lie algebra, but a Lie n-algebra: the Poisson bracket Lie n-algebra. It is natural to call this a Poisson Lie -algebra (see for instance at Poisson Lie 2-algebra).
(Not to be confused with the Lie algebra of a Poisson Lie group, which is a Lie group that itself is equipped with a compatible Poisson manifold structure. It is a bit unfortunate that there is no better established term for the Lie algebra underlying a Poisson algebra apart from “Poisson bracket”.)
Consider the ordinary case in for how a Poisson algebra is obtained from a symplectic manifold .
Here
is an isomorphism.
Given , such that
Define . Then is a Poisson algebra.
We can generalize this to -plectic geometry.
Let be -plectic for .
Observe that then is no longer an isomorphism in general.
Definition
Say
is Hamiltonian precisely if
such that
This makes uniquely defined.
Denote the collection of Hamiltonian forms by .
Define a bracket
by
This satisfies
k
is skew-symmetric;
+ cyclic permutations
.
So the Jacobi dientity fails up to an exact term. This will yield the structure of an L-infinity algebra.
Proposition
Given an -plectic manifold we get a Lie n-algebra structure on the complex
(where the rightmost term is taken to be in degree 0).
where
the unary bracket is ;
the -ary bracket is
This is the Poisson bracket Lie n-algebra.
This appears as (Rogers 11, theorem 3.14).
For this recovers the definition of the Lie algebra underlying a Poisson algebra.
Recall for the mechanism of geometric quantization of a symplectic manifold.
Given a 2-form and the corresponding complex line bundle , consider the Atiyah Lie algebroid sequence
The smooth sections of are the invariant vector fields on the total space of .
Using a connection on we may write such a section as
for a vector field downstairs, a horizontal lift with respect to the given connection and .
Locally on a suitable patch we have that .
We say that preserves the splitting iff we have
One finds that this is the case precisely if
This gives a homomorphism of Lie algebras
We consder now prequantization of 2-plectic manifolds.
Let be a 2-plectic manifold such that the de Rham cohomology class is in the image of integral cohomology (Has integral periods.)
We can form a cocycle in Deligne cohomology from this, encoding a bundle gerbe with connection.
On a cover of this is given in terms of Cech cohomology by data
;
satisfying a cocycle condition.
Now recall that an exact Courant algebroid is given by the following data:
a vector bundle ;
an anchor morphism to the tangent bundle;
an inner product on the fibers of ;
a bracket on the sections of .
Satisfying some conditions.
The fact that the Courant algebroid is exact means that
is an exact sequence.
The standard Courant algebroid is the example where
;
;
the bracket is the skew-symmetrization of the Dorfman bracket
Now with respect to the above Deligne cocycle, build a Courant algebroid as follows:
on each patch is is the standard Courant algebroid ;
glued together on double intersections using the
This gives an exact Courant algebroid as well as a splitting given by the .
The bracket on this is given by the skew-symmetrization of
Here a section preserves the splitting precisely if
for all we have
exactly if is Hamiltonian and .
Theorem
Recall that to every Courant algebroid is associated a Lie 2-algebra .
Then: we have an embedding of L-infinity algebras
given by .
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for -principal ∞-connection)
(extension are listed for sufficiently connected )
duality between algebra and geometry
in physics:
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
The observation that the would-be Poisson bracket induced by a higher degree closed form extends to the Poisson bracket Lie n-algebra is due to
Chris Rogers, algebras from multisymplectic geometry , Letters in Mathematical Physics 100 1 (2012) 29-50 [arXiv:1005.2230, journal]
Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)
with first discussion of application to prequantization in
Chris Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization , arXiv:1009.2975.
Chris Rogers, Higher geometric quantization, talk at Higher Structures 2011 in Göttingen (pdf slides)
Discussion in the more general context of higher differential geometry/extended prequantum field theory is in
Domenico Fiorenza, Chris Rogers, Urs Schreiber,
Higher geometric prequantum theory,
L-∞ algebras of local observables from higher prequantum bundles
See also
See also the references at multisymplectic geometry and n-symplectic manifold.
A higher differential geometry-generalization of contact geometry in line with multisymplectic geometry/-plectic geometry is discussed in
Some more references on application, on top of those mentioned in the articles above.
A survey of some (potential) applications of 2-plectic geometry in string theory and M2-brane models is in section 2 of
and in
On -plectic formulation of gravity, Maxwell theory and Einstein-Maxwell theory:
Petr Hořava, On a covariant Hamilton-Jacobi framework for the Einstein-Maxwell theory Classical and Quantum Gravity 8 11 (1991) 2069 [doi:10.1088/0264-9381/8/11/016]
Dmitri Vey, -Plectic Maxwell Theory [arxiv:1303.2192]
Dmitri Vey, Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian -forms, Classical and Quantum Gravity 32 9 (2015) [arxiv:1404.3546, doi:10.1088/0264-9381/32/9/095005[
Patrick Cabau, Fernand Pelletier. Partial Nambu-Poisson structures on a convenient manifold (2024). (arXiv:2404.08688).
Last revised on April 18, 2024 at 17:02:08. See the history of this page for a list of all contributions to it.