Symplectic manifold

Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system.^[1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential ${\displaystyle dH}$  of a Hamiltonian function ${\displaystyle H}$ .^[2] So we require a linear map ${\displaystyle TM\rightarrow T^{*}M}$  from the tangent manifold ${\displaystyle TM}$  to the cotangent manifold ${\displaystyle T^{*}M}$ , or equivalently, an element of ${\displaystyle T^{*}M\otimes T^{*}M}$ . Letting ${\displaystyle \omega }$  denote a section of ${\displaystyle T^{*}M\otimes T^{*}M}$ , the requirement that ${\displaystyle \omega }$  be non-degenerate ensures that for every differential ${\displaystyle dH}$  there is a unique corresponding vector field ${\displaystyle V_{H}}$  such that ${\displaystyle dH=\omega (V_{H},\cdot )}$ . Since one desires the Hamiltonian to be constant along flow lines, one should have ${\displaystyle \omega (V_{H},V_{H})=dH(V_{H})=0}$ , which implies that ${\displaystyle \omega }$  is alternating and hence a 2-form. Finally, one makes the requirement that ${\displaystyle \omega }$  should not change under flow lines, i.e. that the Lie derivative of ${\displaystyle \omega }$  along ${\displaystyle V_{H}}$  vanishes. Applying Cartan's formula, this amounts to (here ${\displaystyle \iota _{X}}$  is the interior product):

${\displaystyle {\mathcal {L}}_{V_{H}}(\omega )=0\;\Leftrightarrow \;\mathrm {d} (\iota _{V_{H}}\omega )+\iota _{V_{H}}\mathrm {d} \omega =\mathrm {d} (\mathrm {d} \,H)+\mathrm {d} \omega (V_{H})=\mathrm {d} \omega (V_{H})=0}$ 

so that, on repeating this argument for different smooth functions ${\displaystyle H}$  such that the corresponding ${\displaystyle V_{H}}$  span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of ${\displaystyle V_{H}}$  corresponding to arbitrary smooth ${\displaystyle H}$  is equivalent to the requirement that ω should be closed.

A symplectic form on a smooth manifold ${\displaystyle M}$  is a closed non-degenerate differential 2-form ${\displaystyle \omega }$ .^[3][4] Here, non-degenerate means that for every point ${\displaystyle p\in M}$ , the skew-symmetric pairing on the tangent space ${\displaystyle T_{p}M}$  defined by ${\displaystyle \omega }$  is non-degenerate. That is to say, if there exists an ${\displaystyle X\in T_{p}M}$  such that ${\displaystyle \omega (X,Y)=0}$  for all ${\displaystyle Y\in T_{p}M}$ , then ${\displaystyle X=0}$ . Since in odd dimensions, skew-symmetric matrices are always singular, the requirement that ${\displaystyle \omega }$  be nondegenerate implies that ${\displaystyle M}$  has an even dimension.^[3][4] The closed condition means that the exterior derivative of ${\displaystyle \omega }$  vanishes. A symplectic manifold is a pair ${\displaystyle (M,\omega )}$  where ${\displaystyle M}$  is a smooth manifold and ${\displaystyle \omega }$  is a symplectic form. Assigning a symplectic form to ${\displaystyle M}$  is referred to as giving ${\displaystyle M}$  a symplectic structure.

Let ${\displaystyle \{v_{1},\ldots ,v_{2n}\}}$  be a basis for ${\displaystyle \mathbb {R} ^{2n}.}$  We define our symplectic form ω on this basis as follows:

${\displaystyle \omega (v_{i},v_{j})={\begin{cases}1&j-i=n{\text{ with }}1\leqslant i\leqslant n\\-1&i-j=n{\text{ with }}1\leqslant j\leqslant n\\0&{\text{otherwise}}\end{cases}}}$ 

In this case the symplectic form reduces to a simple quadratic form. If I_n denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the 2n × 2n block matrix:

${\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}.}$ 

Let ${\displaystyle Q}$  be a smooth manifold of dimension ${\displaystyle n}$ . Then the total space of the cotangent bundle ${\displaystyle T^{*}Q}$  has a natural symplectic form, called the Poincaré two-form or the canonical symplectic form

${\displaystyle \omega =\sum _{i=1}^{n}dp_{i}\wedge dq^{i}}$ 

Here ${\displaystyle (q^{1},\ldots ,q^{n})}$  are any local coordinates on ${\displaystyle Q}$  and ${\displaystyle (p_{1},\ldots ,p_{n})}$  are fibrewise coordinates with respect to the cotangent vectors ${\displaystyle dq^{1},\ldots ,dq^{n}}$ . Cotangent bundles are the natural phase spaces of classical mechanics. The point of distinguishing upper and lower indexes is driven by the case of the manifold having a metric tensor, as is the case for Riemannian manifolds. Upper and lower indexes transform contra and covariantly under a change of coordinate frames. The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momenta ${\displaystyle p_{i}}$  are "soldered" to the velocities ${\displaystyle dq^{i}}$ . The soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.

A Kähler manifold is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of complex manifolds. A large class of examples come from complex algebraic geometry. Any smooth complex projective variety ${\displaystyle V\subset \mathbb {CP} ^{n}}$  has a symplectic form which is the restriction of the Fubini—Study form on the projective space ${\displaystyle \mathbb {CP} ^{n}}$ .

Riemannian manifolds with an ${\displaystyle \omega }$ -compatible almost complex structure are termed almost-complex manifolds. They generalize Kähler manifolds, in that they need not be integrable. That is, they do not necessarily arise from a complex structure on the manifold.

There are several natural geometric notions of submanifold of a symplectic manifold ${\displaystyle (M,\omega )}$ :

• Symplectic submanifolds of ${\displaystyle M}$  (potentially of any even dimension) are submanifolds ${\displaystyle S\subset M}$  such that ${\displaystyle \omega |_{S}}$  is a symplectic form on ${\displaystyle S}$ .
• Isotropic submanifolds are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called co-isotropic.
• Lagrangian submanifolds of a symplectic manifold ${\displaystyle (M,\omega )}$  are submanifolds where the restriction of the symplectic form ${\displaystyle \omega }$  to ${\displaystyle L\subset M}$  is vanishing, i.e. ${\displaystyle \omega |_{L}=0}$  and ${\displaystyle {\text{dim }}L={\tfrac {1}{2}}\dim M}$ . Lagrangian submanifolds are the maximal isotropic submanifolds.

One major example is that the graph of a symplectomorphism in the product symplectic manifold (M × M, ω × −ω) is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.

Let ${\displaystyle \mathbb {R} _{{\textbf {x}},{\textbf {y}}}^{2n}}$  have global coordinates labelled ${\displaystyle (x_{1},\dotsc ,x_{n},y_{1},\dotsc ,y_{n})}$ . Then, we can equip ${\displaystyle \mathbb {R} _{{\textbf {x}},{\textbf {y}}}^{2n}}$  with the canonical symplectic form

${\displaystyle \omega =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\dotsb +\mathrm {d} x_{n}\wedge \mathrm {d} y_{n}.}$ 

There is a standard Lagrangian submanifold given by ${\displaystyle \mathbb {R} _{\mathbf {x} }^{n}\to \mathbb {R} _{\mathbf {x} ,\mathbf {y} }^{2n}}$ . The form ${\displaystyle \omega }$  vanishes on ${\displaystyle \mathbb {R} _{\mathbf {x} }^{n}}$  because given any pair of tangent vectors ${\displaystyle X=f_{i}({\textbf {x}})\partial _{x_{i}},Y=g_{i}({\textbf {x}})\partial _{x_{i}},}$  we have that ${\displaystyle \omega (X,Y)=0.}$  To elucidate, consider the case ${\displaystyle n=1}$ . Then, ${\displaystyle X=f(x)\partial _{x},Y=g(x)\partial _{x},}$  and ${\displaystyle \omega =\mathrm {d} x\wedge \mathrm {d} y}$ . Notice that when we expand this out

${\displaystyle \omega (X,Y)=\omega (f(x)\partial _{x},g(x)\partial _{x})={\frac {1}{2}}f(x)g(x)(\mathrm {d} x(\partial _{x})\mathrm {d} y(\partial _{x})-\mathrm {d} y(\partial _{x})\mathrm {d} x(\partial _{x}))}$ 

both terms we have a ${\displaystyle \mathrm {d} y(\partial _{x})}$  factor, which is 0, by definition.

The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let

${\displaystyle X=\{(x,y)\in \mathbb {R} ^{2}:y^{2}-x=0\}.}$ 

Then, we can present ${\displaystyle T^{*}X}$  as

${\displaystyle T^{*}X=\{(x,y,\mathrm {d} x,\mathrm {d} y)\in \mathbb {R} ^{4}:y^{2}-x=0,2y\mathrm {d} y-\mathrm {d} x=0\}}$ 

where we are treating the symbols ${\displaystyle \mathrm {d} x,\mathrm {d} y}$  as coordinates of ${\displaystyle \mathbb {R} ^{4}=T^{*}\mathbb {R} ^{2}}$ . We can consider the subset where the coordinates ${\displaystyle \mathrm {d} x=0}$  and ${\displaystyle \mathrm {d} y=0}$ , giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions ${\displaystyle f_{1},\dotsc ,f_{k}}$  and their differentials ${\displaystyle \mathrm {d} f_{1},\dotsc ,df_{k}}$ .

Consider the canonical space ${\displaystyle \mathbb {R} ^{2n}}$  with coordinates ${\displaystyle (q_{1},\dotsc ,q_{n},p_{1},\dotsc ,p_{n})}$ . A parametric submanifold ${\displaystyle L}$  of ${\displaystyle \mathbb {R} ^{2n}}$  is one that is parameterized by coordinates ${\displaystyle (u_{1},\dotsc ,u_{n})}$  such that

${\displaystyle q_{i}=q_{i}(u_{1},\dotsc ,u_{n})\quad p_{i}=p_{i}(u_{1},\dotsc ,u_{n})}$ 

This manifold is a Lagrangian submanifold if the Lagrange bracket ${\displaystyle [u_{i},u_{j}]}$  vanishes for all ${\displaystyle i,j}$ . That is, it is Lagrangian if

${\displaystyle [u_{i},u_{j}]=\sum _{k}{\frac {\partial q_{k}}{\partial u_{i}}}{\frac {\partial p_{k}}{\partial u_{j}}}-{\frac {\partial p_{k}}{\partial u_{i}}}{\frac {\partial q_{k}}{\partial u_{j}}}=0}$ 

for all ${\displaystyle i,j}$ . This can be seen by expanding

${\displaystyle {\frac {\partial }{\partial u_{i}}}={\frac {\partial q_{k}}{\partial u_{i}}}{\frac {\partial }{\partial q_{k}}}+{\frac {\partial p_{k}}{\partial u_{i}}}{\frac {\partial }{\partial p_{k}}}}$ 

in the condition for a Lagrangian submanifold ${\displaystyle L}$ . This is that the symplectic form must vanish on the tangent manifold ${\displaystyle TL}$ ; that is, it must vanish for all tangent vectors:

${\displaystyle \omega \left({\frac {\partial }{\partial u_{i}}},{\frac {\partial }{\partial u_{j}}}\right)=0}$ 

for all ${\displaystyle i,j}$ . Simplify the result by making use of the canonical symplectic form on ${\displaystyle \mathbb {R} ^{2n}}$ :

${\displaystyle \omega \left({\frac {\partial }{\partial q_{k}}},{\frac {\partial }{\partial p_{k}}}\right)=-\omega \left({\frac {\partial }{\partial p_{k}}},{\frac {\partial }{\partial q_{k}}}\right)=1}$ 

and all others vanishing.

As local charts on a symplectic manifold take on the canonical form, this example suggests that Lagrangian submanifolds are relatively unconstrained. The classification of symplectic manifolds is done via Floer homology—this is an application of Morse theory to the action functional for maps between Lagrangian submanifolds. In physics, the action describes the time evolution of a physical system; here, it can be taken as the description of the dynamics of branes.

Another useful class of Lagrangian submanifolds occur in Morse theory. Given a Morse function ${\displaystyle f:M\to \mathbb {R} }$  and for a small enough ${\displaystyle \varepsilon }$  one can construct a Lagrangian submanifold given by the vanishing locus ${\displaystyle \mathbb {V} (\varepsilon \cdot \mathrm {d} f)\subset T^{*}M}$ . For a generic Morse function we have a Lagrangian intersection given by ${\displaystyle M\cap \mathbb {V} (\varepsilon \cdot \mathrm {d} f)={\text{Crit}}(f)}$ .

In the case of Kähler manifolds (or Calabi–Yau manifolds) we can make a choice ${\displaystyle \Omega =\Omega _{1}+\mathrm {i} \Omega _{2}}$  on ${\displaystyle M}$  as a holomorphic n-form, where ${\displaystyle \Omega _{1}}$  is the real part and ${\displaystyle \Omega _{2}}$  imaginary. A Lagrangian submanifold ${\displaystyle L}$  is called special if in addition to the above Lagrangian condition the restriction ${\displaystyle \Omega _{2}}$  to ${\displaystyle L}$  is vanishing. In other words, the real part ${\displaystyle \Omega _{1}}$  restricted on ${\displaystyle L}$  leads the volume form on ${\displaystyle L}$ . The following examples are known as special Lagrangian submanifolds,

1. complex Lagrangian submanifolds of hyperkähler manifolds,
2. fixed points of a real structure of Calabi–Yau manifolds.

The SYZ conjecture deals with the study of special Lagrangian submanifolds in mirror symmetry; see (Hitchin 1999).

The Thomas–Yau conjecture predicts that the existence of a special Lagrangian submanifolds on Calabi–Yau manifolds in Hamiltonian isotopy classes of Lagrangians is equivalent to stability with respect to a stability condition on the Fukaya category of the manifold.

A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibres are Lagrangian submanifolds. Since M is even-dimensional we can take local coordinates (p₁,...,p_n, q¹,...,qⁿ), and by Darboux's theorem the symplectic form ω can be, at least locally, written as ω = ∑ dp_k ∧ dq^k, where d denotes the exterior derivative and ∧ denotes the exterior product. This form is called the Poincaré two-form or the canonical two-form. Using this set-up we can locally think of M as being the cotangent bundle ${\displaystyle T^{*}\mathbb {R} ^{n},}$  and the Lagrangian fibration as the trivial fibration ${\displaystyle \pi :T^{*}\mathbb {R} ^{n}\to \mathbb {R} ^{n}.}$  This is the canonical picture.

Let L be a Lagrangian submanifold of a symplectic manifold (K,ω) given by an immersion i : L ↪ K (i is called a Lagrangian immersion). Let π : K ↠ B give a Lagrangian fibration of K. The composite (π ∘ i) : L ↪ K ↠ B is a Lagrangian mapping. The critical value set of π ∘ i is called a caustic.

Two Lagrangian maps (π₁ ∘ i₁) : L₁ ↪ K₁ ↠ B₁ and (π₂ ∘ i₂) : L₂ ↪ K₂ ↠ B₂ are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form.^[4] Symbolically:

${\displaystyle \tau \circ i_{1}=i_{2}\circ \sigma ,\ \nu \circ \pi _{1}=\pi _{2}\circ \tau ,\ \tau ^{*}\omega _{2}=\omega _{1}\,,}$ 

where τ^∗ω₂ denotes the pull back of ω₂ by τ.

• A symplectic manifold ${\displaystyle (M,\omega )}$  is exact if the symplectic form ${\displaystyle \omega }$  is exact. For example, the cotangent bundle of a smooth manifold is an exact symplectic manifold if we use the canonical symplectic form. The area 2-form on the 2-sphere is a symplectic form that is not exact.
• A symplectic manifold endowed with a metric that is compatible with the symplectic form is an almost Kähler manifold in the sense that the tangent bundle has an almost complex structure, but this need not be integrable.
• Symplectic manifolds are special cases of a Poisson manifold.
• A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form.^[5]
• A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued ${\displaystyle (n+2)}$ -form; it is utilized in Hamiltonian field theory.^[6]

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• Alan Weinstein (1971). "Symplectic manifolds and their lagrangian submanifolds". Advances in Mathematics. 6 (3): 329–46. doi:10.1016/0001-8708(71)90020-X.
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• Dunin-Barkowski, Petr (2022). "Symplectic duality for topological recursion". arXiv:2206.14792 [math-ph].
• "How to find Lagrangian Submanifolds". Stack Exchange. December 17, 2014.
• Lumist, Ü. (2001) [1994], "Symplectic Structure", Encyclopedia of Mathematics, EMS Press
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• Hitchin, Nigel (1999). "Lectures on Special Lagrangian Submanifolds". arXiv:math/9907034.