Lagrangian Grassmannian and the universal Maslov class

Intuition Beginner

A Lagrangian plane is a half-dimensional flat slice through phase space on which the symplectic pairing of position and momentum vanishes. The Lagrangian Grassmannian is the space whose points are all such planes through the origin. As the plane rotates, you can watch it turn relative to a fixed reference plane.

The universal Maslov class is the rule that turns this rotation into an integer. Walk around a closed loop of Lagrangian planes and count how many net half-turns the plane completes against the reference. That signed integer is the Maslov number of the loop.

This idea matters because semiclassical wave optics, the Floer homology grading, and the metaplectic correction in quantisation all use this single turning count to keep track of phase.

Visual Beginner

The diagram shows a Lagrangian plane rotating inside a small symplectic vector space. A reference plane is fixed; an arrow marks the angle, and a turning counter ticks each time the plane lines up with the reference again.

The picture is a mnemonic, not a coordinate proof. It records two things at once: a point in the Lagrangian Grassmannian is a plane, and the Maslov class measures total turning of a loop of planes.

Worked example Beginner

Use the plane with axes called position and momentum. The lines through the origin are exactly the one-dimensional Lagrangian subspaces of this two-dimensional symplectic space.

Pick a moving line by an angle $\theta$ measured from the position axis. So at $\theta =0$ the line is the position axis, at $\theta =\pi /2$ it is the momentum axis, and at $\theta =\pi$ it is the position axis again. The line at angle $\theta +\pi$ is the same line as at $\theta$, because a line is unoriented.

Take a loop that runs $\theta$ from $0$ to $2\pi$. The line passes through every direction twice. Against a fixed reference line, the turning counter advances by 2.

What this tells us: a once-around loop in the space of Lagrangian lines has Maslov number 2, not 1. The doubling is the appearance of the ${\det }^2$ map in disguise.

Check your understanding Beginner

Formal definition Intermediate+

Let $({\mathbb{R}}^{2n},{\omega }_0)$ be the standard symplectic vector space, with ${\omega }_0={\sum }_{i=1}^{n}d{q}_i\wedge d{p}_i$ in coordinates $(q_1,\dots ,q_n,p_1,\dots ,p_n)$ and the sign convention ${\omega }_0(X_H,-)=dH$. A subspace $L\subset {\mathbb{R}}^{2n}$ is Lagrangian when $\dim L=n$ and ${\omega }_0{|}_L=0$. The Lagrangian Grassmannian is the set

$$\Lambda (n)=\{L\subset {\mathbb{R}}^{2n}:\dim L=n,{\omega }_0{|}_L=0\},$$

regarded as a smooth manifold via the homogeneous-space presentation below. The prerequisite unit 05.05.01 supplies the Lagrangian condition; 05.01.01 supplies the underlying linear symplectic structure.

Equip ${\mathbb{R}}^{2n}$ with the compatible complex structure $J_0$ given by $J_0(q,p)=(-p,q)$ and the inner product $⟨(q,p),(q^{\prime },p^{\prime })⟩=q\cdot q^{\prime }+p\cdot p^{\prime }$, so that ${\omega }_0(v,w)=⟨J_{0}v,w⟩$. The unitary group $U(n)$ acts on ${\mathbb{R}}^{2n}={\mathbb{C}}^n$, sending the real subspace ${\mathbb{R}}^n\subset {\mathbb{C}}^n$ to a Lagrangian subspace. The map

$$U(n)\to \Lambda (n),A\mapsto A\cdot {\mathbb{R}}^n$$

is surjective; its stabiliser at ${\mathbb{R}}^n$ is exactly $O(n)$, since a unitary matrix preserves ${\mathbb{R}}^n$ if and only if it is real orthogonal. The induced bijection

$$\Lambda (n)\cong U(n)/O(n)$$

is the canonical diffeomorphism. The dimension is $\dim \Lambda (n)=\dim U(n)-\dim O(n)=n^2-n(n-1)/2=n(n+1)/2$.

The determinant homomorphism $\det :U(n)\to U(1)$ takes the value $\pm 1$ on $O(n)\subset U(n)$, so it does not descend to $\Lambda (n)$. Its square does descend: the map

$$\det {}^2:U(n)/O(n)\to U(1),[A]\mapsto (\det A{)}^2$$

is well-defined and smooth ^{[Arnold 1967]}. The pulled-back generator of $H^1(U(1);\mathbb{Z})\cong \mathbb{Z}$ along this map is the universal Maslov class

$$\mu \in H^1(\Lambda (n);\mathbb{Z}).$$

A counterexample to a common slip: the squared determinant is not the determinant of $A^2$, and the doubling is genuinely necessary because $\det A$ is only defined modulo $\{\pm 1\}$ once $A$ is taken modulo $O(n)$.

A morphism between two Lagrangian Grassmannians arising from symplectic linear isomorphisms $\Phi :({\mathbb{R}}^{2n},{\omega }_0)\to ({\mathbb{R}}^{2n},{\omega }_0)$ is the induced diffeomorphism $L\mapsto \Phi (L)$; this map preserves $\mu$ because the squared-determinant map is functorial in the unitary frame.

Key theorem with proof Intermediate+

Theorem (Arnold 1967). The Lagrangian Grassmannian $\Lambda (n)$ has fundamental group ${\pi }_1(\Lambda (n))=\mathbb{Z}$, and the squared-determinant map ${\det }^2:\Lambda (n)\to U(1)=S^1$ induces an isomorphism

$${\pi }_1(\Lambda (n))\stackrel{\cong }{\to }{\pi }_1(S^1)=\mathbb{Z}.$$

In particular, $H^1(\Lambda (n);\mathbb{Z})\cong \mathbb{Z}$, generated by the pull-back $\mu =({\det }^2{)}^{\ast }d\theta /(2\pi )$, the universal Maslov class. The Maslov index of a Lagrangian loop $\gamma :S^1\to \Lambda (n)$ is the integer

$$\mathrm{Maslov}(\gamma )=\mathrm{deg}(\det {}^2\circ \gamma )\in \mathbb{Z},$$

the winding number of ${\det }^2\circ \gamma$ around $S^1$.

Proof. Apply the long exact sequence of homotopy groups for the fibre bundle $O(n)\to U(n)\to \Lambda (n)$:

$${\pi }_1(O(n))\to {\pi }_1(U(n))\to {\pi }_1(\Lambda (n))\to {\pi }_0(O(n))\to {\pi }_0(U(n)).$$

The classical computations are ${\pi }_1(U(n))=\mathbb{Z}$ (detected by $\det$), ${\pi }_0(U(n))=0$ (connected), ${\pi }_0(O(n))=\mathbb{Z}/2$ (two components $O^{\pm }(n)$), and ${\pi }_1(O(n))=\mathbb{Z}/2$ for $n\ge 2$ with generator the spin loop (and ${\pi }_1(O(1))=0$ since $O(1)$ is discrete). The map ${\pi }_1(O(n))\to {\pi }_1(U(n))$ sends the $\mathbb{Z}/2$ generator to twice the generator of $\mathbb{Z}$, because the spin loop in $O(n)\subset U(n)$ has determinant winding $2$ in $\det :U(n)\to U(1)$. The image in $\mathbb{Z}$ is therefore $2\mathbb{Z}$, and the cokernel of ${\pi }_1(O(n))\to {\pi }_1(U(n))$ is $\mathbb{Z}/2\mathbb{Z}$ at the level of $\det \mathrm{mod}2$.

Together with the surjective connecting map onto $\ker ({\pi }_0(O(n))\to {\pi }_0(U(n)))=\mathbb{Z}/2$, this gives the short exact sequence

$$0\to \mathbb{Z}/(2\mathbb{Z})\stackrel{\det \mathrm{mod}2}{\to }{\pi }_1(\Lambda (n))\to \mathbb{Z}/2\to 0,$$

which splits to $\mathbb{Z}$ via the homomorphism ${\det }^2$. Concretely, ${\det }^2:\Lambda (n)\to S^1$ is well-defined on $U(n)/O(n)$ because $\det (O(n))\subset \{\pm 1\}$, and the induced map on ${\pi }_1$ pairs with the generator $[\gamma ]\in {\pi }_1(\Lambda (n))$ by its $S^1$-winding number. A direct check on the loop ${\gamma }_0(t)=e^{i\pi t/n}\cdot I_n\cdot {\mathbb{R}}^n$, $t\in [0,1]$, gives ${\det }^2\circ {\gamma }_0(t)=e^{2\pi it}$, winding once around $S^1$. So ${\det }_{\ast }^2$ takes a generator to a generator and is an isomorphism. Therefore ${\pi }_1(\Lambda (n))=\mathbb{Z}$, generated by $[{\gamma }_0]$, and $H^1(\Lambda (n);\mathbb{Z})=\mathrm{Hom}({\pi }_1,\mathbb{Z})=\mathbb{Z}$ by Hurewicz and universal coefficients ^{[Arnold 1967]}. The class $\mu =({\det }^2{)}^{\ast }(d\theta /2\pi )$ is the named generator. $\square$

Bridge. The construction here builds toward 05.08.03 (Maslov index of a Lagrangian path or loop in Floer-theoretic contexts), where the universal class is pulled back along path families to assign the integer grading. The defining pattern appears again in 05.08.04 (Conley-Zehnder index), in a sharpened form for paths of symplectic matrices acting on a reference Lagrangian. Putting these together, the foundational insight is that $\Lambda (n)$ is the universal carrier of phase information for Lagrangian objects, and every Maslov-type invariant is a pull-back of the single class $\mu$.

Exercises Intermediate+

Advanced results Master

Fix the standard symplectic space $({\mathbb{R}}^{2n},{\omega }_0)$ and let $\Lambda (n)=U(n)/O(n)$. The squared-determinant fibration ${\det }^2:\Lambda (n)\to S^1$ is a homotopy equivalence onto its image at the level of ${\pi }_1$, and the integral pull-back

$$\mu =(\det {}^2{)}^{\ast }\frac{d\theta }{2\pi }\in H^1(\Lambda (n);\mathbb{Z})$$

is a generator. The Maslov index of a Lagrangian loop $\gamma :S^1\to \Lambda (n)$ is the integer $\mathrm{deg}({\det }^2\circ \gamma )$. The class $\mu$ is invariant under the natural $\mathrm{Sp}(2n,\mathbb{R})$-action on $\Lambda (n)$, because the diagonal unitary frame can be chosen functorially and the squared determinant is conjugation-invariant on $U(n)$ ^{[Arnold-Givental EMS Vol 4 Ch 1]}.

For a path between transverse Lagrangians the Robbin-Salamon refinement gives a half-integer-valued generalisation. Let $\gamma :[0,1]\to \Lambda (n)$ be a smooth path with $\gamma (0),\gamma (1)$ transverse to a fixed reference Lagrangian $L_0$. At each interior crossing $t\in (0,1)$ where $\gamma (t)\cap L_0\ne 0$, the crossing form

$$\Gamma (\gamma ,L_0,t):\gamma (t)\cap L_0\to \mathbb{R},v\mapsto {\omega }_0(v,\frac{d}{dt}\gamma (t)\cdot v)$$

is a quadratic form on the intersection, depending only on the first-order data of the path. A crossing is regular if $\Gamma$ is nondegenerate. The Robbin-Salamon index is

$${\mu }_{\mathrm{RS}}(\gamma ,L_0)=\frac{1}{2}\mathrm{sign}\Gamma (\gamma ,L_0,0)+\sum _{t\in {\mathrm{cross}}^{\circ }}\mathrm{sign}\Gamma (\gamma ,L_0,t)+\frac{1}{2}\mathrm{sign}\Gamma (\gamma ,L_0,1),$$

where each interior crossing is regular and the boundary half-signature terms compensate for non-transversality at the endpoints. For a loop, the boundary terms cancel and ${\mu }_{\mathrm{RS}}$ reduces to the integer Maslov index of the loop ^{[Robbin-Salamon 1993]}.

The Maslov cycle $\Sigma \subset \Lambda (n)$ relative to a reference Lagrangian $L_0$ is the locus of Lagrangians non-transverse to $L_0$. Set ${\Sigma }_k=\{L\in \Lambda (n):\dim (L\cap L_0)=k\}$. Each ${\Sigma }_k$ is a smooth submanifold of codimension $k(k+1)/2$; the top stratum ${\Sigma }_1$ is a codimension-one hypersurface and its Poincaré dual in compactly-supported cohomology represents the universal Maslov class up to a choice of co-orientation ^{[Arnold 1967]}. For a Lagrangian loop $\gamma$ intersecting $\Sigma$ transversally and missing ${\Sigma }_{\ge 2}$, the Maslov index equals the signed count $\gamma \cdot {\Sigma }_1$ of crossings, with sign at each crossing determined by the sign of the crossing form.

The relation to the metaplectic group is the structural source of the doubling. The maximal compact subgroup $U(n)\subset \mathrm{Sp}(2n,\mathbb{R})$ induces ${\pi }_1(\mathrm{Sp}(2n))\cong {\pi }_1(U(n))=\mathbb{Z}$ via $\det$. The metaplectic group $\mathrm{Mp}(2n)\to \mathrm{Sp}(2n)$ is the unique connected double cover; its existence corresponds to the index-$2$ subgroup $2\mathbb{Z}\subset {\pi }_1(\mathrm{Sp}(2n))$. The universal Maslov class on $\Lambda (n)$ is the obstruction to choosing a continuous square root of $\det$ along a Lagrangian loop, and the metaplectic correction in Blattner-Kostant-Sternberg geometric quantisation absorbs this obstruction by tensoring with a square-root half-density bundle.

For a Lagrangian submanifold $L\subset M$ in a symplectic manifold $(M,\omega )$ with compatible almost complex structure $J$, the Maslov class of $L$ is the integral cohomology class ${\mu }_L\in H^1(L;\mathbb{Z})$ obtained by pulling back $\mu \in H^1(\Lambda (n);\mathbb{Z})$ along the Gauss-type map $L\to \Lambda (n)$, $x\mapsto T_{x}L\subset T_{x}M$ (after trivialising $TM{|}_L$ as a complex bundle, using $J$). For monotone Lagrangians (those with ${\mu }_L=\lambda \cdot [\omega {|}_L]$ for some $\lambda \in \mathbb{R}$) Lagrangian Floer homology is well-defined over $\mathbb{Z}/2$ without bubbling corrections, and the integer-graded version exists when ${\mu }_L=0$ (the Maslov-vanishing case used in the original Floer-Arnold conjecture proof for monotone tori in $\mathbb{C}{P}^n$).

Full proof set Master

Proposition. The squared-determinant map ${\det }^2:U(n)/O(n)\to U(1)$ is well-defined and induces an isomorphism ${\pi }_1(U(n)/O(n))\to {\pi }_1(U(1))=\mathbb{Z}$.

Well-definedness: if $A\in U(n)$ and $R\in O(n)$, then $\det (AR)=\det (A)\det (R)$ and $\det (R)\in \{\pm 1\}$, so $\det (AR{)}^2=\det (A{)}^2\det (R{)}^2=\det (A{)}^2$. Hence ${\det }^2$ depends only on the class $[A]\in U(n)/O(n)$. The induced map on ${\pi }_1$ is computed on the explicit generator $[{\gamma }_0]$ of ${\pi }_1(\Lambda (n))$ given by ${\gamma }_0(t)=e^{i\pi t/n}{I}_n\cdot {\mathbb{R}}^n$. The composition ${\det }^2\circ {\gamma }_0(t)=(e^{i\pi t/n}{)}^{2n}=e^{2\pi it}$ winds once around $S^1$, so the induced map sends a generator to a generator. To see that $[{\gamma }_0]$ generates ${\pi }_1(\Lambda (n))$, apply the long exact sequence of $O(n)\to U(n)\to \Lambda (n)$: the boundary $\partial :{\pi }_1(\Lambda (n))\to {\pi }_0(O(n))=\mathbb{Z}/2$ is surjective via $\det \mathrm{mod}2$, and the image of ${\pi }_1(U(n))=\mathbb{Z}$ in ${\pi }_1(\Lambda (n))$ has index $2$, fitting into the short exact sequence $0\to \mathbb{Z}\to {\pi }_1(\Lambda (n))\to \mathbb{Z}/2\to 0$ split by ${\det }^2$. The split extension is $\mathbb{Z}$, with $[{\gamma }_0]$ as generator. $\square$

Proposition. The crossing form $\Gamma (\gamma ,L_0,t)$ is well-defined as a quadratic form on $\gamma (t)\cap L_0$.

Let $\gamma :[0,1]\to \Lambda (n)$ be smooth and let $L_0$ be a reference Lagrangian. Near a crossing time $t=t_0$, choose a Lagrangian complement $L_0^{\prime }\in \Lambda (n)$ transverse to both $L_0$ and $\gamma (t_0)$. Identify a neighbourhood of $L_0$ in $\Lambda (n)$ with the space of self-adjoint linear maps $L_0\to L_0^{\prime }$ via the graph construction: $L\mapsto S_L$ where $L=\mathrm{graph}(S_L)\subset L_0\oplus L_0^{\prime }$. Under this identification, $\gamma (t)$ corresponds to a smooth path $S(t)$ of self-adjoint operators, and $L\cap L_0=\ker (S_L)$. The derivative $\dot{S}(t_0)$ restricted to $\ker (S(t_0))=\gamma (t_0)\cap L_0$ is a self-adjoint endomorphism, hence defines a quadratic form. A computation using ${\omega }_0(v,w)=⟨v,S(t)w⟩$ on the graph identifies $⟨v,\dot{S}(t_0)v⟩={\omega }_0(v,\dot{\gamma }(t_0)\cdot v)$ for $v\in \gamma (t_0)\cap L_0$. The right-hand side is intrinsic to $\gamma$ and $L_0$ (the choice of complement $L_0^{\prime }$ drops out), so $\Gamma$ is well-defined. $\square$

Proposition. For any Lagrangian loop $\gamma :S^1\to \Lambda (n)$, the Robbin-Salamon index equals the integer Maslov index $\mathrm{deg}({\det }^2\circ \gamma )$.

For a loop the boundary terms in the Robbin-Salamon formula vanish (the endpoint conditions are identical), so ${\mu }_{\mathrm{RS}}(\gamma )={\sum }_{t\in {\mathrm{cross}}^{\circ }}\mathrm{sign}\Gamma (\gamma ,L_0,t)$. Perturb $\gamma$ to a loop $\tilde{\gamma }$ intersecting ${\Sigma }_1$ transversally and missing ${\Sigma }_{\ge 2}$; this can be done by Sard since ${\Sigma }_{\ge 2}$ has codimension $\ge 3$ in $\Lambda (n)$. At each crossing $\tilde{\gamma }(t)\cap L_0$ is one-dimensional and the crossing form is a nonzero real number, contributing $\pm 1$ to the Robbin-Salamon sum. On the other hand, the integer Maslov index $\mathrm{deg}({\det }^2\circ \tilde{\gamma })$ is the algebraic intersection number $\tilde{\gamma }\cdot {\Sigma }_1$ with the same signs, because ${\Sigma }_1$ is Poincaré-dual to $\mu$ with a co-orientation matching the crossing-form sign convention ^{[Robbin-Salamon 1993]}. Hence the two integers agree. Homotopy invariance of both sides extends the equality from $\tilde{\gamma }$ to $\gamma$. $\square$

Connections Master

• The Lagrangian-subspace prerequisite is 05.05.01; the linear symplectic framework is 05.01.01; the smooth-manifold language comes from 03.02.01, with differential forms entering through 03.04.02.

• The Floer-theoretic specialisations of the universal Maslov class are 05.08.03 (Maslov index of a loop or path used in Floer homology) and 05.08.04 (Conley-Zehnder index of a path of symplectic matrices), each a pull-back of the universal class along a different naturally arising family.

• The Maslov class controls index theory for Lagrangian-boundary-value problems and serves as the grading source for Lagrangian Floer homology 05.08.02, with the monotonicity condition ${\mu }_L=\lambda \cdot [\omega {|}_L]$ entering the well-definedness of the Floer differential.

• The metaplectic group and its half-density correction appear in geometric quantisation 05.11.02, where the universal Maslov class is the obstruction class controlling polarisation-independence of the quantum Hilbert space.

• In semiclassical asymptotics and Fourier integral operator theory, the Maslov index counts focal-point passages of bicharacteristic curves and enters the WKB phase factor as the Maslov correction ^{[Hörmander 1971]}.

Historical & philosophical context Master

Maslov introduced the index in 1965 in the context of semiclassical asymptotics and the WKB method, as the integer correction to the phase of an oscillatory integral that arises when an integration contour crosses a caustic in the underlying Lagrangian projection ^{[Maslov 1965]}. Arnold gave the geometric interpretation in 1967, identifying the integer as a characteristic class of the Lagrangian Grassmannian and computing ${\pi }_1(\Lambda (n))=\mathbb{Z}$ via the squared determinant map ^{[Arnold 1967]}. Hörmander 1971 incorporated the Maslov class into the symbol calculus of Fourier integral operators, making it the central topological invariant of Lagrangian distributions ^{[Hörmander 1971]}. Guillemin and Sternberg's Geometric Asymptotics (AMS 1977) and Arnold-Givental's EMS Vol 4 Ch 1 are the canonical mid-period expositions of the universal class; Robbin and Salamon 1993 produced the path-version refinement now used throughout symplectic topology and Floer theory ^{[Robbin-Salamon 1993]}.

Bibliography Master

@article{Maslov1965,
  author = {Maslov, V. P.},
  title = {Theory of Perturbations and Asymptotic Methods},
  publisher = {Izdat. Moskov. Univ.},
  address = {Moscow},
  year = {1965},
  note = {French translation: Dunod / Gauthier-Villars, Paris, 1972}
}

@article{Arnold1967Maslov,
  author = {Arnold, V. I.},
  title = {On a characteristic class entering into conditions of quantization},
  journal = {Functional Analysis and its Applications},
  volume = {1},
  number = {1},
  year = {1967},
  pages = {1--14}
}

@article{Hormander1971FIO,
  author = {H{\"o}rmander, L.},
  title = {Fourier integral operators. I},
  journal = {Acta Mathematica},
  volume = {127},
  year = {1971},
  pages = {79--183}
}

@book{GuilleminSternberg1977,
  author = {Guillemin, V. and Sternberg, S.},
  title = {Geometric Asymptotics},
  series = {Mathematical Surveys},
  volume = {14},
  publisher = {American Mathematical Society},
  year = {1977}
}

@article{RobbinSalamon1993,
  author = {Robbin, J. and Salamon, D.},
  title = {The Maslov index for paths},
  journal = {Topology},
  volume = {32},
  number = {4},
  year = {1993},
  pages = {827--844}
}

@incollection{ArnoldGiventalEMS,
  author = {Arnold, V. I. and Givental, A. B.},
  title = {Symplectic Geometry},
  booktitle = {Dynamical Systems IV: Symplectic Geometry and its Applications},
  editor = {Arnol'd, V. I. and Novikov, S. P.},
  series = {Encyclopaedia of Mathematical Sciences},
  volume = {4},
  publisher = {Springer},
  year = {2001},
  edition = {2nd}
}