S-Symplectomorphism: Structure and Applications

• S-symplectomorphism is a specialized class of symplectic diffeomorphisms that preserve the symplectic form along with additional structural constraints.
• It plays a pivotal role in areas such as surface dynamics, singular quotient theory, and scattering processes by providing canonical mappings and symmetry operations.
• Research in this area utilizes invariant theory, isotopy classifications, and diagrammatic methods to establish graded regular symplectomorphisms and self-duality in integrable systems.

An S-symplectomorphism is a specialized class of symplectic diffeomorphisms characterized by strong structural or algebraic constraints depending on context—ranging from symplectic surface dynamics to graded isomorphisms of singular quotients, canonical geodesic symmetries in Fedosov manifolds, scattering mappings on phase space, and self-duality transformations in integrable systems. The notion subsumes both classical canonical transformations satisfying scatter-theoretic limits and regularity or grading properties for quotient singularities, as well as distinguished geometric symmetries in analytic symplectic manifolds.

1. Foundational Definitions and Main Constructions

Most generally, an S-symplectomorphism is a morphism in the category of symplectic manifolds preserving additional structure or satisfying refined invariance conditions:

• On compact orientable surfaces $(M,\omega)$ carrying a $C^\infty$ Morse map $f:M\to P$ (with $P=\mathbb R$ or $S^1$), the group $$S(f,\omega)=\{h\in\mathrm{Diff}(M):h^*\omega=\omega$$, and $f\circ h=f\}$ consists of diffeomorphisms that preserve both the symplectic form and the function $f$ (Maksymenko, 2017).
• The identity component $S_{\mathrm{id}}(f,\omega)$ comprises isotopies through $f$ and $\omega$-preserving diffeomorphisms.
• A canonical homeomorphism

$$\varphi:\mathcal Z_\omega(f)\to S_{\mathrm{id}}(f,\omega)$$

exists, where $$\mathcal Z_\omega(f)=\{\alpha\in C^\infty(M,\mathbb R): H(\alpha)=0\}$$ (functions constant along Hamiltonian vector field $H$-orbits) parametrizes the identity component via "shift maps".

In symplectic quotient theory, an S-symplectomorphism is a homeomorphism between quotients $M_0=Z/G$ and $M_0'=Z'/G'$ such that the pullback $F^*:C^\infty(M_0')\to C^\infty(M_0)$ is a graded regular Poisson algebra isomorphism respecting the real algebra grading and semialgebraic inequalities (Herbig et al., 2019).

For affine symplectic manifolds $(M,\omega,\nabla)$, an S-type connection is one for which geodesic symmetry $s_p$ near each $p$ is a local symplectomorphism, i.e., $s_p^*\omega=\omega$ in normal coordinates. This is equivalent, for analytic connections, to an infinite system of algebraic curvature identities on the $P_r$ endomorphisms (Bieliavsky et al., 2024).

In classical and quantum scattering, the S-symplectomorphism is the canonical phase space map relating incoming and outgoing asymptotics: $$S=\lim_{t_\pm\to\pm\infty}\left(U^\circ(t_0,t_+) \circ U(t_+, t_-) \circ U^\circ(t_-, t_0)\right),$$ where $U^\circ$ and $U$ denote the free and full time-evolution symplectomorphism, respectively, and $S$ acts via exponentiation of the eikonal Hamiltonian vector field (Kim, 28 Dec 2025).

2. Parametrization, Homotopy Types, and Surface Theory

The parametrization theorem for compact oriented surfaces establishes a bijective correspondence between the functions constant along $H$-orbits and the identity component of $f$-preserving symplectomorphisms. The behavior depends critically on the critical point structure of $f$:

• If $f$ has at least one saddle, $$\varphi: \mathcal Z_\omega(f)\to S_{\mathrm{id}}(f,\omega)$$ is a homeomorphism and both spaces are contractible.
• If $f$ only has extremal (max/min) points, $\varphi$ is an infinite cyclic covering and $S_{\mathrm{id}}(f,\omega)\simeq S^1$.

The Kronrod–Reeb graph construction encodes this structure: $S_{\mathrm{id}}(f,\omega)$ is parametrized by continuous functions on the Reeb graph $K$ of $f$, with global contractibility arising from the smoothness requirement for shift-functions in the presence of trivalent vertices (saddle points) (Maksymenko, 2017).

3. Graded Regular Symplectomorphisms and Singular Quotients

In symplectic quotient theory, S-symplectomorphism means a graded regular Poisson algebra isomorphism between algebras of regular functions on symplectic quotients $M_0(A)$ and $M_0(B)$, arising from faithful torus and circle representations with Type II$_k$ weight matrices. The construction proceeds via invariant theory, Seshadri sections, and explicit symplectic embeddings preserving the shell and grading (Herbig et al., 2019).

• The main classification asserts that for these Type II$_k$ quotients, such graded regular symplectomorphisms exist and can be constructed explicitly.
• The counterexamples demonstrate the rigidity: Hilbert series or complex Poisson algebra isomorphisms do not guarantee a real graded regular symplectomorphism; preservation of semialgebraic inequalities is essential.

Quotient Type	Existence of S-symplectomorphism	Algebraic Invariant
Type II$_k$ (faithful)	Yes (explicit classification)	$(k, \alpha(A), \beta(A))$
Hilbert series match only	No, in general	Counterexample via semialgebraic inequality

4. S-Type Connections and Geodesic Symmetries

S-symplectomorphisms as geodesic symmetries generalize the local symmetry of Riemannian spaces to the symplectic category. For a Fedosov manifold $(M,\omega,\nabla)$, S-type connections require that for every point $p$, the geodesic involution $s_p$ is a (local) symplectomorphism, equivalently,

$s_p^*\omega=\omega$

in a symmetric neighborhood. This condition is characterized by a recursive system of curvature identities: $$P_r(p)\in\mathfrak{sp}(T_pM,\omega_p), \quad (P_r)^T=-P_r, \quad r=3,5,7,\dots$$ with $P_r$ built from the Jacobi endomorphism and covariant derivatives (Bieliavsky et al., 2024). All Ricci-type analytic connections and locally symmetric spaces satisfy these S-type conditions, but non-symmetric analytic examples also exist, confirming the condition is strictly weaker than local symmetricity.

5. S-Symplectomorphism in Scattering Theory and Phase Space

In Hamiltonian mechanics, time-evolution is a family of symplectomorphisms, and in scattering, the classical S-symplectomorphism maps incoming to outgoing phase space data. The Magnus expansion expresses the classical eikonal generator,

$$\chi =\int V(t_1)dt_1 + \frac{1}{2}\int_{t_1>t_2}\{V(t_1),V(t_2)\}dt_1dt_2 + \cdots,$$

so that the S-symplectomorphism acts on observables as $S^*=\exp(X_\chi)$, where $X_\chi[g]=\{\chi,g\}$ (Kim, 28 Dec 2025).

In the quantum setting, the adjoint action of the S-matrix translates to a fuzzy symplectic diffeomorphism via the star product, with the quantum eikonal $\chi^\star$ built from deformed Poisson brackets. The classical map arises precisely when $\hbar\to 0$.

Diagrammatic techniques (Penrose arrow notation, Feynman–Magnus graphs) enable efficient computation of higher-order corrections and explicit visualization of tree and loop contributions to $\chi$ and $\chi^\star$.

6. Self-Duality and Mapping Class S-Symplectomorphism

Within integrable systems, the Ruijsenaars self-duality map is realized as an S-symplectomorphism, explicitly as the action of the mapping class generator $S$ on the quasi-Hamiltonian reduction of $SU(n)\times SU(n)$: $S_D(A,B) = (B^{-1},\,BAB^{-1}).$ Under reduction, this descends to a symplectomorphism $S_P$ of $P(\mu_0)$, which is symplectomorphic to $\mathbb{C}P^{n-1}$ with Fubini-Study form. The mapping class $S$ exchanges the two commuting toric moment maps (particle-positions and action-variables), preserves the symplectic structure, and satisfies $S^2=(ST)^3=Q$, with $Q$ acting trivially on the reduced space (Feher et al., 2012).

Geometrically, this describes the duality symmetry of the moduli space of flat $SU(n)$-connections on a one-holed torus, where $S$ maps holonomies along the fundamental cycles and realizes an involutive Dehn twist duality.

7. Sobolev Geometry of the Symplectomorphism Group

The group of Sobolev $H^s$ symplectomorphisms, $\mathrm{Symp}^s(M,\omega)$, forms an infinite-dimensional Hilbert Lie group for $s>n+1$, with Lie algebra elements satisfying $\mathcal{L}_v\omega=0$. Equipped with the right-invariant $H^1$ metric,

$$\langle u,v\rangle_{H^1} =\int_M \left(g(u,v)+\alpha^2 g(\nabla u,\nabla v)\right) d\mu_g,$$

the corresponding geodesic equation takes Euler-Poincaré form. The exponential map is a nonlinear Fredholm operator of index zero for $s>n+3$, and the group is geodesically complete. Conjugate points arise, for example, on $\mathbb{C}P^n$ along the isometric $H^1$ geodesics, revealing rich global geometry in the S-symplectomorphism context (Benn et al., 2017).

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S-symplectomorphisms represent deep intersections of dynamical systems, geometric analysis, singular quotient theory, and quantum-classical correspondences. Their rigid classification and universal presence in both symplectic geometry and scattering theory underscore their centrality as symmetry objects in modern mathematical physics.