Infinite Dimensional Hamiltonian Reduction

• Infinite-dimensional Hamiltonian reduction is a process that simplifies systems with infinite-dimensional phase spaces by using advanced functional-analytic methods.
• It utilizes normal form theorems, slice theorems, and momentum maps to derive local models that elucidate the structure of moduli spaces in gauge theories and PDEs.
• The framework reveals stratified and singular structures, enabling effective analysis through Dirac, Poisson, and Kuranishi reductions in distributed and gauge field systems.

Infinite-dimensional Hamiltonian reduction refers to the process of systematically reducing the complexity of Hamiltonian systems where the phase space, the symmetry group, or both are infinite-dimensional—typically modeled as Fréchet manifolds or function spaces. Such reductions generalize finite-dimensional symplectic reduction, incorporate gauge symmetries, and, via rigorous functional-analytic and geometric methods, reveal rich stratified structures in moduli spaces crucial for field theories, PDEs, and distributed physical systems (Diez, 2019, Seslija et al., 2012).

1. Normal Form Theorems and Equivariant Maps

Normal form theorems underpin infinite-dimensional Hamiltonian reduction by providing a canonical coordinate structure for smooth, group-equivariant maps between Fréchet manifolds. Let $G$ be a locally convex Lie group acting smoothly on $M,N$, with $f: M\to N$ a $G$-equivariant map. Under functional-analytic regularity conditions on $T_{m_0}f$, one achieves invariant decompositions of coordinate spaces:

$$X = \ker\,T_{m_0}f \oplus \operatorname{Coim}\,T_{m_0}f,\qquad Y = \operatorname{Coker}\,T_{m_0}f \oplus \operatorname{Im}\,T_{m_0}f$$

yielding a local normal form,

$$\psi\circ f\circ\varphi^{-1}(x_1,x_2) = (f_{\text{lin}}(x_2)+f_{\text{sing}}(x_1,x_2))\in \operatorname{Coker}\oplus \operatorname{Im}$$

where $f_{\text{lin}}$ is a topological isomorphism (the linear core), and $f_{\text{sing}}$ vanishes at zero and admits zero derivative at $(0,0)$.

Multiple versions exist for Banach, tame Fréchet, and elliptic operators, each with technical nuances regarding slices and regularity (Theorem 2.2.6, 2.2.9, 2.2.13, 2.2.14 of (Diez, 2019)).

2. Slice Theorem and Kuranishi Reduction

The slice theorem constructs $H$-invariant submanifolds (slices) $S\subset M$ near $m_0$ (with $H=G_{m_0}$) such that the orbit $G\times_H S$ locally describes $M$. For Fréchet manifolds:

• Compact $G$ acting linearly admits a slice at every point
• Proper $G$-actions admit slices under additional conditions

The normal form of $f$ on $S$ leads to local models for moduli spaces $Q=M//f = f^{-1}(0)/G$ as $(s^{-1}(0)/H)$, where $s=f_{\text{sing}}(\cdot,0)$ acts between finite-dimensional $H$-modules, imparting a Kuranishi space structure (Theorem 3.2.3 (Diez, 2019)).

3. Infinite-dimensional Momentum Maps and Marle-Guillemin-Sternberg Normal Form

In the infinite-dimensional setting, a weakly symplectic form $\omega$ on $M$ is closed and fiberwise nondegenerate, though typically not surjective onto $T^*_mM$. A symplectic $G$-action, with momentum map $J:M\to\mathfrak{g}^*$, satisfies

$\iota_{\xi_M}\omega + d\langle J,\xi\rangle = 0$

assuming the pairing above extends to $\mathfrak{g}^*$. Group-valued momentum maps are also considered to encode topological data.

Under sufficient conditions (slice existence, split image for $\omega_b$), the Marle-Guillemin-Sternberg (MGS) normal form in infinite dimensions admits a local symplectic model:

$M \approx G\times_H (\mathfrak{h}^*\oplus E)$

where $$E = \ker\,dJ(m_0)\cap (\operatorname{Im}\,\omega_b)$$ is the symplectic slice, and the reduced form takes:

$$J([g,\eta,e]) = \operatorname{Ad}_g^*(\mu+\eta+J_{\text{sing}}(e)),\qquad \omega = \omega_{G\times H}(\eta) + \omega_E(e)$$

with $J_{\text{sing}}$ quadratic in $e$ (Theorems 4.2.25, 4.2.27 (Diez, 2019)).

4. Stratification and Singular Reduction

The reduced phase space $M_\mu = J^{-1}(\mu)/G_\mu$ decomposes into orbit-type submanifolds $J^{-1}(\mu)\cap M(H)$, each projecting onto quotient strata $M_\mu(H)$ carrying closed and nondegenerate induced forms $\omega_\mu(H)$. When normal forms are “strong” and satisfy approximation, the space is stratified—the lower orbit-type strata lie in the frontier of higher ones, and $G$-invariant Hamiltonian dynamics descends to each stratum.

For $M=T^*Q$, cotangent-bundle reduction yields a finer stratification:

• Primary strata: $P(K) = (J^{-1}(0)\cap (T^*Q)(K))/G$, symplectic manifolds
• Secondary strata (“seams”): $P(K;H) = (J^{-1}(0)\cap (T^*Q)(K)\cap T^*Q(H))/G$, which fiber over $Q(H)$ by symplectic maps of the reduced cotangent fibers

Top stratum $P(e)$ is symplectomorphic to $T^*(Q/G)$, while seams act as coisotropic submanifolds, mediating projection discontinuities (see (Diez, 2019), Sections 5.3–5.4).

5. Dirac Structures, Gauge Symmetry, and Poisson Reduction

Hamiltonian systems with boundary energy flow employ infinite-dimensional Dirac structures, notably the Stokes-Dirac structure. For $Q$ (possibly infinite-dimensional) and $F$ (external flows):

$D_{T^*Q\times F^*} = \{(\#(a,e),(a,e))\}$

with $\#$ a bundle map (Section 2, (Seslija et al., 2012)). On manifolds with boundary, the flow and effort spaces are defined by differential forms, and the canonical pairing by integrals over $M$ and $\partial M$.

The Stokes-Dirac structure is characterized by the system:

$$\begin{cases} f_p = (-1)^r d e_q \ f_q = d e_p \ f_b = \operatorname{tr}\,e_p \end{cases} \quad r = pq+1$$

and corresponding maximally isotropic subspace $D\subset F_{p,q}\times E_{p,q}$.

Gauge symmetries are introduced via abelian group actions $G$, typically by addition of exact forms ($p\mapsto p+d\alpha$), with reduction covered by forming the quotient $Q/G$ and mapping tangent/cotangent bundles accordingly. The reduced Dirac structure inherits coisotropic distributions encoded by the orbits of $G$.

Poisson reduction proceeds by pushing the anchor map through the quotient, yielding reduced operators and, in the case of simplicial complexes, discrete analogues via primal-dual cochains and discrete exterior calculus (Section 6, (Seslija et al., 2012)).

6. Kuranishi Structures and Stratified Moduli Spaces

Equivariant maps $f:M\to N$ with momentum maps lead to local models:

$M(H)\cap J^{-1}(\mu) \approx s^{-1}(0)/H$

for $s$—an obstruction map between finite-dimensional $H$-modules—giving rise to Kuranishi space structure in moduli spaces of solutions, where each stratum is symplectic (Theorem 3.2.3 (Diez, 2019)).

7. Applications in Gauge Theories and Distributed Hamiltonian Systems

Infinite-dimensional Hamiltonian reduction is pivotal in gauge field theory and port-Hamiltonian systems:

• Anti-self-dual connections: Moduli space modeled on Kuranishi charts, singular points can be cones over complex projective space $\mathbb{CP}^2$ (for SU(2), $k=1$ instantons)
• 2D Yang-Mills: Stratified moduli space identified with $\operatorname{Hom}(\pi_1(\Sigma),G)/G$
• Yang-Mills-Higgs: Stratification by stabilizers; singularity structure matches harmonic oscillators with $U(1)$ symmetry

Stokes-Dirac and simplicial Dirac structures offer frameworks for distributed-parameter systems, with port-Hamiltonian equations derived via reduction and exemplified by the vibrating string model, where gauge symmetry corresponds to addition of constants to displacement.

A plausible implication is that direct reduction to Stokes-Dirac structures without intermediary Poisson structures remains open and may generalize to electromagnetism and elastodynamics (Seslija et al., 2012).

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References:

• (Diez, 2019): "Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory"
• (Seslija et al., 2012): "Reduction of Stokes-Dirac structures and gauge symmetry in port-Hamiltonian systems"