Ito-Wiener chaos and the Hodge decomposition on an abstract Wiener space

Using the structure of the Boson-Fermion Fock space and an argument taken from [2], we give a new proof of the triviality of the $L^2$ cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [9]. We apply some representation theory of the symmetric group to characterise the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.

💡 Research Summary

The paper presents a new proof of the triviality of the $L^{2}$ de Rham cohomology on an abstract Wiener space, using the algebraic structure of the Boson‑Fermion Fock space and representation theory of the symmetric group. After recalling the classical setting of an abstract Wiener space $(E,H,\mu)$, the author reviews Shigekawa’s results: the Hodge‑Kodaira Laplacian $\Delta_{q}=L+qI$ (with $L=D^{*}D$) satisfies a Weitzenböck formula, and the $L^{2}$ space of $q$‑forms decomposes into exact, co‑exact and harmonic parts, the latter being zero for $q\ge1$.

The core of the new approach is the isometric isomorphism $\Psi$ between the symmetric (Boson) Fock space $F_{s}(H)=\bigoplus_{k\ge0}H^{\odot k}$ and the Gaussian $L^{2}$ space. Extending $\Psi$ to tensor products with exterior powers $H^{\wedge q}$ yields maps $\Psi_{q}:F_{s}(H)\otimes H^{\wedge q}\to L^{2}(E,\mu)\otimes H^{\wedge q}$. Under $\Psi_{q}$ the exterior differential $d_{q}$ and its adjoint $d^{}{q}$ correspond to explicit operators $\breve d{q}$ and $\breve d^{}{q}$ acting on the Fock side. These operators are defined by simple insertion‑deletion rules on symmetric tensors and wedge products, and satisfy $\breve d{q+1}\circ\breve d_{q}=0$, $\breve d^{}_{q}\circ\breve d^{}_{q+1}=0$, and the crucial identity
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