Dolbeault cohomology of a loop space
This paper studies complex analysis on loop spaces. The idea is that the study of Dolbeault cohomology of the loop space \(LM\) of a complex manifold \(M\) will give information about the manifold. Here the loop spaces \(L\mathbb{P}_1\) of the Riemann sphere \(\mathbb{P}_1\) are considered; with different requirements of smoothness \(C^k\) or Sobolev class \(W^{k,p}\) of the loops. The Dolbeault cohomology groups \(H^{0,1}(L\mathbb{P}_1)\) of these spaces are computed. Let \(\mathcal{F}\) be the space of holomorphic functions \(F:\mathbb{C}\times L\mathbb{C}\to\mathbb{C}\) which satisfy (1) \(F(\frac{\zeta}{\lambda},\lambda^2y)=O(\lambda^2)\) as \(\mathbb{C}\ni\lambda\to 0\); (2) \(F(\zeta,x+y)=F(\zeta,x)+F(\zeta,y)\) if \(\text{supp}(x)\cap \text{supp}(y)=\emptyset\); (3) \(F(\zeta,y+const)=F(\zeta,y)\). Theorem 0.1 states \(H^{0,1}(L\mathbb{P}_1)\approx \mathbb{C}\oplus\mathcal{F}\). The isomorphism is topological, and equivariant with respect to the obvious actions of the \(C^k\) diffeomorphism group of \(S^1\). The topology on the cohomology groups is induced by the compact-\(C^{\infty}\) topology on the space of smooth \((r,p)\)-forms. The cohomology groups \(H^{0,1}(L\mathbb{P}_1)\) are acted on by the group \(G=\text{PSL}(2,\mathbb{C})\) of holomorphic automorphisms of \(\mathbb{P}_1\) and theorems 0.2--0.4 are on the structure of \(H^{0,1}(L\mathbb{P}_1)\) as a \(G\)-module. In section 2, an action of \(G\) on \(\mathcal{F}\) is introduced, making \(\mathcal{F}\) a holomorphic \(G\)-module. The \(n\)-th isotypical subspace \(\mathcal{F}^n\) is the subspace of \((n+1)\)-posthomogeneous functions. Theorem 0.2 calculates \(H^{0,1}(L_{\infty}\mathbb{P}_1)\): If \(n\geq 1\), the \(n\)-th isotypical subspace of \(H^{0,1}(L_{\infty}\mathbb{P}_1)\) is isomorphic to the space \(\mathcal{F}^n\), which is spanned by functions of the form \(F(\zeta,y)=\zeta^{\nu} \langle\Phi, \prod_{j=0}^{n}y^{(d_j)}\rangle\), where \(\Phi\) is a distribution on \(S^1\), \(y^d\) is the 0'th derivative, \(d_j\geq d_0=1\) and \(0\leq\nu\leq 2n\). The isomorphism is a \(G\)-map. The fixed point sets are calculated in Theorem 0.3: \(H^{0,1}(L\mathbb{P}_1)^G\) is isomorphic to the dual space \(C^{k-1}(S^1)^*\) (resp. \(W^{k-1,p}(S^1)^*\)) where the dual spaces have the compact-open topology. For low regularity loop spaces, \(L_{1,p}\mathbb{P}_1\) a very concrete representation is given via \(S^1\times \text{Diff}(S^1)\) equivariant isomorphisms. It follows that each irreducible representation of \(S^1\) occurs in \(H^{0,1}(L_{1,p}\mathbb{P}_1)\) with the same finite multiplicity for higher regularity loops -- at least \(C^1\) -- each representation occurs with infinite multiplicity in \(H^{0,1}(L\mathbb{P}_1)\) and hence the authors conclude that for general loop spaces, one should study Dolbeault groups as \(\text{Diff}(S^1)\) representations rather than \(S^1\)-representations.
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