ANALYTIC NON-ABELIAN HODGE THEORY

J.P.PRIDHAM

Abstract. The pro-algebraic fundamental group can be understood as a completion with respect to finite-dimensional non-commutative algebras. We introduce finer invariants by looking at completions with respect to Banach and C^{∗}-algebras, from which we can recover analytic and topological representation spaces, respectively. For a compact Kähler manifold, the C^{∗}-completion also gives the natural setting for non-abelian Hodge theory; it has a pure Hodge structure, in the form of a pro-C^{∗}-dynamical system. Its representations are pluriharmonic local systems in Hilbert spaces, and we study their cohomology, giving a principle of two types, and splittings of the Hodge and twistor structures.

Introduction

In [Sim5] and [Sim6], Simpson defined the coarse Betti, de Rham and Dolbeault moduli spaces of a smooth projective complex variety. These are all algebraic spaces, with a complex analytic isomorphism between the Betti and de Rham moduli spaces. The non-abelian Hodge Theorem of [Sim6, Theorem 7.18] is a homeomorphism between the de Rham and Dolbeault moduli spaces, the key to which was the correspondence between semisimple local systems, pluriharmonic bundles and Higgs bundles.

The reductive pro-algebraic fundamental group π_{1}(X, x)^{red} introduced in [Sim4] encapsulates, in a single object, all the information about the category of semisimple local systems. When X is a compact Kähler manifold, the group scheme π_{1}(X, x)^{red} also has a pure Hodge structure in the form of a discrete circle action, and a description in terms of Higgs bundles.

However, it has long been realised that the reductive pro-algebraic fundamental group is slightly inadequate. From it we can recover the points of the Betti moduli space, and from the full pro-algebraic fundamental group we can even recover their infinitesimal neighbourhoods, but in general these groups convey no information about how the neighbourhoods glue together. A further source of dissatisfaction is the discontinuity of the circle action on π_{1}(X, x)^{red}, since it is continuous on moduli spaces.

The key idea behind this paper is that we can produce finer and more satisfactory invariants by looking at representations with analytic structure, an approach previously considered by Sullivan in [Sul, §9]. The group scheme π_{1}(X, x)^{red} can be recovered from representations in finite-dimensional matrix algebras, but the Riemann–Hilbert correspondence between Betti and de Rham moduli holds with coefficients in any Banach algebra. We accordingly construct Betti, de Rham and Dolbeault moduli functors on Banach algebras, and recover the analytic moduli spaces from these functors. The framed Betti and de Rham functors are represented by a Fréchet algebra which we regard as the analytic completion of ℝ[π_{1}(X, x)].

This work was supported by the Engineering and Physical Sciences Research Council [grant numbers EP/F043570/1 and EP/I004130/1].

To understand the topological structure underlying these analytic spaces, we then restrict to C^{∗}-algebras rather than Banach algebras. There are notions of unitary and pluriharmonic representations with coefficients in any C^{∗}-algebra, and the homeomorphism of moduli spaces above extends to an isomorphism between the semisimple de Rham functor and the polystable Dolbeault functor on polynormal C^{∗}-algebras, via isomorphisms with the pluriharmonic functor. The C^{∗}-algebra of bounded operators gives us a notion of pluriharmonic local systems in Hilbert spaces, and there is a form of Hodge decomposition for these local systems.

Lurking behind these comparisons is the twistor moduli functor on multiplicatively convex Fréchet 𝒪_{ℙ^{1}}^{hol}-algebras. Its fibres at ±i ∈ ℙ^{1} are the Dolbeault functor and its conjugate, while all other fibres are isomorphic to the de Rham functor. The Deligne– Hitchin twistor space can be recovered as an analytic space from the twistor moduli functor, and pluriharmonic torsors give a splitting of the twistor moduli functor on C^{∗}- algebras over C(ℙ^{1}(ℂ)). Twistor cochains then admit a Hodge decomposition on pulling back along the Hopf fibration SU_{2} → ℙ^{1}(ℂ), and a continuous circle action serves to promote twistor structures to Hodge structures.

The structure of the paper is as follows. In §1, we cover some background material on pro-representability. Proposition 1.19 then establishes a topological analogue of Tannaka duality for polynormal C^{∗}-algebras and unitary representations, while Lemma 1.28 gives a similar result for non-unitary representations.

In §2, we introduce the framed Betti and de Rham functors R_{X,x}^{B}, R_{X,x}^{dR} on Banach algebras for any manifold X. In Proposition 2.3, we establish an isomorphism R_{X,x}^{B}(A) ≅ R_{X,x}^{dR}(A) for any pro-Banach algebra A. We can even recover the analytic structure of moduli spaces of G-bundles from these symmetric monoidal functors (Remark 2.6). Proposition 2.10 then shows that R_{X,x}^{B} is represented by a Fréchet algebra completion E_{X,x}^{B} of ℝ[π_{1}(X, x)]. This is a Fréchet bialgebra from which π_{1}(X, x) can be recovered (Lemma 2.11).

In Definition 2.18, we introduce a symmetric monoidal functor R_{X,x}^{J} on C^{∗}-algebras, parametrising pluriharmonic bundles on a compact Kähler manifold X. This is representable by a pro-C^{∗}-bialgebra E_{X,x}^{J} (Proposition 2.22 and Lemma 2.24). There are also a symmetric monoidal functor R_{X,x}^{Dol} on Banach algebras associated to Dolbeault moduli, which is seldom representable, and a harmonic functor extending the definition of R_{X,x}^{J} to all Riemannian manifolds X, but with substantial loss of functoriality.

In §3 we establish relations between the various functors. We can recover the topology on moduli spaces of semisimple representations from E_{X,x}^{J} (Theorem 3.6). Proposition 3.8 then gives a Tannakian description of the polynormal completion of E_{X,x}^{J}, while Corollary 3.11 gives a simple characterisation of continuous morphisms from E_{X,x}^{J} to polynormal C^{∗}-algebras. §3.3 then gives similar results for R_{X,x}^{Dol}. Lemma 3.19 shows that grouplike elements G((E_{X,x}^{J})^{ab}) of the abelianisation of E_{X,x}^{J} are just H_{1}(X, ℤ⊕ℝ), with consequences for complex tori.

There is a continuous circle action on E_{X,x}^{J}, so it is a pro-C^{∗} dynamical system (Proposition 3.31). This allows us to regard E_{X,x}^{J} as an analytic non-abelian Hodge structure of weight 0 (Remark 3.32). In Example 3.33, we see that the circle action on G((E_{X,x}^{J})^{ab}) is just given by the Hodge structure on H^{1}(X, ℝ). Proposition 3.38 then characterises pure Hilbert variations of Hodge structure as representations of E_{X,x}^{J} ⋊S^{1}.

§4 is concerned with Hilbert space representations of E_{X,x}^{J}, which correspond to pluriharmonic local systems 𝕍 in Hilbert spaces. We can identify reduced cohomology H̄^{∗}(X, 𝕍) with the space of smooth 𝕍-valued harmonic forms, as well as establishing the principle of two types and a formality results (§4.2.1). There are analogous, but weaker, results for non-reduced cohomology (§4.2.2). The same is true of direct limits of Hilbert space representations, and Corollary 4.22 shows that the universal such is the continuous dual (E_{X,x}^{J})′ (which can be regarded as the predual of the W^{∗}-envelope of E_{X,x}^{J}).

In §5, these results are extended to show that the natural twistor structure on H̄^{n}(X, 𝕍) is pure of weight n (Corollary 5.5), with a weaker result for non-reduced cohomology (Proposition 5.6). If 𝔼 is the local system associated to the π_{1}(X, x)- representation E_{X,x}^{J}, then Proposition 5.10 shows that this twistor structure can be enhanced to a form of analytic Hodge filtration on the de Rham algebra A^{•}(X, 𝔼′). In §5.4, we re-interpret splittings of twistor structures and Archimedean monodromy in terms of the Hopf fibration.

Finally, in §6, we introduce a whole twistor family of framed moduli functors on multiplicatively convex Fréchet algebras over ℙ^{1}(ℂ), from which we can recover both de Rham and Dolbeault functors. The coarse quotient of the framed twistor space is just the Deligne–Hitchin twistor space (Remark 6.17). The twistor family carries a natural involution σ, and we show in Proposition 6.13 that σ-equivariant sections of the framed twistor space are just framed pluriharmonic bundles. Theorem 6.20, Proposition 6.22 and Corollary 6.24 then give analogues of Theorem 3.6, Proposition 3.8 and Corollary 3.11 in the twistor setting, describing twistors with C^{∗}-algebra coefficients and the topology of the twistor space.

I would like to thank Carlos Simpson for originally posing the problem of finding finer invariants, and for helpful discussions. I would also like to thank the anonymous referee for spotting erroneous statements about Fréchet algebras.

Notation. We will use k to denote either of the fields ℝ, ℂ.

Definition 0.1. Given a k-Hilbert space H, write L(H) for the space of k-linear bounded operators on H, with the norm topology.

Definition 0.2. Given topological spaces X, Y, we write C(X, Y) for the set of continuous maps from X to Y.

Definition 0.3. Given a group G acting on a set X, write [X/G] for the groupoid with objects X and morphisms X × G, where the source of (x, g) is x and the target is xg. Composition of morphisms is given by (xg, h) ∘ (x, g) = (x, gh).

Definition 0.4. Given a group G acting on sets S, T, write S ×_{G} T for the quotient of S × T by the G-action g(s, t) = (gs, g^{−1}t).

Contents

Introduction 1 Notation 3 1. Pro-representability of functors on unital Banach algebras and C^{∗}-algebras 4 1.1. Banach algebras 6 1.2. C^{∗}-algebras 6 1.3. Representations and polynormal C^{∗}-algebras 7 1.4. The category of C^{∗}-algebras with completely bounded morphisms 8 2. The Betti, de Rham and harmonic functors on Banach algebras 10 2.1. The Riemann–Hilbert correspondence 10 2.2. Representability of the de Rham functor 12 2.3. The pluriharmonic functor 14 2.4. Higgs bundles 17 2.5. The harmonic functor 18 3. Analytic non-abelian Hodge theorems 19 3.1. The de Rham projection 19 3.2. Residually finite-dimensional completion, products and complex tori 23 3.3. The Dolbeault projection 25 3.4. Circle actions and C^{∗}-dynamical systems 26 4. Hodge decompositions on cohomology 29 4.1. Sobolev spaces 29 4.2. The Hodge decomposition and cohomology 32 4.3. The W^{∗}-enveloping algebra 33 5. Twistor and Hodge structures on cochains, and SU_{2} 35 5.1. Preliminaries on non-abelian twistor and Hodge filtrations 35 5.2. The twistor structure on cochains 36 5.3. The analytic Hodge filtration on cochains 38 5.4. SU_{2} 38 6. The twistor family of moduli functors 41 6.1. Pro-Banach algebras on projective space 41 6.2. The twistor functors 42 6.3. Universality and σ-invariant sections 43 6.4. Topological twistor representation spaces 45 Bibliography 46 References 46

1. Pro-representability of functors on unital Banach algebras and C^{∗}-algebras

Definition 1.1. Given a functor F: C → Set, an object A ∈ C and an element ξ ∈ F (A), we follow [Gro, §A.3] in saying that the pair (A, ξ) is minimal if for any pair (A′, ξ′) and any strict monomorphism f: A′ → A with F (f)(ξ′) = ξ, f must be an isomorphism.

We say that a pair (A′′, ξ′′) is dominated by a minimal pair if there exists a minimal pair (A, ξ) and a morphism g: A → A′′ with F (g)(ξ) = ξ′′.

Definition 1.2. As in [Gro, §A.3], we say that a functor F: C → Set on a category C containing all finite limits is left exact if it preserves all finite limits. This is equivalent to preserving finite products and equalisers, or to preserving fibre products and the final object.

Lemma 1.3. Let C be a category containing finite limits, and take a left exact functor F: C → Set. Assume that for any cofiltered inverse system {A_{i}}_{i} of strict subobjects of any object A ∈ C, the limit \underleftarrow{lim}_{i} A_{i} exists, and that the map

F (\underleftarrow{lim}_{i} A_{i}) → \underleftarrow{lim}_{i} F (A_{i})

is an isomorphism.

Then every pair (A, ξ ∈ F (A)) is dominated by a minimal pair.

Proof. Given the pair (A, ξ), let I be the full subcategory of the overcategory C ↓ A consisting of strict monomorphisms B → A for which ξ lifts to F (B). Note that this lift must be unique by the monomorphism property. If f ∘ g is a strict monomorphism, then so is g, which implies that all morphisms in I must be strict monomorphisms in C. Moreover, left-exactness of F guarantees that I is closed under the fibre product ×_{A}. the monomorphism properties imply that parallel arrows in I are equal, so I is a cofiltered category.

By hypothesis, the limit L := \underleftarrow{lim}_{B∈I} B exists in C. It is necessarily a strict subobject of A, since it is the limit of all parallel maps sourced at A and equalised by some B ∈ I. The unique lifts of ξ to each F (B) define an element of

\underleftarrow{lim}_{B∈I} F (B),

so by hypothesis we have a corresponding element η ∈ F (L). Therefore L is an object of I and is in fact the initial object of I. The pair (L, η) is therefore minimal, and dominates (A, ξ) as required. □

Definition 1.4. Recall from [Gro, §A.2] that a pro-object X ∈ pro(C) is said to be strict if it is isomorphic to a pro-object of the form {X_{i}}, where each map X_{i} → X_{j} is an epimorphism.

A functor F: C → Set is said to be strictly pro-representable if there exists a strict pro-object X with F ≅ Hom_{pro(C)}(X, −).

Proposition 1.5. Let C be a category containing finite limits and limits of cofiltered inverse systems of strict subobjects. Then a functor F: C → Set is strictly prorepresentable if and only if

(1) F is left exact;

(2) F preserves limits of cofiltered inverse systems of strict subobjects.

Proof. If F satisfies the conditions, then it is left-exact, and by Lemma 1.3 every pair is dominated by a minimal pair. It therefore satisfies the conditions of [Gro, Proposition A.3.1], so is strictly pro-representable.

Conversely, every pro-representable functor F is left-exact, so we need only show that the second condition holds. Write F = \underrightarrow{lim}_{α} Hom(R_{α}, −) for a strict inverse system {R_{α}}_{α}, and take a cofiltered inverse system {A_{i}}_{i} of strict subobjects of some object A ∈ C.

Given an element x ∈ \underleftarrow{lim}_{i} F (A_{i}) with image x_{i} in F (A_{i}), by definition there exist objects R_{α_{i}} and maps y_{i}: R_{α_{i}} → A_{i} lifting x_{i}. Now fix i; the liftings are compatible in the sense that for j > i (increasing α_{j} if necessary), there is a commutative diagram

R_{α_{j}} ⟶^{y_{j}} A_{j}
↓  ↓
R_{α_{i}} ⟶^{y_{i}} A_{i}.

Since A_{j} → A_{i} is a strict monomorphism and R_{α_{j}} → R_{α_{i}} an epimorphism, y_{i} must lift to a map y_{ij}: R_{α_{i}} → A_{j}. Since A_{j} → A_{i} is a monomorphism, the lifting y_{ij} is unique. Considering all j > i together, this gives us a unique map x̃: R_{α_{i}} → \underleftarrow{lim}_{j} A_{j}, which gives rise to a unique pre-image x̃ ∈ F (\underleftarrow{lim}_{j} A_{j}), as required. □

1.1. Banach algebras.

Definition 1.6. Write BanAlg_{k} for the category of unital (not necessarily commutative) Banach algebras over k, with bounded morphisms.

Proposition 1.7. Take a functor F: BanAlg_{k} → Set such that

(1) F preserves all finite limits (equivalently: preserves fibre products and the final object),

(2) F preserves monomorphisms (i.e. maps closed subalgebras to subsets), and (3) for all inverse systems S of closed subalgebras, the map

F ( ⋂_{s∈S} A_{s}) → ⋂_{s∈S} F (A_{s})

is an isomorphism. Then F is strictly pro-representable.

Proof. Given a Banach algebra B, every strict subobject of B is a closed subalgebra. For any cofiltered inverse system {A_{i}}_{i} of strict subobjects of any object B, the limit \underleftarrow{lim}_{i} A_{i} exists in BanAlg_{k} and is given by ⋂_{i} A_{i}. The result now follows from Proposition 1.5. □

1.2. C^{∗}-algebras.

Definition 1.8. Write C^{∗}Alg_{k} for the category of unital (not necessarily commutative) C^{∗}-algebras over k, with bounded involutive morphisms.

Explicitly, a complex C^{∗}-algebra is a complex Banach algebra equipped with an antilinear involution ∗ satisfying (ab)^{∗} = b^{∗}a^{∗} and ∥a^{∗}a∥ = ∥a∥^{2} for all a ∈ A.

A real C^{∗}-algebra is a real Banach algebra equipped with a linear involution ∗ satisfying the conditions above, and having the additional property that 1 + a^{∗}a is invertible for all a ∈ A.

A Banach ∗-algebra over k is a C^{∗}-algebra if and only if it is isometrically ∗-isomorphic to a self-adjoint norm-closed algebra of bounded operators on a Hilbert k-space; for k = ℝ, this is Ingelstam’s Theorem [Goo, 8.2 and 15.3].

Proposition 1.9. Take a functor F: C^{∗}Alg_{k} → Set such that

(1) F preserves all finite limits (equivalently: preserves fibre products and the final object),

(2) F preserves monomorphisms (i.e. maps C^{∗}-subalgebras to subsets), and

(3) for all inverse systems S of nested C^{∗}-subalgebras, the map

F ( ⋂_{s∈S} A_{s}) → ⋂_{s∈S} F (A_{s})

is an isomorphism. Then F is strictly pro-representable.

Proof. The proof of Proposition 1.7 carries over. □

Lemma 1.10. Every complex C^{∗}-algebra becomes a real C^{∗}-algebra by forgetting the multiplication by ℂ.

Proof. We just need to show that 1+a^{∗}a is invertible for all a ∈ A. Now, x ↦ (1+|x|)^{−1} is a continuous function on ℝ and a^{∗}a is positive self-adjoint, so the continuous functional calculus implies that (1 + a^{∗}a)^{−1} ∈ A, as required. □

Lemma 1.11. The category C^{∗}Alg_{ℝ} is equivalent to the category of pairs (A, τ), for A ∈ C^{∗}Alg_{ℂ} and an involution τ: A → A satisfying

(1) τ(ab) = τ(a)τ(b),

(2) τ(a)^{∗} = τ(a^{∗}), and

(3) τ(λ) = λ̄ for λ ∈ ℂ.

Proof. Given B ∈ C^{∗}Alg_{ℝ}, set A := B ⊗_{ℝ} ℂ; this is a complex C^{∗}-algebra, with involution (b ⊗ λ)^{∗} = b^{∗} ⊗ λ̄. The involution τ is then given by complex conjugation, with τ(b⊗λ) = b⊗λ̄. For the quasi-inverse construction, we send a pair (A, τ) to the algebra A^{τ} of τ-invariants. That this is a real C^{∗}-algebra follows from Lemma 1.10.

To see that these are quasi-inverse functors, first note that (B ⊗_{ℝ} ℂ)^{τ} = B. Next, observe that because τ is antilinear, we can write A = A^{τ} ⊕ iA^{τ} ≅ A^{τ} ⊗_{ℝ} ℂ for all pairs (A, τ) as above. □

Definition 1.12. For a complex (resp. real) ∗-algebra A, write U(A) for the group of unitary (resp. orthogonal) elements

{a ∈ A : a^{∗}a = aa^{∗} = 1}.

Write 𝔲(A) for the Lie algebra of anti-self-adjoint elements

{a ∈ A : a^{∗} + a = 0},

and write S(A) for the self-adjoint elements of A, noting that 𝔲(A) = iS(A) when A is complex.

1.3. Representations and polynormal C^{∗}-algebras.

1.3.1. Representation spaces. Fix a real unital C^{∗}-algebra A.

Definition 1.13. Write Rep_{n}^{∗}(A) for the space of unital continuous ∗-homomorphisms ρ: A → Mat_{n}(ℂ) equipped with the topology of pointwise convergence. Write Irr_{n}^{∗}(A) ⊂ Rep_{n}^{∗}(A) for the subspace of irreducible representations.

Definition 1.14. Define FDHilb to be the groupoid of complex finite-dimensional Hilbert spaces and unitary isomorphisms.

Definition 1.15. Write FD^{∗}Rep(A) for the groupoid of pairs (V, ρ) for V ∈ FDHilb and ρ: A → End(V) a unital continuous ∗-homomorphism. Morphisms are given by unitary isomorphisms intertwining representations. The set of objects of FD^{∗}Rep(A) is given the topology of pointwise convergence.

Definition 1.16. Define A_{PN} to be the ring of Gal(ℂ/ℝ)-equivariant continuous additive endomorphisms of the fibre functor from FD^{∗}Rep(A) to the category of vector spaces. Explicitly, a ∈ A_{PN} associates to each pair (V, ρ) an element a(V, ρ) ∈ End(V), subject to the conditions:

(1) For any unitary isomorphism u: V → W, we have a(W, uρu^{−1}) = ua(V, ρ)u^{−1}.

(2) For any (V_{1}, ρ_{1}), (V_{2}, ρ_{2}) ∈ FD^{∗}Rep(A), we have a(V_{1} ⊕V_{2}, ρ_{1} ⊕ρ_{2}) = a(V_{1}, ρ_{1})⊕ a(V_{2}, ρ_{2}).

(3) The maps a: Rep_{n}^{∗}(A) → Mat_{n}(ℂ) are continuous and Gal(ℂ/ℝ)-equivariant.

Lemma 1.17. The ring A_{PN} has the structure of a pro-C^{∗}-algebra over A.

Proof. We may describe A_{PN} as the categorical limit of a diagram of ∗-homomorphisms between the C^{∗}-algebras C(Rep_{n}^{∗}(A), Mat_{n}(ℝ)), thus making it into a pro-C^{∗}-algebra. The ∗-homomorphism A → A_{PN} is given by mapping a to the transformation a(V, ρ) = ρ(a). □

1.3.2. Polynormal C^{∗}-algebras.

Definition 1.18. Recall from [Pea] that an algebra A is said to be n-normal if for all a_{1}, …, a_{2n} ∈ A, we have

∑_{σ∈S_{2n}} sgn(σ)a_{σ(1)}a_{σ(2)} ⋯ a_{σ(2n)} = 0.

Call an algebra polynormal if it is n-normal for some n.

By the Amitsur–Levitzki theorem, the algebra of n × n-matrices over a commutative ring is n-normal. Also note that an n-normal algebra is k-normal for all k ≥ n.

Note that by restricting to n-dimensional representations, we get that for any real C^{∗}-algebra A, the ring A_{PN} of Definition 1.16 is an inverse limit A_{PN} = \underleftarrow{lim}_{n} A_{PN,n} of n-normal C^{∗}-algebras.

We now have a result combining aspects of Tannaka and Takesaki duality:

Proposition 1.19. If A is a polynormal unital C^{∗}-algebra, then the morphism A → A_{PN} of Lemma 1.17 is an isomorphism.

Proof. Since A is N-normal for some integer N, [Pea, §3] implies that A is of type I, with all complex irreducible representations of A having dimension at most N.

For a sufficiently large cardinal α, [Bic] characterises the W^{∗}-envelope A′′⊗ℂ of A⊗ℂ as the ring defined analogously to A_{PN} by replacing “continuous” with “bounded” and “finite-dimensional” with “of dimension at most α”. Since all irreducible representations are at most N-dimensional, the direct integral decomposition of A-representations gives us an injective map A_{PN} → A′′, with boundedness following because any a ∈ A_{PN} is bounded on ∐_{k≤N} Rep_{k}^{∗}(A).

Now, [AS] defines a ring A_{c} ⊂ A′′ to consist of those b for which the functions b, b^{∗}b, bb^{∗} are weakly ∗-continuous on the space P (A) of pure states of A. Since all irreducible representations arise as subrepresentations of N-dimensional representations, continuity on Rep_{N}^{∗}(A) suffices to give continuity on P (A), so the inclusion A_{PN} → A_{c} is an isomorphism.

By [BD], the spectrum of A is Hausdorff, and since A is type I, [AS] then observes that A is perfect, which means that the inclusion A → A_{c} is in fact an isomorphism. Thus the map A → A_{PN} is an isomorphism. □

1.4. The category of C^{∗}-algebras with completely bounded morphisms.

1.4.1. Basic properties.

Lemma 1.20. If A is a C^{∗}-algebra and f: A → B a morphism of Banach algebras, then the image of f has the natural structure of a C^{∗}-algebra, with f: A → Im (f) becoming a C^{∗}-homomorphism.

Proof. The kernel of f is a closed two-sided ideal. Thus by [Seg, Theorem 3], A/ ker f is a C^{∗}-algebra, as required. □

Definition 1.21. Recall that a homomorphism π: A → B of Banach algebras is said to be completely bounded if

sup_{n∈ℕ}∥M_{n}(π)∥ < ∞,

where M_{n}(π): M_{n}(A) → M_{n}(B) is the morphism on n × n matrices given by π.

Given a pro-Banach algebra A = \underleftarrow{lim}_{i} A_{i} with completely bounded structure maps A_{j} → A_{i}, and a Banach algebra B, any morphism π: A → B factors through some A_{i}, and we say that π is completely bounded if the map A_{j} → B is so for some sufficiently large j ≥ i.

Write Hom(A, B)_{cb} for the set of completely bounded homomorphisms from A to B.

Lemma 1.22. If A is a C^{∗}-algebra, then any completely bounded homomorphism f: A → L(H) is conjugate to a ∗-homomorphism of C^{∗}-algebras.

Proof. This is the main result of [Pau]. □

Remark 1.23. Kadison’s similarity problem asks whether all bounded (non-involutive) homomorphisms between C^{∗}-algebras are in fact completely bounded. The answer is affirmative in a wide range of cases, but the general problem remains open. Note that Gardner showed ([Gar, Theorem A]) that all Banach isomorphisms of C^{∗}-algebras are conjugate to C^{∗}-homomorphisms (and hence completely bounded).

Also note that by [Seg, Theorem 3], every closed two-sided ideal of a C^{∗} algebra is a ∗-ideal; combined with Gardner’s result, this implies that any bounded surjective map A → B between C^{∗}-algebras must be conjugate to a C^{∗}-homomorphism, hence completely bounded. The same is true of any bounded map between C^{∗}-algebras whose image is a C^{∗}-subalgebra.

Definition 1.24. Let C^{∗}BAlg_{k} be the category of unital C^{∗}-algebras over k, with completely bounded morphisms (which need not preserve ∗).

Lemma 1.25. For complex C^{∗}-algebras A, B, giving a U(B)-equivariant function f: Hom_{C^{∗}Alg_{ℂ}}(A, B) → B (for the adjoint action on B) is equivalent to giving a B^{×}- equivariant function f̃: Hom_{C^{∗}BAlg_{ℂ}}(A, B) → B.

Proof. There is a canonical inclusion ι: Hom_{C^{∗}Alg_{ℂ}}(A, B) → Hom_{C^{∗}BAlg_{ℂ}}(A, B), so given f̃, we just set f to be f̃∘ ι.

The polar decomposition allows us to write B^{×} = B_{++}U(B), where B_{++} ⊂ S(B) is the subset of strictly positive self-adjoint elements. Given f, there is thus an associated B^{×}-equivariant function f̃: Hom_{C^{∗}Alg_{ℂ}}(A, B) × B_{++} → B given by f̃(p, g) = g^{−1}f(p)g. By Lemma 1.22, the map Hom_{C^{∗}Alg_{ℂ}}(A, B) × B_{++} → Hom_{C^{∗}BAlg_{ℂ}}(A, B) is surjective, and we need to check that f̃ descends.

Now, if ξ ∈ S(B) has the property that exp(ξ) fixes p(A) under conjugation, then exp(iξt) commutes with f(p) for all t, so iξ must also. Thus ξ and hence exp(ξ) commute with f(p), so f̃ does indeed descend. □

1.4.2. Representations. We now fix a real unital C^{∗}-algebra A.

Definition 1.26. Define FDVect to be the category of complex finite-dimensional vector spaces and linear maps.

Definition 1.27. Write FDRep(A) for the category of pairs (V, ρ) for V ∈ FDVect and ρ: A → End(V) a unital continuous morphism of Banach algebras. Morphisms are given by linear maps intertwining representations. The set of objects of FDRep(A) is given the topology of pointwise convergence.

Note that the objects of FDRep(A) decompose into direct sums of irreducibles.

Lemma 1.28. The ring A_{PN} of Lemma 1.17 is isomorphic to the ring A_{PN}′ of Gal(ℂ/ℝ)-equivariant continuous endomorphisms of the fibre functor η: FDRep(A) → FDVect. Explicitly, A_{PN}′ consists of elements a such that

(1) For any linear map f: (V, ρ_{1}) → (W, ρ_{2}), we have a(W, ρ_{2})f = fa(V, ρ_{1}).

(2) The maps a: Rep_{n}^{∗}(A) → Mat_{n}(ℂ) are continuous and Gal(ℂ/ℝ)-equivariant.

Proof. First note that condition (1) applied to the projections V_{1} ⊕V_{2} → V_{i} ensures that for any (V_{1}, ρ_{1}), (V_{2}, ρ_{2}) ∈ FD^{∗}Rep(A), we have a(V_{1}⊕V_{2}, ρ_{1}⊕ρ_{2}) = a(V_{1}, ρ_{1})⊕a(V_{2}, ρ_{2}).

Restriction to ∗-representations then gives us a map ψ: A_{PN}′ → A_{PN}. For a commutative C^{∗}-algebra C, the C^{∗}-algebra Mat_{k}(C) is of type I. This means that any bounded map A → Mat_{k}(C) is completely bounded, so taking B = Mat_{k}(C) in Lemma 1.25 for all k ensures that ψ is an isomorphism. □

Definition 1.29. Given a commutative unital real C^{∗}-algebra A and a ∗- homomorphism A → B of real C^{∗}-algebras, write FDRepÂ(B) for the category of triples (f, V, ρ) for f ∈ Â (the spectrum of A), V ∈ FDVect and ρ: B → End(V) a unital continuous morphism of Banach algebras for which ρ(a) = f(a)id for all a ∈ A. Morphisms are given by linear maps intertwining representations. The set of objects of FDRepÂ(B) is given the topology of pointwise convergence.

The category FDRepÂ(B) has an additive structure over Â, given by (f, V_{1}, ρ_{1}) ⊕ (f, V_{2}, ρ_{2}) = (f, V_{1} ⊕ V_{2}, ρ_{1} ⊕ ρ_{2}).

Lemma 1.30. Given a commutative unital C^{∗}-algebra A and a ∗-homomorphism A → B of real C^{∗}-algebras, the ring B_{PN} of Lemma 1.17 is isomorphic to the ring of Gal(ℂ/ℝ)-equivariant continuous endomorphisms of the fibre functor η: FDRepÂ(B) → FDVect.

Proof. This just combines the proofs of Lemma 1.28 and Proposition 1.19. The only modification is to observe that A = C(Â, ℂ)^{Gal(ℂ/ℝ)}, and that for any irreducible representation ρ: B → End(V), we necessarily have ρ|_{A} = fid, for some f ∈ Â. □

2. The Betti, de Rham and harmonic functors on Banach algebras

2.1. The Riemann–Hilbert correspondence.

Definition 2.1. Given a path-connected topological space X with basepoint x and a unital ℝ-algebra B, define the Betti representation space R_{X,x}^{B}(B) by

R_{X,x}^{B}(B) := Hom_{Gp}(π_{1}(X, x), B^{×}),

where B^{×} is the multiplicative group of units in B.

Define the representation groupoid R_{X}^{B}(B) by R_{X}^{B}(B) := [R_{X,x}^{B}(B)/B^{×}], where B^{×} acts by conjugation. Note that this is independent of the choice of basepoint (being equivalent to the groupoid of B^{×}-torsors on X).

Definition 2.2. Given a connected manifold X with basepoint x and a Banach algebra B, define the de Rham groupoid R_{X}^{dR}(B) to be the groupoid of smooth B^{×}-bundles with flat connections. Thus R_{X}^{dR}(B) consists of pairs (𝒯 , D), where 𝒯 is a right 𝒜_{X}^{0}(B^{×})- torsor, and D is a flat connection on 𝒯 .

Explicitly, write 𝒜_{X}^{n}(ad𝒯 ) := 𝒯 ×_{𝒜_{X}^{0}(B^{×})} 𝒜_{X}^{n}(B), for the adjoint action of B^{×} on B. Then a flat connection on 𝒯 is

D: 𝒯 → 𝒜^{1}(ad𝒯 )

satisfying (1) D is a d-connection: D(pg) = ad_{g}D(p) + g^{−1}dg, for g ∈ 𝒜_{X}^{0}(B^{×});

(2) D is flat: (adD) ∘ D = 0.

Define R_{X,x}^{dR}(B) to be the groupoid of triples (𝒯 , D, f), where (𝒯 , D) ∈ R_{X}^{dR}(B) and f ∈ x^{∗}𝒯 is a distinguished element. Since R_{X,x}^{dR}(B) has no non-trivial automorphisms, we will regard it as a set-valued functor (given by its set of isomorphism classes).

Note that B^{×} acts on R_{X,x}^{dR}(B) by changing the framing, and that the quotient groupoid is then equivalent to R_{X}^{dR}(B).

Proposition 2.3. For any pointed connected manifold (X, x), and any Banach algebra B, there are canonical equivalences

R_{X}^{dR}(B) ≃ R_{X}^{B}(B), R_{X,x}^{dR}(B) ≅ R_{X,x}^{B}(B)

functorial in X, x and B.

Proof. When B is a finite-dimensional matrix algebra, this is [GM, 5.10]. The same proof carries over to Banach algebras, noting that the argument for existence of parallel transport ([KN, §II.3]) holds in this generality, since exp(b) = ∑_{n≥0} b^{n}/n! converges and is invertible for all b ∈ B. □

Remark 2.4. The functor R_{X,x}^{dR} naturally extends to a functor on the category pro(BanAlg) of pro-Banach algebras, by sending any cofiltered inverse system {A_{i}}_{i} to \underleftarrow{lim}_{i} R_{X,x}^{dR}(A_{i}). Since the functor R_{X,x}^{B} commutes with all limits, the equivalences of Proposition 2.3 then extend to pro-Banach algebras. The category pro(BanAlg) contains all multiplicatively convex Fréchet algebras (since they are countable inverse limits of Banach algebras) and indeed all complete LMC algebras via the Arens–Michael decomposition.

For any open subset U ⊂ ℂ^{n}, the ring O(U) := Γ(U, 𝒪_{U}) of holomorphic functions on U can be realised as a pro-Banach algebra by looking at the system of sup norms on compact subspaces. Taking quotients by finitely generated ideals I then gives local models Y := (V (I), 𝒪_{U}/I) for complex analytic spaces, and realises O(Y) as the pro-Banach algebra O(U)/I. A complex analytic morphism from Y to the variety Hom(π_{1}(X, x), GL_{n}(ℂ)) is then just an element of Hom(π_{1}(X, x), GL_{n}O(Y)) = R_{X,x}^{B}(Mat_{n}O(Y)), for the pro-Banach algebra Mat_{n}O(Y) of morphisms Y → Mat_{n}(ℂ).

We can therefore recover the analytic structure of the variety Hom(π_{1}(X, x), GL_{n}(ℂ)) from the set-valued functor R_{X,x}^{B} on Banach algebras, and hence (by Proposition 2.3) from the set-valued functor R_{X,x}^{dR}.

In [Sim6], the varieties R_{X,x}^{B}(Mat_{n}(−)) and R_{X,x}^{dR}(Mat_{n}(−)) are denoted by R_{B}(X, x, n). and R_{DR}(X, x, n).

Lemma 2.5. For any real Banach algebras B, C, there is a canonical map

m: R_{X,x}^{B}(B) × R_{X,x}^{B}(C) → R_{X,x}^{B}(B ⊗_{ℝ}^{π} C),

where ⊗^{π} is the projective tensor product. This makes R_{X,x}^{B} into a symmetric monoidal functor, with unit corresponding to the trivial representation in each R_{X,x}^{B}(B).

Proof. Given representations ρ_{1} : π_{1}(X, x) → B^{×} and ρ_{2} : π_{1}(X, x) → C^{×}, we obtain ρ_{1} ⊗ ρ_{2} : π_{1}(X, x) → (B ⊗ C)^{×}. Taking completion with respect to the projective cross norm gives the required result. □

Remark 2.6. Given any complex affine group scheme G, we may use the tensor structure on R_{X,x}^{B} to recover the affine analytic variety Hom(π_{1}(X, x), G(ℂ)) in the same spirit as Remark 2.4. Explicitly, O(G) is a coalgebra, so can be written as a nested union of finite-dimensional coalgebras. Therefore O(G)^{∨} is a pro-finite-dimensional algebra, and hence a pro-Banach algebra.

Multiplication on G gives us a comultiplication μ: O(G)^{∨} → O(G × G)^{∨}. For any complex analytic space Y, we may then characterise Hom(π_{1}(X, x), G(Y)) as

{ρ ∈ R_{X,x}^{B}(O(G)^{∨} ˆ⊗O(Y)) : μ(ρ) = m(ρ, ρ) ∈ R_{X,x}^{B}(O(G × G)^{∨} ˆ⊗O(Y))}.

2.2. Representability of the de Rham functor.

Lemma 2.7. Given a free group Γ = F (X), the functor

A ↦ Hom_{Gp}(Γ, A^{×})

on the category of real Banach algebras is pro-representable.

Proof. Given a function ν: X → [1, ∞), let ν̄: Γ → [1, ∞) be the largest function subject to the conditions

(1) ν̄(1) = 1;

(2) ν̄(x) = ν̄(x^{−1}) = ν(x) for all x ∈ X;

(3) ν̄(gh) ≤ ν̄(g)ν̄(h). Explicitly, we write any g ∈ Γ as a reduced word g = x_{1}^{n_{1}}x_{2}^{n_{2}} … x_{k}^{n_{k}}, then set ν̄(g) := ∏_{i=1}^{k} ν(x_{i})^{|n_{i}|}. We now define a norm ∥ − ∥_{1,ν} on k[Γ] by setting

∥ ∑_{γ∈Γ} λ_{γ}γ∥_{1,ν} := ∑_{γ∈Γ} |λ_{γ}| · ν̄(γ).

Now, given any representation ρ: Γ → A^{×}, we may define ν: X → [1, ∞) by

ν(x) := max{∥ρ(x)∥, ∥ρ(x^{−1})∥};

this at least 1 because 1 = ρ(x)ρ(x^{−1}), so 1 ≤ ∥ρ(x)∥ · ∥ρ(x^{−1})∥. It follows that for all v ∈ k[Γ] we have ∥ρ(v)∥ ≤ ∥v∥_{1,ν}, so ρ determines a map

k[Γ]^{∧_{ν}} → A,

where k[Γ]^{∧_{ν}} denotes the Banach algebra obtained by completing k[Γ] with respect to the norm ∥ − ∥_{1,ν}.

Next, give [1, ∞)^{X} the structure of a poset by saying ν_{1} ≤ ν_{2} provided ν_{1}(x) ≤ ν_{2}(x) for all x ∈ X. This is in fact a directed set, since we can define max{ν_{1}, ν_{2}} pointwise. There is a canonical morphism

k[Γ]^{∧_{ν_{2}}} → k[Γ]^{∧_{ν_{1}}}

whenever ν_{1} ≤ ν_{2}, which gives us an inverse system k[Γ]^{an} := {k[Γ]^{∧_{ν}}}_{ν} of Banach algebras, indexed by the directed set ([1, ∞)^{X}.

Thus we have shown that

Hom_{Gp}(Γ, A^{×}) ≅ \underrightarrow{lim}_{ν∈[1,∞)^{X}} Hom_{kBanAlg}(k[Γ]^{∧_{ν}}, A),

functorial in Banach k-algebras A. In other words, our pro-representing object is the inverse system k[Γ]^{an}. □

Example 2.8. Take X = {z}, so Γ = ℤ, and let ν(z) = R. Then elements of k[ℤ]^{∧_{ν}} are ∑_{i∈ℤ} λ_{i}z^{i} such that

∑_{i≥0} |λ_{i}|R^{i} < ∞ ∑_{i≤0} |λ_{i}|R^{−i} < ∞.

Thus ℂ[ℤ]^{∧_{R}} is the ring of analytic functions converging on the annulus R^{−1} ≤ |z| ≤ R. Hence \underleftarrow{lim}_{R} ℂ[ℤ]^{∧_{R}} is the ring of analytic functions on ℂ^{∗}, while \underleftarrow{lim}_{R} ℝ[ℤ]^{∧_{R}} is the subring consisting of functions f with \overline{f(z)} = f(z̄)

Contrast this with the isometric Banach completion of ℂ[ℤ], which just gives ℂ[ℤ]^{∧_{1}} = ℓ^{1}(ℤ).

Lemma 2.9. Given a finitely generated free group Γ = F (X), the functor

A ↦ Hom_{Gp}(Γ, A^{×})

on the category of multiplicatively convex Fréchet k-algebras is representable.

Proof. We may embed ℕ_{1} in [1, ∞)^{X} as a subset of the constant functions. Since X is finite, ℕ_{1} is a cofinal subset of [1, ∞)^{X}, giving us an isomorphism

{k[Γ]^{∧_{ν}}}_{ν∈[1,∞)^{X}} ≅ {k[Γ]^{∧_{n}}}_{n∈ℕ_{1}}

in the category of pro-Banach k-algebras. Since ℕ_{1} is countable, k[Γ]^{an} := \underleftarrow{lim}_{n} k[Γ]^{∧_{n}} is a Fréchet algebra.

Applying the proof of Lemma 2.7, we have shown that

Hom(Γ, A^{×}) ≅ Hom_{kFrAlg}(k[Γ]^{an}, A)

for all Banach algebras A. Since any m-convex Fréchet algebra A can be expressed as an inverse limit A = \underleftarrow{lim}_{i} A_{i} of Banach algebras, it follows that the same isomorphism holds for all such algebras, so the functor is representable in m-convex Fréchet algebras. □

Proposition 2.10. Given a finitely generated group Γ, the functor

A ↦ Hom_{Gp}(Γ, A^{×})

on the category of multiplicatively convex Fréchet k-algebras is representable.

Proof. Choose generators X for Γ, so Γ = F (X)/K for some normal subgroup K. Lemma 2.9 gives a Fréchet k-algebra k[F (X)]^{an} governing representations of F (X). Since

Hom_{Gp}(Γ, A^{×}) = {f ∈ Hom_{Gp}(F (X), A^{×}) : f(K) = {1}},

our functor will be represented by a quotient of k[F (X)]^{an}. Specifically, let I be the closed ideal of k[F (X)]^{an} generated by {k − 1 : k ∈ K}, and set k[Γ]^{an} := k[F (X)]^{an}/I. This is a m-convex Fréchet algebra, and

Hom_{Gp}(Γ, A^{×}) ≅ Hom_{kFrAlg}(k[Γ]^{an}, A)

for all such algebras A.

For an explicit description of k[Γ]^{an}, note that the system of norms is given by

∥ ∑ λ_{γ}γ∥_{1,n} = ∑ |λ_{γ}| · n^{w(γ)},

where w(γ) is the minimal word length of γ in terms of X. □

When combined with its tensor structure, this implies that the functor of Proposition 2.10 is a very strong invariant indeed:

Lemma 2.11. The tensor structure of Lemma 2.5 gives k[Γ]^{an} the structure of a mconvex Fréchet bialgebra. The group

G(k[Γ]^{an}) = {a ∈ k[Γ]^{an} : μ(a) = a ⊗ a ∈ k[Γ]^{an} ⊗^{π} k[Γ]^{an}, ε(a) = 1 ∈ k}

of grouplike elements of k[Γ]^{an} is then Γ.

Proof. Applying the map m of Lemma 2.5 to (ξ, ξ), for the canonical element ξ ∈ Hom_{Gp}(Γ, k[Γ]^{an}) gives us a comultiplication μ: k[Γ]^{an} → k[Γ]^{an} ⊗^{π} k[Γ]^{an} = k[Γ × Γ]^{an} and a co-unit ε: k[Γ]^{an} → k. On the topological basis Γ, we must have μ(γ) = (γ, γ) and ε(γ) = 1.

Expressing a ∈ G(k[Γ]^{an}) as ∑_{γ∈Γ} a_{γ}γ, note that the conditions become a_{γ}a_{δ} = 0 for γ ≠ δ, and ∑ a_{γ} = 1; thus a = γ for some γ ∈ Γ. □

Example 2.12. Arguing as in example 2.8, for Γ abelian and finitely generated, ℂ[Γ]^{an} is isomorphic to the ring of complex analytic functions on Hom_{Gp}(Γ, ℂ^{∗}), while ℝ[Γ]^{an} ⊂ ℂ[Γ]^{an} consists of Gal(ℂ/ℝ)-equivariant functions. The multiplicative analytic functions are of course just Γ itself.

Proposition 2.3 then implies:

Corollary 2.13. The functors R_{X,x}^{dR} and R_{X,x}^{B} on real multiplicatively convex Fréchet algebras are representable.

Remark 2.14. Adapting the ideas of [Pri1], the functor R_{X,x}^{B} has a natural extension to those simplicial Banach algebras B_{•} for which B_{n} → π_{0}B is a nilpotent extension for each n. Explicitly, we could set R_{X,x}^{B}(B) to be the set of homotopy classes of maps G(Sing(X, x)) → B_{•}^{×} of simplicial groups, where G is Kan’s loop group. This functor admits a tensor structure extending Lemma 2.5,

The functor R_{X,x}^{dR} has a natural extension to those differential graded Banach algebras B_{•} for which B_{0} → H_{0}B is a nilpotent extension. Explicitly, R_{X,x}^{dR}(B) would consist of pairs (𝒯_{0}, D), where 𝒯_{0} is a 𝒜_{X}^{0}(B_{0}^{×})-torsor and D: T_{0} → ∏_{n} 𝒜_{X}^{n+1} ⊗_{𝒜_{X}^{0}} ad𝒯_{n}(n + 1) is a flat hyperconnection, where ad𝒯_{n} := 𝒯 ×_{𝒜_{X}^{0}(B_{0}^{×})} 𝒜_{X}^{0}(B_{n}).

It then seems likely that [Pri1, Corollary 4.41] should adapt to give natural isomorphisms R_{X,x}^{B}(B) ≅ R_{X,x}^{dR}(NB), where N is Dold–Kan normalisation.

2.3. The pluriharmonic functor. Fix a compact connected Kähler manifold X, with basepoint x ∈ X.

Definition 2.15. Given a real Banach space B, denote the sheaf of B-valued C^{∞} nforms on X by 𝒜_{X}^{n}(B), and let 𝒜_{X}^{•} be the resulting complex. Write A^{•}(X, B) := Γ(X, 𝒜_{X}^{•}(B)). We also write 𝒜_{X}^{•} := 𝒜_{X}^{•}(ℝ) and A^{•}(X) := A^{•}(X, ℝ).

Definition 2.16. Define S to be the real algebraic group ∏_{ℂ/ℝ} 𝔾_{m} obtained as in [Del] 2.1.2 from 𝔾_{m,ℂ} by restriction of scalars. Note that there is a canonical inclusion 𝔾_{m} ↪ S.

The following is a slight generalisation of [Pri2, Definition 2.49]:

Definition 2.17. For any real Banach space B, there is an action of S on 𝒜_{X}^{∗}(B), which we will denote by a ↦ λ ⋄ a, for λ ∈ ℂ^{∗} = S(ℝ). For a ∈ (A^{∗}(X) ⊗ ℂ)^{pq}, it is given by

λ ⋄ a := λ^{p}λ̄^{q}a.

Definition 2.18. Given a real C^{∗}-algebra B, define R_{X}^{J}(B) to be the groupoid of pairs (U(𝒫), D), where U(𝒫) is a right 𝒜_{X}^{0}(U(B))-torsor, and D is a pluriharmonic connection on U(𝒫).

Explicitly, write 𝒫 := U(𝒫) ×_{𝒜_{X}^{0}(U(B))} 𝒜_{X}^{0}(B^{×}), and

ad𝒫 := 𝒫 ×_{𝒜_{X}^{0}(B^{×})} 𝒜_{X}^{0}(B)
= U(𝒫) ×_{𝒜_{X}^{0}(U(B))} 𝒜_{X}^{0}(B)
= [U(𝒫) ×_{𝒜_{X}^{0}(U(B))} 𝒜_{X}^{0}(𝔲(B))] ⊕ [U(𝒫) ×_{𝒜_{X}^{0}(U(B))} 𝒜_{X}^{0}(S(B))],

where U(B) and B^{×} act on B by the adjoint action. Then a pluriharmonic connection on 𝒫 is

D: U(𝒫) → ad𝒫

satisfying (1) D is a d-connection: D(pu) = ad_{u}D(p) + u^{−1}du, for u ∈ 𝒜_{X}^{0}(U(B));

(2) D is flat: (adD) ∘ D = 0;

(3) D is pluriharmonic: (adD) ∘ D^{c} + (adD^{c}) ∘ D = 0. Here, D = d^{+} + ϑ comes from the decomposition of ad𝒫 into anti-self-adjoint and self-adjoint parts, and D^{c} = i ⋄ d^{+} − i ⋄ ϑ.

Define R_{X,x}^{J}(B) to be the groupoid of triples (U(𝒫), D, f), where (U(𝒫), D) ∈ R_{X}^{J}(B) and f ∈ x^{∗}U(𝒫) is a distinguished element. Since R_{X,x}^{J}(B) has no non-trivial automorphisms, we will regard it as a set-valued functor (given by its set of isomorphism classes).

Remarks 2.19. Note that there is a natural action of U(B) on R_{X,x}^{J}(B), given by changing the framing. The quotient groupoid [R_{X,x}^{J}(B)/U(B)] is thus equivalent to R_{X}^{J}(B). In [Sim6, Lemma 7.13], the set R_{X,x}^{J}(Mat_{n}(ℂ)) is denoted by R_{DR}^{J}(X, x, n).

Also note that the definition of R_{X,x}^{J}(B) can be extended to any real Banach ∗-algebra B. However, this will not be true of the harmonic functor of §2.5.

Example 2.20. When V is a real Hilbert space, the algebra L(V) of bounded operators on V is a real C^{∗}-algebra. Then R_{X}^{J}(B) is equivalent to the groupoid of pluriharmonic local systems 𝕍 in Hilbert spaces on X, fibrewise isometric to V. The connection D: 𝒜^{0}(𝕍) → 𝒜^{1}(𝕍) must satisfy the pluriharmonic condition that DD^{c} + D^{c}D = 0, for D^{c} defined with respect to the smooth inner product 𝕍 × 𝕍 → 𝒜_{X}^{0}. Isomorphisms in R_{X}^{J}(B) preserve the inner product.

Definition 2.21. Define the de Rham projection

π_{dR} : R_{X,x}^{J}(B) → R_{X,x}^{dR}(B)

by mapping (U(𝒫), D, f) to the framed flat torsor (𝒫, D, f) = (U(𝒫) ×_{𝒜_{X}^{0}(U(B))} 𝒜_{X}^{0}(B^{×}), D, f ×_{U(B)} B^{×}).

Proposition 2.22. The functor R_{X,x}^{J}: C^{∗}Alg → Set is strictly pro-representable, by an object E_{X,x}^{J} ∈ pro(C^{∗}Alg).

Proof. The final object in C^{∗}Alg is 0, and R_{X,x}^{J}(0) is the one-point set, so R_{X,x}^{J} preserves the final object.

Given maps A → B ← C in C^{∗}Alg and (p_{A}, p_{B}) ∈ R_{X,x}^{J}(A) ×_{R_{X,x}^{J}(B)} R_{X,x}^{J}(C), we get

π_{dR}(p_{A}, p_{C}) ∈R_{X,x}^{dR}(A) ×_{R_{X,x}^{dR}(B)} R_{X,x}^{dR}(C)
≅ R_{X,x}^{B}(A) ×_{R_{X,x}^{B}(B)} R_{X,x}^{B}(C)
≅ R_{X,x}^{B}(A ×_{B} C).

Thus we have a flat torsor (𝒫, D) ∈ R_{X,x}^{dR}(A ×_{B} C).

It follows that p_{A} ≅ (U(𝒫_{A}), D) for some orthogonal form U(𝒫_{A}) ⊂ 𝒫_{A} = 𝒫 ×_{𝒜_{X}^{0}((A×_{B}C)^{×})} 𝒜_{X}^{0}(A^{×}), and similarly for p_{C}. Since the images of p_{A} and p_{C} are equal in R_{X,x}^{dR}(B), there is a framed orthogonal isomorphism α: U(𝒫_{A}) ×_{𝒜_{X}^{0}(U(A))} 𝒜_{X}^{0}(U(B)) → U(𝒫_{C}) ×_{𝒜_{X}^{0}(U(C))} 𝒜_{X}^{0}(U(B)), inducing the identity on 𝒫_{B}. Hence α must itself be the identity, so both U(𝒫_{A}) and U(𝒫_{C}) give the same unitary form U(𝒫_{B}) for 𝒫_{B}. It is easy to check the pluriharmonic conditions, giving an element

(U(𝒫_{A}) ×_{U(𝒫_{B})} U(𝒫_{C}), D) ∈ R_{X,x}^{J}(A ×_{B} C)

over (p_{A}, p_{C}). This is essentially unique, so R_{X,x}^{J} preserves fibre products, and hence finite limits.

Now, given a C^{∗}-subalgebra A ⊂ B, the map R_{X,x}^{J}(A) → R_{X,x}^{J}(B) is injective. This follows because if two framed pluriharmonic bundles 𝒫_{1}, 𝒫_{2} in R_{X,x}^{J}(A) become isomorphic in R_{X,x}^{J}(B), compatibility of framings ensures that the isomorphism f maps x^{∗}𝒫_{1} to x^{∗}𝒫_{2}. Since f is compatible with the connections, it thus gives an isomorphism f: 𝒫_{1} → 𝒫_{2}, by considering the associated local systems.

Finally, given an inverse system {A_{i}}_{i} of nested C^{∗}-subalgebras of a C^{∗}-algebra B and an element of ⋂_{i} R_{X,x}^{J}(A_{i}), we have a compatible system {(𝒫_{i}, D_{i}, f_{i})}_{i}. Set 𝒫 := \underleftarrow{lim}_{i} 𝒫_{i}, with connection D and framing f induced by the D_{i} and f_{i}. This defines a unique element of R_{X,x}^{J}(⋂_{i} A_{i}), showing that

F (⋂_{i} A_{i}) ≅ ⋂_{i} F (A_{i})

Thus all the conditions of Proposition 1.9 are satisfied, so R_{X,x}^{J} is strictly prorepresentable. □

Definition 2.23. Given pro-C^{∗}-algebras B, C over k, define B ˆ⊗_{k}C to be the maximal k-tensor product of B and C, as defined in [Phi, Definition 3.1]; this is again a pro-C^{∗}- algebra.

Lemma 2.24. For any real pro-C^{∗}-algebras B, C, there is a canonical map

m: R_{X,x}^{J}(B) × R_{X,x}^{J}(C) → R_{X,x}^{J}(B ˆ⊗_{ℝ}C),

making R_{X,x}^{J} into a symmetric monoidal functor, with unit corresponding to the trivial torsor in each R_{X,x}^{J}(B).

Proof. Given (U(𝒫), D, f, U(𝒬), E, β) on the left-hand side, we first form the 𝒜_{X}^{0}(U(B ˆ⊗C))-torsor U(ℛ) given by U(ℛ) := (U(𝒫) × U(𝒬))_{𝒜_{X}^{0}(U(B)×U(C))} 𝒜_{X}^{0}(U(B ˆ⊗C)). We then define a connection F on U(ℛ) determined by

F (p, q, 1) = (Dp, q) + (p, Dq) ∈ 𝒜_{X}^{1}(adℛ) = (U(𝒫) × U(𝒬))_{𝒜_{X}^{0}(U(B)×U(C))} 𝒜_{X}^{1}(B ˆ⊗C)

for p ∈ U(𝒫), q ∈ U(𝒬) This is clearly flat and pluriharmonic, and the construction is also symmetric monoidal. □

Note that this gives E_{X,x}^{J} the structure of a pro-C^{∗}-bialgebra, with comultiplication μ: E_{X,x}^{J} → E_{X,x}^{J} ˆ⊗E_{X,x}^{J} coming from m, and counit ε: E_{X,x}^{J} → k coming from the trivial torsor.

The following is immediate:

Lemma 2.25. For any morphism f: X → Y of compact connected Kähler manifolds, there is a natural transformation

f^{∗}: R_{Y,fx}^{J} → R_{X,x}^{J}

of functors.

2.4. Higgs bundles.

Definition 2.26. Given a complex Banach algebra B, write 𝒪_{X}(B) for the sheaf on X given locally by holomorphic functions X → B.

Definition 2.27. For a complex Banach algebra B, a Higgs B-torsor on X consists of an 𝒪_{X}(B)^{×}-torsor 𝒯 , together with a Higgs form θ ∈ ad𝒯 ⊗_{𝒪_{X}} Ω_{X}^{1}, where ad𝒯 := 𝒯 ×_{𝒪_{X}(B)^{×},ad} 𝒪_{X}(B) satisfying

θ ∧ θ = 0 ∈ 𝒯 ⊗_{𝒪_{X}} Ω_{X}^{2}.

Definition 2.28. Let R_{X}^{Dol}(B) be the groupoid of Higgs B-torsors, and R_{X,x}^{Dol}(B) the groupoid of framed Higgs bundles (𝒯 , θ, f), where f: B^{×} → x^{∗}𝒯 is a B^{×}-equivariant isomorphism. Alternatively, we may think of f as a distinguished element of x^{∗}𝒯 .

Note that R_{X,x}^{Dol}(B) is a discrete groupoid, so we will usually identify it with its set of isomorphism classes. Also note that there is a canonical action of B^{×} on R_{X,x}^{Dol}(B) given by the action on the framings. This gives an equivalence

R_{X}^{Dol}(B) ≃ [R_{X,x}^{Dol}(B)/B^{×}]

of groupoids.

The following is immediate:

Lemma 2.29. Giving a Higgs B-torsor X is equivalent to giving an 𝒜_{X}^{0}(B^{×}) torsor 𝒬 equipped with a flat ¯∂-connection, i.e. a map

D′′: 𝒬 → ad𝒬 := 𝒬 ×_{𝒜_{X}^{0}(B^{×}),ad} 𝒜_{X}^{1}(B)

satisfying (1) D′′(pg) = ad_{g}D′′(p) + g^{−1} ¯∂g, for g ∈ 𝒜_{X}^{0}(B^{×});

(2) (adD′′) ∘ D′′ = 0.

Remark 2.30. Note that unlike the Betti, de Rham, and harmonic functors, the Dolbeault functor cannot be pro-representable in general. This is for the simple reason that a left-exact scheme must be affine, but the Dolbeault moduli space is seldom so, since it contains the Picard scheme.

Definition 2.31. Given (U(𝒫), D) ∈ R_{X}^{J}(B), decompose d^{+} and ϑ into (1, 0) and (0, 1) types as d^{+} = ∂ + ¯∂ and ϑ = θ + θ̄. Now set D′ = ∂ + θ̄ and D′′ = ¯∂ + θ. Note that D = D′ + D′′ and D^{c} = iD′ − iD′′.

Definition 2.32. For a complex C^{∗}-algebra B, define the Dolbeault projection map π_{Dol} : R_{X}^{J}(B) → R_{X}^{Dol}(B) by sending (U(𝒫), D) to (𝒫 ×_{𝒜_{X}^{0}(B^{×})} 𝒜_{X}^{0}(B^{×}), D′′).

2.5. The harmonic functor. We now let X be any compact Riemannian real manifold.

Definition 2.33. Given a compact Riemannian manifold X, a real C^{∗}-algebra B, a right 𝒜_{X}^{0}(U(B))-torsor U(𝒫) and a flat connection

D: U(𝒫) → ad𝒫,

say that D is a harmonic connection if (d^{+})^{∗}ϑ = 0 ∈ Γ(X, ad𝒫), for d^{+}, ϑ defined as in Definition 2.18, and the adjoint ∗ given by combining the involution ∗ on ad𝒫 with the adjoint on 𝒜_{X}^{∗} given by the Kähler form.

Lemma 2.34. A flat connection D as above on a compact Kähler manifold is harmonic if and only if it is pluriharmonic.

Proof. The proof of [Sim4, Lemma 1.1] carries over to this generality. □

Definition 2.35. The lemma allows us to extend Definitions 2.18,2.21 to any compact Riemannian manifold X, replacing pluriharmonic with harmonic in the definition of R_{X}^{J}(B) and R_{X,x}^{J}(B).

Proposition 2.36. The functor R_{X,x}^{J}: C^{∗}Alg_{ℝ} → Set is strictly pro-representable, by an object E_{X,x}^{J} ∈ pro(C^{∗}Algℝ).

Proof. The proof of Proposition 2.22 carries over. □

Note that Lemma 2.24 carries over to the functor R_{X,x}^{J} for any compact Riemannian manifold X.

The following is immediate:

Lemma 2.37. For any local isometry f: X → Y of compact connected real Riemannian manifolds, there is a natural transformation

f^{∗}: R_{Y,fx}^{J} → R_{X,x}^{J}

of functors.

Note that this is much weaker than Lemma 2.25, the pluriharmonic functor being a priori functorial with respect to all morphisms.

3. Analytic non-abelian Hodge theorems

3.1. The de Rham projection. Fix a compact connected real Riemannian manifold X, with basepoint x ∈ X.

The argument of [Sim6, Lemma 7.17] (which is only stated for X Kähler) shows that π_{dR} gives a homeomorphism

R_{X,x}^{J}(Mat_{n}(ℂ))/U(n) → Hom(π_{1}(X, x), GL_{n}(ℂ))//GL_{n}(ℂ),

where // denotes the coarse quotient (in this case, the Hausdorff completion of the topological quotient).

As an immediate consequence, note that

R_{X,x}^{J}(ℂ) → Hom_{Gp}(π_{1}(X, x), ℂ^{∗})

is a homeomorphism. Thus the abelianisation of E_{X,x}^{J} ⊗ ℂ is isomorphic to the commutative C^{∗}-algebra C(Hom(π_{1}(X, x), ℂ^{∗}), ℂ), with R_{X,x}^{J} ⊂ R_{X,x}^{J} ⊗ ℂ consisting of Gal(ℂ/ℝ)-equivariant functions.

We now adapt these results to recover a finer comparison between the respective functors.

3.1.1. Harmonic representations.

Proposition 3.1. For a compact Riemannian manifold X and all real C^{∗}-algebras B, the de Rham projection

π_{dR} : R_{X,x}^{J}(B) → R_{X,x}^{dR}(B)

has the property that if p_{1}, p_{2} ∈ R_{X,x}^{J}(B) and if ad_{b}π_{dR}(p_{1}) = π_{dR}(p_{2}) for some strictly positive self-adjoint element b ∈ B, then p_{1} = p_{2}.

Thus

π_{dR} : R_{X,x}^{J}(B)/U(B) → R_{X,x}^{dR}(B)/B^{×}

is injective.

Proof. The first statement above implies the second: it suffices to show that for any (U(𝒫), D, f) ∈ R_{X,x}^{J}(B), there are no other harmonic representations in the B_{++}-orbit of π_{dR}(U(𝒫), D, f). Since B_{++} = exp(S(B)) (by the continuous functional calculus), we can equivalently look at the orbit under the exponential action of the set S(B).

We adapt the proof of [Cor, Proposition 2.3]. The harmonic condition (d^{+})^{∗}ϑ = 0 is equivalent to saying that for all ξ ∈ A^{0}(X, iad𝒫)

⟨ϑ, d^{+}ξ⟩ = 0 ∈ A^{0}(X, B),

where ⟨−, −⟩ is defined using the Riemannian metric.

Now, the set of flat connections on 𝒫 admits a gauge action ⋆ of the smooth automorphism group of 𝒫, and hence via exponentiation an action of the additive group Γ(X, ad𝒫). An isomorphism in B_{++} between two flat connections corresponds to an element of exp(Γ(X, S(ad𝒫))), for S(ad𝒫) ⊂ ad𝒫 consisting of symmetric elements, giving a gauge between the respective connections. Thus the S(B)-orbit above is given by looking at the orbit of D under Γ(X, S(ad𝒫)).

By analogy with [Cor, Proposition 2.3], we fix ξ ∈ A^{0}(X, S(ad𝒫)), let d_{t}^{+}, ϑ_{t} be the anti-self-adjoint and self-adjoint parts of exp(ξt) ⋆ D, and set

f(t) := ⟨ϑ_{t}, ϑ_{t}⟩ ∈ A^{0}(X, B),

Now, \frac{d}{dt}(exp(ξt) ⋆ D) = (exp(ξt) ⋆ D)ξ = d_{t}^{+}ξ + ϑ_{t} ∧ ξ, so \frac{d}{dt}ϑ_{t} = d_{t}^{+}ξ and

f′(t) = 2⟨ξ, (d_{t}^{+})^{∗}ϑ_{t}⟩ ∈ A^{0}(X, B).

In other words, D_{t} is harmonic if and only if f′(t) = 0 for all ξ.

Now, if we set D̂_{t} := d_{t}^{+} − ϑ_{t}, the calculations of [Cor, Proposition 2.3] adapt to give

2f′′(t) = ∥D_{t}ξ + D̂_{t}ξ∥^{2} + ∥D_{t}ξ − D̂_{t}ξ∥^{2} ∈ A^{0}(X, B),

where ∥v∥^{2} := ⟨v, v⟩; unlike Corlette, we are only taking inner product with respect to the Kähler metric, not imposing an additional inner product on B.

Note that f′′(t) an element of A^{0}(X, B_{+}), which lies in A^{0}(X, B_{++}) unless D_{t}ξ = 0. If we start with a harmonic connection D, this implies that exp(ξ) ⋆ D is harmonic if and only if Dξ = 0. However, when Dξ = 0 we have exp(ξ) ⋆ D = D, showing that D is the unique harmonic connection in its B_{++}-orbit. □

Corollary 3.2. For a complex C^{∗}-algebra B and an element p ∈ R_{X,x}^{J}(B^{×}), the centraliser Z(π_{dR}(p), B^{×}) of π_{dR}(p) under the adjoint action of B is given by

Z(π_{dR}(p), B^{×}) = exp({b ∈ S(B) : e^{ibt} ∈ Z(p, U(B))∀t ∈ ℝ}) ⋊ Z(p, U(B));

beware that this is the semidirect product of a set with a group.

Proof. Take g ∈ Z(π_{dR}(p), B^{×}), and observe that the polar decomposition allows us to write g = exp(b)u, for u ∈ U(B) and b ∈ S(B). Since π_{dR} is U(B)-equivariant, we have

ad_{exp(b)}(π_{dR}(ad_{u}(p))) = ad_{g}(π_{dR}(p)) = π_{dR}(p).

Thus Proposition 3.1 implies that ad_{u}(p) = p, so u ∈ Z(p, U(B)).

Since Z(π_{dR}(p), B^{×}) is a group, this implies that exp(b) ∈ Z(π_{dR}(p), B^{×}), and hence that exp(b) commutes with the image of π_{dR}(p). We may apply the continuous functional calculus to take logarithms, showing that b itself commutes with the image of π_{dR}(p), so ibt does also. But then exp(ibt) ∈ Z(p, U(B)) for all t, as required.

Conversely, if exp(ibt) ∈ Z(p, U(B)) for all t, then exp(−ibt)π_{dR}(p) exp(ibt) = π_{dR}(p), and differentiating in t for each element of π_{1}(X, x), we see that ib commutes with π_{dR}(p). Thus exp(b) ∈ Z(π_{dR}(p), B^{×}). □

3.1.2. Topological representation spaces and completely bounded maps.

Lemma 3.3. For the real pro-C^{∗}-algebra E_{X,x}^{J} of Proposition 2.22, there is a canonical map π_{dR} : Hom_{pro(BanAlg)}(E_{X,x}^{J}, B) → R_{X,x}^{dR}(B), functorial in real Banach algebras B.

Proof. Given f: E_{X,x}^{J} → B, Lemma 1.20 factors f as the composition of a surjective C^{∗}-homomorphism g: E_{X,x}^{J} → C and a continuous embedding C ↪ B. The de Rham projection of Definition 2.21 then gives us an element π_{dR}(g) ∈ R_{X,x}^{dR}(C). Combining this with the embedding C^{×} → B^{×} then provides the required element of R_{X,x}^{dR}(B). □

Proposition 3.4. For any real C^{∗}-algebra B, the map of Lemma 3.3 induces an injection

π_{dR} : Hom(E_{X,x}^{J}, B)_{cb} ↪ R_{X,x}^{dR}(B),

for the completely bounded morphisms of Definition 1.21.

Proof. Since B can be embedded as a closed C^{∗}-subalgebra of L(H) for some complex Hilbert space H, we may replace B with L(H). By Lemma 1.22, any completely bounded homomorphism f: E_{X,x}^{J} → L(H) is conjugate to a ∗-morphism, since E_{X,x}^{J} is a pro-C^{∗}-algebra. Therefore Proposition 3.1 shows that

Hom(E_{X,x}^{J}, L(H))_{cb}/GL(H) ↪ R_{X,x}^{dR}(L(H))/GL(H).

Take a homomorphism f: E_{X,x}^{J} → L(H) of C^{∗}-algebras; it suffices to show that the centraliser of f and of π_{dR}(f) are equal. By Corollary 3.2, we know that

Z(π_{dR}(f), GL(H)) = exp({b ∈ S(L(H)) : e^{ibt} ∈ Z(f, U(H))∀t ∈ ℝ}) ⋊ Z(f, U(H)).

If e^{ibt} commutes with f for all f, then e^{ibt}fe^{−ibt} = f, and differentiating in t shows that b commutes with f. Therefore exp(b) commutes with f, showing that

Z(π_{dR}(f), GL(H)) ⊂ Z(f, GL(H)).

The reverse inclusion is automatic, giving the required result. □

Remark 3.5. In [Sim4, Pri2], the pro-reductive fundamental group π_{1}(X, x)_{k}^{red} is studied — this is an affine group scheme over k. By Tannakian duality ([DMOS, Ch. II]), we can interpret the dual O(π_{1}(X, x)_{ℝ}^{red})^{∨} of the ring of functions as the ring of discontinuous Gal(ℂ/ℝ)-equivariant endomorphisms of η_{x}^{dR,ss}.

The group scheme π_{1}(X, x)_{k}^{red} encodes all the information about the sets of finitedimensional representations of π_{1}(X, x). As we will now see, (E_{X,x}^{J})_{PN} encodes all the information about their topologies as well.

Theorem 3.6. For any positive integer n, π_{dR} gives a homeomorphism π_{dR,ss} between the space Hom_{pro(BanAlg)}(E_{X,x}^{J}, Mat_{n}(ℂ)) with the topology of pointwise convergence, and the subspace of R_{X,x}^{dR}(Mat_{n}(ℂ)) whose points correspond to semisimple local systems.

Proof. The isomorphism π_{dR,ss} is given on points by the proof of [Cor, Theorem 3.3], since completely bounded algebra homomorphisms E_{X,x}^{J} → Mat_{n}(ℂ) are those conjugate to ∗-homomorphisms, which in turn correspond to harmonic local systems. We need to show that this is a homeomorphism.

Consider the map π_{dR}^{♯}: E_{X,x}^{dR} → E_{X,x}^{J} of pro-Banach algebras. If T_{i} → T is a convergent net in Hom_{pro(BanAlg)}(E_{X,x}^{J}, Mat_{n}(ℂ)), then T_{i}(π_{dR}^{♯}(γ)) → T (π_{dR}^{♯}(γ)), so π_{dR,ss} is continuous.

Now, the de Rham projection π_{dR} : R_{X,x}^{J}(B) → R_{X,x}^{dR}(B) is automatically injective for C^{∗}-algebras B. Thus the inclusion in E_{X,x}^{J} of the pro-C^{∗}-subalgebra generated by π_{dR}^{♯}(π_{1}(X, x)) must be an epimorphism, since ℝ[π_{1}(X, x)] is dense in E_{X,x}^{dR}. By [Rei, Proposition 2], an epimorphism of C^{∗}-algebras is surjective, so E_{X,x}^{J} must be generated as a pro-C^{∗}-algebra by π_{dR}^{♯}(π_{1}(X, x)), and as a pro-Banach algebra by π_{dR}^{♯}(π_{1}(X, x)) ∪ π_{dR}^{♯}(π_{1}(X, x))^{∗}.

Now, if we have a convergent sequence π_{dR}(T_{i}) → π_{dR}(T), then T_{i}(π_{dR}^{♯}γ) → T (π_{dR}^{♯}γ) for all γ ∈ π_{1}(X, x), so it suffices to show that the same holds for (π_{dR}^{♯}γ)^{∗}. Given T ∈ Hom_{pro(BanAlg)}(E_{X,x}^{J}, Mat_{n}(ℂ)), define T^{∗} by T^{∗}(e) := T (e^{∗})^{∗}. We wish to show that T_{i}^{∗}(π_{dR}^{♯}γ) → T^{∗}(π_{dR}^{♯}γ), which will follow if π_{dR}(T_{i}^{∗}) → π_{dR}(T^{∗}).

As in the proof of Proposition 3.4, we can write T = ad_{g}(S), for S: E_{X,x}^{J}, Mat_{n}(ℂ) a ∗-homomorphism and g ∈ GL_{n}(ℂ). Then T^{∗} = ad_{(g^{∗})^{−1}}(S) = ad_{(gg^{∗})^{−1}}(T); this means that if we write π_{dR}(T) = (𝒱 , D, f), then π_{dR}(T^{∗}) = (𝒱 , D, (f^{∗})^{−1}), where f^{∗}: x^{∗}𝒱 → ℂ^{n} is defined using the harmonic metric on 𝒱 and the standard inner product on ℂ^{n}.

Explicitly, this means that we can describe the involution ∗ on semisimple elements of R_{X,x}^{B}(Mat_{n}(ℂ)) by ρ^{∗} = (C(ρ)^{−1})^{†}, where C is the Cartan involution of [Sim4]. If D = d^{+} + ϑ is the decomposition into anti-hermitian and hermitian parts with respect to the harmonic metric, then C(𝒱 , d^{+} + ϑ, f) = (𝒱 , d^{+} − ϑ, f). The proof of [Cor, Theorem 3.3] ensures that the decomposition D ↦ (d^{+}, ϑ) is continuous in D, so C is continuous. Hence π_{dR}(T) ↦ π_{dR}(T^{∗}) is also continuous, which gives the convergence required. □

3.1.3. The polynormal completion and Tannaka duality.

Definition 3.7. Let FDR_{X,x}^{dR} be the category of pairs (V, p) for V ∈ FDVect and p ∈ R_{X,x}^{dR}(End(V)). Morphisms f: (V_{1}, p_{1}) → (V_{2}, p_{2}) are given by linear maps f: V_{1} → V_{2} for which the adjoint action of

(\begin{matrix}id & 0 \\ f & id \end{matrix}) ∈ (\begin{matrix}End(V_{1}) & 0 \\ Hom(V_{1}, V_{2}) & End(V_{2}) \end{matrix})

on F DR_{X,x}^{dR} ( \begin{matrix}End(V_{1}) & 0 \\ Hom(V_{1},V_{2}) & End(V_{2}) \end{matrix} ) fixes p_{1} ⊕ p_{2}.

Write η_{x}^{dR} : FDR_{X,x}^{dR} → FDVect for the fibre functor (V, p) ↦ V. Let FDR_{X,x}^{dR,ss} ⊂ FDR_{X,x}^{dR} be the full subcategory in which objects correspond to semisimple local systems, with fibre functor η_{x}^{dR,ss}.

Proposition 3.8. The ring (E_{X,x}^{J})_{PN} is isomorphic to the ring of continuous Gal(ℂ/ℝ)- equivariant endomorphisms of η_{x}^{dR,ss}.

Proof. This just combines Lemma 1.28 and Theorem 3.6. □

Remark 3.9. This leads us to contemplate the structure of the ring of continuous endomorphisms f of η_{x}^{dR}. Any finite-dimensional ℂ-algebra arises as a subalgebra of some matrix algebra, so any such f induces continuous maps Hom_{Gp}(π_{1}(X, x), B^{×}) → B for all finite-dimensional algebras B. In particular, this holds when B = Mat_{n}(A) for some Artinian ℂ-algebra A, from which it follows that the maps

f_{V} : R_{X,x}^{dR}(End(V)) → End(V)

are all analytic. In other words, any continuous endomorphism of η_{x}^{dR} is automatically analytic.

When π_{1}(X, x) is abelian, this ensures that the ring (E_{X,x}^{dR})_{FD} ⊗ ℂ of such endomorphisms is the ring of complex analytic functions on Hom_{Gp}(π_{1}(X, x), ℂ^{∗}), which by Example 2.12 is just ℂ[π_{1}(X, x)]^{an}.

In general, the ring (E_{X,x}^{dR})_{FD} is an inverse limit of polynormal Banach algebras, but it is not clear to the author whether it is the pro-polynormal completion of the Fréchet algebra E_{X,x}^{dR}. In general, the map π_{1}(X, x) → (E_{X,x}^{dR})_{FD} need not be injective: [ABC^{+}, §6.5] gives examples of Kähler groups with no faithful linear representations.

Definition 3.10. Given a k-normal real C^{∗}-algebra B, define R_{X,x}^{dR,ss}(B) ⊂ R_{X,x}^{dR}(B) to be the subspace consisting of those p for which ψ(p) corresponds to a semisimple local system for all ψ: B → Mat_{k}(ℂ).

Corollary 3.11. For any k-normal real C^{∗}-algebra B, R_{X,x}^{dR,ss}(B) is isomorphic to the set of continuous algebra homomorphisms E_{X,x}^{J} → B.

Proof. Since B is k-normal, any such morphism E_{X,x}^{J} → B factors uniquely through (E_{X,x}^{J})_{PN} By Proposition 3.8, a homomorphism (E_{X,x}^{J})_{PN} → B corresponds to a continuous Gal(ℂ/ℝ)-equivariant functor p^{∗}: FDRep(B) → FDR_{X,x}^{dR,ss} of topological categories fibred over FDVect. An element p ∈ R_{X,x}^{dR}(B) satisfies this condition provided p^{∗} maps to FDR_{X,x}^{dR,ss} ⊂ FDR_{X,x}^{dR}. □

Remark 3.12. It is natural to ask whether the non-abelian Hodge theorem of [Sim6] extends from finite-dimensional matrix algebras to more general C^{∗}-algebras B. Proposition 3.8 can be thought of as an extension of the correspondence to polynormal C^{∗}-algebras, but it seems unlikely to adapt much further, because the arguments of [Sim6, Cor] rely on sequential compactness of U_{n}. However, it is conceivable that suitable solutions of the heat equation might give rise to asymptotic C^{∗}-homomorphisms.

3.2. Residually finite-dimensional completion, products and complex tori.

Definition 3.13. A pro-C^{∗}-algebra A is said to be residually finite-dimensional if it has a separating family of finite-dimensional ∗-representations.

Given a pro-C^{∗}-algebra A, define the pro-C^{∗}-algebra A_{RFD} to be the universal residually finite-dimensional quotient of A. Explicitly, A_{RFD} is the quotient of A with respect to the pro-ideal given by the system of kernels of finite-dimensional ∗-representations of A.

Note that polynormal C^{∗}-algebras are residually finite-dimensional, so we have completions A → A_{RFD} → A_{PN} for general A.

Proposition 3.14. Given compact connected Kähler manifolds X and Y, there is an isomorphism (E_{X×Y,(x,y)}^{J})_{RFD} ≅ (E_{X,x}^{J})_{RFD} ˆ⊗(E_{Y,y}^{J})_{RFD}.

Proof. The projections give canonical elements of R_{X×Y,(x,y)}^{J}(E_{X,x}^{J}) and R_{X×Y,(x,y)}^{J}(E_{Y,y}^{J}), which by Lemma 2.24 give rise to a canonical map

f: E_{X×Y,(x,y)}^{J} → E_{X,x}^{J} ˆ⊗E_{Y,y}^{J}.

By [Cor], every finite-dimensional representation of E_{X×Y,(x,y)}^{J} corresponds to a semisimple representation of π_{1}(X × Y, (x, y)) = π_{1}(X, x) × π_{1}(Y, y), so factors through E_{X}^{J} ˆ⊗E_{Y}^{J}. Since (E_{X×Y,(x,y)}^{J})_{RFD} ⊂ ∏_{i} End(V_{i}) where V_{i} ranges over finite-dimensional irreducible representations, this implies that

f_{RFD} : (E_{X×Y,(x,y)}^{J})_{RFD} → (E_{X,x}^{J})_{RFD} ˆ⊗(E_{Y,y}^{J})_{RFD}

is injective. However, the basepoint y gives us a map X → X × Y, and hence E_{X,x}^{J} → E_{X×Y,(x,y)}^{J}, ensuring that (E_{X}^{J})_{RFD} lies in the image of f_{RFD}; a similar argument applies to Y. Thus f_{RFD} is surjective, and hence an isomorphism. □

Remark 3.15. Note that we have only imposed the hypothesis that the manifolds be Kähler in order to use the functoriality properties of Lemma 2.25, since Lemma 2.37 is too weak to apply to the maps between X × Y and X.

Lemma 3.16. Given a compact connected Kähler manifold X, the commutative quotient (E_{X,x}^{J})^{ab} is given by

(E_{X,x}^{J})^{ab} ⊗ ℂ = C(H^{1}(X, ℂ^{∗}), ℂ)
(E_{X,x}^{J})^{ab} = \{f ∈ C(H^{1}(X, ℂ^{∗}), ℂ) : f(ρ̄) = \overline{f(ρ)}\}.

Proof. Since (E_{X,x}^{J})^{ab} is a commutative C^{∗}-algebra, the Gelfand–Naimark theorem gives (E_{X,x}^{J})^{ab} ⊗ ℂ ≅ C(Hom(E_{X,x}^{J}, ℂ), ℂ), with E_{X,x}^{J})^{ab} given by Gal(ℂ/ℝ)-invariants. By §3.1.3, we have a homeomorphism Hom(E_{X,e}^{J}, ℂ) ≅ H^{1}(X, ℂ^{∗}), which completes the proof.

□

Corollary 3.17. For X a complex torus with identity e and a fixed Riemannian metric, we have

(E_{X,e}^{J})_{RFD} ⊗ ℂ = C(H^{1}(X, ℂ^{∗}), ℂ)
(E_{X,e}^{J})_{RFD} = \{f ∈ C(H^{1}(X, ℂ^{∗}), ℂ) : f(ρ̄) = \overline{f(ρ)}\}.

Proof. Multiplication on X gives a pointed morphism X × X → X and hence by functoriality of P in X and Proposition 3.14, we have a morphism

(E_{X,e}^{J})_{RFD} ˆ⊗(E_{X,e}^{J})_{RFD} → (E_{X,e}^{J})_{RFD}

of real pro-C^{∗}-algebras, and we may apply Lemma 3.16. □

Remark 3.18. If it were the case that all irreducible representations of π_{1}(X, x) were harmonic and similarly for π_{1}(Y, y), then the proof of Proposition 3.14 would adapt to show that E_{X×Y,(x,y)}^{J} ≅ E_{X,x}^{J} ˆ⊗E_{Y,y}^{J}. As in the proof of Corollary 3.17, that would then imply commutativity of E_{X,e}^{J} for complex tori (X, e), giving E_{X,e}^{J}⊗ℂ = C(Hom(π_{1}(X, e), ℂ^{∗})).

Lemma 3.19. Given a compact connected Kähler manifold X, the grouplike elements G((E_{X,x}^{J})^{ab}) (see Lemma 2.11) of the commutative quotient (E_{X,x}^{J})^{ab} are given by

G((E_{X,x}^{J})^{ab} ⊗ ℂ) ≅ H_{1}(X, ℤ ⊕ ℂ)
G((E_{X,x}^{J})^{ab}) ≅ H_{1}(X, ℤ ⊕ ℝ),

with the map π_{1}(X, x)^{ab} → G((E_{X,x}^{J})^{ab}) given by the diagonal map ℤ → ℤ ⊕ ℝ.

Proof. The coalgebra structure on (E_{X,x}^{J})^{ab} corresponds under Lemma 3.16 to the group structure on H^{1}(X, ℂ^{∗}). Thus G((E_{X,x}^{J})^{ab}) consists of continuous functions f: H^{1}(X, ℂ^{∗}) → ℂ with f(1) = 1 and f(ab) = f(a)f(b).

We have an isomorphism ℂ^{∗} ≅ S^{1} × ℝ, given by re^{iϕ} ↦ (ϕ, r). Thus H^{1}(X, ℂ^{∗}) ≅ H^{1}(X, S^{1} × H^{1}(X, ℝ). By Pontrjagin duality, a continuous group homomorphism H^{1}(X, S^{1}) → ℂ^{∗} is just an element of H_{1}(X, ℤ), and a continuous group homomorphism H^{1}(X, ℝ) → ℂ^{∗} is an element of H_{1}(X, ℂ). □

3.3. The Dolbeault projection. Now let X be a compact connected Kähler manifold with basepoint x ∈ X. Proposition 3.20. For all complex C^{∗}-algebras B, the Dolbeault projection

π_{Dol} : R_{X,x}^{J}(B) → R_{X,x}^{Dol}(B)

has the property that if p_{1}, p_{2} ∈ R_{X,x}^{J}(B) and if ad_{b}π_{Dol}(p_{1}) = π_{Dol}(p_{2}) for some strictly positive self-adjoint element b ∈ B, then p_{1} = p_{2}.

Thus

π_{Dol} : R_{X,x}^{J}(B)/U(B) → R_{X,x}^{Dol}(B)/B^{×}

is injective. Proof. The proof of Proposition 3.1 adapts, replacing D with D′′. □ Corollary 3.21. For an element p ∈ R_{X,x}^{J}(B^{×}), the centraliser Z(π_{Dol}(p), B^{×}) of π_{Dol}(p) under the adjoint action of B is given by

Z(π_{Dol}(p), B^{×}) = exp({b ∈ S(B) : e^{ibt} ∈ Z(p, U(B))∀t ∈ ℝ}) ⋊ Z(p, U(B)); beware that this is the semidirect product of a set with a group. Proof. The proof of Corollary 3.2 carries over. □ Proposition 3.22. For the real pro-C^{∗}-algebra E_{X,x}^{J} of Proposition 2.22, there is a canonical map Hom_{pro(BanAlg)}(E_{X,x}^{J}, B) → R_{X,x}^{Dol}(B), functorial in complex Banach algebras B. This induces an injection

Hom(E_{X,x}^{J}, B)_{cb} ↪ R_{X,x}^{Dol}(B)

whenever B is a C^{∗}-algebra. Proof. The proofs of Lemma 3.3 and Proposition 3.4 carry over to this context, replacing Proposition 3.1 and Corollary 3.2 with Proposition 3.20 and Corollary 3.21. □ Theorem 3.23. For any positive integer n, there is a homeomorphism π_{Dol,st} between the space Hom_{pro(BanAlg)}(E_{X,x}^{J}, Mat_{n}(ℂ)) with the topology of pointwise convergence, and the subspace of R_{X,x}^{Dol}(Mat_{n}(ℂ)) consisting of polystable Higgs bundles E with ch_{1}(E) · [ω]^{dim X−1} = 0 and ch_{2}(E) · [ω]^{dim X−2} = 0. Proof. The isomorphism of points is given by [Sim4, Theorem 1]. Replacing Proposition 3.4 with Proposition 3.22, the argument from the proof of Theorem 3.6 shows that the map π_{Dol} : Hom_{pro(BanAlg)}(E_{X,x}^{J}, Mat_{n}(ℂ)) → R_{X,x}^{Dol}(Mat_{n}(ℂ)) is continuous, so we just need to show that it is open.

Now, [Sim6, Proposition 7.9] implies that the isomorphism π_{dR,ss}∘π_{Dol,st}^{−1} is continuous. Since π_{dR,ss} is a homeomorphism by Theorem 3.6, π_{Dol,st} must also be a homeomorphism.

□

Definition 3.24. Let FDR_{X,x}^{Dol} be the category of pairs (V, p) for V ∈ FDVect and p ∈ R_{X,x}^{Dol}(End(V)), with morphisms defined by adapting the formulae of Definition 3.7.

Let FDR_{X,x}^{Dol,st} ⊂ FDR_{X,x}^{Dol} be the full subcategory in which objects correspond to those of Theorem 3.23. Write η_{x}^{Dol} : FDR_{X,x}^{Dol} → FDVect, η_{x}^{Dol,st} : FDR_{X,x}^{Dol,st} → FDVect for the fibre functors (V, p) ↦ V.

Proposition 3.25. The ring (E_{X,x}^{J})_{PN} ⊗ ℂ is isomorphic to the ring of continuous endomorphisms of η_{x}^{Dol,st}.

Proof. The proof of Proposition 3.8 carries over, replacing Theorem 3.6 with Theorem 3.23. □

Definition 3.26. Given a k-normal complex C^{∗}-algebra B, define R_{X,x}^{Dol,st}(B) ⊂ R_{X,x}^{Dol}(B) to be the subspace consisting of those p for which ψ(p) ∈ FDR_{X,x}^{Dol,st} for all ψ: B → Mat_{k}(ℂ).

Corollary 3.27. For any k-normal complex C^{∗}-algebra B, R_{X,x}^{Dol,st}(B) is isomorphic to the set of continuous algebra homomorphisms E_{X,x}^{J} → B.

Proof. The proof of Corollary 3.11 carries over, replacing Proposition 3.8 with Proposition 3.25. □

3.4. Circle actions and C^{∗}-dynamical systems.

Definition 3.28. Define a circle action on a (real or complex) pro-C^{∗}-algebra A to be a continuous group homomorphism from S^{1} to Aut_{pro(C^{∗}Alg)}(A). Here, the topology on Aut_{pro(C^{∗}Alg)}(A) is defined pointwise, so a net f_{i} converges to f if and only if f_{i}(a) → f(a) for all a ∈ A.

The following is immediate:

Lemma 3.29. Giving a circle action on a C^{∗}-algebra A is equivalent to giving a pro-C^{∗}-algebra homomorphism f: A → C(S^{1}, A) satisfying

(1) 1^{∗} ∘ f = id_{A}: A → C({1}, A) = A;

(2) the diagram

A  ⟶^{f} C(S^{1}, A)
f↓  ↓^{C(S^{1},f)}
C(S^{1}, A) ⟶^{m^{∗}} C(S^{1} × S^{1}, A)

commutes, where m: S^{1} × S^{1} → S^{1} is the multiplication.

Lemma 3.30. If a functor F: C^{∗}Alg_{k} → Set is represented by a pro-C^{∗}-algebra A, then to give a circle action on A is equivalent to giving maps

α_{B}: F (B) → F (C(S^{1}, B)),

functorial in B, such that (1) F (1^{∗}) ∘ α_{B} = id_{F (B)} : F (B) → F (B);

(2) the diagram

F (B)  ⟶^{α_{B}} F (C(S^{1}, B))
α_{B} ↓  ↓^{α_{C(S^{1},B)}}
F (C(S^{1}, B)) ⟶^{F (m^{∗})} F (C(S^{1} × S^{1}, B))

commutes, where m: S^{1} × S^{1} → S^{1} is the multiplication.

Proof. If A has a circle action α, then a homomorphism h: A → B gives rise to C(S^{1}, h): C(S^{1}, A) → C(S^{1}, B), and we define α_{B}(h) := C(S^{1}, h) ∘ α. This clearly satisfies the required properties.

Conversely, given maps α_{B} as above, write A = \underleftarrow{lim}_{i} A_{i} as an inverse limit of C^{∗}- algebras, and let h_{i}: A → A_{i} be the structure map. Then α_{A_{i}}(h_{i}) ∈ F (C(S^{1}, A_{i})) is a map A → C(S^{1}, A_{i}). Since the α_{A_{i}}(h_{i}) are compatible, we may take the inverse limit, giving a map

α: A → C(S^{1}, A)

To see that this is a group homomorphism, just observe that the conditions above ensure that h_{i} ∘ 1^{∗} ∘ α = h_{i} and

C(S^{1} × S^{1}, h_{i}) ∘ C(S^{1}, α) ∘ α = C(S^{1} × S^{1}, h_{i}) ∘ m^{∗}α

for all i. Taking the inverse limit over i shows that this satisfies the conditions of Lemma 3.29.

Finally, note that these two constructions are clearly inverse to each other. □

Proposition 3.31. For every compact Kähler manifold X, there is a canonical continuous circle action on E_{X,x}^{J}.

Proof. Given (U(𝒫), D, f) ∈ R_{X,x}^{J}(B), define α(U(𝒫), D, f) ∈ R_{X,x}^{J}(C(S^{1}, B)) as follows.

Decompose D = d^{+} + ϑ into anti-self-adjoint and self-adjoint parts. Set α(U(𝒫)) := C(S^{1}, U(𝒫)) = U(𝒫)×_{𝒜_{X}^{0}(U(B))} 𝒜_{X}^{0}(C(S^{1}, U(B))), and then define α(D) := d^{+} +t⋄ϑ, where t ∈ C(S^{1}, ℂ) is the canonical embedding and ⋄ is from Definition 2.17. Thus we have constructed α(U(𝒫), D, f) := (C(S^{1}, U(𝒫), d^{+} + t ⋄ ϑ, C(S^{1}, f)), and it is easy to check that this satisfies the conditions of Lemma 3.30. □

Remark 3.32. By considering finite dimensional quotients of E_{X,x}^{J}, the circle action induces a continuous map

S^{1} × E_{X,x}^{J} → O(π_{1}(X, x)_{ℝ}^{red})′,

for O(π_{1}(X, x)_{ℝ}^{red})^{∨} as in Remark 3.5. This descends to a discontinuous action of S^{1} on O(π_{1}(X, x)_{ℝ}^{red})^{∨}, as in [Sim4] (made explicit in the real case as [Pri2, Lemma 6.5]).

Note, however that the circle action descends to continuous actions on (E_{X,x}^{J})_{RFD}, (E_{X,x}^{J})_{PN} (which are subalgebras of O(π_{1}(X, x)_{ℝ}^{red})^{∨}, though not closed).

Continuity of the circle action ensures that the map

S^{1} × π_{1}(X, x) → E_{X,x}^{J}

is continuous, and hence that the induced map S^{1} × π_{1}(X, x) → π_{1}(X, x)_{ℝ}^{red}(ℝ) ⊂ O(π_{1}(X, x)_{ℝ}^{red})^{∨} is continuous. Thus a continuous circle action on E_{X,x}^{J} gives rise to a pure Hodge structure on π_{1}(X, x)^{red} in the sense of [Sim4, §5], but without needing to refer to π_{1}(X, x) itself. This suggests that the most natural definition of a pure non-abelian Hodge structure is a continuous circle action on a pro-C^{∗}-bialgebra.

Example 3.33. Lemma 3.16 gives an isomorphism

(E_{X,x}^{J})^{ab} = \{f ∈ C(H^{1}(X, ℂ^{∗}), ℂ) : f(ρ̄) = \overline{f(ρ)}\},

and 3.19 then shows that the grouplike elements are G((E_{X,x}^{J})^{ab}) ≅ H_{1}(X, ℤ ⊕ ℝ). To describe the circle action on (E_{X,x}^{J})^{ab}, it thus suffices to describe it on the space H^{1}(X, ℂ^{∗}) of one-dimensional complex representations.

Taking the decomposition D = d^{+} + ϑ of a flat connection D into anti-hermitian and hermitian parts, note that we must have (d^{+})^{2} = ϑ^{2} = 0, because commutativity of ℂ^{∗} ensures that commutators vanish, everything else vanishing by hypothesis. This decomposition therefore corresponds to the isomorphism H^{1}(X, ℂ^{∗}) ≅ H^{1}(X, §^{1}) × H^{1}(X, ℝ). Since the action is given by ϑ ↦ t ⋄ ϑ for t ∈ S^{1}, it follows that the S^{1}-action is just the ⋄-action on H^{1}(X, ℝ).

On G((E_{X,x}^{J})^{ab}) ≅ H_{1}(X, ℤ) ⊕ H_{1}(X, ℝ), this means that the circle action fixes H_{1}(X, ℤ) and acts with the ⋄-action on H_{1}(X, ℝ) = H_{1}(X, ℝ)^{∨}.

Definition 3.34. Recall from [Wil, Definition 2.6] that a C^{∗}-dynamical system is a triple (A, G, α), for G a locally compact topological group, A a C^{∗}-algebra, and α a continuous action of G on a A.

Lemma 3.35. The circle action α of Proposition 3.31 gives rise to a pro-C^{∗}-dynamical system (E_{X,x}^{J}, S^{1}, α), i.e. an inverse system of C^{∗}-dynamical systems.

Proof. Since E_{X,x}^{J} is a pro-C^{∗}-algebra, we may write it as an inverse system E_{X,x}^{J} = \underleftarrow{lim}_{i} E_{i}, for C^{∗}-algebras E_{i}. The circle action then sends the structure map h_{i}: E_{X,x}^{J} → E_{i} to the map C(S^{1}, h_{i}) ∘ α: E_{X,x}^{J} → C(S^{1}, E_{i}), and evaluation at 1 ∈ S^{1} recovers h_{i}. We may therefore set E_{α(i)} to be the closure of the image of E_{X,x}^{J} → C(S^{1}, E_{i}), and observe that E_{α(i)} is S^{1}-equivariant, with E_{X,x}^{J} = \underleftarrow{lim}_{i} E_{α(i)}.

Thus (E_{X,x}^{J}, S^{1}, α) = \underleftarrow{lim}_{i}(E_{α(i)}, S^{1}, α) is a pro-C^{∗}-dynamical system. □

The following is taken from [Wil, Lemma 2.27]:

Definition 3.36. Given a C^{∗}-dynamical system (A, G, α) and f ∈ C_{c}(G, A), define

∥f∥ := sup{∥π ⋊ U(f)∥: (π, U) a covariant representation of (A, G, α)}.

Then ∥ − ∥ is called the universal norm, and dominated by ∥ − ∥_{1}.

The completion of C_{c}(G, A) with respect to ∥ − ∥ is the crossed product of A by G, denoted A ⋊_{α} G.

Definition 3.37. Define a polarised real Hilbert variation of Hodge structures of weight n on X to be a real local system 𝕍, with a pluriharmonic metric on 𝒜_{X}^{0}(𝕍), equipped with a Hilbert space decomposition

𝒜_{X}^{0}(𝕍) ⊗ ℂ = ˆ ⨁_{p+q=n}𝒱 ^{pq},

(where ˆ⨁ denotes Hilbert space direct sum), with \overline{𝒱 ^{pq}} = 𝒱 ^{qp}, and satisfying the conditions

∂ : 𝒱 ^{pq} → 𝒱 ^{pq} ⊗_{𝒜_{X}^{0}(ℂ)} 𝒜_{X}^{10}, θ̄: 𝒱 ^{pq} → 𝒱 ^{p+1,q−1} ⊗_{𝒜_{X}^{0}(ℂ)} 𝒜^{01_{X}} ,

for the decomposition D = ∂ + ¯∂ + θ + θ̄ of Definition 2.31.

Proposition 3.38. Real Hilbert space representations of the non-unital pro-C^{∗}-algebra E_{X,x}^{J} ⋊_{α} S^{1} correspond to framed weight 0 polarised real Hilbert variations of Hodge structure.

Proof. By [Wil, Proposition 2.29], a ∗-representation E_{X,x}^{J} ⋊_{α} S^{1} → L(H) for a Hilbert space H consists of:

(1) a ∗-representation ρ: E_{X,x}^{J} → L(H), and

(2) a continuous representation u: S^{1} → U(H) such that

ρ(α(t, a)) = u(t)ρ(a)u(t)^{−1}

for all a ∈ E_{X,x}^{J}, t ∈ S^{1}.

In other words, in R_{X,x}^{J}(C(S^{1}, L(H))), we have α(ρ) = uρu^{−1}, so α(ρ) and ρ are isomorphic in the groupoid R_{X}^{J}(C(S^{1}, L(H))).

Now, by definition of E_{X,x}^{J}, the representation ρ corresponds to a real local system 𝕍, with a pluriharmonic metric on 𝒜_{X}^{0}(𝕍) and a Hilbert space isomorphism f: 𝕍_{x} → H. The representation α(ρ) corresponds to the connection α(D) := d^{+} + t ⋄ ϑ on 𝒜_{X}^{0}(C(S^{1}, 𝕍)) for the standard co-ordinate t: S^{1} → ℂ, together with framing f.

The condition that α(ρ) and ρ are isomorphic then gives us a unitary gauge transformation g between them. In other words, we have a continuous representation g: S^{1} → Γ(X, U(𝒜_{X}^{0}(𝕍))) with α(D) ∘ g = g ∘ D. We must also have g_{x} = u.

Thus g gives us a Hilbert space decomposition

𝒜_{X}^{0}(𝕍) ⊗ ℂ = ˆ ⨁_{p+q=0}𝒱 ^{pq},

with \overline{𝒱 ^{pq}} = 𝒱 ^{qp}, and g(t) acting on 𝒱 ^{pq} as multiplication by t^{p−q}. The condition α(D) ∘ g = g ∘ D then forces the conditions

∂ : 𝒱 ^{pq} → 𝒱 ^{pq} ⊗_{𝒜_{X}^{0}(ℂ)} 𝒜_{X}^{10}, θ̄: 𝒱 ^{pq} → 𝒱 ^{p+1,q−1} ⊗_{𝒜_{X}^{0}(ℂ)} 𝒜^{01_{X}} ,

as required. □

Remark 3.39. Given any E_{X,x}^{J}-representation V, [Wil, Example 2.14] gives an E_{X,x}^{J}⋊S^{1}- representation Ind_{e}^{S^{1}}V. Its underlying Hilbert space is just the space L^{2}(S^{1}, V) of L^{2}-measurable V -valued forms on the circle with respect to Haar measure. For the pluriharmonic local system 𝕍 associated to V, this therefore gives a weight 0 variation Ind_{e}^{S^{1}}𝕍 of Hodge structures on X, with 𝒜_{X}^{0}(Ind_{e}^{S^{1}}𝕍) = 𝒜_{X}^{0}(L^{2}(S^{1}, 𝕍)).

4. Hodge decompositions on cohomology

Fix a compact Kähler manifold X.

Definition 4.1. Given a pluriharmonic local system 𝕍 in real Hilbert spaces on X (as in Example 2.20), the inner product on 𝕍 combines with the Kähler metric on X to give inner products ⟨−, −⟩ on the spaces A^{n}(X, 𝕍) for all n. Given an operator F on A^{∗}(X, 𝕍), we denote the adjoint operator by F^{∗}. Let Δ = DD^{∗} + D^{∗}D.

4.1. Sobolev spaces. Note that in general, the Laplacian Δ is not a bounded operator in the L^{2} norm. We therefore introduce a system of Sobolev norms:

Definition 4.2. Define L_{(2),s}^{n}(X, 𝕍) to be the completion of A^{n}(X, 𝕍) with respect to the inner product ⟨v, w⟩_{s} := ⟨v, (I + Δ)^{s}w⟩.

Note that we then have bounded operators D, D^{c}: L_{(2),s}^{n}(X, 𝕍) → L_{(2),s−1}^{n+1}(X, 𝕍), D^{∗}, D^{c∗}: L_{(2),s}^{n}(X, 𝕍) → L_{(2),s−1}^{n−1}(X, 𝕍) and Δ: L_{(2),s}^{n}(X, 𝕍) → L_{(2),s−2}^{n}(X, 𝕍).

Proposition 4.3. The maps (I +Δ)^{k}: L_{(2),s}^{n}(X, 𝕍) → L_{(2),s−2k}^{n}(X, 𝕍) are Hilbert space isomorphisms, and there are canonical inclusions L_{(2),s}^{n}(X, 𝕍) ⊂ L_{(2),s−1}^{n}(X, 𝕍).

Proof. The proofs of [Dod, Proposition 2.3 and Lemma 2.4] carry over to this generality. □

Definition 4.4. Define H^{n}(X, 𝕍) ⊂ A^{n}(X, 𝕍) to consist of forms with Δα = 0. Regard this as a pre-Hilbert space with the inner product ⟨−, −⟩.

The following implies that H^{n}(X, 𝕍) is in fact a Hilbert space:

Lemma 4.5. The inclusions

H^{n}(X, 𝕍) → {α ∈ L_{(2),0}^{n} : Dα = D^{∗}α = 0} → {α ∈ L_{(2),0}^{n} : Δα = 0}

are Hilbert space isomorphisms.

Proof. When 𝕍 is the local system associated to the π_{1}(X)-representation ℓ^{2}(π_{1}(X, x)), this is [Dod, Lemma 2.5], but the same proof carries over. □

4.1.1. Decomposition into eigenspaces.

Definition 4.6. Define T to be the composition of L_{(2),0}^{n}(X, 𝕍) ⟶^{(I+Δ)^{−1}} L_{(2),2}^{n}(X, 𝕍) ↪ L_{(2),0}^{n}(X, 𝕍). This is bounded and self-adjoint, with spectrum σ(T) ⊂ (0, 1]. Thus the spectral decomposition gives T = ∫_{(0,1]} λdπ_{λ} for some projection-valued measure π on (0, 1].

For S ⊂ [0, ∞) measurable, write

ν(S) := π_{{(1+ρ)^{−1} : ρ∈S}}.

Thus

T = ∫_{ρ∈[0,∞)}(1 + ρ)^{−1}dν_{ρ},

and for v ∈ L_{(2),s+2}(X, 𝕍) we have

Δv = ∫_{ρ∈[0,∞)} ρdν_{ρ}v ∈ L_{(2),s}(X, 𝕍).

If we set E^{n}(S) := ν(S)L_{(2),0}^{n}(X, 𝕍), observe that E^{n} defines a measurable family of Hilbert spaces on [0, ∞), and that we have direct integral decompositions

L_{(2),s}^{n}(X, 𝕍) = ∫_{ρ∈[0,∞)}^{⊕}(1 + ρ)^{−s/2}E_{ρ}^{n}.

4.1.2. Harmonic decomposition of eigenspaces. Since the operators D, D^{c}, D^{∗}, D^{c∗} commute with Δ, they descend to each graded Hilbert space E^{n}(S), provided S is bounded above.

If S also has a strictly positive lower bound, then Δ is invertible on E^{n}(S), so

E^{n}(S) = ΔE^{n}(S).

As Δ = DD^{∗} + D^{∗}D = D^{c}D^{c∗} + D^{c∗}D^{c}, with [D, D^{c}] = [D^{∗}, D^{c}] = [D^{c∗}, D] = 0, this implies that

E^{n}(S) = DE^{n−1}(S) ⊕ D^{∗}E^{n+1}(S) = D^{c}E^{n−1}(S) ⊕ D^{c∗}E^{n+1}(S)
= DD^{c}E^{n−2}(S) ⊕ D^{∗}D^{c}E^{n}(S) ⊕ DD^{c∗}E^{n}(S) ⊕ D^{∗}D^{c∗}E^{n+2}(S).

Furthermore, D: D^{∗}E^{n}(S) → DE^{n−1}(S) and D^{∗}: DE^{n}(S) → D^{∗}E^{n+1}(S) are isomorphisms, with similar statements for D^{c}.

If S is just bounded above and does not contain 0, then the statements above still hold if we replace subspaces with their closures:

Proposition 4.7. There are Hilbert space decompositions

L_{(2),s}^{n}(X, 𝕍) = H^{n}(X, 𝕍) ⊕ \overline{ΔL_{(2),s+2}^{n}(X, 𝕍)}
= H^{n}(X, 𝕍) ⊕ \overline{DL_{(2),s+1}^{n−1}(X, 𝕍)} ⊕ \overline{D^{∗}L_{(2),s+1}^{n+1}(X, 𝕍)}
= H^{n}(X, 𝕍) ⊕ \overline{D^{c}L_{(2),s+1}^{n−1}(X, 𝕍)} ⊕ \overline{D^{c∗}L_{(2),s+1}^{n+1}(X, 𝕍)}
= H^{n}(X, 𝕍) ⊕ \overline{DD^{c}L_{(2),s+2}^{n−2}(X, 𝕍)} ⊕ \overline{D^{∗}D^{c}L_{(2),s+2}^{n}(X, 𝕍)}
⊕ \overline{DD^{c∗}L_{(2),s+2}^{n}(X, 𝕍)} ⊕ \overline{D^{∗}D^{c∗}L_{(2),s+2}^{n+2}(X, 𝕍)}.

for all s.

Proof. This is essentially the Hodge Theorem, and we can construct the decomposition by a slight modification of [GH, pp94–96].

We can define an approximate Green’s function by

G_{ϵ} := ∫_{(0,1−ϵ]} \frac{λ}{1 − λ}dπ_{λ}.

Now, note that (I +Δ)G_{ϵ} = ∫_{(0,1−ϵ]} \frac{1}{1−λ}dπ_{λ}, which is bounded, so G_{ϵ} is the composition of the inclusion L_{(2),s+2}^{n}(X, 𝕍) ↪ L_{(2),s}^{n}(X, 𝕍) with a map

G_{ϵ}: L_{(2),s}^{n}(X, 𝕍) → L_{(2),s+2}^{n}(X, 𝕍).

Also note that G_{ϵ} commutes with Δ, and that ΔG_{ϵ} = π(0, 1 − ϵ] = I − π[1 − ϵ, 1]. As ϵ → 0, this means that π(1) + ΔG_{ϵ} converges weakly to I. Since π(1) is projection onto H^{n}(X, 𝕍), this gives the decomposition required (noting that norm closure and weak closure of a subspace are the same, by the Hahn–Banach Theorem). □

Now, for v ∈ D^{∗}E^{n}(S), ⟨Dv, Dv⟩ = ⟨Δv, v⟩, so ∥Dv∥^{2}/∥v∥^{2} lies in S.

Definition 4.8. Define Δ^{\frac{1}{2}} : L_{(2),s+1}^{n} → L_{(2),s}^{n} by

Δ^{\frac{1}{2}} := ∫_{ρ∈[0,∞)} ρ^{\frac{1}{2}} dν_{ρ}.

This gives

DL_{(2),s+1}^{n−1}(X, 𝕍) = Δ^{\frac{1}{2}} \overline{DL_{(2),s+2}^{n−1}(X, 𝕍)},
ker D ∩ L_{(2),s}^{n}(X, 𝕍) = \overline{DL_{(2),s+1}^{n−1}(X, 𝕍)} ⊕ H^{n}(X, 𝕍).

Thus

H^{n}L_{(2),s}^{•}(X, 𝕍) ≅ H^{n}(X, 𝕍) ⊕ (\overline{DL_{(2),s−1}^{n−1}(X, 𝕍)}/Δ^{\frac{1}{2}} \overline{DL_{(2),s−2}^{n−1}(X, 𝕍)}).

There are similar statements for the operators D^{c}, D^{∗}, D^{c∗}.

Definition 4.9. Define Δ^{− \frac{1}{2}} D: \overline{D^{∗}L_{(2),s+1}^{n}(X, 𝕍)} → L_{(2),s(X,𝕍)}^{n} by

Δ^{− \frac{1}{2}} := ∫_{ρ∈(0,∞)} ρ^{− \frac{1}{2}} dν_{ρ} ∘ D,

and define Δ^{− \frac{1}{2}} D^{c}: \overline{D^{c∗}L_{(2),s+1}^{n}(X, 𝕍)} → L_{(2),s}^{n}(X, 𝕍) similarly.

Proposition 4.10. The operator Δ^{− \frac{1}{2}} D gives a Hilbert space isomorphism from the closed subspace \overline{D^{∗}L_{(2),s+1}^{n}(X, 𝕍)} of L_{(2),s}^{n−1}(X, 𝕍) to the closed subspace \overline{DL_{(2),s+1}^{n−1}(X, 𝕍)} of L_{(2),s}^{n}(X, 𝕍).

Likewise, Δ^{− \frac{1}{2}} D^{c}: \overline{D^{c∗}L_{(2),s+1}^{n}(X, 𝕍)} → \overline{D^{c}L_{(2),s+1}^{n−1}(X, 𝕍)} is a Hilbert space isomorphism. Proof. We prove this for the first case, the second being entirely similar. Given a, b ∈ D^{∗}A^{n}(X, 𝕍), we have

⟨Δ^{− \frac{1}{2}} Da, Δ^{− \frac{1}{2}} Db⟩_{s} = ⟨(I + Δ)^{s}Δ^{− \frac{1}{2}} Da, Δ^{− \frac{1}{2}} Db⟩
= ⟨(I + Δ)^{s}Δ^{−1}D^{∗}Da, b⟩
= ⟨(I + Δ)^{s}a, b⟩ = ⟨a, b⟩_{s},

since D^{∗}a = 0 gives D^{∗}Da = Δa. Taking Hilbert space completions with respect to ⟨−, −⟩_{s} then gives the required result. □ 4.2. The Hodge decomposition and cohomology. Proposition 4.11. The nested intersection ⋂_{s} L_{(2),s}^{p}(X, 𝕍) is the space A^{p}(X, 𝕍) of C^{∞} 𝕍-valued p-forms. Proof. When 𝕍 is finite-dimensional, this is the Global Sobolev Lemma, but the same proof applies for Hilbert space coefficients. □ Theorem 4.12. There are pre-Hilbert space decompositions

A^{n}(X, 𝕍) = H^{n}(X, 𝕍) ⊕\overline{ΔA^{n}(X, 𝕍)}
= H^{n}(X, 𝕍) ⊕\overline{DA^{n−1}(X, 𝕍)} ⊕ \overline{D^{∗}A^{n+1}(X, 𝕍)}
= H^{n}(X, 𝕍) ⊕\overline{D^{c}A^{n−1}(X, 𝕍)} ⊕ \overline{D^{c∗}A^{n+1}(X, 𝕍)}
= H^{n}(X, 𝕍) ⊕\overline{DD^{c}A^{n−2}(X, 𝕍)} ⊕ \overline{D^{∗}D^{c}A^{n}(X, 𝕍)} ⊕ \overline{DD^{c∗}A^{n}(X, 𝕍)} ⊕ \overline{D^{∗}D^{c∗}A^{n+2}(X, 𝕍)}.

for all n. Proof. We just take the inverse limit \underleftarrow{lim}_{s} of the decomposition in Proposition 4.7, and then make the substitution of Proposition 4.11. □ 4.2.1. Reduced cohomology. Definition 4.13. Given a cochain complex C^{•} in topological vector spaces, write

H̄^{n}(C^{•}) := H^{n}(C^{•})/\overline{\{0\}},

where \overline{\{0\}} is the closure of 0. Note that we could equivalently define H̄^{∗} as the quotient of the space of cocycles by the closure of the space of coboundaries.

Given a local system 𝕍 in topological vector spaces on X, define

H̄^{n}(X, 𝕍) := H̄^{n}(A^{•}(X, 𝕍)).

Corollary 4.14. The maps

H^{n}(X, 𝕍) → H̄^{n}(X, 𝕍)

are all topological isomorphisms.

Corollary 4.15 (Principle of two types). As subspaces of A^{n}(X, 𝕍),

ker D ∩ ker D^{c} ∩ (\overline{DA^{n−1}(X, 𝕍)} + \overline{D^{c}A^{n−1}(X, 𝕍)}) = \overline{DD^{c}A^{n−2}(X, 𝕍)}.

Lemma 4.16 (Formality). The morphisms

(H̄_{D^{c}}^{∗}(X, 𝕍), 0) ← (Z_{D^{c}}^{∗}(X, 𝕍), D) → (A^{∗}(X, 𝕍), D)

induce isomorphisms on reduced cohomology.

Proof. The proof of [Sim4, Lemma 2.2] carries over to this generality. □

Remark 4.17. Usually, formality statements such as Lemma 4.16 lead to isomorphisms on deformation functors (see [GM] for the original case, and [Pri1, Proposition 5.3] for the case closest to our setting).

However, there does not appear to be a natural deformation functor associated to topological DGLAs L with obstruction space H̄^{2}(L). Thus, in contrast to the proalgebraic case, it is not clear whether there are natural completions of the homotopy groups which can be described in terms of the reduced cohomology ring.

The description of the Archimedean monodromy in [Pri2, Theorem 9.13] is even less likely to adapt, since it features the Green’s operator G, which we have had to replace with a non-convergent sequence of operators.

4.2.2. Non-reduced cohomology. Taking the inverse limit \underrightarrow{lim}_{s} of the decompositions of §4.1.2, we obtain

DA^{n−1}(X, 𝕍) = Δ^{\frac{1}{2}} \overline{DA^{n−1}(X, 𝕍)},
D^{∗}A^{n+1}(X, 𝕍) = Δ^{\frac{1}{2}} \overline{D^{∗}A^{n+1}(X, 𝕍)},

with similar statements for D^{c}, D^{c∗}.

Thus:

Proposition 4.18.

H^{n}A^{•}(X, 𝕍) ≅ H^{n}(X, 𝕍) ⊕ (\overline{DA^{n−1}(X, 𝕍)}/Δ^{\frac{1}{2}} \overline{DA^{n−1}(X, 𝕍)}).

Applying the operator ∗ then gives

H^{n}A^{•}(X, 𝕍) ⊕_{H^{n}(X,𝕍)} H^{2d−n}A^{•}(X, 𝕍′) ≅ A^{n}(X, 𝕍)/Δ^{\frac{1}{2}} A^{n}(X, 𝕍).

Moreover, we have topological isomorphisms

Δ^{− \frac{1}{2}} D: \overline{D^{∗}A^{n}(X, 𝕍)} → \overline{DA^{n−1}(X, 𝕍)},
Δ^{− \frac{1}{2}} D^{c}: \overline{D^{c∗}A^{n}(X, 𝕍)} → \overline{D^{c}A^{n−1}(X, 𝕍)},

for Δ^{− \frac{1}{2}} D, Δ^{− \frac{1}{2}} D^{c} as in Definition 4.9.

4.3. The W^{∗}-enveloping algebra.

4.3.1. E^{J}(X, x)′.

Definition 4.19. Given a C^{∗}-algebra B and a positive linear functional f, define B_{f} to be the Hilbert space completion of B with respect to the bilinear form ⟨a, b⟩_{f} := f(a^{∗}b). We define π_{f} to be the representation of B on B_{f} by left multiplication. Note that this is a cyclic representation, generated by 1 ∈ B_{f}.

Lemma 4.20. Given a C^{∗}-algebra B, the topological dual is given by B′ = \underrightarrow{lim}_{f} B′_{f}, where f ranges over the filtered inverse system of positive linear functionals on B.

Proof. This amounts to showing that B^{∨∨} = \underleftarrow{lim}_{f} B_{f}. Since the system is filtered (with

f + g ≥ f, g and B_{g} → B_{f} for g ≥ f), B̂ := \underleftarrow{lim}_{f} B_{f} is the completion of B with

respect to the seminorms ∥b∥_{f} := f(b^{∗}b)^{\frac{1}{2}} . The space B_{f} is the strong closure of B in the cyclic representation π_{f}, which is just the image of B^{∨∨}, by the von Neumann bicommutant theorem. Since B^{∨∨} is the completion of B with respect to the system of weak seminorms for all representations, this implies that the map B̂ → B^{∨∨} is an equivalence. □

Lemma 4.21. For a C^{∗}-algebra B, and a B-representation V in Hilbert spaces,

Hom_{B}(V, B′) ≅ V′.

Proof. The space Hom_{B}(V, B′) consists of continuous B-linear maps V → B′, and hence to continuous (A, k)-bilinear maps A × V → k. These correspond to continuous linear maps V → k, as required □

Considering smooth morphisms from X then gives:

Corollary 4.22. For any E := E^{J}(X, x)-representation V in real Hilbert spaces, with corresponding local system 𝕍 on X, there is a canonical topological isomorphism

A^{•}(X, 𝕍) ≅ Hom_{E}(V′, A^{•}(X, 𝔼′)),

where 𝔼′ is the direct system of pluriharmonic local systems corresponding to the ind-E-representation E′ given by left multiplication.

Of course, all the cohomological decomposition results of this section extend to direct limits, so apply to 𝔼′. Conversely, Corollary 4.22 all such results for local systems 𝕍 can be inferred from the corresponding results with 𝔼′-coefficients.

Remark 4.23. The comultiplication E^{J}(X, x) → E^{J}(X, x) ˆ⊗E^{J}(X, x) of Lemma 2.24 induces a multiplication

E^{J}(X, x)′ ¯⊗E^{J}(X, x)′ → E^{J}(X, x)′

on continuous duals, where (\underleftarrow{lim}_{i} E_{i})′ ¯⊗(\underleftarrow{lim}_{j} E_{j})′ := \underrightarrow{lim}_{i,j} E_{i} ¯⊗E_{j} for C^{∗}-algebras E_{i} and the dual tensor product ¯⊗ of [Tak, p. 210]. In particular, ¯⊗ is a crossnorm, so we have a jointly continuous multiplication on E^{J}(X, x)′

Thus A^{•}(X, 𝔼′) is also equipped with a jointly continuous (graded) multiplication, so has the structure of a differential graded topological algebra.

4.3.2. Failure of continuity. Since direct integrals of harmonic representations must be harmonic, Corollary 4.14 and Proposition 4.18 provide us with information about the behaviour of cohomology in measurable families. In particular, they allow us to recover space of measures on the topological spaces of cohomology groups fibred over the moduli spaces of local systems. Thus E_{X,x}^{J} is a much finer invariant than the pro-algebraic completion of π_{1}(X, x).

It is natural to ask whether we can strengthen the Hodge decomposition to incorporate finer topological data. The following example indicates that it does not hold for coefficients in 𝔼 itself:

Example 4.24. Let X be a complex torus, so π_{1}(X, e) ≅ ℤ^{2g}. By Proposition 3.17, (E_{X,e}^{J})_{RFD} ⊗ ℂ = C(Hom(π_{1}(X, e), ℂ^{∗}), ℂ) ≅ C((ℂ^{∗})^{2g}, ℂ).

The complex A^{•}(X, 𝔼_{RFD} ⊗ ℂ) is then quasi-isomorphic to H^{∗}(ℤ^{2g}, C((ℂ^{∗})^{2g}, ℂ)). This is given by taking the completed tensor product of 2g copies of the complex F

given by C(ℂ^{∗}, ℂ) ⟶^{z−1} C(ℂ^{∗}, ℂ), so

H̄^{∗}(X, 𝔼_{RFD} ⊗ ℂ) ≅ ℂ[−2g].

However, D^{∗}D + DD^{∗} = |z − 1|^{2} on the complex F, so harmonic forms are given by

H^{∗}(X, 𝔼_{RFD} ⊗ ℂ) ≅ 0.

5. Twistor and Hodge structures on cochains, and SU_{2}

5.1. Preliminaries on non-abelian twistor and Hodge filtrations. The following is [Pri2, Definition 2.3]:

Definition 5.1. Define C to be the real affine scheme ∏_{ℂ/ℝ} 𝔸^{1} obtained from 𝔸_{ℂ}^{1} by restriction of scalars, so for any real algebra A, C(A) = 𝔸_{ℂ}^{1}(A⊗_{ℝ}ℂ) ≅ A⊗_{ℝ}ℂ. Choosing i ∈ ℂ gives an isomorphism C ≅ 𝔸_{ℝ}^{2}, and we let C^{∗} be the quasi-affine scheme C − {0}.

We let the real algebraic group S = ∏_{ℂ/ℝ} 𝔾_{m} of Definition 2.16 act on C and C^{∗} by inverse multiplication, i.e.

S × C → C
(λ, w) ↦ (λ^{−1}w).

Fix an isomorphism C ≅ 𝔸^{2}, with co-ordinates u, v on C so that the isomorphism C(ℝ) ≅ ℂ is given by (u, v) ↦ u + iv. Thus the algebra O(C) associated to C is the polynomial ring ℝ[u, v]. S is isomorphic to the scheme 𝔸_{ℝ}^{2}−{(α, β) : α^{2} +β^{2} = 0}, with the group isomorphism S(ℝ) ≅ ℂ^{∗} given by (α, β) ↦ α+iβ, and the group isomorphism S(ℂ) ≅ (ℂ^{∗})^{2} given by (α, β) ↦ (α + iβ, α − iβ).

By [Pri2, Corollary 2.6 and Proposition 2.28], real Hodge filtrations (resp. real twistor structures) correspond to S-equivariant (resp. 𝔾_{m}-equivariant) flat vector bundles on C^{∗}. The latter arises because [C^{∗}/𝔾_{m}] ≃ ℙ_{ℝ}^{1}, so 𝔾_{m}-equivariant sheaves on C^{∗} correspond to sheaves on ℙ^{1}.

The following is [Pri2, Definition 2.15]:

Definition 5.2. Define an S-action on the real affine scheme SL_{2} by

(α, β, A) ↦ ( \begin{matrix}1 & 0 \\ 0 & α^{2}+β^{2} \end{matrix} ) A ( \begin{matrix}α & β \\ −β & α \end{matrix} )^{−1} = ( \begin{matrix}α^{2}+β^{2} & 0 \\ 0 & 1 \end{matrix} )^{−1} A ( \begin{matrix}α & −β \\ β & α \end{matrix} ) .

Let row_{1} : SL_{2} → C^{∗} be the S-equivariant map given by projection onto the first row.

The subgroup scheme 𝔾_{m} ⊂ S is given by β = 0 in the co-ordinates above, and there is a subgroup scheme S^{1} ⊂ S given by α^{2} + β^{2} = 1. These induce an isomorphism (𝔾 × S^{1})/(−1, −1) ≅ S. On these subgroups, the action on SL_{2} simplifies as follows:

Lemma 5.3. The action of 𝔾_{m} ⊂ S on SL_{2} is given by

(α, A) ↦ ( \begin{matrix}α^{−1} & 0 \\ 0 & α \end{matrix} ) A

and the action of S^{1} ⊂ 𝔾_{m} is given by

(α, β, A) ↦ A ( \begin{matrix}α & β \\ −β & α \end{matrix} )^{−1} .

The action of S^{1} ⊂ S descends via the maps above to an action on ℙ_{ℝ}^{1}, which is just given by identifying S^{1} with the real group scheme SO_{2}.

5.2. The twistor structure on cochains. Fix a compact Kähler manifold X.

Definition 5.4. Define D̃: A^{n}(X, 𝕍) ⊗ 𝒪_{C^{∗}} → A^{n+1}(X, 𝕍) ⊗ 𝒪_{C^{∗}} by

D̃ = uD + vD^{c},

and write Ã^{•}(X, 𝕍) for the resulting complex. Put a 𝔾_{m}-action on Ã^{•}(X, 𝕍) by letting A^{n}(X, 𝕍) have weight n, and giving 𝒪_{C^{∗}} the action of 𝔾_{m} ⊂ S from Definition 5.1.

Define Z̃^{n}(X, 𝕍) := ker(D̃: A^{n}(X, 𝕍) ⊗ 𝒪_{C^{∗}} → A^{n+1}(X, 𝕍) ⊗ 𝒪_{C^{∗}}), B̃^{n}(X, 𝕍) := Im (D̃: A^{n−1}(X, 𝕍) ⊗ 𝒪_{C^{∗}} → A^{n}(X, 𝕍)) ⊗ 𝒪_{C^{∗}} and

¯H̃^{n}(X, 𝕍) := Z̃^{n}(X, 𝕍)/\overline{B̃^{n}(X, 𝕍)}.

By analogy with [Pri2, Proposition 2.28 and Theorem 7.1], we regard the 𝔾_{m}- equivariant complex Ã^{•}(X, 𝕍) over C^{∗} as a twistor filtration on A^{n}(X, 𝕍).

Corollary 5.5. The canonical inclusion H^{n}(X, 𝕍)(n) ⊗ 𝒪_{C^{∗}} → ¯ H̃^{n}(X, 𝕍) is a 𝔾_{m}- equivariant topological isomorphism.

Proof. It suffices to prove this on pulling back along the flat cover row_{1} : SL_{2} → C^{∗}. We may define D̃^{∗}: A^{n}(X, 𝕍) ⊗ O(SL_{2}) → A^{n−1}(X, 𝕍) ⊗ O(SL_{2}) by D̃^{∗} = yD^{∗} − xD^{c∗}. Then [D̃, D̃^{∗}] = Δ, and since Δ commutes with D and D^{c} it also commutes with D̃.

The result now follows with the same proof as that of Corollary 4.14, replacing D, D^{∗} with D̃, D̃^{∗}. □

Proposition 5.6. If we write H^{n} = H^{n}(X, 𝕍) and M^{m} = \overline{DD^{c}A^{m−2}(X, 𝕍)}, then there is a 𝔾_{m}-equivariant isomorphism

H̃^{n}(X, 𝕍) ≅ [(H^{n} ⊕ M/Δ^{\frac{1}{2}} M^{n})(n) ⊕ (M^{n+1}/Δ^{\frac{1}{2}} M^{n+1})(n − 1)] ⊗ 𝒪_{C^{∗}}

of quasi-coherent sheaves on C^{∗}.

Proof. Writing A^{m} := A^{m}(X, 𝕍), we have a commutative diagram

\xymatrix{ & \overline{DD^{c}A^{n−1}}(n + 1) ⊗ 𝒪_{C^{∗}} \\ \overline{DD^{c∗}A^{n}}(n) ⊗ 𝒪_{C^{∗}} \ar@{-->}[ur]^{vD^{c}} &  & \overline{D^{∗}D^{c}A^{n}}(n) ⊗ 𝒪_{C^{∗}} \ar@{-->}[ul]_{−uD} \\  & \overline{D^{∗}D^{c∗}A^{n+1}}(n − 1) ⊗ 𝒪_{C^{∗}}, \ar@{-->}[ul]^{uD} \ar@{-->}[ur]_{vD^{c}}}

which we may regard as a bicomplex. By Theorem 4.12, the complex Ã^{•}(X, 𝕍) decomposes into a direct sum of H^{n}’s and total complexes of the bicomplexes above.

Arguing as in Proposition 4.10, we have topological isomorphisms

Δ^{−1}DD^{c}: \overline{D^{∗}D^{c∗}A^{n+1}} → \overline{DD^{c}A^{n−1}}
Δ^{− \frac{1}{2}} D: \overline{D^{∗}D^{c}A^{n}} → \overline{DD^{c}A^{n−1}}
Δ^{− \frac{1}{2}} D^{c}: \overline{DD^{c∗}A^{n}} → \overline{DD^{c}A^{n−1}},

so the bicomplex above is linearly isomorphic to

\xymatrix{ & \overline{DD^{c}A^{n−1}}(n + 1) ⊗ 𝒪_{C^{∗}} \\ \overline{DD^{c}A^{n−1}}(n) ⊗ 𝒪_{C^{∗}} \ar@{-->}[ur]^{vΔ^{\frac{1}{2}}} &  & \overline{DD^{c}A^{n−1}}(n) ⊗ 𝒪_{C^{∗}} \ar@{-->}[ul]_{−uΔ^{\frac{1}{2}}} \\  & \overline{DD^{c}A^{n−1}}(n − 1) ⊗ 𝒪_{C^{∗}}. \ar@{-->}[ul]^{uΔ^{\frac{1}{2}}} \ar@{-->}[ur]_{vΔ^{\frac{1}{2}}}}

Since the ideal (u, v) generates 𝒪_{C^{∗}}, cohomology of the top level of the associated total complex is just

(\overline{DD^{c}A^{n−1}}/Δ^{\frac{1}{2}} \overline{DD^{c}A^{n−1}})(n + 1) ⊗ 𝒪_{C^{∗}},

while cohomology of the bottom level is 0. Moreover, the map

\overline{DD^{c}A^{n−1}}(n − 1) ⊗ 𝒪_{C^{∗}} ⟶^{(u,v)} \overline{DD^{c}A^{n−1}}(n)^{2} ⊗ 𝒪_{C^{∗}}

is an isomorphism to the kernel of (v, −u), so cohomology of the middle level is isomorphic to

(\overline{DD^{c}A^{n−1}}(n − 1)/Δ^{\frac{1}{2}} \overline{DD^{c}A^{n−1}}(n − 1)) ⊗ 𝒪_{C^{∗}} ⊗ 𝒪_{C^{∗}},

which completes the proof. □ Remark 5.7. In particular, note that H̃^{n}(X, 𝕍) is of weights n, n − 1 in general, unlike ¯H̃^{n}(X, 𝕍) which is pure of weight n. In particular, this means that the weight filtration given good truncation cannot define a mixed twistor structure on H̃^{n}(X, 𝕍).

We now have the following generalisation of the principle of two types: Lemma 5.8. As subspaces of A^{n}(X, 𝕍) ⊗ O(SL_{2}),

ker D̃ ∩ ker D̃^{c} ∩ (\overline{D̃(A^{n−1}(X, 𝕍) ⊗ O(SL_{2}))} + \overline{D̃^{c}(A^{n−1}(X, 𝕍) ⊗ O(SL_{2}))})

= \overline{D̃D̃^{c}(A^{n−2}(X, 𝕍)O(SL_{2}))}.

Proof. This follows from Lemma 4.15, with the same reasoning as [Pri2, Lemma 1.3]. □ Lemma 5.9 (Formality). The morphisms

(H̄∗ ; D̃^{c}(A^{∗}(X, 𝕍)⊗O(SL_{2})), 0) ← (Z∗ ; D̃^{c}(A(X, 𝕍)⊗O(SL_{2})), D̃) → (A^{∗}(X, 𝕍)⊗O(SL_{2}), D̃)

induce isomorphisms on reduced cohomology. Proof. The proof of [Sim4, Lemma 2.2] carries over to this generality, using Lemma 5.8. □

Following Corollary 4.22 and Remark 4.23, the results above can all be regarded as statements about the topological differential graded algebra Ã^{•}(X, 𝔼′).

5.3. The analytic Hodge filtration on cochains. Recall from §4.3.1 that the local system 𝔼′ on X is defined to correspond to the π_{1}(X, x)-representation given by left multiplication on E^{J}(X, x)′.

Proposition 5.10. The topological cochain complex Ã^{•}(X, 𝔼′) is equipped with a continuous circle action, satisfying:

(1) the S^{1}-action and 𝔾_{m}-actions on Ã^{•}(X, 𝔼′) commute,

(2) the action of S^{1} ⊂ ℂ^{∗} = S(ℝ) on C^{∗} makes Ã^{•}(X, 𝔼′) into an S^{1}-equivariant sheaf on C^{∗}, and

(3) −1 ∈ S^{1} acts as −1 ∈ 𝔾_{m}.

Proof. Since S^{1} acts on E^{J}(X, x), it acts on 𝔼′, and we denote this action by v ↦ t ⊛ v, for t ∈ S^{1}. We may now adapt the proof of [Pri2, Theorem 6.12], defining an S^{1}- action on 𝒜^{∗}(X, ℝ) ⊗_{ℝ} 𝔼′ by setting t ⧆ (a ⊗ v) := (t ⋄ a) ⊗ (t^{2} ⊛ v) for t ∈ S^{1} and ⋄ as in Definition 2.17. Passing to the completion 𝒜^{∗}(X, 𝔼′) completes the proof, with continuity following from Proposition 3.31. □

Remarks 5.11. If the circle action of Proposition 5.10 were algebraic, then by [Pri2, Lemma 2.33] it would correspond to a Hodge filtration on A^{•}(X, 𝔼′). Since finitedimensional circle representations are algebraic, we may regard Proposition 5.10 as the natural structure of an infinite-dimensional Hodge filtration.

In Proposition 5.10, note that we can of course replace E^{J}(X, x) with any inverse system B of C^{∗}-algebra quotients of E^{J}(X, x) to which the S^{1}-action descends, provided we replace 𝔼 with the local system associated to B.

As observed in §4.3.1, we may substitute 𝕍 = 𝔼 in Proposition 5.6 and Lemma 5.9. Note that the resulting isomorphisms are then equivariant with respect to the circle action of Proposition 5.10.

5.4. SU_{2}. As we saw in Corollary 5.5, in order to define the adjoint operator D̃^{∗} to D̃, it is necessary to pull Ã^{•}(X, 𝕍) back along the morphism row_{1} : SL_{2} → C^{∗}. This gives us the complex

row_{1}^{∗}Ã^{•}(X, 𝕍) = (A^{∗}(X, 𝕍) ⊗ O(SL_{2}), D̃),

where D̃ = uD + vD^{c}, with adjoint D̃^{∗} = yD^{∗} − xD^{c∗}.

This leads us to consider the ∗-structure on O(SL_{2}) determined by u^{∗} = y, v^{∗} = −x. This implies x^{∗} = −v, y^{∗} = u, so

(\begin{matrix}u & v \\ x & y \end{matrix})^{∗} = (\begin{matrix}y & −x \\ −v & u \end{matrix}) ,

or A^{∗} = (A^{−1})^{t}.

Lemma 5.12. The real C^{∗}-enveloping algebra C^{∗}(O(SL_{2})) of the real ∗-algebra O(SL_{2}) is the ring of continuous complex functions f on SU_{2} for which

f(Ā) = \overline{f(A)}.

Proof. A ∗-morphism O(SL_{2}) → ℂ is a matrix A ∈ SL_{2}(ℂ) with Ā = A^{∗} = (A^{−1})^{t}, so A ∈ SU_{2}. Thus the Gelfand representation gives C^{∗}(O(SL_{2})) ⊗ ℂ ≅ C(SU_{2}, ℂ).

Now, writing Gal(ℂ/ℝ) = ⟨τ⟩, and taking f ∈ O(SL_{2}) ⊗ ℂ and A ∈ SU_{2}, we have τ(f)(A) = \overline{f(Ā)}. This formula extends to give a Gal(ℂ/ℝ)-action on C(SU_{2}, ℂ), and Lemma 1.11 then gives

C^{∗}(O(SL_{2})) = C(SU_{2}, ℂ)^{τ} .
□

Note that complex conjugation on SU_{2} is equivalent to conjugation by the matrix

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ).

5.4.1. The Hopf fibration. The action of S(ℂ) on SL_{2}(ℂ) from Definition 5.2 does not preserve SU_{2}. However, Lemma 5.3 ensures that for the subgroup schemes 𝔾_{m}, S^{1} ⊂ S, the groups S^{1} = S^{1}(ℝ) ⊂ S^{1}(ℂ) ≅ ℂ^{∗} and S^{1} ⊂ ℂ^{∗} ≅ 𝔾_{m}(ℂ) both preserve SU_{2}.

Thus in the C^{∗} setting, the S-action becomes an action of (S^{1} × S^{1})/(−1, −1) on SU_{2}, given by

(s, t, A) ↦ ( \begin{matrix}s^{−1} & 0 \\ 0 & s \end{matrix} ) A ( \begin{matrix}ℜt & ℑt \\ −ℑt & ℜt \end{matrix} )^{−1} .

Moreover, there is a Gal(ℂ/ℝ)-action on this copy of S^{1} × S^{1}, with the non-trivial automorphism τ given by τ(s, t) = (s^{−1}, t). The action of (S^{1} × S^{1})/(−1, −1) is then τequivariant. Alternatively, we may characterise our group as S^{1} ×S^{1} ⊂ ℂ^{∗} ×ℂ^{∗} ≅ S(ℂ), by sending (s, t) to (st, st^{−1}). On this group S^{1} × S^{1}, the Gal(ℂ/ℝ)-action is then given by τ(w′, w′′) = (\overline{w′′}, \overline{w′}).

Now, consider the composition

SL_{2} ⟶^{row_{1}} C^{∗} → [C^{∗}/𝔾_{m}] ≅ ℙ^{1}.

On taking Gelfand representations of C^{∗}-enveloping algebras, this gives rise to the map

SU_{2} → ℙ^{1}(ℂ),

which is just the Hopf fibration p: S^{3} → S^{2}, corresponding to the quotient by the action of S^{1} ⊂ 𝔾_{m}(ℂ) by diagonal matrices. The action of τ on SU_{2} and on ℙ^{1}(ℂ) is just given by complex conjugation.

5.4.2. Smooth sections. If we write ρ_{n} for the weight n action of 𝔾_{m} on 𝔸^{1}, then we may consider the topological vector bundle

SU_{2} ×_{S^{1},ρ_{n}} ℂ

on ℙ^{1}(ℂ), for the action of S^{1} ⊂ 𝔾_{m}(ℂ) above.

Definition 5.13. Define 𝒜_{ℙ^{1}}^{0}ℂ(n) to be the sheaf of smooth sections of SU_{2} ×_{S^{1},ρ_{n}} ℂ → ℙ^{1}(ℂ), and write A^{0}(ℙ^{1}, ℂ(n)) := Γ(ℙ^{1}(ℂ), 𝒜_{ℙ^{1}}^{0}ℂ(n)). Beware that for n ≠ 0, there is no local system generating 𝒜_{ℙ^{1}}^{0}ℂ(n).

For U ⊂ ℙ^{1}(ℂ), observe that Γ(U, 𝒜_{ℙ^{1}}^{0}ℂ(n)) consists of smooth maps

f: p^{−1}(U) → ℂ

satisfying f(( \begin{matrix}s^{−1} & 0 \\ 0 & s \end{matrix} ) A) = s^{n}f(A) for all s ∈ S^{1}.

For the quotient map q: C^{∗} → ℙ^{1}, we may characterise Γ(U, 𝒪_{ℙ^{1}}^{hol}(n)) as the space of holomorphic maps

f: q^{−1}(U) → ℂ

satisfying f(su, sv) = s^{n}f(u, v) for all s ∈ ℂ^{∗}. The embedding S^{3} ⊂ C^{∗}(ℂ) thus yields

𝒪_{ℙ^{1}}^{hol}(n) ⊂ 𝒜_{ℙ^{1}}^{0}ℂ(n),

and indeed 𝒜_{ℙ^{1}}^{0}ℂ(n) = 𝒪_{ℙ^{1}}^{hol}(n) ⊗_{𝒪_{ℙ^{1}}^{hol}} 𝒜_{ℙ^{1}}^{0}ℂ.

Now, for the conjugate sheaf \overline{𝒪_{ℙ^{1}}^{hol}(n)} , note that Γ(U, \overline{𝒪_{ℙ^{1}}^{hol}(n)} ) is the space of antiholomorphic maps

f: q^{−1}(U) → ℂ

satisfying f(su, sv) = s̄^{n}f(u, v) for all s ∈ ℂ^{∗}. Thus we have a canonical embedding

\overline{𝒪_{ℙ^{1}}^{hol}(n)} ⊂⊂ 𝒜_{ℙ^{1}}^{0}ℂ(−n),

with 𝒜_{ℙ^{1}}^{0}ℂ(−n) = \overline{𝒪_{ℙ^{1}}^{hol}(n)} ⊗\overline{𝒪_{ℙ^{1}}^{hol}} 𝒜_{ℙ^{1}}^{0}ℂ.

Note that the inclusion O(SL_{2}) ⊂ C(SU_{2}, ℂ) gives

u, v ∈ A^{0}(ℙ^{1}, ℂ(1))^{τ} , ū, v̄ ∈ A^{0}(ℙ^{1}, ℂ(−1))^{τ}

and

O(SL_{2}) ⊂ ⨁_{n∈ℤ}A^{0}(ℙ^{1}, ℂ(n))^{τ} .

Definition 5.14. By [Pri2, Definition 2.22], there is a derivation N on O(SL_{2}) given by Nx = u, Ny = v, Nu = Nv = 0, for co-ordinates ( \begin{matrix}u & v \\ x & y \end{matrix}) on SL_{2}. Since this annihilates u, v, it is equivalent to the O(SL_{2})-linear map

Ω(SL_{2}/C^{∗}) → O(SL_{2})

given by dx ↦ u, dy ↦ v.

Note that N has weight 2, and extends (by completeness) to give τ-equivariant differentials

N: 𝒜_{ℙ^{1}}^{0}ℂ(n) → 𝒜_{ℙ^{1}}^{0}ℂ(n + 2).

Note that N is the composition of the anti-holomorphic differential

¯∂_{ℙ^{1}} : 𝒜_{ℙ^{1}}^{0}ℂ(n) → 𝒜_{ℙ^{1}}^{0}ℂ(n) ⊗\overline{𝒪_{ℙ^{1}}^{hol}} \overline{Ω_{ℙ^{1}}}

with the canonical isomorphism Ω_{ℙ^{1}} ≅ 𝒪(−2).

5.4.3. Splittings of the twistor structure. As in [Pri2, Remark 2.16], we can characterise the map row_{1} : SL_{2} → C^{∗} as the quotient C^{∗} = [SL_{2}/𝔾_{a}], where 𝔾_{a} acts on SL_{2} as left multiplication by ( \begin{matrix}1 & 0 \\ 𝔾_{a} & 1 \end{matrix} ). Here, the S-action on 𝔾_{a} has λ acting as multiplication by λλ̄.

Therefore the map q ∘ row_{1} : SL_{2} → ℙ^{1} is given by taking the quotient of SL_{2} by the Borel subgroup B = 𝔾_{m} ⋉ 𝔾_{a}, for the action above. The action of 𝔾_{m} corresponds to weights, while the action of 𝔾_{a} corresponds to the derivation N above, which we regard as the Archimedean monodromy operator as in [Pri2, §3.4].

Definition 5.15. Given a pluriharmonic local system 𝕍, define Ă^{•}(X, 𝕍) to be the sheaf of 𝒪_{ℙ^{1}}^{hol}-modules associated to the 𝔾_{m}-equivariant sheaf Ã^{•}(X, 𝔼^{∨}) on C^{∗}. Explicitly,

Ă^{•}(X, 𝕍) = (⨁_{n}(q_{∗}Ã^{•}(X, 𝕍)) ⊗_{𝒪_{ℙ^{1}}^{alg}} 𝒪_{ℙ^{1}}^{hol}(n))^{𝔾_{m}},

so

Ă^{n}(X, 𝕍) = A^{n}(X, 𝕍) ⊗_{ℝ} 𝒪_{ℙ^{1}}^{hol}(n),

with differential D̆ = uD + vD^{c}, for u, v ∈ Γ(ℙ^{1}, 𝒪_{ℙ^{1}}(1)).

Definition 5.16. Write 𝒜̊^{•}(X, 𝕍) := 𝒜_{ℙ^{1}}^{0} ⊗_{𝒪_{ℙ^{1}}^{hol}} Ă^{∗}(X, 𝕍), and observe that this admits an operator

D̆^{c} := −v̄D + uD^{c}: 𝒜̊^{n} (X, 𝕍) → 𝒜̊^{n+1} (X, 𝕍)(−2).

Now, applying the map O(SL_{2}) ↪ ⨁_{n∈ℤ} A^{0}(ℙ^{1}, ℂ(n))^{τ} to Lemmas 5.8,5.9 yields the following:

Lemma 5.17. As subspaces of 𝒜̊^{n} (X, 𝕍),

ker D̆ ∩ ker D̆^{c} ∩ (\overline{D̆𝒜̊^{n−1} (X, 𝕍)} + \overline{D̆^{c}𝒜̊^{n−1} (X, 𝕍)(2)}) = \overline{D̆D̆^{c}(𝒜̊^{n−2} (X, 𝕍)(2))}.

Thus the morphisms

(ℋ̄ ∗ ; D̆^{c} (𝒜̊^{∗}(X, 𝕍)), 0) ← (𝒵 ∗ ; D̆^{c}(𝒜̊^{∗}(X, 𝕍)), D̆) → 𝒜̊^{n} (X, 𝕍)

induce isomorphisms on reduced cohomology sheaves.

Now, Ã^{•}(X, 𝔼′) can be recovered from row_{1}^{∗}Ã^{•}(X, 𝔼′) and its nilpotent monodromy operator N, and Lemma 4.16 says that row_{1}^{∗}Ã^{•}(X, 𝔼′) is equivalent to H^{∗}(X, 𝔼′) ⊗ O(SL_{2}) up to reduced quasi-isomorphism.

Under the base change above, we have N = ¯∂_{ℙ^{1}} giving an exact sequence

0 → Ă^{•}(X, 𝕍) → 𝒜̊^{•}(X, 𝕍) ⟶^{N} 𝒜̊^{•}(X, 𝕍)(2) → 0.

In other words, we can recover the topological DGA Ă^{•}(X, 𝔼′) from the differential ¯∂ = N on the topological DGA 𝒜̊^{•}(X, 𝔼′), and the latter is just ⨁_{n} 𝒜_{ℙ^{1}}(H^{n}(X, 𝔼′)(n)) up to reduced quasi-isomorphism.

Also note that when we substitute 𝕍 := 𝔼′ in Lemma 5.17, the morphisms all become equivariant with respect to the circle action of Proposition 5.10. This action makes 𝒜̊^{n} (X, 𝔼′) into an S^{1}-equivariant sheaf over ℙ^{1}, where the action on ℙ^{1} is given by t^{2} ∈ S^{1} sending (u : v) to t ⋄ (u : v) = (au − bv : av + bu) for t = a + ib ∈ S^{1}.

6. The twistor family of moduli functors

6.1. Pro-Banach algebras on projective space. On the complex manifold ℙ^{1}(ℂ), we have a sheaf 𝒪_{ℙ^{1}}^{hol} of holomorphic functions, which we may regard as a sheaf of pro-Banach algebras. As a topological space, ℙ^{1}(ℂ) is equipped with a Gal(ℂ/ℝ)-action, the non-trivial element τ acting on points by complex conjugation. There is also an isomorphism

τ_{𝒪}^{♯} : τ^{−1}𝒪_{ℙ^{1}}^{hol} → 𝒪_{ℙ^{1}}^{hol},

given by

τ_{𝒪}^{♯} (f)(z) = \overline{f(z̄)},

and satisfying

τ_{𝒪}^{♯} ∘ τ^{−1}(τ_{𝒪}^{♯}) = id: 𝒪_{ℙ^{1}}^{hol} → 𝒪_{ℙ^{1}}^{hol}.

Definition 6.1. Define FrAlg_{ℙ^{1},ℂ} to be the category of sheaves ℱ of unital multiplicatively convex Fréchet 𝒪_{ℙ^{1}}^{hol}-algebras (i.e. countable pro-Banach algebras equipped with a morphism from 𝒪_{ℙ^{1}}^{hol}), quasi-coherent in the sense that the maps

ℱ (U) ⊗_{𝒪_{ℙ^{1}}^{hol}(U)}^{π} 𝒪_{ℙ^{1}}^{hol}(V) → ℱ (V)

are isomorphisms for all open subspaces V ⊂ U, where ⊗^{π} here denotes projective tensor product.

Define FrAlg_{ℙ^{1},ℝ} to be the category of pairs (ℱ , τ_{ℱ}^{♯} ) for ℱ ∈ FrAlg_{ℙ^{1},ℂ} and

τ_{ℱ}^{♯} : τ^{−1}ℱ → ℱ

an (𝒪_{ℙ^{1}}^{hol}, τ_{𝒪}^{♯})-linear isomorphism satisfying

τ_{ℱ}^{♯} ∘ τ^{−1}(τ_{ℱ}^{♯} ) = id_{ℱ} .

Note that for any m-convex Fréchet k-algebra B, the sheaf B⊗_{k}^{π} 𝒪_{ℙ^{1}}^{hol} lies in FrAlg_{ℙ^{1},ℂ}. When k = ℝ, the involution id_{B} ⊗ τ_{𝒪}^{♯} makes B ⊗_{k}^{π} 𝒪_{ℙ^{1}}^{hol} an object of FrAlg_{ℙ^{1},ℝ}.

The forgetful functor FrAlg_{ℙ^{1},ℝ} → FrAlg_{ℙ^{1},ℂ} has a right adjoint, given by ℱ ↦ ℱ ⊕τ^{−1}ℱ, with the involution τ^{♯} given by swapping summands, and the 𝒪_{ℙ^{1}}^{hol}-structure on τ^{−1}ℱ defined using τ_{𝒪} .

6.2. The twistor functors.

Definition 6.2. Define the groupoid-valued functor R_{X}^{𝕋,ℂ} on FrAlg_{ℙ^{1},ℂ} by letting R_{X}^{𝕋,ℂ}(ℬ) consist of pairs (𝒯 , D̃), for 𝒜_{X}^{0}(pr_{2}^{−1}ℬ^{×})-torsors 𝒯 on X × ℙ^{1}(ℂ) with x^{∗}𝒯 trivial as a ℬ^{×}-torsor on ℙ^{1}(ℂ), and flat ud + vd^{c}-connections

D̃: 𝒯 → 𝒜_{X}^{1} ⊗_{𝒜_{X}^{0}} ad𝒯 (1).

Here u, v are the basis of Γ(ℙ_{ℝ}^{1}, 𝒪(1)) given by the co-ordinates u, v of C^{∗} and the canonical map C^{∗} → ℙ^{1}.

Define the set-valued functor R_{X,x}^{𝕋,ℂ} on FrAlg_{ℙ^{1},ℂ} by letting R_{X,x}^{𝕋,ℂ}(ℬ) be the groupoid of triples (𝒯 , D̃, f), with (𝒯 , D̃) ∈ R_{X}^{𝕋,ℂ}(ℬ) and framing f ∈ Γ(ℙ^{1}(ℂ), x^{∗}𝒯 ).

Definition 6.3. Define the groupoid-valued functor R_{X}^{𝕋} on FrAlg_{ℙ^{1},ℝ} by letting R_{X}^{𝕋}(ℬ) consist of triples (𝒯 , D̃, τ_{𝒯}^{♯} ), for (𝒯 , D̃) ∈ R_{X}^{𝕋,ℂ}(ℬ) and isomorphism τ_{𝒯}^{♯} : (id_{X} × τ)^{−1}𝒯 → 𝒯 satisfying the following conditions. The isomorphism

(τ_{ℬ}^{♯} , τ_{𝒯}^{♯} ): 𝒜_{X}^{1}(pr_{2}^{−1}ℬ^{×}) ×_{τ_{ℬ}^{♯} ,𝒜_{X}^{1}(pr_{2}^{−1}τ^{−1}ℬ^{×})} (id_{X} × τ)^{−1}𝒯 → 𝒯

must be a morphism of 𝒜_{X}^{1}(pr_{2}^{−1}ℬ^{×})-torsors, and the diagram

τ^{−1}𝒯 ⟶^{τ^{−1}D̃} 𝒜_{X}^{1} ⊗_{𝒜_{X}^{0}} τ^{−1}𝒯 (1)
τ_{𝒯}^{♯} ↓  ↓^{τ_{𝒯}^{♯}}
𝒯 ⟶^{D̃} 𝒜_{X}^{1} ⊗_{𝒜_{X}^{0}} 𝒯 (1)

must commute.

Define the functor R_{X,x}^{𝕋} on FrAlg_{ℙ^{1},ℝ} by letting R_{X,x}^{𝕋}(ℬ) be the groupoid of quadruples (𝒯 , D̃, τ_{𝒯}^{♯} , f), for (𝒯 , D̃, τ_{𝒯}^{♯} ) in R_{X}^{𝕋}(ℬ) and

f ∈ Γ(ℙ^{1}(ℂ), x^{∗}𝒯 )^{τ_{𝒯}^{♯}}

a framing.

Remark 6.4. Observe that the groupoids R_{X,x}^{𝕋,ℂ}(ℬ), R_{X,x}^{𝕋}(ℬ) are equivalent to discrete groupoids, so we will regard them as set-valued functors (given by isomorphism classes of objects). Also note that R_{X}^{𝕋,ℂ}(ℬ) and R_{X}^{𝕋}(ℬ) are equivalent to the groupoid quotients [R_{X,x}^{𝕋,ℂ}(ℬ)/Γ(ℙ^{1}(ℂ), ℬ^{×})] and [R_{X,x}^{𝕋}(ℬ)/Γ(ℙ^{1}(ℂ), ℬ^{×})^{τ_{𝒯}^{♯}} ] respectively, with the action given by changing the framing.

The following is straightforward:

Lemma 6.5. The functors R_{X,x}^{𝕋,ℂ} and R_{X}^{𝕋,ℂ} can be recovered from R_{X,x}^{𝕋} and R_{X}^{𝕋}, respectively, via isomorphisms

R_{X,x}^{𝕋,ℂ}(ℬ) ≅ R_{X,x}^{𝕋}(ℬ ⊕ τ^{−1}ℬ) R_{X}^{𝕋,ℂ}(ℬ) ≅ R_{X}^{𝕋}(ℬ ⊕ τ^{−1}ℬ).

Note that the proof of Lemma 2.5 carries over to give canonical comultiplication

R_{X,x}^{𝕋}(ℬ_{1}) × R_{X,x}^{𝕋}(ℬ_{2}) → R_{X,x}^{𝕋}(ℬ_{1} ⊗^{π} ℬ_{2}).

6.3. Universality and σ-invariant sections.

Proposition 6.6. The functors R_{X}^{dR} and R_{X}^{Dol} can be recovered from R_{X}^{𝕋} and R_{X}^{𝕋,ℂ}, respectively. Likewise, R_{X,x}^{dR} and R_{X,x}^{Dol} can be recovered from R_{X,x}^{𝕋} and R_{X,x}^{𝕋,ℂ}.

Proof. Given a point p ∈ ℙ^{1}(ℂ) and a complex m-convex Fréchet algebra B, we may regard the skyscraper sheaf p_{∗}B as an object of FrAlg_{ℙ^{1},ℂ}. For a real m-convex Fréchet algebra B and p ∈ ℙ^{1}(ℝ), we may regard p_{∗}B ⊗ ℂ as an object of FrAlg_{ℙ^{1},ℝ}, with τ^{♯} given by complex conjugation.

Now, just observe that on pulling back to (1 : 0) ∈ ℙ^{1}(ℝ), the differential ud + vd^{c} is just d. At (1 : −i) ∈ ℙ^{1}(ℝ), we have ud + vd^{c} = d − id^{c} = 2∂. Uncoiling the definitions, this gives

R_{X,x}^{dR}(B) = R_{X,x}^{𝕋}((1 : 0)_{∗}B), R_{X,x}^{Dol}(B) = R_{X,x}^{𝕋,ℂ}((1 : −i)_{∗}B),

and similarly for R. □

Lemma 6.7. For a real m-convex Fréchet algebra B, the groupoid R_{X}^{𝕋}(B ⊗_{ℝ}^{π} 𝒪_{ℙ^{1}}^{hol}) is equivalent to the groupoid of triples (𝒫, D, E), where 𝒫 is an 𝒜_{X}^{0}(B^{×})-torsor on X, D, E: 𝒫 → 𝒜_{X}^{1} ⊗_{𝒜_{X}^{0}} ad𝒫 are flat d- and d^{c}-connections, respectively, and DE +ED = 0.

Proof. Take an object (𝒯 , D̃) ∈ R_{X}^{𝕋}(B ⊗_{ℝ} 𝒪_{ℙ^{1}}^{hol}). Triviality of the (B ⊗_{ℝ}^{π} 𝒪_{ℙ^{1}}^{hol})^{×}-torsors x^{∗}𝒯 for x ∈ X ensures that 𝒯 must be of the form 𝒜_{X}^{0}(B ⊗_{ℝ}^{π}B)^{×} ×_{𝒜_{X}^{0}(B^{×})} 𝒫 for some 𝒫 as above.

Now, since Γ(ℙ^{1}, 𝒪_{ℙ^{1}}^{hol}(1))^{τ^{♯}} = ℝu⊕ℝv, the connection D̃ can be regarded as a ud+vd^{c} connection

D̃: 𝒫 → (𝒜_{X}^{1} ⊗_{𝒜_{X}^{0}} ad𝒫) ⊗_{ℝ} (ℝu ⊕ ℝv),

which we write as uD + vE.

Flatness of D̃ is now a statement about Γ(ℙ^{1}, 𝒪_{ℙ^{1}}^{hol}(2))^{τ^{♯}} = ℝu^{2} ⊕ ℝuv ⊕ ℝv^{2}, with the u^{2} and v^{2} terms giving flatness of D and E, and the uv term giving the condition [D, E] = 0. □

Definition 6.8. Define the involution τσ_{ℙ} of the polarised scheme (ℙ^{1}, 𝒪_{ℙ^{1}}(1)) to be the map induced by the action of i ∈ S(ℝ) on C^{∗} from Definition 5.1. In particular, τσ_{ℙ}(u : v) = (v : −u).

Remark 6.9. On [Sim2, p12], the co-ordinate system (u + iv : u − iv) on ℙ^{1}(ℂ) is used, and antiholomorphic involutions σ, τ are defined. In our co-ordinates, these become σ(u : v) = (v̄, −ū) and τ(u : v) = (ū, v̄). This justifies the notation τσ used above. Also note that the 𝔾_{m,ℂ}-action on ℙ_{ℂ}^{1} given in [Sim2, p4] is just the complex form of our circle action on ℙ_{ℝ}^{1} from §§5.3 and 5.4.3.

Definition 6.10. Given a real Banach algebra B, define the involution τσ′ of R_{X}^{𝕋}(B ⊗_{ℝ}^{π} 𝒪_{ℙ^{1}}^{hol}) by sending the pair (𝒯 , D̃) to (τσ_{ℙ}^{−1}𝒯 , Jτσ_{ℙ}^{−1}D̃). Note that this is well-defined because τσ_{ℙ}^{−1}D̃ is a τσ_{ℙ}(ud + vd^{c}) = (vd − ud^{c})-connection, and Jd = d^{c}, Jd^{c} = −d, so Jτσ_{ℙ}^{−1}D̃ is a (ud + vd^{c})-connection.

Definition 6.11. Given a C^{∗}-algebra B, define the Cartan involution C of B^{×} to be given by C(g) = (g^{−1})^{∗}. Note that this induces a Lie algebra involution adC: b ↦ −b^{∗} on the tangent space B of B^{×}.

If B = A ⊗ ℂ for a real C^{∗}-algebra A, we write τ for complex conjugation, so Cτ is the involution Cτ(g) = (ḡ^{−1})^{∗}. Note that adCτ is the ℂ-linear extension of adC on A.

Since R_{X}^{𝕋}(ℬ) only depends on the group of units ℬ^{×} of ℬ, and its tangent Lie algebra ℬ, the Cartan involution induces an involution Cτ of R_{X}^{𝕋}(B ⊗_{ℝ}^{π} 𝒪_{ℙ^{1}}^{hol}).

Definition 6.12. Given a real Banach algebra B, define the involution σ of R_{X}^{𝕋}(B ⊗_{ℝ}^{π} 𝒪_{ℙ^{1}}^{hol}) by σ := (Cτ)(τσ′).

Proposition 6.13. For a real C^{∗}-algebra B, there is a canonical isomorphism

R_{X,x}^{𝕋}(𝒪_{ℙ^{1}}^{hol}(B))^{σ} ≅ R_{X,x}^{J}(B).

Proof. Since R_{X,x}^{𝕋} is the groupoid fibre of R_{X}^{𝕋} → R_{{x}}^{𝕋} over the trivial torsor, Lemma 6.7 shows that an object (𝒯 , D̃, f) of R_{X,x}^{𝕋}(𝒪_{ℙ^{1}}^{hol}(B)) is a quadruple (𝒫, D, E, f) for (𝒫, D, E) as in that lemma, and f our framing. We therefore begin by describing the σ-action on such data. For (𝒫, D, E, f) to be σ-invariant, we must have an isomorphism α: (𝒫, D, E, f) → σ(𝒫, D, E, f).

The torsor 𝒫 maps under σ to C(𝒫), with f ∈ x^{∗}𝒫 mapping to C(f) (since σ′ and τ affect neither). The isomorphism α then gives α: 𝒫 → C(𝒫) such that α(f) = C(f) ∈ x^{∗}C(𝒫). Let U(𝒫) ⊂ 𝒫 consist of sections q with α(q) = C(q). This is non-empty (since its fibre at x contains f), so it must be an 𝒜_{X}^{0}(U(B))-torsor, noting that U(B) is the group of C-invariants in B^{×}. Moreover 𝒫 = 𝒜_{X}^{0}(B^{×})×_{𝒜_{X}^{0}(U(B))} U(𝒫).

Meanwhile, τσ′(uD + vE) = vJD − uJE, so τσ′(D, E) = (−JE, JD). Thus the isomorphism α gives

D|_{𝒬} = −JCE, E|_{𝒬} = JCD.

In other words, E = D^{c} and E^{c} = −D (which are equivalent conditions).

Thus R_{X,x}^{𝕋}(𝒪_{ℙ^{1}}^{hol}(B))^{σ} is equivalent to the groupoid of triples (U(𝒫), D, f), with D flat and [D, D^{c}] = 0. □

Remark 6.14. When B = Mat_{n}(ℂ), this shows that framed pluriharmonic local systems correspond to framed σ-invariant sections of the twistor functor. Without the framings, this will not be true in general, since a σ-invariant section of the coarse moduli space will give a non-degenerate bilinear form which need not be positive definite. Note that for U ⊂ ℙ^{1}, the set of isomorphism classes of R_{X}^{𝕋,ℂ}(Mat_{n}(𝒪_{U}^{hol})) is the set of sections over U of the twistor space T W → ℙ^{1} of [Sim1, §3].

Remark 6.15. Although we have seen that R_{X}^{𝕋} together with its comultiplication encodes all the available information about twistor structures on moduli spaces of local systems, it does not carry information about higher homotopy and cohomology groups. There is, however, a natural extension of R_{X}^{𝕋} to differential graded m-convex Fréchet algebras, by analogy with [Pri1, Pri2]. This would involve taking D̃ to be a hyperconnection D̃: T_{0} → ∏_{n} 𝒜_{X}^{n+1} ⊗_{𝒜_{X}^{0}} ad𝒯_{n}(n + 1). The structures of §4 can all be recovered from this functor.

6.4. Topological twistor representation spaces. In this section, we will show that by considering continuous homomorphisms rather than ∗-homomorphisms, we can describe the entire semisimple locus of the twistor family from (E_{X,x}^{J})_{PN}, rather than just the σ-equivariant sections.

Given a point (a : b) ∈ ℙ^{1}(ℂ) and a complex Banach algebra B, we can generalise the construction of Proposition 6.6 and consider the set R_{X,x}^{𝕋,ℂ}((a : b)_{∗}B). This consists of torsors with flat (ad + bd^{c})-connections.

Definition 6.16. Define T_{X,x,n} := ∐_{(a:b)∈ℙ^{1}(ℂ)} R_{X,x}^{𝕋,ℂ}((a : b)_{∗}Mat_{n}(ℂ)).

Note that T_{X,x,n} inherits a Gal(ℂ/ℝ)-action from R_{X,x}^{𝕋,ℂ}. We can also give T_{X,x,n} a complex analytic structure, by saying that a map f: U → T_{X,x}(B) from an analytic space U consists of an analytic map f: U → ℙ^{1}(ℂ) together with an element of R_{X,x}^{𝕋,ℂ}(f_{∗}(Mat_{n}𝒪_{U})). We will now investigate the underlying topological structure.

Remark 6.17. The adjoint action of GL_{n}(ℂ) on T_{X,x,n} is continuous, and indeed compatible with the complex analytic structure. This allows us to consider the coarse quotient T_{X,x,n}//GL_{n}(ℂ), which is the Hausdorff completion of the topological quotient, equipped with a natural complex analytic structure over ℙ^{1}(ℂ). A straightforward calculation shows that this coarse moduli space is precisely the Deligne-Hitchin twistor space, as constructed in [Hit] and described in [Sim3] §3.

Now, Proposition 6.13 induces a map π_{𝕋}: R_{X,x}^{J}(Mat_{n}(ℂ)) × ℙ^{1}(ℂ) → T_{X,x,n}. For an explicit characterisation, note that an (ad + bd^{c})-connection D̃ lies in R_{X,x}^{J}(Mat_{n}(ℂ)) if and only if

[D̃, JCD̃] = 0,

and that JCD̃ is a (−b̄d + ād^{c}) = σ_{ℙ}^{−1}(ad + bd^{c})-connection.

Definition 6.18. Given a flat (ad+bd^{c})-connection D̃ on a finite-dimensional C^{∞} vector bundle 𝒱 on X for (a : b) ∈ ℙ^{1}(ℂ) − {±i}, we say that (𝒱 , D̃) is semisimple if the local system ker D̃ is so.

Definition 6.19. Define T_{X,x,n}^{st} ⊂ T_{X,x,n} by requiring that the fibre over any point of ℙ^{1}(ℂ)−{±i} consist of the semisimple objects, that the fibre over i be R_{X,x}^{Dol,st}(Mat_{n}(ℂ)) (§3.3), and that the fibre over −i be its conjugate.

We give T_{X,x,n}^{st} the subspace topology, so a map K → T_{X,x,n}^{st} is continuous if the projection f: K → ℙ^{1}(ℂ) is so, and the map lifts to an element of R_{X,x}^{𝕋,ℂ}(f_{∗}C(K, Mat_{n}(ℂ))).

Theorem 6.20. For any positive integer n, there is a natural homeomorphism π_{𝕋,st} over ℙ^{1}(ℂ) between the space Hom_{pro(BanAlg)}(E_{X,x}^{J}, Mat_{n}(ℂ)) × ℙ^{1}(ℂ) with the topology of pointwise convergence, and the space T_{X,x,n}^{st}.

Proof. The homeomorphism is given on the fibre over (a : b) ∈ ℙ^{1}(ℂ) by π_{𝕋,st}(U(𝒫), D, f) = (𝒫, aD + bD^{c}, f). The proofs of Theorems 3.6 and 3.23 adapt to show that π_{𝕋,st} induces homeomorphisms on fibres over ℙ^{1}(ℂ), and in particular is an isomorphism on points. The same arguments also show that π_{𝕋,st} and π_{dR}∘π_{𝕋,st}^{−1}: π_{𝕋,st} → R_{X,x}^{dR,ss}(Mat_{n}(ℂ)) are continuous, so the result follows from Theorem 3.6. □

Definition 6.21. Let FDT_{X,x}^{st} be the category of pairs (V, p, (a : b)) for V ∈ FDVect and (p, (a : b)) ∈ T_{X,x,n}^{st} where n = dim V. Morphisms are defined by adapting the formulae of Definition 3.7. Write η_{x}^{𝕋,st} : FDT_{X,x}^{st} → FDVect×ℙ^{1}(ℂ) for the fibre functors (V, p, (a : b)) ↦ (V, (a : b)).

Proposition 6.22. The C^{∗}-algebra (E_{X,x}^{J})_{PN} ˆ⊗_{ℝ}C(ℙ^{1}(ℂ), ℂ) is isomorphic to the ring of continuous endomorphisms of η_{x}^{𝕋,st}, and this isomorphism is Gal(ℂ/ℝ)-equivariant.

Proof. The proof of Proposition 3.8 carries over, replacing Theorem 3.6 with Theorem 6.20, and Lemma 1.28 with Lemma 1.30. □

Definition 6.23. Given a k-normal real C^{∗}-algebra B over C(ℙ^{1}(ℂ), ℂ)^{Gal(ℂ/ℝ)}, we may regard B as an 𝒪_{ℙ^{1}}^{hol}-algebra via the inclusion of holomorphic functions in continuous functions. Then define R_{X,x}^{𝕋,st}(B) ⊂ R_{X,x}^{𝕋}(B) to be the subspace consisting of those p for which

(ψ(p), (a : b)) ∈ T_{X,x,k}^{st}

for all (a : b) ∈ ℙ^{1}(ℂ) and ψ: B → Mat_{k}(ℂ) with ψ|_{C(ℙ^{1}(ℂ),ℂ)^{Gal(ℂ/ℝ)}} = ev_{(a:b)}id.

Corollary 6.24. For any k-normal real C^{∗}-algebra B over C(ℙ^{1}(ℂ), ℂ)^{Gal(ℂ/ℝ)}, there is a natural isomorphism between R_{X,x}^{𝕋,st}(B)^{Gal(ℂ/ℝ)} and the set of continuous algebra homomorphisms E_{X,x}^{J} → B.

Proof. The proof of Corollary 3.11 carries over, replacing Proposition 3.8 with Proposition 6.22. □

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