2.1 Representations on rigged Hilbert spaces

In this paper, we work with rigged Hilbert spaces and rigged C$^{*}\!$-algebras as introduced in § 2 of our paper [Reference Kristel and WaldorfKW20]; we briefly review all required notions and results, and refer the reader to that paper for more motivation and context.

Given a rigged Hilbert space $E$, one obtains an honest Hilbert space, denoted $\hat E$, by completion with respect to the inner product. The prototypical example of a rigged Hilbert space is the Fréchet space of smooth functions on the circle, equipped with the $L^2$-inner product; its Hilbert completion is the space of square-integrable functions on the circle.

By a smooth representation of a Fréchet Lie group $\mathcal {G}$ on a rigged Hilbert space $E$ we mean an action of $\mathcal {G}$ on $E$ by unitary morphisms of rigged Hilbert spaces, such that the map $\mathcal {G} \times E \to E$ is smooth. The typical example in this paper is the rigged Hilbert space $F_L^{\mathrm {s}}$ of smooth vectors in a Fock space of a Lagrangian $L\subset V$, which carries a smooth representation of a Lie group $\operatorname {Imp}_L^{\theta }(V)$ of implementers, see Proposition 3.2.2.

By definition, an actual C$^{*}\!$-algebra $\hat A$ can be obtained from a rigged C$^{*}\!$-algebra $A$ by norm completion. By a smooth representation of a Fréchet Lie group $\mathcal {G}$ on a rigged C$^{*}\!$-algebra $A$ we mean an action of $\mathcal {G}$ on $A$ by isometric isomorphisms of rigged C$^{*}\!$-algebras, such that the map $\mathcal {G} \times A \to A$ is smooth. The typical example in this paper is the Fréchet algebra $\operatorname {Cl}(V)^{\mathrm {s}}$ of smooth vectors in the Clifford algebra of a real Hilbert space $V$, which carries a smooth representation of the orthogonal group $\operatorname {O}(V)$ via Bogoliubov automorphisms, see Proposition 3.2.4.

Of major importance for this article is a certain type of representations of rigged C$^{*}\!$-algebras on rigged Hilbert spaces.

Definition 2.1.3 Let $A$ be a rigged C$^{*}\!$-algebra. A rigged $A$-module is a rigged Hilbert space $E$ together with a representation $\rho$ of (the underlying algebra of) $A$ on (the underlying vector space of) $E$, such that the map $\rho :A \times E \to E$ is smooth and the following conditions hold for all $a \in A$ and all $v,w \in E$:

(1)\begin{equation} \langle \rho(a,v), \rho(a,v) \rangle \leqslant \| a \|^{2} \langle v, v \rangle \quad \text{and} \quad \langle \rho(a,v),w \rangle = \langle v, \rho(a^{*},w) \rangle. \end{equation}

A (unitary) intertwiner from a rigged $A_1$-module $E_1$ to a rigged $A_2$-module $E_2$ is a pair $(\phi,\psi )$ of a morphism $\phi :A_1 \to A_2$ of rigged C$^{*}\!$-algebras and a bounded (unitary) morphism $\psi :E_1\to E_2$ of rigged Hilbert spaces that intertwines the representations along $\phi$.

We recall that if a map $\rho :A \times E \to E$ is bilinear, it is smooth if and only if it is continuous.

The typical example is the rigged Hilbert space $F_L^{\mathrm {s}}$, which becomes under Clifford multiplication a rigged $\operatorname {Cl}(V)^{\mathrm {s}}$-module, see Proposition 3.2.5. A couple of remarks are in order.

Remark 2.1.4 (a) Conditions (1) in Definition 2.1.3 are chosen such that a rigged $A$-module $E$ with underlying representation $\rho$ induces a $\ast$-homomorphism $\hat \rho : \hat A \to \mathcal {B}(\hat E)$, i.e. a representation of the C$^{*}\!$-algebra $\hat A$ on the Hilbert space $\hat E$, see [Reference Kristel and WaldorfKW20, Remark 2.2.11].

(b) If $\phi :A \to B$ is a morphism of rigged C$^{*}\!$-algebras, and $E$ is a rigged $B$-module with representation $\rho$, then $E$ becomes a rigged $A$-module under the induced representation $(a,v) \mapsto \rho (\phi (a),v)$. Moreover, the pair $(\phi,\mathbb {1})$ is a unitary intertwiner.

(c) Rigged C$^{*}\!$-algebras and rigged modules are compatible with dualization. If $A$ is a rigged C$^{*}\!$-algebra, then its opposite algebra, $A^{\mathrm {opp}}$, is a rigged C$^{*}\!$-algebra in a natural way, and its completion $(A^{\operatorname {opp}})^{\|\cdot \|}$ is the usual opposite C$^{*}\!$-algebra. Let $E$ be a rigged $A$-module. The inner product on $E$ gives us a complex anti-linear injection $\iota : E \rightarrow E^{*}$ mapping $E$ into its continuous linear dual $E^{*}$. Denote the image of $\iota$ by $E^{\sharp }$. We turn $E^{\sharp }$ into a Fréchet space using the identification with $E$. If $\rho$ denotes the representation of $A$ on $E$, then the map

\[ \rho^{\sharp}: A^{\mathrm{opp}} \times E^{\sharp} \rightarrow E^{\sharp}, (a,\varphi) \mapsto \varphi \circ \rho(a,-), \]

is a representation of $A^{\mathrm {opp}}$ on $E^{\sharp }$, and it is straightforward to show that it turns $E^{\sharp }$ into a rigged $A^{\operatorname {opp}}$-module.

Next, we extend our framework, which we have so far recalled from [Reference Kristel and WaldorfKW20], by introducing new structures that ultimately lead to the notion of rigged bimodules over rigged von Neumann algebras.

Under completion, a rigged $A$–$B$-bimodule $E$ becomes a Hilbert space $\hat E$ with commuting representations of the C$^{*}\!$-algebras $\hat A$ and $\hat B^{\operatorname {opp}}=\widehat {B^{\operatorname {opp}}}$. Our main example of a rigged bimodule is again the space $F_L^{\mathrm {s}}$ of smooth vectors in a Fock space, which we equip with a rigged bimodule structure over a certain subalgebra of $\operatorname {Cl}(V)^{\mathrm {s}}$, see § 3.3. Next, we develop the setting of rigged von Neumann algebras.

Remark 2.1.7 If $N=(A,E)$ is a rigged von Neumann algebra, and $\rho$ its defining representation, then the assumption that $\hat \rho : \hat A \rightarrow \mathcal {B}(\hat E)$ is faithful implies that it is an isometry and, hence, a homeomorphism onto its image. We may thus identify $\hat \rho (\hat A)$ with $\hat A$. Then, we define the von Neumann algebra

\[ N'':=(\hat A)'' \subseteq \mathcal{B}(\hat E), \]

and conclude that it contains $\hat A$ as a weakly dense (and, thus, also $\sigma$-weakly dense) subspace. In particular, any rigged von Neumann algebra $N$ gives rise to an ordinary von Neumann algebra $N''$.

An example of a rigged von Neumann algebra is the Clifford algebra $\operatorname {Cl}(V)^{\mathrm {s}}$ with its defining representation on Fock space $F^{\mathrm {s}}_L$, see Proposition 3.2.5. The associated von Neumann algebra is $\mathcal {B}(F_L)$. More interesting examples will be constructed in Proposition 3.2.9; these give rise to type III$_1$-factors.

We recall that ‘normal’ means to be continuous in the $\sigma$-weak topologies, and we recall that these topologies are generated by semi-norms $p_{\rho }(T) := | \mathrm {tr}( \rho T) |$, where $\rho$ is a trace-class operator. Since $\hat A\subset N''$ is $\sigma$-weakly dense, it is clear that the extension $N_1'' \to N_2''$ is unique, if it exists. The following lemma gives us a useful sufficient criterion for the extendability.

Proof. All that needs to be shown is that $\phi :A_{1} \rightarrow A_{2}$ extends to a normal $*$-homomorphism $N_{1}'' \rightarrow N_{2}''$. According to Remark 2.1.7 we view $A_{1}$ as a subset of $\mathcal {B}(\hat E_1)$ and $A_{2}$ as a subset of $\mathcal {B}(\hat E_2)$. Now, we have, for all $a \in A_{1}$ and all $v \in \hat E_2$,

\[ \phi(a)v = \phi(a) \psi \psi^{*} v = \psi a\psi^{*}v. \]

Here, $\psi \psi ^{*}=\mathbb {1}$ since $\psi$ is unitary and, thus, extends to a unitary operator $\psi :\hat E_1 \to \hat E_2$ on the completions. It follows that $\phi (a) = \psi a \psi ^{*}$ as elements of $\mathcal {B}(\hat E_2)$, which implies that the map $C_{\psi }:\mathcal {B}(\hat E_1) \rightarrow \mathcal {B}(\hat E_2), a \mapsto \psi a \psi ^{*}$ extends $\phi$. Next, we prove that $C_{\psi }$ is normal. If $\rho$ is a trace-class operator on $\hat E_2$, then $\psi ^{*} \rho \psi$ is a trace-class operator on $\hat E_1$, and we have

\[ p_{\psi^{*} \rho \psi}(a) = p_{\rho}(C_{\psi}(a)), \]

for all $a \in \mathcal {B}(\hat E_1)$. This implies that $C_{\psi }$ is $\sigma$-weakly continuous. Now, because $A_{1}$ is $\sigma$-weakly dense in $N_{1}''$, and $A_{2}$ is $\sigma$-weakly dense in $N_{2}''$ it follows that $C_{\psi }(N_{1}'') \subseteq N_{2}''$, and that, moreover, $C_{\psi }: N_{1}'' \rightarrow N_{2}''$ is a $*$-homomorphism.

Next is the discussion of modules and bimodules for rigged von Neumann algebras.

We will need the following lemma, which will be used later in the proof of Lemma 2.1.16.

Finally, we come to bimodules for rigged von Neumann algebras. One of the fundamental results of this work exhibits the Fock space $F^{\mathrm {s}}$ as a rigged von Neumann bimodule, see Proposition 3.3.4.

Here, Lemma 2.1.10 implies again that spatial intertwiners are in particular (unitary) intertwiners. We assure by the following result that completion brings us into the classical setting, see Appendix A.1.

We remark that ordinary von Neumann algebras, bimodules and intertwiners form a bicategory, in which the composition of morphisms is given by the Connes fusion of bimodules [Reference BrouwerBro03, Reference Stolz and TeichnerST04]. We have, unfortunately, not yet been able to lift Connes fusion to the setting of rigged von Neumann bimodules and, thus, there is no corresponding bicategory of rigged von Neumann algebras. This is an important issue that we are going to address in future work.

2.2 Locally trivial rigged bundles

In this section we introduce locally trivial bundles of rigged von Neumann algebras and rigged von Neumann bimodules over Fréchet manifolds. The prerequisite notions of rigged Hilbert space bundles, rigged C$^{*}\!$-algebra bundles, and rigged module bundles have been introduced in § 2 of [Reference Kristel and WaldorfKW20], and are discussed there in more detail. Throughout this section, we let $\mathcal {M}$ be a Fréchet manifold.

It is straightforward to see that the fibres of a rigged Hilbert space bundle $\mathcal {E}$ are rigged Hilbert spaces [Reference Kristel and WaldorfKW20, Remark 2.1.10], and that the fibrewise completion of $\mathcal {E}$ is a locally trivial continuous Hilbert space bundle over $\mathcal {M}$ with typical fibre $\hat E$ (see [Reference Kristel and WaldorfKW20, Lemma 2.1.13]). Here, a locally trivial continuous Hilbert space bundle with fibre $H$ has continuous local transition functions $U \times H \rightarrow H$ or, equivalently, strongly continuous maps $U \rightarrow \operatorname {U}(H)$. Likewise, on the level of morphisms, any locally bounded morphism of rigged Hilbert space bundles extends uniquely to a continuous morphism of the corresponding continuous Hilbert space bundles. The following lemma [Reference Kristel and WaldorfKW20, Proposition 2.1.15] shows that our notion of smooth representations (see § 2.1) fits well into the context of rigged Hilbert space bundles.

The spinor bundle on loop space is a rigged Hilbert space bundle over $\mathcal {M}=LM$, and will be defined via Lemma 2.2.2, see Definition 4.1.4. We proceed similarly for rigged C$^{*}\!$-algebra bundles.

One can show that each fibre of a rigged C$^{*}\!$-algebra bundle $\mathcal {A}$ is a rigged C$^{*}\!$-algebra, and that the opposite multiplication produces a rigged C$^{*}\!$-algebra bundle $\mathcal {A}^{\operatorname {opp}}$ with typical fibre $A^{\operatorname {opp}}$. Further, the fibrewise norm completion gives a locally trivial continuous bundle of C$^{*}\!$-algebras with typical fibre $\hat A$, and strongly continuous transition functions [Reference Kristel and WaldorfKW20, Lemma 2.2.6]. Likewise, any morphism of rigged C$^{*}\!$-algebra bundles extends uniquely to a continuous morphism of continuous bundles of C$^{*}\!$-algebras. We remark that every locally trivial continuous bundle of C$^{*}\!$-algebras yields a continuous field of C$^{*}\!$-algebras by taking its continuous sections.

Rigged C$^{*}\!$-algebra bundles can be obtained by associating a smooth representation to a principal bundle, as the following result [Reference Kristel and WaldorfKW20, Proposition 2.2.8] shows.

The Clifford bundle on loop space is a rigged C$^{*}\!$-algebra bundle, and it will be defined via Lemma 2.2.4, see Definition 4.1.5. Next, we discuss module bundles and bimodule bundles for rigged C$^{*}\!$-algebra bundles.

Definition 2.2.5 Let $A$ be a rigged C$^{*}\!$-algebra and $E$ be a rigged $A$-module, with representation $\rho _0$, and let $\mathcal {A}$ be a rigged C$^{*}\!$-algebra bundle over $\mathcal {M}$ with typical fibre $A$. A rigged $\mathcal {A}$-module bundle with typical fibre $E$ is a rigged Hilbert space bundle $\mathcal {E}$ with typical fibre $E$, and a fibre-preserving map

\[ \rho: \mathcal{A} \times_{\mathcal{M}} \mathcal{E} \rightarrow \mathcal{E} \]

with the property that around every point in $\mathcal {M}$ there exist local trivializations $\Phi$ of $\mathcal {A}$ and $\Psi$ of $\mathcal {E}$ that fibrewise intertwine $\rho$ with $\rho _0$, i.e. we have $\Psi _x(\rho (a,v))=\rho _0(\Phi _x(a),\Psi _x(v))$ for all $x\in \mathcal {M}$ over which $\Phi$ and $\Psi$ are defined, and all $a\in \mathcal {A}_x$ and $v\in \mathcal {E}_x$. A pair $(\Phi,\Psi )$ of local trivializations with this property is called compatible.

One can easily show that $\rho$ is automatically a morphism of Fréchet vector bundles. Furthermore, for each $x \in \mathcal {M}$, the map $\rho _{x}$ turns $\mathcal {E}_x$ into a rigged $\mathcal {A}_x$-module; and every pair of compatible local trivializations $(\Phi,\Psi )$ around $x$ yields an invertible unitary intertwiner $(\Phi _x,\Psi _x)$ between the rigged $\mathcal {A}_x$-module $\mathcal {E}_x$ and the rigged $A$-module $E$.

The definition of a rigged bimodule bundle now follows naturally; it is, however, important enough that we give it in full detail.

It is straightforward to see that the fibres $\mathcal {E}_x$ are rigged $(\mathcal {A}_1)_x$–$(\mathcal {A}_2)_x$-bimodules, for each $x\in \mathcal {M}$, and that compatible local trivializations around $x$ establish an invertible unitary intertwiner between $\mathcal {E}_x$ and $E$. For completeness, we also note the bundle version of a bimodule intertwiner.

Later, we will exhibit the spinor bundle on loop space as a rigged module bundle for the Clifford bundle, see § 4. Next we come to the definition of rigged von Neumann algebra bundles and rigged von Neumann bimodule bundles. These definitions are a novelty introduced in this article; we are not aware of any other treatment of smooth locally trivial bundles of von Neumann algebras, or von Neumann bimodules.

Thus, the difference between a rigged $\mathcal {A}$-module bundle $\mathcal {E}$ and a rigged von Neumann algebra bundle $\mathcal {N}=(\mathcal {A},\mathcal {E})$ only lies in fact that the typical fibre is required to be a rigged von Neumann algebra. Via compatible local trivializations (Definition 2.2.5), this property extends to all fibres; thus, every fibre $\mathcal {N}_x=(\mathcal {A}_x,\mathcal {E}_x)$ is a rigged von Neumann algebra. Moreover, Lemma 2.1.10 guarantees that any choice of compatible local trivializations induces fibrewise spatial isomorphisms from $\mathcal {N}_x$ to $N$.

The rigged von Neumann algebra bundles that appear in this paper are certain Clifford algebra bundles over loop spaces, and they appear in §§ 4.1 and 5.1. For now, it remains to define rigged von Neumann bimodule bundles.

The following lemma is to ensure that Definition 2.2.13 gives the correct structure in each fibre.

Proof. Because $\mathcal {F}$ is a rigged $\mathcal {A}_{1}$–$\mathcal {A}_{2}$-bimodule bundle, we have that $\mathcal {F}_{x}$ is a rigged $(\mathcal {A}_{1})_{x}$–$(\mathcal {A}_{2})_{x}$-bimodule. Let $\widehat {(\rho _{1})_{x}}: \widehat {(\mathcal {A}_{1})_{x}} \rightarrow \mathcal {B}(\widehat {\mathcal {F}_{x}})$ be the corresponding $*$-homomorphism. The statement that $\mathcal {F}_{x}$ is a rigged $(\mathcal {N}_{1})_{x}$-module is then equivalent to the statement that $\widehat {(\rho _{1})_{x}}$ extends to $(\mathcal {N}_1)_x''$. But, since the typical fibre $F$ is a rigged von Neumann bimodule, its corresponding maps $\hat A_1 \to \mathcal {B}(\hat F)$ extend to $N_1''$. Next, consider local trivializations $(\Phi ^1,\Psi ^1)$ and $\Psi$ around $x$ as in Definition 2.2.13. Then, we have the extensions $\Phi ^1_x:(\mathcal {N}_1)_x'' \to N_1''$, and the conjugation by the unitary operator $\Psi _x$, which yields a normal $\ast$-homomorphism $\mathcal {B}(\hat F) \to \mathcal {B}(\widehat {\mathcal {F}_{x}})$. Using the compatibility of $\Phi ^{1}$ with $\Psi$ it then follows that the map

\[ (\mathcal{N}_1)_x'' \to N_1'' \to \mathcal{B}(\hat F) \to \mathcal{B}(\widehat{\mathcal{F}_{x}}) \]

is an extension of $\widehat {(\rho _{1})_{x}}$, which is moreover the composition of normal $*$-homomorphisms and, thus, a normal $*$-homomorphism itself. A similar argument for the $*$-homomorphism $\widehat {(\rho _{2})_x}$ proves that $\mathcal {F}_{x}$ is a rigged von Neumann $(\mathcal {N}_{2})_{x}^{\mathrm {opp}}$-module.

In the next section, we will glue rigged von Neumann bimodules bundles; therefore, we also need to introduce (spatial) intertwiners between them.

The definitions guarantees full compatibility with the fibrewise notions: any (spatial) intertwiner induces in the fibre over each point $x\in \mathcal {M}$ a (spatial) intertwiner between rigged von Neumann bimodules in the sense of Definition 2.1.15. Moreover, if $\mathcal {F}$ is a rigged von Neumann $\mathcal {N}_1$–$\mathcal {N}_2$-bimodule bundle with typical fibre $F$, any choice of local trivializations $(\Phi ^1,\Psi ^1)$, $(\Phi ^2,\Psi ^2)$, and $\Psi$ over an open subset $U \subset \mathcal {M}$ as in Definition 2.2.13, assemble into an invertible spatial intertwiner $(\Phi ^1,\Psi ^1,\Phi ^2,\Psi ^2,\Psi )$ from $\mathcal {F}|_U$ to $F \times U$. In particular, rigged von Neumann bimodule bundles are locally trivial in this strong sense of invertible spatial intertwiners.

We would like to make clear and summarize the following important feature of our definitions. Rigged von Neumann algebra bundles and bimodule bundles over $\mathcal {M}$ have in the fibre over each point $x\in \mathcal {M}$ rigged von Neumann algebras and bimodules, as introduced in § 2.1 (see Lemma 2.2.14). Likewise, (spatial) morphisms between rigged von Neumann algebra bundles, and (spatial) intertwiners between rigged von Neumann bimodules restrict in each fibre to (spatial) morphisms of rigged von Neumann algebras and (spatial) intertwiners of rigged von Neumann bimodules, as introduced in § 2.1. In turn, we have shown in Remark 2.1.7 and Lemma 2.1.16 that these fibrewise structures induce, respectively, ordinary von Neumann algebras, ordinary (spatial) morphisms, ordinary bimodules between von Neumann algebras, and ordinary bounded intertwiners between these, in the classical sense. Thus, one can pass, at any time and over any point, to this classical setting.

2.3 Rigged bundles over diffeological spaces

For the discussion of fusion on loop space, we need to treat bundles over diffeological spaces. We recall briefly that a diffeology on a set $X$ consists of a set of maps $c:U \to X$ called ‘plots’, where $U \subset \mathbb {R}^{k}$ is open and $k\in \mathbb {N}_0$ can be arbitrary, subject to three natural axioms; see [Reference Iglesias-ZemmourIgl13] for a comprehensive textbook. A map $f:X \to Y$ between diffeological spaces is called smooth, if its composition with any plot of $X$ results in a plot of $Y$. We let $\mathcal {D}i\!f\!\!f$ denote the category of diffeological spaces. Any smooth manifold $M$ or Fréchet manifold $\mathcal {M}$ becomes a diffeological space by saying that every smooth map $c: U \to M$, for every open subset $U \subset \mathbb {R}^{k}$ and any $k$, is a plot. Maps between smooth manifolds or Fréchet manifolds are then smooth in the classical sense if and only if they are smooth in the diffeological sense. In other words, the category $\mathcal {F}\!r\acute {e}ch$ of Fréchet manifolds fully faithfully embeds into $\mathcal {D}i\!f\!\!f$ (see [Reference LosikLos92]).

Next we describe a general procedure to extend a presheaf $\mathcal {F}$ of categories on $\mathcal {F}\!r\acute {e}ch$ to one on $\mathcal {D}i\!f\!\!f$, and then apply this to various presheaves of bundles defined in the previous section. We briefly recall that a presheaf $\mathcal {F}$ of categories on a category $\mathcal {C}$ is a weak functor $\mathcal {F}:\mathcal {C}^{\operatorname {opp}} \to \mathcal {C}\!at$ to the bicategory of categories, functors, and natural transformations. That is, $\mathcal {F}$ assigns to each object $X$ of $\mathcal {C}$ a category $\mathcal {F}(X)$, to each morphism $f:X \to Y$ in $\mathcal {C}$ a functor $f^{*}: \mathcal {F}(Y) \to \mathcal {F}(X)$, and to each pair $(f,g)$ of composable morphisms $f:X \to Y$ and $g:Y \to Z$ a natural equivalence $\eta _{f,g}: (g \circ f)^{*} \Rightarrow f^{*} \circ g^{*}$, such that the natural equivalences respect the associativity of the composition.

Our aim is to assign to any presheaf $\mathcal {F}$ of categories over $\mathcal {F}\!r\acute {e}ch$ a presheaf $\mathcal {F}^{\mathcal {D}i\!f\!\!f}$ of categories over $\mathcal {D}i\!f\!\!f$. There is an abstract canonical procedure how to do this, which we explain later in Remark 2.3.3. In the following definition we spell out directly the result of this procedure, emphasising the concrete perspective.

It is straightforward to complete the assignment $X \mapsto \mathcal {F}^{\mathcal {D}i\!f\!\!f}(X)$ given in Definition 2.3.2 to a presheaf of categories on $\mathcal {D}i\!f\!\!f$. This is how we extend presheaves of categories from $\mathcal {F}\!r\acute {e}ch$ to $\mathcal {D}i\!f\!\!f$. If $\mathcal {M}$ is a Fréchet manifold, which we may consider as a diffeological space, then there is a faithful functor

(2)\begin{align} \mathcal{F}(\mathcal{M})\to \mathcal{F}^{\mathcal{D}i\!f\!\!f}(\mathcal{M}), \end{align}

which takes an object $\mathcal {E}$ of $\mathcal {F}(\mathcal {M})$ to the pair $((\mathcal {E}_c),(\phi _{c_1,c_2,f}))$, where $\mathcal {E}_c := c^{*}\mathcal {E}$ is just the pullback in $\mathcal {F}$ (recall that any plot $c$ is a smooth map between Fréchet manifolds here) and $\phi _{c_1,c_2,f} : c_1^{*}\mathcal {E} \to f^{*}c_2^{*}\mathcal {E}$ is the isomorphism $\eta _{f,c_2}$ provided by $\mathcal {F}$. A morphism $\psi :\mathcal {E} \to \mathcal {E}'$ is sent to the family with $\psi _c := c^{*}\psi$; it is obvious that this preserves composition and is faithful.

Remark 2.3.3 The extension of presheaves of categories from $\mathcal {F}\!r\acute {e}ch$ to $\mathcal {D}i\!f\!\!f$ that we defined in Definition 2.3.2 fits into a more conceptual perspective, which we want to outline in this remark. To this end, we infer that any diffeological space $X$ can be viewed as a presheaf $\underline {X}$ (of sets) on the category $\mathcal {O}pen$ of open subsets of cartesian spaces, see [Reference Baez and HoffnungBH11]. Namely, the presheaf $\underline {X}$ assigns to an object $U$ in $\mathcal {O}pen$ the set $\underline {X}(U)$ of all plots $c:U \to X$ with domain $U$. In the following we regard $\underline {X}$ even as a presheaf of categories on $\mathcal {O}pen$, and we do this simply by regarding the sets $\underline {X}(U)$ as categories with only identity morphisms. Now, let $\mathcal {F}$ be a presheaf of categories on $\mathcal {F}\!r\acute {e}ch$, which we may restrict to $\mathcal {O}pen \subset \mathcal {F}\!r\acute {e}ch$. Presheaves of categories on $\mathcal {O}pen$ form a bicategory $\mathrm {PSh}(\mathcal {O}pen)$, the bicategory of weak functors $\mathcal {O}pen^{\operatorname {opp}} \to \mathcal {C}\!at$. Now, we consider the functor represented by $\mathcal {F}|_{\mathcal {O}pen}$ and restricted along $\mathcal {D}i\!f\!\!f \to \mathrm {PSh}(\mathcal {O}pen)$,

\[ \mathrm{Hom}_{\mathrm{PSh}(\mathcal{O}pen)}(\,\underline{\,\cdot\,}\,,\mathcal{F}|_{\mathcal{O}pen}):\mathcal{D}i\!f\!\!f^{\operatorname{opp}} \to \mathcal{C}\!at. \]

Explicitly, it assigns to a diffeological space $X$ the category $\mathrm {Hom}_{\mathrm {PSh}(\mathcal {O}pen)}(\underline {X},\mathcal {F}|_{\mathcal {O}pen})$ of morphisms between the objects $\underline {X}$ and $\mathcal {F}|_{\mathcal {O}pen}$ of $\mathrm {PSh}(\mathcal {O}pen)$. It is a straightforward exercise to show that

\begin{align*} \mathrm{Hom}_{\mathrm{PSh}(\mathcal{O}pen)}(\underline{X},\mathcal{F}|_{\mathcal{O}pen})=\mathcal{F}^{\mathcal{D}i\!f\!\!f}(X); \end{align*}

this embeds our Definition 2.3.2 into a proper topos-theoretical framework. The bicategorical version of the Yoneda lemma (see, e.g., [Reference Johnson and YauJY21]) implies now that $\mathcal {F}^{\mathcal {D}i\!f\!\!f}(U) \cong \mathcal {F}(U)$ for every object $U$ in $\mathcal {O}pen$. This exhibits $\mathcal {F}^{\mathcal {D}i\!f\!\!f}$ as the (left) Kan extension of $\mathcal {F}|_{\mathcal {O}pen}$ along $\mathcal {O}pen^{\operatorname {opp}} \to \mathcal {D}i\!f\!\!f^{\operatorname {opp}}$, which is precisely the standard way to extend presheaves from dense subsites to sites [Reference JohnstoneJoh02]. This is our main justification for Definition 2.3.2.

We apply Definition 2.3.2 to the presheaves listed in Remark 2.3.1, obtaining neat definitions of rigged Hilbert space bundles, rigged C$^{*}\!$-algebra bundles, rigged bimodule bundles, rigged von Neumann algebra bundles, and rigged von Neumann bimodule bundles over diffeological spaces. Most examples of such bundles that appear in this article are obtained via the functor (2), in the following way: consider a diffeological space $X$, a Fréchet manifold $\mathcal {M}$, and a smooth map $f: X \to \mathcal {M}$. Then, the composite

produces objects in $\mathcal {F}^{\mathcal {D}i\!f\!\!f}(X)$ from objects in $\mathcal {F}(\mathcal {M})$.

3.1 Lagrangians and Fock spaces

We recall some aspects of infinite-dimensional Clifford algebras and their representations on Fock spaces, for which the book [Reference Plymen and RobinsonPR94] is an excellent reference. The results we require on implementers are more spread out across the literature; see, for instance, [Reference ArakiAra87, Reference OttesenOtt95, Reference NeebNee10b, Reference Kristel and WaldorfKW22]. In §§ 3.1 and 3.2 we consider in some generality a complex Hilbert space $V$ equipped with a real structure $\alpha$, i.e. an anti-unitary involution $\alpha : V \rightarrow V$. From § 3.3 on, we restrict to a specific example, described at the beginning of § 3.3.

Given a unital C$^{*}\!$-algebra $A$, we say that a map $f:V \rightarrow A$ is a Clifford map if the following equations are satisfied, for all $v,w \in V$:

(3)\begin{equation} f(v)f(w) + f(w)f(v) = 2 \langle v, \alpha(w) \rangle \mathbb{1}, \quad f(v)^{*} = f(\alpha(v)). \end{equation}

The Clifford C$^{*}\!$-algebra $\operatorname {Cl}(V)$ is the unique (up to unique isomorphism) unital C$^{*}\!$-algebra equipped with a Clifford map $\iota : V \rightarrow \operatorname {Cl}(V)$, such that for each unital C$^{*}\!$-algebra $A$ and each Clifford map $f: V \rightarrow A$, there exists a unique unital C$^{*}\!$-algebra homomorphism $\operatorname {Cl}(f): \operatorname {Cl}(V) \rightarrow A$ with the property that $\operatorname {Cl}(f) \circ \iota = f$. An explicit construction of $\operatorname {Cl}(V)$ is given in [Reference Plymen and RobinsonPR94, § 1.2].

The orthogonal group of $V$, denoted by $\operatorname {O}(V)$, consists of those unitary transformations of $V$ that commute with the real structure. If $g \in \operatorname {O}(V)$, then $\iota g: V \rightarrow \operatorname {Cl}(V)$ is a Clifford map. We write $\theta _{g} := \operatorname {Cl}( \iota g): \operatorname {Cl}(V) \rightarrow \operatorname {Cl}(V)$ for its extension to $\operatorname {Cl}(V)$. The map $\theta _{g}$ is called the Bogoliubov automorphism associated to $g$. The map $\theta : g \mapsto \theta _{g}$ is a continuous homomorphism from $\operatorname {O}(V)$ to $\operatorname {Aut}(\operatorname {Cl}(V))$, where $\operatorname {O}(V)$ is equipped with the operator norm topology, and $\operatorname {Aut}(\operatorname {Cl}(V))$ is equipped with the strong operator topology; see, e.g., [Reference AmblerAmb12, Proposition 4.35].

A Lagrangian in $V$ is a subspace $L \subset V$ such that $V$ splits as the orthogonal direct sum $V = L \oplus \alpha (L)$. If $L$ is a Lagrangian, then the Fock space $F_{L}$ is the Hilbert completion of the exterior algebra, $\Lambda L$, of $L$. We identify $\alpha (L)$ with the dual of $L$ by identifying $w \in \alpha (L)$ with the linear map $L \ni v \mapsto \langle v, \alpha (w) \rangle$. If $v \in L$ and $w \in \alpha (L) \simeq L^{*}$, then we write $c(v): F_{L} \rightarrow F_{L}$ for left multiplication with $v$, and $a(w) : F_{L} \rightarrow F_{L}$ for contraction with $w$. The maps $c(v)$ and $a(v)$ are bounded operators on $F_{L}$, and the map

\[ \rho_{L}:V = L \oplus \alpha(L) \rightarrow \mathcal{B}(F_{L}), \quad (v,w) \mapsto \sqrt{2}(c(v) + a(w)) \]

is a Clifford map. This means that $\operatorname {Cl}(\rho _{L}): \operatorname {Cl}(V) \rightarrow \mathcal {B}(F_{L})$ is a unital C$^{*}\!$-algebra homomorphism, i.e. a representation of $\operatorname {Cl}(V)$ on $F_{L}$. This representation is irreducible [Reference Plymen and RobinsonPR94, Theorem 2.4.2] and faithful; hence, we may identify $\operatorname {Cl}(V)$ with its image in $\mathcal {B}(F_{L})$. Whenever convenient, we adopt the notation $a \triangleright v = \operatorname {Cl}(\rho _{L})(a)(v)$ for $a \in \operatorname {Cl}(V)$ and $v \in F_{L}$.

If $g \in \operatorname {O}(V)$, then we say that $g$ is implementable, if there exists $U \in \operatorname {U}(F_{L})$ with the property that

(4)\begin{equation} \theta_{g}(a) = UaU^{*} \end{equation}

for all $a \in \operatorname {Cl}(V)$; the operator $U$ is said to implement $g$. The problem to decide which $g\in \operatorname {O}(V)$ are implementable is called the ‘implementability problem’. It is completely solved: an element $g \in \operatorname {O}(V)$ is implementable if and only if the operator $P_{L} g P_{L}^{\perp }$ is Hilbert–Schmidt, where $P_L$ is the orthogonal projection to $L$, see [Reference Plymen and RobinsonPR94, Theorem 3.3.5] or [Reference ArakiAra87, Theorem 6.3]. We write $\operatorname {O}_{L}(V)$ for the set consisting of those $g \in \operatorname {O}(V)$ which are implementable; the set $\operatorname {O}_{L}(V)$ is, in fact, a subgroup of $\operatorname {O}(V)$.

The group $\operatorname {O}(V)$ can be equipped with the structure of Banach Lie group in the standard way, with underlying topology the operator norm topology. The Lie algebra of $\operatorname {O}(V)$ is

\[ \mathfrak{o}(V) = \{ X \in \mathcal{B}(V) \mid [X,\alpha] = 0, X^{*} = -X \}. \]

The subgroup $\operatorname {O}_L(V)$ can also be equipped with the structure of a Banach Lie group, whose underlying topology is given by the norm $\| g\|_{\mathcal {J}} = \| g\| + \| P_{L} g P_{L}^{\perp } \|_{2}$, where $\|g\|$ is the operator norm of $g$, and where $\| \cdot \|_{2}$ is the Hilbert–Schmidt norm [Reference Kristel and WaldorfKW22, § 3.4]. The inclusion $\operatorname {O}_L(V) \to \operatorname {O}(V)$ is smooth. The Lie algebra of $\operatorname {O}_{L}(V)$ is

\[ \mathfrak{o}_{L}(V) = \{ X \in \mathfrak{o}(V) \mid \| P_{L} X P_{L}^{\perp} \|_{2} < \infty \}. \]

The group of implementers, $\operatorname {Imp}_{L}(V)$, is defined to be the subgroup of $\operatorname {U}(F_{L})$ consisting of those operators $U \in \operatorname {U}(F_{L})$, for which there exists a $g \in \operatorname {O}_{L}(V)$ such that (4) holds. If $U \in \operatorname {Imp}_{L}(V)$, then the element $g \in \operatorname {O}_{L}(V)$ that it implements is determined uniquely, and we obtain a group homomorphism $q:\operatorname {Imp}_{L}(V) \rightarrow \operatorname {O}_{L}(V)$. Using the irreducibility of the representation of $\operatorname {Cl}(V)$ on $F_{L}$ together with Schur's lemma, we see that, for each $g \in \operatorname {O}_{L}(V)$, the fibre $q^{-1}\{g\}$ is a $\operatorname {U}(1)$-torsor. We equip the group $\operatorname {Imp}_{L}(V)$ with the structure of Banach Lie group as in [Reference Kristel and WaldorfKW22, Theorem 3.15], see also [Reference WurzbacherWur01]. We then have that the exact sequence

\[ \operatorname{U}(1) \rightarrow \operatorname{Imp}_{L}(V) \xrightarrow{q} \operatorname{O}_{L}(V) , \]

is a central extension of Banach Lie groups. It is important to note that the topology underlying the Banach Lie group structure on $\operatorname {Imp}_{L}(V)$ is not the operator norm topology, and that the inclusion map $\operatorname {Imp}_{L}(V) \rightarrow \operatorname {U}(F_{L})$ is not continuous, let alone smooth.

Let us assume that we are given a further orthogonal decomposition $V=V_{-} \oplus V_{+}$ of $V$ into two Hilbert spaces $V_{\pm }$ which are preserved under $\alpha$. In the sequel, such a splitting will implement the splitting of a loop into two paths. Given such a splitting, we are interested in the operators on $V$ which preserve this splitting. We write $P_{\pm }$ for the projection onto $V_{\pm }$ and define

\[ \mathfrak{o}^{\theta}(V) := \{ X \in \mathfrak{o}(V) \mid P_{\pm}X P_{\mp} = 0 \}, \quad \mathfrak{o}^{\theta}_{L}(V) := \{ X \in \mathfrak{o }_{L}(V) \mid P_{\pm}X P_{\mp} = 0 \}. \]

We observe that the maps $X \mapsto P_{\pm } X P_{\mp }$ are linear and continuous (with respect to the operator norm), which implies that $\mathfrak {o}^{\theta }(V)$ is a closed linear subspace of $\mathfrak {o}(V)$ and, hence, a Banach space. The subspace $\mathfrak {o}_{L}^{\theta }(V)$ is the pre-image of $\mathfrak {o}^{\theta }(V)$ under the continuous inclusion $\mathfrak {o}_{L}(V) \rightarrow \mathfrak {o}(V)$, and hence a Banach space as well. The spaces $\mathfrak {o}^{\theta }(V)$ and $\mathfrak {o}_{L}^{\theta }(V)$ are Lie subalgebras of $\mathfrak {o}(V)$ and $\mathfrak {o}_{L}(V)$, respectively. Next, we define the metric groups

\[ \operatorname{O}^{\theta}(V) := \{ g \in \operatorname{O}(V) \mid P_{\pm}gP_{\mp} = 0 \}, \quad \operatorname{O}^{\theta}_{L}(V) := \{ g \in \operatorname{O}_{L}(V) \mid P_{\pm}gP_{\mp} = 0 \}. \]

Using standard techniques (see, e.g., [Reference Kristel and WaldorfKW22, Lemma 3.13]), one may then show that the exponential maps $\mathfrak {o}^{\theta }(V) \rightarrow \operatorname {O}^{\theta }(V)$ and $\mathfrak {o}_{L}^{\theta }(V) \rightarrow \operatorname {O}_{L}^{\theta }(V)$ are local homeomorphisms, which allows us to equip $\operatorname {O}^{\theta }(V)$ and $\operatorname {O}_{L}^{\theta }(V)$ with the structure of Banach Lie groups, with Lie algebras $\mathfrak {o}^{\theta }(V)$ and $\mathfrak {o}^{\theta }_{L}(V)$, respectively. It is then clear from the construction that $\operatorname {O}^{\theta }(V)$ and $\operatorname {O}_{L}^{\theta }(V)$ are closed submanifolds of $\operatorname {O}(V)$ and $\operatorname {O}_{L}(V)$, respectively. If $g \in \operatorname {O}^{\theta }(V)$, then we set $g_{\pm } := P_{\pm }gP_{\pm } \in \operatorname {O}(V_{\pm })$. We observe that the maps $g \mapsto g_{\pm }$ are smooth group homomorphisms. Finally, we restrict the group of implementers to the split setting, and define the Banach Lie group

\[ \operatorname{Imp}_{L}^{\theta}(V) := \operatorname{Imp}_{L}(V)|_{\operatorname{O}_{L}^{\theta}(V)}, \]

which then is a central extension of $\operatorname {O}_{L}^{\theta }(V)$ by $\operatorname {U}(1)$.

Next, we consider the Clifford algebras $\operatorname {Cl}(V_{\pm })$. We observe that extending by zero gives isometries $V_{\pm } \rightarrow V$ which moreover intertwine the real structures, and thus induce isometric $*$-homomorphisms $\iota _{\pm }: \operatorname {Cl}(V_{\pm }) \rightarrow \operatorname {Cl}(V)$. The algebra product

\[ \operatorname{Cl}(V_{-}) \times \operatorname{Cl}(V_{+}) \subset \operatorname{Cl}(V) \times \operatorname{Cl}(V) \to \operatorname{Cl}(V) \]

induces a unital isomorphism $\operatorname {Cl}(V_{-}) \otimes \operatorname {Cl}(V_{+}) \cong \operatorname {Cl}(V)$ of C$^{*}\!$-algebras. (Here, $\otimes$ stands for any choice of tensor product of C$^{*}\!$-algebras. The choice is immaterial, because the Clifford C$^{*}\!$-algebra is uniformly hyperfinite and, hence, nuclear.) Under these identifications, we have $\iota _{-}(a_{-}) = a_{-} \otimes \mathbb {1}$ and $\iota _{+}(a_{+}) = \mathbb {1} \otimes a_{+}$, for $a_{\pm } \in \operatorname {Cl}(V_{\pm })$. The following result expresses that the Bogoliubov automorphisms are compatible with the splitting; this will be used later in the proofs of Lemmas 3.4.2 and 5.2.1.

Lemma 3.1.1 For all $g_{-} \oplus g_{+} \in \operatorname {O}^{\theta }(V)$ and all $a_{\pm } \in \operatorname {Cl}(V_{\pm })$ we have

\[ \theta_{g_{-} \oplus g_{+}}(a_{-} \otimes a_{+}) = \theta_{g_{-}}(a_{-}) \otimes \theta_{g_{+}}(a_{+}), \]

where $\theta _{g_{\pm }}$ are the Bogoliubov automorphisms of the Clifford algebras $\operatorname {Cl}(V_{\pm })$.

3.2 Smooth Fock spaces

In §§ 4 and 5 we construct bundles of rigged Hilbert spaces and rigged C$^{*}\!$-algebras over Fréchet manifolds. This requires a detailed study of smoothness properties of the representations obtained in § 3.1; this is the goal of this section. We continue working with a complex Hilbert space $V$ with a real structure $\alpha$, additionally equipped with an orthogonal decomposition into two Hilbert spaces $V_{\pm }$ which are preserved under $\alpha$. Our first goal is to equip the Fock space $F_L$ and the Clifford C$^{*}\!$-algebra $\operatorname {Cl}(V)$ with the structure of a rigged Hilbert space and a rigged C$^{*}\!$-algebra, respectively. We have done this already in our earlier paper [Reference Kristel and WaldorfKW20]; however, there we have not taken the splitting $V=V_{-}\oplus V_{+}$ into account. For the purpose of this article, the splitting is essential, and it leads to a finer rigging (we compare them in Remark 3.2.6 below). Therefore, we describe the important steps again.

As a subgroup of $\operatorname {U}(F_{L})$, the group $\operatorname {Imp}^{\theta }_{L}(V)$ comes equipped with a unitary representation on $F_{L}$. As is typical for infinite-dimensional representations, the action map $\operatorname {Imp}_{L}^{\theta }(V) \times F_{L} \rightarrow F_{L}$ is not smooth. The subspace of smooth vectors in $F_{L}$ is defined as usual to be

\[ F^{\mathrm{s}}_{L} := \{ v \in F_{L} \mid \operatorname{Imp}^{\theta}_{L}(V) \rightarrow F_{L}, U \mapsto Uv \text{ is smooth} \}. \]

The following result follows directly from [Reference Kristel and WaldorfKW22, Proposition 3.17], see also [Reference NeebNee10b, § 10.1].

By definition of smooth vectors, the Lie algebra $\mathfrak {imp}^{\theta }(V)$ of $\operatorname {Imp}_L^{\theta }(V)$ acts infinitesimally on $F_L^{\mathrm {s}}$, i.e. for $X \in \mathfrak {imp}^{\theta }(V)$ and $v \in F^{\mathrm {s}}_{L}$, we may define

\[ Xv := \frac{{d} }{{d} t} \bigg|_{t=0} \exp(t X)(v). \]

Let $\mathcal {P}(F_{L})$ be the set of all continuous semi-norms on $F_{L}$. We define a topology on $F_{L}^{\mathrm {s}}$ by the following family of semi-norms [Reference NeebNee10a, § 4]:

\[ p_{n}(v) = \sup \{ p( X_{1} \ldots X_{n} v) \mid X_{i} \in \mathfrak{imp}^{\theta}(V), \|X_{i}\| \leqslant 1 \}, \quad p \in \mathcal{P}(F_{L}), n \in \mathbb{N}_{0}. \]

The following result is proved completely analogously to [Reference Kristel and WaldorfKW20, Proposition 3.2.4].

Just like the action map $\operatorname {Imp}^{\theta }_{L}(V) \times F_{L} \rightarrow F_{L}$ is not smooth, the map $\operatorname {O}^{\theta }(V) \times \operatorname {Cl}(V) \rightarrow \operatorname {Cl}(V)$ for the action by Bogoliubov automorphisms is not smooth either, an issue that we handle in a similar way. We write $\operatorname {Cl}(V)^{\mathrm {s}}$ for the subspace of smooth vectors in $\operatorname {Cl}(V)$, i.e.

\[ \operatorname{Cl}(V)^{\mathrm{s}} := \{ a \in \operatorname{Cl}(V) \mid \operatorname{O}^{\theta}(V) \rightarrow \operatorname{Cl}(V), g \mapsto \theta_{g}(a) \text{ is smooth} \}. \]

The Lie algebra $\mathfrak {o}_{res}^{\theta }(V)$ acts on $\operatorname {Cl}(V)^{\mathrm {s}}$, which allows us to proceed as follows. Let $\mathcal {R}(\operatorname {Cl}(V))$ be the set of continuous semi-norms on $\operatorname {Cl}(V)$. The topology on $\operatorname {Cl}(V)^{\mathrm {s}}$ is then defined by the family of semi-norms

\[ r_{n}(a) = \sup \{ r( Y_{1} \ldots Y_{n} a) \mid Y_{i} \in \mathfrak{o}^{\theta}(V), \|Y_{i}\| \leqslant 1 \}, \quad r \in \mathcal{R}(\operatorname{Cl}(V)), n \in \mathbb{N}_{0}. \]

In analogy with [Reference Kristel and WaldorfKW20, Proposition 3.2.7] we then have the following proposition.

Finally, we adapt to this situation, and obtain the following result.

We remark that the rigged von Neumann algebra $\operatorname {Cl}_{\mathrm {vN}}(V)^\mathrm {s}$ determines an ordinary von Neumann algebra $\operatorname {Cl}_{\mathrm {vN}}(V):= (\operatorname {Cl}_{\mathrm {vN}}(V)^\mathrm {s})''$ as the completion of $\operatorname {Cl}(V)$ acting on $F_L$ (Remark 2.1.7). It is well-known that $\operatorname {Cl}_{\mathrm {vN}}(V)=\mathcal {B}(F_L)$.

Remark 3.2.6 In our earlier paper [Reference Kristel and WaldorfKW20, § 3.2] we considered different Fréchet spaces as the riggings on $F_L$ and $\operatorname {Cl}(V)$, obtained without assuming a splitting of $V$:

\begin{align*} F^{\infty}_{L} &:= \{ v \in F_{L} \mid \operatorname{Imp}_{L}(V) \rightarrow F_{L}, U \mapsto Uv \text{ is smooth} \},\\ \operatorname{Cl}(V)^{\infty} &:= \{ a \in \operatorname{Cl}(V) \mid \operatorname{O}(V) \rightarrow \operatorname{Cl}(V), g \mapsto \theta_{g}(a) \text{ is smooth} \}. \end{align*}

Because of the fact that $\operatorname {Imp}_{L}^{\theta }(V)$ is contained in $\operatorname {Imp}_{L}(V)$ we have that $F_L^{\infty } \subset F_{L}^{\mathrm {s}}$. This means that we improve the rigging by passing to the subgroup $\operatorname {Imp}_L^{\theta }(V)$. The analogous statement is true for $\operatorname {Cl}(V)^{\infty }$ and $\operatorname {Cl}(V)^{\mathrm {s}}$. The riggings $F_L^{\infty }$ and $\operatorname {Cl}(V)^{\infty }$ have been appropriate for the construction of the spinor bundle on loop space and its Clifford action. For the construction of the fusion product we need the ‘finer’ riggings $F_L^{\mathrm {s}}$ and $\operatorname {Cl}(V)^{\mathrm {s}}$; in particular, we need these in Lemma 3.2.7, see Remark 3.2.8.

We now move beyond the analogy with [Reference Kristel and WaldorfKW20, § 3.2] and lift the separate Clifford C$^{*}\!$-algebras $\operatorname {Cl}(V_{\pm })$ to the setting of rigged C$^{*}\!$-algebras, by defining

\[ \operatorname{Cl}(V_{\pm})^{\mathrm{s}} := \{ a \in \operatorname{Cl}(V_{\pm}) \mid \operatorname{O}(V_{\pm}) \rightarrow \operatorname{Cl}(V_{\pm}),\; g \mapsto \theta_{g}(a) \text{ is smooth} \}. \]

Just like $\operatorname {Cl}(V)^{\mathrm {s}}$ these are rigged C$^{*}\!$-algebras.

Proof. Let $a \in \operatorname {Cl}(V_{-})$ be arbitrary. We then see from Lemma 3.1.1 that the map

(5)\begin{equation} \operatorname{O}^{\theta}(V) \rightarrow \operatorname{Cl}(V), \quad g \mapsto \theta_{g}(a \otimes \mathbb{1}) \end{equation}

is the composition of the Lie group homomorphism $\operatorname {O}^{\theta }(V) \to \operatorname {O}(V_{-}),\;g \mapsto g_{-}$ with the action map $\operatorname {O}(V_{-}) \to \operatorname {Cl}(V_{-}),\; g_{-}\mapsto \theta _{g_{-}}(a)$, followed by the smooth map $\iota _{-}:\operatorname {Cl}(V_{-})\to \operatorname {Cl}(V)$. This proves that if $a \in \operatorname {Cl}(V_{-})^{\mathrm {s}}$, then (5) is smooth, and we have $a \otimes \mathbb {1} \in \operatorname {Cl}(V)^{\mathrm {s}}$; and thus $\iota _{-}$ restricts to a map $\iota _{-}:\operatorname {Cl}(V_{-})^{\mathrm {s}} \rightarrow \operatorname {Cl}(V)^{\mathrm {s}}$. What remains to be shown is that this restriction $\iota _{-}: \operatorname {Cl}(V_{-})^{\mathrm {s}} \rightarrow \operatorname {Cl}(V)^{\mathrm {s}}$ is continuous with respect to the Fréchet space structures. Because $\iota _{-}$ is linear, it suffices to show continuity at $0$. First, observe that if $r \in \mathcal {R}(\operatorname {Cl}(V))$, then $r \circ \iota _{-} \in \mathcal {R}(\operatorname {Cl}(V_{-}))$. Moreover, we claim that

(6)\begin{align} r_{n}(a \otimes \mathbb{1}) = (r \circ \iota_{-})_{n}(a), \end{align}

indeed

\begin{align*} r_{n}(a \otimes \mathbb{1}) &= \sup \{ r(Y_{1} \ldots Y_{n}(a\otimes \mathbb{1})) \mid Y_{i} \in \mathfrak{o}^{\theta}(V), \| Y_{i} \| \leqslant 1 \}\\ &= \sup \{ r(Y_{1}|_{V_{-}} \ldots Y_{n}|_{V_{-}}(a)\otimes \mathbb{1}) \mid Y_{i} \in \mathfrak{o}^{\theta}(V), \| Y_{i} \| \leqslant 1 \}\\ &=\sup \{ r \circ \iota_{-}(Y_{-,1} \ldots Y_{-,n}(a)) \mid Y_{-,i} \in \mathfrak{o}(V_{-}), \| Y_{i} \| \leqslant 1 \}\\ &= (r \circ \iota_{-})_{n}(a). \end{align*}

From (6) it follows that the preimage of the subbasis open neighbourhood of zero given by $\{ x \in \operatorname {Cl}(V) \mid r_{n}(x) < \varepsilon \} \subset \operatorname {Cl}(V)$ is the subbasis open neighbourhood of zero given by $\{ a \in \operatorname {Cl}(V_{-}) \mid (r \circ \iota _{-})_{n}(a) < \varepsilon \}$. The discussion of $\operatorname {Cl}(V_{+})$ is analogous.

We obtain the following result about the rigged C$^{*}\!$-algebras $\operatorname {Cl}(V_{\pm })^{\mathrm {s}}$.

The rigged von Neumann algebra $N_{\pm }(V)^{\mathrm {s}} = (\operatorname {Cl}(V_{\pm })^{\mathrm {s}},F_{L}^{\mathrm {s}})$ determines a von Neumann algebra $N_{\pm }(V) := (N_{\pm }(V)^{\mathrm {s}})''$ in the classical sense, as the von Neumann closure of the C$^{*}\!$-algebra $\operatorname {Cl}(V_{\pm })$ acting on $F_{L}$, see Remark 2.1.7. It is well known that these von Neumann algebras are III$_{1}$-factors (see [Reference Stolz and TeichnerST04, Example 4.3.2] and [Reference WassermannWas98, § 16]).

3.3 Free fermions as a rigged bimodule

We now fix a concrete Hilbert space $V$, a real structure $\alpha$, a Lagrangian $L$, and a splitting $V=V_{-}\oplus V_{+}$, which will remain the same for the rest of the paper. Let $\mathbb {S} \rightarrow S^{1}$ be the odd spinor bundle on the circle, i.e. that associated to the odd (i.e. the connected) spin structure on the circle. We set $V := L^{2}(S^{1}, \mathbb {S} \otimes \mathbb {C}^{d})$, where $d$ is a natural number (later it is the spacetime dimension). Pointwise complex conjugation gives a real structure $\alpha : V \rightarrow V$. In [Reference Kristel and WaldorfKW22, § 2] it is explained how the space of smooth 2$\pi$-antiperiodic functions on the real line can be identified with the dense subspace $\Gamma (S^{1},\mathbb {S} \otimes \mathbb {C}^{d}) \subset V$ of smooth sections. Under this identification, $V$ has an orthonormal basis $\{ \xi _{n,j} \}_{n \in \mathbb {N}, j =1,\ldots,d}$ by setting

\[ \xi_{n,j}(t) = e^{-i (n+ {1}/{2} ) t} e_{j}, \]

where $\{ e_{j} \}_{j=1,\ldots,d}$ is the standard basis of $\mathbb {C}^{d}$. It is further shown that $\alpha (\xi _{n,j}) = \xi _{-n-1,j}$ for all $n$ and all $j$. It follows that the closed linear span

\[ L := \operatorname{span} \{ \xi_{n,j} \mid n \geqslant 0, \ j=1,\ldots,d \} \]

is a Lagrangian in $V$. The corresponding Fock space $F:= F_L$ is called the free fermions. Let us write $I_{+} \subset S^{1}$ for the open upper semicircle and $I_{-} \subset S^{1}$ for the open lower semicircle. If $f \in V$, then we write $\operatorname {Supp}(f)$ for the support of $f$. We consider the subspaces

\[ V_{\pm} := \{ f\in V \mid \operatorname{Supp}(f) \subseteq I_{\pm} \}; \]

they yield a decomposition $V = V_{-} \oplus V_{+}$, and $\alpha$ restricts to real structures on $V_{\pm }$. This puts us in the setting of § 3.2, and we thus obtain rigged von Neumann algebras $N_{\pm }^{\mathrm {s}} := N_{\pm }(V)^{\mathrm {s}} := (\operatorname {Cl}(V_{\pm })^{\mathrm {s}},F^{\mathrm {s}})$.

We denote by $\tau$ the complex-linear extension of the map $\xi _{n,j}\mapsto \xi _{-n-1,j}$. The map $\tau$ is orthogonal, exchanges $L$ with $\alpha (L)$, and exchanges $V_{+}$ with $V_{-}$. We remark that while $\tau \in \operatorname {O}(V)$, one can show that it is not implementable. Since $\alpha$ interchanges $L$ with $\alpha (L)$ as well, the anti-unitary isomorphism $\alpha \tau$ preserves both $L$ and $\alpha (L)$. In particular, it induces an anti-unitary operator $\Lambda _{\alpha \tau }: F \rightarrow F$.

If $g \in \operatorname {O}(V)$, then we write $\tau (g) := \tau \circ g \circ \tau \in \operatorname {O}(V)$. With this notation we have, for all $f \in V$ and all $g \in \operatorname {O}(V)$, that $\tau (g(f)) = \tau (g) (\tau (f))$. Because $\tau$ interchanges $L$ with $\alpha (L)$ we have that conjugation by $\tau$ yields an isometric (hence, smooth) group homomorphism from $\operatorname {O}_{L}(V)$ into $\operatorname {O}_{L}(V)$. Finally, because $\tau$ exchanges $V_{+}$ with $V_{-}$ we have that $\tau$ preserves $\operatorname {O}_{L}^{\theta }(V)$, and moreover exchanges $\operatorname {O}(V_{-})$ with $\operatorname {O}(V_{+})$.

We recall that the Fock space $F$ is equipped with the left action of $\operatorname {Cl}(V_{-})$ induced by the inclusion $\iota _{-}:\operatorname {Cl}(V_{-}) \to \operatorname {Cl}(V)$. Next, we equip it with a compatible right action of $\operatorname {Cl}(V_{-})$. We note that $F$ is naturally graded, as it is the completion of an exterior algebra. We let $\operatorname {k}: F \rightarrow F$ be the ‘Klein transformation’, that is, $\operatorname {k}$ acts on $F$ as the identity on the even part, and as multiplication by $i$ on the odd part. Note that $\operatorname {k}$ is unitary. Then we define the operator

(7)\begin{equation} J := \operatorname{k}^{-1} \Lambda_{\alpha \tau}, \end{equation}

which is an anti-unitary operator on $F$ with $J^2=1$. Now, we define a right action as

(8)\begin{equation} F \times \operatorname{Cl}(V_{-}) \rightarrow F, \quad (v,a) \mapsto Ja^{*}Jv. \end{equation}

We shall then write $v \triangleleft a$ for the right action of $a$ on $v$.

Next, we pass to the rigged setting. In Propositions 3.2.5 and 3.2.9 we have seen that $F^{\mathrm {s}}$ is a rigged $\operatorname {Cl}(V_{-})^{\mathrm {s}}$-module under the left action. The analog is true for the right action of (8), as the following result shows.

Proof. The proof is mostly standard, and analogous to that of [Reference Kristel and WaldorfKW20, Proposition 3.2.8]. To carry out the necessary computations one must make use of the formulas

(9)\begin{equation} J\theta_{\tau(g)}(a)J = \theta_{g}(JaJ) \quad (a \in \operatorname{Cl}(V), g \in \operatorname{O}(V)), \end{equation}

which is [Reference Kristel and WaldorfKW22, Lemma 4.8], and

\[ X(v \triangleleft a) = X(v) \triangleleft a + v \triangleleft \tau(X)(a) \quad (X \in \mathfrak{o}_{L}(V)), \]

where $\tau (X) = \tau \circ X \circ \tau \in \mathfrak {o}_{L}(V)$, which follows from (9). From this, one obtains a non-commutative binomial expansion, cf. [Reference Kristel and WaldorfKW20, Equation (12)], at which point the proof is completely analogous to that of Proposition 3.2.5.

We recall from Proposition 3.2.9 that $\operatorname {Cl}(V_{-})^{\mathrm {s}}$ is not just a rigged C$^{*}\!$-algebra but also forms a rigged von Neumann algebra $N^{\mathrm {s}}=N^{\mathrm {s}}_{-}=(\operatorname {Cl}(V_{-})^{\mathrm {s}},F^{\mathrm {s}})$, whose defining representation is the left action.

Proof. It remains to show that the right action, $(\operatorname {Cl}(V_{-})^{\mathrm {s}})^{\operatorname {opp}} \times F^{\mathrm {s}} \to F^{\mathrm {s}}$, makes $F^{\mathrm {s}}$ a rigged von Neumann $(N^{\mathrm {s}})^{\mathrm {opp}}$-module, where $(N^{\mathrm {s}})^{\operatorname {opp}} = ((\operatorname {Cl}(V_{-})^{\mathrm {s}})^{\mathrm {opp}},(F^{\mathrm {s}})^{\sharp })$, see Definition 2.1.12. For this, all that needs to be shown is that the induced map

\[ \hat\rho: \operatorname{Cl}(V_{-})^{\mathrm{opp}} \rightarrow \mathcal{B}(F), a \mapsto Ja^{*}J \]

extends to a normal $*$-homomorphism

\[ (\operatorname{Cl}(V_{-})^{\mathrm{opp}})'' \rightarrow \mathcal{B}(F). \]

Now, let $\rho _{0}$ be the defining representation of $(N^{\mathrm {s}})^{\mathrm {opp}}$, let $c: F^{\sharp } \rightarrow F$ be the canonical anti-linear isomorphism, and let $\sigma : \mathcal {B}(F^{\sharp }) \rightarrow \mathcal {B}(F)$ be the $*$-homomorphism

\[ \sigma: T \mapsto Jc T c^{-1}J, \]

then one checks that the following diagram commutes.

Thus, $\sigma$ extends $\hat \rho$. Moreover, since $Jc:F^{\sharp } \rightarrow F$ is an isometric isomorphism, it follows that $\sigma$ is a normal $*$-homomorphism.

We recall that the rigged von Neumann algebra $N^{\mathrm {s}} =N^{\mathrm {s}}_{-}= (\operatorname {Cl}(V_{-})^{\mathrm {s}},F^{\mathrm {s}})$ determines a von Neumann algebra $N=(N^{\mathrm {s}})''$, see Remark 2.1.7. Moreover, the statement that $F^{\mathrm {s}}$ is a rigged von Neumann $N^{\mathrm {s}}$–$N^{\mathrm {s}}$-bimodule implies that its completion, the Fock space $F$, is an $N$–$N$-bimodule in the classical sense, see Lemma 2.1.16.

As we recall in Appendix A.1, a cyclic and separating vector for a representation of a von Neumann algebra equips that representation with the structure of a so-called standard form of the von Neumann algebra. Using this in our case for $\Omega \in F$ (see [Reference Kristel and WaldorfKW22, § 4.2]), we note the following result.

A standard form of a von Neumann algebra, viewed as a bimodule, is neutral with respect to Connes fusion, as we recall in Proposition A.2.6 and Corollary A.2.7. More precisely, there are unitary intertwiners $\lambda _K: F \boxtimes K \to K$ and $\rho _K: K \boxtimes F \to K$ for any $N$–$N$-bimodule $K$, and for $K=F$ we have coincidence $\lambda _{F}=\rho _{F}$. We will denote this unitary intertwiner by

\[ \chi: F \boxtimes F \rightarrow F. \]

We shall need a more explicit description of $\chi$ to prove one of our key results, Theorem 4.3.3. To make sense of this explicit description, we first recall the basic definition of the Connes fusion product, see Appendix A, and in particular Definition A.2.1, for more details. We start by writing $\mathcal {D}(F,\Omega ) := \operatorname {Hom}_{-,N}(L^{2}_{\Omega }(N), F)$ for the space of bounded linear maps that intertwine the right $N$ actions. We define a map

\[ p_{\Omega}: \operatorname{Hom}_{-,N}(L^{2}_{\Omega}(N),L^{2}_{\Omega}(N)) \rightarrow N \]

by requiring $p_{\Omega }(x) \triangleright v = x(v)$ for all $v \in L^{2}_{\Omega }(N)$. Next, we consider the space $\mathcal {D}(F,\Omega ) \otimes F$ equipped with the degenerate inner product

\[ \langle x \otimes v, y \otimes w \rangle_{\Omega} := \langle v, p_{\Omega}( x^{*} y) \triangleright w \rangle. \]

The Connes fusion $F\boxtimes F$ is now the Hilbert completion of $\mathcal {D}(F,\Omega ) \otimes F/ \ker \langle \cdot, \cdot \rangle _{\Omega }$. On the space $\mathcal {D}(F, \Omega ) \otimes F$ the map $\chi :F \boxtimes F \to F$ is given by (see Proposition A.2.6 and Corollary A.2.7)

(10)\begin{equation} x \otimes v \mapsto p_{\Omega}(u\circ x) \triangleright v, \end{equation}

where $u$ is the invertible intertwiner $u: F \to L^{2}_{\Omega } (N)$ of Remark 3.3.6.

We remark that Connes fusion is coherently associative (Proposition A.2.5), which allows us to omit bracketing of multiple Connes fusions. The following lemma follows then directly from Corollary A.2.7.

Lemma 3.3.8 The isomorphism $\chi$ is associative in the sense that the following diagram commutes.

3.4 Connes fusion of implementers

The goal of this section is to use Connes fusion and the unitary intertwiner $\chi : F \boxtimes F \to F$ in order to define a product of certain implementers. Since Connes fusion is only available in an ungraded setting, we need to restrict the discussion to even implementers, i.e. to operators $U \in \operatorname {Imp}_L(V) \subset \mathcal {B}(F)$ that preserve the natural grading of $F$. The collection of all even implementers is, in fact, easy to find. Namely, the group $\operatorname {O}_{L}(V)$ has two connected components, thus $\operatorname {Imp}_{L}(V)$ has two connected components as well. All the elements of the identity component of $\operatorname {Imp}_{L}(V)$ are even, and all the remaining elements are odd [Reference ArakiAra87, Theorem 6.7]. Later, in § 4, we will pull back all this structure to a connected Lie group, and hence anyway only see the connected component of $\operatorname {O}_L(V)$.