Quantization of Hamiltonian loop group spaces

Abstract

We prove a Fredholm property for spin-c Dirac operators $\mathsf {D}$ on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group $K\ltimes \Gamma$, with K compact and $\Gamma$ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing $[{\mathcal {X}}] \cap [\mathsf {D}]$ yields an element of the formal completion $R^{-\infty }(T)$ of the representation ring of a maximal torus $T \subset H$; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating $[{\mathcal {X}}] \cap [\mathsf {D}]$ corresponds to an element of the ring of projective positive energy representations of the loop group.

Appendices

Appendix A. Group automorphisms and the descent map

Throughout this section G will be a locally compact group equipped with a left-invariant Haar measure, $\sigma \in \text{ Aut }(G)$ will be a group automorphism preserving the Haar measure, and $\langle \sigma \rangle \ltimes G$ the semi-direct product group. We write $g \mapsto g^\sigma$ for the action of $\sigma$ on $g \in G$.

Let A be a $\langle \sigma \rangle \ltimes G$-$C^*$ algebra; A is also a G-$C^*$ algebra by restricting the group action. Define the G-$C^*$ algebra $A^\sigma$ to be the $C^*$ algebra A equipped with the new G-action $\pi ^\sigma (g)=\pi (g^\sigma )$, where $\pi$ is the G-action on A. Since A is a $\langle \sigma \rangle \ltimes G$-$C^*$ algebra, there is an action map

$$\begin{aligned} \alpha ^A_\sigma :A \rightarrow A. \end{aligned}$$

Viewed as a map $A \rightarrow A^\sigma$, $\alpha ^A_\sigma$ is an isomorphism of G-$C^*$ algebras. In particular, it induces maps

$$\begin{aligned} (\alpha ^A_\sigma )_*:\text{ K }_0^G(A) \rightarrow \text{ K }_0^G(A^\sigma ), \qquad (\alpha ^A_\sigma )^*:\text{ K }^0_G(A^\sigma ) \rightarrow \text{ K }^0_G(A). \end{aligned}$$

Below we usually omit the $*$ from the notation.

For any group homomorphism $\sigma :G_1 \rightarrow G_2$ one has a restriction homomorphism (cf. [23])

$$\begin{aligned} \sigma :\text{ KK }_{G_2}(A,B) \rightarrow \text{ KK }_{G_1}(A,B), \end{aligned}$$

where A, B become $G_1$-$C^*$ algebras by pre-composing the $G_2$ action with $\sigma$. As a special case, if $\sigma \in \text{ Aut }(G)$ is a group automorphism, one obtains a map

$$\begin{aligned} \sigma :\text{ KK }_G(A,B) \rightarrow \text{ KK }_G(A^\sigma ,B^\sigma ). \end{aligned}$$

Definition A.1

Let A, B be $\langle \sigma \rangle \ltimes G$-$C^*$ algebras, with $\alpha _\sigma ^A$, $\alpha _\sigma ^B$ the corresponding action maps. Define

$$\begin{aligned} \tau _\sigma :\text{ KK }_G(A,B) \rightarrow \text{ KK }_G(A,B) \end{aligned}$$

to be the composition

$$\begin{aligned} \text{ KK }_G(A,B) \xrightarrow {\alpha _\sigma ^B} \text{ KK }_G(A,B^\sigma ) \xrightarrow {(\alpha _\sigma ^A)^{-1}} \text{ KK }_G(A^\sigma ,B^\sigma ) \xrightarrow {\sigma ^{-1}} \text{ KK }_G(A,B). \end{aligned}$$

In general $\tau _\sigma$ is not the identity map (we saw an example in Sect.  4.4), but it does act as the identity on the image of the restriction map $\text{ KK }_{\langle \sigma \rangle \ltimes G}(A,B) \rightarrow \text{ KK }_G(A,B)$.

We next describe the relationship between $\tau _\sigma$ and the descent map $j_G$. The G-equivariant $^*$-homomorphism $\alpha ^A_\sigma :A \rightarrow A^\sigma$ induces a $^*$-homomorphism

$$\begin{aligned} \alpha ^A_\sigma :G \ltimes A \rightarrow G \ltimes A^\sigma \end{aligned}$$

(27)

(on $C_c(G,A)$ it is given by applying $\alpha ^A_\sigma$ point-wise). Recall that a $^*$-homomorphism $\alpha :A \rightarrow B$ determines an element $\alpha \in \text{ KK }(A,B)$, such that push-forward (resp. pull-back) by $\alpha$ in K-theory (resp. K-homology) are given by an appropriate $\text{ KK }$-product. Thought of in this way, the corresponding element in $\text{ KK }(G\ltimes A,G\ltimes A^\sigma )$ is the image of $\alpha ^A_\sigma \in \text{ KK }_G(A,A^\sigma )$ under the descent map $j_G$.

The group automorphism $\sigma :G \rightarrow G$ also induces a $^*$-homomorphism

$$\begin{aligned} \sigma ^A :G \ltimes A \rightarrow G \ltimes A^\sigma , \end{aligned}$$

(28)

given on $C_c(G,A)$ by $a \mapsto a^\sigma$, where $a^\sigma (g):=a(g^\sigma )$.

Proposition A.2

Let A,B be $\langle \sigma \rangle \ltimes G$-$C^*$ algebras. For any $\mathsf {x} \in \text{ KK }_G(A^\sigma ,B^\sigma )$ one has the following equality in $\text{ KK }(G\ltimes A,G\ltimes B)$:

$$\begin{aligned} j_G(\sigma ^{-1}(\mathsf {x}))=\sigma ^A \otimes j_G(\mathsf {x})\otimes (\sigma ^B)^{-1}. \end{aligned}$$

Proof

Let $({\mathcal {E}},\rho ,F)$ be a cycle representing $\mathsf {x}$, thus in particular ${\mathcal {E}}$ is an $(A^\sigma ,B^\sigma )$-bimodule with $B^\sigma$-valued inner product $(\cdot ,\cdot )$. To avoid confusion between the G-$C^*$ algebras B and $B^\sigma$, we will write all formulas in terms of the action map $(g^\prime ,b)\mapsto g^\prime .b$ for B, not$B^\sigma$. Thus, for example, the $G\ltimes B^\sigma$-valued inner product for $j_G(\mathsf {x})$ is expressed as

$$\begin{aligned} (e_1,e_2)_{G\ltimes B^\sigma }(g)=\int _G (g_1^\sigma )^{-1}.\big (e_1(g_1),e_2(g_1g)\big )\,\,dg_1, \end{aligned}$$

i.e. $(g_1^\sigma )^{-1}$ appears in the formula instead of $g_1^{-1}$. The $(G\ltimes A,G\ltimes B)$-bimodule structure for $\sigma ^A \otimes j_G(\mathsf {x})\otimes (\sigma ^B)^{-1}$ is given by the formulas

$$\begin{aligned} (a\cdot e)(g)=\int _G a(g_1^\sigma )g_1.e(g_1^{-1}g)\,\,dg_1, \qquad (e\cdot b)(g)=\int _G e(g_1)g_1^\sigma .b\big ((g_1^\sigma )^{-1}g^\sigma \big )\,\,dg_1, \end{aligned}$$

and the $G\ltimes B$-valued inner product is

$$\begin{aligned} (e_1,e_2)_{G\ltimes B}(g)=\int _G (g_1^\sigma )^{-1}.\big (e_1(g_1),e_2(g_1g^{\sigma ^{-1}})\big )\,\, dg_1. \end{aligned}$$

On the other hand, the $(G\ltimes A,G\ltimes B)$-bimodule structure for $j_G(\sigma ^{-1}(\mathsf {x}))$ is given by the formulas

$$\begin{aligned} (a \star e)(g)=\int _G a(g_1)g_1^{\sigma ^{-1}}.e(g_1^{-1}g)\,\,dg_1, \qquad (e \star b)(g)=\int _G e(g_1)g_1.b(g_1^{-1}g)\,\,dg_1, \end{aligned}$$

and the $G\ltimes B$-valued inner product is

$$\begin{aligned} (e_1,e_2)^{\star }_{G\ltimes B}(g)=\int _G g_1^{-1}.\big (e_1(g_1),e_2(g_1g)\big )\,\,dg_1. \end{aligned}$$

The formulas are not the same, but there is a linear map

$$\begin{aligned} e \in C_c(G,{\mathcal {E}}) \mapsto e^\sigma \in C_c(G,{\mathcal {E}}), \qquad e^\sigma (g):=e(g^\sigma ) \end{aligned}$$

which intertwines the bimodule structures and $C_c(G,B)$-valued inner products, i.e. $(a \star e)^\sigma =a\cdot e^\sigma$, $(e \star b)^\sigma =e^\sigma \cdot b$ and $(e_1^\sigma ,e_2^\sigma )_{G\ltimes B}=(e_1,e_2)^{\star }_{G\ltimes B}$ (one uses the $\sigma$-invariance of the Haar measure). Thus, the map extends to an isometric isomorphism between the completions, intertwining the bimodule structures. $\square$

Definition A.3

Let A be a $\langle \sigma \rangle \ltimes G$-$C^*$ algebra. Let

$$\begin{aligned} \tau ^A_\sigma =(\alpha ^A_\sigma )^{-1} \circ \sigma ^A :G\ltimes A \rightarrow G\ltimes A, \end{aligned}$$

where $\alpha ^A_\sigma$ (resp. $\sigma ^A$) is as in (27) (resp. (28)). Then $\tau ^A_\sigma$ is an automorphism of $G\ltimes A$. On $C_c(G,A)$ it is given by the formula

$$\begin{aligned} \tau ^A_\sigma (f)(g)=(\alpha ^A_\sigma )^{-1}(f(g^\sigma )). \end{aligned}$$

The corresponding element of $\text{ KK }(G\ltimes A,G\ltimes A)$ is the $\text{ KK }$-product

$$\begin{aligned} \tau ^A_\sigma =\sigma ^A \otimes (\alpha ^A_\sigma )^{-1} \in \text{ KK }(G\ltimes A,G\ltimes A). \end{aligned}$$

The following is a corollary of Proposition A.2.

Corollary A.4

Let A, B be $\langle \sigma \rangle \ltimes G$-$C^*$ algebras. For any $\mathsf {x} \in \text{ KK }_G(A,B)$,

$$\begin{aligned} j_G(\tau _\sigma (\mathsf {x}))=\tau ^A_\sigma \otimes j_G(\mathsf {x}) \otimes (\tau ^B_\sigma )^{-1}. \end{aligned}$$

Similar to $\tau _\sigma$, $\tau ^A_\sigma$ acts trivially on elements which are $\langle \sigma \rangle \ltimes G$-equivariant:

Proposition A.5

Let $\widetilde{G}=\langle \sigma \rangle \ltimes G$. Let A be a $\widetilde{G}$-$C^*$ algebra. Let

$$\begin{aligned} \iota _G :G\ltimes A \rightarrow \widetilde{G} \ltimes A \end{aligned}$$

denote the $^*$-homomorphism induced by the open inclusion $G \hookrightarrow \widetilde{G}$. Then

$$\begin{aligned} \tau ^A_\sigma \otimes \iota _G = \iota _G \in \text{ KK }(G\ltimes A,\widetilde{G} \ltimes A). \end{aligned}$$

Proof

The automorphism $\tau ^A_\sigma \in \text{ Aut }(G\ltimes A)$ extends to an automorphism of $\widetilde{G} \ltimes A$, given by the same formula with $\widetilde{g}^\sigma :=\text{ Ad }_\sigma (\widetilde{g}) \in \widetilde{G}$. Thus

$$\begin{aligned} \iota _G \circ \tau ^A_\sigma =\widetilde{\tau }^A_\sigma \circ \iota _G, \end{aligned}$$

where $\widetilde{\tau }^A_\sigma$ denotes the extended automorphism. In fact $\widetilde{\tau }^A_\sigma$ is an inner automorphism, i.e. there exists a unitary $u_\sigma$ in the multiplier algebra of $\widetilde{G} \ltimes A$ such that $\widetilde{\tau }^A_\sigma =\text{ Ad }_{u_\sigma }$, cf. [10, II.10.3.10]. The result follows from the fact that any inner automorphism of a $C^*$ algebra D induces the identity element in $\text{ KK }(D,D)$. $\square$

Appendix B. Schrödinger-type operators

If A is a self-adjoint operator on a Hilbert space H with domain $\text{ dom }(A)$ and spectrum in $[1,\infty )$, then one defines an associated positive definite quadratic form

$$\begin{aligned} q_A(u_1,u_2)=(Au_1,u_2) \end{aligned}$$

for all $u_1,u_2 \in \text{ dom }(A)$. The completion of $\text{ dom }(A)$ using the inner product $q_A$ is a Hilbert space $\text{ dom }(q_A)$ which can be identified with $\text{ dom }(A^{1/2})$, and is known as the form domain of A (cf. [40, VIII.6]). Given self-adjoint operators A, B with spectrum in $[1,\infty )$ one writes

$$\begin{aligned} A \ge B \end{aligned}$$

if

$$\begin{aligned} \text{ dom }(q_A) \subset \text{ dom }(q_B) \quad \text {and} \quad q_A(v,v)\ge q_B(v,v) \quad \forall v \in \text{ dom }(q_A) \end{aligned}$$

(cf. [39, XIII.2, p. 85]). Equivalently, $A \ge B$ if the inclusion mapping

$$\begin{aligned} (\text{ dom }(q_A),q_A) \hookrightarrow (\text{ dom }(q_B),q_B) \end{aligned}$$

is norm-decreasing. It is enough to check the inequality $q_A(v,v)\ge q_B(v,v)$ on a core for A. More generally if A, B are self-adjoint operators with spectrum in $[-c,\infty )$ for some $c \ge 0$ then one writes $A \ge B$ if $A+c+1 \ge B+c+1$.

Proposition B.1

Let ${\mathcal {Y}}$ be a complete Riemannian manifold. Let $\mathsf {D}$ be a symmetric $1^{st}$ order elliptic differential operator with finite propagation speed acting on sections of a Hermitian vector bundle S, and let $H=L^2({\mathcal {Y}},S)$. Let $\rho$, V be continuous functions such that $\rho$ is bounded, V is bounded below, and V is proper on the support of $\rho$. Let A be a self-adjoint operator with spectrum in $(0,\infty )$, and suppose

$$\begin{aligned} A \ge \mathsf {D} ^2+V. \end{aligned}$$

Then the operator $\rho A^{-1}$ is compact.

Remark B.2

Consider the special case when V is proper and bounded below. The operator $\mathsf {D}^2+V$ is a Schrödinger-type operator with potential going to infinity at infinity. In this case, it is known that $\mathsf {D}^2+V$ has discrete spectrum. Many proofs of this appear in the literature, cf. [41] (for $\mathsf {D}^2=-\Delta$), [25] (for a proof based on a method of Gromov-Lawson). It is also closely related to a Fredholm criterion of Anghel [4], and to the property of being ‘$\kappa$-coercive’ for all $\kappa >0$ in [7, Corollary 5.6].

Proof

Note that if $\rho (A+c)^{-1}$ is compact for some $c>0$, then so is $\rho A^{-1}$ since

$$\begin{aligned} \rho A^{-1}-\rho (A+c)^{-1}=c\rho (A+c)^{-1}A^{-1} \end{aligned}$$

and the right hand side is compact since $\rho (A+c)^{-1}$ is compact and $A^{-1}$ is bounded. Therefore we may as well assume that $V \ge 1$ and the spectrum of A is contained in $[1,\infty )$.

The operator $\mathsf {D}^2+V$ is self-adjoint (cf. [15, Theorem 4.6]) and positive. Let q denote the corresponding quadratic form, defined initially on the domain of $\mathsf {D}^2+V$ by

$$\begin{aligned} q(u_1,u_2)=\big ((\mathsf {D}^2+V)u_1,u_2\big ) \end{aligned}$$

and then extended to the completion $H(q)=\text{ dom }(q) \subset H$ of the domain of $\mathsf {D}^2+V$ with respect to the inner product q. Since

$$\begin{aligned} q(u,u)=\Vert \mathsf {D}u\Vert ^2+\Vert \sqrt{V}u\Vert ^2 \end{aligned}$$

the Hilbert space H(q) can be described more simply as

$$\begin{aligned} H(q)=\{u \in H^1|\sqrt{V}u \in H\}, \end{aligned}$$

where $H^1:=\text{ dom }(\mathsf {D})$ (a Sobolev space). The operator $A^{-1}$ defines a bounded linear map $H \rightarrow \text{ dom }(A)$, where the domain $\text{ dom }(A)$ of A is equipped with the A-norm $\Vert u\Vert _A:=\Vert Au\Vert$ (cf. [40, VIII]). The Cauchy-Schwartz inequality and $q_A(u,u)\ge q(u,u)$ imply that the inclusion maps

$$\begin{aligned} \text{ dom }(A) \subset \text{ dom }(q_A) \hookrightarrow H(q) \end{aligned}$$

are bounded. Thus $\rho A^{-1}$ factors as a composition of bounded linear maps:

$$\begin{aligned} H \xrightarrow {A^{-1}}\text{ dom }(A) \hookrightarrow H(q) \xrightarrow {M_\rho } H \end{aligned}$$

where $M_\rho$ denotes the operator given by multiplication by the function $\rho$. It therefore suffices to show that the map

$$\begin{aligned} M_\rho :H(q) \rightarrow H \end{aligned}$$

is a compact mapping. Without loss of generality assume $|\rho |\le 1$. For $n \in {\mathbb {Z}}_{\ge 0}$ let $W_n=V^{-1}((-\infty ,n])$; by assumption $\text{ supp }(\rho )\cap W_n$ is compact. For each n, let $g_n:{\mathcal {Y}}\rightarrow [0,1]$ be a bump function equal to 1 on $W_n$ and supported in $W_{n+1}$. Since $g_n\rho$ has compact support,

$$\begin{aligned} M_{g_n\rho }:H(q) \rightarrow H \end{aligned}$$

is compact, by the Rellich lemma. We claim that the difference

$$\begin{aligned} M_\rho -M_{g_n\rho }=M_{(1-g_n)\rho } \end{aligned}$$

converges to zero in norm, hence $M_\rho$ is also compact. Indeed, if $q(u,u)\le 1$ then

$$\begin{aligned} 1 \ge q(u,u) \ge \int _{{\mathcal {Y}}\setminus W_n} n|u|^2 \qquad \Rightarrow \qquad \int _{{\mathcal {Y}}\setminus W_n}|u|^2 \le \tfrac{1}{n}. \end{aligned}$$

Thus

$$\begin{aligned} \Vert M_{(1-g_n)\rho }\Vert ^2=\sup _{q(u,u)\le 1} \, \, \int _{{\mathcal {Y}}} (1-g_n)^2|\rho u|^2 \le \sup _{q(u,u)\le 1} \, \, \int _{{\mathcal {Y}}\setminus W_{n}}|u|^2 \le \tfrac{1}{n}, \end{aligned}$$

which goes to 0 as $n \rightarrow \infty$. $\square$

Remark B.3

More generally, suppose a compact group K acts isometrically on ${\mathcal {Y}}$, commuting with A, $\mathsf {D}$ and that V is K-invariant. The Hilbert space H decomposes into isotypic components

$$\begin{aligned} H=\bigoplus _{\pi \in \text{ Irr }(K)} H_\pi , \end{aligned}$$

and the operator A decomposes accordingly into self-adjoint operators $A_\pi$ with $\text{ dom }(A_\pi )=\text{ dom }(A)\cap H_\pi \subset H_\pi$, and similarly for $\mathsf {D}^2+V$. Suppose the conditions in Proposition B.1 are satisfied except that the inequality holds only on $H_\pi$, that is

$$\begin{aligned} A_\pi \ge (\mathsf {D}^2+V)_\pi . \end{aligned}$$

Essentially the same argument, with $\rho A_\pi ^{-1}$ factored as

$$\begin{aligned} H_\pi \xrightarrow {A_\pi ^{-1}} \text{ dom }(A_\pi ) \hookrightarrow H(q)\xrightarrow {M_\rho } H \end{aligned}$$

shows that $\rho A_\pi ^{-1}:H_\pi \rightarrow H$ is a compact operator.