2.1 Construction of $\boldsymbol{D}_{\boldsymbol{K}}$

Let K be a compact torus with Lie algebra ${\mathfrak {k}}$ . We fix an inner product on $\mathfrak {k}$ and identify ${\mathfrak {k}}^*={\mathfrak {k}}$ . We often identify $\textrm {Irr}(K)$ with $\Lambda ^*$ , where we put $\Lambda :=\textrm {ker}(\textrm {exp}:\mathfrak {k}\to K)$ . Let M be a complete Riemannian manifold and W a $\mathbb {Z}/2$ -graded Clifford module bundle over M. Suppose that K acts on M in an isometric way and the action lifts to W as a unitary action. Take a K-invariant Hermitian connection $\nabla $ of W.

For $\xi \in \mathfrak {k}$ , we denote the induced infinitesimal action of $\xi $ on M by $\underline {\xi }^M$ . Let $\mathcal {L}_{\xi }:\Gamma (W)\to \Gamma (W)$ be the induced derivative defined by

$$ \begin{align*} \mathcal{L}_{\xi}s : x\mapsto\left.\frac{d}{dt}\right|_{t=0}\textrm{exp}(t\xi)s(\textrm{exp}(-t\xi)x) \end{align*} $$

for $s\in \Gamma (W)$ . Let $\mu :M\to \textrm {End}(W)\otimes {\mathfrak {k}}^*$ be the map defined by Kostant’s formula:

(2.1) $$ \begin{align} \mathcal{L}_{\xi}-\nabla_{\underline{\xi}^M}=\sqrt{-1}\mu(\xi)=\sqrt{-1}\mu_{\xi} \quad (\xi\in{\mathfrak{k}}). \end{align} $$

Fix an orthonormal basis $\{\xi _1, \ldots , \xi _n\}$ of $\mathfrak {k}$ .

Definition 2.1 We define the orbital Dirac-type operator $D_K:\Gamma _c(W)\to \Gamma _c(W)$ by

$$ \begin{align*} D_K:=\sum_{i=1}^nc(\underline{\xi_i}^M)(\mathcal{L}_{\xi_i}-\sqrt{-1}\mu_{\xi_i}). \end{align*} $$

Proof (1) follows from the definition of $D_K$ . (2) follows from the anticommutativity between $c(dh)$ and $c(\underline {\xi _i}^M)$ for any K-invariant function h. By using (2), one can show (3) by the computation of the anticommutator:

$$ \begin{align*} (DD_K+D_KD)h=c(dh)D_K+hDD_K+D_Kc(dh)+hD_KD=h(DD_K+D_KD). \end{align*} $$

▪

Remark 2.2 The differential term $\displaystyle \sum _{i=1}^nc(\underline {\xi _i^M})\mathcal {L}_{\xi _i}$ in $D_K$ is the orbital Dirac operator in the sense of Kasparov [Reference Kasparov14]. On the other hand, the multiplication term $\displaystyle \sum _{i=1}^nc(\underline {\xi _i^M})\mu _{\xi _i}$ is equal to $c(\underline {\mu })$ for $\underline {\mu }:=\displaystyle \sum _{i=1}^n\underline {\xi _i^M}\mu _{\xi _i}$ , which gives the deformation studied by Braverman [Reference Braverman6]. For each $\rho \in \textrm {Irr}(K)$ , one has $\mathcal {L}_{\xi _i}=\sqrt {-1}\rho (\xi _i)$ on each isotypic component $L^2(W_L)^{(\rho )}$ , and hence,

$$ \begin{align*} D_K^{(\rho)}=\sqrt{-1}\sum_{i=1}^nc(\underline{\xi_i^M})(\rho(\xi_i)-\mu_{\xi_i})= \sqrt{-1}c(\underline{\rho}-\underline{\mu}), \end{align*} $$

and

$$ \begin{align*} (D_K^{(\rho)})^2=|\underline{\rho}-\underline{\mu}|^2, \end{align*} $$

where $\underline {\rho }$ is the infinitesimal action induced by $\rho \in \mathfrak {k}^*=\mathfrak {k}$ . In other words, $D_K$ gives a kind of shift of Braverman’s deformation. We investigate the relation between our deformation and Braverman’s deformation in the next section.

For $\rho \in \textrm {Irr}(K)$ , let $Z_{\rho }:=\textrm {Zero}(\underline {\rho }-\underline {\mu })$ be the set of points in M at which the vector field $\underline {\rho }-\underline {\mu }$ vanishes. Note that $Z_\rho $ coincides with the set of critical points of $|{\rho }-{\mu }|^2$ in M, and it contains $M^K\cup \mu ^{-1}(\rho )$ . The above description of $D_K$ implies the following.

Proposition 2.3 For $x\in M$ and $\rho \in \textrm {Irr}(K)$ , we have

$$ \begin{align*} \ker(D_K|_{K\cdot x})^{(\rho)}\neq 0 \Longleftrightarrow x\in Z_{\rho}. \end{align*} $$

Let D be the Dirac operator acting on $\Gamma (W)$ , which is defined by the connection $\nabla $ . For each $\rho \in \textrm {Irr}(K)$ , we put

$$ \begin{align*} V_\rho:=M\setminus Z_{\rho}. \end{align*} $$

Then, because $(D_K^{(\rho )}|_{K\cdot x})^2$ is a strictly positive operator on $\Gamma (W|_{K\cdot x})^{(\rho )}$ for any $x\in V_\rho $ , there exists a constant $C_{\rho , x}$ such that

$$ \begin{align*} |( (DD_{K}+D_{K}D)s, s )_W|\leq C_{\rho,x}( D_{K}^2 s,s)_W \end{align*} $$

and

$$ \begin{align*} |( (D_{K}s, s )_W|\leq C_{\rho,x}( D_{K}^2 s,s)_W \end{align*} $$

hold for any $s\in \Gamma (W_L|_{K\cdot x})^{(\rho )}$ .

By using the above K-acyclic orbital Dirac-type operator, we have a family of deformations of the Dirac operator D,

$$ \begin{align*} \hat D_\rho=D+f_\rho^4D_K, \end{align*} $$

or

$$ \begin{align*} D_{\rho,t}=D+t\varphi_\rho^4 D_K \quad (t \gg 0) \end{align*} $$

as in Corollary A.3, Definition A.2, and Corollary A.6. As a consequence of Theorem 2.4, Corollary A.3, Definition A.2, and Corollary A.6, we have the following.

For latter convenience, we investigate the case of Hermitian manifold in detail. We assume that the metric on M is induced from a K-invariant Hermitian structure $(g, J)$ and the Clifford module bundle W is given by

$$ \begin{align*} W=\wedge^{\bullet}T_{\mathbb{C}}M\otimes L \end{align*} $$

for a K-equivariant Hermitian line bundle with Hermitian connection $(L,\nabla ^L)$ over M, where $T_{\mathbb {C}}M=TM$ is the vector bundle regarded as a complex vector bundle by J. This W carries a structure of $\mathbb {Z}/2$ -graded $\textrm {Cl}(TM)$ -module bundle with the Clifford multiplication $c:TM\to \textrm {End}(W)$ defined by the exterior product and its adjoint. In this case, $\mu $ is a map to $\mathfrak {k}^*$ determined by

$$ \begin{align*} \mathcal{L}_{\xi}^L-\nabla_{\underline{\xi}^M}^L=\sqrt{-1}\mu(\xi)=\sqrt{-1}\mu_{\xi} \quad (\xi\in{\mathfrak{k}}), \end{align*} $$

and we have

$$ \begin{align*} \mathcal{L}_\xi= \mathcal{L}_{\xi}^M\otimes\textrm{id}+\textrm{id}\otimes\mathcal{L}_{\xi}^L. \end{align*} $$

For $x\in M$ , let $H^0(K\cdot x; L|_{K\cdot x})$ be the space of global parallel sections on $(L, \nabla ^L)|_{K\cdot x}$ , which is a vector space of dimension at most one. Suppose that $H^0(K\cdot x; L|_{K\cdot x})\neq 0$ and s is its nontrivial element, then we have

$$ \begin{align*} 0=\nabla^L_{\underline{\xi}^M}s=(\mathcal{L}_{\xi}^L-\sqrt{-1}\mu_{\xi})s \end{align*} $$

for all $\xi \in \mathfrak {k}$ . This equation implies that $\mu _\xi (x)$ is an integer for all $\xi $ , and hence, we have the following.

2.2 Noncomplete case and localization formula

As we will mention in the end of Appendix A.4, the index associated with the K-acyclic orbital Dirac-type operator can be defined for noncomplete situation. For instance, suppose that the first conditionFootnote ⁴ in Theorem 2.4 is satisfied. We take a K-invariant compact submanifold $X_\rho $ with boundary as a neighborhood of $Z_{\rho }$ and attach a cylinder $\partial X_\rho \times [0,\infty )$ to $\partial X_\rho $ , so that we have a K-invariant complete Riemannian manifold $\tilde X_\rho $ with K-invariant cylindrical end. Let $\tilde \mu $ , $\tilde W$ , and $\tilde D_K$ be the extensions of $\mu $ , W, and $D_K$ on $\tilde X_\rho $ such that they have translational invariance and $\ker (\tilde D_K^{(\rho )}|_{K\cdot x})=\ker ((\tilde D_K^{(\rho )}|_{K\cdot x})^2)=\ker (|\underline {\rho }-\underline {\mu }|^2)=0$ for any $x\in \partial X_\rho \times (0,\infty )$ . These data define a Fredholm operator on $L^2(\tilde W)^{(\rho )}$ as in Corollary A.3. Though we agree that it is a little bit strange notation,Footnote ⁵ we denote this index by

(2.2) $$ \begin{align} [Z_\rho]\in \mathbb{Z}. \end{align} $$

We decompose

$$ \begin{align*} Z_\rho=\mu^{-1}(\rho)\cup\left(\bigcup_{\alpha} Z_{\rho,\alpha}\right) \end{align*} $$

into the disjoint union of the connected components, where $Z_{\rho ,\alpha }$ is a connected component other than $\mu ^{-1}(\rho )$ . This description enable us to get more refined decomposition of (2.2) into the summation of local contributions from each component, which we denote by

$$ \begin{align*} [Z_\rho]=[\mu^{-1}(\rho)]+\sum_{\alpha} [Z_{\rho,\alpha}]. \end{align*} $$

The excision formula implies the following localization formula.

Theorem 2.8 If the conditions in Theorem 2.4 are satisfied, then the index $[\hat D]=[M, W, D_K]\in R^{-\infty }(K)$ defined by the K-acyclic orbital Dirac-type operator $D_K$ satisfies

$$ \begin{align*} [M,W,D_K](\rho)=[Z_\rho]=[\mu^{-1}(\rho)]+\sum_{\alpha} [Z_{\rho,\alpha}] \end{align*} $$

for each $\rho \in \textrm {Irr}(K)$ .

In Section 5, we discuss the case of Hamiltonian circle action and symplectic toric case. In these cases, one has the vanishing $[Z_{\rho ,\alpha }]=0$ , and hence, we realize the localization of the index $[M,W,D_K]$ into the lattice points $\textrm {Irr}(K)=\Lambda ^*$ .

2.3 Relation with Braverman’s deformation

In [Reference Braverman6], Braverman studied a Witten-type deformation of the Dirac operator and its equivariant index on noncompact K-manifold. In a symplectic geometric setting, Braverman’s deformation is given by the Clifford multiplication of the Hamiltonian vector field of the norm square of the moment map. In particular, in the setting in Section 2.1 (not necessarily K is a torus), we can consider the Braverman’s deformation as

$$ \begin{align*} D_{\mu}:=D-h\sqrt{-1}c(\underline{\mu}), \end{align*} $$

where $h:M\to \mathbb {R}$ is a K-invariant function called an ${\textit {admissible function}}$ , which satisfies a suitable growth condition. Braverman showed several fundamental properties of $D_{\mu }$ . In particular, he showed that $D_{\mu }$ is a K-Fredholm operator and the resulting index in $R^{-\infty }(K)$ is independent of a choice of the admissible function. Moreover, the index is equal to Atiyah’s transverse index. After that, his equivariant index has been applied in several directions, for instance, a solution to Vergne’s conjecture by Ma and Zhang [Reference Ma and Zhang20].

In this section, we consider the same setup in Section 2.1 and assume the followings to make the situation simple.

We show the following.

Theorem 2.10 Under Assumption 2.9, we have

$$ \begin{align*} \textrm{index}_K(D_{\mu})=[\hat D]\in R^{-\infty}(K). \end{align*} $$

As a corollary of Braverman’s index theorem [Reference Braverman6, Theorem 5.5] we also have the following.

We first note that under Assumption 2.9, we can take f as in Appendix A, so that $f=|\mu |$ on the outside of a compact neighborhood of the compact subset $\mu ^{-1}(0)$ . Moreover, we can take an admissible function h to be $f_{\rho }^4=\varphi _{\rho }^4f^4$ for each $\rho \in \textrm {Irr}(K)$ , where $\varphi _\rho $ is the cutoff function for $V_{\rho }=M\setminus Z_{\rho }$ as in (A.1).

To show Theorem 2.10, we fix $\rho \in \textrm {Irr}(K)$ and consider the following one-parameter family in the setting in Section 2.1:

$$ \begin{align*} {\mathbb{D}}_\epsilon:=D+\epsilon f_{\rho}^4D_K-(1-\epsilon)\sqrt{-1}f_\rho^4c(\underline{\mu}) \end{align*} $$

for $\epsilon \in [0,1]$ . We show that for each $\rho $ , an unbounded operator $\mathbb {D}_\epsilon ^{(\rho )}$ on $L^2(W)^{(\rho )}$ gives a norm-continuous family of the bounded transformations, such as $\displaystyle \frac {\mathbb {D}_\epsilon }{\sqrt {1+\mathbb {D}_\epsilon ^2}}$ , and hence, the equality

$$ \begin{align*} \textrm{index}_K(D_\mu)(\rho)=\textrm{index}((\mathbb{D}_\epsilon)^{(\rho)})=\textrm{index}(\hat D_\rho)= [\hat D](\rho) \end{align*} $$

holds. We use the following criteria.

As in [Reference Loizides and Song19, Remark 4.10] it suffices to show the third condition in Lemma 2.13 in our situation.

Hereafter, we mainly consider the isotypic component of operators. Even if so, we often omit the superscript $(\cdot )^{(\rho )}$ of the isotypic component for simplicity and use the notation as $D:L^2(W)^{(\rho )}\to L^2(W)^{(\rho )}$ and so on. As we noted in Remark 2.2, one can write as $\displaystyle D_K=\sqrt {-1}c(\underline {\rho }-\underline {\mu })$ on $L^2(W)^{(\rho )}$ , and hence, we have

$$ \begin{align*} \mathbb{D}_\epsilon&=D+ f_\rho^4\sqrt{-1}\left(\epsilon c(\underline{\rho}-\underline{\mu})-(1-\epsilon)c(\underline{\mu})\right) \\ &= D+f_{\rho}^4\sqrt{-1}c(\epsilon\underline{\rho}-\underline{\mu}) \\ &=D-f_\rho^4\sqrt{-1}c(\underline{\mu})+\epsilon f_\rho^4\sqrt{-1}c(\underline{\rho}) . \end{align*} $$

Then, the third condition in Lemma 2.13 is equivalent to

$$ \begin{align*} C'\left(f_\rho^4\sqrt{-1}c(\underline{\rho})\right)^2 \leq (\mathbb{D}_\epsilon)^2+C. \end{align*} $$

for some constants $C, C'>0$ . Because

$$ \begin{align*} \left(\sqrt{-1}c(\underline{\rho})\right)^2= \left|\underline{\rho}\right|^2\leq \sum_{i}|\rho(\xi_i)\underline{\xi_i^M}|^2 \end{align*} $$

by using an orthonormal basis $\{\xi _1,\ldots , \xi _n\}$ of $\mathfrak {k}$ , and our boundedness condition on $|\underline {\xi _i^M|}$ , it suffices to show the following.

Lemma 2.14 There exist constants $C, C'>0$ such that

$$ \begin{align*} C'f_{\rho}^8\leq (\mathbb{D}_\epsilon)^2+C \end{align*} $$

holds for all $\epsilon \in [0,1]$ .

Proof On $L^2(W)^{(\rho )}$ , we have

$$ \begin{align*} (\mathbb{D}_\epsilon)^2&=D^2+\sqrt{-1}\left(Df_\rho^4c(\epsilon\underline{\rho}-\underline{\mu})+f_\rho^4c(\epsilon\underline{\rho}-\underline{\mu})D\right)+ f_\rho^8|\epsilon\underline{\rho}-\underline{\mu}|^2 \\ &=D^2+\sqrt{-1}\left(4f_\rho^3 c(df_\rho)c(\epsilon\underline{\rho}-\underline{\mu})+ f_{\rho}^2(Dc(\epsilon\underline{\rho}-\underline{\mu})+c(\epsilon\underline{\rho}-\underline{\mu})D)f_\rho^2\right)\\ &\quad +f_\rho^8|\epsilon\underline{\rho}-\underline{\mu}|^2. \end{align*} $$

On the other hand, there exist constants $C_1>0$ and $C_2>0$ such that

$$ \begin{align*} |c(df_\rho)c(\epsilon\underline{\rho}-\underline{\mu})|\leq \|df_\rho\||\epsilon\underline{\rho}-\underline{\mu}| \leq C_1 |\epsilon\rho-\mu| \end{align*} $$

and

$$ \begin{align*} |f_{\rho}^2(Dc(\epsilon\underline{\rho}-\underline{\mu})+c(\epsilon\underline{\rho}-\underline{\mu})D)f_\rho^2| \leq C_2|\epsilon\rho-\mu|f_{\rho}^4D_K^2=C_2f_\rho^4|\epsilon\rho-\mu|^3, \end{align*} $$

where we get the inequality in a similar way as in the proof of Proposition A.2 and we use the assumption on cylindrical end, so that we can take $C_2$ uniformly. So we have

$$ \begin{align*} (\mathbb{D}_\epsilon)^2\geq -4C_1 f_{\rho}^3|\epsilon\rho-\mu|-C_2f_\rho^4|\epsilon\rho-\mu|^3+ f_\rho^8|\epsilon\underline{\rho}-\underline{\mu}|^2. \end{align*} $$

On the other hand, because $\mu $ is proper and $M^K$ is compact, $|\epsilon \underline {\rho }-\underline {\mu }|$ is uniformly positive on the outside of a compact subset, and hence, there exists $C'>0$ such that

$$ \begin{align*} f_{\rho}^8|\epsilon\underline{\rho}-\underline{\mu}|+1>2C'f_{\rho}^8. \end{align*} $$

Because $f_\rho =|\mu |$ on the outside of a compact subset, there exists $C>0$ independent from $\epsilon \in [0,1]$ such that

$$ \begin{align*} -4C_1 f_{\rho}^3|\epsilon\rho-\mu|-C_2f_\rho^4|\epsilon\rho-\mu|^3-1+C'f_{\rho}^8>-C. \end{align*} $$

Finally, we have

$$ \begin{align*} (\mathbb{D}_\epsilon)^2>(1-C'f_{\rho}^8-C)+(2C'f_{\rho}^8-1)=C'f_{\rho}^8-C, \end{align*} $$

and hence, $(\mathbb {D}_\epsilon )^2+C>C'f_\rho ^8$ . ▪

3.1 Direct product

For $i=0,1$ , let $K_i$ be a torus. Let $M_i$ be a complete Riemannian manifold and $W_i\to M_i$ a $\mathbb {Z}/2$ -graded Clifford module bundle on which $K_i$ acts in an isometric way. Suppose that there exists a $K_i$ -acyclic orbital Dirac-type operator $(D_{K_i}, \{V_{i,\rho _i}\}_{\rho _i\in \textrm {Irr}(K_i)})$ on $(M_i, W_i)$ . Put $M:=M_0\times M_1$ and define a Clifford module bundle W over M by the outer tensor product

$$ \begin{align*} W:=W_0\boxtimes W_1 \end{align*} $$

for the projections onto the first and second factors of M. For $\rho =(\rho _0, \rho _1)\in \textrm {Irr}(K_0)\times \textrm {Irr}(K_1)$ , we define $V_{\rho }$ by

$$ \begin{align*} V_{\rho}:={V_{0,\rho_0}}\times V_{1,\rho_1} \end{align*} $$

whose complement in M is compact. Let $D_K:\Gamma (W)\to \Gamma (W)$ be an operator defined by

$$ \begin{align*} D_K:= D_{K_0}\otimes \textrm{id}+\varepsilon_{W_0}\otimes D_{K_1}=D_{K_0}+\varepsilon_{W_0}D_{K_1}, \end{align*} $$

where $\varepsilon _{W_0}: W_0\to W_0$ is the grading operator on $W_0$ . Because $D_{K_0}(\varepsilon _{W_0}D_{K_1})+(\varepsilon _{W_0}D_{K_1})D_{K_0}=0$ , one has the following.

Dirac operators $D_i$ on $W_i$ give rise to the Dirac operator D on W:

$$ \begin{align*} D:= D_{0}\otimes \textrm{id}+\varepsilon_{ W_0}\otimes D_{1}= D_{0}+\varepsilon_{ W_0}D_{1}. \end{align*} $$

For each $\rho _i\in \textrm {Irr}(K_i)$ , we take a $K_i$ -invariant cutoff function $\varphi _{i, \rho _i}$ on $M_i$ with $\varphi _{i,\rho _i}|_{M_i\setminus V_{i,\rho _i}}\equiv 0$ as in (A.1). For $\rho =(\rho _1, \rho _2)\in \textrm {Irr}(K)$ , define a function $\varphi _\rho :M\to [0,1]$ by $\varphi _\rho :=\varphi _{0,\rho _0} \varphi _{1,\rho _1}$ , which gives a cut-off function with $\varphi _{\rho }|_{\widetilde M\setminus \widetilde V_{\rho }}\equiv 0$ . Then, we have a Fredholm operator on $L^2(W)^{(\rho )}$ as the deformation

$$ \begin{align*} \hat D_\rho= D+t\varphi_\rho^4 D_K \quad (t\gg 0). \end{align*} $$

In particular, we have the index

$$ \begin{align*} \textrm{index}(\hat D_\rho)=[M](\rho)\in \mathbb{Z}. \end{align*} $$

On the other hand, we have the sum of the deformations

$$ \begin{align*} \hat D_\rho'=(D_0+t\varphi_{0,\rho_0}^4D_{K_0})+\varepsilon_{W_0}(D_1+t\varphi_{1,\rho_1}^4D_{K_1})= D+t(\varphi_{0,\rho_0}^4D_{K_0}+\varepsilon_{W_0}\varphi_{1,\rho_1}^4D_{K_1}), \end{align*} $$

which is also Fredholm on $L^2(W)^{(\rho )}$ . In fact, by using the similar estimate in the proof of Proposition A.7, one can see that $\hat D_\rho '$ is coercive (see [Reference Anghel2] or Proposition A.7) on the outside of a compact subset containing $\varphi _{0,\rho _0}^{-1}(0)\cup \varphi _{1,\rho _1}^{-1}(0)=\varphi _\rho ^{-1}(0)$ .

Proof This follows from the fact that the deformation of D by

$$ \begin{align*} \varphi_{0,\rho_0}^4\varphi_{1,\rho_1}^{4\delta}D_{K_0}+\varepsilon_{W_0}\varphi_{0,\rho_0}^{4\delta}\varphi_{1,\rho_1}^4D_{K_1} \quad (0\leq\delta\leq 1) \end{align*} $$

gives a family of coercive operators by using the similar argument in the proof of Proposition A.7. ▪

Now, consider the Fredholm operator $D_1+t\varphi _{1,\rho _1}^4D_{K_1}$ on $L^2(W_1)^{(\rho _1)}$ and we put

$$ \begin{align*} E_{\rho_1}:=\ker(D_1+t\varphi_{1,\rho_1}^4D_{K_1})=E^+_{\rho_1}\oplus E^-_{\rho_1} \end{align*} $$

as the $\mathbb {Z}/2$ -graded finite-dimensional vector space. Then, there is a natural embedding

$$ \begin{align*} L^2(W_0\otimes E_{\rho_1})^{(\rho_0)}\to L^2(W)^{(\rho)} \end{align*} $$

whose image is preserved by $(D_0+t\varphi _{\rho _0}^4D_{K_0})\otimes \textrm {id}$ . Let $D_{\rho _0,E_{\rho _1}}$ be the restriction of $(D_0+t\varphi _{\rho _0}^4D_{K_0})\otimes \textrm {id}$ on this image, which gives a Fredholm operator on $L^2(W_0\otimes E_{\rho _1})^{(\rho _0)}$ .

Proposition 3.3 We have

$$ \begin{align*} [M](\rho)=\textrm{index}(D_{\rho_0,E_{\rho1}}). \end{align*} $$

If we write $\textrm {index}(D_0+t\varphi _{\rho _0}^4D_{K_0})=E_{\rho _0}^+-E_{\rho _0}^-$ as an element in the K-group K(pt) $\cong \mathbb {Z}$ , then we have

$$ \begin{align*} [M](\rho)=(E_{\rho_0}^+-E_{\rho_0}^-)\otimes (E_{\rho_1}^+-E_{\rho_1}^-). \end{align*} $$

Hereafter, we exhibit examples and useful formulas. These examples give local models in the computation in Section 5.

Example 3.4 (Cylinder) Let $M_1$ be the cotangent bundle of the circle $T^*S^1\cong \mathbb {R}\times S^1$ equipped with the standard symplectic structure, almost complex structure, and the natural $S^1$ -action on the $S^1$ -factor. Let $(r,\theta )$ be the coordinate on $M_1$ . Fix $\rho \in \textrm {Irr}(S^1)\cong \mathbb {Z}$ and put

$$ \begin{align*} L_\rho:=M_1\times \mathbb{C}_{(\rho)}, \end{align*} $$

where $\mathbb {C}_{(\rho )}$ is the one-dimensional Hermitian vector space with $S^1$ -action of weight $\rho $ . We take a connection $\nabla $ on $L_\rho $ defined by

$$ \begin{align*} \nabla=d-2\pi\sqrt{-1}\mu(r)dr, \end{align*} $$

where $\mu :\mathbb {R}\to \mathbb {R}$ is a smooth nondecreasing $S^1$ -invariant function such that

$$ \begin{align*} \mu(r)= \begin{cases} r+\rho \quad \left(|r|<\frac{1}{4}\right) \\ \frac{1}{2}+\rho \quad \left(|r|>\frac{3}{4}\right). \end{cases} \end{align*} $$

We take a Clifford module bundle $W_{1,\rho }$ as

$$ \begin{align*} W_{1,\rho}=\wedge^{\bullet}T_{\mathbb{C}}M_1\otimes L_\rho=(\mathbb{C}\oplus\mathbb{C})\otimes L_\rho, \end{align*} $$

with the Clifford action $c:T^*M_1\to \textrm {End}(W_{1,\rho })$ given by

$$ \begin{align*} c(dr)= \begin{pmatrix} 0 & -\sqrt{-1} \\ -\sqrt{-1} & 0 \end{pmatrix}, \quad c(d\theta)= \begin{pmatrix} 0 & {-1} \\ 1 & 0 \end{pmatrix}. \end{align*} $$

These structures give rise to a Dolbeault–Dirac operator D and an $S^1$ -acyclic orbital Dirac-type operator $(D_{1,\rho }, \{V_{1,\rho , \tau }\}_{\tau })$ with

$$ \begin{align*} V_{1,\rho, \tau}= \begin{cases} M_1\setminus (\{0\}\times S^1) \quad (\tau=\rho) \\ M_1 \quad (\tau\neq\rho), \end{cases} \end{align*} $$

and all the data satisfy the condition in Theorem 2.4. In particular, we have the resulting index as an element in $R^{-\infty }(S^1)$ . We denote it by $[M_{1,\rho }]$ . By the direct computation, one has the following.

Proposition 3.5 $[M_{1,\rho }]$ is the delta function supported at $\rho \in \textrm {Irr}(S^1)$ . Namely, we have

$$ \begin{align*} [M_{1,\rho}] : R(S^1)\to \mathbb{Z}, \quad \tau\mapsto \delta_{\rho\tau}. \end{align*} $$

Example 3.6 (Vector space) Consider $M_2=\mathbb {C}$ with the standard $S^1$ -action. Let $B_{\delta }(0)$ be the open disc centered at the origin with radius $\delta>0$ . Here, we take an $S^1$ -invariant metric on $M_2$ , so that it is standard on $B_{\frac {1}{4}}(0)$ and isometric on the outside of $B_{\frac {3}{4}}(0)$ to that on the subset $\left \{r\geq \frac {3}{4}\right \}\times S^1$ of $M_1$ . Put

$$ \begin{align*} L_\rho:=M_2\times \mathbb{C}_{(\rho)}. \end{align*} $$

We take a connection $\nabla $ on $L_\rho $ and a Clifford module bundle $W_{2,\rho }$ , so that they are standard on $B_{\frac {1}{4}}(0)$ and isomorphic to those on $\left \{r>\frac {3}{4}\right \}\times S^1\subset M_1$ in Example 3.4 under the identification between $M_2\setminus B_{\frac {3}{4}}(0)$ . These structures give rise to a Dirac operator D and an $S^1$ -acyclic orbital Dirac-type operator $(D_{2,\rho }, \{V_{2,\rho ,\tau }\}_{\tau })$ with

$$ \begin{align*} V_{2, \rho,\tau}=\mathbb{C}\setminus\{0\}, \end{align*} $$

and all the data satisfy the condition in Theorem 2.4. We denote the resulting index by $[M_{2,\rho }]$ . By the direct computation, one has the following.

Proposition 3.7 $[M_{2,\rho }]$ is the delta function supported at $\rho \in \textrm {Irr}(S^1)$ . Namely, we have

$$ \begin{align*} [M_{2,\rho}] : R(S^1)\to \mathbb{Z}, \quad \tau\mapsto \delta_{\rho\tau}. \end{align*} $$

Example 3.8 (Product of cylinders and discs)

Let $l,m$ be nonnegative integers and M the product of l copies of the cylinder $M_1$ and m copies of the disc $M_2$ in the previous examples:

$$ \begin{align*} M:=M_1\times\cdots \times M_1\times M_2\times\cdots \times M_2=(M_1)^l\times (M_2)^m. \end{align*} $$

There is the natural induced action of $K:=(S^1)^{l+m}$ on M. We use the natural identifications

$$ \begin{align*} \textrm{Irr}(K)=\left(\textrm{Irr}(S^1)\right)^{l+m}, \end{align*} $$

and

$$ \begin{align*} R(K)=R(S^1)^{\otimes (l+m)}. \end{align*} $$

Take $\rho =(\rho _1, \ldots , \rho _{l},\rho _1^{\prime } ,\ldots , \rho _{k}^{\prime })\in \textrm {Irr}(K)$ and consider the corresponding structures $(M_1, W_{1,\rho _i}, D_{1,\rho _i}, \{V_{1,\rho _i,\tau }\}_{\tau \in \textrm {Irr}(S^1)})$ and $(M_2, W_{2,\rho _j^{\prime }}, D_{2,\rho _j^{\prime }}, \{V_{2,\rho _j^{\prime }, \tau }\}_{\tau \in \textrm {Irr}(S^1)})$ . Using the outer tensor product, we can define the product of the Clifford module bundle

$$ \begin{align*} W_\rho:=W_{1,\rho_1}\boxtimes \cdots\boxtimes W_{1, \rho_l} \boxtimes W_{2, \rho_1^{\prime}} \boxtimes \cdots \boxtimes W_{2, \rho_m^{\prime}}, \end{align*} $$

which is a Clifford module bundle over M. The products

$$ \begin{align*} D_K:=D_{1,\rho_1}\boxtimes \cdots\boxtimes D_{1, \rho_l} \boxtimes D_{2, \rho_1^{\prime}} \boxtimes \cdots \boxtimes D_{2, \rho_m^{\prime}} \end{align*} $$

and

$$ \begin{align*} & V_\tau:=V_{1,\rho_1, \tau_1}\times \cdots \times V_{1,\rho_l, \tau_l}\times V_{2,\rho_1^{\prime},\tau_1^{\prime}}\times \cdots \times V_{2,\rho_m^{\prime},\tau_m^{\prime}} \quad \\ &\qquad\qquad (\tau=(\tau_1,\ldots, \tau_l,\tau^{\prime}_1,\ldots, \tau_m^{\prime})\in \textrm{Irr}(K)) \end{align*} $$

induce a K-acyclic orbital Dirac-type operator on $(M,W)$ , where for operators $A:\mathcal {H}_0\to \mathcal {H}_0$ and $B:\mathcal {H}_1\to \mathcal {H}_1$ on $\mathbb {Z}/2$ -graded Hilbert spaces, their product $A\boxtimes B :\mathcal {H}_0\otimes \mathcal {H}_1\to \mathcal {H}_0\otimes \mathcal {H}_1$ is defined by

$$ \begin{align*} A\boxtimes B:=A\otimes\textrm{id}+ \varepsilon_0 \otimes B \end{align*} $$

with the grading operator $\varepsilon _0$ of $\mathcal {H}_0$ . In fact, the data $(D_K, \{V_\tau \}_\tau )$ satisfy the conditions in Theorem 2.4, in particular, we have the resulting index $[M_\rho ]\in R^{-\infty }(K)$ . The product formula (Proposition 3.3) implies the following equality.

Proposition 3.9 We have

$$ \begin{align*} [M_\rho]=[M_{1,\rho_1}]\otimes \cdots \otimes [M_{1,\rho_l}]\otimes [M_{2,\rho_1^{\prime}}]\otimes \cdots \otimes[M_{2,\rho_m^{\prime}}]. \end{align*} $$

Namely, $[M_{\rho }]$ is the delta function supported at $\rho \in \textrm {Irr}(K)$ .

This structure serves as a local model of a neighborhood of the fiber of the moment map of symplectic toric manifold in Section 5.3.

3.2 Fiber bundle over a closed manifold

Let X be a closed Riemannian manifold, $E\to X$ a $\mathbb {Z}/2$ -graded Clifford module bundle over X, and $P\to X$ a principal G-bundle for a compact Lie group G. Consider a K-acyclic orbital Dirac-type operator $(D_K, \{V_{\rho }\}_{\rho \in \textrm {Irr}(K)})$ on $(M,W)$ as in Theorem 2.4. Suppose that $G\times K$ acts on $W\to M$ in an isometric way and $(D_K, \{V_{\rho }\}_{\rho \in \textrm {Irr}(K)})$ is G-invariant. Consider the diagonal action of G on $P\times M$ and the quotient manifold

$$ \begin{align*} \widetilde M:=(P\times M)/G, \end{align*} $$

which has a structure of M-bundle $\pi :\widetilde M\to X$ . Let $\widetilde W\to \widetilde M$ be the vector bundle defined by

$$ \begin{align*} \widetilde W:=\pi^*E\otimes\left((P\times W)/G\right), \end{align*} $$

which has a structure of a Clifford module bundle over $\widetilde M$ by using an appropriate connection of P. One can define operators $\widetilde D_W$ and $\widetilde D_E$ on $\widetilde W$ as lifts (by using a trivialization of P and a partition of unity if necessary) of Dirac operators $D_W$ on W and $D_E$ on E. Then,

$$ \begin{align*} \widetilde D:=\widetilde D_E+\widetilde D_W \end{align*} $$

is a Dirac operator on $\widetilde W$ .

For $\rho \in \textrm {Irr}(K)$ , let $\widetilde V_{\rho }$ be the open subset defined by

$$ \begin{align*} \widetilde V_{\rho}:=(P\times V_{\rho})/G \end{align*} $$

whose complement in $\widetilde M$ is compact. $D_K$ induces an operator $\widetilde D_K$ on $\widetilde W$ . One can see that $(\widetilde D_K, \{\widetilde V_{\rho }\}_{\rho \in \textrm {Irr}(K)})$ is a K-acyclic orbital Dirac-type operator on $(\widetilde M, \widetilde W)$ . In particular, we have a Fredholm operator

$$ \begin{align*} \widetilde D_{\rho}=\widetilde D+t{\widetilde\varphi_\rho}^4 \widetilde D_K \end{align*} $$

on $L^2(\widetilde W)^{(\rho )}$ , where $\widetilde \varphi _\rho :\widetilde M\to [0,1]$ is the cut-off function induced from the cut-off function $\varphi _\rho $ on M as in (A.1). In this way, we have an element $[\widetilde M]\in R^{-\infty }(K)$ defined by

$$ \begin{align*} [\widetilde M](\rho):=\textrm{index}(\widetilde D_\rho). \end{align*} $$

Now, consider the Fredholm operator $D_W+t\varphi _{\rho }^4D_{K}$ on $L^2(W)^{(\rho )}$ and we put

$$ \begin{align*} E_{\rho}:=\ker(D_W+t\varphi_{\rho}^4D_{K})=E^+_{\rho}\oplus E^-_{\rho} \end{align*} $$

as the $\mathbb {Z}/2$ -graded finite-dimensional vector space. Then, there is a natural embedding

$$ \begin{align*} L^2(E\otimes E_{\rho})\to L^2(\widetilde W)^{(\rho)} \end{align*} $$

whose image is preserved by $\widetilde D_E$ . Let $D_{E, {\rho }}$ be the restriction of $\widetilde D_E$ on this image, which gives a Fredholm operator on $L^2(E\otimes E_{\rho })$ , because the symbol of $D_{E, {\rho }}$ is equal to the tensor product of $\textrm {id}_{E_{\rho }}$ and the symbol of $D_E$ , in particular, it is an elliptic operator on the closed manifold X.

Proposition 3.10 For each $\rho \in \textrm {Irr}(K)$ , we have

$$ \begin{align*} [\widetilde M](\rho)=\textrm{index}(D_{E, \rho}). \end{align*} $$

If we write $\textrm {index}(D_E)=E_{0}^+-E_{0}^-$ as an element in the K-group K(pt) $\cong \mathbb {Z}$ , then we have

$$ \begin{align*} [\widetilde M](\rho)=(E_{0}^+-E_{0}^-)\otimes (E_{\rho}^+-E_{\rho}^-). \end{align*} $$

Example 3.11 Let K be a torus. Consider $M=T^*K$ with the K-acyclic orbital Dirac-type operator $(D_K, \{V_{\rho }\}_{\rho \in \textrm {Irr}(K)})$ defined as the product of Example 3.4. Suppose that we take a Clifford module bundle by using $\mathbb {C}_{(\rho )}$ for a fixed $\rho \in \textrm {Irr}(K)$ . Then, we have

$$ \begin{align*} [M] : R(K)\to \mathbb{Z}, \quad \rho^{\prime}\mapsto \delta_{\rho\rho^{\prime}}. \end{align*} $$

Let X be a closed Riemannian manifold, $E\to X$ a Clifford module bundle, and $P\to X$ a principal K-bundle. Let $\widetilde M$ be the M-bundle over X defined by

$$ \begin{align*} \widetilde M=(P\times M)/K. \end{align*} $$

Proposition 3.10 ensures us that

$$ \begin{align*} [\widetilde M] : R(K)\to \mathbb{Z}, \quad \rho^{\prime}\mapsto \textrm{index}(E)\delta_{\rho\rho^{\prime}}, \end{align*} $$

where $\textrm {index}(E)$ is the index of a Dirac operator on E. This example serves as a local model of a neighborhood of the inverse image of the moment map of Hamiltonian torus action in Section 5.2.

5.1 Definition: general case

Let K be a compact torus and M a symplectic manifold equipped with Hamiltonian K-action. Suppose that there exists a K-equivariant prequantizing line bundle $(L,\nabla )$ , and let $\mu :M\to \mathfrak {k}^*$ be the associated moment map. We use the Clifford module bundle $W=\wedge ^\bullet T_{\mathbb {C}} M\otimes L$ for a K-invariant compatible almost complex structure. We assume the following for the moment:

Definition 5.1 We define its quantization $\mathcal {Q}_K(M)\in R^{-\infty }(K)$ by

(5.1) $$ \begin{align} \mathcal{Q}_K(M)(\rho):=[\widetilde X_{\rho}](\rho) \in \mathbb{Z} \quad (\rho\in \textrm{Irr}(K)), \end{align} $$

where $\widetilde X_{\rho }$ is a complete manifold containing $Z_{\rho }$ as its neighborhood on which the Dirac operator along orbits defined as in Definition 2.1 gives a $\rho $ -acyclic orbital Dirac-type operator for $\widetilde X_\rho \setminus Z_{\rho }$ .

The excision property guarantees that the number $\mathcal {Q}_K(M)(\rho )$ is independent from the choice of such $\widetilde X_{\rho }$ . Theorem 2.8 enables us to describe $\mathcal {Q}_K(M)(\rho )$ into the sum of local contributions

$$ \begin{align*} \mathcal{Q}_K(M)(\rho)=[\mu^{-1}(\rho)]+\sum_{\alpha\in\textrm{Irr}(K)\setminus\{\rho\}}[Z_{\rho, \alpha}]. \end{align*} $$

It would be natural to expect the vanishing of $[Z_{\rho , \alpha }]$ . One possible way to show this vanishing is using a combination of the coincidence of $[Z_{\rho ,\alpha }]$ with the transverse index and vanishing results for it, e.g., by Paradan [Reference Paradan22]. In the subsequent subsections, instead of using them, we have the vanishing of $[Z_{\rho , \alpha }]$ for the circle action case and the toric case based on Theorem 4.2, and we define the quantization $\mathcal {Q}_K(M)$ under a weaker assumption than Assumption 5.1.

The quantization $\mathcal {Q}_K(M)$ is a generalization of K-equivariant spin $^c$ quantization using the index of Dolbeault–Dirac operator in the compact case, which is often denoted by $RR_K(M)$ and called the equivariant Riemann–Roch number or Riemann–Roch character.

5.2 [Q,R]=0 for noncompact Hamiltonian torus manifolds

In this subsection, we consider the case $K=S^1$ . Because, in this case, one has

$$ \begin{align*} Z_{\rho}=\mu^{-1}(\rho)\cup M^{K}, \end{align*} $$

for each $\rho \in \textrm {Irr}(K)=\Lambda ^*$ and $\mu (M^{K})\subset \Lambda ^*$ , the quantization $\mathcal {Q}_K(M)$ has a localization property to $\Lambda ^*$ . Moreover, one has a decomposition

$$ \begin{align*} M^K=\bigcup_{\alpha\in\Lambda^*} M^K\cap \mu^{-1}(\alpha), \end{align*} $$

which gives us a decomposition of the index

$$ \begin{align*} [Z_{\rho}]=[\mu^{-1}(\rho)]+\sum_{\alpha\in\Lambda^*\setminus\{\rho\}}[M^{K}\cap\mu^{-1}(\alpha)]\in\mathbb{Z}, \end{align*} $$

where we use the notation as in Section 2.2. Proposition 2.6 and Theorem 4.2 imply that we have

$$ \begin{align*} [M^{K}\cap\mu^{-1}(\alpha)]=0 \quad (\alpha\in\Lambda^*\setminus\{\rho\}). \end{align*} $$

This observation enables us to define $\mathcal {Q}_K(M)$ by

$$ \begin{align*} \mathcal{Q}_K(M)(\rho):=[\mu^{-1}(\rho)] \end{align*} $$

without Assumption 5.1. We only need the following assumption.

This definition leads us to a proof of [Q,R]=0, the principal of “quantization commutes with reduction,”as in [Reference Fujita, Furuta and Yoshida11] in the noncompact case.

For a regular value $\xi \in {\mathfrak {k}}^*$ of $\mu :M\to {\mathfrak {k}}^*$ , let $M_\xi $ be the symplectic quotient at $\xi $ :

$$ \begin{align*} M_\xi:=\mu^{-1}(\xi)/K, \end{align*} $$

which is a closed symplectic manifold (orbifold) under Assumption 5.2. Moreover, if a regular value $\rho $ is an element of $\textrm {Irr}(K)$ , then there exists a natural prequantizing line bundle over $M_\rho $ , and hence, one can define the Riemann–Roch number $RR(M_\rho )$ as the index of the Dolbeault–Dirac operator associated with a K-invariant compatible almost complex structure.

Theorem 5.3 Suppose that $\rho \in \textrm {Irr}(K)$ is a regular value of the moment map $\mu :M\to {\mathfrak {k}}^*$ . Then, we have

$$ \begin{align*} \mathcal{Q}_K (M)(\rho)=RR(M_\rho). \end{align*} $$

Proof A neighborhood of $\mu ^{-1}(\rho )$ in M can be identified with the product

$$ \begin{align*} (T^*K\times \mu^{-1}(\rho))/K \end{align*} $$

by the Darboux-type theorem (see [Reference Fujita, Furuta and Yoshida11, Lemma 7.1] for example), which has a structure of $T^*K$ -bundle over $M_\rho $ . By applying the product formula in Example 3.11, we have

$$ \begin{align*} [\mu^{-1}(\rho)]=RR(M_\rho). \end{align*} $$

 ▪

5.3 A Danilov-type formula for noncompact toric manifolds

Now, we focus on the symplectic toric case. Namely, we assume that K is a torus with $2\dim (K)=\dim (M)$ . In this case, Assumption 5.2 is automatically satisfied, because the preimage of each point is a single orbit. We can define the quantization $\mathcal {Q}_K(M),$ as it is noted in the previous section. In fact, for each $\rho \in \textrm {Irr}(K)$ , the image $\mu (Z_{\rho ,\alpha })$ of the component of $Z_{\rho }$ is contained in the boundary of the momentum polytope $\mu (M)$ , and one can see $[\mu ^{-1}(\rho )]=1$ and $[Z_{\rho ,\alpha }]=0$ by the same argument in [Reference Fujita8, Section 6.1] together with Proposition 2.6 and Theorem 4.2. These observations enable us to define $\mathcal {Q}_K(M)\in R^{-\infty }(K)$ and give the following description, which is a noncompact generalization of Danilov’s formula.

Theorem 5.5 Under the above setup, we have

$$ \begin{align*} \mathcal{Q}_K(M)=\sum_{\rho\in \mu(M)\cap \Lambda^*}\mathbb{C}_{(\rho)}, \end{align*} $$

where the right-hand side is an element in $R^{-\infty }(K)$ , which is characterized by

$$ \begin{align*} \textrm{Irr}(K)\ni \rho' \mapsto \begin{cases} 1 \quad (\rho'\in \mu(M)\cap \Lambda^*) \\ 0 \quad (\rho'\notin\mu(M)\cap \Lambda^*). \end{cases} \end{align*} $$

6.1 Application to quantization of Hamiltonian loop group spaces

Quantization of Hamiltonian loop group spaces is studied in various directions. In particular, Loizides and Song [Reference Loizides and Song18] studied it from the viewpoint of index theory and KK-theory. Their construction is based on their previous work [Reference Loizides, Meinrenken and Song16] with Meinrenken in which they constructed a spinor bundle over a proper Hamiltonian loop group space and a nice finite-dimensional noncompact submanifold in it, which is transverse to the orbits of the loop group action. One key ingredient in [Reference Loizides and Song18] is to associate a K-homology cycle to such a noncompact manifold. They established an index theory using the $\textrm {C}^*$ -algebraic condition, which they call the $(\Gamma , K)$ -admissibility, where K is a compact Lie group and $\Gamma $ is a countable discrete group with proper length function. They showed that in the proper Hamiltonian loop group space case, the $(\Lambda , T)$ -admissibility is satisfied for a maximal torus T of K, and the resulting K-homology class has an antisymmetric property with respect to some Weyl group action of K, which gives rise to quantization as an element in the fusion ring of K.

In this paper, we constructed a similar K-homology cycle without using $(\Gamma , K)$ -admissibility. In the subsequent research, we will investigate an approach of quantization of Hamiltonian loop group spaces by incorporating the action of the integral lattice $\Lambda $ in our construction appropriately. In such an approach, it would be interesting to understand how the localization phenomenon of our index is reflected in the quantization of loop group spaces.

There is another related work by Takata. In [Reference Takata24], an $LS^1$ -equivariant index is constructed as an element in the fusion ring from the viewpoint of KK-theory and noncommutative geometry. He also developed an index theorem in infinite-dimensional setting in [Reference Takata23, Reference Takata25]. It would be also interesting to investigate how our construction is positioned in Takata’s theory.

6.2 Deformation as KK-products

Motivated by the pioneering work by Kasparov [Reference Kasparov14], Loizides et al. showed in [Reference Loizides, Rodsphon and Song17] that the K-homology class which is obtained by Braverman’s deformation factors as a KK-product of the Dirac class and a KK-class arising from the deformation. It is desirable to understand our deformation using the acyclic orbital Dirac-type operator as a KK-product.