Witten Deformation and Divergence-Free Symmetric Killing 2-Tensors

Abstract

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for $p=2$.

1. Introduction

Let $(M,g)$ be a connected closed Riemannian n-manifold and let ${\mathrm{Sym}}^p\left(T^{*}M\right)$ denote the p-th symmetric product of the cotangent bundle $T^{*}M$. The space of smooth sections of ${\mathrm{Sym}}^p\left(T^{*}M\right)$ is denoted by

$$S^p\left(M\right)=\mathsf{\Gamma }\left({\mathrm{Sym}}^p\left(T^{*}M\right)\right).$$

Here, the symmetric product is given by

$${\alpha }_1\cdots {\alpha }_p=\sum _{\sigma \in {\mathfrak{S}}_p}{\alpha }_{\sigma \left(1\right)}\otimes \cdots \otimes {\alpha }_{\sigma \left(p\right)},$$

where ${\alpha }_1,\cdots ,{\alpha }_p\in S^1\left(M\right)={\mathsf{\Omega }}^1\left(M\right)$ and ${\mathfrak{S}}_p$ denote the symmetric groups on p letters. (There is a different convention for the symmetric product such that a factor of $1/p!$ appears on the right-hand side (cf. [1]).) For simplicity, we will often write $\alpha \beta$ instead of $\alpha ·\beta$.

Let ∇ be the Levi-Civita connection on $TM$, and its induced connection on $S^p\left(T^{*}M\right)$ is denoted by the same ∇. Then, the symmetric covariant derivative is the first-order operator

$$d^s:S^p\left(M\right)\to S^{p+1}\left(M\right):\alpha \mapsto \sum _k{e}^k{\nabla }_{e_k}\alpha ,$$

(1)

where $\left\{e_k\right\}$ is an orthonormal frame on $TM$ and its dual coframe $\left\{e^k\right\}$.

The Riemannian metric on M induces a pointwise inner product on $S^p\left(M\right)$ using the formula

$$\langle {\alpha }_1\cdots {\alpha }_p,{\beta }_1\cdots {\beta }_p\rangle =\sum _{\sigma \in {\mathfrak{S}}_p}\langle {\alpha }_1,{\beta }_{\sigma \left(1\right)}\rangle \cdots \langle {\alpha }_p,{\beta }_{\sigma \left(p\right)}\rangle ,$$

where ${\alpha }_i,{\beta }_j\in S^1\left(M\right)$. Its induced $L^2$-inner product on $S^p\left(M\right)$ is denoted by

$$(\alpha ,\beta )={\int }_M\langle \alpha ,\beta \rangle .$$

Then, the formal adjoint of $d^s$ is the divergence operator:

$$d^s*:S^{p+1}\left(M\right)\to S^p\left(M\right):\alpha \mapsto -\sum _k{e}_k⌟{\nabla }_{e_k}\alpha$$

(2)

(cf. Theorem 3.3 of [2]).

A symmetric tensor $\alpha \in S^p\left(M\right)$ is called Killing tensor if $d^s\alpha =0$, and it is divergence-free if $d^{s*}\alpha =0$. Consider the following self-adjoint second-order operator:

$$D=d^{s*}{d}^s+d^s{d}^{s*}:S^p\left(M\right)\to S^p\left(M\right).$$

(3)

It is an elliptic operator on $S^p\left(M\right)$ (see Lemma 2) such that

$${\mathcal{D}}^p(M,g):=kerD=ker{d}^s\cap ker{d}^{s*},$$

hence, ${\mathcal{D}}^p(M,g)$ is the space of divergence-free symmetric Killing p-tensors.

There is a well-known dimension formula for symmetric Killing p-tensors:

$$dimker{d}^s\le {\displaystyle \frac{1}{n}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+p}{p+1}}\right){\displaystyle \left(\genfrac{}{}{0pt}{}{n+p-1}{p}\right)}$$

(4)

(see [3,4,5,6,7]). The equality holds when M is the n-sphere $S^n$ with the standard metric. When $p=1$, Killing 1-forms are traceless and hence divergence-free (cf. Section 2 of [8]). These are dual to Killing vector fields, and (4) specializes to

$$dim{\mathcal{D}}^1(M,g)\le \left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right).$$

(5)

This is the well-known bound on the dimension of the Lie algebra of the space of infinitesimal isometries of a Riemannian n-manifold.

The aim of this paper is to find a dimension bound analogous to (4) for the space of divergence-free symmetric Killing p-tensors. Our approach is to use a Morse function f to find a deformation $D_t$ of the operator D satisfying the following properties:

(P1)

$dimker{D}_t=dimkerD$ for all t.

(P2)

$dimker{D}_t\le {\sum }_{i}dimker\overline{K}\left(x_i\right)$ for sufficiently large t, where the sum is over the critical points $\left\{x_i\right\}$ of f and $\overline{K}\left(x_i\right)$ is a model operator (see Section 3.2.2).

(P3)

$dimker\overline{K}\left(x_i\right)$ can be calculated.

For (P1), we follow Witten’s method in [9]; for (P2), we use a semiclassical analysis of model operators in [10] (see also [11]); and for (P3), we make the technical assumption that the metric g is adapted to f, that is, $g_{ij}={\delta }_{ij}$ in a Morse chart around each critical point. This assumption, however, imposes a limitation that our analysis applies only to metrics adapted to f.

Witten deformations and semiclassical analysis are extensively studied and well-understood (see, for example, [12,13,14] and references therein). However, it is generally difficult to explicitly calculate the kernel or spectrum of model operators $\overline{K}\left(x_i\right)$. Thus, for a deformation $D_t$ satisfying (P1) and (P2), the main task is to calculate $dimker\overline{K}\left(x_i\right)$.

Theorem 1.

Let $(M,g)$ be a connected closed n-manifold and f be a Morse function such that the metric g is adapted to f. Then,

$$dim{\mathcal{D}}^p(M,g)\le \sum dimker\overline{K}\left(x_i\right),$$

(6)

where the sum is over all critical points $x_i$ of f. Furthermore, for the critical point x of f with index m, the integers

$$K_m^{n,p}=dimker\overline{K}\left(x\right)$$

depend only on n, p, and m and satisfy

$$\begin{array}{c}\left(a\right)K_m^{n,p}=0\mathit{if}m\ge p\mathit{or}\mathit{if}m=n.\hfill \\ \left(b\right)K_0^{n,1}={\displaystyle \left(\genfrac{}{}{0pt}{}{n+1}{2}\right)}\left(c\right)K_0^{n,2}=\left\{\begin{array}{cc}1& \mathit{if}n=1\hfill \\ 3& \mathit{if}n=2\hfill \\ {\displaystyle \frac{1}{n}}{\displaystyle \left(\genfrac{}{}{0pt}{}{n+2}{3}\right)}{\displaystyle \left(\genfrac{}{}{0pt}{}{n+1}{2}\right)}-{\displaystyle \frac{n(n+3)}{2}}& \mathit{if}n\ge 3\hfill \end{array}\right.\hfill \end{array}$$

The inequality (6) is a consequence of (P1) and (P2), which are proven in Proposition 1(a) and Proposition 2, respectively, and the integers $K_m^{n,p}$ depend only on n, p, and m by (13). When $p=1$, this theorem implies inequality (5) under the adapted metric assumption because every connected closed manifold admits a Morse function with a unique critical point of index zero. When $p=2$, Theorem 1 shows the following:

Corollary 1.

If M admits a Morse function with a unique minimum and no critical point of index 1, then, for an adapted metric,

$$dim{\mathcal{D}}^2(M,g)\le K_0^{n,2},$$

where $K_0^{n,2}$ is given above.

A simply connected closed manifold admitting a perfect Morse function satisfies the condition of Corollary 1. Examples of such a manifold include $S^n$ ($n\ge 2$), $\mathbb{C}{\mathbb{P}}^n$, complex Grassmannian manifolds, and their products (cf. [15]).

At the moment, the authors do not know whether the adapted metric assumption can be eliminated and whether $K_m^{n,p}=0$ when $m\ne 0$.

The rest of this paper is organized as follows. Section 2 reviews basic operators on ${\mathrm{Sym}}^p\left(TM\right)$ and calculates the symbol of D. In Section 3, we define a deformation $D_t$ satisfying (P1) and find its model operators $\overline{K}\left(x\right)$ satisfying (P2). We then employ Hermite polynomials and operators, satisfying the canonical commutation relations to prove Theorem 1(a) in Section 4 and to compute $K_0^{n,1}$ and $K_0^{n,2}$ in Section 5; this provides proof for Theorem 1, hence showing (P3).

2. Preliminaries

This section reviews canonical commutation relation (7) below, fixes notations, and calculates the symbol of the operator D defined in (3).

2.1. Canonical Commutation Relations

Definition 1.

For each 1-form $\gamma \in S^1\left(M\right)={\mathsf{\Omega }}^1\left(M\right)$, define operators

$$\begin{array}{cc}\hfill c_{\gamma }:S^p\left(M\right)& \to S^{p+1}\left(M\right)\mathrm{by}{c}_{\gamma }\left(\alpha \right)=\gamma \alpha ,\hfill \\ \hfill a_{\gamma }:S^p\left(M\right)& \to S^{p-1}\left(M\right)\mathrm{by}{a}_{\gamma }\left(\alpha \right)={\gamma }^{*}⌟\alpha ,\hfill \end{array}$$

where ${\gamma }^{*}$ is the metric dual of γ, i.e., $\gamma \left(v\right)=g({\gamma }^{*},v)$.

Let $\left\{e^k\right\}$ be an orthonormal frame on $T^{*}M$. If we regard $S^p\left(M\right)$ as the space of polynomials of degree p in the variables $e^1,\cdots ,e^n$, then $c_{\gamma }$ is the multiplication by $\gamma$ and $a_{\gamma }$ is the directional derivative in the direction of $\gamma$. For instance, if $\gamma ={\sum }_k{\gamma }_k{e}^k$, $\alpha ={\left(e^1\right)}^3{e}^2$, and $\beta ={\left(e^1\right)}^2{e}^2$ then

$$\langle a_{\gamma }\alpha ,\beta \rangle =\langle 3{\gamma }_1{\left(e^1\right)}^2{e}^2+{\gamma }_2{\left(e^1\right)}^3,\beta \rangle =6{\gamma }_1=\langle \alpha ,c_{\gamma }\beta \rangle .$$

The following well-known fact plays an essential role in our discussions.

Lemma 1.

For any $\gamma ,\mu \in S^1\left(M\right)$, we have the following:

(a) 

The adjoint of $a_{\gamma }$ is $a_{\gamma }^{*}=c_{\gamma }$.

(b) 

Operators $c_{\gamma }$ and $a_{\gamma }$ satisfy the canonical commutation relations.

$$c_{\gamma }{c}_{\mu }=c_{\mu }{c}_{\gamma },a_{\gamma }{a}_{\mu }=a_{\mu }{a}_{\gamma },c_{\gamma }{a}_{\mu }-a_{\mu }{c}_{\gamma }=-\langle \gamma ,\mu \rangle Id.$$

(7)

Let ∇ denote the connection on ${\mathrm{Sym}}^p\left(T^{*}M\right)$ induced from the Levi-Civita connection. Given an orthonormal frame $\left\{e_k\right\}$ on $TM$ and its dual coframe $\left\{e^k\right\}$, we set

$${\nabla }_k={\nabla }_{e_k},c_k=c_{e^k},a_k=a_{e^k}.$$

Then, we can write the symmetric covariant derivative $d^s$ in (1) and the divergence operator $d^{s*}$ in (2) as

$$d^s=\sum _k{c}_k{\nabla }_k\mathrm{and}{d}^{s*}=-\sum _k{a}_k{\nabla }_k.$$

2.2. The Symbol Map

Let E and F be vector bundles over M. Recall that the (principal) symbol of a $k^{th}$-order liner differential operator $D:\mathsf{\Gamma }\left(E\right)\to \mathsf{\Gamma }\left(F\right)$ is the section

$${\sigma }_D\in \mathsf{\Gamma }\left(\mathrm{Hom}({\pi }^{*}E,{\pi }^{*}F)\right)(\mathrm{where}\pi :T^{*}M\to M)$$

such that for $(x,w)\in T^{*}M$, the symbol map ${\sigma }_D(x,w):E_x\to F_x$ is given by

$${\sigma }_D(x,w)\left(\xi \right)={\displaystyle \frac{1}{k!}}D\left(f^k\xi \right)\left(x\right),$$

where f is a function satisfying $f\left(x\right)=0$ and $df\left(x\right)=w$. (Some authors use a different convention in which a factor ${(-i)}^k$ appears on the right-hand side (cf. [16]).) A linear differential operator D is called elliptic if the symbol map ${\sigma }_D(x,w)$ is an isomorphism for every $(x,w)\in T^{*}M\setminus \left\{0\right\}$.

Lemma 2.

(a) 

$d^{*s}{d}^s$ is elliptic; hence, $ker{d}^s=ker{d}^{s*}{d}^s$ has a finite dimension.

(b) 

$d^{s*}{d}^s\pm d^s{d}^{s*}$ are elliptic.

Proof. 

The symbols of $d^s$, $d^{s*}$, $d^{s*}{d}^s$, and $d^s{d}^{s*}$ are given as

$$\begin{array}{cc}{\sigma }_{d^s}(x,w)=c_w,\hfill & {\sigma }_{d^{s^{*}}}(x,w)=-a_w,\hfill \\ {\sigma }_{d^{s*}{d}^s}(x,w)=-a_w{c}_w,\hfill & {\sigma }_{d^s{d}^{s*}}(x,w)=-c_w{a}_w,\hfill \end{array}$$

(8)

where $(x,w)\in T^{*}M\setminus \left\{0\right\}$. Since $c_w$ is injective and $a_w^{*}=c_w$, the symbol $-a_w{c}_w$ is also injective and hence an isomorphism. This proves (a). For (b), first note that

$$c_w{a}_w-a_w{c}_w=-{\left|w\right|}^{2}Id$$

by (7). Also, note that $c_w{a}_w+a_w{c}_w$ is an isomorphism because

$$〈(a_w{c}_w+c_w{a}_w)\alpha ,\alpha 〉=\langle c_w\alpha ,c_w\alpha \rangle +\langle a_w\alpha ,a_w\alpha \rangle$$

and $c_w$ is injective. Therefore, (b) follows. □

Remark 1.

In general, the divergence operator $d^{s*}$ has an infinite-dimensional kernel since its symbol is given by contraction $a_w$. One can find such an example in Section 3.3 of [17] in the context of symplectic Dirac operators.

Remark 2.

The elliptic operator $d^{s*}{d}^s-d^s{d}^{s*}$ on symmetric tensors, first studied by Sampson [18], is a zeroth-order perturbation of the trace Laplacian; hence, it is a symmetric analog of the Hodge Laplacian on differential forms.

3. Witten Deformation

This section applies Witten’s method in [9] to perturb the operator $D=d^{s*}{d}^s+d^s{d}^{s*}$ in (3) via a Morse function f and then uses Corollary 1.3 of [10] to obtain an upper bound for $dimkerD$ given by local data (or model operators) at the critical points of f. To obtain this upper bound explicitly, we assume the metric is adapted to f.

3.1. Perturbation

We will choose a deformation $D_t$ of D satisfying (P1) and (P2) in the Introduction as follows. We fix a smooth function f on M, and for $t\in \mathbb{R}$, we define

$$d_t:S^p\left(M\right)\to S^{p+1}\left(M\right)\mathrm{by}{d}_t=e^{-tf}{d}^s{e}^{tf}.$$

Definition 2.

For $t\in \mathbb{R}$, let $D_t=d_t^{*}{d}_t+d_{-t}{d}_{-t}^{*}$.

Since $d_t=d^s+t{c}_{df}$ and $d_t^{*}=e^{tf}{d}^{s*}{e}^{-tf}=d^{s*}+t{a}_{df}$, we have

$$\begin{array}{cc}\hfill & D_t=D+tB+t^{2}V,\mathrm{where}\hfill \\ \hfill & B=d^{s*}{c}_{df}-c_{df}{d}^{s*}+a_{df}{d}^s-d^s{a}_{df},V=a_{df}{c}_{df}+c_{df}{a}_{df}.\hfill \end{array}$$

(9)

Note that $D_t$, D, B, and V are all self-adjoint. The proposition below shows (P1) and is crucial for obtaining (P2).

Proposition 1.

(a) 

$dimker{D}_t=dimkerD$ for all $t\in \mathbb{R}$.

(b) 

B is a bundle map, i.e., its symbol is ${\sigma }_B=0$.

(c) 

$V=2a_{df}^{*}{a}_{df}+{\left|df\right|}^{2}Id$; hence, $\langle V\alpha ,\alpha \rangle \ge {\left|df\right|}^2{\left|\alpha \right|}^2$, $\forall \alpha \in S^p\left(M\right)$.

Proof. 

Define an isomorphism $L_t:S^p\left(M\right)\to S^p\left(M\right)$ by

$$L_t\left(\alpha \right)=e^{-tf}\alpha .$$

Then, for $d^s,d_t:S^p\left(M\right)\to S^{p+1}\left(M\right)$ and $d^{s*},d_{-t}^{*}:S^p\left(M\right)\to S^{p-1}\left(M\right)$,

$$L_t(ker{d}^s)=ker{d}_t\mathrm{and}{L}_t(ker{d}^{s*})=ker{d}_{-t}^{*}.$$

Now, (a) follows from

$$ker{D}_t=ker{d}_t\cap ker{d}_{-t}^{*}=L_t\left(ker{d}^s\cap ker{d}^{s*}\right)=L_t\left(kerD\right).$$

(b) follows from (8) and (7):

$${\sigma }_B(x,w)=-a_w{c}_{df}+c_{df}{a}_w+a_{df}{c}_w-c_w{a}_{df}=-\langle df,w\rangle Id+\langle w,df\rangle Id=0.$$

Lastly, (c) follows from Lemma 1. □

Because B is a bundle map, $D_t$ is a zeroth-order perturbation of D such that as $\langle D\alpha ,\alpha \rangle \ge 0$ and $\langle V\alpha ,\alpha \rangle \ge {\left|df\right|}^2{\left|\alpha \right|}^2$, elements of $ker{D}_t$ concentrate at critical points of f as $t\to \infty$ (cf. Lemma 9.2 of [19] or Proposition 2.4 of [20]). For further examples of concentration by perturbing elliptic operators, see Section 7 of [21] and Section 9 of [19,20,22].

3.2. Local Data

In this subsection, we compute model operators under the adapted metric assumption below and apply Corollary 1.3 of [10] to show (P2).

3.2.1. Adapted Metric

Suppose f is a Morse function and x is a critical point of index m. Then, by Morse Lemma, there exists local coordinates $\left\{y_k\right\}$ around x with $y_k\left(x\right)=0$ such that

$$f\left(y\right)=f\left(0\right)-{\displaystyle \frac{1}{2}}{y}_1^2-\cdots -{\displaystyle \frac{1}{2}}{y}_m^2+{\displaystyle \frac{1}{2}}{y}_{m+1}^2+\cdots +{\displaystyle \frac{1}{2}}{y}_n^2.$$

Now assume that the Riemannian metric g on M is adapted to f, that is, the metric g is given by

$$g={\left(d{y}_1\right)}^2+\cdots +{\left(d{y}_n\right)}^2.$$

(cf. [15]). Such a metric always exists and the 1-forms $e^k=d{y}_k$ form an orthonormal frame of $T^{*}M$.

Let $\alpha ={\sum }_J{\alpha }_J{e}^J$, where $J=(j_1,\cdots ,j_n)$ is a multi-index, that is, $e^J={\left(e^1\right)}^{j_1}\cdots {\left(e^n\right)}^{j_n}.$ Then, in the local coordinates $\left\{y_k\right\}$,

$$\begin{array}{cc}\hfill & d^s\alpha =\sum _J\sum _k{\displaystyle \frac{\partial {\alpha }_J}{\partial y_k}}{c}_k{e}^J,d^{s*}\alpha =-\sum _J\sum _k{\displaystyle \frac{\partial {\alpha }_J}{\partial y_k}}{a}_k{e}^J,\hfill \\ \hfill & df=-y^1{e}^1\cdots -y^m{e}^m+y^{m+1}{e}^{m+1}+\cdots +y^n{e}^n,\hfill \\ \hfill & {\nabla }_{k}df=\sum _k{s}_k{e}^k\mathrm{where}{s}_k=\left\{\begin{array}{cc}\hfill -1& k\le m,\hfill \\ \hfill 1& k>m\hfill \end{array}\right.\hfill \\ \hfill & V\left(y\right)=\sum _{k,\ell }\left(s_k{s}_{\ell }\right)y_k{y}_{\ell }(a_k{c}_{\ell }+c_k{a}_{\ell }).\hfill \end{array}$$

(10)

Observe that by Proposition 1(c), we have

$$V\left(0\right)=0\mathrm{and}\langle V\left(y\right)\alpha ,\alpha \rangle \ge {\left|y\right|}^2{\left|\alpha \right|}^2.$$

(11)

3.2.2. Model Operators

Recall that the rank of ${\mathrm{Sym}}^p\left(T^{*}M\right)$ is the dimension of the space of homogeneous polynomials of degree p on ${\mathbb{R}}^n$, which is

$$n_p:=\left({\displaystyle \genfrac{}{}{0pt}{}{n+p-1}{p}}\right).$$

(12)

In the local coordinates $\left\{y_k\right\}$ and a trivialization of ${\mathrm{Sym}}^p\left(T^{*}M\right)$ given by the local frame $\{e^k=d{y}_k\}$, the operator $D_t=D+tB+t^{2}V$ becomes an $n_p\times n_p$ matrix differential operator in ${\mathbb{R}}^n$ and D becomes a matrix differential operator of the form

$$D=\sum _{k,\ell }{\displaystyle \frac{{\partial }^2}{\partial y_k\partial y_{\ell }}}{D}_{k\ell }\left(y\right)+\mathrm{lower}\text{-}\mathrm{order}\mathrm{terms}.$$

The model operator $\overline{K}\left(x\right)$ at the critical point x is the sum of operators

$$\overline{K}\left(x\right)=\overline{D}+\overline{B}+\overline{V},$$

where operators $\overline{D}$, $\overline{B}$, and $\overline{V}$ are defined as follows:

• ${\displaystyle \overline{D}=\sum _{k,\ell }{\displaystyle \frac{{\partial }^2}{\partial y_k\partial y_{\ell }}}{D}_{k\ell }\left(0\right)}$ is the second-order homogeneous matrix differential operator obtained by taking the second-order terms of D with the coefficients frozen at the point x.
• $\overline{B}=B\left(0\right)$ is an endomorphism of the fiber ${\mathrm{Sym}}^p\left(T_x^{*}M\right)$.
• ${\displaystyle \overline{V}={\displaystyle \frac{1}{2}}\sum _{k,\ell }{\displaystyle \frac{{\partial }^{2}V}{\partial y_k\partial y_{\ell }}}\left(0\right)y_k{y}_{\ell }}$ is the quadratic part of V near x.

(See [10] and also [11].) In our case, using (7) and (10),

$$\begin{array}{ll}\hfill \overline{D}& =-\sum _{k,\ell }{\partial }_k{\partial }_{\ell }(a_k{c}_{\ell }+c_k{a}_{\ell }),\overline{B}=-\sum _k{s}_k(a_k{c}_k+c_k{a}_k),\hfill \\ \hfill \overline{V}& =\sum _{k,\ell }\left(s_k{s}_{\ell }\right)y_k{y}_{\ell }(a_k{c}_{\ell }+c_k{a}_{\ell }).\hfill \end{array}$$

(13)

The model operator $\overline{K}\left(x\right)$ is a self-adjoint second-order elliptic operator acting in $L^2\left({\mathbb{R}}^n\right)\otimes {\mathrm{Sym}}^p\left(T_x^{*}M\right)$, as a closure from the domain $C_0^{\infty }\left({\mathbb{R}}^n\right)\otimes {\mathrm{Sym}}^p\left(T_x^{*}M\right)$. Moreover, $\overline{K}\left(x\right)$ is positive semi-definite (see Remark 3 below). Observe that (13) implies the dimension $K_m^{n,p}=dimker\overline{K}\left(x\right)$ and depends only on n, p, and m, while it is independent of the critical point x.

3.2.3. Semiclassical Analysis

Let $x_1,\cdots ,x_N$ be the critical points of the Morse function f. Then, the model operator for $D_t$ on M is the elliptic operator

$$K=\underset{i=1}{\overset{N}{⨁}}\overline{K}\left(x_i\right)$$

acting in the orthogonal direct sum ${⨁}_i\left(L^2\left({\mathbb{R}}^n\right)\otimes S^p\left(T_{x_i}^{*}M\right)\right)$. Since $\langle D\alpha ,\alpha \rangle \ge 0$ and B is a bundle map, (11) implies that as $t\to \infty$, the eigenvalues (multiplicities counted) of the operator $t^{-1}{D}_t$ concentrate near the eigenvalues of the model operator K (see Proposition 1.2 of [10]). As a corollary, we obtain (P2) in the Introduction as follows.

Proposition 2

(Corollary 1.3 of [10]). There exists $t_0>0$ such that for any $t>t_0$, we have

$$dimker{D}_t=dimker{t}^{-1}{D}_t\le dimkerK=\sum _{i}dimker\overline{K}\left(x_i\right).$$

4. Hermite Polynomials and Operator Formalism

Throughout the remaining sections, let x denote a critical point of the Morse function f with Morse index m as in Section 3.2.1. For notational simplicity, we set

$$\overline{K}=\overline{K}\left(x\right)\mathrm{and}{S}^p={\mathrm{Sym}}^p\left(T_x^{*}M\right).$$

As the model operator $\overline{K}$ acts in $L^2\left({\mathbb{R}}^n\right)\otimes S^p$, Hermite functions on ${\mathbb{R}}^n$ are useful for calculating $ker\overline{K}$ because they constitute an orthogonal basis for $L^2\left({\mathbb{R}}^n\right)$ (cf. [23]). However, we will use Hermite polynomials on ${\mathbb{R}}^n$ and relevant weighted operators for computational convenience. In this section, we write weighted operators in terms of operators convenient for handling Hermite polynomials and use this operator formalism to prove Theorem 1(a).

Recall that for a multi-index $I=(i_1,\cdots ,i_n)$,

$$I!=i_1!\cdots i_n!\mathrm{and}\left|I\right|=\sum i_k.$$

Also recall that $\left\{e^J\right|e^k=d{y}_k,\left|J\right|=p\}$ is an orthogonal basis for $S^p$. Let $1_k$ denote the multi-index with i-th component ${\delta }_{ik}$. Note that $I=\sum i_k{1}_k$.

4.1. Hermite Polynomials on ${\mathbb{R}}^n$

The classical Hermite polynomials on $\mathbb{R}$ are given by

$$H_k\left(t\right)={(-1)}^k{e}^{t^2}{\displaystyle \frac{d^k}{d{t}^k}}{e}^{-t^2}.$$

Here, the degree of $H_k\left(t\right)$ is k. These Hermite polynomials on $\mathbb{R}$ satisfy

$$H_k^{\prime }\left(t\right)=2k{H}_{k-1}\mathrm{and}t{H}_k\left(t\right)=k{H}_{k-1}\left(t\right)+{\displaystyle \frac{1}{2}}{H}_{k+1}.$$

(14)

Definition 3.

(a) 

The Hermite polynomials on ${\mathbb{R}}^n$ are products of Hermite polynomials on $\mathbb{R}$, that is,

$$H_I\left(y\right)=H_{i_1}\left(y_1\right)\cdots H_{i_n}\left(y_n\right),$$

where $I\in {\mathbb{Z}}^n$. If some $i_k<0$, we set $H_I=0$.

(b) 

Denote the space generated by degree q Hermite polynomials by

$${\mathcal{H}}^q=\mathrm{span}\{H_I:\left|I\right|=q\}.$$

Below are well-known properties of the Hermite polynomials $H_I$ that are frequently used for our subsequent discussions.

• For $d\mu =e^{-{\left|y\right|}^2}dy$, $\left\{H_I\right\}$ is an orthogonal basis for $L^2({\mathbb{R}}^n,d\mu )$; hence, the vectors $H_I{e}^J$ form an orthogonal basis for $L^2({\mathbb{R}}^n,d\mu )\otimes S^p$ such that

  $$\left(H_I{e}^J,H_{I^{\prime }}{e}^{J^{\prime }}\right)={\int }_{{\mathbb{R}}^n}{H}_I{H}_{I^{\prime }}d\mu \langle e^J,e^{J^{\prime }}\rangle ={\delta }_{I,I^{\prime }}{\delta }_{J,J^{\prime }}{\left(\sqrt{\pi }\right)}^n{2}^{\left|I\right|}I!J!.$$

  (15)
• The dimension of the space ${\mathcal{H}}^q$ is the number of multi-indices I with $\left|I\right|=q$, so $dim{\mathcal{H}}^q=n_q$, where $n_q$ is given in (12).
• Using (14), the Hermite polynomials $H_I$ satisfy

  $${\partial }_k{H}_I=2i_k{H}_{I-1_k}\mathrm{and}{y}_k{H}_I=i_k{H}_{I-1_k}+{\displaystyle \frac{1}{2}}{H}_{I+1_k}.$$

  (16)

4.2. Weighted Operators

Noting that $\varphi \in L^2({\mathbb{R}}^n,d\mu )$ if and only if $e^{-{\displaystyle \frac{1}{2}}{\left|y\right|}^2}\varphi \in L^2\left({\mathbb{R}}^n\right)$, we define

Definition 4.

For the exponential weight function $w=e^{-{\displaystyle \frac{1}{2}}{\left|y\right|}^2}$, let

$${\overline{K}}_w=w^{-1}\overline{K}w,\mathit{that}\mathit{is},K_w\left(\beta \right)=w^{-1}\overline{K}\left(w\beta \right).$$

The weighted operator ${\overline{K}}_w$ acts in $L^2({\mathbb{R}}^n,d\mu )\otimes S^p$ such that

$$dimker{\overline{K}}_w=dimker\overline{K}\mathrm{and}{\overline{K}}_w=w^{-1}\overline{D}w+\overline{B}+\overline{V},$$

where by a simple calculation,

$$w^{-1}\overline{D}w=-\sum _{k,\ell }(-{\delta }_{k\ell }+y_k{y}_{\ell }-y_k{\partial }_{\ell }-y_{\ell }{\partial }_k+{\partial }_k{\partial }_{\ell })(a_k{c}_{\ell }+c_k{a}_{\ell }).$$

(17)

4.3. Canonical Commutation Relations

Without further confusion, we will often regard $y_k$ as an operator:

$$y_k:g\left(y\right)\mapsto y_{k}g\left(y\right).$$

Noting (16), we obtain handy operators as follows.

Definition 5.

For $k=1,\cdots ,n$, set

$$C_k=2y_k-{\partial }_k\mathit{and}{A}_k={\displaystyle \frac{1}{2}}{\partial }_k.$$

$C_k$ and $A_k$ are analogous to $c_k$ and $a_k$ in the following sense. Recall that

$$\begin{array}{cc}\hfill c_k:S^p& \to S^{p+1}:e^J\mapsto e^{J+1_k},\hfill \\ \hfill a_k:S^p& \to S^{p-1}:e^J\mapsto j_k{e}^{J-1_k},\hfill \end{array}$$

(18)

Analogously, when restricted to ${\mathcal{H}}^q$, using (16), $C_k$ and $A_k$ are given as

$$\begin{array}{cc}\hfill & C_k:{\mathcal{H}}^q\to {\mathcal{H}}^{q+1}:H_I\mapsto H_{I+1_k},\hfill \\ \hfill & A_k:{\mathcal{H}}^q\to {\mathcal{H}}^{q-1}:H_I\mapsto i_k{H}_{I-1_k}.\hfill \end{array}$$

(19)

Furthermore, one has

Lemma 3.

(a) 

$A_k^{*}={\displaystyle \frac{1}{2}}{C}_k$ with respect to the inner product of $L^2({\mathbb{R}}^n,d\mu )$.

(b) 

The operators $C_k$ and $A_k$ satisfy the canonical commutation relations:

$$C_k{C}_{\ell }=C_{\ell }{C}_k,A_k{A}_{\ell }=A_{\ell }{A}_k,C_k{A}_{\ell }-A_{\ell }{C}_k=-{\delta }_{k\ell }I.$$

(20)

The proof is immediate from definitions.

4.4. Operator Formalism

Definition 6.

(a) 

Recalling $s_k$ in (10) is defined by the Morse index m, we set

$${\left(Cc\right)}^{-}=\sum _{k\le m}{C}_k{c}_k,{\left(Cc\right)}^{+}=\sum _{m<\ell }{C}_{\ell }{c}_{\ell },{\left(Ca\right)}^{-}=\sum _{k\le m}{C}_k{a}_k,$$

and likewise for ${\left(Ca\right)}^{+},{\left(Ac\right)}^{-},{\left(Ac\right)}^{+},{\left(Aa\right)}^{-},and{\left(Aa\right)}^{+}$.

(b) 

Let $P={\left(Ca\right)}^{-}-2{\left(Aa\right)}^{+}$ and $Q={\left(Cc\right)}^{-}-2{\left(Ac\right)}^{+}$.

The proposition below is our key fact for analyzing $ker{\overline{K}}_w$. In particular, it shows ${\overline{K}}_w$ is self-adjoint and positive semi-definite. Observe that as $A_k^{*}={\displaystyle \frac{1}{2}}{C}_k$ and $a_k^{*}=c_k$, we have

$$P^{*}=2{\left(Ac\right)}^{-}-{\left(Cc\right)}^{+}\mathrm{and}{Q}^{*}=2{\left(Aa\right)}^{-}-{\left(Ca\right)}^{+}.$$

Proposition 3.

${\overline{K}}_w=P^{*}P+Q^{*}Q$ and

$$Q^{*}Q-Q{Q}^{*}=2\sum _k{C}_k{A}_k+2\sum _{k\le m}{c}_k{a}_k-2\sum _{m<\ell }{c}_{\ell }{a}_{\ell }+2m.$$

(21)

Proof. 

Since ${\partial }_k=2A_k$ and $y_k=A_k+{\displaystyle \frac{1}{2}}{C}_k$, using (20) and (17), one has

$$\begin{array}{cc}\hfill {\overline{K}}_w& =\sum _{k,\ell \le m}(C_k{A}_{\ell }+C_{\ell }{A}_k)(a_k{c}_{\ell }+c_k{a}_{\ell })+\sum _{m<k,\ell }(C_k{A}_{\ell }+C_{\ell }{A}_k)(a_k{c}_{\ell }+c_k{a}_{\ell })\hfill \\ \hfill & +2\sum _{k\le m}(a_k{c}_k+c_k{a}_k)-2\sum _{k\le m<\ell }(4A_k{A}_{\ell }+C_k{C}_{\ell })(a_k{c}_{\ell }+c_k{a}_{\ell }).\hfill \end{array}$$

(22)

Since $C_{\ell }{A}_k(a_k{c}_{\ell }+c_k{a}_{\ell })=C_{\ell }{A}_k(c_{\ell }{a}_k+a_{\ell }{c}_k)$ by (7), the Equation (22) becomes

$$\begin{array}{cc}\hfill {\overline{K}}_w& =2{\left(Ca\right)}^{-}{\left(Ac\right)}^{-}+2{\left(Cc\right)}^{-}{\left(Aa\right)}^{-}+2{\left(Ca\right)}^{+}{\left(Ac\right)}^{+}+2{\left(Cc\right)}^{+}{\left(Aa\right)}^{+}\hfill \\ \hfill & -4{\left(Aa\right)}^{-}{\left(Ac\right)}^{+}-{\left(Cc\right)}^{-}{\left(Ca\right)}^{+}-4{\left(Ac\right)}^{-}{\left(Aa\right)}^{+}-{\left(Ca\right)}^{-}{\left(Cc\right)}^{+}\hfill \\ \hfill & +2\sum _{k\le m}(a_k{c}_k+c_k{a}_k).\hfill \end{array}$$

(23)

On the other hand, by (7) and (20), one obtains

$$\begin{array}{cc}\hfill {\left(Ca\right)}^{-}{\left(Ac\right)}^{-}& ={\left(Ac\right)}^{-}{\left(Ca\right)}^{-}+\sum _{k\le m}{A}_k{C}_k-\sum _{k\le m}{c}_k{a}_k-m,\hfill \\ \hfill {\left(Cc\right)}^{-}{\left(Aa\right)}^{-}& ={\left(Aa\right)}^{-}{\left(Cc\right)}^{-}-\sum _{k\le m}{A}_k{C}_k-\sum _{k\le m}{a}_k{c}_k+m,\hfill \\ \hfill {\left(Ca\right)}^{-}{\left(Cc\right)}^{+}& ={\left(Cc\right)}^{+}{\left(Ca\right)}^{-},{\left(Cc\right)}^{-}{\left(Ca\right)}^{+}={\left(Ca\right)}^{+}{\left(Cc\right)}^{-}.\hfill \end{array}$$

(24)

Plugging (24) into Equation (23) and then factoring gives the first assertion. For the second assertion, apply (7) and (20) to obtain

$$2{\left(Ca\right)}^{+}{\left(Ac\right)}^{+}=2{\left(Ac\right)}^{+}{\left(Ca\right)}^{+}+2\sum _{m<\ell }{A}_{\ell }{C}_{\ell }-2\sum _{m<\ell }{c}_{\ell }{a}_{\ell }.$$

This and the second equation of (24) then yield the second assertion. □

Remark 3.

The weighted operator ${\overline{K}}_w=w^{-1}\overline{K}w$ is positive semi-definite with respect to the inner product on $L^2({\mathbb{R}}^n,d\mu )\otimes S^p$, so the model operator $\overline{K}$ is positive semi-definite with respect to the inner product on $L^2\left({\mathbb{R}}^n\right)\otimes S^p$.

As a corollary to Proposition 3, we have

Corollary 2.

If $m=n$, then $ker{\overline{K}}_w=0$.

Proof. 

Let $m=n$. Then, ${\left(Aa\right)}^{+}={\left(Ac\right)}^{+}=0$; hence, we have

$${\overline{K}}_w={\left(Ca\right)}^{*}Ca+{\left(Cc\right)}^{*}Cc,$$

where $Ca={\sum }_k{C}_k{a}_k$ and $Cc={\sum }_k{C}_k{c}_k$. As $Cc$ is injective, we conclude that ${\overline{K}}_w$ is positive definite, hence $ker{\overline{K}}_w=0$. □

4.5. Proof of Theorem 1(a)

By Corollary 2 and $dimker\overline{K}=dimker{\overline{K}}_w$, the proposition below completes the proof of Theorem 1(a).

Proposition 4.

If $m\ge p$, then $ker{\overline{K}}_w=0$.

Proof. 

Suppose $\beta \in ker{\overline{K}}_w$. Then, by Proposition 3, $Q^{*}Q\left(\beta \right)=0$. Noting that $Q{Q}^{*}$ is positive semi-definite, using (21), we obtain

$$\begin{array}{cc}\hfill 0& \ge \left(\left(\sum _k{C}_k{A}_k+\sum _{k\le m}{c}_k{a}_k-\sum _{m<\ell }{c}_{\ell }{a}_{\ell }+m\right)\beta ,\beta \right)\hfill \\ \hfill & =2\sum _k(A_k\beta ,A_k\beta )+\sum _{k\le m}(a_k\beta ,a_k\beta )-\sum _{m<\ell }(a_{\ell }\beta ,a_{\ell }{\beta )+m|\left|\beta \right||}^2.\hfill \end{array}$$

(25)

Since $\beta$ can be written as $\beta ={\sum }_{q,J}{\beta }_J^q{e}^J$, where ${\beta }_J^q{e}^J\in {\mathcal{H}}^q\otimes S^p$, by (15), (18), and (19), the inequality (25) implies that

$$0\ge \sum _{q,J}\left(q+\sum _{k\le m}{j}_k-\sum _{m<\ell }{j}_{\ell }+m\right)\left|\right|{\beta }_J^q{e}^J{\left|\right|}^2.$$

As $\sum j_{\ell }\le \left|J\right|=p\le m$, it follows that

$$0\ge \sum _{q,J}\left(q+\sum _{k\le m}{j}_k\right)\left|\right|{\beta }_J^q{e}^J{\left|\right|}^2.$$

This inequality implies that ${\beta }_J^q=0$ for all $q\ne 0$; hence, $\beta ={\sum }_J{\beta }_J^0{e}^J\in {\mathcal{H}}^0\otimes S^p$ such that

$$0=Q\left(\beta \right)={\left(Cc\right)}^{-}\beta -2{\left(Ac\right)}^{+}\beta ={\left(Cc\right)}^{-}\beta .$$

Since ${\left(Cc\right)}^{-}$ is injective on ${\mathcal{H}}^0\otimes S^p$ (as $m\ne 0$), we conclude that $\beta =0$. □

5. Dimension Count

This section proves Theorem 1(b) and (c) in the Introduction.

5.1. Weighted Operator with Morse Index Zero

Let ${\overline{K}}_w$ be the weighted operator as in Section 4. From now on, we assume that the Morse index is $m=0$. In this case, as ${\left(Ca\right)}^{-}={\left(Cc\right)}^{-}=0$, using Proposition 3,

$${\overline{K}}_w={\left(2Aa\right)}^{*}2Aa+{\left(2Ac\right)}^{*}2Ac,$$

where $Aa={\sum }_k{A}_k{a}_k$ and $Ac={\sum }_k{A}_k{c}_k$. Consider

and observe that

(a)

${\overline{K}}_w({\mathcal{H}}^q\otimes S^p)\subset {\mathcal{H}}^q\otimes S^p$.

(b)

If $\beta ={\sum }_q{\beta }_q\in ker{\overline{K}}_w$, where ${\beta }_q\in {\mathcal{H}}^q\otimes S^p$, then by (a),

$${\beta }_q\in ker{\overline{K}}_w,\forall q\ge 0.$$

(c)

$ker{\overline{K}}_w=ker\left(Aa\right)\cap ker\left(Ac\right)$. In particular, ${\mathcal{H}}^0\otimes S^p\subset ker{\overline{K}}_w$.

(d)

Using (18) and (19), it follows from (21) that

$${(CaAc-AcCa)|}_{{\mathcal{H}}^q\otimes S^p}=(q-p)Id.$$

(26)

(e)

Since both $CaAc$ and $AcCa$ are positive semi-definite, (26) shows

$${q>p\Rightarrow kerAc|}_{{\mathcal{H}}^q\otimes S^p}{=0\mathrm{and}q<p\Rightarrow kerCa|}_{{\mathcal{H}}^q\otimes S^p}=0.$$

(27)

With this understood, we provide the following definition:

Definition 7.

Let ${\overline{K}}_w$ be as above with $m=0$ and define

$${\mathbb{K}}^{q,p}=ker\left(Aa{|}_{{\mathcal{H}}^q\otimes S^p}\right)\cap ker\left(Ac{|}_{{\mathcal{H}}^q\otimes S^p}\right).$$

One may then use the above facts (a)–(e) to conclude the following:

Lemma 4.

Let ${\overline{K}}_w$ be as above with $m=0$. Then,

$$ker{\overline{K}}_w=\underset{q\le p}{⨁}{\mathbb{K}}^{q,p}\mathit{and}{\mathbb{K}}^{0,p}={\mathcal{H}}^0\otimes S^p.$$

5.2. Proof of Theorem 1(b)

Let $p=1$ and $m=0$. In this case, by Lemma 4,

$$K_0^{n,1}=dimker\overline{K}=dimker{\overline{K}}_w=dim{\mathcal{H}}^0\otimes S^1+dim{\mathbb{K}}^{1,1}.$$

Since $dim{\mathcal{H}}^0\otimes S^1=n$, Theorem 1(b) follows from the following lemma.

Lemma 5.

${\displaystyle dim{\mathbb{K}}^{1,1}=\left(\genfrac{}{}{0pt}{}{n}{2}\right)}$.

Proof. 

Write $\beta \in {\mathcal{H}}^1\otimes S^1$ as $\beta ={\sum }_{k,\ell }{\beta }_{\ell }^k{H}_k{e}^{\ell }$. Then,

$$Ac\left(\beta \right)=\sum _{k,\ell }{\beta }_{\ell }^k{H}_0{e}^k{e}^{\ell }=0⟺{\beta }_{\ell }^k+{\beta }_k^{\ell }=0,\forall k,\ell .$$

This implies that ${kerAc|}_{{\mathcal{H}}^1\otimes S^1}$ is spanned by the vectors $H_k{e}^{\ell }-H_{\ell }{e}^k$. The Lemma now follows since $Aa(H_k{e}^{\ell }-H_{\ell }{e}^k)=0$ for all $k\ne \ell$. □.

5.3. Proof of Theorem 1(c)

For $p=2$ and $m=0$, by Lemma 4,

$$K_0^{n,2}=dimker\overline{K}=dimker{\overline{K}}_w=dim{\mathcal{H}}^0\otimes S^2+dim{\mathbb{K}}^{1,2}+dim{\mathbb{K}}^{2,2}.$$

Since ${\displaystyle dim{\mathcal{H}}^0\otimes S^2=\left(\genfrac{}{}{0pt}{}{n+1}{2}\right)}$, Theorem 1(c) follows from Lemmas 6 and 7 below.

Lemma 6.

${\mathbb{K}}^{1,2}=0$ for $n=1$, and if $n\ge 2$, we have

$$dim{\mathbb{K}}^{1,2}=n\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{3}}\right)-n.$$

Proof. 

Since the map $Ca:{\mathcal{H}}^0\otimes S^3\to {\mathcal{H}}^1\otimes S^2$ is injective by (27), its adjoint map is surjective. Hence,

$${dimkerAc|}_{{\mathcal{H}}^1\otimes S^2}=n\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{3}}\right).$$

(28)

If $n=1$, ${kerAc|}_{{\mathcal{H}}^1\otimes S^2}$ is trivial by (28), and hence, so is ${\mathbb{K}}^{1,2}$. For $n\ge 2$, by dimension Formula (28), it suffices to show that the restriction map ${Aa:kerAc|}_{{\mathcal{H}}^1\otimes S^2}\to {\mathcal{H}}^0\otimes S^1$ is surjective. Let $u_{k\ell }=H_k{e}^k{e}^{\ell }-H_{\ell }{e}^k{e}^k$. Then,

$$Ac\left(u_{k\ell }\right)=0\mathrm{and}Aa\left(u_{k\ell }\right)=e^{\ell }.$$

This shows that the restriction map ${Aa:kerAc|}_{{\mathcal{H}}^1\otimes S^2}\to {\mathcal{H}}^0\otimes S^1$ is surjective. □

Lemma 7.

${\mathbb{K}}^{2,2}=0$ for $n=1,2$, and if $n\ge 3$, we have

$$dim{\mathbb{K}}^{2,2}=\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)-n\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{3}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right).$$

(29)

Proof. 

For notational simplicity, let ${Ac=Ac|}_{{\mathcal{H}}^2\otimes S^2}$. Since the map

$$Ca:{\mathcal{H}}^1\otimes S^3\to {\mathcal{H}}^2\otimes S^2$$

is injective by (27), its adjoint $Ac$ is surjective; hence, we have

$$dimkerAc=\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)-n\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{3}}\right).$$

This shows ${\mathbb{K}}^{2,2}=0$ when $n=1$. Let $n\ge 2$. One may choose, by inspection, linearly independent vectors in $kerAc$ as follows. For distinct $i,j,k,\ell$, let

$$\begin{array}{cc}\hfill u_{k\ell }& =2H_{k\ell }{e}^{k\ell }-H_{\ell \ell }{e}^{kk}-H_{kk}{e}^{\ell \ell }(n\ge 2),\hfill \\ \hfill v_{k\ell j}& =H_{kk}{e}^{\ell j}-H_{k\ell }{e}^{kj}-H_{kj}{e}^{k\ell }+H_{\ell j}{e}^{kk}(n\ge 3),\hfill \\ \hfill w_{ijk\ell }& =H_{ij}{e}^{k\ell }-2H_{ik}{e}^{j\ell }+H_{i\ell }{e}^{jk}+H_{jk}{e}^{i\ell }-2H_{j\ell }{e}^{ik}+H_{k\ell }{e}^{ij}(n\ge 4).\hfill \end{array}$$

Here, $H_{k\ell }=H_{(k,\ell )}$ and $e^{k\ell }=e^{(k,\ell )}$. These vectors form a basis for $kerAc$ because they span a subspace of dimension:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)+3\left({\displaystyle \genfrac{}{}{0pt}{}{n}{3}}\right)+2\left({\displaystyle \genfrac{}{}{0pt}{}{n}{4}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)-n\left({\displaystyle \genfrac{}{}{0pt}{}{n+2}{3}}\right)=dimkerAc.$$

Observing that

$$Aa\left(u_{k\ell }\right)=2H_{\ell }{e}^{\ell }+2H_k{e}^k,Aa\left(v_{k\ell j}\right)=-H_{\ell }{e}^j-H_j{e}^{\ell },Aa\left(w_{ijk\ell }\right)=0,$$

we define a subspace of ${\mathcal{H}}^1\otimes S^1$ by

$$S=\langle H_k{e}^k+H_{\ell }{e}^{\ell }|k\ne \ell \rangle \oplus \langle H_k{e}^{\ell }+H_{\ell }{e}^k|k\ne \ell \rangle \subset {\mathcal{H}}^1\otimes S^1.$$

Obviously, $Aa(kerAc)\subset S$ and the restriction map $Aa:kerAc\to S$ is injective for $n=2$ and surjective for $n\ge 3$. Therefore, the Lemma follows from the counting dimension of S: for $n\ge 3$,

$$dim\langle H_k{e}^k+H_{\ell }{e}^{\ell }|k\ne \ell \rangle =n\mathrm{and}dim\langle H_k{e}^{\ell }+H_{\ell }{e}^k|k\ne \ell \rangle =\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right),$$

where the first equality follows since $Aa(u_{k\ell }+u_{kj}-u_{\ell j})=2H_k{e}^k$. □