2.1. Preliminaries on the Witten’s perturbations

2.1.1. Basic notation

Let $M\equiv (M,g)$ be a closed Riemannian n-manifold. For any smooth Euclidean/Hermitian vector bundle E over M, let $C^m(M;E)$ , $C^\infty (M;E)$ , $L^2(M;E)$ , $L^\infty (M;E)$ and $H^m(M;E)$ denote the spaces of distributional sections that are $C^m$ , $C^\infty $ , $L^2$ , $L^\infty $ and of Sobolev order m, respectively; as usual, E is removed from this notation if it is the trivial line bundle. Consider the induced scalar product $\langle \ ,\ \rangle $ and norm $\|\ \|$ on $L^2(M;E)$ , and the induced norm $\|\ \|_{L^\infty }$ on $L^\infty (M;E)$ . Fix also norms, $\|\ \|_m$ on every $H^m(M;E)$ and $\|\ \|_{C^m}$ on $C^m(M;E)$ . If P is the orthogonal projection of $L^2(M;E)$ to some closed subspace V, then $P^\perp $ denotes the orthogonal projection to $V^\perp $ . Let $o(E)$ denote the flat real orientation line bundle of E. It is said that E is orientable when $o(E)$ is trivial. In this case, an orientation of E is described by a (necessarily smooth) nonvanishing flat section $\mathcal {O}_E$ of $o(E)$ ; for simplicity, it will be said that $\mathcal {O}_E$ itself is an orientation. In particular, an orientation of M is described using $o(M):=o(TM)$ . The flat line bundle $o(E)\otimes o(E)$ is always trivial.

Let $T_{\mathbb {C}} M=TM\otimes \mathbb {C}$ and $T_{\mathbb {C}}^*M=T^*M\otimes \mathbb {C}$ . The exterior bundle with coefficients in $\mathbb {K}=\mathbb {R},\mathbb {C}$ is denoted by $\Lambda _{\mathbb {K}}=\Lambda _{\mathbb {K}} M$ , and let $\Omega (M,\mathbb {K})=C^\infty (M;\Lambda _{\mathbb {K}})$ ; in particular, $C^\infty (M,\mathbb {K})=\Omega ^0(M,\mathbb {K})$ . The Levi–Civita connection is denoted by $\nabla =\nabla ^M$ . As usual, d and $\delta $ denote the de Rham derivative and coderivative, and let $D=d+\delta $ and $\Delta =D^2=d\delta +\delta d$ (the Laplacian). Let $Z(M,\mathbb {K})$ and $B(M,\mathbb {K})$ denote the kernel and image of d in $\Omega (M,\mathbb {K})$ . Thus, $H^{\bullet }(M,\mathbb {K})=Z(M,\mathbb {K})/B(M,\mathbb {K})$ is the de Rham cohomology with coefficients in $\mathbb {K}$ . We typically consider complex coefficients, so we will omit $\mathbb {K}$ from all of the above notation just when $\mathbb {K}=\mathbb {C}$ . Take $\|\ \|_m$ and $\|\ \|_{C^m}$ given on $\Omega (M)$ by

$$\begin{align*}\|\alpha\|_m=\sum_{k=0}^m\|D^k\alpha\|\;,\quad\|\alpha\|_{C^m}=\sum_{k=0}^m\|\nabla^k\alpha\|_{L^\infty}. \end{align*}$$

In particular, we take $\|\ \|=\|\ \|_0$ and $\|\ \|_{C^0}=\|\ \|_{L^\infty }|_{C^0(M;E)}$ .

On any graded vector space $V^{\bullet }$ , let $\mathsf {w}$ and ${\mathsf {N}}$ be the degree involution and number operator; that is, $\mathsf {w}=(-1)^k$ and ${\mathsf {N}}=k$ on $V^k$ . For any homogeneous linear operator between graded vector spaces, $T:V^{\bullet }\to W^{\bullet }$ , the notation $T_k$ means its precomposition with the canonical projection of $V^{\bullet }$ to $V^k$ . If T is of degree l ( $T(V^k)\subset W^{k+l}$ for all k), then

(2.1) $$ \begin{align} \mathsf{w} T=(-1)^lT\mathsf{w}\;,\quad {\mathsf{N}} T=T({\mathsf{N}}+l). \end{align} $$

For any $\eta \in \Omega ^1(M,\mathbb {R})$ with $\eta ^\sharp =X\in \mathfrak {X}(M):=C^\infty (M;TM)$ ( $\eta =g(X,{\cdot })$ ), let $\mathcal {L}_X$ and $\iota _X$ denote the Lie derivative and interior product with respect to X, and let ${\eta \lrcorner }=-(\eta \wedge )^*=-\iota _X$ . Using the identity $\operatorname {Cl}(T^*M)\equiv \Lambda _{\mathbb {R}} M$ defined by the symbol of filtered algebras, the left Clifford multiplication by $\eta $ is $c(\eta )={\eta \wedge }+{\eta \lrcorner }$ , and the composition of $\mathsf {w}$ with the right Clifford multiplication by $\eta $ is $\hat c(\eta )={\eta \wedge }-{\eta \lrcorner }$ ; in particular, $c(\eta )^*=-c(\eta )$ and $\hat c(\eta )^*=\hat c(\eta )$ . Recall that, for any $h\in C^\infty (M,\mathbb {R})$ ,

(2.2) $$ \begin{align} [D,h]=\hat c(dh). \end{align} $$

In the whole paper, unless otherwise indicated, we will use the following notation without further comment. We use constants $C,c>0$ without even mentioning their existence, and their precise values may change from line to line. We may add subindices or primes to these constants if needed. We also use a complex parameter $z=\mu +i\nu \in \mathbb {C}$ ( $\mu ,\nu \in \mathbb {R}$ and $i=\sqrt {-1}$ ). Recall that $\partial _z=(\partial _\mu -i\partial _\nu )/2$ and $\partial _{\bar z}=(\partial _\mu +i\partial _\nu )/2$ .

2.1.2. Perturbations defined by a closed real 1-form

For any $\omega \in Z^1(M)$ , we have the Witten’s type perturbations $d_\omega $ , $\delta _\omega $ , $D_\omega $ and $\Delta _\omega $ of d, $\delta $ , D and $\Delta $ . Given $\eta \in Z^1(M,\mathbb {R})$ and $z\in \mathbb {C}$ , we write $d_z=d_{z\eta }$ , $\delta _z=\delta _{z\eta} $ , $D_z=D_{z\eta }$ and $\Delta _z=\Delta _{z\eta }$ . These operators have the following expressions:

(2.3) $$ \begin{align} \left. \begin{aligned} d_z&=d+z\,\eta\wedge\;,\quad \delta_z=d_z^*=\delta-\bar z\,{\eta\lrcorner}\;,\\ D_z&=d_z+\delta_z=D+\mu\hat c(\eta)+i\nu c(\eta) =D_{i\nu}+\mu\hat c(\eta)\;,\\ \Delta_z&=D_z^2=d_z\delta_z+\delta_zd_z =\Delta+\mu\mathsf{H}_\eta+i\nu\mathsf{J}_\eta+|z|^2|\eta|^2\\ &=\Delta_{i\nu}+\mu\mathsf{H}_\eta+\mu^2|\eta|^2\;, \end{aligned} \right\} \end{align} $$

where, for $X=\eta ^\sharp $ ,

$$\begin{align*}\mathsf{H}_\eta=D\hat c(\eta)+\hat c(\eta)D=\mathcal{L}_X^*+\mathcal{L}_X\;,\quad \mathsf{J}_\eta=Dc(\eta)+c(\eta)D=\mathcal{L}_X^*-\mathcal{L}_X. \end{align*}$$

Note that $\mathsf {H}_\eta $ is of order zero and $\mathsf {J}_\eta $ of order one.

As families of operators, $d_z$ and $\delta _z$ are holomorphic and antiholomorphic functions of z, respectively. More precisely, it follows from Equation (2.3) that

(2.4) $$ \begin{align} \left. \begin{alignedat}{3} \partial_zd_z&={\eta\wedge}\;,&\quad\partial_z\delta_z&=0\;,&\quad\partial_z\Delta_z&={\eta\wedge}\,\delta_z+\delta_z\,{\eta\wedge}\;,\\ \partial_{\bar z}d_z&=0\;,&\quad\partial_{\bar z}\delta_z&=-{\eta\lrcorner}\;,&\quad\partial_{\bar z}\Delta_z&=-{\eta\lrcorner}\,d_z-d_z\,{\eta\lrcorner}. \end{alignedat} \right\} \end{align} $$

The operator $d_z$ defines an elliptic complex on $\Omega (M)$ , whose cohomology is denoted by $H_z^{\bullet }(M)$ . Since $d_z$ has the same principal symbol as d, it is a generalized Dirac complex and $\Delta _z$ a self-adjoint generalized Laplacian [Reference Berline, Getzler and Vergne7, Definition 2.2]. If $\theta =\eta +dh$ for some $h\in C^\infty (M,\mathbb {R})$ , then the multiplication operator

(2.5) $$ \begin{align} e^{zh}:(\Omega(M),d_{z\theta})\to(\Omega(M),d_{z\eta}) \end{align} $$

is an isomorphism of differential complexes, and therefore it induces an isomorphism $H_{z\theta }^{\bullet }(M)\cong H_{z\eta }^{\bullet }(M)$ . Thus, the isomorphism class of $H_z^{\bullet }(M)$ only depends on $\xi :=[\eta ]\in H^1(M,\mathbb {R})$ and $z\in \mathbb {C}$ . By ellipticity, $D_z$ and $\Delta _z$ have a discrete spectrum, and there is a decomposition, equalities and isomorphism of Hodge type,

(2.6) $$ \begin{align} \left. \begin{gathered} \Omega(M)=\ker\Delta_z\oplus\operatorname{im} d_z\oplus\operatorname{im}\delta_z\;,\\ \ker\Delta_z=\ker D_z=\ker d_z\cap\ker\delta_z\;,\quad \operatorname{im}\Delta_z=\operatorname{im} D_z=\operatorname{im} d_z\oplus\operatorname{im}\delta_z\;,\\ H_z^{\bullet}(M)\cong\ker\Delta_z\;, \end{gathered} \right\} \end{align} $$

as topological vector spaces. The orthogonal projections of $\Omega (M)$ to $\ker \Delta _z$ , $\operatorname {im} d_z$ and $\operatorname {im}\delta _z$ are denoted by $\Pi _z=\Pi ^0_z$ , $\Pi ^1_z$ and $\Pi ^2_z$ , respectively; thus, $\Pi _z^\perp =\Pi ^1_z+\Pi ^2_z$ . The restrictions $d_z:\operatorname {im}\delta _z\to \operatorname {im} d_z$ , $\delta _z:\operatorname {im} d_z\to \operatorname {im}\delta _z$ and $D_z:\operatorname {im} D_z\to \operatorname {im} D_z$ are topological isomorphisms, and therefore the compositions $d_z^{-1}\Pi _z^1$ , $\delta _z^{-1}\Pi _z^2$ and $D_z^{-1}\Pi _z^\perp $ are defined and continuous on $\Omega (M)$ . For every degree k, the diagram

(2.7)

is commutative. The twisted Betti numbers $\beta _z^k=\beta _z^k(M,\xi )=\dim H_z^k(M)$ give rise to the usual Euler characteristic [Reference Farber28, Proposition 1.40],

(2.8) $$ \begin{align} \sum_k(-1)^k\beta_z^k=\chi(M). \end{align} $$

(This is also a consequence of the index theorem.) For every degree k, $\beta _z^k$ is independent of z outside a discrete subset of $\mathbb {C}$ , where $\beta _z^k$ jumps (Mityagin and Novikov [Reference Novikov57, Theorem 1]). This ground value of $\beta _z^k$ is called the k-th Novikov Betti number, denoted by $\beta _{\mathrm {No}}^k=\beta _{\mathrm {No}}^k(M,\xi )$ . It will be shown in Section 6.2.4 that

(2.9) $$ \begin{align} \beta_z^k=\beta_{\mathrm{No}}^k\quad\text{for}\quad|\mu|\gg0. \end{align} $$

(When z is real, this is proved in [Reference Farber27, Theorem 2.8], [Reference Braverman and Farber14, Lemma 1.3], [Reference Burghelea and Haller18, Proposition 4].) Thus, the discrete set of parameters $z\in \mathbb {C}$ with $\beta _z^k(M,\xi )>\beta _{\mathrm {No}}^k(M,\xi )$ for some degree k is contained in a strip $|\mu |\le C$ .

By Equation (2.3) and since $\eta $ is real, for all $\alpha \in \Omega (M)$ ,

(2.10) $$ \begin{align} \overline{d_z\alpha}=d_{\bar z}\bar\alpha\;,\quad\overline{\delta_z\alpha}=\delta_{\bar z}\bar\alpha\;,\quad \overline{D_z\alpha}=D_{\bar z}\bar\alpha\;,\quad\overline{\Delta_z\alpha}=\Delta_{\bar z}\bar\alpha. \end{align} $$

So conjugation induces $\mathbb {C}$ -antilinear isomorphisms

$$\begin{align*}H_z^k(M)\cong H_{\bar z}^k(M)\;,\quad\ker\Delta_{z,k}\cong\ker\Delta_{\bar z,k}\;, \end{align*}$$

yielding $\beta _z^k=\beta _{\bar z}^k$ .

2.1.3. Case of an exact form

When $\eta =dh$ for some $h\in C^\infty (M,\mathbb {R})$ , we have the original Witten’s perturbations, which satisfy

(2.11) $$ \begin{align} \left. \begin{gathered} d_z=e^{-zh}\,d\,e^{zh}=e^{-i\nu h}\,d_\mu\,e^{i\nu h}\;,\quad \delta_z=e^{\bar zh}\,\delta\,e^{-\bar zh}=e^{-i\nu h}\,\delta_\mu\,e^{i\nu h}\;,\\ D_z=e^{-i\nu h}\,D_\mu\,e^{i\nu h}\;,\quad \Delta_z=e^{-i\nu h}\,\Delta_\mu\,e^{i\nu h}. \end{gathered} \right\} \end{align} $$

Thus, the multiplication operator

(2.12) $$ \begin{align} e^{zh}:(\Omega(M),d_z)\to(\Omega(M),d) \end{align} $$

is an isomorphism of differential complexes. Therefore, $H_z^{\bullet }(M)\cong H^{\bullet }(M)$ , yielding $\beta _z^k=\beta ^k=\beta ^k(M)$ (the kth Betti number) in this case. Moreover multiplication by $e^{i\nu h}$ defines a unitary isomorphism $\ker \Delta _z\cong \ker \Delta _\mu $ .

2.1.4. Interpretation of the closed form as a flat connection

There is a unique flat connection $\nabla ^{M\times \mathbb {C}}$ on the trivial complex line bundle $M\times \mathbb {C}$ so that $\nabla ^{M\times \mathbb {C}}1=\eta $ . The corresponding flat complex line bundle is denoted by $\mathcal {L}=\mathcal {L}_\eta $ . Note that $\mathcal {L}_{z\eta }=\mathcal {L}^z$ . Let $(\Omega (M,\mathcal {L}^z)=(\Omega (M),d^{\mathcal {L}^z})$ be the de Rham complex with coefficients in $\mathcal {L}^z$ . It is well known that $d_z=d^{\mathcal {L}^z}$ on $\Omega (M)=\Omega (M,\mathcal {L}^z)$ , and therefore $H^{\bullet }(M,\mathcal {L}^z)=H_z^{\bullet }(M)$ . Since every $\mathcal {L}^z$ is canonically trivial as a line bundle, it has a canonical Hermitian structure $g^{\mathcal {L}^z}$ . An easy local computation shows that (see the example given in [Reference Bismut, Zhang and Laudenbach10, pp. 11–12])

(2.13) $$ \begin{align} \nabla^{\mathcal{L}^z}g^{\mathcal{L}^z}=-2\mu\eta\otimes g^{\mathcal{L}^z}. \end{align} $$

2.1.5. Perturbed operators on oriented manifolds

The mappings $(\alpha ,\beta )\mapsto \alpha \wedge \beta $ and $(\alpha ,\beta )\mapsto \alpha \wedge \bar \beta $ induce respective bilinear and sesquilinear maps,

$$\begin{align*}H_z^k(M)\times H_{-z}^l(M) \to H^{k+l}(M)\;,\quad H_z^k(M)\times H_{-\bar z}^l(M) \to H^{k+l}(M)\;, \end{align*}$$

as follows from the interpretation of $d_z$ given in Section 2.1.4, or by a direct check.

Now, assume M is oriented. Then the above maps and integration on M define respective nondegenerate bilinear and sesquilinear pairings

$$\begin{align*}H_z^k(M)\times H_{-z}^{n-k}(M)\to\mathbb{C}\;,\quad H_z^k(M)\times H_{-\bar z}^{n-k}(M)\to\mathbb{C}. \end{align*}$$

Thus

(2.14) $$ \begin{align} \beta_z^k=\beta_{-z}^{n-k}=\beta_{-\bar z}^{n-k}=\beta_{\bar z}^k. \end{align} $$

Let $\star $ and $\bar \star $ denote the $\mathbb {C}$ -linear and $\mathbb {C}$ -antilinear extensions to $\Lambda M$ of the Hodge operator $\star $ on $\Lambda _{\mathbb {R}} M$ , respectively. These operators are determined by the conditions

$$\begin{align*}\alpha\wedge\overline{\star\beta}=g(\alpha,\beta)\,\operatorname{dvol}=\alpha\wedge\bar\star\beta \end{align*}$$

for $\alpha ,\beta \in \Omega (M)$ , where $\operatorname {dvol}=\star 1$ is the volume form. The following equalities on $\Omega ^k(M)$ follow from Equation (2.3) and the usual equalities relating $\star $ , d, $\delta $ , ${\eta \wedge }$ and ${\eta \lrcorner }$ (see, e.g., [Reference Roe63, Chapters 1 and 3], [Reference Gilkey31, Section 1.5.2], [Reference Berline, Getzler and Vergne7, Section 3.6]):

(2.15) $$ \begin{align} \left. \begin{alignedat}{3} d_z\,\star&=(-1)^k\star\,\delta_{-\bar z}\;,&\quad\delta_z\,\star&=(-1)^{k+1}\star\,d_{-\bar z}\;,&\quad \Delta_z\,\star&=\star\,\Delta_{-\bar z}\;,\\ d_z\,\bar\star&=(-1)^k\;\bar\star\;\delta_{-z}\;,&\quad\delta_z\;\bar\star&=(-1)^{k+1}\,\bar\star\;d_{-z}\;,&\quad \Delta_z\;\bar\star&=\bar\star\;\Delta_{-z}. \end{alignedat} \right\} \end{align} $$

Then we get a linear isomorphism $\star :\ker \Delta _z\to \ker \Delta _{-\bar z}$ and an antilinear isomorphism $\bar \star :\ker \Delta _z\to \ker \Delta _{-z}$ , inducing a linear isomorphism $H_z^k(M)\cong H_{-\bar z}^{n-k}(M)$ and an antilinear isomorphism $H_z^k(M)\cong H_{-z}^{n-k}(M)$ by Equation (2.6).

2.2. Perturbation of the Sobolev norms

For $m\in \mathbb {N}_0$ and $\omega \in Z^1(M)$ , define the norm $\|\ \|_{m,\omega }$ on $H^m(M;\Lambda )$ by

$$\begin{align*}\|\alpha\|_{m,\omega}=\sum_{k=0}^m\big\|D_\omega^k\alpha\big\|. \end{align*}$$

Proposition 2.1. For all $\omega \in Z^1(M)$ and $\alpha \in H^m(M;\Lambda )$ ,

$$ \begin{gather*} \|\alpha\|_{m,\omega}\le C_m\sum_{k=0}^m\|\omega\|_{C^k}^{m-k}\|\alpha\|_k\;,\quad \|\alpha\|_m\le C_m\sum_{k=0}^m\|\omega\|_{C^k}^{m-k}\|\alpha\|_{k,\omega}. \end{gather*} $$

Proof. We proceed by induction on m. We have $\|\ \|_{0,\omega }=\|\ \|$ . Now, take $m>0$ and assume these inequalities hold for $m-1$ . For $\eta \in Z^1(M,\mathbb {R})$ and $\alpha \in \Omega (M)$ , we have

(2.16) $$ \begin{align} \|\hat c(\eta)\alpha\|_m,\|c(\eta)\alpha\|_m\le C^{\prime}_m\|\eta\|_{C^m}\|\alpha\|_m. \end{align} $$

Applying these inequalities to the real and imaginary parts of $\omega $ and using the induction hypothesis and Equation (2.3), we get

$$ \begin{align*} \|\alpha\|_{m,\omega}&=\|\alpha\|+\|D_{\omega}\alpha\|_{m-1,\omega} \le\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\|\omega\|_{C^k}^{m-1-k}\|D_{\omega}\alpha\|_k\\ &\le\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\|\omega\|_{C^k}^{m-1-k}\big(\|D\alpha\|_k+C^{\prime}_k\|\omega\|_{C^k}\|\alpha\|_k\big)\\ &\le\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\|\omega\|_{C^k}^{m-1-k}\big(\|\alpha\|_{k+1}+C^{\prime}_k\|\omega\|_{C^k}\|\alpha\|_k\big)\\ &\le C_m\sum_{l=0}^m\|\omega\|_{C^l}^{m-l}\|\alpha\|_l\;, \end{align*} $$

$$ \begin{align*} \|\alpha\|_m&=\|\alpha\|+\|D\alpha\|_{m-1} \le\|\alpha\|+\|D_{\omega}\alpha\|_{m-1}+C^{\prime}_{m-1}\|\omega\|_{C^{m-1}}\|\alpha\|_{m-1}\\ &\le\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\big(\|\omega\|_{C^k}^{m-1-k}\|D_{\omega}\alpha\|_{k,\omega} +C^{\prime}_{m-1}\|\omega\|_{C^k}^{m-k}\|\alpha\|_{k,\omega}\big)\\ &\le\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\big(\|\omega\|_{C^k}^{m-1-k}\|\alpha\|_{k+1,\omega} +C^{\prime}_{m-1}\|\omega\|_{C^k}^{m-k}\|\alpha\|_{k,\omega}\big)\\ &\le C_m\sum_{l=0}^m\|\omega\|_{C^l}^{m-l}\|\alpha\|_{l,\omega}. \; \end{align*} $$

Let $Z(M,\mathbb {Z})\subset Z(M,\mathbb {R})$ denote the graded additive subgroup of forms that represent cohomology classes in the image of the canonical homomorphism $H^{\bullet }(M,\mathbb {Z})\to H^{\bullet }(M,\mathbb {R})$ . Recall that we can consider $H^1(M,\mathbb {Z})$ as a lattice in $H^1(M,\mathbb {R})$ by the universal coefficient theorem for cohomology. Let $\theta $ be the multivalued angle function on $\mathbb {S}^1$ . Then $d\theta $ is the angular form on $\mathbb {S}^1$ with $\int _{\mathbb {S}^1}d\theta =2\pi $ . For $\eta \in Z^1(M,\mathbb {R})$ , we have $\eta \in 2\pi Z^1(M,\mathbb {Z})$ if and only if there is some smooth map $h:M\to \mathbb {S}^1$ such that $\eta =h^*d\theta $ (see, e.g., [Reference Farber28, Lemma 2.1]).

In Proposition 2.1, the dependence of the constants on $\omega $ cannot be avoided. For instance, for $M=\mathbb {S}^1$ with the standard metric $g=(d\theta )^2$ , we have $\|1\|_m=\sqrt {2\pi }$ , whereas $\|1\|_{m,i\eta }=\sqrt {2\pi }\sum _{k=0}^m|\nu |^k$ for $\eta =\nu \,d\theta $ ( $\nu \in \mathbb {R}$ ). However, the following version of a Sobolev inequality for $\|\ \|_{m,i\eta }$ involves a constant independent of $\eta $ .

Proposition 2.2. If $m>n/2$ , for all $\eta \in Z^1(M,\mathbb {R})$ and $\alpha \in H^m(M;\Lambda )$ ,

$$\begin{align*}\|\alpha\|_{L^\infty}\le C_m\|\alpha\|_{m,i\eta}. \end{align*}$$

Proof. By the Sobolev embedding theorem, we have

$$\begin{align*}C_{m,i\eta}:=\sup_{0\ne\alpha\in\Omega(M)}\frac{\|\alpha\|_{L^\infty}}{\|\alpha\|_{m,i\eta}}>0. \end{align*}$$

Take any $\eta \in Z^1(M,\mathbb {R})$ and $\omega \in 2\pi Z^1(M,\mathbb {Z})$ , and let $\eta '=\eta +\omega $ . Then $\omega =h^*d\theta $ for some smooth function $h:M\to \mathbb {S}^1$ . Since the difference between the multiple values of $\theta $ at every point of $\mathbb {S}^1$ are in $2\pi \mathbb {Z}$ , the functions $e^{\pm ih^*\theta }$ are well defined and smooth on M. Moreover, applying Equation (2.11) locally, we get $D_{i\eta '}=e^{-ih^*\theta }\,D_{i\eta }\,e^{ih^*\theta }$ . So, for $0\ne \alpha \in \Omega (M)$ ,

$$ \begin{align*} \|\alpha\|_{L^\infty}&=\|e^{i\,h^*\theta}\alpha\|_{L^\infty}\le C_{m,i\eta}\|e^{ih^*\theta}\alpha\|_{m,i\eta}\\ &=C_{m,i\eta}\sum_{k=0}^m\|D_{i\eta}^k\,e^{ih^*\theta}\alpha\| =C_{m,i\eta}\sum_{k=0}^m\|e^{-ih^*\theta}\,D_{i\eta}^k\,e^{ih^*\theta}\alpha\|\\ &=C_{m,i\eta}\sum_{k=0}^m\|D_{i\eta'}^k\alpha\| =C_{m,i\eta}\|\alpha\|_{m,i\eta'}. \end{align*} $$

This shows that

(2.17) $$ \begin{align} \eta-\eta'\in2\pi Z^1(M,\mathbb{Z})\Rightarrow C_{m,i\eta}=C_{m,i\eta'}. \end{align} $$

Since $2\pi H^1(M,\mathbb {Z})$ is a lattice in $H^1(M,\mathbb {R})$ , there is a compact subset $K\subset H^1(M,\mathbb {R})$ such that

(2.18) $$ \begin{align} K+2\pi H^1(M,\mathbb{Z})=H^1(M,\mathbb{R}). \end{align} $$

Take a linear subspace $V\subset Z^1(M,\mathbb {R})$ such that the canonical projection $V\to H^1(M,\mathbb {R})$ is an isomorphism, and let $L\subset V$ be the compact subset that corresponds to K. By Equation (2.18),

(2.19) $$ \begin{align} L+2\pi Z^1(M,\mathbb{Z})=Z^1(M,\mathbb{R}). \end{align} $$

Moreover, L is bounded with respect to $\|\ \|_{C^m}$ . Therefore, by Proposition 2.1, for all $\eta \in L$ and $\alpha \in \Omega (M)$ ,

$$\begin{align*}\|\alpha\|_{L^\infty}\le C_{m,0}\|\alpha\|_m\le C_m\|\alpha\|_{m,i\eta}\;, \end{align*}$$

yielding

(2.20) $$ \begin{align} \sup_{\eta\in L}C_{m,i\eta}\le C_m. \end{align} $$

The result follows from Equations (2.17), (2.19) and (2.20).

Given $\eta \in Z^1(M,\mathbb {R})$ , we write $\|\ \|_{m,z}=\|\ \|_{m,z\eta }$ . Proposition 2.1 has the following direct consequence.

Corollary 2.3. For all $\alpha \in H^m(M;\Lambda )$ and $z\in \mathbb {C}$ ,

$$\begin{align*}\|\alpha\|_{m,z}\le C_m\sum_{k=0}^m|z|^{m-k}\|\alpha\|_k\;,\quad \|\alpha\|_m\le C_m\sum_{k=0}^m|z|^{m-k}\|\alpha\|_{k,z}. \end{align*}$$

Proposition 2.4. For all $\alpha \in H^1(M;\Lambda )$ and $z\in \mathbb {C}$ ,

$$\begin{align*}\|\alpha\|_{1,z}\le C\big(\|\alpha\|_{1,i\nu}+|\mu|\|\alpha\|\big)\;,\quad \|\alpha\|_{1,i\nu}\le C\big(\|\alpha\|_{1,z}+|\mu|\|\alpha\|\big). \end{align*}$$

Proof. By Equations (2.3) and (2.16),

$$ \begin{align*} \|\alpha\|_{1,z}&=\|\alpha\|+\|D_z\alpha\| \le\|\alpha\|+\|D_{i\nu}\alpha\|+C'|\mu|\|\alpha\| \le C\big(\|\alpha\|_{1,i\nu}+|\mu|\|\alpha\|\big)\;,\\ \|\alpha\|_{1,i\nu}&=\|\alpha\|+\|D_{i\nu}\alpha\| \le\|\alpha\|+\|D_z\alpha\|+C'|\mu|\|\alpha\| \le C\big(\|\alpha\|_{1,z}+|\mu|\|\alpha\|\big). \; \end{align*} $$

3.1. Preliminaries on asymptotic expansions of heat kernels

Let A be a positive semidefinite symmetric elliptic differential operator of order a, and B a differential operator of order b; both of them are defined in $C^\infty (M;E)$ for some Hermitian vector bundle E over M. Then $Be^{-tA}$ is a smoothing operator with Schwartz kernel $K_t(x,y)$ in $C^\infty (M^2;E\boxtimes E^*)$ (omitting the Riemannian density $\operatorname {dvol}(y)$ of the second factor). On the diagonal, there is an asymptotic expansion (as $t\downarrow 0$ ) with respect to the seminorms $\|\ \|_{C^m}$ ( $m\in \mathbb {N}_0$ ) on $C^\infty (M;E\otimes E^*)$ [Reference Gilkey31, Lemma 1.9.1], [Reference Berline, Getzler and Vergne7, Theorem 2.30, Proposition 2.46 and the paragraph that follows],

(3.1) $$ \begin{align} K_t(x,x)\sim\sum_{l=0}^\infty e_l(x)t^{(l-n-b)/a}\;, \end{align} $$

with $e_l\in C^\infty (M;E\otimes E^*)$ . Moreover, using a local system of coordinates, a local trivialization of E and standard multi-index notation, if $B=\sum _\alpha b_\alpha (x)D^\alpha _x$ , then $e_l(x)=\sum _\alpha b_\alpha (x)e_{l,\alpha }(x)$ , where the $e_{l,\alpha }(x)$ are smooth local invariants of the symbol of A which are homogeneous of degree $l+|\alpha |-b$ . They vanish if $l+b$ is odd or if $l+|\alpha |-b<0$ . Hence, the function

$$\begin{align*}h(t)=\operatorname{Tr}\big(Be^{-tA}\big)=\int_M\operatorname{tr} K_t(x,x)\,\operatorname{dvol}(x) \end{align*}$$

has an asymptotic expansion

(3.2) $$ \begin{align} h(t)\sim\sum_{l=0}^\infty a_lt^{(l-n-b)/a}\;, \end{align} $$

where

(3.3) $$ \begin{align} a_l=\int_M\operatorname{tr} e_l(x)\,\operatorname{dvol}(x)\;, \end{align} $$

which vanishes if $l+b$ is odd.

The case of truncated heat kernels, in the following sense, is also needed. Given any $\lambda \ge 0$ , let $P_{A,\lambda }$ be the spectral projection of A corresponding to $[0,\lambda ]$ ; thus, $P_{A,\lambda }^\perp $ is the spectral projection corresponding to $(\lambda ,\infty )$ . By ellipticity, $P_{A,\lambda }$ is of finite rank, and $Be^{-tA}P_{A,\lambda }$ is a smoothing operator defined for all $t\in \mathbb {R}$ . Take any orthonormal frame $\phi _1,\dots ,\phi _\kappa $ of $\operatorname {im} P_{A,\lambda }$ , consisting of eigensections with corresponding eigenvalues $0\le \lambda _1\le \dots \le \lambda _\kappa \le \lambda $ . Then the Schwartz kernel $H_t(x,y)$ of $Be^{-tA}P_{A,\lambda }$ ( $t\ge 0$ ) is given by

$$\begin{align*}H_t(x,y)=\sum_{j=1}^\kappa e^{-t\lambda_j}(B\phi_j)(x)\otimes\phi_j(y)\;, \end{align*}$$

using the isomorphism $E\cong E^*$ given by the Hermitian structure. Thus, $H_t(x,y)$ is defined for all $t\in \mathbb {R}$ and smooth. So

$$\begin{align*}\operatorname{Tr}(Be^{-tA}P_{A,\lambda})=\int_M\operatorname{tr} H_t(x,x)\,\operatorname{dvol}(x). \end{align*}$$

In particular, for $t=0$ , we have

(3.4) $$ \begin{align} \; \; H_0(x,x)&=\sum_{j=1}^\kappa(B\phi_j)(x)\otimes\phi_j(x)\;, \end{align} $$

(3.5) $$ \begin{align} \operatorname{Tr}(BP_{A,\lambda})&=\int_M\operatorname{tr} H_0(x,x)\,\operatorname{dvol}(x). \end{align} $$

The Schwartz kernel of $Be^{-tA}P_{A,\lambda }^\perp $ is $\widetilde K_t(x,y)=K_t(x,y)-H_t(x,y)$ ( $t>0$ ), which has an asymptotic expansion

(3.6) $$ \begin{align} \widetilde K_t(x,x)\sim\sum_{l=0}^\infty\tilde e_l(x)t^{(l-n-b)/a}\;, \end{align} $$

where the first $n+b$ sections $\tilde e_l$ are given by

$$\begin{align*}\tilde e_l(x)= \begin{cases} e_l(x) & \text{if}\ l<n+b\\ e_l(x)-H_0(x,x) & \text{if}\ l=n+b. \end{cases} \end{align*}$$

Then the function

(3.7) $$ \begin{align} \tilde h_\lambda(t)=\operatorname{Tr}\big(Be^{-tA}P_{A,\lambda}^\perp\big)=\operatorname{Tr}\big(Be^{-tA}\big)-\operatorname{Tr}(Be^{-tA}P_{A,\lambda}) \end{align} $$

has an asymptotic expansion

(3.8) $$ \begin{align} \tilde h_\lambda(t)=\int_M\widetilde K_t(x,x)\,\operatorname{dvol}(x)\sim\sum_{l=0}^\infty\tilde a_lt^{(l-n-b)/a}\;, \end{align} $$

where the first $n+b$ coefficients $\tilde a_l$ are given by

(3.9) $$ \begin{align} \tilde a_l= \begin{cases} a_l & \text{if}\ l<n+b\\ a_l-\operatorname{Tr}(BP_{A,\lambda}) & \text{if}\ l=n+b. \end{cases} \end{align} $$

Consider also smooth families of such operators, $\{A_\epsilon \}$ and $\{B_\epsilon \}$ , for $\epsilon $ in some parameter space. Then $\operatorname {Tr}(B_\epsilon e^{-tA_\epsilon })$ is smooth in $(t,\epsilon )$ , and we add $\epsilon $ to the above notation, writing for instance $K_t(x,y,\epsilon )$ , $e_l(x,\epsilon )$ , $h(t,\epsilon )$ , $a_l(\epsilon )$ , $\widetilde K_t(x,y,\epsilon )$ , $\tilde e_l(x,\epsilon )$ , $\tilde h(t,\epsilon )$ and $\tilde a_l(\epsilon )$ in Equations (3.1), (3.2), (3.6) and (3.8). The operator $B_\epsilon P_{A_\epsilon ,\lambda }$ may not be smooth in $\epsilon $ when some nonconstant spectral branch of $\{A_\epsilon \}$ reaches the value $\lambda $ . If the values of all nonconstant spectral branches of $\{A_\epsilon \}$ stay away from some neighborhood of $\lambda $ , then $\tilde h_\lambda (t,\epsilon )$ is smooth in $(t,\epsilon )$ .

3.2. Preliminaries on zeta functions of operators

The last expression of Equation (3.10) is the Mellin transform of the function $\tilde h_\lambda (t)$ divided by $\Gamma (s)$ . This function $\zeta (s,A,B,\lambda )$ is called the zeta function of $(A,B,\lambda )$ . If $B=1$ or $\lambda =0$ , they may be omitted from the notation.

We will also use $\zeta (s,A,B,\lambda )$ when B is not a differential operator, with the same definition. Then the asymptotic expansion (3.8) and the properties stated in Proposition 3.1 need to be checked. With this generality, we can write

$$ \begin{align*} \zeta(s,A,B,\lambda) &=\zeta(s,A,BP_{A,\lambda}^\perp)=\zeta(s,A,P_{A,\lambda}^\perp B)\;,\\ \zeta(s,A,B) &=\zeta(s,A,BP_{A,\lambda})+\zeta(s,A,B,\lambda). \end{align*} $$

Since $P_{A,\lambda }$ is of finite rank, $\zeta (s,A,BP_{A,\lambda })$ is always defined and holomorphic on $\mathbb {C}$ .

3.3. Zeta invariants of closed real 1-forms

According to Proposition 3.1 (i), let

$$\begin{align*}\zeta(s,z)=\zeta(s,z,\eta)=\zeta(s,\Delta_z,{\eta\wedge}\,D_z\mathsf{w})\;, \end{align*}$$

which is a meromorphic function of $s\in \mathbb {C}$ . For $\Re s\gg 0$ ,

$$ \begin{align*} \zeta(s,z)&=\operatorname{Str}\big({\eta\wedge}\,D_z\Delta_z^{-s}\Pi_z^\perp\big) =\operatorname{Str}\big({\eta\wedge}\,\delta_z\Delta_z^{-s}\Pi_z^1\big)\\ &=\operatorname{Str}\big({\eta\wedge}\,D_z^{-1}\Delta_z^{-s+1}\Pi_z^\perp\big)=\operatorname{Str}\big({\eta\wedge}\,d_z^{-1}\Delta_z^{-s+1}\Pi^1_z\big)\;, \end{align*} $$

using that ${\eta \wedge }\,d_z$ and ${\eta \wedge }\,\delta _z^{-1}$ change the degree of homogeneous forms. So, when $\zeta (s,z)$ is regular at $s=1$ , the value $\zeta (1,z)$ is a renormalized version of the super-trace of ${\eta \wedge }\,d_z^{-1}\Pi ^1_z$ , which is called the zeta invariant of $(M,g,\eta ,z)$ for the scope of this paper. According to Proposition 3.1 (ii) and since $\Gamma (s)$ is regular at $s=1$ , $\zeta (s,z)$ might have a simple pole at $s=1$ . But it will be shown that $\zeta (s,z)$ is regular at $s=1$ for all $\eta \in Z^1(M,\mathbb {R})$ and $z\in \mathbb {C}$ (Corollary 3.9).

3.4. Heat invariants of perturbed operators

Consider the notation of Section 2.1.2. For $k=0,\dots ,n$ , let $K_{z,k,t}(x,y)$ denote the Schwartz kernel of $e^{-t\Delta _{z,k}}$ . By Equation (3.1), its restriction to the diagonal has an asymptotic expansion (as $t\downarrow 0$ ),

$$\begin{align*}K_{z,k,t}(x,x)\sim\sum_{l=0}^\infty e_{k,l}(x,z)t^{(l-n)/2}\;, \end{align*}$$

where every $e_{k,l}(x,z)$ is a smooth local invariant of z and the jets of the local coefficients of g and $\eta $ , which is homogeneous of degree l, and vanishes if l is odd. According to Equations (3.2) and (3.3),

$$\begin{align*}h_k(t,z):=\operatorname{Tr}\big(e^{-t\Delta_{z,k}}\big)\sim\sum_{l=0}^\infty a_{k,l}(z)t^{(l-n)/2}\;, \end{align*}$$

where

$$\begin{align*}a_{k,l}(z)=\int_M\operatorname{str} e_{k,l}(x,z)\,\operatorname{dvol}(x). \end{align*}$$

The Schwartz kernel of $e^{-t\Delta _z}\mathsf {w}$ is

$$\begin{align*}K_{z,t}(x,y)=\sum_{k=0}^n(-1)^kK_{z,k,t}(x,y). \end{align*}$$

We have induced asymptotic expansions,

$$ \begin{gather*} K_{z,t}(x,x)\sim\sum_{l=0}^\infty e_l(x,z)t^{(l-n)/2}\;,\\ h(t,z):=\operatorname{Str}\big(e^{-t\Delta_z}\big)\sim\sum_{l=0}^\infty a_l(z)t^{(l-n)/2}\;, \end{gather*} $$

where

$$\begin{align*}e_l(x,z)=\sum_{k=0}^n(-1)^ke_{k,l}(x,z)\;,\quad a_l(z)=\sum_{k=0}^n(-1)^ka_{k,l}(z). \end{align*}$$

3.5. Derived heat invariants of perturbed operators

The following are sometimes called the derived heat density and derived heat invariant of order l of $d_z$ or $\Delta _z$ [Reference Günther and Schimming33], [Reference Ray and Singer61], [Reference Gilkey31, page 181], [Reference Álvarez López and Gilkey3]:

$$ \begin{gather*} \mathfrak{e}_l(x,z)=\sum_{k=0}^n(-1)^kke_{k,l}(x,z)\;,\\ \mathfrak{a}_l(z)=\sum_{k=0}^n(-1)^kka_{k,l}(z)=\int_M\operatorname{str}\mathfrak{e}_l(x,z)\,\operatorname{dvol}(x). \end{gather*} $$

We have

(3.11) $$ \begin{align} \operatorname{Str}\big({\mathsf{N}} e^{-t\Delta_z}\big)\sim\sum_{l=0}^\infty\mathfrak{a}_l(z)t^{(l-n)/2}. \end{align} $$

3.6. Regularity

By Equations (3.2) and (3.3), we have an asymptotic expansion of the form

(3.12) $$ \begin{align} \operatorname{Str}\big({\eta\wedge}\,D_ze^{-t\Delta_z}\big)\sim\sum_{l=0}^\infty b_l(z)t^{(l-n-1)/2}\;, \end{align} $$

where $b_l(z)=0$ if l is even.

Proof. For all k, we have [Reference Berline, Getzler and Vergne7, Corollary 2.50]

$$\begin{align*}\partial_z\operatorname{Tr}\big(e^{-t\Delta_{z,k}}\big)=-t\operatorname{Tr}\big((\partial_z\Delta_{z,k})e^{-t\Delta_{z,k}}\big). \end{align*}$$

So, by Equations (2.1) and (2.4),

$$ \begin{align*} \partial_z\operatorname{Str}\big({\mathsf{N}} e^{-t\Delta_z}\big)&=-t\operatorname{Str}\big({\mathsf{N}}(\partial_z\Delta_z)e^{-t\Delta_z}\big)\\ &=-t\operatorname{Str}\big({\mathsf{N}}\,{\eta\wedge}\,\delta_ze^{-t\Delta_z}\big)-t\operatorname{Str}\big({\mathsf{N}}\delta_z\,{\eta\wedge}\,e^{-t\Delta_z}\big)\\ &=-t\operatorname{Str}\big({\mathsf{N}}\,{\eta\wedge}\,\delta_ze^{-t\Delta_z}\big)-t\operatorname{Str}\big(\delta_z({\mathsf{N}}-1)\,{\eta\wedge}\,e^{-t\Delta_z}\big)\\ &=-t\operatorname{Str}\big({\mathsf{N}}\,{\eta\wedge}\,\delta_ze^{-t\Delta_z}\big)+t\operatorname{Str}\big(({\mathsf{N}}-1)\,{\eta\wedge}\,\delta_ze^{-t\Delta_z}\big)\\ &=-t\operatorname{Str}\big({\eta\wedge}\,D_ze^{-t\Delta_z}\big). \; \end{align*} $$

Proof. By Equations (3.11) and (3.12); Theorem 3.4; and Proposition 3.7, for $l\le n-1$ ,

$$\begin{align*}b_l(z)=-\partial_z\mathfrak{a}_{l+1}(z)=0. \; \end{align*}$$

Corollary 3.9. If n is even and $\Re s>0$ , or n is odd and $\Re s>1/2$ , then

$$\begin{align*}\zeta(s,z)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\operatorname{Str}\big({\eta\wedge}\,D_ze^{-t\Delta_z}\big)\,dt\;, \end{align*}$$

where the integral is absolutely convergent, and therefore $\zeta (s,z)$ is smooth in this half-plane.

Proof. By Equation (3.12) and Corollary 3.8,

(3.13) $$ \begin{align} \operatorname{Str}\big({\eta\wedge}\,D_ze^{-t\Delta_z}\big)= \begin{cases} O(1) & \text{if}\ n\ \text{is even}\\ O\big(t^{-1/2}\big) & \text{if}\ n\ \text{is odd} \end{cases} \quad(t\downarrow0). \end{align} $$

On the other hand, there is some $c>0$ such that

(3.14) $$ \begin{align} \operatorname{Str}\big({\eta\wedge}\,D_ze^{-t\Delta_z}\big)=O(e^{-ct})\quad(t\uparrow+\infty). \end{align} $$

So the stated integral is absolutely convergent for $\Re s>0$ if n is even, or for $\Re s>1/2$ if n is odd, defining a holomorphic function of s on this half-plane. Then the stated equality is true because it holds for $\Re s\gg 0$ .

Corollary 3.11. For all $z\in \mathbb {C}$ ,

$$\begin{align*}\zeta(1,z)=\lim_{t\downarrow0}\operatorname{Str}\big({\eta\wedge}\,D_z^{-1}e^{-t\Delta_z}\Pi_z^\perp\big). \end{align*}$$

Proof. By Corollary 3.9, Equation (3.13) and Equation (3.14), and since

$$\begin{align*}\operatorname{Str}\big({\eta\wedge}\,D_z^{-1}e^{-t\Delta_z}\Pi_z^\perp\big)=O(e^{-ct})\quad(t\uparrow+\infty)\;, \end{align*}$$

we get

$$ \begin{align*} \zeta(1,z)&=\int_0^\infty\operatorname{Str}\big({\eta\wedge}\,D_ze^{-u\Delta_z}\Pi_z^\perp\big)\,du =\lim_{t\downarrow0}\int_t^\infty\operatorname{Str}\big({\eta\wedge}\,D_ze^{-u\Delta_z}\Pi_z^\perp\big)\,du\\ &=\lim_{t\downarrow0}\operatorname{Str}\big({\eta\wedge}\,D_z^{-1}e^{-t\Delta_z}\Pi_z^\perp\big). \; \end{align*} $$

Corollaries 3.9 and 3.11 give Theorem 1.1.

3.7. The case of the differential of a function

Let us consider the special case where $\eta =dh$ for a smooth real-valued function h.

Lemma 3.12. We have

$$\begin{align*}\operatorname{Str}\big({\eta\wedge}\,d_z^{-1}e^{-t\Delta_z}\Pi^1_z\big)=-\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi_z^\perp\big). \end{align*}$$

Proof. Since ${\eta \wedge }=[d,h]$ ,

$$ \begin{align*} \operatorname{Str}\big({\eta\wedge}\,d_z^{-1}e^{-t\Delta_z}\Pi^1_z\big) &=\operatorname{Str}\big([d_z,h]\,d_z^{-1}e^{-t\Delta_z}\Pi^1_z\big)\\ &=\operatorname{Str}\big(d_z\,h\,d_z^{-1}e^{-t\Delta_z}\Pi^1_z\big)-\operatorname{Str}\big(h\,d_zd_z^{-1}e^{-t\Delta_z}\Pi^1_z\big)\\ &=-\operatorname{Str}\big(h\,d_z^{-1}e^{-t\Delta_z}\Pi^1_zd_z\big)-\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi^1_z\big)\\ &=-\operatorname{Str}\big(h\,d_z^{-1}d_ze^{-t\Delta_z}\Pi^2_z\big)-\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi^1_z\big)\\ &=-\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi^2_z\big)-\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi^1_z\big)\\ &=-\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi_z^\perp\big). \; \end{align*} $$

Corollary 3.13. We have

$$\begin{align*}\zeta(1,z)=-\lim_{t\downarrow0}\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi_z^\perp\big). \end{align*}$$

Proof. By Corollary 3.13, it is enough to prove that $\operatorname {Str}(h\,e^{-t\Delta _z}\Pi _z^\perp )\in \mathbb {R}$ . But, taking adjoints,

$$\begin{align*}\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi_z^\perp\big)=\overline{\operatorname{Str}\big(\Pi_z^\perp e^{-t\Delta_z}\,h\big)} =\overline{\operatorname{Str}\big(h\,\Pi_z^\perp e^{-t\Delta_z}\big)} =\overline{\operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi_z^\perp\big)}. \; \end{align*}$$

Corollary 3.15. If M is oriented, then

$$\begin{align*}\zeta(1,z)=\zeta(1,-\bar z)=\zeta(1,-z)=\zeta(1,\bar z). \end{align*}$$

Proof. By Equation (2.15),

$$ \begin{align*} \operatorname{Str}\big(h\,e^{-t\Delta_z}\Pi_z^\perp\big)&=\operatorname{Str}\big(\star\star^{-1}h\,e^{-t\Delta_z}\Pi_z^\perp\big) =\operatorname{Str}\big(\star^{-1}h\,e^{-t\Delta_z}\Pi_z^\perp\star\big)\\ &=\operatorname{Str}\big(\star^{-1}\star h\,e^{-t\Delta_{-\bar z}}\Pi_{-\bar z}^\perp\big) =\operatorname{Str}\big(h\,e^{-t\Delta_{-\bar z}}\Pi_{-\bar z}^\perp\big). \end{align*} $$

Thus, the first equality of the statement holds by Corollary 3.13. The second equality follows with a similar argument, using $\bar \star $ instead of $\star $ . The third equality is equivalent to the first one.

4.1. Preliminaries on Morse forms

Recall that a critical point p of any $h\in C^\infty (M,\mathbb {R})$ is called nondegenerate if the symmetric bilinear form $\operatorname {Hess}_ph$ on $T_pM$ is nondegenerate; then the index of $\operatorname {Hess}_ph$ is denoted by $\operatorname {ind}(p)$ . By the Morse lemma [Reference Milnor, Spivak and Wells49, Lemma 2.2], this means that

(4.1) $$ \begin{align} h-h(p)=\frac{1}{2}\sum_{j=1}^n\epsilon_{p,j}(x_p^j)^2=\frac{1}{2}\big(|x_p^+|^2-|x_p^-|^2\big)\;, \end{align} $$

where

(4.2) $$ \begin{align} \epsilon_{p,j}= \begin{cases} -1 & \text{if}\ j\le\operatorname{ind}(p)\\ 1 & \text{if}\ j>\operatorname{ind}(p)\;, \end{cases} \end{align} $$

on some chart $(U_p,x_p=(x_p^1,\dots ,x_p^n))$ (centered) at p (Morse coordinates), where $x_p^-=(x_p^1,\dots ,x_p^{\operatorname {ind}(p)})$ and $x_p^+=(x_p^{\operatorname {ind}(p)+1},\dots ,x_p^n)$ .

Recall that h is called a Morse function when all of its critical points are nondegenerate. Then its critical points form a finite set denoted by $\operatorname {Crit}(h)$ . The Morse functions form an open and dense subset of $C^\infty (M,\mathbb {R})$ [Reference Hirsch36, Theorem 6.1.2]. On every $U_p$ , we can assume the metric is Euclidean with respect to Morse coordinates:

(4.3) $$ \begin{align} g=\sum_{j=1}^n(dx_p^j)^2. \end{align} $$

Now, take any $\eta \in Z^1(M,\mathbb {R})$ . We can show that if p is a zero of $\eta $ , then $(\nabla \eta )_p$ is independent of the choice of the connection $\nabla $ , and is symmetric. The zero p is called nondegenerate of index k if $(\nabla \eta )_p$ is nondegenerate of index k. In this case, any local primitive $h_{\eta ,p}$ of $\eta $ near p is a Morse function, and we can choose it so that $h_{\eta ,p}(p)=0$ . On a domain $U_p$ of Morse coordinates $x_p=(x_p^1,\dots ,x_p^n)$ for $h_{\eta ,p}$ at p, also called Morse coordinates for $\eta $ at p, $h_{\eta ,p}$ is given by the center and right-hand side of Equation (4.1), and

(4.4) $$ \begin{align} \eta=\sum_{j=1}^n\epsilon_{p,j}x_p^j\,dx_p^j. \end{align} $$

If all zeros are nondegenerate, then $\eta $ is called a Morse form. In this case, its zeros form a finite set, $\mathcal {X}=\operatorname {Zero}(\eta )$ ; subsets of $\mathcal {X}$ defined by conditions on the index are denoted by writing the conditions as subscripts; for instance, $\mathcal {X}_k$ , $\mathcal {X}_+$ and $\mathcal {X}_{<k}$ are the subsets of zeros of index k, of positive index and of index ${}<k$ , respectively. For any $\xi \in H^1(M,\mathbb {R})$ , the Morse representatives of $\xi $ form a dense open subset of $\xi $ , considering $\xi \subset \Omega ^1(M,\mathbb {R})$ with the $C^\infty $ topology (see, e.g., [Reference Pajitnov59, Theorem 2.1.25]). If $\xi =0$ , this is just the classical property of Morse functions mentioned before.

From now on, unless otherwise stated, we will use some $\eta \in Z^1(M,\mathbb {R})$ and a Riemannian metric g on M satisfying (a) (Section 1.1).

The Hopf index of $\eta ^\sharp $ at any $p\in \mathcal {X}_k$ is $(-1)^k$ (Section 6.1.1). Thus, by the Hopf index theorem,

(4.5) $$ \begin{align} \sum_{k=0}^n(-1)^k|\mathcal{X}_k|=\chi(M). \end{align} $$

4.2. The small and large spectrum

Consider the perturbed operators (2.3) defined by $\eta $ and g. We can suppose the closures $\overline {U_p}$ ( $p\in \mathcal {X}$ ) are disjoint from each other, and $x_p(U_p)=(-4r,4r)^n$ for some $r>0$ independent of p with $4r<1$ . Let $U=\bigcup _{p\in \mathcal {X}}U_p$ .

Denoting also the coordinates of $\mathbb {R}^n$ by $(x_p^1,\dots ,x_p^n)$ , consider the function $h_p\in C^\infty (\mathbb {R}^n)$ defined by the center and right-hand side of Equation (4.1). Let $d^{\prime }_{p,z}$ , $\delta ^{\prime }_{p,z}$ , $D^{\prime }_{p,z}$ and $\Delta ^{\prime }_{p,z}$ ( $z\in \mathbb {C}$ ) denote the corresponding Witten’s operators on $\mathbb {R}^n$ , whose restrictions to $(-4r,4r)^n$ agree via $x_p$ with $d_z$ , $\delta _z$ , $D_z$ and $\Delta _z$ on $U_p$ .

For any $z\in \mathbb {C}$ with $\mu>0$ , let $\Delta ^{\prime }_{p,z}=e^{-i\nu h_p}\Delta ^{\prime }_{p,\mu }e^{i\nu h_p}$ . Since the operator of multiplication by $e^{-i\nu h_p}$ is unitary, $\Delta ^{\prime }_{p,z}$ is also selfadjoint and nonnegative in $L^2(\mathbb {R}^n;\Lambda )$ , it has a discrete spectrum with the same eigenvalues and multiplicities as $\Delta ^{\prime }_{p,\mu }$ , and its kernel is generated by the normalized form $e^{\prime }_{p,z}:=e^{-i\nu h_p}e^{\prime }_{p,\mu }$ . We will also use the notation

$$\begin{align*}e^{\prime}_{p,z}=x_p^*e^{\prime}_{p,z}\in C^\infty\big(U_p;\Lambda^{\operatorname{ind}(p)}\big). \end{align*}$$

The function $x_p^*h_p\in C^\infty (U_p)$ agrees with $h_{\eta ,p}$ , which is also denoted by $h_p$ in this section.

Fix an even $C^\infty $ function $\rho :\mathbb {R}\to [0,1]$ such that $\rho =1$ on $[-r,r]$ and $\operatorname {supp}\rho \subset [-2r,2r]$ . For every $p\in \mathcal {X}$ , let

(4.8) $$ \begin{align} \rho_p&=\rho(x_p^1)\cdots\rho(x_p^n)\in C^\infty_{\mathrm{c}}(U_p)\;, \end{align} $$

(4.9) $$ \begin{align} e_{p,\mu}&=\frac{\rho_p}{a_\mu}e^{\prime}_{p,\mu}\in C^\infty_{\mathrm{c}}\big(U_p;\Lambda^{\operatorname{ind}(p)}\big)\;, \end{align} $$

(4.10) $$ \begin{align} e_{p,z}&=e^{-i\nu h_p}e_{p,\mu}=\frac{\rho_p}{a_\mu}e^{\prime}_{p,z}\in C^\infty_{\mathrm{c}}\big(U_p;\Lambda^{\operatorname{ind}(p)}\big)\;, \end{align} $$

where

(4.11) $$ \begin{align} a_\mu=\bigg(\int_{-2r}^{2r}\rho(x)^2e^{-\mu x^2}\,dx\bigg)^{\frac n2}=\Big(\frac\pi\mu\Big)^{\frac n4}+O(e^{-c\mu})\;, \end{align} $$

as $\mu \to +\infty $ . The extensions by zero of the forms $e_{p,z}$ to M are also denoted by $e_{p,z}$ . They form an orthonormal basis of a graded subspace $E_z\subset \Omega (M)$ with $\dim E_z=|\mathcal {X}|$ . Observe that $d_z$ does not preserve $E_z$ so that $E_z$ is not a subcomplex of $(\Omega (M), d_z)$ . Let $P_z$ be the orthogonal projection of $L^2(M;\Lambda )$ to $E_z$ .

Proposition 4.3. If $\mu \gg 0$ and $\beta \in H^1(M;\Lambda )$ with $\operatorname {supp}\beta \subset M\setminus U$ , then

$$\begin{align*}\|D_z\beta\|\ge C\mu\,\|\beta\|. \end{align*}$$

Proposition 4.5. For all $m\in \mathbb {N}_0$ , if $\mu \gg 0$ , then

$$\begin{align*}\|D_ze_{p,z}\|_m\le |\nu|^me^{-c_m\mu}\;,\quad\|D_ze_{p,z}\|_{m,i\nu}\le e^{-c_m\mu}. \end{align*}$$

Proof. From Proposition 4.1 (iii) and Equations (2.2), (4.9) and (4.10), we get

(4.12) $$ \begin{align} D_ze_{p,z}=D_z\Big(\frac{\rho_p}{a_\mu}e^{\prime}_{p,z}\Big) =e^{-i\nu h_p}\frac1{a_\mu}\Big(\frac{\pi}4\Big)^{n/4}\hat c(d\rho_p)e^{\prime}_{p,\mu}. \end{align} $$

Thus, the stated estimate of $\|D_ze_{p,z}\|_m$ is true by Equations (4.9) and (4.11), since $d\rho _p=0$ around p, and using the definition of $h_p$ and the condition $4r<1$ . (When $\nu =0$ , this is indicated in [Reference Zhang75, Eq. (6.17)].)

By Equation (2.11), for all $k\in \mathbb {N}_0$ and $p\in \mathcal {X}$ , the form $D_{i\nu }^kD_ze_{p,z}$ is the extension by zero of the form $e^{-i\nu h_p}D^kD_\mu e_{p,\mu }$ on $U_p$ . Then the stated estimate of $\|D_ze_{p,z}\|_{m,i\nu }$ follows from the case $\nu =0$ .

Proposition 4.6. If $\mu \gg 0$ , then

$$\begin{align*}\|D_ze_{p,z}\|_{L^\infty}\le e^{-c\mu}. \end{align*}$$

Consider the partition of $\operatorname {spec}\Delta _z$ into its intersections with $[0,1]$ and $(1,\infty )$ , called the small and large spectrum; the term small/large eigenvalues may be also used. Let $E_{z,\mathrm {sm}}\subset \Omega (M)$ denote the graded finite-dimensional subspace generated by the eigenforms of the small eigenvalues, let $E_{z,\mathrm {la}}=E_{z,\mathrm {sm}}^\perp $ in $L^2(M;\Lambda )$ , and let $P_{z,\mathrm {sm/la}}$ be the orthogonal projection to $E_{z,\mathrm {sm/la}}$ , called small/large projection. Moreover, $(\Omega (M),d_z)$ splits into a topological direct sum of the subcomplexes $E_{z,\mathrm {sm}}$ and $E_{z,\mathrm {la}}\cap \Omega (M)$ , called the small and large complexes, and Equation (2.6) gives

(4.13) $$ \begin{align} H^{\bullet}(E_{z,\mathrm{sm}},d_z)\cong H_z^{\bullet}(M)\;,\quad H^{\bullet}(E_{z,\mathrm{la}}\cap \Omega(M),d_z)=0. \end{align} $$

For any operator B defined on $\Omega (M)$ or $L^2(M;\Lambda )$ , let $B_{z,\mathrm {sm/la}}=BP_{z,\mathrm {sm/la}}$ .

Proposition 4.7. For all $m\in \mathbb {N}_0$ , $\mu \gg 0$ and $\alpha \in E_z$ ,

$$\begin{align*}\|\alpha-P_{z,\mathrm{sm}}\alpha\|_{m,i\nu}\le e^{-c_m\mu}\|\alpha\|. \end{align*}$$

Proof. This follows like [Reference Zhang75, Lemma 5.8 and Theorem 6.7], using $\|\ \|_{m,i\nu }$ instead of $\|\ \|_m$ . The following are the main steps of the proof.

Let $\mathbb {S}^1=\{\,\omega \in \mathbb {C}\mid |\omega |=1\,\}$ . With the argument of the proof of [Reference Zhang75, Eq. (5.27)], using Proposition 4.4, we get that, for all $\alpha \in H^1(M;\Lambda )$ , $w\in \mathbb {S}^1$ and $\mu \gg 0$ ,

$$\begin{align*}\|(w-D_z)\alpha\|\ge C\|\alpha\|. \end{align*}$$

Thus, $w-D_z:H^1(M;\Lambda )\to L^2(M;\Lambda )$ is bijective, and, for all $\beta \in L^2(M;\Lambda )$ , $w\in \mathbb {S}^1$ and $\mu \gg 0$ ,

(4.14) $$ \begin{align} \big\|(w-D_z)^{-1}\beta\big\|\le C^{-1}\|\beta\|. \end{align} $$

On the other hand, arguing like in the proof of [Reference Zhang75, Eq. (6.18)], it follows that, for all $\gamma \in H^m(M;\Lambda )$ , $w\in \mathbb {S}^1$ and $\mu \gg 0$ ,

$$\begin{align*}\|\gamma\|_{m,i\nu}\le C_m\big(\|(w-D_z)\gamma\big\|_{m-1,i\nu}+\mu\|\gamma\|_{m-1,i\nu}+\|\gamma\|\big). \end{align*}$$

Continuing by induction on $m\in \mathbb {N}_0$ , we obtain

$$\begin{align*}\|\gamma\|_{m,i\nu}\le C_m\Big(\mu^m\|\gamma\|+\sum_{k=1}^m\mu^{k-1}\|(w-D_z)\gamma\big\|_{m-k,i\nu}\Big). \end{align*}$$

In other words, for all $\beta \in H^{m-1}(M;\Lambda )$ ,

$$\begin{align*}\big\|(w-D_z)^{-1}\beta\big\|_{m,i\nu}\le C_m\Big(\mu^m\big\|(w-D_z)^{-1}\beta\big\|+\sum_{k=1}^m\mu^{k-1}\|\beta\|_{m-k,i\nu}\Big). \end{align*}$$

Applying Equation (4.14) to this inequality, we get, for $m\ge 1$ ,

(4.15) $$ \begin{align} \big\|(w-D_z)^{-1}\beta\big\|_{m,i\nu}\le C_m\mu^m\|\beta\|_{m-1,i\nu}. \end{align} $$

From Equations (4.14) and (4.15) and Proposition 4.5, it follows that, for $m\in \mathbb {N}_0$ ,

(4.16) $$ \begin{align} \big\|(w-D_z)^{-1}D_z e_{p,z}\big\|_{m,i\nu}=O\big(e^{-c_m\mu}\big) \end{align} $$

as $\mu \to +\infty $ , uniformly on $w\in \mathbb {S}^1$ . But, endowing $\mathbb {S}^1$ with the counterclockwise orientation, basic spectral theory gives (see, e.g., [Reference Dunford and Schwartz25, Section VII.3])

(4.17) $$ \begin{align} P_{z,\mathrm{sm}} e_{p,z}- e_{p,z}&=\frac{1}{2\pi i}\int_{\mathbb{S}^1}\big((w-D_z)^{-1}-w^{-1}\big) e_{p,z}\,dw \nonumber \\ &=\frac{1}{2\pi i}\int_{\mathbb{S}^1}w^{-1}(w-D_z)^{-1}D_z e_{p,z}\,dw. \end{align} $$

The result follows using Equation (4.16) in Equation (4.17).

Corollary 4.8. For $\mu \gg 0$ and $\alpha \in E_z$ ,

$$\begin{align*}\|\alpha-P_{z,\mathrm{sm}}\alpha\|_{L^\infty}\le e^{-c\mu}\|\alpha\|. \end{align*}$$

Proof. Apply Propositions 2.2 and 4.7.

Alternatively, the proof of Proposition 4.7 can be modified as follows to get this result (some step of this alternative argument will be used later). Iterating Equation (4.15), we get

$$\begin{align*}\big\|(w-D_z)^{-1}\beta\big\|_{m,i\nu}\le C^{\prime}_m\mu^{(m+1)m/2}\|\beta\|\;, \end{align*}$$

for all $\beta \in L^2(M;\Lambda )$ . Then, by Proposition 2.2,

(4.18) $$ \begin{align} \big\|(w-D_z)^{-1}\beta\big\|_{L^\infty}\le C\mu^{(m+1)m/2}\|\beta\|. \end{align} $$

Thus, by Proposition 4.5,

$$\begin{align*}\big\|(w-D_z)^{-1}D_z e_{p,z}\big\|_{L^\infty}=O\big(e^{-c_m\mu}\big) \end{align*}$$

as $\mu \to +\infty $ . Finally, apply this expression in Equation (4.17).

When $\mu \gg 0$ , Equation (4.5) also follows from Corollary 4.9 and Equations (2.8) and (4.13).

Theorem 4.10 (Cf. [Reference Burghelea and Haller17, Theorem 3])

We have

$$\begin{align*}\operatorname{spec}\Delta_z\subset\big[0,e^{-c|\mu|}\big]\cup\big[C|\mu|,\infty\big). \end{align*}$$

Proof. First, we establish the theorem for $|\mu |\gg 0$ , and then the constants will be changed to cover all $\mu $ .

We can assume $\mu \ge 0$ according to Remark 4.2. By Propositions 4.4, 4.7 and 2.4, for all $\alpha \in E_z$ ,

$$ \begin{align*} \|D_z P_{z,\mathrm{sm}}\alpha\| &\le\|D_z\alpha\|+\|D_z(\alpha- P_{z,\mathrm{sm}}\alpha)\|\le\|D_z\alpha\|+\|\alpha- P_{z,\mathrm{sm}}\alpha\|_{1,z}\\ &\le\| P_z^\perp D_z\alpha\|+C(\mu\|\alpha- P_{z,\mathrm{sm}}\alpha\|+\|\alpha- P_{z,\mathrm{sm}}\alpha\|_{1,i\nu})\\ &\le\big(e^{-c\mu}+C\big(\mu e^{-c_0\mu}+e^{-c_1\mu}\big)\big)\|\alpha\|. \end{align*} $$

Hence, by Corollary 4.9, for all $\beta \in E_{z,\mathrm {sm}}$ ,

$$\begin{align*}0\le\langle\Delta_z\beta,\beta\rangle=\|D_z\beta\|^2\le e^{-c\mu}\,\|\beta\|^2. \end{align*}$$

This shows that

(4.19) $$ \begin{align} \operatorname{spec}\Delta_z\cap[0,1]\subset\big[0,e^{-c\mu}\big]. \end{align} $$

Now, let $\phi \in E_{z,\mathrm {la}}\cap H^1(M;\Lambda )$ , and write $\alpha = P_z\phi \in E_z$ and $\beta = P_z^\perp \phi \in E_z^\perp \cap H^1(M;\Lambda )$ . By Proposition 4.7,

$$\begin{align*}\|\alpha\|^2=\langle\alpha,\phi\rangle=\langle\alpha- P_{z,\mathrm{sm}}\alpha,\phi\rangle\le\|\alpha- P_{z,\mathrm{sm}}\alpha\|\|\phi\|\le e^{-c_0\mu}\|\alpha\|\|\phi\|\;, \end{align*}$$

yielding

$$\begin{align*}\|\alpha\|\le e^{-c_0\mu}\|\phi\|. \end{align*}$$

So

$$\begin{align*}\|\beta\|=\|\phi-\alpha\|\ge\|\phi\|-\|\alpha\|\ge\big(1-e^{-c_0\mu}\big)\|\phi\|. \end{align*}$$

Then, by Proposition 4.4,

$$ \begin{align*} \|D_z\phi\|&\ge\|D_z\beta\|-\|D_z\alpha\| \ge\| P_z^\perp D_z\beta\|-e^{-c\mu}\|\alpha\|\\ &\ge C\sqrt{\mu}\,\|\beta\|-e^{-c\mu}\|\phi\| \ge\big(C\sqrt{\mu}\big(1-e^{-c_0\mu}\big)-e^{-c\mu}\big)\|\phi\|. \end{align*} $$

Therefore, for all $\phi \in E_{z,\mathrm {la}}\cap H^1(M;\Lambda )$ ,

$$\begin{align*}\langle\Delta_z\phi,\phi\rangle=\|D_z\phi\|^2\ge C\mu\|\phi\|^2. \end{align*}$$

This proves that

(4.20) $$ \begin{align} \operatorname{spec}\Delta_z\cap(1,\infty)\subset[C\mu,\infty). \end{align} $$

The inclusions (4.19) and (4.20) give the result for $\mu \gg 0$ . But, in those inclusions, we can take c and C so small that, if one of them is not true for some $\mu \ge 0$ , then $C\mu \le e^{-c\mu }$ .

4.3. Ranks of some projections in the small complex

Recall that $(\Pi _z^\perp )_{\mathrm {sm},k}$ , $\Pi ^1_{z,\mathrm {sm},k}$ and $\Pi ^2_{z,\mathrm {sm},k}$ denote the orthogonal projections to the images of $\Delta _{z,\mathrm {sm},k}$ , $d_{z,\mathrm {sm},k-1}$ and $\delta _{z,\mathrm {sm},k+1}$ , respectively. Let $m_{z,k}$ , $m_{z,k}^1$ and $m_{z,k}^2$ be the corresponding ranks (or traces) of these projections. They satisfy

(4.21) $$ \begin{align} m_{z,k}=m_{z,k}^1+m_{z,k}^2\;,\quad m_{z,0}^1=m_{z,n}^2=0\;,\quad m_{z,k}^2=m_{z,k+1}^1\;, \end{align} $$

where the last equality is true because $d_z:\operatorname {im}\delta _z\to \operatorname {im} d_z$ is an isomorphism. For $\mu \gg 0$ , we have $m_{z,k},m_{z,k}^j\le |\mathcal {X}_k|$ by Corollary 4.9 and Equation (4.21).

Lemma 4.11. The numbers $m_{z,k}^j$ are determined by the numbers $m_{z,k}$ :

$$\begin{align*}m_{z,k+1}^1=m_{z,k}^2=\sum_{p=0}^k(-1)^{k-p}m_{z,p} =\sum_{q=k+1}^n(-1)^{q-k-1}m_{z,q}. \end{align*}$$

Proof. By Equations (2.8) and (4.5) and Lemma 4.12,

$$\begin{align*}\operatorname{Str}\big((\Pi_z^\perp)_{\mathrm{sm}}\big)=\sum_k(-1)^k|\mathcal{X}_k|-\sum_k(-1)^k\beta_z^k=\chi(M)-\chi(M)=0. \; \end{align*}$$

Lemma 4.14. If M is oriented, then, for $k=0,\dots ,n$ ,

$$\begin{align*}m_{z,k}=m_{-\bar z,n-k}=m_{-z,n-k}\;,\quad m_{z,k}^1=m_{-\bar z,n-k}^2=m_{-z,n-k}^2. \end{align*}$$

Proof. This is true because, by Equation (2.15),

$$\begin{align*} (\Pi_z^\perp)_{\mathrm{sm},k}\,\star&=\star\,(\Pi_{-\bar z}^\perp)_{\mathrm{sm},n-k}\;,\quad \Pi^1_{z,\mathrm{sm},k}\,\star=\star\,\Pi^2_{-\bar z,\mathrm{sm},n-k}\;,\\ (\Pi_z^\perp)_{\mathrm{sm},k}\,\bar\star&=\bar\star\,(\Pi_{-z}^\perp)_{\mathrm{sm},n-k}\;,\quad \Pi^1_{z,\mathrm{sm},k}\,\bar\star=\bar\star\,\Pi^2_{-z,\mathrm{sm},n-k}. \; \end{align*}$$

By Corollary 4.15, we write $m_k=m_k(\eta )=m_{z,k}$ and $m_k^j=m_k^j(\eta )=m_{z,k}^j$ for $\mu \gg 0$ .

Corollary 4.16. If M is oriented, then, for $k=0,\dots ,n$ ,

$$\begin{align*}m_k(\eta)=m_{n-k}(-\eta)\;,\quad m_k^1(\eta)=m_{n-k}^2(-\eta)=m_{n-k+1}^1(-\eta). \end{align*}$$

Corollary 4.17. For $\mu \gg 0$ ,

$$\begin{align*}\operatorname{Str}(\Pi_{z,\mathrm{sm}}^1)=-\operatorname{Str}(\Pi_{z,\mathrm{sm}}^2)=\sum_{k=0}^n(-1)^kkm_k. \end{align*}$$

If moreover M is oriented and n is even, then

$$\begin{align*}\sum_{k=0}^n(-1)^kkm_k=\sum_{k=0}^n(-1)^kk|\mathcal{X}_k|-\frac n2\chi(M). \end{align*}$$

Proof. Corollary 4.13 gives the first equality. By Lemma 4.11 and Corollary 4.13,

$$ \begin{align*} \operatorname{Str}(\Pi_{z,\mathrm{sm}}^1)=\sum_{k=0}^n(-1)^k\sum_{q=k}^n(-1)^{q-k}m_q =\sum_{q=0}^n(-1)^q(q+1)m_q=\sum_{q=0}^n(-1)^qqm_q. \end{align*} $$

Now, assume M is oriented and n is even. Then, by Equations (2.8), (2.9) and (2.14),

$$ \begin{align*} \sum_{k=0}^n(-1)^kk\beta_{\mathrm{No}}^k&=\sum_{l=0}^n(-1)^{n-l}(n-l)\beta_{\mathrm{No}}^{n-l} =\sum_{l=0}^n(-1)^l(n-l)\beta_{\mathrm{No}}^l\\ &=n\chi(M)-\sum_{l=0}^n(-1)^ll\beta_{\mathrm{No}}^l. \end{align*} $$

Hence, the last equality of the statement follows from Lemma 4.12.

4.4. Asymptotic properties of the small projection

Proposition 4.19. For every $\tau \in \mathbb {R}$ , on $L^2(M;\Lambda )$ , as $\mu \to +\infty $ ,

$$\begin{align*}P_{z,\mathrm{sm}}=P_z+O\big(e^{-c\mu}\big) =P_{z,\mathrm{sm}}P_{z+\tau,\mathrm{sm}}P_{z,\mathrm{sm}}+O\big(\mu^{-2}\big) =P_{z+\tau,\mathrm{sm}}+O\big(\mu^{-1}\big). \end{align*}$$

Proof. By Corollary 4.9, for $\mu \gg 0$ , the elements $P_{z,\mathrm {sm}} e_{p,z}$ ( $p\in \mathcal {X}$ ) form a base of $E_{z,\mathrm {sm}}$ . Applying the Gram–Schmidt process to this base, we get an orthonormal base $\tilde e_{p,z}$ . By Proposition 4.7,

(4.22) $$ \begin{align} \tilde e_{p,z}=e_{p,z}+O\big(e^{-c\mu}\big). \end{align} $$

This gives the first equality of the statement: for any $\alpha \in L^2(M;\Lambda )$ ,

$$\begin{align*}P_z\alpha=\sum_{p\in\mathcal{X}}\langle\alpha, e_{p,z}\rangle e_{p,z}=\sum_{p\in\mathcal{X}}\langle\alpha,\tilde e_{p,z}\rangle\tilde e_{p,z}+O\big(e^{-c\mu}\big)\|\alpha\| =P_{z,\mathrm{sm}}\alpha+O\big(e^{-c\mu}\big)\|\alpha\|. \end{align*}$$

Since the sets $U_p$ ( $p\in \mathcal {X}$ ) are disjoint one another, for $p\ne q$ in $\mathcal {X}$ ,

(4.23) $$ \begin{align} \langle e_{p,z},e_{q,z+\tau}\rangle=0. \end{align} $$

On the other hand, by Equations (4.8)–(4.11), we can also assume

(4.24) $$ \begin{align} \langle e_{p,z}, e_{p,z+\tau}\rangle&=\langle e^{-i\nu h_p}e_{p,\mu},e^{-i\nu h_p} e_{p,\mu+\tau}\rangle =\langle e_{p,\mu}, e_{p,\mu+\tau}\rangle\nonumber\\ &=\frac{(\mu(\mu+\tau))^{n/4}}{\pi^{n/2}}\big\langle\rho_pe^{-\mu|x_p|^2/2},\rho_pe^{-(\mu+\tau)|x_p|^2/2}\big\rangle +O\big(e^{-c\mu}\big)\nonumber\\ &=\frac{(\mu(\mu+\tau))^{n/4}}{\pi^{n/2}}\int_{\mathbb{R}^n}e^{-(\mu+\tau/2)|x_p|^2}\,dx_p+O\big(e^{-c\mu}\big)\nonumber\\ &=\frac{(\mu(\mu+\tau))^{n/4}}{(\mu+\tau/2)^{n/2}}+O\big(e^{-c\mu}\big)=1+O\big(\mu^{-2}\big)\;, \end{align} $$

where $dx_p=dx_p^1\dots dx_p^n=\operatorname {dvol}(x_p)$ . Combining Equation (4.22) for z and $z+\tau $ with Equations (4.23) and (4.24), we obtain

(4.25) $$ \begin{align} P_{z+\tau,\mathrm{sm}}\tilde e_{p,z}&=\sum_{q\in\mathcal{X}}\langle \tilde e_{p,z},\tilde e_{q,z+\tau}\rangle\tilde e_{q,z+\tau} =\sum_{q\in\mathcal{X}}\langle e_{p,z}, e_{q,z+\tau}\rangle e_{q,z+\tau}+O\big(e^{-c\mu}\big)\nonumber\\ &= e_{p,z+\tau}+O\big(\mu^{-2}\big)=\tilde e_{p,z+\tau}+O\big(\mu^{-2}\big). \end{align} $$

Repeating Equation (4.25) interchanging the roles of z and $z+\tau $ , we get

$$\begin{align*}P_{z,\mathrm{sm}}P_{z+\tau,\mathrm{sm}}\tilde e_{p,z}=P_{z,\mathrm{sm}}\tilde e_{p,z+\tau}+O\big(\mu^{-2}\big)=\tilde e_{p,z}+O\big(\mu^{-2}\big). \end{align*}$$

This gives the second equality of the statement: For any $\alpha \in L^2(M;\Lambda )$ ,

$$ \begin{align*} P_{z,\mathrm{sm}}\alpha&=\sum_{p\in\mathcal{X}}\langle\alpha,\tilde e_{p,z}\rangle\tilde e_{p,z} =P_{z,\mathrm{sm}}P_{z+\tau,\mathrm{sm}}\sum_{p\in\mathcal{X}}\langle\alpha,\tilde e_{p,z}\rangle\tilde e_{p,z} +O\big(\mu^{-2}\big)\|\alpha\|\\&=P_{z,\mathrm{sm}}P_{z+\tau,\mathrm{sm}}P_{z,\mathrm{sm}}\alpha+O\big(\mu^{-2}\big)\|\alpha\|. \end{align*} $$

By Equation (4.25),

$$ \begin{align*} \|\tilde e_{p,z}-\tilde e_{p,z+\tau}\big\|^2 &=\|\tilde e_{p,z}\|^2-2\Re\langle\tilde e_{p,z},\tilde e_{p,z+\tau}\rangle+\|\tilde e_{p,z+\tau}\|^2=2-2\Re\langle P_{z+\tau,\mathrm{sm}}\tilde e_{p,z},\tilde e_{p,z+\tau}\rangle\\ & =2-2\Re\langle\tilde e_{p,z+\tau},\tilde e_{p,z+\tau}\rangle+O\big(\mu^{-2}\big)=O\big(\mu^{-2}\big)\;, \end{align*} $$

which means

(4.26) $$ \begin{align} \tilde e_{p,z}=\tilde e_{p,z+\tau}+O\big(\mu^{-1}\big). \end{align} $$

The last stated equality follows from Equations (4.25) and (4.26): For any $\alpha \in L^2(M;\Lambda )$ ,

$$ \begin{align*} P_{z,\mathrm{sm}}\alpha &=\sum_{p\in\mathcal{X}}\langle\alpha,\tilde e_{p,z}\rangle\tilde e_{p,z} =\sum_{p\in\mathcal{X}}\langle\alpha,\tilde e_{p,z+\tau}\rangle\tilde e_{p,z+\tau}+O\big(\mu^{-1}\big)\alpha\\ &=P_{z+\tau,\mathrm{sm}}\alpha+O\big(\mu^{-1}\big)\alpha. \; \end{align*} $$

Corollary 4.20. For every $\tau \in \mathbb {R}$ , on $L^2(M;\Lambda )$ ,

$$\begin{align*}d_{z+\tau,\mathrm{sm}}-d_{z+\tau}P_{z,\mathrm{sm}}=O\big(\mu^{-1}\big)\quad(\mu\to+\infty). \end{align*}$$

Proof. Since $d_{z+\tau }=d_z+\tau \,{\eta \wedge }$ , it follows from Theorem 4.10 that $d_{z+\tau }$ is bounded on $E_{z,\mathrm {sm}}+E_{z+\tau ,\mathrm {sm}}$ , uniformly on $\mu \gg 0$ . Hence, by Proposition 4.19,

$$\begin{align*}d_{z+\tau,\mathrm{sm}}-d_{z+\tau}P_{z,\mathrm{sm}}=d_{z+\tau}(P_{z+\tau,\mathrm{sm}}-P_{z,\mathrm{sm}})=O\big(\mu^{-1}\big). \; \end{align*}$$

Proposition 4.21. On $L^2(M;\Lambda )$ ,

$$\begin{align*}P_{z,\mathrm{sm}}\,{\eta\wedge},{\eta\wedge}\,P_{z,\mathrm{sm}}=O\big(\mu^{-1}\big)\quad(\mu\to+\infty). \end{align*}$$

Proof. By Theorem 4.10, for all $\alpha \in \Omega (M)$ ,

$$\begin{align*}\|d_zP_{z,\mathrm{sm}}\alpha\|^2=\langle\delta_zd_zP_{z,\mathrm{sm}}\alpha,P_{z,\mathrm{sm}}\alpha\rangle \le\langle\Delta_zP_{z,\mathrm{sm}}\alpha,P_{z,\mathrm{sm}}\alpha\rangle \le O\big(e^{-c\mu}\big)\;, \end{align*}$$

yielding $d_zP_{z,\mathrm {sm}}=O\big (e^{-c\mu }\big )$ . This is also true with the parameter $z+1$ . So, by Corollary 4.20,

$$\begin{align*}{\eta\wedge}\,P_{z,\mathrm{sm}}=(d_{z+1}-d_z)P_{z,\mathrm{sm}} =d_{z+1}P_{z+1,\mathrm{sm}}-d_zP_{z,\mathrm{sm}}+O\big(\mu^{-1}\big) =O\big(\mu^{-1}\big). \; \end{align*}$$

4.5. Derivatives of the small projection

Proposition 4.23. We have

$$\begin{align*}\operatorname{rank}\partial_zP_{z,\mathrm{sm}}\le2|\mathcal{X}|\quad(\mu\gg0)\;,\quad\partial_zP_{z,\mathrm{sm}}=O\big(\mu^{-1}\big)\quad(\mu\to+\infty). \end{align*}$$

Proof. By Equation (2.4) and Theorem 4.10, for $\mu \gg 0$ and every $\omega \in \mathbb {S}^1$ , a standard computation gives

(4.27) $$ \begin{align} \partial_z\big((w-D_z)^{-1}\big)=(w-D_z)^{-1}\,{\eta\wedge}\,(w-D_z)^{-1}. \end{align} $$

Then, by Equation (4.14), $\partial _z\big ((w-D_z)^{-1}\big )$ defines an operator on $L^2(M;\Lambda )$ , bounded uniformly on $w\in \mathbb {S}^1$ and $z\in \mathbb {C}$ . By Equation (4.14) and Proposition 4.21, we also get

$$ \begin{align*} & P_{z,\mathrm{la/sm}}\partial_z\big((w-D_z)^{-1}\big)\,P_{z,\mathrm{sm/la}}\\ &\quad =(w-D_z)^{-1}P_{z,\mathrm{la/sm}}\,{\eta\wedge}\,P_{z,\mathrm{sm/la}}(w-D_z)^{-1} =O\big(\mu^{-1}\big)\;, \end{align*} $$

uniformly on $w\in \mathbb {S}^1$ .

On the other hand, applying again basic spectral theory, we obtain

$$\begin{align*}P_{z,\mathrm{sm}}=\frac{1}{2\pi i}\int_{\mathbb{S}^1}(w-D_z)^{-1}\,dw \end{align*}$$

for $\mu \gg 0$ , yielding

(4.28) $$ \begin{align} \dot P_{z,\mathrm{sm}}=\frac{1}{2\pi i}\int_{\mathbb{S}^1}\partial_z\big((w-D_z)^{-1}\big)\,dw\;, \end{align} $$

which defines an operator on $L^2(M;\Lambda )$ , bounded uniformly on z.

Using that $P_{z,\mathrm {sm}}$ is an orthogonal projection, the argument of the proof of [Reference Berline, Getzler and Vergne7, Proposition 9.37] shows that

(4.29) $$ \begin{align} \dot P_{z,\mathrm{sm}}=P_{z,\mathrm{la}}\dot P_{z,\mathrm{sm}}P_{z,\mathrm{sm}} +P_{z,\mathrm{sm}}\dot P_{z,\mathrm{sm}}P_{z,\mathrm{la}}. \end{align} $$

So $\operatorname {rank}\dot P_{z,\mathrm {sm}}\le 2\operatorname {rank} P_{z,\mathrm {sm}}\le 2|\mathcal {X}|$ by Corollary 4.9, and

$$ \begin{align*} \dot P_{z,\mathrm{sm}}&=\frac{1}{2\pi i}\int_{\mathbb{S}^1}P_{z,\mathrm{la}}\partial_z\big((w-D_z)^{-1}\big)\,P_{z,\mathrm{sm}}\,dw\\ &\phantom{=\text{}}\text{}+\frac{1}{2\pi i}\int_{\mathbb{S}^1}P_{z,\mathrm{sm}}\partial_z\big((w-D_z)^{-1}\big)\,P_{z,\mathrm{la}}\,dw =O\big(\mu^{-1}\big). \; \end{align*} $$

Lemma 4.24. For all $p\in \mathcal {X}$ ,

$$\begin{align*}\partial_ze_{p,z}=\bigg(\frac n{8\mu}-\frac{|x^+_p|^2}{2}+O(e^{-c\mu})\bigg)e_{p,z}\quad(\mu\to+\infty). \end{align*}$$

Proof. Using integration by parts, and since $\rho $ is an even function and $\rho '$ vanishes on $[-r,r]$ , we obtain

(4.30) $$ \begin{align} \int_{-2r}^{2r}\rho(x)^2x^2e^{-\mu x^2}\,dx &=\frac1{2\mu}\int_{-2r}^{2r}(2\rho(x)\rho'(x)x+\rho(x)^2)e^{-\mu x^2}\,dx \nonumber\\ &=\frac1{2\mu}\Big(\frac\pi\mu\Big)^{\frac12}+O(e^{-c\mu}). \end{align} $$

So

$$ \begin{align*} \partial_\mu a_\mu&=\partial_\mu\bigg(\bigg(\int_{-2r}^{2r}\rho(x)^2e^{-\mu x^2}\,dx\bigg)^{\frac n2}\bigg)\\ &=-\frac n2\bigg(\int_{-2r}^{2r}\rho(x)^2e^{-\mu x^2}\,dx\bigg)^{\frac n2-1} \int_{-2r}^{2r}\rho(x)^2x^2e^{-\mu x^2}\,dx\\ &=-\frac n2\Big(\frac\pi\mu\Big)^{\frac n4-\frac12}\frac1{2\mu}\Big(\frac\pi\mu\Big)^{\frac12}+O(e^{-c\mu}) =-\frac n{4\mu}\Big(\frac\pi\mu\Big)^{\frac n4}+O(e^{-c\mu}). \end{align*} $$

Hence, by Equation (4.11),

(4.31) $$ \begin{align} \partial_\mu\Big(\frac1{a_\mu}\Big) =-\frac{\partial_\mu a_\mu}{a_\mu^2} =\frac n{4\mu}\Big(\frac\pi\mu\Big)^{\frac n4} \Big(\frac\mu\pi\Big)^{\frac n2}+O(e^{-c\mu}) =\frac n{4\mu}\Big(\frac\mu\pi\Big)^{\frac n4}+O(e^{-c\mu}). \end{align} $$

It also follows from Proposition 4.1 (iii) and Equations (4.9), (4.11) and (4.31) that

(4.32) $$ \begin{align} \partial_\mu e_{p,\mu} &=\partial_\mu\Big(\frac{\rho_p}{a_\mu}e^{-\mu|x_p|^2/2}\,dx_p^1\wedge\dots\wedge dx_p^{\operatorname{ind}(p)}\Big) \nonumber\\ &=\bigg(\partial_\mu\Big(\frac1{a_\mu}\Big)a_\mu-\frac{|x_p|^2}2\bigg)e_{p,\mu} =\bigg(\frac n{4\mu}-\frac{|x_p|^2}2+O(e^{-c\mu})\bigg)e_{p,\mu}. \end{align} $$

So, by Equation (4.10),

(4.33) $$ \begin{align} \partial_\mu e_{p,z}=\bigg(\frac n{4\mu}-\frac{|x_p|^2}2+O(e^{-c\mu})\bigg)e_{p,z}\;,\quad \partial_\nu e_{p,z}=-ih_pe_{p,z}. \end{align} $$

Then the result follows using the right-hand side of Equation (4.1).

Proposition 4.25. For all $p\in \mathcal {X}$ ,

$$\begin{align*}\|\partial_z(D_ze_{p,z})\|_{L^\infty}=O(e^{-c\mu})\quad(\mu\to+\infty). \end{align*}$$

Proof. From Equation (4.12), we get

(4.34) $$ \begin{align} \partial_z(D_ze_{p,z}) &=\frac12\bigg(e^{-i\nu h_p} \partial_\mu\Big(\frac1{a_\mu}\Big(\frac\pi\mu\Big)^{\frac n4}\Big)\hat c(d\rho_p)e_{p,\mu} \nonumber\\ &\quad {}+e^{-i\nu h_p}\frac1{a_\mu}\Big(\frac\pi\mu\Big)^{\frac n4}\hat c(d\rho_p)\partial_\mu e_{p,\mu}-h_pe^{-i\nu h_p}\frac1{a_\mu}\Big(\frac\pi\mu\Big)^{\frac n4}\hat c(d\rho_p)e_{p,\mu}\bigg). \end{align} $$

By Equations (4.11) and (4.31),

(4.35) $$ \begin{align} \partial_\mu\Big(\frac1{a_\mu}\Big(\frac\pi\mu\Big)^{\frac n4}\Big) &=\partial_\mu\Big(\frac1{a_\mu}\Big)\Big(\frac\pi\mu\Big)^{\frac n4} -\frac{n\pi}{4a_\mu\mu^2}\Big(\frac\pi\mu\Big)^{\frac n4-1}\nonumber\\ &=\frac n{4\mu}\Big(\frac\mu\pi\Big)^{\frac n4}\Big(\frac\pi\mu\Big)^{\frac n4} -\frac{n\pi}{4\mu^2}\Big(\frac\pi\mu\Big)^{\frac n4-1}\Big(\frac\mu\pi\Big)^{\frac n4}+O(e^{-c\mu}) =O(e^{-c\mu}). \end{align} $$

The result follows applying Proposition 4.1 (iii) and Equations (4.9), (4.11), (4.32) and (4.35) to Equation (4.34), and using that $d\rho _p=0$ around p.

Proposition 4.26. For every $p\in \mathcal {X}$ ,

$$\begin{align*}\|\partial_z(P_{z,\mathrm{sm}}e_{p,z}-e_{p,z})\|_{L^\infty}=O(e^{-c\mu})\quad(\mu\to+\infty). \end{align*}$$

Proof. By Equation (4.17),

$$ \begin{align*} \partial_z(P_{z,\mathrm{sm}}e_{p,z}-e_{p,z}) &=\frac1{2\pi i}\int_{\mathbb{S}^1}w^{-1}\partial_z\big((w-D_z)^{-1}\big)D_ze_{p,z}\,dw\\ &\quad {}+\frac1{2\pi i}\int_{\mathbb{S}^1}w^{-1}(w-D_z)^{-1}\partial_z(D_ze_{p,z})\,dw. \end{align*} $$

Now, apply Equations (4.18) and (4.27) and Propositions 4.6 and 4.25.

5.1. Small and large zeta invariants

According to Sections 3.2 and 4.2, if B is an operator in $L^2(M;\Lambda )$ so that $\zeta (s,\Delta _z,B)$ is defined, we have

$$\begin{align*}\zeta(s,\Delta_z,B)=\zeta_{\mathrm{sm}}(s,\Delta_z,B)+\zeta_{\mathrm{la}}(s,\Delta_z,B)\;, \end{align*}$$

where

$$\begin{align*}\zeta_{\mathrm{sm/la}}(s,\Delta_z,B)=\zeta(s,\Delta_z,B_{z,\mathrm{sm/la}}). \end{align*}$$

These are the contributions from the small/large spectrum to $\zeta (s,\Delta _z,B)$ , which are called the small/large zeta functions of $(\Delta _z,B)$ . In particular, we can write

$$\begin{align*}\zeta(s,z)=\zeta_{\mathrm{sm}}(s,z)+\zeta_{\mathrm{la}}(s,z)\;, \end{align*}$$

where $\zeta _{\mathrm {sm/la}}(s,z)=\zeta _{\mathrm {sm/la}}(s,z,\eta )$ is the small/large zeta function of $(\Delta _z,{\eta \wedge }\,D_z\mathsf {w})$ . Since $\zeta _{\mathrm {sm}}(s,z)$ is an entire function, $\zeta _{\mathrm {la}}(s,z)$ has the same poles as $\zeta (s,z)$ (Remark 3.10), with the same residues. The value $\zeta _{\mathrm {sm/la}}(1,z)$ will be called the small/large zeta invariant of $(M,g,\eta ,z)$ . The following results follow like Corollaries 3.9 and 3.11.

Corollary 5.1. If $\Re s>1/2$ , then

$$\begin{align*}\zeta_{\mathrm{la}}(s,z)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\operatorname{Str}\big({\eta\wedge}\,D_ze^{-t\Delta_z}P_{z,\mathrm{la}}\big)\,dt\;, \end{align*}$$

where the integral is absolutely convergent.

Corollary 5.2. We have

$$ \begin{align*} \zeta_{\mathrm{sm}}(1,z)&=\operatorname{Str}({\eta\wedge}\,D_z^{-1}(\Pi_z^\perp)_{\mathrm{sm}})\;,\\ \zeta_{\mathrm{la}}(1,z)&=\lim_{t\downarrow0}\operatorname{Str}\big({\eta\wedge}\,D_z^{-1}e^{-t\Delta_z}P_{z,\mathrm{la}}\big). \end{align*} $$

5.2. Truncated heat invariants of perturbed operators

For $k=0,\dots ,n$ , let $K^{\prime }_{z,k,t}(x,y)$ and $\widetilde K_{z,k,t}(x,y)$ denote the Schwartz kernels of $e^{-t\Delta _{z,k}}\Pi _z^\perp $ and $e^{-t\Delta _{z,k}}P_{z,\mathrm {la},k}$ , respectively. According to Section 3.1, their restrictions to the diagonal have asymptotic expansions (as $t\downarrow 0$ ),

(5.1) $$ \begin{align} K^{\prime}_{z,k,t}(x,x)\sim\sum_{l=0}^\infty e^{\prime}_{k,l}(x,z)t^{(l-n)/2}\;,\quad \widetilde K_{z,k,t}(x,x)\sim\sum_{l=0}^\infty\tilde e_{k,l}(x,z)t^{(l-n)/2}. \end{align} $$

We have

(5.2) $$ \begin{align} e^{\prime}_{k,l}(x,z)&= \begin{cases} e_{k,l}(x,z) & \text{if}\ l<n\\ e_{k,n}(x,z)-\beta_z^k & \text{if}\ l=n\;, \end{cases}\nonumber\\ \tilde e_{k,l}(x,z)&= \begin{cases} e_{k,l}(x,z) & \text{if}\ l<n\\ e_{k,n}(x,z)-H_{z,k,0}(x,x) & \text{if}\ l=n\;, \end{cases} \end{align} $$

where $H_{z,k,t}(x,y)$ is the Schwartz kernel of $e^{-t\Delta _{z,k}}P_{z,\mathrm {sm},k}$ , which is defined for all $t\in \mathbb {R}$ and is smooth. We also have asymptotic expansions

(5.3) $$ \begin{align} h^{\prime}_k(t,z)&:=\operatorname{Tr}\big(e^{-t\Delta_{z,k}}\Pi_z^\perp\big)\sim\sum_{l=0}^\infty a^{\prime}_{k,l}(z)t^{(l-n)/2},\quad\, \end{align} $$

(5.4) $$ \begin{align} \tilde h_k(t,z)&:=\operatorname{Tr}\big(e^{-t\Delta_{z,k}}P_{z,\mathrm{la},k}\big)\sim\sum_{l=0}^\infty\tilde a_{k,l}(z)t^{(l-n)/2}. \end{align} $$

By Equations (3.4), (3.5) and (3.9),

(5.5) $$ \begin{align} a^{\prime}_{k,l}(z)&=\int_M\operatorname{str} e^{\prime}_{k,l}(x,z)\,\operatorname{dvol}(x) = \begin{cases} a_{k,l}(z) & \text{if}\ l<n\\ a_{k,l}(z)-\beta_z^k & \text{if}\ l=n. \end{cases} \end{align} $$

(5.6) $$ \begin{align} \tilde a_{k,l}(z)&=\int_M\operatorname{str}\tilde e_{k,l}(x,z)\,\operatorname{dvol}(x)= \begin{cases} a_{k,l}(z) & \text{if}\ l<n\\ a_{k,l}(z)-\dim E_{z,\mathrm{sm}}^k & \text{if}\ l=n. \end{cases} \end{align} $$

The operators $e^{-t\Delta _z}\Pi _z^\perp \mathsf {w}$ and $e^{-t\Delta _z}P_{z,\mathrm {la}}\mathsf {w}$ have Schwartz kernels

$$\begin{align*}K^{\prime}_{z,t}(x,y)=\sum_{k=0}^n(-1)^kK^{\prime}_{z,k,t}(x,y)\;,\quad \widetilde K_{z,t}(x,y)=\sum_{k=0}^n(-1)^k\widetilde K_{z,k,t}(x,y)\;, \end{align*}$$

with induced asymptotic expansions

$$\begin{align*}K^{\prime}_{z,t}(x,x)\sim\sum_{l=0}^\infty e^{\prime}_l(x,z)t^{(l-n)/2}\;,\quad \widetilde K_{z,t}(x,x)\sim\sum_{l=0}^\infty\tilde e_l(x,z)t^{(l-n)/2}\;, \end{align*}$$

where

$$ \begin{gather*} e^{\prime}_l(x,z)=\sum_{k=0}^n(-1)^ke^{\prime}_{k,l}(x,z)\;,\quad \tilde e_l(x,z)=\sum_{k=0}^n(-1)^k\tilde e_{k,l}(x,z). \end{gather*} $$

We also have induced asymptotic expansions,

$$ \begin{gather*} h'(t,z):=\operatorname{Str}\big(e^{-t\Delta_z}\Pi_z^\perp\big)\sim\sum_{l=0}^\infty a^{\prime}_l(z)t^{(l-n)/2}\;,\\ \tilde h(t,z):=\operatorname{Str}\big(e^{-t\Delta_z}P_{z,\mathrm{la}}\big)\sim\sum_{l=0}^\infty\tilde a_l(z)t^{(l-n)/2}\;, \end{gather*} $$

where

$$\begin{align*}a^{\prime}_l(z)=\sum_{k=0}^n(-1)^ka^{\prime}_{k,l}(z)\;,\quad \tilde a_l(z)=\sum_{k=0}^n(-1)^k\tilde a_{k,l}(z). \end{align*}$$

If $\mu \gg 0$ , by Equation (2.9), Corollary 4.9 and Theorem 4.10, $e^{\prime }_{k,l}(x,z)$ and $\tilde e_{k,l}(x,z)$ depend smoothly on z (Section 3.1), and therefore so do $h^{\prime }_k(t,z)$ , $\tilde h_k(t,z)$ , $a^{\prime }_{k,l}(z)$ , $\tilde a_{k,l}(z)$ , $e^{\prime }_l(x,z)$ , $\tilde e_l(x,z)$ , $h'(t,z)$ , $\tilde h(t,z)$ , $a^{\prime }_l(z)$ and $\tilde a_l(z)$ .

5.3. Truncated derived heat invariants of perturbed operators

For $k=0,\dots ,n$ and $j=1,2$ , let

$$ \begin{gather*} h^j_k(t,z)=\operatorname{Tr}\big(e^{-t\Delta_{z,k}}\Pi^j_{z,k}\big)\;,\quad \tilde h^j_k(t,z)=\operatorname{Tr}\big(e^{-t\Delta_{z,k}}\Pi^j_{z,\mathrm{la},k}\big). \end{gather*} $$

Lemma 5.3. We have

$$\begin{align*}h_{k+1}^1(t,z)=h_k^2(t,z)=\sum_{p=0}^k(-1)^{k-p}h^{\prime}_p(t,z) =\sum_{q=k+1}^n(-1)^{q-k-1}h^{\prime}_q(t,z). \end{align*}$$

Proof. This follows by induction on k, using that

$$\begin{align*}h_0^1(t,z)=h_n^2(t,z)=0\;,\quad h^{\prime}_k(t,z)=h_k^1(t,z)+h_k^2(t,z)\;,\quad h_k^2(t,z)=h_{k+1}^1(t,z). \end{align*}$$

The last equality holds because the diagram of Equation (2.7) is commutative.

Let

$$ \begin{gather*} h^j(t,z)=\operatorname{Str}\big(e^{-t\Delta_z}\Pi^j_z\big)=\sum_{k=0}^n(-1)^kh^j_k(t,z)\;,\\ \tilde h^j(t,z)=\operatorname{Str}\big(e^{-t\Delta_z}\Pi^j_{z,\mathrm{la}}\big)=\sum_{k=0}^n(-1)^k\tilde h^j_k(t,z). \end{gather*} $$

Thus,

(5.7) $$ \begin{align} h'(t,z)=h^1(t,z)+h^2(t,z)\;,\quad\tilde h(t,z)=\tilde h^1(t,z)+\tilde h^2(t,z). \end{align} $$

Corollary 5.5. We have

$$\begin{align*}h^1(t,z)=-h^2(t,z)=\sum_{k=0}^n(-1)^kkh^{\prime}_k(t,z)=\operatorname{Str}\big({\mathsf{N}} e^{-t\Delta_z}\Pi_z^\perp\big). \end{align*}$$

Proof. Corollary 5.4 and Equation (5.7) give the first equality. By Lemma 5.3 and Corollary 5.4,

$$ \begin{align*} h^1(t,z)&=\sum_{k=0}^n(-1)^k\sum_{q=k}^n(-1)^{q-k}h^{\prime}_q(t,z)=\sum_{q=0}^n(-1)^q(q+1)h^{\prime}_q(t,z)\\ &=h'(t,z)+\sum_{q=0}^n(-1)^qqh^{\prime}_q(t,z)=\sum_{q=0}^n(-1)^qqh^{\prime}_q(t,z). \; \end{align*} $$

Applying Equation (5.3) and Lemma 5.3, we get

(5.8) $$ \begin{align} h^j_k(t,z)\sim\sum_{l=0}^\infty a_{k,l}^j(z)t^{(l-n)/2}\;,\quad h^j(t,z)\sim\sum_{l=0}^\infty a^j_l(z)t^{(l-n)/2}\;, \end{align} $$

where

$$ \begin{gather*} a^1_{k+1,l}(z)=a^2_{k,l}(z)=\sum_{p=0}^k(-1)^{k-p}a^{\prime}_{p,l}(t,z) =\sum_{q=k+1}^n(-1)^{q-k-1}a^{\prime}_{q,l}(t,z)\;,\\ a^1_l(z)=-a^2_l(z)=\sum_{k=0}^n(-1)^kka^{\prime}_{k,l}(z). \end{gather*} $$

Lemma 5.3, Corollary 5.4 and Equation (5.8) have obvious versions for $\tilde h^j_k(t,z)$ and $\tilde h^j(t,z)$ , with similar proofs. The coefficients of the corresponding asymptotic expansions are denoted by $\tilde a^j_{k,l}(z)$ and $\tilde a^j_l(z)$ .

5.4. Zeta function versus theta function

Consider also the meromorphic function

(5.9) $$ \begin{align} \theta(s,z)=\theta(s,z,\eta)=-\zeta(s,\Delta_z,{\mathsf{N}}\mathsf{w})\;, \end{align} $$

called theta function of $\Delta _z$ , and write

$$\begin{align*}\theta(s,z)=\theta_{\mathrm{sm}}(s,z)+\theta_{\mathrm{la}}(s,z)\;,\\ \end{align*}$$

where

(5.10) $$ \begin{align} \theta_{\mathrm{sm/la}}(s,z)=\theta_{\mathrm{sm/la}}(s,z,\eta)=-\zeta_{\mathrm{sm/la}}(s,\Delta_z,{\mathsf{N}}\mathsf{w}). \end{align} $$

By Corollary 5.5,

(5.11) $$ \begin{align} -\zeta(s,\Delta_z,\Pi_z^1\mathsf{w})&=\zeta(s,\Delta_z,\Pi_z^2\mathsf{w})=\theta(s,z)\;,\nonumber\\ -\zeta_{\mathrm{sm/la}}(s,\Delta_z,\Pi_z^1\mathsf{w})&=\zeta_{\mathrm{sm/la}}(s,\Delta_z,\Pi_z^2\mathsf{w})=\theta_{\mathrm{sm/la}}(s,z). \end{align} $$

Recall that $\zeta (s,z)$ is smooth at $s=1$ (Corollary 3.9). Moreover, $\theta (s,z)$ is smooth at $s=0$ [Reference Seeley66]. The same is true for $\zeta _{\mathrm {la}}(s,z)$ and $\theta _{\mathrm {la}}(s,z)$ .

Proposition 5.8. If $\mu \gg 0$ , then

$$\begin{align*}\partial_z\theta_{\mathrm{la}}(s,z)=s\zeta_{\mathrm{la}}(s+1,z). \end{align*}$$

Proof. Recall that a dot may be used to denote $\partial _z$ . Like in Equation (4.29),

$$\begin{align*}\dot\Pi^1_z=\big(\Pi^1_z\big)^\perp\dot\Pi^1_z\Pi^1_z+\Pi^1_z\dot\Pi^1_z\big(\Pi^1_z\big)^\perp. \end{align*}$$

Therefore, since $\Pi ^1_z$ and $(\Pi ^1_z)^\perp $ commute with $\Delta _z^{-s}$ and $P_{z,\mathrm {la}}$ , for $\Re s\gg 0$ ,

$$\begin{align*}\zeta_{\mathrm{la}}(s,\Delta_z,\dot\Pi^1_z\mathsf{w})=\operatorname{Str}\big(\dot\Pi^1_z\Delta_z^{-s}P_{z,\mathrm{la}}\big)=0\;, \end{align*}$$

yielding $\zeta _{\mathrm {la}}(s,\Delta _z,\dot \Pi ^1_z\mathsf {w})=0$ for all s because this is a meromorphic function. Hence, since $\Delta _z$ and $\Pi ^1_{z,\mathrm {la}}\mathsf {w}$ commute, Proposition 3.1 (i),(v) gives

(5.12) $$ \begin{align} \partial_z\zeta_{\mathrm{la}}(s,\Delta_z,\Pi^1_z\mathsf{w})=-s\zeta_{\mathrm{la}}(s+1,\Delta_z,\dot\Delta_z\Pi^1_z\mathsf{w}) =-s\operatorname{Str}\big(\dot\Delta_z\Delta_z^{-s-1}\Pi^1_{z,\mathrm{la}}\big). \end{align} $$

Next, by Equation (2.4),

(5.13) $$ \begin{align} \dot\Delta_z\Pi^1_{z,\mathrm{la}} =({\eta\wedge}\,\delta_z+\delta_z\,{\eta\wedge})\Pi^1_{z,\mathrm{la}} ={\eta\wedge}\,\delta_z\Pi^1_{z,\mathrm{la}}+\delta_z\,{\eta\wedge}\,\Pi^1_{z,\mathrm{la}}. \end{align} $$

But, since $\Pi ^1_z\delta _z=0$ ,

(5.14) $$ \begin{align} \operatorname{Str}\big(\delta_z\,{\eta\wedge}\,\Delta_z^{-s-1}\Pi^1_{z,\mathrm{la}}\big) =-\operatorname{Str}\big({\eta\wedge}\,\Delta_z^{-s-1}\Pi^1_{z,\mathrm{la}}\delta_z\big)=0. \end{align} $$

From Equations (5.11)–(5.14) and Proposition 3.1 (i), we get

$$ \begin{align*} \partial_z\theta_{\mathrm{la}}(s,z)&=-\partial_z\zeta_{\mathrm{la}}(s,\Delta_z,\Pi^1_z\mathsf{w}) =s\operatorname{Str}\big({\eta\wedge}\,\delta_z\Delta_z^{-s-1}\Pi^1_{z,\mathrm{la}}\big)\\ &=s\operatorname{Str}\big({\eta\wedge}\,D_z\Delta_z^{-s-1}\Pi^1_{z,\mathrm{la}}\big) =s\zeta_{\mathrm{la}}(s+1,z). \; \end{align*} $$

5.5. The case of the differential of a Morse function

Let us consider the special case where $\eta =dh$ for a Morse function h. The following four results follow like Lemma 3.12 and Corollaries 3.13 to 3.15.

Lemma 5.11. For $\mu \gg 0$ ,

$$ \begin{align*} \operatorname{Str}\big({\eta\wedge}\,d_z^{-1}\Pi^1_{z,\mathrm{sm}}\big)&=-\operatorname{Str}\big(h\,(\Pi_z^\perp)_{\mathrm{sm}}\big)\;,\\ \operatorname{Str}\big({\eta\wedge}\,d_z^{-1}e^{-t\Delta_z}\Pi^1_{z,\mathrm{la}}\big)&=-\operatorname{Str}\big(h\,e^{-t\Delta_z}P_{z,\mathrm{la}}\big). \end{align*} $$

Corollary 5.12. For $\mu \gg 0$ ,

$$ \begin{align*} \zeta_{\mathrm{sm}}(1,z)&=-\operatorname{Str}\big(h\,(\Pi_z^\perp)_{\mathrm{sm}}\big)\;,\\ \zeta_{\mathrm{la}}(1,z)&=-\lim_{t\downarrow0}\operatorname{Str}\big(h\,e^{-t\Delta_z}P_{z,\mathrm{la}}\big). \end{align*} $$

Corollary 5.14. If M is oriented and $|\mu |\gg 0$ , then

$$\begin{align*}\zeta_{\mathrm{sm/la}}(1,z)=\zeta_{\mathrm{sm/la}}(1,-\bar z) =\zeta_{\mathrm{sm/la}}(1,-z)=\zeta_{\mathrm{sm/la}}(1,\bar z). \end{align*}$$