1.1 Finite-dimensional Hodge theory and some estimates

Let

(1.1) $$ \begin{align} (W^{\bullet}, \partial) : 0 \rightarrow W^0 \rightarrow \cdots \rightarrow W^n \rightarrow 0 \end{align} $$

be a chain complex of finite-dimensional complex vector spaces. Let $H^{\bullet }(W^{\bullet }, \partial )=\operatorname {\mathrm {Ker}}(\partial |_{W^{\bullet }})/\mathrm {Im}(\partial |_{W^{{\bullet }-1}})$ be the cohomology of $(W^{\bullet }, \partial )$ . Let $h^{W^{\bullet }} = \bigoplus _{k=0}^n h^{W^k}$ be a Hermitian metric on $W^{\bullet }$ . Let $\partial ^*$ be the adjoint of $\partial $ . Set $D = \partial + \partial ^*$ , which is self-adjoint with respect to $h^{W^{\bullet }}$ .

We state the finite-dimensional Hodge theorem without proof.

Let $W^{\bullet } = \bigoplus _{\lambda \geqslant 0} W^{\bullet }_\lambda $ be such that $D^2\big |_{W^{\bullet }_\lambda } = \lambda \operatorname {\mathrm {Id}}$ . We denote ${W^{\bullet }_\lambda }' = W^{\bullet }_\lambda \cap \operatorname {\mathrm {Ker}} \partial $ and ${W^{\bullet }_\lambda }" = W^{\bullet }_\lambda \cap \operatorname {\mathrm {Ker}} \partial ^*$ . For $\lambda>0$ , we have an orthogonal decomposition $W^{\bullet }_\lambda = {W^{\bullet }_\lambda }' \oplus {W^{\bullet }_\lambda }"$ . Let $\big \lVert \cdot \big \rVert $ be the norm on $W^{\bullet }$ induced by $h^{W^{\bullet }}$ . For $w'\in {W^{\bullet }_\lambda }'$ and $w"\in {W^{\bullet }_\lambda }"$ , we have

(1.2) $$ \begin{align} \big\lVert \partial^* w' \big\rVert^2 = \lambda \big\lVert w'\big\rVert^2 \;,\hspace{5mm} \big\lVert \partial w" \big\rVert^2 = \lambda \big\lVert w"\big\rVert^2 \;. \end{align} $$

For $\Lambda \subseteq \mathbb {R}$ , let $P^\Lambda : W^{\bullet } \rightarrow \bigoplus _{\lambda \in \Lambda } W^{\bullet }_\lambda $ be the orthogonal projection.

The following elementary results (Proposition 1.2 and Corollary 1.3) will be used mainly in §5.2 and §5.3, to prove that some operators’ range is closed to be contained in the range of some spectral projector. See also the explanation in the beginning of §5.

Proof. Let $w = \sum _\lambda w_\lambda $ and $v = \sum _\lambda v_\lambda $ with $w_\lambda ,v_\lambda \in W^{\bullet }_\lambda $ . For $\lambda>0$ , let $w_\lambda = w^{\prime }_\lambda + w^{\prime \prime }_\lambda $ and $v_\lambda = v^{\prime }_\lambda + v^{\prime \prime }_\lambda $ with $w^{\prime }_\lambda ,v^{\prime }_\lambda \in {W^{\bullet }_\lambda }'$ and $w^{\prime \prime }_\lambda ,v^{\prime \prime }_\lambda \in {W^{\bullet }_\lambda }"$ . By (1.2), we have

(1.4) $$ \begin{align} \big\lVert \partial w \big\rVert^2 = \sum_{\lambda>0} \lambda \big\lVert w^{\prime\prime}_\lambda \big\rVert^2 \;,\hspace{5mm} \big\lVert \partial^* v \big\rVert^2 = \sum_{\lambda>0} \lambda \big\lVert v^{\prime}_\lambda \big\rVert^2 \;. \end{align} $$

Assume $\gamma>0$ , then by (1.4) and our assumptions, we have

(1.5) $$ \begin{align} \Big\lVert \sum_{\lambda>\gamma}w_\lambda \Big\rVert^2 & = \Big\lVert \sum_{\lambda>\gamma}w^{\prime\prime}_\lambda \Big\rVert^2 + \Big\lVert \sum_{\lambda>\gamma}w^{\prime}_\lambda \Big\rVert^2 \nonumber\\ & \leqslant \Big\lVert \sum_{\lambda>\gamma}w^{\prime\prime}_\lambda \Big\rVert^2 + 2 \bigg( \Big\lVert \sum_{\lambda>\gamma}v^{\prime}_\lambda \Big\rVert^2 + \Big\lVert \sum_{\lambda>\gamma}w^{\prime}_\lambda - \sum_{\lambda>\gamma}v^{\prime}_\lambda \Big\rVert^2 \bigg) \nonumber \\ & \leqslant \frac{1}{\gamma}\big\|\partial w\big\|^2+\frac{2}{\gamma} \big\|\partial^*v\big\|^2+2\big\|w-v\big\|^2 \nonumber\\ &\leqslant 3\alpha+2\beta. \end{align} $$

On the other hand, if $\gamma =0$ , then $w_\lambda "=v_\lambda '=0$ for any $\lambda>0$ , so from the first two lines of (1.5) we find $\Big \lVert \sum _{\lambda>0}w_\lambda \Big \rVert ^2 \leqslant 2\beta $ . Thus, the first inequality in (1.3) is proved.

We get the second inequality in (1.3) using a similar argument as in (1.5) for $\Sigma _{\lambda>\gamma } v_\lambda $ .

Finally, the second point of Proposition 1.2 is clear if $\gamma =0$ , and if $\gamma>0$ , we have $\big \lVert w - P^{[0,\gamma ]}w\big \rVert ^2 \leqslant \frac {1}{\gamma }\big \lVert D w \big \rVert ^2 \leqslant \alpha $ .

For $w\in W^{\bullet }$ , we define $\big \lVert w \big \rVert _1^2 = \big \lVert w \big \rVert ^2 + \big \lVert Dw \big \rVert ^2$ .

1.2 Torsion forms and some estimates

Let S be a compact manifold. Let

(1.6) $$ \begin{align} (W^{\bullet},\nabla^{W^{\bullet}}, \partial) : 0 \rightarrow W^0 \rightarrow \cdots \rightarrow W^n \rightarrow 0 \end{align} $$

be a flat chain complex of complex vector bundles over S, i.e., $\nabla ^{W^{\bullet }} = \bigoplus _{k=0}^n \nabla ^{W^k}$ is a flat connection on $W^{\bullet }=\bigoplus _{k=0}^n {W^k}$ , $\partial : W^{\bullet } \rightarrow W^{{\bullet }+1}$ is a linear map between complex vector bundles satisfying $\partial ^2=0$ , and $\partial \nabla ^{W^{\bullet }} + \nabla ^{W^{\bullet }} \partial = 0$ .

We extend the action of $\partial $ to $\Omega ^{\bullet }(S,W^{\bullet })$ as follows: for $\tau \in \Omega ^k(S)$ and $w\in {\mathscr {C}^\infty }(S,W^{\bullet })$ , $\partial \big (\tau \otimes w\big ) = (-1)^k \tau \otimes \partial w$ . We extend the action of $\nabla ^{W^{\bullet }}$ to $\Omega ^{\bullet }(S,W^{\bullet })$ in the same way as in (0.34).

Since $\partial $ is covariantly constant with respect to the connection $\nabla ^{W^{\bullet }}$ , there is a $\mathbb {Z}$ -graded complex vector bundle $H^{\bullet }$ over S whose fibre over $s\in S$ is the cohomology of $\big (W^{\bullet }_s,\partial \big |_{W^{\bullet }_s}\big )$ (see [Reference Bismut and Lott12, p. 307]). Let $\nabla ^{H^{\bullet }}$ be the connection on $H^{\bullet }$ induced by $\nabla ^{W^{\bullet }}$ in the sense of [Reference Bismut and Lott12, Def. 2.4]. By [Reference Bismut and Lott12, Prop. 2.5], $(H^{\bullet },\nabla ^{H^{\bullet }})$ is a $\mathbb {Z}$ -graded flat complex vector bundle.

Set

(1.7) $$ \begin{align} A" = \partial + \nabla^{W^{\bullet}} \;. \end{align} $$

We have $\big ( A" \big )^2 = 0$ , i.e., $A"$ is a flat superconnection in the sense of [Reference Bismut and Lott12, §1].

Let $h^{W^{\bullet }} = \bigoplus _{k=0}^n h^{W^k}$ be a Hermitian metric on $W^{\bullet }$ . In the sequel, we call such a data $(W^{\bullet },\nabla ^{W^{\bullet }},\partial , h^{W^{\bullet }})$ a flat chain complex of Hermitian vector bundles.

Let $\omega ^{W^{\bullet }} \in \Omega ^1\big (S,\mathrm {End}(W^{\bullet })\big )$ be the Bismut-Zhang curvature 1-form of $(W^{\bullet },\nabla ^{W^{\bullet }},h^{W^{\bullet }})$ as in (0.1). Let $\partial ^*$ be the adjoint of $\partial $ . Let $A'$ be the adjoint superconnection of $A"$ in the sense of [Reference Bismut and Lott12, §1]. By [Reference Bismut and Lott12, §2(b)], we have

(1.8) $$ \begin{align} A' = \partial^* + \nabla^{W^{\bullet}} + \omega^{W^{\bullet}} \;. \end{align} $$

Set

(1.9) $$ \begin{align} X = \frac{1}{2} (A' - A") = \frac{1}{2}(\partial^*-\partial) + \frac{1}{2} \omega^{W^{\bullet}} \in \Omega^{\bullet}\big(S,\mathrm{End}(W^{\bullet})\big) \;. \end{align} $$

Let $N^{W^{\bullet }}$ be the number operator on $W^{\bullet }$ , i.e., $N^{W^{\bullet }}\big |_{W^k} = k\operatorname {\mathrm {Id}}$ . For $t>0$ , set $h^{W^{\bullet }}_t = t^{N^{W^{\bullet }}}h^{W^{\bullet }}$ . Let $X_t$ be the operator X associated with $h^{W^{\bullet }}_t$ . For convenience, we also introduce $\mathfrak {X}_t = t^{N^{W^{\bullet }}/2} X_t t^{-N^{W^{\bullet }}/2}$ . We have

(1.10) $$ \begin{align} X_t = \frac{1}{2}(t \partial^*-\partial) + \frac{1}{2} \omega^{W^{\bullet}} \;,\hspace{5mm} \mathfrak{X}_t = \frac{\sqrt{t}}{2}(\partial^*-\partial) + \frac{1}{2} \omega^{W^{\bullet}} \;. \end{align} $$

We define $\varphi : \Omega ^{\mathrm {even}}(S) \rightarrow \Omega ^{\mathrm {even}}(S)$ as follows,

(1.11) $$ \begin{align} \varphi \omega = (2\pi i)^{-k }\omega \hspace{2.5mm} \text{for }\omega\in\Omega^{2k}(S) \;. \end{align} $$

Recall that $f(z)=ze^{z^2}$ , so that $f'(z) = (1+2z^2) e^{z^2}$ . We define

(1.12) $$ \begin{align} f^\wedge(A",h^{W^{\bullet}}_t) = \varphi \operatorname{\mathrm{Tr}} \left[(-1)^{N^{W^{\bullet}}}\frac{N^{W^{\bullet}}}{2} f'(\mathfrak{X}_t) \right] \in \Omega^{\mathrm{even}}(S) \;. \end{align} $$

We denote $\chi '(W^{\bullet }) = \sum _k (-1)^kk \,\mathrm {rk}\big (W^k\big )$ and $\chi '(H^{\bullet }) = \sum _k (-1)^kk \,\mathrm {rk}\big (H^k\big )$ .

Definition 1.4 (see [Reference Bismut and Lott12, Def. 2.20])

The torsion form associated with $(\nabla ^{W^{\bullet }},\partial ,h^{W^{\bullet }})$ is defined by

(1.13) $$ \begin{align} \mathscr{T}\big(\nabla^{W^{\bullet}},\partial,h^{W^{\bullet}}\big) & = - \int_0^{+\infty} \bigg[ f^\wedge(A",h^{W^{\bullet}}_t) - \frac{1}{2}\chi'(H^{\bullet})\nonumber \\ & \hspace{35mm} - \frac{1}{2}\big(\chi'(W^{\bullet}) - \chi'(H^{\bullet})\big) f'\Big(\frac{i\sqrt{t}}{2}\Big) \bigg] \frac{dt}{t} \;. \end{align} $$

By [Reference Bismut and Lott12, Thm. 2.13, Prop. 2.18], the integrand in (1.13) is integrable and is in $Q^S$ , the vector space of real even forms on S.

Let $h^{H^{\bullet }}$ be the Hermitian metric on $H^{\bullet }$ induced by $h^{W^{\bullet }}$ via the identification $H^{\bullet } \simeq \operatorname {\mathrm {Ker}}\big ((\partial +\partial ^*)^2\big ) \hookrightarrow W^{\bullet }$ defined by Theorem 1.1. Let $f\big (\nabla ^{W^{\bullet }},h^{W^{\bullet }}\big ), f\big (\nabla ^{H^{\bullet }},h^{H^{\bullet }}\big ) \in \Omega ^{\mathrm {odd}}(S)$ be as in (0.3). By [Reference Bismut and Lott12, Thm. 2.22], we have

(1.14) $$ \begin{align} d \mathscr{T}\big(\nabla^{W^{\bullet}},\partial,h^{W^{\bullet}}\big) = f\big(\nabla^{W^{\bullet}},h^{W^{\bullet}}\big) - f\big(\nabla^{H^{\bullet}},h^{H^{\bullet}}\big) \;. \end{align} $$

Let $(\widetilde {W}^{\bullet } = \bigoplus _{k=0}^n \widetilde {W}^k,\nabla ^{\widetilde {W}^{\bullet }},\widetilde {\partial })$ be another flat chain complex of complex vector bundles over S. Let $\widetilde {H}^{\bullet }$ be its cohomology. We assume that for $k=0,\cdots ,n$ ,

(1.15) $$ \begin{align} \mathrm{rk}\big(W^k\big) = \mathrm{rk}\big(\widetilde{W}^k\big) \;,\hspace{5mm} \mathrm{rk}\big(H^k\big) = \mathrm{rk}\big(\widetilde{H}^k\big) \;. \end{align} $$

Let $h^{\widetilde {W}^{\bullet }} = \bigoplus _{k=0}^n h^{\widetilde {W}^k}$ be a Hermitian metric on $\widetilde {W}^{\bullet }$ .

Let $g^{TS}$ be a Riemannian metric on $TS$ . Let $\big |\cdot \big |$ be the norm on $TS$ induced by $g^{TS}$ . For $\omega \in \Omega ^{\bullet }(S)$ , we denote by $\big | \omega \big |$ its $\mathscr {C}^0$ -norm. For an operator A on $W^{\bullet }$ , we denote by $\big \lVert A \big \rVert $ its operator norm with respect to $h^{W^{\bullet }}$ . For $A\in \Omega ^{\bullet }(S,\mathrm {End}(W^{\bullet }))$ , we denote

(1.16) $$ \begin{align} \big\lVert A \big\rVert = \sup_{k\in\mathbb{N},\; x\in S,\; v_1,\cdots,v_k\in T_xS,\; |v_1|,\cdots,|v_k|\leqslant 1} \big\lVert A(v_1,\cdots,v_k) \big\rVert \;. \end{align} $$

Let $0<\lambda _{\mathrm {min}}\leqslant \lambda _{\mathrm {max}}$ and $l>0$ be such that

(1.17) $$ \begin{align} \mathrm{Sp}\big((\partial^*+\partial)^2\big) \subseteq \{0\} \cup [\lambda^2_{\mathrm{min}},\lambda^2_{\mathrm{max}}] \;,\hspace{5mm} \big\lVert \omega^{W^{\bullet}} \big\rVert \leqslant l \;. \end{align} $$

The following elementary quantitative comparison results on torsion forms – Proposition 1.5, Remark 1.6 and Corollary 1.7 – will be used in the proof of Theorems 3.5 (see (6.96)) and 3.6 (see Propositions 7.3 and 7.4), to prove that torsion forms converge to torsion of models in our two-parameter deformation.

Proposition 1.5. There exists a function $C: \mathbb {N} \times \mathbb {N} \times \mathbb {R}_+ \times \mathbb {R}_+ \rightarrow \mathbb {R}_+$ such that for any $(W^{\bullet },\nabla ^{W^{\bullet }},\partial ,h^{W^{\bullet }})$ , $(\widetilde {W}^{\bullet },\nabla ^{\widetilde {W}^{\bullet }},\widetilde {\partial },h^{\widetilde {W}^{\bullet }})$ , $\lambda _{\mathrm {min}}$ , $\lambda _{\mathrm {max}}$ and l as above, if there exist an isomorphism of graded complex vector bundles $\alpha : W^{\bullet }\rightarrow \widetilde {W}^{\bullet }$ and $0< \delta < 13^{-1}\lambda _{\mathrm {min}}\lambda _{\mathrm {max}}^{-1}$ satisfying

(1.18) $$ \begin{align} \big\lVert \alpha^*\widetilde{\partial} - \partial \big\rVert \leqslant \lambda_{\mathrm{min}}\delta \;,\hspace{5mm} \big\|\alpha^*h^{\widetilde{W}^{\bullet}} - h^{W^{\bullet}}\big\|_{h^{W^{\bullet}}} \leqslant \delta\;,\hspace{5mm} \big\lVert \alpha^*\omega^{\widetilde{W}^{\bullet}} - \omega^{W^{\bullet}} \big\rVert \leqslant \delta \;, \end{align} $$

then

(1.19) $$ \begin{align} \Big| \mathscr{T}\big(\nabla^{W^{\bullet}},\partial,h^{W^{\bullet}}\big) - \mathscr{T}\big(\nabla^{\widetilde{W}^{\bullet}},\widetilde{\partial},h^{\widetilde{W}^{\bullet}}\big) \Big| \leqslant C\big(\dim S, \mathrm{rk}(W^{\bullet}),l,\lambda_{\mathrm{max}}/\lambda_{\mathrm{min}}\big) \delta^{1/2} \;. \end{align} $$

Proof. Replacing $\partial $ by $\lambda _{\mathrm {min}}^{-1}\partial $ and replacing $\widetilde {\partial }$ by $\lambda _{\mathrm {min}}^{-1}\widetilde {\partial }$ , we may assume that $\lambda _{\mathrm {min}} = 1$ . Then (1.17) and the first inequality in (1.18) become

(1.20) $$ \begin{align} \mathrm{Sp}\big((\partial^*+\partial)^2\big) \subseteq \{0\} \cup [1,\lambda^2_{\mathrm{max}}] \;, \hspace{5mm} \big\lVert \omega^{W^{\bullet}} \big\rVert \leqslant l \;, \hspace{5mm} \big\lVert \alpha^*\widetilde{\partial} - \partial \big\rVert \leqslant \delta \;. \end{align} $$

By (1.20), we have

(1.21) $$ \begin{align} \big\lVert \alpha^*\widetilde{\partial} \big\rVert \leqslant \delta + \big\lVert \partial \big\rVert \leqslant \delta + \lambda_{\mathrm{max}} \;. \end{align} $$

Since $\big \lVert A \big \rVert = \big \lVert A^* \big \rVert $ for any operator A on $W^{\bullet }$ , we have

(1.22) $$ \begin{align} \big\lVert \big(\alpha^*\widetilde{\partial}\big)^* - \partial^* \big\rVert = \big\lVert \alpha^*\widetilde{\partial} - \partial \big\rVert \;. \end{align} $$

Let $\widetilde {\partial }^*$ be the adjoint of $\widetilde {\partial }$ with respect to $h^{\widetilde {W}^{\bullet }}$ . Note that $\alpha ^*\widetilde {\partial }^*$ is the adjoint of $\alpha ^*\widetilde {\partial }$ with respect to $\alpha ^*h^{\widetilde {W}^{\bullet }}$ and $\big (\alpha ^*\widetilde {\partial }\big )^*$ is the adjoint of $\alpha ^*\widetilde {\partial }$ with respect to $h^{W^{\bullet }}$ , by the second inequality in (1.18), we have

(1.23) $$ \begin{align} \big\lVert \alpha^*\widetilde{\partial}^* - \big(\alpha^*\widetilde{\partial}\big)^* \big\rVert \leqslant 3\delta \big\lVert \alpha^*\widetilde{\partial} \big\rVert \;. \end{align} $$

By (1.20)-(1.23) and the assumption $0 < \delta < 13^{-1}\lambda ^{-1}_{\mathrm {max}}$ , we have

(1.24) $$ \begin{align} \big\lVert \alpha^*\widetilde{\partial}^* - \partial^* \big\rVert & \leqslant \big\lVert \alpha^*\widetilde{\partial}^* - \big(\alpha^*\widetilde{\partial}\big)^* \big\rVert + \big\lVert \big(\alpha^*\widetilde{\partial}\big)^* - \partial^* \big\rVert \nonumber \\ & \leqslant 3\delta(\delta + \lambda_{\mathrm{max}}) + \delta \leqslant 5 \delta \lambda_{\mathrm{max}} \;. \end{align} $$

By (1.20), (1.24) and the assumption $0 < \delta < 13^{-1}\lambda ^{-1}_{\mathrm {max}}$ , we have

(1.25) $$ \begin{align} \big\lVert \partial^*\pm\partial-\alpha^*(\widetilde{\partial}^*\pm\widetilde{\partial}) \big\rVert \leqslant 6 \delta \lambda_{\mathrm{max}} \leqslant \frac{6}{13} \;. \end{align} $$

By (1.20) and (1.25), we have

(1.26) $$ \begin{align} \mathrm{Sp}\big((\widetilde{\partial}^*+\widetilde{\partial})^2\big) \subseteq \Big[0,\frac{6^2}{13^2}\Big] \cup \Big[\frac{7^2}{13^2},3\lambda^2_{\mathrm{max}}\Big] \;. \end{align} $$

Moreover, the dimension of the eigenspace of $(\widetilde {\partial }^*+\widetilde {\partial })^2$ associated with eigenvalues in $\big [0,\frac {6^2}{13^2}\big ]$ equals the dimension of $\operatorname {\mathrm {Ker}}\big ((\partial ^*+\partial )^2\big )$ . On the other hand, by Theorem 1.1 and the second identity in (1.15), we have $\dim \operatorname {\mathrm {Ker}}\big ((\partial ^*+\partial )^2\big ) = \dim \operatorname {\mathrm {Ker}}\big ((\widetilde {\partial }^*+\widetilde {\partial })^2\big )$ . As a consequence, the only possible eigenvalue of $(\widetilde {\partial }^*+\widetilde {\partial })^2$ in $\big [0,\frac {6^2}{13^2}\big ]$ is zero, i.e.,

(1.27) $$ \begin{align} \mathrm{Sp}\big((\widetilde{\partial}^*+\widetilde{\partial})^2\big) \subseteq \big\{0\big\} \cup \Big[\frac{7^2}{13^2},3\lambda^2_{\mathrm{max}}\Big] \;. \end{align} $$

Let $\omega ^{\widetilde {W}^{\bullet }}$ be the Bismut-Zhang curvature 1-form of $(\widetilde {W}^{\bullet },\nabla ^{\widetilde {W}^{\bullet }},h^{\widetilde {W}^{\bullet }})$ as in (0.1). For $t>0$ , we denote

(1.28) $$ \begin{align} \mathfrak{X}_t = \frac{\sqrt{t}}{2}(\partial^*-\partial) + \frac{1}{2} \omega^{W^{\bullet}} \;,\hspace{5mm} \widetilde{\mathfrak{X}}_t = \alpha^*\Big( \frac{\sqrt{t}}{2}(\widetilde{\partial^*}-\widetilde{\partial}) + \frac{1}{2} \omega^{\widetilde{W}^{\bullet}} \Big) \;. \end{align} $$

Set $U = \big \{\lambda \in \mathbb {C}\;:\; -1<\mathrm {Re}(z)<1\big \}$ . In the sequel, we will use $C_1,C_2,\cdots $ to denote constants depending on $\dim S$ , $\mathrm {rk}W^{\bullet }$ , l and $\lambda _{\mathrm {max}}/\lambda _{\mathrm {min}}$ . By (1.17), the third inequality in (1.18), (1.20), (1.25) and (1.27), for $\lambda \in \partial U$ and $t>0$ , we have

(1.29) $$ \begin{align} \big\lVert (\lambda - \mathfrak{X}_t )^{-1} \big\rVert \leqslant C_1 \;,\hspace{5mm} \big\lVert (\lambda - \widetilde{\mathfrak{X}}_t )^{-1} \big\rVert \leqslant C_1 \;,\hspace{5mm} \big\lVert \mathfrak{X}_t - \widetilde{\mathfrak{X}}_t \big\rVert \leqslant C_1(1+\sqrt{t})\delta \;. \end{align} $$

Let $f^\wedge \big (\widetilde {A}",h^{\widetilde {W}^{\bullet }}_t\big )$ be as in (1.12) with $(\nabla ^{W^{\bullet }},\partial ,h^{W^{\bullet }})$ replaced by $(\nabla ^{\widetilde {W}^{\bullet }},\widetilde {\partial },h^{\widetilde {W}^{\bullet }})$ . For $t>0$ , by (1.12), we have

(1.30) $$ \begin{align} & f^\wedge\big(A",h^{W^{\bullet}}_t\big) - f^\wedge\big(\widetilde{A}",h^{\widetilde{W}^{\bullet}}_t\big)\nonumber \\ & = \frac{1}{2\pi i} \int_{\partial U} \varphi \operatorname{\mathrm{Tr}}\bigg[ (-1)^{N^{W^{\bullet}}}\frac{N^{W^{\bullet}}}{2} \Big( (\lambda - \mathfrak{X}_t )^{-1} - (\lambda - \widetilde{\mathfrak{X}}_t )^{-1} \Big) \bigg] f'(\lambda) d\lambda\nonumber \\ & = \frac{1}{2\pi i} \int_{\partial U} \varphi \operatorname{\mathrm{Tr}}\bigg[ (-1)^{N^{W^{\bullet}}}\frac{N^{W^{\bullet}}}{2} (\lambda - \mathfrak{X}_t )^{-1} (\mathfrak{X}_t-\widetilde{\mathfrak{X}}_t) (\lambda - \widetilde{\mathfrak{X}}_t )^{-1} \bigg] f'(\lambda) d\lambda \;. \end{align} $$

By (1.29) and (1.30), we have

(1.31) $$ \begin{align} \Big| f^\wedge\big(A",h^{W^{\bullet}}_t\big) - f^\wedge\big(\widetilde{A}",h^{\widetilde{W}^{\bullet}}_t\big) \Big| \leqslant C_2 (1+\sqrt{t}) \delta \;. \end{align} $$

Proceeding the same way as in the proofs of [Reference Bismut and Lott12, Thm. 2.13, Prop. 2.18] and applying (1.17), the third inequality in (1.18), (1.20) and (1.27), we obtain the following estimates: for $0<t<1$ ,

(1.32) $$ \begin{align} \Big| f^\wedge\big(A",h^{W^{\bullet}}_t\big) - \frac{1}{2} \chi'(W^{\bullet}) \Big| \leqslant C_3 t \;,\hspace{5mm} \Big| f^\wedge\big(\widetilde{A}",h^{\widetilde{W}^{\bullet}}_t\big) - \frac{1}{2} \chi'(W^{\bullet}) \Big| \leqslant C_3 t \;; \end{align} $$

for $t>1$ ,

(1.33) $$ \begin{align} \Big| f^\wedge\big(A",h^{W^{\bullet}}_t\big) - \frac{1}{2} \chi'(H^{\bullet}) \Big| \leqslant \frac{C_4}{\sqrt{t}} \;,\hspace{5mm} \Big| f^\wedge\big(\widetilde{A}",h^{\widetilde{W}^{\bullet}}_t\big) - \frac{1}{2} \chi'(H^{\bullet}) \Big| \leqslant \frac{C_4}{\sqrt{t}} \;. \end{align} $$

By (1.13), (1.15) and (1.31)-(1.33), we have

(1.34) $$ \begin{align} & \Big| \mathscr{T}\big(\nabla^{W^{\bullet}},\partial,h^{W^{\bullet}}\big) - \mathscr{T}\big(\nabla^{\widetilde{W}^{\bullet}},\widetilde{\partial},h^{\widetilde{W}^{\bullet}}\big) \Big| \nonumber\\ & \leqslant \; 2 \int_0^{\delta} C_3 t \frac{dt}{t} + \int_{\delta}^{\delta^{-1}} C_2 (1+\sqrt{t}) \delta \frac{dt}{t} + 2 \int_{\delta^{-1}}^{+\infty} \frac{C_4}{\sqrt{t}} \frac{dt}{t} \leqslant\; C_5 \delta^{1/2} \;. \end{align} $$

Let $(F,\nabla ^F)$ be a flat complex vector bundle over S. Let $h^F_1$ and $h^F_2$ be Hermitian metrics on F. Let $\omega ^F_1$ (resp. $\omega ^F_2$ ) be the Bismut-Zhang curvature 1-form as in (0.1) with $h^F$ replaced by $h^F_1$ (resp. $h^F_2$ ). We consider the chain complex $F \xrightarrow {\mathrm {Id}} F$ , where the first F is equipped with Hermitian metric $h^F_1$ and the second F is equipped with Hermitian metric $h^F_2$ . Let $\mathscr {T}\big (\nabla ^F,h^F_1,h^F_2\big )$ be its torsion form. Let $l>0$ be such that $\big \lVert \omega ^F_1 \big \rVert \leqslant l$ .

Corollary 1.7. There exists a function $C: \mathbb {N} \times \mathbb {N} \times \mathbb {R}_+ \rightarrow \mathbb {R}_+$ such that for any $(F,\nabla ^F,h^F_1,h^F_2)$ and l as above, if there exists $\delta \in (0,13^{-1})$ satisfying

(1.35) $$ \begin{align} \big\|h^F_2 - h^F_1\big\|_{h^F_1} \leqslant \delta \;,\hspace{5mm} \big\lVert \omega^F_2 - \omega^F_1 \big\rVert \leqslant \delta \;, \end{align} $$

then

(1.36) $$ \begin{align} \Big| \mathscr{T}\big(\nabla^F,h^F_1,h^F_2\big) \Big| \leqslant C\big(\dim S, \mathrm{rk}(F),l\big) \delta^{1/2} \;. \end{align} $$

1.3 Analytic torsion forms

Let $\pi : M_1 \rightarrow S$ be a smooth fibration with compact fibre $Z_1$ . Let $N = \partial M_1$ . We assume that $\pi \big |_N: N \rightarrow S$ is a smooth fibration with fibre Y. Then we have $Y = \partial Z_1$ . We identify a tubular neighbourhood of $N \subseteq M_1$ with $[-1,0]\times N$ such that N is identified with $\{0\}\times N$ and the following diagram commutes,

(1.37)

where $\mathrm {pr}_2: [-1,0]\times N \rightarrow N$ is the projection to the second factor.

Let $T^HM_1 \subseteq TM_1$ be a horizontal subbundle of $TM_1$ which has product structure on $[-1,0]\times N$ , in the sense of (0.5) and (0.8). Then $T^HN := T^HM_1\big |_N \subseteq TN$ is a horizontal subbundle of $TN$ , i.e., $TN = T^HN \oplus TY$ . Moreover, we have

(1.38) $$ \begin{align} \Lambda^{\bullet}(T^*M_1) = \Lambda^{\bullet}(T^{H,*}M_1) \otimes \Lambda^{\bullet}(T^*Z_1) \simeq \pi^*\big(\Lambda^{\bullet}(T^*S)\big) \otimes \Lambda^{\bullet}(T^*Z_1) \;. \end{align} $$

Let $(F,\nabla ^F)$ be a flat complex vector bundle over $M_1$ . We trivialize $F\big |_{[-1,0]\times N}$ similarly to (0.9). Let $g^{TZ_1}$ be a metric on $TZ_1$ , and let $g^{TY}$ be the induced metric on $TY$ via the embedding $N = \partial M_1 \hookrightarrow M_1$ . Let $h^F$ be a Hermitian metric on F. We assume that $g^{TZ_1}$ and $h^F$ have a product structure on $[-1,0]\times N$ , in the sense of (0.10).

Set $\mathscr {F} = \Omega ^{\bullet }(Z_1,F)$ , which is a $\mathbb {Z}$ -graded complex vector bundle of infinite dimension over S. By (1.38), we have the formal identity $\Omega ^{\bullet }(M_1,F) = \Omega ^{\bullet }(S,\mathscr {F})$ .

For $U\in TS$ , let $U^H \in T^HM_1$ be its horizontal lift, i.e., $\pi _*U^H = U$ . For $U\in {\mathscr {C}^\infty }(S,TS)$ , let $L_{U^H}$ be the Lie differentiation operator acting on $\Omega ^{\bullet }(M_1,F)$ . For $U\in {\mathscr {C}^\infty }(S,TS)$ and $s\in {\mathscr {C}^\infty }(S,\mathscr {F})$ , we define

(1.39) $$ \begin{align} \nabla^{\mathscr{F}}_U s = L_{U^H} s \;. \end{align} $$

Then $\nabla ^{\mathscr {F}}$ is a connection on $\mathscr {F}$ preserving the grading.

Let $P^{TZ_1}: TM_1 = T^HM_1 \oplus TZ_1\rightarrow TZ_1$ be the canonical projection. We define

(1.40) $$ \begin{align} \mathcal{T}(U,V) = - P^{TZ_1}[U^H,V^H] \in {\mathscr{C}^\infty}(M_1,TZ_1) \hspace{2.5mm} \text{for } U,V\in{\mathscr{C}^\infty}(S,TS) \;. \end{align} $$

Then $\mathcal {T}\in {\mathscr {C}^\infty }\big (M_1,\pi ^*\big (\Lambda ^2(T^*S)\big )\otimes TZ_1\big )$ . Let $i_{\mathcal {T}}\in {\mathscr {C}^\infty }\big (M_1,\pi ^*\big (\Lambda ^2(T^*S)\big )\otimes \mathrm {End}\big (\Lambda ^{\bullet }(T^*Z_1)\big )\big )$ be the interior multiplication by $\mathcal {T}$ in the vertical direction.

The flat connection $\nabla ^F$ (resp. $\nabla ^F\big |_{Z_1}$ ) naturally extends to an exterior differentiation operator on $\Omega ^{\bullet }(M_1,F)$ (resp. $\Omega ^{\bullet }(Z_1,F) = \mathscr {F}$ ), which we denote by $d^{M_1}$ (resp. $d^{Z_1}$ ). In the sense of [Reference Bismut and Lott12, §2(a)], the operator $d^{M_1}$ is a superconnection of total degree $1$ on $\mathscr {F}$ . By [Reference Bismut and Lott12, Prop. 3.4], we have $d^{M_1} = d^{Z_1} + \nabla ^{\mathscr {F}} + i_{\mathcal {T}}$ .

Let $\mathcal {T}^*\in {\mathscr {C}^\infty }\big ({M_1},\pi ^*\big (\Lambda ^2(T^*S)\big )\otimes T^*Z_1\big )$ be the dual of $\mathcal {T}$ with respect to $g^{TZ_1}$ . Let $h^{\mathscr {F}}$ be the $L^2$ -metric on $\mathscr {F}$ with respect to $g^{TZ_1}$ and $h^F$ . Let $d^{{M_1},*}, d^{{Z_1},*},\nabla ^{\mathscr {F},*}$ be the formal adjoints of $d^{M_1},d^{Z_1},\nabla ^{\mathscr {F}}$ with respect to $h^{\mathscr {F}}$ in the sense of [Reference Bismut and Lott12, Def. 1.6]. By [Reference Bismut and Lott12, Prop. 3.7], we have $d^{{M_1},*} = d^{{Z_1},*} + \nabla ^{\mathscr {F,*}} - \mathcal {T}^*\!\wedge $ .

We denote

(1.41) $$ \begin{align} \omega^{\mathscr{F}} = \nabla^{\mathscr{F},*} - \nabla^{\mathscr{F}} \in \Omega^1(S,\mathrm{End}(\mathscr{F})) \;. \end{align} $$

For $X\in T{Z_1}$ , let $X^*\in T^*{Z_1}$ be its dual with respect to $g^{T{Z_1}}$ . We define

(1.42) $$ \begin{align} \hat{c}(X) = X^*\!\wedge + i_X \in \mathrm{End}(\Lambda^{\bullet}(T^*{Z_1})) \hspace{2.5mm} \text{for } X\in T{Z_1} \;. \end{align} $$

Let $N^{T{Z_1}}$ be the number operator on $\Lambda ^{\bullet }(T^*{Z_1})$ , i.e., $N^{T{Z_1}}\big |_{\Lambda ^p(T^*{Z_1})} = p \operatorname {\mathrm {Id}}$ . Then $N^{T{Z_1}}$ acts on $\mathscr {F}$ in the obvious way. For $t>0$ , let $d^{{M_1},*}_t$ be the formal adjoint of $d^{M_1}$ with respect to $h^{\mathscr {F}}_t := t^{N^{T{Z_1}}}h^{\mathscr {F}}$ . Set $\mathscr {D}_t = \frac {1}{2}t^{N^{T{Z_1}}/2} \big ( d^{{M_1},*}_t - d^{M_1} \big ) t^{-N^{T{Z_1}}/2}$ . We have

(1.43) $$ \begin{align} \mathscr{D}_t & = \frac{\sqrt{t}}{2} \big(d^{{Z_1},*}-d^{Z_1}\big) + \frac{1}{2}\omega^{\mathscr{F}} - \frac{1}{2\sqrt{t}}\hat{c}(\mathcal{T}) \;, \nonumber \\ \mathscr{D}_t^2 & = - \frac{t}{4} \big(d^{{Z_1},*}d^{Z_1}+d^{Z_1}d^{{Z_1},*}\big) + \text{nilpotent operator} \;. \end{align} $$

The operator $\Delta ^{Z_1} := d^{{Z_1},*}d^{Z_1}+d^{Z_1}d^{{Z_1},*}$ is fibrewise essentially self-adjoint with respect to the absolute boundary condition. Its self-adjoint extension will still be denoted by $\Delta ^{Z_1}$ . More precisely, $\mathrm {Dom}\big (\Delta ^{Z_1}\big )$ is given by the first equation in [Reference Puchol, Zhang and Zhu55, (1.5)]. And the extension of $\mathscr {D}_t^2$ with $\mathrm {Dom}\big ( \mathscr {D}_t^2 \big ) = \mathrm {Dom}\big (\Delta ^{Z_1}\big )$ will still be denoted by $\mathscr {D}_t^2$ . Let $\mathrm {End}_{\mathrm {tr}}(\mathscr {F}) \subseteq \mathrm {End}(\mathscr {F})$ be the sub vector bundle of trace class operators. Recall that $f'(z) = (1+2z^2)e^{z^2}$ . By (1.43), we have $f'\big (\mathscr {D}_t^2\big )\in \Omega ^{\bullet }\big (\mathrm {End}_{\mathrm {tr}}(\mathscr {F})\big )$ .

Let $\operatorname {\mathrm {Tr}}: \mathrm {End}_{\mathrm {tr}}(\mathscr {F}) \rightarrow \mathbb {C}$ be the trace map, which extends to $\operatorname {\mathrm {Tr}}: \mathrm {End}_{\mathrm {tr}}(\mathscr {F})\otimes \Lambda ^{\bullet }(T^*S) \rightarrow \Lambda ^{\bullet }(T^*S)$ . Let $\varphi $ be as in (1.11).

Let $H^{\bullet }(Z_1,F)=\bigoplus _{p=0}^{\dim Z_1}H^p(Z_1,F)$ be the fibrewise absolute cohomology of $Z_1$ with coefficients in F. We denote

(1.44) $$ \begin{align} \chi'({Z_1},F) = \sum_{p=0}^{\dim {Z_1}} (-1)^pp\, \mathrm{rk}\big(H^p({Z_1},F)\big) \;. \end{align} $$

We recall the definition of analytic torsion forms [Reference Zhu65, Def. 2.18], [Reference Bismut and Lott12, Def. 3.22].

Definition 1.8. The analytic torsion form for $(T^H{M_1},g^{T{Z_1}},h^F)\in Q^S$ is defined by

(1.45) $$ \begin{align} \mathscr{T}(T^H{M_1},g^{T{Z_1}},h^F) = & - \int_0^{+\infty} \bigg\{ \varphi \operatorname{\mathrm{Tr}}\Big[(-1)^{N^{T{Z_1}}}\frac{N^{T{Z_1}}}{2}f'\big(\mathscr{D}_t\big)\Big] \nonumber\\ & - \frac{\chi'({Z_1},F)}{2} - \Big(\frac{\dim ({Z_1}) \mathrm{rk}(F)\chi({Z_1})}{4} - \frac{\chi'({Z_1},F)}{2}\Big)f'\Big(\frac{i\sqrt{t}}{2}\Big) \bigg\} \frac{dt}{t} \;. \end{align} $$

The convergence of the integral in (1.45) follows from the family local index theorem [Reference Bismut and Lott12, Thm 3.21], [Reference Zhu65, Thm 2.17]. And $d \mathscr {T}(T^H{M_1},g^{T{Z_1}},h^F)$ is given by (0.14) for $j=1$ .

Recall that $Q^{S,0} \subseteq Q^S$ is the subspace of exact forms. By the de Rham theorem (see [Reference Brüning and Ma16, Thm. 1.1 (d)]), $Q^{S,0} \subseteq Q^S$ is closed for the $\mathscr {C}^0$ topology. We will view $\mathscr {T}(T^H{M_1},g^{T{Z_1}},h^F)$ as an element in $Q^S/Q^{S,0}$ .

2.1 Chain complexes from a pair of linear maps

Let $W_1$ , $W_2$ and V be finite-dimensional complex vector spaces. Let $\tau _1: W_1\rightarrow V$ and $\tau _2: W_2\rightarrow V$ be linear maps. We define a chain complex $\big (C^{\bullet }(\tau _1,\tau _2),\partial \big )$ as follows,

(2.1) $$ \begin{align} 0 \rightarrow C^0(\tau_1,\tau_2) := W_1\oplus W_2 & \xrightarrow{\partial} C^1(\tau_1,\tau_2) := V \rightarrow 0 \nonumber \\ (w_1,w_2) & \mapsto \tau_2(w_2) - \tau_1(w_1) \;. \end{align} $$

We denote

(2.2) $$ \begin{align} V_j = \mathrm{Im}(\tau_j) \subseteq V \quad \text{ and }\quad K_j = \mathrm{Ker}(\tau_j) \subseteq W_j \quad\text{ for } j=1,2. \end{align} $$

We define a chain complex $\big (C^{\bullet }_{\mathrm {r}}(\tau _1,\tau _2),\partial \big )$ as follows,

(2.3) $$ \begin{align} 0 \rightarrow C^0_{\mathrm{r}}(\tau_1,\tau_2) := V_1\oplus V_2 & \xrightarrow{\partial} C^1_{\mathrm{r}}(\tau_1,\tau_2) := V \rightarrow 0 \nonumber \\ (v_1,v_2) & \mapsto v_2 - v_1 \;. \end{align} $$

We have a short exact sequence of chain complexes,

(2.4)

where $K_1\oplus K_2 \rightarrow C^0(\tau _1,\tau _2) = W_1 \oplus W_2$ is the direct sum of the embeddings in (2.2).

Set

(2.5) $$ \begin{align} C^{\bullet}_0 = C^{\bullet}(\tau_1,\tau_2) \;,\; C^{\bullet}_1 = C^{\bullet}(\tau_1,\operatorname{\mathrm{Id}}_V) \;,\; C^{\bullet}_2 = C^{\bullet}(\operatorname{\mathrm{Id}}_V,\tau_2) \;,\; C^{\bullet}_3 = C^{\bullet}(\operatorname{\mathrm{Id}}_V,\operatorname{\mathrm{Id}}_V)\;. \end{align} $$

We define $\alpha _1: C^{\bullet }_0 \rightarrow C^{\bullet }_1$ , $\alpha _2: C^{\bullet }_0 \rightarrow C^{\bullet }_2$ , $\beta _1: C^{\bullet }_1 \rightarrow C^{\bullet }_3$ and $\beta _2: C^{\bullet }_2 \rightarrow C^{\bullet }_3$ as follows,

(2.6) $$ \begin{align} & \alpha_1\big|_{C^0_0} = \operatorname{\mathrm{Id}}_{W_1} \oplus \tau_2 \;,\hspace{5mm} \alpha_2\big|_{C^0_0} = \tau_1 \oplus \operatorname{\mathrm{Id}}_{W_2} \;,\hspace{5mm} \alpha_1\big|_{C^1_0} = \alpha_2\big|_{C^1_0} = \operatorname{\mathrm{Id}}_V \;, \nonumber\\ & \beta_1\big|_{C^0_1} = \tau_1 \oplus \operatorname{\mathrm{Id}}_V \;,\hspace{5mm} \beta_2\big|_{C^0_2} = \operatorname{\mathrm{Id}}_V \oplus \tau_2 \;,\hspace{5mm} \beta_1\big|_{C^1_1} = \beta_2\big|_{C^1_2} = \operatorname{\mathrm{Id}}_V \;. \end{align} $$

We have a short exact sequence of chain complexes,

(2.7)

For $j=0,1,2,3$ , let $H^k\big (C^{\bullet }_j,\partial \big )$ be the k-th cohomology group of $\big (C^{\bullet }_j,\partial \big )$ . From (2.7), we get a long exact sequence of cohomology groups,

(2.8)

We denote

(2.9) $$ \begin{align} W_{12} = \Big\{(w_1,w_2)\in W_1 \oplus W_2 \;:\;\tau_1(w_1)=\tau_2(w_2)\Big\} \;. \end{align} $$

Let $C^{\bullet }_{\mathrm {r}} = C^{\bullet }_{\mathrm {r}}(\tau _1,\tau _2)$ . A direct calculation yields

(2.10) $$ \begin{align} & H^0\big( C^{\bullet}_0,\partial \big) = W_{12} \simeq H^0\big( C^{\bullet}_{\mathrm{r}},\partial \big) \oplus K_1\oplus K_2 \;,\hspace{5mm} \nonumber\\ &H^0\big( C^{\bullet}_1 \oplus C^{\bullet}_2,\partial \big) = W_1 \oplus W_2 \;,\hspace{5mm} H^0\big( C^{\bullet}_3,\partial \big) = V \;, \nonumber\\ & H^1\big( C^{\bullet}_0,\partial \big) = H^1\big( C^{\bullet}_{\mathrm{r}},\partial \big)= V/\big(V_1 + V_2\big) \;,\hspace{5mm} H^1\big( C^{\bullet}_1 \oplus C^{\bullet}_2,\partial \big) = H^1\big( C^{\bullet}_3,\partial \big) = 0 \;. \end{align} $$

Thus the long exact sequence (2.8) is

(2.11)

2.2 Flat family of complexes

Now let $W_1$ , $W_2$ and V be flat complex vector bundles over a smooth manifold S. Let $\tau _1: W_1\rightarrow V$ and $\tau _2: W_2\rightarrow V$ be morphisms between flat complex vector bundles. Then the chain complexes $\big (C^{\bullet }_j,\partial \big )$ ( $j=0,1,2,3$ ) considered in §2.1 become flat chain complexes of complex vector bundles over S, and the exact sequence (2.7) becomes an exact sequence of flat chain complexes.

Let $h^{W_1}$ , $h^{W_2}$ and $h^V$ be Hermitian metrics on $W_1$ , $W_2$ and V. For $j=0,1,2,3$ , we construct a Hermitian metric $h^{C^{\bullet }_j}= h^{C^0_j} \oplus h^{C^1_j}$ on $C^{\bullet }_j$ as follows,

(2.12) $$ \begin{align} & h^{C^0_0} = h^{W_1} \oplus h^{W_2} \;;\hspace{5mm} h^{C^0_3} = \frac{1}{2} h^V \oplus \frac{1}{2} h^V \;;\hspace{5mm} h^{C^0_j} = h^{W_j} \oplus \frac{1}{2} h^V \hspace{5mm} \text{for } j=1,2 \;; \nonumber\\ & h^{C^1_j} = h^V \hspace{5mm}\text{for } j=0,1,2,3 \;. \end{align} $$

Let $\mathscr {T}_j \in Q^S$ be the torsion form (see §1.2) associated with $\big (C^{\bullet }_j,\partial ,h^{C^{\bullet }_j}\big )$ .

The exact sequence (2.8) becomes an exact sequence of flat complex vector bundles. Let $\mathscr {T}_{\mathscr {H}} \in Q^S$ be the torsion form (see §1.2) associated with the exact sequence (2.8) equipped with Hermitian metrics induced by $h^{C^{\bullet }_j}$ via Theorem 1.1.

Theorem 2.1. The following equation holds in $Q^S/Q^{S,0}$ ,

(2.13) $$ \begin{align} \mathscr{T}_0 - \mathscr{T}_1 - \mathscr{T}_2 + \mathscr{T}_3 + \mathscr{T}_{\mathscr{H}} = 0. \end{align} $$

To prove this theorem, we will need the following lemma, which is a consequence of [Reference Goette27, Thm. 7.37]Footnote ² .

Lemma 2.2. Let $(E^{{\bullet },j},\nabla ^{E^{{\bullet },j}},\partial ^j_V,h^{E^{{\bullet },j}})$ , $j=0,1,2$ , be three flat chain complexes of Hermitian vector bundles over S. Assume that there is an exact sequence of flat complexes

(2.14) $$ \begin{align} 0\to E^{{\bullet},0} \overset{\partial_H}{\longrightarrow} E^{{\bullet},1}\overset{\partial_H}{\longrightarrow} E^{{\bullet},2}\to 0, \end{align} $$

which means that $\partial _H$ is compatible with the connections and boundary operators of our complexes. We view these data as a double complex $(E^{{\bullet },{\bullet }}, \partial _V,\partial _H)$ .

Let $(MV^{\bullet },\nabla ^{MV^{\bullet }},\partial _{MV},h^{MV^{\bullet }})$ be the Mayer-Vietoris sequence associated with (2.14):

(2.15) $$ \begin{align} \cdots \to H^k(E^{{\bullet},0},\partial_V^0) \overset{a^k}{\longrightarrow} H^k(E^{{\bullet},1},\partial_V^1)\overset{b^k}{\longrightarrow} H^k(E^{{\bullet},2},\partial_V^2)\overset{c^k}{\longrightarrow} H^{k+1}(E^{{\bullet},0},\partial_V^0)\to \cdots \end{align} $$

Finally, let $(E^{i,{\bullet }},\nabla ^{E^{i,{\bullet }}},\partial _H^i,h^{E^{i,{\bullet }}})$ be the flat chain complex of Hermitian vector bundle given by (2.14) in (vertical) degree i.

Then in $Q^S/Q^{S,0}$ , we have

(2.16) $$ \begin{align} & \mathscr{T}(\nabla^{MV^{\bullet}},\partial_{MV},h^{MV^{\bullet}}) + \sum_{j=0}^2 (-1)^j \mathscr{T}(\nabla^{E^{{\bullet},j}},\partial_V^j,h^{E^{{\bullet},j}}) \nonumber \\ = & \sum_i (-1)^i \mathscr{T}(\nabla^{E^{i,{\bullet}}},\partial_H^i,h^{E^{i,{\bullet}}}). \end{align} $$

Proof. Let $(E_{tot},\partial _{tot})$ be the total complex of the double complex $(E^{{\bullet },{\bullet }}, \partial _V,\partial _H)$ , and let $(E_r^{{\bullet },{\bullet }},\partial _r)_r$ be the spectral sequence associated with the filtration on $E_{tot}$ given by the horizontal degree. Note that the double complex $E^{{\bullet },{\bullet }}$ has exact rows, so $(E_{tot},\partial _{tot})$ is acyclic. One computes that for $a^{\bullet }$ and $b^{\bullet }$ in (2.15),

(2.17)

Next, $c^{\bullet }$ in (2.15) induces an isomorphism $H^{\bullet }(E^{{\bullet },2},\partial _V^2)/\mathrm {Im}(b^{\bullet }) = H^{\bullet }(E^{{\bullet },2},\partial _V^2)/\operatorname {\mathrm {Ker}}(c^{\bullet }) \simeq \mathrm {Im}(c^{\bullet })=\operatorname {\mathrm {Ker}}(a^{{\bullet }+1})$ , and

(2.18) $$ \begin{align} \begin{aligned} &E_2^{p,0}=\operatorname{\mathrm{Ker}}(a^p), \quad E_2^{p,1}=0 \quad \text{and} \quad E_2^{p,2}=H^{p}(E^{{\bullet},2},\partial_V^2)/\mathrm{Im}(b^{p}),\\ & \partial_2\colon E_2^{p,0} \to E_2^{p-1,2} \quad \text{is the inverse of said isomorphism.} \end{aligned} \end{align} $$

Finally, $(E_3,\partial _3)=0$ .

In this proof, to simplify the notations, and because the connections, metrics and boundary operators on every complex we consider are canonically defined from $\nabla ^{E^{{\bullet },{\bullet }}}$ , $h^{E^{{\bullet },{\bullet }}}$ , $\partial _V$ and $\partial _H$ , we will denote $\mathscr {T}(\nabla ^{W^{\bullet }},\partial _W,h^{W^{\bullet }})$ simply as $\mathscr {T}(W^{\bullet })$ for any flat chain complex of Hermitian vector bundles $(W^{\bullet },\nabla ^{W^{\bullet }},\partial _W,h^{W^{\bullet }})$ . When it helps to follow the computations, we will also denote this torsion by $\mathscr {T}(W^0\to \cdots \to W^n)$ if $W^{\bullet }=\bigoplus _{k=0}^nW^k$ . In the same way, we will denote $H^k(W^{\bullet },\partial _W)$ simply by $H^k(W^{\bullet })$ .

By [Reference Goette27, Thm. 7.37], we then find

(2.19) $$ \begin{align} \mathscr{T}(E_{tot})= \mathscr{T}(E_0)+\mathscr{T}(E_1)+\mathscr{T}(E_2). \end{align} $$

By (2.17), we have

(2.20) $$ \begin{align} \mathscr{T}(E_0) = \sum_{j=0}^2 (-1)^j \mathscr{T}(E^{{\bullet},j}). \end{align} $$

To compute $\mathscr {T}(E_1)+\mathscr {T}(E_2)$ , let us recall [Reference Ma44, (3.9)]: for any flat chain complex of Hermitian vector bundles $(W^{\bullet },\nabla ^{W^{\bullet }},\partial _W,h^{W^{\bullet }})$ , we have

(2.21) $$ \begin{align} \mathscr{T}(W^{\bullet}) = &\sum_k (-1)^k\bigg[ \mathscr{T}\Big(\operatorname{\mathrm{Ker}}(\partial_W^k)\hookrightarrow W^k \overset{\partial_W^k}{\longrightarrow}\mathrm{Im}(\partial_W^k)\Big) \nonumber\\ & +\mathscr{T}\Big(\mathrm{Im}(\partial_W^{k-1})\hookrightarrow \operatorname{\mathrm{Ker}}(\partial_W^k) \twoheadrightarrow H^k(W^{\bullet})\Big) \bigg], \end{align} $$

where $\partial _W^k=\partial _W|_{W^k}$ . Applying this here, we find on the one hand

(2.22) $$ \begin{align} \begin{aligned} \mathscr{T}\Big( H^k(E^{{\bullet},0}) &\overset{a^k}{\longrightarrow} H^k(E^{{\bullet},1})\overset{b^k}{\longrightarrow} H^k(E^{{\bullet},2}) \Big) \\ = &\,\mathscr{T}\Big(\operatorname{\mathrm{Ker}}(a^k)\hookrightarrow H^k(E^{{\bullet},0}) \overset{a^k}{\longrightarrow} \mathrm{Im}(a^k)\Big) \\ &- \mathscr{T}\Big(\operatorname{\mathrm{Ker}}(b^k)\hookrightarrow H^k(E^{{\bullet},1}) \overset{b^k}{\longrightarrow} \mathrm{Im}(b^k)\Big) \\ &+\mathscr{T}\Big(\mathrm{Im}(b^k)\hookrightarrow H^k(E^{{\bullet},2}) \twoheadrightarrow H^k(E^{{\bullet},2})/\mathrm{Im}(b^k) \Big), \end{aligned} \end{align} $$

and on the other hand

(2.23) $$ \begin{align} \begin{aligned} \mathscr{T}(MV^{\bullet})= \sum_k (-1)^k\Bigg[&\mathscr{T}\Big(\operatorname{\mathrm{Ker}}(a^k)\hookrightarrow H^k(E^{{\bullet},0}) \overset{a^k}{\longrightarrow} \mathrm{Im}(a^k)\Big) \\ &- \mathscr{T}\Big(\operatorname{\mathrm{Ker}}(b^k)\hookrightarrow H^k(E^{{\bullet},1}) \overset{b^k}{\longrightarrow} \mathrm{Im}(b^k)\Big) \\ &+\mathscr{T}\Big(\operatorname{\mathrm{Ker}}(c^k)\hookrightarrow H^k(E^{{\bullet},2}) \overset{c^k}{\longrightarrow} \mathrm{Im}(c^k) \Big)\Bigg]. \end{aligned} \end{align} $$

By (2.17), (2.22) and (2.23) we get

(2.24) $$ \begin{align} \mathscr{T}(MV^{\bullet})-\mathscr{T}(E_1) = & \,\sum_k (-1)^k\Bigg[\mathscr{T}\Big(\operatorname{\mathrm{Ker}}(c^k)\hookrightarrow H^k(E^{{\bullet},2}) \overset{c^k}{\longrightarrow} \mathrm{Im}(c^k) \Big)\nonumber \\ &- \mathscr{T}\Big(\mathrm{Im}(b^k)\hookrightarrow H^k(E^{{\bullet},2}) \twoheadrightarrow H^k(E^{{\bullet},2})/\mathrm{Im}(b^k) \Big) \Bigg]. \end{align} $$

Now, applying [Reference Bismut and Lott12, Thm. A1.4] to the exact double complex

we find from (2.18) and (2.24) that

(2.25) $$ \begin{align} \mathscr{T}(MV^{\bullet})-\mathscr{T}(E_1)=\mathscr{T}(E_2). \end{align} $$

By (2.19), (2.20) and (2.25), we finally obtain

(2.26) $$ \begin{align} \mathscr{T}(E_{tot})= \mathscr{T}(MV^{\bullet})+\sum_{j=0}^2 (-1)^j \mathscr{T}(E^{{\bullet},j}). \end{align} $$

To conclude the proof of (2.16), we compute $\mathscr {T}(E_{tot})$ by using the spectral sequence associated with the filtration on $E_{tot}$ given by the vertical degree, denoted by $({E^{\prime }}_r^{{\bullet },{\bullet }},\partial ^{\prime }_r)_r$ . This sequence is computed as follows:

(2.27)

By [Reference Goette27, Thm. 7.37] and (2.27) we get

(2.28) $$ \begin{align} \mathscr{T}(E_{tot})= \mathscr{T}(E^{\prime}_0) = \sum_i (-1)^i \mathscr{T}(E^{i,{\bullet}}). \end{align} $$

Lemma 2.15 follows from (2.26) and (2.28).

Proof of Theorem 2.1

Applying Lemma 2.15 to the setting of Theorem 2.1, we get that

(2.29) $$ \begin{align} \mathscr{T}_0 - \mathscr{T}_1 - \mathscr{T}_2 + \mathscr{T}_3 + \mathscr{T}_{\mathscr{H}} = \mathscr{T}^{\{0\}} - \mathscr{T}^{\{1\}}, \end{align} $$

where $\mathscr {T}^{\{k\}}$ is the torsion of the sequence (2.7) in degree k.

Let $(\underline {W},\nabla ^{\underline {W}},h^{\underline {W}})$ and $(\underline {V},\nabla ^{\underline {V}},h^{\underline {V}})$ be two flat complex vector bundles and let $\underline {\tau }\colon \underline {W}\to \underline {V}$ be a morphism of flat complex vector bundles. Let $(E^{\bullet },\partial )$ be the complex

(2.30)

We endow $E^{\bullet }$ with the connection $\nabla ^{E^{\bullet }}$ and metric $h^{E^{\bullet }}$ induced by those of $\underline {W}$ and $\underline {V}$ . Then $(E^{\bullet },\nabla ^{E^{\bullet }},\partial ,h^{E^{\bullet }})$ is flat chain complex of Hermitian vector bundles.

By the anomaly formula [Reference Bismut and Lott12, Thm. 2.24], we see that $\mathscr {T}(\nabla ^{E^{\bullet }},\partial ,h^{E^{\bullet }})$ does not depend on the metric $h^{\underline {V}}$ . For $s>0$ , set $h^{\underline {V}}_s=sh^{\underline {V}}$ , and let $h^{E^{\bullet }}_s$ be the induced metric on $E^{\bullet }$ . Let $q\colon E^{\bullet } \to E^{\bullet }$ be such that q is the identity on the factors $\underline {V}$ and 0 on the factors $\underline {W}$ . Then we have an isomorphism of flat chain complexes of Hermitian vector bundles $s^{q/2} \colon (E^{\bullet },\nabla ^{E^{\bullet }},\partial ,h^{E^{\bullet }}_s) \to (E^{\bullet },\nabla ^{E^{\bullet }},\partial _s,h^{E^{\bullet }})$ , with $\partial _s =s^{q/2} \partial s^{-q/2}$ . Observe that $\partial _s$ is given by the same formula as $\partial $ in (2.30) but with $\underline {\tau }$ replaced by $\sqrt {s}\underline {\tau }$ , so $\partial _s$ has a limit, denoted by $\partial _0$ , as $s\to 0$ . As the cohomology of $(E^{\bullet },\partial _s)$ is constant for $s\geq 0$ , we get that

(2.31) $$ \begin{align} \mathscr{T}(\nabla^{E^{\bullet}},\partial,h^{E^{\bullet}}_s)=\mathscr{T}(\nabla^{E^{\bullet}},\partial_s,h^{E^{\bullet}}) \underset{s\to 0}{\longrightarrow}\mathscr{T}(\nabla^{E^{\bullet}},\partial_0,h^{E^{\bullet}}). \end{align} $$

Moreover, $(E^{\bullet },\nabla ^{E^{\bullet }},\partial _0,h^{E^{\bullet }})$ splits in the sense of [Reference Ma44, §3.1], so its torsion vanishes. We conclude from the above discussion that

(2.32) $$ \begin{align} \mathscr{T}(\nabla^{E^{\bullet}},\partial,h^{E^{\bullet}})=0. \end{align} $$

Now, in degree one, (2.7) is the complex $E^{\bullet }$ for $(\underline {W},\nabla ^{\underline {W}},h^{\underline {W}})=(\underline {V},\nabla ^{\underline {V}},h^{\underline {V}})=(V,\nabla ^V,h^V)$ and $\tau =\operatorname {\mathrm {Id}}_V$ , so (2.32) gives

(2.33) $$ \begin{align} \mathscr{T}^{\{1\}} = 0. \end{align} $$

In degree 0, we take $(\underline {W},\nabla ^{\underline {W}},h^{\underline {W}})=(W_1\oplus W_2,\nabla ^{W_1}\oplus \nabla ^{W_2},h^{W_1}\oplus h^{W_2})$ , $(\underline {V},\nabla ^{\underline {V}},h^{\underline {V}})=(V\oplus V,\nabla ^V\oplus \nabla ^V,\frac {1}{2} h^V \oplus \frac {1}{2} h^V)$ and $\underline {\tau }=\tau _1\oplus \tau _2$ . Let $\psi $ be the morphism from (2.7) in degree 0 to $E^{\bullet }$ given by

(2.34) $$ \begin{align} \begin{aligned} &\psi|_{C_0^0} = \operatorname{\mathrm{Id}}_{W_1}\oplus \operatorname{\mathrm{Id}}_{ W_2}, &\psi|_{C_1^0\oplus C_2^0} \colon (w_1,v,v',w_2) \mapsto (w_1,w_2,v',v), \\ &\psi|_{C_3^0} = \operatorname{\mathrm{Id}}_{V}\oplus(- \operatorname{\mathrm{Id}}_{V}). \end{aligned} \end{align} $$

Then $\psi $ is an isomorphism of flat chain complexes of Hermitian vector bundles, and by (2.32) we have

(2.35) $$ \begin{align} \mathscr{T}^{\{0\}}= \mathscr{T}(\nabla^{E^{\bullet}},\partial,h^{E^{\bullet}})= 0. \end{align} $$

From (2.29), (2.33) and (2.35), we obtain Theorem 2.1.

3.1 A two-parameter deformation

Recall that $M_{j,R}$ , $j=0,1,2,3$ , were defined in (0.20) and that $\pi _R: M_R \rightarrow S$ was constructed in the paragraph containing (0.21). We may view $M_1"$ , $M_2"$ and $IN_R$ as subsets of $M_R$ . Set

(3.1) $$ \begin{align} \pi_{j,R} = \pi_R\big|_{M_{j,R}}: M_{j,R} \rightarrow S \hspace{2.5mm} \text{for } j=0,1,2,3 \;. \end{align} $$

Let $Z_{j,R}$ be the fibre of $\pi _{j,R}$ . We denote $Z_R = Z_{0,R}$ .

Recall that the diffeomorphism $\varphi _R: M \rightarrow M_R$ was constructed in (0.22). Recall that $T^HM \subseteq TM$ and $T^HN \subseteq TN$ were introduced in (0.5) and (0.8). Set

(3.2) $$ \begin{align} T^HM_R = \varphi_{R,*}\big(T^HM\big) \subseteq TM_R \;. \end{align} $$

Then we have $TM_R = T^HM_R \oplus TZ_R$ . By (0.8), (0.22) and (3.2), we have $T^HM_R \big |_{IN_R} = \mathrm {pr}_2^*\big (T^HN\big )$ , where $\mathrm {pr}_2: IN_R = [-R,R]\times N \rightarrow N$ is the canonical projection. For $j=0,1,2,3$ , set

(3.3) $$ \begin{align} T^HM_{j,R} = T^HM_R\big|_{M_{j,R}} \subseteq TM_{j,R} \quad \text{and} \quad g^{TZ_{j,R}} = g^{TZ_R}\big|_{M_{j,R}}, \end{align} $$

with $g^{TZ_R}$ in (0.24).

We still denote by $(F,\nabla ^F, h^F)$ the flat complex vector bundle with Hermitian metric $h^F$ on $M_R$ induced by the product structure (0.9) and (0.10) of $(F,\nabla ^F, h^F)$ on M.

Let $f_\infty : [-1,1] \rightarrow [0,1]$ be a smooth even function as in (0.25). We further assume that

(3.4) $$ \begin{align} & \big| f^{\prime}_\infty(s) \big| \leqslant 2 \hspace{2.5mm} \text{for } s \in [-1,1] \;; \nonumber\\ & f_\infty(s) = 1 - s^2/2 \hspace{2.5mm} \text{for } |s|\leqslant \frac{1}{4} \;; \hspace{2.5mm} f_\infty(s) = (s-b)^2/2 \hspace{2.5mm} \text{for } b = \pm 1 ,\; |s-b|\leqslant \frac{1}{4} \;. \end{align} $$

Let $\chi : \mathbb {R} \rightarrow \mathbb {R}$ be a smooth function such that

(3.5) $$ \begin{align} 0 \leqslant \chi \leqslant 1 \;,\hspace{5mm} \chi\big|_{]-\infty,1/4]} = 0 \;,\hspace{5mm} \chi\big|_{[1/2,+\infty[} = 1 \;,\hspace{5mm} 0 \leqslant \chi' \leqslant 8 \;. \end{align} $$

Figure 2

The graph of $\chi $ .

For $T\geqslant 0$ , as $f^{\prime }_\infty $ is odd, there is a unique smooth function $f_T: [-1,1] \rightarrow [0,1]$ satisfying

(3.6) $$ \begin{align} f_T(-1) = f_T(1) = 0 \;,\hspace{5mm} f^{\prime}_T(s) = f^{\prime}_\infty(s) \chi\big(e^{T^2}(1-|s|)\big) \;. \end{align} $$

Figure 3

The graphs of $f_\infty $ (dashed) and of $f_T$ (solid).

By (3.4)-(3.6), we have the following uniform estimates,

(3.7) $$ \begin{align} f_T(s) = f_\infty(s) + \mathscr{O}\big(e^{-T^2}\big) \;,\hspace{5mm} f_T'(s) = f^{\prime}_\infty(s) + \mathscr{O}\big(e^{-T^2}\big) \;,\hspace{5mm} f_T"(s) = \mathscr{O}\big(1\big) \;. \end{align} $$

Moreover, we have $\mathrm {supp}(f_T) \subseteq \big [-1+e^{-T^2}/4,1-e^{-T^2}/4\big ]$ . We will view $f_T$ as a smooth function on $M_R$ in the sense of (0.27). Set

(3.8) $$ \begin{align} h^F_T = e^{-2Tf_T}h^F \;. \end{align} $$

For $j=0,1,2,3$ , we define the analytic torsion form (see Definition 1.8)

(3.9) $$ \begin{align} \mathscr{T}_{j,R,T} \in Q^S \end{align} $$

associated with $\pi _{j,R}$ , $T^HM_{j,R}$ , $g^{TZ_{j,R}},F\big |_{M_{j,R}}$ , $\nabla ^F\big |_{M_{j,R}}$ and $h^F_T\big |_{M_{j,R}}$ .

For $j=0,1,2,3$ , let $d^{Z_{j,R}}$ be the de Rham operator on $\Omega ^{\bullet }(Z_{j,R},F)$ . Let $\big \lVert \cdot \big \rVert _{Z_{j,R}}$ be the $L^2$ -norm on $\Omega ^{\bullet }(Z_{j,R},F)$ with respect to $g^{TZ_{j,R}}$ and $h^F$ . Let $d^{Z_{j,R},*}$ be the formal adjoint of $d^{Z_{j,R}}$ with respect to $\big \lVert \cdot \big \rVert _{Z_{j,R}}$ . Set

(3.10) $$ \begin{align} d^{Z_{j,R}}_T = e^{-Tf_T}d^{Z_{j,R}}e^{Tf_T} \;,\hspace{5mm} d^{Z_{j,R},*}_T = e^{Tf_T}d^{Z_{j,R},*}e^{-Tf_T} \;,\hspace{5mm} D^{Z_{j,R}}_T = d^{Z_{j,R}}_T + d^{Z_{j,R},*}_T \;. \end{align} $$

We remark that $e^{Tf_T} D^{Z_{j,R}}_T e^{-Tf_T}$ is the Hodge de Rham operator with respect to $g^{TZ_{j,R}}$ and $h^F_T$ . The self-adjoint extension of $D^{Z_{j,R}}_T$ with $\mathrm {Dom}\big (D^{Z_{j,R}}_T\big ) = \Omega ^{\bullet }_{\mathrm {abs}}(Z_{j,R},F)$ (see [Reference Puchol, Zhang and Zhu55, (1.4)]) will still be denoted by $D^{Z_{j,R}}_T$ . By the Hodge theorem (see [Reference Brüning and Ma15, Thm 3.1], [Reference Puchol, Zhang and Zhu55, Thm. 1.1]), the following map is bijective,

(3.11) $$ \begin{align} \operatorname{\mathrm{Ker}}\big(D^{Z_{j,R}}_T\big) & \rightarrow H^{\bullet}(Z_{j,R},F)\nonumber \\ \omega & \mapsto \big[e^{Tf_T}\omega\big] \;. \end{align} $$

For $j=0,1,2,3$ , let $h^{H^{\bullet }(Z_j,F)}_{R,T}$ be the Hermitian metric on $H^{\bullet }(Z_j,F) = H^{\bullet }(Z_{j,R},F)$ induced by $\big \lVert \cdot \big \rVert _{Z_{j,R}}$ via the isomorphism (3.11). Let

(3.12) $$ \begin{align} \mathscr{T}_{\mathscr{H},R,T}\in Q^S \end{align} $$

be the torsion form ([Reference Bismut and Lott12, §2], see §1.2) associated with the exact sequence (0.12) equipped with Hermitian metrics $\big (h^{H^{\bullet }(Z_j,F)}_{R,T}\big )_{j=0,1,2,3}$ .

3.2 Several intermediate results

We fix a constant $0<\kappa <1/3$ .

Theorem 3.1. There exists $\alpha>0$ such that for $j=0,1,2,3$ and $T = R^\kappa \gg 1$ , we have

(3.13) $$ \begin{align} \mathrm{Sp} \Big(RD^{Z_{j,R}}_T\Big) \subseteq \big]-\infty,-\alpha\sqrt{T}\big] \cup \big[-1,1\big] \cup \big[\alpha\sqrt{T},+\infty\big[ \;. \end{align} $$

Let $\mathscr {E}_{j,R,T} \subseteq \Omega ^{\bullet }(Z_{j,R},F)$ be eigenspace of $RD^{Z_{j,R}}_T$ associated with eigenvalues in $[-1,1]$ . Since $d^{Z_{j,R}}_T$ commutes with $D^{Z_{j,R},2}_T$ , we have a finite-dimensional complex

(3.14) $$ \begin{align} \big(\mathscr{E}_{j,R,T}\,,\,d^{Z_{j,R}}_T\big) \;. \end{align} $$

We will show in Theorem 3.3 that for $T=R^\kappa \gg 1$ , $\dim \mathscr {E}_{j,R,T}$ is independent of $R,T$ .

Recall that the chain complexes $\big (C^{\bullet }_j,\partial \big )$ with $j=0,1,2,3$ were constructed in §2.2. The constructions depend on the morphisms $\tau _1: W_1 \rightarrow V$ and $\tau _2: W_2 \rightarrow V$ . For $j=1,2$ , let $\alpha _j: Y \simeq \partial Z_j \hookrightarrow Z_j$ be the canonical embedding. In the sequel, we take

(3.15) $$ \begin{align} V = H^{\bullet}(Y,F) \;,\hspace{2.5mm} W_j = H^{\bullet}(Z_j,F) \;,\hspace{2.5mm} \tau_j = \alpha_j^* \hspace{2.5mm} \text{ for } \; j=1,2 \;. \end{align} $$

Then $W_1$ , $W_2$ and V are $\mathbb {Z}$ -graded flat vector bundles. We will use the notations $W^{\bullet }_1$ , $W^{\bullet }_2$ , $V^{\bullet }$ and $\big (C^{{\bullet },{\bullet }}_j,\partial \big )$ to emphasize the grading, i.e., $C^{0,k}_0 = W^k_1 \oplus W^k_2$ , $C^{1,k}_0 = V^k$ , etc. Note that the complex $C_0^{0,k}\to C_0^{1,k}$ appears in the Mayer-Vietoris sequence (0.12).

We will construct a Hermitian metric on $C^{{\bullet },{\bullet }}_j$ . For that, we need several notations. We define $Z_{1,\infty } = Z_{1,0} \cup [0,+\infty [ \times Y$ and $Z_{2,\infty } = Z_{2,0} \cup ]-\infty ,0] \times Y$ , where $\partial Z_{1,0} \simeq \partial Z_{2,0} \simeq Y$ are identified with $\{0\}\times Y$ . Then $Z_{1,\infty }$ and $Z_{2,\infty }$ may be viewed as the limit of $Z_{1,R}$ and $Z_{2,R}$ as $R\rightarrow \infty $ . For $j=1,2$ , let $d^{Z_{j,\infty }}$ be the de Rham operator on $\Omega ^{\bullet }(Z_{j,\infty },F)$ , let $d^{Z_{j,\infty },*}$ be its formal adjoint, let $D^{Z_{j,\infty }} = d^{Z_{j,\infty }} + d^{Z_{j,\infty },*}$ be the Hodge de Rham operator on $\Omega ^{\bullet }(Z_{j,\infty },F)$ . Let $D^Y$ be the Hodge de Rham operator on $\Omega ^{\bullet }(Y,F)$ . We denote $\mathscr {H}^{\bullet }(Y,F) = \operatorname {\mathrm {Ker}}\big (D^Y\big )$ . Following [Reference Puchol, Zhang and Zhu55, (2.52)], we define

(3.16) $$ \begin{align} \mathscr{H}^{\bullet}_{\mathrm{abs}}(Z_{1,\infty},F) & = \Big\{ (\omega,\hat{\omega}) \in \Omega^{\bullet}(Z_{1,\infty},F) \times \mathscr{H}^{\bullet}(Y,F) \;: \nonumber\\ & d^{Z_{1,\infty}}\omega = d^{Z_{1,\infty},*}\omega = 0 \;,\hspace{5mm} \omega\big|_{[0,+\infty[\times Y} - \hat{\omega} \text{ is } L^2 \text{-integrable} \Big\} \;, \end{align} $$

where $\hat {\omega }$ is viewed as a constant section in ${\mathscr {C}^\infty }\big ([0,+\infty [,\mathscr {H}^{\bullet }(Y,F)\big ) \subseteq \Omega ^{\bullet }([0,+\infty [\times Y,F)$ . We also define $\mathscr {H}^{\bullet }_{\mathrm {abs}}(Z_{2,\infty },F)$ in the same way. By [Reference Puchol, Zhang and Zhu55, Prop. 3.16, Thm. 3.19],

(3.17) $$ \begin{align} \mathscr{H}^{\bullet}_{\mathrm{abs}}(Z_{j,\infty},F) & \rightarrow H^{\bullet}(Z_{j,\infty},F) = W^{\bullet}_j\nonumber \\ (\omega,\hat{\omega}) & \mapsto [\omega] \end{align} $$

is bijective. Moreover, we claim that the following diagram commutes,

(3.18)

Let us prove this claim. For $(\omega ,\hat {\omega }) \in \mathscr {H}^{\bullet }_{\mathrm {abs}}(Z_{j,\infty },F)$ , we denote by $i_u \colon Y \hookrightarrow Z_{j,\infty }$ the identification of Y with $\{u\}\times Y$ . Then, as on the cylinder $0= d^{Z_{j,\infty }}\omega = (du\wedge \frac {\partial }{\partial u} + d^{Y})\omega $ , we get that the cohomology class $[i_u^*\omega ]$ is constant in $V^{\bullet }$ , and in particular $\tau _j([\omega ]) = [i_u^*\omega ]$ . On the other hand, by the structure of eigensections on a cylinder [Reference Puchol, Zhang and Zhu55, (2.10)] (see also (4.28)), we see that $i_u^*\omega \underset {u\to (-1)^j\infty }{\longrightarrow } \hat {\omega }$ , so $\tau _j([\omega ])=[\hat {\omega }]$ .

By [Reference Puchol, Zhang and Zhu55, (2.40)], we have

(3.19) $$ \begin{align} & W^{\bullet}_j = K^{\bullet}_j \oplus K^{{\bullet},\perp}_j \qquad\text{with} \nonumber\\ &K^{\bullet}_j = \Big\{ [\omega] \;:\; (\omega,\hat{\omega})\in\mathscr{H}^{\bullet}_{\mathrm{abs}}(Z_{j,\infty},F)\;,\; \hat{\omega} = 0 \Big\} \;, \nonumber\\ &K^{{\bullet},\perp}_j = \Big\{ [\omega] \;:\; (\omega,\hat{\omega})\in\mathscr{H}^{\bullet}_{\mathrm{abs}}(Z_{j,\infty},F)\;, \nonumber\\ & \hspace{20mm} \omega \text{ is a generalized eigensection of } D^{Z_{j,\infty}} \Big\} \;. \end{align} $$

By (3.18), the definition of $K^{\bullet }_j$ in (3.19) is compatible with (2.2). We construct a Hermitian metric $h^{K^{\bullet }_j}$ on $K^{\bullet }_j$ as follows: for $(\omega ,\hat {\omega })\in \mathscr {H}^{\bullet }_{\mathrm {abs}}(Z_{j,\infty },F)$ with $\hat {\omega }=0$ ,

(3.20) $$ \begin{align} h^{K^{\bullet}_j}\big([\omega],[\omega]\big) = \big\lVert\omega\big\rVert^2_{Z_{j,\infty}} \;. \end{align} $$

We construct a Hermitian metric $h^{K^{{\bullet },\perp }_j}$ on $K^{{\bullet },\perp }_j$ as follows: for $(\omega ,\hat {\omega })\in \mathscr {H}^{\bullet }_{\mathrm {abs}}(Z_{j,\infty },F)$ with $\omega $ a generalized eigensection of $D^{Z_{j,\infty }}$ ,

(3.21) $$ \begin{align} h^{K^{{\bullet},\perp}_j}\big([\omega],[\omega]\big) = \big\lVert\hat{\omega}\big\rVert^2_Y \;. \end{align} $$

By [Reference Puchol, Zhang and Zhu55, (2.53)], $h^{K^{\bullet }_j}$ and $h^{K^{{\bullet },\perp }_j}$ are well-defined. Set

(3.22) $$ \begin{align} h^{W^{\bullet}_j}_{R,T} = h^{K^{\bullet}_j} \oplus \frac{\sqrt{\pi}}{2} R T^{-1/2} h^{K^{{\bullet},\perp}_j} \;. \end{align} $$

Let $h^{V^{\bullet }}$ be the Hermitian metric on $V^{\bullet }$ induced by $\big \lVert \cdot \big \rVert _Y$ via the identification $V^{\bullet } = \mathscr {H}^{\bullet }(Y,F)$ induced by the Hodge theory. Set

(3.23) $$ \begin{align} h^{V^{\bullet}}_{R,T} = \sqrt{\pi} R T^{-1/2} h^{V^{\bullet}} \;. \end{align} $$

We construct a Hermitian metric $h^{C^{{\bullet },{\bullet }}_j}_{R,T}$ on $C^{{\bullet },{\bullet }}_j$ as follows,

(3.24) $$ \begin{align} & h^{C^{0,{\bullet}}_0}_{R,T} = h^{W^{\bullet}_1}_{R,T} \oplus h^{W^{\bullet}_2}_{R,T} \;;\hspace{5mm} h^{C^{0,{\bullet}}_3}_{R,T} = \frac{1}{2} h^{V^{\bullet}}_{R,T} \oplus \frac{1}{2} h^{V^{\bullet}}_{R,T} \;; \nonumber\\ & h^{C^{0,{\bullet}}_j}_{R,T} = h^{W^{\bullet}_j}_{R,T} \oplus \frac{1}{2} h^{V^{\bullet}}_{R,T} \hspace{2.5mm}\text{for } j=1,2 \;; \hspace{5mm} h^{C^{1,{\bullet}}_j}_{R,T} = h^{V^{\bullet}}_{R,T} \hspace{2.5mm}\text{for } j=0,1,2,3 \;. \end{align} $$

For a positive function $G(R,T)$ and a two-parameter family $A_{R,T}\in \Omega ^{\bullet }\Big (S,\mathrm {End}\big (C^{{\bullet },{\bullet }}_j\big )\Big )$ (resp. $\tau _{R,T} \in \Omega ^{\bullet }(S,E)$ with $\big (E,\lVert \cdot \rVert _E\big )$ a Hermitian vector bundle on S) depending on $R,T\geqslant 1$ , we write

(3.25) $$ \begin{align} A_{R,T} = \mathscr{O}_{R,T}\big(G(R,T)\big) \qquad \Big(\;\text{resp.}\quad \tau_{R,T} = \mathscr{O}_E\big(G(R,T)\big)\;\Big) \;, \end{align} $$

if there exists $a>0$ such that the $\mathscr {C}^0$ -norm of $A_{R,T}$ induced by the operator norm with respect to $h^{C^{{\bullet },{\bullet }}_j}_{R,T}$ (resp. the $\mathscr {C}^0$ -norm of $\tau _{R,T}$ induced by $\lVert \cdot \rVert _E$ ) is dominated by $a G(R,T)$ for $R,T\geqslant 1$ . We remark that $\mathscr {O}_E(\cdot )$ is independent of the fixed $\lVert \cdot \rVert _E$ . If E is a trivial line bundle, we abbreviate (3.25) as $\tau _{R,T} = \mathscr {O}\big (G(R,T)\big )$ .

The following result is the core of our proof. It is an analogue of [Reference Bismut and Zhang14, Thms. 6.9 and 6.12] and [Reference Bismut and Goette9, Thm. 11.4] in our setting.

For ease of notations, we denote $\partial _T = \pi ^{-1/2} T^{1/2} e^{-T} \partial : C^{0,{\bullet }}_j \rightarrow C^{1,{\bullet }}_j$ .

Let $\widehat {\mathscr {T}}^k_{j,R,T}\in Q^S$ be the torsion form (see §1.2) associated with $\big (C^{{\bullet },k}_j,R^{-1}\partial _T,h^{C^{{\bullet },k}_j}_{R,T}\big )$ . We view $\big (C^{{\bullet },{\bullet }}_j,R^{-1}\partial _T\big )$ as a complex, whose component of degree k is $\bigoplus _{p+q=k}C^{p,q}_j$ .

Lemma 3.4. Let $\widehat {\mathscr {T}}_{j,R,T}$ be the torsion form associated with $\big (C^{{\bullet },{\bullet }}_j,R^{-1}\partial _T,h^{C^{{\bullet },{\bullet }}_j}_{R,T}\big )$ , then

(3.28) $$ \begin{align} \widehat{\mathscr{T}}_{j,R,T} = \sum_{k=0}^{\dim Z} (-1)^k \widehat{\mathscr{T}}^k_{j,R,T} \;. \end{align} $$

Proof. For any flat Hermitian complex $(W^{\bullet },\nabla ^{W^{\bullet }},\partial , h^{W^{\bullet }})$ , we define its $\ell $ th right shift $W_{[\ell ]}^{\bullet }$ by $W_{[\ell ]}^k = W^{k-\ell }$ , endowed with the induced structures $(\nabla ^{W_{[\ell ]}^{\bullet }},\partial _{[\ell ]}, h^{W_{[\ell ]}^{\bullet }})$ . Recall that $\mathfrak {X}_t$ is defined in (1.10), and set $f'(\partial + \nabla ^{W^{\bullet }},h^{W^{\bullet }}_t) = (2i\pi )^{1/2}\varphi \operatorname {\mathrm {Tr}} \left [(-1)^{N^{W^{\bullet }}}f'(\mathfrak {X}_t) \right ]$ . By (1.12), we have

(3.29) $$ \begin{align} \begin{aligned} f^\wedge(\partial_{[\ell]} + \nabla^{W_{[\ell]}^{\bullet}},h^{W_{[\ell]}^{\bullet}}_t) &=(-1)^\ell f^\wedge(\partial + \nabla^{W^{\bullet}},h^{W^{\bullet}}_t) + \frac{(-1)^\ell\ell}{(2i\pi)^{1/2}}f'(\partial + \nabla^{W^{\bullet}},h^{W^{\bullet}}_t), \\ \chi'(W_{[\ell]}^{\bullet}) &= (-1)^\ell \chi'(W^{\bullet}) + (-1)^\ell\ell\chi(W^{\bullet}). \end{aligned} \end{align} $$

By [Reference Bismut and Lott12, Prop. 1.3 and Thm. 2.11], $f'(\partial + \nabla ^{W^{\bullet }},h^{W^{\bullet }}_t) = \chi (W^{\bullet })$ . Note also that $H^{\bullet }(W_{[\ell ]}^{\bullet }) = H^{\bullet }(W^{\bullet })_{[\ell ]}$ and $\chi (W^{\bullet })=\chi (H^{\bullet }(W^{\bullet }))$ , so using (3.29) we find

(3.30) $$ \begin{align} \mathscr{T}\big(\nabla^{W_{[\ell]}^{\bullet}},\partial_{[\ell]}, h^{W_{[\ell]}^{\bullet}}\big) = (-1)^\ell \mathscr{T}\big(\nabla^{W^{\bullet}},\partial,h^{W^{\bullet}}\big). \end{align} $$

This implies (3.28) because, as a flat Hermitian complex, $C^{{\bullet },{\bullet }}_j = \bigoplus _k (C_j^{{\bullet },k})_{[k]}$ .

We equip $Q^S$ with the $\mathscr {C}^0$ -norm and $Q^S/Q^{S,0}$ with quotient norm. For a family of elements in $Q^S/Q^{S,0}$ parameterized by $R,T\geqslant 1$ , we use the same notation as in (3.25).

Theorem 3.5. For $T = R^\kappa \gg 1$ , the following identity holds in $Q^S/Q^{S,0}$ ,

(3.31) $$ \begin{align} \sum_{j=0}^3 (-1)^{j(j-3)/2} \mathscr{T}_{j,R,T} = \sum_{j=0}^3 (-1)^{j(j-3)/2} \widehat{\mathscr{T}}_{j,R,T} + \mathscr{O}\big(R^{-\kappa/4}\big) \;. \end{align} $$

We have a Mayer-Vietoris exact sequence of flat complex vector bundles over S,

(3.32) $$ \begin{align} 0 \rightarrow H^0\big( C^{{\bullet},k}_0,R^{-1}\partial_T \big) & \rightarrow H^0\big( C^{{\bullet},k}_1 \oplus C^{{\bullet},k}_2,R^{-1}\partial_T \big) \nonumber\\ & \rightarrow H^0\big( C^{{\bullet},k}_3,R^{-1}\partial_T \big) \rightarrow H^1\big( C^{{\bullet},k}_0,R^{-1}\partial_T \big) \rightarrow 0 \;, \end{align} $$

which is induced by (2.7) with $\partial $ replaced by $R^{-1}\partial _T$ . We equip (3.32) with Hermitian metrics induced by $h^{C^{{\bullet },{\bullet }}_j}_{R,T}$ via Theorem 1.1. Let $\widehat {\mathscr {T}}_{\mathscr {H},R,T}^k\in Q^S$ be the torsion form (see §1.2) associated with the exact sequence (3.32).

Theorem 3.6. For $T = R^\kappa \gg 1$ , the following identity holds in $Q^S/Q^{S,0}$ ,

(3.33) $$ \begin{align} \mathscr{T}_{\mathscr{H},R,T} = \sum_k (-1)^k \widehat{\mathscr{T}}_{\mathscr{H},R,T}^k + \mathscr{O}\big(R^{-1/4+\kappa/8}\big) \;. \end{align} $$

Proof of Theorem 0.1

By Theorem 2.1 and (3.28), we have

(3.34) $$ \begin{align} \sum_{j=0}^3 (-1)^{j(j-3)/2} \widehat{\mathscr{T}}_{j,R,T} + \sum_k (-1)^k \widehat{\mathscr{T}}_{\mathscr{H},R,T}^k = 0\hspace{2.5mm} \in Q^S/Q^{S,0} \;. \end{align} $$

By Theorems 3.5, 3.6 and (3.34), as $T = R^\kappa \rightarrow +\infty $ , we have

(3.35) $$ \begin{align} \sum_{j=0}^3 (-1)^{j(j-3)/2} \mathscr{T}_{j,R,T} + \mathscr{T}_{\mathscr{H},R,T} = \mathscr{O}\big(R^{-\kappa/4}\big) \hspace{2.5mm} \in Q^S/Q^{S,0} \;. \end{align} $$

On the other hand, using the anomaly formula [Reference Bismut and Lott12, Thm. 3.24], [Reference Zhu66, Thm. 1.5] in the same way as in [Reference Zhu66, §1.7], we can show that the left-hand side of (3.35) is independent of R and T. Hence, for any $R\geqslant 1$ and $T\geqslant 0$ , we have

(3.36) $$ \begin{align} \sum_{j=0}^3 (-1)^{j(j-3)/2} \mathscr{T}_{j,R,T} + \mathscr{T}_{\mathscr{H},R,T} \in Q^{S,0} \;. \end{align} $$

Taking $R=1$ and $T=0$ in (3.36), we obtain (0.15).

§5 is devoted to the proof of Theorems 3.1 and 3.3, §6 to the proof of Theorem 3.5, and §7 to the proof of Theorem 3.6.

4.1 Hodge theory for an interval

We denote $I = [-1,1]$ . Let $u\in I$ be the coordinate. Let V be a finite-dimensional complex vector space. Let $V_1,V_2 \subseteq V$ be vector subspaces. We construct a sheaf $\mathscr {V}$ on I as follows: for any open subset $U\subseteq I$ ,

(4.1) $$ \begin{align} \mathscr{V}(U) = \Big\{ & \text{ locally constant function } \alpha: U \rightarrow V \;:\; \nonumber\\ & \hspace{20mm} \alpha(-1) \in V_1 \; \text{ if } -1\in U \;,\; \alpha(1) \in V_2 \; \text{ if } 1\in U \Big\} \;. \end{align} $$

Let $H^{\bullet }(I,\mathscr {V})$ be the cohomology of I with coefficients in $\mathscr {V}$ . We have

(4.2) $$ \begin{align} H^0(I,\mathscr{V}) = V_1 \cap V_2 \;,\hspace{5mm} H^1(I,\mathscr{V}) = V/(V_1+V_2) \;. \end{align} $$

Let $h^V$ be a Hermitian metric on V. We denote $V[du] = V \oplus V du = V \otimes \Lambda ^{\bullet }(T^*I)$ . Let $\big \lVert \cdot \big \rVert _{V[du]}$ be the norm on $V[du]$ induced by $h^V$ and the metric on $\Lambda ^{\bullet }(T^*I)$ such that $\big |du\big |=1$ . We introduce the following Clifford actions on $V[d u]$ ,

(4.3) $$ \begin{align} c = d u \wedge - \; i_{\frac{\partial}{\partial u}} \;,\hspace{5mm} \hat{c} = d u \wedge + \; i_{\frac{\partial}{\partial u}} \;. \end{align} $$

Then c (resp. $\hat {c}$ ) is skew-adjoint (resp. self-adjoint). Moreover,

(4.4) $$ \begin{align} c^2 = -1 \;,\hspace{5mm} {\hat{c}}^2 = 1 \;,\hspace{5mm} c\hat{c} + \hat{c}c = 0 \;. \end{align} $$

Let $\big \lVert \cdot \big \rVert _{[-1,1]}$ be the $L^2$ -norm on $\Omega ^{\bullet }(I,V)={\mathscr {C}^\infty }(I,V[du])$ induced by $\big \lVert \cdot \big \rVert _{V[du]}$ . Let $d^V$ be the de Rham operator on $\Omega ^{\bullet }(I,V)$ , let $d^{V,*}$ be its formal adjoint, let $D^V = d^V + d^{V,*}$ . Then we have

(4.5) $$ \begin{align} D^V = c \frac{\partial}{\partial u} \qquad \text{and} \qquad D^{V,2} = - \frac{\partial^2}{\partial u^2}. \end{align} $$

For $u\in I$ and $\omega \in \Omega ^{\bullet }(I,V)={\mathscr {C}^\infty }(I,V[du])$ , we denote by $\omega _u\in V[d u]$ the value of $\omega $ at u. For $j=1,2$ , let $V_j^\perp \subseteq V$ be the orthogonal complement of $V_j\subseteq V$ . Set

(4.6) $$ \begin{align} \Omega^{\bullet}_{\textbf{bd}}(I,V) = \Big\{ \omega\in\Omega^{\bullet}(I,V) \;:\; \omega_{-1}\in V_1 \oplus V_1^\perp d u \;,\; \omega_{1}\in V_2 \oplus V_2^\perp d u \Big\} \;. \end{align} $$

Let $D^V_{\textbf {{bd}}}$ be the self-adjoint extension of $D^V$ with $\mathrm {Dom}\big (D^V_{\textbf {{bd}}}\big ) = \Omega ^{\bullet }_{\textbf {{bd}}}(I,V)$ . We will also consider $D^{V,2}_{\textbf {{bd}}}$ with $\mathrm {Dom}\big (D^{V,2}_{\textbf {{bd}}}\big ) = \Big \{ \omega \in \Omega ^{\bullet }_{\textbf {{bd}}}(I,V) \;:\; D^V\omega \in \Omega ^{\bullet }_{\textbf {{bd}}}(I,V) \Big \}$ . We have

(4.7) $$ \begin{align} \operatorname{\mathrm{Ker}}\big(D^{V,2}_{\textbf{{bd}}}\big)\Big|_{\Omega^0_{\textbf{{bd}}}(I,V)} = V_1 \cap V_2 \;,\hspace{5mm} \operatorname{\mathrm{Ker}}\big(D^{V,2}_{\textbf{{bd}}}\big)\Big|_{\Omega^1_{\textbf{{bd}}}(I,V)} = \big(V_1^\perp \cap V_2^\perp\big) d u \;, \end{align} $$

where the right-hand sides are viewed as constant functions on I with values in $V[d u]$ . From (4.2) and (4.7), we get an isomorphism $\operatorname {\mathrm {Ker}}\big (D^{V,2}_{\textbf {{bd}}}\big ) \simeq H^{\bullet }(I,\mathscr {V})$ .

4.2 Witten-type deformation on an interval

For $T \geqslant 0$ , set

(4.8) $$ \begin{align} d^V_{T} = e^{-Tf_T} d^V e^{Tf_T} \;,\hspace{5mm} d^{V,*}_{T} = e^{Tf_T} d^{V,*} e^{-Tf_T} \;,\hspace{5mm} D^V_T = d^V_T + d^{V,*}_T \;, \end{align} $$

where $f_T$ was defined by (3.6). We have

(4.9) $$ \begin{align} D^V_T = D^V + Tf^{\prime}_T\hat{c} = c \frac{\partial}{\partial u} + Tf^{\prime}_T\hat{c} \;,\hspace{5mm} D^{V,2}_T = -\frac{\partial^2}{\partial u^2}+ Tf^{\prime\prime}_Tc\hat{c} + T^2|f^{\prime}_T|^2 \;. \end{align} $$

Let $D^V_{T,\textbf {{bd}}}$ be the self-adjoint extension of $D^V_T$ with $\mathrm {Dom}\big (D^V_{T,\textbf {{bd}}}\big ) = \Omega ^{\bullet }_{\textbf {{bd}}}(I,V)$ .

Theorem 4.1. There exist $\beta>\alpha >0$ such that for $T \gg 1$ , we have

(4.10) $$ \begin{align} \mathrm{Sp}\big(D^V_{T,\textbf{{bd}}}\big) \subseteq \big]-\infty,-\alpha\sqrt{T}\big] \cup \big[-\beta\sqrt{T}e^{-T},\beta\sqrt{T}e^{-T}\big] \cup \big[\alpha\sqrt{T},+\infty\,\big[ \;. \end{align} $$

Proof. For $T\geqslant 0$ , set $\widetilde {D}^V_T = D^V + Tf^{\prime }_\infty \hat {c}$ , with $f_\infty $ in (3.4). Let $\widetilde {D}^V_{T,\textbf {{bd}}}$ be the self-adjoint extension of $\widetilde {D}^V_T$ with $\mathrm {Dom}\big (\widetilde {D}^V_{T,\textbf {{bd}}}\big ) = \Omega ^{\bullet }_{\textbf {{bd}}}(I,V)$ . Then, by (3.7) and (4.9), it is sufficient to show that there exist $\beta>\alpha >0$ such that

(4.11) $$ \begin{align} \mathrm{Sp}\Big(\widetilde{D}^V_{T,\textbf{{bd}}}\Big) \subseteq \big]-\infty,-2\alpha\sqrt{T}\big] \cup \big[-\beta\sqrt{T}e^{-T}/2,\beta\sqrt{T}e^{-T}/2\big] \cup \big[2\alpha\sqrt{T},+\infty\,\big[ \;. \end{align} $$

For $T\geqslant 0$ , we define smooth functions $\phi _{1,T}, \phi _{2,T}, \phi _{3,T}: I \rightarrow \mathbb {R}$ as follows, with $\chi $ in (3.5),

(4.12) $$ \begin{align} \phi_{1,T}(u) = \phi_{2,T}(-u) & = \big(1-\chi(4u+4)\big)\exp\big(-T(u+1)^2/2\big) \;, \nonumber\\ \phi_{3,T}(u) & = \big(1-\chi(4|u|)\big)\exp\big(-Tu^2/2\big). \end{align} $$

Let $\big (C^{\bullet }_{\mathrm {r}},\partial \big )$ be the complex in (2.3) associated with $V_1,V_2\subseteq V$ . For $T\geqslant 0$ , we construct a linear map $J_T: C^{\bullet }_{\mathrm {r}} \rightarrow \Omega ^{\bullet }_{\textbf {{bd}}}(I,V)$ as follows,

(4.13) $$ \begin{align} \text{for } (v_1,v_2)\in V_1 \oplus V_2 & = C^0_{\mathrm{r}} \;,\hspace{2.5mm} J_T(v_1,v_2) = \phi_{1,T} v_1 + \phi_{2,T} v_2 \in {\mathscr{C}^\infty}(I,V) \,;\nonumber\\ \text{for } v\in V & = C^1_{\mathrm{r}} \;,\hspace{2.5mm} J_T(v) = \phi_{3,T} du \otimes v \in \Omega^1(I,V) \;. \end{align} $$

We can now obtain (4.11) by proceeding in the same way as in [Reference Bismut and Zhang14, §6] with $J_T$ in [Reference Bismut and Zhang14, Def. 6.5] replaced by the $J_T$ constructed in (4.13). The only difference is that in [Reference Bismut and Zhang14], the manifold have no boundary, but it causes no harm because the analysis is the same away from the boundary, and the same computations as in [Reference Bismut and Zhang14] show that $J_T$ still gives a local model for the kernel of the harmonic oscillator, with our boundary condition, near the critical points $\pm 1$ of $f_\infty $ .

For $\Lambda \subseteq \mathbb {R}$ , let $E^\Lambda _T$ be the eigenspace of $D^V_{T,\textbf {{bd}}}$ associated with eigenvalues in $\Lambda $ .

In the following proposition, we prove that an eigenform of $D^V_T$ for a small eigenvalue which almost satisfies the boundary conditions in (4.6) is almost an eigenform of $D^V_{T,\textbf {{bd}}}$ . This will be used in §5 to link the zero-mode (see (5.19)) of certain forms to eigenforms of $D^V_{T,\textbf {{bd}}}$ . For $j=1,2$ , let $P_j : V[d u] \rightarrow V_j \oplus V_j^\perp du$ be the orthogonal projection. We denote $P_j^\perp = \mathrm {Id} - P_j$ . Let $P^\Lambda _T : \Omega ^{\bullet }(I,V) \rightarrow E^\Lambda _T$ be the orthogonal projection with respect to $\big \lVert \cdot \big \rVert _{[-1,1]}$ . For $\omega \in \Omega ^{\bullet }(I,V)$ , we denote $\big \lVert \omega \big \rVert _{V[du],\mathrm {max}} = \max \Big \{\big \lVert \omega _u\big \rVert _{V[du]}\;:\;u\in [-1,1]\Big \}$ .

Proposition 4.3. For $T \gg 1$ , $\varepsilon>0$ , $0<\epsilon <\sqrt {T}$ , $-\sqrt {T}<\lambda <\sqrt {T}$ and $\omega \in \Omega ^{\bullet }(I,V)$ , if

(4.14) $$ \begin{align} D^V_T\omega = \lambda\omega \hspace{5mm} \text{and} \hspace{5mm} \big\lVert P_1^\perp\omega_{-1} \big\rVert_{V[du]} + \big\lVert P_2^\perp\omega_1 \big\rVert_{V[du]} \leqslant \epsilon\varepsilon \big\lVert\omega\big\rVert_{V[du],\mathrm{max}} \;, \end{align} $$

then we have

(4.15) $$ \begin{align} \Big\lVert \omega - P^{[\lambda-\epsilon,\lambda+\epsilon]}_T\omega \Big\rVert_{[-1,1]} = \mathscr{O}\big(T^{3/2}\big) \varepsilon \big\lVert \omega \big\rVert_{[-1,1]} \;. \end{align} $$

Proof. Set $\chi _T(u) = \chi (Tu-T+1)$ , with $\chi $ in (3.5). We define $\omega '\in \Omega ^{\bullet }_{\textbf {{bd}}}(I,V)$ by:

(4.16) $$ \begin{align} \omega^{\prime}_u = \left\{ \begin{array}{rl} \omega_u - \chi_T(-u)P_1^\perp\omega_{-1} & \text{if } u < 0 \;,\\ \omega_u - \chi_T(u)P_2^\perp\omega_1 & \text{if } u \geqslant 0 \;. \end{array} \right. \end{align} $$

By the inequality in (4.14), (4.16) and the assumption $0<\epsilon <\sqrt {T}$ , we have

(4.17) $$ \begin{align} \big\lVert \omega - \omega' \big\rVert_{[-1,1]} = \mathscr{O}\big(T^{-1/2}\big) \epsilon\varepsilon \big\lVert\omega\big\rVert_{V[du],\mathrm{max}} = \mathscr{O}\big(1\big) \varepsilon \big\lVert\omega\big\rVert_{V[du],\mathrm{max}} \;. \end{align} $$

A direct calculation yields

(4.18) $$ \begin{align} & \Big(\big(D^V_{T,\textbf{{bd}}}-\lambda\big)\omega'\Big)_u \nonumber \\ & = \left\{ \begin{array}{rl} \chi_T'(-u) c P_1^\perp\omega_{-1} + \big(\lambda-Tf^{\prime}_T\hat{c}\big) \chi_T(-u) P_1^\perp\omega_{-1} & \text{if } u < 0\;,\\ -\chi_T'(u) c P_2^\perp\omega_1 + \big(\lambda-Tf^{\prime}_T\hat{c}\big) \chi_T(u) P_2^\perp\omega_1 & \text{if } u \geqslant 0\;. \end{array} \right. \end{align} $$

By the inequality in (4.14), (4.18) and the construction of $\chi _T$ , we get

(4.19) $$ \begin{align} \Big\lVert \big(D^V_{T,\textbf{{bd}}}-\lambda\big)\omega'\Big\rVert_{[-1,1]} = \mathscr{O}\big(\sqrt{T}\big) \epsilon\varepsilon \big\lVert\omega\big\rVert_{V[du],\mathrm{max}} \;. \end{align} $$

By Proposition 1.2 and (4.19), we have

(4.20) $$ \begin{align} \Big\lVert \omega' - P^{[\lambda-\epsilon,\lambda+\epsilon]}_T\omega' \Big\rVert_{[-1,1]} = \mathscr{O}\big(\sqrt{T}\big) \varepsilon \big\lVert\omega\big\rVert_{V[du],\mathrm{max}} \;. \end{align} $$

Let $\big \lVert \cdot \big \rVert _{H^1,[-1,1]}$ be the $H^1$ -norm on $\Omega ^{\bullet }(I,V)$ . By (4.4), (4.9) and the identity in (4.14), we have $\frac {\partial }{\partial u}\omega = \big ( Tf^{\prime }_Tc\hat {c} - \lambda c \big ) \omega $ , which yields $\big \lVert \omega \big \rVert _{H^1,[-1,1]} = \mathscr {O}\big (T\big ) \big \lVert \omega \big \rVert _{[-1,1]}$ . Then, by the Sobolev embedding theorem, we have

(4.21) $$ \begin{align} \big\lVert\omega\big\rVert_{V[du],\mathrm{max}} = \mathscr{O}\big(1\big) \big\lVert\omega\big\rVert_{H^1,[-1,1]} = \mathscr{O}\big(T\big) \big\lVert\omega\big\rVert_{[-1,1]} \;. \end{align} $$

From (4.17), (4.20) and (4.21), we obtain (4.15).

4.3 Two-parameter deformation on a cylinder

Let $(Y,g^{TY})$ be a closed Riemannian manifold. For $R>0$ , we denote $I_R=[-R,R]$ and $IY_R= I_R\times Y$ . Let $(u,y)\in [-R,R]\times Y$ be the coordinates. We will also use the coordinates $(s,y) = (u/R,y)\in [-1,1]\times Y$ . We equip $T(IY_R)$ with the Riemannian metric $d u^2 + g^{TY}$ .

Let $(F, \nabla ^F)$ be a flat complex vector bundle over Y with Hermitian metric $h^F$ . The pullback of F (resp. $h^F$ ) via the canonical projection $IY_R \rightarrow Y$ will still be denoted by F (resp. $h^F$ ). Let $D^Y$ be the Hodge de Rham operator on $\Omega ^{\bullet }(Y,F)$ . Under the identification $\Omega ^{\bullet }(I_R,\Omega ^{\bullet }(Y,F)) \ni \sigma + du \otimes \tau \mapsto \sigma + du \wedge \tau \in \Omega ^{\bullet }(IY_R,F)$ for $\sigma ,\tau \in {\mathscr {C}^\infty }(I_R,\Omega ^{\bullet }(Y,F))$ , the Hodge de Rham operator on $\Omega ^{\bullet }(IY_R,F)$ is given by

(4.22) $$ \begin{align} D^{IY_R} = \hat{c}c D^Y + c \; \frac{\partial}{\partial u} = R^{-1} \Big( \hat{c}c RD^Y + c \; \frac{\partial}{\partial s} \Big) \;, \end{align} $$

where $\hat {c}c = i_{\frac {\partial }{\partial u}} du\wedge - du\wedge i_{\frac {\partial }{\partial u}}$ appears since $D^Y$ anti-commutes with $du\wedge $ .

We will view $f_T$ in (3.6) as a function on $IY_R$ , i.e., $f_T(s,y) = f_T(s)$ , $f_T(u,y) = f_T(u/R)$ . We denote

(4.23) $$ \begin{align} f_T' = \frac{\partial}{\partial s} f_T = R \frac{\partial}{\partial u} f_T \;. \end{align} $$

For $T\geqslant 0$ , let $\widetilde {D}^{IY_R}_T$ be the Hodge de Rham operator with respect to $d u^2 + g^{TY}$ and $h^F_T := e^{-2Tf_T}h^F$ . Set $D^{IY_R}_T = e^{-Tf_T} \widetilde {D}^{IY_R}_T e^{Tf_T}$ . We have

(4.24) $$ \begin{align} R D^{IY_R}_T &= \hat{c}c RD^Y + c \; \frac{\partial}{\partial s} + Tf^{\prime}_T\hat{c} \;, \nonumber \\ R^2 D^{IY_R,2}_T& = R^2 D^{Y,2} - \frac{\partial^2}{\partial s^2} + Tf^{\prime\prime}_Tc\hat{c} + T^2|f^{\prime}_T|^2 \;. \end{align} $$

Let $\Omega ^{\bullet }(Y,F) = \bigoplus _\mu \mathscr {E}^\mu (Y,F)$ be such that $D^Y\big |_{\mathscr {E}^\mu (Y,F)} = \mu \operatorname {\mathrm {Id}}$ . We denote $\mathscr {H}^{\bullet }(Y,F) = \operatorname {\mathrm {Ker}}\big (D^Y\big ) = \mathscr {E}^0(Y,F)$ . We have the formal decomposition

(4.25) $$ \begin{align} \Omega^{\bullet}(IY_R, F) = \bigoplus_\mu \Omega^{\bullet}\big(I_R,\mathscr{E}^\mu(Y,F)\big) \;. \end{align} $$

Let $D^{\mathscr {E}^\mu (Y,F)}_T$ be the operator $D^V_T$ in §4.2 with V replaced by $\mathscr {E}^\mu (Y,F)$ . We have

(4.26) $$ \begin{align} RD^{IY_R}_T\Big|_{\Omega^{\bullet}(I_R,\mathscr{E}^\mu(Y,F))} = R\mu \,\hat{c}c + D^{\mathscr{E}^\mu(Y,F)}_T = R\mu \,\hat{c}c + c\frac{\partial}{\partial s} + Tf^{\prime}_T\hat{c} \;. \end{align} $$

As a consequence, we have

(4.27) $$ \begin{align} R^2D^{IY_R,2}_T\Big|_{\Omega^{\bullet}(I_R,\mathscr{E}^\mu(Y,F))} & = R^2\mu^2 -\frac{\partial^2}{\partial s^2} + T^2|f^{\prime}_T|^2 + Tf^{\prime\prime}_Tc\hat{c} \;, \nonumber\\ RD^{IY_R}_T\Big|_{\Omega^{\bullet}(I_R,\mathscr{H}^{\bullet}(Y,F))} & = D^{\mathscr{H}^{\bullet}(Y,F)}_T = c\frac{\partial}{\partial s} + Tf^{\prime}_T\hat{c} \;. \end{align} $$

Now we give an explicit formula for the eigensections of $D^{IY_R} = D^{IY_R}_T\big |_{T=0}$ . Let $\delta _Y>0$ be such that $\mathrm {Sp}(D^Y) \, \cap \, ]-\delta _Y,\delta _Y[ \, \subseteq \{0\}$ . By (4.26) with $T=0$ , for $-\delta _Y < \lambda < \delta _Y$ and $\omega \in \Omega ^{\bullet }(IY_R,F)$ satisfying $D^{IY_R} \omega =\lambda \omega $ , we have