2.1 Eta invariant of an elliptic operator

Let $E\rightarrow X $ be a complex vector bundle over a smooth, compact, Riemannian manifold X of dimension d. Let $D\colon C^{\infty }(X, E)\rightarrow C^{\infty }(X, E)$ be an elliptic differential operator of order $m\geq 1$ . Let $\sigma (D)$ be its principal symbol.

Definition 2.1.1. Let $R_{\theta }:=\{\rho e^ {i\theta }: \rho \in [0,\infty ]\}$ . The angle $\theta \in [0,2\pi )$ is a principal angle for D if

$$ \begin{align*} \operatorname{\mathrm{spec}}(\sigma_{D}(x,\xi))\cap R_{\theta}=\emptyset,\quad \forall x\in X,\forall\xi\in T_{x}^{*}X,\xi\neq 0. \end{align*} $$

Definition 2.1.2. Let $I\subset [0,2\pi )$ . Let $L_{I}$ be a solid angle defined by

$$ \begin{align*} L_{I}:=\{\rho e^ {i\theta}: \rho\in(0,\infty),\theta\in I\}. \end{align*} $$

The angle $\theta $ is an Agmon angle for D if it is a principal angle for D and there exists an $\varepsilon>0$ such that

$$ \begin{align*} \operatorname{\mathrm{spec}}(D)\cap L_{[\theta-\varepsilon,\theta+\varepsilon]}=\emptyset. \end{align*} $$

We define here the eta function associated with non-self-adjoint operators D with elliptic, self-adjoint principal symbol (see [Reference Gilkey21]). By [Reference Markus26, §I.6], the space $L^{2}(X,E)$ of square integrable sections of E is the closure of the algebraic direct sum of finite-dimensional D-invariant subspaces

$$ \begin{align*} L^{2}(X,E)=\overline{\bigoplus\limits_{k\geq 1} \Lambda_{k}} \end{align*} $$

such that the restriction of D to $\Lambda _{k}$ has a unique eigenvalue $\lambda _{k}$ and $\lim _{k\rightarrow \infty }\vert \lambda _{k}\vert =\infty $ . In general, the above sum is not a sum of mutually orthogonal subspaces. The space $\Lambda _{k}$ is called the space of root vectors of D with eigenvalue $\lambda _{k}$ . The algebraic multiplicity $m_{k}$ of the eigenvalue $\lambda _{k}$ is defined as the dimension of the space $\Lambda _{k}$ . Since the principal symbol of D is self-adjoint, the angles $\pm \pi /2$ are principal angles for D. Hence, D also possesses an Agmon angle (see [Reference Braverman and Kappeler6, Section 3.10]). Let $\theta $ be an Agmon angle for D. Denote by $\log _{\theta }\lambda _{k}$ the branch of the logarithm in $\mathbb {C}\setminus R_{\theta }$ with $\theta <\operatorname {\mathrm {Im}}(\log _{\theta }\lambda _{k})<\theta +2\pi $ . Let $(\lambda _{k})_{\theta }:=e^{\log _{\theta }\lambda _{k}}$ .

Definition 2.1.3. For Re $(s)\gg 0$ , we define the eta function $\eta _{\theta }(s,D)$ of D by

$$ \begin{align*} \eta_{\theta}(s,D):=\sum_{\text{{Re}}(\lambda_{k})>0} m_{k}(\lambda_{k})_{\theta}^{-s}-\sum_{\text{{Re}}(\lambda_{k})<0} m_{k}(-\lambda_{k})_{\theta}^{-s}. \end{align*} $$

Note that since the angles $\pm \pi /2$ are principal angles for D, there are at most finitely many eigenvalues of D on or near the imaginary axis. Hence, the eigenvalues of D that are purely imaginary do not contribute to the definition of the eta function. It has been shown by Grubb and Seeley ([Reference Grubb and Seeley23, Theorem 2.7]) that $\eta _{\theta }(s,D)$ has a meromorphic continuation to the whole complex plane $\mathbb {C}$ with isolated simple poles and that is regular at $s=0$ . Moreover, the number $\eta _{\theta }(0,D)$ is independent of the Agmon angle $\theta $ . Hence, we write $\eta (0,D)$ instead of $\eta _{\theta }(0,D)$ .

Definition 2.1.4. Let $m_{+}$ , respectively $m_{-}$ , denote the number of the eigenvalues of D on the positive, respectively negative, part of the imaginary axis. We define the eta invariant $\eta (D)$ of the operator D by

$$ \begin{align*} \eta(D)=\frac{\eta(0,D)+m_{+}-m_{-}}{2}. \end{align*} $$

By [Reference Braverman and Kappeler6, (3.25)], $\eta (D)$ is independent of the Agmon angle $\theta $ .

2.2 Regularized determinant of an elliptic operator

We recall here the definition of the regularized determinant of an elliptic operator. For more details, we refer the reader to [Reference Braverman and Kappeler6, Subsection 3.5 and Definition 3.6]. Let $X,E$ and D be as in the previous subsection. We assume, in addition, that D has a self-adjoint principal symbol and it is invertible. Let $\theta $ be an Agmon angle for D. We define the zeta function $\zeta _{\theta }(s,D)$ by

$$ \begin{align*} \zeta_{\theta}(s,D):=\operatorname{\mathrm{Tr}}(D_{\theta}^{-s}),\quad \text{Re}(s)>\frac{\dim(X)}{m}, \end{align*} $$

where $D_{\theta }^{-s}$ is a pseudodifferential operator with continuous kernel (see [Reference Braverman and Kappeler6, (3.15)]). The zeta function $\zeta _{\theta }(s,D)$ admits a meromorphic continuation to $\mathbb {C}$ , and it is regular at zero ([Reference Seeley45]). We define the regularized determinant of D by

$$ \begin{align*} {\det}_{\theta}:=\exp\bigg( -\frac{d}{ds}\bigg\vert_{s=0} \zeta_{\theta}(s,D) \bigg). \end{align*} $$

We denote by $L{\det }_{\theta }$ the particular value of the logarithm of the determinant such that

$$ \begin{align*} L{\det}_{\theta}(D)=-\zeta_{\theta}'(0,D). \end{align*} $$

By [Reference Braverman and Kappeler6, Subsection 3.10], the regularized determinant ${\det }_{\theta }(D)$ is independent of the Agmon angle $\theta $ .

2.3 Graded regularized determinant of an elliptic operator

We recall here the notion of the graded regularized determinant of an elliptic differential operator. Let $E=E_{+}\oplus E_{-}$ be a $\mathbb {Z}_{2}$ -graded vector bundle over a compact, Riemannian manifold X. Let $D\colon C^{\infty }(X,E)\rightarrow C^{\infty }(X,E)$ be an elliptic differential operator, which is bounded from below. We assume that D preserves the grading, that is, we assume that with respect to the decomposition

$$ \begin{align*} C^{\infty}(X,E)=C^{\infty}(X,E^{+})\oplus C^{\infty}(X,E^{-}), \end{align*} $$

D takes the form,

$$ \begin{align*} D= \left( {\begin{array}{@{}cc@{}} D{+} & 0 \\ 0 & D_{-} \\ \end{array} } \right). \end{align*} $$

Then, the graded determinant $\det _{\operatorname {\mathrm {gr}}}(D)$ of D is defined by

$$ \begin{align*} {\det}_{\operatorname{\mathrm{gr}}}(D)=\frac{\det(D_{+})}{\det(D_{-})}, \end{align*} $$

where $\det (D_{+})$ and $ \det (D_{-})$ denote the regularized determinants of the operators $D_{+}$ and $D_{-}$ , respectively.

3.1 Odd signature operator

Following the idea of Braverman and Kappeler in [Reference Braverman and Kappeler6], we consider representations of $\Gamma $ , which are nonunitary, but belong to a neighbourhood of the set of acyclic, unitary representations in a suitable topology such that the odd signature operator B (see Definition 3.1.1 below) is bijective. For these representations, Braverman and Kappeler defined the refined analytic torsion for a compact, oriented, odd-dimensional manifold X. Having applications in mind, instead of considering the regularized determinant of the (flat) Laplacian they consider the graded determinant of the even part of its square root, the odd signature operator B. We recall here some definitions from [Reference Braverman and Kappeler6].

Let X be a compact, oriented, Riemannian manifold of odd dimension $d=2r-1$ . Denote by g the Riemannian metric on X. Let $E\rightarrow X$ be a complex vector bundle over X, endowed with a flat connection $\nabla $ . Let $\Lambda ^{k}(X,E)$ be the space of k-differential forms on X with values in E. We denote by $\nabla $ the induced differential $\nabla \colon \Lambda ^{k}(X,E)\rightarrow \Lambda ^{k+1}(X,E)$ . Let $\Gamma \colon \Lambda ^{k}(X,E) \rightarrow \Lambda ^{d-k}(X,E)$ be the chirality operator defined by the formula

$$ \begin{align*} \Gamma\omega:=i^{(d+1)/2}(-1)^{k(k+1)/2}\ast' \omega, \end{align*} $$

where $\ast '$ denotes the operator acting on sections of $\Lambda ^{k}T^{\ast }X\otimes E$ as $\ast \otimes \operatorname {\mathrm {Id}}$ , and $\ast $ is the usual Hodge $\ast $ -operator.

Definition 3.1.1. We define the odd signature operator $B=B(\nabla , g)$ acting on $\Lambda ^{*}(X,E)$ by

(3.1) $$ \begin{align} B:=\Gamma \nabla+\nabla \Gamma. \end{align} $$

We set

$$ \begin{align*} \Lambda^{\operatorname{\mathrm{even}}}(X,E):=\bigoplus_{p=0}^{r-1} \Lambda^{2p}(X,E). \end{align*} $$

Let $B^{\operatorname {\mathrm {even}}}$ be the even part of the odd signature operator B, defined by $B^{\operatorname {\mathrm {even}}}:=B\colon \Lambda ^{\operatorname {\mathrm {even}}}(X,E)\rightarrow \Lambda ^{\operatorname {\mathrm {even}}}(X,E)$ .

3.2 Spaces of connections

We consider the following two assumptions.

Assumption 1. The connection $\nabla $ is acyclic, that is, the twisted de Rham complex

$$ \begin{align*} 0\rightarrow \Lambda^{0}(X,E) \overset{\nabla}{\longrightarrow} \Lambda^{1}(X,E) \overset{\nabla}{\longrightarrow}\cdots \overset{\nabla}{\longrightarrow}\Lambda^{d}(X,E) \rightarrow 0 \end{align*} $$

is acyclic,

$$ \begin{align*} \operatorname{\mathrm{Im}}(\nabla|_{\Lambda^{k-1}(X,E)})=\operatorname{\mathrm{Ker}}(\nabla|_{\Lambda^{k}(X,E)}), \end{align*} $$

for every $k=1,\ldots ,d$ .

Suppose that there exists a Hermitian metric h on E, which is preserved by $\nabla $ . In such a case, we call $\nabla $ a Hermitian connection. For such connections, one can easily see that Assumption 1 implies Assumption 2 and vice versa (see [Reference Braverman and Kappeler6, Subsection 6.6]). Hence, all acyclic, Hermitian connections satisfy Assumptions 1 and 2.

Following [Reference Braverman and Kappeler6], we denote by $\Lambda ^{1}(X,\operatorname {\mathrm {End}}(E))$ the space of $1$ -forms with values in $\operatorname {\mathrm {End}}(E)$ . We define the sup-norm on $\Lambda ^{1}(X,\operatorname {\mathrm {End}}(E))$ by

$$ \begin{align*} \lVert\omega\rVert_{\sup}:= \max_{x\in X}\lvert \omega(x)\rvert, \end{align*} $$

where the norm $\lvert \cdot \rvert $ is induced by a Hermitian metric on E and a Riemannian metric on X. The topology defined by this norm is called the $C^{0}$ -topology and it is independent of the metrics. We identify the space of connections on E with $\Lambda ^{1}(X,\operatorname {\mathrm {End}}(E))$ by choosing a connection $\nabla _{0}$ and associating to a connection $\nabla $ the $1$ -form $\nabla -\nabla _{0}\in \Lambda ^{1}(X,\operatorname {\mathrm {End}}(E))$ . Hence, by this identification, the $C^{0}$ -topology on $\Lambda ^{1}(X,\operatorname {\mathrm {End}}(E))$ provides a topology on the space of connections on E, which is independent of the choice of $\nabla _{0}$ . This topology is called the $C^{0}$ -topology on the space of connections.

Let $\operatorname {\mathrm {Flat}}(E)$ be the set of flat connections on E and $\operatorname {\mathrm {Flat}}'(E,g)\subset \operatorname {\mathrm {Flat}}(E)$ be the set of flat connections on E, satisfying Assumptions 1 and 2. The topology induced on these sets by the $C^{0}$ -topology on the space of connections on E is also called the $C^{0}$ -topology. The following proposition is proved in [Reference Braverman and Kappeler6].

3.3 Spaces of representations

Let $\operatorname {\mathrm {Rep}}(\pi _{1}(X),\mathbb {C}^{n})$ be the space of all n-dimensional, complex representations of $\pi _{1}(X)$ . This space has a natural structure of a complex algebraic variety. Let $\{\gamma _{1},\ldots ,\gamma _{L}\}$ be a finite set of generators of $\pi _{1}(X)$ . The elements $\gamma _{i}$ satisfy finitely many relations. Then, a representation $\chi \in \operatorname {\mathrm {Rep}}(\pi _{1}(X),\mathbb {C}^{n})$ is given by $2L$ invertible $n\times n$ -matrices $\chi (\gamma _{1}),\ldots ,\chi (\gamma _{L}),\chi (\gamma _{1}^{-1}),\ldots ,\chi (\gamma _{L}^{-1})$ , which satisfy finitely many polynomial equations. Hence, we view $\operatorname {\mathrm {Rep}}(\pi _{1}(X),\mathbb {C}^{n})$ as an algebraic subset of $\operatorname {\mathrm {Mat}}_{n\times n}(\mathbb {C})^{2L}$ with the induced topology.

For a representation $\chi \in \operatorname {\mathrm {Rep}}(\pi _{1}(X),\mathbb {C}^{n})$ , we consider the associated flat vector bundle $E_{\chi }\rightarrow X$ . Let $\nabla _{\chi }$ be the flat connection on $E_{\chi }$ . Then, the monodromy of $\nabla _{\chi }$ is isomorphic to $\chi $ . We denote also by $\nabla _{\chi }$ the induced differential $\nabla _{\chi }\colon \Lambda ^{*}(X,E_{\chi })\rightarrow \Lambda ^{*+1}(X,E_{\chi })$ . We denote by $B_{\chi }$ the odd signature operator $B_{\chi }=B(\nabla _{\chi },g)$ acting on $\Lambda ^{*}(X,E_{\chi })$ . Let $B_{\chi }^{\operatorname {\mathrm {even}}}$ be the restriction of $B_{\chi }$ to $\Lambda ^{\operatorname {\mathrm {even}}}(X,E_{\chi })$ .

Let $\operatorname {\mathrm {Rep}}_{0}(\pi _{1}(X),\mathbb {C}^{n})$ be the set of all acyclic representations of $\pi _{1}(X)$ , that is, the set of all representations $\chi $ such that $\nabla _{\chi }$ is acyclic. Let $\operatorname {\mathrm {Rep}}^{u}(\pi _{1}(X),\mathbb {C}^{n})$ be the set of all unitary representations of $\pi _{1}(X)$ , that is, the set of all representations $\chi $ such that there exists a Hermitian scalar product $(\cdot ,\cdot )$ on $\mathbb {C}^{n}$ , which is preserved by the matrices $\chi (\gamma )$ , for all $\gamma \in \pi _{1}(X)$ . $\operatorname {\mathrm {Rep}}^{u}(\pi _{1}(X),\mathbb {C}^{n})$ can be viewed as the real locus of the complex algebraic variety $\operatorname {\mathrm {Rep}}(\pi _{1}(X),\mathbb {C}^{n})$ . We set

$$ \begin{align*} \operatorname{\mathrm{Rep}}^{u}_{0}(\pi_{1}(X),\mathbb{C}^{n}):=\operatorname{\mathrm{Rep}}_{0}(\pi_{1}(X),\mathbb{C}^{n})\cap \operatorname{\mathrm{Rep}}^{u}(\pi_{1}(X),\mathbb{C}^{n}). \end{align*} $$

Suppose that for some representation $\chi _{0}\in \operatorname {\mathrm {Rep}}_{0}(\pi _{1}(X),\mathbb {C}^{n})$ the operator $B_{\chi _{0}}$ is bijective. Then, there exists an open neighbourhood (in classical topology) $V \subset \operatorname {\mathrm {Rep}}(\pi _{1}(X),\mathbb {C}^{n})$ of the set $\operatorname {\mathrm {Rep}}^{u}_{0}(\pi _{1}(X),\mathbb {C}^{n})$ of acyclic, unitary representations such that, for all $\chi \in V$ , the pair $(\nabla _{\chi },g)$ satisfies Assumptions 1 and 2 of Subsection 6.2 (see [Reference Braverman and Kappeler6, Subsection 13.7]).

4.1 Definition

Let $\nabla $ be acyclic. Let $\cal {M}(\nabla )$ be the set of all Riemannian metrics g on X such that $B^{\operatorname {\mathrm {even}}}=B^{\operatorname {\mathrm {even}}}(\nabla ,g)$ is bijective. Then, by [Reference Braverman and Kappeler6, Proposition 6.8], $\cal {M}(\nabla )\neq \emptyset $ , for all flat connections in an open neighbourhood, in $C^{0}$ -topology, of the set of acyclic Hermitian connections. Let V be an open neighbourhood of $\operatorname {\mathrm {Rep}}^{u}_{0}(\pi _{1}(X),\mathbb {C}^{n})$ as in Subsection 3.3.

Definition 4.1.1. Let $\chi \in V$ . We define the refined analytic torsion $T_{\chi }=T(\nabla _{\chi })$ by

(4.1) $$ \begin{align} T_{\chi}:={\det}_{\operatorname{\mathrm{gr}},\theta}(B_{\chi}^{\operatorname{\mathrm{even}}})e^{i\pi \operatorname{\mathrm{rank}}(E_{\chi})\eta_{tr}}\in \mathbb{C}\backslash\{0\}, \end{align} $$

where $\eta _{\operatorname {\mathrm {tr}}}=\eta _{tr}(g)=\frac {1}{2}\eta (0, B_{tr}(g))$ , and $\eta (0, B_{tr}(g))$ denotes the eta invariant of the even part of the odd signature operator corresponding to the trivial line bundle, endowed with the trivial connection. The refined analytic torsion is independent of the Agmon angle $\theta \in (-\pi ,0)$ for the operator $B_{\chi }^{\operatorname {\mathrm {even}}}=B_{\chi }^{\operatorname {\mathrm {even}}}(g)$ and the Riemannian metric $g\in \cal {M}(\nabla )$ .

Remark 4.1.2. The graded determinant of the operator $B_{\chi }^{\operatorname {\mathrm {even}}}$ is slightly different than the usual expression of the graded determinant as in Subsection 2.2.3. We recall here briefly the definition of ${\det }_{\operatorname {\mathrm {gr}}}(B_{\chi }^{\operatorname {\mathrm {even}}})$ . For more details, we refer the reader to [Reference Braverman and Kappeler6, Subsection 2.2 and 6.11]. We set

$$ \begin{align*} &\Lambda^{k}_{+}(X,E):=\operatorname{\mathrm{Ker}}(\nabla\Gamma)\cap\Lambda^{k}(X,E)\\ &\Lambda^{k}_{-}(X,E):=\operatorname{\mathrm{Ker}}(\Gamma\nabla)\cap\Lambda^{k}(X,E). \end{align*} $$

Assumption 2 in Subsection 3.2 implies that $\Lambda ^{k}(X,E)=\Lambda ^{k}_{+}(X,E)\oplus \Lambda ^{k}_{-}(X,E)$ (see [Reference Braverman and Kappeler6, Subsection 6.9]). We set

$$ \begin{align*} \Lambda^{\operatorname{\mathrm{even}}}_{\pm}(X,E)=\bigoplus_{p=0}^{r-1} \Lambda^{2p}_{\pm}(X,E). \end{align*} $$

We denote by $B_{\chi ,\pm }^{\operatorname {\mathrm {even}}}$ the restriction of $B_{\chi }^{\operatorname {\mathrm {even}}}$ to $\Lambda ^{\operatorname {\mathrm {even}}}_{\pm }(X,E)$ . Then, $B_{\chi }^{\operatorname {\mathrm {even}}}$ leaves the subspaces $\Lambda ^{\operatorname {\mathrm {even}}}_{\pm }(X,E)$ invariant. By Assumption 2, the operators $B_{\chi ,\pm }^{\operatorname {\mathrm {even}}}$ are bijective. Hence, we define the graded determinant ${\det }_{\operatorname {\mathrm {gr}}}(B_{\chi }^{\operatorname {\mathrm {even}}})\in \mathbb {C} \backslash \{0\}$ of $B_{\chi }^{\operatorname {\mathrm {even}}}$ by

$$ \begin{align*} {\det}_{\operatorname{\mathrm{gr}},\theta}(B_{\chi}^{\operatorname{\mathrm{even}}}):=\frac{\det_{\theta}(B_{\chi,+}^{\operatorname{\mathrm{even}}})} {\det_{\theta}(-B_{\chi,-}^{\operatorname{\mathrm{even}}})}, \end{align*} $$

where $\theta $ is an Agmon angle for $B_{\chi }^{\operatorname {\mathrm {even}}}$ (and an Agmon angle for $B_{\chi ,\pm }^{\operatorname {\mathrm {even}}}$ as well).

4.2 Independence of the refined analytic torsion of the Riemannian metric and the Agmon angle

In [Reference Braverman and Kappeler6, Theorem 9.3], it is proved that $T_{\chi }$ is independent of the Agmon angle $\theta $ and the Riemannian metric g. We set

(4.2) $$ \begin{align} \xi=\xi(\chi,g,\theta):=\frac{1}{2}\sum_{k=0}^{d}(-1)^{k+1}k L{\det}_{2\theta}(B_{\chi}^{2}|_{\Lambda^{k}(X,E_{\chi})}). \end{align} $$

The number $\xi $ is defined in [Reference Braverman and Kappeler6, (7.79)] (see also (8.102) in the same paper). Let $\eta =\eta (\chi ,g)=\eta (B^{\operatorname {\mathrm {even}}}_{\chi })$ denote the eta invariant of the even part of the odd signature operator $B_{\chi }$ . By [Reference Braverman and Kappeler6, Theorem 7.2], for a suitable choice of an Agmon angle for $B_{\chi }$ , the following equality holds

(4.3) $$ \begin{align} {\det}_{\operatorname{\mathrm{gr}},\theta}(B_{\chi}^{\operatorname{\mathrm{even}}})=e^{\xi(\chi,g,\theta)}e^{-i\pi\eta(\chi,g)}. \end{align} $$

The graded determinant ${\det }_{\operatorname {\mathrm {gr}}}(B_{\chi }^{\operatorname {\mathrm {even}}})$ above does depend on the Riemannian metric g. This is the reason why the additional exponential $e^{i\pi \operatorname {\mathrm {rank}}(E_{\chi })\eta _{tr}(g)}$ is considered in the definition of the refined analytic torsion in [Reference Braverman and Kappeler6], to remove the metric anomaly. By results in [Reference Gilkey21, p. 52], modulo $\mathbb {Z}$ , the difference

$$ \begin{align*} \eta(\chi,g)-\operatorname{\mathrm{rank}}(E_{\chi})\eta_{tr}(g) \end{align*} $$

is independent of g. In addition, by [Reference Braverman and Kappeler6, Proposition 9.7], for $g_{1},g_{2}\in \mathcal {M}(\nabla )$ and suitable choices of Agmon angles $\theta _{0},\theta _{1}$ ,

$$ \begin{align*} \xi(\chi,g_{1},\theta_{1})=\xi(\chi,g_{2},\theta_{2})\quad \mod\pi i. \end{align*} $$

Hence, for different choices of Agmon angles and Riemannian metrics, the corresponding expressions in Equation (4.1) coincide up to a sign. To see that the two expressions coincide see [Reference Braverman and Kappeler6, p. 238].

Remark 4.2.1.

$$ \begin{align*} \rho_{\chi}:=\eta(\chi,g)-\operatorname{\mathrm{rank}}(E_{\chi})\eta_{tr}(g), \end{align*} $$

is called the $rho$ invariant $\rho _{\chi }$ of the operator $B_{\chi }^{\operatorname {\mathrm {even}}}$ .

7.1 Representation theory of Lie groups

Let

$$ \begin{align*} G=KAN \end{align*} $$

be the standard Iwasawa decomposition of G, and let M be the centralizer of A in K. Then, $M=\operatorname {\mathrm {Spin}}(d-1)$ . Let $\mathfrak {a}$ and $\mathfrak {m}$ be the Lie algebras of A and M, respectively. Let $\mathfrak {b}$ be a Cartan subalgebra of $\mathfrak {m}$ . Let $\mathfrak {h}$ be a Cartan subalgebra of $\mathfrak {g}$ . We consider the complexifications $\mathfrak {g}_{\mathbb {C}}:=\mathfrak {g}\oplus i\mathfrak {g}$ , $\mathfrak {h}_{\mathbb {C}}:=\mathfrak {h}\oplus i\mathfrak {h}$ and $\mathfrak {m}_{\mathbb {C}}:=\mathfrak {m}\oplus i\mathfrak {m}$ .

Let $\Delta ^{+}(\mathfrak {g},\mathfrak {a})$ be the set of positive roots of $(\mathfrak {g},\mathfrak {a})$ and $\mathfrak {g}_{\alpha }$ the corresponding root spaces. In the present case, $\Delta ^{+}(\mathfrak {g},\mathfrak {a})$ consists of a single root $\alpha $ . Let $M'=\operatorname {\mathrm {Norm}}_{K}(A)$ be the normalizer of A in K. Let $H\in \mathfrak {a}$ such that $\alpha (H)=1$ . With respect to the inner product, induced by (2.1) on $\mathfrak {a}$ , H has norm 1. We define

(7.2) $$ \begin{align} A^{+}:=\{\exp(tH)\colon t\in\mathbb{R}^{+}\}; \end{align} $$

(7.3) $$ \begin{align} \rho:=\frac{1}{2}\dim(\mathfrak{g}_\alpha)\alpha; \end{align} $$

(7.4) $$ \begin{align} \rho_{\mathfrak{m}}:=\frac{1}{2}\sum_{\alpha\in \Delta^+(\mathfrak{m}_{\mathbb{C}},\mathfrak{b})}\alpha. \end{align} $$

The inclusion $i\colon M\hookrightarrow K$ induces the restriction map $i^{*}\colon R(K)\rightarrow R(M)$ , where $R(K),R(M)$ are the representation rings over $\mathbb {Z}$ of K and M, respectively. Let $\widehat {K},\widehat {M}$ be the sets of equivalent classes of irreducible unitary representations of K and M, respectively. For the highest weight $\nu _{\tau }$ of $\tau \in \widehat {K}$ , we have $\nu _{\tau }=(\nu _{1},\ldots ,\nu _{n})$ , where $\nu _{1}\geq \ldots \geq \nu _{n}$ and $\nu _{i},i=1,\ldots ,n$ are all half integers (i.e., $\nu _{i}=q_{i}+\frac {1}{2}$ , $q_{i}\in \mathbb {Z}$ ). For the highest weight $\nu _{\sigma }$ of $\sigma \in \widehat {M}$ , we have

(7.5) $$ \begin{align} \nu_{\sigma}=(\nu_{1},\ldots,\nu_{n-1},\nu_{n}), \end{align} $$

where $\nu _{1}\geq \ldots \geq \nu _{n-1}\geq \lvert \nu _{n}\rvert $ and $\nu _{i},i=1,\ldots ,n$ are all half integers ([Reference Bunke and Olbrich7, p. 20]). We denote by $(s,S)$ be the spin representation of K and by $(s^{+},S^{+})$ , $(s^{-},S^{-})$ the half spin representations of M ([Reference Friedrich20, p. 22]).

7.2 The Casimir element

Let $Z_{i}$ be a basis of $\mathfrak {g}$ , and let $Z^{j}$ be the basis of $\mathfrak {g}$ , which is determined by $\langle Z_{i}, Z^{j} \rangle =\delta _{ij}$ , where $\langle \cdot , \cdot \rangle $ is as in Equation (7.1). Let $\mathcal {U}(\mathfrak {g}_{\mathbb {C}})$ be the universal enveloping algebra of $\mathfrak {g}_{\mathbb {C}}$ , and let $Z(\mathfrak {g}_{\mathbb {C}})$ be its center. Then, $\Omega \in Z(\mathfrak {g}_{\mathbb {C}})$ is given by

$$ \begin{align*} \Omega=\sum_{i}Z_{i}Z^{i}. \end{align*} $$

By Kuga’s Lemma ([Reference Matsushima and Murakami28, (6.9)]), the Hodge Laplacian, acting on the space $\Lambda ^{*}(G/K)$ of differential forms on $G/K$ , coincides with $-R(\Omega )$ , where $R(\Omega )$ is the Casimir operator on $\Lambda ^{*}(G/K)$ , induced by $\Omega $ .

7.3 Definition of the twisted dynamical zeta functions

We consider the twisted Ruelle and Selberg zeta functions associated with the geodesic flow on the sphere vector bundle $S(X)$ of $X=\Gamma \backslash G/ K$ . Since K acts transitively on the unit vectors in $\mathfrak {p}$ , $S(\widetilde {X})$ can be represented by the homogeneous space $G/M$ . Therefore $S(X)=\Gamma \backslash G/M$ .

We recall the Cartan decomposition $G=KA^{+}K$ of G, where $A^{+}$ is as in Equation (7.2). Then, every element $g\in G$ can be written as $g=ha_{+}k$ , where $h,k\in K$ and $a_{+}=\exp (tH)$ , for some $t\in \mathbb {R}^{+}$ . The positive real number t equals $d(eK,gK)$ , where $d(\cdot ,\cdot )$ denotes the geodesic distance on $\widetilde {X}$ . It is a well-known fact that there is a one-to-one correspondence between the closed geodesics on a manifold X with negative sectional curvature and the nontrivial conjugacy classes of the fundamental group $\pi _{1}(X)$ of X ([Reference Gromoll, Klingenberg and Meyer22]).

The hyperbolic elements of $\Gamma $ can be realized as the semisimple elements of this group, that is, the diagonalizable elements of $\Gamma $ . Since $\Gamma $ is a cocompact subgroup of G, every element $\gamma \in \Gamma -\{e\}$ hyperbolic. We denote by $c_{\gamma }$ the closed geodesic on X, associated with the hyperbolic conjugacy class $[\gamma ]$ . We denote by $l(\gamma )$ the length of $c_{\gamma }$ . Since $\Gamma $ is torsion-free, $l(\gamma )$ is always positive and therefore we can obtain an infimum for the length spectrum $\operatorname {\mathrm {spec}}(\Gamma ):=\{l(\gamma )\colon \gamma \in \Gamma \}$ . An element $\gamma \in \Gamma $ is called primitive if there exists no $n\in \mathbb {N}$ with $n>1$ and $\gamma _{0}\in \Gamma $ such that $\gamma =\gamma _{0}^{n}$ . A primitive element $\gamma _{0}\in \Gamma $ corresponds to a geodesic on X. The prime geodesics correspond to the periodic orbits of minimal length. Hence, if a hyperbolic element $\gamma $ in $\Gamma $ is generated by a primitive element $\gamma _{0}$ , then there exists a $n_{\Gamma }(\gamma )\in \mathbb {N}$ such that $\gamma =\gamma _{0}^{n_{\Gamma }(\gamma )}$ and the corresponding closed geodesic is of length $l(\gamma )=n_{\Gamma }(\gamma )l(\gamma _{0})$ .

We lift now the closed geodesic $c_{\gamma }$ to the universal covering $\widetilde {X}$ . For $\gamma \in \Gamma $ , $l(\gamma ):=\inf \{d(x,\gamma x)\colon x\in \widetilde {X}\}$ , or $l(\gamma )=\inf \{d(eK,g^{-1}\gamma gK)\colon g\in G\}$ . Hence, we see that the length of the closed geodesic $l(\gamma )$ depends only on $\gamma \in \Gamma $ . Let $\gamma \in \Gamma $ , with $ \gamma \neq e$ . Then, by [Reference Wallach51, Lemma 6.5] there exist a $g\in G$ , a $m_{\gamma } \in M$ , and an $a_{\gamma } \in A^{+}$ such that $ g^{-1}\gamma g=m_{\gamma }a_{\gamma }$ . The element $m_{\gamma }$ is determined up to conjugacy in M, and the element $a_{\gamma }$ depends only on $\gamma $ . As in [Reference Bunke and Olbrich7, Section 3.1], we consider the geodesic flow $\phi $ on $S(X)$ , given by the map $ \phi \colon \mathbb {R}\times S(X)\ni (t,\Gamma gM)\rightarrow \Gamma g\exp (-tH) M \in S(X) $ . A closed orbit of $\phi $ is described by the set $c:=\{\Gamma g\exp (-tH) M\colon t\in \mathbb {R}\}$ , where $g\in G$ is such that $g^{-1}\gamma g:=m_{\gamma }a_{\gamma }\in MA^{+}$ . The Anosov property of the geodesic flow $\phi $ on $S(X)$ can be expressed by the following $d\phi $ -invariant splitting of $TS(X)$

(7.6) $$ \begin{align} TS(X)=T^sS(X)\oplus T^cS(X)\oplus T^uS(X), \end{align} $$

where $T^sS(X)$ consists of vectors that shrink exponentially, $T^uS(X)$ consists of vectors that expand exponentially, and $T^{c}S(X)$ is the one-dimensional subspace of vectors tangent to the flow, with respect to the Riemannian metric, as $t\rightarrow \infty $ . The splitting in Equation (7.6) corresponds to the splitting

$$ \begin{align*} TS(X)=\Gamma\backslash G\times_{\operatorname{\mathrm{Ad}}}(\overline{\mathfrak{n}}\oplus \mathfrak{a} \oplus \mathfrak{n}), \end{align*} $$

where $\operatorname {\mathrm {Ad}}$ denotes the adjoint action of $\operatorname {\mathrm {Ad}}(\exp (-tH))$ on $\overline {\mathfrak {n}}, \mathfrak {a},\mathfrak {n}$ , and $\overline {\mathfrak {n}}=\theta \mathfrak {n}$ is the sum of the negative root spaces of the system $(\mathfrak {g},\mathfrak {a})$ .

Definition 7.3.1. Let $\chi \colon \Gamma \rightarrow \operatorname {\mathrm {GL}}(V_{\chi })$ be a finite-dimensional, complex representation of $\Gamma $ and $\sigma \in \widehat {M}$ . The twisted Selberg zeta function $Z(s;\sigma ,\chi )$ is defined by the infinite product

$$ \begin{align*} Z(s;\sigma,\chi):=\prod_{\substack{[\gamma]\neq{e}\\ [\gamma]\operatorname{\mathrm{prime}}}} \prod_{k=0}^{\infty}\det\big(\operatorname{\mathrm{Id}}-(\chi(\gamma)\otimes\sigma(m_\gamma)\otimes S^k(\operatorname{\mathrm{Ad}}(m_\gamma a_\gamma)|_{\overline{\mathfrak{n}}})) e^{-(s+|\rho|)\lvert l(\gamma)}\big), \end{align*} $$

where $s\in \mathbb {C}$ , $S^k(\operatorname {\mathrm {Ad}}(m_\gamma a_\gamma )_{\overline {\mathfrak {n}}})$ denotes the k-th symmetric power of the adjoint map $\operatorname {\mathrm {Ad}}(m_\gamma a_\gamma )$ restricted to $\mathfrak {\overline {n}}$ and $\rho $ is as in Equation (7.3).

By [Reference Spilioti48, Proposition 3.4], there exists a positive constant c such that the twisted Selberg zeta function $Z(s;\sigma ,\chi )$ converges absolutely and uniformly on compact subsets of the half-plane $\operatorname {\mathrm {Re}}(s)>c$ .

Definition 7.3.2. Let $\chi \colon \Gamma \rightarrow \operatorname {\mathrm {GL}}(V_{\chi })$ be a finite-dimensional, complex representation of $\Gamma $ and $\sigma \in \widehat {M}$ . The twisted Ruelle zeta function $ R(s;\sigma ,\chi )$ is defined by the infinite product

$$ \begin{align*} R(s;\sigma,\chi):=\prod_{\substack{[\gamma]\neq{e}\\ [\gamma]\operatorname{\mathrm{prime}}}}\det\big(\operatorname{\mathrm{Id}}-(\chi(\gamma)\otimes\sigma(m_{\gamma}))e^{-sl(\gamma)}\big)^{(-1)^{d-1}}, \end{align*} $$

where $s\in \mathbb {C}$ .

By [Reference Spilioti48, Proposition 3.5], there exists a positive constant r such that the twisted Selberg zeta function $R(s;\sigma ,\chi )$ converges absolutely and uniformly on compact subsets of the half-plane $\operatorname {\mathrm {Re}}(s)>r$ .

The twisted dynamical zeta functions associated with an arbitrary representation of $\Gamma $ has been studied in [Reference Spilioti48] and [Reference Spilioti49]. Specifically, by [Reference Spilioti48, Theorem 1.1 and Theorem 1.2], the twisted dynamical zeta functions admit a meromorphic continuation to the whole complex plane.

8.1 Twisted Bochner–Laplace operator

In this section, we define the twisted Bochner–Laplace operator as it is introduced in [Reference Müller35, Section 4]. In addition, we describe this operator in the locally symmetric space setting.

Let $E_{0}\rightarrow X$ be a complex vector bundle with covariant derivative $\nabla $ . We define the second covariant derivative $\nabla ^2$ by

$$ \begin{align*} \nabla^2_{V,W}:=\nabla_{V}\nabla_{W}-\nabla_{\nabla^{LC}_{V}W}, \end{align*} $$

where $V,W\in C^{\infty }(X,TX)$ and $\nabla ^{LC}$ denotes the Levi–Civita connection on $TX$ . We define the connection Laplacian $\Delta _{E_{0}}$ to be the negative of the trace of the second covariant derivative, that is,

$$ \begin{align*} \Delta_{E_{0}}:=-\operatorname{\mathrm{Tr}}\nabla^2. \end{align*} $$

If $E_{0}$ is equipped with a Hermitian metric compatible with the connection $\nabla $ , then by [Reference Lawson and Michelsohn25, p. 154], the connection Laplacian is equal to the Bochner–Laplace operator, that is,

$$ \begin{align*} \Delta_{E_{0}}=\nabla^{*}\nabla. \end{align*} $$

In terms of a local orthonormal frame field $(e_{1},\ldots ,e_{d})$ of $T_{x}X$ , for $x\in X$ , the connection Laplacian is given by

$$ \begin{align*} \Delta_{E_{0}}=-\sum_{j=1}^{d}\nabla^ 2{{}_{e_j,e_j}}. \end{align*} $$

The principal symbol of $\Delta _{E_{0}}$ equals

$$ \begin{align*} \sigma_{\Delta_{E_{0}}}(x,\xi) =\lVert \xi \rVert^ {2} \operatorname{\mathrm{Id}}_{({E_{0}})_{x}}, \end{align*} $$

where $x\in X$ and $\xi \in \operatorname {\mathrm {T}}_{x}^{*}X$ . $\Delta _{E_{0}}$ acts in $L^{2}(X,E_{0})$ with domain $C^{\infty }(X,E_{0})$ . The operator $\Delta _{E_{0}}\colon C^{\infty }(X,E_{0})\circlearrowright $ is a second order, elliptic, formally self-adjoint differential operator.

Let $\chi \colon \Gamma \rightarrow \operatorname {\mathrm {GL}}(V_{\chi })$ be a finite-dimensional, complex representation of $\Gamma $ . Let $E_{\chi }\rightarrow X$ be the associated flat vector bundle over X, equipped with a flat connection $\nabla ^{E_{\chi }}$ . We specialize to the twisted case $E=E_{0}\otimes E_{\chi }$ , where $E_{0}\rightarrow X$ is a complex vector bundle equipped with a connection $\nabla ^{E_{0}}$ and a metric, which is compatible with this connection. Let $\nabla ^{E}=\nabla ^{E_{0}\otimes E_{\chi }}$ be the product connection, defined by

$$ \begin{align*} \nabla^{E_{0}\otimes E_{\chi}}:=\nabla^{E_{0}}\otimes 1+1\otimes\nabla^{E_{\chi}}. \end{align*} $$

We define the operator $\Delta _{E_{0},\chi }^\sharp $ by

(8.1) $$ \begin{align} \Delta_{E_{0},\chi}^{\sharp}:=-\operatorname{\mathrm{Tr}}\big((\nabla^{E_{0}\otimes E_{\chi}})^2\big). \end{align} $$

We choose a Hermitian metric in $E_{\chi }$ . Then, $\Delta _{E_{0},\chi }^{\sharp }$ acts on $L^{2}(X,E_{0}\otimes E_{\chi })$ . However, it is not a formally self-adjoint operator in general. We want to describe this operator locally. Following the analysis in [Reference Müller35], we consider an open subset U of X such that $E_{\chi }\lvert _{U}$ is trivial. Then, $E_{0}\otimes E_{\chi }\lvert _{U}$ is isomorphic to the direct sum of m-copies of $E_{0}\lvert _{U}$ , that is,

$$ \begin{align*} (E_{0}\otimes E_{\chi})\lvert_{U}\cong\oplus_{i=1}^{m}E_{0}\lvert_{U}, \end{align*} $$

where $m:=\operatorname {\mathrm {rank}}(E_\chi )=\dim V_{\chi }$ . Let $(e_{i}),i=1,\ldots ,m$ be any basis of flat sections of $E_{\chi }\lvert _{U}$ . Then, each $\phi \in C^{\infty }(U,(E_{0}\otimes E_{\chi })\lvert _{U})$ can be written as

$$ \begin{align*} \phi=\sum_{i=1}^{m}\phi_{i} \otimes e_{i}, \end{align*} $$

where $\phi _{i}\in C^{\infty }(U, E_{0}\lvert _{U}), i=1,\ldots ,m$ . The product connection is given by

$$ \begin{align*} \nabla_{Y}^{E_{0}\otimes E_{\chi}}\phi=\sum_{i=1}^{m}(\nabla_{Y}^{E_{0}})(\phi_{i})\otimes e_{i}, \end{align*} $$

where $Y\in C^{\infty }(X,TX)$ . The local expression above is independent of the choice of the base of flat sections of $E_{\chi }\vert _{U}$ since the transition maps comparing flat sections are constant. By Equation (8.1), we obtain the twisted Bochner–Laplace operator acting on $C^{\infty }(X,E_{0}\otimes E_{\chi })$ given by

(8.2) $$ \begin{align} \Delta_{E_{0},\chi}^{\sharp}\phi=\sum_{i=1}^{m}(\Delta_{E_{0}}\phi_{i})\otimes e_{i}, \end{align} $$

where $\Delta _{E_{0}}$ denotes the Bochner–Laplace operator $\Delta ^{E_{0}}=(\nabla ^{E_{0}})^*\nabla ^{E_{0}}$ associated to the connection $\nabla ^{E_{0}}$ . Let now $\widetilde {E}_{0}, \widetilde {E}_{\chi }$ be the pullbacks to $\widetilde {X}$ of $E_{0},E_{\chi }$ , respectively. Then,

$$ \begin{align*} \widetilde{E}_{\chi}\cong \widetilde{X}\times V_{\chi}, \end{align*} $$

and

(8.3) $$ \begin{align} C^{\infty}(\widetilde{X}, \widetilde{E}_{0}\otimes \widetilde{E}_{\chi})\cong C^{\infty}(\widetilde{X}, \widetilde{E}_{0})\otimes V_{\chi}. \end{align} $$

With respect to the isomorphism (8.3), it follows from Equation (8.2) that the lift of $\Delta _{E_{0},\chi }^{\sharp }$ to $\widetilde {X}$ takes the form

(8.4) $$ \begin{align} \widetilde{\Delta}^{\sharp}_{E_{0},\chi}=\widetilde{\Delta}_{E_{0}}\otimes \operatorname{\mathrm{Id}}_{V_{\chi}}, \end{align} $$

where $\widetilde {\Delta }_{E_{0}}$ is the lift of $\Delta _{E_{0}}$ to $\widetilde {X}$ . By Equation (8.2), $\Delta _{E_{0},\chi }^{\sharp }$ has principal symbol

$$ \begin{align*} \sigma_{\Delta_{E_{0},\chi}^\sharp}(x,\xi)=\lVert \xi \rVert^ {2}_{x} \operatorname{\mathrm{Id}}_{({E_{0}\otimes E_{\chi})_{x}}}, \quad x\in X, \xi\in\operatorname{\mathrm{T}}_{x}^{*}X. \end{align*} $$

Hence, it has nice spectral properties, that is, its spectrum is discrete and contained in a translate of a positive cone $C\subset \mathbb {C}$ such that $\mathbb {R}^{+}\subset C$ ([Reference Spilioti48, Lemma 8.6]).

We specialize now the twisted Bochner–Laplace operator $\Delta _{E_{0},\chi }^{\sharp }$ to the case of the operator $\Delta ^{\sharp }_{\tau ,\chi }$ acting on smooth sections of the twisted vector bundle $E_{\tau }\otimes E_{\chi }$ . Here, $E_{\tau }$ is the locally homogeneous vector bundle associated with a finite-dimensional, unitary representation $\tau $ of K. The keypoint is that when we consider the lift $\widetilde {\Delta }^{\sharp }_{\tau ,\chi }$ of the twisted Bochner–Laplace operator $\Delta ^{\sharp }_{\tau ,\chi }$ to the universal covering $\widetilde {X}$ , it acts as the identity operator on $V_{\chi }$ . By Equation (8.1), we have

$$ \begin{align*} \widetilde{\Delta}^{\sharp}_{\tau,\chi}=\widetilde{\Delta}_{\tau}\otimes \operatorname{\mathrm{Id}}_{V_{\chi}}, \end{align*} $$

where $\widetilde {\Delta }_\tau $ is the lift to $\widetilde {X}$ of the Bochner–Laplace operator $\Delta _{\tau }$ , associated with the representation $\tau $ of K.

8.2 The twisted operator $A_{\chi }^{\sharp }(\sigma )$

In this section, we define the twisted operators $A_{\chi }^{\sharp }(\sigma )$ , associated with $\sigma \in \widehat {M}$ and representations $\chi $ of $\Gamma $ , acting on smooth sections of twisted vector bundles. These operators are first introduced in [Reference Bunke and Olbrich7]. For more details, we refer the reader to [Reference Spilioti48, Section 5].

We define the restricted Weyl group as the quotient $ W_{A}:=M'/M$ . Recall from Section 7.1 that $M'=\operatorname {\mathrm {Norm}}_{K}(A)$ is the normalizer of A in K. Then, $W_{A}$ has order 2. Let $w\in W_{A}$ be the nontrivial element of $W_{A}$ , and $m_{w}$ a representative of w in $M'$ . The action of $W_{A}$ on $\widehat {M}$ is defined by $(w\sigma )(m):=\sigma (m_{w}^{-1}mm_{w})$ , $m\in M, \sigma \in \widehat {M}$ . Following the proof of Proposition 1.1 in [Reference Bunke and Olbrich7] (see also [Reference Pfaff39, Proposition 2.3]), there exist unique integers $m_{\tau }(\sigma )\in \{-1,0,1\}$ , which are equal to zero except for finitely many $\tau \in \widehat {K}$ such that there exist unique integers $m_{\tau }(\sigma )\in \{-1,0,1\}$ , which are equal to zero except for finitely many $\tau \in \widehat {K}$ such that

• • if $\sigma $ is Weyl invariant, $ \sigma =\sum _{\tau \in \widehat {K}}m_{\tau }(\sigma )i^{*}(\tau ); $

• • if $\sigma $ is non-Weyl invariant, $ \sigma +w\sigma =\sum _{\tau \in \widehat {K}}m_{\tau }(\sigma )i^{*}(\tau ). $

We define a locally homogeneous vector bundle $E(\sigma )$ associated to $\sigma $ by

$$ \begin{align*} E(\sigma):=\bigoplus_{\substack{\tau\in\widehat{K}\\m_{\tau}(\sigma)\neq 0}}E_{\tau}, \end{align*} $$

where $E_{\tau }$ is the locally homogeneous vector bundle associated with $\tau \in \widehat {K}$ . The vector bundle $E(\sigma )$ has a grading

(8.5) $$ \begin{align} E(\sigma)=E(\sigma)^{+}\oplus E(\sigma)^{-}. \end{align} $$

This grading is defined exactly by the positive or negative sign of $m_{\tau }(\sigma )$ . Let $\widetilde {E}(\sigma )$ be the pullback of $E(\sigma )$ to $\widetilde {X}$ . Then,

$$ \begin{align*} \widetilde{E}(\sigma)=\bigoplus_{\substack{\tau\in\widehat{K}\\ m_{\tau}(\sigma)\neq0}}\widetilde{E}_{\tau}. \end{align*} $$

We assume now that $\tau \in \widehat {K}$ is irreducible. Recall that $ \widetilde {\Delta }_{\tau }=-R(\Omega )+\lambda _{\tau }\operatorname {\mathrm {Id}}. $ ([Reference Miatello29, Proposition 1.1], see also [Reference Spilioti48, (5.4)]) We put

$$ \begin{align*} \widetilde{A}_{\tau}:=\widetilde{\Delta}_{\tau}-\lambda_{\tau}\operatorname{\mathrm{Id}}. \end{align*} $$

Hence, the operator $\widetilde {A}_{\tau }$ acts like $-R(\Omega )$ on the space of smooth sections of $\widetilde {E}_{\tau }$ . It is an elliptic, formally self-adjoint operator of second order. By [Reference Chernoof12], it is an essentially self-adjoint operator. Its self-adjoint extension will be also denoted by $\widetilde {A}_{\tau }$ . We get then the operator $\widetilde {A}^{\sharp }_{\tau ,\chi }$ acting on the space $C^{\infty }(\widetilde {X},\widetilde {E}_{\tau }\otimes \widetilde {E}_{\chi })$ , defined by

$$ \begin{align*} \widetilde{A}^{\sharp}_{\tau,\chi}=\widetilde{A}_{\tau}\otimes\operatorname{\mathrm{Id}}_{V_{\chi}}. \end{align*} $$

We put

(8.6) $$ \begin{align} c(\sigma):=-\lvert \rho \rvert^{2}-\lvert\rho_{m}\rvert^{2}+\lvert \nu_{\sigma}+\rho_{m}\rvert^{2}, \end{align} $$

where $\nu _{\sigma }$ is as in Equations (7.5) and $\rho ,\rho _{m}$ are defined by Equations (7.3) and (7.4), respectively. We define the operator $A_{\chi }^{\sharp }(\sigma )$ acting on $C^{\infty }(X,E(\sigma )\otimes E_{\chi })$ by

(8.7) $$ \begin{align} A_{\chi}^{\sharp}(\sigma):=\bigoplus_{m_{\tau}(\sigma)\neq 0}A_{\tau,\chi}^{\sharp}+c(\sigma). \end{align} $$

The operator $A_{\chi }^{\sharp }(\sigma )$ preserves the grading and it is a non-self-adjoint, elliptic operator of order two.

11.1 Refined torsion of a finite-dimensional complex with a chirality operator

We consider now the chirality operator $\Gamma : C^{*}\rightarrow C^{*}$ such that $\Gamma : C^{j}\rightarrow C^{d-j}$ , for $j=0,\cdots ,d$ . For $c_{j}\in \det (C^{j})$ , $\Gamma c_{j}\in \det (C^{d-j})$ is the image of $c_{j}$ under the isomorphism $\det (C^{j})\rightarrow \det (C^{d-j})$ , induced by $\Gamma $ . Let $d=2r-1$ be an odd integer. We fix nonzero elements $c_{j}\in \det (C^{j})$ , $j=0,\cdots , r-1$ and consider the element

$$ \begin{align*} c_{\Gamma}:= (-1)^{\mathcal{R}(C^{*})}c_{0}\otimes & c_{1}^{-1}\otimes \cdots c_{r-1}^{(-1)^{r-1}}\\ &\otimes(\Gamma c_{r-1})^{(-1)^{r}} \otimes(\Gamma c_{r-2})^{(-1)^{r-1}} \otimes\cdots \otimes (\Gamma c_{0}^{-1})\notag \end{align*} $$

of $\det (C^{*})$ , where

$$ \begin{align*} \mathcal{R}(C^{*})=\frac{1}{2}\sum_{j=0}^{r-1}\dim C^{j}(\dim C^{j}+(-1)^{r+j}). \end{align*} $$

It follows from the definition of $c_{j}^{-1}$ that $c_{\Gamma }$ is independent of the choice of $c_{j}$ , $j=0,\ldots ,r-1$ .

Definition 11.1.1. We define the refined torsion $\rho _{\Gamma }$ of the pair $(C^{*},\Gamma )$ by

(11.2) $$ \begin{align} \rho_{\Gamma}=\rho_{C^{*},\Gamma}:=\phi_{C^{*}}(c_{\Gamma}), \end{align} $$

where $\phi _{C^{*}}$ is as in Equation (11.1).

We consider now the finite-dimensional analogue of the odd signature operator B, defined in a Riemannian manifold-setting by Equation (3.1). Let $B\colon C^{*}\rightarrow C^{*}$ be the signature operator, defined by

$$ \begin{align*} B:=\partial\Gamma+\Gamma\partial. \end{align*} $$

We consider its square, $B^{2}=(\partial \Gamma )^{2}+(\Gamma \partial )^{2}$ . Let $I\subset [0,\infty )$ . Let $C^{j}_{I}\subset C^{j}$ be the span of the generalized eigenvectors of the restriction of $B^{2}$ to $C^{j}$ , corresponding to eigenvalues r with $\vert r \vert \in I$ . Both $\Gamma $ and $\partial $ commute with B and hence with $B^{2}$ . Then, we have $\Gamma :C^{j}_{I}\rightarrow C^{d-j}_{I}$ , and $\partial : C^{j}_{I}\rightarrow C^{j+1}_{I}$ . Hence, we obtain a subcomplex $C_{I}^{*}$ of $C^{*}$ and the restriction $\Gamma _{I}$ of $\Gamma $ to $C_{I}^{*}$ is the chirality operator on this complex. Let $\partial _{I}, B_{I}, B^{\operatorname {\mathrm {even}}}_{I}$ be the restriction to $C_{I}^{*}$ of $\partial , B, B^{\operatorname {\mathrm {even}}}$ , correspondingly. Here, we denote $ B^{\operatorname {\mathrm {even}}}\colon C^{\operatorname {\mathrm {even}}}\rightarrow C^{\operatorname {\mathrm {even}}}$ , where $C^{\operatorname {\mathrm {even}}}:=\bigoplus _{j \operatorname {\mathrm {even}}} C^{j}$ . By Lemma 5.8 in [Reference Braverman and Kappeler4], if $0 \notin I$ , then the complex $(C^{*}_{I},\partial _{I})$ is acyclic. For every $\lambda \geq 0$ , we have

$$ \begin{align*} C^{*}=C^{*}_{[0,\lambda]}\oplus C^{*}_{(\lambda,\infty)}. \end{align*} $$

In particular,

$$ \begin{align*} H^{*}_{(\lambda,\infty)}(\partial)=0, \quad H^{*}_{[0,\lambda]}(\partial)\cong H^{*}(\partial). \end{align*} $$

Hence, there are canonical isomorphisms

$$ \begin{align*} \Phi\colon\det(H^{*}_{(\lambda,\infty)}(\partial))\rightarrow\mathbb{C}, \quad \Psi\colon\det(H^{*}_{[0,\lambda]}(\partial))\rightarrow\det(H^{*}(\partial)). \end{align*} $$

By [Reference Braverman and Kappeler4, Proposition 5.10], for each $\lambda \geq 0$ ,

$$ \begin{align*} \rho_{\Gamma}={\det}_{\operatorname{\mathrm{gr}}}(B^{\operatorname{\mathrm{even}}}_{(\lambda,\infty)})\rho_{\Gamma_{[0,\lambda]}}, \end{align*} $$

where $\rho _{\Gamma _{[0,\lambda ]}}$ is an element of $\det (H^{*}(\partial ))$ via the canonical isomorphism $\Psi $ .

11.2 Refined analytic torsion as an element of the determinant line for compact odd-dimensional manifolds

We consider now a compact, oriented, Riemannian manifold $(X,g)$ of odd dimension $d=2r-1$ and a complex vector bundle E over X, endowed with a flat connection $\nabla $ . Let $\Lambda ^{k}(X,E)$ be the space of k-differential forms on X with values in E. Let B be the odd signature operator acting on $\Lambda ^{k}(X,E)$ as in Definition 3.1.1. Let $I\subset [0,\infty )$ . Let $\Lambda ^{k}_{I}(X,E)$ be the image of $\Lambda ^{k}(X,E)$ under the spectral projection of $B^{2}$ , corresponding to eigenvalues whose absolute value lie in I. If I is bounded, then the subspace $\Lambda ^{k}_{I}(X,E)$ is finite-dimensional ([Reference Braverman and Kappeler4, Section 6.10]). Let $B_{I}$ , $\Gamma _{I}$ be the restrictions to $\Lambda ^{k}_{I}(X,E)$ of B and $\Gamma $ , respectively. For every $\lambda \geq 0$ , we have

$$ \begin{align*} \Lambda^{*}(X,E)=\Lambda^{*}_{[0,\lambda]}(X,E)\oplus \Lambda^{*}_{(\lambda,\infty)}(X,E). \end{align*} $$

Since for every $\lambda \geq 0$ the complex $\Lambda ^{*}_{(\lambda ,\infty )}(X,E)$ is acyclic, we get

$$ \begin{align*} H^{*}_{[0,\lambda]}(X,E)\cong H^{*}(X,E). \end{align*} $$

Definition 11.2.1. The refined analytic torsion $T=T(\nabla )$ is an element of $\det (H^{*}(X,E))$ , defined by

$$ \begin{align*} T(\nabla)=\rho_{\Gamma_{[0,\lambda]}}{\det}_{\operatorname{\mathrm{gr}}}(B^{\operatorname{\mathrm{even}}}_{(\lambda,\infty)})e^{i\pi \operatorname{\mathrm{rank}}(E_{\chi})\eta_{tr}(g)}, \end{align*} $$

where $\rho _{\Gamma _{[0,\lambda ]}}$ is as in Equation (11.2).

By [Reference Braverman and Kappeler4, Proposition 7.8] and [Reference Braverman and Kappeler4, Theorem 9.6], the refined analytic torsion is independent the choice of $\lambda \geq 0$ , the choice of the Agmon angle $\theta \in (-\pi ,0)$ for $B^{\operatorname {\mathrm {even}}}$ , and the Riemannian metric g.