Perturbative BF Theory in Axial, Anosov Gauge

Abstract

The twisted Ruelle zeta function of a contact, Anosov vector field, is shown to be equal, as a meromorphic function of the complex parameter $\hbar \in \mathbb {C}$ and up to a phase, to the partition function of an $\hbar$-linear quadratic perturbation of BF theory, using an “axial” gauge fixing condition given by the Anosov vector field. Equivalently, it is also obtained as the expectation value of the same quadratic, $\hbar$-linear, perturbation, within a perturbative quantisation scheme for BF theory, suitably generalised to work when propagators have distributional kernels.

1 Introduction

Topological (quantum) field theories (TQFTs) are a valuable resource of insight for both pure and applied mathematics, for they are able to package relevant information, in both their classical and quantum formulation. Typically, the field theories “know” about the topology of the underlying manifold, e.g. through topological invariants as well as group representations and cohomology, and provide numerous deep links between seemingly unrelated areas of mathematics (see [3, 7, 39, 43] just to name a few).

An example of this paradigm is given by (Abelian) BF theory [8, 13, 39], a model of great importance in its simplicity, for it serves as a nexus that links to several other scenarios. In certain cases, it can be seen as an example of Chern–Simons theory in three dimensions, thus connecting to knot invariants; it is a particular case of the Poisson sigma model in two dimensions, which in turn is related to Kontsevich’s formality and deformation quantisation; and it can be seen as the “topological backbone” of theories such as Yang–Mills and even theories of gravity, which can be thought of as perturbations of BF theory.

A first result that exemplifies the importance of TQFTs for pure mathematics, and the main motivator for this work, is derived from the quantisation of Abelian BF theory: The partition function of the theory (in the “Lorenz gauge”, see below) is the analytic torsion of Ray and Singer [36], as shown by [39, 40] (see also [15, 30] for two alternative approaches).

To properly address the quantisation of Abelian BF theory, and rigorously phrase the previous claim, one needs to address the fact that it is a gauge theory, i.e. it is invariant under the action of an infinite-dimensional group of local symmetries. This degeneracy is nonphysical, and it needs to be “removed” in the appropriate way in order to extract meaningful observable (i.e. physical) information from the theory. This procedure is called gauge fixing, and it is conveniently addressed within a rigorous approach to quantisation of the theory by means of the Batalin–Vilkovisky formalism [5, 6].

A gauge fixing can be thought of as an arbitrary choice to present the quotient of the space of fields by the group action within the space of fields itself,¹ and it is expected to be an immaterial choice on physical grounds. Note that all mentioned results that link Abelian BF theory to the analytic torsion are obtained by choosing a particular gauge fixing condition that goes under the name of “Lorenz gauge”, which depends on the choice of an arbitrary metric on the underlying manifold.

Recently, one of the authors noted how a different choice of gauge fixing condition, which instead depends on the choice of an Anosov-contact structure on an odd-dimensional manifold, leads to the statement that the partition function for Abelian BF theory returns the (absolute value of the) value at zero² of a meromorphic function called (twisted) Ruelle zeta [37], when it is computed using this gauge fixing condition [30]. This result is immediately tied to a conjecture due to Fried, stating that the value at zero of the (twisted) Ruelle zeta function over a manifold that admits an Anosov vector field should compute exactly the analytic torsion of the manifold [26, 27]. The conjecture is true if the quantisation procedure can be shown not to depend on the arbitrary choice of gauge fixing.³

While in this paper we will not discuss the invariance of the theory on the choice of gauge fixing (which is done in the companion paper [38]), we will complete the result of [30] by showing that the whole Ruelle zeta function, i.e. as a meromorphic function on the complex plane, can be given a rigorous field-theoretic presentation, either by means of the expectation value of a suitably chosen quadratic functional of the fields, or as the partition function of a perturbation of BF theory by means of the same functional. This, together with the result in [30], completely reconstructs the Ruelle zeta function perturbatively, by means of Feynman diagrams.

In order to do this, despite BF theory being “simple” from the perturbative point of view, a significant adaptation of the standard methods used to compute Feynman integrals (e.g. those presented in [18]) becomes necessary. This is due to the simple, but crucial, reason that the operators that appear naturally after imposing the contact, Anosov, gauge fixing condition are not elliptic, and thus, their heat kernels are not smooth. As a matter of fact, BF theory in this gauge features the operator $\mathcal {L}_X$ acting on differential forms, where X is a contact, Anosov vector field, and perturbative quantisation of the theory only makes sense via a careful manipulation of the wavefront sets of the propagator of the theory. This requires phrasing perturbative quantisation within the setting of microlocal analysis. Hence, besides discussing the particular example of BF theory, we employ a generalisation of the perturbative quantisation techniques to a much larger class of operators.

The paper is structured as follows. In Sect. 2, we provide an introduction to perturbative quantisation of gauge field theories in the BV formalism and set the main definitions and methods we will use in the remainder of the paper.

In Sect. 3, we define the geometric setting we will need to define BF theory (Sect. 3.1) and the twisted Ruelle zeta function (Sect. 3.2).

The rest of the paper is devoted to recovering the twisted Ruelle zeta function from the quantisation of BF theory. To this end, we introduce a functional with which we perturb the action of BF theory at first order in $\hbar$.

We first show (Sect. 3.4) that, up to a phase, the full Ruelle zeta function (as a function of $\hbar \in \mathbb {C}$) is the partition function of the perturbed action in the contact, Anosov gauge (Theorem 3.20). This requires a regularisation scheme for determinants of operators on infinite-dimensional vector spaces that is a generalisation of the (more widespread) zeta-function regularisation. We call it flat-regularisation, since it uses the flat trace and flat determinant tools.

Then, we change perspective slightly and consider the perturbing functional as an observable, of which we compute the expectation value with respect to free, Abelian, BF theory, in the contact, Anosov gauge (Sect. 3.5). This time we obtain that the ratio of Ruelle zeta function by its value at zero can be obtained as the perturbative expectation value of our perturbing functional, thus effectively reconstructing the whole zeta function (Theorem 3.27).

2 Perturbation Theory and Effective Field Theory

In this section, we introduce the relevant notions of classical field theory with local symmetries and discuss its perturbative quantisation in the effective field theory sense. Our framework of choice to handle perturbative quantisation of gauge theories is the Batalin–Vilkovisky (BV) cohomological framework. We will outline the basics of perturbative quantisation for gauge theories in the BV formalism, starting from a finite-dimensional scenario.

2.1 Finite-Dimensional Perturbation Theory

Let V be a finite-dimensional vector space. The stationary phase formula provides a small $\hbar$ asymptotic expansion of the oscillatory integral

$$\begin{aligned} \int _V e^{\frac{i}{\hbar }f(x)}\,\textrm{d}x. \end{aligned}$$

We apply this perturbative expansion to a function

$$\begin{aligned} f(x) = \frac{1}{2}{\textbf{Q}}(x,x) + I(x), \end{aligned}$$

where ${\textbf{Q}}(x,x)$ is a non-degenerate quadratic form on V, the free action, and I(x) is a polynomial, called the interaction term, and thought of as a perturbation of the free action. One way to write the stationary phase formula in this case (see, for example, [35]) is

$$\begin{aligned} \int _V e^{\frac{i}{\hbar }(\frac{1}{2}{\textbf{Q}}(x,x)+I(x))}\,\textrm{d}x \sim \int _V e^{\frac{i}{2\hbar }{\textbf{Q}}(x,x)}\,\textrm{d}x \cdot e^{\frac{1}{\hbar }\Gamma ({\textbf{Q}}^{-1},\,I)(0)}, \quad \textrm{as}\quad \hbar \rightarrow 0, \end{aligned}$$

(1)

where the right-hand side is interpreted as a formal power series in the parameter $\hbar$. The term $\Gamma ({\textbf{Q}}^{-1},\,I)$ is defined as a sum over connected Feynman diagrams:

$$\begin{aligned} \Gamma ({\textbf{Q}}^{-1},\,I)(0) = \sum _\gamma \frac{\hbar ^{l(\gamma )}}{|\textrm{Aut}(\gamma )|}\Phi _\gamma (i{\textbf{Q}}^{-1},\{iI_d\})(0), \end{aligned}$$

(2)

where a Feynman diagram $\gamma$ is a graph specified by the following data:

• a finite set $V(\gamma )$ of vertices,

• a finite set $H(\gamma )$ of half-edges,

• a map $i: H(\gamma ) \rightarrow V(\gamma )$, assigning each half-edge to the vertex, to which it is attached,

• an involution $\sigma : H(\gamma ) \rightarrow H(\gamma )$, whose set of fixed points, $T(\gamma )$, is the set of tails of the diagram, and whose set of two-element orbits, $E(\gamma )$, is the set of edges.

A Feynman diagram is connected if its underlying graph is connected, and the sum in (2) is over connected diagrams $\gamma$. Each term is multiplied by a power of $\hbar$, where the exponent $l(\gamma )$ is the number of loops occurring in the graph $\gamma$. The other prefactor, $|\textrm{Aut}(\gamma )|$, called the symmetry factor of $\gamma$, is the cardinality of the automorphism group of the graph.

To each Feynman diagram $\gamma$ we associate its weight $\Phi _\gamma (i{\textbf{Q}}^{-1},\{iI_d\})$, which is defined as a function on the vector space V, depending on the quadratic form ${\textbf{Q}}$ and the interaction term I of the action functional.

Let us write $I(x)=\sum _d I_d(x)$, where $I_d$ is a homogeneous polynomial of degree d, viewed as elements of the d-th tensor power of the dual space: $I_d \in (V^*)^{\otimes d}$. Then, denote by ${\textbf{Q}}^{-1}$ the inverse of (the matrix representing) the quadratic form ${\textbf{Q}}$. Note that, in finite dimensions, the inverse of ${\textbf{Q}}$ can canonically be viewed as an element ${\textbf{Q}}^{-1} \in V\otimes V$. This is the propagator of the theory.

Given a Feynman diagram $\gamma$ and a vector $v\in V$, the weight of (the Feynman diagram) $\gamma$ is constructed as follows:

• 1. To each tail in $T(\gamma )$ we associate a factor of v and to each edge in $E(\gamma )$ we associate a factor of $i{\textbf{Q}}^{-1}$. Taking the tensor product of the factors associated with tails and edges returns an element of $V^{\otimes |H(\gamma )|}$.

• 2. To each vertex of order d, i.e. a vertex with d half-edges attached to it, we assign a factor of $iI_d$. Taking the tensor product similarly produces an element of $(V^*)^{\otimes |H(\gamma )|}$.

• 3. The weight $\Phi _\gamma (i{\textbf{Q}}^{-1},\{iI_d\})(v)$ is the contraction of these two elements of the respective tensor power.

Remark 2.1

The integral in (1) is not necessarily convergent, even for finite-dimensional V. In order for the stationary phase formula to be an actual asymptotic expansion, the integral should be taken with respect to a compactly supported measure. Although there are ways to make rigorous sense of the formula (1), we ignore these subtleties here, since in infinite dimensions we just take Equation (1) as a definition. $\diamondsuit$

Remark 2.2

The factors of the imaginary unit i are introduced since we are considering an oscillatory integral.⁴ The vector v can be viewed as an “external field”, and in the expression (1) we are setting this external field to zero, which is equivalent to only considering Feynman diagrams with no tails. However, we will need the formulation with the external field shortly, when we consider effective interactions. $\diamondsuit$

The idea of perturbation theory is to use the perturbative expansion to assign a value to oscillatory integrals, even when the vector space V is infinite-dimensional. Going back to Equation (1): from the perturbation theory perspective, one needs to normalise expectation values of interaction polynomials using the free theory (given by the quadratic form ${\textbf{Q}}$) as a benchmark.

In other words, one is interested in the quantity

$$\begin{aligned} \biggl \langle e^{\frac{i}{\hbar }I} \biggr \rangle \doteq \frac{\int _V e^{\frac{i}{\hbar }(\frac{1}{2}{{\textbf {Q}}}(x,x)+I(x))}\,\text {d}x}{\int _V e^{\frac{i}{2\hbar }{{\textbf {Q}}}(x,x)}\,\text {d}x} = e^{\frac{1}{\hbar }\Gamma ({{\textbf {Q}}}^{-1},\,I)(0)}, \end{aligned}$$

(3)

interpreted as the expectation value of $\exp (\frac{i}{\hbar }I)$ with respect to the free theory. Note that this can also be interpreted as the interacting partition function, with the free partition function set to one. Observe furthermore that the right-hand side is now expressed solely in terms of the Feynman diagram expansion, which one can hope to make sense of even in infinite dimensions.

Remark 2.3

In the case where the action functional is quadratic, which is often referred to as a free theory, a simplification of the perturbative approach is available: The partition function for such a quadratic functional can formally be interpreted as an oscillatory, Gaussian, integral. In finite dimensions, the result of this integral can be expressed in terms of the determinant of the matrix in the quadratic form (up to a phase)

$$\begin{aligned} Z=\int _V e^{\frac{i}{2\hbar }{\textbf{Q}}(x,x)}\textrm{d}x \propto \textrm{det}({\textbf{Q}})^{-\frac{1}{2}} \end{aligned}$$

Given the appropriate generalisation of the determinant to infinite-dimensional operators, which requires a choice of regularisation, one can define the partition function of a quadratic functional by extending the oscillatory, Gaussian, integral formula to the infinite-dimensional case. In particular, when we have access to the value of the free partition function one can use the perturbative framework to compute the interacting partition function. Cf. Remark 2.5. $\diamondsuit$

In Appendix B, we will introduce an appropriate notion of determinant for operators on function spaces, called flat determinant. Foreshadowing what will come, we phrase the following definition in terms of $\textrm{det}^\flat$.

Definition 2.4

Let $S: \mathcal {V}\rightarrow \mathbb {R}$ be a functional of the form $S(v) = \left\langle v,\, Av \right\rangle$, where $\mathcal {V}$ is a (possibly infinite-dimensional) graded vector space endowed with an inner product $\left\langle \cdot ,\, \cdot \right\rangle$ and $A:\mathcal {V}\rightarrow \mathcal {V}$ is a degree-preserving linear operator with well-defined flat determinant. Then, we define the value of the oscillatory, Gaussian, “integral” as

$$\begin{aligned} \int _V e^{iS} \doteq |\textrm{sdet}^\flat (A)|^{-\frac{1}{2}}= \bigl |\prod _k \textrm{det}^\flat (A_k)^{(-1)^k}\bigr |^{-\frac{1}{2}}, \end{aligned}$$

(4)

where $A_k=A\vert _{\mathcal {V}^{(k)}}$ acts on homogeneous elements of degree k.

Remark 2.5

The integral over summands of odd degree in $\mathcal {V}$ results in a determinant raised to the opposite sign of what is expected from the usual oscillatory, Gaussian, integral, see, for example, [35]. Compared to an n-dimensional oscillatory, Gaussian, integral, we have dropped the constant factor of $(2\pi )^{\frac{n}{2}}$ and also ignored the phase factor $\exp (i\frac{\pi }{4}\textrm{sign}(A))$, which will not matter for the applications considered in the present paper. $\diamondsuit$

2.2 Effective Field Theory

A local, Lagrangian, field theory is the data of a space of fields $\mathcal {F}=C^\infty (M,F)$, modelled on sections of vector bundles,⁵ and a local action functional $S: \mathcal {F}\rightarrow \mathbb {R}$, of the form⁶

$$\begin{aligned} S(\varphi ) = \int _M ((j^\infty )^*\textsf{L})(\varphi )(x), \end{aligned}$$

(5)

where $\textsf{L}\in \Omega ^{0,\textrm{top}}(J^\infty F)$—often called a Lagrangian density—only depends on a finite-order jet of a section $\varphi \in \mathcal {F}$.

Since the space of fields is infinite dimensional, a measure-theoretic interpretation of the typical expressions used in quantum theory, like:

$$\begin{aligned} Z = ``\int _\mathcal {F}e^{\frac{i}{\hbar }S(\varphi )}\,\mathcal {D}\varphi \,'', \qquad \langle \mathcal {O}\rangle = `` \int _\mathcal {F}\mathcal {O}(\varphi ) e^{\frac{i}{\hbar }S(\varphi )}\mathcal {D}\varphi \,'', \end{aligned}$$

(6)

is generally not available. (Here, $\mathcal {O}$ is a functional on $\mathcal {F}$, the parameter $\hbar$ is formal, and $\mathcal {D}\varphi$ is intended to be a measure on $\mathcal {F}$.) However, one can make sense of these expressions perturbatively, i.e. as formal power series in $\hbar$, as described in detail in Sect. 2.1.

While for quadratic functionals, like the free part of a general action, we could generalise the finite-dimensional scenario, there still remain problems in evaluating Feynman diagrams in infinite dimensions. We illustrate this with a simple example. Assume that our space of fields is $\mathcal {F}\doteq C^\infty (M)$, the space of smooth functions on some compact manifold M, and we are given a free theory defined by the functional

$$\begin{aligned} S_0(\varphi ) = \frac{1}{2}\int _M \varphi (x) D \varphi (x)\,\textrm{d}x, \end{aligned}$$

where D is an injective differential operator on M. (In terms of the picture outlined above the quadratic functional $S_0$ corresponds to $\frac{1}{2}{\textbf{Q}}(x,x)$.) The propagator of the theory is then the Schwartz kernel of $D^{-1}$, which in general will only be a distribution on $M\times M$, denoted⁷$P \in \mathcal {D}'(M\times M)$.

Given an interaction term $I(\varphi )$, homogeneous of degree d, we can view it as an element $I\in \mathcal {D}'(M^d)$ and attempt to compute the Feynman diagram expansion $\Gamma (P,I)(\varphi )$ for some external field $\varphi$. However, we run into problems when trying to apply the Feynman rules described above. Since both the propagator P and the interaction I are distributions, there is, in general, no way to contract them and thus construct the Feynman weight $\Phi _\gamma (i{\textbf{Q}}^{-1},\{iI_d\})(\varphi )$.

One way to deal with this problem is through the framework of effective field theory. Here, we follow Costello’s formulation of effective quantum field theory [18].⁸ The idea is to replace the interaction term I with a family of effective interactions⁹

$$\begin{aligned} \{I[L]\in C^\infty (\mathcal {F}), L\in \mathbb {R}_+\}, \end{aligned}$$

with L determining the length scale. The offending propagator P is replaced by its heat kernel regularised counterpart, that is, to $L_1, L_2 \in \mathbb {R}_+$ (interpreted as two length scales) we associate the propagator:

$$\begin{aligned} P_{L_1,L_2} = \int _{L_1}^{L_2}K_t\,\textrm{d}t, \end{aligned}$$

where $K_t$ is the heat kernel for D, i.e. the Schwartz kernel of the operator $\exp (-tD)$. When D is an elliptic operator with strictly positive spectrum, this propagator is a smooth function, i.e. $P_{L_1,L_2} \in C^\infty (M\times M)$ for all $L_1,L_2 > 0$, where we can even set $L_2=\infty$. Thus, $P_{L_1,L_2}$ can be contracted with distributions and the Feynman rules are well defined for this propagator.

In order to link together the behaviours at different values of the parameter L, the effective interactions I[L] are required to satisfy the renormalisation group equation:

$$\begin{aligned} iI[L_2](\varphi ) = \Gamma (P_{L_1,L_2},\, I[L_1])(\varphi ), \end{aligned}$$

(7)

relating the scale-$L_2$ to the scale-$L_1$ interactions. Then, the expectation value in (3) can be computed as

$$\begin{aligned} \biggl \langle e^{\frac{i}{\hbar }I} \biggr \rangle = e^{\frac{1}{\hbar }\Gamma (P_{L,\infty }, I[L])(0)}, \end{aligned}$$

(8)

and the renormalisation group equation implies that this is independent of the length scale-L used in the expression. Once again, this expectation value can also be interpreted as the partition function of the interacting theory, where the undefined free partition function has been set to 1.

Remark 2.6

Starting from a local interaction I one can obtain a family of effective interactions by renormalisation. The scale-L interaction is given by

$$\begin{aligned} iI[L](\varphi ) = \lim _{\varepsilon \rightarrow 0}\Gamma (P_{\varepsilon ,L},I)(\varphi ), \end{aligned}$$

(9)

provided the limit exists. When this is not the case, one subtracts counterterms $I^{CT}(\varepsilon )$ from the interaction to make the limit of the renormalised interaction well defined:

$$\begin{aligned} iI^R[L](\varphi ) = \lim _{\varepsilon \rightarrow 0}\Gamma (P_{\varepsilon ,L},I-I^{CT}(\varepsilon ))(\varphi ). \end{aligned}$$

We will not make use of this technique, since the example we will study in Sect. 3.3 does not require any counterterms for the limit in (9) to exist. $\diamondsuit$

2.3 Effective Field Theory in the BV Formalism

Perturbative, effective, field theory needs to be amended when dealing with gauge theories. These are models that admit local symmetries, i.e. a set of (local) vector fields preserving the action functional (possibly up to total derivatives). In this case, the field-theoretic description is characterised by redundancies that must be removed, in order to extract sensible information from the model. The removal of such symmetry is called gauge fixing.

In the presence of gauge symmetries, the quadratic form ${\textbf{Q}}$ becomes degenerate [31] and the asymptotic expansion (2) no longer makes sense. As a consequence, for the perturbative methods to be well defined, one needs to implement gauge fixing in perturbative effective field theory.

Indeed, the construction outlined above can be combined with the Batalin–Vilkovisky (BV) formalism for gauge field theories [5, 6]. Here, we provide a brief summary of the construction of effective theories within the BV formalism, following [18, 19], and [35] (see also [44]).

The BV formalism deals with gauge theories by expanding the original space of fields $\mathcal {F}$ to a graded $(-1)$-symplectic manifold $\mathcal {F}_{BV}$, and replacing the original action functional with a functional $S_{BV}$ on $\mathcal {F}_{BV}$. More precisely:

Definition 2.7

(Classical BV Theory). A classical BV theory is given by the data $(\mathcal {F}_{BV}, \Omega _{BV}, S_{BV}, Q_{BV})$, where

• the space of BV fields $\mathcal {F}_{BV}$ is a $\mathbb {Z}$-graded manifold,

• the BV form $\Omega _{BV}$ is a degree $-1$, local, (weakly) symplectic¹⁰ form on $\mathcal {F}_{BV}$,

• the classical BV operator $Q_{BV}$ is a degree 1, local, vector field on $\mathcal {F}_{BV}$ with the cohomological property $[Q_{BV},Q_{BV}]=0$,

• the BV action $S_{BV}$ is a local function on $\mathcal {F}_{BV}$ of degree 0, satisfying the Hamiltonian condition and the classical master equation:

  $$\begin{aligned} \iota _{Q_{BV}}\Omega _{BV} = \delta S_{BV}, \qquad \{S_{BV},S_{BV}\}_{\Omega _{BV}} = 0. \end{aligned}$$

Here, $\{\,\cdot ,\,\cdot \,\}_{\Omega _{BV}}$ is the Poisson bracket on (Hamiltonian) functions induced by the (weak) symplectic structure.

In our applications, $\mathcal {F}_{BV}$ is the space of sections of some graded vector bundle¹¹$\mathbb {F}\rightarrow M$ over a compact, connected, orientable manifold M without boundary, i.e. $\mathcal {F}_{BV} = C^\infty (M,\mathbb {F})$. Since $\mathcal {F}_{BV}$ is a vector space, the graded, weakly symplectic structure can be defined directly as a graded anti-symmetric, bilinear, pairing on the space of fields:

$$\begin{aligned} \Omega _{BV}: \mathcal {F}_{BV}\times \mathcal {F}_{BV} \rightarrow \mathbb {R}. \end{aligned}$$

The pairing then induces a map $\Omega _{BV}^\flat : \mathcal {F}_{BV} \rightarrow \mathcal {F}_{BV}^*$ into the (strong) dual space to the space of fields, and we require this map to be injective. (See [33, Section 48] for a treatment of infinite-dimensional symplectic manifolds.)

In this setting, the notion of Lagrangian subspace requires an appropriate generalisation. We employ the following (cf. [16, 42]):

Definition 2.8

Let $(\mathcal {V},\Omega )$ be a (weak) symplectic vector space. A subspace $\textsf{L}\subset \mathcal {V}$ is said to be Lagrangian iff it is isotropic, i.e. $\Omega \vert _{{\textsf{L}} \times {\textsf{L}}}=0$ and there is a splitting $\mathcal {V}= \textsf{L}\oplus \textsf{K}$, where $\textsf{K}$ is also isotropic.

We write the quadratic, free, part of the (BV) action as

$$\begin{aligned} S_0(\varphi ) = \frac{1}{2}\Omega _{BV}(\varphi ,Q\varphi ), \quad \varphi \in \mathcal {F}_{BV}, \end{aligned}$$

where Q is a differential operator on $\mathcal {F}_{BV}$, of degree 1 with respect to the grading, which squares to zero. Note that $Q^2 = 0$ is equivalent to the classical master equation for $S_0$.

Remark 2.9

(Q vs. $Q_{BV}$) Note that Q and $Q_{BV}$ may differ. In full generality, $Q_{BV}$ may contain nonlinear terms, owing to the fact that it is the Hamiltonian vector field of the possibly non-quadratic action $S_{BV}$. A typical example is given by $Q=d$ the de Rham differential, while $Q_{BV}$ may act as $d_A$ for some connection A, which could be a field configuration. In the perturbative quantisation setting, it may be convenient (but not strictly necessary) to work with Q and treat nonlinear contributions as part of the interaction. In our main application, Abelian BF theory, there is no distinction between Q and $Q_{BV}$. $\diamondsuit$

To apply the perturbative framework to a gauge theory in the BV Formalism, as mentioned, we need a gauge fixing condition.

Definition 2.10

A gauge fixing operator is a linear, degree $-1$, local map $Q_{GF}:\mathcal {F}_{BV} \rightarrow \mathcal {F}_{BV}$ such that $Q_{GF}^2=0$, and such that $\textsf{L}\doteq \textrm{im}(Q_{GF})$ is a Lagrangian subspace. Moreover, we require that

$$\begin{aligned} D=[Q_{GF},Q]= QQ_{GF} + Q_{GF}Q \end{aligned}$$

be a differential operator, and that the Lagrangian splitting $\mathcal {F}=\textsf{L}\oplus \textsf{K}$ be D-invariant.

Remark 2.11

(Comparison with [18]) In the framework outlined ibidem, a gauge fixing operator is instead required to induce a splitting of the form

$$\begin{aligned} \mathcal {F}_{BV} = {{\,\textrm{im}\,}}(Q)\oplus {{\,\textrm{im}\,}}(Q_{GF})\oplus \ker (D). \end{aligned}$$

(10)

We will only deal with the case $\ker (D)=\{0\}$, although we will not be able to achieve this splitting in the space of smooth sections for the particular gauge fixing condition we will be interested in (see Remark 3.22). The idea of extracting the kernel of D from $\mathcal {F}_{BV}$ and choosing a Lagrangian splitting of the complement is explored in [14] as well, where it relates to a choice of “residual fields”, on which the resulting effective field theory will depend. We will not pursue this idea further, as it lies beyond the scope of this work. We note here that both Q and $Q_{GF}$ commute with D.

Another restrictive condition that is imposed on the free action and the gauge fixing operator by the framework of [18] is the requirement that D be a generalised Laplacian. Among other things, this guarantees the existence of a smooth heat kernel. We shall shortly dispose of this requirement and work with some alternative regularisation procedures to make sense of the Feynman rules. $\diamondsuit$

When working with a quadratic BV theory, for which we can write $S_{BV} = \frac{1}{2} \Omega _{BV}(\varphi , Q_{BV}\varphi )$, we can look at the restriction $S_{BV}\vert _\textsf{L}$ of the quadratic functional onto the gauge fixing subspace. One can then consider the gauge-fixed partition function, as the “would-be oscillatory, Gaussian, integral” for the gauge-fixed action functional. Notice that restricting to $\textsf{L}$ involves the Jacobian of the gauge fixing condition (see, for example, [30, Remark 31]). We will work with the following generalisation of the Gaussian-integral interpretation of the partition function:

Definition 2.12

Let $(\mathcal {F}_{BV}, \Omega _{BV}, S_{BV}, Q_{BV})$ be a BV theory with quadratic action functional $S_{BV}=\Omega _{BV}(\varphi ,Q_{BV}\varphi )$ and with gauge fixing operator $Q_{GF}$, so that $D=[Q_{GF},Q_{BV}]$ and denote ${{\,\textrm{im}\,}}(Q_{GF})=\textsf{L}$. The gauge-fixed partition function is¹²:

$$\begin{aligned} Z(S_{BV},\textsf{L}) = |\textrm{sdet}^\flat (Q_{GF}|_\textsf{K})|^{\frac{1}{2}} \cdot |\textrm{sdet}^\flat (D\vert _{\textsf{L}})|^{-\frac{1}{2}}. \end{aligned}$$

This approach generalises to BV actions that are not quadratic, for which the decomposition $S_{BV} = S_0 + I$ holds, with $S_0= \frac{1}{2}\Omega _{BV}(\varphi , Q \varphi )$ as above. The free partition function of the theory is

$$\begin{aligned} Z^{\text {free}}(S_{BV},\textsf{L}) \doteq Z(S_0,\textsf{L}). \end{aligned}$$

We will define the kernel of an operator using the symplectic pairing $\Omega _{BV}$. To this end, note that the pairing $\Omega _{BV}$ induces a map

$$\begin{aligned} 1\otimes \Omega _{BV}: \mathcal {F}_{BV}\otimes \mathcal {F}_{BV}\otimes \mathcal {F}_{BV} \rightarrow \mathcal {F}_{BV}. \end{aligned}$$

For $\mathcal {F}_{BV} = C^\infty (M,\mathbb {F})$ this can be extended to a map:

$$\begin{aligned} 1\otimes \Omega _{BV}: \mathcal {D}'(M\times M, \mathbb {F}\boxtimes \mathbb {F})\otimes C^\infty (M,\mathbb {F}) \rightarrow \mathcal {D}'(M,\mathbb {F}). \end{aligned}$$

Given an operator B on $\mathcal {F}_{BV}$, we can then define its kernel $K_B\in \mathcal {D}'(M\times M, \mathbb {F}\boxtimes \mathbb {F})$ by

$$\begin{aligned} K_B * \varphi \doteq (-1)^{|B|}(1\otimes \Omega _{BV})(K_B\otimes \varphi ) = B\varphi , \quad \forall \,\varphi \in \mathcal {F}_{BV}, \end{aligned}$$

(11)

where |B| is the degree of the operator with respect to the grading. Note that $K_B$ is akin to the Schwartz kernel of B, but instead of using integration with respect to a density on M to define the pairing with functions, we use the symplectic structure $\Omega _{BV}$. In this sense, it can be thought of as a generalisation of the Schwarz kernel for symmetric pairings to anti-symmetric bilinear forms. See [18] for further details. In the context of BF theory, introduced in Sect. 3.1, this definition of the kernel through the symplectic pairing should be compared to Remark B.2.

Definition 2.13

The heat kernel of $D=[Q_{GF},Q]$ is the distribution $K_t$, such that

$$\begin{aligned} K_t * \varphi = (1\otimes \Omega _{BV})(K_t\otimes \varphi ) = e^{-tD}\varphi . \end{aligned}$$

(12)

The propagator from scale $L_1$ to $L_2$ is

$$\begin{aligned} P_{L_1,L_2} \doteq \int _{L_1}^{L_2}(Q_{GF}\otimes 1)K_t\,\textrm{d}t, \end{aligned}$$

(13)

so that

$$\begin{aligned} P_{L_1,L_2} * \varphi = \int _{L_1}^{L_2} Q_{GF}\,e^{-tD}\varphi \,\textrm{d}t = \int _{L_1}^{L_2} e^{-tD}Q_{GF}\varphi \,\textrm{d}t. \end{aligned}$$

Remark 2.14

We can think of $P_{0,\infty }$, as the kernel of the inverse of Q restricted to the gauge fixing subspace, since for $\varphi \in {{\,\textrm{im}\,}}(Q_{GF})$ we have

$$\begin{aligned} P_{0,\infty } * Q\varphi = \int _0^\infty e^{-tD}Q_{GF}Q\varphi \,\textrm{d}t = \int _0^\infty e^{-tD}D\varphi \,\textrm{d}t = -\int _0^\infty \frac{\textrm{d}}{\textrm{d}t}\left( e^{-tD}\varphi \right) \,\textrm{d}t = \varphi , \end{aligned}$$

assuming that $\exp (-tD)\varphi$ vanishes as $t\rightarrow \infty$. $\diamondsuit$

Using this propagator, we can introduce effective interactions for the BV theory. As above, these are families of functionals $I[L] \in C^\infty (\mathcal {F}_{BV})$, satisfying the renormalisation group equation (7). (Note that the renormalisation group equation now depends on the choice of gauge fixing through the $Q_{GF}$ entering the propagator, as does the expectation value defined in (8).)

The family of functionals $\{I[L],L\in \mathbb {R}_+\}$ must satisfy certain consistency conditions in order for the perturbative field theory to make sense. Since these are not going to be relevant to this work, we refer to the definition of an effective BV theory in [18]. (A version of these consistency conditions adapted to the case at hand is given in [41].)

As mentioned in Remark 2.11, the effective field theory framework is often restricted to the case that $D=[Q_{GF},Q]$ is an elliptic operator, e.g. [18] requires D to be a generalised Laplacian. This ensures that the propagator $P_{L_1,L_2}$ of Definition 2.13 is a smooth function for all finite nonzero scales $0<L_1,L_2<\infty$. Thus, as explained in Sect. 2.2, the evaluation of the weight $\Phi _\gamma (iP_{L_1,L_2},\{iI_d\})(\varphi )$ of a Feynman diagram $\gamma$ involves the contraction of a distribution formed from the interaction terms $\{I_d\}$ with a smooth function formed from the propagator and the external fields. So the weight of any Feynman diagram is well defined for finite nonzero scales $L_1, L_2$ and it is only the UV and IR limits, $L_1\rightarrow 0$ and $L_2\rightarrow \infty$, that can pose problems.

In what follows, we will be interested in applying the effective field theory framework to a gauge fixing that gives rise to a non-elliptic operator D. This results in a distributional propagator $P_{L_1,L_2}$, and some care must be taken in applying the Feynman rules described in Sect. 2.1.

To evaluate a Feynman diagram with N half-edges, a distribution $u_I$ on $M^N$ given by a tensor power of interaction functionals must be contracted with a tensor product of propagators and external fields, which is now also a distribution $u_P$ on $M^N$. Such an extended notion of contraction is possible provided that the distributions satisfy a condition on their wavefront set.

Definition 2.15

Given a connected Feynman diagram $\gamma$ with N half-edges, let ${u_I \in \mathcal {D}'(M^N)}$, ${u_P \in \mathcal {D}'(M^N)}$ be the distributions constructed from the interactions, respectively from the propagator and external fields, according to the Feynman rules of Sect. 2.1. If the wavefront sets of $u_I$ and $u_P$ satisfy

$$\begin{aligned} \big ({{\,\textrm{WF}\,}}(u_P)+{{\,\textrm{WF}\,}}(u_I)\big ) \cap \{0_{M^N}\} = \emptyset , \end{aligned}$$

(14)

where $\{0_{M^N}\}$ is the graph of the zero section in $T^*(M^N)$ and the sum on the left is the fibrewise sum in the cotangent bundle $T^*(M^N)$, then we define the weight of the Feynman diagram to be

$$\begin{aligned} \Phi _\gamma (iP_{L_1,L_2},\{iI_d\})(\varphi ) = \left\langle u_I,\, u_P \right\rangle \doteq \left\langle u_I u_P,\, 1 \right\rangle \end{aligned}$$

(15)

Here, $u_I u_P$ is the product of distributions,¹³ which is well defined as a distribution on $M^N$ under the wavefront set condition (14), see [32, Theorem 8.2.10]. Note that this extended notion of contraction is the unique continuous extension of the contraction of a distribution with a smooth function, and can be viewed as a universal regularisation procedure. For the case at hand, $u_I$ and $u_P$ are given explicitly in Equation (44).

Remark 2.16

One could of course attempt to apply Definition 2.15 directly to the Schwartz kernel of $Q_{GF}D^{-1}$ as distributional propagator. We choose to introduce the regularisation procedure of Definition 2.15 within the context of effective field theory, in order to regularise the Feynman diagrams for the propagator $P_{L_1,L_2}$ at finite nonzero scales. In this way, the $L_1\rightarrow 0$ (UV) and $L_2\rightarrow \infty$ (IR) limits, which may still be problematic, can be dealt with separately by further regularisation and by renormalisation as needed. The advantage of this approach is that, whereas the wavefront set condition (14) must hold for each diagram individually for the weight to be well defined, the UV and IR limits are only enforced on the sum over all Feynman diagrams $\Gamma (P_{L_1,L_2},I)(\varphi )$ and cancellations may occur. $\diamondsuit$

3 Ruelle Zeta Function in Perturbative Field Theory

In this work, we will be interested in interpreting certain quantities defined in the world of dynamical systems as the results of perturbative quantisation of a topological field theory called twisted, Abelian, BF theory, formulated in the Batalin–Vilkovisky formalism.

The following topological setup will be used throughout. Let M be an n-dimensional compact, connected, orientable manifold, and let $\pi :E\rightarrow M$ be a rank-r, complex, vector bundle over M. We further assume that E is equipped with a smooth Hermitian inner product, which we denote by $\left\langle \,\cdot \,,\, \cdot \, \right\rangle _E$, and a flat connection $\nabla$ compatible with the inner product. Note that the flat connection induces a unitary representation of the fundamental group $\rho : \pi _1(M) \rightarrow U(\mathbb {C}^r)$. Denote by $\Omega ^k(M,E)$ the space of smooth differential k-forms on M with values in E. Since the connection is flat, the exterior covariant differential associated with $\nabla$,

$$\begin{aligned} d^\nabla :\Omega ^k(M,E) \rightarrow \Omega ^{k+1}(M,E), \end{aligned}$$

satisfies $d^\nabla \! \circ \, d^\nabla = 0$ and thus defines a cochain complex, called twisted de Rham complex, with cohomology groups $H^\bullet (M,E)$. We require this complex to be acyclic, i.e. $H^k(M,E) = 0$, for all $0\le k\le n$.

Definition 3.1

(Twisted topological data). We call $(M,E,\nabla ,\rho )$ the twisted topological data.¹⁴

3.1 Twisted, Abelian, BF Theory

Given the twisted topological data of Definition 3.1, we can define a classical BV theory in the sense of Definition 2.7. We view $\Omega ^\bullet (M,E)$ as a $\mathbb {Z}$-graded vector space, where k-forms are assigned degree $-k$. Denote by $\Omega ^\bullet (M,E)[j]$ the j-shift of this graded vector space, that is, the degree of a vector in $\Omega ^\bullet (M,E)[j]$ is shifted up by j compared to the degree of the corresponding vector in $\Omega ^\bullet (M,E)$. Thus, a k-form in $\Omega ^\bullet (M,E)[j]$ has degree $j-k$. The space of fields for (twisted) Abelian BF theory is defined as

$$\begin{aligned} \mathcal {F}_{BF} = \Omega ^\bullet (M,E)[1] \oplus \Omega ^\bullet (M,E)[n-2] \end{aligned}$$

(16)

Thus, a field $(\mathbb {A},\mathbb {B}) \in \mathcal {F}_{BF}$ consists of a pair of inhomogeneous differential forms

$$\begin{aligned} \mathbb {A}= \mathbb {A}^{(0)} + \cdots + \mathbb {A}^{(n)}, \quad \mathbb {B}= \mathbb {B}^{(0)} + \cdots + \mathbb {B}^{(n)}, \end{aligned}$$

where $\mathbb {A}^{(k)}$ is an E-valued k-form, of total degree $|\mathbb {A}^{(k)}| = 1-k$, and $\mathbb {B}^{(k)}$ is an E-valued k-form of total degree $|\mathbb {B}^{(k)}| = n-2-k$. The symplectic pairing $\Omega _{BF}: \mathcal {F}_{BF}\times \mathcal {F}_{BF} \rightarrow \mathbb {R}$, is given by

$$\begin{aligned} \Omega _{BF}((0,\mathbb {B}),(\mathbb {A},0)) = \int _M \langle \mathbb {B}\wedge \mathbb {A}\rangle _E, \end{aligned}$$

(17)

and extended by graded anti-symmetry and linearity to all of $\mathcal {F}_{BF}\times \mathcal {F}_{BF}$. Here, $\langle \mathbb {B}\wedge \mathbb {A}\rangle _E$ denotes taking the inner product in E and the exterior product in $\wedge ^\bullet T^*M$, and the integral is only of the top-form part of the resulting expression. Finally, the action functional for (twisted) Abelian BF theory is

$$\begin{aligned} S_{BF}\equiv S_{d^\nabla }(\mathbb {A},\mathbb {B}) \doteq \int _M \langle \mathbb {B}\wedge d^\nabla \mathbb {A}\rangle _E. \end{aligned}$$

(18)

Remark 3.2

Note that we can view $d^\nabla$ as acting on $\mathcal {F}_{BV}$:

$$\begin{aligned} d^\nabla (\mathbb {A},\mathbb {B}) = (d^\nabla \mathbb {A},d^\nabla \mathbb {B}), \quad (\mathbb {A},\mathbb {B}) \in \Omega ^\bullet (M,E)[1] \oplus \Omega ^\bullet (M,E)[n-2]. \end{aligned}$$

In this way, $d^\nabla$ becomes a degree 1 operator on $\mathcal {F}_{BF}$ and the action functional¹⁵ can be written as

$$\begin{aligned} S_{d^\nabla }(\mathbb {A},\mathbb {B}) = \frac{1}{2}\Omega _{BF}((\mathbb {A},\mathbb {B}), d^\nabla (\mathbb {A},\mathbb {B})). \end{aligned}$$

Thanks to the compatibility of $\nabla$ with the inner product on E, Stokes’ theorem shows that $d^\nabla$ is graded anti-symmetric with respect to the symplectic pairing. $\diamondsuit$

Remark 3.3

One can show that the Poisson bracket with respect to $\Omega _{BF}$ is well defined for local functionals and that

$$\begin{aligned} \{S_{d^\nabla },S_{d^\nabla }\}_{\Omega _{BF}} = \pm 2\int _M \langle d^\nabla \mathbb {B}\wedge d^\nabla \mathbb {A}\rangle _E = \pm 2\int _M \langle \mathbb {B}\wedge (d^\nabla )^2\mathbb {A}\rangle _E = 0, \end{aligned}$$

where ± denotes a sign depending on the degree of $\mathbb {A}$ and $\mathbb {B}$. Thus, $S_{d^\nabla }$ satisfies the classical master equation, see Definition 2.7, and $(\mathcal {F}_{BF},\Omega _{BF},S_{d^\nabla },d^\nabla )$ defines a classical BV theory. $\diamondsuit$

We conclude this section by considering a slightly more general setting, where we replace the operator $d^\nabla$ by some operator Q with similar properties as $d^\nabla$. Namely, consider a differential operator on the space of E-valued forms

$$\begin{aligned} Q: \Omega ^\bullet (M,E) \rightarrow \Omega ^{\bullet +1}(M,E), \end{aligned}$$

satisfying $Q^2 = 0$ and graded anti-symmetry with respect to the pairing $\Omega _{BF}$. We then consider the following theory, which can be thought of as a generalisation of Abelian BF theory, with the same space of fields and symplectic structure, but action functional

$$\begin{aligned} S_Q(\mathbb {A},\mathbb {B}) = \int _M \langle \mathbb {B}\wedge Q\mathbb {A}\rangle _E. \end{aligned}$$

This will be used in Sect. 3.4 to view the perturbed action functional as a quadratic theory for a perturbed differential. Note that the arguments of Remark 3.3 still apply, showing that this defines a classical BV theory.

Definition 3.4

(Generalised BF theory). We call generalised BF theory the classical BV theory specified by the data $(\mathcal {F}_{BF},\Omega _{BF},S_Q,Q)$.

3.2 Ruelle Zeta Function

Consider again the twisted topological data of Definition 3.1. Assume, additionally, that M is a contact manifold with contact form $\alpha$, and that it is equipped with a contact, Anosov vector field, i.e.

Definition 3.5

(Contact Anosov vector fields). The flow $\varphi _t:M\rightarrow M$ of a vector field $X\in C^\infty (M,TM)$ is Anosov if there exists a $d\varphi _t$-invariant, continuous, splitting of the tangent bundle:

$$\begin{aligned} T_xM = T_0(x) \oplus T_s(x) \oplus T_u(x), \quad \text {with }\, T_0(x) = \mathbb {R}X(x), \end{aligned}$$

i.e. such that $d\varphi _t(T_\bullet (x)) = T_\bullet (\varphi _t(x))$ with $\bullet \in \{0, s, u\}$, and such that there exist constants $C, \theta >0$ for which we have, $\forall t \ge 0$:

$$\begin{aligned} \forall v \in T_s(x):\, \Vert d\varphi _t(x)v\Vert \le Ce^{-\theta t}\Vert v\Vert , \quad \forall v \in T_u(x):\, \Vert d\varphi _{-t}(x)v\Vert \le Ce^{-\theta t}\Vert v\Vert . \end{aligned}$$

Here, $\Vert \cdot \Vert$ is the norm induced by some Riemannian metric on M.¹⁶ We call $T_s/T_u$ the stable/unstable bundles, and we assume that they are orientable. A vector field is Anosov iff its flow is. An Anosov vector field is contact, on a contact manifold M, if there exists a contact form $\alpha$ such that X is its Reeb vector field

Definition 3.6

(Twisted Ruelle zeta function). Let X be an Anosov vector field on M with flow $\varphi _t$. The Ruelle zeta function for the flow $\varphi$, twisted by a (unitary) representation $\rho$ of $\pi _1(M)$, is

$$\begin{aligned} \zeta _\rho (\lambda ) := \prod _{\gamma \in \mathcal {P}} \textrm{det}(I - \rho ([\gamma ])e^{-\lambda \ell (\gamma )}), \end{aligned}$$

(19)

where $[\gamma ]\in \pi _1(M)$ is the conjugacy class of a single-winding, closed, orbit¹⁷$\gamma \in \mathcal {P}$ of the flow $\varphi _t$ of the Anosov vector field X and $\lambda \in \mathbb {C}$ and $\ell (\gamma )$ is its length (or period).

Remark 3.7

The (twisted) Ruelle zeta function was shown to be analytic in a half-plane of large enough $\Re (\lambda )$, and admit a meromorphic continuation to $\mathbb {C}$ [12, 22, 23, 28, 34]. $\diamondsuit$

Let the dimension of the contact manifold be $\dim (M)=2m+1$. In the contact Anosov case the rank of the stable/unstable bundles satisfy $\textrm{rank}(T_s)=\textrm{rank}(T_u)=m$, see [25]. In order to express the Ruelle zeta function in terms of certain regularised determinants, we start from the observation [22, 30] that the (twisted) Ruelle zeta decomposes as

$$\begin{aligned} \log \zeta _\rho (\lambda ) = (-1)^m\sum _{k=0}^{2m} (-1)^k \log \zeta _{\rho ,k}(\lambda ). \end{aligned}$$

(20)

An important interpretation of this comes from the Guillemin trace formula, which allows us to provide an integral representation of $\zeta _{\rho ,k}(\lambda )$:

$$\begin{aligned} \log \zeta _{\rho , k}(\lambda ) = -\int _0^\infty t^{-1}e^{-t\lambda } \textrm{tr}^\flat e^{-t\mathcal {L}_{X,k}} \,\textrm{d}t. \end{aligned}$$

(21)

where $\mathcal {L}_{X,k} = \mathcal {L}_X\bigr |_{\Omega ^k(M,E)\cap \ker (\iota _X)}$ is the Lie derivative restricted to k-forms lying in the kernel of $\iota _X$, and the flat trace evaluates to [29]

$$\begin{aligned} \textrm{tr}^\flat e^{-t\mathcal {L}_{X,k}} = \sum _{\gamma \in \mathcal {P}} \sum _{j=1}^\infty \ell (\gamma ) \delta (t - j\ell (\gamma )) \frac{\textrm{tr}(\rho ([\gamma ])^j)\textrm{tr} (\wedge ^k P(\gamma )^j)}{| \textrm{det}^\flat (I - P(\gamma )^j) |}, \end{aligned}$$

(22)

as a distribution on $\mathbb {R}_+$¹⁸. Hence, we can conclude that

Theorem 3.8

([22, 30]). The Ruelle zeta function is the flat-superdeterminant of the (degree-preserving) operator $\mathcal {L}_X + \lambda$ restricted to the subspace $\ker (\iota _X) \subset \Omega ^\bullet (M,E)$:

$$\begin{aligned} \zeta _\rho (\lambda )^{(-1)^m} = \textrm{sdet}^\flat \bigl ((\mathcal {L}_{X} + \lambda )\bigr |_{\ker (\iota _X)}\bigr ) = \prod _{k=0}^{2m} {\textrm{det}^\flat } (\mathcal {L}_{X,k} + \lambda )^{(-1)^k}. \end{aligned}$$

(23)

Remark 3.9

(Anisotropic Sobolev spaces) An important observation in this context is the relation between the poles of the meromorphic extension¹⁹ of the resolvent, defined (for large $\Re (\lambda )$) as

$$\begin{aligned} R_X(\lambda )=(\mathcal {L}_X+\lambda )^{-1} = \int _0^\infty e^{-t(\mathcal {L}_X+\lambda )}\,\textrm{d}t, \end{aligned}$$

and the eigenvalues of the operator $\mathcal {L}_X$ acting on certain Hilbert spaces specially tailored to the dynamics. On such anisotropic Sobolev spaces $H_{sG}$, $\mathcal {L}_X$ can be shown to have discrete spectrum. The vector field X (or sometimes the Lie derivative operator $\mathcal {L}_X$) is said to have a Pollicott–Ruelle resonance at $\lambda$, if there are eigenvectors in some $H_{sG}$ for the eigenvalue $\lambda$. A resonance for the eigenvalue zero is then related to the resolvent having a pole at $\lambda =0$. When this is not the case, one can “invert” $\mathcal {L}_X$ with an important caveat: it is inverted as an operator from a domain that is dense in some anisotropic Sobolev space of sufficient regularity s, so that $\mathcal {L}_X:{\mathfrak {D}}(\mathcal {L}_X) \subset H_{sG} \rightarrow H_{sG}$. Henceforth we will forgo the explicit construction²⁰ of $H_{sG}$, and refer to [22, 24] for details. $\diamondsuit$

Remark 3.10

It can be shown that the flat determinants appearing in the alternating product (23) are entire functions of $\lambda \in \mathbb {C}$, whose zeros coincide with the Pollicott–Ruelle resonances in the respective form degree, see [22]. Thus, as befits a determinant, ${\textrm{det}^\flat } (\mathcal {L}_{X,k} + \lambda )$ vanishes precisely when $\mathcal {L}_{X,k} + \lambda$ is not invertible on some anisotropic Sobolev space of sufficient regularity. $\diamondsuit$

3.3 Ruelle Zeta and Field Theory

In this section, we apply the field theory framework, outlined in Sect. 2, to BF Theory (see Sect. 3.1). We will employ a particular gauge fixing, introduced in [30] and studied in our companion paper [38], which is available—given the twisted topological data of Definition 3.1—on contact manifolds that admit a contact Anosov flow. In this setting, the effective field theory philosophy has a nice interpretation in terms of said flow.

In order to obtain a well-defined nonzero value for the gauge-fixed free partition function, we impose the following additional condition on the Anosov flow:

Assumption A

The Anosov vector field X has no Pollicott–Ruelle resonance at 0, which is then in the resolvent set of the closed densely defined operator $\mathcal {L}_X: {\mathfrak {D}}(\mathcal {L}_X)\subset H_{sG} \rightarrow H_{sG}$, acting on some anisotropic Sobolev space $H_{sG}$, see [22, Section 3.1].

Definition 3.11

(Twisted, contact, Anosov data). Let $(M,E,\nabla ,\rho )$ be the twisted topological data of Definition 3.1, and let additionally M be a contact manifold of dimension $2m+1$ endowed with a contact Anosov vector field X satisfying Assumption A. We call the data $(M,E,\nabla ,\rho ,X)$ the twisted, contact, Anosov data.

Recall that in (twisted) Abelian, BF theory the operator $Q_{BV}\equiv Q$ of Definition 2.7 (cf. Remark 2.9) is simply given by:

$$\begin{aligned} Q(\mathbb {A},\mathbb {B}) = (d^\nabla \mathbb {A},d^\nabla \mathbb {B}), \quad (\mathbb {A},\mathbb {B}) \in \mathcal {F}_{BF} = \Omega ^\bullet (M,E)[1] \oplus \Omega ^\bullet (M,E)[n-2]. \end{aligned}$$

Proposition 3.12

([30, 38]). The operator

$$\begin{aligned} Q_{GF} \circlearrowright \mathcal {F}_{BF}, \qquad Q_{GF} (\mathbb {A},\mathbb {B}) = (\iota _X\mathbb {A},\iota _X\mathbb {B}), \end{aligned}$$

is a gauge fixing operator. In particular, defining²¹

$$\begin{aligned} Q_{GF}^* \circlearrowright \mathcal {F}_{BF}, \qquad Q_{GF}^* (\mathbb {A},\mathbb {B}) = (\alpha \wedge \mathbb {A},\alpha \wedge \mathbb {B}) \end{aligned}$$

we have the Lagrangian splitting:

$$\begin{aligned} \mathcal {F}_{BF}=\textrm{im}(Q_{GF}) \oplus \textrm{im}(Q_{GF}^*) = \textrm{ker}(\iota _X\oplus \iota _X) \oplus \textrm{im}(\alpha \wedge \oplus \alpha \wedge ), \end{aligned}$$

which is invariant under the action of the graded commutator:

$$\begin{aligned}D(\mathbb {A},\mathbb {B}) = [Q_{GF},Q](\mathbb {A},\mathbb {B}) = \mathcal {L}_X(\mathbb {A},\mathbb {B}).\end{aligned}$$

For the sake of simplicity, we will abuse notation and simply write: $\textsf{L}_X:= \ker (\iota _X) = {{\,\textrm{im}\,}}(\iota _X)$ to denote the associated gauge fixing Lagrangian, which will be referred to as “the axial, Anosov gauge”.

Remark 3.13

The heat kernel for D is the kernel of $\varphi _{-t}^*$, the pullback by the flow of X. Thus, we can think of the propagator $P_{L_1,L_2}$ as propagating particles along the flow of X, from time $L_1$ to time $L_2$. The scale-L effective interaction has an interpretation as the average interaction that particles feel as they flow along X for a time scale L. In other words, the interpretation of L as a length scale described in Sect. 2.2 has given way to an interpretation of L as the time scale for the flow by X. $\diamondsuit$

As was shown in [30], we can use field theory to interpret the (absolute value of the) Ruelle zeta function, in its determinant form (Theorem 3.8) in terms of the partition function of BF theory. Notice that in order to compare with Definition 2.12, one needs to recall that $\textrm{sdet}^\flat (D\vert _{\textsf{L}_X}) = \textrm{sdet}^\flat (\mathcal {L}_X\vert _{{{\,\textrm{im}\,}}(\iota _X)})^2$, and the grading on $\mathcal {F}_{BF}$ differs by 1 w.r.t. the natural grading on $\Omega ^\bullet (M,E)$ (see also [30, Remarks 11 and 12]).

Theorem 3.14

Let $(M,E,\nabla ,\rho ,X)$ be the twisted, contact, Anosov data. Up to a phase, the partition function of BF theory in the axial, Anosov gauge is the value of the Ruelle zeta function at zero:

$$\begin{aligned} Z(S_{d^\nabla },\textsf{L}_X) = |\zeta _\rho (0)|^{(-1)^m}. \end{aligned}$$

(24)

3.4 The Perturbing Functional

In this section, we will consider a perturbation of the free BF action by a functional $\digamma _X$, which enters at order $\hbar$:

$$\begin{aligned} S_{d^\nabla }(\mathbb {A},\mathbb {B}) = \int _M \langle \mathbb {B}\wedge d^\nabla \mathbb {A}\rangle _E \leadsto S_{Q_X} \doteq S_{d^\nabla } + \hbar \digamma _X. \end{aligned}$$

Remark 3.15

Assumption A implies that the inverse $\mathcal {L}_X^{-1}: H_{sG} \rightarrow H_{sG}$ is well defined on anisotropic Sobolev spaces. Note that both $\iota _X$ and $d^\nabla$ commute with $\mathcal {L}_X^{-1}$. $\diamondsuit$

Definition 3.16

Under Assumption A, the perturbing functional is:

$$\begin{aligned} \digamma _X(\mathbb {A},\mathbb {B}) = \int _M \langle \mathbb {B}\wedge \mathcal {L}_X^{-1}d^\nabla \mathbb {A}\rangle _E. \end{aligned}$$

(25)

This functional exhibits some peculiarities. The operator $W_X:= \mathcal {L}_{X}^{-1}d^\nabla$ appearing in the functional does not necessarily map smooth sections to smooth sections. Indeed, the condition $\mathcal {L}_X^{-1}(\omega )$ smooth for all $\omega \in \Omega ^\bullet (M,E)$ is equivalent to the Lie derivative $\mathcal {L}_X: \Omega ^\bullet (M,E)\rightarrow \Omega ^\bullet (M,E)$ being surjective, which is not necessarily the case. Thus, $\mathcal {L}_X^{-1}d^\nabla \mathbb {A}\in H_{sG}$ may only be a distributional section in the anisotropic Sobolev space (see Remark 3.9). Nevertheless, we can still integrate it against the smooth section $\mathbb {B}$, so the functional $\digamma _X$ is well defined on the space of fields. However, it is not a local functional in the sense of Sect. 2.

Remark 3.17

In some sense, we can view $W_X=\mathcal {L}_X^{-1}d^\nabla$ as a chain contraction for the chain complex defined by our gauge fixing operator $\iota _X$. Indeed,

$$\begin{aligned} \iota _X\mathcal {L}_X^{-1}d^\nabla + \mathcal {L}_X^{-1}d^\nabla \iota _X = \mathcal {L}_X^{-1}(\iota _Xd^\nabla + d^\nabla \iota _X) = \mathcal {L}_X^{-1}\mathcal {L}_X = \textrm{id}_{{\mathfrak {D}}(\mathcal {L}_X)} \end{aligned}$$

is the identity operator on the domain of $\mathcal {L}_X$ in $H_{sG}$. Since smooth sections are in ${\mathfrak {D}}(\mathcal {L}_X)$, this gives the identity on the chain complex $\Omega ^\bullet (M,E)$. However, as mentioned above, $W_X$ may not actually define a map into the chain complex of smooth sections, so $W_X$ defines a chain contraction only on appropriately defined distributional sections of the twisted cotangent bundle. $\diamondsuit$

Treating $\hbar$ as a formal parameter, and denoting $Q_X \doteq (1 + \hbar \mathcal {L}_X^{-1})d^\nabla = d^\nabla +\hbar W_X$, we define our perturbed action as

$$\begin{aligned} S_{Q_X}(\mathbb {A},\mathbb {B}) = S_{d^\nabla }(\mathbb {A},\mathbb {B}) + \hbar \digamma _X(\mathbb {A},\mathbb {B}) = \int _M \langle \mathbb {B}\wedge (1+\hbar \mathcal {L}_X^{-1})d^\nabla \mathbb {A}\rangle _E. \end{aligned}$$

(26)

Applying the Poisson bracket to $S_{Q_X}$, we find

$$\begin{aligned} \{S_{Q_X},S_{Q_X}\} = \pm 2\int _M \langle (1+\hbar \mathcal {L}_X^{-1})d^\nabla \mathbb {B}\wedge (1+\hbar \mathcal {L}_X^{-1})d^\nabla \mathbb {A}\rangle _E = 0, \end{aligned}$$

where we used the fact that $d^\nabla$ and $\mathcal {L}_X^{-1}$ commute, so that both forms in the wedge product are in the image of $d^\nabla$. Thus, the action functional S satisfies the classical master equation.

Proposition 3.18

The critical locus of $S_{Q_X}$ on the gauge fixing subspace of Proposition 3.12 is given by Pollicott–Ruelle resonant states with resonance $\hbar$.

Proof

The dynamical content of the perturbed classical action $S_{Q_X}$ lies in the Euler–Lagrange equations. For the field $\mathbb {A}$, the Euler–Lagrange equation gives

$$\begin{aligned} Q_X\mathbb {A}=(1+\hbar \mathcal {L}_X^{-1})d^\nabla \mathbb {A}= 0. \end{aligned}$$

Factoring out the $\mathcal {L}_X^{-1}$, this can equivalently be written as

$$\begin{aligned} \mathcal {L}_X^{-1}(\mathcal {L}_X+\hbar )d^\nabla \mathbb {A}= \mathcal {L}_X^{-1}d^\nabla (\mathcal {L}_X+\hbar )\mathbb {A}= 0. \end{aligned}$$

In particular, solutions to the equations of motion of the BV theory are obtained from

$$\begin{aligned} (\mathcal {L}_X+\hbar )\mathbb {A}= 0. \end{aligned}$$

In other words, Pollicott–Ruelle resonant states with resonance $\hbar$ provide non-trivial solutions. In fact, if we restrict $\mathbb {A}$ to satisfy our gauge fixing condition $\mathbb {A}\in {{\,\textrm{im}\,}}(\iota _X)$, then these are the only non-trivial solutions to the equations of motion. Indeed, if $d^\nabla (\mathcal {L}_X+\hbar )\mathbb {A}= 0$, then $(\mathcal {L}_X+\hbar )\mathbb {A}\in \ker (d^\nabla )$. Since $\mathcal {L}_X$ leaves the gauge fixing subspace ${{\,\textrm{im}\,}}(\iota _X)$ invariant, we also have $(\mathcal {L}_X+\hbar )\mathbb {A}\in {{\,\textrm{im}\,}}(\iota _X)$. But ${{\,\textrm{im}\,}}(\iota _X)\cap \ker (d^\nabla )=\{0\}$ by the injectivity of $\mathcal {L}_X$, so $(\mathcal {L}_X+\hbar )\mathbb {A}=0$. Thus, on the gauge fixing subspace the equations of motion only have non-trivial solutions if $\hbar$ is a Pollicott–Ruelle resonance and the solutions are given by the resonant states. The Euler–Lagrange equation for the field $\mathbb {B}$ can be analysed similarly. $\square$

Remark 3.19

(Dependence on X). Note that the Anosov vector field X enters the perturbed action functional $S_{Q_X}$ in (26) via the perturbing functional $\digamma _X$. In addition, we will use the vector field X to define our gauge fixing condition. This is akin to what happens in Yang–Mills theory, where the action functional depends on a Riemannian metric, and the same metric is used to define the Lorenz gauge. Somewhat unusual here is that the original action functional of BF theory depends solely on topological data, whereas the perturbation introduces a dependence on the additional datum X. As observed above, after gauge fixing the perturbed action has critical locus given by the Pollicott–Ruelle resonances with respect to X. Of course, the vector field entering the gauge fixing condition is a priori independent of the vector field entering $S_{Q_X}$. One can consider a family of action functionals $S_{Q_X}$ depending on a vector field X, to which we can apply a family of gauge fixing conditions defined by another vector field Y. The most sensible result for the partition function of the theory is obtained when we set $Y=X$. An analysis of the case when the two vector fields are allowed to vary independently of each other is given in [41]. $\diamondsuit$

The perturbation $\hbar \digamma _X$ is actually a quadratic functional in the fields. Hence, it is worthwhile to treat the full action $S_{Q_X} = S_{d^\nabla } + \hbar \digamma _X$ as a free action and compute the partition function for the gauge fixing operator $\iota _X$, using the formalism of infinite-dimensional Gaussian integrals employed in Definition 2.12.

Theorem 3.20

The partition function of perturbed BF theory $S_{Q_X} = S_{d^\nabla } + \hbar \digamma _X$, when computed in the axial, Anosov gauge $\textsf{L}_X = \ker (\iota _X)$ on a manifold equipped with the twisted, contact, Anosov data of Definition 3.11, is the twisted Ruelle zeta function up to a phase:

$$\begin{aligned} Z(S_{Q_X},\textsf{L}_X) = |\zeta _X(\hbar )|^{(-1)^m}. \end{aligned}$$

Proof

Being quadratic, we can apply the definition of partition function (Definition 2.12) to the functional S, where the operator Q is now $d^\nabla +\hbar \mathcal {L}_X^{-1}d^\nabla$. Thus, for the gauge fixing condition $\textsf{L}_X = \ker (\iota _X) = {{\,\textrm{im}\,}}(\iota _X)$, the evaluation of the partition function yields²²

$$\begin{aligned} Z(S,\textsf{L}_X){} & {} \doteq \big |\textrm{sdet}^\flat (\iota _X(d^\nabla +\hbar \mathcal {L}_X^{-1}d^\nabla )|_{{{\,\textrm{im}\,}}(\iota _X)})\big | = \big |\textrm{sdet}^\flat ((\mathcal {L}_X+\hbar )|_{{{\,\textrm{im}\,}}(\iota _X)})\big | \\{} & {} = |\zeta _X(\hbar )|^{(-1)^m}, \end{aligned}$$

where $\iota _Xd^\nabla \vert _{{{\,\textrm{im}\,}}(\iota _X)} = \mathcal {L}_X\vert _{{{\,\textrm{im}\,}}(\iota _X)}$ cancels with the $\mathcal {L}_X^{-1}$ in the second term. $\square$

To gain more insight on this result, we will treat the perturbation functional as an interaction term instead in what follows. This is justified by the fact that it enters at a higher order correction in $\hbar$ w.r.t. the free BF action $S_{d^\nabla }$. In the perturbative framework, we look at the expectation value of the functional $\exp (iF)$ for the gauge fixing $\iota _X$. Dividing the result above by the value of the free partition function, as in (3), this immediately gives

Corollary 3.21

$$\begin{aligned} \biggl \langle e^{i\digamma _X} \biggr \rangle _{\textsf{L}_X}\doteq & {} \frac{Z(S_{Q_X},\textsf{L}_X)}{Z^{\text {free}}(S_{Q_X},\textsf{L}_X)} = \frac{Z(S_{Q_X},\textsf{L}_X)}{Z(S_{d^\nabla },\textsf{L}_X)} \nonumber \\= & {} \frac{\big |\textrm{sdet}^\flat ((\mathcal {L}_X+\hbar )|_{{{\,\textrm{im}\,}}(\iota _X)})\big |}{\big |\textrm{sdet}^\flat (\mathcal {L}_X|_{{{\,\textrm{im}\,}}(\iota _X)})\big |} = \left| \frac{\zeta _X(\hbar )}{\zeta _X(0)}\right| ^{(-1)^m}. \end{aligned}$$

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In the next section, we will show that the same result can be obtained from perturbation theory, and therefore without the use of Definition 2.12 for the partition function, as a generalisation of infinite-dimensional Gaussian integrals.

3.5 Feynman Diagrams for the Axial, Anosov Gauge Fixing

In the rest of this section, we will treat the functional $\hbar \digamma _X$ as an interaction term for the free action $S_{d^\nabla }$, as in Sect. 2. This is worthwhile, since $D=\mathcal {L}_X$ is not elliptic, and thus, a non-trivial adaptation of the perturbative quantisation setup of [18] is required.

The operators $Q, Q_{GF}$ and D, introduced for BF theory in Sect. 3.3, can be represented as

$$\begin{aligned} Q = \begin{pmatrix} d^\nabla &{} 0 \\ 0 &{} d^\nabla \end{pmatrix}, \quad Q_{GF} = \begin{pmatrix} \iota _X &{} 0 \\ 0 &{} \iota _X \end{pmatrix}, \quad D = \begin{pmatrix} \mathcal {L}_X &{} 0 \\ 0 &{} \mathcal {L}_X \end{pmatrix}, \end{aligned}$$

with a copy of the operators acting on each component of the direct sum

$$\begin{aligned} \mathcal {F}_{BF} = \Omega ^\bullet (M,E)[1] \oplus \Omega ^\bullet (M,E)[n-2]. \end{aligned}$$

Remark 3.22

(A comparison of gauge fixing requirements) An operator $Q_{GF}$ is “gauge fixing” in the sense of [18] iff the space of fields splits as $\mathcal {F}_{BF} = {{\,\textrm{im}\,}}(Q)\oplus {{\,\textrm{im}\,}}(Q_{GF})$ (when $\ker (D)=\{0\}$). This definition is a priori different from Definition 2.10. Indeed, by assumption on the vector field X, the operator $D=\mathcal {L}_X$ has trivial kernel; by its injectivity, we have

$$\begin{aligned} {{\,\textrm{im}\,}}(d^\nabla )\cap {{\,\textrm{im}\,}}(\iota _X) \subset \ker (\mathcal {L}_X) = \{0\}. \end{aligned}$$

Furthermore, we can write a smooth section $\omega \in \Omega ^\bullet (M,E)$, as

$$\begin{aligned} \omega = \mathcal {L}_X\mathcal {L}_X^{-1}\omega = d^\nabla \iota _X\mathcal {L}_X^{-1}\omega + \iota _Xd^\nabla \mathcal {L}_X^{-1}\omega . \end{aligned}$$

In this sense, $\omega$ can be written as the sum of two sections

$$\begin{aligned} \omega _1 = d^\nabla \iota _X\mathcal {L}_X^{-1}\omega \in {{\,\textrm{im}\,}}(d^\nabla ), \quad \omega _2 = \iota _Xd^\nabla \mathcal {L}_X^{-1}\omega \in {{\,\textrm{im}\,}}(\iota _X). \end{aligned}$$

However, $\omega _1$ and $\omega _2$ are not necessarily smooth, belonging instead to anisotropic Sobolev spaces. So we cannot speak of an actual splitting of the space of smooth forms $\Omega ^\bullet (M,E)$ in this sense. Note, however, that $Q_{GF}$ is a gauge fixing operator in the sense of Definition 2.10, as shown in Proposition 3.12. This leads to problems in applying some of the results of [18]. A Hodge-type decomposition leading to a gauge fixing compatible with the above considerations is explored in [20, Section 4.2]. We will not give these issues any further consideration for now. $\diamondsuit$

The heat kernel $K_t^{tot}$ of D also consists of two components for each of the spaces making up $\mathcal {F}_{BF}$. Recall from Sect. 2.3 that the kernel is defined using the symplectic pairing $\Omega _{BF}$. For a section $\mathbb {A}\in \Omega ^\bullet (M,E)[1]$, we have

$$\begin{aligned} e^{-tD}\mathbb {A}(x) = e^{-t\mathcal {L}_X}\mathbb {A}(x) = (K_t^{tot}*\mathbb {A})(x) = \int _M K_t^{(1)}(x,y)\wedge \mathbb {A}(y), \end{aligned}$$

where the integral over the y-coordinate is actually the pairing of a distribution with a smooth function and we suppressed the inner product in the fibres E from our notation. Due to the definition of $*$ using $\Omega _{BF}$, the y part of the kernel $K_t^{(1)}(x,y)$ is actually over the space $\Omega ^\bullet (M,E)[n-2]$. Similarly, for $\mathbb {B}\in \Omega ^\bullet (M,E)[n-2]$ we have

$$\begin{aligned} e^{-tD}\mathbb {B}(x) = e^{-t\mathcal {L}_X}\mathbb {B}(x) = (K_t^{tot}*\mathbb {B})(x) = \int _M K_t^{(2)}(x,y)\wedge \mathbb {B}(y), \end{aligned}$$

Thus, the kernels $K_t^{(1)}$ and $K_t^{(2)}$ are identical as distributions, being equal to the kernel of the operator $\varphi _{-t}^* = \exp (-t\mathcal {L}_X)$, which we just write as $K_t$. But, with respect to the grading, the part of $K_t^{(1)}$ acting on k-forms must be viewed as an element of

$$\begin{aligned} \mathcal {D}'(M,\wedge ^kT^*M\otimes E)[1]\otimes \mathcal {D}'(M,\wedge ^{n-k}T^*M\otimes E)[n-2], \end{aligned}$$

and similarly for $K_t^{(2)}$. The propagator of the theory is then obtained as

$$\begin{aligned} P^{tot}_{L_1,L_2} = \int _{L_1}^{L_2} (Q_{GF}\otimes 1)K^{tot}_t\,\textrm{d}t \end{aligned}$$

and thus consists of two copies of

$$\begin{aligned} P_{L_1,L_2} = \int _{L_1}^{L_2} (\iota _X\otimes 1)K_t\,\textrm{d}t. \end{aligned}$$

(28)

The two copies of this propagator, one connecting the field $\mathbb {A}$ to $\mathbb {B}$ and the other $\mathbb {B}$ to $\mathbb {A}$, will lead to a factor of two in all the Feynman diagrams we compute below.

The Lie derivative is not an elliptic operator and its heat kernel fails to be smooth. Thus, the Feynman diagrams for the axial, Anosov gauge fixing must be evaluated using the extended notion of contraction in Definition 2.15. Our quadratic interaction functional $\digamma _X$ leads to particularly simple Feynman diagrams. At each order of $\hbar$, there are two Feynman diagrams, which are described in Sect. 3.6. In Appendix A, we show that theses diagrams are indeed well defined according to Definition 2.15.

Proposition 3.23

Let $P_{L_1,L_2}$ be the propagator for the axial, Anosov gauge fixing and $\hbar \digamma _X$ the interaction introduced in (25). Then for any $0\le L_1< L_2 < \infty$ and any connected Feynman graph $\gamma$ of the theory, we have $({{\,\textrm{WF}\,}}(u_P)+{{\,\textrm{WF}\,}}(u_I)) \cap \{0\} = \emptyset ,$ and thus, the weight

$$\begin{aligned} \Phi _\gamma (iP_{L_1,L_2},i\hbar \digamma _X)(\mathbb {A},\mathbb {B}) \end{aligned}$$

is well defined in the sense of Definition 2.15.

In the next section, we will compute the scale-L effective interaction corresponding to $\digamma _X$:

$$\begin{aligned} i\hbar \digamma _X[L](\mathbb {A},\mathbb {B}) = \Gamma (P_{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}). \end{aligned}$$

Note that by Proposition 3.23 we can evaluate the Feynman diagrams directly for the propagator $P_{0,L}$. Following the philosophy outlined in Remark 2.16, one could also first compute the diagrams for the propagator $P_{\varepsilon ,L}$ and then take the limit $\varepsilon \rightarrow 0$. This leads to the same result, i.e.

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Gamma (P_{\varepsilon ,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}) = \Gamma (P_{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}). \end{aligned}$$

Thus, in contrast to the case where D is a generalised Laplacian, our propagator $P_{\varepsilon ,L}$ behaves nicely as $\varepsilon \rightarrow 0$, and we do not need to introduce counterterms, which could otherwise be necessary to compensate divergences at small length scales in diagrams that contain a loop. This can also be understood from the fact that, as we will see, all loop diagrams in our example involve the flat trace $\textrm{tr}^\flat (\exp (-t\mathcal {L}_X))$, which vanishes for t small enough. This is due to the fact that there exists a shortest non-trivial closed geodesic.

In the IR limit, however, our propagator is badly behaved. This is because the Lie Derivative operator $\mathcal {L}_X$ does not have strictly positive spectrum. Thus, we introduce a regularisation procedure, in order to regularise the propagators $P_{L_1,L_2}$ as $L_2$ goes to $\infty$. We do this by introducing a parameter $\lambda \in \mathbb {C}$ and defining

$$\begin{aligned} P_{L_1,L_2}^\lambda = \int _{L_1}^{L_2} e^{-\lambda t}(\iota _X\otimes 1)K_t\,\textrm{d}t. \end{aligned}$$

To compute Feynman diagrams we first insert $P_{L_1,L_2}^\lambda$ for $\Re (\lambda ) \gg 0$ as our propagator and then take the limit $\lambda \rightarrow 0$, applying analytic continuation in $\lambda$ as necessary. For any finite $L_2$, the limit²³$\lambda \rightarrow 0$ just returns our original propagator $P_{L_1,L_2}$. It is only when setting $L_2 =\infty$ that analytic continuation becomes necessary.

Remark 3.24

Note that the renormalisation group equation (7) should be slightly modified in the presence of this regularisation procedure. Denoting

$$\begin{aligned} i\hbar \digamma _X^\lambda [L](\mathbb {A},\mathbb {B}) = \Gamma (P^\lambda _{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}), \end{aligned}$$

i.e. the scale-L effective interaction in the presence of the regularising parameter, the renormalisation group equation should now read

$$\begin{aligned} i\hbar \digamma _X[L_2](\mathbb {A},\mathbb {B}) = \lim _{\lambda \rightarrow 0}\Gamma (P^\lambda _{L_1,L_2},\hbar \digamma _X^\lambda [L_1])(\mathbb {A},\mathbb {B}). \end{aligned}$$

Once again, this distinction is only relevant when setting $L_2=\infty$. $\diamondsuit$

3.6 Computation of the Scale-L Effective Interaction

We now compute the family of effective interactions arising from our functional $\digamma _X$. That is, we view $\hbar \digamma _X$ as the scale-0 interaction and use the perturbative expansion to compute the scale-L interaction $\hbar \digamma _X[L]$ as in (9). Let us stress, once again, that we are treating a quadratic functional as an interaction because it is inserted at order 1 in $\hbar$.

As noted above, the propagator behaves well at small scales, and so the effective interaction at scale-L can be obtained directly from the Feynman diagram expansion as

$$\begin{aligned} i\hbar \digamma _X[L](\mathbb {A},\mathbb {B}) =\lim _{\lambda \rightarrow 0} \Gamma (P^\lambda _{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}) = \Gamma (P_{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}), \end{aligned}$$

without the need for renormalisation (i.e. the addition of counterterms). Here, we take $L<\infty$, so that we can ignore the regularisation procedure $\lambda \rightarrow 0$. To this end, we rewrite the interaction functional as

$$\begin{aligned} \digamma _X(\mathbb {A},\mathbb {B}) = \int _M \langle \mathbb {B}\wedge W_X\mathbb {A}\rangle _E = \int _M \langle W_X^*\mathbb {B}\wedge \mathbb {A}\rangle _E , \end{aligned}$$

where $W_X=\mathcal {L}_X^{-1}d^\nabla$ and $W_X^*$ is the adjoint of $W_X$ with respect to the symplectic pairing, which is just equal to $d^\nabla \mathcal {L}_X^{-1}$ up to a sign depending on the form degree.

The perturbative expansion

$$\begin{aligned} \Gamma (P_{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}) = \sum _{\gamma \, \textrm{connected}} \frac{\hbar ^{l(\gamma )}}{|\textrm{Aut}(\gamma )|}\Phi _\gamma (iP_{0,L},i\hbar \digamma _X)(\mathbb {A},\mathbb {B}) \end{aligned}$$

is a formal power series in the parameter $\hbar$. In terms of Feynman diagrams, there is a factor of $\hbar$ attached to each vertex due to the interaction $i\hbar \digamma _X$, and also a factor of $\hbar$ for each loop in the diagram. Note that since our interaction is quadratic, we must attach exactly two propagators to each vertex. Thus, the possible connected Feynman diagrams are as follows.

At order $\hbar$, we just get back our original interaction $i\hbar \digamma _X$. At order $\hbar ^{N}$ for $N > 1$, there are two Feynman diagrams to consider. The first connects the external fields $\mathbb {A}$ and $\mathbb {B}$ by a chain of $N-1$ propagators, the diagram below on the left. The second is independent of the external fields and features $N-1$ propagators arranged in a loop, the diagram below on the right.

We will call the series of diagrams depending on the external fields the interaction term, $\Gamma _{int}(P_L,\hbar \digamma _X)$, and the series independent of the external fields the trace term, $\Gamma _{tr}(P_L,\hbar \digamma _X)$. In the following, we suppress the wedge product and the inner product over E from our notation, since the expressions are already somewhat unwieldy as is. Evaluating the Feynman diagrams in the interaction term gives

$$\begin{aligned} \begin{aligned} \Gamma _{int}(P_{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B})\,\,&=\sum _{N=1}^\infty \hbar ^N(-1)^{N-1}i \\&\int _{M_1}\cdots \int _{M_N} W_X^*\mathbb {B}(x_1)(1\otimes W_X^*)P_{0,L}(x_1,x_2)\\&\cdots (1\otimes W_X^*)P_{0,L}(x_{N-1},x_N)\mathbb {A}(x_N) \\&= \sum _{N=1}^\infty \hbar ^N(-1)^{N-1}i\\&\quad \int _0^L\,\textrm{d}t_1\cdots \int _0^L\,\textrm{d}t_{N-1} \int _{M_1}\cdots \int _{M_N} \\&\quad W_X^*\mathbb {B}(x_1)(\iota _X\otimes W_X^*)K_{t_1}(x_1,x_2)\\&\quad \cdots (\iota _X\otimes W_X^*)K_{t_{N-1}}(x_{N-1},x_N)\mathbb {A}(x_N). \end{aligned}\nonumber \\ \end{aligned}$$

(29)

The Feynman diagram above on the left has N vertices and $N-1$ propagators, so the Feynman rules lead to a factor of $i^N\cdot i^{N-1} = (-1)^{N-1}i$. Also, the diagram has a symmetry factor of $\frac{1}{2}$, but this cancels with the factor of 2 coming from the two copies of $P_{0,L}$ in the full propagator. We have decorated the integrals over the manifold M with subscripts to denote which variable is being integrated. These integrals should actually be understood as distributional pairings. Note that $(\iota _X\otimes W_X^*)K_t$ is precisely the kernel of the operator $\iota _X e^{-t\mathcal {L}_X} W_X$, since applying the adjoint $W_X^*$ to the second variable of $K_t$ corresponds to precomposing with the operator $W_X$. Thus, the integrals over the manifold in the above expression are just application of this operator, e.g.

$$\begin{aligned} \int _{M_N} (\iota _X\otimes W_X^*)K_{t_{N-1}}(x_{N-1},x_N)\mathbb {A}(x_N) = \big (\iota _X e^{-t_{N-1}\mathcal {L}_X} W_X\mathbb {A}\big )(x_{N-1}). \end{aligned}$$

Applying this observation, rewriting $W_X=\mathcal {L}_X^{-1}d^\nabla$, and using the fact that $\iota _X$ and $\mathcal {L}_X$ commute, we get

$$\begin{aligned} \Gamma _{int}(P_{0,L},\hbar \digamma _X){} & {} = \sum _{N=1}^\infty \hbar ^N(-1)^{N-1}i \int _0^L\,\textrm{d}t_1\cdots \int _0^L\,\textrm{d}t_{N-1} \int _M (\mathcal {L}_X^{-1}d^\nabla )^*\mathbb {B}\, \nonumber \\{} & {} \quad \wedge \big ( e^{-t_1\mathcal {L}_X} \mathcal {L}_X^{-1}\iota _Xd^\nabla \cdots e^{-t_{N-1}\mathcal {L}_X} \mathcal {L}_X^{-1}\iota _Xd^\nabla \mathbb {A}\big ). \end{aligned}$$

(30)

We proceed similarly for the diagrams depicted on the right above, the trace term, and find

$$\begin{aligned} \Gamma _{tr}(P_{0,L},\hbar \digamma _X)(\mathbb {A},\mathbb {B}){} & {} = \sum _{N=1}^\infty \hbar ^{N+1}\frac{(-1)^N}{N} \int _{M_1}\cdots \int _{M_N} \nonumber \\{} & {} \quad \sum _{k=0}^n (-1)^{k+1} (1\otimes W_X^*)P_{0,L}^{(k)}(x_1,x_2)\nonumber \\{} & {} \quad \cdots (1\otimes W_X^*)P_{0,L}^{(k)}(x_N,x_1) \nonumber \\{} & {} = \sum _{N=1}^\infty \hbar ^{N+1}\frac{(-1)^N}{N} \int _0^L\,\textrm{d}t_1\cdots \int _0^L\,\textrm{d}t_N \nonumber \\{} & {} \quad \sum _{k=0}^n (-1)^{k+1} \int _{M_1}\cdots \int _{M_N} (\iota _X\otimes W_X^*)K_{t_1}^{(k)}(x_1,x_2)\nonumber \\{} & {} \quad \cdots (\iota _X\otimes W_X^*)K_{t_n}^{(k)}(x_N,x_1). \end{aligned}$$

(31)

The diagram at order $\hbar ^{N+1}$ has N vertices and N propagators, with the vertices contributing a factor of $\hbar ^N$ and an extra factor of $\hbar$ coming from the loop. The vertices and propagators each contribute a factor of $i^N$ giving rise to the sign $(-1)^N$. The symmetry of the diagram is that of an N-gon with a symmetry factor of $\frac{1}{2N}$, but again the $\frac{1}{2}$ gets cancelled by the two copies of the propagator. We denoted by $P_{0,L}^{(k)}$ the part of the propagator acting on k-forms. The closed loop leads to a minus sign in odd degrees with respect to the grading of the fields. Recall that this grading is shifted with respect to the form degree, thus the sign of $(-1)^{k+1}$. The integrals should again be interpreted in terms of distributional pairing. Noting once more the presence of the kernel for $\iota _X e^{-t\mathcal {L}_X} W_X= e^{-t\mathcal {L}_X}\iota _Xd^\nabla \mathcal {L}_X^{-1}$, we recognise in the last line of equation (31) the flat trace of an operator, namely

$$\begin{aligned} \Gamma _{tr}(P_{0,L},\hbar \digamma _X){} & {} =\sum _{N=1}^\infty \hbar ^{N+1}\frac{(-1)^N}{N} \int _0^L\,\textrm{d}t_1\cdots \int _0^L\,\textrm{d}t_N\, \nonumber \\{} & {} \quad \sum _{k=0}^n (-1)^{k+1}\, \textrm{tr}^\flat \big (e^{-t_1\mathcal {L}_X} \mathcal {L}_X^{-1}\iota _Xd^\nabla \cdots e^{-t_N\mathcal {L}_X} \mathcal {L}_X^{-1}\iota _Xd^\nabla \big |_{\Omega ^k(M,E)}\big )\nonumber \\ \end{aligned}$$

(32)

Remark 3.25

In Appendix A, we check the wavefront set condition, which is necessary to apply the Feynman rules with distributional propagators. For the Feynman diagrams arising from the interaction term, this condition is in fact equivalent to the requirement that the composition of the operators in Equation (30) be well defined. For the Feynman diagrams of the trace term, the wavefront set condition in addition ensures that the flat trace in (32) is well defined. $\diamondsuit$

3.7 Computation of the Expectation Value

The expectation value of the functional $\exp (i\digamma _{X})$ or, from a different perspective, the partition function of the interacting theory with the free partition function set to 1, is computed in the perturbative framework as

$$\begin{aligned} \biggl \langle e^{i\digamma _X} \biggr \rangle _{\textsf{L}_X} = e^{\frac{1}{\hbar }\Gamma (P_{0,\infty }, \hbar \digamma _X)(0,0)}, \end{aligned}$$

where $\textsf{L}={{\,\textrm{im}\,}}(\iota _X)$ is the image of the gauge fixing operator entering the propagator, and we omitted the limit $\lambda \rightarrow 0$ and the regularisation $P_{0,\infty }^\lambda$, which is necessary here. The exponent on the right is the scale infinity interaction with the external fields set to 0. As we saw in the previous section, the effective interaction consists of two terms

$$\begin{aligned} \Gamma (P^\lambda _{0,\infty }, \hbar \digamma _X)(\mathbb {A},\mathbb {B}) = \Gamma _{int}(P^\lambda _{0,\infty }, \hbar \digamma _X)(\mathbb {A},\mathbb {B}) + \Gamma _{tr}(P^\lambda _{0,\infty }, \hbar \digamma _X). \end{aligned}$$

(33)

The trace term, $\Gamma _{tr}$, is a constant functional independent of the external fields $\mathbb {A}$ and $\mathbb {B}$. On the other hand, the external fields enter the interaction term, $\Gamma _{int}$, as in (30). Thus, when we set $\mathbb {A}=\mathbb {B}=0$ in (33), the interaction term vanishes. We are left with the trace term, where the scale L is taken to infinity. The regularising parameter $\lambda \in \mathbb {C}$, discussed in Sect. 3.5, involves a factor of $e^{-t_i\lambda }$ for each of the propagators (cf. Equation (32)). With this regularisation procedure, we have

$$\begin{aligned} \Gamma (P_{0,\infty }, \hbar \digamma _X)(0,0)= & {} \Gamma _{tr}(P_{0,\infty },\hbar \digamma _X)\nonumber \\= & {} \lim _{\lambda \rightarrow 0}\, \sum _{N=1}^\infty \hbar ^{N+1}\frac{(-1)^N}{N} \nonumber \\{} & {} \quad \int _0^\infty \,\textrm{d}t_1\,e^{-t_1\lambda }\cdots \int _0^\infty \,\textrm{d}t_N\,e^{-t_N\lambda } \nonumber \\{} & {} \quad \sum _{k=0}^n (-1)^{k+1}\, \textrm{tr}^\flat \big (e^{-t_1\mathcal {L}_X} (\mathcal {L}_X^{-1}\iota _Xd^\nabla ) \nonumber \\{} & {} \quad \cdots e^{-t_N\mathcal {L}_X} (\mathcal {L}_X^{-1}\iota _Xd^\nabla )\big |_{\Omega ^k(M,E)}\big ) \end{aligned}$$

(34)

Note the presence of the operator $\mathcal {L}_X^{-1}\iota _Xd^\nabla$ in the above expression. Since $\iota _Xd^\nabla = \mathcal {L}_X$ on ${{\,\textrm{im}\,}}(\iota _X)$, this operator acts as the identity on ${{\,\textrm{im}\,}}(\iota _X)$. In contrast, it annihilates the subspace ${{\,\textrm{im}\,}}(d^\nabla )$. Thus, $\mathcal {L}_X^{-1}\iota _Xd^\nabla$ acts like a projection onto the subspace ${{\,\textrm{im}\,}}(\iota _X)$. The operator $\mathcal {L}_X^{-1}\iota _Xd^\nabla$ is, however, not a true projection operator, since it fails to map smooth functions to smooth functions, see the discussion in Sect. 3.4. However, inside the flat trace it plays the role of a projection operator, as the next lemma shows.

Lemma 3.26

Let $B: \Omega ^k(M,E) \rightarrow \mathcal {D}'(M,\wedge ^k T^*M \otimes E)$ be a continuous operator leaving the subspace ${{\,\textrm{im}\,}}(\iota _X)\subset \Omega ^k(M,E)$ invariant. Assume that the flat traces $\textrm{tr}^\flat (B\mathcal {L}_X^{-1}\iota _Xd^\nabla )$ and $\textrm{tr}^\flat (\mathcal {L}_X^{-1}\iota _Xd^\nabla B)$ are well defined. Then,

$$\begin{aligned} \textrm{tr}^\flat (B\mathcal {L}_X^{-1}\iota _Xd^\nabla ) = \textrm{tr}^\flat (B|_{{{\,\textrm{im}\,}}(\iota _X)}). \end{aligned}$$

Proof

We have a fibrewise splitting of the vector bundle $\wedge ^kT^*M$ of the form

$$\begin{aligned} \wedge ^kT^*M = \wedge ^kT_0^*M \oplus \alpha \wedge (\wedge ^{k-1}T_0^*M), \end{aligned}$$

where $T_0^*M$ are the cotangent vectors annihilating X. This leads to a splitting of the smooth E-valued k-forms into

$$\begin{aligned} \Omega ^k(M,E) = \Omega _0^k(M,E)\oplus \alpha \wedge \Omega _0^{k-1}(M,E). \end{aligned}$$

The projection onto $\Omega _0^k(M,E) = \Omega ^k(M,E)\cap {{\,\textrm{im}\,}}(\iota _X)$ with respect to this splitting can be written as

$$\begin{aligned} \prod \nolimits _{{{\,\textrm{im}\,}}(\iota _X)} = \iota _X \circ \alpha \wedge . \end{aligned}$$

Thus,

$$\begin{aligned} \textrm{tr}^\flat (B|_{{{\,\textrm{im}\,}}(\iota _X)}) = \textrm{tr}^\flat (B\circ \iota _X\circ \alpha \wedge ). \end{aligned}$$

Since the image of $\mathcal {L}_X^{-1}\iota _Xd^\nabla$ is contained in ${{\,\textrm{im}\,}}(\iota _X)$, we have

$$\begin{aligned} B\mathcal {L}_X^{-1}\iota _Xd^\nabla = B(\iota _X\circ \alpha \wedge )\mathcal {L}_X^{-1}\iota _Xd^\nabla . \end{aligned}$$

Using the cyclicity of the flat trace, Proposition B.4, this shows that

$$\begin{aligned} \textrm{tr}^\flat (B\mathcal {L}_X^{-1}\iota _Xd^\nabla ) = \textrm{tr}^\flat (\mathcal {L}_X^{-1}\iota _Xd^\nabla B (\iota _X\circ \alpha \wedge )). \end{aligned}$$

Finally, since $\mathcal {L}_X^{-1}\iota _Xd^\nabla$ acts as the identity on ${{\,\textrm{im}\,}}(\iota _X)$, and B leaves $\textrm{im}(\iota _X)$ invariant, we can drop this operator from the flat trace above to get

$$\begin{aligned} \textrm{tr}^\flat (B\mathcal {L}_X^{-1}\iota _Xd^\nabla ) = \textrm{tr}^\flat (B (\iota _X\circ \alpha \wedge )) = \textrm{tr}^\flat (B|_{{{\,\textrm{im}\,}}(\iota _X)}) \end{aligned}$$

$\square$

Using Lemma 3.26, we see that the first occurrence of $\mathcal {L}_X^{-1}\iota _Xd^\nabla$ in (34) projects down to ${{\,\textrm{im}\,}}(\iota _X)$, and since this subspace is left invariant by $\exp (-t\mathcal {L}_X)$, the further occurrences just act as the identity and can be ignored. Taking this into account, equation (34) becomes

$$\begin{aligned}{} & {} \Gamma (P_{0,\infty }, \hbar \digamma _X)(0) = \lim _{\lambda \rightarrow 0}\, \sum _{N=1}^\infty \hbar ^{N+1}\frac{(-1)^N}{N}\sum _{k=0}^n (-1)^{k+1}\int _0^\infty \,\textrm{d}t_1\nonumber \\{} & {} \quad \cdots \int _0^\infty \,\textrm{d}t_N\, e^{-(t_1+\cdots +t_N)\lambda }\, \textrm{tr}^\flat \big (e^{-(t_1+\cdots +t_N)\mathcal {L}_X}\big |_{\Omega _0^k(M,E)}\big ). \end{aligned}$$

(35)

Changing variables in the integral over $t_N$ to $t = t_1+\cdots +t_N$ we find

$$\begin{aligned}{} & {} \int _0^\infty \,\textrm{d}t_1\cdots \int _0^\infty \,\textrm{d}t_N\,e^{-(t_1+\cdots t_N)\lambda }\, \textrm{tr}^\flat \big (e^{-(t_1+\cdots +t_N)\mathcal {L}_X}\big |_{\Omega _0^k(M,E)}\big )\quad \nonumber \\{} & {} \quad = \int _0^\infty \,\textrm{d}t\,\int _0^t\,\textrm{d}t_1\int _0^{t-t_1}\,\textrm{d}t_2 \cdots \int _0^{t-t_1-\cdots -t_{N-2}}\,\textrm{d}t_{N-1}\,e^{-t\lambda }\,\textrm{tr}^\flat \big (e^{-t\mathcal {L}_X}\big |_{\Omega _0^k(M,E)}\big )\nonumber \\ \end{aligned}$$

(36)

Evaluating the integrals over $t_1,\dots ,t_{N-1}$, we get a factor of $t^{N-1}/(N-1)!$. So equation (36) becomes

$$\begin{aligned} \begin{aligned}&\int _0^\infty \,\textrm{d}t\,\frac{t^{N-1}}{(N-1)!}\, e^{-t\lambda }\,\textrm{tr}^\flat \big (e^{-t\mathcal {L}_X}\big |_{\Omega _0^k(M,E)}\big ) \\&\qquad = \frac{(-1)^{N-1}}{(N-1)!}\,\frac{\textrm{d}^{N-1}}{\textrm{d}\lambda ^{N-1}}\int _0^\infty \,\textrm{d}t\, e^{-t\lambda }\,\textrm{tr}^\flat \big (e^{-t\mathcal {L}_X}\big |_{\Omega _0^k(M,E)}\big ). \end{aligned} \end{aligned}$$

For $\Re (\lambda ) \gg 0$, the integral featured above converges to

$$\begin{aligned} \int _0^\infty \,\textrm{d}t\, e^{-t\lambda }\,\textrm{tr}^\flat \big (e^{-t\mathcal {L}_X}\big |_{\Omega _0^k(M,E)}\big ) = \frac{\textrm{d}}{\textrm{d}\lambda }\log \textrm{det}^\flat \big ((\mathcal {L}_X+\lambda )|_{\Omega _0^k(M,E)}\big ), \end{aligned}$$

so for $\lambda$ in this domain of the complex plane (36) becomes

$$\begin{aligned} \frac{(-1)^{N-1}}{(N-1)!}\,\frac{\textrm{d}^N}{\textrm{d}\lambda ^N}\log \textrm{det}^\flat \big ((\mathcal {L}_X+\lambda )|_{\Omega _0^k(M,E)}\big ). \end{aligned}$$

Since the flat determinant is a well-defined analytic function for all $\lambda \in \mathbb {C}$, see Remark 3.10, this expression provides an analytic continuation of the integral in (36). Inserting this expression into (35), and noting that the factors of $(-1)^N$ and $(-1)^{N-1}$ combine to an overall minus sign, we find

$$\begin{aligned} \begin{aligned} \Gamma (P_{0,\infty }, \hbar \digamma _X)(0)&= \lim _{\lambda \rightarrow 0}\,\sum _{k=0}^n (-1)^k\, \\&\sum _{N=1}^\infty \,\frac{\hbar ^{N+1}}{N!} \frac{\textrm{d}^N}{\textrm{d}\lambda ^N}\log \textrm{det}^\flat \big (\mathcal {L}_X+\lambda )|_{\Omega _0^k(M,E)}\big ) \\&= \hbar \sum _{k=0}^n (-1)^k\lim _{\lambda \rightarrow 0} \big (\log \textrm{det}^\flat \big ((\mathcal {L}_X+\lambda +\hbar )|_{\Omega _0^k(M,E)}\big )\\&-\log \textrm{det}^\flat \big ((\mathcal {L}_X+\lambda )|_{\Omega _0^k(M,E)}\big )\big ), \end{aligned} \end{aligned}$$

where we recognised the Taylor expansion of $\log \textrm{det}^\flat \big ((\mathcal {L}_X+\lambda +\hbar )|_{\Omega _0^k(M,E)}\big )$ with respect to $\hbar$ up to the missing zeroth order term, which we then added and subtracted again. Since this flat determinant is an entire function on the complex plane, the Taylor expansion in $\hbar$ converges in a neighbourhood of $\lambda$ for each $\lambda \in \mathbb {C}$. Taking $\lambda$ to zero and exponentiating, we finally find

$$\begin{aligned} \biggl \langle e^{i\digamma _X} \biggr \rangle _{\textsf{L}_X} = e^{\frac{1}{\hbar }\Gamma (P_{0,\infty }, \hbar \digamma _X)(0)} = \prod _{k=0}^n\biggl (\frac{\textrm{det}^\flat \big ((\mathcal {L}_X+\hbar )|_{\Omega _0^k(M,E)}\big )}{\textrm{det}^\flat \big (\mathcal {L}_X|_{\Omega _0^k(M,E)}\big )}\biggr )^{(-1)^k} \end{aligned}$$

(37)

In terms of the Ruelle zeta function, see Sect. 3.2, this result can be formulated as follows.

Theorem 3.27

Consider Abelian BF theory for a vector bundle E over a $(2m+1)$-dimensional manifold M with acyclic flat connection. Let X be a contact Anosov vector field with no Pollicott–Ruelle resonance at 0, and let $\digamma _X$ be the functional

$$\begin{aligned} \digamma _X(\mathbb {A},\mathbb {B}) = \int _M \langle \mathbb {B}\wedge \mathcal {L}_X^{-1}d^\nabla \mathbb {A}\rangle _E. \end{aligned}$$

Then, the expectation value of $\exp (i\digamma _X)$ with respect to the gauge fixing $\textsf{L}_X={{\,\textrm{im}\,}}(\iota _X)$ satisfies

$$\begin{aligned} \Bigl \langle e^{i\digamma _X} \Bigr \rangle _{\textsf{L}_X} = \left( \frac{\zeta _X(\hbar )}{\zeta _{X}(0)}\right) ^{(-1)^m}, \end{aligned}$$

(38)

where $\zeta _X$ is the Ruelle zeta function.

Remark 3.28

(Ruelle zeta from perturbation theory). As we had observed in Corollary 3.21, this expectation value can also be framed as the perturbative evaluation of the interacting partition function, with interaction given by $\hbar \digamma _X$. Indeed, in the perturbation theory framework, the free partition function is generally considered to be a not-accessible quantity (and is just set to 1), but adding the interaction term $\hbar \digamma _X$ leads to an interacting partition function that can be evaluated in the axial, Anosov gauge as a power series in $\hbar$. This perturbed partition function provides information on the ratio $\zeta _X(\hbar ) / \zeta _{X}(0)$.

However, the free partition function of BF theory in the axial, Anosov gauge is accessible, when interpreted as an infinite-dimensional oscillatory, Gaussian, integral as in Theorem 3.14. Using Theorem 3.8, this provides information on the value at 0 of the Ruelle zeta function. Recalling that $\hbar \in \mathbb {C}$ is just an arbitrary complex number, we see that these two descriptions combined allow for the recovery of the full Ruelle zeta function (up to a phase) in quantum field theory. $\diamondsuit$

3.8 A Comment on Our Assumptions

Two are the main assumptions that play a major role in our constructions and results. First of all, we require the twisted de Rham complex to be acyclic. Observe that this requirement enabled the original investigations on the analytic torsion as well, before it was realised that one can make sense of the analytic torsion as a more sophisticated algebraic object (a half density on cohomology or a metric on its determinant line), and that the equality between analytic and Reidemeister torsions can be phrased in that language [9]. To extend our results to the non-acyclic case, one would need to generalise the Ruelle zeta function in a similar fashion. Particularly interesting for this purpose is the generalisation provided in [17], and its relation to Turaev torsion. From the point of view of BV quantisation, such a generalisation is quite natural. If a Hodge-type decomposition is given so that the acyclic part of the complex is separated from the cohomology (for example that investigated in [20, Section 4.2]), one can proceed with what is often referred to as a BV push-forward, i.e. partial integration of the classical fields. The computation of the partition function using BV push-forward leaves one with precisely a half density on cohomology (or more generally some space of “zero modes”), and any further statement can be conveniently phrased in this language. (This was performed for BF theory in the metric (Lorenz) gauge exactly in this spirit in [15].)

Regarding Assumption A, it is in general non-trivial to find Anosov vector fields that satisfy it. However, there are some results for low-dimensional manifolds. As shown in [21, Section 7], when the twisted de Rham complex is acyclic, Assumption A holds for any contact Anosov vector field in dimension 3. In dimension 5, the same is true if X is the geodesic vector field on the sphere bundle of a 3-dimensional hyperbolic manifold. In general, we note that in virtue of [21, Theorem 2], there exists at least an open subset worth of Anosov vector fields that satisfy the assumption, and the value of the Ruelle zeta function at zero is locally constant on that open set. As a matter of fact, Assumption A is stronger than (in that it implies) the acyclicity condition of $\rho$, due to [20, Theorem 2.1]. In fact, for any Anosov vector field, the Pollicott–Ruelle resonant states at 0 form a complex that is quasi-isomorphic to the (twisted) de Rham complex. Whether the opposite is true and the two assumptions coincide is not known. In any case, to relax Assumption A might require once again a generalisation of the concept of Ruelle zeta “function”, in order to accommodate for residual factors coming from the cohomology of the Pollicott–Ruelle resonance complex.

Appendices

Appendix A. Proof of Proposition 3.23

We begin by giving an explicit description of the wavefront sets of the propagator and the interaction functional. Let $0\le L_1<L_2<\infty$ and consider the propagator

$$\begin{aligned} P_{L_1,L_2} = \int _{L_1}^{L_2}(\iota _X\otimes 1)K_t\,\textrm{d}t, \end{aligned}$$

(39)

where $K_t$ is the Schwartz kernel of $\exp (-t\mathcal {L}_X)$. Viewing $K_t$ for $t\in (L_1,L_2)$ as a distribution on $(L_1,L_2)\times M\times M$, we have by [22, Appendix B]

$$\begin{aligned} {{\,\textrm{WF}\,}}(K_t) \subset \{(t,-\xi (X(x)),e^{tH_p}(x,\xi ),x,-\xi )\,\,|\,\, t \in (L_1,L_2),\, (x,\xi )\in T^*M\},\nonumber \\ \end{aligned}$$

(40)

where

$$\begin{aligned} e^{tH_p}(x,\xi ) = (\varphi _t(x),d\varphi _{-t}(\varphi _t(x))^T\cdot \xi ) \end{aligned}$$

is the Hamiltonian flow of the principal symbol $p(x,\xi )=\xi (X(x))$. The propagator in (39) can be written as

$$\begin{aligned} \pi _*((\iota _X\otimes 1)K_t), \end{aligned}$$

using the push-forward of the map

$$\begin{aligned} \pi : (t,x,y)\in (L_1,L_2)\times M\times M \rightarrow (x,y)\in M\times M. \end{aligned}$$

See [11, Section 6.1] for an introduction to the notion of push-forward of distributions. In virtue of the behaviour of the wavefront set under push-forward [11, Theorem 6.3], we find that the wavefront set of the propagator can be bounded by

$$\begin{aligned} {{\,\textrm{WF}\,}}(P_{L_1,L_2})\subset & {} \{(e^{tH_p}(x,\xi ),x,-\xi )\,\,|\,\, t \nonumber \\\in & {} (L_1,L_2),\, (x,\xi )\in T^*M,\, \xi (X(x))=0\}. \end{aligned}$$

(41)

Note that application of the bundle endomorphism $(\iota _X\otimes 1)$ to $K_t$ does not increase the wavefront set.

The wavefront set of the interaction functional $\digamma _X$, which is the Schwartz kernel of $W_X = \mathcal {L}_X^{-1}d^\nabla$, can be calculated from the explicit description of ${{\,\textrm{WF}\,}}(\mathcal {L}_X^{-1})$ given in [22, Proposition 3.3], since composition with the differential operator $d^\nabla$ does not increase the wavefront set [32, Theorem 8.2.14]. We have

$$\begin{aligned} {{\,\textrm{WF}\,}}(\digamma _X) \subset N^*\Delta \cup \Omega '_+ \cup (E_u^*\times E_s^*), \end{aligned}$$

(42)

where

$$\begin{aligned} N^*\Delta = \{(x,\xi ,x,-\xi ) \,\,|\,\, (x,\xi )\in T^*M\} \end{aligned}$$

and

$$\begin{aligned} \Omega '_+ = \{(e^{tH_p}(x,\xi ),x,-\xi )\,\,|\,\, t \in \mathbb {R}_+,\, (x,\xi )\in T^*M,\, \xi (X(x))=0\}. \end{aligned}$$

$E_s^*$ and $E_u^*$ are the dual stable and unstable bundles.

The Feynman diagrams for our interaction are described in Sect. 3.6, where they are split into an interaction term and a trace term. In the notation of Definition 2.15, we will now compute the wavefront set of the distributions $u_P$ and $u_I$ for these diagrams, given as tensor products of propagators and interaction functionals, and check condition (14). For this, one needs to know how the wavefront set behaves under tensor product, see [32, Theorem 8.2.9]. For distributions u, v, we have

$$\begin{aligned} {{\,\textrm{WF}\,}}(u\otimes v) \subset {{\,\textrm{WF}\,}}(u)\times {{\,\textrm{WF}\,}}(v){} & {} \cup \, {{\,\textrm{WF}\,}}(u)\times ({{\,\textrm{supp}\,}}(v)\times \{0\}) \nonumber \\{} & {} \cup \, ({{\,\textrm{supp}\,}}(u)\times \{0\})\times {{\,\textrm{WF}\,}}(v). \end{aligned}$$

(43)

We will ignore factors of $\hbar$ and i in what follows.

For the diagram with N vertices in the interaction term, we have

$$\begin{aligned} u_I&= \digamma _X\otimes \cdots \otimes \digamma _X, \end{aligned}$$

(44a)

$$\begin{aligned} u_P&= \mathbb {B}\otimes P_{L_1,L_2}\otimes \cdots \otimes P_{L_1,L_2}\otimes \mathbb {A}, \end{aligned}$$

(44b)

for smooth external fields $\mathbb {A},\mathbb {B}$. Applying (43) we find that

$$\begin{aligned} {{\,\textrm{WF}\,}}(u_P){} & {} \subset \big \{(x_0,0,e^{t_1H_p}(x_1,\xi _1),x_1,-\xi _1,\dots ,e^{t_{N-1}H_p}(x_{N-1},\xi _{N-1}),x_{N-1},\\ {}{} & {} -\xi _{N-1},x_N,0)\big \}, \end{aligned}$$

where $t_i\in (L_1,L_2)$, $x_i\in M$ and $\xi _i\in T_{x_i}^*M$ with $\xi _i(X(x))=0$. We can include the sets ${{\,\textrm{WF}\,}}(u)\times ({{\,\textrm{supp}\,}}(v)\times \{0\})$ and $({{\,\textrm{supp}\,}}(u)\times \{0\})\times {{\,\textrm{WF}\,}}(v)$ arising from the application of (43) by allowing $\xi _i=0$. However, note that $\xi _1=\dots =\xi _{N-1}=0$ is not included in ${{\,\textrm{WF}\,}}(u_P)$.

Consider now that part of ${{\,\textrm{WF}\,}}(u_I)$ arising from the $\Omega _+$ part of ${{\,\textrm{WF}\,}}(\digamma _X)$. We denote this by $\mathcal {S}$. By (43), we have

$$\begin{aligned} \mathcal {S} \subset \{(e^{s_1H_p}(y_1,\eta _1),y_1,-\eta _1,\dots ,e^{s_{N}H_p}(y_{N},\eta _{N}),y_{N},-\eta _{N})\}, \end{aligned}$$

with $s_i>0$, $(y_i,\eta _i)\in T^*M$ and $\eta _i(X(x))=0$.

For $({{\,\textrm{WF}\,}}(u_P)+\mathcal {S})$ to intersect the zero section, there must be $s_i$, $t_i$, $(x_i,\xi _i)$, $(y_i,\eta _i)$ satisfying

$$\begin{aligned} \begin{aligned} (x_i,\xi _i)&= e^{s_{i+1}H_p}(y_{i+1},\eta _{i+1}), \quad \textrm{for}\quad 1\le i\le N-1, \\ (y_i,\eta _i)&= e^{t_{i}H_p}(x_{i},\xi _{i}), \quad \textrm{for}\quad 1\le i\le N-1, \\ y_N&=x_N, \quad \eta _N=0 \\ x_0&=\varphi _t(y_1), \quad \eta _1=0 \end{aligned} \end{aligned}$$

(45)

Together this implies that

$$\begin{aligned} \xi _1=\dots =\xi _{N-1}=\eta _1=\dots =\eta _N=0, \end{aligned}$$

(46)

which is impossible as the wavefront sets of $u_P$ and $u_I$ do not intersect the zero section.

The set $\mathcal {S}$ came from only considering the $\Omega '_+$ part of the wavefront set for each factor of $\digamma _X$ in the tensor product $u_I$ when applying (43). If we replace $\Omega '_+$ with $N^*\Delta$ for the j-th factor in the tensor product, this changes the first equation in (45) to $(x_j,\xi _j)=(y_{j+1},\eta _{j+1})$, and the conclusion (46) still holds. Similarly, if we replace any number of the $\Omega '_+$ by $N^*\Delta$ in the formula for the wavefront set of the tensor product, this conclusion remains true.

Finally, we now also consider the set $E^*_u\times E^*_s$, when taking the tensor product in $u_I$. If $E_u^*\times E_s^*$ only enters in one factor, say the j-th factor, then the first equation in (45) is replaced by the condition

$$\begin{aligned} (x_{j-1},\xi _{j-1})\in E^*_u, \quad e^{t_jH_p}(x_j,\xi _j)\in E^*_s \end{aligned}$$

(47)

However, working from both sides, i.e. from $\eta _1=0$ and $\eta _N=0$, and using the remaining equations in (45) still leads to the conclusion (46). If we now allow $E^*_u\times E^*_s$ to appear in both the j-th and $(j+k)$-th factors of the tensor product, then (47) for j and $j+k$, together with the fact that the Hamiltonian flow $\exp (tH_p)$ leaves $E^*_s$ and $E^*_u$ invariant, implies that for all i with $j \le i \le j+k-1$, $\xi _i \in E^*_s\cap E^*_u = \{0\}$. This again recovers (46). We have now treated the full wavefront set of $u_I$, and conclude that ${{\,\textrm{WF}\,}}(u_P)+{{\,\textrm{WF}\,}}(u_I)$ does not intersect the zero section, so the Feynman diagrams of the interaction term are well defined according to Definition 2.15.

Turning to the Feynman diagram of the trace term with N vertices, we again have

$$\begin{aligned} u_I = \digamma _X\otimes \cdots \otimes \digamma _X. \end{aligned}$$

The distribution $u_P$ is less simple to write as a tensor product, since it involves a permutation of the configuration space $M^{2N}$. Letting $\sigma$ be the diffeomorphism

$$\begin{aligned} \sigma : M^{2N}\rightarrow M^{2N}, \quad \sigma (z_1,\dots ,z_{2N}) = (z_{2},\dots ,z_{2N},z_1), \end{aligned}$$

we have

$$\begin{aligned} u_P = \sigma ^*(P_{L_1,L_2}\otimes \cdots \otimes P_{L_1,L_2}). \end{aligned}$$

The wavefront set of $u_P$ is bounded by

$$\begin{aligned} {{\,\textrm{WF}\,}}(u_P){} & {} \subset \big \{(x_N,-\xi _N,e^{t_1H_p}(x_1,\xi _1),x_1,-\xi _1,\dots ,e^{t_{N-1}H_p}(x_{N-1}, \xi _{N-1}),\\{} & {} \qquad x_{N-1},-\xi _{N-1}, e^{t_NH_p}(x_N,\xi _N))\big \}, \end{aligned}$$

where $t_i\in (L_1,L_2)$, $x_i\in M$ and $\xi _i\in T_{x_i}^*M$ with $\xi _i(X(x))=0$.

Again considering first $\mathcal {S}$, the part of ${{\,\textrm{WF}\,}}(u_I)$ arising from taking $\Omega '_+$ in each factor of $\digamma _X$, we find, similarly to (45), that a point in $({{\,\textrm{WF}\,}}(u_P)+\mathcal {S})\cap \{0\}$ must satisfy

$$\begin{aligned} \begin{aligned} (x_i,\xi _i)&= e^{s_{i+1}H_p}(y_{i+1},\eta _{i+1}), \quad \textrm{for}\quad 1\le i\le N-1, \\ (y_i,\eta _i)&= e^{t_{i}H_p}(x_{i},\xi _{i}), \quad \textrm{for}\quad 1\le i\le N, \\ (x_N,\xi _N)&= e^{s_{1}H_p}(y_{1},\eta _{1}) \end{aligned} \end{aligned}$$

(48)

Writing $t=\sum _i (t_i+s_i)$ and $(x,\xi )=(x_1,\xi _1)$, these equations together imply $e^{tH_p}(x,\xi )=(x,\xi )$, i.e.

$$\begin{aligned} \varphi _t(x) = x, \quad d\varphi _{-t}(x)^T\cdot \xi =\xi \end{aligned}$$

(49)

Note that we also have $\xi (X(x))=0$; that is, $\xi$ is an element of $E^*_s \oplus E^*_u$. Writing $\xi = (\xi _s,\xi _u)$, for the stable and unstable part, (49) implies

$$\begin{aligned} (\xi _s,\xi _u) = (d\varphi _{kt}(x)^T\cdot \xi _s,\, d\varphi _{kt}(x)^T\cdot \xi _u), \quad \forall k\in \mathbb {Z}. \end{aligned}$$

But by the Anosov property of the flow, we now have

$$\begin{aligned} \Vert \xi _s\Vert = \Vert d\varphi _{kt}(x)^T\cdot \xi _s\Vert \le Ce^{-\theta kt}\Vert \xi _s\Vert , \quad \forall k\in \mathbb {N}, \end{aligned}$$

for some $C,\theta >0$, and similarly for $\xi _u$ with $-k\in \mathbb {N}$. Since $t>0$, this forces $\xi =\xi _1$ to be 0, and together with (48) this again leads to the conclusion (46).

Including the sets $N^*\Delta$ and $E_u^*\times E_s^*$ in the wavefront set of the tensor product $u_I$, can be dealt with exactly as for the interaction term discussed above. We conclude that $({{\,\textrm{WF}\,}}(u_P)+{{\,\textrm{WF}\,}}(u_I))\cap \{0\}=\emptyset$, so the Feynman diagrams of the trace term are also well defined.

Appendix B. The Flat Determinant

In this appendix, we provide a brief introduction to the flat trace and flat determinant. We follow the microlocal approach of [22], see also [4] for an approach using mollifiers. The flat determinant can be viewed as a generalisation of the more well-known zeta function regularised determinant, which is best suited for elliptic operators, whereas the flat determinant can deal with a larger class of operators.

1.1 B.1. The Flat Trace

Let $B:C^\infty (M,F) \rightarrow \mathcal {D}'(M,F)$ be a continuous operator mapping smooth sections of a vector bundle F over a compact orientable manifold M into distributional sections of F. We view the Schwartz kernel of B as a distributional section²⁴

$$\begin{aligned} K_B \in \mathcal {D}'(M\times M, F\boxtimes (F^*\otimes \wedge ^n T^*M)), \end{aligned}$$

(50)

where $F^*$ is the dual bundle to F. The pullback by the diagonal map, $\iota : x\in M \rightarrow (x,x)\in M\times M$, when it is well defined, produces a distributional section

$$\begin{aligned} \iota ^*K_B \in \mathcal {D}'(M,F\otimes F^*\otimes \wedge ^n T^*M). \end{aligned}$$

The flat trace is obtained from this expression by taking the fibrewise trace in F and integrating over M. This procedure reproduces the usual trace when B has smooth Schwartz kernel and is thus trace class.

In the following, we will also need a slightly extended notion of flat trace for semigroups of operators. We can view the Schwartz kernel of a strongly continuous semigroup $A_t: C^\infty (M,F)\rightarrow C^\infty (M,F)$ with $t\in \mathbb {R}_+$ as a distributional section on $\mathbb {R}_+\times M\times M$. We then use the extended diagonal map

$$\begin{aligned} \tilde{\iota }: (t,x) \in \mathbb {R}_+\times M \rightarrow (t,x,x) \in \mathbb {R}_+\times M\times M, \end{aligned}$$

to pull back the Schwartz kernel, which, after integration over M, produces a distribution on $\mathbb {R}_+$.

Definition B.1

(Flat trace). Let $B :C^\infty (M,F) \rightarrow \mathcal {D}'(M,F)$ be a continuous operator. Assume that its Schwartz kernel satisfies

$$\begin{aligned} {{\,\textrm{WF}\,}}(K_B) \cap N^*\iota = \emptyset , \end{aligned}$$

(51)

where $N^*\iota$ is the set of conormals to $\iota$:

$$\begin{aligned} N^*\iota = \{(x,\xi ,x,-\xi ) \in T^*(M\times M) \,\mid \, (x,\xi ) \in T^*M\}. \end{aligned}$$

(52)

Then, we define the flat trace of B as

$$\begin{aligned} \textrm{tr}^\flat (B) = \langle \textrm{Tr}_F(\iota ^*K_B), 1 \rangle . \end{aligned}$$

(53)

Let $A_t: C^\infty (M,F)\rightarrow C^\infty (M,F)$ be a strongly continuous semigroup, and let $K_A$ be the Schwartz kernel of $A_t$, viewed as a distribution on $\mathbb {R}_+\times M\times M$. If

$$\begin{aligned} {{\,\textrm{WF}\,}}(K_A) \cap N^*\tilde{\iota } = \emptyset , \end{aligned}$$

(54)

where

$$\begin{aligned} N^*\tilde{\iota } = \{(t,0,x,\xi ,x,-\xi ) \in T^*(M\times M) \,\mid \, t\in \mathbb {R}_+,\, (x,\xi ) \in T^*M\}, \end{aligned}$$

we define the flat trace of $A_t$, $\textrm{tr}^\flat (A_t) \in \mathcal {D}'(\mathbb {R}_+)$, as

$$\begin{aligned} \left\langle \textrm{tr}^\flat (A_t),\, \chi \right\rangle = \left\langle \textrm{Tr}_F(\tilde{\iota }^*K_A),\, \chi \otimes 1 \right\rangle , \quad \forall \,\chi \in C^\infty _c(\mathbb {R}_+). \end{aligned}$$

(55)

Remark B.2

In our applications, we will mostly consider degree-preserving operators on the space of vector bundle valued forms $\Omega ^\bullet (M,E)$. In this case, the vector bundle F in (50) is of the form $E\otimes \wedge ^kT^*M$. Since $(\wedge ^kT^*M)^*\otimes \wedge ^nT^*M \cong \wedge ^{n-k}T^*M$, we can view the Schwartz kernel as a distributional section

$$\begin{aligned} K_B \in \mathcal {D}'(M\times M, (E\otimes \wedge ^k T^*M)\boxtimes (E^*\otimes \wedge ^{n-k} T^*M)). \end{aligned}$$

Under this identification, the action of $K_B$ on $\Omega ^k(M,E)$ is given in terms of the exterior product, and the formula for the flat trace can be replaced by

$$\begin{aligned} \textrm{tr}^\flat (B) = \left\langle \textrm{Tr}_E(\iota ^*K_B),\, 1 \right\rangle , \end{aligned}$$

where the pullback already gives a top degree exterior form without the need for a fibrewise trace in $\wedge ^kT^*M$ $\diamondsuit$

Note that the assumptions on the wavefront set of the Schwartz kernel, (51) and (54), are precisely the conditions for the respective pullbacks to be well defined, see [32, Theorem 8.2.4]. Identifying operators with their Schwartz kernel, [32, Theorem 8.2.4] also implies:

Proposition B.3

For any closed conic set $\Gamma \subset T^*(M\times M)$ that satisfies ${\Gamma \cap N^*\iota = \emptyset }$, the flat trace is a linear functional $\textrm{tr}^\flat :\mathcal {D}'_\Gamma (M\times M) \rightarrow \mathbb {C}$, which is sequentially continuous with respect to the Hörmander topology, see [32, Definition 8.2.2].

Similarly, for the case of semigroups, the flat trace is a linear map $\textrm{tr}^\flat :\mathcal {D}'_\Gamma (M\times M) \rightarrow \mathcal {D}'(R_+)$, sequentially continuous with respect to the Hörmander topology for any closed conic set $\Gamma \subset T^*(\mathbb {R}_+\times M\times M)$ that satisfies ${\Gamma \cap N^*\tilde{\iota } = \emptyset }$.

Note that by the density of $C^\infty (M\times M)$ in $\mathcal {D}'_\Gamma (M\times M)$, $\textrm{tr}^\flat$ is in fact the unique continuous extension of the usual trace for smoothing operators to $\mathcal {D}'_\Gamma (M\times M)$. In addition to being a continuous linear functional, the flat trace possesses another characteristic property of a trace, namely cyclicity, see [17, Section 4.5] for a proof.

Proposition B.4

Let $A,B: C^\infty (M,F) \rightarrow \mathcal {D}'(M,F)$ be two continuous operators. Assume that the compositions $A\circ B$ and $B\circ A$ are well defined, and both satisfy the wavefront condition in (51). Then,

$$\begin{aligned} \textrm{tr}^\flat (A\circ B) = \textrm{tr}^\flat (B\circ A). \end{aligned}$$

Remark B.3

We defined the flat trace for continuous operators from $C^\infty (M)$ to $\mathcal {D}'(M)$. However, if X, Y are topological vector spaces with continuous inclusions $i: C^\infty (M) \rightarrow X$ and $j: Y \rightarrow \mathcal {D}'(M)$, then the definition automatically extends to continuous operators $B:X\rightarrow Y$. Indeed, any such B induces an operator $j\circ B\circ i :C^\infty (M) \rightarrow \mathcal {D}'(M)$, whose flat trace can be computed if the wavefront condition holds. In this way, we can, for example, take the flat trace of bounded operators between $L^p$ spaces or Sobolev spaces and this flat trace will actually only depend on the restriction of the operator to smooth functions. $\diamondsuit$

1.2 B.2. The Flat Determinant

We will define the flat determinant of an operator B through the heat kernel of B, that is, the semigroup $\exp (-tB)$ generated by B. The formula to have in mind is

$$\begin{aligned} \log (\textrm{det}^\flat (B)) = -\frac{\textrm{d}}{\textrm{d}s}\Bigl (\frac{1}{\Gamma (s)}\int _0^\infty t^{s-1}e^{-\lambda t}\,\textrm{tr}^\flat (e^{-tB}) \, \textrm{d}t\Bigr )\Bigr |_{s=0,\,\lambda =0}, \end{aligned}$$

(56)

where $\Gamma (s)$ is the gamma function. This is akin to the characterisation of the zeta regularised determinant via the Mellin transform. However, the above formula requires some care, as we will allow the flat trace of $\exp (-tB)$ to take values in the space of distributions, as in Definition B.1, and the evaluation at $s=0$, $\lambda =0$ may necessitate analytic continuation. Thus, we choose cut-off functions $\chi _N\in C^\infty _c(\mathbb {R}_+)$ with $\chi _N=1$ on $[\frac{1}{N},N]$ and define the following function of two complex variables:

$$\begin{aligned} F_B(s,\lambda ) = \frac{1}{\Gamma (s)} \lim _{N\rightarrow \infty }\left\langle \textrm{tr}^\flat (e^{-tB}),\, \chi _N(t) t^{s-1} e^{-\lambda t} \right\rangle , \end{aligned}$$

(57)

assuming for the moment that the limit is well defined. If $F_B$ has an analytic continuation to $s=0$, $\lambda =0$, we define the flat determinant of B as:

$$\begin{aligned} \textrm{det}^\flat (B) = \exp \Bigr (-\frac{\textrm{d}}{\textrm{d}s} F_B(s,\lambda )\Big |_{s=0,\,\lambda =0}\,\Bigr ). \end{aligned}$$

(58)

Note that letting $\lambda$ take on a nonzero value in the above formula provides a definition of $\textrm{det}^\flat (B+\lambda )$.

In summary, we have the following definition.

Definition B.6

(Flat determinant). Let $B :C^\infty (M,F) \rightarrow \mathcal {D}'(M,F)$ satisfy the following conditions:

• B generates a strongly continuous semigroup of operators $e^{-tB} :C^\infty (M,F) \rightarrow C^\infty (M,F)$.

• The Schwartz kernel of $e^{-tB}$ viewed as a distribution on $\mathbb {R}_+\times M\times M$, satisfies the wavefront condition (54).

• For any smooth bounded function f on $\mathbb {R}_+$, the sequence $\left\langle \textrm{tr}^\flat (e^{-tB}),\, \chi _N(t)f(t) t^{s-1} e^{-\lambda t} \right\rangle$ converges uniformly in s and $\lambda$ as $N\rightarrow \infty$ for $\Re (s), \Re (\lambda )$ large enough.

• The function $F_B(s,\lambda )$ in (57) has an analytic continuation to $s=0,\,\lambda =0$.

Then, the flat determinant of B is well defined and given by equation (58).

Remark B.7

Although the introduction of a bounded function in the third point above may seem ad hoc, this is just the analog of taking the absolute value, which is not available for distributions. That is, requiring convergence in the presence of a bounded function f is akin to requiring absolute convergence of the integral $\int _0^\infty |\textrm{tr}^\flat (e^{-tB})t^{s-1}e^{-\lambda t}|\,\textrm{d}t$. This ensures that the limit in (57) is independent of the choice of cut-off functions $\chi _N$. $\diamondsuit$

Remark B.8

Note that we can think of s as regularising the integral in $F_B(s,\lambda )$ at small t and $\lambda$ as regularising the integral at large t. In the context of quantum field theory, where $\int _0^\infty \exp (-tB)\,\textrm{d}t$ has an interpretation as the propagator associated with B, the parameters s and $\lambda$ can be thought of as regularisation procedures for UV and IR divergences, respectively. $\diamondsuit$

Remark B.9

If the operator $\exp (-tB)$ is smoothing, then the flat trace agrees with the usual trace and the flat determinant reduces to the zeta regularised determinant, see, for example, [4, Proposition 6.8]. This applies in particular when B is a generalised Laplacian. One can thus think of the flat determinant as an extension of the zeta regularised determinant to operators whose heat kernel has non-trivial wavefront set. $\diamondsuit$

Remark B.10

If $\textrm{tr}^\flat (\exp (-tB))$ behaves well as $t\rightarrow 0$, so that the $N\rightarrow \infty$ limit in Definition B.6 is well defined already for s in a neighbourhood of $s=0$, then the derivative at $s=0$ in equation (58) can be evaluated directly. This leads to the formula

$$\begin{aligned} \log (\textrm{det}^\flat (B+\lambda )) = -\int _0^\infty t^{-1}e^{-\lambda t}\,\textrm{tr}^\flat (e^{-tB}) \, \textrm{d}t, \end{aligned}$$

which should be interpreted in terms of distributional pairing and analytic continuation in $\lambda$ as in Definition B.6. This is, for instance, the case when B is the Lie derivative with respect to an Anosov vector field, as considered in the main part of this text. $\diamondsuit$

In our applications, the function space $C^\infty (M,F)$ has the additional structure of a graded vector space. In this context, the determinant of an operator must take into account the grading of the underlying vector space. This leads to the definition of the super determinant for degree-preserving operators.

Definition B.11

(Flat super determinant). Let $F=\bigoplus _k F^{(k)}$ be a graded vector bundle. Let $B: C^\infty (M,F)\rightarrow C^\infty (M,F)$ be a degree-preserving operator, with components $B^{(k)}: C^\infty (M,F^{(k)})\rightarrow C^\infty (M,F^{(k)})$ acting in degree k. Assuming that each component $B^{(k)}$ satisfies the requirements of Definition B.6, we define the flat super determinant of B as

$$\begin{aligned} \textrm{sdet}^\flat (B) = \prod _k \textrm{det}^\flat (B^{(k)})^{(-1)^k}. \end{aligned}$$

(59)

Equivalently, we can obtain the flat super determinant directly form equation (58) if we replace the flat trace in (57) by the flat super trace:

$$\begin{aligned} \textrm{str}^\flat (e^{-tB}) = \sum _k (-1)^k \textrm{tr}^\flat (e^{-tB^{(k)}}). \end{aligned}$$

(60)

Finally, we require a notion of the flat determinant restricted to an invariant subspace L of the operator B.

Definition B.12

Let $L\subset C^\infty (M)$ be a closed subspace left invariant under the action of the semigroup $\exp (-tB)$. Furthermore, assume that there exists a continuous projection operator $\Pi _L: C^\infty (M)\rightarrow C^\infty (M)$ with image L satisfying

$$\begin{aligned} {{\,\textrm{WF}\,}}(\Pi _L) \subset N^*\iota , \end{aligned}$$

(61)

see (52). Then, we define the flat determinant of B restricted to L by replacing $\textrm{tr}^\flat (\exp (-tB))$ in Definition B.6 with $\textrm{tr}^\flat (\exp (-tB)\Pi _L)$, and we say that the flat determinant of B restricts to L.

Remark B.13

Note that condition (61) together with the behaviour of wavefront sets under composition of operators [32, Theorem 8.2.14] implies that $\textrm{tr}^\flat (\exp (-tB)\Pi _L)$ is well defined if $\textrm{tr}^\flat (\exp (-tB))$ is. Moreover, if $\Pi '_L$ is some other projection onto L satisfying the requirements of Definition B.12, then the cyclicity of the flat trace implies

$$\begin{aligned} \textrm{tr}^\flat (\exp (-tB)\Pi _L){} & {} = \textrm{tr}^\flat (\exp (-tB)\Pi '_L\Pi _L) = \textrm{tr}^\flat (\Pi _L\exp (-tB)\Pi '_L)\\{} & {} = \textrm{tr}^\flat (\exp (-tB)\Pi '_L), \end{aligned}$$

i.e. Definition B.12 is independent of the choice of projection operator. $\diamondsuit$