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 \(\rho : \pi _1(M) \rightarrow U(\mathbb ^r)\). 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 \(\nabla \),

$$\begin d^\nabla :\Omega ^k(M,E) \rightarrow \Omega ^(M,E), \end$$

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.Footnote 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 \(\mathbb \)-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

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

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Thus, a field \((\mathbb ,\mathbb ) \in \mathcal _\) consists of a pair of inhomogeneous differential forms

$$\begin \mathbb = \mathbb ^ + \cdots + \mathbb ^, \quad \mathbb = \mathbb ^ + \cdots + \mathbb ^, \end$$

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

$$\begin \Omega _((0,\mathbb ),(\mathbb ,0)) = \int _M \langle \mathbb \wedge \mathbb \rangle _E, \end$$

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and extended by graded anti-symmetry and linearity to all of \(\mathcal _\times \mathcal _\). Here, \(\langle \mathbb \wedge \mathbb \rangle _E\) 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

$$\begin S_\equiv S_(\mathbb ,\mathbb ) \doteq \int _M \langle \mathbb \wedge d^\nabla \mathbb \rangle _E. \end$$

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Remark 3.2

Note that we can view \(d^\nabla \) as acting on \(\mathcal _\):

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

In this way, \(d^\nabla \) becomes a degree 1 operator on \(\mathcal _\) and the action functionalFootnote 15 can be written as

$$\begin S_(\mathbb ,\mathbb ) = \frac\Omega _((\mathbb ,\mathbb ), d^\nabla (\mathbb ,\mathbb )). \end$$

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 \(\Omega _\) is well defined for local functionals and that

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

where ± denotes a sign depending on the degree of \(\mathbb \) and \(\mathbb \). Thus, \(S_\) satisfies the classical master equation, see Definition 2.7, and \((\mathcal _,\Omega _,S_,d^\nabla )\) 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

$$\begin Q: \Omega ^\bullet (M,E) \rightarrow \Omega ^(M,E), \end$$

satisfying \(Q^2 = 0\) and graded anti-symmetry with respect to the pairing \(\Omega _\). 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

$$\begin S_Q(\mathbb ,\mathbb ) = \int _M \langle \mathbb \wedge Q\mathbb \rangle _E. \end$$

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 \((\mathcal _,\Omega _,S_Q,Q)\).

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:

$$\begin T_xM = T_0(x) \oplus T_s(x) \oplus T_u(x), \quad \text \, T_0(x) = \mathbb X(x), \end$$

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

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

Here, \(\Vert \cdot \Vert \) is the norm induced by some Riemannian metric on M.Footnote 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

$$\begin \zeta _\rho (\lambda ) := \prod _} \textrm(I - \rho ([\gamma ])e^), \end$$

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where \([\gamma ]\in \pi _1(M)\) is the conjugacy class of a single-winding, closed, orbitFootnote 17\(\gamma \in \mathcal \) of the flow \(\varphi _t\) of the Anosov vector field X and \(\lambda \in \mathbb \) 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 \(\mathbb \) [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 \(\textrm(T_s)=\textrm(T_u)=m\), 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

$$\begin \log \zeta _\rho (\lambda ) = (-1)^m\sum _^ (-1)^k \log \zeta _(\lambda ). \end$$

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

$$\begin \log \zeta _(\lambda ) = -\int _0^\infty t^e^ \textrm^\flat e^_} \,\textrmt. \end$$

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where \(\mathcal _ = \mathcal _X\bigr |_\) is the Lie derivative restricted to k-forms lying in the kernel of \(\iota _X\), and the flat trace evaluates to [29]

$$\begin \textrm^\flat e^_} = \sum _} \sum _^\infty \ell (\gamma ) \delta (t - j\ell (\gamma )) \frac(\rho ([\gamma ])^j)\textrm (\wedge ^k P(\gamma )^j)}^\flat (I - P(\gamma )^j) |}, \end$$

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as a distribution on \(\mathbb _+\)Footnote 18. Hence, we can conclude that

Theorem 3.8

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

$$\begin \zeta _\rho (\lambda )^ = \textrm^\flat \bigl ((\mathcal _ + \lambda )\bigr |_\bigr ) = \prod _^ ^\flat } (\mathcal _ + \lambda )^. \end$$

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Remark 3.9

(Anisotropic Sobolev spaces) An important observation in this context is the relation between the poles of the meromorphic extensionFootnote 19 of the resolvent, defined (for large \(\Re (\lambda )\)) as

$$\begin R_X(\lambda )=(\mathcal _X+\lambda )^ = \int _0^\infty e^_X+\lambda )}\,\textrmt, \end$$

and the eigenvalues of the operator \(\mathcal _X\) acting on certain Hilbert spaces specially tailored to the dynamics. On such anisotropic Sobolev spaces \(H_\), \(\mathcal _X\) can be shown to have discrete spectrum. The vector field X (or sometimes the Lie derivative operator \(\mathcal _X\)) is said to have a Pollicott–Ruelle resonance at \(\lambda \), if there are eigenvectors in some \(H_\) 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” \(\mathcal _X\) 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 \(\mathcal _X:}(\mathcal _X) \subset H_ \rightarrow H_\). Henceforth we will forgo the explicit constructionFootnote 20 of \(H_\), 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 \(\lambda \in \mathbb \), whose zeros coincide with the Pollicott–Ruelle resonances in the respective form degree, see [22]. Thus, as befits a determinant, \(^\flat } (\mathcal _ + \lambda )\) vanishes precisely when \(\mathcal _ + \lambda \) 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 \(\mathcal _X: }(\mathcal _X)\subset H_ \rightarrow H_\), acting on some anisotropic Sobolev space \(H_\), 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 \(Q_\equiv Q\) of Definition 2.7 (cf. Remark 2.9) is simply given by:

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

Proposition 3.12

([30, 38]). The operator

$$\begin Q_ \circlearrowright \mathcal _, \qquad Q_ (\mathbb ,\mathbb ) = (\iota _X\mathbb ,\iota _X\mathbb ), \end$$

is a gauge fixing operator. In particular, definingFootnote 21

$$\begin Q_^* \circlearrowright \mathcal _, \qquad Q_^* (\mathbb ,\mathbb ) = (\alpha \wedge \mathbb ,\alpha \wedge \mathbb ) \end$$

we have the Lagrangian splitting:

$$\begin \mathcal _=\textrm(Q_) \oplus \textrm(Q_^*) = \textrm(\iota _X\oplus \iota _X) \oplus \textrm(\alpha \wedge \oplus \alpha \wedge ), \end$$

which is invariant under the action of the graded commutator:

$$\beginD(\mathbb ,\mathbb ) = [Q_,Q](\mathbb ,\mathbb ) = \mathcal _X(\mathbb ,\mathbb ).\end$$

For the sake of simplicity, we will abuse notation and simply write: \(\textsf_X:= \ker (\iota _X) = \,}}(\iota _X)\) 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 \(\varphi _^*\), the pullback by the flow of X. Thus, we can think of the propagator \(P_\) 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 \(\textrm^\flat (D\vert __X}) = \textrm^\flat (\mathcal _X\vert _\,}}(\iota _X)})^2\), and the grading on \(\mathcal _\) 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:

$$\begin Z(S_,\textsf_X) = |\zeta _\rho (0)|^. \end$$

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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 \(\hbar \):

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

Remark 3.15

Assumption A implies that the inverse \(\mathcal _X^: H_ \rightarrow H_\) is well defined on anisotropic Sobolev spaces. Note that both \(\iota _X\) and \(d^\nabla \) commute with \(\mathcal _X^\). \(\diamondsuit \)

Definition 3.16

Under Assumption A, the perturbing functional is:

$$\begin \digamma _X(\mathbb ,\mathbb ) = \int _M \langle \mathbb \wedge \mathcal _X^d^\nabla \mathbb \rangle _E. \end$$

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This functional exhibits some peculiarities. The operator \(W_X:= \mathcal _^d^\nabla \) appearing in the functional does not necessarily map smooth sections to smooth sections. Indeed, the condition \(\mathcal _X^(\omega )\) smooth for all \(\omega \in \Omega ^\bullet (M,E)\) is equivalent to the Lie derivative \(\mathcal _X: \Omega ^\bullet (M,E)\rightarrow \Omega ^\bullet (M,E)\) being surjective, which is not necessarily the case. Thus, \(\mathcal _X^d^\nabla \mathbb \in H_\) 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 \(\mathbb \), 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 \(W_X=\mathcal _X^d^\nabla \) as a chain contraction for the chain complex defined by our gauge fixing operator \(\iota _X\). Indeed,

$$\begin \iota _X\mathcal _X^d^\nabla + \mathcal _X^d^\nabla \iota _X = \mathcal _X^(\iota _Xd^\nabla + d^\nabla \iota _X) = \mathcal _X^\mathcal _X = \textrm_}(\mathcal _X)} \end$$

is the identity operator on the domain of \(\mathcal _X\) in \(H_\). Since smooth sections are in \(}(\mathcal _X)\), 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 \(Q_X \doteq (1 + \hbar \mathcal _X^)d^\nabla = d^\nabla +\hbar W_X\), we define our perturbed action as

$$\begin S_(\mathbb ,\mathbb ) = S_(\mathbb ,\mathbb ) + \hbar \digamma _X(\mathbb ,\mathbb ) = \int _M \langle \mathbb \wedge (1+\hbar \mathcal _X^)d^\nabla \mathbb \rangle _E. \end$$

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Applying the Poisson bracket to \(S_\), we find

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

where we used the fact that \(d^\nabla \) and \(\mathcal _X^\) 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 \(S_\) 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 \(S_\) lies in the Euler–Lagrange equations. For the field \(\mathbb \), the Euler–Lagrange equation gives

$$\begin Q_X\mathbb =(1+\hbar \mathcal _X^)d^\nabla \mathbb = 0. \end$$

Factoring out the \(\mathcal _X^\), this can equivalently be written as

$$\begin \mathcal _X^(\mathcal _X+\hbar )d^\nabla \mathbb = \mathcal _X^d^\nabla (\mathcal _X+\hbar )\mathbb = 0. \end$$

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

$$\begin (\mathcal _X+\hbar )\mathbb = 0. \end$$

In other words, Pollicott–Ruelle resonant states with resonance \(\hbar \) provide non-trivial solutions. In fact, if we restrict \(\mathbb \) to satisfy our gauge fixing condition \(\mathbb \in \,}}(\iota _X)\), then these are the only non-trivial solutions to the equations of motion. Indeed, if \(d^\nabla (\mathcal _X+\hbar )\mathbb = 0\), then \((\mathcal _X+\hbar )\mathbb \in \ker (d^\nabla )\). Since \(\mathcal _X\) leaves the gauge fixing subspace \(\,}}(\iota _X)\) invariant, we also have \((\mathcal _X+\hbar )\mathbb \in \,}}(\iota _X)\). But \(\,}}(\iota _X)\cap \ker (d^\nabla )=\\) by the injectivity of \(\mathcal _X\), so \((\mathcal _X+\hbar )\mathbb =0\). 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 \(\mathbb \) can be analysed similarly. \(\square \)

Remark 3.19

(Dependence on X). Note that the Anosov vector field X enters the perturbed action functional \(S_\) 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 \(S_\). One can consider a family of action functionals \(S_\) 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 \(S_ = S_ + \hbar \digamma _X\) 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 \(S_ = S_ + \hbar \digamma _X\), when computed in the axial, Anosov gauge \(\textsf_X = \ker (\iota _X)\) on a manifold equipped with the twisted, contact, Anosov data of Definition 3.11, is the twisted Ruelle zeta function up to a phase:

$$\begin Z(S_,\textsf_X) = |\zeta _X(\hbar )|^. \end$$

Proof

Being quadratic, we can apply the definition of partition function (Definition 2.12) to the functional S, where the operator Q is now \(d^\nabla +\hbar \mathcal _X^d^\nabla \). Thus, for the gauge fixing condition \(\textsf_X = \ker (\iota _X) = \,}}(\iota _X)\), the evaluation of the partition function yieldsFootnote 22

$$\begin Z(S,\textsf_X) & \doteq \big |\textrm^\flat (\iota _X(d^\nabla +\hbar \mathcal _X^d^\nabla )|_\,}}(\iota _X)})\big | = \big |\textrm^\flat ((\mathcal _X+\hbar )|_\,}}(\iota _X)})\big | \\ & = |\zeta _X(\hbar )|^, \end$$

where \(\iota _Xd^\nabla \vert _\,}}(\iota _X)} = \mathcal _X\vert _\,}}(\iota _X)}\) cancels with the \(\mathcal _X^\) 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 \(S_\). 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

$$\begin \biggl \langle e^ \biggr \rangle __X}\doteq & \frac,\textsf_X)}}(S_,\textsf_X)} = \frac,\textsf_X)},\textsf_X)} \nonumber \\= & \frac^\flat ((\mathcal _X+\hbar )|_\,}}(\iota _X)})\big |}^\flat (\mathcal _X|_\,}}(\iota _X)})\big |} = \left| \frac\right| ^. \end$$

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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 \(S_\), as in Sect. 2. This is worthwhile, since \(D=\mathcal _X\) is not elliptic, and thus, a non-trivial adaptation of the perturbative quantisation setup of [18] is required.

The operators \(Q, Q_\) and D, introduced for BF theory in Sect. 3.3, can be represented as

$$\begin Q = \begin d^\nabla & 0 \\ 0 & d^\nabla \end, \quad Q_ = \begin \iota _X & 0 \\ 0 & \iota _X \end, \quad D = \begin \mathcal _X & 0 \\ 0 & \mathcal _X \end, \end$$

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

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

Remark 3.22

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

$$\begin \,}}(d^\nabla )\cap \,}}(\iota _X) \subset \ker (\mathcal _X) = \. \end$$

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

$$\begin \omega = \mathcal _X\mathcal _X^\omega = d^\nabla \iota _X\mathcal _X^\omega + \iota _Xd^\nabla \mathcal _X^\omega . \end$$

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

$$\begin \omega _1 = d^\nabla \iota _X\mathcal _X^\omega \in \,}}(d^\nabla ), \quad \omega _2 = \iota _Xd^\nabla \mathcal _X^\omega \in \,}}(\iota _X). \end$$

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

$$\begin e^\mathbb (x) = e^_X}\mathbb (x) = (K_t^*\mathbb )(x) = \int _M K_t^(x,y)\wedge \mathbb (y), \end$$

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 \(\Omega _\), the y part of the kernel \(K_t^(x,y)\) is actually over the space \(\Omega ^\bullet (M,E)[n-2]\). Similarly, for \(\mathbb \in \Omega ^\bullet (M,E)[n-2]\) we have

$$\begin e^\mathbb (x) = e^_X}\mathbb (x) = (K_t^*\mathbb )(x) = \int _M K_t^(x,y)\wedge \mathbb (y), \end$$

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

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

and similarly for \(K_t^\). The propagator of the theory is then obtained as

$$\begin P^_ = \int _^ (Q_\otimes 1)K^_t\,\textrmt \end$$

and thus consists of two copies of

$$\begin P_ = \int _^ (\iota _X\otimes 1)K_t\,\textrmt. \end$$

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The two copies of this propagator, one connecting the field \(\mathbb \) to \(\mathbb \) and the other \(\mathbb \) to \(\mathbb \), 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 \(P_\) 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 \((\,}}(u_P)+\,}}(u_I)) \cap \ = \emptyset ,\) and thus, the weight

$$\begin \Phi _\gamma (iP_,i\hbar \digamma _X)(\mathbb ,\mathbb ) \end$$

is well defined in the sense of Definition 2.15.

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

$$\begin i\hbar \digamma _X[L](\mathbb ,\mathbb ) = \Gamma (P_,\hbar \digamma _X)(\mathbb ,\mathbb ). \end$$

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

$$\begin \lim _\Gamma (P_,\hbar \digamma _X)(\mathbb ,\mathbb ) = \Gamma (P_,\hbar \digamma _X)(\mathbb ,\mathbb ). \end$$

Thus, in contrast to the case where D is a generalised Laplacian, our propagator \(P_\) 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 \(\textrm^\flat (\exp (-t\mathcal _X))\), 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 \(\mathcal _X\) does not have strictly positive spectrum. Thus, we introduce a regularisation procedure, in order to regularise the propagators \(P_\) as \(L_2\) goes to \(\infty \). We do this by introducing a parameter \(\lambda \in \mathbb \) and defining

$$\begin P_^\lambda = \int _^ e^(\iota _X\otimes 1)K_t\,\textrmt. \end$$

To compute Feynman diagrams we first insert \(P_^\lambda \) 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 limitFootnote 23