Pollicott-Ruelle spectrum and Witten Laplacians

We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey--Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten-Helffer-Sj\"ostrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.


Introduction
Let M be a smooth (C ∞ ), compact, oriented, boundaryless manifold of dimension n ≥ 1. Let f : M → R be a smooth Morse function whose set of critical points is denoted by Crit(f ). In [60], Witten introduced the following semiclassical deformation of the de Rham coboundary operator: where Ω • (M) denotes smooth differential forms on M. Then, fixing a smooth Riemannian metric g on M, he considered the adjoint of this operator with respect to the induced scalar product on the space of L 2 forms L 2 (M, Λ(T * M)) : where V f is the gradient vector field associated with the pair (f, g), i.e. the unique vector field satisfying ∀x ∈ M, df (x) = g x (V f (x), .).
The operator d f,h + d * f,h is the analog of a Dirac operator and its square is usually defined to be the Witten Laplacian [60]. In the present paper, we take a different convention and we choose to rescale it by a factor h 2 . Hence, the Witten Laplacian will be defined as where L V f is the Lie derivative along the gradient vector field. This defines a selfadjoint, elliptic operator whose principal symbol coincides with the principal symbol of the Hodge-De Rham Laplace operator acting on forms. It has a discrete spectrum on L 2 (M, Λ k (T * M)) 1 that we denote, for every 0 ≤ k ≤ n, by j (h) → +∞ as j → +∞. It follows from the works of Witten [60] and Helffer-Sjöstrand [43] that there exists a constant ǫ 0 > 0 such that, for every 0 ≤ k ≤ n and for every h > 0 small enough, there are exactly c k (f ) eigenvalues inside the interval [0, ǫ 0 ], where c k (f ) is the number of critical points of index k -see e.g. the recent proof of Michel and Zworski in [49,Prop. 1]. Building on the strategy initiated by Witten, Helffer and Sjöstrand also showed that one can associate to these low energy eigenmodes an orientation complex whose Betti numbers are the same as the Betti numbers of the manifold [43,Th. 0.1]. Another approach to this question was developped by Bismut and Zhang in their works on the Reidemeister torsion [5,6,62]: following Laudenbach [47], they interpreted the Morse complex in terms of currents.
The aim of our article is to describe the convergence of all the spectral data (meaning both eigenvalues and eigenmodes) of the Witten Laplacian. This will be achieved by using microlocal techniques that were developped in the context of dynamical systems [17,18]. Note that part of these results could probably be obtained by more classical methods in the spirit of the works of Simon [55] and Helffer-Sjöstrand [43] on harmonic oscillators. We refer the reader to the book of Helffer and Nier [42] for a detailed account of the state of the art on these aspects. Regarding the convergence of the spectrum, Frenkel, Losev and Nekrasov [29] did very explicit computations of the Witten's spectrum for the case of the height function on the sphere, and they implicitely connect this spectrum to a dynamical spectrum as we shall do here. They also give a strategy to derive asymptotic expansions for dynamical correlators of holomorphic gradient flows acting on compact Kähler manifolds. Yet, unlike [29], we attack the problem from the dynamical viewpoint rather than from the semiclassical perspective. Also, we work in the C ∞ case instead of the compact Kähler case and we make use of tools from microlocal analysis to replace tools from complex geometry.
The main purpose of the present work is to propose an approach to these problems having a more dynamical flavour than these references. We stress that our study of the limit operator is self-contained and that it does not make use of the tools developped in the above references. It is more inspired by the study of the so-called transfer operators in dynamical systems [37,2,65], and this dynamical perspective allows us to make some explicit connection between the spectrum of the Witten Laplacian and the dynamical results from [17,19].
Conventions. All along the article, we denote by Ω k (M) the set of C ∞ differential forms of degree k, i.e. smooths sections M → Λ k (T * M). The topological dual of Ω n−k (M) is the set of currents of degree k and it will be denoted by D ′k (M), meaning k-differential forms with coefficients in the set of distributions [54]. Harvey-Lawson. In order to illustrate our results, we let ϕ t f be the flow induced by the gradient vector field V f , and, given any critical point a of f of index k, we introduce its unstable manifold

Semiclassical versus dynamical convergence and a question by
Recall from the works of Smale [56] that this defines a smooth embedded submanifold of M whose dimension is equal to n − k and whose closure is a union of unstable manifolds under the so-called Smale transversality assumption. Then, among other results, we shall prove the following Theorem: Theorem 2.1 (Semiclassical versus dynamical convergence). Let f be a smooth Morse function and g be a smooth Riemannian metric such that V f is C 1 -linearizable near every critical point and such that V f satisfies the Smale transversality assumption. Let 0 ≤ k ≤ n.
Then, for every a ∈ Crit(f ) of index k, there exists (U a , S a ) in D ′k (M) × D ′n−k (M) such that the support of U a is equal to W u (a) and such that Moreover, there exists ǫ 0 > 0 small enough such that, for every (ψ 1 , ψ 2 ) ∈ Ω k (M) × Ω n−k (M) and for every 0 < ǫ < ǫ 0 , is the spectral projector on [0, ǫ] for the self-adjoint elliptic operator W (k) f,h . Remark 2.2. The Smale transversality assumption means that the stable and unstable manifolds satisfy some transversality conditions [56] -see paragraph 3.1.1 for a brief reminder. Recall that, given a Morse function f , this property is satisfied by a dense open set of Riemannian metrics thanks to the Kupka-Smale Theorem [46,57]. The hypothesis of being C 1 -linearizable near every critical point means that, near every a in Crit(f ), one can find a C 1 -chart such that the vector field can be written locally as V f (x) = L f (a)x∂ x , where L f (a) is the unique (symmetric) matrix satisfying d 2 f (a) = g a (L f (a)., .). By fixing a finite number of nonresonance conditions on the eigenvalues of L f (a), the Sternberg-Chen Theorem [51] ensures that, for a given f , one can find an open and dense subset of Riemannian metrics satisfying this property.
Let us now comment the several statements contained in this Theorem. First, as we shall see in Lemma 5.10, the current U a coincides with the current of integration [W u (a)] when restricted to the open set M \ ∂W u (a) with ∂W u (a) = W u (a) \ W u (a). Hence, the first part of the Theorem shows how one can extend [W u (a)] into a globally defined current which still satisfies the transport equation L V f (U a ) = 0. This extension was done by Laudenbach in the case of locally flat metrics in [47] by analyzing carefully the structure of the boundary ∂W u (a). Here, we make this extension for more general metrics via a spectral method and the analysis of the structure of the boundary is in some sense hidden in the construction of our spectral framework [17,18]. We emphasize that Laudenbach's construction shows that these extensions are currents of finite mass while our method does not say a priori anything on that aspect.
The second part of the Theorem shows that several quantities that appeared in previous analytical works on Morse theory coincide. In the case of a locally flat metric, the fact that the first and the third quantity in equation (1) are equal was shown by Harvey and Lawson [41]. In [17], we showed how to prove this equality when the flow satisfies some more general (smooth) linearization properties than the ones appearing in [47,41]. Here, we will extend the argument from [17] to show that this equality remains true under the rather weak assumptions of Theorem 2.1. The last equality tells us that the low eigenmodes of the Witten Laplacian converge in a weak sense to the same quantities. In particular, it recovers the fact that the number of small eigenvalues in degree k is equal to the number of critical points of index k.
In a nutshell, our Theorem identifies a certain semiclassical limit of scalar product of quasimodes for the Witten Laplacian with a large time limit of some dynamical correlation for the gradient flow which converges to equilibrium : Dynamical object , for every (ψ 1 , ψ 2 ) ∈ Ω k (M) 2 . From this point of view, this Theorem gives some insights on a question raised by Harvey and Lawson in [40,Intro.] who asked about the connection between their approach to Morse theory and Witten's one. It is then tempting to exchange the two limits, and Theorem 2.1 shows that intertwining these two limits requires to take into account the small eigenvalues of the Witten Laplacian.
Proving the second part of this Theorem amounts to determining the limit of the spectral projectors of the Witten Laplacian (after conjugation by e f h ) viewed as operators from Ω k (M) to D ′k (M). Recall that Helffer-Sjöstrand [43, §1] and Bismut-Zhang [6,Def. 6.6] constructed explicit bases for the bottom of the spectrum of the Witten Laplacian. Using the approach of these references, we would have to verify that these quasimodes (after renormalization by e f h ) converge to the currents constructed by Laudenbach [47] -see [13,Chap. 9] for a related discussion. As far as we know, this question has not been addressed explicitely in the literature. This convergence will come out naturally of our spectral analysis. We will in fact show the convergence of all the spectral projectors (not only at the bottom of the spectrum) and identify their limits in terms of dynamical quantitiessee Theorem 2.4 below.
2.2. Asymptotics of Witten spectral data. Before stating our results on the convergence of the spectral data of the Witten Laplacian, we need to describe a dynamical question which was studied in great details in [17] in the case of Morse-Smale gradient flows -see also [3,22] for earlier related results. Recall that a classical question in dynamical systems is to study the asymptotic behaviour of the correlation function which already appeared in the statement of Theorem 2.1. Following [52,53], it will in fact be simpler to consider the Laplace transform of t → ϕ −t * f , i.e. for Re(z) large enough, One of the consequence of our results from [17,18] is that this Laplace transform admits a meromorphic extension from Re(z) > C 0 (with C 0 > 0 large enough) to C under the assumptions of Theorem 2.1. In [17,19], we also gave an explicit description of the poles and residues of this function under C ∞ -linearization properties of the vector field V f . These assumptions were for instance verified as soon as infinitely many nonresonance assumptions are satisfied. We shall explain in Theorem 5.1 how to recover this result under the weaker assumptions of Theorem 2.1. Proving such a meromorphic extension is part of the study of Pollicott-Ruelle resonances in the theory of hyperbolic dynamical systems. We refer for instance to the book of Baladi [2] or to the survey article of Gouëzel [37] for detailed accounts and references related to these dynamical questions. More specifically, we used a microlocal approach to deal with these spectral problems. We also refer to the survey of Zworski for the relation of these questions with scattering theory [65] from the microlocal viewpoint. Coming back to dynamical systems, the Pollicott-Ruelle resonances are interpreted as the spectrum of −L V f on appropriate Banach spaces of currents. In the following, we shall denote by R k the poles of the meromorphic continuation of R k (z), and by π (k) z 0 the residue at each z 0 ∈ C. These poles are the so-called Pollicott-Ruelle resonances while the range of the residues are the resonant states. They correspond to the spectral data of −L V f on appropriate anisotropic Sobolev spaces of currents and they describe in some sense the structure of the long time dynamics of the gradient flow. Our main spectral result shows that the spectral data of the Witten Laplacian converges to this Pollicott-Ruelle spectrum. More precisely, one has: Theorem 2.4 (Convergence of Witten spectral data). Suppose that the assumptions of Theorem 2.1 are satisfied. Let 0 ≤ k ≤ n. Then, the following holds (2) conversely, any z 0 ∈ R k is the limit of a sequence (−λ (k) j (h)) h→0 + . Moreover, for any z 0 ∈ R, there exists ǫ 0 > 0 small enough such that, for every (ψ 1 , ψ 2 ) ∈ Ω k (M) × Ω n−k (M), Following Theorem 5.1 below, this result shows that the Witten eigenvalues converge, as h → 0, to integer combinations of the Lyapunov exponents. Small eigenvalues are known to be exponentially small in terms of h [43,42,49] but our proof does not say a priori anything on this aspect of the Witten-Helffer-Sjöstrand result. The convergence of Witten eigenvalues could be recovered from the techniques of [43,55] but the convergence of spectral projectors would be more subtle to prove with these kind of semiclassical methods as one would first need to identify the limit. In fact, this Theorem also tells us that, up to renormalization by e f h , the spectral projectors of the Witten Laplacian converge to the residues of the dynamical correlation function. In section 5, we will describe more precisely the properties of these limiting spectral projectors.
2.3. Witten-Helffer-Sjöstrand's instanton formulas. Following [20], we can verify that ((U a ) a∈Crit(f ) , d) generate a finite dimensional complex which is nothing else but the Thom-Smale-Witten complex [60,Eq. (2.2)]. In section 6, we will explain how to prove some topological statement which complements what was proved in [20], namely : Theorem 2.5 (Witten's instanton formula). Suppose that the assumptions of Theorem 2.1 are satisfied. Then, for every pair of critical points (a, b) with ind(b) = ind(a) + 1, there exists 1 n ab ∈ Z such that where n ab counts algebraically the number of instantons connecting a and b.
In particular, the complex ((U a ) a∈Crit(f ) , d) can be defined over Z and realizes in the space of currents the Morse homology over Z.
In the case of locally flat metrics, this relation between the formula for the boundary of unstable currents and Witten's instanton formula follows for instance from the works of Laudenbach in [47], and his proof could probably be revisited to deal with more general metrics. Yet, the proof we will give of this result is of completely different nature and it will be based on our spectral approach to the problem. The main difference with [17,20] is that, in these references, we were able to prove that the complex ((U a ) a∈Crit(f ) , d) forms a subcomplex of the De Rham complex of currents which was quasi-isomorphic to the De Rham complex (Ω • (M), d) but we worked in the (co)homology with coefficients in R. The instanton formula (2) allows us to actually consider ((U a ) a∈Crit(f ) , d) as a Zmodule and directly relate it to the famous Morse complex defined over Z appearing in the litterature whose integral homology groups contain more information than working with real coefficients [35, p. 620].
Coming back to the bottom of the spectrum of the Witten Laplacian, we can define an analogue of Theorem 2.5 at the semiclassical level. For that purpose, we need to introduce analogues of the Helffer-Sjöstrand WKB states for the Witten Laplacian [43,42]. We fix ǫ 0 > 0 small enough so that the range of f,h ) in every degree k is equal to the number of critical points of index k. Then, for h > 0 small enough, we define the following WKB states: where k is the index of the point a. We will show in Proposition 7.5 that, for every critical point a, the sequence (e . As a corollary of Theorem 2.5, these WKB states also verify the following exact tunneling formulas : Corollary 2.6 (Witten-Helffer-Sjöstrand tunneling formula). Suppose that the assumptions of Theorem 2.1 are satisfied. Then, for every critical point a of f and for every h > 0 small enough, where n ab is the same integer as in Theorem 2.5.
The formula we obtain may seem slightly different from the one appearing in [43,Eq. (3.27)] -see also 2 [6, Th. 6.12] when f is a Morse function verifying f (a) = ind(a) for every critical point a. This is mostly due to the choice of normalization, and we will compare more precisely our quasimodes with the ones of Helffer-Sjöstrand in Section 8.

2.4.
A conjecture by Fukaya. As a last application of our analysis, we would like to show that our families of WKB states (U(h)) h→0 + also verifies Fukaya's asymptotic formula for Witten's deformation of the wedge product [32,Conj. 4.1]. This approach could probably be adapted to treat the case of higher order products. Yet, this would be at the expense of a more delicate combinatorial work that would be beyond the scope of the present article and we shall discuss this elsewhere. Recall that this conjecture was recently solved by Chan-Leung-Ma in [12] via WKB approximation methods [43] which are different from our approach.
Let us now describe precisely the framework of Fukaya's conjecture for products of order 2 which corresponds to the classical wedge product ∧ -see also paragraph 7.6 for more details. Consider three smooth real valued functions (f 1 , f 2 , f 3 ) on M, and define their differences : We assume the functions (f 12 , f 23 , f 31 ) to be Morse. To every such pair (ij), we associate a Riemannian metric g ij , and we make the assumption that the corresponding gradient vector fields V f ij satisfy the Morse-Smale property 3 and that they are C 1 -linearizable. In particular, they are amenable to the above spectral analysis, and, for any critical point a ij of f ij and for every 0 < h ≤ 1, we can associate a WKB state U a ij (h). From elliptic regularity, these are smooth differential forms on M and Fukaya predicted that the integral has a nice asymptotic formula whenever the intersection W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ) consists of finitely many points. Note that, for this integral to make sense, we implicitely supposed that Let us explain the difficulty behind this question. After renormalization, a way to solve this conjecture amounts first to prove the convergence of the family of smooth differential formsŨ in the space of currents as h → 0 + . As was already said, we are not aware of a place where this convergence of (renormalized) Witten quasimodes is handled in the literature as this is not the approach followed in [43,6,12] to prove tunneling formulae. Our construction shows that these smooth forms indeed converge to U a ij in the space of currents. The additional difficulty one has to treat in order to answer Fukaya's question is the following. Even if the currents lim h→0 +Ũ a ij (h) exist and even if the wedge product of their limits makes sense, it is not clear that we can interchange the limits as follows: In order to justify this, the second difficulty of Fukaya's question is to show that convergence holds in the appropriate topology involving a control of the wavefront set of the currents.
Without additional assumptions, there is no reason why all this would be true. Fukaya thus requires that the triple (f 12 , g 12 ), (f 23 , g 23 ) and (f 31 , g 31 ) satisfies the generalized Morse-Smale property [45, §6.8]. That is, for every and similarly for any permutation of (12,23,31). Note that, for every Morse function, we associate a priori a different metric. As for the Morse-Smale property, the Kupka-Smale method [46,57] defines an element of D ′n (M) satisfying M U a 12 ∧ U a 23 ∧ U a 31 ∈ Z, and lim h→0 + e − f 12 (a 12 )+f 23 (a 23 )+f 31 (a 31 ) Recall that the integers M U a 12 ∧ U a 23 ∧ U a 31 defined by triple intersections of unstable currents have a deep geometrical meaning. On the one hand, these integers actually count the number of Y shaped gradient flow trees [30, p. 8] as described in subsection 7.5. On the other hand, they give a representation of the cup-product in Morse cohomology at the cochain level -see subsection 7.6 for a brief reminder. Hence, the second part of this Theorem shows that, if we define an analogue of the cup-product at the semiclassical level, then it converges, up to some renormalization factors, to the usual cup-product on the Morse cohomology. We emphasize that the main new property in this Theorem is really the asymptotic formula as h → 0 + . Up to some normalization factors, this is exactly the asymptotic formula conjectured by Fukaya for the WKB states of Helffer-Sjöstrand [43]. Here, our states are constructed in a slightly different manner. Yet, they belong to the same eigenspaces as the ones from [43] -see Section 8 for a comparison. Finally, we note that, going through the details of the proof, we would get that the rate of convergence is in fact of order O(h). However, for simplicity of exposition, we do not keep track of this aspect in our argument. in Crit(f ) 2 , we can define a disc D whose boundary ∂D ⊂ Λ f ∪ 0 is a 2-gon made of the union of two smooth curves e 1 and e 2 joining a and b, e 1 and e 2 are respectively contained in the Lagrangian submanifolds Λ f and 0. Denote by θ the Liouville one form and by ω = dθ the canonical symplectic form on T * M. Then, by the Stokes formula where we choose e 1 to be oriented from b to a. Hence, the exponents in the asymptotic formula of Theorem 2.6 can be interpreted as the symplectic area of the disc D defined by Λ f and the zero section of T * M 5 . In the semiclassical terminology of [42, §15], this quantity is controlled by the Agmon distance associated with the potential d x f 2 g * (x) . Yet, it does not seem to have an interpretation as the action along some Hamiltonian trajectory.
A similar geometric interpretation holds in the case of Theorem 2.7 and the picture goes as follows. We consider a triangle (3-gon) T inside T * M with vertices (v 12 , v 23 , v 31 ) ∈ (T * M) 3 whose projection on M are equal to (a 12 , a 23 , a 31 ). The edges (e 1 , e 2 , e 3 ) are contained in the three Lagrangian submanifolds Λ f 1 , Λ f 2 and Λ f 3 . To go from v 23 to v 12 , we follow some smooth curve e 2 in Λ f 2 , from v 31 to v 23 follow some line e 3 in Λ f 3 and from v 12 to v 31 , we follow some line e 1 in Λ f 1 . These three lines define the triangle T and we can compute which is (up to the sign) the term appearing in the exponential factor of Theorem 2.7. Note that the triangle T does not necessarily bound a disk.
2.6. Convergence of the Witten Laplacian to the gradient vector field. The key observation to prove our different results is the following exact relation [29, equation (3.6)]: where L V f is the Lie derivative along the gradient vector field and ∆ g = dd * + d * d ≥ 0 is the Laplace Beltrami operator. Indeed, one has [29, equations (3.4), (3.5)] : ,h d , which yields (6) thanks to the Cartan formula. Hence, the rough idea is to prove that the spectrum of the Witten Laplacian converges to the spectrum of the Lie derivative, provided that it makes sense. This kind of strategy was used by Frenkel, Losev and Nekrasov [29] to compute the spectrum of L V f in the case of the height function on the canonical 2-sphere. However, their strategy is completely different from ours. Frenkel, Losev and Nekrasov computed explicitely the spectrum of the Witten Laplacian and show how to take the limit as h → 0 + . Here, we will instead compute the spectrum of the limit operator explicitely and show without explicit computations why the spectrum of the Witten Laplacian should converge to the limit spectrum. In particular, our proof makes no explicit use of the classical results of Helffer and Sjöstrand on the Witten Laplacian [43].
Our first step will be to define an appropriate functional framework where one can study the spectrum of L V f + h∆g 2 for 0 ≤ h ≤ h 0 . Recall that, following the microlocal strategy of Faure and Sjöstrand for the study of the analytical properties of hyperbolic dynamical 5 Hence the name disc instantons. systems [28], we constructed in [17] some families of anisotropic Sobolev spaces H m Λ (M) indexed by a parameter Λ > 0 and such that : has discrete spectrum on the half plane {Re(z) > −Λ}. This spectrum is intrinsic and it turns out to be the correlation spectrum appearing in Theorem 5.1. For an Anosov vector field V , Dyatlov and Zworski proved that the correlation spectrum is in fact the limit of the spectrum of an operator of the form L V + h∆g 2 [25] -see also [7,27,64,21] for related questions. We will thus show how to adapt the strategy of Dyatlov and Zworski to our framework. It means that we will prove that the family of operators has nice spectral properties on the anisotropic Sobolev spaces H m Λ (M) constructed in [17]. This will be the object of section 3. Once these properties will be established, we will verify in which sense the spectrum of H h converges to the spectrum of H 0 in the semiclassical limit h → 0 + -see section 4 for details. In [17], we computed explicitely the spectrum of H 0 on these anisotropic Sobolev spaces. Under some (generic) smooth linearization properties, we obtained an explicit description of the eigenvalues and a rather explicit description of the generalized eigenmodes. Here, we generalize the results of [17] by relaxing these smoothness assumptions and by computing the spectrum under the more general assumptions of Theorem 2.1. For that purpose, we will make crucial use of some earlier results of Baladi and Tsujii on hyperbolic diffeomorphisms [3] in order to compute the eigenvalues. Compared to [17,19], we will however get a somewhat less precise description of the corresponding eigenmodes. This will be achieved in section 5. Then, in section 6, we combine these results to prove Theorems 2.1 to 2.6. In section 7, we describe the wavefront set of the generalized eigenmodes and we show how to use this information to prove Theorem 2.7. Finally, in section 8, we briefly compare our quasimodes with the ones appearing in [43]. The article ends with two appendices. Appendix A shows how to prove the holomorphic extension of the dynamical Ruelle determinant in our framework. Appendix B contains the proof of a technical lemma needed for our analysis of wavefront sets.

2.7.
Conventions. In all this article, ϕ t f is a Morse-Smale gradient flow which is C 1linearizable acting on a smooth, compact, oriented and boundaryless manifold of dimension n ≥ 1.
Acknowledgements. When we gave talks on our previous work [17], a recurrent question was about the relation between our results and the Witten Laplacian. We thank the many people who asked us this question as it motivated us to explore seriously this problem. We also warmly thank J.F. Bony, A. Drouot, C. Guillarmou, F. Hérau, F. Laudenbach and B. Merlet for useful discussions related to this work. Finally, we warmly thank the two anonymous referees for their careful and detailed reading: their suggestions allowed us to improve the presentation of this work. The second author is partially supported by the Agence Nationale de la Recherche through the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR project GERASIC (ANR-13-BS01-0007-01).

Anisotropic Sobolev spaces and Pollicott-Ruelle spectrum
In [17,18], we have shown how one can build a proper spectral theory for the operator −L V f . In other words, we constructed some anisotropic Sobolev spaces of currents on which we could prove that the spectrum of −L V f is discrete in a certain half-plane Re(z) > −C 0 . The corresponding discrete eigenvalues are intrinsic and are the so-called Pollicott-Ruelle resonances. Our construction was based on a microlocal approach that was initiated by Faure and Sjöstrand in the framework of Anosov flows [28] and further developped by Dyatlov and Zworski in [23]. As was already explained in paragraph 2.6, we will try to relate the spectrum of the Witten Laplacian to the spectrum of −L V f by the use of the relation (6). Hence, our first step will be to show that our construction from [17] can be adapted to fit (in a uniform manner) with the operator : Note that we changed the sign so that ϕ −t * f will correspond to the propagator in positive times of H 0 . In the case of Anosov flows, this perturbation argument was introduced by Dyatlov and Zworski in [25]. As their spectral construction is slightly different from the one of Faure and Sjöstrand in [28] and as our proof of the meromorphic extension ofĈ ψ 1 ,ψ 2 in [17] is closer to [28] than to [25], we need to slightly revisit some of the arguments given in [28,17] to fit the framework of [25]. This is the purpose of this section where we will recall the definition of anisotropic Sobolev spaces and of the corresponding Pollicott-Ruelle resonances. More precisely, among other useful things, we will prove Proposition 3.1. There exists some constant C 0 > 0 such that, for every 0 ≤ h ≤ 1, the Schwartz kernel of ( H h − z) −1 is holomorphic on Re(z) > C 0 . Moreover, it has a meromorphic extension from Re(z) > C 0 to C whose poles coincide with the Witten eigenvalues for h > 0.
The poles of this meromorphic extension are called the resonances of the operator H h and, for h = 0, they are called Pollicott-Ruelle resonances.
3.1. Anisotropic Sobolev spaces. In [17,18], one of the key difficulty is the construction of an order function adapted to the Morse-Smale dynamics induced by the flow ϕ t f . Before defining our anisotropic Sobolev spaces, we recall some of the properties proved in that reference and we also recall along the way some properties of Morse-Smale gradient flows. We refer to [59] for a detailed introduction on that topic.

Stable and unstable manifolds.
Similarly to the unstable manifold W u (a), we can define, for every a ∈ Crit(f ), A remarkable property of gradient flows is that, given any x in M, there exists an unique Equivalently, the unstable manifolds form a partition of M. It is known from the works of Smale [56] that these submanifolds are embedded inside M [59, p. 134] and that their dimension is equal to n − r(a) where r(a) is the Morse index of a. The Smale transversality assumption is the requirement that, given any x in M, one has Equivalently, it says that the intersection of is empty, where N * (W) ⊂ T * M\0 denotes the conormal of the manifold W. In the proofs of section 3, an important role is played by the Hamiltonian vector field generated by Recall that the corresponding Hamiltonian flow can be written .
The corresponding vector field are denoted by X H f andX H f .

Escape function.
In all this paragraph, V f satisfies the assumption of Theorem 2.1.
We recall the following result [28, Lemma 2.1]: This Lemma was proved by Faure and Sjöstrand in [28] in the case of Anosov flows and its extension to gradient flows require some results on the Hamiltonian dynamics that were obtained in [17,Sect. 3] -see also [18,Sect. 4] in the more general framework of Morse-Smale flows.
As we have a functionm(x; ξ) defined on S * M, we introduce a smooth function m defined on T * M which satisfies We set the order function of our escape function to be It was shown in [17, Lemma 4.1] that it satisfies the following properties (for V u , V s and ǫ > 0 small enough 6 ): 2 on a conic neighborhood of Γ + (for ξ x ≥ 1), and such that there exists R 0 > 0 for which the escape function The order function. We can now define our anisotropic Sobolev space in the terminology of [27,28]. First of all, such spaces require the existence of an order function m N 0 ,N 1 (x; ξ) in C ∞ (T * M) with bounded derivatives which is adapted to the dynamics of ϕ t f . Once we are given an escape function by Lemma 3.3, we set belongs to the class of symbols S ǫ (T * M) for every ǫ > 0. We shall denote this property by G 0 N 0 ,N 1 ∈ S +0 (T * M). We emphasize that the construction below will require to deal with symbols of variable order m N 0 ,N 1 whose pseudodifferential calculus was described in [27,Appendix].

Anisotropic Sobolev currents.
Let us now define the spaces we shall work with. Let 0 ≤ k ≤ n. We consider the vector bundle This allows to define the Hilbert space L 2 (M, Λ k (T * M)) and to introduce an anisotropic Sobolev space of currents by setting m . We refer to [23, App. C.1] for a brief reminder of pseudodifferential operators with values in vector bundles -see also [4]. In particular, adapting the proof of [27,Cor. 4] to the vector bundle valued framework, one can verify that A Mimicking the proofs of [27], we can deduce some properties of these spaces of currents. First of all, they are endowed with a Hilbert structure inherited from the L 2 -structure on where the injections are continuous.

Hodge star and duality for anisotropic Sobolev currents. Recall now that the Hodge star operator [4, Part I.4] is the unique isomorphism
where ., .
(k) g * (x) is the induced Riemannian metric on Λ k (T * M) and where ω g is the Riemannian volume form. In particular, ⋆ k induces an isomorphism from H , whose Hilbert structure is given by the scalar product where, for every ψ 1 in Ω k (M) and ψ 2 in Ω n−k (M), one has the following duality relation: Pollicott-Ruelle resonances. Now that we have defined the appropriate spaces, we have to explain the spectral properties of . Following Faure and Sjöstrand in [28], we introduce the following conjugation of the operator i H h : For similar reasons, we also omit the dependence in k. In any case, the spectral properties of H h acting on the anisotropic Sobolev space H m N 0 ,N 1 k (M) are the same as those of the operator P h acting on the simpler Hilbert space L 2 (M, Λ k (T * M)). We will now apply the strategy of Faure and Sjöstrand in order to derive some spectral properties of the above operators. Along the way, we keep track of the dependence in h which is needed to apply the arguments from Dyatlov and Zworski [25] on the convergence of the spectrum. In all this section, we follow closely the proofs from [28, Sect. 3] and we emphasize the differences.
3.2.1. The conjugation argument. The first step in Faure-Sjöstrand's proof consists in computing the symbol of the operator P h . Starting from this operator, we separate it in two terms which we will treat separately for the sake of simplicity. The key ingredient of [28] is the following lemma [28, Lemma 3.2]: ) whose symbol in any given system of coordinates is of the form 7 , where X H f is the Hamiltonian vector field generating the characteristic flow of L V f in T * M whose definition is recalled in paragrap 3.1.1. The operator Q 2 is a pseudodifferential operator in Ψ 2 (M, Λ k (T * M)) whose symbol in any given system of coordinates is of the form Note that, compared to [28], we study H h rather than i H h . In this Lemma, the notation O(.) means that the remainder is independent of the order function m N 0 ,N 1 , while the notation O m (.) means that it depends on m N 0 ,N 1 . As all the principal symbols are proportional to Id Λ k (T * M ) , the proof of [28] can be adapted almost verbatim to encompass the case of a general vector bundle and of the term corresponding to the Laplace-Beltrami operator. Hence, we shall omit it and refer to this reference for a detailed proof. Recall that more general symbols with values in Λ k (T * M) do not commute and the composition formula does not work as in the scalar case for more general symbols.
In particular, this Lemma says that Q 1 is an element in Ψ 1 (M, Λ k (T * M)). We can consider that it acts on the domain Ω k (M) which is dense in L 2 (M, Λ k (T * M)). In particular, according to [28,Lemma A.1], it has a unique closed extension as an unbounded operator on L 2 (M, Λ k (T * M)). For Q 2 , this property comes from the fact that the symbol is elliptic [61,Chapter 13 p. 125]. In other words, for h > 0, the domain of P h is the domain of Q 2 (namely H 2 (M, Λ k T * M)), while it is given by the domain of Q 1 for h = 0. The same properties also hold for the adjoint operator.
3.2.2. The adjoint part of the operator and its symbol. We now verify that this operator has a discrete spectrum in a certain half-plane in C. Following [28], this will be done by arguments from analytic Fredholm theory. Compared to that reference, note that one aspect of the proof is simpler as, in the Anosov case, the escape function does not decay in the flow direction and one has to use the ellipticity of the symbol in that direction. Here, the escape function decays everywhere. Hence this extra difficulty does not appear. Recall that the strategy from [28] consists in studying the properties of the adjoint part of the operator whose symbol (for every h > 0) is, according to Lemma 3.4, given in any given system of coordinates by where the first three terms correspond to the contribution of Q 1 and the last two terms to the contribution of Q 2 . Here the remainder O(S 0 ) comes from Lemma 3.4 and more precisely from the subprincipal symbol of −L V f in our choice of quantization. We already note that, according to Lemma 3.3, there exists some constant C > 0 independent of m N 0 ,N 1 such that, in the sense of quadratic forms, We can now follow the proof of [28]. First of all, arguing as in [28,Lemma 3.3], we can show that P h has empty spectrum for Re(z) > C 0 , where C 0 is some positive constant that may depend on m but which can be made uniform in terms of h ∈ [0, 1). In other words, the resolvent defines a bounded operator for Im(z) > C 0 . In particular, this shows the first part of Proposition 3.1. Now, we will show how to extend it meromorphically to some half-plane for any choice of N 0 , N 1 large enough.

3.2.3.
From resolvent to the semigroup. Before doing that, we already note that the proof of [28,Lemma 3.3] implicitely shows that, for every z in C satisfying Re(z) > C 0 , one has which will allow to relate the spectrum of the generator to the spectrum of the corresponding semigroup ϕ −t * f . In particular, combining this observation with [26, Cor. 3.6, p.76], we know that, for t ≥ 0 (M) generates a strongly continuous semigroup whose norm verifies (11) ∀t ≥ 0,

Resolvent construction and meromorphic continuation.
We fix some large integer L > dim(M)/2 to ensure that the operator (1 + ∆ g ) −L is trace class. As a first step towards our proof of the meromorphic continuation, we show the following Lemma: Lemma 3.5. There exists some R > 0 such that, if we set defines a bounded operator for Re(z) > C−C N 2 and its operator norm satisfies the estimate At this point of our argument, the fact that the operators are trace class is not that important but it will be useful later on when we will consider determinants.
Proof. For every u in C ∞ (M) and for every 0 ≤ h ≤ 1, we combine (8) and (9) with the Sharp Gårding inequality. This yields: Hence, one gets . Now, observe that, for every ǫ > 0, there exists some constant C ǫ > 0 such that For R = C m C ǫ , let us now set and we find We can now argue like in [28,Lemma 3.3] to conclude that P h + χ R − z is invertible for Applying the Cauchy-Schwarz inequality, we find that This implies that P h + χ R − z is injective. We can argue similarly for the adjoint operator and obtain that ]. Hence, we can conclude that defines a bounded operator for Re(z) > C−C N 2 and that its operator norm satisfies the estimate We now write the following identity: is trace-class for every h 0 as composition of a trace class operator and a bounded one. Moreover, it depends holomorphically on z in the domain {Re(z) > − C N −C 2 } implying that P h − z is a holomorphic family of Fredholm operators for z in the same domain. Finally, we can apply arguments from analytic Fredholm theory to P h − z [63, Th. D.4 p. 418] which yields the analytic continuation of ( P h − z) −1 as a meromorphic family of Fredholm Arguing as in [28,Lemma 3.5], we can conclude that P h has discrete spectrum with finite multiplicity on Re(z) > C−C N 2 . To summarize, one has Lemma 3.6. The operator Since P h is conjugated to H h , the above discussion implies that H h has a discrete spectrum with finite multiplicity on Re(z) > C−C N 2 as an operator acting on H . In particular, this shows the meromorphic continuation of the Schwartz kernel (in the sense [28,Sect. 4] for more details. In the case h > 0, the poles of this meromorphic continuation are exactly the Witten eigenvalues. In particular, they are of the form Our next step will be to show that this Witten spectrum indeed converges to the Pollicott-Ruelle spectrum. 3.2.5. Convention. In the following, we shall denote this intrinsic discrete spectrum by R k (h). They correspond to the eigenvalues of H h acting on an appropriate Sobolev space of currents of degree k. When h > 0, these are the Witten eigenvalues (up to a factor −1) while, for h = 0, they represent the correlation spectrum of the gradient flow, which is often referred as the Pollicott-Ruelle spectrum. Given z 0 in R k (0), we will denote by π (k) z 0 the spectral projector associated with the eigenvalue z 0 which can be viewed as linear map from Ω k (M) to D ′k (M) with finite rank. Recall from [28,Sect. 4] that this operator is intrinsic.
3.2.6. Boundedness on standard Sobolev spaces. Denote by H s (M, Λ k (T * M)) the standard Sobolev space of index s > 0, i.e.
The above construction shows that defines a bounded operator for Re(z) > C−C N 2 which depends holomorphically on z. For the sequel, we will in fact need something slightly stronger: and for every s ∈ [−s 0 , s 0 ], the resolvent : ). Moreover, one has, for every 0 ≤ h ≤ 1, The argument is the same as before except that the order function has to be replaced by m N 0 ,N 1 + s, and a direct inspection of the proof allows to verify that all the constants can be made uniform for s in some fixed interval s ∈ [−s 0 , s 0 ].

3.3.
Pollicott-Ruelle resonances as zeros of a Fredholm determinant. From expression (13), we know that, for Re(z) > C−C N 3 , z belongs to the spectrum of P h if and only if the operator Id + χ R (P h − χ R − z) −1 is not invertible. As we have shown that Moreover, the multiplicity of z as an eigenvalue of P h coincides with the multiplicity of z as a zero of D m (h, z) [24, Prop. B.29].

From the Witten spectrum to the Pollicott-Ruelle spectrum
Now that we have recalled the precise notion of resonance spectrum for the limit operator −L V f , we would like to explain how the Witten spectrum converges to the resonance spectrum of the Lie derivative. This will be achieved by an argument due to Dyatlov and Zworski [25] in the context of Anosov flows -see also [64]. In this section, we briefly recall their proof adapted to our framework.
Remark 4.1. In [25], Dyatlov and Zworski prove something slightly stronger as they obtain smoothness in h. Here, we are aiming at something simpler and we shall not prove smoothness which would require some more work that would be beyond the scope of the present article -see [25] for details in the Anosov case.

4.1.
Convergence of the eigenvalues. We fix N 0 , N 1 , s 0 > 2 and R as in the statement of Lemma 3.7. Using the conventions of this paragraph, we start by studying the regularity of the operator Applying Lemma 3.7 with s 0 > 2, we can compose with the two resolvents and get Still from Lemma 3.7 with s 0 > 2, we find that (15) is bounded for Re(z) > C−C N 3 and uniformly for h ∈ [0, 1] as an operator from L 2 to H −2 . Hence, we have verified that    N 1 (h, .) is a continuous map from [0, 1] to the space of holomorphic functions on Re(z) > C−C N 3 . Fix now z 0 an eigenvalue of P 0 lying in the half-plane Re(z) > C−C N 3 and having algebraic multiplicity m z 0 . This corresponds to a zero of multiplicity m z 0 of the determinant map D m N 0 ,N 1 (0, .) evaluated at h = 0. As the spectrum of P 0 is discrete with finite multiplicity on this half plane, we can find a small enough r 0 > 0 such that the closed disk centered at z 0 of radius r 0 contains only the eigenvalue z 0 . The map }) being continuous, we know that, for every 0 < r 1 ≤ r 0 , for h ≥ 0 small enough (which depends on z 0 and on r 1 ) and for |z − z 0 | = r 1 , Hence, from the Rouché Theorem and for h ≥ 0 small enough, the number of zeros counted with multiplicity of D m N 0 ,N 1 (h) lying on the disk {z : |z − z 0 | ≤ r 1 } equals m z 0 . As, for h > 0, the Witten eigenvalues lie on the real axis, we have shown the following Theorem : . Moreover, given any z 0 in (−∞, 0], there exists r 0 > 0 such that, for every 0 < r 1 ≤ r 0 , for h > 0 small enough (depending on z 0 and r 1 ), the number of elements (counted with algebraic multiplicity) inside In fact, the result also holds for any (ψ 1 , 9 For z 0 / ∈ R k , one has π (k) Together with Theorem 4.4, this Theorem shows that all the spectral data of the Witten Laplacian converge to the ones of −L V f . In particular, it concludes the proof of Theorem 2.4.
Proof. Using Theorem 4.4, it is enough to show the existence of r 0 and to prove convergence for r 1 = r 0 . As before, it is also enough to prove this result for the conjugated operators acting on the standard Hilbert space L 2 (M, Λ k (T * M)). Fix z 0 in R and N 0 , N 1 large enough to ensure that Re(z 0 ) > C−C N

3
. The spectral projector 10 . associated with z 0 can be written [24, Th. C.6] : where C(z 0 , r 0 ) is a small circle of radius r 0 centered at z 0 such that z 0 is the only eigenvalue of P 0 inside the closed disk surrounded by C(z 0 , r 0 ). When z 0 is not an eigenvalue, we choose the disk small enough to ensure that there is no eigenvalues inside it. If we denote by m z 0 the algebraic multiplicity of z 0 (which is eventually 0 if z 0 ∈ R k ), then, for h small enough, the spectral projector associated to P h , has rank m z 0 from Theorem 4.4. We can now argue as in [25,Prop. 5.3] and we will show that, for every ψ 1 in Ω k (M) and every ψ 2 in Ω n−k (M), Once this equality will be proved, we will be able to conclude by recalling that the generalized eigenmodes are independent of the choice of the order function m N 0 ,N 1 used to define A N and by observing that Hence, it remains to prove (16). For that purpose, we use the conventions of Lemma 3.7 and write By construction of the compact operator χ R , the family ( P h + χ R − z) −1 is holomorphic and has no poles in some neighborhood of z 0 as z 0 > C−C N 3 . Therefore, only the term 10 Note that this is eventually 0 if z 0 ∈ R k ( P h − z) −1 χ R ( P h + χ R − z) −1 contributes to the contour integral defining the spectral projector Π (k) λ 0 (h): From Theorem 4.4, we know that, for |z − z 0 | = r 0 and for h small enough, the operator ( P h − z) −1 is uniformly bounded as an operator in B (L 2 (M), L 2 (M)). Moreover, we have seen that the map  (1), The term underbraced in factor of h being uniformly bounded as an operator from L 2 to L 2 for |z − z 0 | = r 0 (as h → 0), we finally find that, for every ψ 1 in L 2 (M, Λ k (T * M)), which concludes the proof of (16).

4.3.
Properties of the semigroup. We would now like to relate the spectral properties of −L V f to the properties of the propagator ϕ −t * f = e −tL V f . To that aim, the classical approach is to prove some resolvent estimate and to use some contour integral to write the inverse Laplace transform. Here, we proceed slightly differently (due to the specific nature of our problem) and we rather study the spectral properties of the time one 11 map ϕ −1 * acting on the anisotropic Sobolev spaces that we have defined. More precisely, using the conventions of Section 3, one has: is a bounded operator whose essential spectral radius is ≤ e are given by Moreover, the spectral projector of λ = e z 0 is given by the projector π (k) z 0 that was defined in paragraph 3.2.5.
Recall that π (k) z 0 corresponds to the spectral projector of −L (k) V f associated with the eigenvalue z 0 and that it is intrinsic (i.e. independent of the choice of the order function m N 0 ,N 1 ) as its Schwartz kernel corresponds to the residue at z 0 of the meromorphic continuation of the Schwartz kernel of (−L V f − z) −1 -see also [28,Th. 1.5].
Proof. Rather than stuying the time one map of the flow, we will study the spectral properties of the hyperbolic diffeomorphism ϕ q := ϕ − 1 q f for every fixed q ≥ 1. The reason for doing this is that we aim at relating the spectral data of ϕ −1 * f to the ones of the generator −L V f -see below.
In the rest of the proof, we verify that ϕ * q has discrete spectrum, with arguments similar to those used for the generator. More precisely, we follow the arguments of [27,Th. 1] applied to the hyperbolic diffeomorphism ϕ q . Precisely, following this reference, we can verify that the order function m N 0 ,N 1 from Lemma 3.2 satisfies the assumptions of [27, Lemma 2]. Then, following almost verbatim [27, section 3.2], we can deduce that the transfer operator defines a bounded operator on the anisotropic space H m N 0 ,N 1 k (M) which can be decomposed as (17) ϕ * q =r m,q +ĉ m,q , whereĉ m,q is a compact operator and the remainderr m,q has small operator norm bounded as : r m,q ≤ e C− C N q 2 (for some uniform C that may be slightly larger than before). Taking q = 1, this shows the first part of the Proposition.
Note that, for every q ∈ N, we can make r m,q arbitrarily small by choosing N large enough. Again, we can verify that the discrete spectrum is intrinsic, i.e. independent of the choice of order function. This is because the eigenvalues and associated spectral projectors correspond to the poles and residues of a discrete resolvent defined as an operator from Ω k (M) to D ′,k (M) as follows. Consider the series +∞ l=0 e −lz ϕ l * q . Then, by the direct bound : we deduce that, for Re(z) large enough, the series  (17) is that the resolvent of ϕ * q (λ − ϕ * q ) −1 : Ω k (M) → D ′,k (M) has a meromorphic extension from |λ| > e C 0 to λ ∈ C with poles of finite multiplicity which correspond to the eigenvalues of the operator ϕ * q [27, Corollary 1]. In other words, (Id − e −z ϕ * q ) −1 : Ω k (M) → D ′,k (M) has a meromorphic extension from Re(z) > C 0 (with C 0 > 0 large enough) to z ∈ C with poles of finite multiplicity. Denote byπ (k) λ,q the spectral projector of ϕ * q associated to the eigenvalue λ which is obtained from the contour integral formula :π where γ is a small circle around λ. This corresponds to the residues of the discrete resolvent at e z = λ.
We will now verify that the second part of the Proposition, i.e. that the spectral data of the diffeomorphism coincide with the ones of the generator. As ϕ * q commutes with −L (k) V f , we can deduce that the range ofπ V f is real, we can deduce that the poles of (Id − e −z/q ϕ * q ) −1 belong to R k ⊂ R modulo 2iπZ. In particular, taking q = 1, this shows that eigenvalues of ϕ −1 * f are exactly given by the expected set. Take now z 0 in R k : it remains to show that (18)π (k) e z 0 ,1 = π (k) z 0 , i.e. that the spectral projectors are the same for both problems. This is where we will crucially use that q is arbitrary. Note already that, as ϕ q q = ϕ 1 , one hasπ λ q ,1 for every q ≥ 1. Recall also that eigenvalues were shown to be real for every q ≥ 1. Hence, π (k) e z 0 /q ,q =π (k) e z 0 ,1 . We now decompose the resolvent (z + L (k) V f ) −1 as follows: For Re(z) large enough, this expression makes sense viewed as an operator from Ω k (M) to D ′,k (M). We have seen that it can be meromorphically continued to C by using the fact that we have built a proper spectral framework and that we may pick N 0 and N 1 arbitrarily large. Consider now a small contour γ around z 0 containing no other elements of R k . Integrating over this contour tells us that, for every q ≥ 1 : As an operator on Ω k (M), we can observe that As a direct Corollary of Proposition 4.6, we get the following result on the asymptotics of the correlation function of the time-one map ϕ −1 * f : Corollary 4.7. Let 0 ≤ k ≤ n then for any z 0 ∈ R k = R k (0), there is an integer d (k) z 0 ≥ 1 s.t. for any Λ > 0, there exist N 0 , N 1 large enough so that for every (ψ 1 , ψ 2 ) ∈ Ω k (M) × Ω n−k (M) and for every integer p ≥ 0, where π Note that together with (11), this expansion could be applied to more general times t which are not necessarily in Z + .

Computation of the Pollicott-Ruelle resonances
In [17], we gave a full description of the Pollicott-Ruelle spectrum of a Morse-Smale gradient flow under certain nonresonance assumptions. Our proof was based on an explicit construction of the generalized eigenmodes and we shall now give a slightly different proof based on the works of Baladi and Tsujii on Axiom A diffeomorphisms [3,2]. In order to state the result, we define the dynamical Ruelle determinant [2, p. 65-68], for every 0 ≤ k ≤ n, as : This quantity is related to the notion of distributional determinants [39, p. 313]. This function is well defined for Re(z) large enough, and, from appendix A, it has a holomorphic extension to C. The zeros of this holomorphic extension can be explicitely described in terms of the Lyapunov exponents of the flow ϕ t f at the critical points of f : where the numbers (χ j (a)) n j=1 are the eigenvalues of L f (a) which is the unique (symmetric) matrix satisfying d 2 f (a) = g a (L f (a)., .). Using the conventions of Section 2, one has : Theorem 5.1. Suppose that the assumptions of Theorem 2.1 are satisfied. Then, for every 0 ≤ k ≤ n, the set of Pollicott-Ruelle resonances is given by Moreover, for every z 0 ∈ R, the rank of the spectral projector π (k) z 0 : Ω k (M) → D ′k (M), is equal to the multiplicity 12 of z 0 as a zero of ζ (k) R (z). Among other things, this result shows that the correlation spectrum depends only on the Lyapunov exponents of the flow. In other words, the global correlation spectrum of a gradient flow depends only on the 0-jet of the metric at the critical points. Thus, this result gives in some sense some insights on Bowen's first problem in [9] from the perspective of the global dynamic of the flow instead of the local one. Note that, if we were interested in the local dynamics near critical points, this could be recovered from the results of Baladi and Tsujii in [3]. Still regarding Bowen's question, we will also verify below that the range of the residues are generated by families of currents carried by the unstable manifolds of the gradient flows. Besides this support property, we do not say much things on the structure of these residues except in the case z 0 = 0 -see Lemma 5.10.
Regarding Section 3, the only thing left to prove is that the eigenvalues and their algebraic multiplicities are given by the zeros of the Ruelle dynamical determinant. As was already mentionned, this result was already proved in [17,19] under stronger linearization assumptions. Our new proof will only make use of the assumptions that the gradient flow is C 1 -linearizable which is necessary to construct our anisotropic Sobolev space but not for the results from [3]. Yet, in some sense, it will be less self-contained as we shall use the results of [3] as a "black-box" while, in the proof of [17], we determined the spectrum by hands even if it was under more restrictive assumptions. Another advantage of the proof from [17] was that it gave an explicit local form of the eigenmodes and some criteria under which we do not have Jordan blocks -see also [19] for slightly more precise results. The key idea compared with [17,19] is to use the localized results of Baladi-Tsujii to guess the global resonance spectrum from the one near each critical point. To go from local to global, we will use the geometry of the stratification by unstable manifolds to glue together, in some sense, these local spectras and make them into a global spectrum.
Before starting our proof, let us recall the following classical result of Smale which will be useful to organize our induction arguments [56] -see [18] for a brief reminder of Smale's works: Theorem 5.2 (Smale partial order relation). Suppose that ϕ t f is a Morse-Smale gradient flow. Then, for every a in Crit(f ), the closure of the unstable manifold W u (a) is the union of certain unstable manifolds W u (b) for some critical points in Crit(f ). Moreover, is a partial order relation on Crit(f ).
In this section, we use the results of Baladi and Tsujii [3]. For that purpose, we treat near every critical point the time one map ϕ 1 := ϕ −1 f of the flow ϕ t f as a hyperbolic 12 When z 0 / ∈ R k , one has π (k) diffeomorphism with only one fixed point. Recall that ϕ t f is a Morse-Smale gradient flow which is C 1 -linearizable, hence amenable to the analysis of the previous sections.

5.1.
Local spectra from the work of Baladi-Tsujii. We start by recalling the results of [3]. Fix 0 ≤ k ≤ n, the degree of the differential forms we are going to consider and a critical point a of f . Note that the reference [3] mostly deals with 0-forms i.e. functions on M, which corresponds to k = 0. General results for transfer operators acting on vector bundles are given in [3, section 2] and [2, section 6.4]. In this paragraph, we consider the transfer operator acting on sections of the bundle Λ k T * M → M of k-forms on M by pull-back : For any open subset U ⊂ M, we will denote by Ω • c (U) the differential forms with compact support in U. Then, one can find a small enough open neighborhood V a of a in M such that, for every has a meromorphic extension to C. This is a straightforward consequence of 13 [3, Theorem 2.1] and [2, Theorem 6.12 p. 178] once we note that smooth differential forms are contained in the Banach spaces of distributional sections of Λ k T * M used in these references. The result of [3] is in fact much more general as it holds for any Axiom A diffeomorphism provided that the observables are supported in the neighborhood of a basic set (which is here reduced to the critical point a). Note that this result could also be deduced from the analysis in [38]. .
Recall that we show in appendix A that these functions are holomorphic in C and that we can compute their zeros. Actually, it has been proved in the litterature [48,36,23,22] for various classes of dynamical systems that the poles of dynamical correlations correspond to the zeros of the dynamical Ruelle determinant. Moreover, for any such pole z 0 , one can find a continuous linear map which is of finite rank equal to the multiplicity of z 0 as a zero of ζ (k) R,a and such that the residue ofĉ ψ 1 ,ψ 2 ,a (z) at z = z 0 is equal to Again, π (k) a,z 0 corresponds to the spectral projector of ϕ * 1 acting on a certain anisotropic Banach space of currents in D ′k (V a ). Now the key observation is to note that the spectral projector π (k) a,z 0 : Ω k c (V a ) → D ′k (V a ), whose existence follows from the work [3], is just the localized version of the global spectral projector π (k) z 0 that was defined in paragraph 3.2.5. Indeed, using Corollary 4.7, we find that, for ψ 1 ∈ Ω k c (V a ), (19) ∀ψ 1 ∈ Ω k c (V a ), π (k) z 0 (ψ 1 ) = π (k) a,z 0 (ψ 1 ), where equality holds in the sense of currents in D ′k (V a ). To see this, one sums over p ≥ 1 in Corollary 4.7 in order to recover the local correlation functionĉ ψ 1 ,ψ 2 ,a and to identify its residues at z = z 0 .
The above means that every element of {z 0 ∈ R : ζ (k) R,a (z 0 ) = 0} contributes to the set R k of Pollicott-Ruelle resonances of the transfer operator acting on k-forms. The objective is to show that there is no other contributions to the set R k . More precisely, we shall prove that R k exactly equals the union over Crit(f ) of local spectras where the zeros are counted with multiplicity.

5.2.
Gluing local spectras. The main purpose of this section is to prove the following statement from which Theorem 5.1 follows: Proposition 5.3. Let 0 ≤ k ≤ n and let z 0 ∈ R. Then, one has a,z 0 .
In particular, as was already explained, one can deduce from [3,2] that Rk π (k) z 0 is equal to the multiplicity of z 0 as a zero of ζ (k) R,a . Note that this may be equal to 0 if z 0 does not belong to the set R k of resonances. 5.2.1. Construction of a "good" basis of Pollicott-Ruelle resonant states. Let z 0 be an element in R k . We fix (U j ) j=1,...mz 0 to be a basis of the range of π (k) z 0 . These are generalized eigenstates of eigenvalue z 0 for −L V f acting on suitable anisotropic Sobolev space of currents of degree k. We aim at showing that we can choose this family in such a way that supp(U j ) ⊂ W u (a) for some critical point a of f (depending on j). Intuitively, we are looking for a "good" basis of generalized eigencurrents with minimal possible support which by some propagation argument should be at least the closure of an unstable manifolds.
We also warn the reader that the notion of linear independence we need for our basis is a little bit subtle and depends on the open subset in which we consider our current. Indeed, we may have some currents which are linearly independent as elements in D ′,k (M) but become dependent when we restrict them to smaller open subsets U ⊂ M. We define: are independent germs at a.

We denote by
In particular, the spectral projector π (k) z 0 can be written as follows: The currents (S j (a, z 0 )) j,a,z 0 are generalized eigenmodes for the dual operator (−L (k) acting on the anisotropic Sobolev space H −m n−k (M). Also, from the definition of the dual basis, one has, for every critical points (a, b), for every indices (j, k) and for every (z, z ′ ) in R k , The purpose of this paragraph is now to prove Lemma 5.5. To that aim, we begin with the following preliminary : Lemma 5.6. Let U 1 ∈ D ′,k (M) be an element inside the range of π (k) z 0 . Then, for every a ∈ Crit(f ), there existsŨ 1 (a) inside the range of π where eachŨ 1 (a) is supported in W u (a).
Proof. By [19,Lemma 7.7] which is a propagation Lemma aimed at controlling supports of generalized eigencurrents, we know that if a current U 1 is identically 0 on a certain open set V then this vanishing property propagates by the flow and U 1 vanishes identically on ∪ t∈R ϕ t f (V ). We set Max(U 1 ) to be the set of critical points a of f such that U 1 ∈ Ran π (k) z 0 is not identically zero near a and such that, for every b ≻ a, U 1 identically vanishes near b. In particular, this means that, for every a in Max(U 1 ), the current U 1 is supported by W u (a) in a neighborhood of a by [19,Lemma 7.8] which gives a control on the support of generalized eigencurrents near maximal elements of Crit(f ).
Remark 5.7. We will implicitely use the fact that anisotropic Sobolev spaces of currents are C ∞ (M)-modules which can be seen as follows : u ∈ H where we used the composition for pseudodifferential operators [27,Th. 8,p. 39] and elements in Ψ 0 (M) are bounded in L 2 .
Let us now decompose U 1 into currents with minimal support. For every critical point a, we set χ a to be a smooth cutoff function which is identically equal to 1 near a and χ a vanishes away from a. Then, for every a in Max(U 1 ), we definẽ U 1 (a) := π (k) z 0 (χ a U 1 ) , and we want to verify thatŨ 1 (a) is supported in W u (a) and that it is equal to U 1 near a.
To that aim, we apply Proposition 4.6 to the test current χ a U 1 (belonging to H m N 0 ,N 1 k for N 0 , N 1 large enough) and to some test form ψ 2 in Ω n−k (M): On the other hand, if we choose ψ 2 compactly supported in M − W u (a), then we can verify that In particular, we find that This implies thatŨ 1 (a) is supported by W u (a). If we now choose ψ 2 to be compactly supported in the neighborhood of a where χ a = 1, then one has where we used the fact that U 1 is supported by W u (a). Applying the asymptotic expansion of Proposition 4.6 one more time to the left hand side of the above equality, we find that z 0 (U 1 ) in a neighborhood of a. We can now definẽ which by construction still belongs to the range of π (k) z 0 and which is now identically 0 in a neighborhood of each b satisfying b a for every a in Max(U 1 ). Then, eitherŨ 1 = 0 in which case U 1 = aŨ 1 (a) is decomposed with this minimal support property and we are done. Otherwise, we repeat the above argument withŨ 1 instead of U 1 and deal with critical points which are smaller for Smale's partial order relation. As there is only a finite number of critical points to exhaust, this procedure will end after a finite number of steps and we will find that where the support ofŨ 1 (a) is contained in W u (a) with some of theŨ 1 (a) that may be taken equal to 0.
We can now turn to the proof of Lemma 5.5. Thanks to Lemma 5.6, we obtain m (k) a (z 0 ) ∈ N, a ∈ Crit(f ) and some family of nontrivial currents (U j,a (z 0 )) a∈Crit(f ),1 j m (k) a (z 0 ) which spans the image of π (k) z 0 and such that each U j,a (z 0 ) is supported in W u (a). Note that our family of currents may not be linearly independent and we can extract a subfamily to make it into a basis of Ran(π (k) z 0 ). However, we recall that we are aiming at some stronger linear independence property than linear independence in D ′k (M). To fix this problem, we start from a critical point a such that (U j (a, z 0 )) j=1,...,m (k) a (z 0 ) are not independent germs at a and, for every b ≻ a, (U j (b, z 0 )) j=1,...,m (k) b (z 0 ) are linearly independent germs at b. We next define a method to localize the linear dependence near a as follows.
Definition 5.8 (Local rank of germs at some point). Consider the family of currents (U j (a, z 0 )) j=1,...,m (k) a (z 0 ) . Define a sequence B a (l) of balls of radius 1 l around a. Consider the sequence r l = Rank(U j (a, z 0 )| Ba(l) ) j=1,...,m (k) a (z 0 ) where each U j (a, z 0 )| Ba(l) ∈ D ′,k (B a (l)) is the restriction of U j (a, z 0 ) ∈ D ′,k (M) to the ball B a (l).We call lim l→+∞ r l the rank of the germs (U j (a, z 0 )) j=1,...,m (k) a (z 0 ) at a.
If lim l→+∞ r l < m  (U j (a, z 0 )) j=1,...,m (k) a (z 0 ) . It means that one can find some j (say j = 1) such that, on the open set V a , Then, we setŨ which is equal to 0 near a. Hence, by propagation [19,Lemma 7.7],Ũ(z 0 ) is supported inside W u (a) − W u (a). Thus, proceeding by induction on Smale's partial order relation, we can without loss of generality suppose that, for every critical point a, the currents (U j (a, z 0 )) j=1,...,m (k) a (z 0 ) are linearly independent germs at a and not only as elements of D ′,k (M). This concludes the proof of Lemma 5.5.

Support of the dual basis.
We would like to show that the dual basis a (z 0 ) defined above contains only currents with minimal support. In fact, we will prove that Lemma 5.9. For all z 0 ∈ R k , the above dual basis satisfies the condition : The above bound on the support of the dual basis actually shows that : (22) supp (S j (a, z 0 )) ∩ supp (U j (a, z 0 )) = {a}.
Proof. Let 0 ≤ k ≤ n and let z 0 ∈ R k . We shall prove this Lemma by induction on Smale's partial order relation . In that manner, it is sufficient to prove that, for every a ∈ Crit(f ) such that the conclusion of the Lemma holds for all 14 b ≻ a, one has . Fix such a critical point a and ψ 1 compactly supported in M −W s (a). Then, we consider V a to be a small enough neighborhood of a which does not interesect the support of ψ 1 and we fix ψ 2 in Ω k c (V a ). From [18, Remark 4.5 p. 17], we know that, if V a is chosen small enough, then ϕ −t f (V a ) remains inside the complementary of supp(ψ 1 ) for t ≥ 0. In particular, for every t ≥ 0, ϕ −t * f (ψ 1 ) ∧ ψ 2 = 0. Applying the asymptotic expansion of Proposition (4.6), we then find that z 0 ,a is spanned by the currents (U j (a, z 0 )| Va ) j=1,...,m (k) a (z 0 ) which concludes the proof of Proposition 5.3.

5.4.
No Jordan blocks for z 0 = 0. Let 0 ≤ k ≤ n. Thanks to Remark A.1, we know that the multiplicity of 0 as a zero of ζ whose support is equal to W u (a). We would now like to verify that we can pick N = 1, or equivalently that there is no Jordan blocks in the kernel. Suppose by contradiction that we have a nontrivial Jordan block, i.e. there exist u 0 = 0 and u 1 = 0 such that We fix a to be a critical point of index k such that u 0 is not equal to 0 near a. Such a point exists as u 0 is a linear combination of the (U b ) b:dimW s (b)=k . Recall from Smale's Theorem that, for every b in Crit(f ), W u (b) − W u (b) is the union of unstable manifolds whose dimension is < dim W u (b). Hence, as u 1 is also a linear combination of the (U b ) b:dimW s (b)=k , we necessarily have that u 1 is proportional to U a near a. In a neighborhood of a, we then have u 0 = α 0 U a (with α 0 = 0) and u 1 = α 1 U a . If we use the eigenvalue equation, we find that, in a neighborhood of a: As U a is not identically 0 near a, we find the expected contradiction.
We next prove the following Lemma on the local structure of eigencurrents in Ker(L V f ) near critical points : Proof. Recall from [56,59] that W u (a) is an embedded submanifold inside M. Then, there is a local system of coordinates (x 1 , . . . x n ) such that W u (a) is given locally near y 0 by {x 1 = . . . = x r = 0} , where r is the index of a. The current of integration on W u (a), for the choice of orientation given by [dx 1 ∧ . . . ∧ dx r ] (see [16, appendix D] for a discussion about orientations of integration currents), reads in this system of coordinates : [W u (a)] = δ 0 (x 1 , . . . , x r )dx 1 ∧ . . . ∧ dx r by [16,Corollary D.4]. Moreover, for all test form ω whose support does not meet the boundary ∂W u (a) = W u (a) \ W u (a), one has for all t ∈ R the identity : , ω since ϕ t f : M → M is an orientation preserving diffeomorphism which leaves W u (a) invariant. This implies that in the weak sense To end this section and as a consequence of (11), Corollary 4.7 and Lemma 5.10, let us record the following improvement of the results from [41,17]: Theorem 5.11 (Vacuum states). Suppose that the assumptions of Theorem 2.1 are satisfied and fix 0 ≤ k ≤ n. Then, for every In this section, we collect the different informations we proved so far and prove the main statements of the introduction except for Theorem 2.7 that will be proved in section 7.
6.1. Proof of Theorem 2.5. Regarding the limit operator, it now remains to show the Witten's instanton formula of Theorem 2.5. For that purpose, we first discuss some orientations issues on curves connecting some pair (a, b) of critical points of f . Choosing some orientation of every unstable manifolds (W u (a)) a∈Crit(f ) defines a local germ of current [W u (a)] near every critical point a and some integration current in D ′,• (M \ ∂W u (a)). Both Theorem 5.11 and Lemma 5.10 show us that each germ [W u (a)] extends into a globally well-defined current U a on M which coincides with [W u (a)] on M \ ∂W u (a). As M is oriented, the orientation of W u (a) induces a canonical coorientation on W s (a) so that the intersection pairing at the level of currents gives M χ[W u (a)] ∧ [W s (a)] = χ(a) for every a ∈ Crit(f ) and for all smooth χ compactly supported near a. Given any two critical points (a, b) verifying ind(a) = ind(b) + 1, recall from [59,Prop. 3.6] that there exists finitely many flow lines connecting a and b. These curves are called instantons and we shall denote them by γ ab . Such a curve is naturally oriented by the gradient vector field V f , hence defines a current of integration of degree n − 1, [γ ab ] ∈ D ′,n−1 (M). Definition 6.1. We define an orientation coefficient σ(γ ab ) ∈ {±1} by the following relation : (24) [ in the neighborhood of some x ∈ γ ab where x differs from both (a, b).
Let us verify that this definition makes sense. From the Smale transversality assumption -see paragraph 3.1.1, one has, for x ∈ γ ab \ {a, b}, the intersection of the conormals N * (W u (a)) and N * (W s (b)) is empty. Hence, according to [44, p. 267] (see also [11]  Proof. Thanks to the transversality assumption, we can use local coordinates (x, y, h) where locally X = {x = 0}, Y = {y = 0} and Z = {x = 0, y = 0} Hence, one has Altogether, this shows that the coefficient σ(γ ab ) is well defined. In fact, using the flow, we see that the formula holds true on M \ {a, b}. We are now ready to prove Theorem 2.5 by setting where the sum runs over instantons from a to b. In other words, the integer n ab counts with sign the number of instantons connecting a and b. We first recall that, as d commutes with L V f and as the currents (U a ) a∈Crit(f ) are elements in 15 Ker(L V f ), we already know that dU a ∈ Ker(L V f ). Hence, one has where the coefficients n ′ ab are a priori real numbers. The goal is to prove that they are indeed equal to the integer coefficients n ab we have just defined. Let a be some critical point of f of index k. Choose some arbitrary cutoff function χ such that χ = 1 in a small neighborhood of a and χ = 0 outside some slightly bigger neighborhood of a. Then the following identity holds true in the sense of currents : where we used the fact that [W u (a)] = U a on the support of χ, Smale's Theorem 5.2 and the fact that χ ∧ dU a = 0 since dU a is a linear combination of the U b with ind(b) = ind(a) + 1. In other words, we used the fact that the current dU a is supported by ∂W u (a). Choose where the sum runs over instantons γ ab connecting a and b. Recall from above that the wedge product makes sense thanks to Smale's transversality assumption. We choose O in such a way that O does not meet the support of dχ, then the following identity holds true : We just proved that, for any function χ such that χ = 1 near a and χ = 0 outside some slightly bigger neighborhood of a, one has Note that this equality remains true for any χ such that χ = 1 near a and χ = 0 in some neighborhood of ∂W u (a) = W u (a) \ W u (a). In particular, it applies to the pull-back ] on the support of ϕ −t * f (χ) by Lemma 5.10. Still from this Lemma, one knows that S b = [W s (b)] on the support of d(ϕ −t * f (χ)). Therefore, one has also For every smooth test (n − k)-form ψ 2 compactly supported in M \ W u (a), we can verify that ) ∧ ψ 2 = 0, which implies that the above reduces to This shows that the complex Ker(L V f ), d generated by the currents (U a ) a∈Crit(f ) is welldefined as a Z-module. Then, we note that tensoring the above complex with R This ends our proof of Theorem 2.5.

6.2.
Proof of the results on the Witten Laplacian. First of all, we note that the result from Theorem 2.1 : is a direct consequence of Theorem 2.4 which yields a convergence of spectral projectors 0 (ψ 1 )∧ψ 2 combined with Theorem 5.11 where the limit term lim t→+∞ M ϕ −t * f (ψ 1 ) ∧ ψ 2 is identified with the term M π (k) z 0 (ψ 1 ) ∧ ψ 2 coming from the spectral projector corresponding to the eigenvalue 0 -see section 5. Hence, it now remains to recall that Theorem 2.4, which claims that the spectral data of the Witten Laplacian converge to the spectral data of −L V f , follows straightforwardly from the content of Theorems 4.4 and 4.5.
We now prove Corollary 2.6 about the Witten-Helffer-Sjöstrand tunnelling formula for our WKB states which becomes a direct corollary of Theorem 2.5. Indeed, our WKB states were defined by using the spectral projector on the small eigenvalues of the Witten Laplacian, i.e.
where k is the index of the critical point. Thanks to Theorem 2.5, we already know Recall now that d f, W f, = W f, d f, and that the spectral projector has the following integral expression f,h ). It is then sufficient to apply the spectral projector to both sides of (27) in order to conclude.
Hence, the (Ũ a (h)) a∈Crit(f ) are linearly independent for h > 0 small enough as the (U a ) a∈Crit(f ) are. After multiplication by e − f h , the same holds for the family (U a (h)) a∈Crit(f ) . Note that the linear independence would also follow from the arguments of paragraph 8 below but our argument here is independent of the Helffer-Sjöstrand construction of quasimodes. Finally, it seems to us that determining the limit of the Helffer-Sjöstrand quasimodes would probably be a delicate task via the semiclassical methods from [43] -see Remark 8.1 below.

Proof of Theorem 2.7
In this section, we give the proof of Theorem 2.7 which states that our WKB states verify the Fukaya's instanton formula. Using the conventions of Theorem 2.7, we start with the following observation: where ǫ 0 > 0 is small enough and where ij belongs to {12, 23, 31}. Hence, we can deduce that  converges as h → 0 + to M U a 12 ∧U a 23 ∧U a 31 , and that this limit is an integer. In particular, we will already have to justify that U a 12 ∧ U a 23 ∧ U a 31 is well defined. The proof will be divided in two steps. First, we will show that (Ũ a ij (h)) h→0 + defines a bounded sequence in some space of currents D ′ Γ ij (M) with prescribed wavefront sets. Then, we will apply theorems on the continuity of wedge products for currents with transverse wavefront sets.
7.1. Background on Fukaya's conjecture. Before proving Fukaya's conjecture on Witten Laplacians, we start with a brief overview of the context in which they appear. These problems are related to symplectic topology and Morse theory, and it goes without saying that the reader is strongly advised to consult the original papers of Fukaya for further details [30,31,32]. In symplectic topology, one would like to attach invariants to symplectic manifolds in particular to Lagrangian submanifolds since they play a central role in symplectic geometry. Motivated by Arnold's conjectures on Lagrangian intersections, Floer constructed an infinite dimensional generalization of Morse homology named Lagrangian Floer homology which is the homology of some chain complex (CF (L 0 , L 1 ), ∂) associated to pairs of Lagrangians (L 0 , L 1 ) and generated by the intersection points of L 0 and L 1 [1, Def. 1.4, Th. 1.5]. Then, for several Lagrangians satisfying precise geometric assumptions, it is possible to define some product operations on the corresponding Floer complexes [1,Sect. 2] and the collection of all these operations and the relations among them form a so called A ∞ structure first described by Fukaya. The important result is that the A ∞ structure, up to some natural equivalence relation, does not depend on the various choices that were made to define it in the same way as the Hodge-De Rham cohomology theory of a compact Riemannian manifold does not depend on the choice of metric g.  (N). This means that the space of cocycles is an invariant of our space. However, there are manifolds which have the same cohomology groups, hence the same homology groups by Poincaré duality, and which are not homeomorphic hence (co)homology is not enough to specify the topology of a given manifold. One direction to get more invariants would be to give some informations on relations among (co)cycles. Recall that (Ω • (M), d, ∧) is a differential graded algebra where the algebra structure comes from the wedge product ∧, and the fact that ∧ satisfies the Leibniz rule w.r.t. the differential d readily implies that ∧ : Ω Poincaré duality, this operation on cohomology geometrically encodes intersection theoretic informations among cycles and gives more informations than the usual (co)homology groups. Algebras of A ∞ type are far reaching generalizations of differential graded algebras where the wedge product is replaced by a sequence of k-multilinear products for all k 2 with relations among them generalizing the Leibniz rule [58].
In perfect analogy with symplectic topology, Fukaya introduced A ∞ structures in Morse theory [30,Chapter 1]. In that case, the role of Lagrangian pairs (L 0 , L 1 ) is played by a pair of smooth functions (f 0 , f 1 ) such that f 0 −f 1 is Morse. Note that it is not a priori possible to endow the Morse complex with the wedge product ∧ of currents since currents carried by the same unstable manifold cannot be intersected because of the lack of transversality. The idea is to perturb the Morse functions to create transversality. Thus, we should deal with several pairs of smooth functions. In that context, Fukaya formulated conjectures [32,Sect. 4.2] related to the A ∞ structure associated with the Witten Laplacian. He predicted that the WKB states of Helffer and Sjöstrand should verify more general asymptotic formulas than the tunneling formulas associated with the action of the twisted coboundary operator d f,h [32, Conj. 4.1 and 4.2]. Indeed, after twisting the De Rham coboundary operator d and getting tunneling formulas for d f,h , the next natural idea is to find some twisted version of Cartan's exterior product ∧ and see if one can find some analogue of the tunneling formulas for twisted products. At the semiclassical limit h → 0 + , Fukaya conjectured that this twisted product should converge to the Morse theoretical analogue of the wedge product modulo some exponential corrections related to disc instantons [33,34]. Hence, as for the coboundary operator, the cup product in Morse cohomology would appear in the asymptotics of the Helffer-Sjöstrand WKB states. The purpose of the next paragraphs is to show that our quasimodes also satisfy the asymptotic formula conjectured by Fukaya for the wedge product. 7.2. Wavefront set of eigencurrents. In this paragraph, we fix V f to be a smooth Morse-Smale gradient vector field which is C 1 -linearizable. Fix 0 ≤ k ≤ n and Λ > 0. Then, following section 3, choose some large enough integers N 0 , N 1 to ensure that for every 0 ≤ h < h 0 , the operator has a discrete spectrum with finite multiplicity on the domain Re(z) > −Λ. Recall from [28, Th. 1.5] that the eigenmodes are intrinsic and that they do not depend on the choice of the order function. Recall also from section 3 that, up to some uniform constants, the parameter Λ has to be smaller than c 0 min{N 0 , N 1 } which is the quantity appearing in Lemma 3.3. Hence, if we choose N ′ 1 ≥ N 1 , we do not change the spectrum on Re(z) > −Λ. In particular, any generalized eigenmode U ∈ H m N 0 ,N 1 k (M) associated with an eigenvalue z 0 belongs to any anisotropic Sobolev space H We also note from the proof of Theorem 4.5 that, for every N ′ 1 ≥ N 1 , Remark 7.1. Note that the proof in section 4 shows that the convergence is of order O(h) but we omit this information for simplicity of exposition.
We now have to recall a few facts on the topology of the space D ′,k Γ − (V f ) (M) of currents whose wavefront set is contained in the closed conic set Γ − (V f ) = ∪ a∈Crit(f ) N * (W u (a)) ⊂ T * M \0 which is defined in paragraph 3.1.1. Note that we temporarily omit the dependence in V f as we only deal with one Morse function for the moment. Recall that on some vector space E, given some family of seminorms P , we can define a topology on E which makes it a locally convex topological vector space. A neighborhood basis of the origin is defined by the subsets {x ∈ E s.t. P (x) < A} with A ∈ R * + and with P a seminorm. In the particular case of currents, we will use the strong topology: where χ is supported on some chart U, where u = |α|=k u α dx α where α is a multi-index, where F is the Fourier transform calculated in the local chart and C is a closed cone such that (supp We emphasize that this definition is given purely in terms of local charts without loss of generality. The above topology is in fact intrinsic as a consequence of the continuity of the pull-back [11, Prop 5.1 p. 211] as emphasized by Hörmander [44, p. 265]. Note that it is the same to consider currents or distributions when we define the relevant topologies since currents are just elements of the form u i 1 ,...,i k dx i 1 ∧ · · · ∧ dx i k in local coordinates (x 1 , . . . , x n ) where the coefficients u i 1 ,...,i k are distributions.
Note from (28) that (Ũ a (h)) 0≤h<1 is a bounded family in the anisotropic Sobolev space H m N 0 ,N 1 k (M) and is thus bounded in H −s (M, Λ k (T * M)) for s large enough. In particular, from definition 7.2, it is a bounded family in D ′,k (M). We would now like to verify that it is a bounded family in D ′,k Γ − (M) which converges in the normal topology as h goes to 0 in order to apply the results from [11]. For that purpose, we can already observe that, for some → 0 as h → 0. In particular, it converges for the strong topology in D ′,k (M). Hence, it remains to discuss the boundedness and the convergence with respect to the seminorms . N,C,χ,α,U . We note that these seminorms involve the L ∞ norm while the anisotropic spaces we deal with so far are built from L 2 norms. This problem is handled by the following Lemma : Lemma 7.4 (L 2 vs L ∞ ). Let N,Ñ be some positive integers and let W 0 be a closed cone in R n * . Then, for every closed conic neighborhood W of W 0 , one can find a constant C = C(N,Ñ, W ) > 0 such that, for every u in C ∞ c (B R n (0, 1)), one has sup We postpone the proof of this Lemma to appendix B and we show first how to use it in our context. We consider the family of currents (Ũ a (h)) 0≤h<h 0 in D ′,k (M) and we would like to show that it is a bounded family in D ′,k Γ − (M) and thatŨ a (h) converges to U a in the normal topology we have just defined. Recall that this family is bounded and that we have convergence in every anisotropic Sobolev space with H Note that, up to shrinking the neighborhood used to define the order function in paragraph 3.1.2 and up to increasing N 1 , we can suppose that m N 0 ,N ′ 1 (x; ξ) is larger than N/2 for any N ′ 1 and for every (x; ξ) in a small conical neighborhood W of (x 0 , ξ 0 ).
Fix now a smooth test function χ supported near x 0 and a closed cone W 0 which is strictly contained in the conical neighborhood W we have just defined. Thanks to Lemma 7.4 and to the Plancherel equality, the norm we have to estimate is where χ 1 ∈ C ∞ is identically equal to 1 on the conical neighborhood W and equal to 0 outside a slightly bigger neighborhood. ForÑ large enough, we can already observe that the second term Ũ a (h) H −Ñ in the upper bound is uniformly bounded asŨ a (h) is uniformly bounded in some fixed anisotropic Sobolev space. Hence, it remains to estimate By composition of pseudodifferential operators and as we chose N ′ 1 large enough to ensure that the order function m N 0 ,N 1 is larger than N/2 on supp(χ 1 ), we can deduce that this quantity is bounded (up to some constant) by Op(A (k) N 0 ,N ′ 1 )Ũ a (h) L 2 which is exactly the norm on the anisotropic Sobolev space. To summarize, this argument shows the following Proposition 7.5. Let V f be a Morse-Smale gradient flow which is C 1 -linearizable. Then, there exists h 0 > 0 such that, for every 0 ≤ k ≤ n and for every a ∈ Crit(f ) of index k, the family (Ũ a (h)) 0≤h<h 0 is bounded in D ′,k Γ − (V f ) (M). Moreover,Ũ a (h) converges to U a for the normal topology in D ′,k Γ − (V f ) (M) as h → 0 + . This proposition is the key ingredient we need in order to apply the theoretical results from [11]. Before doing that, we can already observe that, if we come back to the framework of Theorem 2.7, then the generalized Morse-Smale assumptions ensure that the wavefront sets of the three family of currents are transverse. In particular, we can define the wedge product even for 16 h = 0, i.e. U a 12 ∧ U a 23 ∧ U a 31 defines an element in D ′,n (M) -see below.
7.3. Convergence of products. Given two closed conic sets (Γ 1 , Γ 2 ) which have empty intersection, the usual wedge product of smooth forms extends uniquely as an hypocontinuous map for the normal topology [11, Th. 6.1] . The notion of hypocontinuity is a strong notion of continuity adapted to bilinear maps from E × F → G where E, F, G are locally convex spaces [11, p. 204-205]. It is weaker than joint continuity but implies that the bilinear map is separately continuous in each factor uniformly in the other factor in a bounded subset, which is enough for our purposes 17 .
Remark 7.6. The proof in [11] was given for product of distributions and it extends to currents as D ′,k Γ (M) = D ′ Γ (M)⊗ C ∞ (M ) Ω k (M). Recall that the fact that ∧ is hypocontinuous means that, for every neighborhood W ⊂ D ′,k+l s(Γ 1 ,Γ 2 ) (M) of zero and for every bounded set B 2 ⊂ D ′,l Γ 2 (M ) , there is some open neighborhood U 1 ⊂ D ′,k Γ 1 of zero such that ∧(U 1 ×B 2 ) ⊂ W . The same holds true if we invert the roles of 1 and 2. We note that hypocontinuity implies boundedness, in the sense that any bounded subset of D ′k Γ 1 (M) × D ′l Γ 2 (M) is sent to a bounded subset of D ′k+l s(Γ 1 ,Γ 2 ) (M). This follows from the observation that a set B is bounded iff for every open neighborhood U of 0, B can be rescaled by multiplication by λ > 0 such that λB ⊂ U.
Let us now come back to the proof of Theorem 2.7. This is where we will crucially use the generalized transversality assumptions (5) introduced before Theorem 2.7. We start by considering two points a 12 and a 23 . In order to make the wedge product of U a 12 and U a 23 , one needs to verify that Γ − (V f 12 ) ∩Γ − (V f 23 ) = ∅. To see this, recall that Γ − (V f 12 ) ∩Γ − (V f 23 ) is equal to which is a subset of (a,b,c)∈Crit(f 12 )×Crit(f 23 )×Crit(f 31 ) (T W u (a) ∩ T W u (c) + T W u (b)) ⊥ = ∅, where the last equality is the content of our generalized Morse-Smale transversality assumption. Combining Proposition 7.5 with the hypocontinuity of the wedge product, we find that (Ũ a 12 (h) ∧Ũ a 23 (h)) 0≤h<h 0 is a bounded family in D ′,k+l s(Γ − (V f 12 ),Γ − (V f 23 )) (M), where k is the index of a 12 and l is the one of a 23 . Moreover, as h → 0, one has U a 12 (h) ∧Ũ a 23 (h) →Ũ a 12 (0) ∧Ũ a 23 (0) = U a 12 ∧ U a 23 , for the normal topology of D ′,k+l Then, as our three vector fields verify the generalized Morse-Smale assumptions (5), we can repeat this argument with the spaces D which concludes the proof of Theorem 2.7 up to the fact that we need to verify that M U a 12 ∧ U a 23 ∧ U 31 is an integer. 7.4. End of the proof. In order to conclude the proof of Theorem 2.7, we will show that W u (a 12 ), W u (a 23 ) and W u (a 31 ) intersect transversally at finitely many points belonging to the intersection W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ). Then, the fact that M U a 12 ∧ U a 23 ∧ U 31 is an integer will follow from Lemma 6.2. Let us start by showing that any point in the intersection W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ) must belong to W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ). We proceed by contradiction. Suppose that x belongs to W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ) but not to W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ). From Smale's Theorem 5.2, it means that there exists critical points b 1 , b 2 and b 3 such that x belongs to W u (b 1 ) ∩ W u (b 2 ) ∩ W u (b 3 ) with W u (b i ) ⊂ W u (a ij ) and at least one of the i verifies dim(W u (b i )) < dim(W u (a ij )). This implies that dim(W u (b 1 )) + dim(W u (b 2 )) + dim(W u (b 3 )) < 2n.
Using transversality one more time, we get that dim(W u (b 1 ) ∩ W u (b 2 ) ∩ W u (b 3 )) = dim(W u (b 1 )) + dim(W u (b 2 )) + dim(W u (b 3 )) − 2n < 0, which contradicts the fact that W u (b 1 ) ∩ W u (b 2 ) ∩ W u (b 3 ) is not empty. Hence, we have already shown that W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ) = W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ), and it remains to show that this intersection consists of finitely many points. For that purpose, observe that as the intersection W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ) is transverse, it defines a 0-dimensional submanifold of M. Thus, x belonging to W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ) is an isolated point inside W u (a ij ) for the induced topology by the embedding of W u (a ij ) in M, for every ij in {12, 23, 31}. Moreover, as this intersection coincides with its closure, we can deduce there can only be finitely many points in it and this concludes the proof of Theorem 2.7.
7.5. Morse gradient trees. Now that we have shown that the limit in Theorem 2.7 is an integer, let us give a geometric interpretation to this integer in terms of counting gradient flow trees. From the above proof, we count with orientation the number of points in W u (a 12 ) ∩ W u (a 23 ) ∩ W u (a 31 ). In dynamical terms, such a point x 0 corresponds to the intersection of three flow lines starting from a 12 , a 23 and a 31 and passing through x 0 . This represents a one dimensional submanifold having the form of a Y shaped tree whose edges are gradient lines. Hence, the integral M U a 12 ∧ U a 23 ∧ U a 31 counts the number of such Y shaped gradient tree given by a triple of Morse-Smale gradient flows. 7.6. Cup products. These triple products can be interpreted in terms of the cup-products appearing in Morse theory [30,31]. Indeed, we can naturally define a bilinear map: where f 13 = −f 31 . This can be done as follows. Its coefficients in the basis (U a 12 , U a 23 , U a 13 ) are given by M U a 12 ∧ U a 23 ∧ U a 31 ∈ Z. Note that, as f 13 = −f 31 , one has 18 U a 31 = S a 13 . Hence, these coefficients can be written in a more standard way as M U a 12 ∧ U a 23 ∧ S a 13 ∈ Z.
As we defined a map on all the generators of the Morse complex, this endows the whole Morse complex with a product which is defined, for every (U 1 , U 2 ) ∈ Ker(−L Note that, compared with the classical theory where these maps are defined in an algebraic manner [31], our formulation is of purely analytical nature. Thanks to the remark made in paragraph 7.5, one can verify that these algebraic and analytical maps are exactly the same. It is already known that the map m 2 induces a cup product on the cohomology. Let us reprove this fact using our analytical approach rather than an algebraic proof. Recall from [17,20] that the Morse complex is quasi-isomorphic to the De Rham complex (Ω(M), d) via the spectral projector associated with the eigenvalue 0. Hence, it is sufficient to show that m 2 induces a well-defined map on the cohomology of the Morse complex. To see this, we fix (U 1 , U 2 ) in Ker(−L In particular, the spectral projector can be written as Hence, from the definition of our WKB state U a (h), one has which can be expanded as follows: Everything now boils down to the calculation of More precisely, if we are able to prove that (30) α abb ′ (h) = δ ab δ bb ′ α a (h)(1 + O(h)) + O(e −C 0 /h ), for a certain α a (h) = 0 depending polynomially on h (that has to be determined), then, after gathering all the equalities, we will find that showing that our quasimodes are at leading order equal to the ones of Helffer and Sjöstrand (up to some normalization factor). Let us now prove (30) by making use of the results from [43]. First of all, we write that According to [43,Eq. (1.38)], we know that where Ψ b (h) is a certain "Gaussian state" centered at b defined by [43,Eq. (1.35)] and C 0 is some positive constant. Thus, as f (x) ≥ f (a) on the support of U a , we have with C 0 > 0 which is slightly smaller than before. We now introduce a smooth cutoff function χ a which is equal to 1 in a neighborhood of a and we write Thanks to [43,Th. 1.4] and to the fact that the support of U a is equal to W u (a), we know that the second term, which corresponds to the points which are far from a, is also exponentially small. Hence Thanks to Lemma 5.10, this can be rewritten as Using  . This shows that our eigenmodes are not a priori normalized in L 2 . To fix this, we would need to set, for every critical point a of f ,