Witten Deformation and analytic continuation

Transcription

1 Witten Deformation and analytic continuation Dan Burghelea Department of Mathematics Ohio State University, Columbus, OH Bonn, June 27, 2013

2 Based on joint work with STEFAN HALLER Departments of Mathematics Univ of Vienna, Austria YOONEON LEE Department of Mathematics Inha University Seoul, Koreea.

3 Motivation Witten deformation permits to select a FINITE SEGMENT of the spectrum of the Laplace-Beltrami operators (eigenvalues, eigenforms) of a closed Riemannian manifold (M, g) when additional data (Morse function/closed one form) are given. It is hoped that this segment is TOPOLOGICALLY RELEVANT and COMPUTABLE.

4 Notation When one writes A(t) one understands a real, or complex or vector valued analytic function in t R. When one writes A(z) one understands a complex or vector valued holomorphic=analytic function in an open neighborhood of R in C.

5 Witten deformation The Witten Deformation is a one parameter family (t R or C) of de Rham type complexes (Ω (M), d (t)) where d (t) = e tf d (e tf ) = d + tdf f : M R or more general d (t) = d +tω ω Ω 1 (M), dω = 0 (1) (2)

6 The Witten Laplacians In the presence of a Riemannian metric g, the family (Ω (M), d (t)) produces the family q (z) of elliptic operators on Ω (M) defined by q (z) = q + z(l X + L X ) + z2 X 2.

7 Essential features: q (z) is a polynomial with coefficients self-adjoint operators For t R, q (t) = (d q+1 (t) d q (t) + d q 1 (t) (d q (t)) is zero order perturbation of the standard laplacian q.

8 Here L X is the Lie derivative w. r. to the vector field X = grad g f, L X its formal adjoint. L X + L X is of order zero. q (t) is zero order perturbation of q

9 Theorem (Relich, Kato) There exist the real valued analytic functions λ q i (t) and of vector valued analytic functions ω i (t) Ω q (M) i = 1, 2,, t R, each with a holomorphic extension λ q i (z), ω q i (z) to an open neighborhood of R inside C, so that: 1 each λ i (z) is an eigenvalue of q (z) and λ i (z) exhaust all eigenvalues 2 ω i (z) are eigenforms corresponding to the eigenvalue λ i (z), of norm 1 for z = t real number λ q i (z) and ω q i (z) are referred to as branches (of eigenvalues and eigenforms).

10 The case of Morse functions/forms Suppose that f or ω has all critical points non degenerate. Theorem (Witten) If (M n, g) is a closed Riemannian manifold then there exist the constants C 1, C 2, C 3, T 0 > 0 so that for t T 0 (1) spec q (t) (C 1 e C2t, C 3 t) =, and (2) the number of eigenvalues of q (t) counted with multiplicity in the interval [0, C 1 e C2t ] is equal to N q, the number of critical points of index q.

11 Important consequences: 1 Exactely N q branches λ q i (t), i = 1, 2, N q go exponentially fast to 0 = the virtually small branches, while all others go at least linearly fast to = the large branches (but no more than quadratically fast). Of them exactly β q (β N q ) are identically zero. 2 They define a canonical orthogonal decomposition (Ω (M), d (t)) = (Ω (M) sm (t), d (t)) (Ω (M) la (t), d (t)), real analytic in t.

12 Ultimately Ω sm extends to Ω sm(z), d (z) a holomorphic family of finite dimensional complexes quasi-isomorphic to Ω (M), d (z). (Ω(M) sm, d(z)), is the span of the eigen-forms corresponding to the virtually small branches (Ω la (t), d (t)) is a real analytic family (for t R) of acyclic complexes. (Ω(M) la, d(t)) is the span of the eigen-forms corresponding to large branches.

13 Theorem For a residual set of smooth functions/ closed one form (in C r, r 2)topology 1 each not identically zero virtually small branch of eigenvalues of is simple, 2 exactly β q resp. β N q (Novikov Betti numbers ) are identically zero and 3 there is a canonical (up to multiplication by ±1) choices of virtually small branches of eigenforms, which are orthonormal, hence a canonical base (up to multiplication by ±1) of Ω (M) sm (z), d (z). If one restrict to virtually small branches "residual set" can be replaced by "open and dense set".

14 Integration theory Let f : M R be a Morse function s.t. (f, g) satisfies Morse Smale condition. For x Cr q (f ) let Wx M be the unstable manifold of X = grad f at the rest point x. Proposition For any ω Ω q (M) and any x Cr q (f ) the integral Wx convergent and defines a continuous linear map ω is Int x : Ω q (M) C. for the Frechet topology on Ω q (M).

15 Let (C (M, X), δx ) be the Morse-Thom complex defined by the partition of M in cells ( the unstable manifolds of X.) The integration provides the quasi isomorphism Let Int : (Ω (M), d ) (C (M, X), δ X ) Int (z) := (Ω (M), d (t)) e zf (Ω (M), d ) Int (C (M, X), δ X ) Int (z) is holomorphic in z and quasi-isomorphism for any z.

16 Theorem 1. Int (t) restricted to Ω (M) sm (t), w.r. to the canonical bases satisfies Int q sm(t) = (Id + O(e Ct ))D q (t) where D q (t) = (t/π) 1/4 q/2 Id. 2.The maps a q (z) = det Int q sm(z) is holomorphic in an open neighborhood U of R in C and the a priory rational function a(z) = q (a q (z)) ( 1)q has no zeros and no poles on R.

17 The Virtually small spectral package w.r. to a Morse function f. The restrictions of the branches convergent to 0 define the virtually small spectral package of (M, g) provided by (M, g, f ). VS q (M, g, f )) = {λ q 1, λq 2, λq N q ; ω q 1, ωq N q ; a q, a}. a q (z) is derived from integration of ω q i (z) on the unstable sets.

18 Let T q = (TorH q (M; Z)). There are exactly N q β q positive real eigenvalues (counted with their multiplicity) in VS q (M, g, f ). Theorem T ( 1) q q = q ( λ q i ) ( 1)q a λ q i VS q,λ q i 0 Additional application For a generic vector field X which admits a Lyapunov cohomology class using analytic continuation and results of BH one can regularize the number of closed trajectories (which are actually countably many) and express this number in terms of the topology of the underlying manifold. This number a priory integer is actually real.

19 Conjectures. 1 a q (t) 0 for any t R. 2 If f s, 0 s 1 is a smooth family of Morse functios with (f s, g) Morse Smale for any s them VS(M, g, f s ) is constant in s. Item (2) of the above conjecture became recently a theorem. This implies that the virtually small package VS(M, g, f ) can be calculated with arbitrary accuracy.