Wittens Laplacian and the Morse Inequalities Background Morse - PowerPoint PPT Presentation

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Witten’s Laplacian and the Morse Inequalities Background Morse Inequalities Witten’s Idea Gianmarco Molino Local Approximation University of Connecticut Weak Morse Inequalities December 1, 2017 Strong and Polynomial Morse Inequalities References

Witten’s Laplacian and the Morse Inequalities 1 Background Gianmarco Molino 2 Morse Inequalities Background Morse Inequalities 3 Witten’s Idea Witten’s Idea Local 4 Local Approximation Approximation Weak Morse Inequalities 5 Weak Morse Inequalities Strong and Polynomial Morse Inequalities 6 Strong and Polynomial Morse Inequalities References

Morse Theory Witten’s Laplacian and the Morse Inequalities Morse Theory is the study of critical points of a smooth Gianmarco function f : M → R . Molino A smooth manifold M is a topological manifold with Background compatible smooth atlas (in the following all manifolds are Morse Inequalities assumed to be n -dimensional, smooth, oriented, closed, Witten’s Idea and without boundary.) Local A critical point q ∈ M of a smooth function f : M → R is Approximation a zero of the differential df . Weak Morse Inequalities The Hessian H f ( q ) of f at a critical point q ∈ M is the Strong and Polynomial matrix of second derivatives. (Independent of coordinate Morse Inequalities system at critical points.) References

Morse Functions Witten’s Laplacian and the Morse Inequalities A smooth function f : M → R is called Morse if its critical Gianmarco points are isolated and nondegenerate (that is, the Hessian Molino of f is nonsingular.) Background Remark: Nondegenerate critical points are necessarily Morse Inequalities isolated. Witten’s Idea The Morse index m q of a critical point q is the dimension Local of the negative eigenspace of H f ( q ). Approximation The i -th Morse number M i is the number of critical points Weak Morse Inequalities with Morse index i . Strong and Polynomial Remark: The Morse numbers are invariant under Morse diffeomorphism. Inequalities References

Betti Numbers Witten’s Laplacian and the Morse Inequalities Associated to every smooth manifold is the sequence of Gianmarco Betti numbers β i , 0 ≤ i ≤ n defined as Molino dR ( M ) = dim { α ∈ Ω i : d α = 0 } Background β i = dim H i Morse { d β : β ∈ Ω i − 1 } Inequalities Witten’s Idea where Ω i is the space of differential i -forms. This sequence is a Local Approximation topological invariant, and notably Weak Morse Inequalities n Strong and � ( − 1) i β i χ ( M ) = Polynomial Morse i =0 Inequalities References

Morse Inequalities Witten’s Laplacian and the Morse Inequalities The Weak Morse Inequalities are a classical result, proved using Gianmarco geometric techniques by Milnor [1]. Molino Theorem (Weak Morse Inequalities) Background Morse Let f : M → R be Morse. Then for any 0 ≤ i ≤ n Inequalities Witten’s Idea β i ≤ M i Local Approximation Weak Morse and moreover Inequalities n � Strong and ( − 1) i M i χ ( M ) = Polynomial Morse i =0 Inequalities References

Morse Inequalities Witten’s Theorem (Polynomial Morse Inequalities) Laplacian and the Morse Inequalities Let f : M → R be Morse. Then for any t ∈ R there exists a Gianmarco sequence of nonnegative integers Q i such that Molino Background n n n − 1 M i t i − β i t i = (1 + t ) � � � Q i t i M t − P t := Morse Inequalities i =0 i =0 i =0 Witten’s Idea Local Approximation Theorem (Strong Morse Inequalities) Weak Morse Inequalities Let f : M → R be Morse. Then for any 0 ≤ k ≤ n Strong and Polynomial Morse Inequalities k k � ( − 1) i + k β i ≤ � ( − 1) i + k M i References i =0 i =0

Edward Witten Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background In 1982, Edward Witten published a proof [2] of the Morse Morse Inequalities, essentially using the idea of the flow generated by Inequalities a Morse functions, with an intuition deriving from Quantum Witten’s Idea Mechanics. He was awarded a Fields Medal in 1990, partially Local Approximation for this work. Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Supersymmetry Witten’s Laplacian and the Morse Inequalities Gianmarco Molino A Hilbert space H is called supersymmetric if there exists a decomposition H = H + ⊕ H − and maps Background Morse Q 1 : H + → H − Inequalities Witten’s Idea Q 2 : H − → H + Local H , ( − 1) F : H → H Approximation Weak Morse Inequalities that obey certain symmetry rules. Strong and Polynomial Morse Inequalities References

Supersymmetry Witten’s Laplacian and the Morse Inequalities Notice that the space of differential forms is supersymmetric, Gianmarco Molino splitting into even and odd forms, with Background n / 2 n / 2 − 1 Morse Ω ∗ = Ω 2 p ⊕ Inequalities � � Ω 2 p +1 Witten’s Idea i =0 i =0 Local Q 1 = d + δ Approximation Weak Morse Q 2 = i ( d − δ ) Inequalities H = ∆ = d δ + δ d Strong and Polynomial Morse Inequalities References

Witten Deformation Witten’s Laplacian and the Morse Inequalities Gianmarco Witten generalizes this idea, conjugating d with the flow e tf Molino for t ≥ 0 , f Morse. Background d t = e − tf de tf Morse Inequalities δ t = e tf δ e − tf Witten’s Idea Local ∆ t = d t δ t + δ t d t Approximation Weak Morse It is easily verifiable that Ω ∗ is still a supersymmetric space Inequalities Strong and using these deformed operators. Polynomial Morse Inequalities References

Hodge Theory Witten’s To understand the motivation for the Witten Laplacian, we Laplacian and the Morse need to look to Hodge Theory. Inequalities Gianmarco Theorem (Hodge Theorem) Molino Background For 0 ≤ i ≤ n, the maps Morse Inequalities h i : ker ∆ i → H i dR ( M ) Witten’s Idea ω �→ [ ω ] Local Approximation Weak Morse are isomorphisms. Inequalities Strong and Polynomial Corollary Morse Inequalities For 0 ≤ i ≤ n, References β i = dim ker ∆ i

Hodge Theory Witten’s Laplacian and The corollary is the starting point for analytic approaches to the Morse Inequalities the Betti numbers. We will prove the Hodge Theorem using a Gianmarco heat flow argument, the following lemma will be necessary. Molino Lemma Background Morse For all smooth differential forms ω , Inequalities Witten’s Idea ∆ e − t ∆ ω = e − t ∆ ∆ ω Local Approximation and Weak Morse Inequalities de − t ∆ ω = e − t ∆ d ω Strong and Polynomial Morse The first claim follows from self-adjointness of ∆, while the Inequalities second can be proved using the uniqueness of solutions to the References heat equation in L 2 .

Hodge Theory Witten’s Laplacian and the Morse Inequalities Proof (Hodge Theorem). Gianmarco Let { ω i } i ∈ N be an orthonormal basis for Ω p with ∆ ω i = λ i ω i . Molino This can be done since M is compact, and follows from the Background Spectral Theorem for compact, self-adjoint operators applied to Morse Inequalities the heat operator e − t ∆ . Then Witten’s Idea Local N Approximation t →∞ e − t ∆ ω = lim � a i e − t λ i ω i = � lim a i ω i Weak Morse t →∞ Inequalities i i =0 Strong and Polynomial where { ω i , . . . , ω N } is an orthonormal basis for ker ∆ p . Thus Morse Inequalities as t → ∞ , ω flows to its harmonic component. References

Hodge Theory Witten’s Laplacian and the Morse Proof (Hodge Theorem). Inequalities Gianmarco Then for any closed differential form ω , Molino � t Background e − t ∆ ω − ω = ∂ t ( e − t ∆ ω ) dt Morse Inequalities 0 � t � � Witten’s Idea e − t ∆ δω dt = d − Local 0 Approximation Weak Morse so � t Inequalities � � e − t ∆ ω = ω + d e − t ∆ δω dt − ∈ [ ω ] Strong and Polynomial 0 Morse Inequalities which implies that heat flow preserves the cohomology class of References a form.

Hodge Theory Witten’s Proof (Hodge Theorem). Laplacian and the Morse Inequalities We see that each cohomology class contains a harmonic form, Gianmarco � t Molino t →∞ e − t ∆ ω = ω − lim e − t ∆ δω dt = ω − d ∆ − 1 δω lim t →∞ d Background 0 Morse Inequalities (which is well-defined, after showing δω is independent of Witten’s Idea ker ∆) , now finally we show that the form is unique. Assume Local there exist harmonic forms η 1 � = η 2 with [ η 1 ] = [ η 2 ] Then Approximation Weak Morse Inequalities η 1 = η 2 + d θ Strong and Polynomial δη 1 = δη 2 + δ d θ Morse Inequalities 0 = δ d θ References so 0 = � θ, δ d θ � = � d θ, d θ � = � d θ � 2

Witten Laplacian Witten’s Lemma Laplacian and the Morse For any t ≥ 0 , Inequalities β i = dim ker ∆ i Gianmarco t Molino Background Proof. Morse Inequalities Observe that d t e − tf = ( e − tf de tf ) e − tf = e − tf d , which implies Witten’s Idea that e − tf : Ω i → Ω i is an isomorphism making the following Local Approximation diagram commute, Weak Morse Inequalities d d d Ω i +1 Ω i · · · · · · Strong and Polynomial Morse e − tf e − tf Inequalities d t d t d t Ω i Ω i +1 · · · · · · References so β i = dim ker ∆ i = dim ker ∆ i t .