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Witten's Proof of Morse Inequalities ApPENDIX Witten's Proof of Morse Inequalities o. Introduction In his paper "Supersymmetry and Morse Theory," E. Witten [Witl) pre­ sented an analytic proof of Morse inequalities. It is the purpose of this appendix to introduce his proof. According to de Rham-Hodge Theory, the Betti numbers of a differential manifold M are related to the dimensions of harmonic forms. In the first section, we shall briefly review Hodge theory. The idea of Witten's proof is to introduce a perturbed elliptic complex for a given Morse function f as follows: dp_l dp ••• ---+ AP-l(M) -----+ AP(M) -----+ AP+1(M) --+ ••. e-t! 1 e-t ! 1 e-t! 1 P - 1 P d t d ••• ---+ AP-l(M) -----+ AP(M) ~ AP(M) --+ •.. with df = e-t ! dpetf, p = 0,1,2, ... ,n - 1, and to compute the perturbed Laplacian We range the eigenvalues of D.f as follows 0::; Af(t) ::; A~(t) ::; ... ::; A~(t) ::; .... The Hodge theory implies that there are j3p eigenvalues equal to 0, where j3p is the pth Betti number, p = 0, 1, ... ,n - 1. In local coordinates, 274 Witten's Proof of Morse Inequalities Assume that x· is a critical point of f. We approximate ~f in a neighbor­ hood of x· and obtain the approximate perturbed Laplacian: where {J.'i} are the eigenvalues of the Hessian d? f(x·). If we put all these ~t,,,,' together (in the product space) for all critical 1 points {xi} as a new operator, and range all eigenvalues as follows: o :::; tef :::; t~ :::; ... , the number of zero-eigenvalues is then proved to be the number of critical points with Morse indices p. The simple version of Morse inequalities {3p:::; mp := #{xi I ind(f,xi) = p}, then will be proved if we have the following asymptotics: lim A~(t) =~, k = 1,2, .... t->oo t k A revised elliptic complex is used to prove the final version of the Morse inequalities. The material of this appendix is based on [Witl), F. Annik [Annl), G. Henniart [Henl) and B. Helffer [Hell). 1. A Review of Hodge Theory Let (M, g) be a compact, connected, Coo, n-Riemannian manifold without boundary. T M denotes the tangent bundle; T",M denotes the tangent space at x EM; T· M denotes the cotangent bundle; T; M denotes the cotangent space at x EM; APT· M denotes the anti-symmetric tensor product of T· M. The section of APT· M is called a p-form over M. Coo (APT· M), the set of all Coo p-forms, is denoted by AP(M). V w E AP(M), in the local coordinates (Xl, ... ,xn), is expressed as follows: W= We write gii = 9 (a~i' a~,)' gii = g(dxi, dxi ), i :::; i,j :::; n. Then (gii) = (gii )-1 is positive definite. 1. Hodge Theory 275 We may also extend 9 to p-forms: where (k1 , ••• ,kp) runs over (1, ... ,p), and is even ck,,,·k,, = ±1 if (k1, ... ,Pk ) 1, .. · , p is odd. The differential operator d: AP(M) ...... AP+l(M) is defined to be (1) a linear operator, i i &a" ... ,p i it i " (2) d (a·., ....." (x)dx ' !I. ••• !I. dx ,,) = "'~L.Jl=1 &x) dx !I. dx !I. •.. !I. dx • From the definition, it is easily seen that (i) rP = O. (ii) d(w !I. B) = dw !I. B + (-I)Pw !I. dB, 'r/ wE AP(M), 'r/ BE Aq(M). The Hodge star operator *: AP(M) ...... An-p(M) is defined as follows: (1) *(a(x)w + b(x)8) = a(x) * w + b(x) * B, (2) * (dxi, !I. .•• !I. dxip) = '" .,. ".. .... gi,;>, ... gi"j>" dxi,,+, !I. L.Jk, .. ·k" "10, Jop ',,+1 'n .•. !I. dxin , where 1 ::; j1 < ... < jp ::; n, 1 ::; ip+l < ... < in ::; n, {ill'" ,jp, ip+l .. , in} is a permutation of {I, ... n}, {k1, ... ,kp} is a per- mutation of {I, ... ,p}, 'fJl,."ln = Igj1/ 2cl, ..... In' Igl = det(gij), and 1,2, ... , n \ is even Cl,,,.ln = ±1 if ( ill i2,' .. , in) is odd. Then we have (i) *1 = 'fJ, *'fJ = 1 where 'fJ = Igl 1/2dx1 !I. ..• !I. dxn, (ii) * * w = (-I)p(n-p)w 'r/ w E AP(M), (iii) g(w, 8)'fJ = wll. (*8) 'r/ w, 8 E AP(M). Claim. We only want to verify this identity for w = dxi, !I. ••• !I. dxi " , 8 = dxi1 !I. ... !I. dxip , with 1 ::; i1 < ... < ip ::; n, 1 ::; it < ... < jp ::; n. LHS = L ck, ... k"gi';O, ... gi"io" . 'fJ, RHS= 276 Witten's Proof of Morse Inequalities Since {i1, ... ,in} must be a permutation of {I, ... ,n}, and {t1···tp , ip+l'" in} is a permutation of {I, ... , n} with t1 < t2 < ... < tp, we have i1 = tt, ... , ip = tp. Therefore, 1 2 idk in RHS = IgI / ci''''in L ck,,,.kpg , ... gipjkPdxit /\ ... /\ dx k,,,.k p = Ig1 1/ 2 L ck,,,.kpgidkt ... gipjkpdx1 /\ ... /\ dxn k,,,.k p =LHS. The scalar product on AP(M) is defined by (w.8) = 1M g(w. 8)1/ = 1M w /\ (*8). It is real, symmetric, bilinear and positive definite. The completion of AP(M) with respect to ( , ) is denoted by Ai2(M). It is a Hilbert space. The codi./Jerential operator dO: AP(M) -> AP-1(M) is defined to be the adjoint operator of d with respect to ( , ), i.e., (d*w, p) = (w, dp) v wE AP(M), V p E AP-1(M). Note. The scalar products on both sides are different! Basic properties of d*. (i) d* = (_1)n(p-1)+l * d*. Claim. (d"w,p) = (dp,w) = (_I)p(n-p)(dp. ** w) = 1M dp /\ (*w) = (-I)P 1M P /\ (d * w) = (_I)p+(p-1)(n-p+l) 1M p /\ (* * d * w) = (_I)(p-1)(n-p)+1(p,*d*w) (ii) dOdO = O. 1. Hodge Theory 277 The Laplacian. tl.P: AP(M) -. AP(M) is defined to be d*d + dd*. A p­ form w satisfying tl.Pw = 0 is called a p-harmonic form. Denote HP(M) = ker(tl.P ). Example. p = O. V f E COO(M), tl.°f = d*df = -lgl-1/ 2 ~ (lgI1/2gij~f) . E. OXj OXi ',J This is the Laplace-Beltrami operator on (M, g). We have (i) Let D(tl.P), be the space of Wi-Sobolev sections of the vector bundle APT* M. Then tl.P is positive and self-adjoint. Claim. V O,w E AP(M), we have (tl.Pw,O) = (d*d + dd*)w, 0) = (dw, dO) + (d*w, d*O) = (w, (d*d + dd*)O) = (w, tl.PO). Friedrich's extension provides the self adjointness. The positiveness is ob­ vious. (ii) tl.P is an elliptic operator. See (vi) in the following paragraph. (iii) tl.P possesses only discrete spectrum, i.e., it has only eigenvalues O'(tl.P) = {Ai < A~ < ... }, with Ai 2: 0, A~ -. +00 as k -. 00, and each eigenvalue has only finite multiplicity. This follows from Riesz-Schauder theory. Exterior and interior product. V wE A1(M), w/\: dxi1 /\ ... /\ dxip f-+ W /\ dX i1 /\ ... /\ dxip , AP(M) -. AP+1(M), P i",:dxi1/\ .. ·/\dxip f-+ E(-1)i+1g(w,dxij)dxil/\ j=l ... /\ dxij /\ ... /\ dxip , AP(M) -. AP-1(M), are called the exterior and interior product with respect to w respectively. These products are extended to AP(M) linearly. (i) One has V 0 E AP-1(M), V t/J E AP(M), (w /\ 0, t/J) = (0, i",t/J). (ii) V f E AO(M), V 0 E AP(M), d*(fO) = fd*O - idlB. 278 Witten's Proof of Morse Inequalities Claim. V t/J E AP-1(M), (t/J, d*(fO) = (dt/J,IO) = (fdt/J,O) = (d(ft/J) - dl 1\ t/J, 0) = (ft/J, d*O) - (t/J, idlO) = (t/J, Id*O - idfO). Claim. We may verify this for W1 = dx1 and W2 = dx 1 or dx2 in suitable coordinates. (iv) The principal symbol of the differential operator d is O'Ld = if.l\, where f. = '£';=1 f.jdxj , (6, ... ,f.n) E T* M. Therefore Therefore, O'Ldw = i '£ f.jdxj 1\ W, V wE AP(M). Note. For d, the symbol O'd = the principal symbol O'Ld. (v) O'Ld* = i . ie, where f. = '£ f.jdxj . Claim. Letting 8, W denote the Fourier transforms (in local coordinates) for 0 E AP(M) and t/J E AP+1(M) respectively, (0, d*t/J) = (dO, t/J) = (O'Ld· 8,;j) = i(f.1\ (f,;j) = i(8, ie;j). Therefore ((f,O'd*.;j) = i((f,ie;j). (vi) O'LI1 = -1f.12. Claim. O'dd*d + dd*) = O'Ld*O'Ld + O'LdO'Ld* = - (f. 1\ ie + ie . f.1\) = -1f.12. By choosing f. along an axis, say f. = (f.1>'" ,f.n) = 1f.le1> 1. Hodge Theory 279 Elliptic complex. Let Mn be a Riemannian manifold, and let E = {EiHi' be a family of vector bundles over M. Let d = {di } 0- 1 , i = 0,1, ... ,n - 1, be a family of pseudo differential operators (1/JDO) of order r, satisfying (1) di+ 1di = 0, (2) V x E M, V ~ E T;M \ {9}, the sequence is exact, where lTLd(x,~) is the principal symbol of the 1/JDO d. We say that (E, d) is an elliptic complex. Example (de Rham). We define (E, d) as follows: n-l { } E= {N'T"M}np=o' d = dp p=o' where dp is the differential operator. This is an elliptic complex. Claim. We only want to verify the exactness of the sequence °---+ A°T* M _u-=L....:do=---'l AIT* M Since V W E AP(M), lTLd(x, ~)wx = i~ /\ Wx , where ~ = Ej=1 ~idxi' and it is easy to see that (Choose ~ along an axis, say ~ = el> if il = 1 if il > 1 Therefore, ker lTLd(x,~) = Span{ei1 /\ ••• /\ eip 11 = il < ..
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