An alternative stochastic proof for the Atiyah-Singer index theorem

Transcription

1 An alternative stochastic proof for the Atiyah-Singer index theorem Kefeng Liu Center for Mathematics Science, Zhejiang University, Hangzhou 3127, P.R.China Haoran Zhao Center for Mathematics Science, Zhejiang University, Hangzhou 3127, P.R.China Abstract According to Watanabe [6], by generalized Wiener functional we could give a simple stochastic proof of the index theorem for twisted Dirac operator (the Atiyah-Singer index theorem) on Clifford module. MR Subject Classification: 58J2 Chinese Library Classification: O Introduction The purpose of this work is to provide a simple stochastic proof of the index theorem for the twisted Dirac operator by generalized Wiener functional. In 195s and 196s, Atiyah, Singer, Hirzebruch, Bott have established the index theorem([1], [12]), which is an interesting theorem in geomery and analysis. Later on, the refined local index theorems have been established by several authors Atiyah-Bott-Patodi [2], Patodi [19], [2], Gilkey [13], [14]. The proofs of the local index theorems are complicated, which need knowledge from analysis, geometry and algebra. On the classical method proofs of the local index theorem, we could refer to Atiyah-Bott-Patodi [2], Getlzer [4], [5], Berline-Getzler-Vergne [11], Yu [24], [25], Roe [17], Gilkey [14], Ponge [3] etc, where Berline-Getzler-Vergne [11] is a good reference on heat equation method for index theorem. In 1984, Bismut [7] has introduced stochastic analysis method to prove the local index theorem. Bismut s work is significant. After that, several proofs by stochastic method have been provided, for example, Watanabe [6], Baudoin [21], Hsu [28], [29]. In [6], Watanabe has proved the Gauss-Bonnet-Chern theorem, the Signature index theorem by generalized Wiener Keywords: index theorem, Brownian motion, heat equation, stochastic analysis, Dirac operator, Malliavin calculus, Feynman-Kac formula

2 functional which is included in stochastic analysis. In Shigekawa-Ueki [22], Riemann- Roch theorem has been proved by this method. In this work, we prove the twist Dirac index theorem ( the Atiyah-Singer theorem) by Watanabe s method. For the convenience of readers, we state the twisted Dirac index theorem. Let p(t, x, y) is the heat kernel for Dirac heat equation on Clifford module E of M, F E /S is the twisting curvature of E (see [11] p117), then 1 lim Str(p(t, x, x))dx = ( t 2πi ) d 2 { Â(M)ch(F E /S )} d (x), where M is a d-dimensional Riemannian manifold, d is even, E is a Clifford module on M, Â(M) is A genus of M, ch(f E /S ) = Str E /S [exp( F E /S )] is relative Chern character of E. Getzler [5] has used the rescale to get the limit of supertrace of the heat kernel, where an important estimate was obtained by Stroock s Feynman-Kac formula. Most of his methods are classical. Bismut [7] s stochastic method contains the splitting of Wiener space and the implicit function theorem. Our method is from Watanabe [18], [38], different from Bismut [7] s. Of course our idea is motivated by many works which contain Bismut [7], Hsu [29], Baudoin [21] etc. By virtue of the Malliavin calculus and generalized expectation of generalized Wiener functional, we have the probabilistic representation of the heat kernel: p(ε 2, x, y) = E[M ε e ε δ y (X ε )], where δ is a generalized function: Dirac delta function, M, e, X are random variables, δ(x) is generalized Wiener functional got by the composition of δ with the non-degenerate smooth Wiener functional X, E is generalized expectation (On precise definition of these notation, please read the following context or [18], [38].), then we could compute the limit of supertrace by the asymptotic expansion with respect to ε in Sobolev space of Wiener functional. geometry. Our proof doesn t involve the principal bundle and stochastic differential Our paper is organized as follows. In the first part, we simply introduce Clifford algebra and Dirac operator. In the second part, we give Feynman-Kac formula for parabolic systems. In the third part, we state some basic definitions and theorems on the Malliavin calculus. In the fourth part, we give the proof by generalized Wiener functional and its asymptotic expansion. 2

3 2. Clifford algebra and Spin bundle For the convenience of the readers, we would introduce the knowledge on Clifford algebra simply. On details the readers may consult Berline-Getzler-Vergne [11], Jost [26]. Let V be an oriented d dimensional Euclidean space. We assume that d = 2l is even. The tensor algebra is T (V ) = R V (V V ) By the equivalent relations uv + vu = 2 < u, v > in tensor algebra, we could define Clifford algebra Cl(V ). Let {e i } be an oriented basis of V and c i be the image of the e i in the Cl(V ). Then c i1...c ik, k d, 1 i 1 <... < i k d, forms a basis of Cl(V ). Let Cl + (V ) be the subspace of elements of even degrees, Cl (V ) be the subspace of the elements of odd degree. There is a unique graded complex Clifford module S = S + S, called the spinor module, such that Cl(V ) C = End(S), wiith Cl + (V ) S + S +, Cl + (V ) S S, The supertrace on Cl(V ) is defined by Cl (V ) S S +, Cl (V )S + S. Str(a) = tr S +a tr S a, where a is regarded as an element of End(S). If a = a i1...i k c i1...c ik, k d 1 i 1 <...<i k d then Str(a) = ( 2i) l a 12...d. (1.1) Let E = E + E is a superspace, we could define Str(b) = tr + (b) tr (b) for b End(E). Then we could define supertrace on End(S E), such that Str(a b) = Str(a) Str(b). There is a linear isomorphism ( not algebra isomorphism) between the wedge product (V ) and Cl(V )(see [11] Pro 3.5) which is denoted by c. The group Spin(V ) is the group 3

4 obtained by exponentiating the Lie algebra C 2 (V ) inside the Clifford algebra C(V ). Let us assume that M be a Riemannian manifold which has a spin structure, the spin principal bundle Spin(M) has a connection which is induced from SO(M). spinor bundle L is defined to be the associated bundle Spin(M) Spin(d) S, which also has a linear induced connection. The The Clifford bundle Cl(M) has a action on L, and C Cl(M) = End(L). A Clifford module E on an even diemnsional Riemannian manifold M is Z 2 -graded bundle on M with a graded action of the bundle C(M) on it. operator D is defined to be D = c(e i ) ei, where {e i } is a local orthonormal frame, {e i } is its dual base. Dirac Lemma 1 (Lichnerowicz formula): Let E be a Clifford module on M, s be the scalar curvature of M, F E /S be the twisting curvature of E (see [11]), then where s is scalar curvature on M. D 2 = c(f E /S ) s 4, 3. Stroock s version of Feynman-Kac formula for the parabolic systems Stroock has given Feynman-Kac formula for parabolic systems (see [34]). According to his technique, we consider a slightly different parabolic system in form. Let f, φ are C n valued functions, {gij}, 1 i, j d is a real positive symmetric matrix, Γ i, 1 i d, c are bounded complex n n matrixs valued function defined in R d whose all partial derivatives are bounded, b i,1 i d are bounded scalar functions defined in R d whose all partial derivatives are bounded. f t = 1 2 gij ( x i + Γ i)( x j + Γ j)f bi ( x i + Γ i)f cf, f(, x) = φ(x), x R d (t, x) (, ) Rd Lemma 2: We assume that σk i σj k = gij, du i x(t) = σk i (u x(t))dw k bi dt, de(t) = e t Γ i (u x (t)) du i x(t), dm(t) = 1 2 M(t)e(t)c(u(t))e 1 (t)dt, e = 1, M t = 1, u x () = x, 4

5 where {ω k } is a Brownian motion on R d, e t, M(t) is a n n dimensional matrix valued process, stands for Stratonovich integral. Then f(t, x) = E(M t e t φ(u x (t))). Proof: we could apply the Ito formula to M s e s φ(t s, u x (s)). 4. The Malliavin calculus The materials in this section are from ([6], [18], [38]). Due to limited space,we couldn t state too many definitions and theorems on the Malliavin calculus. For more details we could refer to the above works. Let (W, F, P ) be the d-dimensional Wiener space: W = {W : C[, 1] R d ; w() = }. With the supremum norm, W is a Banach space. Let H be the subspace of W consisting of all h which is absolutely continuous and has square integrable derivatives. If we define the inner product: < h, g >= i h i (t) g i (t)dt, then H is a Hilbert space. We define h[w] by h[w] = i dh i dt (s)dwi (s) for h H. h[w] is a Gaussian random variable, E(h 1 [w]h 2 [w]) =< h 1, h 2 >. For a measurable function F (w) on W, if there is a polynomial g such that F (w) = g(h 1 [w], h 2 [w], h 3 [w],..., h n [w]), then we call it polynomial functional, the total space of polynomial functional is denoted by S. Let L 2 (P ) be the linear space of square integrable random variables on W. There is a decomposition called Wiener s homogeneous chaos: L 2 (P ) = C C 1... C n... We denote the progection of L 2 (P ) onto C n by J n. Let operators L, (I L) s be defined on S by LF = ( n)j n F, n (I L) s F = n (1 + n) s J n F, where for polynomial functional F S, the above sum has only finite items. Let p,s be defined by F p,s = (I L) s 2 F p, 5

6 where p is the usual L p norm. Let D s p is the completion of S under the norm p,s. Then D s p Ds p, p p, s s, (D s p) = D s q, p 1 + q 1 = 1. We define D = s> 1<p< D s p, D = s> 1<p< D s p, D = k=1 1<p< D k p, D = k=1 1<p< D k p. Similarly, for a separable Hilbert space E, D (E), D (E), we may define the functional space S(E). For a polynomial functional F, the derivative DF S(H) is given by: DF = n i g(h 1 [w], h 2 [w],..., h n [w]) i h i(s)ds. By computation, we could have: < DF, h >= d dϵ F (ω + ϵh) ϵ=. The operator D could be uniquely extended to a linear operator by Meyer s work D : D (E) D (H E). The following theorem plays an important role in Malliavin calculus. Theorem 1 (Malliavin[36], Ikeda-Watanabe [18] etc): If F D (R d ), let σ ij =< DF i (ω), DF j (ω) > D, i, j = 1,..., d. (σ ij ) is called the Malliavin covariance of F. If [det(σ ij )] 1 1<p< L p, F is called to be nondegenerate in the sense of Malliavin, then the law of F has the C density. 6

7 5. The probabilistic proof In this section we would use generalized Wiener functional and its asymptotic expansion, which are introduced in [6], [18], [38]. The most notations in this section are the same as the above works. The asymptotic expansion of the fundamental solution (heat kernel) of heat equation is determined by local conditions ([11], [18]). We consider the linear space R d, equipped with {g ij }, which is the same as {δ ij } except a bounded set containing the original point. We may choose a normal coordinate around the original point, {e i }, 1 i d, is a basis of R d. Let E = L W be a Clifford module on R d, W is a s-dimensional graded vector bundle. Parallel transport along the rays from the original point gives a trivialization of W. Let ω be the connection of W in this frame, 1 2 F ( i, j )dx i dx j be the curvature of W. We have (see[6], [11]) g ij (x) = δ ij 1 3 R imkj()x m x k + O( x 3 ), and and Γ i jk (x) = 1 3 R ijkm()x m R ikjm()x m + O( x 2 ), σ i k = δi k R imnk()x m x n + O( x 3 ), and ω i (x) = 1 F ( i, j )()x j + O( x 2 ). 2 j Let c i = σi ke k, then {c i } is an oriented orthonormal frame. Let { σ k i } be the inverse of {σ i k }, define Γ i jk = ( e j c k, c i ), then Γ i jk = x j (σl k ) σi l + σl k σi sγ s jl, Γ i jk (x) = (1 6 R imjk() R ijmk())x m R ijkm()x m R ikjm()x m + O( x ) 2 = 1 2 R ikjm()x m + O( x 2 ) Let Cl(R d ) denote Clifford algebra of Euclid space R d. Because of the scaling property of Brownian motion, we consider the following stochastic differential equation on R d (Cl(R d ) End(R s )), with a parameter ε dx i (t) = εσk i (X(t))dwk (t) ε2 2 gjk (X(t))Γ i jk (X(t))dt, de(t) = e(t) Γ i dx i (t), i, j, k = 1, 2,...d. (X(), e() = (, 1), 7 (1)

8 where e(t) is a Clifford algebra valued process, Γ i = 1 4 Γ k ij c jc k ω i. Denote this solution by r ε (t) = (X ε (t), e ε (t)), then r ε (t) r(ε 2 t), where stands for the laws are the same, r(t) is the solution when ε = 1. dm(t) = ε2 2 M(t)e(t)[c(F E /S ) + s 4 ]e 1 (t)dt, M() = 1, For the Dirac heat equation on spin bundle, { f t = 1 2 D2 f, f(, x) = φ(x). we can conclude that the solution of heat equation f(t, x) = E[M(t)e(t)φ(X x (t))], where φ is bounded initial function, the heat kernel p(ε 2,, ) = E[M ε (1)e ε (1)δ (X ε (1))], where δ is the Dirac δ function at the original point which is introduced in generalized functions theory. According to [6], [18], [38], We have X ε (1) = εw(1) + O(ε 2 ) (2) in D as ε. in D as ε. where δ (X ε (1)) = ε d δ (w(1)) + O(ε d+1 ) (3) θ ε (t) = e ε (t) = 1 + t t Γ i dx ε (s) i e ε (s) dθ ε (s), = ε 2 (C ij (t)c i c j F (t)) + O(ε 3 ) in D as ε, where C ij (t) = 1 8 R ijmk() t F (t) = 1 F ( i, j )() 2 j w k (s) dw m (s), t w j (s) dw i (s). 8

9 e ε (1) = 1 + = 1 + θ ε (1) + = 1 + θ ϵ (1) + e ε dθ ε (s) t1 e ε (t 2 ) dθ ε (t 1 ) θ ε (t 1 ) dθ ε (t 1 ) + t1 t2 = 1 + A 1 + A A l + O(ε 2l+2 ) (4) e ε (t 3 ) dθ ε (t 3 ) dθ ε (t 2 ) dθ ε (t 1 ) in D (Cl(R d ) End(R s )) as ε, where assuming C(t) = C ij (t)c i c j F (t), A m = t1 t2 = ε 2m t1 in D (Cl(R d ) End(R s )) tm 1 tm 1 in D (Cl(R d ) End(R s )), where B n = ( 1) n ε 2n dθ ε (t m ) dθ ε (t m 1 )... dθ ε (t 1 ) dc(t m ) dc(t m 1 )... dc(t 1 ) + O(ε 2m+1 ) M ε (1) = 1 + B 1 + B B l + O(ε 2l+2 ) (5) t1... tn 1 = ε2n n! [1 2 ( c(f E /S ()) s() 4 )]n + O(ε 2n+1 ), where b(t) = e(t)[ c(f E /S ) s 4 ]e 1 (t). Lemma 3 ([18]): 1 2 bε (t n ) 1 2 bε (t n 1 ) bε (t 1 )dt n dt n 1...dt 1 E[δ (w(1)) Φ(w)] = (2π) l E[Φ(w) w(1) = ], Φ D. By (3), (4), (5), lemma 3, Str[p(ε 2,, )]e 1 e 2... e d = ( 2i) d 2 = 1 (2iπ) d 2 m+n= d 2 m+n= d 2 E[ 1 m! (1 4 Ω ij E[ 1 m! (1 2 Ω ij as ε where Ω ij = 1 2 R ijkm()e k e m. w i (s) dw j (s)) m δ (w(1))] Str E /S[ F E /S ()] n 2 n + O(ε) n! w i (s) dw j (s)) m w(1) = ] Str E /S[ F E /S ()] n + O(ε) n! It is obvious that the computation result should be the polynomial of Ω ij. Because twoforms are communicated under the product, {Ω ij } could be regarded as a skew symmetric 9

10 scalar matrix. While the Brownian bridge measure is O(d)-invariance, we may assume {Ω ij } is in block diagonal form such that Ω 2i 1,2i = 2x i, Ω 2i,2i 1 = 2x i, i = 1,..., l, and other entries are. simple. The independence in law among {w i } will make the computation There is a stochastic area formula which is given by P Levy ([18]): E{exp[ix( w 1 (s)dw 2 (s) So formally we may get that: E{exp[x( Therefore This is w 1 (s)dw 2 (s) E[exp( 1 2 Ω ij lim Str[p(t,, t )]e1 e 2... e d = perhaps we have completed the proof. Acknowledgements. w 2 (s)dw 1 (s))] w 1 (1) =, w 2 (1) = } = w 2 (s)dw 1 (s))] w 1 (1) =, w 2 (1) = } = 2x e x e x = ix sinh( ix) = w i (s) dw j Ω (s)) w(1) = ] = det( 2 sinh( 1 2 = Â(M). 2 Ω))1 1 (2πi) {Â(M)()str d E /S[exp(F E /S ())]} d, 2 x sinh x x sin x The second author would like to thank to his advisor Professor Kefeng Liu for his support and encouragement and Professor Hongwei Xu for his support and help with him. He is also grateful to maths science center in Tsinghua University for their supports and help. He would also like to thank to Professor Elton Hsu for his help and work, Professor Yu Yanlin for his teachings in Zhejiang University. He also thank to his friends and classmates Xu Hao, Yin Fangliang, Zhao Liang, Zhao Entao, Ma Chao, Rao Sheng, Zhang Dong, Leng Yan, Ru Qiang, Lu Weijun, Yu Baohua in school. Finally we should be grateful to Professor Watanabe for his establishing generalized Wiener functional and its asymptotic expansions. References [1] M. Atiyah, Collected works, Clarendon Press, Oxford, (1988). [2] M. Atiyah, R Bott, V K Patodi. On the Heat Equation and the Index Theorem. Inventiones mathematicae, Volume 19, Number 4 (1973),

11 [3] M. F. Atiyah, R. Bott, A. Shapiro, Clifford modules, Topology, 3, Suppl.1, (1964), [4] E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys. (1983), [5] E. Getzler, A short proof of the local Atiyah-Singer index theorem. Topology vol. 25, no1, (1986) [6] S. Watanabe, Short time asymptotic problems In Wiener functional integration theory. Application to heat kernels and index theorems, In: Korezlioglu H, Ustunel A S ed, Lecture Notes in Mathematics, Volume1444, Springer (199), [7] J. M. Bismut. The Atiyah Singer theorems: A probabilistic approach. Journal of Functional Analysis. I. The index theorem, Volume 57, Issue 1, June (1984), II. The Lefschetz fixed point formulas, Volume 57, Issue 3, July 1984, 329C348. [8] J. M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Inventiones Mathematicae, Volume 83, Number 1 (1986), [9] J. M. Bismut, Large Deviations and the Malliavin Calculus, Birkhäuser, (1984). [1] N. Berline, M. Vergne, A proof of Bismut local index theorem for a family of Dirac operators, Topology vol. 26, no4, (1987), [11] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 24. [12] F. Hirebruch, Topological methods in algebra geometry, Springer, (1966). [13] P. B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Advances in Math, V1 number 3 (1973), [14] P. B. Gilkey, Curvatrue and the eigenvalues of the Dolbeault complex for Kaher manifolds. Advances in Math 11, (1973), [15] Witten, Supersymmetry and Morse theory, Journal of differential geometry, 17, (1982),

12 [16] H. P. McKean, and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 no. 1 (1967), 43C69. [17] J. Roe, Elliptic operators, topology and asymptotic methods, 2nd ed, Pitman Research Notes in Mathematics Series, vol. 395, Longman, Harlow, (1998). [18] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland/Kodansha,(1989). [19] V. K. Patodi, Curvature and the eigenforms of the Laplace operator, Journal of Differential Geometry 5 (1971) [2] V. K. Patodi, An analytic proof of the Riemann-Roch-Hirzebruch theorem, journal of Differential Geometry (1971), [21] F. Baudoin, Chen series and Atiyah-Singer theorem, Journal of Functional Analysis, Vol. 254, 2, (28) [22] I. Shigekawa, and N. Ueki, A stochastic approach to the Riemann-Roch theorem, Osaka journal of Maths 26(1989), [23] L. Alvarez-Gaume, Supersymmetry and the Atiyah-Singer index theorem. Communication.Mathematics Phys, 9, (1983). [24] Y. L. Yu, Local index theorem for Dirac operators. Acta Mathematica Sinica Volume 3, Number 2 (1987), [25] Y. L. Yu, The index theorem and the heat equation method, Peking University press, (1995), In Chinese. [26] J. Jost, Remannian geometry and geometric ananlysis Springer, (22). [27] W. P. Zhang, The local Atiyah-Singer index theorem for families of Dirac operators. Lecture Notes In mathematics, 1369, Springer, (1989). [28] Elton. Hsu, Stochastic analysis On manifolds. American Mathematical Society, (22), [29] Elton. Hsu, Stochastic local Gauss-Bonnet-Chern theorem. Journal of Theoretical Probability, Volume 1, Number 4 (1997)

13 [3] R. Ponge, A new short proof of the local index formula and some of its applications. Comm. Math. Phys. 241 (23), no. 2-3, [31] D. S. Freed, Geometry of Dirac operators, preprint, Downloadable at [32] S. Rosenberg, Laplacian on Riemannian manifolds, Lodon Math Society, (1998). [33] S. S. Chern, Lectures on Differential Geometry. Peking University Press (21). (In Chinese) [34] D. W. Stroock, On certain systens of parabolic equations Communications on pere and applied mathematics VOL.XXIII, (197), [35] D. W. Stroock, S. R. S. Varadhan, Multidimensional diffusion process, Springer, (1979). [36] P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proc. Intern. Symp. SDE Kyoto 1976, Kinokuniya (1978), [37] S. Watanabe, Lectures on stochastic differential equations and Malliavin calculus, Springer-Verlag, (1984). [38] S. Watanabe, Analysis of Wiener functionals (Malliavin Calculus) and its applications to heat kernels, Annals of probability, 15 (1987),1-39. [39] I. Karatzas, and S. E. Shreve, Brownian motion and stochastic calculus, Springer, (1995). [4] D. Nualart, Malliavin calculus and related topics, Springer, (26). [41] Z. Y. Huang, Foundations of stochastic analysis, Acdemic Press, Beijing, (22), (In Chinese). 13