Logarithm of differential forms and regularization of volume forms 1

Transcription

1 Differential Geometry and Its Applications 165 Proc Conf, Opava (Czech Republic), August 27 31, 2001 Silesian University, Opava, 2001, Logarithm of differential forms and regularization of volume forms 1 Akira Asada Abstract By using logarithm of derivation, logarithm of a differential form is defined It allows to define fractional order differential forms on a flat space Regularized volume form on a flat infinite-dimensional space is defined by using fractional order differential forms Applying noncommutative connection, we show regularized volume form can be defined on a mapping space, if its string class vanishes Keywords Logarithm of derivation, fractional order differential forms, spectral ζ function, regularized volume form MS classification 58A10; 58J52, 26A33, 81R60 1 Introduction In our study on regularization of differential operators on a Hilbert space H, infinite products of trigonometric functions appear as proper functions ([6]) It leads to the study of regularized infinite product of coordinate functions of H, see [7] Precisely, to define regularization, we need to equip a positive Schatten class operator G such that ζ (G, s) = tr(g s ) allows analytic continuation to s = 0and holomorphic at s = 0 Such pairing is closely related to Connes spectral triple ([8]) But our approach is more concrete and analytic We assume ζ (G, s) has its first pole at s = d We also set ζ (G, 0) = ν and call it the regularized dimension of H The typical example of such pair is {L 2 (X, E), G}, X a compact Riemannian manifold, E a symmetric (or Hermitian) vector bundle over X and L 2 (X, E) is the Hilbert space of L 2 -sections of E, andg the Green operator of a nondegenerate selfadjoint elliptic differential operator acting on the sections of E ([11], see also [13] when X is noncompact) 1 This paper is in final form and no version of it will be submitted for publication elsewhere

2 166 A Asada The regularized infinite product x n is defined to be the analytic continuation of x µs n n to s = 0, where µ n is a proper value of G with the proper function e n ; Ge n = µ n e n Strictly saying, one cannot define x n if x = x n e n belongs to H, and we need to extend H as follows: Let x k = G k x be the k-th Sobolev norm of x H,andW k the k-th Sobolev space constructed from H and x K We set H = k<0 W k and define the spaces H (finite) and H (0) by { H (finite) = x = } x n e n H lim n µ d/2 n x n exists, (1) { H (0) = x = } x n e n H lim n µ d/2 n x n = 0 (2) By definition, we have H (finite) = H (0) Ce, e = µ d/2 n e n Then x n is defined for the elements of H (finite), ([7]) Since x n is linear in each variable, we may consider ( N / x 1 x N ) x n = n N x n, but we cannot compute lim N ( N / x 1 x N ) x n To overcome this difficulty, we used fractional order derivation and define regularized infinite degree derivation / x n by ( ) 1 µs n = xn Ɣ(1 + µ s n ) x µs n n Then we have / x n x n = 1, see [7] Motivated by these phenomenon, we introduce logarithm of differential forms by using logarithm of derivation, which is defined by ( ) log = lim x n h +0 ( 1 h h x h n ) I There are several alternative definitions of log( / x n )Inthispaper,wedefine log( / x n ) by using Borel transformation ([2]) Then we define logarithms of vector fields and differential forms log(dx 1 ), log(dx 2 ),, by using logarithm of derivation We also impose the commutation relation [log(dx n ), log(dx m )] = πi, n < m Fractional order differential form d a x n is defined to be exp(a log(dx n )), 0 Ra < 2 This definition differs from the definition in [9] (cf [11], see also [1]) Let ω(s) be µ s n log(dx n) Then we define regularized volume form (regularized infinite degree wedge product), dx n by, dx n = e ζ (G,s)(ζ (G,s) 1)/2 e ω(s) s=0 (3) Definitions of logarithm of differential forms and regularized volume form use flatness of H If a Sobolev manifold M is parallelisable, then we can define regularized volume form of M IfM is a mapping space Map(X, M), applying noncommutative connection ([3]), we can show that there is a subvariety N of Map(X, M) such that Map(X, M) N is parallelisable ([4, 5]) Since regularized volume form generates a line bundle, regularized volume form of Map(X, M) is defined if this s=0

3 Logarithm of differential forms 167 line bundle is extended to Map(X, M) Since string class gives the obstruction to this extension, Map(X, M) has the regularized volume form if the string class vanishes 2 Logarithm of derivation I One variable function In this section, we recall the definition and property of log(d/dx) We study log(d/dx) by using Borel transformation For f (x) = c n x n, its Borel transformation B[ f ] = B[ f (t)](x) is defined and has the following properties B[ f ] = c n n! x n = 1 2πi [ d f dx B[ f ] = B t f (t) e x/t dt (4) t ], B[ f g] = B[ f ] B[g], (5) x u v = d u(x t)v(t)dt (6) dx 0 In the rest, we use the notation f ( u) = c n u n, f (u) = c n u n, u n = n {}}{ u u For example, e u = (u n /n!) The following lemma is proved in [1] Lemma 1 The following holds,where γ is the Euler constant e t log x = e γ t Ɣ(1 + t) x t (7) If B[log x] is defined, it must be x n /n! = e n log x So by this lemma, we define B[log x] by B[log x] = log x + γ We denote the algebra of finite exponential type entire functions with the variable w = log x by F Since w w n 1 = w n P n 1 (w), where n n k (n + 1)! P n (w) = ( 1) ζ (n + 2 k)w k, k! k=0 ([2]), if f (w) F, then f ( w) acts on F by the -product We set F = { f ( w) f F} Definition 1 We define fractional derivation and logarithm of derivation of f F by d a ( ) f d dx a = e a(w+γ ) f, log f = (w + γ ) f (8) dx

4 168 A Asada Example We have d a ( d log dx dx a x n n! = Ɣ(n + 1 a) x n a, ( )x n = x n log(x) + ( ( γ ))) n Note 1 Lemma 1 shows that e a(w+γ ) can be interpreted as a function, unless a is not a negative integer We consider e n(w+γ ) to be the operator d n /dx n if n is a natural number As for fractional derivation, d a /dx a is nonlocal unless a is an integer We consider its initial to be 0 (cf [9]) Note 2 When a < 0, we consider e a(w+γ ) to be a fractional order indefinite integral operator Similarly, we consider (w + γ ) f to be the logarithm of the indefinite integral operator x 0 f (t) dt 3 Logarithm of derivation II Several variable function Let w n be log x n, where x 1, x 2,, are coordinate functions of H (finite) Then the algebra of finite exponential type functions of w 1,w 2,, is denoted by FWe also define the -product of w n and w m by w n w m = w n w m + m n π i, n m (9) m n 2 To define the action of F to F, we need some preliminaries Let S = {p 1,,p n } be a set of natural numbers (may not be distinct form each other), T a subset of S and T the complement of T in S Letp l T, then we associate with p l a natural number S (p l ) = (p l ) as follows; (p l ) = 1, if p l = min T, (p l ) = 2, if p l = min {p j p j = 1} T, and so on The sign sgnt = sgn{p j1,,p jk } is defined by ( ) 1 k sgn{p j1,,p jk } = sgn (10) (p j1 ) (p jk ) Let w n w m be w n w m if n m and wn 2 if n = m Thenwedefinew p 1 w pn to be w p1 w pn + ( ) πi m sgn {p j1 2,,p jn 2m}w p w j1 p (11) jn 2m 2m m

5 Logarithm of differential forms 169 Definition 2 We define fractional partial derivation a / xn a and logarithm of partial derivation log( / x n ) by a ( ) xn a f = e a(w n+γ ) f, log f = (w n + γ ) f (12) x n By (9), we have [w n,w m ] = sgn{n, m}πi Hence we obtain By (13), we have e a(w n+γ ) e b(w m+γ ) = e (a(w n+γ )+b(w m +γ )+sgn{n,m}abπi/2) (13) Proposition 1 (i) Let n m and 1 be e πi if m > n,and e πi if m < n Then we have e a(w n+γ ) e b(w m+γ ) = ( 1) ab e b(w m+γ ) e a(w n+γ ) (14) (ii) Let 1 be the same as above,a 1 and δ n,m the Kronecker s delta Then we have ( ) e a(w n+γ ) e a(w m+γ ) sin(πa) a2 xm a 1 = ( 1), πa x n (15) e (w n+γ ) e (w m+γ ) 1 = e (w m+γ ) e (w n+γ ) 1 = δ n,m (16) Note 3 Operators such as log( / x 1 + / x 2 ) are not contained in F 4 Regularized infinite product Let Rs > d Then we may consider ω(s) = µ s n (w n + γ ) to be an element both of F and F We mainly consider ω(s) to be an element of F But we use same notation when considered to be an element of F Lemma 2 We have e ω(s) = e (ζ (G,s)(ζ (G(s) 1)/2)πi e µs 1 (w 1+γ ) e µs 2 (w 2+γ ) (17) Proof First we note, if c n converges absolutely, then we have m=n c n c m = m=1 n=m where C = c n Hence we have m=n+1 Hence we obtain c n c m, C(C 1) c n c m = 2 m c n = C e c 1w 1 e c 2w 2 =e (C(C 1)πi)/2 e c n w n n=m+1,

6 170 A Asada Definition 3 Let f F The regularized infinite product, e (w n+γ ) is defined to be the regularization of e (w 1+γ ) e (w 2+γ ), by the formula have, e (w n+γ ) f = e ν(ν 1)πi/2 e ω(s) f s=0 (18) However,, e (w n+γ ) does not act on F But if f is a polynomial, we, e (w n+γ ) f = lim n n x 1 x n f Since log Ɣ(1 + µ s n ) = γµs n + O(µ2s n ),wehave µ s n (w n γ ) log Ɣ(1 + µ s n ) = µs n w n + O(µ 2s n ) Hence Ɣ(1 + µ s n ) 1 e µs n (w n+γ ) = e µs n w n + O(µ 2s n ) Therefore we may consider f = e ν(ν 1)πi/2 e µ s n w n f s=0 (19) xn Note 4 Similarly, we may consider e ν(ν 1)πi/2 e µ s n (w n+γ ) f s=0 to be the regularized infinite-dimensional indefinite integral Q(x) f (x) d x, see [6] Let i 1 < i 2 < < i m be an m-set of integers Then we set ω 11,i 2,,i m (s) = µ s n (w n + γ ) n / {i 1,,i m } Starting ω i1,,i m (s), regularized infinite product, n / {i 1,,i m } e (w n+γ ) is defined, and we have, n / {i 1,,i m } e (w n+γ ) = ( 1) (i 1 1)+ +(i m 1), e (w i 1 γ ) e (w im γ ) e (w n+γ ) (20) 5 Fractional degree differential forms Let F 1 be the submodule of F which consists of the homogeneous elements of degree 1 We set { Rcn F 1, = cn w n F 1 0}, { (21) Rcn F 1,+ = cn w n F 1 0}

7 Logarithm of differential forms 171 The algebras generated by {e u u F 1,± } and 1 are denoted by Exp(F 1,± ), respectively Let φ Exp(F 1, ) and ψ Exp(F 1,+ ) Then φ ψ 1 is a function of x 1, x 2,, x n = r n e iθ n The constant part c 0 (φ ψ 1) of φ ψ 1 is lim N ( lim n 1,,n N 1 2n 1 π 2n1 π 0 dθ 1 1 2n N π 2n N π 0 ) dθ N φ ψ 1(θ 1,) We denote Exp(F 1,± ) k the vector subspaces of Exp(F 1,± ) generated by e u, where u = k j=1 c n j w n j such that no c n j vanishes Definition 4 Let φ Exp(F 1, ) k and ψ Exp(F 1,+ ) k Then we define the pairing φ,ψ by φ,ψ = c 0 (φ ψ 1) (22) For example, by Proposition 1, we have e a(w n+γ ), e b(w m+γ ) = 0, a b, or n m, (23) e a(w n+γ ), e a(w n+γ ) = ( 1), a Z (24) πa Let I + = I be the ideal generated by e 2(w n+γ ) in Exp(F1, +) A similar ideal in Exp(F 1, ) is also denoted by I (a 1)2 sin(πa) Definition 5 The algebra of fractional differential forms H is defined to be H = Exp(F 1,+ )/I Elements of H are said to be fractional order differential forms Let u be the class of u Exp(F 1,+ ) in H Then the wedge product of u and v by (u v) d a x n means the class of e a(w n+γ ) By the definition of I, we may assume 0 Ra < 2 By (14), we have d a x n d b x m = ( 1) ab d b x m d a x n, n m (25) Here 1 = e πi if n < m and e πi if n > m Note that this multiplication rule is different from that of [8] Let ψ = e u, where u = c n w n Thentoset u = c c n w n, c c n = cn, mod 2, 0 R c c n < 2, we define the action ψ f of ψ to f F by ψ f = (e u ) f = e u f (26) For example, ω(s) = ω(s) if s is small So (e ω(s) ) f = e ω(s) f, if s is small

8 172A Asada Definition 6 We define regularized infinite wedge product, dx n, ie, dx 1 dx 2,by, dx n f = e ν(ν 1)πi/2 (e ω(s) ) f s=0 (27) Note 5 Let Q(x) be { t n e n 0 t n x n } Then we may define the integral of 1 = 1, dx n on Q(x) by dx n = 1d x (28) Q(x), 1 Q(x) This suggests if we can give a definition of infinite dimensional integral by using regularized infinite wedge product, then it provides a mathematical justification of the formula such as e ( 2π x,dx ) 1 Dx = det D Note 6 Similarly, by using ω i1,,i m (s), we can define regularized infinite wedge product, n / {i 1,,i m } dx n We regard this form to be a regularized ( m)-form (cf [4, 5]) 6 Regularized volume form on a mapping space Let X be a d-dimensional compact spin manifold, M an n-dimensional smooth manifold Then the mapping space Map(X, M) is a Sobolev manifold modelled by W k (X) R n We fix the Sobolev metric of W k (X) by fixing a nondegenerate selfadjoint elliptic operator D, whose green operator is denoted by G For simplicity, we denote the connected component of Map(X, M) consisting of 0-homotopic maps by the same notation Map(X, M) It allows Clifford extension Map(X, M) C, which is modelled by W k (X, E) R n,w k (X, E) is the k-th Sobolev space of spinor fields of X In this case, we take for D the Dirac operator with a mass term on X tensored by the identity of R n, see [3] But in the framework of this paper, we take for G the Green operator of D 2 Let ξ = {g UV } be a G-bundle over M Then defining g X UV by g X UV ( f )(x) = g UV( f (x) ), wehaveamap(x, G)-bundle ξ X over Map(XM) Especially, we have τ(map(x, M)) = τ(m) X, where τ = τ(m), etc, mean the tangent bundle of M, etc Similar results hold for Map(X, M) C In this case, Map(X, G) is contained in the restricted general linear group GL p, where p > d/2 ([12]) In the rest, we also assume Map(X, M) C has a complex structure

9 Logarithm of differential forms 173 Let G be a Lie group with the Lie algebra g For a Map(X, G)-bundle ξ = { g UV }, a collection of Map(X, g)-valued functions {A U } such that (D + A U ) g UV = g UV (D + A V ), is said to be a connection of ξ with respect to D ([4, 5]) In general, D + A U (x) degenerates at some x If there exists a connection {A U } such that D + A U is nondegenerate at any point, then ξ is trivial as a GL p -bundle ([3]) Therefore, we take a connection {A U } of τ(map(x, M) = C), and set { ( Y = p Map(X, M) C ker D + AU (p) ) } {0} (29) τ(map(x, M) C Y ) is trivial Hence we can take H (finite) as the fibre of τ(map(xm) C Y ) Here, H = L 2 (X, E) R n Then, a section of (Map(X, M) C Y ) ( )H is said to be a fractional order differential form on Map(X, M) C Y Note 7 Considering det(d + A U (p)) to be the analytic continuation of det(d + A U (p) + m) to m = 0, see [3], we may regard Y to be the o-set of det(d + A U ) By using the trivialization of τ(map(x, M) C Y ), we obtain a family of operators D Y whose principal part is D, parametrized by Map(X, M) C Y Let µ 1 (p) µ 2 (p) > 0 be the proper values of G Y, the Green operator of D Y 2, and let e 1(p), e 2 (p),, be their proper functions We define ω(s)(p) to be µn (p) s w n (p), where w n (p) is defined similar to w n By assumption, lim x Y ω(s)(p) diverges Hence, dx n (p) = e ν(p)(ν(p) 1)/2 e ω(s)(p) s=0, has singularities along Y The counter term to this singularity is (det D Y ) 2d However, (det D Y ) 2d, dx n (p) may be discontinuous on Map(X, M) C Because e ν(p)(ν(p) 1)πi/2 may be many valued on Map(X, M) C Summarizing these, we have (cf [4, 5]) Theorem 1 The regularized volume form, dv = dx n (p) on Map(X, M) C Y defines a cross-section of the determinant bundle of Map(X, M) C So regularized volume form exists on Map(X, M) C if its determinant bundle is trivial Note 8 Since we used positive definite G as the metric, regularized volume form exists under more mild condition In fact, starting from positive definite G, determinant bundle is trivial if H 1 (Map(X, M), Z) = 0 Note 9 By the same reason,, n / {i 1,,i m } dx n(p) is defined on Map(X, M) C, if the determinant bundle of Map(X, M) is trivial

10 174 A Asada We can regard this form to be the regularized ( m)-form on Map(X, M) C In fact, since G is trivial, ( m)-forms on Map(X, M) are defined on Map(X, M) under more mild condition We conclude this paper to remark that although we can define regularized volume form and ( m)-forms on a mapping space with vanishing string class, it is not known whether one can define fractional differential forms or logarithm of differential forms on a mapping space with nontrivial tangent bundle References [1] H Ahmedov, A Yildizand Y Ucan, Fractional super Lie algebras and groups, Preprint, arxiv: mathrt/ [2] A Asada, Some extension of Borel transformation, J Fac Sci Shinshu Univ 9 (1974) [3] A Asada, Non commutative geometry of GL p -bundles, Colloq Math Soc János Bolyai 66 (1995) [4] A Asada, Clifford bundles over mapping spaces, in: Differential Geometry and Its Applications, Proc Conf, Brno 1998, (Masaryk Univ, Brno, 1999) [5] A Asada, Spectral invariants and geometry of mapping spaces, Geometric aspects of partial differential equations, (Roskilde, 1998), Contemp Math 242 (Amer Math Soc, Providence, 1999) [6] A Asada, Regularization of differential operators on a Hilbert space and geometric meaning of zeta-regularization, Steps in Differential Geometry, (Debrecen, 2000), (Inst Math Inform, Debrecen, 2001) [7] A Asada, Regularized product of infinitely many independent variables on a Hilbert space and regularization of infinite dimensional indefinite integral via fractional calculus, Proc ISMMS 2001, Kolkata, to appear [8] A Connes, Geometry from the spectral point of view, Lett Math Phys 34 (1995) [9] K Cottrill-Shepherd and M Naber, Fractional differential forms, J Math Phys 42 (2001) [10] PR Gilkey, The residue of the global η functions at the origin, Adv in Math 40 (1981) [11] R Kerner, Z 3 -graded exterior differential calculus and gauge theories of higher order, Lett Math Phys 36 (1996) [12] J Mickelsson and G Rajeev, Current algebras in D-dimensions and determinant bundles over infinite-dimensional Grassmannians, Comm Math Phys 116 (1988) [13] KP Wojciechowski, The ζ -determinant and the additivity of the η-invariant on the smooth, self-adjoint Grassmannian, Comm Math Phys 201 (1999) Akira Asada , Nogami Takarazuka, Japan asada-a@poporonejp