akira zeta reg infinite dims

March 16, 2004 19:14 WSPC/IJGMMP-J043 00007

International Journal of Geometric Methods in Modern PhysicsVol. 1, Nos. 1 & 2 (2004) 107157c World Scientific Publishing Company

REGULARIZED CALCULUS: AN APPLICATION

OF ZETA REGULARIZATION TO INFINITE DIMENSIONAL

GEOMETRY AND ANALYSIS

ASADA AKIRA

Faculty of Science, Sinsyu University,

3-6-21 Nogami Takarazuka, 665-0022 Japan

[email protected]

Received 21 October 2003Accepted 24 December 2003

A method of regularization in infinite dimensional calculus, based on spectral zeta func-tion and zeta regularization is proposed. As applications, a mathematical justificationof appearance of RaySinger determinant in Gaussian Path integral, regularized volumeform of the sphere of a Hilbert space with the determinant bundle, eigenvalue problemsof regularized Laplacian, are investigated. Geometric counterparts of regularization pro-cedure are also discussed applying arguments from noncommutative geometry.

Keywords: -regularization; spectral function; fractional calculus; ( p)-forms; reg-ularized volume form.

Contents

0. Introduction 108

1. Elliptic Operators and their Zeta Functions 112

2. Eta Function of an Elliptic Operator 113

3. The RaySinger Determinant 116

4. Variation of Special Values of (P, s) with respect

to the Mass Term 118

5. Variation of det (P +m) with respect to m 119

6. Spectral Invariants and Loop Group Bundles 121

7. Hilbert space equipped with a Schatten Class Operator 125

8. Regularized Calculus via Fractional Calculus 127

9. Polar Coordinate of Hilbert Space and Regularized Volume Form

of Infinite Dimensional Sphere 130

10. Regularization of the Laplacian on H 132

11. Regularization of the Dirac Operator on H 134

107


108 A. Asada

12. Logarithm of Derivation I. One-Variable Functions 137

13. Logarithm of Derivation II. Several-Variable Functions 139

14. Regularized Volume Form 140

15. Grassmann Algebra with an -degree Element and -forms 14216. Exterior Differential of ( p)-forms 14417. Regularized Exterior Differential 147

18. Regularized Volume Form on a Mapping Space 148

19. Regularized Volume Form and the Determinant Bundle 151

20. Half Infinite Forms 153

Acknowledgment 155

References 155

0. Introduction

This is a survey paper of spectral -function and its applications to infinite

dimensional geometry and analysis. Expositions are mainly focused on regularized

calculus, a systematic way of such application.

In the study of infinite dimensional analysis and geometry, we often meet the

problem of divergence. For example, if ~v(t) = (f1(t), f2(t), . . .) is an infinite dimen-

sional vector, then div (~v) is

n=1 dfn(t)/dt which might be divergent. Another

example is the Laplacian

4 =

n=1

2

x2n,

of a Hilbert space H . 4r, r(x) = x, diverges and we cannot give the polarcoordinate expression of 4. To overcome these difficulties, we propose a systematicuse of -regularization [5, 7, 913].

The spectral -function (P, s) of an elliptic operator P is trPs =sn .

It is known that (P, s) allows analytic continuation and holomorphic at s = 0,

if it is defined on a compact manifold M ([14, 29, 30, 51], for the non-compact

case, cf. [54]). We may consider = (P, 0) to be the regularized dimension of

H = L2(M), because the eigenfunctions of P spans H and formally, (P, 0) counts

the number of eigenvalues. Since (P, s) = logn sn , (P, 0) can be regarded

as the regularized value of log detP [49].To apply these regularizations, we consider a pair {H,G}, H a Hilbert space and

G a Schatten class operator on H , whose -function (G, s) = trGs is holomorphic

at s = 0. A typical example of such a pair is H = L2(M) and G is the Green

operator of P . Such a pairing is closely related to Connes spectral triple [23], and


Regularized Calculus 109

more concrete but narrow. G induces Sobolev norm in H by xk = Gkx. Thecomplete orthonormal basis {en,k} of W k is fixed to be the eigenvectors of G:

en,k = knen; Gen = nen.

By this norm and H , we construct the Sobolev space W k. Since a p-form on W k

is a smooth pWk-valued function, we may define a smooth map to pW k to bean ( p)-form, with the commutation relation

p q = ep(p)iq p,q p = ep(p)ip q .

If we consider only real forms, we cannot apply this rule unless is an integer.

There is no reason to expect that is an integer, which limits the application of

the method to the real valued forms. But if G is the Green operator of an elliptic

operator P , we can select a mass term m so that (G, 0) is an integer, where G

is the Green operator of P +mI [6]. Hence we can assume is an integer, in this

case.

We note that this commutation relation is similar to the commutation relation

of noncomutative torus when / Q.But this construction lacks information on volume form or its regularization.

To treat regularization of volume form, we use logarithm of differential forms [10].

Since dxn is the dual of /xn, wn = log(dxn) must be the dual of log(/xn),

which can be defined by using Laplace or Borel transformation. Regularized volume

form is defined by using (s) =snwn. Geometrically, regularized volume form

is a cross-section of the determinant bundle which is defined by using the Ray

Singer determinant of G. Analytically, the integral of the regularized volume form

is interpreted via fractional calculus. Intuitively, this calculation performs infinite

dimensional integral by counting contributions from all dimensions, but weighted by

sn. The answer is an analytic function of s. Then consider its analytic continuation

to s = 0, which is considered to be the (regularized) value of the integral.

This regularization procedure justifies the appearance of the RaySinger deter-

minant in the Gaussian path integral:

He(x,Px)Dx = 1

detP,

where P is positive non-degenerate and H is some extension of H . It is also shown

that the polar coordinate expression of the regularized volume form (volume ele-

ment) : dx : on H takes the form

: dx := r1 dr

n=1

(sin n)n1 dn d.

Note that the polar coordinate of H lacks the longitude (), and latitudes

(1, 2, . . .) needed to satisfy the constraints

limn

sin 1 sin n = 0.


110 A. Asada

The above volume form is defined on the space on which latitudes are independent

and longitude is added. It is also suggested that the regularized volume of the sphere

of this extended space is 2/2

(/2) .

Owing to the constraint of latitudes, regularized volume form diverges on the

sphere of H . But to define regularized Laplacian : 4 : by

: 4 : f = 4(s)f |s=0, 4(s) =

n=1

2sn2

x2n,

: 4 : allows polar coordinate expression [13], and its spherical part, considered onan extended space of H , has

sinn+1 n as an eigenfunction.

This paper aims to review these results as applications of spectral -functions.

To be self-contained, definitions and elementary properties of spectral - and

-functions and the RaySinger determinant are reviewed in Secs. 13. Sections 4

and 5 deal with the variation of special values of -function and the RaySinger

determinant with respect to mass-term [6]. Section 6 deals with applications of

-function to the geometry of loop group bundles and related bundles [24]. This

section is also a preparation to the study of the geometric counterpart of regularized

calculus. Readers may skip Sec. 6, till they have read Sec. 18.

The pair {H,G} is introduced in Sec. 7. Useful spaces such as

W k0(finite) =

{

xnen,k

l



The rest of the paper is devoted to explaining the geometric counterpart of

the regularized calculus. For this purpose, the logarithm of derivation is exposed in

Secs. 12 and 13. Our definition of logarithm of derivation uses Borel transformation:

B[f ](z) = 12i

f()

e

z d

(= cn

n!zn, f =

cnz

n).

To define Borel transformation of log z, we use the formula

e]t log z

=

tnn!

n (log z)] ](log z)

= e

t

(1 + t)zt,

where f]g = ddz z0 f(z )g() d, and is the Euler constant [1]. This is proved

in Sec. 12. By this equality, Borel transformation of log z should be log z + , and

the following definitions are justified.

log

(d

dx

)f = (w + )]f, d

af

dxa= e]a(w+)]f,

where w = logx. To define ordinary differential form from logarithm of derivation,

we use noncommutative underlying algebra of noncommutative Hilbert space in

Sec. 13. Then by using (s) =sn(wn +), we define regularized volume form on

a flat space in Sec. 14. Before we define regularized volume form on curved space,

we review the elementary properties of ( p)-forms on a flat space in Sec. 15.Roughly speaking, the exterior differential of an ( 1)-form is given by

d =

n=1

(1)n1 fnxn

dx, =

n=1

fnd{n}x.

Hence a (p)-form may not be exterior differentiable. In Sec. 16, we prove globalexactness of exterior differentiable (p)-forms [5]. This result shows the exteriordifferential operator on the space of ( p)-forms, is not nilpotent. So it gives ageometric example of Kerners higher-order gauge theory [15, 27, 34].

For ( p)-forms, exterior differentiability is a strong constraint. For example,(x) =

(1)n1xnd{n}x is not exterior differentiable. So we introduce the

regularized exterior differential : d : by

: d : = d(s)|s=0, (s) =

I

sI(detG)sfId

Ix, =

I

fIdIx.

For example, we have

: d : (ra) = ( + a)ra, r(x) = x.

Regularized exterior differential is a kind of Paychas -regularized trace [19, 43, 44]

and regularized Laplacian and Dirac operator are written by using regularized exte-

rior differential. But the underlying concepts of regularized exterior differential and

regularized calculus seems to be different. It shall be a future problem how to

relate them.


112 A. Asada

In Sec. 18, we construct regularized volume form on a mapping space. Then it

is expressed as a cross-section of the determinant bundle in Sec. 19. Our definition

of the determinant bundle is not yet related to the central or abelian extension of

the structure group of the tangent bundle of Map(X,M) (Map(X,G) or GLp, cf.

[28, 38, 40]). This will be a future problem. ( p)-forms on Map(X,M) are alsodefined. If X is a compact spin manifold, we can define half -forms on Map(X,M)under suitable assumptions. This is discussed in Sec. 20. Calculus of half -formsare not developed. It will be the next problem.

1. Elliptic Operators and their Zeta Functions

Let M be a d-dimensional (compact) Riemannian manifold, E a (Hermitian) vector

bundle over M , P : C(M ;E) C(M ;E), a differential operator acting on thesections of E. The symbol (P ) : (E) (E) is defined by

(P ) =

||=m

P(x), P =

||m

P(x)

x.

P is said to be elliptic if and only if (P ) is an isomorphism (except on 0-section).

In the rest, we assume P has no 0-mode, for simplicity.

If a (pseudo) differential operator P allows spectral decomposition, then an angle

is said to be the Agmon angle of P , if aug does not belong to {z| < aug z < + }, for some , where s are the eigenvalues of P . If P is non-degenerate andhas an Agmon angle, then taking to be a path in C \ {z|aug z = } surroundingthe eigenvalues of P , the -function (P, s) = (P, s) of P is defined by [42]

(P, s) = tr

(1

2i

s(I P )1d).

If P is self-adjoint, then eigenvalues of P are real, so P has Agmon angles. If P

is positive, then we can take 0 < < 2, and (P, s) =sn . While if P has

infinitely many positive and negative eigenvalues, then

(P, s) = +(P, s) =

n>0

sn + eis

n0

sn + eis

n



for the correction of residue formula in [51], cf. [53]. For further results,

cf. [17, 31, 39]. Since

0

ts1et dt = s

0

(t)s1etd(t) = (s)s,

we get

0

tr etP ts1 dt = (s)(P, s). (3)

Hence by the above asymptotic expansion, we have

(s)(P, s) =

1

ts1tr(etP ) dt+

0nN

an(P )

1

0

t(nd)/m+s1 dt

+

1

0

tr(etP )

0nN

an(P )t(nd)/m

ts1 dt.

The first term of the right-hand side exists for all s, the third term exists if

0, the second term is continued meromorphically on thewhole complex plane with possible poles at s = (d n)/m, n 0, of order 1.Hence (P, s) is meromorphic on the whole complex plane with possible poles at

(d n)/m, n 0, with the order 1.Note. Although P is not elliptic, if its heat kernel allows asymptotic expansion

involving fractional powers of log t (and t), then its -function is meromorophic on

the complex plane, but may have higher-order poles.

2. Eta Function of an Elliptic Operator

If P is self-adjoint with infinitely many positive and negative eigenvalues, then its

-function (P, s) is defined by

(P, s) =

n>0

sn

n


114 A. Asada

where (s) is the Riemann zeta function. Since lims0 s(s+ 1) = 1, we get

lims0

(P, s)(= (P, 0)) = 1 a. (4)

Since we have

tr (PetP2

) =

net2n ,

sgn||s = (2)(s+1)/2 = (s+ 1

2

)1

0

t(s+1)/21et2

dt,

we obtain

(s+ 1

2

)(P, s) =

0

t(s+1)/2tr(PetP2

) dt. (5)

We know the following estimate and asymptotic expansion

|tr (PetP 2)| Ctjet,tr (PetP

2

)

n0

cn(P )t(n3m)/d.

Hence (P, s) is meromorphic on the whole complex plane.

If m = 1, we define an operator D on M R+ by

D =

s+ P,

(D =

s+ P

).

We impose the boundary condition (A.P.S. condition)

M

f(x, 0)(x) dx = 0, 0, i.e. +f(, 0) = 0, (6)

where + is the projection to the positive eigenspace of P . The adjoint condition

of this condition is

(I +)f(, 0) = 0.

Let 41 = DD, 42 = DD. Then 41 is the operator 2

s2 +P2, with the boundary

condition (A.P.S. condition [14], cf. [41])

+f(, 0) = 0, (I +){(

f

s+ Pf

)s=0

}= 0. (7)

Setting f(x, s) =f(s)(x), this condition becomes

f(x) = 0, 0,(dfds

+ f

)s=0

= 0, < 0. (8)

Under these conditions, the fundamental solutions of t 2

s2 + 2 are

e2t

4t

{exp

( (s )

2

4t

) exp

( (s+ )

2

4t

)},



for 0, ande

2t

4t

{exp

( (s )

2

4t

)+ exp

( (s+ )

2

4t

)}

+e(s+)erfc

((s+ )

2t

t

), erfc(x) =

2

x

e2

d

for < 0.

As for 42, we impose the boundary condition

f(0) = 0, < 0,

(dfds

+ f

)s=0

= 0, 0. (9)

Then we have

K(t, x, s) = (et41 et42)|(x,s;x,s)

=

sgn

(e

2t

t

es2

t + ||e2||serfc(st

+ ||t

))|(x)|2

=

sgn

s

(1

2e2||serfc

(st

+ ||t

))|(x)|2,

K(t) =

0

M

K(t, x, s) dx ds = sgn

2erfc(||

t).

Hence we get

dK(t)

dt=

14t

n

nent.

We also have

limt

K(t) = 12h, h = dim KerP,

0

(K(t) +

1

2h

)ts1 dt =

(s+ 12 )

2s

n 6=0

sgnn|n|2s

=(s+ 12 )

2s

(2s).

Hence if K(t) has the asymptotic expansion of the form

kn aktk/2, t +0, we

obtain

(P, 2s) = 2s

(s+ 12 )

(h

2s+

N

k=n

ak

(k2 + s)+ N (s)

), (10)

(P, 0) = (2a0 + h). (11)Hence (P, s) is holomorphic at s = 0 in this case.

In general, it is known that (P, s) is holomorphic at s = 0, if P is an elliptic

(pseudo) differential operator on a compact manifold [14, 29, 30]. For the operators

on a manifold with boundary, we refer to [16] and [54].


116 A. Asada

3. The RaySinger Determinant

If (P, s) is holomorphic at s = 0, we define the RaySinger (-regularized) deter-

minant detP = det P by [42, 49]

detP = e(P,0), (12)

Since (P, s) =

lognsn , we may write

detP =

n

snn |s=0,

where the arguments of the powers of ns are determined by the Agmon angle.

If P and Pn have a common Agmon angle , then (Pn, s) = (P, ns). Hence

we get

det Pn = (det P )

n. (13)

But in general, det (PQ) is different from detP detQ, although P , Q and PQ have

a common Agmon angle.

If P is positive, detP is unique. In this case, (P, s) takes a real value on the

real axis, because (P, s) takes a real value if s > d/m. Since the residue at s = d/n

is given by

Ress=d/m(P, s) = limsd/m+0

(s d

m

)(P, s),

it is a real number. Hence the coefficients of the Laurent expansion of (P, s) at

s = d/m are real numbers. Therefore (P, s) takes a real value if s > (d 1)/m.Repeating this discussion, (P, s) takes a real value if s is a real number provided

(P, s) is finite. So detP is positive. We note that the residues of (P, s) are also

real numbers [6].

Since (tP, s) = ts(P, s), we get

(tP, s) = log t ts(P, s) + ts (P, s).

Hence if P is positive and t > 0, denoting = (P, 0), we have

det (tP ) = t detP. (14)

Note. Formally, this formula holds for arbitrary t. But if is not an integer, t

depends on the selected Agmon angle.

For the rest, we assume P is self-adjoint, and set

P+ =P + |P |

2, P =

P |P |2

, (15)

where |P | is ||(, ). Note that (|P |, s) =

||s is written as (P 2, s/2).We also set

(P, s) =(P, s) (P, s)

2, (16)



= (P, 0) =(P, 0) (P, 0)

2. (17)

By definition, we have

(P, s) = (P+, s) + eis(P, s).

Since we have

(P, s) = (P+) + ieis(P, s) + eis (P, s),

we obtain

detP = ei detP+ detP = e

i det |P |. (18)

By this formula, det+ P and det P are conjugate to each other. Hence | detP | isunique. We also obtain [6]:

Theorem 1. Let P be a self-adjoint elliptic (pseudo) differential operator. Then

the following are equivalent:

(1) detP is unique.

(2) detP is a real number.

(3) is an integer.

Note 1. If P is the Dirac operator D/ with the proper values {n}, then to define

sym det (D/+ im) = (1)(D/2++m2,0) det (D/2+ +m2),sym det (iD/+m) = det (D/2+ +m2),

we have

sym det (iD/+m) =

det (D/2+m2),

sym det (D/+ im) =

det (D/2+m2), (D/

2+m2, 0) 0 mod 4,

sym det (D/+ im) =

det (D/2

+m2), (D/2

+m2, 0) 2 mod 4.

Hence sym det (D/+ im) 6=

det (D/2

+m2) unless (D/2

+m2, 0) 0mod 4 [6].Note 2. There is another definition of the determinant of differential operator due

to Quillen [46]. But it gives essentially the same value [50]. RaySinger determinant

is used to the calculus of path-integral [21, 22, 32], central (or abelian) extension

of the algebra of pseudo-differential operators [35, 47] and representation of the

fundamental group [18, 20, 49].


118 A. Asada

4. Variation of Special Values of (P, s) with respect

to the Mass Term

First we assume P is positive. Denoting P +m instead of P +mI , we get

(P +m, s) =

(n +m)s

=

sn

(1 s m

n+ (1)k s(s 1) (s k + 1)

k!

(m

n

)k+

)

= (P, s) s m(P, s + 1) +

+(1)k s(s 1) (s k + 1)k!

mk(P, s + k) + .

Hence denoting (P, 0) = , we have

(P +m, 0) = Ress=1(P, s)m Ress=k(P, s)mk

k . (19)

Note. Originally, this formula was proved under the assumption |m| < 1. Butdenoting N (P, s) =

n>N

sn , we have

N (P, 0) = N,N (P +m, 0) = (P +m, 0) N,

Ress=kN (P +m, s) = Ress=k(P, s),

and (19) holds for arbitrary m.

Next we assume P is self-adjoint. Since P +m = (P+ +m) (P m), we have(P +m, s) = (P+ +m, s) + (1)s(P m, s).

As for (P + m, s), we get (P + m, s) = (P+ + m, s) (P m, s). By (18),we have

(P+ +m, 0) = +

k1

Ress=k(P+, s)mk

k,

(P m, 0) =

k1

(1)kRess=k(P, s)mk

k.

Since (P +m, 0) = (P+ +m, 0) + (Pm, 0) and

Ress=k(P +m, s) = Ress=k(P+ +m, s) + (1)kRess=k(P m, s),we obtain [6]

(P +m, 0) =

1k[d/m]

Ress=k(P, s)mk

k. (20)

Similarly, we can compute (P +m, c) in terms of special values (and residues)

of (P, s). For example, if m = 1, c < 0 is not an integer, then

(P +m, c) = (P, c) +

k=1

(1)k c(c+ 1) (c+ k 1)k!

(P, c+ k)mk.



If c is a negative integer and (P +m, s) is finite at s = c, then

(P +m, c) = (P, c) +

c

k=1

(1)k c(c+ 1) (c+ k 1)k!

+

c+d

j=c+1

(1)j(

1

j+

jk

k=1

c (c+ k 1)k!

1

j k

)Ress=j+c(P, s)m

j .

Note. As stated in the Introduction, sometimes, we need integrity of (P, 0), which

cannot be expected in general. But by (20), (P + m, 0) becomes an integer, by

selecting a mass term.

5. Variation of det (P + m) with respect to m

We arrange eigenvalues of a self-adjoint P as 0 < |1| < |2| < . The multiplicityof i is denoted by i,+ if it is positive and i, if it is negative. We also use the

notations

(P,+, s) = (|P |, s), (P,, s) = (P, s).Their Laurent expansions at s = k are denoted by

(P,, s+ k) = ak,,1s1 + ak,,0 + ak,,1s+ ,where ak,,1 = 0 if k > d/m. Since

d

ds((s+ 1) (s+ k 1))|s=0 =

k1

i=1

(k 1)!(

1

i

),

we get

d

ds

(s (s+ k 1)

k!(P,, s+ k)

)s=0

=1

k

(1 + 1

k 1

)ak,,1 +

1

kak,,0.

Since |P +m| = (P+ +m) + (P m), we have

(|P +m|, 0) = (|P |, 0) mRess=1 (P, s) +m2

2Ress=2 (|P |, s) + .

Hence we obtain

(|P +m|, 0) = (|P |, 0) +d

k=1

(1)k 1k

(1 + 1

k 1

)ak,(1)k ,1m

k

+

k=1

(1)k 1kak,(1)k ,0m

k, |m| < |1|. (21)

Let ak,N,,i be the coefficients of the Laurent expansion of N (|P +m|,, s). Thenwe have

ak,N,,1 = ak,,1,

ak,N,,0 = ak,,0 +

(i,+ + (1)ki,)|i|k,


120 A. Asada

N (|P |, 0) = (|P |, 0) +N

i=1

i log |i|,

N (|P +m|, 0) = (|P +m|, 0) +N

i=1

i log |i|.

Applying (21) to N (|P +m|, 0) and using the above formulas, the following expan-sion holds if |m| < |N+1|:

(|P +m|, 0) = (|P |, 0) +N

i=1

i log |i| N

i=1

i log |i +m|

+

d

k=1

(1)kmk

k

(1 + + 1

k 1

)ak,(1)k,1

+

k=1

(1)kmk

k

(ak,(1)k ,0

N

i=1

(i,+ + (1)ki,)|i|k).

We denote this right-hand side by (|P +m|, 0)N . Then, since

(|P +m|, 0)N = (|P +m|, 0)N1 + N,+(log |N | log |N +m|)+N,(log |N | log |N m|)

k=1

(1)kN,+|N |kmk

k

k=1

N,|N |kmk

k

= (|P +m|, 0)N1 N,+ log(

1 +m

|N |

) N, log

(1 m|N |

)

+N,+ log

(1 +

m

|N |

)+ N, log

(1 m|N |

),

on |m| < |N |, (|P +m|, 0) continued on a whole complex plane.By (|P +m|, 0)N , on |m| < |N+1|, we obtain

det |P +m| = det |P | N

i=1

(1 + sgni

m

|i|

)i,sgn i

exp(

k=1

(1)kmk

k

(ak,(1)k,0

N

i=1

(i,+ + (1)ki,)|i|k))

exp(

d

k=1

(1)kmk

k

(1 + + 1

k 1

)ak,(1)k ,1

). (22)

As a consequence, we have

Lemma 2 ([5]). Det|P +m| vanishes at m = i with the order i,sgn i .

This lemma will be used in the construction of the determinant bundle and the

regularized volume form on a mapping space in Sec. 19.



Note 1. For |m| is large, this analytic continuation of det |P+m| gives det(P+m),rather than | det(P +m)|.Note 2. If k > d/m, we get

limN

N

i=1

(i,+ + (1)ki,)|i|k = ak,(1)k,0.

We set |P +m| = |P ||I +mG|. On |m| < |N+1|, we may regard

det |I +mG|

=

N

i=1

(1 + sgni

m

|i|

)sgn i

exp(

k=1

(1)kmk

k

(ak,(1)k,0

N

i=1

(i,+ + (1)ki,)|i|k))

.

Hence we may interpret the term

exp

(

d

k=1

(1)kmk

k

(1 + + 1

k 1

)ak,(1)k ,1

),

as the multiplicative anomaly of det |P +m| = det (|P ||I +mG|).Note 3. If P is self-adjoint but not positive, we use e(P+mI) det (|P | + mI) asthe analytic continuation of det(P +mI).

6. Spectral Invariants and Loop Group Bundles

This section deals with results on loop group bundles obtained by using spectral

invariants [3, 4]. They are used in Sec. 18.

Let X be a compact d-dimensional spin manifold, D/ the Dirac operator acting

on the spinor fields on X . For simplicity, we assume D/ has no 0-modes. Let G be

U(n) or SO(n), V a representation space of G, and let H the Hilbert space of

V -valued spinor fields on X . J = D/|D/|1 defines an involution on H . Let A(H) andG(H) be the algebra and group of bounded and invertible operators on H , and set

glp = {T A(H)|[T, J ] Ip}, GLp = {T G(H)|[T, J ] Ip}, (23)

where Ip is the pth Schatten ideal [52]. Then it is known [40] that

Map(X,G) GLp, p > d/2. (24)

Let = {gUV } be a Map(X,G)-bundle over a Hilbert manifold M . Then,besides the usual connection taking the values in Map(X, g)-valued 1-forms, g

is the Lie algebra of G, has two different connections; noncommutative con-

nection and connection with respect to the Dirac operator ([3, 4], cf. [48]).


122 A. Asada

They are defined as follows:

Definition 3. A collection of smooth (continuous) maps U : U Ip, p > d/2, issaid to be a noncommutative connection of , if it satisfies

(J + U )gUV = gUV (J + V ). (25)

Definition 4. A collection of smooth (continuous) functions AU on U taking values

in V -valued spinor fields on X is said to be a connection of with respect to the

Dirac operator D/, if it satisfies

(D/+AU )gUV = gUV (D/+AV ). (26)

It is shown that is recovered by its noncommutative connection (connection

with respect to the Dirac operator respectively [4]). So these connections contain

all topological informations of as a GLp-bundle.

Note 1. We can define a connection with respect to P . P is a differential operator,

similar to a connection with respect to the Dirac operator. But in a connection with

respect to P , P is not the Dirac operator, and does not contain as much information

as a connection with respect to the Dirac operator does (cf. [5]).

Note 2. f : U Map(X,G) induces the map f [ : U X G;

f [(p, x) = (f(p))(x), p U, x X.

Let = {gUV } be a Map(X,G)-bundle over M , then [ = {g[UV } is a G-bundleover M X . A connection {U} of induces a collection of (g-valued) 1-forms{A[U} on M X . But this collection is not a connection of [, because the exteriordifferential d on M X divides dM + dX and we only have

(g[UV )1dMg[UV = A

[V (g[UV )1A[Ug[UV .

Hence to get a connection of [, we need to supply additional terms {BU} such that

(g[UV )1dXg[UV = BV (g[UV )1BUg[UV .

Connection with respect to the Dirac operator supplies {BU} on M .On a based mapping space, this supplied term contains more information than

an ordinary connection. For example, let G be the based loop group and LG the

free loop group, then associated to the exact sequence

e G LG G e, (f) = f(),

we obtain the exact sequence of cohomology sets

H1(M,Gd)

H1(M,LGd)

H1(M,Gd),



where Gd, etc. mean the sheaves of germs of smooth map to G, etc. Hence the-image of a G-bundle is trivial. Let be a LG-bundle, then its pth Chern class

cp([) is written

cp([) = cp,M + cp,S1

,

cp,M H2p(M,Z) e0, cp,S1 H2p1(M,Z) e1,

where ei is the generator of Hi(S1,Z), i = 0, 1. To describe cp,S

1

, we need {BU}.Since () is trivial, if is an G-bundle, only cp,S

1

is meaningful. So connection

with respect to the Dirac operator (and noncommutative connection, the quantum

version of connection with respect to the Dirac operator) is more useful than an

ordinary connection for G-bundles (cf. [2, 44]).

The curvature of a noncommutative connection {U} is defined by

{RU}, RU = JU + UJ + 2U .

Proposition 5 ([3]). The curvature of a noncommutative connection has the fol-

lowing properties:

(1) If a GLp-bundle has a noncommutative connection whose curvature takes the

values in Iq, q > p/2, then the structure group of is reduced to GLq.

(2) A GLp-bundle always has a noncommutative connection whose curvature takes

the values in Ip/2.

By these facts, we obtain

Theorem 6 ([3, 4]). A GLp-bundle is equivalent to a GL1-bundle as a smooth

(or topological) bundle.

The proof of this fact suggests the link of this fact and KatoRellichs Theorem

of perturbation theory [33].

By this theorem, we may consider a GLp-bundle as a loop group bundle. But if

we consider extensions of their structure groups, there are big differences between

loop group bundles and GLp-bundles, p 3 [28, 40].Let {AU} be a connection of with respect to D/. Then the spectra of D/+AU (x),

x U does not depend on U . We set

A(x) = (D/+AU (x), 0).

A(x) is smooth at x U if D/+AU (x) has no zero mode. We set

Yk = {y M |dim.ker(D/+AU (y)) = k, }, Y =

k1

Yk.

Let : [1, 1] M be a path crossing Yk at s = 0. Then we have

limt0

A((t)) = limt+0

A((t)) 2k.


124 A. Asada

Hence exp(iA) is smooth on M . Since dA =1ie

iAd(eiA), dA is a closed

1-form on M . It may not be exact, because as for the current TA [] =

MA ,

we get

dTA [] =

k1

2k

Yk

M

dA .

If {AU} is another connection of with respect to D/, to set AU = AU + BU ,AU,t = AU + tBU gives a connection (0 t 1). We set A(t, x) = At(x). Thenon M I , we get

dA = dA + d

( 1

0

tA(t, x) dt

).

Hence the de Rham class of dA does not depend on the choice of AU . If M = S1

and X = S1, takeD/+AU =1i

(ddt

)+m , we have dA = 2md . Hence the de Rham

class of dA is 2me, where e is the generator of H1(S1,Z). This class is 2m ind,

where ind : H1(S1,U) H1(S1,Z) is the induced map of ind : U Z. Wecan reduce to the case M = S1 for arbitrary M . So denoting the de Rham class of

dA by dA, we obtain

Theorem 7 ([4]). dA does not depend on the choice of A, and we have

dA = 2ind. (27)

Example. By Bott periodicity, the characteristic map of a U -bundle over M is

a map g : M U(N). Canonical transition function gUV is

gUV (t) = ethU ethV , g|U = ehU .

For this transition function, we can take {ihU} as a connection. Since the Greenoperator GU of

1i

(ddt + hU

)is

GU = iethU

( t

0

ehU() d + (I g)1g 1

0

ehU() d

)

= iethU(

(I g)1 1

0

ehU() d 1

t

ethU() d

),

is trivial if and only if 1 is not an eigenvalue of g(x), for all x M . String classesof are realized by {x| det (g(x) I) = 0} [4, 5].

As an application of this example, we have the following realization of the string

class on S3 whose characteristic map is the identity map of S3.

Let g be the identity map S3 SU(2). Then det (g(x) I) = 0 at x = I , theidentity matrix. Hence the string class of the bundle with the characteristic map g

is the dual class of a point (the genertor of H3(S3,Z)).



Appendix to Sec. 6: Free loop group bundles and DO-bundles

Let LG be the free loop group over G. Then by the exact sequence

0G iLG jG 0, j() = (0),

we have the exact sequence of cohomology sets

H1(M,Gd)iH1(M,LGd)

jH1(M,Gd),

where Gd, etc. mean the sheaves of germs of smooth maps from M to G, etc. If

G = U(n) and chp(E) is the pth Chern class of a G-bundle E, then

chp(j()) = cp0() e0 + cp1() e1,

where cp0() H2p(M,Z), cp1() H2p1(M,Z), and ei are the generators ofH i(S1,Z). Since

cp0() = chp(j()),

cp0() is expressed by using (Lg-valued) connection and curvature of . But toexpress cp1(), we need additional data which recover information from {BU}.

If is a G-bundle, then cp0(j(i())) = 0, so cp1s are only characteristic classes.

These are the string classes [2].

If = {gUV } is a DO-bundle over M [44], denoting (g) the principal symbolof g, {(gUV )} defines a G-bundle over M \M (or on ,S

m1

M). Here M is

the cotangent bundle over M and ,Sm1

M is its associated Sm1-bundle. Since

M is a vector bundle over M , we can divide

H2p(M \M,C) = (H2p(M,C) res1(H2pm+1(M,C)).

Since H(M,C) = H(M,C), we may set

chp(()) = cp0() + res1(cp1()), c

p0 H2p(M,C), cp1 H2pm+1(M,C).

The (DO-valued) curvature of only express cp0().

Note. Let {UV } be the transition function of astM , considered acting on theunit sphere of the fibre. Then by the cocycle rule

(gUV )(x, UV (p))(gV W )(x, V W (p)) = (gUW )(x, V W (p)),

() defines a twisted Map(Sm1, G)-bundle over M . Hence DO-bundles are

closely related to Map(Sm1, G)-bundles.

7. Hilbert space equipped with a Schatten Class Operator

Let H be a Hilbert space, G a Hermitian non-degenerate Schatten class operator

on H (cf. [52]), such that (G, s) = tr(Gs) allows analytic continuation and holo-

morphic at s = 0. The typical example of such a pair {H,G} is H = L2(M,E),where E is a Hermitian vector bundle over M and G is the Green operator of a


126 A. Asada

non-degenerate self-adjoint elliptic (pseudo) differential operator P acting on the

sections of E. In this case, we have (G, s) = (P, s), and detG = exp( (G, 0)) is

equal to (detP )1.

In the pair {H,G}, we fix the complete orthonormal basis of H to be the eigen-vectors {en} of G: Gen = nen. {H,G} has the following numerical invariants:

(1) The regularized dimension = (G, 0) of H .

(2) The location d of the first pole of (G, s).

(3) The regularized volume detG of the virtual cube of H .

By using the norm of H and G, we define the kth Sobolev norm xkof x H by xk = Gkxk if Gkx is defined. We denote the completion of{x|xk l. We also set [7, 9]

W k+0 =

l>k

W l, W k0 =

l 0. Thenwe define the spaces W k0(0) and W k0(finite) by

W k0(0) ={

xnen,k W k0 lim

nd/2n xn = 0

}(28)

W k0(finite) ={

xnen,k W k0 lim

nd/2n xn exists

}. (29)

If k = 0, we use the notations H(0), H(finite) and e instead of W00,

W k0(finite) and e,0. By definition, we have

W k0(finite) = W k0(0) Ke,k, K = R or C. (30)

As a topological space, we consider W k0(0) to be a subspace of W k0, but

W k0(finite) is considered to be the product space of W k0(0) and K.Let x =

xnen,k, xn > 0, n = 1, 2, . . . , be an element of W

k0(finite).

Then we set

Q(x) ={

ynen,k W k0(finite)||yn| xn},

Q(x,+) ={

ynen,k W k0(finite)|0 yn xn}.



By definition, if W k0(finite) is a real vector space, then

t>0

Q(te,k) = Wk0(finite). (31)

t>0

Q(te,k,+) = Wk0(finite)+. (32)

Here W k0(finite) means {xnen,k W k0(finite)|xn 0}.By (30), x W k0(finite) is written uniquely as x = y + te,k. Let x =xnen,k, y =

ynen,k, assuming t 6= 0, we have

n=1

xsnn =

n=1

(yn + d/2n t)

sn = t

(P

n=1 sn

)

n=1

(d/2n )sn

n=1

(1 +

d/2n ynt

)sn.

Hence if

(1 + d/2n yn/t) converges, we get

n=1

xsnn |s=0 = t(detG)d/2

n=1

(1 +

d/2n ynt

). (33)

Definition 9. The right-hand side of (33) is called the regularized infinite product

of x1, x2, . . . , and denoted by :xn :.

The regularized infinite product is linear in each variable xn and positive if each

xn is positive [9]. We define regularized infinite product :

n/{i1,,ip}xn : by

:

n/{i1,,ip}

xn :=p

xi1 xip:

n=1

xn : . (34)

As an application, we may define :xnn : by

:

n=1

xnn :=

n=1

n

x1 xn:

j=1

xj :

.

8. Regularized Calculus via Fractional Calculus

To define the limit limn n/x1 xn, we use fractional differential a/xa,

which is defined by (cf. [45])

af(x)

xa=

1

(a)

x

0

f()

(x )a+1 d,


128 A. Asada

transformation, or by using the translation operator i,h : i,hf(. . . , xi, . . .) =

f(. . . , xi + h, . . .) and define

af

xai= lim

h0

(i,h I)afha

,

where (i,h I)a is defined by using the binomial expansion.

Example. We have

da

dxaxn =

n!

(n a+ 1)xna.

This formula is valid although n is not an integer.

Definition 10. We define the regularization

Q

n xnof

n

xnby

n=1 xn

=

n=1

1

(1 + sn)

sn

xsnn

f

s=0

. (35)

Note. The factor 1/(1+sn) is not natural. But since

(1+sn) has singularities

at {d/n|n N}, so 0 is an essential singularity of (1 + sn), we need this factor.

If f is sufficiently regular, we have

lims

n=1

1

(1 + sn)

sn

xsnn

f = f.

So the above definition is meaningful.

By definition, f

Qn=1 xn

= 0 if f does not depend on some xn, and

n=1 xn

;

n=1

xn := 1. (36)

Fractional indefinite integral Ia(f) of order a with the initial condition

Ia(f)(0) = 0 is defined by

Ia(f)(x) =

x

0

f(t)dat =1

(a)

x

0

(x t)a1f(t) dt. (37)

Let

n (1 + sn)I

snt=xnf be

limn

(1 + sn)Isn( (1 + sn)I

s1(f |t=x1) )|T=xn .

If f is smooth, then lima+0 Iaf = f , a > 0. Hence if 1 < 1, we have

lims

n=1

(1 + sn)Isnt=xnf = f(x).



Definition 11. We define the regularized integral

Q(x,+)f : dx; of f on

Q(x,+) by

Q(x,+)

f : dx :=

n=1

(1 + sn)Isnt=xnf |s=0. (38)

By definition, we have

Q(x,+)

1 : dx : = :

n=1

xn :, (39)

: d(tx) : = t : dx :, (40)

where t > 0 is a scalar.

The following example is one of the main result of this paper.

Example. We compute the Gaussian integral

He(x,Dx)Dx, where D is a posi-

tive non-degenerate elliptic operator whose Green operator is G and assume 1 < 1

[11]. Since e2(x,Dx) =e2

1n x

2n , we calculate

Q(re,+)

e

1n x

2n : dx := lim

N

N

n=1

sn

d/2n r

0

(d/2n r t)sn1e

1n t

2

dt.

Let u be1n t and bn =

(d+1)/2n r, then

sn

d/2n r

0

(d/2n r t)sn1e

1n t

2

dt = snsn/2n

bn

0

(bn u)sn1eu

2

du.

Since lims sn

bn0 (bn t)

sn1et

s

dt = eb2n , we have

lims

(lim

N

N

n=1

sn

d/2n r

0

(d/2n r t)sn1e

1n t

2

dt

)= e

P

b2n .

Hence this limit exists, becauseb2n = r

2(d+ 1).

Since the RaySinger determinant detD of D issnn |s=0, to derive

H(finite)

e(x,Dx)Dx = 1detD

, (41)

it is sufficient to show

limr

lims0

2

bn

0

(bn t)sn1et

2

dt = 1. (42)

For simplicity, we assume


130 A. Asada

Then we have gr(t) (bn t)

sn1. Hence by Lebesgues convergence theorem, we

obtain (40). Hence by (31), (39) holds provided the integral is taken overH(finite).

Therefore our regularization provides a mathematical justification of the appear-

ance of the RaySinger determinant in the calculus of Gaussian Path integral.

Note. This justification is done calculating the integral on H(finite). On H , we

cannot give such justification, at least by using regularized integral.

9. Polar Coordinate of Hilbert Space and Regularized Volume

Form of Infinite Dimensional Sphere

Let r = x, x =xnen H . The polar coordinate of H is defined by

x1 = r cos 1, x2 = r sin 1 cos 2, . . . ,

xn = r sin 1 sin n1 cos n, . . . , 0 i .

This polar coordinate has only latitudes, and lacks longitude [13]. Since

r2 = x21 + + x2n + r2 sin2 1 sin2 n, latitudes 1, 2, . . . are not independent,but satisfy the constraints

limn

sin 1 sin n = 0. (43)

Since 0 sin n 1, if 0 n , the limit

t = limn

sin 1 sin n(

=

n=1

sin n

), (44)

exists, although 1, 2, . . . are independent. We consider t to be a new variable

added to H . We also introduce an angle ; 0 2 , and set

H = H R2 = {(x, y, z)|x H, y = t cos, z = t sin}. (45)

H is a Hilbert space with the norm x2 = x2 +y2 +z2, x = (x, y, z), and has thelongitude . The subspace of H with the longitudes = 0, = , i.e. the subspace

defined by z = 0 is denoted by H0. By definition, H0 = H R. Identifying this Rto Re of H(finite), we may relate H0 and H(finite).

We compute the regularized volume form of the sphere of H by using the regu-

larized integral as follows.

We regard Rn to be the subspace of H spanned by e1, . . . , en. The cubeQ(x,+) Rn is denoted by Q(x,+)n. Then we have

n

i=1

(1 + si )Isnt=ai (f) =

n

i=1

si

Q(a,+)n

rn1n

i=1

(ai xi)si 1

n2

i=1

sinni1 i dr d1 dn2 d.



Here is the longitude of Rn and a =aiei. Since

(ai xi)si 1 = r

si 1(sin 1 sin i1 cos i)

si 1

(aixi

1)si1

,

for 1 i n 2, we getn2

i=1

(ai xi)si1 = r1++

sn2n+2

n2

i=1

(aixi

1)si1

n2

i=1

(sin i)si ++

sn2n+i+2(cos i)

si1.

Hence we obtain

n

i=1

(ai xi)si 1rn1

n2

i=1

sinni1 i

=n

i=1

(aixi

1)si 1

rs1++

sn1

n2

i=1

(sin i)i++

sn1(cos i)

si 1(sin cos)sn1. (46)

Defining Q(a,+) in H0, we perform regularization of the integral of a function f

on Q(a,+). Then, symbolically, we can write

Q(a,+)

f(x) : dx : =

Q(a,+)

f(x)

n=1

(anxn

1)sn1

rP

i2 si dr

n=1

sn(sin n)P

in+1 si (cos n)

sn1 dn|s=0. (47)

(47) shows that for suitable test functions f , we may consider

: dx := r1 dr

n=1

(sin n)n1 dn. (48)

Hence we have

Theorem 12 ([12]). The regularized volume form of the sphere S of H0 is

: d :=: d :=

n=1

(sin n)n1 dn. (49)

Similarly, the regularized volume form of the sphere S is

: d :=: d :=

n=1

(sin n)n1 d d.


132 A. Asada

Note that since t =

sin n, we have

: d := t : d : . (50)

Note 1. Since

n=1 sin n = 0 on H , : d : is singular on the sphere of H . In other

words, : d : is meaningful only when considered on S, that is only considered

on H0.

Note 2. By (48), we may write

1

2

H(finite)

e(x,Dx) : dx :=

0

r1 dr

Se(x,Dx) : d : .

Here the factor 1/2 comes from the lack of longitude in H(finite). According to

our regularization scheme, we may write

0

r1 dr

Se(x,Dx) : d :=

1

/2

detD

0

r1er2

dr

S: d : .

Since0s/21er

2

dr = (/2)/2, we may conclude

2

S: d :=

2/2

( 2 ), i.e.

S: d : d =

2/2

( 2 ). (51)

Note that this answer cannot be obtained via the direct calculus of the following

limit

limn

0

0

sin2 1 sinn1 n d1 dn.

10. Regularization of the Laplacian on H

This section and the next deal with eigenvalue problems of regularized Laplacian

and Dirac operator [7, 13]. There are other applications of -regularization to infinite

dimensional analysis. But they use the same framework as previous sections.

Let 4 =n=1 2/x2n be the Laplacian onH . SinceH is an infinite dimensionalspace, 4r diverges and we cannot give the polar coordinate expression of 4.

We consider 4 on {H,G}. Sincexnen =

xn,ken,k implies xn,k =

sn xn,

we have

xn,k= sn

xn, i.e. 4 (k) =

n=1

2kn2

x2n.



Here 4(k) means the Laplacian of W k. In general, we set

4(s) =

n=1

2sn2

x2n. (52)

Definition 13. We define the regularized Laplacian : 4 : by: 4 : f = 4(s)f |s=0. (53)

Example. Since 2ra/xn = ara2 + a(a 2)ra4x2n, we get

: 4 : ra =(a(G, s)ra2 +

n=1

a(a 2)2sn x2nra4)

s=0

= a(a+ 2)ra2.

Proposition 14. : 4 : allows the following polar coordinate expression, whichdepends only on the regularized dimension .

: 4 : = 2

r2+ 1r

r+

1

r2[](= 4[]), (54)

[] =

n=1

1

sin2 1 sin2 n1

(2

2n+ ( n 1)cos n

sin n

n

). (55)

Note. This polar coordinate expression seems similar to the finite dimensional

case. But contrary to the finite dimensional case, most of the coefficients (n1)of the 1st order terms of (55) are negative.

We say [] is the regularized spherical Laplacian of H [13]. It is defined not

only on the unit sphere on H but also on S, the unit sphere of H0. To solve the

eigenvalue problem

[] = a0, (56)we set (1, 2, . . .) =

n Tn(n). Then each Tn should satisfy the equation

sinn+1 nd

dn

(sinn+1 n

dTndn

)+

(an1 +

an

sin2 n

)Tn = 0, (57)

where a0, a1, . . . is a series of real numbers. Tns need to be continuous at the

boundary n = 0, , and the sereis a0, a1, . . . should be choosen to satisfy this

demand. An example of such a series and solutions are

an = ln(ln + n 2), ln N, l1 l2 0,with Tns as Gegenbauer polynomials or

n0

sinn+12l n dn, where l =

limn ln, provided is an integer [13]. It was noted that these eigenfunctions

have finite dimensional analogy if they are considered on the sphere on H . But

on S, most of these eigenfunctions have no finite dimensional analogy. Another

problem of these eigenfunctions is they seem not to behave well with the regularized

volume form on S.


134 A. Asada

But there exist eigenfunctions of [] which behave well with the regularized

volume form on S as explained below.

To get an eigenfunction of [] to behave well with the regualrized volume form

of S, we note

d2 sinc

d2= c(c 1) sinc2 c2 sinc .

Hence taking an = n+ 1 , sinan n satisfies (57). Therefore

(1, 2, . . . , ) =

n=1

sinn+1 n, (58)

satisfies

[] = ( 1), : d :=

n=1

dn. (59)

Note that vanishes on the sphere of H. So it is meaningful only on S.

Note. The limit limn 0 0d1 dn diverges. But by (40), the following

regularization may be allowed.

S : d := .

We can also compute eigenvalues and eigenfunctions of the periodic boundary

value problems of : 4 : of the following types

f |xn=

d/2n

= f |xn=

d/2n,

f

xn

xn=d/2

=f

xn

xn=

d/2n

, (60)

f |xn=0 = f |xn=d/2n = 0, (61)

which are considered on H(finite). But the discussions and answers are similar to

the case of the regularized Dirac operator [7] which will be explained in the next

section. So we omit the discussion on this boundary condition for : 4 :.

11. Regularization of the Dirac Operator on H

Let Cl(H) be the Clifford algebra generated by {en} with the relation eiej +ejei =2ij over K. Here en is the corresponding element of en H . The infinite spinore thought to be 5 of Cl(H), is defined to be the element satisfying the following

relations ([8], cf. [36]).

eei = (1)1eie, ee = (1)(1)/2. (62)

Definition 15. We say Cl[e] is the Clifford algebra (over H) with an infinite

spinor.

Note. If Cl[e] is an R-module, it is an associative algebra if and only if is aninteger. While if it is a C-module, to define

eie = e(1)ieei, eei = e

(1)ieie,



Cl[e] becomes an associative algebra for arbitrary . Note that the commutation

relation of e and ei is the same as that of noncommutative torus.

We introduce L2-norms on Cl(H) and Cl(H)[e] by the following type of inner

product, and consider them to be Hilbert spaces.

(ei1 eip , ej1 ejp) = i1j1 ipjp .

Note. Similarly, we can define Clifford algebra Cl(W k) over W k to be the Clifford

algebra generated by en,k and the infinite spinor e,k of cl(Wk) [8]. By using en

and e, they are the algebra generated by the relations

eiej + ejei = 2ki ij , e2 = (1)(1)/2(detG)2k.

Justifications of multiplication formulas of e and e,k will be given in Sec. 14.

We consider Cl(W k) and Cl(W k)[e,k] to be Hilbert spaces. A spinor field on

a subset of H or H(finite) (resp. on a subset of W k or W k0(finite)) is a smooth

map from a subset of H H(finite) (resp. from a subset of W k or W k0(finite)) to

Cl(H)[e] (resp. to Cl(Wk)[e,k]).

The Dirac operator D/ on H is defined by

n=1 en

xn. It acts on the space of

spinor fields. The regularized Dirac operator : D/ : is defined by

: D/ : f = D/(s)f |s=0, D/(s) =

n=1

snen

xn. (63)

By definition, we have : D/ :2= : 4 : on the space of smooth functions.

We impose the periodic boundary condition (60) to : D/ : considered on

H(finite), and solve its eigenvalue problem as follows [7, 13]:

To set f(x1, x2, . . .) =

n=1 fn(xn) and fn(xn) = un(xn)+ envn(xn), the equa-

tion D/(s)f = (s)f with the boundary condition (60) induces the equations

unxn

= snn(s)vn,vnxn

= snn(s)un,

with the boundary conditions

fn(d/2n ) = fn(d/2n ),fnxn

(d/2n ) =fnxn

(d/2n ).

Hence we have

n(s) = mnsn, un = An cos(mn

d/2n xn) +Bn sin(mn

d/2n xn).

Here mns are integers. Since it must be

n=1

mnd/2n en H(finite),


136 A. Asada

we have

mn = mn+1 = = m, n N + 1, (64)

for large N . Hence iffn is meaningful, we obtain

D/(s)f = (m(G, s d/2) +

N

n=1

(mn m)sd/2n

)f.

Therefore, if (G,d/2) is finite, we have

: D/ : f = (m(G,d/2) +

N

n=1

(mn m)d/2n

)f. (65)

The eigen spinor belonging to this eigenvalue takes the form

n=1 fn. We assume

An = Bn = 1. Then we only need to consider the following two cases:

Case 1: limn | cos(mnd/2n xn)| = 1.In this case, the product of infinite numbers of sin(mn

d/2n xn) vanishes. Hence

n=1

(cos(mnd/2n xn) + en sin(mn

d/2n xn)) Cl(H).

Case 2: limn | sin(mnd/2n xn)| = 1.In this case, the product of infinite numbers of cos(mn

d/2n xn) vanishes. Since

n=1

(cos(mnd/2n xn) + en sin(mn

d/2n xn))

= e

n=1

(sin(mnd/2n xn) + (1)nen cos(mnd/2n xn)),

we have f Cl(H) e, in this case.

Therefore we conclude f Cl(H)[e]. Note that in both cases, we use theconvention (1) = (1) . Hence we have

Theorem 16 ([7, 13]). If m = 0, then the eigen spinor fields of the periodic

boundary value problem of the regularized Dirac operator are finite sum of finite

products of trigonometric functions, so they come from eigen spinor fields of finite

dimensional Dirac equations together with their eigenvalues.

If m 6= 0, the periodic boundary value problem of the regularized Dirac operatorhas the eigenvalue (G,d/2) which has no finite dimensional analogy. Its eigenspinor field also does not have finite dimensional analogy.

Note. The eigen spinor field belonging to the eigenvalue (G,d/2) vanishes onH . But it is non-trivial when considered on H(finite).



12. Logarithm of Derivation I. One-Variable Functions

To treat the geometric counterpart of a regularized integral, we use the logarithm

of derivation log(/xn) = limh0(h/xhn I), which will be studied by using

Borel transfomation [1].

First we consider 1-variable functions. In this case, for a holomorphic function

f at the origin, its Borel transformation B[f ] is defined by

B[f ](x) = 12i

f()

ex/ d

(=

n=0

cnn!xn

),

and satisfies

d

dxB[f ] = B

[f

t

], B[fg] = B[f ]]B[g], f]g = d

dx

x

0

f(x )g() d.

We also use the notation u]n =

n u] ]u, f(]u) =

cnu

]n, f(u) =cnu

n.

Lemma 17 ([1]). Let be the Euler constant. Then we have

e]t log x =et

(1 + t)xt. (66)

Proof. Since log (1 + t) = t+m=2(1)m(m)/mtm, we get

et

(1 + t)= 1 +

n=2

[n/2]

s=1

2j1js,j1++js=n

(1)ns (j1) (js)j1 js

tn.

Byn

k=11

j1jk js= j1++jsj1js ,

(s)=k

1

j(1) (j(1) + + j(s))=

1

j1 jk js(j(1) + + j(s))

=1

n j1 jk js,

holds for fixed j1, . . . , js, 2 j1 js, j1 + + js, if

St

1

j(1)(j(1) + j(2)) (j(1) + + j(t))=

1

j1j2 jt, (67)

holds for t < s. Since (67) is true if t = 1, (67) is true for all t.

Since

x

0

log(x )(log )n1 d = logx x

0

(log )n1 d

n=1

1

mxm x

0

m(log )n1 d,


138 A. Asada

to set logx](log x)n1 =n

k=0 an,k(logx)k , we get

an,n = 1, an,n1 = 0, an,0 = (1)n1(n 1)!(n),

an,K =(n 1)!

k!(n k 1)!ank,0, 2 k n 1.

Hence, to set (logx)]n =n

k=0(log x)n, we obtain

bn,n = 1, bn,n1 = 0, bn,k =n!

k!(n k)!bnk,0, 2 k n 1,

bn,0 =

[n/2]

s=1

ji2,j1++js=n

(1)ns n!(j1) (js)j1(j1 + j2) (j1 + + js)

, n 2.

By (67), bn,0/n! is the coefficient of tn of the Taylor expansion of the entire function

et/(1 + t). Hence for arbitrary c > 0, we get |bn,0/n!| = o(cn). Therefore weobtain

n=0

tn

n!(log x)]n =

n=0

tn

n!

(n

k=0

bn,k(log x)k

)=

k=0

(

n=k

k!

n!bn,kt

nk

)tk

k!(logx)k

=

(1 +

n=2

bn,0n!

tn

)(

n=0

tn

n!(logx)n

)=

et

(1 + t)xt.

By this lemma, we define the Borel transformation of logarithm by

B[log ](x) = logx+ . (68)

Definition 18. We define the logarithm of derivation log(d/dx) by

log

(d

dx

)f(x) = B

[log

(1

)u

](x)

= (w + )]f(x), f(x) = B[u](x), w = logx. (69)

Example. We have

log

(d

dx

)xn = xn

(logx+

(

(1 + + 1

n

))), n 1.

For n = 0, we have (log(d/dx))1 = B[log ] = logx+ .

Note. By using ]-product, we can define fractional derivation by

daf

dxa= e]a(w+)]f (= B[au]).

Appropriate domain of log(d/dx) is the space of finite exponential type functions

of w, which is denoted by F . If f, g F , we can define their ]-product. Explicitly,we obtain w]wn1 = wn Pn1(w), where

Pn(w) =

n

k=0

(1)nk (n+ 1)!k!

(n+ 2 k)wk .



If f F , f(]x) acts on F by the ]-product. This algebra is denoted by F.Although e]nw = 0 as an element of F , it is (n), the nth derivative of the -

function as an element of F, so it is not equal to 0 in F.

13. Logarithm of Derivation II. Several-Variable Functions

Let F be the algebra of finite exponential functions of w1, w2, . . .. Here wi = logxiand x1, x2, . . . are the coordinate functions of H (or H

(finite)). We define the

]-product of wn and wm, n 6= m, by

wn]wm = wn wm +m n|m n|

i

2. (70)

Let S = {p1, . . . , pm} be a set of natural numbers. Here we allow pi = pj , i 6= j.T S and {T is the complement of T in S. For an element pi of T , a naturalnumber \(pi) = \S(pi) is corresponded by

\(pi) = 1, pi = min T, \(pi) = 2, pi = min {{pj |\pj = 1} T,

and so on. Then we define the sign sgnT of T = {pj1 , . . . , pjk} by

sgn {pj1 , . . . , pjk} = sgn(

1, . . . , k

\(pj1 ), . . . , \(pjk )

). (71)

Let wn]wm be wn]wm if n 6= m and w2n if n = m.

Definition 19 ([10]). The product wp1 ] ]wpn is defined by

wp1 wpn +

2mn

(i

2

)msgn {{pj1 , . . . , pjn2m}wpj1 wpjn2m .

The algebra generated by w1, w2, . . . with the ]-product is denoted by F. It actson F by ]-product.

Definition 20. Let f F . Then we define

a

xanf = e]a(wn+)]f, log

(

xn

)f = (wn + )]f. (72)

By (70), we have

[wn, wm] = sgn (m n)i, (73)

where [wn, wm] = [wn, wm]] = wn]wm wm]wn. Hence we may regard F to be theunderlying noncommutative space of H.

By (73), we have

e]a(wn+)]e]b(wm+) = e(a(wn+)+b(wm+)+sgn(mn)abi/2).


140 A. Asada

By this equality, assuming 1 = ei if m > n and ei if m < n, we have:

Proposition 21. We have the following formulas:

e]a(wn+)]e]b(Wm+) = (1)abe]b(wm+)]e]a(wn+), n 6= m, (74)

e]a(wn+)]e]a(wm+)]1 = (1)a2 sinaa

(xmxn

)a, a 6= 1, (75)

e](wn+)]e](wm+)]1 = e](wm+)]e](wn+)]1 = n,m. (76)

Note. Operators such as log(/x1 + /x2) are not contained in F.

If

n=1 cn converges absolutely to C, then

n=1

m=n+1 = C(C1)/2. Hence

we have

e]c1w1]e]c2w2] = eC(C1)i/2e]P

cnwn .

Therefore, to set (s) =

n=1 sn(wn + ), we get

e](s) = e((G,s)((G,s)1)/2)ie]s1(w1+)]e]

s2(w2+)] , d. (77)

Definition 22. Regularized infinite product :,

n=1 ]e](wn+) : thought to be

the infinite product e](w1+)]e](w2+)] is defined by

:

,

n=1

]e](wn+) : ]f = e(1)i/2e](s)]f |s=0. (78)

:,

n=1 ]e](wn+) : is not defined on F . But if limn

nfx1xn

exists, we have

:

,

n=1

]e](wn+) : ]f = limn

nf

x1 xn.

Since log (1 + ) = + O(2), we have (w ) log((1 + ) =w+O(2). Hence (1+sn)1e]

sn(wn+) = e]

snwn+O(2sn ). Therefore we have

:

n=1 xn: f = e(1)i/2e](

P

snwn)]f |s=0. (79)

14. Regularized Volume Form

Let F1 be the submodule of F generated by homogeneous element of degree 1. Wedenote

F1, ={

cnwn F1|



Denoting the constant term of u]v]1, where u exp(F1,), and v exp(F1,+),by c0(u]v]1), we define the pairing u, v by

u, v = c0(u]v]1). (80)

It has the integral expression

limn

(lim

i1,...,in

1

2i1

2i1

0

d1 1

2in

2in

0

dnu]v]]1(1, . . .)

).

Hence this pairing is possible on the dense subspaces of exp(F1,). For example,we have

e]a(wn+), e]b(wm+) = 0, a 6= b, or n 6= m,

e]a(wn+), e]a(wn+) = (1)(a1)2 sin(a)a

, a / Z.

The sign of this second formula is (1)a2 1a, where 1a = e2ai [10].Let I+,c = Ic be the ideal generated by e]2(wn+) + c in exp(F1,+). Similar ideal

in exp(F1,) is also denoted by Ic.

Definition 23 ([10]). We say H = exp(F1,+)/I0 is the algebra of fractionaldifferential forms on H . Its elements are said to be the fractional differential forms.

Fractional differential form was also defined by Cottrill-Shephered and Naber

[24, 25] without using logarithm of differential forms. They also defined matrix

order diferential forms [25, II] and developed their calculus on finite dimensional

space.

Note. Similarly, we say Cl(H) = exp(F1,+)/I1 is the algebra of fractional spinorson H .

We can define the algebras W k and Cl(W k) in the same way. In these cases,

we replace wn by wn,k = wn k logn.

Let u[ be the class of u exp(F1,+) in H . Then we define the wedge productof u[ v[ of u[ and v[ by

u[ v[ = (u]v)[. (81)

We also set

daxn = (e]a(wn+))[, 0 < m.

Let u becnwn and = e

]u. We set

u\ =

cnwn, cn cn, mod 2, 0


142 A. Asada

Then we define the action [ f of [ to f F by

[ f = (e]u\)]f.

Since (s)\ = (s) if |s| is small, (e](s))[ f = e](s)]f if |s| is small.

Definition 24 ([10]). The regularized infinite wedge product : ,n=1 dxn :thought to be dx1 dx2 , is defined by

: ,n=1 dxn : f = e(1)/2(e](s))[ f |s=0. (84)

By definition, we may define

Q(x,+)

f : ,n=1 :=

Q(x),+)

f(x) : dx : . (85)

That is, we may consider : ,n=1 dxn : to be the regularized volume form of H

(or rather of H(finite)). In the rest, we denote dx instead of : ,n=1 dxn :, for

simplicity.

Let I = {i1, . . . , ip}, 1 i1 < < ip, be a set of natural numbers. Thendenoting

I(s) =

j /I

sj(wj + ),

we can define : n/Idxn : similar to : ,n=1 dxn :. We can regard :

n/Idxn : to be

the regularization of the infinite wedge product

dx1 . . . dxi11 dxi1+1 . . . dxip1 dxip+1 dxip+2 . . . .

For simplicity, we denote : n/Idxn : by dIx.

Note 1. Applying the same process to Cl(H), we obtain -spinor. Then we canjustify our calculations in Sec. 11.

Note 2. Taking a matrix A, we can define the matrix order differential form

dAxn by

dAxn = e]A(wn+) = I +

n1

An

n!(wn + )

]n.

But unless the Lie algebra generated by A, B is nilpotent, we cannot get a practical

commutation rule of the wedge product of dAxn and dBxm (cf. [25, II]).

15. Grassmann Algebra with an -degree Element and -forms

In Sec. 14, we defined -form dx and ( I)-form dIx, where

I = {i1, . . . , ip}, 1 i1 < , ip, .



Then, denoting dIx = dxi1 . . . dxip , we have by definition

dJx dIx = d{IJ}x, J I, dJx dIx = 0, J * I. (86)

We call the degree of dIx as p, where p = |I |, the number of elements of I .The Grassmann algebra over H (resp. over W k) with ( I)-forms is denoted

by Gr(H) (resp. Gr(W k)). By (83), we have

Theorem 25. The commutation relations of the wedge products of a p-form p

and an ( q)-form q are

p q = (1)p(q)q p, 1 = ei, (87)q p = (1)(q)pp q , 1 = ei. (88)

Hence Gr(H) is an associative algebra if it is considered to be a C-algebra.While Gr(H) is associative if and only if is an integer, if it is considered to

be an R-algebra. Note that similar to the Clifford algebra with an -spinor, thiscommutation relation is the same as that of noncommutative torus, if / Q.

We denote Gr(H) the Grassmann algebra over H . The module of ( p)-forms in Gr(H) is denoted by Gr(H). Then by (86), we can define the pairing of

p Gr(H) and q Gr(H) by

p, p = c, p p = cdx, p, q = 0, p 6= q. (89)

By this pairing, we regard Gr(H) to be the dual space of Gr(H) and Gr(H) to

be the dual space of Gr(H). Since a differential form on an open set U of W k

takes the value in Gr(Wk), we may consider an (p)-form on U to be a smoothmap from U to p(W k), the submodule of pth degree elements of Gr(W k) [5]. In

other words, we can regard

Gr(Wk) = Gr(Wk) Gr(W k).

Note. In this case, the pairing of p Gr(Wk) and p Gr(W k) should be

, = c, = c det (G)2kdx.

By the definitions of Wk and W k, we know G2k : Wk = W k. Regarding anelement of Gr(Wk) to be an alternative tensor, we define

G2k,]u1 . . . up = G2ku1 . . . G2kup.

Then we have G2k,] : Gr(Wk) = Gr(W k). We also define the map G2k,] :Gr(W k) = Gr(Wk) similarly.

Definition 26 ([5]). The Hodge -operator : Gr(Wk) Gr(Wk) isdefined by

up = (1)p det(G)kG2k,]up, (90)p = (1)p det(G)kG2k,]p. (91)


144 A. Asada

16. Exterior Differential of ( p)-forms

A smooth map from U to Gr(Wk) is the finite degree form and to Gr(W k) is

the infinite degree form. By this interpretation, we may regard an ( p)-form on

U to be a smooth alternative map f from U to

p W k W k. Then regarding

x1, . . . , xp1 to be parameters and denote df the Frechet differential of f , we define

the exterior differential df of f by

df(x, x1, . . . , xp1) = (1)p1tr(df(x, x1, . . . , xp1, x)), (92)

provided df(x, x1, . . . , xp1, x) is a trace class operator on Wk.

By using coordinate expression, we have

d

(

I

fidIx

)=

J

(

i/J

(1)sgn(i,J) f{i,J}xi

)dJx,

sgn (i, J) = i 1, i < j1, J = {j1, . . . , jp1},sgn (i, J) = j1 + + jk1 k, jk1 < i < jk,sgn (i, J) = j1 + + jp1 p, jp1 < i.

Since this right-hand side contains infinite sum, ( p)-forms may not be exteriordifferentiable in general. In fact, we have [5, 11]:

Theorem 27. An exterior differentiable ( p)-form is exact.

Proof. Since the theorem is true if p = 0, we assume p = 1 and set =fnd

nx. Then we have d =(

n=1(1)n1 fnxn)dx. Since is exterior

differentiable,

n=1(1)n1 fnxn converges. We assume

= d, =

n=1

gn,n+1d{n,n+1}x.

Then it must be

g1,2x2

= f1, (1)n2(gn1,nxn1

gn,n+1xn+1

)= fn, n 2.

We set g1,2 = x10f1 dt, and gn,n+1 =

xn+10

((1)n1fn + gn1,nxn+1

)dt. Then,

since

g2,3 =

x3

0

(f2 +

x1

x2

0

f1 d

)dt =

x3

0

(f2 +

x2

0

f1x1

d

)dt,



we getg2,3x2

= x30

( f2x2 +

f1x1

)dt. We assume

gn1,nxn1

=

xn

0

(n1

i=1

(1)i1 fixi

)dt. (93)

Then, since

gn,n+1xn+1

= (1)n1fn +gn1,nxn1

= (1)n1fn + xn

0

(n1

i=1

(1)i1 fixi

)dt,

we have

gn+1,n+2xn+2

= (1)nfn+1 +gn,n+1xn

= (1)nfn+1 +

xn

xn+1

0

((1)n1fn +

xn

0

n1

i=1

(1)i1 fixi

d

)dt

= (1)nfn+1 + xn+1

0

(n

i=1

(1)i1 fixi

)dt.

Hence we havegn,n+1

xn= xn+10

(ni=1(1)i1 fixi

)dt. Therefore (93) holds for any

n 1.Since limn(1)i1 fnxn converges,

gn,n+1xn C|xn+1|, holds for some C > 0.

Hence

n=1 gn,n+1d{n,n+1}x converges and we have the theorem for p = 1.

Let p 2 and J = {j1, . . . , jp}, J = {j1, . . . , jp1}.

=

J

i>jp1

f{J,i}d{J,i}x,

be an ( p)-form. Formal exterior differential d of is given by

d =

J

kJ

f{J,j}xk

dyk d{J,j}x+

j>jp1

(1)j+p f{J,j}

xjd{J

,j}x

.

sumj>jp1 (1)jf{J,j}/xj and the sum of these sums with respect to J convergesif is exterior differentiable.

If = d,

=

J

i>jp1

g{J,i,i+1}d{J,i,i+1}x,


146 A. Asada

is an exterior differentiable ( p 1)-form. Then since d is given by

J

(

kjp1+1

(1)j+p(g{J,j+1,j+2}

xj+2 g{J

,j,j+1}

xj

)d{J

,j+1}x

),

it must be

f{J,jp1+1} =

k jp1 + 1.

Since the right-hand sides of these equalities are finite sums, we can replace

f{J,jp1+1} by

f{J,1} = f{J,jp1+1}

k jp1 + 1.

Since this formula is the same as the case p = 1, converges if is exterior

differentiable. Hence we have the theorem.

Note 1. d2 6= 0 on the space of ( p)-forms, by this theorem. For example, take =

(1 1/2n)xnxn+1d{n,n+1}x, we have

d =

n=1

(1)nxn2nd{n}x, d2 = dx 6= 0.

Therefore we can construct a geometric model of Kerners higher order gauge

theory based on differential d such that [15, 27, 34]

dN1 6= 0, dN = 0, N 3,by using suitable subspace of the space of ( p)-forms.Note 2. If f is a function, then d(f) = df + fd. Hence

d2(f) = d2f df d+ df d+ fd2 = fd2.



In general, we obtain

d2n(f) = fd2n, d2n+1(f) = df d2n+ fd2n+1. (94)

If an ( p)-form is exterior differentiable, then = d, so is written as dv,that is = d2v. By (94), by using smooth partition of unity, we can construct v

globally. Hence also exists globally.

17. Regularized Exterior Differential

Since dxn,k = kn dxn, as an element of Gr

(H), the ( p)-form dI dx;k inGr(Wk) is

ki1 kip(det(G)

kdIx, I = {i1, . . . , ip}.

So for an ( I)-form = I fIdIx, we set

(s) =

I

si1 sip det(G)

sfIdIx. (95)

Since dxn(s) should be sn dxn, we can define (s) for a finite degree form , by

the same way. Note that d(s) 6= (d)(s), in general.

Definition 28 ([11, 12]). We define the regularized exterior differential : d :

of by

: d : = d((s))|s=0. (96)

The factor det(G)s has no effect in the definition of regularized exterior differ-

ential. This factor should be meaningful when we consider regularized differential

on curved spaces.

Note. Except for the factor det(G)s, this definition is the same as the definitions

of regularized Laplacian and Dirac operator given in Secs. 10 and 11.

If is exterior differentiable, then we have : d : = d. Since a finite degree

form is always exterior differentiable, we need not consider regularized exterior dif-

ferential for finite degree forms. While there exist regularized exterior differentiable,

but not exterior differentiable ( p)-forms, so regularized exterior differential ismeaningful only on the space of ( p)-forms.

Example. Let =

n=1(1)n1xnd{n}x. Then (s) is not exterior differen-tiable, but since

(s) =

n=1

(1)n1sn det(G)sd{n}x,

we have d(s) = (G, s) det(G)sdx, if d. Hence we obtain

: d : = dx, : d : (ra) = (a+ )dx.


148 A. Asada

By this second equality, r is not exterior differentiable, but closed with

respect to regularized exterior differential. We may regard r to be the (regu-

larized) volume form of the sphere of H (or H(finite)). But at present, we cannot

yet relate r and the regularized volume form of the sphere obtained in Sec. 9.

Note. We have

(s) = d(s), (s) =

n=1

n

k=1

skxkxk+1d{k,k+1}x,

if 0. Because |xn xn+1| < Cdn for some C > 0. Hence (s) is not exteriordifferentiable, but exact if 0 < n/2. Under this assumption, elements of W k(X) or

W k(X,E), the space of Sobolev k-class spinors, are continuous by the Sobolev

embedding theorem.

In the rest, we only consider the connected component of constant maps in

Map(X,M) which is also denoted by Map(X,M). Then the graph

Gr(f) = {(x, f(x))|x X} X M,

of f Map(X,M) is homotopic to X, M . Since the normal bundle of Xin XM is trivial, the normal bundle of Gr(f) in XM is trivial. Hence there is a



neighborhood U(Gr(f)) of Gr(f) such that U(Gr(f)) is diffeomorphic to Gr RN .Hence Map(X,M) is a Sobolev manifold modeled by W k(x) RN , where W k(X)is the kth Sobolev space of scalar fields on X .

If M is a real algebraic manifold defined by the equations

F1(x1, . . . , xm) = = Fj(x1, . . . , xm) = 0,

in Rm, then Map(X,M) is a connected component of the algebraic set inW k(X) Rm determined by

F1(f1, . . . , fm) = = Fj(f1, . . . , fm) = 0, fi W k(X). (100)Since F1, . . . , Fj are polynomials, Eqs. (100) are meaningful lifting W

k(X) to

W k(X,E), the k-Sobolev space of spinor fields on X , under a fixed ordering of

variables X1, . . . , Xm. Hence Map(X,M) is contained in a connected component

C of an algebraic set in W k(X,E) Rm. Taking a sufficiently small neighborhoodM of Map(X,M) in C, M is a deformation retract of Map(X,M) and a Sobolevmanifold modeled by W k(X,E) [5]. For simplicity, we use M instead of Map(X,M)and consider Map(X,M) to be a Sobolev manifold modeled by W k(X,E).

In the rest, D stands for 4 IN or D/ IN acting on Rn-valued functions on Xor on the (vector of) spinor field on X added mass-terms if necessary. The Green

operator of D is denoted by G.

Since the tangent space T of Map(X,M) is either of Wk(X) RN orWk(X,E) RN , we equip G with T . But for the later discussions, it is moreconvenient to use D instead of G. So we consider T as being equipped with D.

Let {gUV } be the transition functions of the tangent bundle = (M) ofM , then the transition functions of the tangent bundle X = (Map(X,M)) of

Map(X,M) are given by {gXUV }, where gXUV means

(gXUV (f))(x) = gUV (f(x)), f : X U V.

Hence X is a Map(X,G)-bundle, where G is the structure group of . Since X is

a spin manifold, we may regard Map(X,G) GLp, p > n/2. Hence it is a loopgroup bundle as a smooth bundle.

Since Map(X,M) is a Hilbert manifold, by using smooth partition of unity, we

can construct a connection {AU} of X with respect to D. Since D is self-adjoint,we can take AU so that AU (x) is Hermitian for any x U . But if D = D/ IN andD +AU (x) is always non-degenerate, then we obtain global polarization J on

X .

Since gXUV J = JgXUV , there exists {hU}, hU : U Map(X,G) such that

hUgUV h1V =

(aUV 0

0 bUV

).

{aUV } and {bUV } define Hilbert bundles, so they are trivial by Kuipers Theo-rem [37]. Therefore X is trivial. On the other hand, if D = 4 IN , we canchoose AU so that D+AU (x) is non-degenerate for any x U , because we can addmass-term mI to AU . Hence we obtain


150 A. Asada

Proposition 29. (i) If Map(X,M) is modeled by W k(X) RN , then X has aconnection {AU} with respect toD = 4IN such thatD+AU (x) is non-degeneratefor any x U and U .(ii) If Map(X,M) is modeled by W k(X,E)RN , then X has a connection {AU}with respect to D = D/ IN such that D+AU (x) is non-degenerate for any x Uand U , if and only if X is trivial as a GLp-bundle.

Since D + AU (x) and D + AV (x) are unitary equivalent, they have the same

spectra. Hence (D+AU (x), s), or (D+AU (x), s) does not depend on U . We set

A(x, s) = (D +AU (x), s), A(x, s) = (D +AU (x), s).

The function detA(x) = det(D +AU (x)) is defined by

det(D +AU (x)) = det(D +AU (x) +mI)|m=0. (101)

By (22), if D + AU (x) is non-degenerate, then this definition coincides with the

RaySinger determinant of D+AU (x), and if D+AU (x) degenerates, then det(D+

AU (x)) = 0. Note that although D+AU (x) and D+AV (x) have the same spectra,

det(D+AU (x)) and det(D+AV (x)) may be different if they are defined by (101).

We set

Y = {x Map(X,M)| ker(D +AU (x)) 6= {0}}= {x Map(X,M)| det

A(x) = 0}.

Then on Map(X,M) \ Y , X is trivial.Let {hU} be the trivialization of X |(Map(X,M) \ Y ). Then to set

DY (x) = hU (x)1(D +AU (x))hU (x) (= D +A

U (x)),

{AU} is a connection of X |(Map(X,M) \ Y ) with respect to D. Since DY (x)is non-degenerate, D2Y (x) has the Green operator GY (x) whose eigenvalues and

eigenfiunctions are

1(x) 2(x) > 0, GY (x)ei(x) = ie(x).

On an arcwise-connected component of Map(X,M) \ Y , we can take both i(x)and ei(x) to be continuous in x.

As in Sec. 14, we set

(s)(x) =

n(x)s(wn(x) + ),

where wn(x) is thought to be logxn(x), x1(x), x2(x), . . . are the coordinates of the

fibre of X given by e1(x), e2(x), . . .. Since we may consider (s)\ = (s), if |s| is

small, we set

e(x)((x)1)/2e](s)|s=0 =: ,n=1 dxn(x) :, (102)

Definition 30 ([10]). We say : ,n=1 dxn : is the regularized volume form ofMap(X,M).



Note. This volume form depends on the choice of DY . Its regularization uses

spectres of D2Y . So it must be called to be the regularized volume form with respect

to D2Y . But for simplicity, we only call it regularized volume form.

19. Regularized Volume Form and the Determinant Bundle

Since limn 1(xn) = , if limnXn Y , limn (s)(xn) diverges. Hencethe regularized volume form

: ,n=1 dxn(x) := e(x)((x)1)/2e](x)|s=0,

has singularities along Y . Since n(x)1 is an eigenvalue of D2Y (x), the counter

term to this singularity is (detDY )2.

(detDY (x))2 : ,n=1 dxn(x) : also has the factor e(x)((x)1)/2. This factor

may be a multiple-valued function on Map(X,Y ). But this ambuigity is resolved

by using the covering of Map(X,M).

The singularity that comes from Y is resolved by constructing a line bundle

over Map(X,M). The regularized volume form is a cross-section of 2. Hence can

be interpreted as the determinant bundle of Map(X,M).

Let {AU} be a connection of X with respect to D. det(D + AU (x) is definedby (101). Note that it may be

det(D +AU (x) +mI)|m=0 6= det((D +AU (x) +mI) +mI)|m=m.But by (22), we have

det(D +AU (x) +mI)|m=0det((D +AU (x) +mI) +mI)m=m

C(= C \ {0}),

although det(D + AU (x)) = 0. Since D + AU (x) and D + AV (x) have the same

spectra, we get

det(D +AU (x))

det(D +AV (x)): U V C. (103)

Lemma 31. As a smooth bundle, {det(D + AU )/ det(D + AV )} does not dependon the choice of connection {AU}.

Proof. If {AU} is another connection of X with respect to D, then to setAU,t = AU +BU , 0 t 1, BU = AU AU ,

{AU,t} becomes a connection of X with respect to D. Since det(D+AU,t) dependssmoothly on t, the bundles {det(D+AU )/ det(d+AV )} and {det(D+AU )/ det(D+AV )} belong to the same connected components of H2(Map(X,M),Z), which isdiscrete. Hence they are the same as smooth bundles.

Since det(h1U (D + AU )hU )/ det(D + AU ) : U C, the equivalence class ofthe line bundle {det(D+AU )/ det(D+AV )} is not influenced by the change of thetransition function of X . Hence we have the lemma.


152 A. Asada

Definition 32. We denote the bundle {det(D+AV )/ det(D+AU )} by and callthe determinant bundle of Map(X,M).

By the definition of and the regularized volume form : ,n=1 dxn :, the regu-larized volume form (transformed by {hU}) is a cross-section of 2. We denote thiscross-section by dvol(D2Y ).

Definition 33. We call dvol(D2Y ) the regularized volume form of Map(X,M) with

respect to D2Y .

For simplicity, we denote dvol instead of dvol(D2Y ) and call the regularized

volume form of Map(X,M).

Note. The power 2 of comes from our definition of (s). Because its coefficients

are the square of eigenvalues of D.

Let X, be the cotangent bundle of Map(X,M). By the Sobolev duality, the

fibre of X, is Wk, where Wk stands for either Wk(X)RN or Wk(X,E)RN .

Definition 34. We call a smooth cross-section of p(Wk) 2 to be an ( p)-form on Map(X,M).

By definition, if {U} is an ( p)-form, then

U = f1U , fU = det(D +AU )

2, (104)

where is a cross-section of X,. By (104), (D+Au(x))2k gives a pairing of the

fibres of pX and pX, 2 which may be considered as the Sobolev duality.Therefore we can apply the definition of the Hodge -operator (90), (91) in this

case. Hence we can define the exterior differential d for ( p)-forms. Its localproperties are the same as in Secs. 16 and 17. Especially, any exterior differentiable

( p)-form on Map(X,M) is exact on Map(X,M). Hence we cannot expect( p)-dimensional de Rham theory on Map(X,M).

Precisely, let

Bpd .: The sheaf of germs of smooth closed ( p)-forms,Cpex.d .: The sheaf of germs of smooth exterior differentiable ( p)-forms.

Then we have the exact sequences

0 BpdiCpex.d

d dCpex.d 0,0 Bpd

i dCp1ex.dd d2Cp1ex.d 0.

Since Cpex.d and d2Cp1ex.d are both fine sheaves, we get

Hp(Map(X,M),Bqd ) = Hp1(Map(X,M), dCq+1ex.d ),

Hp(Map(X,M),Bqd ) = Hp(Map(X,M), dCq1ex.d ),



by these exact sequences. Hence we have

Hp(Map(X,M),Bq)= H0(Map(X,M), dCq+p1ex.d )/d(H0(Map(X,M), C

q+p1ex.d ).

By this isomorphism and Note 2 of Sec. 16, we conclude

Hp(Map(X,M),Bq) = 0, p 1.

Note. Owing to the existence of the factor detG in the definition of regularized

exterior differential, we may define regularized exterior differential on Map(X,M).

20. Half Infinite Forms

In this section, we assume Map(X,M) is modeled by W k(X,E) RN , anddimY > 1, that is Map(X,M) \ Y is arcwise-connected.

Let Wk(X)x,+ (resp. Wk(X)x,) be the positive (resp. negative) eigen space

of D +AU (x) in Wkx , the fiber of

X at x Map(X,M). Then we have

Wk(X) = Wk(X)x,+ Wk(X)x,, x / Y.

Since Map(X,Y ) \ Y is arcwise connected, there is a path such that

Map(X,M) \ Y, (0) = x, (1) = y,

for any x, y / Map(X,M) \ Y . By the definition of Y , D + AU ((t)), 0 t 1does not have 0-mode. Hence it induces a unitary operator U,x,y : W

k(X)x, =Wk(X)y,. Hence we have

X |(Map(X,M) \ Y ) = X+ X , X,x = Wk(X)x,. (105)

Here X,x means the fibre of X at x.

Let n,(x) and en,(x) be the proper value and function of D +AU (x);

(D +AU (x))en,(x) = n,(x)en,(x), x / Map(X,M) \ Y.

Let xn, be the coordinate of Wk(X)x, determined by en, and wn,(x) is the

same as wn(x). Then we set

(s)(x) =

n=1

sn,(wn,(x) + ),

and define : ,n=1 dxn, : by

: ,n=1 dxn, := e(x)((x)1)/2e](s)(x)|s=0. (106)

We define the operator (D+AU (x)) by (15). Then we can define det(D+AU (x))similar to det(D +AU (x)). By definition, we have

det(D +AU (x)) 6= 0, x / Y, det(D +AU (x)) = 0, x Y.


154 A. Asada

Since det(D + AV (x))/ det(D + AU (x)) C, x Map(X,M), we define theline bundles by

=

{det(D +AV (x))det(D +AU (x))

}.

By definitions, : ,n=1 dxn, : define cross-sections dvol(D +A) of .

Definition 35. We call dvol+(D +A) (resp. dvol(D +A)) to be the regularized

positive (resp. negative) volume form of Map(X,M) with respect to D +A.

For simplicity, we write dvol instead of dvolpm(D + A) and say regularized

positive (or negative) half volume form of Map(X,M).

Since are line bundles, + is a line bundle. dvol+ dvol is across-section of this bundle. Giving the commutation rule

u v = (1)+v u,u a cross-section of + and v a cross-section of , we interprete this tensor product

to be the wedge product. Then

dvol(D +A) = dvol+(D +A) dvol(D +A), (107)should be the regularized volume form of Map(X,M) with respect to D +A.

The bundles X are not extended to bundles over Map(X,M). But X are

extended to bundles X on Map(X,M). Similarly, denoting X, the dual bundles

of X ,

X, = (X ) 2,are extended to bundles X, over Map(X,M). Since + might be differentfrom 2, dvol(D + a) might be different from dvol(D2Y ).

Note. If + and are equivalent, then we have

X+ X = X +. (108)Hence under the same assumption, we get

p(X+ X ) =

(pX

) p+.

Definition 36. (i) We call a smooth cross-section (p,0) of pX+ (resp. (0,p) ofpX ) to be a (p,+)-form (resp. (p,)-form) on Map(X,M).(ii) We call a smooth cross-section (p,0) of pX,+ p+1+ (resp. (0,p) ofpX, p+1 ) to be an ( p,+)-form (resp. ( p,)-form) on Map(X,M).

( p,)-forms are said to be half infinite forms. For example, dvol are(,)-forms, so they are half infinite forms. By using p(X+ X ) and so on, wecan define several mixed type forms. But (108) shows unless assuming flatness of

, we cannot define nilpotent exterior differential for these forms.



Acknowledgement

Just before starting to write this paper, I was informed that Prof. Gr. Tsagas of

Aristotole University of Thessaloniki passed away. Many parts of this paper were

first reported at a workshop on Global Analysis, Differential Geometry and Lie

Algebras at Thessaloniki, organized by Prof. Tsagas. Here, I express my heartfelt

thanks to Prof. Tsagas and dedicate this paper to his memory.

References

[1] A. Asada, Some extension of Borel transformatio

akira zeta reg infinite dims

Documents

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elliptic operator p

infinite forms

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method of regularization

hilbert space h