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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
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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.
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 111
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.
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 113
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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
)},
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 115
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].
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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)
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 117
= (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].
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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.
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 119
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,
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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.
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Regularized Calculus 121
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]).
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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),
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Regularized Calculus 123
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.
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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)).
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 125
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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}.
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 127
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,
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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).
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 129
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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.
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Regularized Calculus 131
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.
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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.
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Regularized Calculus 133
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.
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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,
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 135
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),
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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).
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 137
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,
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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 .
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 139
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).
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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|
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Regularized Calculus 141
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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, .
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 143
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)
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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,
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 145
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,
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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.
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Regularized Calculus 147
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.
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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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
Regularized Calculus 149
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
March 16, 2004 19:14 WSPC/IJGMMP-J043 00007
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).
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Regularized Calculus 151
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.
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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 ),
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Regularized Calculus 153
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.
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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.
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Regularized Calculus 155
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