a) Scaling of metrics on X and ^.
b) A basic closed 1-form on R* x R*.
c) A change of coordinates.
d) A contour integral.
The purpose of this Section is to construct a closed 1-form P on R* x R* associa-ted to the two parameters semi-group u > 0, T > 0->exp(—(u(DX-^-rT'V))2), and a contour r in R* X R * depending on three parameters s. A, To, so that P=0. To
Jr
prove Theorem 0.1, we will then push the contour F to the boundary ofR* x R*, and take the obvious limit in the previous identity.
The construction of the form P is directly related to results of [B2] concerning double transgression formulas for Quillen's superconnection forms, and their depen-dence on the considered metrics. In fact we interpret our result in terms of the scaling of the metrics ^Tx, /^o, . . ., h^ by the factors l/u2, 1/T2, . . ., 1/T2"1 respectively.
This scaling will also play a key role in Section 10.
This Section is organized as follows. In a), we describe the scaling of the metrics on TX, ^o, . . . . ^. In b), we construct a basic closed 1-form, a, on R* x R * . In c) we obtain our form P by a change of coordinates. In d), we describe the contour r i n R * x R * .
a) Scaling of metrics on X and ^
Take u > 0, T > 0. Suppose that the metrics ^Tx, /^°, . . ., /^m are replaced by the metrics g^/u2, /^°, A^/T2, . . . . h^/T2"1. Then the adjoints of the operators ^x, v become u2^, T2^*.
The basic idea of the paper is to study the deformation of various supertraces as u and T vary. However at a technical level, it is easier to scale ^x, ^x* and v, v* in the same way.
Our supertraces will be calculated on the Z^-graded vector space E.
Proposition 3 . 1 . — For any u > 0, T > 0, the following identities hold (3.1) Tr, [N^ exp (- (^ + v + u2 ^ + T2 v)2)]
= Tr, [N^ exp (- (u ^x + ^) + T (p + z;*))2)], Tr, [NH exp (- (^x + v + u2 9^ + T2 z;*)2)]
= Tr, [NH exp (- (u ^ + ^) + T (z. + z;*))2)].
Proof. - Observe that
(3.2) u^^-^^x+v+u2y^^2^)^^u~^=u(9x+8x')^^(v+v^
[N^NH]=O.
Using the fact that super-traces vanish on supercommutators [Ql], (3.1) follows. D Remark 3.2. - The key fact is that in the left-hand side of (3.1), the coboundary operator ^x + v does not change, only the metrics on TX and ^ are changing. In the right-hand side, the coboundary operator y-^-v is changed into M^+TZ;, and the metrics on TX, ^ do not vary. The correct geometric picture is the one given by the left-hand side.
b) A basic closed 1-form on R* x R*.
Set
(3.3) Dx-^x+^x••
v / V=7;+Z;*.
For u > 0, T ^ 0, set
(3.4) A^^D^TV.
Then Ay y 1s an elliptic first order differential operator.
Theorem 3.3. - Let o^ y be the smooth \-form on R* x R*
(3.5) au'T = ^ Trs [N^ exp (~ Au2 T)] ~ % Tr- ^H exp (~ A"2 T)]-Then tty -p is a closed form.
Proof. - For u > 0, T ^ 0, the operator exp(-A^ -p) is given by a smooth kernel on the manifold X. By preceding as in [Bl, Proposition 2.8] i.e. by expressing the above supertraces as integrals on the manifold X of integrals of supertraces of heat kernels evaluated on the diagonal, one can easily justify the following manipulations of supertraces.
We have the identity
(3.6) '-TrJN^exp^A^)]
01
= 9„ < irj iNyexpi A,, T t) A T , ——— I \\ ^Ti-rN^xo^-A2 -h^A ^•'^IVIl ob (. L \ L 5T J/Di,=o
Now 8A^ T/5T = V. Since supertraces vanish on supercommutators, we rewrite (3.6) in the form
(3.7) ^Tr,[N^exp(-A^)]= - ^ {Tr,[[<^ N^]exp(-A^-6V)]},.o.
Clearly
(3.8) [A^N^-M^+M^'.
Using (3.6)-(3.8), we get
(3.9) ^TrJN^exp(-A^)]
=^{Tr,[(^x-^)exp(-A^-6(z,+^))]},^.
Now
(3.10) A^2 T == [u y + T v, u y + T z;*],
and so A^ preserves the total degree in E. Also ^x, v increase the total degree by one and ^x*, v* decrease the total degree by one. The degree counting argument of [BGS1, Proposition 1.8] gives
(3.11) ^{Tr^x-3xt)exp(-A^-^z,+^))]}^
= ^ {Tr^^expC-A^-^^-TrJ^'expC-A^-^)]},^.
Using (3.9), (3.11), we obtain
(3.12) -^TrJN^xp^A^)]}
8T iu ' J 9 f l
, M
__a_
Tb^ {Tr, [^ exp (- A^2 ^ - &^*)] - Tr, px* exp ( - A^2 ^ - ^)]},. „•
By interchanging the roles of u and T, bearing in mind that the analogue of N^
is -NH and using again the degree counting argument of [BGS1, Proposition 1.8], we also get
(3.13) |J^Tr,[NHexp(-A^)]lu } A •T" nk.T - / A 7 \i i
=-a{TrJ^exp(-A,2T-&^)]-TrJz;*exp(-A^-6SX)]}„o.
00
Since supertraces vanish on supercommutators, we find that
(3.14) ^{TrJ^expC-A^-^M-TrJ^expC-A^-^n^o ob
=^{Tr,hexp(-A^-63x•)]-Tr,h*exp(-A„2^.-63x)]}^o•
00
From (3.12), (3.14), we deduce that the form o^ y is closed. D
Remark 3 . 4 . — Theorem 3.3 is in fact a consequence of a general result established in [B2, Theorem 2.2] on double transgression formulas for Quillen superconnection Chern character forms, which extend corresponding formulas of Bott and Chern [BoC, 3.28] for ordinary connections. Here we consider E as a vector bundle over a base S consisting of a point. In fact if /4 is the metric on i; which is the direct sum of the metrics /^o, /^i/T2, . . ., /^/T2^ then
(3.15) ^rl^=-rNH. 5T T
Similarly if g^011^ denotes the metric induced by the metric g^ju2 on ACT^^X), then
3 ^(0,i)x x
(3.16) g^—— =2 N V. ou u
By [B2, Theorem 2.2], since the base S is just a single point, we find that the form 5^ T given by
auT=Tr,^f^N^-rfTNH)exp(-(^x+z;+^3x'+T2^)2)1 L\ u T / J is closed. By Proposition 3.1, ^u,T=(lu,T^ ^d so ^T 1s closed.
c) A change of coordinates For u > 0, T > 0, set
(3.17) B^^D^TV).
Theorem 3.5. - Let P^y be the \-form on R* x R*
(3.18) P».T= ^Tr,[(N^-NH)exp(-B^)]- ^TrJNHexp(-B^)].
u T Then py y w a closed form.
Proof. — In Theorem 3.3, we make the change of variables u -> u, T ->• u T.
Theorem 3.5 follows. D
d) A contour integral
We now fix constants e. A, To such that 0 < e < l ^ A < + o o , l ^ T o < + o o . Let r = Fg A, TO ^e the oriented contour in R* x R*.
u
r.
A
r, r,
r.
Tr T
pie. i
As shown in Figure 1, the contour r is made of four oriented pieces:
r i : T = T o ; £ ^ M ^ A , F^: 1 ^ T ^ T o ; M = A , r 3 : T = l ; s ^ M ^ A , T^: 1 i$T <$T(); y=s.
The orientation of r\, . . ., Y^ is indicated in Figure 1.
For 1 ^ k ^ 4, set (3.19) I,°=f P,T.T O -~ | ^,1
^rfc
J i f r
Theorem 3.6. — The following identity holds
4
(3.20) ^ I,°=0.
f c = l
Proo/. - Theorem 3.6 is a trivial consequence of Theorem 3.5. D
Remark 3.7. - We now will make A - ^ + o o , To ->- + oo, £ -> 0 in this order in identity (3.20). Typically each term 1^ (1 ^ k ^ 4) will diverge at one or several stages of this process. However because of the identity (3.20), the divergences will cancel, often for non trivial reasons. Once the divergences in each term will have been substracted off, we will ultimately obtain an identity which is exactly our main Theorem 6.1.
Roughly speaking, in this process,
• I? will calculate the Ray-Singer torsion of the complex (F, 3^.
• 1^ will calculate the ratio of the metrics | | ~ J | [^.
• 1^ will calculate the Ray-Singer torsion of the complex (E, (F+Z;).
• 1^ will produce highly non trivial local terms, which include the Bott-Chern current T(^, /^) of Bismut-Gillet-Soule [BGS4] and the class B(TY, TX|v, g^^) of Bismut [B3].
Remark 3.8. - Take toe]0, 1[. Replacing To by to, we then obtain a contour r^A,ro- Again P=0. In principle by making A - > + o o , to->0, £ - ^ 0 in this
^e.A^o
order, we may reprove Theorem 2.1. However, to carry this out, one then has to deal analytically with spectral sequences which in general do not degenerate. The proof of Theorem 2.1 has exactly consisted in carefully avoiding this difficulty.