The Zeta-Invariant of Dirac Operators Coupled to ... - CiteSeerX

The Zeta-Invariant of Dirac Operators Coupled to Instantons

Helga Baum Abstract We calculate the zeta-invariant of Dirac operators

DA2 coupled to instantons A.

1 Introduction

Let (M n; g) be a closed Riemannian spin manifold and denote by S the spinor bundle of (M n ; g). Furthermore, let P be a G-principal bundle on M , % : G ! GL(V ) a faithful unitary representation and E := P % V the associated vector bundle. Each connection A of P induces a Dirac operator DA : ?(S E ) ?! ?(S E ) with values in E . DA2 is a non-negative, elliptic, selfadjoint dierential operator. The zeta-function  (DA2 ; s) of DA2 is de ned by X ?s:  (DA2 ; s) := 2spec(DA2 ) >0

 (DA2 ;
) is holomorphic on the half plane Re(s) > n2 and has a meromorphic extension to the complex plane with only simple poles. In s = 0 the zeta-function is regular,  (DA2 ) :=  (DA2 ; 0) is called the zeta-invariant of DA2 . In the present paper we calculate the zeta-invariant  (DA2 ) and the residua in the only poles s = 1 and s = 2 on 4-dimensional manifolds using the relation between the zetafunction and the coecients in the heat expansion. Since DA2 is of Laplace type, these coecients can be expressed in geometrical terms by Gilkey's formulas (Theorem 1). This implies the following relation between the zeta-invariant and the Yang-Mills functional L% (A) on 4-dimensional manifolds:  (DA2 ) = r(M )
dim E + 1212 L% (A) ? dim Ker DA; where 1 Z (5R2 ? 7k0

 (D; s) is holomorphic on the half plane Re(s) > n2 . Let denote by ak (D) the coecients in the asymptotic expansion of the trace of e?tD (t > 0) Trace(e?tD ) t#0 (4t)? n 2

1 X

k=0

ak (D)tk :

(1)

The coecients ak (D) are local quantities, they can be expressed as integrals Z

ak (D) = Trxak (D; x)dM;

(2)

M

where ak (D;
) are sections in Hom(F; F ) determined by the total symbol of D. Let denote by H : ?(F ) ! ?(F ) the projection on Ker D. D + H is positive and  (D; s) + dim Ker D =  (D + H; s): Using the Mellin transform for the ?-function ?(s) = and

Z1

0

ts?1e?t dt = ?s

Trace e?t(D+H) =

one obtains

Z1

0

X

2spec(D+H)

ts?1e?t dt;  2 IR;  > 0 e?t

Z1 1  (D; s) + Ker D = ?(s) ts?1Tr(e?t(D+H) )dt for Re(s) > n2 : 0

2

From the asymptotic expansion (1) follows ?(s)( (D; s) + dim Ker D)) = Z1

= ts?1(4t)? n 2

0

= (4)? n 2

r X tk ak ( k=0

D + H)dt + Rr (s)

D + H) + R (s); r n k=0 s ? ( 2 ? k) r a ( X k

(3)

where Rr (s) is holomorphic for Re(s) > n?2 r (r > n). H is a smoothing operator, hence D and D + H has the same total symbol and ak (D) = ak (D + H). The ?-function is meromorphic and has only poles of 1st order in the non-positive integers. Therefore, (3) gives a meromorphic extension of  (D; s) to the complex plane. The poles of  (D; s) are simple and ly in the set f n2 ? k; k 2 IN [ f0gg. The residuum in sk = n2 ? k is Ress=sk  (D; s) = (4)? n ak (D)?(sk )?1: (4) In the non-positive integers  (D;
) is regular and for ` 2 IN [ f0g ( (5)  (D; ?`) + dim KerD = 0(4)? n a (D)(Res ?)?1 nn =odd 2m: m+` s=?` If the operator D : ?(F ) ! ?(F ) is a generalized Laplacian (that means that D has metric principal symbol) then there exists a Weitzenbock formula for D D = (rF )rF ? H; where rF is an unique determined metric connection and H a homomorphism on F . Then, by Gilkey's method ([4], Chap. 4.6), the coecients ak (D) in the asymptotic expansion (1) can be calculated in the following way: 9 > > ao (D;
) = idF > > > > > 1 > a1 (D;
) = 6 R idF + H = ; (6) 1 1 1 1 a2 (D;
) = f? 30
R + 72 R2 ? 180 kRick2 + 180 k > > > > > > ^ H) > + 16 RH + 12 H 2 + 121 P Wk`Wk` ? 16
( ; 2

2

k;`

where R is the scalar curvature, Ric the Ricci tensor and < the Riemannian curvature of (M; g). Wk` are the coecients of the curvature tensor induced by rF Wk` = [rFsk ; rFs` ] ? rF[sk;s`] with respect to an ON-basis (s1 ; : : : ; sn).
^ denotes the Bochner-Laplace operator of the covariant derivative on the space of selfadjoint endomorphisms on F , induced by rF . Hence, for generalized Laplacians on 4-dimensional manifolds it is possible (using (2) and (6)) to calculate the zeta-invariant  (D) :=  (D; 0)  (D) = (41 )2 a2 (D) ? dim Ker D (7)

and the residua in the only poles s = 1 and s = 2 occuring in the meromorphic expansion (8) Ress=1 (D;
) = (41 )2 a1 (D); Ress=2 (D;
) = (41 )2 ao (D): 3

3 The coecients in the heat expansion for coupled Dirac operators

Let (M n ; g) be a closed connected Riemannian spin manifold and let S denote the spinor bundle of (M; g). Let G be a compact connected semi-simple Lie group, P a G-principal bundle on M and % : G ! GL(V ) a faithful unitary representation of G. Each connection A of P induces a Dirac operator DA on M with values in the associated vector bundle E = P % V DA : ?(S E ) ?! ?(S E ) de ned by

DA =

n X i=1

si
(rSsi 1 + 1 rAsi );

where rS is the spinor derivative, rA is the covariant derivative on E given by A,
is the Cliord multiplication and (s1 ; : : : ; sn) is a local ON-basis on (M n ; g). For DA2 we have the following Weitzenbock formula DA2 = rr + 41 R + QA; (9) where S 1 + 1 rA and r = r X si
sj