a kastler kalau walze type theorem and the spectral action for perturbations of dirac operators on manifolds with boundary

Research ArticleA Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary Yong Wang School of Mathematics and Statistics,

Research Article

A Kastler-Kalau-Walze Type Theorem and

the Spectral Action for Perturbations of Dirac Operators on

Manifolds with Boundary

Yong Wang

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

Correspondence should be addressed to Yong Wang; wangy581@nenu.edu.cn

Received 23 October 2013; Accepted 13 January 2014; Published 17 March 2014

Academic Editor: Jaume Gin´e

Copyright © 2014 Yong Wang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary

As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds

1 Introduction

The noncommutative residue found in [1,2] plays a

promi-nent role in noncommutative geometry In [3], Connes

used the noncommutative residue to derive a conformal

4-dimensional Polyakov action analogy In [4], Connes proved

that the noncommutative residue on a compact manifold

𝑀 coincided with Dixmier’s trace on pseudodifferential

operators of order − dim 𝑀 Several years ago, Connes

made a challenging observation that the noncommutative

residue of the square of the inverse of the Dirac operator

was proportional to the Einstein-Hilbert action, which is

called the Kastler-Kalau-Walze theorem now In [5], Kastler

gave a brute-force proof of this theorem In [6], Kalau and

Walze proved this theorem by the normal coordinates way

simultaneously In [7], Ackermann gave a note on a new proof

of this theorem by the heat kernel expansion way The

Kastler-Kalau-Walze theorem had been generalized to some cases, for

example, Dirac operators with torsion [8], CR manifolds [9],

andR𝑛[10] (see also [11,12])

On the other hand, Fedosov et al defined a

noncommu-tative residue on Boutet de Monvel’s algebra and proved that

it was the unique continuous trace in [13] In [14], Schrohe

gave the relation between the Dixmier trace and the

non-commutative residue for manifolds with boundary In [15,16],

we gave an operator-theoretic explanation of the gravitational action for manifolds with boundary and proved a Kastler-Kalau-Walze type theorem for Dirac operators and signature operators on manifolds with boundary

Perturbations of Dirac operators were investigated by several authors In [17], Sitarz and Zajac investigated the spectral action for scalar perturbations of Dirac operators

In [18, p 305], Iochum and Levy computed the heat kernel coefficients for Dirac operators with one-form perturbations

In [19], Hanisch et al derived a formula for the gravita-tional part of the spectral action for Dirac operators on 4-dimensional spin manifolds with totally antisymmetric torsion and this is a perturbation with three forms of Dirac operators On the other hand, in [20], Connes and Moscovici considered the conformal perturbations of Dirac operators Investigating the perturbations of Dirac operators has some significance (see [18,19,21]) Motivated by [17–19], we study the Dirac operators with general form perturbations We prove a Kastler-Kalau-Walze type theorem for general form perturbations and the conformal perturbations of Dirac operators for compact manifolds with or without boundary

We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds and give detailed computations of spectral action for scalar perturbations of Dirac operators in [17]

Abstract and Applied Analysis

Volume 2014, Article ID 619120, 13 pages

http://dx.doi.org/10.1155/2014/619120

This paper is organized as follows In Section 2, we

prove the Lichnerowicz formula for perturbations of Dirac

operators and prove a Kastler-Kalau-Walze type theorem for

perturbations of Dirac operators on4-dimensional compact

manifolds with or without boundary InSection 3, we prove

a Kastler-Kalau-Walze type theorem for conformal

perturba-tions of Dirac operators on compact manifolds with or

with-out boundary InSection 4, we compute the spectral action

for Dirac operators with scalar and two-form perturbations

on4-dimensional compact manifolds

2 A Kastler-Kalau-Walze Type Theorem for

Perturbations of Dirac Operators

2.1 A Kastler-Kalau-Walze Type Theorem for Perturbations

be a smooth compact Riemannian𝑛-dimensional manifold

without boundary and let𝑉 be a vector bundle on 𝑀 Recall

that a differential operator𝑃 is of Laplace type if it has locally

the form

𝑃 = − (𝑔𝑖𝑗𝜕𝑖𝜕𝑗+ 𝐴𝑖𝜕𝑖+ 𝐵) , (1) where𝜕𝑖is a natural local frame on𝑇𝑀, 𝑔𝑖,𝑗 = 𝑔(𝜕𝑖, 𝜕𝑗) and

(𝑔𝑖𝑗)1≤𝑖,𝑗≤𝑛 is the inverse matrix associated with the metric

matrix(𝑔𝑖,𝑗)1≤𝑖,𝑗≤𝑛on𝑀, and 𝐴𝑖and𝐵 are smooth sections

of End(𝑉) on 𝑀 (endomorphism) If 𝑃 is a Laplace type

operator of the form (1), then (see [22]) there is a unique

connection∇ on 𝑉 and a unique endomorphism 𝐸 such that

𝑃 = − [𝑔𝑖𝑗(∇𝜕𝑖∇𝜕𝑗− ∇∇𝐿

𝜕𝑖 𝜕 𝑗) + 𝐸] , (2) where∇𝐿 denotes the Levi-Civita connection on𝑀

More-over (with local frames of𝑇∗𝑀 and 𝑉), ∇𝜕𝑖 = 𝜕𝑖+ 𝜔𝑖and𝐸

are related to𝑔𝑖𝑗,𝐴𝑖, and𝐵 through

𝜔𝑖= 1

2𝑔𝑖𝑗(𝐴𝑗+ 𝑔𝑘𝑙Γ

𝑗

𝑘𝑙𝐼𝑑) ,

𝐸 = 𝐵 − 𝑔𝑖𝑗(𝜕𝑖(𝜔𝑗) + 𝜔𝑖𝜔𝑗− 𝜔𝑘Γ𝑖𝑗𝑘) ,

(3)

whereΓ𝑘

𝑖𝑗are the Christoffel coefficients of∇𝐿

Now, we let𝑀 be an 𝑛-dimensional oriented spin

man-ifold with Riemannian metric 𝑔 We recall that the Dirac

operator𝐷 is locally given as follows in terms of orthonormal

frames𝑒𝑖, 1 ≤ 𝑖 ≤ 𝑛, and natural frames 𝜕𝑖of𝑇𝑀:

𝐷 = ∑

𝑖,𝑗

𝑔𝑖𝑗𝑐 (𝜕𝑖) ∇𝜕𝑆𝑗 = ∑

𝑖

𝑐 (𝑒𝑖) ∇𝑒𝑆𝑖, (4)

where 𝑐(𝑒𝑖) denotes the Clifford action which satisfies the

relation

𝑐 (𝑒𝑖) 𝑐 (𝑒𝑗) + 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑖) = −2𝛿𝑗𝑖,

∇𝜕𝑆𝑖 = 𝜕𝑖+ 𝜎𝑖,

𝜎𝑖= 1

4∑𝑗,𝑘⟨∇𝜕𝐿𝑖𝑒𝑗, 𝑒𝑘⟩ 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑘)

(5)

Let

𝜕𝑗= 𝑔𝑖𝑗𝜕𝑖, 𝜎𝑖= 𝑔𝑖𝑗𝜎𝑗, Γ𝑘= 𝑔𝑖𝑗Γ𝑖𝑗𝑘 (6)

By(6a) in [5], we have

𝐷2= − 𝑔𝑖𝑗𝜕𝑖𝜕𝑗− 2𝜎𝑗𝜕𝑗+ Γ𝑘𝜕𝑘

− 𝑔𝑖𝑗[𝜕𝑖(𝜎𝑗) + 𝜎𝑖𝜎𝑗− Γ𝑖𝑗𝑘𝜎𝑘] +1

4𝑠,

(7)

where𝑠 is the scalar curvature Let Ψ be a smooth differential form on𝑀 and we also denote the associated Clifford action

byΨ We will compute 𝐷2Ψ:= (𝐷 + Ψ)2 We note that

(𝐷 + Ψ)2= 𝐷2+ 𝐷Ψ + Ψ𝐷 + Ψ2, (8)

𝐷Ψ + Ψ𝐷 = ∑

𝑖𝑗

𝑔𝑖𝑗(𝑐 (𝜕𝑖) Ψ + Ψ𝑐 (𝜕𝑖)) 𝜕𝑗

+ ∑

𝑖𝑗

𝑔𝑖𝑗(𝑐 (𝜕𝑖) 𝜕𝑗(Ψ) + 𝑐 (𝜕𝑖) 𝜎𝑗Ψ + Ψ𝑐 (𝜕𝑖) 𝜎𝑗)

(9)

By (7)–(9), we have

𝐷2Ψ= − 𝑔𝑖𝑗𝜕𝑖𝜕𝑗+ (−2𝜎𝑗+ Γ𝑗+ 𝑐 (𝜕𝑗) Ψ + Ψ𝑐 (𝜕𝑗)) 𝜕𝑗 + 𝑔𝑖𝑗[−𝜕𝑖(𝜎𝑗) − 𝜎𝑖𝜎𝑗+ Γ𝑖𝑗𝑘𝜎𝑘+ 𝑐 (𝜕𝑖) 𝜕𝑗(Ψ) + 𝑐 (𝜕𝑖) 𝜎𝑗Ψ + Ψ𝑐 (𝜕𝑖) 𝜎𝑗] +14𝑠 + Ψ2

(10)

By (10) and (3), we have

𝜔𝑖= 𝜎𝑖−1

2[𝑐 (𝜕𝑖) Ψ + Ψ𝑐 (𝜕𝑖)] ,

𝐸 = − 𝑐 (𝜕𝑖) 𝜕𝑖(Ψ) − 𝑐 (𝜕𝑖) 𝜎𝑖Ψ − Ψ𝑐 (𝜕𝑖) 𝜎𝑖

−1

4𝑠 − Ψ2+

1

2𝜕𝑗[𝑐 (𝜕𝑗) Ψ + Ψ𝑐 (𝜕𝑗)]

−1

2Γ𝑘[𝑐 (𝜕𝑘) Ψ + Ψ𝑐 (𝜕𝑘)] +

1

2𝜎𝑗[𝑐 (𝜕𝑗) Ψ + Ψ𝑐 (𝜕𝑗)] +1

2[𝑐 (𝜕𝑗) Ψ + Ψ𝑐 (𝜕𝑗)] 𝜎𝑗

−𝑔𝑖𝑗

4 [𝑐 (𝜕𝑖) Ψ + Ψ𝑐 (𝜕𝑖)] [𝑐 (𝜕𝑗) Ψ + Ψ𝑐 (𝜕𝑗)]

(11) For a smooth vector field𝑋 on 𝑀, let 𝑐(𝑋) denote the Clifford action So,

∇𝑋= ∇𝑋𝑆 −1

2[𝑐 (𝑋) Ψ + Ψ𝑐 (𝑋)] (12) Since 𝐸 is globally defined on 𝑀, so we can perform computations of 𝐸 in normal coordinates Taking normal

coordinates about 𝑥0, then, 𝜎𝑖(𝑥0) = 0, 𝜕𝑗[𝑐(𝜕𝑗)](𝑥0) =

0, Γ𝑘(𝑥0) = 0, 𝑔𝑖𝑗(𝑥0) = 𝛿𝑗𝑖, so that

𝐸 (𝑥0) = −14𝑠 − Ψ2+12[𝜕𝑗(Ψ) 𝑐 (𝜕𝑗) − 𝑐 (𝜕𝑗) 𝜕𝑗(Ψ)]

−1

4[𝑐 (𝜕𝑖) Ψ + Ψ𝑐 (𝜕𝑖)]

2(𝑥0)

= −1

4𝑠 − Ψ2+

1

2[𝑒𝑗(Ψ) 𝑐 (𝑒𝑗) − 𝑐 (𝑒𝑗) 𝑒𝑗(Ψ)]

−14[𝑐 (𝑒𝑖) Ψ + Ψ𝑐 (𝑒𝑖)]2(𝑥0)

= −14𝑠 − Ψ2+12[∇𝑒𝑆𝑗(Ψ) 𝑐 (𝑒𝑗) − 𝑐 (𝑒𝑗) ∇𝑒𝑆𝑗(Ψ)]

−1

4[𝑐 (𝑒𝑖) Ψ + Ψ𝑐 (𝑒𝑖)]

2(𝑥0)

(13)

We get the following Lichnerowicz formula

Proposition 1 Let Ψ be a smooth differential form on 𝑀 and

𝐷Ψ:= 𝐷 + Ψ; then

𝐷2Ψ= − [𝑔𝑖𝑗(∇𝜕𝑖∇𝜕𝑗− ∇∇𝐿

𝜕𝑖 𝜕 𝑗)] + 14𝑠 + Ψ2

−1

2[∇𝑒𝑆𝑗(Ψ) 𝑐 (𝑒𝑗) − 𝑐 (𝑒𝑗) ∇𝑒𝑆𝑗(Ψ)]

+1

4[𝑐 (𝑒𝑖) Ψ + Ψ𝑐 (𝑒𝑖)]

2,

(14)

where∇𝜕𝑖is defined by (12) and𝑋 = 𝜕𝑖.

We see two special cases ofProposition 1 WhenΨ = 𝑓,

where𝑓 is a smooth function on 𝑀, we have

∇𝑋= ∇𝑋𝑆 − 𝑓𝑐 (𝑋) , 𝐸 = −14𝑠 + (𝑛 − 1) 𝑓2 (15)

Corollary 2 When Ψ = 𝑓, one has

𝐷2𝑓= − [𝑔𝑖𝑗(∇𝜕𝑖∇𝜕𝑗− ∇∇𝐿

𝜕𝑖 𝜕 𝑗)] +1

4𝑠 + (1 − 𝑛) 𝑓2 (16)

Let 𝜂 = 𝑎𝑖𝑒𝑖 be a one form, where 𝑎𝑖 is a smooth real

function, let 𝑒𝑖 be a dual orthonormal frame by parallel

transport along geodesic, and let 𝑋 = 𝑎𝑖𝑒𝑖 be the dual

vector field of 𝜂 When Ψ = √−1𝑐(𝜂), by (12), we have

∇𝑌= ∇𝑆

𝑌+ √−1𝑔(𝑋, 𝑌), where 𝑌 is a smooth vector field on

𝑀 By 𝑒𝑗(𝑐(𝑒𝑖)) = 0 and 𝑑𝑒𝑙(𝑥0) = 0 (see [15]), we have

𝐸 (𝑥0) = −14𝑠 − |𝑋|2+√−1

2

× [𝑒𝑗(𝑎𝑘) 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑗) − 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑘) 𝑒𝑗(𝑎𝑘)] +14[𝑐 (𝑒𝑖) 𝑐 (𝑋) + 𝑐 (𝑋) 𝑐 (𝑒𝑖)]2

= −14𝑠 +√−1

2 𝑒𝑗(𝑎𝑘) [𝑐 (𝑒𝑘) 𝑐 (𝑒𝑗) − 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑘)]

= −1

4𝑠 + √−1 ∑𝑘 ̸= 𝑗𝑒𝑗(𝑎𝑘) 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑗) (𝑥0)

= −1

4𝑠 − √−1𝑐 (𝑑𝜂) (𝑥0)

(17)

Corollary 3 For a one-form 𝜂 and the Clifford action 𝑐(𝜂), one

has

(𝐷 + √−1𝑐 (𝜂))2= − [𝑔𝑖𝑗(∇𝜕𝑖∇𝜕𝑗− ∇∇𝐿

𝜕𝑖 𝜕𝑗)]

+1

4𝑠 + √−1𝑐 (𝑑𝜂)

(18)

WhenΨ is a two-form, we let Ψ = 2 ∑𝑘<𝑙𝑎𝑘𝑙𝑒𝑘 ∧ 𝑒𝑙 =

∑ 𝑎𝑘𝑙𝑒𝑘∧ 𝑒𝑙, where𝑎𝑘𝑙 = −𝑎𝑙𝑘, and𝑐(Ψ) = ∑ 𝑎𝑘𝑙𝑐(𝑒𝑘)𝑐(𝑒𝑙) So,

∇𝑒𝑖 = 𝑒𝑖+14∑

𝑠,𝑡𝜔𝑠𝑡(𝑒𝑖) 𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) − ∑

𝑘,𝑙 ̸= 𝑖

𝑎𝑘𝑙𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒𝑖) ,

(19) where𝜔𝑠𝑡(𝑒𝑖) denotes the connection coefficient By (13),

𝐸 = −1

4𝑠 − [𝑎𝑘𝑙𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙)]

2

+1

2{𝑒𝑗(𝑎𝑘𝑙) [𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒𝑗) − 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙)]}

−14[𝑎𝑘𝑙(𝑐(𝑒𝑖)𝑐(𝑒𝑘)𝑐(𝑒𝑙) + 𝑐(𝑒𝑘)𝑐(𝑒𝑙)𝑐(𝑒𝑖))]2

(20) Let𝑆(𝑇𝑀) be the spinor bundle on 𝑀 and dim(𝑆(𝑇𝑀)) = 𝑑 and Tr(𝐴) denote the trace of 𝐴, for 𝐴 ∈ End(𝑆(𝑇𝑀)) Since, for𝑘 ̸= 𝑙, ̃𝑘 ̸=̃𝑙,

Tr[𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙)] = 𝑑 (−𝛿̃𝑘𝑘𝛿̃𝑙𝑙+ 𝛿̃𝑙𝑘𝛿̃𝑘𝑙) , (21)

we have

Tr{[𝑎𝑘𝑙𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙)]2} = −2𝑑𝑎𝑘𝑙2 (22)

Since the trace of the product of odd Clifford elements is zero,

we have

Tr[1

2{𝑒𝑗(𝑎𝑘𝑙) [𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒𝑗) − 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙)]}] = 0,

(23)

Tr{[𝑎𝑘𝑙(𝑐 (𝑒𝑖) 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) + 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒𝑖))]2}

= 𝑎𝑘𝑙𝑎̃𝑘̃𝑙Tr[𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙) 𝑐(𝑒𝑖)2

+ 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐(𝑒𝑖)2𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙) + 𝑐 (𝑒𝑖) 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒𝑖) 𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙) + 𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒𝑖) 𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙) 𝑐 (𝑒𝑖) ]

= −2𝑛𝑑𝑎𝑘𝑙𝑎̃𝑘̃𝑙(−𝛿̃𝑘𝑘𝛿̃𝑙𝑙+ 𝛿̃𝑙𝑘𝛿̃𝑘𝑙)

− 2 ∑

𝑖 ̸= 𝑘,𝑙

𝑎𝑘𝑙𝑎̃𝑘̃𝑙Tr[𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙)]

+ 2∑

𝑖=𝑘

𝑎𝑘𝑙𝑎̃𝑘̃𝑙Tr[𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙)]

+ 2∑

𝑖=𝑙

𝑎𝑘𝑙𝑎̃𝑘̃𝑙Tr[𝑐 (𝑒𝑘) 𝑐 (𝑒𝑙) 𝑐 (𝑒̃𝑘) 𝑐 (𝑒̃𝑙)]

= 4𝑛𝑑𝑎𝑘𝑙2 + (𝑛 − 2) 4𝑑𝑎2𝑘𝑙− 4𝑑𝑎2𝑘𝑙− 4𝑑𝑎𝑘𝑙2

= 8 (𝑛 − 2) 𝑑𝑎𝑘𝑙2

(24)

By (20) and (22)–(24), we have

Tr𝐸 = 𝑑 (−1

4𝑠 + (6 − 2𝑛) |Ψ|2) (25) and we get the following

Corollary 4 Let Ψ = ∑ 𝑎𝑘𝑙𝑒𝑘∧ 𝑒𝑙and𝑎𝑘𝑙= −𝑎𝑙𝑘; then tr𝐸 =

𝑑(−(1/4)𝑠 + (6 − 2𝑛)|Ψ|2).

For a general differential formΨ, by (13) and Tr(𝐴𝐵) =

Tr(𝐵𝐴), we have

Tr(𝐸) = Tr [−14𝑠 − Ψ2−14[𝑐 (𝑒𝑖) Ψ + Ψ𝑐 (𝑒𝑖)]2]

= Tr [−14𝑠 −12Ψ𝑐 (𝑒𝑖) Ψ𝑐 (𝑒𝑖) + (𝑛2 − 1) Ψ2]

(26)

By the Kastler-Kalau-Walze theorem (see [5,6]), we have

Wres(𝐷−𝑛+2Ψ ) = (2𝜋)𝑛/2

(𝑛/2 − 2)!∫𝑀Tr[

6𝑠 + 𝐸] 𝑑V𝑜𝑙𝑀, (27) where Wres denotes the noncommutative residue (see [2])

By (26) and (27), we have the following

Theorem 5 For even 𝑛-dimensional compact spin manifolds

holds:

Wres (𝐷−𝑛+2

Ψ )

= (2𝜋)𝑛/2 (𝑛/2 − 2)!

× ∫

𝑀Tr[−1

12𝑠

−1

2𝑐 (Ψ) 𝑐 (𝑒𝑖) 𝑐 (Ψ) 𝑐 (𝑒𝑖) + (𝑛/2 − 1) 𝑐(Ψ)2] 𝑑V𝑜𝑙𝑀

(28)

ByCorollary 2, we have the following

Corollary 6 For even 𝑛-dimensional compact spin manifolds

equality holds:

Wres (𝐷−𝑛+2𝑓 ) = (2𝜋)𝑛/2𝑑

(𝑛/2 − 2)!∫𝑀[−1

12𝑠 + (𝑛 − 1) 𝑓2] 𝑑V𝑜𝑙𝑀

(29)

ByCorollary 3, we have the following

Corollary 7 For even 𝑛-dimensional compact spin manifolds

holds:

Wres (𝐷−𝑛+2Ψ ) = − (2𝜋)𝑛/2𝑑

12 × (𝑛/2 − 2)!∫𝑀𝑠 𝑑V𝑜𝑙𝑀 (30)

ByCorollary 4and (27), we have the following

Corollary 8 For even 𝑛-dimensional compact spin manifolds

holds:

Wres (𝐷−𝑛+2Ψ ) = (2𝜋)𝑛/2𝑑

(𝑛/2 − 2)!

× ∫

𝑀Tr[− 1

12𝑠 + (6 − 2𝑛) |Ψ|2] 𝑑V𝑜𝑙𝑀

(31)

2.2 A Kastler-Kalau-Walze Type Theorem for Perturbations of

be a compact 4-dimensional spin manifold with boundary

𝜕𝑀 and let 𝑈 ⊂ 𝑀 be the collar neighborhood of 𝜕𝑀 which is diffeomorphic to𝜕𝑀 × [0, 1) And we will compute the noncommutative residue for manifolds with boundary of (𝜋+𝐷Ψ−1)2 That is, we will compute ̃Wres[(𝜋+𝐷−1Ψ)2] (for the related definitions, see [15]) and we take the metric as in [15]

Let(𝑥󸀠, 𝑥𝑛) ∈ 𝑈, where 𝑥󸀠 ∈ 𝜕𝑀 and 𝑥𝑛denotes the normal

direction coordinate By (2.2.4) in [15], we have

̃

Wres[(𝜋+𝐷Ψ−1)2]

= ∫

𝑀∫

|𝜉|=1trace𝑆(𝑇𝑀)[𝜎−4(𝐷−2Ψ)] 𝜎 (𝜉) 𝑑𝑥

+ ∫

𝜕𝑀Φ,

(32)

where

Φ

= ∫

|𝜉 󸀠 |=1∫+∞

−∞

∞

∑

𝑗,𝑘=0

∑ (−𝑖)|𝛼|+𝑗+𝑘+1

𝛼! (𝑗 + 𝑘 + 1)!

× trace𝑆(𝑇𝑀)[𝜕𝑗

𝑥 𝑛𝜕𝛼

𝜉 󸀠𝜕𝑘

𝜉 𝑛𝜎+

𝑟 (𝐷−1

Ψ) (𝑥󸀠, 0, 𝜉󸀠, 𝜉𝑛)

× 𝜕𝛼

𝑥 󸀠𝜕𝑗+1𝜉

𝑛 𝜕𝑘

𝑥 𝑛𝜎𝑙(𝐷−1

Ψ)

× (𝑥󸀠, 0, 𝜉󸀠, 𝜉𝑛) ] 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠,

(33)

where the sum is taken over𝑟−𝑘−|𝛼|+𝑙−𝑗−1 = −4, 𝑟, 𝑙 ≤ −1,

and𝜎+

𝑟(𝐷−1

Ψ) = 𝜋+

𝜉 𝑛𝜎𝑟(𝐷−1

Ψ) (for the definition of 𝜋+, see [15])

ByTheorem 5, we have

∫

𝑀∫

|𝜉|=1tr[𝜎−4(𝐷−2

Ψ)] 𝜎 (𝜉) 𝑑𝑥

= 4𝜋2∫

𝑀Tr[−1

12𝑠 −

1

2Ψ𝑐 (𝑒𝑖) Ψ𝑐 (𝑒𝑖) + (𝑛2− 1) Ψ2] 𝑑V𝑜𝑙𝑀

(34)

So, we only need to compute∫𝜕𝑀Φ In analogy with Lemma

2.1 of [15], we can prove the following useful result

Lemma 9 The symbolic calculus of pseudodifferential

opera-tors yields

𝑞−1(𝐷−1Ψ) =√−1𝑐 (𝜉)

󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨2 ,

𝑞−2(𝐷−1Ψ) = 𝑞−2(𝐷−1) +𝑐 (𝜉) Ψ𝑐 (𝜉)

󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨4

(35)

Similar to the computations in Section2.2.2 in [15], we

can split Φ into the sum of five terms Since 𝑞−1(𝐷−1Ψ) =

𝑞−1(𝐷−1), then terms (a) (I), (a) (II), and (a) (III) in our case

are the same as the terms (a) (I), (a) (II), and (a) (III) in [15], so

term (a) (I)

= − ∫

|𝜉 󸀠 |=1∫+∞

−∞ ∑

|𝛼|=1

trace[𝜕𝜉𝛼󸀠𝜋+𝜉𝑛𝑞−1

× 𝜕𝑥𝛼󸀠𝜕𝜉𝑛𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= 0, term (a) (II)

= −12∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜕𝑥𝑛𝜋𝜉+𝑛𝑞−1

× 𝜕𝜉2𝑛𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= −3

8𝜋ℎ󸀠(0) Ω3𝑑𝑥󸀠, term (a) (III)

= −1

2∫|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜕𝜉𝑛𝜋+𝜉𝑛𝑞−1

× 𝜕𝜉𝑛𝜕𝑥𝑛𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= 38𝜋ℎ󸀠(0) Ω3𝑑𝑥󸀠

(36)

Then, we only need to compute term (b) and term (c) By Lemma 9,

term (b) := −𝑖 ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜋𝜉+𝑛𝑞−2(𝐷−1)

× 𝜕𝜉𝑛𝑞−1(𝐷−1)]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

− 𝑖 ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜋𝜉+𝑛(𝑐 (𝜉) Ψ𝑐 (𝜉)

󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨4 )

× 𝜕𝜉𝑛𝑞−1(𝐷−1) ]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

(37)

By term (b) in [15], we have

− 𝑖 ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜋𝜉+𝑛𝑞−2(𝐷−1)

× 𝜕𝜉𝑛𝑞−1(𝐷−1)]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= 9

8𝜋ℎ󸀠(0) Ω3𝑑𝑥󸀠,

(38)

where Ω3 is the canonical volume of 3-dimensional unit

sphere Moreover,

𝜋+𝜉𝑛(𝑐 (𝜉) Ψ𝑐 (𝜉)

󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨4 ) (𝑥0)󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨

󵄨󵄨|𝜉 󸀠 |=1

= 𝜋+𝜉𝑛[[𝑐 (𝜉

󸀠) + 𝜉𝑛𝑐 (𝑑𝑥𝑛)] Ψ [𝑐 (𝜉󸀠) + 𝜉𝑛𝑐 (𝑑𝑥𝑛)]

= 2𝜋𝑖1 ∫

Γ +(( (𝑐 (𝜉󸀠) Ψ𝑐 (𝜉󸀠) + 𝑐 (𝑑𝑥𝑛) Ψ𝑐 (𝜉󸀠) 𝜂𝑛

+ 𝑐 (𝜉󸀠) Ψ𝑐 (𝑑𝑥𝑛) 𝜂𝑛 + 𝑐 (𝑑𝑥𝑛) Ψ𝑐 (𝑑𝑥𝑛) 𝜂2𝑛)

× ((𝜂𝑛+ 𝑖)2(𝜉𝑛− 𝜂𝑛))−1)

× ((𝜂𝑛− 𝑖)2)−1) 𝑑𝜂𝑛

= [ (𝑐 (𝜉󸀠) Ψ𝑐 (𝜉󸀠) + 𝑐 (𝑑𝑥𝑛) Ψ𝑐 (𝜉󸀠) 𝜂𝑛

+ 𝑐 (𝜉󸀠) Ψ𝑐 (𝑑𝑥𝑛) 𝜂𝑛+ 𝑐 (𝑑𝑥𝑛) Ψ𝑐 (𝑑𝑥𝑛) 𝜂2𝑛)

× ((𝜂𝑛+ 𝑖)2(𝜉𝑛− 𝜂𝑛))−1](1)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨𝜂 𝑛 =𝑖

= − 𝑖𝜉𝑛+ 2

4(𝜉𝑛− 𝑖)2𝑐 (𝜉

󸀠) Ψ𝑐 (𝜉󸀠)

4(𝜉𝑛− 𝑖)2[𝑐 (𝑑𝑥𝑛) Ψ𝑐 (𝜉

󸀠) + 𝑐 (𝜉󸀠) Ψ𝑐 (𝑑𝑥𝑛)]

− 𝑖𝜉𝑛

4(𝜉𝑛− 𝑖)2𝑐 (𝑑𝑥𝑛) Ψ𝑐 (𝑑𝑥𝑛) ,

𝜕𝜉𝑛𝑞−1󵄨󵄨󵄨󵄨󵄨|𝜉󸀠 |=1= √−1 [ 1 − 𝜉𝑛2

(1 + 𝜉2

𝑛)2𝑐 (𝑑𝑥𝑛) −

2𝜉𝑛 (1 + 𝜉2

𝑛)2𝑐 (𝜉

󸀠)] (39)

By (39) and

𝑐(𝜉󸀠)2󵄨󵄨󵄨󵄨󵄨󵄨|𝜉󸀠|=1= −1, 𝑐(𝑑𝑥𝑛)2= −1,

𝑐 (𝜉󸀠) 𝑐 (𝑑𝑥𝑛) = −𝑐 (𝑑𝑥𝑛) 𝑐 (𝜉󸀠) , Tr(𝐴𝐵) = Tr (𝐵𝐴) ,

(40)

we get trace[𝜋+𝜉𝑛(𝑐 (𝜉) Ψ𝑐 (𝜉)

󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨4 ) × 𝜕𝜉𝑛𝑞−1(𝐷−1)] (𝑥0)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨|𝜉 󸀠 |=1

2(1 + 𝜉2)2Tr[𝑐 (𝑑𝑥𝑛) Ψ]

2(1 + 𝜉2)2 Tr[𝑐 (𝜉

󸀠) Ψ]

(41)

Considering, for𝑖 < 𝑛, ∫|𝜉󸀠 |=1𝜉𝑖𝜎(𝜉󸀠) = 0, then

− 𝑖 ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜋𝜉+𝑛(𝑐 (𝜉) Ψ𝑐 (𝜉)

󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨4 )

× 𝜕𝜉𝑛𝑞−1(𝐷−1) ]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

=𝜋

4Ω3Tr[𝑐 (𝑑𝑥𝑛) Ψ] 𝑑𝑥󸀠, term (b)= 98𝜋ℎ󸀠(0) Ω3𝑑𝑥󸀠+𝜋4Ω3Tr[𝑐 (𝑑𝑥𝑛) Ψ] 𝑑𝑥󸀠

(42) Similarly, we have

term (c)= −9

8𝜋ℎ󸀠(0) Ω3𝑑𝑥󸀠−

𝜋

4Ω3Tr[𝑐 (𝑑𝑥𝑛) Ψ] 𝑑𝑥󸀠.

(43) Then, the sum of terms (b) and (c) is zero andΦ is zero Then,

we get the following

Theorem 10 Let 𝑀 be a 4-dimensional compact spin

Let Ψ be a general differential form on 𝑀 Then,

̃Wres [(𝜋+𝐷−1Ψ)2]

= 4𝜋2∫

𝑀Tr[−1

12𝑠 −

1

2𝑐 (Ψ) 𝑐 (𝑒𝑖) 𝑐 (Ψ) 𝑐 (𝑒𝑖) + 𝑐(Ψ)2] 𝑑V𝑜𝑙𝑀

(44)

In [16], we proved a Kastler-Kalau-Walze theorem associ-ated with Dirac operators for 6-dimensional spin manifolds with boundary In fact, our computations hold for general Laplacians This implies the following

Proposition 11 (see [16]) Let 𝑀 be a 6-dimensional compact

(see (1.3) in [ 15 ]) Let Δ be a general Laplacian acting on

̃Wres [(𝜋+Δ−1)2] = 8𝜋3∫

𝑀Tr[𝑠

6+ 𝐸] 𝑑V𝑜𝑙𝑀. (45) Since𝐷2

Ψis a general Laplacian, then we get the following

Corollary 12 Let 𝑀 be a 6-dimensional compact spin

̃Wres [(𝜋+𝐷−2

Ψ)2]

= 8𝜋3∫

𝑀Tr[− 1

12𝑠 −

1

2Ψ𝑐 (𝑒𝑖) Ψ𝑐 (𝑒𝑖) + 2Ψ2] 𝑑V𝑜𝑙𝑀.

(46)

In the above two cases, the boundary terms vanish In the

following, we will give a boundary term nonvanishing case

and compute Wres((𝐷Ψ𝐷)−1) We have

𝐷Ψ𝐷 = − 𝑔𝑖𝑗𝜕𝑖𝜕𝑗+ (−2𝜎𝑗+ Γ𝑗+ Ψ𝑐 (𝜕𝑗)) 𝜕𝑗

+ 𝑔𝑖𝑗[−𝜕𝑖(𝜎𝑗) − 𝜎𝑖𝜎𝑗+ Γ𝑖𝑗𝑘𝜎𝑘+ Ψ𝑐 (𝜕𝑖) 𝜎𝑗]

+14𝑠,

𝜔𝑖= 𝜎𝑖−12Ψ𝑐 (𝜕𝑖) ,

𝐸 = − Ψ𝑐 (𝜕𝑖) 𝜎𝑖−1

4𝑠 +

1

2𝜕𝑗[Ψ𝑐 (𝜕𝑗)] −

1

2Γ𝑘Ψ𝑐 (𝜕𝑘) + 𝑔𝑖𝑗[1

2𝜎𝑖Ψ𝑐 (𝜕𝑗) +

1

2Ψ𝑐 (𝜕𝑖) 𝜎𝑗−

1

4Ψ𝑐 (𝜕𝑖) Ψ𝑐 (𝜕𝑗)]

(47) Similar to the proof of (13), we have

𝐸 = −14𝑠 +12∇𝑒𝑆𝑖(Ψ) 𝑐 (𝑒𝑖) −14Ψ𝑐 (𝑒𝑖) Ψ𝑐 (𝑒𝑖) (48)

So,

𝐷Ψ𝐷 = − [𝑔𝑖𝑗(∇𝜕𝑖∇𝜕𝑗− ∇∇𝐿

𝜕𝑖 𝜕 𝑗)]

+1

4𝑠 −

1

2∇𝑒𝑆𝑖(Ψ) 𝑐 (𝑒𝑖) +1

4Ψ𝑐 (𝑒𝑖) Ψ𝑐 (𝑒𝑖)

(49)

Then, we get the following

Proposition 13 Let 𝑀 be a 4-dimensional compact spin

manifold without boundary Then,

Wres [(𝐷Ψ𝐷)−1]

= 4𝜋2∫

𝑀Tr[−121 𝑠 +12∇𝑒𝑆𝑖(Ψ) 𝑐 (𝑒𝑖)

−14Ψ𝑐 (𝑒𝑖) Ψ𝑐 (𝑒𝑖)] 𝑑V𝑜𝑙𝑀

(50)

WhenΨ is a one-form, we can get the following corollary

Corollary 14 Let 𝑀 be a 4-dimensional compact spin

Wres [(𝐷Ψ𝐷)−1]

= 16𝜋2∫

𝑀[−1

12𝑠 +

1

2𝛿 (Ψ) − 2|Ψ|2] 𝑑V𝑜𝑙𝑀 (51) Now, we compute ̃Wres[𝜋+𝐷−1

Ψ𝜋+𝐷−1] We have that terms (a) and (b) are the same as inTheorem 10, and since term(c) = −(9/8)𝜋ℎ󸀠(0)Ω3𝑑𝑥󸀠, we get

∫

𝜕𝑀Φ = 𝜋4Ω3∫

𝜕𝑀Tr[𝑐 (𝑑𝑥𝑛) Ψ] 𝑑V𝑜𝑙𝜕𝑀 (52) and the following

Proposition 15 Let 𝑀 be a 4-dimensional compact spin

manifold with boundary Then,

̃Wres [𝜋+𝐷−1

Ψ𝜋+𝐷−1]

= 4𝜋2∫

𝑀Tr[−121 𝑠 +12𝑒𝑖(Ψ) 𝑐 (𝑒𝑖)

−14Ψ𝑐 (𝑒𝑖) Ψ𝑐 (𝑒𝑖)] 𝑑V𝑜𝑙𝑀 +𝜋

4Ω3∫𝜕𝑀Tr[𝑐 (𝑑𝑥𝑛) Ψ] 𝑑V𝑜𝑙𝜕𝑀

(53)

term vanishes WhenΨ = 𝐾𝑑𝑥𝑛near the boundary, where

𝐾 is the extrinsic curvature, then the boundary term is proportional to the gravitational action on the boundary In fact, the reason for the boundary term being not zero is that

𝜋+𝐷Ψand𝜋+𝐷 are not symmetric

3 A Kastler-Kalau-Walze Type Theorem for Conformal Perturbations of Dirac Operators

In [20], Connes and Moscovici defined a twisted spectral triple and considered the conformal Dirac operator𝑒ℎ𝐷𝑒ℎ, where ℎ is a smooth function on a manifold 𝑀 without boundary We want to compute Wres[(𝑒ℎ𝐷𝑒ℎ)−2] We know that

Wres[(𝑒ℎ𝐷𝑒ℎ)−2] = Wres [𝑒−ℎ𝐷−1𝑒−2ℎ𝐷−1𝑒−ℎ]

= Wres [𝑒−2ℎ𝐷−1𝑒−2ℎ𝐷−1]

(54)

In the following, we will compute the more general case, that

is, Wres[𝑓𝐷−1𝑔𝐷−1], for nonzero smooth functions 𝑓 and 𝑔, and prove a Kastler-Kalau-Walze type theorem for conformal

Dirac operators When𝑓 = 𝑔 = 𝑒−2ℎ, we get the expression

of Wres[(𝑒ℎ𝐷𝑒ℎ)−2] We have

Wres[𝑓𝐷−1𝑔𝐷−1]

= Wres [(𝑓−1𝐷𝑔−1𝐷)−1]

= Wres {(𝑓−1𝑔−1𝐷2+ 𝑓−1[𝐷, 𝑔−1] 𝐷)−1}

= ∫

𝑀𝑓𝑔 wres [(𝐷2− 𝑔−1𝑐 (𝑑𝑔) 𝐷)−1] ,

(55)

where Wres denotes the residue density, and we note that

the Kastler-Kalau-Walze theorem holds at the residue density

level Some computations show that

𝐷2− 𝑔−1𝑐 (𝑑𝑔) 𝐷

= −𝑔𝑖𝑗𝜕𝑖𝜕𝑗+ [−2𝜎𝑗+ Γ𝑗− 𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑗)] 𝜕𝑗

+ [−𝜕𝑗𝜎𝑗− 𝜎𝑗𝜎𝑗+ Γ𝑘𝜎𝑘+1

4𝑠 − 𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑗) 𝜎𝑗] ,

𝜔𝑖= 𝜎𝑖+12𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑖) ,

𝐸 = − 𝑠

4+ 𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑗) 𝜎𝑗

− 𝜕𝑗[12𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑗)]

−1

2𝜎𝑗𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑗) −

1

2𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑖) 𝜎𝑖

−1

4𝑔𝑖𝑗𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑖) 𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑗)

+12𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑘) Γ𝑘

(56)

Since 𝐸 is globally defined, we can compute it in the

normal coordinates Then, we have

Tr(𝐸) (𝑥0)

= Tr [−4𝑠−12𝜕𝑗(𝑔−1𝑐 (𝑑𝑔)) 𝑐 (𝜕𝑗)

−14𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑖) 𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑖)] (𝑥0) ,

Tr[𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑖) 𝑔−1𝑐 (𝑑𝑔) 𝑐 (𝜕𝑖)] (𝑥0)

= 𝑔−2Tr[∑

𝑖,𝑘,𝑙

𝜕𝑔

𝜕𝑥𝑘

𝜕𝑔

𝜕𝑥𝑙𝑐 (𝜕𝑘) 𝑐 (𝜕𝑖) 𝑐 (𝜕𝑙) 𝑐 (𝜕𝑖)]

= 𝑔−2Tr[∑

𝑖,𝑘

(𝜕𝑔

𝜕𝑥𝑘)

2

𝑐 (𝜕𝑘) 𝑐 (𝜕𝑖) 𝑐 (𝜕𝑘) 𝑐 (𝜕𝑖)]

= 𝑔−2Tr[∑

𝑖 ̸= 𝑘

(𝜕𝑔

𝜕𝑥𝑘)

2

𝑐 (𝜕𝑘) 𝑐 (𝜕𝑖) 𝑐 (𝜕𝑘) 𝑐 (𝜕𝑖)

+ ∑

𝑘

( 𝜕𝑔

𝜕𝑥𝑘)

2

𝑐(𝜕𝑘)4]

= −2𝑔−2∑

𝑘

(𝜕𝑥𝜕𝑔

𝑘)2Tr[𝐼𝑑]

(57) Similarly,

Tr[𝜕𝑗(𝑔−1𝑐 (𝑑𝑔)) 𝑐 (𝜕𝑗)] = ∑

𝑗

[ 1

𝑔2(𝜕𝑔

𝜕𝑥𝑗)

2

− 𝑔−1𝜕2𝑔

𝜕𝑥2 𝑗

] (58) So,

Tr[6𝑠+ 𝐸] = −𝑠3+ 2𝑔−1∑

𝑗

𝜕2𝑔

𝜕𝑥2

𝑗 = −3𝑠− 2𝑔−1Δ (𝑔) (59) By

∫

𝑀𝑓Δ (𝑔) 𝑑V𝑜𝑙𝑀= ∫

𝑀⟨𝑑𝑓, 𝑑𝑔⟩ 𝑑V𝑜𝑙𝑀, (60)

we get the following

Theorem 17 Let 𝑀 be a 4-dimensional compact spin manifold

without boundary; then

Wres [𝑓𝐷−1𝑔𝐷−1] = −4𝜋2∫

𝑀[𝑓𝑔𝑠3 + 2 ⟨𝑑𝑓, 𝑑𝑔⟩] 𝑑V𝑜𝑙𝑀

(61)

Kastler-Kalau-Walze theorem for conformal Dirac operators

In fact,Theorem 17holds true for any choice of the smooth functions 𝑓 and 𝑔, since we can prove (61) by means of the symbolic calculus of pseudodifferential operators without using (55), and it is not essential that𝑓 and 𝑔 are nowhere vanishing

Now, we consider manifolds with boundary and we will compute ̃Wres[𝜋+(𝑓𝐷−1)𝜋+(𝑔𝐷−1)] As in [15], we have five terms as follows:

term (a) (I) = −𝑓𝑔 ∫

|𝜉 󸀠 |=1∫+∞

−∞ ∑

|𝛼|=1

trace[𝜕𝜉𝛼󸀠𝜋+𝜉𝑛𝑞−1

× 𝜕𝛼𝑥󸀠𝜕𝜉𝑛𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

− 𝑓∑

𝑗<𝑛𝜕𝑗(𝑔) ∫

|𝜉 󸀠 |=1∫+∞

−∞ ∑

|𝛼|=1

trace[𝜕𝜉𝑗𝜋+𝜉𝑛

× 𝑞−1𝜕𝜉𝑛

× 𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= 0, term (a) (II) = − 12𝑓𝑔 ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜕𝑥𝑛𝜋+

𝜉 𝑛𝑞−1

× 𝜕𝜉2𝑛𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

−1

2𝑔𝜕𝑥𝑛𝑓

× ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜋+𝜉𝑛𝑞−1

× 𝜕𝜉2𝑛𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= −3

8𝜋ℎ󸀠(0) Ω3𝑓𝑔 𝑑𝑥󸀠

−𝜋𝑖2Ω3𝑔𝜕𝑥𝑛(𝑓) 𝑑𝑥󸀠, term (a) (III)= 3

8𝜋ℎ󸀠(0) Ω3𝑓𝑔 𝑑𝑥󸀠

+𝜋𝑖2Ω3𝑓𝜕𝑥𝑛(𝑔) 𝑑𝑥󸀠

(62)

As in [15], we have

term (b) = −𝑖 ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜋𝜉+𝑛𝑞−2

× 𝜕𝜉𝑛𝑞−1]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= 98𝑓𝑔𝜋ℎ󸀠(0) Ω3𝑑𝑥󸀠,

term (c) = −𝑖 ∫

|𝜉 󸀠 |=1∫+∞

−∞ trace[𝜋+𝜉𝑛𝑞−1

× 𝜕𝜉𝑛𝑞−2]

× (𝑥0) 𝑑𝜉𝑛𝜎(𝜉󸀠) 𝑑𝑥󸀠

= −98𝑓𝑔𝜋ℎ󸀠(0) Ω3𝑑𝑥󸀠

(63)

So, the sum of terms (b) and (c) is zero Then, we obtain

∫

𝜕𝑀Φ = 𝜋𝑖Ω2 3∫

𝜕𝑀[𝑓𝜕𝑥𝑛(𝑔) − 𝑔𝜕𝑥𝑛(𝑓)]𝑥

𝑛 =0𝑑V𝑜𝑙𝜕𝑀 (64)

By the definition of the noncommutative residue for man-ifolds with boundary, we have that the interior term of

̃ Wres[𝜋+(𝑓𝐷−1)𝜋+(𝑔𝐷−1)] equals Wres[𝑓𝐷−1𝑔𝐷−1] Then,

byTheorem 17and (64), we get the following

Theorem 19 Let 𝑀 be a 4-dimensional compact spin manifold

with boundary Then,

̃Wres [𝜋+(𝑓𝐷−1) 𝜋+(𝑔𝐷−1)]

= −4𝜋2∫

𝑀[𝑓𝑔𝑠3 + 2 ⟨𝑑𝑓, 𝑑𝑔⟩] 𝑑V𝑜𝑙𝑀 +𝜋𝑖Ω3

2 ∫𝜕𝑀[𝑓𝜕𝑥𝑛(𝑔) − 𝑔𝜕𝑥𝑛(𝑓)]󵄨󵄨󵄨󵄨󵄨𝑥 𝑛 =0𝑑V𝑜𝑙𝜕𝑀

(65)

When𝑓 = 1 and 𝑔 = 𝑥𝑛𝐾 near the boundary, we have that the boundary term is proportional to the gravitational action on the boundary

4 The Spectral Action for Perturbations of Dirac Operators

In [18], Iochum and Levy computed heat kernel coefficients for Dirac operators with one-form perturbations and proved that there are no tadpoles for compact spin manifolds without boundary In [17], they investigated the spectral action for scalar perturbations of Dirac operators In [19], Hanisch

et al derived a formula for the gravitational part of the spectral action for Dirac operators on4-dimensional spin manifolds with totally antisymmetric torsion In fact, Dirac operators with totally antisymmetric torsion are three-form perturbations of Dirac operators In this section, we will give some details on the spectral action for Dirac operators with scalar perturbations We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact spin manifolds without boundary For the perturbed self-adjoint Dirac operator𝐷Ψ, we will calculate the bosonic part of the spectral action It is defined

to be the number of eigenvalues of𝐷Ψin the interval[−∧, ∧] with∧ ∈ R+ It is expressed as

𝐼 = Tr 𝐹 (𝐷2Ψ

Here, Tr denotes the operator trace in the𝐿2completion of

Γ(𝑀, 𝑆(𝑇𝑀)) and 𝐹 : R+ → R+is a cut-off function with support in the interval[0, 1] which is constant near the origin Let dim𝑀 = 𝑛 By Lemma 1.7.4 in [22], we have the heat trace asymptotics, for𝑡 → 0,

Tr(𝑒−𝑡𝐷2Ψ) ∼ ∑

𝑚≥0𝑡𝑚−𝑛/2𝑎2𝑚(𝐷2Ψ) (67)

One uses the Seeley-DeWitt coefficients𝑎2𝑚(𝐷2

Ψ) and 𝑡 =

∧−2 to obtain an asymptotics for the spectral action when

dim𝑀 = 4

𝐼 = tr 𝐹 (𝐷2Ψ

∧2) ∼ ∧4𝐹4𝑎0(𝐷2Ψ)

+ ∧2𝐹2𝑎2(𝐷2Ψ) + ∧0𝐹0𝑎4(𝐷2Ψ) as ∧ 󳨀→ ∞

(68)

with the first three moments of the cut-off function which are

given by𝐹4 = ∫0∞𝑠𝐹(𝑠)𝑑𝑠, 𝐹2 = ∫0∞𝐹(𝑠)𝑑𝑠, and 𝐹0 = 𝐹(0)

Let

Ω𝑖𝑗= ∇𝑒𝑖∇𝑒𝑗− ∇𝑒𝑗∇𝑒𝑖− ∇[𝑒𝑖,𝑒𝑗] (69)

We use [22, Thm 4.1.6] to obtain the first three coefficients of

the heat trace asymptotics:

𝑎0(𝐷Ψ) = (4𝜋)−𝑛/2∫

𝑎2(𝐷Ψ) = (4𝜋)−𝑛/2∫

𝑀Tr[𝑠

6+ 𝐸] 𝑑V𝑜𝑙, (71)

𝑎4(𝐷Ψ)

= (4𝜋)−𝑛/2

360

× ∫

𝑀Tr[−12𝑅𝑖𝑗𝑖𝑗,𝑘𝑘+ 5𝑅𝑖𝑗𝑖𝑗𝑅𝑘𝑙𝑘𝑙

− 2𝑅𝑖𝑗𝑖𝑘𝑅𝑙𝑗𝑙𝑘+ 2𝑅𝑖𝑗𝑘𝑙𝑅𝑖𝑗𝑘𝑙− 60𝑅𝑖𝑗𝑖𝑗𝐸 + 180𝐸2+ 60𝐸,𝑘𝑘+ 30Ω𝑖𝑗Ω𝑖𝑗] 𝑑V𝑜𝑙

(72)

WhenΨ = 𝑓, by (15) and (71),

𝑎2(𝐷𝑓) = (2𝜋)−𝑛/2[−12𝑠 + (𝑛 − 1) 𝑓2] , (73)

5𝑠2+ 60𝑠𝐸 + 180𝐸2= 5

4𝑠2− 30 (𝑛 − 1) 𝑠𝑓2 + 180(𝑛 − 1)2𝑓4

(74)

Tr[Ω𝑖𝑗Ω𝑖𝑗] is globally defined; thus we only compute it in

normal coordinates about 𝑥0 and the local orthonormal

frame𝑒𝑖obtained by parallel transport along geodesics from

𝑥0 Then,

𝜔𝑠𝑡(𝑥0) = 0, 𝜕𝑖(𝑐 (𝑒𝑗)) = 0, [𝑒𝑖, 𝑒𝑗] (𝑥0) = 0

(75)

We know that the curvature of the canonical spin connection

is

𝑅𝑆(𝑒𝑖, 𝑒𝑗) = −1

4

𝑛

∑

𝑠,𝑡=1

𝑅𝑀𝑖𝑗𝑠𝑡𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) (76)

Then, we have

Ω (𝑒𝑖, 𝑒𝑗) (𝑥0)

= [𝑒𝑖+1

4∑𝑠,𝑡𝜔𝑠𝑡(𝑒𝑖) 𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) − 𝑓𝑐 (𝑒𝑖)]

× [𝑒𝑗+14∑

𝑠,𝑡𝜔𝑠𝑡(𝑒𝑗) 𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) − 𝑓𝑐 (𝑒𝑗)]

− [𝑒𝑗+1

4∑𝑠,𝑡𝜔𝑠𝑡(𝑒𝑗) 𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) − 𝑓𝑐 (𝑒𝑖)]

× [𝑒𝑖+1

4∑𝑠,𝑡𝜔𝑠𝑡(𝑒𝑖) 𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) − 𝑓𝑐 (𝑒𝑖)]

= −1 4

𝑛

∑

𝑠,𝑡=1

𝑅𝑀𝑖𝑗𝑠𝑡𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) − 𝑒𝑖(𝑓) 𝑐 (𝑒𝑗) + 𝑒𝑗(𝑓) 𝑐 (𝑒𝑖) + 2𝑓2𝑐 (𝑒𝑖) 𝑐 (𝑒𝑗) , for 𝑖 ̸= 𝑗

(77)

So,

Tr[Ω𝑖𝑗Ω𝑖𝑗] (𝑥0)

= ∑

𝑖 ̸= 𝑗

Tr{1 16

𝑛

∑

𝑠,𝑡,𝑠 1 ,𝑡 1 =1

𝑅𝑀𝑖𝑗𝑠𝑡𝑅𝑀𝑖𝑗𝑠1𝑡1𝑐 (𝑒𝑠)

× 𝑐 (𝑒𝑡) 𝑐 (𝑒𝑠1) 𝑐 (𝑒𝑡1) + 𝑒𝑖(𝑓)2𝑐(𝑒𝑗)2+ 𝑒𝑗(𝑓)2𝑐(𝑒𝑖)2 + 4𝑓4𝑐 (𝑒𝑖) 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑖) 𝑐 (𝑒𝑗)

−𝑓2 2

𝑛

∑

𝑠,𝑡=1

𝑅𝑀𝑖𝑗𝑠𝑡[𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) 𝑐 (𝑒𝑖) 𝑐 (𝑒𝑗) + 𝑐 (𝑒𝑖) 𝑐 (𝑒𝑗) 𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡)]

− 𝑒𝑖(𝑓) 𝑒𝑗(𝑓) [𝑐 (𝑒𝑗) 𝑐 (𝑒𝑖) + 𝑐 (𝑒𝑖) 𝑐 (𝑒𝑗)] }

(78)

By (21), we obtain

𝑛

∑

𝑖,𝑗,𝑠,𝑡,𝑠1,𝑡1=1

Tr[1

16𝑅𝑀𝑖𝑗𝑠𝑡𝑅𝑀𝑖𝑗𝑠1𝑡1𝑐 (𝑒𝑠) 𝑐 (𝑒𝑡) 𝑐 (𝑒𝑠1) 𝑐 (𝑒𝑡1)]

= −𝑑 8

𝑛

∑

𝑖,𝑗,𝑠,𝑡=1

(𝑅𝑀𝑖𝑗𝑠𝑡)2,

(79)

∑

𝑖 ̸= 𝑗

Tr[𝑒𝑖(𝑓)2𝑐(𝑒𝑗)2+ 𝑒𝑗(𝑓)2𝑐(𝑒𝑖)2]

= 2𝑑 (1 − 𝑛) ∑

𝑖

𝑒𝑖(𝑓)2= 2𝑑 (1 − 𝑛) 󵄨󵄨󵄨󵄨𝑑𝑓󵄨󵄨󵄨󵄨2,

(80)