S.G. Scott and K.P. Wojciechowski

GAFA Geom funct anal c

Birkhauser Verlag Basel

Vol

X GAFA Geometric And Functional Analysis

THE DETERMINANT AND QUILLEN

DETERMINANT FOR A DIRAC OPERATOR ON A

MANIFOLD WITH BOUNDARY

SG Scott and KP Wojciechowski

Abstract

Let D denote a Dirac op erator on a compact o dddimensional mani

fold M with b oundary Y The elliptic b oundary value problem D is

the op erator D with domain determined by a b oundary condition P

from the smo oth selfadjoint Grassmannian Gr D It has a well

dened determinant see Wo The determinant line bundle over

Gr D has a natural trivialization in which the canonical Quillen

determinant section b ecomes a function denoted by det D equal

C P

to the Fredholm determinant of a naturally asso ciated op erator on

the space of b oundary sections In this pap er we show that the

regularized determinantdet D is equal to det D mo dulo a natural

P C P

multiplicative constant

Intro duction

Since the early stages of Quantum Mechanics there has b een a fundamental

need for a rigorous and workable denition of the determinan tofaninvert

ible linear op erator acting on an innitedimensional space The Fredholm

determinant provides a natural extension from the niterank case to a lin

ear op erator T HHacting on a separable Hilb ert space such that

T Id is an op erator of trace class It is dened in a small neighb ourho o d

of the identity Id by the formula

det e e

where denotes an op erator of trace class and for general T Id with

traceclass by the absolutely convergent series

det Id Tr

In particular the Fredholm determinant retains the characteristic multi

plicative prop erty of the niterank algebraic determinant

det T T det T det T

Fr Fr Fr

SG SCOTT AND KPWOJCIECHOWSKI GAFA

The traceclass condition is however very restrictive and the class of op

erators with Fredholm determinants certainly do es not include any class

of elliptic dierential op erators Nevertheless imp ortant applications in

P hy sics particularly in Quantum Field Theory see the fundamental pa

p ers MS led to the idea that the Fredholm determinantallows one to

study the ratio of the determinant of an elliptic op erator to the determi

nant of a comparable basep oint op erator for instance the ratio of the

determinant of a Hamiltonian with p otential and the determinant of the

free Hamiltonian In this waya regularized determinant relativetoachoice

of basep oint op erator may b e dened

On the other hand in many imp ortant problems such as in quantizing

gauge theories it is necessary to discuss directly a regularized determinant

of an elliptic op erator The Heuristic Approach to the determinant in this

context was rst prop osed by mathematicians for the case of a p ositive

denite secondorder elliptic dierential op erator

L C M S C M S

acting on the sections of a smo oth vector bundle S over a closed mani

M The op erator L has a discrete sp ectral resolution and so formally fold

has determinant equal to the innite pro duct of its eigenvalues The start

ing p oint in dening a regularized pro duct is the following formula for an

invertible niterank linear op erator T

ln det T fTr T gj

For large Res the function of the op erator L is just the trace o ccurring

on the right side of

s tL s

t Tr e dt sTr L

It is a holomorphic function of s for Res dim M and has a mero

morphic extension to the whole complex plane with only simple p oles see

Se In particular s is not a p ole Hence f sgj is

L s

welldened and wemay dene the determinant by

det L e

This denition was intro duced in in a famous pap er of Rayand

Singer RS in order to dene Analytic Torsion the analytical counterpart

of the top ological invariant FranzReidemeister Torsion The equalityof

the two torsions was subsequently proved indep endently by Je Cheeger

and Werner Muller see C Mu Since then there have b een numerous

applications of the determinantinphysics and mathematics b eginning

with the Hawking pap er H on quantum gravity

Vol PROJECTIVE EQUALITY OF DETERMINANTS

For p ositivedenite op erators of Laplace typ e over a closed manifold

the determinantprovides a generally satisfactory regularization metho d

Though the fundamental multiplicative prop erty of the Fredholm deter

minant no longer holds if L and L denote two p ositive elliptic

op erators of p ositive order on a Hilb ert space H then in general

det L L det L det L

We refer to KV for a detailed study of the socalled Multiplicative

Anomaly In manyphysical applications however suchasthequantization

of Fermions one encounters the more problematic task of dening the de

terminant of a rstorder Dirac operator These are not p ositive op erators

and it is at this p oint that anomalies may arise due to the phase of the deter

minantAtS For a Dirac op erator D C M S C M S acting on

the sections of a bundle of Cliord mo dules over a closed o dddimensional

manifold M one pro ceeds in the following way see BoW for an intro

duction and notation The op erator D is an elliptic selfadjoint rstorder

op erator and hence has innitely many p ositive and negative eigenvalues

Let f g denote the set of p ositive eigenvalues and f g denote

k k N k k N

the set of negativeeigenvalues Once again sTrD iswelldened

and holomorphic for Res dim M and wehave

X X

s s

k k

s s s s s s s s

X X

k k k k k k k k

k k

which can b e written as

s s s s

D D

P P

s s

is the function of the op erator D intro where s

k k

duced byAtiyah Pato di and Singer see AtPS Once again it is holomor

phic for Res large and has a meromorphic extension to the whole complex

plane with only simple p oles There is no p ole at s and therefore we

can study the derivativeof sats Wehave

f g

ds s

The ambiguity in dening ie a choice of sp ectral cut now leads

to an ambiguity in the phase of the determinant Wehave

s i s

and we pickthe sign This leads to the following formula for the

SG SCOTT AND KPWOJCIECHOWSKI GAFA

determinant of the Dirac op erator D

det D e e

We come back to the discussion of the choice of sign in in the nal

section of the pap er

The purp ose of this pap er is to explain a direct and precise identitybe

tween the determinant of a selfadjoint elliptic b oundary value problem

for the Dirac op erator over an o dddimensional manifold with b oundary

and a regularization of the determinant as the Fredholm determinantofa

canonically asso ciated op erator over the b oundaryWe consider an innite

dimensional Grassmannian of elliptic b oundary conditions commensurable

with the AtiyahPato diSinger condition The latter regularization is natu

rally understo o d in the sense explained b elow as the ratio of the determi

nant of the elliptic b oundary value problem to the determinant of a base

p oint elliptic b oundary value problem It is a regularization canonically

constructed from the top ology of the asso ciated determinant line bundle

and hence called the canonical determinant The canonical determinantis

a robust algebraic op eratortheoretic ob ject while the determinantisa

highly delicate analytic ob ject and so it is surprising that they coincide

Though the equality of the torsions mentioned ab ove at least suggests that

the determinantmay b e somehow related to Fredholm determinants

To formalise the construction of taking the ratios of determinants used

to dene the canonical determinantwe use the machinery of the determi

nant line bundle This was intro duced in a fundamental pap er of Quillen

Q for a family of CauchyRiemann op erators acting on a Hermitian bundle

over a Riemann surface as the pullbac k of the corresp onding universal

determinant bundle over the space of Fredholm op erators on a separable

Hilb ert space Without making further choices the determinant arises not

as a function on the parameter space of op erators but as a canonical section

A det A of the asso ciated determinant line bundle DE T More precisely

max max

det A lives in the complex line Det A Ker A Coker A and

is nonzero if and only if A is invertible Using function regularization

Quillen constructed a natural Hermitian metric on the determinant bun

dle for a family of Cauchy Riemann op erators and computed its curvature

This was extended by Bismut and Freed to the context of general families of

Dirac op erators on closed manifolds see BF and the curvature identied

with the form comp onent of the families index density It did not though

provide a straightforward corresp ondence b etween the determinantand

the Quillen determinant section More precisely given that det A is de

Vol PROJECTIVE EQUALITY OF DETERMINANTS

ned the problem is to identify the nonzero section of DE T such that

det A det A A Clearly the global existence of such a section

is equivalent to the triviality of the determinant line bundle From this

view p oint the principal result of this pap er is the exact identication of

the basep oint section for the class of elliptic b oundary value problems

considered here In order to link this up with Fredholm determinants we

use a construction of the determinant line bundle due to Segal Seg

We do not discuss in this pap er the corresp onding problem for a closed

manifold Perhaps surprisingly it is easier to discuss the relation b etween

the determinant and the Quillen determinant section on a manifold with

b oundary b ecause of reduction to the boundary integral see BoW

In early the second author as the followup to his work on the

invariant on a manifold with b oundary see Wo showed the existence

of the determinant on the Grassmannian of generalized AtiyahPato di

Singer b oundary conditions A little earlier the rst author using the Se

gal construction of the determinant line bundle intro duced the canonical

C determinant on this Grassmannian and showed that it is equal to the

determinant in the one dimensional case see S see also BoSW for

a discussion of the one dimensional case in the spirit of this exp osition

The present pap er contains the result of join twork the pro of of the equal

ityofthe determinantand C determinant up to a natural multiplicative

constantinany o dd dimension Early progress was rep orted in the note

SW and the results of this pap er were announced in SW We refer to

BoMSW for a discussion of related topics in the evendimensional case see

also the review WoSMB The construction of a metric and compatible

connection on the determinant line bundle using the canonical regulariza

tion for a family of Dirac op erators over a closed manifold endowed with a

partition is explained in S

Wenow give a more detailed presentation of the situation discussed in

this pap er Let D C M S C M S denote a compatible Dirac

op erator acting on the space of sections of a bundle of Cliord mo dules S

over a compact connected manifold M with b oundary Y It is not actually

necessary to assume that D is a compatible Dirac op erator further tech

nical comments are made in the nal section of the pap er In the present

pap er we always assume that M is an o dddimensional manifold the even

dimensional case will b e discussed separately And we discuss only the

Product Case Namely we assume that the Riemannian metric on M and

the Hermitian structure on S are pro ducts in a certain collar neighborhood

SG SCOTT AND KPWOJCIECHOWSKI GAFA

of the b oundary Let us x a parameterization N Y of the collar

Then in N the op erator D has the form

D G B

where G S jY S jY is a unitary bundle isomorphism Cliord multi

plication by the unit normal vector and B C Y S jY C Y S jY

is the corresp onding Dirac op erator on Y which is an elliptic selfadjoint

op erator of rst order Furthermore G and B do not dep end on the normal

co ordinate u and they satisfy the identities

G Id and GB BG

Since Y has dimension m the bundle S jY decomp oses into its p ositiveand

S and wehave a corresp ond negativechirality comp onents S jY S

ing splitting of the op erator B into B C Y S C Y S where

B B Equation can b e rewritten in the form

i B

B i

In order to obtain a Fredholm op erator with go o d elliptic regularity

prop erties wehave to imp ose a b oundary condition on the op erator D Let

denote the sp ectral pro jection of B onto the subspace of L Y S jY

spanned b y the eigenvectors corresp onding to the nonnegative eigenvalues

of B Itiswell known that is an elliptic b oundary condition for the

op erator D see AtPS BoW The meaning of the ellipticityisas

follows Weintro duce the unb ounded op erator D equal to the op erator

D with domain

dom D s H M S j sjY

where H denotes the rst Sob olev space Then the op erator

D D domD L M S

is a Fredholm op erator with kernel and cokernel consisting only of smo oth

sections

The orthogonal pro jection is a pseudo dierential op erator of order

see BoW In fact we can takeany pseudo dierential op erator R of order

with principal symb ol equal to the principal symbol of and obtain an

op erator D which satises the aforementioned prop erties In order to

explain this phenomenon we give a short exp osition of the necessary facts

from the theory of elliptic b oundary problems In contrast to the case of

an elliptic op erator on a closed manifold the op erator D has an innite

dimensional space of solutions More precisely the space

D s in M n Y s C M S

Vol PROJECTIVE EQUALITY OF DETERMINANTS

is innitedimensional Weintro duce the Calderon pro jection whichis

the pro jection onto the L closure of the Cauchy Data space HD of the

op erator D

HD f C Y S jY s C M S D s in M n Y and sjY f

The pro jection P D is a pseudo dierential op erator with principal symbol

equal to the symbol of see BoW Moreover in our situation P D

is an orthogonal pro jection in L Y S jY This is not true in more general

situations for instance in the case of nonpro duct metric structures near

the b oundary The op erator D has the Unique Continuation Property

and hence wehave a one to one corresp ondence b etween solutions of the

op erator D and the traces of the solutions on the b oundary Y at least in

the case of a connected manifold M This explains roughlywhy only the

pro jection P onto the kernel of the b oundary conditions R matters If the

dierence P P D is an op erator of order then it follows that we

cho ose the domain of the op erator D in suchaway that we throwaway

almost all solutions of the op erator D on M n Y with the p ossible exception

of a nitedimensional subspace The ab ove condition on P allows us also

to construct a parametrix for the op erator D hence we obtain regularity

of the solutions of the op erator D We refer to the monograph BoW

for more details

In the following we do not discuss the determinant on the total space

of elliptic b oundary conditions for the op erator D wecho ose a smaller

and more convenient space mostly in order to avoid unpleasanttechnical

D questions We restrict ourselves to the study of the Grassmannian Gr

of all pseudo dierential pro jections which dier from by an op erator

of order The space Gr D has innitely many connected comp onents

and two b oundary conditions P and P b elong to the same connected

comp onent if and only if

index D index D

P P

We are interested however in selfadjoint realizations of the op erator D

The antiinvolution G S jY S jY equips L Y S jY with a symplectic

structure and using Greens formula

D s s s D s Gs jY s jY dy

it is shown in BoW that the b oundary condition R provides a selfadjoint

realization D of the op erator D if and only if ker R is a Lagrangian sub

space of L Y S jY see BoW BoW DoW Wemay therefore

restrict our attention to those elements of Gr D which are pro jections

SG SCOTT AND KPWOJCIECHOWSKI GAFA

onto Lagrangian subspaces of L Y S jY More preciselyweintro duce

Gr D the Grassmannian of orthogonal pseudo dierential pro jections P

such that P is an op erator of order and

GP G Id P

The space Gr D iscontained in the connected comp onentof Gr D pa

rameterizing pro jections P with index D

For analytical reasons asso ciated with the existence of the determinant

in this pap er we discuss only the Smooth Selfadjoint Grassmanniana

dense subset of the space Gr D dened by

Gr ernel P has a smo oth k D P Gr D

The sp ectral pro jection is an elementofGr D if and only if

ker B fg On the other hand it is well known that the orthogo

nal Calderon pro jection P D is an elementof Gr D see for instance

BoW Moreover it was proved in S Prop osition see also DK

App endix that P D is a smo othing op erator and hence that P D

is an elementof Gr D The nitedimensional p erturbations of see

also DoW LW and Wo provide further examples of b oundary con

ditions from Gr D The latter were intro duced by Je Cheeger who

called them Ideal Boundary Conditions see C

For any P Gr D the op erator D has a discrete sp ectrum nicely

distributed on the real line see BoW DoW It was shown by the

second author that for any P Gr D s and s are welldened

functions holomorphic for Res large and with meromorphic extensions

to the whole complex plane with only simple p oles In particular b oth

functions are holomorphic in a neighborhood of s Therefore det D

dened byformula is a welldened smo oth function on Gr D see

The canonical determinant is dened in the following way The family

of elliptic b oundary value problems fD j P Gr D g parameterized by

Gr D P Gr D P has a smo oth kernel

DE T D Gr D with deter has an asso ciated determinant line bundle

minant section

max max

P det D DetD KerD CokerD

P P P P

see section On the other hand relative to the basep oint Calderon pro

jection P D Gr D wehave the smo oth family of Fredholm op erators

over the b oundary

P Gr D g S P PPD HD Ran P

Vol PROJECTIVE EQUALITY OF DETERMINANTS

with asso ciated Segal determinant line bundle DE T again equipp ed

P D

with its determinant section P det S P From S weknow that

there is a canonical line bundle isomorphism preserving these sections

DE T D DE T det D det S P

P D

and therefore since DE T is a nontrivial complex line bundle whose

P D

rst Chern class generates H Gr D Z Z no global nonzero section

of DE T D exists However from the computation of homotopy groups in

BoW DoWwehave H Gr D Z and so the determinantline

bundle do es restrict to a trivializable complex line bundle over the smo oth

Grassmannian of selfadjoint b oundary conditions

The problem nowistoidentify which trivialization denes the deter

minant det D Tomake the presentation smo other we assume henceforth

that ker B fg this is in fact not a serious restriction and it will b e ex

plained in section that we can easily relax this condition The correct

choice of trivialization is indicated by the fact that any elliptic b ound

ary condition P Gr D is describ ed precisely by the prop erty that

its range is the graph of an elliptic unitary isomorphism T F F

such that T B B B has a smo oth kernel S where F are

the spaces of chiral spinor elds over the b oundary In section of this

pap er we explain how this denes a preferred nonzero basep oint section

P P Det D Thecanonical determinant is then dened to b e the

en in Det D quotient tak

det D

C P

and this turns out to b e the Fredholm determinant of an op erator living

on the b oundary Y constructed from S P The main result of the pap er

is the following Theorem

D Theorem The following equality holds over Gr

det D det D det D

P C P

P D

Remark Theorem shows that at least on Gr D the

determinant is an ob ject which is a natural extension of the welldened

algebraic concept of the determinant

The identication of det D with a regularized Fredholm de

P D

terminant of the op erator S P living on the b oundary extends the cor

resp onding result for the index whichiswell known see BoW Theo

rem

Theorem suggests a new approach to the pasting formula for

the determinant with resp ect to a partitioning of a closed manifold The

SG SCOTT AND KPWOJCIECHOWSKI GAFA

pasting formula for det was intro duced in S It is hop ed that a new

insightinto the pasting mechanism of the determinant will b e obtained

by combining results of S and formula For a recent application

of the results of this pap er to an adiabatic pasting formula see PW

To prove Theorem wefollow a basic idea of Robin Forman F and

study the variation of the relative determinants More preciselygiven

two pro jections P P Gr D we dene two oneparameter families of

b oundary conditions P and compute the relativevariation

fln det D ln det D gj

P P r

r r

for b oth the canonical determinant and the determinant and show that

they coincide Here we face the technical problem of dealing with a family of

unb ounded op erators with varying domain To circumvent this and make

sense of the variation with resp ect to the b oundary condition we follow

Douglas and Wo jciechowski DoW and apply their Unitary Trick see

section Finally using the fact that the space of pro jections P in Gr D

such that D is invertible is actually path connected see section and N

weintegrate the variational equality in order to obtain formula of

Theorem

The pap er is organized as follows In section we explain the construc

tion of the canonical determinant We follow here the exp osition of S

Assume that for given P Gr D the op erator D is invertible In

section we present our construction of an inverse To do that we D

have to discuss certain asp ects of the theory of elliptic b oundary problems

We also intro duce K the Poisson map of the op erator D andK P the

Poisson map of the op erator D The rst is used in the construction

of the Calderon pro jection The op erator K P app ears in several crucial

places in our computation of the variation of the canonical determinant

In section we discuss the variation of the determinant and in section

we study the variation of the canonical determinant It has already

b een mentioned that the work Wo is crucial for the study here of the

determinant while in the calculation of the variation of the canonical

determinantwewere inuenced by the work of Robin Forman F

With the variational equality at hand section contains the nal steps

of the pro of of Theorem

In section we discuss an immediate application of our result to the

mo dulus of the determinant regarded as a function on the Grassmannian

Gr D We show that the Calderon pro jection is the only critical p oint

of this function on the space Gr D of pro jections P Gr D such that

Vol PROJECTIVE EQUALITY OF DETERMINANTS

the op erator D is invertible

In section we make some nal comments on several technical issues

wehave to deal with in this pap er We discuss the choice of the sign of the

phase of determinant We also explain the necessary changes required

for the case of a noninvertible tangential op erator B A more detailed

explanation of the top ological structure of Gr D is given

Acknowledgements Wewant to thank our friends and collab orators

Bernhelm Bo ossBavnb ek and Ryszard Nest for constant supp ort and valu

able discussions The concept of the Canonical Determinant is implicit in

the work of Graeme Segal and his work has greatly inuenced our eorts in

understanding this b eautiful sub ject We are grateful also to JeanMichel

Bismut for helpful comments and to the referee for a careful reading of the

manuscript

Canonical Determinant on the Grassmannian Gr D

In this section we review briey the construction of the determinantline

bundle and give an explicit construction of the canonical determinant

The determinant line bundle over the space of Fredholm op erators was

rst intro duced in a seminal pap er of Quillen Q An equivalent construc

tion which is b etter suited to our purp oses here w as subsequently given

by Segal see Seg and so we follow his approach Let FredH denote

the space of Fredholm op erators on a separable Hilb ert space HWework

rst in the connected comp onent Fred H of this space parameterizing

op erators of index zero For A Fred H dene

Fred S FredH S A is traceclass

Fix a traceclass op erator A such that S A A is an invertible op erator

Then the determinant line of the op erator A is dened as

Det A Fred C

where the equivalence relation is dened by

det S R R z S S Rz S z

The Fredholm determinant of the op erator S R is welldened as it is of

the form Id plus a trace class op erator Denoting the equivalence class of

a pair R z byR z complex multiplication is dened on Det A by

R z R z

The determinant element is dened by

det A A

and is nonzero if and only if A is invertible

SG SCOTT AND KPWOJCIECHOWSKI GAFA

The complex lines t together over Fred H to dene a complex line

bundle L the determinant line bundle To see this observe rst that over

the op en set U in Fred Hdenedby

U F Fred H F A is invertible

the assignment F det F denes a trivializing nonvanishing section of

L The transition map b etween the canonical determinant elements over

U U is the smo oth holomorphic function

A B

g F det F AF B

AB Fr

This denes L globally as a complex line bundle over Fred H endowed

with its determinant section A det AIfind A d we dene Det A to

b e the determinant line of A as an op erator HHC if d or

HC H if d and the construction extends in the obvious way

to the other comp onents of FredH Note that the determinant section is

zero outside of Fred H

We use this construction in order to dene the determinant line bun

dle over Gr D For each pro jection P Gr D wehave the Segal

determinantlineDetS P of the op erator

S P PPD HD Ran P

and the determinant line Det D of the b oundary value problem D

P P

domD L M S These lines t together in the manner explained

ab ove to dene determinant line bundles DE T and DE T D over the

P D

Grassmannian some care has to b e taken as the op erator acts b etween two

dierent Hilb ert spaces but with the obvious notational mo dications we

once again obtain welldened determinant line bundles The canonical

isomorphism identies the two line bundles and preserves the deter

minant elements The bundle DE T is a nontrivial line bundle over

P D

Gr D but when restricted to the Grassmannian Gr D it is canoni

cally trivial

We use the sp ecic trivialization intro duced in S Recall that we

work here with orthogonal pro jections onto the Lagrangian subspaces of

hy data space L Y S jY which are a compact p erturbation of the Cauc

HD Wehave assumed that ker B fg and hence B is an element

of Gr D The range of B is actually the graph of the unitary

op erator V F F given by the formula

V B B B

This identication extends to the whole Grassmannian Gr D elements

are in to corresp ondence with unitary maps V F F such that the

Vol PROJECTIVE EQUALITY OF DETERMINANTS

dierence V V is an op erator with a smo oth kernel The corresp onding

orthogonal pro jection P is given by the formula

Id V

V Id

By cho osing a basep oint the corresp ondence dened ab oveallows us to

establish an isomorphism b etween Gr D and the group U F of uni

taries acting on F L Y S which dier from Id by an op erator

with a smo oth kernel It is convenient for us to work with the Calderon

pro jection as basep oint hence let K C Y S C Y S denote

the unitary such that HD is equal to g r aphK For any pro jection P

Gr D there exists T T P F F such that Ran P g r aphT

and so wehave a natural isomorphism Gr D U F dened by the

map P TK This is expressed in terms of the homogeneous structure

of the Grassmannian by

Id Id

F F

P P D

TK KT

Nowwe can dene a nonvanishing section l of the determinant line bundle

DE T over Gr D The value of l at the pro jection P is the class in

P D

S P of the couple Det

U P

where the op erator U P acts from HD toRanP That is l P

det U P The fact that l P is an element of DetS P follows from the

following elementary result

Lemma The dierence b etween U P and the op erator S P

PPD HD Ran P is an op erator with a smo oth kernel hence

det U P U P is an elementof Det S P

Proof The op erator U P acts from g r aphK HD tog r aphT

RanP and acts by

x x Id x

Tx Kx TK Kx

The op erator PPD is given by the following formula

Id T K T K

PPD

T K Id TK

leading to the following expression for the op erator S P PPD

HD RanP

T K T K Id Id

F F

x x x

S P

Id TK Id TK

F F Kx Kx

SG SCOTT AND KPWOJCIECHOWSKI GAFA

Because T K resp TK diers from Id resp Id by a smo oth

F F

ing op erator it is nowobvious that the dierence U P SP isanoper

ator with a smo oth kernel

The discussion ab ove allows us to now dene the Canonical Determinant

over Gr D Let A HD Ran P denote an invertible Fredholm

op erator such that A SP is an op erator of trace class Wehave

det A A

U P U P A

U P det U P A

det U P A U P

U P A det U P det

where we use equations and The ab ove identity means we

can dene the determinant of the op erator A as the ratio in Det A of the

nonvanishing canonical elements det A and det U P or equivalently as

the Fredholm determinant of the op erator U P A This leads to the

following denition of the canonical determinant of the op erator D

Definition We dene the Canonical Determinant of the elliptic

b oundary value problem D by

det D det S P det U P S P

C P C Fr

The naturality of this denition lies in the identication of the abstract

determinants of the Fredholm op erators D and S P by the isomorphism

the section in is just the image of det U P under

In fact from the pro of of Lemma we see that the determinant on the

right side of the equality is the Fredholm determinant of the op erator

Id T K

TK Id

acting on the graph of the op erator K Hence we obtain

Lemma

Id KT

det D det

C P Fr

where the Fredholm determinant on the right side is taken on F

Wemay therefore reformulate Theorem as

Theorem The following equality holds over Gr D

Id KT

det D det D det

P Fr

P D

Vol PROJECTIVE EQUALITY OF DETERMINANTS

Equivalently since det D

P D

det D

C P

det D det D

P D P D

Remark In section we also use determinants of op erators of the

form

S P S P Ran P Ran P

under the assumption that the op erator S P isinvertible From the dis

cussion presented ab ove it follows that for anyFredholm op erator A

Ran P Ran P such that the dierence b etween A and the op erator

P P Ran P Ran P is of trace class we can dene the canonical

determinantof A using the formula

A det A det

C Fr

T T

where Ran P is equal to graph T

i i

Boundary Problems Dened by Gr D Inverse

Op erator and Poisson Maps

For any P Gr D the op erator D isaFredholm op erator hence it has

closed range As a consequence we can dene an inverse to the induced

op erator dom D ker D L M S coker D Ifwe assume that P is

P P P

an elementofGr D then the op erator D is selfadjoint and ker D

P P

coker D It follows that if we assume ker D fg then there exists an

P P

inverse D to the op erator D

This In this section wegive an explicit formula for the op erator D

formula plays a key role in the pro of of the main result of the pap er The

op erator D is a sum of two op erators The rst is the interior inverse

of D The second is a correction term which lives on the b oundary

We start with the interior part of the inverse Let M M M

denote the closed double of the manifold M M is a copyofM with

reversed orientation The bundle of Cliord mo dules S extends to a bundle

S of Cliord mo dules over M and the op erator D determines a compatible

Dirac op erator D over M equal to D on M and D on M We refer

to Wo DoW for the details of these constructions and applications to

the analytic realization of K homol og y The op erator

M S C M S D C

is an invertible selfadjoint op erator hence its inverse D isawelldened

elliptic op erator of order over the manifold M We also have natural

SG SCOTT AND KPWOJCIECHOWSKI GAFA

extension and restriction maps acting on sections of S and S The extension

by zero op erator e L M S L M S is given by the formula

f on M n M

e f

on M

s s s

The restriction op erator r H M S H M S where H denotes

the s S obol ev spaceisgiven by f f f jM To simplify the notation

in the following wealways write

D e D r

The op erator D is the interior part of the inverse It is used in several

crucial constructions in the theory of b oundary problems It maps L M S

into H M S however the range is not necessarily inside the domain

of D For this reason wehavetointro duce an additional term to obtain

an op erator with the correct range To do this we need to study the

situation in a neighb orho o d of the b oundary Y The restriction of smo oth

sections to the b oundary extends to a continuous map

s s

H M S H Y S jY

whichiswelldened for s see BoW The corresp onding adjoint

op erator in the distributional sense provides us with a welldened

map

s s

H Y S jY H M S

for s Now for any real s the mapping

K r D G C Y S jY C M S

s s

extends to a continuous map K H Y S jY H M S with range

equal to the space

D f in M n Y kerD s f H M S

In fact the map

s s

D s Ran P D H Y S jY H Y S jY K H

kerD s

is an isomorphism see BoW Wehave the following equality

D D Id K

which holds on the space of smo oth sections see BoW Lemma

The op erator K is called the Poisson operator of D It denes the

Calderon pro jection

P D K

see BoW Theorem

Vol PROJECTIVE EQUALITY OF DETERMINANTS

Remark Formula gives a priori only a pro jector not an

orthogonal pro jection onto HD In the situation discussed in this pap er

however the resulting pro jector is orthogonal We refer to BoW for the

details of the construction which is originally due to Calderon and Seeley

To construct the correction term to the op erator D we use the inverse

of the op erator S P The invertibilityof S P is equivalenttoD being

invertible

Lemma The op erator D is an invertible op erator if and only if the

op erator S P PPD HD Ran P is invertible

Proof The Grassmannian Gr D is a subspace of the big Grassmannian

D has Gr D see Wo BoW DoW App endix B The space Gr

countably many connected comp onents distinguished by the index of the

op erator S P ie P and P b elong to the same connected comp onentof

Gr D if and only if index S P index S P The space Gr D is

contained in the index z er o comp onentof Gr D Nowwehave

ker S P f P D f f and P f

and

coker S P g Pg g and P D g

P is an elementofGr D then index S P index D We see If

that the op erator S P isinvertible if and only if it has trivial kernel

Similarly a selfadjointFredholm op erator D is invertible only if it has

trivial kernel On the other hand the op erator K denes an isomorphism

K ker S P ker D

This ends the pro of of lemma

Remark Note that the lemma proves a somewhat stronger state

ment via the Poisson op erator K constructing solutions for the op erator

S P is equivalent to constructing solutions to the elliptic b oundary value

problem D and the same for the adjoints In particular this implies that

the index of the two op erators coincide This is the underlying reason why

it is easier to compute determinants on manifolds with b oundary than on

closed manifolds

From nowonwe assume that D is invertible The op erator S P is

not a pseudo dierential op erator as it acts from Ran P into HD How

ever we can show that it is a restriction of an elliptic pseudo dierential op

erator of order More precisely the op erator PPD Id P Id P D

is an elliptic pseudo dierential op erator which can b e written as

S P Id P Id P D HD HD W W

SG SCOTT AND KPWOJCIECHOWSKI GAFA

where W Ran P It can b e seen that

ker S P cokerId P Id P D

and

coker S P kerId P Id P D

Therefore if we assume that ker S P fg then the op erator PPD

Id P Id P D is invertible Its inverse is an elliptic op erator see

for instance BoW and it follows that

Id P Id P D P S P P D PPD

We can now present the formula for the inverse of the op erator D

Theorem Assume that the op erator D dom D L M S is

P P

invertible then its inverse is given by the formula

D D KSP P D

ProofFrom wehave that DK is equal to in M n Y and hence that

DD is equal to Id on L M S Now let f L M S then

P D f P D f P KS P P D f

P D f PPD S P P D f

P D f P D f

and hence D f dom D Wehaveshown that D D L M S

P P

L M S is equal to Id and that D L M S dom D hence

is indeed a rightinverse of D and obviously since the op erator D

index D it is also a left inverse

Corollary Let P P Gr D such that the op erators D and

D are invertible Then the dierence D D is an op erator with

P P

smo oth kernel

Proof It follows from Theorem that

D D K S P P SP P D

P P

Now the fact that P P is an op erator with a smo oth kernel and equation

implies that the op erator S P P SP P also has a smo oth

kernel

D For the rest of this section we take a closer lo ok at the op erator D

as it allows us to intro duce another imp ortan t op erator K P the Poisson

op erator of the op erator D From formula wehave that

D D Id K KSP P Id K

Id K KS P PPD KSP P Id KSP P

Vol PROJECTIVE EQUALITY OF DETERMINANTS

Hence if f dom D then P f and

f f D f Id KSP P D

We dene the Poisson operator of D by

K P KS P P

The op erator K P app eared in the second term of the op erator D D

Let g denote an element in the range of the pro jection P More precisely

assume that g H Y S jY and Pg g Then K P g is a solution of D

which b elongs to the Sob olev space H M S and K P g is an ele

ment of the space of CauchydataofD in general not equal to g However

the part of K P g along P is in fact equal to the original element g

P K P g P KS P Pg PPD S P Pg Pg

In section we also need the following results

Lemma Let P P Gr D such that the op erators D and D

P P

are invertible Let f f Ran P and assume that

P K P f P K P f

Then f f and K P f K P f

ProofWehave

P K P f S P S P f

i i

hence the rst equality follows from the invertibility of the op erators S P

and S P The second is a consequence of the Unique Continuation Prop

erty for Dirac op erators Wehave

K P f P D S P f S P S P S P f

S P S P S P f K P f

and hence two solutions of D with the same C auchy data hence they are

equal

Proposition

K P P D D D

P P P

ProofWexf L M S Let

f h K P P D

Observe that the section h is the unique solution of D with b oundary data

f Indeed along P equal to P D

P hP f g K P P D

f f P D P P D S P P D

P P

SG SCOTT AND KPWOJCIECHOWSKI GAFA

and uniqueness is a consequence of Lemma Now the section g

f is also a solution of D and the restriction of g to the b oundary D D

P P

has P comp onent equal to

P g P K S P P SP P D f

P P D S P P SP P D f

D f KS P P P D f P

P D f

and therefore h and g are the same section

Remark The construction of the inverse presented in this section

gives in fact a parametrix for any elliptic b oundary problem for the Dirac

op erator First of all if P is an elementof Gr D we can still use formula

in order to construct the aforementioned parametrix The op erator

S P has to b e replaced by an op erator RP of the form

P D RP Ran P HD

where R denotes any parametrix of the elliptic op erator PPD

Id P Id P D The formula

C D KRP P D

nowgives an op erator such that D C Id and C D Id have smo oth

P P P P

kernels

More generally this formula gives a parametrix for any elliptic

b oundary problem D as dened in BoW where the authors were fol

lowing Seeleys exp osition Se The reason is that N the kernel of

the b oundary condition T and HD formaFredholm pair of subspaces

of L Y S jY which allows a parametrix R to b e constructed This fact

was well known to Bo oss and W o jciechowski and is implicit in their work

BoW and BoW see also Prop osition in BoMSW Last but not

least we are not really restricted in this construction of the parametrix

only to Dirac op erators This construction holds for any rst order elliptic

dierential op erator on a compact manifold with b oundary The details

will b e presented elsewhere

The explicit construction of the parametrix presented in this pap er

may b e found elsewhere in the literature in related contexts It was used for

instance byPeter Gilkey and Lance Smith in their work on the invariant

for a class of lo cal elliptic b oundary problems see GS See also the

work of Forman F

Vol PROJECTIVE EQUALITY OF DETERMINANTS

Variation of the determinantonGr D

In this section we study the variation of the determinant of the op era

tor D whereP Gr D under a change of b oundary condition From

section we know that the Grassmannian Gr D can b e identied with

the group U F If we x a base pro jection for instance P D then

any other pro jection is of the form

Id Id

F F

P P D

g g

where g F F is a unitary op erator such that g Id has a smo oth

ernel k

Weintro duce a smo oth oneparameter family fg g of op erators

r r

from U F with g Id Let fP g denote the corresp onding family

of pro jections

Id Id

F F

P P

g g

Wewant to compute the variation

fln det D gj

P r

For the purp oses of this pap er it is enough to solve an easier problem

Let us x two elements of the Grassmannian P and P such that D and

D are invertible op erators The family fg g determines two parameter

P r

families of pro jections

Id Id

F F

P P

ir i

g g

with resp ect to whichwemay study the relativevariation

fln det D ln det D gj

P P r

r r

The rst obstacle here is that the domains of the unb ounded op erators

D vary with the parameter r Itwas explained in DoW and LWhow

to solve this problem by applying a Unitary Twist The p oint is that

wemay extend the family of unitary isomorphisms fg g on the b oundary

sections to a family of unitary transformations fU g on L M S Todo

that x a smo oth nondecreasing function u such that

u for u and u for u

and for each r intro duce the parameter family

for u g g

u r

Nowwe dene a transformation U as follows

on fug Y N Y

Id on M n N

We then have the following elementary result

SG SCOTT AND KPWOJCIECHOWSKI GAFA

Lemma The op erators D and U D U are unitarily equivalent

P r P

ir i

op erators

Clearly the op erators U dep end up on the choice of the extension func

tion however by Lemma the determinant do es not whichisallthat

we need The canonical determinant is also indep endent of the choice of

the family fU g This follows from the fact that P P D and the op erator

r ir

P P D are unitarily equivalent

i r

D g g P P D g P g g P P D g P D g P g P

r ir r i r i r r i

r r r r

In the following we use the notation

D j D U D U and D

r r r r

The main result of this section is the following theorem

Theorem For any pair of pro jections P P Gr D such that D

and D are invertible op erators and for any smo oth parameter family

fg g of op erators from U F with g Id the following equality holds

r o

fln det D ln det D gj Tr D D D

P P r

r r

P P

Notice that although the variation D is lo calized in N the variation

of the determinant is not It dep ends on global data b ecause of the term

and it is this that makes the determinant a more dicult D D

P P

sp ectral invariant than the invariant which corresp onds to the phase of

the determinant Indeed formula contains only a variation of the

dierence of logarithms of the mo dulus of the determinant The reason

is that the variation of the phase of the determinantofD is equal to the

variation of the phase of the determinantof D

Theorem The variation of the phase of the determinant of the

op erator D dep ends only on the family of the unitaries fg g on F such

P r

that

Id Id

F F

P P

g g

not on the choice of the basep oint pro jection P More sp ecically

is a constant function of the pro jection P and the variation of the

invariant dep ends only on the family fg g

Proof The theorem follows from two technical results proved in the work

of the second author Wo The phase of the determinant is equal to

exp

It was shown in Wo Prop osition that is constanton Gr D

hence the variation of the logarithm of the phase is equal to the variation

Vol PROJECTIVE EQUALITY OF DETERMINANTS

of the invariant times i The formula for the variation of the

invariantwas derived in the pro of of Theorem in Wo Wehave

i d d g

j du Tr g

D r

dr dr u

In particular the right side of do es not dep end on P

Remark A sp ecial case of the formula was discussed in the

pap er SW

Next we study the logarithm of the mo dulus of the determinant

lnj det D j

It is well known see section of Wo that

s tD

t Tr e dt lim

where denotes the Euler constant The fact that do es not dep end

on P allows us to study just the variation of the integral in formula

and with the help of Duhamels Principle we obtain

j TrtD D e dt

r P

dr t

Z Z

tD tD

P P

i i

Tr D D e dt Tr D D D e dt

P P

i i

tD D

P P

i i

lim Tr D D e j lim Tr D D e

P P

i i

We therefore have

Lemma

lim Tr D D e

dr D

In general the limit on the right hand of the equation is just the

constant term in the asymptotic expansion of the trace However since we

discuss the dierence in this situation we actually obtain the true

op erator trace

fln det D ln det D gj

P P r

r r

D D

P P

e lim Tr D D e lim Tr D D

P P

lim Tr D D D e

P P

D Tr D D

P P

where for the nal step we use Corollary This completes the pro of of Theorem

SG SCOTT AND KPWOJCIECHOWSKI GAFA

Variation of the Canonical Determinant

In this section we prove that the relativevariation of the canonical deter

minant coincides with the relativevariation of the determinant We

b egin with the following result

Proposition The following formula holds for any P P Gr D

such that D and D are invertible op erators

P P

S P S P det D det D det

r r C P C P Fr

r r

T T

where S P denotes the op erator P P D HD Ran P

r i i r r i

Proof Let

U Ran P Ran P

T T

and observe that

U U U U U

T T T T T T T T

T T

and that if A Ran P Ran P is of the form Id plus traceclass then

det A det U AU

Fr Fr T T

T T

where the determinant on the leftside is taken on Ran P and the deter

minant on the rightside is taken on Ran P Then since U P U

wehave using the invariance of the canonical determinant under a

unitary twist and the multiplicativity of the Fredholm determinant

det D det D det U S P det U S P

C P C P Fr r Fr r

r r

K T K T

r r

det U S P S P U

Fr r K T

K T

det i U U S P S P U

Fr T T r K T

K T

U S P S P det

T T r r Fr

where the last two lines use and resp ectively

Hence setting

S P S P P Ran P Ran P S U

T r r r T

wehave proved

Corollary

d d

Tr S S fln det D ln det D gj Tr S S

C P C P r r

r r

dr dr

Lemma

Tr S S Tr P K P P K P

Vol PROJECTIVE EQUALITY OF DETERMINANTS

ProofWe compute

d d

ln det S Tr S S Tr S S

r r

dr dr

r r

Tr P P D P P D P

r r

T T

P P D P P D

T T

P KS P P P KS P P Tr

P K P P K P Tr

Tr P K P P K P

The lemma is proved

The next lemma takes care of the variation of the op erator K P

Lemma The following formula holds at r

K P K P j D D K P

r r P

Proof Let us x f Ran P and let s K P f Wehave

r r

D s and P s f

r r r

hence dierentiation with resp ect to r gives

d d d d

D s D s and P s P s

r r r r r r

dr dr dr dr

hence s dom D We obtain

r P

d d d d

K P f s D D D s D K P f

r r r r r r

rP rP

dr dr dr dr

This gives at r

K P D D K P

The trace of S S is therefore given by the following formula

Tr S S Tr P D D K P P K P

The next imp ortantstepistochange the order of the op erators under

the trace

D D K P P K P Tr P

Tr P D D K P P P K P

Tr P K P P D D K P P

The exchange is justied by the fact that

P K P P P D S P P

is a pseudo dierential op erator of order with the symb ol equal to the

symbol of P D and hence that it is a b ounded op erator on L Y S jY

Thus wehave

D K P P Tr S S Tr P K P P D

SG SCOTT AND KPWOJCIECHOWSKI GAFA

Now from equation and Prop osition wehave that

Tr S S Tr P D D D K P P

P P

The op erator on the right side of has a smo oth kernel see Corol

lary and so we can again switch the order of op erators

D D D Tr P D D D K P P Tr K P P P

P P P P

Tr K S P P K S P P SP P D D

Tr K S P P SP P D D TrD D D

P P

where wehave used and

Thus wehave

Tr S S TrD D D

P P

This completes the pro of of the following theorem

Theorem With the assumptions of Theorem one has

fln det D ln det D gj

P P r

r r

fln det D ln det D gj

C P C P r

r r

Equality of the Determinants

In this section weprove the main result of the pap er First we need the

following elementary result which allows us to integrate the equality

We refer to N for a more detailed discussion of the top ology of Gr D