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Criterion for the Index Theorem on the lattice. Pedro Bicudo Dep Física IST Lisboa. Coimbra, 25-29 September 2002. 0. Introduction. Ginsparg-Wilson relation.

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criterion for the index theorem on the lattice

Criterion for the Index Theorem on the lattice

Pedro Bicudo

Dep Física IST Lisboa

Coimbra, 25-29 September 2002

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide2

0. Introduction

Ginsparg-Wilson relation

From the onset of Lattice theory it was realized that the finite difference Dirac action would suffer from doubling of the fermion spectrum and from the cancellation of the axial anomaly. LaterGinspargand Wilson derived a relation to recover the axial anomaly,

D g5 + g5 D =D g5 RD

where R is proportional to the lattice spacing a. This is not strictly chiral invariant. Recently

Lusherproved that chiral invariance can be recovered in an extended form,

dY= g5 (1- 1 R D) Y , dY= g5 (1- 1 D R) Y

and Hasenfratz Laliena and Niedermayer showed that the Atiyah-Singer Index Theorem is also recovered on the lattice,

n--n+=q , r= 1 tr{g5 RD}

where n-is the number of zero modes of with negative chirality. The Index Teorem on the lattice is confirmed by the Fixed point action and by the overlap action of Neuberger .

2

2

2

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide3

Charge

Indeed1 tr{g5 RD} is a charge density , and it is local if the product RD is local.

Nevertheless the topological charge on the lattice is neither unique nor conserved.

2

Another possible choice for the topological density is the simplest simple gauge definition. In d=2 QED ( the discrete version of the Schwinger model) r is defined with the plaquette,

r1= 1 Arg ( P ) F12

continuum

P(n) = U1(n) U2(n+1) U*1(n+2) U*2(n)

Moreover gauge configurations with

topological charge can be explicitely

constructed with a topological parameter

Q which coincides with the charge Q1

when Q is integer

2 p

0

0

0

0

Ex. Of a 3x3 sub-square

with chargeQ:

2pQ3/9

2pQ6/9

0

-2pQ2/9

-2pQ2/9

-2pQ2/9

0

-2pQ1/9

-2pQ1/9

0

0

0

0

U2

U1

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide4

Question ?

The topological charge on the lattice is neither unique nor conserved, and the discrete version of theDirac action is not unique. In this talk we address the extremum question,

‘’ What is the lattice Dirac action with the best index ? ‘’

Putting it in other words, which is the one with the best topological charge?

Because this is a vast problem, we specialize to Dirac operators constructed with the Wilson action, and to local lattice operators. We will also perform our tests in d=2.

The summary of the study presented here is,

1. Topological properties of the Wilson action

2. The index of the Neuberger overlap action

3. A criterion

4. Conclusion

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide5

1. Topological properties of the Wilson action

Wilson included in the first and simplest lattice version of the Dirac action,

wD(n,n’)=Sigi(Ui(n) dn+i,n’- Ui(n)+dn,n’+i ) + r a Si( 2 dn,n’ - Ui(n) dn+i,n’- Ui(n)+dn,n’+i ) + m I

a 2 a2

a Gauge invariant first derivative , and a Gauge invariant Laplacian

which fixes the doubling problem of the finite difference first derivative.

In the free limit the first derivative has the Fourier Transform,

(dn+i,n’- dn,n’+i ) sin(k)

a

p

0

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide6

A short footage 15 s film of the topology of the Wilson action

showing the

eigenvalues of

the 4x4 Wilson

action as a

function of the

arbitrary

topological

parameter of

the gauge

configuration,

Q= 0 2.

Im l

Re l

UNITS:

a =1

r =1

R=1

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide7

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide8

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide9

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide10

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide11

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide12

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide13

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide14

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide15

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide16

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide17

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide18

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide19

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide20

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide21

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide22

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide23

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide24

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide25

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide26

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide27

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide28

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide29

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide30

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide31

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide32

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide33

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide34

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide35

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide36

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide37

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide38

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide39

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide40

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide41

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide42

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide43

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide44

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide45

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide46

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide47

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide48

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide49

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide50

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide51

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide52

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide53

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide54

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide55

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide56

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide57

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide58

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide59

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide60

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide61

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide62

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide63

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide64

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide65

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide66

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide67

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide68

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide69

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide70

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide71

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide72

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide73

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide74

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide75

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide76

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide77

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide78

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide79

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide80

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide81

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide82

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide83

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide84

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide85

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide86

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide87

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide88

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide89

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide90

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide91

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide92

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide93

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide94

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide95

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide96

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide97

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide98

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide99

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide100

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide101

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide102

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide103

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide104

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide105

Im l

Re l

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide106

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide107

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide108

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide109

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide110

Im l

Re l

End of film

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide111

Re l

Q

This shows the real eigenvalues of the 4x4 Wilson action as a function of the topological parameter Q.

Quadruplets of eigenvalues appear from the complex plane at discrete values of Q.

When the density r is small the at all points of the lattice, the real eigenvalues are close to

1, 2 or 4

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide112

Re c

Q

This shows the chirality c of the real eigenvalues of the 4x4 Wilson action as a function of the topological parameter Q.

v+ g5 v

c =

v+v

When the density r is small at all points of the lattice, the real eigenvalues have a chirality close to

1 or -1

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide113

# l

*signc

Q1

Q

This shows the number of real eigenvalues with

l <1.

of the Wilson action in a 4x4 lattice, compared with the topological charge Q1

When the density r is small at all points of the lattice, # l and Q1 are equal.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide114

General NXN result

Back to the general NXN case we find that for small topological charge densities, that

the Wilson action is close to comply with the Index Theorem. The difference is proportional

to a small number ,

e = | q2 | / N2

Where is the q2 number of small real eigenvalues.

Actually the real eigenvalues and the respective chiralities are,

l = 0 + 3.0 e +o(e 2) , c = [-1 +o(e 2) ] sign (q2)

l = 2 +o(e 2) , c = [+1 -2 .0 e +o(e 2) ] sign (q2)

l = 4 - 3.0 e +o(e 2) , c = [-1 +o(e 2) ] sign (q2)

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide115

2. The index of the Neuberger overlap action

The Neuberger overlap action,

nD = 1 1 + X , X = nD -m0 I , X+ = g5 X g5

R X+X

is a solution of the Ginsparg-Wilson relation. Moreover it produces the trace,

tr { nD} = 1 tr g5 X

R |g5 X |

= 1S sign( l’i )

R i

where l’iare the eigenvalues of g5 X , which is hermitean .

Now let us start at a low parameter m0 and increase it continuously. Whenever g5 X passes by a root, l’i changes sign, and the charge R tr { nD}/2 has a discrete step of +1.

These steps occur at the zero modes of X, and this coincides with m0=li, an eigenvalue of the Wilson action.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide116

q3

m0

This shows the index of the Neuberger Action when we run the parameter

m0= -1 5

This checks

that the index jumps precisely

at the location

of the real eigenvalues

of the Wilson

action.

...

Position of

Wilson real

eigenvalues

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide117

Im l,l’

Re l,l ’

This figure

shows an

arbitrary

interpolation

between the

eigenvalues of

the Neuberger

action and the

eigenvalues

of the Wilson

action in a 4x4

lattice for Q=1.

The Neuberger

Action projects

the eigenvalues

of the Wilson

action on the

trigonometric circle.

In particular the real eingenvalues are projected on +1.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide118

3. A criterion

For low density the real eigenvalues of the Wilson action wD have chirality ~ +1. This implies that the corresponding eigenvectors of wD are very close to eigenvectors of wD+ .

It is then possible to derive the criterion,

‘’ A hermitean solution of the Ginsparg-Wilson relation wgrD with constant R and constructed with the Wilson action wD , and wD+, complies maximally with the Index Theorem if and only if

wgrD2q( l- l0 ) (q is the Heaviside step function)

R

when we replace wD,wD+l where l is a real number 0< l < 4 and where l0 belongs to a narrow subinterval of ]0,2[ ‘’.

In particular for the d=2 lattice Schwinger model 1.0< l0 < 1.2

The Neuberger action is the simplest one that produces the desired step function, therefore it is the solution that maximises our criterion.

2

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide119

4. Conclusion

.The Wilson action approximately complies with the Index Theorem.

.The Neuberger overlap action with parameter m0=1.0 to 1.2 is the action with the best index, among the actions constructed with the Wilson action.

.This should not depend on the dimension, nevertheless I ought to extend that to d=4, with a better computer.

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide120

References

Atiyah Singer AnMa87, 485;546;531 (1968)

Wilson PRD10, 2445 (1974)

Nielsen Ninomiya NPB185, 20 (1981); NPB193, 173 (1981)

Ginsparg Wilson PRD25, 2649 (1982)

Neuberger PLB417, 141 (1998); PLB427, 353 (1998)

Hasenfratz laliena Niedermayer PLB427, 125 (1998)

Luscher NPB538, 515 (1999)

Luscher NPB428, 342 (1998)

Chiu PRD58,074511 (1998)

The end

Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)

slide121
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide122
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide123
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide124
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide125
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide126
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide127
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide128
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide129
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)
slide130
Wilson PRD10, 2445 (1974); Ginsparg Wilson PRD25, 2649 (1982); Neuberger PLB417, 141 (1998), Hasenfratz laliena Niedermayer PLB427, 125 (1998)