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)

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)

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)

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)

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)

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)

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

Criterion for the Index Theorem on the lattice

Criterion for the Index Theorem on the lattice

Presentation Transcript

The Problem of the Criterion

Classical Antiferromagnets On The Pyrochlore Lattice

The Index for Inclusion

Chiral Magnetic Effect on the Lattice

Anomalous transport on the lattice

Hosotani mechanism on the lattice

A nomalous transport on the lattice

Baryons (and Mesons) on the Lattice

Hadron Physics on the Lattice

Rayleigh Criterion Contents: Diffraction patterns The Rayleigh criterion

Why Extra Dimensions on the Lattice?

Quarkonium correlators on the lattice

Nuclear Physics on the Lattice

Lattice QCD and flavour physics (II): Impact of lattice on the

Pentaquarks on the Lattice

Near Light Cone QCD On The Lattice

Atiyah-Singer Index Theorem

Searching for light scalar tetraquarks on the lattice

The Kelly Criterion

Criterion D: Reflecting on the impacts of science