Fermions

To introduce spinors on fuzzyS² ×S², we will rst have to have a look at the commutative case. There, we will calculate the Dirac operator and bring it into a form which is more suitable for the fuzzy case. The formulation of fuzzy gauge theory using the SO(6)-Cliord algebra will proove very usefull, and the fuzzy Dirac operator will be a simple generalization of the commutative one. But this Dirac operator (because it is based onSO(6) instead ofSO(3)×SO(3)) will be reducible, which is why we will have to introduce projectors onto the physical Dirac fermions. Chirality can be introduced either using the chirality operator inherited fromSO(6) or using a Ginsparg-Wilson system.

9.4.1 The commutative Dirac operator on S

²

× S

²

To nd a form of the commutative Dirac operator onS²×S² which is suitable for the fuzzy case, one can generalize the approach of [51] forS², which is carried out in detail in Appendix D.3.3: One can write the atSO(6) Dirac operator D6 in 2 dierent forms, using the usual at Euclidean coordinates and also using the spherical coordinates of the spheres. Then one can relateD₆ with the curved four-dimensional Dirac operatorD₄ onS²×S² in the same spherical coordinates.

This leads to an explicit expression forD₄ involving only the angular momen-tum generators, which is easy to generalize to the fuzzy case. In terms of these tangential derivativesJµ,the result becomes the simple expression

D₄ = Γ^µJ_µ+

µ 0 1 1 0

¶ +i

µ 0 1

−1 0

¶

= Γ^µJ_µ+ 2C₀, (9.43) which is clearly aSO(3)×SO(3)-covariant Hermitian rst-oder dierential oper-ator. HereΓ^µ generate theSO(6) Cliord algebra (9.22),C₀ is dened in (9.25), and we putR = 1 for simplicity here. However this Dirac operator is reducible, acting on 8-dimensional spinorsΨ8 corresponding to theSO(6) Cliord algebra.

HenceΨ₈ should be a combination of two independent 4-component Dirac spinors on the 4-dimensional spaceS²×S². To see this, we will construct explicit pro-jectors projecting onto these 4-dimensional spinors, and identify the appropriate 4-dimensional chirality operators. This will provide us with the desired physical Dirac or Weyl fermions.

9.4.2 Chirality and projections for the spinors

There are 3 obvious operators which anti-commute withD₄. One is the usual 6-dimensional chirality operator

Γ :=iΓ^L₁Γ^L₂Γ^L₃Γ^R₁Γ^R₂Γ^R₃ =

µ −1 0 0 1

¶

, (9.44)

which satises

{D₄,Γ}= 0, Γ^† = Γ, Γ² = 1. (9.45) The 8-component spinorsΨ₈split accordingly into two 4-component spinorsΨ₈ = µ ψ_α

ψ_β

¶

, which transform as 4 resp. 4 under so(6) ∼= su(4); recall the related discussion in chapter 9.2. The other operators of interest are

χL = Γ^iLxiL and χR= Γ^iRxiR. (9.46) They preserveSO(3)×SO(3) ⊂SO(6), and satisfy

{D₄, χ_L,R}= 0 ={χ_L, χ_R} (9.47) as well as

χ²_L,R = 1. (9.48)

We will also use

χ= 1

√2 Γ^µx_µ = 1

√2 (χ_L+χ_R) (9.49)

which satises similar relations. This means that P_± = 1

2(1±iχ_Lχ_R) (9.50)

with

P_±² =P_±, P₊+P₋= 1 and P₊P₋ = 0 (9.51) are Hermitian projectors commuting with the Dirac operator onS²×S² as well as with Γ,

P_±^† =P_± and [P_±, D₄] = [P_±,Γ] = 0. (9.52) Therefore they project onto subspaces which are preserved byD₄ and Γ. Hence the spinor Lagrangian can be written as

Ψ^†₈D₄Ψ₈ = Ψ^†₊D₄Ψ₊+ Ψ^†₋D₄Ψ₋ (9.53) involving two Dirac spinorsΨ± =P±Ψ8. In order to get one 4-component Dirac spinor, we can e.g. impose the constraint

P+Ψ8 = Ψ8, (9.54)

or equivalently give one of the two components a large mass, by adding a term

MΨ^†₈P₋Ψ₈ (9.55)

to the action with M → ∞. The physical chirality operator is now identied using (9.52) and (9.45) asΓacting onΨ₊. It can be used to dene 2-component Weyl spinors onS²×S².

To make the above more explicit, consider the a pole of the spheres, i.e.

x_L=



 1 0 0



 and x_R =



 1 0 0



. (9.56)

In the basis (9.21) for the Cliord algebra we then get explicitly P_±= 1

2(1±i

µ −γ_L¹γ_R¹ 0 0 γ_L¹γ_R¹

¶ ) = 1

2(1±σ₃⊗σ₃⊗σ₃). (9.57) This means that

P+ = diag(1,0,0,1,0,1,1,0) (9.58) projects onto a 4-dimensional subspace exactly as expected.

9.4.3 Gauged fuzzy Dirac and chirality operators

To nd a fuzzy analogue of the Dirac operator (9.43) coupled to the gauge elds, we recall the connection between the gauge theory onS² ×S² and the SO(6) Gamma matrices established in chapter 9.2. In the spirit of that chapter a natural fuzzy spinor action would involve

Ψ^†CΨ, (9.59)

where Ψ is now a 8N × N-matrix (with Grassman entries). Of course, (9.59) does not have the appropriate commutative limit, but we can splitC into a fuzzy Dirac operatorDb and the operatorχbdened by

b χΨ =

√2

N (Γ^µΨλ_µ−C₀Ψ), (9.60)

which generalizes (9.49); here we used the denition (9.24,9.25) ofC₀. This operator satises

b

χ² = 1, (9.61)

and reduces to (9.49) in the commutative limit. Note also thatχb commutes with gauge transformations, since the coordinatesλ_µ are acting from the right in (9.60). Setting

Jb_µΨ = [λ_µ,Ψ], (9.62) we get for the fuzzy Dirac operator

Db =C− N

√2 χb= Γ^µ(Jb_µ+A_µ) + 2C₀ = Γ^µD_µ+ 2C₀. (9.63) Here³

Db_µ:=Jb_µ+A_µ (9.64)

is a covariant derivative operator, i.e. Db_µψ → UDb_µψ which is easily veried using (9.6). ThisDb clearly has the correct commutative limit (9.43) for vanishing A, and the gauge elds are coupled correctly. In particular, this denition of Db applies also to the topologically non-trivial solutions of chapter 9.5 without any modications. Moreover, the chirality operatorΓ as dened in (9.44) anti-commutes withDb also in the fuzzy case,

{D,b Γ}= 0. (9.65)

3We setR= 1 in this chapter for simplicity.

Furthermore, using some identities given at the beginning of chapter 9.2 we obtain for Db²ψ:

Db²ψ = (Σ^µνF_µν+Db_µDb_µ+{Γ^µ, C₀}Db_µ+ 2)ψ (9.66)

=: (Σ^µνFµν+¤b + 2)ψ,

dening the covariant 4-dimensional Laplacian¤b acting on the spinors. This corresponds to the usual expression forDb² on curved spaces, and the constant 2 is due to the curvature scalar. SinceDb² and Σ^µνFµν are both Hermitian and commute withΓ andPb± as dened in (9.69) in the largeN limit, it follows that

¤b satises these properties as well.

9.4.4 Projections for the fuzzy spinors

For the fuzzy case, we can again consider the following operators b

χLΨ = 2

N(Γ^iLΨλiL+C₀^LΨ), (9.67) b

χRΨ = 2

N(Γ^iRΨλiR+C₀^RΨ) which satisfy

b

χ²_L,R= 1, {bχ_L,χb_R}= 0. (9.68) This implies(bχ_Lχb_R)² =−1, and we can write down the projection operators

Pb_± = 1

2(1±ibχ_Lχb_R) (9.69) which have the commutative limit (9.50) and the properties (9.51). However, the projector no longer commutes with the fuzzy Dirac operator (9.63):

[D,b χb_Lχb_R] = {D,b χb_L}bχ_R−χb_L{D,b χb_R} (9.70)

= −2 N

³¡2(λ_iL+A_iL)Jb_iL−2A_iLλ_iL+ 2C₀^LΓ^iLDb_iL+ 1¢ b χ_R

−bχ_L¡

2(λ_iR+A_iR)Jb_iR−2A_iRλ_iR+ 2C₀^R Γ^iRDb_iR+ 1¢´

, which only vanishes forN → ∞ and tangentialA_µ (9.9). To reduce the degrees of freedom to one Dirac 4-spinor, we should therefore add a mass term

MΨ^†₈Pb−Ψ8 (9.71)

which for M → ∞ suppresses one of the spinors, rather than impose an exact constraint as in (9.54). This is gauge invariant sincePb± commutes with gauge transformations,

Pb_±ψ →UPb_±ψ . (9.72)

The complete action for a Dirac fermion on fuzzyS_N² ×S_N² is therefore given by SDirac =

Z

Ψ^†₈(Db+m)Ψ8+MΨ^†₈Pb−Ψ8 (9.73) with M → ∞. The physical chirality operator is given byΓ (9.44), which allows to consider Weyl spinors as well.

9.4.5 The Ginsparg-Wilson relations

There is an alternative approach to introduce chirality on fuzzy spaces, using the Ginsparg-Wilson relations. These were initially designed to study chiral fermions on the lattice [43], but they proved to be applicable to fuzzy fermions as well [9, 10]. On the fuzzy sphere, the Dirac and the chirality operator can be cast into a form in which they fulll these relations. This makes it possible to study issues such as topological properties and index theory [6, 101]. We will see that the same relations can be formulated for our model, too.

A Ginsparg-Wilson system consists of two involutionsΓ andΓ⁰, i.e.

Γ² = 1 ; Γ^†= Γ and Γ⁰² = 1 ; Γ^0† = Γ⁰. (9.74) In our case, these two involutions are dened as two dierent noncommutative versions of chirality, one acting from the left, the other one acting from the right

ΓΨ =

√2

N (Γ^µλ_µ+C₀)Ψ, (9.75)

Γ⁰Ψ =

√2

N (Γ^µΨλµ−C0Ψ). (9.76) We recognizeΓ⁰ as the fuzzy operator (9.60). But alsoΓ has the commutative operator (9.49) as its limit.

In the Ginsparg-Wilson system, the Dirac operator was initially dened to be d= 1

aΓ(Γ−Γ⁰), (9.77)

where a is the lattice spacing, but here we will choose D= N

2

√2(Γ−Γ⁰), (9.78)

as this reproduces our fuzzy Dirac operator (9.63) (with gauge elds switched o). We can now dene an alternative chirality operator

χ= 1

2(Γ + Γ⁰). (9.79)

It fullls

{D, χ} = 0, (9.80)

2N²χ²+D² = 2N².

Thereforeχexactly anticommutes withD, but it vanishes on the top modes ofD, i.e. for |D|=√

2N. But at least for every eigenstateΨ_E with positive eigenvalue E <√

2N

DΨ_E =EΨ, (9.81)

the ungauged fuzzy Dirac operator has also an eigenstateΨ_−E =χΨ_E with the negative eigenvalue−E because of

DΨ_−E =DχΨ =−χDΨ =−χEΨ = −EΨ_−E. (9.82) This can be used [6] to derive the following index theorem forD

Ind(D) = n₊−n₋ =T r(χ). (9.83) To include gauge elds, we can write

Γ_A=

√2

N (Γ^µ(λ_µ+A_µ) +C₀) =

√2

N C. (9.84)

With

D_A = N 2

√2(Γ_A−Γ⁰), (9.85)

χ_A = 1

2(Γ_A+ Γ⁰) (9.86)

we now get

{D, χ} = N 2

√2(Γ²_A−1) (9.87)

= N

2

√2( 2

N²(B_µB_µ+1

2 + Σ^µν₈ F_µν)−1)

=

√2

N (B_µB_µ+N²−1

2 + Σ^µν₈ F_µν),

which corresponds exactly to the result of [101] for the fuzzy sphere. Other results of [101] are therefore expected to hold in our case, too.

Alternatively, the gauge elds could also be introduced in a way that is closer to the Ginsparg-Wilson setting by normalizingΓ_A.

Im Dokument Noncommutative Gauge Theory beyond the Canonical Case (Seite 104-111)