EL CAJÓN: VERSATILIDAD Y SÍNTESIS MUSICAL

Of course we can switch to the?_{-product (3.63) corresponding to normal ordering}

by using the transformation (3.65)

ρ=e−i

2ax∂x∂y. (4.45)

The vectorelds are then mapped to the dierential operators by applying (4.34). For the coordinates we get

δy =ρyρ−1 =y−

i

2ax∂x and δx =ρxρ

−1 _=xe−i

2a∂y, (4.46) revealing the derivative nature of multiplication of coordinates from the left. The derivative in they-direction stays undeformed

δ∂y =ρ∂yρ−1 =∂y, (4.47)

the derivative in thex-direction becomes

δ∂x =ρ∂xρ−1 =e i

2a∂y∂_x. (4.48)

We can combine (4.46) and (4.48) to give

δx∂x =ρx∂xρ−1 =ρxρ−1ρ∂xρ−1 =x∂x. (4.49) Note that the deformationδacts trivially onx∂x, as it does commute withρ. For the other vectorelds linear in the coordinates we get from (4.46,4.47,4.48)

δx∂y = x∂ye− i 2a∂y, δy∂y = y∂y− i 2ax∂x∂y, (4.50) δy∂x = y∂xe i 2a∂y − i 2ax∂ 2 xe i 2a∂y.

Chapter 5

Gauge theory on curved NC spaces

One hope associated with the application of noncommutative geometry in physics is a better description of quantized gravity. At least it should be possible to construct eective actions where traces of this unknown theory remain. If one believes that quantum gravity is in a sense a quantum eld theory, then its ob- servables are operators on a Hilbert space and therefore elements of an algebra. Some properties of this algebra should be reected in the noncommutative geom- etry the eective actions are constructed on. As the noncommutativity should be induced by background gravitational elds, the classical limit of the eective actions should reduce to actions on curved spacetimes [75, 29].

In the canonical case, the gauge theory reduces in the commutative limit to a theory on at spacetime. Therefore it is necessary to develop concepts working with more general algebras, since one would expect that curved backgrounds are related to algebras with nonconstant commutation relations. We will use the derivations of?_{-product algebras we studied in chapter 4 to build covariant}

derivatives for noncommutative gauge theory. We will be able to write down a noncommutative action by linking these derivations to a frame eld induced by a nonconstant metric. In the commutative limit, this action reduces to gauge theory on a curved manifold. As an example we will again studyκ-deformed spacetime, where the action reduces in the commutative limit to scalar electrodynamics on a manifold with constant curvature.

We will also introduce Seiberg-Witten maps to do noncommutative gauge theory with arbitrary gauge groups. A proof of the existence of the Seiberg- Witten-map for an Abelian gauge potential will be given for the formality?- product. We will also give explicit formulas for the Weyl ordered?-product up to second order.

5.1 The general formalism

5.1.1 Noncommutative gauge theory

To do gauge theory on the noncommutative spaces equipped with the more com- plicated?-products of chapter 3, we will try to follow the formalism of the canon- ical case as much as possible.

Fields in the fundamental representation will again transform as

δΛΨ = iΛ?Ψ. (5.1)

The commutator of two such gauge transformations should again be a gauge transformation, i.e we again want

(δΛδΞ−δΞδΛ)Ψ =δi[Ξ?_,Λ]Ψ. (5.2) As in the canonical case, this is only possible for gauge groupsU(N)_{. The rst}

dierence to the canonical case occurs when we look at the transformation prop- erties of a derivative

δΛ(∂iΨ) =∂i(iΛ?Ψ) =i(∂iΛ)?Ψ +iΛ?(∂iΨ) +iΛ(∂i?)Ψ. (5.3) The additional term iΛ(∂i?)Ψ is in general no longer zero, corresponding to a nontrivial coproduct of the derivative. If we now want to add a gauge eldAi to the derivative to make it gauge invariant, i.e.

DiΨ =∂iΨ−iAi?Ψ, (5.4)

the transformation properties ofAi also have to oset this new term to get

δΛ(DiΨ) = iΛ? DiΨ. (5.5)

From this we get

δΛ(Ai)?Ψ = ∂iΛ?Ψ +i[Λ?, Ai]?Ψ + Λ(∂i?)Ψ, (5.6) which means that the gauge potential can no longer be a function, it has to be derivative valued. To see this better, we take as an example the?-product (3.64) for theκ-deformed plane. The above formula then reads

δΛ(Ax)?asΨ = (∂xΛ)?as(e− i

2a∂yΨ) (5.7)

+((e2ia∂y−1)Λ)?_as(∂_xΨ) +i[Λ?as, A_x]?_asΨ.

To oset the terms coming from the deformed Leibniz rule for∂x(where additional derivatives act on the right hand side), the gauge eldAx has to become derivative

valued. Gauge theory using such derivative valued gauge elds was constructed in [34, 33, 35], but we will try a dierent approach here.

We saw in chapter 2.3 that there is a dierent formulation for noncommutative gauge theory in terms of covariant coordinates. So let us see what happens if we try to gauge the coordinates with a more complicated?-product. We want to have

δΛ(Xi?Ψ) =δΛ((xi+Aei)?Ψ) =iΛ? Xi?Ψ. (5.8) Therefore the gauge eld_Ae_{i has to transform as}

δΛAei =i[Λ?, xi] +i[Λ?,Aei]. (5.9) This means thatAei is still a function, because the commutator with a coordinate of course has an undeformed Leibniz rule. But there is a problem with this Ansatz: the gauge eldAei vanishes in the commutative limit. In the canonical case, this could be solved by dening a new eld(θ−1₎ij_Ae

j, but this is no longer possible as the now coordinate dependentθ−1 _{would spoil the transformation properties}

of the new object.

This is why we introduced derivationsδX in chapter 4. They do have both an undeformed Leibniz rule and a nonvanishing commutative limit. So we introduce covariant derivations as

DX =δX −iAX, (5.10)

where X is a Poisson vector eld. The gauge eldAX will transform as

δΛAX =δXΛ +i[Λ ?, AX]. (5.11) Then, a eld strengthFX,Y can be dened as

−iFX,Y = [DX ?, DY]−D[X,Y]?, (5.12) the properties ofD and [·, ·]? making sure that the eld strength is function- valued and transforms covariantly1_.

1_{This can also be expressed in the language of the noncommutative forms introduced in}

appendix A.2. AX is the connection one form evaluated on the vector eldX. It transforms

like

δΛA=δΛ +iΛ∧A−iA∧Λ. (5.13)

The covariant derivative of a eld is now

DΨ =δΨ−iA∧Ψ, (5.14) and the eld strength becomes