Inverse Seiberg-Witten-map

clearly fullls the rst property due to the properties of the trace. Note that symmetrization is only necessary for space dimension bigger than 4 due to the cyclicity of the trace. In dimensions 2 and 4 we may replacestrby the ordinary trace tr. F_ij(k) is the Fourier transform of a closed form if

k_[iF_jk]= 0 (6.20)

or if the current

J^{i1···i2n−2} =strF ,Xˆ

³Fˆ^[i1i2· · ·Fˆ^{i2n−3i2n−2]}e^ikjFˆj

´ (6.21)

is conserved, respectively

k_iJ^i···= 0. (6.22)

This is easy to show, if one uses strF ,ˆXˆ

³

[k_iXˆⁱ,Xˆ^l]e^ikjXˆj· · ·

´ (6.23)

=strF ,ˆXˆ

³

[ ˆX^l, e^ikjXˆj]· · ·

´

=strF ,ˆXˆ

³

e^ikjXˆj[ ˆX^l,· · ·]

´ , which can be calculated by simple algebra.

To prove thatF has the right commutative limit, we have to switch to the

?-product formalism and expand the formula inθ^ij. The expression (6.19) now becomes

F[X]_ij(k) =

Z d²ⁿx P f(θ)

³

(F_?ⁿ⁻¹)_ij ? e^ik?^jXj

´

sym F,X. (6.24)

The expression in brackets has to be symmetrized inF^ij and Xⁱ for n > 2. Up to second order inθ^ij, the commutatorF^ij of two covariant coordinates is

F^ij =−i[X^{i ?}, X^j] =θ^ij −θ^ikf_klθ^lj−θ^kl∂_lθ^ija_k+O(3) (6.25) with f_ij =∂_ia_j −∂_ja_i the ordinary eld strength. Furthermore we have

e^ik_?^iXi =e^ikixi(1 +ik_iθ^ija_j) +O(2). (6.26) If we choose an antisymmetric?-product, the symmetrization will annihilate all the rst order terms of the?-products between theF^ij andXⁱ, and therefore we get

−F[X]_ij(k) (6.27)

=−2n

Z d²ⁿx

²θⁿ

³

²_ijθⁿ⁻¹−(n−1)²_ijθⁿ⁻²θf θ

−θ^kl∂_l(²_ijθⁿ⁻¹)a_k

´

e^ikixi+O(1)

=−2n

Z d²ⁿx

²θⁿ

³

²_ijθⁿ⁻¹−(n−1)²_ijθⁿ⁻²θf θ

−1

2²_ijθⁿ⁻¹f_klθ^kl

´

e^ikixi +O(1)

=d²ⁿx

³

θ⁻¹_ij + 2n(n−1)²_ijθⁿ⁻²θf θ

²θⁿ

−1

2θ_ij⁻¹fklθ^kl

´

e^ikixi +O(1),

using partial integration and∂_i(²θⁿθ^ij) = 0. To simplify notation we introduced

²_ijθⁿ⁻¹ =²_{iji1j1···in−1jn−1}θ^i1j1· · ·θ^in−1jn−1etc. (6.28) In the last line we have used

θ_ij⁻¹ =−(θⁿ⁻¹)_ij

P f(θ) =−2n²_ijθⁿ⁻¹

²θⁿ . (6.29)

We will now have a closer look at the second term, noting that θ^ij²_ijθⁿ⁻²θf θ

²θⁿ =− 1

2nθ_kl⁻¹θ^krf_rsθ^sl =− 1

2nf_rsθ^rs (6.30) and therefore

²_ijθⁿ⁻²θf θ

²θⁿ =a²_ijθⁿ⁻¹

²θⁿ frsθ^rs+bfij (6.31) with a+b=−_2n¹ . Taking e. g. i= 1, j = 2 we see that

²_12···klθⁿ⁻²θ^krf_rsθ^sl = ²_12···klθⁿ⁻²(θ^k1θ^2l−θ^k2θ^1l)f₁₂ (6.32) +terms withoutf₁₂.

Especially there are no terms involvingf12θ¹² and we get for the two terms on the right hand side of (6.31)

2a²12θⁿ⁻¹f12θ¹² =−2nb²12θ¹²θⁿ⁻¹f12 (6.33) and thereforeb=−_n^a. This has the solution

a=− 1

2(n−1) and b= 1

2n(n−1). (6.34)

With the resulting

2n(n−1)²_ijθⁿ⁻²θf θ

²θⁿ = 1

2θ⁻¹_ij fklθ^kl+fij (6.35)

we nally get

−F[X]ij(k) = Z

d²ⁿx¡

θ⁻¹_ij +fij

¢e^ikixi+O(1). (6.36) Therefore

f[X]_ij =F[X]_ij(k)− F[x]_ij(k) (6.37) is a closed form that reduces in the commutative limit to the commutative Abelian eld strength. We have found an expression for the inverse Seiberg-Witten map.

Part II

Matrix model approach

Though?-products are a convenient tool for studying noncommutativity, their strength lies mainly in the perturbative regime. For other purposes, especially nonperturbative ones, a dierent approach using a dierent representation of the algebra of functions on noncommutative space is better suited.

If we take the simple example of a noncommutative plane with canonical noncommutativity

[x, y] =iθ, (6.38)

we see immediately that this is nothing but the Heisenberg algebra, for which we can use the well known Fock-space representation. In2n dimensions, we can usen such pairs of coordinates which upon complexication become creation and annihilation operators on the Fock-space. Using this approach, it was possible to study many nonperturbative features of noncommutative eld theory such as solitons and instantons (see e.g. [37] for references).

We will call this approach matrix model approach, as the gauge theory can be described as a matrix model having the noncommutative space as its ground state, the uctuations creating the gauge theory. But noncommutative spacetime with canonical commutation relations has to be represented on an innite-dimensional vectorspace, leading to a number of problems. First of all, there are the well known divergencies of noncommutative gauge theory. Then, the rank of the gauge group can't be xed in this model [37]. Therefore we are looking for spaces that can be represented as nite-dimensional matrix algebras, where everything is well dened. The space on which we will base our constructions will be the fuzzy sphere [73], anN-dimensional matrix algebra corresponding to a truncation of the spherical harmonics on the sphere at angular momentumN −1. To go to 4 dimensions, we will use the product of two such fuzzy spheresS_N² ×S_N², generated by N²-dimensional matrices. In one limit, this fuzzy space goes over to the product of two commutative spheres, but in a dierent limit, it also goes to noncommutativeR⁴ with canonical commutation relations. Our interest will therefore be twofold: On one hand we will study gauge theory on this fuzzy space as the deformation of commutative gauge theory, on the other hand as a regularization of gauge theory onR⁴_θ.

Chapter 7

The canonical case

Before we study gauge theory on a nite-dimensional fuzzy space, we rst want to present the usual matrix model approach to noncommutative gauge theory on R⁴_θ. After a quick look at the innite-dimensional Fock-space representation of R⁴_θ, we will show how gauge theory can be formulated as a matrix model with ground stateR⁴_θ. The uctuations around this ground state will create the gauge theory. Finally we will have a look at a certain class of instantons, the so called uxon solutions.

7.1 The Heisenberg algebra

In two dimensions, the coordinate algebra with canonical deformation

[x, y] = iθ (7.1)

is nothing but the well known Heisenberg algebra. But now the noncommuta-tivity isn't between the coordinates and momenta, but between the coordinates themselves. Of course we can use the usual Fock space representation for this algebra by rst dening

x_±:=x±iy (7.2)

with

[x₊, x₋] = 2θ. (7.3)

The Fock space is given by

H ={|ni, n²N₀}, (7.4) where the creator and annihilator operators act as

x₋|ni=√ 2θ√

n+ 1|n+ 1i, x₊|ni=√ 2θ√

n|n−1i. (7.5)

This can be generalized to higher dimensions. Any2n-dimensional algebra with canonical commutation relations can by suitable rotations be brought into a form where it consists ofn pairs of noncommuting variables (7.1). As we will mostly be concerned with the 4-dimensional case in the following, we will present it here in more detail.

The most general noncommutativeR⁴_θ is generated by coordinates subject to the commutation relations

[x_µ, x_ν] =iθ_µν, (7.6)

where µ, ν ∈ {1, . . . ,4}. Using suitable rotations,θ_µν can always be cast into the form

θ_µν =







0 θ12 0 0

−θ₁₂ 0 0 0

0 0 0 θ₃₄

0 0 −θ34 0





 . (7.7)

To simplify the following formulas, we restrict our discussion from now on to the selfdual case

θµν = 1

2εµνρσθρσ (7.8)

and denote

θ:=θ₁₂=θ₃₄; (7.9)

the generalizations to the antiselfdual and the general case are obvious. In terms of the complex coordinates

x_±L:=x₁±ix₂ , x_±R :=x₃±ix₄, (7.10) the commutation relations (7.6) take the form

[x_+a, x_−b] = 2θδ_ab, [x_+a, x_+b] = [x_−a, x_−b] = 0, (7.11) where a, b∈ {L, R}. The Fock-space representationH of (7.11) has the standard basis

H={|n1, n2i, n1, n2 ∈N0}, (7.12) with

x_−L|n₁, n₂i=√ 2θ√

n₁+1|n₁+1, n₂i, x_+L|n₁, n₂i=√ 2θ√

n₁|n₁−1, n₂i x_−R|n₁, n₂i=√

2θ√

n₂+1|n₁, n₂+1i, x_+R|n₁, n₂i=√ 2θ√

n₂|n₁, n₂−1i. (7.13) The derivatives on this space are inner, i.e. they are produced by the commutator with a coordinate

−iθ^µν∂ν = [b ·, x^µ], (7.14) just as in the?-product formalism.

Im Dokument Noncommutative Gauge Theory beyond the Canonical Case (Seite 79-89)