N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions
N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions
N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Chapter 1<br />
Gauging the Central Charge<br />
There exist basically two approaches to theories <strong>with</strong> N = 2 supersymmetry. While<br />
<strong>with</strong>out doubt the more sophisticated harmonic superspace [1] offers some advantages<br />
over ordinary superspace, in this thesis we shall nevertheless employ the latter only,<br />
which makes it easier to switch back and forth between superfields and components.<br />
For a treatment of theories <strong>with</strong> gauged central charge <strong>with</strong>in the framework of harmonic<br />
superspace we refer to [2], where several results presented here have already been<br />
published.<br />
The reader might want to have a look at the appendix first to become acquainted <strong>with</strong><br />
our conventions concerning Lorentz and spinor indices.<br />
1.1 The <strong>N=2</strong> Supersymmetry Algebra<br />
Extended supersymmetry algebras in four spacetime dimensions involve in addition<br />
to the Poincaré generators Pµ and Mµν two-component Weyl spinor charges Q i α and<br />
their hermitian conjugates Q †<br />
˙αi<br />
, which are Grassmann-odd and generate supersymmetry<br />
transformations. The index i belongs to a representation of an internal symmetry group<br />
and runs from 1 to some number N that counts the supersymmetries. In [3] Haag et al.<br />
have determined the most general supersymmetry algebra compatible <strong>with</strong> reasonable<br />
requirements on relativistic quantum field theories. It contains an invariant subalgebra<br />
that is spanned by the generators of translations and supersymmetry transformations,<br />
and for N > 1 additional bosonic generators, denoted by Z ij , may also occur. These<br />
must commute <strong>with</strong> every element of the supersymmetry algebra and for this reason<br />
are called central charges. The odd part of the subalgebra reads<br />
{Q i α , Q †<br />
˙αj } = δi jσ µ<br />
α ˙α Pµ , {Q i α , Q j<br />
β } = εαβZ ij , {Q †<br />
˙αi , Q† ˙βj } = ε ˙α ˙ βZ† ij<br />
, (1.1)<br />
while all commutators vanish. It is evident that the central charges Z ij must be antisymmetric<br />
in the pair ij. For N = 2 this implies that there are at most two hermitian<br />
central charges,<br />
N = 2 ⇒ Z ij = ε ij (Z1 + iZ2) , Z †<br />
ij = −εij(Z1 − iZ2) . (1.2)<br />
When central charges are absent the above algebra is, among others, invariant under<br />
unitary transformations<br />
(Q i α) ′ = U i jQ j α , (Q †<br />
˙αi )′ = U ∗ i j Q †<br />
˙αj , P ′ µ = P µ , U ∈ U(N) . (1.3)<br />
3