This thesis is primarily concerned with the study of five-dimensional N = 2 super- gravity theories coupled to supersymmetric matter multiplets. Such theories, apart from being interesting in their own right, can give us insights into the non-perturbative structure of string and M-theory through study of the various solitonic objects which they admit.
In this section we first introduce the relevant field content for studying supersym- metric field theories in five dimensions, before analysing the action describing an ar- bitrary number of vector multiplets coupled to supergravity. This action will be our starting point for much of the work presented in this thesis.
3.2.1 The field content
We start with the fermionic content [28]. In five dimensions (assuming Lorentzian space- time signature) the minimal spinor representations are symplectic Majorana spinorsλi, withi= 1,2, which transform as a doublet under theSU(2) R-symmetry group of the five-dimensionalN = 2 superalgebra. Introducing the totally antisymmetric tensorij,
the symplectic Majorana condition just tells us that the spinors λi satisfy the reality condition [28]
(λi)∗ =−Bλj ij,
whereB is the charge conjugation matrix.
Counting the degrees of freedom, we see that the minimal spinor in five dimensions has 8 real components. Hence, the “N = 2” theory, which admits 8 real supercharges, is really the minimal allowed in five dimensions.
We now turn to the on-shell massless multiplets of theN = 2 theory. The ones we will need for our present purposes are the N = 2 vector and gravity multiplets.
The N = 2 vector multiplet in five dimensions consists of a U(1) gauge field Aµˆ, a real scalar φ, and an SU(2) doublet of symplectic Majorana spinors λi. Counting the on-shell degrees of freedom, we find 3 + 1 = 4 on the bosonic side, and 4 on the fermionic side, which matches as required. For the off-shell multiplets, we would need to add a further auxiliary bosonic fieldYij =Yji transforming as a triplet ofSU(2) [28]. However, for our purposes the on-shell multiplets will suffice.
The N = 2 gravity multiplet in five dimensions consists of the vielbein emˆ ˆ
µ, a
gauge fieldAµˆ called the ‘graviphoton’, and anSU(2) doublet of symplectic Majorana gravitiniψµiˆ. Again, counting the on-shell degrees of freedom we find 3 + 5 = 8 on the bosonic side and 2×4 = 8 on the fermionic side.
For the matter-coupled supergravity theories that we will be interested in we want to take n(5)V copies of the vector multiplets and a single copy of the gravity multiplet. Indexing the vector multiplets byx= 1, . . . , n(5)V , our full field content is
(emµˆ, ψµiˆ,Aµˆ, Axµˆ, λi|x, φx).
At this stage we should point out a bit of a sleight-of-hand. The φx, it will turn out, parametrise an n(5)V -dimensional manifold H, which is not necessarily flat. Therefore the spinorsλi|xshould really be vectorsλi|awitha= 1, . . . , n(5)V a flatSO(n(5)V ) tangent space index introduced via the vielbeins fa
x of H [30]. However, since we will only be
refer the reader to the appropriate places in the literature for a full treatment.
The most general supersymmetry transformation rules for this field content are given in equation (2.6) of [30], and we do not repeat them here. Taking them as granted we now turn our attention to constructing a five-dimensional action describing the dynamics of this field content, which should be gauge-invariant and supersymmetric.
3.2.2 The five-dimensional action
The action describing the coupling ofn(5)V five-dimensional N = 2 vector multiplets to supergravity was first analysed in [30]. We here concentrate only on the bosonic sector, since this will be all we need to construct black brane solutions, and refer to [30] for the fermionic completion. Using the conventions of [32], the five-dimensional action is given by S5 = Z d5x " p ˆ g Rˆ 2 − 3 4gxy(φ)∂µˆφ x∂µˆφy−1 4aij(h)F i ˆ µˆνFj|µˆˆν ! + 1 6√6cijkε ˆ µνˆρˆσˆλˆFi ˆ µˆνF j ˆ ρˆσA k ˆ λ . (3.10)
Here x = 1, . . . , n(5)V labels the vector multiplet scalars, while i = 1, . . . , n(5)V + 1 la- bels1 the gauge fields: n(5)V from the vector multiplets, and one (the graviphoton) from the supergravity multiplet. Our convention is to use ‘hats’ for the five-dimensional spacetime indices ˆµ= 0, . . . ,4.
In [30] the authors showed that requirements of gauge invariance (which imposes that the coefficients cijk be constant) and supersymmetry restricts the scalar target
space of the five-dimensional theory to be an n(5)V -dimensional projective special real (PSR) manifold, as described in Section 2.1.4. That is, the scalar fieldsφx parametrise the hypersurface H(h) = 1 within a conic affine special real (CASR) manifold with Hesse potential
H(h) =cijkhihjhk.
The gauge coupling matrixaij(h) is then given by the components (2.13) of the tensor
fieldaon the CASR manifold, while gxy(φ) =aij(h) ∂hi ∂φx ∂hj ∂φx,
defines the metric on the PSR manifold. Hence, the full dynamics of the Lagrangian (3.10) is determined once we specify the number of vector multiplets, n(5)V , and the coefficients cijk.
The five-dimensional matter-coupled supergravity theories described (for the vector multiplet sector) by (3.10) can be obtained from 11-dimensional supergravity [91] by compactification on a Calabi-Yau three-fold X [92]. In this description, the Hodge numbers (h1,1, h2,1) of the Calabi-Yau determine the number of vector and hypermulti- plets present in the five-dimensional theory, while the coefficientscijk are given by the
intersection matrix
cijk ≡
Z
X
Vi∧Vj ∧Vk,
where i, j, k = 1, . . . , h1,1 and Vi is a basis of the cohomology H1,1(X). We will come
to the structure of the hypermultiplet sector in Section 3.4.