Manifolds of generalised G-structures in string compactifications

Abstract

A G-structure on a differentiable manifold M of dimension n can be described as a reduction of the linear frame bundle L(M) of M to a Lie subgroup G of $GL(n,\mathbb{R})$. Such a reduction is equivalent to the existence of certain geometric structures on M, depending on what the subgroup G is. For example, an O(n)-structure corresponds to the existence of a Riemannian metric g. Similarly, by the existence of an almost complex structure J, the structure group reduces to $GL(n/2,\mathbb{C})$. If a Riemannian metric and an almost complex structure are compatible and the metric is hermitian then the structure group reduces to SU(n/2). In a similar fashion, a generalized G-structure can be described as a reduction of the structure group of the principal bundle associated with the generalized tangent bundle $TM\oplus T^*M$. The natural structure group of $TM\oplus T^*M$ is O(n,n). The generalized G-structures also correspond to the existence of certain geometrical objects. For example, the reduction of the structure group from O(n,n) to $O(n)\times O(n)$ corresponds to the existence of a generalized metric. Similarly, on an even-dimensional real manifold $M$ a generalized almost complex structure is given by a reduction of the structure group from O(n,n) to U(n/2,n/2). A generalized almost complex structure is defined by the existence of a pure spinor which is a section of the exterior bundle $\bigwedge^\bullet T^* M$. The SU(n/2,n/2)-structure is equivalent to the existence of a globally defined pure spinor of non-vanishing norm. Furthermore, $SU(n/2)\times SU(n/2)$-structure is given by the existence of two compatible pure spinors. The main theme of this thesis is the study of manifolds of generalized G-structure relevant to string compactifications. Superstring theory is a quantum theory of gravity consistent in 10 dimensions. There are five consistent superstring theories and the low energy dynamics of massless space-time fields are governed by ten-dimensional supergravity theories. The supergravity field equations are nonlinear partial differential equations that can be regarded as a generalization of field equations of Einstein's theory of general relativity (GR). In a supersymmetric compactification of Type II string theory down to 4 dimensions, it is required that the structure group of the generalized tangent bundle $TM \oplus T^*M$ of the six-dimensional internal manifold M is reduced from SO(6,6) to $SU(3) \times SU(3)$. This is equivalent to the existence of two globally defined compatible pure spinors $\Phi_1$ and $\Phi_2$. Furthermore, these pure spinors should satisfy certain first-order differential equations, namely supersymmetry equations. We show that these equations are covariant under certain Pin(d,d) transformations. We also show that Non-Abelian T-duality (NATD) which is generated by a coordinate-dependent Pin(d,d) transformation is a particular solution generating transformation for these pure spinor equations. Our method is demonstrated by studying the NATD of a specific class of geometries with SU(2) isometry and SU(3)-structure. Some of the manifolds belonging to this class are $AdS_5\times T^{1,1}$, $AdS_5\times Y^{p,q}$ and $AdS_5\times S^5$. It is interesting to note that in each case, the internal manifold is a Sasaki-Einstein manifold. We show that the transformed pure spinors are associated with an SU(2)-structure. The plan of the thesis is as follows: in section 2, we study principal fiber bundles, vector bundles, and linear frame bundles. Then, we study the concept of the reduction of the structure groups. We also give familiar examples of G-structures in detail. In section 3, we briefly review the relation between G-holonomy and torsion-free G-structures. In section 4, we study the basic concepts regarding the geometry of the generalized tangent bundle $TM\oplus T^*M$. This leads us to the definition of a generalized G-structure. Since our main interest is in $SU(3)\times SU(3)$-structures we give in a separate subsection the description of $SU(3)\times SU(3)$-structures and the associated pure spinors in detail. In section 5, we focus on the differential equations to be satisfied by the pure spinors for preservation of ${\cal{N}}=1$ supersymmetry. We study the covariance of these equations under constant and non-constant Pin(d,d) transformations. Then, we study Non-Abelian T-duality (NATD) transformations in detail, and we show the invariance of pure spinor equations under NATD. In section 6, we consider a specific class of geometries. We transform the pure spinors associated with the SU(3)-structure and show that the resulting pure spinors determine an SU(2) structure. We also study the NATD transformation of the metric, the B field, and the Ramond-Ramond fields.