Cumrun Vafa discusses the construction of compact examples of D-manifolds for type IIB strings, which can be interpreted as compactifications of a 12-dimensional 'F-theory'. He provides evidence for a more natural reformulation of type IIB theory in terms of F-theory. Key points include: 1. **Compactification of M-theory**: Compactification of M-theory on a manifold admitting an elliptic fibration is equivalent to compactification of F-theory on the manifold times a circle. 2. **N = 1 Theories**: A large class of N = 1 theories in 6 dimensions are obtained by compactifying F-theory on Calabi-Yau threefolds. 3. **Spin(7) Holonomy Manifolds**: Compactification of F-theory on Spin(7) holonomy manifolds down to 4 dimensions may provide a solution to the cosmological constant problem, as proposed by Witten. 4. **New Vacua for Type IIB Strings**: Vafa constructs a compact example of type IIB strings on a D-manifold, involving 24 7-branes on a sphere, and argues that it is dual to a heterotic string compactification on a torus. 5. **Type IIB and F-theory**: The strong/weak duality of type IIB and the existence of a 12-dimensional theory with signature (10,2) are discussed, leading to the proposal of F-theory. 6. **Compactifications of F-theory**: Vafa considers compactifications of F-theory on Calabi-Yau, G2 holonomy, and Spin(7) holonomy manifolds to 6, 5, and 4 dimensions, respectively, and discusses the resulting theories and their potential phenomenological implications. The paper highlights the geometric and mathematical structures underlying string theory and provides concrete examples of compactifications, which are crucial for understanding the dualities and vacua in string theory.Cumrun Vafa discusses the construction of compact examples of D-manifolds for type IIB strings, which can be interpreted as compactifications of a 12-dimensional 'F-theory'. He provides evidence for a more natural reformulation of type IIB theory in terms of F-theory. Key points include: 1. **Compactification of M-theory**: Compactification of M-theory on a manifold admitting an elliptic fibration is equivalent to compactification of F-theory on the manifold times a circle. 2. **N = 1 Theories**: A large class of N = 1 theories in 6 dimensions are obtained by compactifying F-theory on Calabi-Yau threefolds. 3. **Spin(7) Holonomy Manifolds**: Compactification of F-theory on Spin(7) holonomy manifolds down to 4 dimensions may provide a solution to the cosmological constant problem, as proposed by Witten. 4. **New Vacua for Type IIB Strings**: Vafa constructs a compact example of type IIB strings on a D-manifold, involving 24 7-branes on a sphere, and argues that it is dual to a heterotic string compactification on a torus. 5. **Type IIB and F-theory**: The strong/weak duality of type IIB and the existence of a 12-dimensional theory with signature (10,2) are discussed, leading to the proposal of F-theory. 6. **Compactifications of F-theory**: Vafa considers compactifications of F-theory on Calabi-Yau, G2 holonomy, and Spin(7) holonomy manifolds to 6, 5, and 4 dimensions, respectively, and discusses the resulting theories and their potential phenomenological implications. The paper highlights the geometric and mathematical structures underlying string theory and provides concrete examples of compactifications, which are crucial for understanding the dualities and vacua in string theory.