Two Views on Gravity: F-Theory and Holography

Abstract

We investigate two different views towards gravitational theories using mathematical frameworks. First, we study gravitational theories with supersymmetries in various dimensions from top-down approach via F-theory or M-theory compactifications. We utilize the geometry of elliptic fibrations to investigate such compactifications. The other viewpoint does not require to study theories with supersymmetries. With the framework of von Neumann algebras, we study gravitational theories in the bulk and its boundary conformal field theories. We consider the construction of supergravity theories in three to six dimensions via the compactifcation of M-theory and F-theory on elliptically-fibered manifolds. Interesting gauge theory sectors arise when such manifolds have singularities. We study the resolutions of singularities of these spaces which give the window onto the low energy physics of effective supergravity. We consider elliptically-fibered Calabi--Yau threefolds that give rise to supergravities with simple gauge groups, with a particular emphasis on F$_4$, G$_2$, Spin($7$), and Spin($8$), or semi-simple gauge groups of the form SO($4$), Spin($4$), \sug , \susu , \susp , \susp /$\mathbb{Z}_2$, \susuf , and \susuf /$\mathbb{Z}_2$. For such models we enumerate the spectra in five-dimensions and six-dimensions with eight supercharges via M-theory and F-theory compactifications and determine the structure of the Coulomb branch for these 5d theories. Furthermore we verify that all local anomalies in 6d are canceled. For theories with an abelian gauge group we introduced a new, general model for an elliptic fibration that realizes this $U(1)$ symmetry. The physical spectra often depends on topological invariants of the elliptic fibration. In particular, when the effective theory is required to be supersymmetric, the elliptic fibration must be Calabi--Yau. In the more general case, when the fibration is not assumed to be Calabi--Yau, we utilized the resolution of singularities to determine a host of topological invariants and characteristic numbers for elliptic fibrations that correspond to a physical gauge group with a simple non-abelian factor. These include the Euler characteristic, Hodge numbers, Chern numbers, Pontryagin numbers, Todd genus, holomorphic genera, L-genus, A-genus, and the M-theory curvature invariant. In a different vein, infinite-dimensional von Neumann algebras of various types are used to understand the local algebras in quantum field theories. Utilizing such von Neumann algebras, one can study holographic quantum field theories and their gravity duals by incorporating toy models from quantum error correction. We reformulate the entanglement wedge reconstruction in the language of infinite-dimensional von Neumann algebras. Using the frame of Tomita--Takasaki theory, we can also write the infinite-dimensional analog of the relative entropies. Using these techniques, we show that for a general infinite-dimensional Hilbert space, the entanglement wedge reconstruction is identical to the equivalence in relative entropies between the boundary and the bulk.