UC Davis

The dissertation is concerned with various physical properties and mathematical applications of conformal field theories. The relevant conformal field theories include both holographic CFTd with d ≥ 2 and generic non-holographic CFT 2 . The dissertation is divided into three parts.

In Part I, we study information-theoretic properties of entanglement entropies of holographic CFT d . In particular, we study holographic entropy inequalities and their structural properties by making use of a judicious grouping of terms into certain multipartite information quantities. This results in the discovery of both new inequalities and considerably simpler forms for known inequalities. We also obtain a negative result on the interpretation of quantum information quantities associated with certain holographic entropy inequalities as correlation measures.

In Part II, we study twist operators in CFT 2 and their relation with both physical and mathematical objects. In Chapter 3, we formulate a generalized stress-tensor method for twist operator correlators (TOCs) and utilize it to establish a precise relation between TOCs and certain tau functions associated with branched covers of P 1 . This bypasses certain issues in the conventional stress-tensor method and further clarifies the mathematical nature of TOCs. In Chapter 4, we introduce a novel representation for TOCs and their associated tau functions by utilizing properties of ground state modular Hamiltonians in CFT 2 . The connection with modular Hamiltonian originates from the formal path integral representation of the ground state reduced density matrix in CFT 2 . For a class of genus-zero TOCs, we also argue an approximate factorization property, utilizing the known ground state correlation structure of large-c holographic CFT 2 and the universality of genus-zero TOCs. We provide numerical checks of our statements in various examples.

In Part III, we study an exact CFT 2 method for black hole perturbation problem in AdS d+1 , which is in turn dual to thermal correlators of holographic CFT d . In particular, we refine and

viii further develop a recent exact analytic approach to black hole perturbation problem based on the semiclassical Virasoro blocks, or equivalently via AGT relation, the Nekrasov partition functions in the Nekrasov-Shatashvili limit. Focusing on asymptotically AdS 5 black hole backgrounds, we derive new universal exact expressions for holographic thermal two-point functions and quantization conditions for the associated quasinormal modes (QNMs). Our expressions for the holographic CFT 4 closely resemble the well-known results for 2d thermal CFTs on R 1,1 . This structural similarity stems from the locality of fusion transformation for Virasoro blocks. We provide numerical checks of our quantization conditions for QNMs. Additionally, we discuss the application of our results to understand specific physical properties of QNMs, including their near-extremal and asymptotic limits. The latter is related to a certain large-momentum regime of semiclassical Virasoro blocks dual to Seiberg-Witten prepotentials.