We present a simple prescription for computing conformal blocks and correlation functions holographically in AdS$_3$ in terms of Wilson lines merging at a bulk vertex. This is shown to reproduce global conformal blocks and heavy-light Virasoro blocks. In the case of higher spin theories the space of vertices is in one-to-one correspondence with the space of ${\cal W}_N$ conformal blocks, and we show how the latter are obtained by explicit computations.Comment: 40 pages, 1 figur

We continue the study of the Wilson line representation of conformal blocks in twodimensional conformal field theory; these have an alternative interpretation as gravitational Wilson lines in the context of the AdS 3 /CFT 2 correspondence. The gravitational Wilson line involves a path-ordered exponential of the stress tensor, and its expectation value can be computed perturbatively in an expansion in inverse powers of the central charge c. The short-distance singularities which occur in the associated stress tensor correlators require systematic regularization and renormalization prescriptions, whose consistency with conformal Ward identities presents a subtle problem. The regularization used here combines dimensional regularization and analytic continuation. Representation theoretic arguments, based on SL(2, R) current algebra, predict an exact result for the Wilson line anomalous dimension and, by building on previous work, we verify that the perturbative calculations using our regularization and renormalization prescriptions reproduce the exact result to order 1/c 3 included. We also discuss a related, but somewhat simpler, Wilson line in Wess-Zumino-Witten models that yields current algebra conformal blocks, and we emphasize the distinction between Wilson lines constructed out of non-holomorphic and purely holomorphic currents.

We initiate a systematic study of high energy matrix elements of local operators in 2D CFT. Knowledge of these is required in order to determine whether the generalized eigenstate thermalization hypothesis (ETH) can hold in such theories. Most high energy states are high level Virasoro descendants, and by employing an oscillator representation of the Virasoro algebra we develop an efficient method for computing matrix elements of primary operators in such states. In parameter regimes where we expect (e.g. from AdS/CFT intuition) thermalization to occur, we observe striking patterns in the matrix elements: diagonal matrix elements are smoothly varying and off-diagonal elements, while nonzero, are power-law suppressed compared to the diagonal elements. We discuss the implications of these universal properties of 2D CFTs in regard to their compatibility with generalized ETH.

Virasoro conformal blocks are expected to exponentiate in the limit of large central charge c and large operator dimensions h i , with the ratios h i /c held fixed. We prove this by employing the oscillator formulation of the Virasoro algebra and its representations. The techniques developed are then used to provide new derivations of some standard results on conformal blocks.Here c is the central charge, h i are conformal dimensions of the external operators, h is the conformal dimension of the exchanged primary and z is the cross-ratio. Although there is compelling evidence for (1.2), a first principles derivation of this well-known formula is lacking. The aim of this paper is to close this gap.An intuitively appealing, but somewhat heuristic, argument for exponentiation is provided by Liouville theory. At large c, correlation functions of heavy primary operators may be computed using the saddle point approximation to the Liouville path integral. Assuming that the saddle point picks out a particular Virasoro block, together with the large c behavior of the DOZZ structure constants in this regime [2], the result follows. A strong check of (1.2) comes