Abstract
We present a unified framework for the holographic computation of Virasoro conformal blocks at large central charge. In particular, we provide bulk constructions that correctly reproduce all semiclassical Virasoro blocks that are known explicitly from conformal field theory computations. The results revolve around the use of geodesic Witten diagrams, recently introduced in arXiv:1508.00501, evaluated in locally AdS$_3$ geometries generated by backreaction of heavy operators. We also provide an alternative computation of the heavy-light semiclassical block -- in which two external operators become parametrically heavy -- as a certain scattering process involving higher spin gauge fields in AdS$_3$; this approach highlights the chiral nature of Virasoro blocks. These techniques may be systematically extended to compute corrections to these blocks and to interpolate amongst the different semiclassical regimes.
Highlights
Given a consistent theory of gravity in AdS, one can compute correlation functions that obey CFT axioms, and admit a decomposition into conformal blocks
The results revolve around the use of geodesic Witten diagrams, recently introduced in [1], evaluated in locally AdS3 geometries generated by backreaction of heavy operators
The main result is that a conformal block — more precisely, a conformal partial wave — is obtained from a “geodesic Witten diagram.”
Summary
We consider a four-point function of Virasoro primary operators Oi(zi, zi) on the plane, O1(z1, z1)O2(z2, z2)O3(z3, z3)O4(z4, z4). A basis for the Hilbert space of the CFT consists of the set of primary states |Op (equivalently, local primary operators Op) and their Virasoro descendants, i.e. the set of irreducible highest weight representations of the Virasoro algebra This implies the existence of a Virasoro conformal block decomposition of the four-point function, O1(∞, ∞)O2(0, 0)O3(z, z)O4(1, 1) = C12pCp34F (hi, hp, c; z − 1)F (hi, hp, c; z − 1) , p (2.3) where the sum runs over all irreducible representations of the Hilbert space. If all ratios hi/c are held fixed in the limit, one can apply Zamolodchikov’s monodromy method (well reviewed in [17, 27]) to determine the semiclassical Virasoro block. In [18] it was explained why this prescription works, by thinking about the relationship between Zamolodchikov’s monodromy method and the linearized backreaction produced by these worldlines
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