Abstract
In the AdS3/CFT2 correspondence, physical interest attaches to understanding Virasoro conformal blocks at large central charge and in a kinematical regime of large Lorentzian time separation, t ∼ c. However, almost no analytical information about this regime is presently available. By employing the Wilson line representation we derive new results on conformal blocks at late times, effectively resumming all dependence on t/c. This is achieved in the context of “light-light” blocks, as opposed to the richer, but much less tractable, “heavy-light” blocks. The results exhibit an initial decay, followed by erratic behavior and recurrences. We also connect this result to gravitational contributions to anomalous dimensions of double trace operators by using the Lorentzian inversion formula to extract the latter. Inverting the stress tensor block provides a pedagogical example of inversion formula machinery.
Highlights
JHEP04(2019)026 does not so far extend to the regime of cross ratio space corresponding to large Lorentzian time
The results exhibit an initial decay, followed by erratic behavior and recurrences. We connect this result to gravitational contributions to anomalous dimensions of double trace operators by using the Lorentzian inversion formula to extract the latter
Inverting the stress tensor block provides a pedagogical example of inversion formula machinery
Summary
We recall the basic construction of the Wilson line, and how it provides a representation of Virasoro conformal blocks that admits a convenient expansion at large central charge. The Wilson line is conjectured to provide a representation of the Virasoro vacuum OPE block [10], O(x1)O(x2) = h; out|P e C a(z)|h; in + [non-pure stress tensor terms]. Let the CFT state be such that T (z) has a classical expectation value at large c, T (z) ∼ c In this regime the T (z) operator appearing in the Wilson line can be replaced by its expectation value, which we continue to denote by T (z). If we take the SL(2) matrix element using (2.5) we find h|P e (z)L−1)dz This makes perfect sense as it says that if we generate a stress tensor by performing a conformal transformation z → f (z), the Wilson line result takes the form of a primary two-point function transformed by f (z). The notion of taking T (z) to be an operator corresponds to performing the Chern-Simons path integral, rather than restricting to a fixed classical background
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