Abstract
Classical conformal blocks appear in the large central charge limit of 2D Virasoro conformal blocks. In the AdS3/CFT2 correspondence, they are related to classical bulk actions and used to calculate entanglement entropy and geodesic lengths. In this work, we discuss the identification of classical conformal blocks and the Painlevé VI action showing how isomonodromic deformations naturally appear in this context. We recover the accessory parameter expansion of Heun’s equation from the isomonodromic τ -function. We also discuss how the c = 1 expansion of the τ -function leads to a novel approach to calculate the 4-point classical conformal block.
Highlights
0, x, 1, ∞, are the conformal dimensions of the chiral primary operators V∆i, with δi being the classical dimensions
We discuss the identification of classical conformal blocks and the Painlevé VI action showing how isomonodromic deformations naturally appear in this context
We show that this can be consistently done only if c = 1. This has a two-fold purpose: first, to argue that the isomonodromic τ -function can be understood as a c = 1 correlator and, second, to show that the monodromy data of c = ∞ conformal blocks and c = 1 correlators can be encoded in the same Fuchsian system
Summary
We review how the semiclassical limit of a special 6-point conformal block leads to a Fuchsian equation with 4 singular points and one apparent singularity. Assuming heavy-light factorization and exponentiation2 [2, 3, 16], we write the semiclassical limit of the 6-point conformal block when b → 0 as. Fusing the light degenerate field with any of the other fields, we end up with the semiclassical limit of the 5-point block with all insertions being heavy. The semiclassical limit of this equation gives a constraint on the accessory parameters (2.14) This is exactly the condition for z = λ to be an apparent singularity of (2.13) [41, 42, 50]. (2.13) is the isomonodromic deformation of a 4-point Fuchsian equation, a deformed Heun’s equation, with z = λ being an apparent singularity and not contributing to the monodromy data [41, 42, 50] This means that isomonodromic deformations naturally emerge in CFT. We discuss how isomonodromic deformations relate the monodromy group of the 4-punctured sphere and the moduli space of Fuchsian equations ( called opers in the literature [51])
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
