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Register a new userLightcone expansions of conformal blocks in closed form
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Authors
Wenliang Li
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We present new closed-form expressions for 4-point scalar conformal blocks in the s- and t-channel lightcone expansions. Our formulae apply to intermediate operators of arbitrary spin in general dimensions. For physical spin $ell$, they are composed of at most $(ell+1)$ Gaussian hypergeometric functions at each order.
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We present a factorized decomposition of 4-point scalar conformal blocks near the lightcone, which applies to arbitrary intermediate spin and general spacetime dimensions. Then we discuss the systematic expansion in large intermediate spin and the resummations of the large-spin tails of Regge trajectories. The basic integrals for the Lorentzian inversion are given by Wilson functions.
We study large $c$ conformal blocks outside the known limits. This work seems to be hard, but it is possible numerically by using the Zamolodchikov recursion relation. As a result, we find new some properties of large $c$ conformal blocks with a pair of two different dimensions for any channel and with various internal dimensions. With light intermediate states, we find a Cardy-like asymptotic formula for large $c$ conformal blocks and also we find that the qualitative behavior of various large $c$ blocks drastically changes when the dimensions of external primary states reach the value $c/32$. And we proceed to the study of blocks with heavy intermediate states $h_p$ and we find some simple dependence on heavy $h_p$ for large $c$ blocks. The results in this paper can be applied to, for example, the calculation of OTOC or Entanglement Entropy. In the end, we comment on the application to the conformal bootstrap in large $c$ CFTs.
We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of $log t$ term. Our analysis covers the entire parameter regions with respect to the replica number $n$ and the conformal dimension $h_O$ of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikovs recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge $c$, drastically changes when the dimensions of external primary states reach the value $c/32$. In particular, when $h_Ogeq c/32$ and $ngeq 2$, we find a new universal formula $Delta S^{(n)}_Asimeq frac{nc}{24(n-1)}log t$. Our numerical results also confirm existing analytical results using the HHLL approximation.
We interpret aspects of the Schur indices, that were identified with characters of highest weight modules in Virasoro $(p,p)=(2,2k+3)$ minimal models for $k=1,2,dots$, in terms of paths that first appeared in exact solutions in statistical mechanics. From that, we propose closed-form fermionic sum expressions, that is, $q, t$-series with manifestly non-negative coefficients, for two infinite-series of Macdonald indices of $(A_1,A_{2k})$ Argyres-Douglas theories that correspond to $t$-refinements of Virasoro $(p,p)=(2,2k+3)$ minimal model characters, and two rank-2 Macdonald indices that correspond to $t$-refinements of $mathcal{W}_3$ non-unitary minimal model characters. Our proposals match with computations from 4D $mathcal{N} = 2$ gauge theories textit{via} the TQFT picture, based on the work of J Song arXiv:1509.06730.
For conformal field theories in arbitrary dimensions, we introduce a method to derive the conformal blocks corresponding to the exchange of a traceless symmetric tensor appearing in four point functions of operators with spin. Using the embedding space formalism, we show that one can express all such conformal blocks in terms of simple differential operators acting on the basic scalar conformal blocks. This method gives all conformal blocks for conformal field theories in three dimensions. We demonstrate how this formalism can be applied in a few simple examples.
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