SciPost Phys. 7, 003 (2019) ·
published 4 July 2019

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We use the monodromy method to compute expectation values of an arbitrary
number of light operators in finitely excited ("heavy") eigenstates of
holographic 2D CFT. For eigenstates with scaling dimensions above the BTZ
threshold, these behave thermally up to small corrections, with an effective
temperature determined by the heavy state. Below the threshold we find
oscillatory and not decaying behavior. As an application of these results we
compute the expectation of the outoftime order arrangement of four light
operators in a heavy eigenstate, i.e. a sixpoint function. Above the threshold
we find maximally scrambling behavior with Lyapunov exponent $2\pi T_{\rm
eff}$. Below threshold we find that the eigenstate OTOC shows persistent
harmonic oscillations.
SciPost Phys. 5, 022 (2018) ·
published 11 September 2018

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We consider unitary, modular invariant, twodimensional CFTs which are
invariant under the parity transformation $P$. Combining $P$ with modular
inversion $S$ leads to a continuous family of fixed points of the $SP$
transformation. A particular subset of this locus of fixed points exists along
the line of positive left and rightmoving temperatures satisfying $\beta_L
\beta_R = 4\pi^2$. We use this fixed locus to prove a conjecture of Hartman,
Keller, and Stoica that the free energy of a large$c$ CFT$_2$ with a suitably
sparse lowlying spectrum matches that of AdS$_3$ gravity at all temperatures
and all angular potentials. We also use the fixed locus to generalize the
modular bootstrap equations, obtaining novel constraints on the operator
spectrum and providing a new proof of the statement that the twist gap is
smaller than $(c1)/12$ when $c>1$. At large $c$ we show that the operator
dimension of the first excited primary lies in a region in the
$(h,\overline{h})$plane that is significantly smaller than
$h+\overline{h}<c/6$. Our results for the free energy and constraints on the
operator spectrum extend to theories without parity symmetry through the
construction of an auxiliary parityinvariant partition function.
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in Report on Parity and the modular bootstrap