Abstract
We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h, overline{h} ) → (∞, ∞) for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on (h, overline{h} ) plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary 2D CFT without any conserved current Agle frac{pi left(c-1right){r}^2}{24} is satisfied, where g is the twist gap over vacuum and A is the minimal “areal gap”, generalizing the minimal gap in dimension to (h′, overline{h}^{prime } ) plane and r=frac{4sqrt{3}}{pi}simeq 2.21 . We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.
Highlights
Integrated density of statesWe prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory
(∞, ∞) for any arbitrary spin is derived using the theorem
We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory
Summary
We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h, h) → (∞, ∞) is derived using the theorem. We remark that the regime of validity depends on whether we put in the assumption of having a twist gap. We mean there exists a number τ∗ > 0 such that there is no operator with twist τ ∈ (0, τ∗) and there are finite number of zero twist operators with dimension less than c/12. We make two remarks: a) the fact that there are finite number of zero twist operators with dimension less than c/12 is always true since there are finite number of operators with dimension less than c/12 for finite central charge, b) Not having any operator with twist τ ∈ (0, τ∗) disallows having 0 as twist accumulation point
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