Tauberian-Cardy formula with spin

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,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,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 Ag ≤ π(c−1)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 ,h ) plane and r = 4 √ 3 π 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.

Universal dynamics of heavy operators in CFT2

“…We expect, however, that a much stronger version is true, where one needs to integrate only over a small window; results that establish this kind of behaviour go under the general name of Tauberian theorems (see e.g. [16,[33][34][35][36][37] for recent applications of Tauberian theorems in this context). In the present case we would require new results for several variables, adapted to the Virasoro crossing transforms.…”

“…Rather, the sum converges in the sense of distributions (it should converge when integrated against any test function), which requires some 'smearing', and the asymptotic formulas should be interpreted accordingly. The most conservative statement is that the formula applies in an integrated sense: the total number of states below a given energy or spin is asymptotic to the integral of the Cardy formula (see [35][36][37] for a more detailed discussion and rigorous results). In the particular case of the Cardy formula, a very interesting recent paper [35] has shown that if the averaging window is of fixed width in the large dimension limit, corrections due to the finite size of the averaging window only affect the order-one term in the expansion of the logarithm of the density of states at large dimension.…”

Analytic functional bootstrap for CFTs in d > 1

“…We will not be able to obtain such functionals, and as we discuss in section 6, it's not even clear to us that the above even is achievable. 8 Nevertheless, let us try to emulate this structure and see how far we can go.…”

“…In the last decade, a ruthless siege of the crossing equation has yielded a number of detailed insights into the structure of conformal field theories (CFTs) in general spacetime dimension d. The surprising observations that even simple proddings of crossing equations [1,2] yield strong constraints on CFT spectra have since led to the development of numerical (see [3] for an extensive, but still far from complete review) and analytical methods to bound and determine CFT data. An incomplete list of the latter includes applications of Tauberian theory [4][5][6][7][8], large spin expansions and systematics [9][10][11][12][13][14][15][16] and the Polyakov bootstrap [17][18][19][20][21].…”

Lessons from the Ramond sector

“…Hellerman then demonstrated in [4] that modular constraints could not only produce asymptotic statements but also universal bounds applicable at intermediate energies. Subsequent explorations, fueled by new analytic ideas and powerful numerical methods, led to an explosion of exciting new results [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”