Modular invariance, tauberian theorems and microcanonical entropy

Abstract

We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.April 2019 †

Tauberian-Cardy formula with spin

“…We further remark that for CFTs where the partition function nicely factorizes into holomorphic and antiholomorphic pieces, the leading result directly follows from the analogous result for large ∆ = h +h, proven in [2], nonetheless the error term in analogues of eq. (1.1) and eq.…”

“…So we can see that lim ∆→∞ F (∆/2, ∆/2) is power law suppressed compared to lim Figure 1: Approaching to infinity on (h ,h ) plane: The blue lines denote how the number of operators with (h ,h ) is counted such that h +h ≤ ∆ and then we let ∆ → ∞, this is given by the function F MZ (∆), originally calculated in [2]. On the other hand, the black lines denote how the number of operator is counted upto some value of h =h = ∆/2 i.e.…”

“…The Cardy formula [1] for the asymptotic density of states has recently been rigorously derived with an estimate for the error term in [2,3]. A natural question is to ask whether one can generalize the formalism so as to make it sensitive to the spin or equivalently to the conformal weights h,h separately.…”

“…(1.1) and eq. (1.2) is that they have leading 2 If we say f = O(1), we mean |f | < M for a fixed positive number M . 3 A cautionary remark is that here in this paper unless otherwise mentioned, the twist is NOT kept finite while taking this limit.…”

Analytic functional bootstrap for CFTs in d > 1

“…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].…”