Virasoro conformal blocks in closed form

Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide three closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge $c$. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. Another is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large $c$. In both cases, the $1/c$ expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.Comment: 24 pages + appendices. v2: added refs, minor corrections, improved discussion of Sec.

Comments on Renyi entropy in AdS3_3/CFT2_2

We extend and refine recent results on Renyi entropy in two-dimensional conformal field theories at large central charge. To do so, we examine the effects of higher spin symmetry and of allowing unequal left and right central charges, at leading and subleading order in large total central charge. The result is a straightforward generalization of previously derived formulae, supported by both gravity and CFT arguments. The preceding statements pertain to CFTs in the ground state, or on a circle at unequal left- and right-moving temperatures. For the case of two short intervals in a CFT ground state, we derive certain universal contributions to Renyi and entanglement entropy from Virasoro primaries of arbitrary scaling weights, to leading and next-to-leading order in the interval size; this result applies to any CFT. When these primaries are higher spin currents, such terms are placed in one-to-one correspondence with terms in the bulk 1-loop determinants for higher spin gauge fields propagating on handlebody geometries.Comment: 41 pages. v3: various minor clarifications; added Subsection 4.3 including a result on the entanglement limit; added ref

A universal feature of CFT Renyi entropy

We show that for a d-dimensional CFT in flat space, the Renyi entropy S_q across a spherical entangling surface has the following property: in an expansion around q=1, the first correction to the entanglement entropy is proportional to C_T, the coefficient of the stress tensor vacuum two-point function, with a fixed d-dependent coefficient. This is equivalent to a similar statement about the free energy of CFTs living on S^1 x H^{d-1} with inverse temperature \beta=2\pi q. In addition to furnishing a direct argument applicable to all CFTs, we exhibit this result using a handful of gravity and field theory computations. Knowledge of C_T thus doubles as knowledge of Renyi entropies in the neighborhood of q=1, which we use to establish new results in 3d vector models at large N.Comment: 19 pages + refs; v3: minor revision

Beyond a=ca=c: Gravitational Couplings to Matter and the Stress Tensor OPE

We derive constraints on the operator product expansion of two stress tensors in conformal field theories (CFTs), both generic and holographic. We point out that in large $N$ CFTs with a large gap to single-trace higher spin operators, the stress tensor sector is not only universal, but isolated: that is, $\langle TT{\cal O}\rangle=0$, where ${\cal O}\neq T$ is a single-trace primary. We show that this follows from a suppression of $\langle TT{\cal O}\rangle$ by powers of the higher spin gap, $\Delta_{\rm gap}$, dual to the bulk mass scale of higher spin particles, and explain why $\langle TT{\cal O}\rangle$ is a more sensitive probe of $\Delta_{\rm gap}$ than $a-c$ in 4d CFTs. This result implies that, on the level of cubic couplings, the existence of a consistent truncation to Einstein gravity is a direct consequence of the absence of higher spins. By proving similar behavior for other couplings $\langle T{\cal O}_1{\cal O}_2\rangle$ where ${\cal O}_i$ have spin $s_i\leq 2$, we are led to propose that $1/\Delta_{\rm gap}$ is the CFT "dual" of an AdS derivative in a classical action. These results are derived by imposing unitarity on mixed systems of spinning four-point functions in the Regge limit. Using the same method, but without imposing a large gap, we derive new inequalities on these three-point couplings that are valid in any CFT. These are generalizations of the Hofman-Maldacena conformal collider bounds. By combining the collider bound on $TT$ couplings to spin-2 operators with analyticity properties of CFT data, we argue that all three tensor structures of $\langle TTT\rangle$ in the free-field basis are nonzero in interacting CFTs.Comment: 42+25 pages. v2: added refs, minor change

Double-Trace Flows and the Swampland

We explore the idea that large $N$, non-supersymmetric conformal field theories with a parametrically large gap to higher spin single-trace operators may be obtained as infrared fixed points of relevant double-trace deformations of superconformal field theories. After recalling the AdS interpretation and some potential pathologies of such flows, we introduce a concrete example that appears to avoid them: the ABJM theory at finite $k$, deformed by $\int\!{\cal O}^2$, where ${\cal O}$ is the superconformal primary in the stress-tensor multiplet. We address its relation to recent conjectures based on weak gravity bounds, and discuss the prospects for a wider class of similarly viable flows. Next, we proceed to analyze the spectrum and correlation functions of the putative IR CFT, to leading non-trivial order in $1/N$. This includes analytic computations of the change under double-trace flow of connected four-point functions of ABJM superconformal primaries; and of the IR anomalous dimensions of infinite classes of double-trace composite operators. These would be the first analytic results for anomalous dimensions of finite-spin composite operators in any large $N$ CFT$_3$ with an Einstein gravity dual.Comment: 25+13 pages. v2: refs added, minor clarification

M-Theory Reconstruction from (2,0) CFT and the Chiral Algebra Conjecture

We study various aspects of the M-theory uplift of the $A_{N-1}$ series of $(2,0)$ CFTs in 6d, which describe the worldvolume theory of $N$ M5 branes in flat space. We show how knowledge of OPE coefficients and scaling dimensions for this CFT can be directly translated into features of the momentum expansion of M-theory. In particular, we develop the expansion of the four-graviton S-matrix in M-theory via the flat space limit of four-point Mellin amplitudes. This includes correctly reproducing the known contribution of the $R^4$ term from 6d CFT data. Central to the calculation are the OPE coefficients for half-BPS operators not in the stress tensor multiplet, which we obtain for finite $N$ via the previously conjectured relation [arXiv:1404.1079] between the quantum ${\cal W}_N$ algebra and the $A_{N-1}$ $(2,0)$ CFT. We further explain how the $1/N$ expansion of ${\cal W}_N$ structure constants exhibits the structure of protected vertices in the M-theory action. Conversely, our results provide strong evidence for the chiral algebra conjecture.Comment: 30+18 pages. v2: added refs, fixed typos/notatio

Unitarity Methods in AdS/CFT

We develop a systematic unitarity method for loop-level AdS scattering amplitudes, dual to non-planar CFT correlators, from both bulk and boundary perspectives. We identify cut operators acting on bulk amplitudes that put virtual lines on shell, and show how the conformal partial wave decomposition of the amplitudes may be efficiently computed by gluing lower-loop amplitudes. A central role is played by the double discontinuity of the amplitude, which has a direct relation to these cuts. We then exhibit a precise, intuitive map between the diagrammatic approach in the bulk using cutting and gluing, and the algebraic, holographic unitarity method of [1] that constructs the non-planar correlator from planar CFT data. Our analysis focuses mostly on four-point, one-loop diagrams — we compute cuts of the scalar bubble, triangle and box, as well as some one-particle reducible diagrams — in addition to the five-point tree and four-point double-ladder. Analogies with S-matrix unitarity methods are drawn throughout