Abstract An effective theory designed to compute Virasoro identity blocks at large central charge, expressed in terms of the propagation of a reparametrization/shadow mode between bilocal vertices, was recently put forward. In this paper I provide the formal theoretical framework underlying this effective theory by reformulating it in terms of standard concepts: conformal geometry, generating functionals and Feynman diagrams. A key ingredient to this formalism is the bilocal vertex operator, or reparametrized two-point function, which is shown to generate arbitrary stress tensor insertions into a two-point function of reference. I also suggest an extension of the formalism designed to compute generic Virasoro blocks.
Abstract An effective theory designed to compute Virasoro identity blocks at large central charge, expressed in terms of the propagation of a reparametrization/shadow mode between bilocal vertices, was recently put forward. In this paper I provide the formal theoretical framework underlying this effective theory by reformulating it in terms of standard concepts: conformal geometry, generating functionals and Feynman diagrams. A key ingredient to this formalism is the bilocal vertex operator, or reparametrized two-point function, which is shown to generate arbitrary stress tensor insertions into a two-point function of reference. I also suggest an extension of the formalism designed to compute generic Virasoro blocks.
Reparametrization mode Ward Identities and chaos in higher-pt. correlators in CFT2
Recently introduced reparametrization mode operators in CFTs have been shown to govern stress tensor interactions via the shadow operator formalism and seem to govern the effective dynamics of chaotic systems. We initiate a study of Ward identities of reparametrization mode operators i.e. how two dimensional CFT Ward identities govern the behaviour of insertions of reparametrization modes ε in correlation functions: ⟨εεϕϕ⟩. We find that in the semi-classical limit of large c they dictate the leading O(c−1) behaviour. While for the 4pt function this reproduces the same computation as done by Heahl, Reeves \& Rozali in \cite{Haehl:2019eae}, in the case of 6pt function of pair-wise equal operators this provides an alternative way of computing the Virasoro block in stress-tensor comb channel. We compute a maximally out of time ordered correlation function in a thermal background and find the expected behaviour of an exponential growth governed by Lyapunov index λL=2π/β lasting for twice the scrambling time of the system t∗=2πβlogc for the maximally braided type of out-of-time-ordering. However when only the internal operators of the comb channel are \emph{out-of-time-ordered}, the correlator sees no exponential behaviour despite the inclusion of the Virasoro contribution. From a bulk perspective for the \emph{out-of-time-ordered} 4pt function we find that the Casimir equation for the stress tensor block reproduces the linearised back reaction in the bulk.
Abstract We study six-point correlation functions in two dimensional conformal field theory, where the six operators are grouped in pairs with equal conformal dimension. Assuming large central charge c and a sparse spectrum, the leading contribution to this correlation function is the six-point Virasoro identity block — corresponding to each distinct pair of operators fusing into the identity and its descendants. We call this the star channel. One particular term in the star channel identity block is the stress tensor SL(2, ℝ) (global) block, for which we derive an explicit expression. In the holographic context, this object corresponds to a direct measure of nonlinear effects in pure gravity. We calculate additional terms in the star channel identity block that contribute at the same order at large c as the global block using the novel theory of reparametrizations, which extends the shadow operator formalism in a natural way. We investigate these blocks’ relevance to quantum chaos in the form of six-point scrambling in an out-of time ordered correlator. Interestingly, the global block does not contribute to the scrambling mode of this correlator, implying that, to leading order, six-point scrambling is insensitive to the three-point graviton coupling in the bulk dual. Finally, we compare our findings with a different OPE channel, called the comb channel, and find the same result for the chaos exponent in this decomposition.
A bstract We study two novel approaches to efficiently encoding universal constraints imposed by conformal symmetry, and describe applications to quantum chaos in higher dimensional CFTs. The first approach consists of a reformulation of the shadow operator formalism and kinematic space techniques. We observe that the shadow operator associated with the stress tensor (or other conserved currents) can be written as the descendant of a field ε with negative dimension. Computations of stress tensor contributions to conformal blocks can be systematically organized in terms of the “soft mode” ε , turning them into a simple diagrammatic perturbation theory at large central charge. Our second (equivalent) approach concerns a theory of reparametrization modes, generalizing previous studies in the context of the Schwarzian theory and two-dimensional CFTs. Due to the conformal anomaly in even dimensions, gauge modes of the conformal group acquire an action and are shown to exhibit the same dynamics as the soft mode ε that encodes the physics of the stress tensor shadow. We discuss the calculation of the conformal partial waves or the conformal blocks using our effective field theory. The separation of conformal blocks from shadow blocks is related to gauging of certain symmetries in our effective field theory of the soft mode. These connections explain and generalize various relations between conformal blocks, shadow operators, kinematic space, and reparametrization modes. As an application we study thermal physics in higher dimensions and argue that the theory of reparametrization modes captures the physics of quantum chaos in Rindler space. This is also supported by the observation of the pole skipping phenomenon in the conformal energy-energy two-point function on Rindler space.
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.
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 two new 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. The other 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.
We show that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be re-absorbed by placing the CFT in a non-trivial background metric. This leads to a more precise computation of the Virasoro conformal blocks between heavy and light operators, which are shown to be equivalent to global conformal blocks evaluated in the new background. We also generalize to the case where the operators carry U(1) charges. The refined Virasoro blocks can be used as the seed for a new Virasoro block recursion relation expanded in the heavy-light limit. We comment on the implications of our results for the universality of black hole thermality in AdS3, or equivalently, the eigenstate thermalization hypothesis for CFT2 at large central charge.