We revisit AdS2 holography with the Sachdev-Ye-Kitaev models in mind. Our main result is to rewrite a generic theory of gravity near an AdS2 throat as a novel hydrodynamics coupled to the correlation functions of a conformal quantum mechanics. This gives a prescription for the computation of n-point functions in the dual quantum mechanics. We thereby find that the dual is maximally chaotic.Introduction. The Sachdev-Ye-Kitaev (SYK) models [1,2] are quantum mechanical systems with random all-to-all interactions. They have been recently argued to have a gravity dual in two dimensions.The basic SYK model is a theory of 2N Majorana fermions ψ a (a = 1, .., 2N ) perturbed by quenched disorder. The Hamiltonian is

We compute the thermodynamic properties of the Sachdev-Ye-Kitaev (SYK) models of fermions with a conserved fermion number, Q. We extend a previously proposed Schwarzian effective action to include a phase field, and this describes the low temperature energy and Q fluctuations. We obtain higherdimensional generalizations of the SYK models which display disordered metallic states without quasiparticle excitations, and we deduce their thermoelectric transport coefficients. We also examine the corresponding properties of Einstein-Maxwell-axion theories on black brane geometries which interpolate from either AdS 4 or AdS 5 to an AdS 2 × R 2 or AdS 2 × R 3 near-horizon geometry. These provide holographic descriptions of non-quasiparticle metallic states without momentum conservation. We find a precise match between low temperature transport and thermodynamics of the SYK and holographic models. In both models the Seebeck transport coefficient is exactly equal to the Q-derivative of the entropy. For the SYK models, quantum chaos, as characterized by the butterfly velocity and the Lyapunov rate, universally determines the thermal diffusivity, but not the charge diffusivity.

We present a generating functional which describes the equilibrium thermodynamic response of a relativistic system to external sources. A variational principle gives rise to constraints on the response parameters of relativistic hydrodynamics without making use of an entropy current. Our method reproduces and extends results available in the literature. It also provides a technique for efficiently computing n-point zero-frequency correlation functions within the hydrodynamic derivative expansion without the need to explicitly solve the equations of hydrodynamics.

By studying the Euclidean partition function on a cone, we argue that pure and mixed gravitational anomalies generate a "Casimir momentum" which manifests itself as parity violating coefficients in the hydrodynamic stress tensor and charge current. The coefficients generated by these anomalies enter at a lower order in the hydrodynamic gradient expansion than would be naively expected. In 1+1 dimensions, the gravitational anomaly affects coefficients at zeroth order in the gradient expansion. The mixed anomaly in 3+1 dimensions controls the value of coefficients at first order in the gradient expansion.Comment: 37 page

We rewrite the Chern-Simons description of pure gravity on global AdS 3 and on Euclidean BTZ black holes as a quantum field theory on the AdS boundary. The resulting theory is (two copies of) the path integral quantization of a certain coadjoint orbit of the Virasoro group, and it should be regarded as the quantum field theory of the boundary gravitons. This theory respects all of the conformal field theory axioms except one: it is not modular invariant. The coupling constant is 1/c with c the central charge, and perturbation theory in 1/c encodes loop contributions in the gravity dual. The QFT is a theory of reparametrizations analogous to the Schwarzian description of nearly AdS 2 gravity, and has several features including: (i) it is ultraviolet-complete; (ii) the torus partition function is the vacuum Virasoro character, which is one-loop exact by a localization argument; (iii) it reduces to the Schwarzian theory upon compactification; (iv) it provides a powerful new tool for computing Virasoro blocks at large c via a diagrammatic expansion. We use the theory to compute several observables to one-loop order in the bulk, including the "heavy-light" limit of the identity block. We also work out some generalizations of this theory, including the boundary theory which describes fluctuations around two-sided eternal black holes. arXiv:1808.03263v3 [hep-th] 20 Sep 2018 7 Supersymmetry 58 8 Discussion 65 1. Eq. (1.4) is an ultraviolet-complete two-dimensional theory in its own right. It is a perfectly sensible CFT in all respects except one: it is not modular invariant. In fact, the Hilbert space of the model is just the vacuum module, which can be established by computing the torus partition function. The latter is one-loop exact on account of the localization formula, and is simply the vacuum character. (Maloney and Witten obtained the same Hilbert space from their geometric quantization.)2. The action (1.4) may be understood as a Wess-Zumino term for the Weyl and gravitational anomalies of a chiral theory with chiral central charge C. The field φ also contributes 13 to the anomaly (rather than 1, due to the P SL(2; R) quotient, which effectively removes the first oscillator), and we find a total chiral central charge c = C +13 as we explain in Section 5. Moreover, 1/C plays the role of the coupling constant of the model, so that it is weakly coupled at large central charge. Excitations of the φ field correspond to boundary gravitons, and so the model computes loop-level Witten diagrams with exchanges of the gravitational field. The shift c = C + 13 may then be understood as a one-loop renormalization of the central charge, which in fact is already visible in the direct gravitational computation of [2]. (There is no contradiction with the statement that the cosmological constant of AdS 3 gravity is unnrenormalized in the Chern-Simons formulation [6]. The shift comes from the boundary excitations rather than the renormalization of a bulk divergence.)3. The only P SL(2; R)-invariant local operators of the model are ...