One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete-the Bruhat-Tits tree for PGL(2, Q p )-but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat-Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu-Takayanagi formula for the entanglement entropy appears naturally.Much attention has been paid of late to ideas that allow certain features of conformal field theory, such as long-range correlations, to be reproduced in lattice systems or other finitary models. As an example, the multiscale entanglement renormalization ansatz (or MERA), formulated by Vidal in [58], provides an algorithm to compute many-qubit quantum states whose entanglement properties are similar to those of the vacuum state in a conformal field theory. In Vidal's method, the states of progressively more distant qubits are entangled using successive layers of a self-similar network of finite tensors.These proposals can typically be thought of as constructing analogues of the CFT vacuum state using a quantum circuit with an additional "spatial direction," consisting of the successive computational layers of the circuit, and corresponding roughly to the distance scale up to which long-range entanglement has been introduced. As such, they are strongly suggestive of the AdS/CFT correspondence [30,36,61], in which a ddimensional conformal field theory is related to a gravitational theory in d+1-dimensional negatively curved spacetime, and the extra direction can be interpreted as a renormalization scale (or equivalently a length scale) from the perspective of the boundary theory. Furthermore, the construction of the layers (in which the number of tensors scales exponentially with the number of layers) bears comparison with the geometry of hyperbolic space. It was thus natural to search for a connection with ...
We present new formulas for n-particle tree-level scattering amplitudes of sixdimensional N = (1, 1) super Yang-Mills (SYM) and N = (2, 2) supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the (n − 3)! solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of N = (1, 1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2, C) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional N = (2, 2) SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional N = 4 SYM on the Coulomb branch.
We formulate a Euclidean theory of edge length dynamics based on a notion of Ricci curvature on graphs with variable edge lengths. In order to write an explicit form for the discrete analog of the Einstein-Hilbert action, we require that the graph should either be a tree or that all its cycles should be sufficiently long. The infinite regular tree with all edge lengths equal is an example of a graph with constant negative curvature, providing a connection with p-adic AdS/CFT, where such a tree takes the place of anti-de Sitter space. We compute simple correlators of the operator holographically dual to edge length fluctuations. This operator has dimension equal to the dimension of the boundary, and it has some features in common with the stress tensor.
We present tree-level n-particle on-shell scattering amplitudes of various brane theories with 16 conserved supercharges. These include the world-volume theory of a probe D3-brane or D5-brane in 10D Minkowski spacetime as well as a probe M5-brane in 11D Minkowski spacetime, which describes self interactions of an abelian tensor supermultiplet with 6D (2, 0) supersymmetry. Twistor-string-like formulas are proposed for tree-level scattering amplitudes of all multiplicities for each of these theories. The R symmetry of the D3-brane theory is shown to be SU(4) × U(1), and the U(1) factor implies that its amplitudes are helicity conserving. Each of 6D theories (D5-brane and M5-brane) reduces to the D3-brane theory by dimensional reduction. As special cases of the general M5-brane amplitudes, we present compact formulas for examples involving only the self-dual B field with n = 4, 6, 8.
Due to the failure of thermodynamics for low temperature near-extremal black holes, it has long been conjectured that a "thermodynamic mass gap'' exists between an extremal black hole and the lightest near-extremal state. For non-supersymmetric near-extremal black holes in Einstein gravity with an AdS2 throat, no such gap was found. Rather, at that energy scale, the spectrum exhibits a continuum of states, up to non-perturbative corrections. In this paper, we compute the partition function of near-BPS black holes in supergravity where the emergent, broken, symmetry is PSU(1,1|2). To reliably compute this partition function, we show that the gravitational path integral can be reduced to that of a N=4 supersymmetric extension of the Schwarzian theory, which we define and exactly quantize. In contrast to the non-supersymmetric case, we find that black holes in supergravity have a mass gap and a large extremal black hole degeneracy consistent with the Bekenstein-Hawking area. Our results verify a plethora of string theory conjectures, concerning the scale of the mass gap and the counting of extremal micro-states.
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