Quantum Fields and Strings

In this talk, I will connect physical properties of scattering amplitudes to the Riemann zeta function. Specifically, I will construct a closed-form amplitude, describing the tree-level exchange of a tower with masses m^2_n = \mu^2_n, where \zeta(\frac{1}{2}\pm i \mu_n) = 0. Requiring real masses corresponds to the Riemann hypothesis, locality of the amplitude to meromorphicity of the zeta function, and universal coupling between massive and massless states to simplicity of the zeros of \zeta. Unitarity bounds from dispersion relations for the forward amplitude translate to positivity of the odd moments of the sequence of 1/\mu^2_n.

In this talk we discuss aspects of the non-relativistic two-body problems in AdS spacetime and their CFT duals. Specifically, we focus on understanding the spectrum of double-twist operators in the dual CFT as well as realizing the flat space scattering and bound states from the CFT correlator.

To understand the double-twist operator spectrum we use quantum perturbation theory in the bulk. We then match the result with the inversion formula for consistency. Next, we show how to obtain the flat space scattering from the correlator by using Euclidean time evolution to construct the scattering states; finally we demonstrate how to get the bound states through the WKB approximation. Time permitting, we will also discuss how to obtain physical quantities such as precession of the near-circular orbits in AdS from the data of the double twist operators lying on the first two Regge trajectories.

Multi-matrix integrals in  planar, large N limit are genuine functional integrals, in general very difficult to compute.  Apart from some known solvable examples (one-matrix model, two-matrix model with specific interaction, etc) one has to rely on perturbation theory or Monte-Carlo at large enough N.  Matrix bootstrap (MB), initiated by Anderson and Kruczenski, and Lin,  is an interesting alternative to these methods. MB deals with the planar loop equations for loop moments -- the averages of traces of "words" built out of products of matrix variables.   The number of unknowns - loop moments - grows with the length of words quicker then the number of  loop equations. The needed extra conditions come from the positivity of correlation matrix of loop moments. This allows, at a given cut-off on the length of words, to establish the upper and lower limits for particular, lowest loop moments, sometimes with an excellent precision.  The main difficulty of the previous works is the non-linearity of loop equations, leading to a  non-convex  optimization procedure. In our recent paper with Zechuan Zheng, we propose to complete this scheme with the relaxation procedure:   the non-linear loop equations are incorporated into the relaxation matrix  as linear inequalities. The problem becomes the standard SDP, allowing for longer loops and thus a better precision. We demonstrate the relaxed matrix bootstrap (RMB) on the example of an analytically unsolvable 2-matrix model. The RMB for Z_2 symmetric states  gives a very satisfactory precision for generic parameters,  up to 6 digits.  We also managed  to apply   RMB  to  more challenging, Z_2 symmetry breaking solutions, though with  less of precision.  We also prove analytically, using the Hamburger problem, that MB for the 1-matrix model converges to physical solutions, with eigenvalues distributed only on the real axis.

We consider quantum field theory on a rigid de Sitter space. We show that the perturbative expansion of late-time correlation functions to all orders can be equivalently generated by a non-unitary Lagrangian on a Euclidean AdS geometry. We use this to infer the analytic structure of the spectral density that captures the conformal partial wave expansion of a late-time four-point function, to derive an OPE expansion, and to constrain the operator spectrum. Generically, dimensions and OPE coefficients do not obey the usual CFT notion of unitarity. Instead, unitarity of the de Sitter theory manifests itself as the positivity of the spectral density. We illustrate and check these properties by explicit calculations in a scalar theory by computing first tree-level, and then full one-loop-resummed exchange diagrams.

I discuss the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions. I analyze this problem with the help of the tensionless string on AdS3xS3xT4 (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold of T4. I argue that large stringy corrections around a fixed background can be interpreted as different semiclassical geometries, thus making a sum over semi-classical geometries superfluous.

Crossing symmetry asserts that particles are indistinguishable from anti-particles traveling back in time. In quantum field theory, this statement translates to the long-standing conjecture that probabilities for observing the two scenarios in a scattering experiment are described by one and the same function. Why could we expect it to be true? In this talk we examine this question in a simplified setup and take steps towards illuminating a possible physical interpretation of crossing symmetry. To be more concrete, we consider planar scattering amplitudes involving any number of particles with arbitrary spins and masses to all loop orders in perturbation theory. We show that by deformations of the external momenta, one can smoothly interpolate between the future and the past lightcones without encountering any singularities.

I will explain how to compute correlation functions of two heavy operators and a light BPS single-trace operator at strong coupling using a dual description of D-branes absorbing a supergravity mode. Our approach is inspired by the large charge expansion of CFT and resolves some confusions in the literature on the holographic computation involving heavy operators. In particular, we point out two important effects which are often missed; the first one is an average over classical configurations of the heavy state, which physically amounts to projecting the state to an eigenstate of quantum numbers. The second one is the contribution from wave functions of the heavy state. Time permitting I will also comment on possible applications to states dual to black holes and fuzzballs.

This talk will be about entanglement entropy in empty 4-dimensional de Sitter spacetime of a non-conformal QFT [1]. I will first briefly describe the set-up and show how a hydrodynamic plasma dilutes and falls out of equilibrium due to expansion towards empty de Sitter spacetime. Interestingly, in the empty setting we can show that extremal surfaces in the holographic dual of spherical entangling regions on the boundary QFT probe beyond the dual event horizon if and only if the entangling region is larger than the cosmological horizon.

[1] Jorge Casalderrey-Solana, Christian Ecker, David Mateos and WS, Strong-coupling dynamics and entanglement in de Sitter space, 2011.08194