Domain Walls, Triples and Acceleration

We present a construction of domain walls in string theory. The domain walls can bridge both Minkowski and AdS string vacua. A key ingredient in the construction are novel classical Yang-Mills configurations, including instantons, which interpolate between toroidal Yang-Mills vacua. Our construction provides a concrete framework for the study of inflating metrics in string theory. In some cases, the accelerating space-time comes with a holographic description. The general form of the holographic dual is a field theory with parameters that vary over space-time.Comment: 63 pages, 1 figure; LaTe

Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields

Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk showed that for binary Reed-Muller codes of length $n$ and distance $d = O(1)$, one can correct $\operatorname{polylog}(n)$ random errors in $\operatorname{poly}(n)$ time (which is well beyond the worst-case error tolerance of $O(1)$). In this paper, we consider the problem of `syndrome decoding' Reed-Muller codes from random errors. More specifically, given the $\operatorname{polylog}(n)$-bit long syndrome vector of a codeword corrupted in $\operatorname{polylog}(n)$ random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in $\operatorname{polylog}(n)$ time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of Saptharishi et al. 2. The second algorithm is obtained by implementing the steps of the Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.Comment: 24 page

Stress Tensor from the Trace Anomaly in Reissner-Nordstrom Spacetimes

The effective action associated with the trace anomaly provides a general algorithm for approximating the expectation value of the stress tensor of conformal matter fields in arbitrary curved spacetimes. In static, spherically symmetric spacetimes, the algorithm involves solving a fourth order linear differential equation in the radial coordinate r for the two scalar auxiliary fields appearing in the anomaly action, and its corresponding stress tensor. By appropriate choice of the homogeneous solutions of the auxiliary field equations, we show that it is possible to obtain finite stress tensors on all Reissner-Nordstrom event horizons, including the extreme Q=M case. We compare these finite results to previous analytic approximation methods, which yield invariably an infinite stress-energy on charged black hole horizons, as well as with detailed numerical calculations that indicate the contrary. The approximation scheme based on the auxiliary field effective action reproduces all physically allowed behaviors of the quantum stress tensor, in a variety of quantum states, for fields of any spin, in the vicinity of the entire family (0 le Q le M) of RN horizons.Comment: 43 pages, 12 figure

Isospin diffusion in thermal AdS/CFT with flavor

We study the gauge/gravity dual of a finite temperature field theory at finite isospin chemical potential by considering a probe of two coincident D7-branes embedded in the AdS-Schwarzschild black hole background. The isospin chemical potential is obtained by giving a vev to the time component of the non-Abelian gauge field on the brane. The fluctuations of the non-Abelian gauge field on the brane are dual to the SU(2) flavor current in the field theory. For the embedding corresponding to vanishing quark mass, we calculate all Green functions corresponding to the components of the flavor current correlator. We discuss the physical properties of these Green functions, which go beyond linear response theory. In particular, we show that the isospin chemical potential leads to a frequency-dependent isospin diffusion coefficient.Comment: 26 pages, 8 figures, typos correcte

Singular Instantons Made Regular

The singularity present in cosmological instantons of the Hawking-Turok type is resolved by a conformal transformation, where the conformal factor has a linear zero of codimension one. We show that if the underlying regular manifold is taken to have the topology of $RP^4$, and the conformal factor is taken to be a twisted field so that the zero is enforced, then one obtains a one-parameter family of solutions of the classical field equations, where the minimal action solution has the conformal zero located on a minimal volume noncontractible $RP^3$ submanifold. For instantons with two singularities, the corresponding topology is that of a cylinder $S^3\times [0,1]$ with D=4 analogues of `cross-caps' at each of the endpoints.Comment: 23 pages, compressed and RevTex file, including nine postscript figure files. Submitted versio

The symplectic and twistor geometry of the general isomonodromic deformation problem

Hitchin's twistor treatment of Schlesinger's equations is extended to the general isomonodromic deformation problem. It is shown that a generic linear system of ordinary differential equations with gauge group SL(n,C) on a Riemann surface X can be obtained by embedding X in a twistor space Z on which sl(n,C) acts. When a certain obstruction vanishes, the isomonodromic deformations are given by deforming X in Z. This is related to a description of the deformations in terms of Hamiltonian flows on a symplectic manifold constructed from affine orbits in the dual Lie algebra of a loop group.Comment: 35 pages, LATE

Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow

We study constant mean curvature Lorentzian hypersurfaces of $\mathbb{R}^{1,d+1}$ from the point of view of its Cauchy problem. We completely classify the spherically symmetric solutions, which include among them a manifold isometric to the de Sitter space of general relativity. We show that the spherically symmetric solutions exhibit one of three (future) asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like cylinder isometric to some $\mathbb{R}\times\mathbb{S}^d$ and (iii) infinite expansion to the future converging asymptotically to a time translation of the de Sitter solution. For class (iii) we examine the future stability properties of the solutions under arbitrary (not necessarily spherically symmetric) perturbations. We show that the usual notions of asymptotic stability and modulational stability cannot apply, and connect this to the presence of cosmological horizons in these class (iii) solutions. We can nevertheless show the global existence and future stability for small perturbations of class (iii) solutions under a notion of stability that naturally takes into account the presence of cosmological horizons. The proof is based on the vector field method, but requires additional geometric insight. In particular we introduce two new tools: an inverse-Gauss-map gauge to deal with the problem of cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson tensor to obtain natural energy quantities.Comment: Version 2: 60 pages, 1 figure. Changes mostly to fix typographical errors, with the exception of Remark 1.2 and Section 9.1 which are new and which explain the extrinsic geometry of the embedding in more detail in terms of the stability result. Version 3: updated reference