On renormalizability of the massless Thirring model

We discuss the renormalizability of the massless Thirring model in terms of the causal fermion Green functions and correlation functions of left-right fermion densities. We obtain the most general expressions for the causal two-point Green function and correlation function of left-right fermion densities with dynamical dimensions of fermion fields, parameterised by two parameters. The region of variation of these parameters is constrained by the positive definiteness of the norms of the wave functions of the states related to components of the fermion vector current. We show that the dynamical dimensions of fermion fields calculated for causal Green functions and correlation functions of left-right fermion densities can be made equal. This implies the renormalizability of the massless Thirring model in the sense that the ultra-violet cut-off dependence, appearing in the causal fermion Green functions and correlation functions of left-right fermion densities, can be removed by renormalization of the wave function of the massless Thirring fermion fields only.Comment: 17 pages, Latex, the contribution of fermions with opposite chirality is added,the parameterisation of fermion determinant by two parameters is confirmed,it is shown that dynamical dimensions of fermion fields calculated from different correlation functions can be made equal.This allows to remove the dependence on the ultra-violet cut-off by the renormalization of the wave function of Thirring fermion fields onl

Conformal flow on S3S^3 and weak field integrability in AdS4_4

We consider the conformally invariant cubic wave equation on the Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ for small rotationally symmetric initial data. This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS$_4$) and connects it to various questions of AdS stability. We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the weak field regime. It turns out that this effective system, which we call the conformal flow, exhibits some remarkable features, such as low-dimensional invariant subspaces, a wealth of stationary states (for which energy does not flow between the modes), as well as solutions with nontrivial exactly periodic energy flows. Based on these observations and close parallels to the cubic Szego equation, which was shown by Gerard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in AdS$_4$ are integrable as well.Comment: 22 pages, v2: minor revisions, several references added, v3: typos corrected, v4: typos corrected, one reference added, matches version accepted by CM

Convex geometry of quantum resource quantification

We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures and show that several of them are equal on pure states. Further, we investigate the dual formulation of resource quantifiers, characterising sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k, multipartite entanglement, as well as magic states, providing insight into the quantification of the resources and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof-extended negativity, a measure of k-coherence which generalises the L1 norm of coherence, and a hierarchy of norm-based quantifiers of k-partite entanglement generalising the greatest cross norm.Comment: 45 pages, 2 figures. v6: section 4.1 rewritten to simplify proofs and correct inconsistencies; theorem numbering consistent with published versio

Bi-invariant metrics on contactomorphism groups

Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed non-degenerate 2-form. Although in symplectic topology a famous bi-invariant metric, the Hofer metric, has been studied since more than 20 years ago, it is only recently that some somehow analogous bi-invariant metrics have been discovered on the group of diffeomorphisms that preserve a contact structure. In this expository article I will review some constructions of bi-invariant metrics on the contactomorphism group, and how these metrics are related to some other global rigidity phenomena in contact topology which have been discovered in the last few years, in particular the notion of orderability (due to Eliashberg and Polterovich) and an analogue in contact topology (due to Eliashberg, Kim and Poltorovich) of Gromov's symplectic non-squeezing theorem.Comment: Expository article, written in occasion of the 5-th IST-IME meeting in S\~ao Paulo (July 2014) in honor of Prof. Orlando Lopes. To appear in the S\~ao Paulo Journal of Mathematical Science

Approximating Operator Norms via Generalized Krivine Rounding

We consider the $(\ell_p,\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A$ over vectors $x,y$ with $\|x\|_p=\|y\|_r=1$. The problem is equivalent to computing the $p \to r^*$ operator norm of $A$. The case $p=r=\infty$ corresponds to the classical Grothendieck problem. Our main result is an algorithm for arbitrary $p,r \ge 2$ with approximation ratio $(1+\epsilon_0)/(\sinh^{-1}(1)\cdot \gamma_{p^*} \,\gamma_{r^*})$ for some fixed $\epsilon_0 \le 0.00863$. Comparing this with Krivine's approximation ratio of $(\pi/2)/\sinh^{-1}(1)$ for the original Grothendieck problem, our guarantee is off from the best known hardness factor of $(\gamma_{p^*} \gamma_{r^*})^{-1}$ for the problem by a factor similar to Krivine's defect. Our approximation follows by bounding the value of the natural vector relaxation for the problem which is convex when $p,r \ge 2$. We give a generalization of random hyperplane rounding and relate the performance of this rounding to certain hypergeometric functions, which prescribe necessary transformations to the vector solution before the rounding is applied. Unlike Krivine's Rounding where the relevant hypergeometric function was $\arcsin$, we have to study a family of hypergeometric functions. The bulk of our technical work then involves methods from complex analysis to gain detailed information about the Taylor series coefficients of the inverses of these hypergeometric functions, which then dictate our approximation factor. Our result also implies improved bounds for "factorization through $\ell_{2}^{\,n}$" of operators from $\ell_{p}^{\,n}$ to $\ell_{q}^{\,m}$ (when $p\geq 2 \geq q$)--- such bounds are of significant interest in functional analysis and our work provides modest supplementary evidence for an intriguing parallel between factorizability, and constant-factor approximability

Single Scale Analysis of Many Fermion Systems. Part 1: Insulators

We construct, using fermionic functional integrals, thermodynamic Green's functions for a weakly coupled fermion gas whose Fermi energy lies in a gap. Estimates on the Green's functions are obtained that are characteristic of the size of the gap. This prepares the way for the analysis of single scale renormalization group maps for a system of fermions at temperature zero without a gap.Comment: 42 page

Explicit Error Bounds for Carleman Linearization

We revisit the method of Carleman linearization for systems of ordinary differential equations with polynomial right-hand sides. This transformation provides an approximate linearization in a higher-dimensional space through the exact embedding of polynomial nonlinearities into an infinite-dimensional linear system, which is then truncated to obtain a finite-dimensional representation with an additive error. To the best of our knowledge, no explicit calculation of the error bound has been studied. In this paper, we propose two strategies to obtain a time-dependent function that locally bounds the truncation error. In the first approach, we proceed by iterative backwards-integration of the truncated system. However, the resulting error bound requires an a priori estimate of the norm of the exact solution for the given time horizon. To overcome this difficulty, we construct a combinatorial approach and solve it using generating functions, obtaining a local error bound that can be computed effectively.Comment: 32 pages, 3 figure

On order preserving and order reversing mappings defined on cones of convex functions

In this paper, we first show that for a Banach space $X$ there is a fully order reversing mapping $T$ from ${\rm conv}(X)$ (the cone of all extended real-valued lower semicontinuous proper convex functions defined on $X$) onto itself if and only if $X$ is reflexive and linearly isomorphic to its dual $X^*$. Then we further prove the following generalized ``Artstein-Avidan-Milman'' representation theorem: For every fully order reversing mapping $T:{\rm conv}(X)\rightarrow {\rm conv}(X)$ there exist a linear isomorphism $U:X\rightarrow X^*$, $x_0^*, \;\varphi_0\in X^*$, $\alpha>0$ and $r_0\in\mathbb R$ so that \begin{equation}\nonumber (Tf)(x)=\alpha(\mathcal Ff)(Ux+x^*_0)+\langle\varphi_0,x\rangle+r_0,\;\;\forall x\in X, \end{equation} where $\mathcal F: {\rm conv}(X)\rightarrow {\rm conv}(X^*)$ is the Fenchel transform. Hence, these resolve two open questions. We also show several representation theorems of fully order preserving mappings defined on certain cones of convex functions. For example, for every fully order preserving mapping $S:{\rm semn}(X)\rightarrow {\rm semn}(X)$ there is a linear isomorphism $U:X\rightarrow X$ so that \begin{equation}\nonumber (Sf)(x)=f(Ux),\;\;\forall f\in{\rm semn}(X),\;x\in X, \end{equation} where ${\rm semn}(X)$ is the cone of all lower semicontinuous seminorms on $X$

The Burnside problem for DiffVol(S2)\text{Diff}_{\text{Vol}}(\mathbb{S}^2)

Let $S$ be a closed surface and $\text{Diff}_{\text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k \in \mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every periodic group of bounded exponent $G \subset \text{Diff}_{\text{Vol}}(S)$ is a finite group.Comment: Added Alejandro Kocsard and Federico Rodr\'iguez-Hertz as coauthors. Include new results about actions of periodic groups on Tori and hyperbolic manifold

Discrepancy of second order digital sequences in function spaces with dominating mixed smoothness

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^d$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the BMO and exponential Orlicz norms, as well as Sobolev, Besov and Triebel-Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.Comment: A journal requested to split the first version of the paper arXiv:1601.07281 into two parts. This is the second part which contains the results on the BMO and exponential Orlicz norm of the discrepancy function. Further the discrepancy function in Sobolev-, Besov- and Triebel-Lizorkin spaces of dominating mixed smoothness is considere