On scale symmetry in gauge theories

Buchholz and Fredenhagen proved that particles in the vacuum sector of a scale invariant local quantum field theory do not scatter. More recently, Weinberg argued that conformal primary fields satisfy the wave equation if they have nonvanishing matrix elements between the vacuum and one-particle states. These results do not say anything about actual candidates for scale invariant models, which are nonconfining Yang-Mills theories with no one-particle states in their vacuum sector. The BRST quantization of gauge theories is based on a state space with an indefinite inner product, and the above-mentioned results do not apply to such models. However, we prove that, under some assumptions, the unobservable basic fields of a scale invariant Yang-Mills theory also satisfy the wave equation. In ordinary field theories, particles associated with such a dilation covariant hermitian scalar field do not interact with each other. In the BRST quantization of gauge theories, there is no such triviality result.Comment: 33 page

The dimension of the space of Garnier equations with fixed locus of apparent singularities

We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed exponents at the points of a finite set $P$ and apparent singularities at a finite set $Q$ (disjoint from $P$) determine a linear system of maximal rank. In addition, we show that certain auxiliary parameters can also be fixed. This enables us to conclude that the family of such differential equations is of the expected dimension and to define a birational map between an open subset of the moduli space of logarithmic connections with fixed logarithmic points and regular semi-simple residues and the Hilbert scheme of points on a quasi-projective surface.Comment: 17 page

Vector Equilibrium Problems on Dense Sets

In this paper we provide sufficient conditions that ensure the existence of the solution of some vector equilibrium problems in Hausdorff topological vector spaces ordered by a cone. The conditions that we consider are imposed not on the whole domain of the operators involved, but rather on a self segment-dense subset of it, a special type of dense subset. We apply the results obtained to vector optimization and vector variational inequalities.Comment: arXiv admin note: substantial text overlap with arXiv:1405.232

Deformations of Fuchsian equations and logarithmic connections

We give a geometric proof to the classical fact that the dimension of the deformations of a given generic Fuchsian equation without changing the semi-simple conjugacy class of its local monodromies (``number of accessory parameters'') is equal to half the dimension of the moduli space of deformations of the associated local system. We do this by constructing a weight 1 Hodge structure on the infinitesimal deformations of logarithmic connections, such that deformations as an equation correspond to the $(1,0)$-part. This answers a question of Nicholas Katz, who noticed the dimension doubling mentioned above. We then show that the Hitchin map restricted to deformations of the Fuchsian equation is a one-to-one etale map. Finally, we give a positive answer to a conjecture of Ohtsuki about the maximal number of apparent singularities for a Fuchsian equation with given semisimple monodromy, and define a Lagrangian foliation of the moduli space of connections whose leaves consist of logarithmic connections that can be realised as Fuchsian equations having apparent singularities in a prescribed finite set.Comment: 38 pages, content substantially improved, new results and applications adde

A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in points of some domain, where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses into the convex domain K. The other, pluripotential theoretic approach, mainly due to Baran, works for even more general sets, and yields estimates through the use of the pluricomplex Green function (the Zaharjuta -Siciak extremal function). Using the inscribed ellipse method on non-symmetric convex domains, a key role was played by the generalized Minkowski functional a(K,x). With the aid of this functional, our current knowledge is precise within a constant (squareroot 2) factor. Recently L. Milev and the author derived the exact yield of this method in the case of the simplex, and a number of numerical improvements were obtained compared to the general estimates known. Here we compare the yields of this real, geometric method and the results of the complex, pluripotential theoretical approaches on the case of the simplex. In conclusion we can observe a few remarkable facts, comment on the existing conjectures, and formulate a number of new hypothesis

On uniform asymptotic upper density in locally compact abelian groups

Starting out from results known for the most classical cases of N, Z^d, R^d or for sigma-finite abelian groups, here we define the notion of asymptotic uniform upper density in general locally compact abelian groups. Even if a bit surprising, the new notion proves to be the right extension of the classical cases of Z^d, R^d. The new notion is used to extend some analogous results previously obtained only for classical cases or sigma-finite abelian groups. In particular, we show the following extension of a well-known result for Z of Furstenberg: if in a general locally compact Abelian group G a subset S of G has positive uniform asymptotic upper density, then S-S is syndetic

Right order Turan-type converse Markov inequalities for convex domains on the plane

For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Tur\'an, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although partial results provided general lower estimates of lower order, as well as certain classes of domains with lower bounds of order $n$, it was not clear what order of magnitude the general convex domains may admit here. Here we show that for all compact and convex domains $K$ with nonempty interior and polynomials $p$ with all their zeroes in $K$ $||p'|| > c(K) n ||p||$ holds true, while $||p'|| < C(K) n ||p||$ occurs for any $K$. Actually, we determine $c(K)$ and $C(K)$ within a factor of absolute numerical constant

Decomposition as the sum of invariant functions with respect to commuting transformations

Let A be an arbitrary set. For any transformation T (self-map of A) let T(f)(x):=f(T(x)) (for all x in A) be the usual shift operator. A function g is called periodic, i.e., invariant mod T, if Tg=g (=Ig, where I is the identity operator). As a natural generalization of various earlier investigations in different function spaces, we study the following problem. Let T_j (j=1,...,n) be arbitrary commuting mappings -- transformations -- from A into A. Under what conditions can we state that a function f from A to A is the sum of "periodic", that is, T_j-invariant functions f_j? An obvious necessary condition is that the corresponding multiple difference operator annihilates f, i.e., D_1 ... D_n f= 0, where D_j:=T_j-I. However, in general this condition is not sufficient, and our goal is to complement this basic condition with others, so that the set of conditions will be both necessary and sufficient

Gravitational Instabilities and Censorship of Large Scalar Field Excursions

Large, localized variations of light scalar fields tend to collapse into black holes, dynamically "censoring" distant points in field space. We show that in some cases, large scalar excursions in asymptotically flat spacetimes can be UV-completed by smooth Kaluza-Klein bubble geometries, appearing to circumvent 4d censorship arguments. However, these spacetimes also exhibit classical instabilities related to the collapse or expansion of a bubble of nothing, providing a different censorship mechanism. We show that the Kerr family of static KK bubbles, which gives rise to an infinite scalar excursion upon dimensional reduction, is classically unstable. We construct a family of initial data in which the static bubbles sit at a local maximum of the energy, and we give a general argument that such a property indeed indicates mechanical instability in gravity. We also discuss the behavior of wound strings near a bubble, a local probe of the large traversal through moduli space.Comment: 24 page

Potential theoretic approach to rendezvous numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x1,...,xn} in X, there exists some point x in X with the average of the distances d(x,xj) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named "the magic number" of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.Comment: 21 page