Related papers: Extremal polygons in R^3
We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect…
We study Abelian $S$-duality of Maxwell theory on $A$-type asymptotically locally Euclidean (ALE) spaces. Unlike on closed four-manifolds, the Maxwell path integral on an ALE space is not naturally a scalar partition function. Rather, it…
Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the…
We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area…
Multipolar expansions are a foundational tool for describing basis functions in quantum mechanics, many-body polarization, and other distributions on the unit sphere. Progress on these topics is often held back by complicated and competing…
We systematically classify all possible poles of superconformal blocks as a function of the scaling dimension of intermediate operators, for all superconformal algebras in dimensions three and higher. This is done by working out the…
We reduce the computation of three point function of three spinning operators with arbitrary polarizations to a statistical mechanics problem via the hexagon formalism. The central building block of these correlation functions is the…
In this paper, we consider the non-singular extension problem for circle-valued Morse functions on closed orientable surfaces. The problem asks, given a circle-valued Morse function $f\colon M\to S^{1}$ on a closed orientable surface $M$,…
We describe a prescription for constructing conformal blocks in conformal field theories in any space-time dimension with arbitrary quantum numbers. Our procedure reduces the calculation of conformal blocks to constructing certain group…
We develop analytical methods for computing the structure constant for three heavy operators, starting from the recently proposed hexagon approach. Such a structure constant is a semiclassical object, with the scale set by the inverse…
A $configuration$ of a linkage $\Gamma$ is a possible positioning of $\Gamma$ in $\mathbb{R}^d$ and the collection of all such forms the configuration space $\mathcal{C}(\Gamma)$ of $\Gamma$. We here introduce the notion of the $symmetric…
Fold maps are fundamental tools in the theory of singularities of differentiable maps and its applications to geometry. They are higher dimensional variants of Morse functions. Classes of special generic maps and round fold maps are…
The AdS/CFT correspondence relates Wilson loops in $N$=4 SYM theory to minimal area surfaces in AdS space. If the loop is a plane curve the minimal surface lives in hyperbolic space $H_3$ (or equivalently Euclidean AdS$_3$ space). We argue…
We study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a line segment, a…
The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds.…
In Euclidean space, one can use the dot product to give a formula for the area of a triangle in terms of the coordinates of each vertex. Since this formula involves only addition, subtraction, and multiplication, it can be used as a…
In reversible dynamical systems, it is frequently of importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend…
We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions.…
In this paper we show that a notion of non-degeneracy which allows to develop Morse theory is generically satisfied for a large class of $C^2$-functionals defined on Banach spaces. The main element of novelty with respect to the previous…
We discuss the Cardy limit of 3d supersymmetric partition functions which allow the factorization into the hemisphere indices: the generalized superconformal index, the refined topologically twisted index and the squashed sphere partition…