The Polytope of $k$-Star Densities

This paper describes the polytope $\mathbf{P}_{k;N}$ of $i$-star counts, for all $i\le k$, for graphs on $N$ nodes. The vertices correspond to graphs that are regular or as regular as possible. For even $N$ the polytope is a cyclic polytope, and for odd $N$ the polytope is well-approximated by a cyclic polytope. As $N$ goes to infinity, $\mathbf{P}_{k;N}$ approaches the convex hull of the moment curve. The affine symmetry group of $\mathbf{P}_{k;N}$ contains just a single non-trivial element, which corresponds to forming the complement of a graph.The results generalize to the polytope $\mathbf{P}_{I;N}$ of $i$-star counts, for $i$ in some set $I$ of non-consecutive integers. In this case, $\mathbf{P}_{I;N}$ can still be approximated by a cyclic polytope, but it is usually not a cyclic polytope itself.Polytopes of subgraph statistics characterize corresponding exponential random graph models. The elongated shape of the $k$-star polytope gives a qualitative explanation of some of the degeneracies found in such random graph models.

Affine Symmetry Tensors in Minkowski Space

Killing vectors are generators of symmetries in a spacetime. This article defines certain generalizations of Killing vectors, called affine symmetry tensors, or simply affine tensors. While the affine vectors of the Minkowski spacetime are well known, and partial results for valence n = 2 have been discussed, affine tensors of valence n > 2 have never been exhibited. In this article, we discuss a computational algorithm to compute affine tensors in Minkowski spacetime, and discuss the results for affine tensors of valence 2 ≤ n ≤ 7. After comparison with analogous results concerning Killing tensors, we make several conjectures about the spaces of affine tensors in Minkowski spacetime. KEYWORDS: Affine Symmetry Tensors; Affine Vectors; Killing Tensors; Killing Vectors; Minkowski Spacetime; Dimension; Maple CAS; Lie Derivative; Generalized Killing Tensor

COVARIANT INTEGRAL QUANTIZATIONS AND THEIR APPLICATIONS TO QUANTUM COSMOLOGY

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on group representation and probabilistic aspects of these constructions. Simple phase space examples illustrate the procedure: plane (Weyl-Heisenberg symmetry), half-plane (affine symmetry). Interesting applications to quantum cosmology (“smooth bouncing”) for Friedmann-Robertson-Walker metric are presented and those for Bianchi I and IX models are mentioned.

Viruses and fullerenes – symmetry as a common thread?

The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirements dictated by carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affine symmetry approach. This suggests the applicability of affine symmetry in a wider context in nature, as well as offering a novel perspective on the geometric principles underpinning carbon chemistry.

ON THE DESCRIPTION OF SURFACE OPERATORS IN ${\mathcal N} = 2^*$ SYM

Alday and Tachikawa [Lett. Math. Phys.94, 87 (2010)] observed that the Nekrasov partition function of [Formula: see text] superconformal gauge theories in the presence of fundamental surface operators can be associated to conformal blocks of a 2D CFT with affine sl(2) symmetry. This can be interpreted as the insertion of a fundamental surface operator changing the conformal symmetry from the Virasoro symmetry discovered in Ref. 2 to the affine Kac–Moody symmetry. A natural question arises as to how such a 2D CFT description can be extended to the case of non-fundamental surface operators. Motivated by this question, we review the results [Y. Hikida and V. Schomerus, JHEP0710, 064 (2007); S. Ribault, JHEP0805, 073 (2008)] and put them together to suggest a way to address the problem: It follows from this analysis that the expectation value of a non-fundamental surface operator in the SU(2) [Formula: see text] super Yang–Mills (YM) theory would be in correspondence with the expectation value of a single vertex operator in a two-dimensional CFT with reduced affine symmetry and whose central charge is parametrized by the integer number that labels the type of singularity of the surface operator.

SPACE–TIME STRUCTURE IN THE ULTIMATE THEORY

After criticizing the various existing attempts at extending the concept of the Minkowski space, the following problem is considered: If there is the ultimate theory at all, how should the space–time in it be formulated? The principle of "quantum priority" is proposed, and under this principle, it is argued that the ultimate theory should have the affine symmetry in the framework of quantum gravity. It is shown that the Poincaré symmetry of particle physics is realized as a result of spontaneous breakdown of the affine symmetry.