M-theory cosmology, octonions, error correcting codes

We study M-theory compactified on twisted 7-tori with G2-holonomy. The effective 4d supergravity has 7 chiral multiplets, each with a unit logarithmic Kähler potential. We propose octonion, Fano plane based superpotentials, codifying the error correcting Hamming (4, 7) code. The corresponding 7-moduli models have Minkowski vacua with one flat direction. We also propose superpotentials based on octonions/error correcting codes for Minkowski vacua models with two flat directions. We update phenomenological α-attractor models of inflation with 3α = 7, 6, 5, 4, 3, 1, based on inflation along these flat directions. These inflationary models reproduce the benchmark targets for detecting B-modes, predicting 7 different values of $$ r=12\alpha /{N}_e^2 $$ r = 12 α / N e 2 in the range 10−2 ≳ r ≳ 10−3, to be explored by future cosmological observations.

Quantum Computation and Measurements from an Exotic Space-Time R4

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture G Q ( 2 , 2 ) (at index 15), and the combinatorial Grassmannian Gr ( 2 , 8 ) (at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) R 4 s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.

Taxonomy of Three-Qubit Mermin Pentagrams

Given the fact that the three-qubit symplectic polar space features three different kinds of observables and each of its labeled Fano planes acquires a definite sign, we found that there are 45 distinct types of Mermin pentagrams in this space. A key element of our classification is the fact that any context of such pentagram is associated with a unique (positive or negative) Fano plane. Several intriguing relations between the character of pentagrams’ three-qubit observables and ‘valuedness’ of associated Fano planes are pointed out. In particular, we find two distinct kinds of negative contexts and as many as four positive ones.

The Ramsey Number of Fano Plane Versus Tight Path

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer~$N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to determine all $H$ that are $\mathbb{F}$-good.In this short paper we make progress on this problem and prove that the tight path of length $n$ is $\mathbb{F}$-good.

The Structure of Binary Matroids with no Induced Claw or Fano Plane Restriction

A well-known conjecture of András Gyárfás and David Sumner states that for every positive integer m and every finite tree T there exists k such that all graphs that do not contain the clique Km or an induced copy of T have chromatic number at most k. The conjecture has been proved in many special cases, but the general case has been open for several decades. The main purpose of this paper is to consider a natural analogue of the conjecture for matroids, where it turns out, interestingly, to be false. Matroids are structures that result from abstracting the notion of independent sets in vector spaces: that is, a matroid is a set M together with a nonempty hereditary collection I of subsets deemed to be independent where all maximal independent subsets of every set are equicardinal. They can also be regarded as generalizations of graphs, since if G is any graph and I is the collection of all acyclic subsets of E(G), then the pair (E(G),I) is a matroid. In fact, it is a binary matroid, which means that it can be represented as a subset of a vector space over F2. To do this, we take the space of all formal sums of vertices and represent the edge vw by the sum v+w. A set of edges is easily seen to be acyclic if and only if the corresponding set of sums is linearly independent. There is a natural analogue of an induced subgraph for matroids: an induced restriction of a matroid M is a subset M′ of M with the property that adding any element of M−M′ to M′ produces a matroid with a larger independent set than M′. The natural analogue of a tree with m edges is the matroid Im, where one takes a set of size m and takes all its subsets to be independent. (Note, however, that unlike with graph-theoretic trees there is just one such matroid up to isomorphism for each m.) Every graph can be obtained by deleting edges from a complete graph. Analogously, every binary matroid can be obtained by deleting elements from a finite binary projective geometry, that is, the set of all one-dimensional subspaces in a finite-dimensional vector space over F2. Finally, the analogue of the chromatic number for binary matroids is a quantity known as the critical number introduced by Crapo and Rota, which in the case of a graph G turns out to be ⌈log2(χ(G))⌉ -- that is, roughly the logarithm of its chromatic number. One of the results of the paper is that a binary matroid can fail to contain I3 or the Fano plane F7 (which is the simplest projective geometry) as an induced restriction, but also have arbitrarily large critical number. By contrast, the critical number is at most two if one also excludes the matroid associated with K5 as an induced restriction. The main result of the paper is a structural description of all simple binary matroids that have neither I3 nor F7 as an induced restriction.