A note on entanglement edge modes in Chern Simons theory

We elaborate on the extended Hilbert space factorization of Chern Simons theory and show how this arises naturally from a proper regularization of the entangling surface in the Euclidean path integral. The regularization amounts to stretching the entangling surface into a co-dimension one surface which hosts edge modes of the Chern Simons theory when quantized on a spatial subregion. The factorized state is a regularized Ishibashi state and reproduces the well known topological entanglement entropies. We illustrate how the same factorization arises from the glueing of two spatial subregions via the entangling product defined by Donnelly and Freidel.Comment: Some typos fixe

Entanglement branes in a two-dimensional string theory

What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large $N$. The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the Modular hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an E-brane.Comment: 37 pages, 12 figures. v3: Modified title and abstrac

Gluing together Modular flows with free fermions

We revisit the calculation of multi-interval modular Hamiltonians for free fermions using a Euclidean path integral approach. We show how the multi-interval modular flow is obtained by gluing together the single interval modular flows. Using this relation, we obtain an exact expression for the multi-interval modular Hamiltonian and entanglement entropy in agreement with existing results. An essential ingredient in our derivation is the introduction of the \emp{modular action}. This determines the non-local field theory describing the free fermion reduced density matrix, and makes manifest it's non-local conformal symmetry and $U(1)$ Kacs-Moody symmetry.Comment: Substantially revised, errors and typos corrected. 29 pages, 9 figure

Genetic analysis of Brassica carinata

Brassica carinata is being actively pursued as a new industrial oil crop platform for the Canadian Prairies. A genetic assessment of B. carinata was performed to elucidate its evolutionary origins and create a genetic map to assist in locating genes and traits of interest that would help in marker-assisted breeding. First, genetic analysis using simple sequence repeat (SSR) markers, previously tested on B. juncea and B. napus, was performed, to examine the genetic diversity of 37 B. carinata lines. SSR analysis revealed world accessions were more diverse than lines conditioned to grow in the prairies. Diversity analysis revealed that the parental lines of a double haploid (DH) population, 179 and 345, obtained from the John Innes Centre (JIC), were among the more genetically diverse lines, supporting the use of this population for linkage mapping. Genetic markers created from 3’ targeted SNP discovery between 179 and 345, were tested on the DH population resulting in the generation of a B. carinata genetic linkage map essentially with no prior sequence data knowledge. This genetic map contained 341 SNP and 86 SSR loci identifying eight linkage groups belonging to the B genome, nine belonging to the C genome and two unidentified groups spanning 2041 cM. Comparative mapping of polymorphic markers identified in the amphidiploid B. carinata indicated the orientation of B and C genomes coincide with that of other Brassica species, and the two genomes have remained essentially unaltered, with no major chromosomal rearrangements since the formation of B. carinata. A lesser number of polymorphic markers were detected in the C genome, which suggested the B genome is more genetically diverse in B. carinata. Limited field trials of the 179 x 345 DH population were performed during the 2011 and 2012 growing seasons. Preliminary quantitative trait loci (QTLs) for agronomic traits including flowering time (FT), plant height (PH), and seed quality were identified