Black holes admitting a Freudenthal dual

The quantised charges x of four dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U-duality and whose U-invariant quartic norm Delta(x) determines the lowest order entropy. Here we introduce a Freudenthal duality x -> \tilde{x}, for which \tilde{\tilde{x}}=-x. Although distinct from U-duality it nevertheless leaves Delta(x) invariant. However, the requirement that \tilde{x} be integer restricts us to the subset of black holes for which Delta(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantised charges A of five dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N(A) determines the lowest order entropy. We introduce an analogous Jordan dual A*, with N(A) necessarily a perfect cube, for which A**=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift.Comment: 32 pages revtex, 10 tables; minor corrections, references adde

Three fermions with six single particle states can be entangled in two inequivalent ways

Using a generalization of Cayley's hyperdeterminant as a new measure of tripartite fermionic entanglement we obtain the SLOCC classification of three-fermion systems with six single particle states. A special subclass of such three-fermion systems is shown to have the same properties as the well-known three-qubit ones. Our results can be presented in a unified way using Freudenthal triple systems based on cubic Jordan algebras. For systems with an arbitrary number of fermions and single particle states we propose the Pl\"ucker relations as a sufficient and necessary condition of separability.Comment: 23 pages LATE

Is there a Jordan geometry underlying quantum physics?

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of the algebra of observables, in the same way as Lie groups belong to the Lie part. Both the Lie geometry and the Jordan geometry are well-adapted to describe certain features of quantum theory. We concentrate here on the mathematical description of the Jordan geometry and raise some questions concerning possible relations with foundational issues of quantum theory.Comment: 30 page

Observations on Integral and Continuous U-duality Orbits in N=8 Supergravity

One would often like to know when two a priori distinct extremal black p-brane solutions are in fact U-duality related. In the classical supergravity limit the answer for a large class of theories has been known for some time. However, in the full quantum theory the U-duality group is broken to a discrete subgroup and the question of U-duality orbits in this case is a nuanced matter. In the present work we address this issue in the context of N=8 supergravity in four, five and six dimensions. The purpose of this note is to present and clarify what is currently known about these discrete orbits while at the same time filling in some of the details not yet appearing in the literature. To this end we exploit the mathematical framework of integral Jordan algebras and Freudenthal triple systems. The charge vector of the dyonic black string in D=6 is SO(5,5;Z) related to a two-charge reduced canonical form uniquely specified by a set of two arithmetic U-duality invariants. Similarly, the black hole (string) charge vectors in D=5 are E_{6(6)}(Z) equivalent to a three-charge canonical form, again uniquely fixed by a set of three arithmetic U-duality invariants. The situation in four dimensions is less clear: while black holes preserving more than 1/8 of the supersymmetries may be fully classified by known arithmetic E_{7(7)}(Z) invariants, 1/8-BPS and non-BPS black holes yield increasingly subtle orbit structures, which remain to be properly understood. However, for the very special subclass of projective black holes a complete classification is known. All projective black holes are E_{7(7)}(Z) related to a four or five charge canonical form determined uniquely by the set of known arithmetic U-duality invariants. Moreover, E_{7(7)}(Z) acts transitively on the charge vectors of black holes with a given leading-order entropy.Comment: 43 pages, 8 tables; minor corrections, references added; version to appear in Class. Quantum Gra

Cardiovascular disease, cancer and mortality among people with type 2 diabetes and alcoholic or non-alcoholic fatty liver disease hospital admission

OBJECTIVE: To describe associations between alcoholic fatty liver disease (ALD) or non-alcoholic fatty liver disease (NAFLD) hospital admission and cardiovascular disease (CVD), cancer, and mortality in people with T2DM. RESEARCH DESIGN AND METHODS: We performed a retrospective cohort study using linked population-based routine data from the diabetes register, hospital, cancer and death records for people aged 40-89 years, diagnosed with T2DM in Scotland 2004-2013 who had one or more hospital admission records. Liver disease and outcomes were identified using International Classification of Diseases codes. We estimated hazard ratios from Cox proportional hazards models, adjusted for key risk factors (aHRs). RESULTS: There were 134,368 people with T2DM (1707 with ALD and 1452 with NAFLD) with mean follow-up of 4.3 years for CVD and 4.7 years for mortality. Among people with ALD, NAFLD or without liver disease hospital records respectively there were: 378, 320 and 21,873 CVD events, 268, 176 and 15,101 cancers and 724, 221 and 16,203 deaths. For ALD and NAFLD respectively, aHRs (95% CIs) compared to the group with no record of liver disease were: 1.59 (1.43, 1.76) and 1.70 (1.52, 1.90), for CVD; 40.3 (28.8, 56.5) and 19.12(11.71 31.2), for hepatocellular cancer (HCC); 1.28 (1.12, 1.47) and 1.10 (0.94, 1.29) for non-HCC cancer; 4.86 (4.50, 5.24) and 1.60 (1.40, 1.83) for all-cause mortality. CONCLUSIONS: Hospital records of ALD or NAFLD are associated, to varying degrees, with increased risk of CVD, cancer and mortality in people with T2DM

The effect of dapagliflozin on glycaemic control and other cardiovascular disease risk factors in type 2 diabetes mellitus:a real-world observational study

Aims/hypothesis: Dapagliflozin, a sodium–glucose cotransporter 2 (SGLT2) inhibitor, is indicated for improving glycaemic control in type 2 diabetes mellitus. Whether its effects on HbA1c and other variables, including safety outcomes, in clinical trials are obtained in real-world practice needs to be established. Methods: We used data from the comprehensive national diabetes register, the Scottish Care Information-Diabetes (SCI-Diabetes) collaboration database, available from 2004 to mid-2016. Data within this database were linked to mortality data from the General Registrar, available from the Information Services Division (ISD) of the National Health Service in Scotland. We calculated crude within-person differences between pre- and post-drug-initiation values of HbA1c, BMI, body weight, systolic blood pressure (SBP) and eGFR. We used mixed-effects regression models to adjust for within-person time trajectories in these measures. For completeness, we evaluated safety outcomes, cardiovascular disease events, lower-limb amputation and diabetic ketoacidosis, focusing on cumulative exposure effects, using Cox proportional hazard models, though power to detect such effects was limited. Results: Among 8566 people exposed to dapagliflozin over a median of 210 days the crude within-person change in HbA1c was −10.41 mmol/mol (−0.95%) after 3 months’ exposure. The crude change after 12 months was −12.99 mmol/mol (−1.19%) but considering the expected rise over time in HbA1c gave a dapagliflozin-exposure-effect estimate of −15.14 mmol/mol (95% CI −15.87, −14.41) (−1.39% [95% CI −1.45, −1.32]) at 12 months that was maintained thereafter. A drop in SBP of −4.32 mmHg (95% CI −4.84, −3.79) on exposure within the first 3 months was also maintained thereafter. Reductions in BMI and body weight stabilised by 6 months at −0.82 kg/m2 (95% CI −0.87, −0.77) and −2.20 kg (95% CI −2.34, −2.06) and were maintained thereafter. eGFR declined initially by −1.81 ml min−1 [1.73 m]−2 (95% CI −2.10, −1.52) at 3 months but varied thereafter. There were no significant effects of cumulative drug exposure on safety outcomes. Conclusions/interpretation: Dapagliflozin exposure was associated with reductions in HbA1c, SBP, body weight and BMI that were at least as large as in clinical trials. Dapagliflozin also prevented the expected rise in HbA1c and SBP over the period of study

Freudenthal triple classification of three-qubit entanglement

We show that the three-qubit entanglement classes: (0) Null, (1) Separable A-B-C, (2a) Biseparable A-BC, (2b) Biseparable B-CA, (2c) Biseparable C-AB, (3) W and (4) GHZ correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3 and 4 of a Freudenthal triple system defined over the Jordan algebra C+C+C. We also compute the corresponding SLOCC orbits.Comment: 11 pages, 2 figures, 6 tables, revtex; minor corrections, references added; version appearing in Phys. Rev.

Spectrum Generating Conformal and Quasiconformal U-Duality Groups, Supergravity and Spherical Vectors

After reviewing the algebraic structures that underlie the geometries of N=2 Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions with symmetric scalar manifolds, we give a unified realization of their three dimensional U-duality groups as spectrum generating quasiconformal groups. They are F_{4(4)}, E_{6(2)}, E_{7(-5)}, E_{8(-24)} and SO(n+2,4). Our formulation is covariant with respect to U-duality symmetry groups of corresponding five dimensional supergravity theories, which are SL(3,R), SL(3,C), SU*(6), E_{6(6)} and SO(n-1,1)X SO(1,1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character. We also give their quadratic Casimir operators and determine their values. Our work lays the algebraic groundwork for constructing the unitary representations of these groups induced by their geometric quasiconformal actions, which include the quaternionic discrete series. For rank 2 cases, SU(2,1) and G_{2(2)}, corresponding to simple N=2 supergravity in four and five dimensions, this program was carried out in arXiv:0707.1669. We also discuss the corresponding algebraic structures underlying symmetries of matter coupled N=4 and N>4 supergravity theories. They lead to quasiconformal realizations of split real forms of U-duality groups as a straightforward extension of the quaternionic real forms.Comment: Section 4 is split with the addition of a subsection on quadratic Casimir operators; references added; typos corrected. Latex file; 53 page