Conformal Field Theories, Representations and Lattice Constructions

An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and $Z_2$-twisted theories, $H(\Lambda)$ and $\tilde H(\Lambda)$ respectively, which may be constructed from a suitable even Euclidean lattice $\Lambda$. Similarly, one may construct lattices $\Lambda_C$ and $\tilde\Lambda_C$ by analogous constructions from a doubly-even binary code $C$. In the case when $C$ is self-dual, the corresponding lattices are also. Similarly, $H(\Lambda)$ and $\tilde H(\Lambda)$ are self-dual if and only if $\Lambda$ is. We show that $H(\Lambda_C)$ has a natural ``triality'' structure, which induces an isomorphism $H(\tilde\Lambda_C)\equiv\tilde H(\Lambda_C)$ and also a triality structure on $\tilde H(\tilde\Lambda_C)$. For $C$ the Golay code, $\tilde\Lambda_C$ is the Leech lattice, and the triality on $\tilde H(\tilde\Lambda_C)$ is the symmetry which extends the natural action of (an extension of) Conway's group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurman's construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFT's. We find that of the 48 theories $H(\Lambda)$ and $\tilde H(\Lambda)$ with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.Comment: 65 page

Rooted Spiral Trees on Hyper-cubical lattices

We study rooted spiral trees in 2,3 and 4 dimensions on a hyper cubical lattice using exact enumeration and Monte-Carlo techniques. On the square lattice, we also obtain exact lower bound of 1.93565 on the growth constant $\lambda$. Series expansions give $\theta=-1.3667\pm 0.001$ and $\nu = 1.3148\pm0.001$ on a square lattice. With Monte-Carlo simulations we get the estimates as $\theta=-1.364\pm0.01$, and $\nu = 1.312\pm0.01$. These results are numerical evidence against earlier proposed dimensional reduction by four in this problem. In dimensions higher than two, the spiral constraint can be implemented in two ways. In either case, our series expansion results do not support the proposed dimensional reduction.Comment: replaced with published versio

A new transfer-matrix algorithm for exact enumerations: Self-avoiding polygons on the square lattice

We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm. The new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110.Comment: 17 pages, 8 figures, IoP style file

Linking routinely collected social work, education and health data to enable monitoring of the health and health care of school-aged children in state care (‘looked after children’) in Scotland: a national demonstration project

Background and objectives: Children in state care (‘looked after children’) have poorer health than children who are not looked after. Recent developments in Scotland and elsewhere have aimed to improve services and outcomes for looked after children. Routine monitoring of the health outcomes of looked after children compared to those of their non-looked after peers is currently lacking. Developing capacity for comparative monitoring of population based outcomes based on linkage of routinely collected administrative data has been identified as a priority. To our knowledge there are no existing population based data linkage studies providing data on the health of looked after and non-looked after children at national level. Smaller scale studies that are available generally provide very limited information on linkage methods and hence do not allow scrutiny of bias that may be introduced through the linkage process. Study design and methods: National demonstration project testing the feasibility of linking routinely collected looked after children, education, and health data. Participants: All children in publicly funded school in Scotland in 2011/12. Results: Linkage between looked after children data and the national pupil census classified 10,009 (1.5%) and 1,757 (0.3%) of 670,952 children as, respectively, currently and previously looked after. Recording of the unique pupil identifier (Scottish Candidate Number, SCN) on looked after children returns is incomplete, with 66% of looked after records for 2011/12 for children of possible school age containing a valid SCN. This will have resulted in some under-ascertainment of currently and, particularly, previously looked after children within the general pupil population. Further linkage of the pupil census to the NHS Scotland master patient index demonstrated that a safe link to the child’s unique health service (Community Health Index, CHI) number could be obtained for a very high proportion of children in each group (94%, 95%, and 95% of children classified as currently, previously, and non-looked after respectively). In general linkage rates were higher for older children and those living in more affluent areas. Within the looked after group, linkage rates were highest for children with the fewest placements and for those in permanent fostering. Conclusions: This novel data linkage demonstrates the feasibility of monitoring population based health outcomes of school aged looked after and non-looked after children using linked routine administrative data. Improved recording of the unique pupil identifier number on looked after data returns would be beneficial. Extending the range of personal identifiers on looked after children returns would enable linkage to health data for looked after children who are not in publicly funded schooling (i.e. those who are pre- or post-school, home schooled, or in independent schooling)

Size and area of square lattice polygons

We use the finite lattice method to calculate the radius of gyration, the first and second area-weighted moments of self-avoiding polygons on the square lattice. The series have been calculated for polygons up to perimeter 82. Analysis of the series yields high accuracy estimates confirming theoretical predictions for the value of the size exponent, $\nu=3/4$, and certain universal amplitude combinations. Furthermore, a detailed analysis of the asymptotic form of the series coefficients provide the firmest evidence to date for the existence of a correction-to-scaling exponent, $\Delta = 3/2$.Comment: 12 pages 3 figure

Inequalities in the dental health needs and access to dental services among looked after children in Scotland: a population data linkage study

Background: There is limited evidence on the health needs and service access among children and young people who are looked after by the state. The aim of this study was to compare dental treatment needs and access to dental services (as an exemplar of wider health and well-being concerns) among children and young people who are looked after with the general child population. Methods: Population data linkage study utilising national datasets of social work referrals for ‘looked after’ placements, the Scottish census of children in local authority schools, and national health service’s dental health and service datasets. Results: 633 204 children in publicly funded schools in Scotland during the academic year 2011/2012, of whom 10 927 (1.7%) were known to be looked after during that or a previous year (from 2007–2008). The children in the looked after children (LAC) group were more likely to have urgent dental treatment need at 5 years of age: 23%vs10% (n=209/16533), adjusted (for age, sex and area socioeconomic deprivation) OR 2.65 (95% CI 2.30 to 3.05); were less likely to attend a dentist regularly: 51%vs63% (n=5519/388934), 0.55 (0.53 to 0.58) and more likely to have teeth extracted under general anaesthesia: 9%vs5% (n=967/30253), 1.91 (1.78 to 2.04). Conclusions: LAC are more likely to have dental treatment needs and less likely to access dental services even when accounting for sociodemographic factors. Greater efforts are required to integrate child social and healthcare for LAC and to develop preventive care pathways on entering and throughout their time in the care system

The effect of introducing a financial incentive to promote application of fluoride varnish in dental practice in Scotland:a natural experiment

Background: Financial incentives are often used to influence professional practice, yet the factors which influence their effectiveness and their behavioural mechanisms are not fully understood. In keeping with clinical guidelines, Childsmile (Scotland’s oral health improvement programme) advocates twice yearly fluoride varnish application (FVA) for children in dental practice. To support implementation Childsmile offered dental practitioners a fee-per-item payment for varnishing 2–5-year-olds’ teeth through a pilot. In October 2011 payment was extended to all dental practitioners. This paper compares FVA pre- and post-roll-out and explores the financial incentive’s behavioural mechanisms. Methods: A natural experimental approach using a longitudinal cohort of dental practitioners (n = 1090) compared FVA pre- (time 1) and post- (time 2) financial incentive. Responses from practitioners who did not work in a Childsmile pilot practice when considering their 2–5-year-old patients (novel incentive group) were compared with all other responses (continuous incentive group). The Theoretical Domains Framework (TDF) was used to measure change in behavioural mechanisms associated with the incentive. Analysis of covariance was used to investigate FVA rates and associated behavioural mechanisms in the two groups. Results: At time 2, 709 74%, of eligible responders, were followed up. In general, FVA rates increased over time for both groups; however, the novel incentive group experienced a greater increase (β [95% CI] = 0.82 [0.72 to 0.92]) than the continuous incentive group. Despite this, only 33% of practitioners reported ‘always’ varnishing increased risk 2–5-year-olds’ teeth following introduction of the financial incentive, 19% for standard risk children. Domain scores at time 2 (adjusting for time 1) increased more for the novel incentive group (compared to the continuous incentive group) for five domains: knowledge, social/professional role and identity, beliefs about consequences, social influences and emotion. Conclusions: In this large, prospective, population-wide study, a financial incentive moderately increased FVA in dental practice. Novel longitudinal use of a validated theoretical framework to understand behavioural mechanisms suggested that financial incentives operate through complex inter-linked belief systems. While financial incentives are useful in narrowing the gap between clinical guidelines and FVA, multiple intervention approaches are required

On the Relationship between the Uniqueness of the Moonshine Module and Monstrous Moonshine

We consider the relationship between the conjectured uniqueness of the Moonshine Module, ${\cal V}^\natural$, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possible $Z_n$ meromorphic orbifold constructions of ${\cal V}^\natural$ based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster group $M$ together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that ${\cal V}^\natural$ is unique, we then consider meromorphic orbifoldings of ${\cal V}^\natural$ and show that Monstrous Moonshine holds if and only if the only meromorphic orbifoldings of ${\cal V}^\natural$ give ${\cal V}^\natural$ itself or the Leech theory. This constraint on the meromorphic orbifoldings of ${\cal V}^\natural$ therefore relates Monstrous Moonshine to the uniqueness of ${\cal V}^\natural$ in a new way.Comment: 53 pages, PlainTex, DIAS-STP-93-0