Dimer geometry, amoebae and a vortex dimer model

We present a geometrical approach for studying dimers. We introduce a connection for dimer problems on bipartite and non-bipartite graphs. In the bipartite case the connection is flat but has non-trivial ${\bf Z}_2$ holonomy round certain curves. This holonomy has the universality property that it does not change as the number of vertices in the fundamental domain of the graph is increased. It is argued that the K-theory of the torus, with or without punctures, is the appropriate underlying invariant. In the non-bipartite case the connection has non-zero curvature as well as non-zero Chern number. The curvature does not require the introduction of a magnetic field. The phase diagram of these models is captured by what is known as an amoeba. We introduce a dimer model with negative edge weights that give rise to vortices. The amoebae for various models are studied with particular emphasis on the case of negative edge weights which corresponds to the presence of vortices. Vortices gives rise to new kinds of amoebae with certain singular structures which we investigate. On the amoeba of the vortex full hexagonal lattice we find the partition function corresponds to that of a massless Dirac doublet.Comment: 25 pages, 9 figures Latest version: some references added and typos remove

Ray-Singer Torsion, Topological field theories and the Riemann zeta function at s=3

Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large $p$ behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for zeta(3): the ordinary Riemann zeta function evaluated at s=3. The torsion turns out to be trivial for the cases L(6,1), L((10,3) and L(12,5) and is, in general, greater than unity for large p and less than unity for a finite number of p and q.Comment: 10 page

BRST Quantisation and the Product Formula for the Ray-Singer Torsion

We give a quantum field theoretic derivation of the formula obeyed by the Ray-Singer torsion on product manifolds. Such a derivation has proved elusive up to now. We use a BRST formalism which introduces the idea of an infinite dimensional Universal Gauge Fermion, and is of independent interest being applicable to situations other than the ones considered here. We are led to a new class of Fermionic topological field theories. Our methods are also applicable to combinatorially defined manifolds and methods of discrete approximation such as the use of a simplicial lattice or finite elements. The topological field theories discussed provide a natural link between the combinatorial and analytic torsion.Comment: 24 pages. TEX error of first version corrected: a \input is delete

Cylinders with a steel-concrete-steel wall to resist external pressure

[EN] In the 1980’s Manchester University carried out over 110 tests on cylinders with a composite wall (steel-concrete-steel) subjected to external pressure as already reported in the literature. This paper describes further tests on 9 cylinders with a composite wall and a dome end subjected to external pressure and reports the results and compares them with theory. The cylinders were 500 mm diameter and 1250 mm long and four of them had penetrations through the cylinder wall. These tests were carried out under contract for Tecnomare SpA of Italy and have not been previously reported because of confidentiality reasons. The agreement between test behaviour, failure load and the theory developed at Manchester University is good. The philosophy for the design of such vessels for seabed structures is discussed and a ‘depth margin’ method proposed as it is a more realistic way of applying safety. Examples of designs for different depths are given and compared with the predicted failure pressure.Tecnomare SpA of Italy is thanked for providing the financial support for this work. The tests were carried out at Manchester University by Dr. Tom Nash, John Smith & Alan Graham under the direction of the late Professor Peter Montague.Goode, C.; Nash, T. (2018). Cylinders with a steel-concrete-steel wall to resist external pressure. En Proceedings of the 12th International Conference on Advances in Steel-Concrete Composite Structures. ASCCS 2018. Editorial Universitat Politècnica de València. 647-652. https://doi.org/10.4995/ASCCS2018.2018.7066OCS64765

\u3ci\u3eHistory of the war in Affghanistan, from its commencement to its close; including a general sketch of the policy, and the various circumstances which induced the British government to interfere in the affairs of Affghanistan. From the journal and letters of an officer high in rank, and who has served many years in the Indian army. Edited by Charles Nash, esq., with an introductory description of the country, and its political state previous to the war \u3c/i\u3e

History of the war in Affghanistan, from its commencement to its close; including a general sketch of the policy, and the various circumstances which induced the British government to interfere in the affairs of Affghanistan. From the journal and letters of an officer high in rank, and who has served many years in the Indian army. Edited by Charles Nash, esq., with an introductory description of the country, and its political state previous to the war

Zeta function continuation and the Casimir energy on odd- and even-dimensional spheres

The zeta function continuation method is applied to compute the Casimir energy on spheres SN. Both odd and even dimensional spheres are studied. For the appropriate conformally modified Laplacian A the Casimir energy $is shown to be finite for all dimensions even though, generically, it should diverge in odd dimensions due to the presence of a pole in the associated zeta function ζA(s). The residue of this pole is computed and shown to vanish in our case. An explicit analytic continuation of ζA(s) is constructed for all values of N. This enables us to calculate$ exactly and we find that the Casimir energy vanishes in all even dimensions. For odd dimensions δ is never zero but alternates in sign as N increases through odd values. Some results are also derived for the Casimir energy of other operators of Laplacian type

Chiral Fermions and Spinc structures on Matrix approximations to manifolds

The Atiyah-Singer index theorem is investigated on various compact manifolds which admit finite matrix approximations (``fuzzy spaces'') with a view to applications in a modified Kaluza-Klein type approach in which the internal space consists of a finite number of points. Motivated by the chiral nature of the standard model spectrum we investigate manifolds that do not admit spinors but do admit Spinc structures. It is shown that, by twisting with appropriate bundles, one generation of the electroweak sector of the standard model, including a right-handed neutrino, can be obtained in this way from the complex projective space Bbb CBbb P2. The unitary grassmannian U(5)/(U(3) Ã U(2)) yields a spectrum that contains the correct charges for the Fermions of the standard model, with varying multiplicities for the different particle states

The Standard Model Fermion Spectrum from Complex Projective spaces

It is shown that the quarks and leptons of the standard model, including a right-handed neutrino, can be obtained by gauging the holonomy groups of complex projective spaces of complex dimensions two and three. The spectrum emerges as chiral zero modes of the Dirac operator coupled to gauge fields and the demonstration involves an index theorem analysis on a general complex projective space in the presence of topologically non-trivial SU(n)xU(1) gauge fields. The construction may have applications in type IIA string theory and non-commutative geometry

Modular invariance, lattice field theories and finite size corrections

We give a lattice theory treatment of certain one and two dimensional quantum field theories. In one dimension we construct a combinatorial version of a non-trivial field theory on the circle which is of some independent interest in itself while in two dimensions we consider a field theory on a toroidal triangular lattice. We take a continuous spin Gaussian model on a toroidal triangular lattice with periods $L_0$ and $L_1$ where the spins carry a representation of the fundamental group of the torus labeled by phases $u_0$ and $u_1$. We compute the {\it exact finite size and lattice corrections}, to the partition function $Z$, for arbitrary mass $m$ and phases $u_i$. Summing $Z^{-1/2}$ over a specified set of phases gives the corresponding result for the Ising model on a torus. An interesting property of the model is that the limits $m\rightarrow0$ and $u_i\rightarrow0$ do not commute. Also when $m=0$ the model exhibits a {\it vortex critical phase} when at least one of the $u_i$ is non-zero. In the continuum or scaling limit, for arbitrary $m$, the finite size corrections to $-\ln Z$ are {\it modular invariant} and for the critical phase are given by elliptic theta functions. In the cylinder limit $L_1\rightarrow\infty$ the ``cylinder charge'' $c(u_0,m^2L_0^2)$ is a non-monotonic function of $m$ that ranges from $2(1+6u_0(u_0-1))$ for $m=0$ to zero for $m\rightarrow\infty$ but from which one can determine the central charge $c$. The study of the continuum limit of these field theories provides a kind of quantum theoretic analog of the link between certain combinatorial and analytic topological quantities.Comment: 25 pages Plain Te