Heterotic T-folds with a small number of neutral moduli

We discuss non-geometric supersymmetric heterotic string models in D=4, in the framework of the free fermionic construction. We perform a systematic scan of models with four a priori left-right asymmetric Z2 projections and shifts. We analyze some 220 models, identifying 18 inequivalent classes and addressing variants generated by discrete torsions. They do not contain geometrical or trivial neutral moduli, apart from the dilaton. However, we show the existence of flat directions in the form of exactly marginal deformations and identify patterns of symmetry breaking where product gauge groups, realized at level one, are broken to their diagonal at higher level. We also describe an “inverse Gepner map” from Heterotic to Type II models that could be used, in certain non geometric settings, to define “effective” topological invariants.

Topological lattice actions for the 2d XY model

We consider the 2d XY Model with topological lattice actions, which are invariant against small deformations of the field configuration. These actions constrain the angle between neighbouring spins by an upper bound, or they explicitly suppress vortices (and anti-vortices). Although topological actions do not have a classical limit, they still lead to the universal behaviour of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition — at least up to moderate vortex suppression. In the massive phase, the analytically known Step Scaling Function (SSF) is reproduced in numerical simulations. However, deviations from the expected universal behaviour of the lattice artifacts are observed. In the massless phase, the BKT value of the critical exponent ηc is confirmed. Hence, even though for some topological actions vortices cost zero energy, they still drive the standard BKT transition. In addition we identify a vortex-free transition point, which deviates from the BKT behaviour.

High resolution immunoelectron microscopic localization of functional domains of laminin, nidogen, and heparan sulfate proteoglycan in epithelial basement membrane of mouse cornea reveals different topological orientations

Thin and ultrathin cryosections of mouse cornea were labeled with affinity-purified antibodies directed against either laminin, its central segments (domain 1), the end of its long arm (domain 3), the end of one of its short arms (domain 4), nidogen, or low density heparan sulfate proteoglycan. All basement membrane proteins are detected by indirect immunofluorescence exclusively in the epithelial basement membrane, in Descemet's membrane, and in small amorphous plaques located in the stroma. Immunoelectron microscopy using the protein A-gold technique demonstrated laminin domain 1 and nidogen in a narrow segment of the lamina densa at the junction to the lamina lucida within the epithelial basement membrane. Domain 3 shows three preferred locations at both the cellular and stromal boundaries of the epithelial basement membrane and in its center. Domain 4 is located predominantly in the lamina lucida and the adjacent half of the lamina densa. The low density heparan sulfate proteoglycan is found all across the basement membrane showing a similar uniform distribution as with antibodies against the whole laminin molecule. In Descemet's membrane an even distribution was found with all these antibodies. It is concluded that within the epithelial basement membrane the center of the laminin molecule is located near the lamina densa/lamina lucida junction and that its long arm favors three major orientations. One is close to the cell surface indicating binding to a cell receptor, while the other two are directed to internal matrix structures. The apparent codistribution of laminin domain 1 and nidogen agrees with biochemical evidence that nidogen binds to this domain.

On the signature of positive braids and adjacency for torus knots

The objects of study in this thesis are knots. More precisely, positive braid knots, which include algebraic knots and torus knots. In the first part of this thesis, we compare two classical knot invariants - the genus g and the signature σ - for positive braid knots. Our main result on positive braid knots establishes a linear lower bound for the signature in terms of the genus. In the second part of the thesis, a positive braid approach is applied to the study of the local behavior of polynomial functions from the complex affine plane to the complex numbers. After endowing polynomial function germs with a suitable topology, the adjacency problem arises: for a fixed germ f, what classes of germs g can be found arbitrarily close to f? We introduce two purely topological notions of adjacency for knots and discuss connections to algebraic notions of adjacency and the adjacency problem.

O(N) Models with Topological Lattice Actions

A variety of lattice discretisations of continuum actions has been considered, usually requiring the correct classical continuum limit. Here we discuss “weird” lattice formulations without that property, namely lattice actions that are invariant under most continuous deformations of the field configuration, in one version even without any coupling constants. It turns out that universality is powerful enough to still provide the correct quantum continuum limit, despite the absence of a classical limit, or a perturbative expansion. We demonstrate this for a set of O(N) models (or non-linear σ-models). Amazingly, such “weird” lattice actions are not only in the right universality class, but some of them even have practical benefits, in particular an excellent scaling behaviour.

Rasmussen's spectral sequences and the \germslN-concordance invariants

Combining known spectral sequences with a new spectral sequence relating reduced and unreduced slN-homology yields a relations hip between the Homflypt-homology of a knot and its slN-concordance invariants. As an application, some of the slN-concordance invariants are shown to be linearly independent.

Probing the moduli dependence of refined topological amplitudes

With the aim of providing a worldsheet description of the refined topological string, we continue the study of a particular class of higher derivative couplings Fg,n in the type II string effective action compactified on a Calabi–Yau threefold. We analyse first order differential equations in the anti-holomorphic moduli of the theory, which relate the Fg,n to other component couplings. From the point of view of the topological theory, these equations describe the contribution of non-physical states to twisted correlation functions and encode an obstruction for interpreting the Fg,n as the free energy of the refined topological string theory. We investigate possibilities of lifting this obstruction by formulating conditions on the moduli dependence under which the differential equations simplify and take the form of generalised holomorphic anomaly equations. We further test this approach against explicit calculations in the dual heterotic theory.

Comparison of different lattice definitions of the topological charge

We present a comparison of different definitions of the topological charge on the lattice, using a small-volume ensemble with 2 flavours of dynamical twisted mass fermions. The investigated definitions are: index of the overlap Dirac operator, spectral projectors, spectral flow of the HermitianWilson- Dirac operator and field theoretic with different kinds of smoothing of gauge fields (HYP and APE smearings, gradient flow, cooling). We also show some results on the topological susceptibility.