Supersymmetric quantum mechanics on the lattice: III. Simulations and algorithms

In the fermion loop formulation the contributions to the partition function naturally separate into topological equivalence classes with a definite sign. This separation forms the basis for an efficient fermion simulation algorithm using a fluctuating open fermion string. It guarantees sufficient tunnelling between the topological sectors, and hence provides a solution to the fermion sign problem affecting systems with broken supersymmetry. Moreover, the algorithm shows no critical slowing down even in the massless limit and can hence handle the massless Goldstino mode emerging in the supersymmetry broken phase. In this paper – the third in a series of three – we present the details of the simulation algorithm and demonstrate its efficiency by means of a few examples.

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.

Testing for Measurement Equivalence in Surveys: Dimensions of Social Trust across Cultural Contexts

Our study evaluates the dimensionality and equivalence of social trust across cultural contexts, using new data from Switzerland and the World Values Survey 2005–2008. Whereas some scholars assert that trust should be regarded as a coherent concept, others claim that trust is better conceived of as a multidimensional concept. In contrast to the conventional dichotomy of the forms of social trust, we identify three distinct forms of trust, namely, particularized, generalized, and identity-based trust. Moreover, we dispute the view that respondents understand the wording of survey questions regarding social trust differently between different cultural contexts, which would imply that comparative research on trust is a pointless endeavor. Applying multiple-group confirmatory factor analysis to the various constructs of social trust, we conclude that one may study relationships among the three forms of trust and other theoretical constructs as well as compare latent means across cultural contexts. Our analyses therefore provide an optimistic outlook for future comparative analyses that investigate forms of social trust across cultural contexts.

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.

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.

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.