The massive supermembrane on a knot

We obtain the Hamiltonian formulation of the 11D Supermembrane theory non-trivially compactified on a twice punctured torus times a 9D Minkowski space-time. It corresponds to a M2-brane formulated in 11D space with ten non-compact dimensions. The critical points like the poles and the zeros of the fields describing the embedding of the Supermembrane in the target space are treated rigorously. The non-trivial compactification generates non-trivial mass terms appearing in the bosonic potential, which dominate the full supersymmetric potential and should render the spectrum of the (regularized) Supermembrane discrete with finite multiplicity. The behaviour of the fields around the punctures generates a cosmological term in the Hamiltonian of the theory.The massive supermembrane can also be seen as a nontrivial uplift of a supermembrane torus bundle with parabolic monodromy in M9 × T2. The moduli of the theory is the one associated with the punctured torus, hence it keeps all the nontriviality of the torus moduli even after the decompactification process to ten noncompact dimensions. The formulation of the theory on a punctured torus bundle is characterized by the (1, 1) − knots associated with the monodromies.

Shapes of hyperbolic triangles and once-punctured torus groups

Let $$\Delta $$ Δ be a hyperbolic triangle with a fixed area $$\varphi $$ φ . We prove that for all but countably many $$\varphi $$ φ , generic choices of $$\Delta $$ Δ have the property that the group generated by the $$\pi $$ π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$ φ ∈ ( 0 , π ) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $$\mathfrak {C}_\theta $$ C θ of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$ θ = 2 ( π - φ ) , and answer an analogous question for the holonomy map $$\rho _\xi $$ ρ ξ of such a hyperbolic structure $$\xi $$ ξ . In an appendix by Gao, concrete examples of $$\theta $$ θ and $$\xi \in \mathfrak {C}_\theta $$ ξ ∈ C θ are given where the image of each $$\rho _\xi $$ ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.

Two dialects for KZB equations: generating one-loop open-string integrals

Two different constructions generating the low-energy expansion of genus-one configuration-space integrals appearing in one-loop open-string amplitudes have been put forward in refs. [1–3]. We are going to show that both approaches can be traced back to an elliptic system of Knizhnik-Zamolodchikov-Bernard(KZB) type on the twice-punctured torus.We derive an explicit all-multiplicity representation of the elliptic KZB system for a vector of iterated integrals with an extra marked point and explore compatibility conditions for the two sets of algebra generators appearing in the two differential equations.

Complex Langevin analysis of 2D U(1) gauge theory on a torus with a θ term

Monte Carlo simulation of gauge theories with a θ term is known to be extremely difficult due to the sign problem. Recently there has been major progress in solving this problem based on the idea of complexifying dynamical variables. Here we consider the complex Langevin method (CLM), which is a promising approach for its low computational cost. The drawback of this method, however, is the existence of a condition that has to be met in order for the results to be correct. As a first step, we apply the method to 2D U(1) gauge theory on a torus with a θ term, which can be solved analytically. We find that a naive implementation of the method fails because of the topological nature of the θ term. In order to circumvent this problem, we simulate the same theory on a punctured torus, which is equivalent to the original model in the infinite volume limit for |θ| < π. Rather surprisingly, we find that the CLM works and reproduces the exact results for a punctured torus even at large θ, where the link variables near the puncture become very far from being unitary.

Symmetry of topographs of Markoff forms

The Markoff spectrum is defined as the set of normalized values of arithmetic minima of indefinite quadratic forms. In the theory of the Markoff spectrum we observe various kinds of symmetry. Each of Conway’s topographs of quadratic forms which give values in the discrete part of the Markoff spectrum has a special infinite path consisting of edges. It has symmetry with respect to a translation along the path and countable central symmetries by which the path is invariant. We prove that these properties are obtained from the fact that the path is a discretization of a geodesic in the upper half-plane which corresponds to a value of the discrete part of the Markoff spectrum and projects to a simple closed geodesic on the once punctured torus with the highest degree of symmetry.

A classification of (1,1)-positions

In this paper, we describe the equivalence classes of simple arcs between the two punctures on a 2-punctured torus [Formula: see text] up to isotopy by using the given four generators [Formula: see text] and [Formula: see text]. Actually, we show that a class of simple arcs is represented by an ordered sequence of four integers. Also, we introduce an algorithm to check whether or not an ordered sequence of four integers represents a class of simple arcs in [Formula: see text] We want to point out that this result classifies the [Formula: see text]-positions.