The central sphere of an ALE space

We consider the induced metric on the spherical fixed point set of a circle action on an ALE space and describe it by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic.

Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective

We revisit the correspondence between Calabi-Yau (CY) threefold isolated singularities \mathbf{X}𝐗 and five-dimensional superconformal field theories (SCFTs), which arise at low energy in M-theory on the space-time transverse to \mathbf{X}𝐗. Focussing on the case of toric CY singularities, we analyze the “gauge-theory phases” of the SCFT by exploiting fiberwise M-theory/type IIA duality. In this setup, the low-energy gauge group simply arises on stacks of coincident D6-branes wrapping 2-cycles in some ALE space of type A_{M-1}AM−1 fibered over a real line, and the map between the Kähler parameters of \mathbf{X}𝐗 and the Coulomb branch parameters of the field theory (masses and VEVs) can be read off systematically. Different type IIA “reductions” give rise to different gauge theory phases, whose existence depends on the particular (partial) resolutions of the isolated singularity \mathbf{X}𝐗. We also comment on the case of non-isolated toric singularities. Incidentally, we propose a slightly modified expression for the Coulomb-branch prepotential of 5d \mathcal{N}=1𝒩=1 gauge theories.

Stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of nonstationary nonlinear convection-diffusion initial- boundary value problem in a time-dependent domain. The problem is reformulated using the arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so-called ALE derivative and an additional convective term. The problem is discretized with the use of the ALE- space time discontinuous Galerkin method (ALE-STDGM). In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The nonlinear convection terms are discretized with the aid of a numerical flux. The main attention is paid to the proof of the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties.

The enhancement of supersymmetry in M-strings

We study two M5-branes on [Formula: see text] ALE space. We introduce some M2-branes suspended between the M5-branes. Then, the boundaries of M2-branes look like strings. We call them “M-strings.” The M-strings have [Formula: see text] supersymmetry by considering the brane configuration on [Formula: see text] ALE space. We calculate the partition function of M-strings by using the refined topological vertex formalism. We find that the supersymmetry of M-strings gets enhanced to [Formula: see text] by tuning some Kähler parameters. Furthermore, we discuss another possibility of the enhancement of supersymmetry which is different from the above one.