We construct singular solutions to special Lagrangian equations with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced correspondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.

It is argued that soft-collinear effective theory for processes involving both soft and collinear partons, such as exclusive B-meson decays, should include a new mode in addition to soft and collinear ones. These “soft-collinear messengers” can interact with both soft and collinear particles without taking them far off-shell. They thus can communicate between the soft and collinear sectors of t...

For large values of the Higgs mass the low energy structure of the gauged linear sigma model in the spontaneously broken phase can adequately be described by an effective field theory. We present a manifestly gauge-invariant functional technique to explicitly evaluate the corresponding effective Lagrangian from the underlying theory.

Making a global phase symmetry local usually involves the introduction of a rank 1, vector field in the definition of the covariant derivative. In this paper we show how a symmetry can be gauged using fields of various ranks in the definition of the covariant derivative. Some of these higher rank gauge theories share similarities with general relativity in that the covariant derivative has term...

The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to betterknown approaches using moduli spaces of stable maps and special Lagrangian fibrations, this approach follows more closely the original line of thinking that lead to the discovery of mirror symmetry by physicists. The ultimate goal of the vertex algebra a...

We consider the theories obtained by dimensional reduction to D = 1, 2, 3 of 4D supersymmetric Yang–Mills theories and calculate there the effective low-energy lagrangia describing moduli space dynamics — the low-dimensional analogs of the Seiberg–Witten effective lagrangian. The effective theories thus obtained are rather beautiful and interesting from mathematical viewpoint. In addition, thei...

A possible theory of an open membrane is proposed. The space-time coordinates, swept out by the membrane, are denoted as X"(<1°, <1 1, <1 2) where <1" are parameters. A simple Lagrangian which is invariant under there-parametrization 0" 0-7<1°'(<1), <1 1'=<11 and <1 2'=<12 is set up and the equations of motion and the boundary conditions are derived. After the gauge is fixed, they are solved in...

How large can the Lagrangian of an r-graph with m edges be? Frankl and Füredi [1] conjectured that the r-graph of size m formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs of size m. We prove the first ‘interesting’ case of this conjecture, namely that the 3-graph with ( t 3 ) edges and largest Lagrangian is [t](3). We also prove that this...

We define a semi-Hopf algebra which is more general than a Hopf algebra. Then we construct the supersymmetry algebra via the adjoint action on this semi-Hopf algebra. As a result we have a supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the Lagrangian N =1 and N =2 supersymmetry. ∗email : [email protected] 1