Hopf Term, Loop Algebras and Three Dimensional Navier-Stokes Equation

The dynamics of the 3 dimensional perfect fluid is equivalent to the motion of vortex filaments or "strings". We study the action principle and find that it is described by the Hopf term of the nonlinear sigma model. The Poisson bracket structure is described by the loop algebra, for example, the Virasoro algebra or the analogue of O(3) current algebra. As a string theory, it is quite different from the standard Nambu-Goto string in its coupling to the extrinsic geometry. We also analyze briefly the two dimsensional case and give some emphasis on the $w_{1+\infty}$ structure.Comment: 11 pages, LateX file, Some of crucial references are adde

BPS Condition of String Junction from M theory

We give a simple derivation of BPS condition of string junction from M theoryComment: 6 pages, Latex, 2 figure

Plane Partition Realization of (Web of) W-algebra Minimal Models

Recently, Gaiotto and Rapcak (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Prochazka and Rapcak, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR's idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred it as the web of W-algebra (WoW). In this paper, we demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W-algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U(1) charge connecting two PPs is negative. For the simplest nontrivial WoW, N=2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.Comment: 37pages; references and minor comments added in 2nd versio

Matrix Theory, Hilbert Scheme and Integrable System

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relate the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These basis can be related to the eigenfunctions of Calogero-Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.Comment: 13 pages 1 figur

M5 from M2

Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case of infinite dimensional Lie 3-algebras based on the Nambu-Poisson structure of three dimensional manifolds. We show that the model contains self-dual 2-form gauge fields in 6 dimensions, and the result may be interpreted as the M5-brane world-volume action.Comment: 15 pages, minor modificatio

Aspects of Effective Theory for Multiple M5-Branes Compactified On Circle

A supersymmetric non-Abelian self-dual gauge theory with the explicit introduction of Kaluza-Klein modes is proposed to give a classical description of multiple M5-branes on $R^5 \times S^1$. The gauge symmetry is parametrized by Lie-algebra valued 1-forms with the redundancy of a 0-form, and the supersymmetry transformations without gauge-fixing are given. We study BPS configurations involving KK modes, including M-waves and M2-branes with non-trivial distributions around the circle. Finally, this supersymmetric gauge theory of two-forms can be equipped with more general non-Abelian gerbes in five dimensions.Comment: 34 pages, minor modificatio