Integral forms for tensor powers of the Virasoro vertex operator algebra L(12,0)L(\frac{1}{2},0) and their modules

We construct integral forms containing the conformal vector $\omega$ in certain tensor powers of the Virasoro vertex operator algebra $L(\frac{1}{2},0)$, and we construct integral forms in certain modules for these algebras. When a triple of modules for a tensor power of $L(\frac{1}{2},0)$ have integral forms, we classify which intertwining operators among these modules respect the integral forms. As an application, we explore how these results might be used to obtain integral forms in framed vertex operator algebras.Comment: 20 page

Non-negative integral level affine Lie algebra tensor categories and their associativity isomorphisms

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra $\hat{\mathfrak{g}}$ at a fixed level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this category is equivalent as an abelian category to $\mathbf{D}(\mathfrak{g},\ell)$, the category of finite-dimensional modules for the Zhu's algebra $A(L_{\hat{\mathfrak{g}}}(\ell,0))$, which is a quotient of $U(\mathfrak{g})$. Our main result is a direct construction using Knizhnik-Zamolodchikov equations of the associativity isomorphisms in $\mathbf{D}(\mathfrak{g},\ell)$ induced from the associativity isomorphisms constructed by Huang and Lepowsky in $L_{hat{\mathfrak{g}}}(\ell,0)-\mathbf{mod}$. This construction shows that $\mathbf{D}(\mathfrak{g},\ell)$ is closely related to the Drinfeld category of $U(\mathfrak{g})[[\hbar]]$-modules used by Kazhdan and Lusztig to identify categories of $\hat{\mathfrak{g}}$-modules at irrational and most negative rational levels with categories of quantum group modules.Comment: 49 page

Auxin-induced growth inhibition a natural consequence of two-point attachment

It is characteristic of a great number of biologically active substances that the responses which they elicit are twofold, low concentrations of the material promoting a particular activity, and higher concentrations inhibiting it. This is the case with the auxin-induced growth responses of plants. An active auxin such as indole acetic acid (IAA) brings about and is essential to growth in length of stems, hypocotyls and other plant organs including the Avena coleoptile