Related papers: Generalized Pentagon Equations

We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K<x,y,z> over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding…

We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$,…

The path integral description of the Wess-Zumino-Witten $\to$ Liouville reduction is formulated in a manner that exhibits the conformal invariance explicitly at each stage of the reduction process. The description requires a conformally…

We consider a field $F$ and positive integers $n$, $m$, such that $m$ is not divisible by $\mathrm{Char}(F)$ and is prime to $n!$. The absolute Galois group $G_F$ acts on the group $\mathbb{U}_n(\mathbb{Z}/m)$ of all $(n+1)\times(n+1)$…

Homological Projective duality (HP-duality) theory, introduced by Kuznetsov [42], is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete…

The KPZ fixed point is a 2d random field, conjectured to be the universal limiting fluctuation field for the height function of models in the KPZ universality class. Similarly, the periodic KPZ fixed point is a conjectured universal field…

In this paper, we develop the theory of Jacobian rings of open complete intersections, which mean a pair $(X,Z)$ where $X$ is a smooth complete intersection in the projective space and and $Z$ is a simple normal crossing divisor in $X$…

We extend the the definition of Kumjian-Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian-Pask algebras are universally defined and have a graded uniqueness theorem. We also…

To every compact oriented surface that is composed entirely out of 2-dimensional 0- and 1-handles, we construct a dg category using structures arising in Khovanov homology. These dg categories form part of the 2-dimensional layer (a.k.a.…

We introduce a hypergoemetirc series with two complex variables, which generalizes Appell's, Lauricella's and Kemp\'e de F\'eriet's hypergeometric series, and study the system of differential equations that it satisfies. We determine the…

In this self-contained book, following Edward Witten, Maxim Kontsevich, Greg Kuperberg and Dylan Thurston, we define an invariant Z of framed links in rational homology 3-spheres, and we study its properties. The invariant Z, which is often…

Let $A$ be a unital simple separable exact C$^*$-algebra which is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed Cuntz class is path connected. This result applies in particular…

We introduce a notion of holonomy in twistor space and construct a holonomy operator by use of a spinor-momenta formalism in twistor space. The holonomy operator gives a monodromy representation of the Knizhnik-Zamolodchikov (KZ) equation,…

Following the idea of Alekseev and Shatashvili we derive the path integral quantization of a modified relativistic particle action that results in the Feynman propagator of a free field with arbitrary spin. This propagator can be associated…

Let A be a nonassociative algebra such that the associator (A,A^2,A) vanishes. If A is freely generated by an element f, there are commuting derivations delta_n, n=1,2,..., such that delta_n(f) is a nonlinear homogeneous polynomial in f of…

We give an interpretation of the holographic correspondence between two-dimensional $BF$ theory on the punctured disk with gauge group ${\rm PSL}(2,\mathbb R)$ and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The…

This paper aims at generalizing some geometric properties of Grassmannians of finite dimensional vector spaces to the case of Grassmannnians of infinite dimensional ones, in particular for that of $k((z))$. It is shown that the Determinant…

Most of pointed Hopf algebras of dimension $p^m$ with large coradical are shown to be generalized path algebras. By the theory of generalized path algebras it is obtained that the representations, homological dimensions and radicals of…

In 2011, Penttila and Williford constructed an infinite new family of primitive $Q$-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space…

We illustrate the use of the theory of $qq$-characters by deriving the BPZ and KZ-type equations for the partition functions of certain surface defects in quiver ${\mathcal N}=2$ theories. We generate a surface defect in the linear quiver…