[en] We obtain Zakrzewski's deformation of Fun SL(2) through the construction of a *-product on SL(2). We then give the deformation of U(sl(2)) dual to this, as well as a Poincare basis for both algebras. (orig.)

[en] A 2-torsion topological phase exists for Hamiltonians symmetric under the wallpaper group with glide reflection symmetry, corresponding to the unorientable cycle of the Klein bottle fundamental domain. We prove a mod 2 twisted Toeplitz index theorem, which implies a bulk-edge correspondence between this bulk phase and the exotic topological zero modes that it acquires along a boundary glide axis.

[en] We propose a simple approach to formal deformations of associative algebras. It exploits the machinery of multiplicative coresolutions of an associative algebra A in the category of A-bimodules. Specifically, we show that certain first-order deformations of A extend to all orders and we derive explicit recurrent formulas determining this extension. In physical terms, this may be regarded as the deformation quantization of noncommutative Poisson structures on A.

[en] With the aim of better understanding the class of 4D theories generated by compactifications of 6D superconformal field theories (SCFTs), we study the structure of $\mathcal{N}=1$ supersymmetric punctures for class ${\mathcal{S}}_{\mathrm{\Gamma <!--\; ??\; -->}}$ theories, namely the 6D SCFTs obtained from M5-branes probing an ADE singularity. For M5-branes probing a ${\mathbb{C}}^2/{\mathbb{Z}}_k$ singularity, the punctures are governed by a dynamical system in which evolution in time corresponds to motion to a neighboring node in an affine A-type quiver. Classification of punctures reduces to determining consistent initial conditions which produce periodic orbits. The study of this system is particularly tractable in the case of a single M5-brane. Even in this “simple” case, the solutions exhibit a remarkable level of complexity: Only specific rational values for the initial momenta lead to periodic orbits and small perturbations in these values lead to vastly different late-time behavior. Another difference from half BPS punctures of class $\mathcal{S}$ theories includes the appearance of a continuous complex “zero mode” modulus in some puncture solutions. The construction of punctures with higher-order poles involves a related set of recursion relations. The resulting structures also generalize to systems with multiple M5-branes as well as probes of D- and E-type orbifold singularities.

[en] We consider the trigonometric classical r-matrix for ${\mathfrak{g}\mathfrak{l}}_N$ and the associated quantum Gaudin model. We produce higher Hamiltonians in an explicit form by applying the limit $q\to <!--\; ???\; -->1$ to elements of the Bethe subalgebra for the XXZ model.

[en] Vector bundles and double vector bundles, or twofold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of supergeometry and $\mathbb{Z}$-graded manifolds of degree $\le <!--\; ???\; -->2$. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of $\mathbb{Z}$-graded manifolds of degree $\le <!--\; ???\; -->2$ and the category of (double) vector bundles with additional structures. In this paper we study the geometrization process in the case of ${\mathbb{Z}}^r$-graded manifolds of type $\mathrm{\Delta <!--\; ??\; -->}$, where $\mathrm{\Delta <!--\; ??\; -->}$ is a certain weight system and r is the rank of $\mathrm{\Delta <!--\; ??\; -->}$. We establish an equivalence between a subcategory of the category of n-fold vector bundles and the category of graded manifolds of type $\mathrm{\Delta <!--\; ??\; -->}$.

[en] In the spectral and scattering theory for a Schrödinger operator with a time-periodic potential $H(t)=p^2/2+V(t,x)$, the Floquet Hamiltonian $K=-<!--\; ???\; -->i{\mathrm{\partial <!--\; ???\; -->}}_t+H(t)$ associated with H(t) plays an important role frequently, by virtue of the Howland–Yajima method. In this paper, we introduce a new conjugate operator for K in the standard Mourre theory, that is different from the one due to Yokoyama, in order to relax a certain smoothness condition on V.

[en] From ’t Hooft’s argument, one expects that the analyticity domain of an asymptotically free quantum field theory is horn shaped. In the usual Borel summation, the function is obtained through a Laplace transform and thus has a much larger analyticity domain. However, if the summation process goes through the process called acceleration by Écalle, one obtains such a horn-shaped analyticity domain. We therefore argue that acceleration, which allows to go beyond standard Borel summation, must be an integral part of the toolkit for the study of exactly renormalisable quantum field theories. We sketch how this procedure is working and what are its consequences.

[en] Motivated by gauge theory, we develop a general framework for chain complex-valued algebraic quantum field theories. Building upon our recent operadic approach to this subject, we show that the category of such theories carries a canonical model structure and explain the important conceptual and also practical consequences of this result. As a concrete application, we provide a derived version of Fredenhagen’s universal algebra construction, which is relevant e.g. for the BRST/BV formalism. We further develop a homotopy theoretical generalization of algebraic quantum field theory with a particular focus on the homotopy-coherent Einstein causality axiom. We provide examples of such homotopy-coherent theories via (1) smooth normalized cochain algebras on $\mathrm{\infty <!--\; ???\; -->}$-stacks, and (2) fiber-wise groupoid cohomology of a category fibered in groupoids with coefficients in a strict quantum field theory.

[en] We present a reduction theory for first-order Lagrangian field theories which takes into account the conservation of momenta. The relation between the solutions of the original problem with a prescribed value of the momentum and the solutions of the reduced problem is established. An illustrative example is discussed in detail.