We study derived categories arising from quivers with potential associated to a decorated marked surface S_Delta, in the sense taken in a paper by Qiu. We prove two conjectures from Qiu’s paper in which, under a bijection between certain objects in these categories and certain arcs in S, the dimensions of morphisms between these objects equal the intersection numbers between the corresponding arcs.

We study the fundamental groups of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(Gamma_N Q)of the Calabi–Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected. As an application, we show that the Donaldson–Thomas type invariant associated to Q can be calculated as a quantum dilogarithm function on its exchange graph. In the case of D(Gamma_N Q), we show that the faithfulness of the Seidel–Thomas braid group action (which is known for Q of type A or N=2) implies the simply connectedness of the space of stability conditions.

We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters for automorphic forms occurring in the discrete spectrum, given by the first named author.

We show that the Kazhdan-Lusztig category KL_k of level-k finite-length modules with highest-weight composition factors for the affine Lie superalgebra gl(1|1)ˆ has vertex algebraic braided tensor supercategory structure, and that its full subcategory O_k^fin of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple gl(1|1)ˆ-module in KL_k has a projective cover in O_k^fin, and we determine all fusion rules involving simple and projective objects in O_k^fin. Then using Knizhnik-Zamolodchikov equations, we prove that KL_k and O_k^fin are rigid. As an application of the tensor supercategory structure on O_k^fin, we study certain module categories for the affine Lie superalgebra sl(2|1)ˆ at levels 1 and −1/2. In particular, we obtain a tensor category of sl(2|1)ˆ-modules at level −1/2 that includes relaxed highest-weight modules and their images under spectral flow.

Let e be an arbitrary even nilpotent element in the general linear Lie super- algebra glM|N and let We be the associated finite W-superalgebra. Let Ym|n be the super Yangian associated to the Lie superalgebra glm|n. A subalgebra of Ym|n, called the shifted super Yangian and denoted by Ym|n(σ), is defined and studied. Moreover, an explicit iso- morphism between We and a quotient of Ym|n(σ) is established.