The BV formalism: Theory and application to a matrix model

We review the BV formalism in the context of [Formula: see text]-dimensional gauge theories. For a gauge theory [Formula: see text] with an affine configuration space [Formula: see text], we describe an algorithm to construct a corresponding extended theory [Formula: see text], obtained by introducing ghost and anti-ghost fields, with [Formula: see text] a solution of the classical master equation in [Formula: see text]. This construction is the first step to define the (gauge-fixed) BRST cohomology complex associated to [Formula: see text], which encodes many interesting information on the initial gauge theory [Formula: see text]. The second part of this article is devoted to the application of this method to a matrix model endowed with a [Formula: see text]-gauge symmetry, explicitly determining the corresponding [Formula: see text] and the general solution [Formula: see text] of the classical master equation for the model.

Perverse Schobers and Wall Crossing

For a balanced wall crossing in geometric invariant theory (GIT), there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterize when this is an intersection cohomology complex of a local system on the punctured disk.

The Hodge Theory of Maps

This chapter showcases two further lectures on the Hodge theory of maps, and they are mostly composed of exercises. The first lecture details a minimalist approach to sheaf cohomology, and then turns to the intersection cohomology complex, which is limited to the definition and calculation of the intersection complex Isubscript X of a variety of dimension d with one isolated singularity. Finally, this lecture discusses the Verdier duality. The second lecture sets out the Decomposition theorem, which is the deepest known fact concerning the homology of algebraic varieties. It then considers the relative hard Lefschetz and the hard Lefschetz for intersection cohomology groups.

3-Leibniz bialgebras (3-Lie bialgebras)

The aim of this paper is to extend the notion of Leibniz bialgebra (and Lie bialgebra) to [Formula: see text]-Leibniz bialgebra (and [Formula: see text]-Lie bialgebra) by use of the cohomology complex of [Formula: see text]-Leibniz algebras. Some theorems about Leibniz bialgebras are extended and proved in the case of [Formula: see text]-Leibniz bialgebras ([Formula: see text]-Lie bialgebras). Moreover, a new theorem on the correspondence between [Formula: see text]-Leibniz bialgebra and its associated Leibniz bialgebra is proved. Finally, some examples are discussed in detail.

Lefschetz operator and local Langlands mod ℓ: The regular case

Let p and ℓ be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod ℓ cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vignéras’s local Langlands correspondence modulo ℓ for unipotent representations.

Opérateur de Lefschetz sur les tours de Drinfeld et Lubin–Tate

We define and study a Lefschetz operator on the equivariant cohomology complex of the Drinfeld and Lubin–Tate towers. For ℓ-adic coefficients we show how this operator induces a geometric realization of the Langlands correspondence composed with the Zelevinski involution for elliptic representations. Combined with our previous study of the monodromy operator, this suggests a possible extension of Arthur’s philosophy for unitary representations occurring in the intersection cohomology of Shimura varieties to the possibly non-unitary representations occurring in the cohomology of Rapoport–Zink spaces. However, our motivation for studying the Lefschetz operator comes from the hope that its geometric nature will enable us to realize the mod-ℓ Langlands correspondence due to Vignéras. We discuss this problem and propose a conjecture.

LOCAL COMMUTATORS AND DEFORMATIONS IN CONFORMAL CHIRAL QUANTUM FIELD THEORIES

We study the general form of Möbius covariant local commutation relations in conformal chiral quantum field theories and show that they are intrinsically determined up to structure constants, which are subject to an infinite system of constraints. The deformation theory of these commutators is controlled by a cohomology complex, whose cochain spaces consist of linear maps that are subject to a complicated symmetry property, a generalization of the anti-symmetry of the Lie algebra case.