Derived representation schemes and Nakajima quiver varieties

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on $$S^4$$ S 4 . On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.

A note on super Koszul complex and the Berezinian

We construct the super Koszul complex of a free supercommutative A-module V of rank p|q and prove that its homology is concentrated in a single degree and it yields an exact resolution of A. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to $$\Pi ^{p+q} A$$ Π p + q A , with $$\Pi $$ Π the parity changing functor. Finally, we show that, given an automorphism of V, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of V.

Nilpotence Varieties

We consider algebraic varieties canonically associated with any Lie superalgebra, and study them in detail for super-Poincaré algebras of physical interest. They are the locus of nilpotent elements in (the projectivized parity reversal of) the odd part of the algebra. Most of these varieties have appeared in various guises in previous literature, but we study them systematically here, from a new perspective: As the natural moduli spaces parameterizing twists of a super-Poincaré-invariant physical theory. We obtain a classification of all possible twists, as well as a systematic analysis of unbroken symmetry in twisted theories. The natural stratification of the varieties, the identification of strata with twists, and the action of Lorentz and R-symmetry are emphasized. We also include a short and unconventional exposition of the pure spinor superfield formalism, from the perspective of twisting, and demonstrate that it can be applied to construct familiar multiplets in four-dimensional minimally supersymmetric theories. In all dimensions and with any amount of supersymmetry, this technique produces BRST or BV complexes of supersymmetric theories from the Koszul complex of the maximal ideal over the coordinate ring of the nilpotence variety, possibly tensored with any equivariant module over that coordinate ring. In addition, we remark on a natural connection to the Chevalley–Eilenberg complex of the supertranslation algebra, and give two applications related to these ideas: a calculation of Chevalley–Eilenberg cohomology for the (2, 0) algebra in six dimensions, and a degenerate BV complex encoding the type IIB supergravity multiplet.

Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free $${{\mathscr {O}}}_X$$ O X -module $${{\mathscr {A}}}$$ A which has a Lie algebroid structure. Here X is either a complex manifold or a regular scheme over an algebraically closed field k. One spectral sequence can be associated with $${{\mathscr {A}}}$$ A by choosing a global section V of $${{\mathscr {A}}}$$ A , and considering a Koszul complex with a differential given by inner product by V. This spectral sequence is shown to degenerate at the second page by using Deligne’s degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E of a holomolorphic vector bundle $${{\mathscr {E}}}$$ E on a complex manifold. If V is a differential operator on $${{\mathscr {E}}}$$ E with scalar symbol, i.e, a global section of $${{{\mathscr {D}}}_{{{\mathscr {E}}}}}$$ D E , we associate with the pair $$({{\mathscr {E}}},V)$$ ( E , V ) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($${{\mathscr {E}}}=0$$ E = 0 ) case.

DETECTING KOSZULNESS AND RELATED HOMOLOGICAL PROPERTIES FROM THE ALGEBRA STRUCTURE OF KOSZUL HOMOLOGY

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^{R}$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^{R}$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen–Macaulay local ring are rational, sharing a common denominator.

Golod–Shafarevich-Type Theorems and Potential Algebras

Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

Representations of the fundamental group and the torsor of deformations. Global aspects

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.