Related papers: The $\hat{A}$ genus as a projective volume form on…

Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski to describe quantum states as projective families of density matrices over a collection of smaller, simpler…

Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and…

We construct a rigged Hilbert space for the square integrable functions on the line L^2(R) adding to the generators of the Weyl-Heisenberg algebra a new discrete operator, related to the degree of the Hermite polynomials. All together,…

Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts…

We study the relationship between the equations defining a projective variety and properties of its secant varieties. In particular, we use information about the syzygies among the defining equations to derive smoothness and normality…

Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector fields on $M$ with…

This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and polyvector fields. We then introduce the formalism of…

Let $X$ be a prehomogeneous vector space under a connected reductive group $G$ over $\mathbb{R}$. Assume that the open $G$-orbit $X^+$ admits a finite covering by a symmetric space. We study certain zeta integrals involving (i) Schwartz…

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed…

The space of generalized projective structures on a Riemann surface $\Sigma$ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of $W_N$-gravity on…

Let $X$ be a projective variety with an action of a reductive group $G$. Each ample $G$-line bundle $L$ on $X$ defines an open subset $X^{\rm ss}(L)$ of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of…

We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S\, =\, \sum_{i=1}^d x_i$, where $x_i \,\in\, X$ are distinct points. After defining branched ${\rm SO}(3,{\mathbb…

Given a connected real Lie group and a contractible homogeneous proper $G$--space $X$ furnished with a $G$--invariant volume form, a real valued volume can be assigned to any representation $\rho\colon \pi_1(M)\to G$ for any oriented closed…

We introduce volume forms on mapping stacks in derived algebraic geometry using a parametrized version of the Reidemeister-Turaev torsion. In the case of derived loop stacks we describe this volume form in terms of the Todd class. In the…

We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.

We develop a homology theory for directed spaces, based on the semi-abelian category of (non-unital) associative algebras. The major ingredient is a simplicial algebra constructed from convolution algebras of certain trace categories of a…

In this paper we study $A$-projections, i.e. operators of a Hilbert space $\HH$ which act as projections when a seminorm is considered in $\HH$. $A$-projections were introduced by Mitra and Rao \cite{[MitRao74]} for finite dimensional…

We study moduli spaces of twisted quasimaps to a hypertoric variety $X$, arising as the Higgs branch of an abelian supersymmetric gauge theory in three dimensions. These parametrise general quiver representations whose building blocks are…

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall…

Let $G$ be a reductive group and $X$ a smooth projective curve. We prove that, for $G$ classical and $\sigma$ an arbitrary $G$-local system on $X$, the space $\overline{\operatorname{Op}}^{gen}_{G,\sigma}$ of generic extended oper…