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We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps inducing a homology isomorphism. This approach, naturally arising in string…

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and…

We consider homological mirror symmetry in the context of hypertoric varieties, showing that appropriate categories of B-branes (that is, coherent sheaves) on an additive hypertoric variety match a category of A-branes on a Dolbeault…

Let $C_2$ denote the cyclic group of order two. Given a manifold with a $C_2$-action, we can consider its equivariant Bredon $RO(C_2)$-graded cohomology. In this paper, we develop a theory of fundamental classes for equivariant submanifolds…

Strata of $k$-differentials on smooth curves parameterize sections of the $k$-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $\kappa$ and $\psi$…

We investigate the duality structure of quantum lattice systems with topological order, a collective order also appearing in fractional quantum Hall systems. We define electromagnetic (EM) duality for all of Kitaev's quantum double models…

Given a connected, reductive $\mathbf{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and a smooth zip period map $\zeta:X \to \mathop{\text{$G$-{\tt Zip}}}\nolimits^{\mu}$, we study which classes in the Wedhorn-Ziegler tautological rings…

We prove stability of rank two tautological bundles on the Hilbert square of a surface (under a mild positivity condition) and compute their Chern classes.

Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…

We find for g at most 5 a stratification of depth g-2 of the moduli space of curves M_g with the property that its strata are affine and the classes of their closures provide a Q-basis for the Chow ring of M_g. The first property confirms a…

Let $E$ be a vector bundle of rank $r\geq 2$ on a smooth projective curve $C$ of genus $g \geq 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define $${\se}_k(E):=k\deg…

I point out that (BPS saturated) A-type D-branes in superstring compactifications on Calabi-Yau threefolds correspond to {\em graded} special Lagrangian submanifolds, a particular case of the graded Lagrangian submanifolds considered by M.…

In this work we compute the Chen--Ruan cohomology and the stringy Chow ring of the moduli spaces of smooth and stable $n$-pointed curves of genus 1. We suggest a definition for an Orbifold Tautological Ring in genus 1, which is both a…

We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this…

Let $\MC$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree, over a smooth projective curve $C$. This paper identifies the algebraic cohomology ring $\HA^*(\MC)$, i.e. the subring of the…

We discuss graded D-brane systems of the topological A model on a Calabi-Yau threefold, by means of their string field theory. We give a detailed analysis of the extended string field action, showing that it satisfies the classical master…

We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of…

Let M_{g,n} be the moduli space of stable genus g curves with n marked points. M_{g,n} has boundary strata consisting of nodal curves. The fundamental classes of these boundary strata may be linearly dependent in the Chow group…

For a closed oriented surface $ \Sigma $ let $X_{\Sigma,n}$ be the space of isomorphism classes of orientation preserving $n$-fold branched coverings $ \Sigma\rightarrow S^2 $ of the two-dimensional sphere. At a previous paper, the authors…

Let U be a smooth quasi-projective variety over a field k that is finite, the algebraic closure of a finite field or algebraically closed of characteristic 0. Let X be a suitable projective compactification of U, and D an effective divisor…