The Deligne conjecture (many times a theorem) endows Hochschild cochains of a linear category with the structure of an $E_2$-algebra, that is, of an algebra over the little 2-disks operad. In this paper, we prove the cyclic Deligne conjecture, stating that for a linear category equipped with a Calabi-Yau structure (a kind of non-commutative orientation), the Hochschild cochains is endowed with the finer structure of a framed $E_2$-algebra, that is, of a circle-equivariant algebra over the little 2-disks operad. Our approach applies simultaneously to both smooth and proper linear categories, as well as to linear functors equipped with a relative Calabi-Yau structure, and works for a very general notion of linear category, including any dualizable presentable $\infty$-category. As a particular application, given a compact oriented manifold with boundary $\partial M \subset M$, our construction gives chain-level genus zero string topology operations on the relative loop homology $H_{*}(LM,L\partial M)$.

We show that a Calabi-Yau structure of dimension $d$ on a smooth dg category $C$ induces a symplectic form of degree $2-d$ on the moduli space of objects $M_{C}$. We show moreover that a relative Calabi-Yau structure on a dg functor $C \to D$ compatible with the absolute Calabi-Yau structure on $C$ induces a Lagrangian structure on the corresponding map of moduli $M_{D} \to M_{C}$.

We prove a Darboux theorem for derived schemes with symplectic forms of degree k > 0 k>0 , in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme X \mathbfit {X} with symplectic form ω ~ \tilde {\omega } of degree k k is locally equivalent to ( Spec ⁡ A , ω ) (\operatorname {Spec} A,\omega ) for Spec ⁡ A \operatorname {Spec} A an affine derived scheme in which the cdga A A has Darboux-like coordinates with respect to which the symplectic form ω \omega is standard, and in which the differential in A A is given by a Poisson bracket with a Hamiltonian function Φ \Phi of degree k + 1 k+1 . When k = − 1 k=-1 , this implies that a − 1 -1 -shifted symplectic derived scheme ( X , ω ~ ) (\mathbfit {X}, \tilde {\omega }) is Zariski locally equivalent to the derived critical locus Crit ⁡ ( Φ ) \operatorname {Crit}(\Phi ) of a regular function Φ : U → A 1 \Phi :U\rightarrow \mathbb {A}^1 on a smooth scheme U U . We use this to show that the classical scheme X = t 0 ( X ) X=t_0(\mathbfit {X}) has the structure of an algebraic d-critical locus, in the sense of Joyce. In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications to categorified and motivic Donaldson–Thomas theory of Calabi–Yau 3-folds, and to defining new Donaldson–Thomas type invariants of Calabi–Yau 4-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.

We introduce relative noncommutative Calabi-Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi-Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi-Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

This is the fifth in a series of papers on the 'k-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi. We extend our earlier results from (derived) schemes to (derived) Artin stacks. We prove four main results:(a) If (X, omega(X)) is a k-shifted symplectic derived Artin stack for k < 0, then near each x is an element of X we can find a 'minimal' smooth atlas phi: U -> X, such that (U, phi*(omega(X))) may be written explicitly in coordinates in a standard 'Darboux form'.(b) If (X, omega(X)) is a (-1)-shifted symplectic derived Artin stack and X = t(0)(X) the classical Artin stack, then X extends to a 'd-critical stack' (X, s), as by Joyce.(c) If (X, s) is an oriented d-critical stack, we define a natural perverse sheaf (sic)(X,s)(center dot) on X, such that whenever T is a scheme and t : T -> X is smooth of relative dimension n, T is locally modelled on a critical locus Crit (f: U -> A(1)), and t*((sic)(X, s)* )[n] is modelled on the perverse sheaf of vanishing cycles P nu(center dot)(U,f) of f.(d) If (X, s) is a finite-type oriented d-critical stack, we can define a natural motive MFX,s in a ring of motives (M) over bar (st,(mu) over cap)(X) on X, such that if T is a scheme and t: T -> X is smooth of dimension n, then T is modelled on a critical locus Crit (f: U -> A(1)), and L-n/2 circle dot t* (MFX,s) is modelled on the motivic vanishing cycle MFU,Fmot,phi of f.Our results have applications to categorified and motivic extensions of DonaldsonThomas theory of Calabi-Yau 3-folds.

Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of vanishing cycles" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $\Phi:U\to U$ is an isomorphism with $f\circ\Phi=f$ and $\Phi\vert_X=$id$_X$. Then $\Phi$ induces an isomorphism $\Phi_*:PV_{U,f}\to PV_{U,f}$. We show that $\Phi_*$ is multiplication by det$(d\Phi\vert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=f\vert_{X^{(3)}}:X^{(3)}\to\mathbb A^1$. (c) If $U,V$ are smooth $\mathbb C$-schemes, $f:U\to\mathbb A^1$, $g:V\to\mathbb A^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $\Phi:U\to V$ is an embedding with $f=g\circ\Phi$ and $\Phi\vert_X:X\to Y$ an isomorphism, there is a natural isomorphism $\Theta_\Phi:PV_{U,f}\to\Phi\vert_X^*(PV_{V,g})\otimes_{\mathbb Z_2}P_\Phi$, for $P_\Phi$ a natural principal $\mathbb Z_2$-bundle on $X$. (d) If $(X,s)$ is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to\mathbb A^1)$ then $P_{X,s}$ is locally modelled on $PV_{U,f}$. We also generalize our results to replace $U,X$ by complex analytic spaces, and $PV_{U,f}$ by $\mathcal D$-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090, arXiv:1403.2403, arXiv:1404.1329, arXiv:1504.00690.

AbstractWe show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$. As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 2-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.

We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.

Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent bundle T*P^1 and modules over the preprojective algebra of an affine Dynkin quiver. These different equivalences are related by spherical twists, which take the place of the reflection functors for P^1.