Abstract:
In this note we observe that, contrary to the usual lore, string orbifolds do not describe strings on quotient spaces, but rather seem to describe strings on objects called quotient stacks, a result that follows from simply unraveling definitions, and is further justified by a number of results. Quotient stacks are very closely related to quotient spaces; for example, when the orbifold group acts freely, the quotient space and the quotient stack are homeomorphic. We explain how sigma models on quotient stacks naturally have twisted sectors, and why a sigma model on a quotient stack would be a nonsingular CFT even when the associated quotient space is singular. We also show how to understand twist fields in this language, and outline the derivation of the orbifold Euler characteristic purely in terms of stacks. We also outline why there is a sense in which one naturally finds B nonzero on exceptional divisors of resolutions. These insights are not limited to merely understanding existing string orbifolds: we also point out how this technology enables us to understand orbifolds in M-theory, as well as how this means that string orbifolds provide the first example of an entirely new class of string compactifications. As quotient stacks are not a staple of the physics literature, we include a lengthy tutorial on quotient stacks.

Abstract:
Let M be a connected, simply connected, closed and oriented manifold, and G a finite group acting on M by orientation preserving diffeomorphisms. In this paper we show an explicit ring isomorphism between the orbifold string topology of the orbifold [M/G] and the Hochschild cohomology of the dg-ring obtained by performing the smash product between the group G and the singular cochain complex of M.

Abstract:
The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" definition of orbifolds as proper etale Lie groupoids up to Morita equivalence. The second goal is to describe two complementary ways of thinking of orbifolds as a 2-category: 1. the weak 2-category of foliation Lie groupoids, bibundles and equivariant maps between bibundles and 2. the strict 2-category of Deligne-Mumford stacks over the category of smooth manifolds.

Abstract:
We consider worldsheet string theory on $Z_N$ orbifolds of $AdS_3$ associated with conical singularities. If the orbifold action includes a similar twist of $S^3$, supersymmetry is preserved, and there is a moduli space of vacua arising from blowup modes of the orbifold singularity. We exhibit the spectrum, including the properties of twisted sectors and states obtained by fractional spectral flow. A subalgebra of the spacetime superconformal symmetry remains intact after the $Z_N$ quotient, and serves as the spacetime symmetry algebra of the orbifold.

Abstract:
We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. In particular, we give a good notion of a free loop stack, and of a mapping stack $\map(Y,\XX)$, where $Y$ is a compact space and $\XX$ a topological stack, which is functorial both in $\XX$ and $Y$ and behaves well enough with respect to pushouts. We also construct a bivariant (in the sense of Fulton and MacPherson) theory for topological stacks: it gives us a flexible theory of Gysin maps which are automatically compatible with pullback, pushforward and products. Further we prove an excess formula in this context. We introduce oriented stacks, generalizing oriented manifolds, which are stacks on which we can do string topology. We prove that the homology of the free loop stack of an oriented stack and the homology of hidden loops (sometimes called ghost loops) are a Frobenius algebra which are related by a natural morphism of Frobenius algebras. We also prove that the homology of free loop stack has a natural structure of a BV-algebra, which together with the Frobenius structure fits into an homological conformal field theories with closed positive boundaries. Using our general machinery, we construct an intersection pairing for (non necessarily compact) almost complex orbifolds which is in the same relation to the intersection pairing for manifolds as Chen-Ruan orbifold cup-product is to ordinary cup-product of manifolds. We show that the hidden loop product of almost complex is isomorphic to the orbifold intersection pairing twisted by a canonical class. Finally we gave some examples including the case of the classifying stacks $[*/G]$ of a compact Lie group.

Abstract:
This paper is a continuations of the project initiated in the book string topology for stacks. We construct string operations on the SO(2)-equivariant homology of the (free) loop space $L(X)$ of an oriented differentiable stack $X$ and show that $H^{SO(2)}_{*+dim(X) -2}(L(X))$ is a graded Lie algebra. In the particular case where $X$ is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on.

Abstract:
For every smooth projective variety, we construct an action of the Heisenberg algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks which contains the Fock space as a subrepresentation. The action is induced by functors on the level of the derived categories which form a weak categorification of the action.

Abstract:
We show that near closed points with linearly reductive stabilizer, Artin stacks are formally locally quotient stacks by the stabilizer. We conjecture that the statement holds etale locally and we provide some evidence for this conjecture. In particular, we prove that if the stabilizer of a point is linearly reductive, the stabilizer acts algebraically on a miniversal deformation space generalizing results of Pinkham and Rim.

Abstract:
This article contains a slightly expanded version of the lectures given by the author at the summer school "Algebraic stacks and related topics" in Mainz, Germany from August 31 to September 4, 2015. The content of these lectures is purely expository and consists of two main goals. First, we provide a treatment of Artin's approximation and algebraization theorems following the ideas of Conrad and de Jong which rely on a deep desingularization result due to Neron and Popescu. Second, we prove that under suitable hypotheses, algebraic stacks are etale locally quotients stacks in a neighborhood of a point with a linearly reductive stabilizer.

Abstract:
A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of $H^2(X,{\mathbb G}_m)$ is surjective.