Abstract. For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A r−1 .We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.Comment: Exposition improved and additional details provided. To appear in Inventiones Mathematica

We construct a mathematical theory of Witten's Gauged Linear Sigma Model (GLSM). Our theory applies to a wide range of examples, including many cases with non-Abelian gauge group.Both the Gromov-Witten theory of a Calabi-Yau complete intersection X and the Landau-Ginzburg dual (FJRW-theory) of X can be expressed as gauged linear sigma models. Furthermore, the Landau-Ginzburg/Calabi-Yau correspondence can be interpreted as a variation of the moment map or a deformation of GIT in the GLSM. This paper focuses primarily on the algebraic theory, while a companion article [FJR16] will treat the analytic theory.

Abstract. This article treats various aspects of the geometry of the moduli S 1/r g of r-spin curves and its compactification S 1/r g . Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand-Dikii (KdVr) heirarchy. There is also a W -algebra conjecture for these spaces [16] generalizing the the Virasoro conjecture of quantum cohomology.For any line bundle K on the universal curve over the stack of stable curves, there is a smooth stack S 1/r g,n (K) of triples (X, L, b) of a smooth curve X, a line bundle L on X, and an isomorphism b : L ⊗r -K. In the special case that K = ω is the relative dualizing sheaf, then S 1/r g,n (K) is the stack S 1/r g,n of r-spin curves.We construct a smooth compactification S 1/r g,n (K) of the stack S 1/r g,n (K), describe the geometric meaning of its points, and prove that it is projective.We also prove that when r is odd and g > 1, the compactified stack of spin curves S 1/r g and its coarse moduli space S 1/r g are irreducible, and when r is even and g > 1, S 1/r g is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when g = 1, S 1/r 1 is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of S 1/r g [15], and also in the study of the cohomological field theory related to Witten's conjecture [16,34].

Abstract. In this note we give a new, natural construction of a compactification of the stack of smooth r-spin curves, which we call the stack of stable twisted r-spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible Gm-spaces and Q-line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves.We construct representable morphisms from the stacks of stable twisted r-spin curves to the stacks of stable r-spin curves and show that they are isomorphisms. Many delicate features of r-spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the∂-operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.The moduli space of smooth r-spin curves was compactified by the second author, using torsion free sheaves and coherent nets of torsion free sheaves in [4] and [5]. In order to construct a satisfactory compactification it was necessary in those papers to study in detail the behavior of torsion free sheaves with an r-power map. The infinitesimal properties of those compactifications are quite subtle.The purpose of this note is to give a new, natural construction of a compactification of the stack of smooth r-spin curves, which we call the stack of stable twisted r-spin curves, and to describe its properties and relations to other compactifications.In the first section we define the stack of smooth r-spin curves and give an alternate construction in terms of certain principal G m -bundles. To generalize these to stable curves we then recall the twisted curves of [1] and give a natural definition of twisted stable r-spin curves, very much analogous to the situation of [2]. We show that over smooth curves all of these constructions are equivalent to each other. Moreover, we identify the stack of twisted stable r-spin curves with a special case of a stack of twisted stable maps of [1].In the second section we describe the infinitesimal structure of this stack, again in analogy to the treatment of [2]. This is relatively straightforward and is similar to the infinitesimal structure of usual stable curves.