The Lefschetz-Lunts formula for deformation quantization modules

We adapt to the case of deformation quantization modules a formula of V. Lunts who calculates the trace of a kernel acting on Hochschild homology.Comment: 19 pages; Mathematische Zeitschrift 201

Matrix factorizations via Koszul duality

In this paper we prove a version of curved Koszul duality for Z/2Z-graded curved coalgebras and their coBar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations MF(R,W). We show how Dyckerhoff's generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel-Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category MF(R,W). Similar results are also obtained in the orbifold case and in the graded case.Comment: Latex 34 pages, rewritten introduction, deleted an appendix, minor modification on proofs, final versio

PBW for an inclusion of Lie algebras

Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition for the existence of a splitting of this filtration. In turn such a splitting yields an isomorphism between the h-modules U(g)/U(g)h and S(n). For the diagonal embedding h \subset h \oplus h the condition is automatically satisfied and we recover the classical Poincae-Birkhoff-Witt theorem. The main theorem and its proof are direct translations of results in algebraic geometry, obtained using an ad hoc dictionary. This suggests the existence of a unified framework allowing the simultaneous study of Lie algebras and of algebraic varieties, and a closely related work in this direction is on the way.Comment: Major revision, proofs of several results rewritten. Added a section explaining the case of a general representation, as opposed to the trivial one. 20 pages, LaTe

Lagrangian homology spheres in (A_m) Milnor fibres via C^*-equivariant A_infinity modules

We establish restrictions on Lagrangian embeddings of rational homology spheres into certain open symplectic manifolds, namely the (A_m) Milnor fibres of odd complex dimension. This relies on general considerations about equivariant objects in module categories (which may be applicable in other situations as well), as well as results of Ishii-Uehara and Ishii-Ueda-Uehara concerning the derived categories of coherent sheaves on the resolutions of (A_m) surface singularities.Comment: version 2: better results, simpler proofs; version 3: title changed as requested by referee, other very minor modification

One-dimensional Chern-Simons theory and the A^\hat{A} genus

We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X, as such a Chern-Simons theory. Our main result is that the partition function of this theory is naturally identified with the A-genus of X. From the perspective of derived geometry, our quantization construct a volume form on the derived loop space which can be identified with the A-class.Comment: 61 pages, figures, final versio

Non-birational twisted derived equivalences in abelian GLSMs

In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with non-birational Kahler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kahler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov's `homological projective duality.' Along the way, we shall see how `noncommutative spaces' (in Kontsevich's sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.Comment: 54 pages, LaTeX; v2: typo fixe

When is the self-intersection of a subvariety a fibration?

Abstract We provide a necessary and sufficient condition for the derived self-intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme. As a consequence we deduce a generalized HKR isomorphism. We also investigate the relationship of our result to path spaces in homotopy theory, Buchweitz–Flenner formality in algebraic geometry, and draw parallels with similar results in Lie theory and symplectic geometry

Linear spaces of matrices of constant rank and instanton bundles

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P^3 and derived category tools. The method allows one to prove the existence of new examples of size 10x10 and 14x14 via instanton bundles of charge 2 and 4 respectively, and provides an explanation for what used to be the only known example (Westwick 1996). We also give an algorithm to construct explicitly a matrix of size 14 of this type.Comment: Revised version, 22 pages. Brief intro to derived category tools and details to proof of Lemma 3.5 added, some typos correcte

The Hochschild-Kostant-Rosenberg isomorphism for quantized analytic cycles

In this article, we provide a detailed account of a construction sketched by Kashiwara in an unpublished manuscript concerning generalized HKR isomorphisms for smooth analytic cycles whose conormal exact sequence splits. It enables us, among other applications, to solve a problem raised recently by Arinkin and C\u{a}ld\u{a}raru about uniqueness of such HKR isomorphisms in the case of the diagonal injection. Using this construction, we also associate with any smooth analytic cycle endowed with an infinitesimal retraction a cycle class which is an obstruction for the cycle to be the vanishing locus of a transverse section of a holomorphic vector bundle.Comment: Mistake corrected in Section 5.