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Damien Calaque

Publications and source records attributed to Damien Calaque.

At least 37 records · Page 2Linked to original sources
arXiv2019

Formal moduli problems and formal derived stacks

This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by Lurie and Pridham) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy, which gives a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasising the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by Joost Nuiten, and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of Gaitsgory and Rozenblyum.

math.AG↗
arXiv2018

A note on the $(\infty,n)$-category of cobordisms

In this extended note we give a precise definition of fully extended topological field theories à la Lurie. Using complete $n$-fold Segal spaces as a model, we construct an $(\infty,n)$-category of $n$-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms $n\operatorname{Cob}$ and the cobordism bicategory $n\operatorname{Cob}^{ext}$ from it.

math.AT↗
arXiv2018

Derived stacks in symplectic geometry

This is a survey paper on derived symplectic geometry, that will appear as a chapter contribution to the book "New Spaces for Mathematics and Physics", edited by Mathieu Anel and Gabriel Catren. Our goal is to explain how derived stacks can be useful for ordinary symplectic geometry, with an emphasis on examples coming from classical topological field theories. More precisely, we use classical Chern-Simons theory and moduli spaces of flat $G$-bundles and $G$-local systems as leading examples in our journey. We start in the introduction by reviewing various point-of-views on classical Chern--Simons theory and moduli of flat connections. In the main body of the Chapter we try to convince the reader how derived symplectic geometry (after Pantev-Toën-Vaquié-Vezzosi somehow reconciles all these different point-of-views.

math.SG↗
arXiv2017

The Ext algebra of a quantized cycle

Given a quantized analytic cycle $(X, σ)$ in $Y$, we give a categorical Lie-theoretic interpretation of a geometric condition, discovered by Shilin Yu, that involves the second formal neighbourhood of $X$ in $Y$. If this condition (that we call tameness) is satisfied, we prove that the derived Ext algebra $\mathcal{RH}om_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ is isomorphic to the universal enveloping algebra of the shifted normal bundle $\mathrm{N}_{X/Y}[-1]$ endowed with a specific Lie structure, strengthening an earlier result of Căldăraru, Tu, and the first author This approach allows to get some conceptual proofs of many important results in the theory: in the case of the diagonal embedding, we recover former results of Kapranov, Markarian, and Ramadoss about (a) the Lie structure on the shifted tangent bundle $\mathrm{T}_X[-1]$ (b) the corresponding universal enveloping algebra (c) the calculation of Kapranov's big Chern classes. We also give a new Lie-theoretic proof of Yu's result for the explicit calculation of the quantized cycle class in the tame case: it is the Duflo element of the Lie algebra object $\mathrm{N}_{X/Y}[-1]$.

math.AG↗
arXiv2017

Shifted cotangent stacks are shifted symplectic

We prove that shifted cotangent stacks carry a canonical shifted symplectic structure. We also prove that shifted conormal stacks carry a canonical Lagrangian structure. These results were believed to be true but no written proof was available in the Artin case.

math.AG↗
arXiv2014

Lagrangian structures on mapping stacks and semi-classical TFTs

We extend a recent result of Pantev-Toen-Vaquie-Vezzosi, who constructed shifted symplectic structures on derived mapping stacks having a Calabi-Yau source and a shifted symplectic target. Their construction gives a clear conceptual framework for the so-called AKSZ formalism. We extend the PTVV construction to derived mapping stacks with boundary conditions, which is required in most applications to quantum field theories (see e.g. the work of Cattaneo-Felder on the Poisson sigma model, and the recent work of Cattaneo-Mnev-Reshetikhin). We provide many examples of Lagrangian and symplectic structures that can be recovered in this way. We finally give an application to topological field theories (TFTs). We expect that our approach will help to rigorously constuct a 2 dimensional TFT introduced by Moore and Tachikawa. A subsequent paper will be devoted to the construction of fully extended TFTs (in the sense of Baez-Dolan and Lurie) from mapping stacks.

math.AG↗
arXiv2014

On the Lie algebroid of a derived self-intersection

Let $i:X\hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. Denote by $N$ the normal bundle of $X$ in $Y$. We describe the construction of two Lie-type structures on the shifted bundle $N[-1]$ which encode the information of the formal neighborhood of $X$ inside $Y$. We also present applications of classical Lie theoretic constructions (universal enveloping algebra, Chevalley-Eilenberg complex) to the understanding of the geometry of embeddings.

math.AG↗
arXiv2013

Triviality of the higher Formality Theorem

It is noted that the higher version of M. Kontsevich's Formality Theorem is much easier than the original one. Namely, we prove that the higher Hochschild-Kostant-Rosenberg map is already a homotopy e_{n+1}-formality quasi-isomorphism whenever n is at least 2.

math.QA↗
arXiv2013

A PBW theorem for inclusions of (sheaves of) Lie algebroids

Inspired by the recent work of Chen-Stiénon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincaré-Birkhoff-Witt type results to hold. The tools we use (e.g. the first infinitesimal neighbourhood Lie algebroid) are straightforward generalizations of the ones previously developped by Caldararu, Tu and the author for Lie algebra inclusions.

math.QA↗
arXiv2012

Caldararu's conjecture and Tsygan's formality

In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet (the existence of the latter was conjectured by Tsygan). We also rely on the fact - recently proved by the first two authors - that Shoikhet's quasi-isomorphism is compatible with cap product after twisting with a Maurer-Cartan element.

math.AG↗
arXiv2011

Two interacting Hopf algebras of trees

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. In this work we consider a Hopf algebra H by introducing a coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feynman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees by means of a natural H-bicomodule structure on the latter. This enables us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.

math.CO↗
arXiv2011

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.

math.QA↗
arXiv2011

Weak quantization of Poisson structures

In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira. We begin the paper with a recollection of known facts about deformation theory of cosimplicial differential graded Lie algebras.

math.QA↗
arXiv2010

Bimodules and branes in deformation quantization

We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space $X$. The result implies in particular that the Kontsevich deformation quantizations of $\mathrm{S}(X^*)$ and $\wedge(X)$ associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.

math.QA↗
arXiv2010

Hochschild cohomology for Lie algebroids

We define the Hochschild (co)homology of a ringed space relative to a locally free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an algebraic variety. We show that our (co)homology groups can be computed using suitable standard complexes. Our formulae depend on certain natural structures on jetbundles over Lie algebroids. In an appendix we explain this by showing that such jetbundles are formal groupoids which serve as the formal exponentiation of the Lie algebroid.

math.AG↗
arXiv2009

Global formality at the $G_\infty$-level

In this paper we prove that the sheaf of $\Lscr$-poly-differential operators for a locally free Lie algebroid $\Lscr$ is formal when viewed as a sheaf of $G_\infty$-algebras via Tamarkin's morphism of DG-operads $G_\infty\r B_\infty$. In an appendix we prove a strengthening of Halbout's globalization result for Tamarkin's local quasi-isomorphism.

math.QA↗