arXiv subjects
Publications and source records attributed to Damien Calaque.
These notes are intended to be an introduction to shifted symplectic geometry, targeted to Poisson geometers with a serious background in homological algebra. They are extracted from a mini-course given by the first author at the Poisson 2024 summer school that took place at the Accademia Pontaniana in Napoli.
We construct symmetric monoidal higher categories of iterated Calabi-Yau cospans, that are noncommutative analogs of iterated lagrangian correspondences. We actually give a general (and functorial) procedure that applies to iterated nondegenerate cospans on certain comma categories. This allows us to factor the AKSZ fully extended TFT associated with the moduli of objects of a Calabi-Yau category (taking values in iterated lagrangian correspondences) through a fully extended TFT taking values in iterated Calabi-Yau cospans.
This short paper illustrates the general framework introduced in the paper "Not too little discs" (arXiv:2407.18192), joint with Victor Carmona, on yet another one dimensional example. It exhibits a discrete model for the free scalar field on the real line, adapting the treatment from the book of Costello--Gwilliam to the discrete setting.
By the introduction of locally constant prefactorization algebras at a fixed scale, we show a mathematical incarnation of the fact that observables at a given scale of a topological field theory propagate to every scale over euclidean spaces. The key is that these prefactorization algebras over $\mathbb{R}^n$ are equivalent to algebras over the little $n$-disc operad. For topological field theories with defects, we get analogous results by replacing $\mathbb{R}^n$ with the spaces modelling corners $\mathbb{R}^p\times\mathbb{R}^{q}_{\geq 0}$. As a toy example in $1d$, we quantize, once more, constant Poisson structures.
This article generalizes the theory of shifted symplectic structures to the relative context and non-geometric stacks. We describe basic constructions that naturally appear in this theory: shifted cotangent bundles and the AKSZ procedure. Along the way, we also develop the theory of shifted symplectic groupoids presenting shifted symplectic structures on quotients and define a deformation to the normal cone for shifted Lagrangian morphisms.
We show that relative Calabi--Yau structures on noncommutative moment maps give rise to (quasi-)bisymplectic structures, as introduced by Crawley-Boevey-Etingof-Ginzburg (in the additive case) and Van den Bergh (in the multiplicative case). We prove along the way that the fusion process (a) corresponds to the composition of Calabi-Yau cospans with "pair-of-pants" ones, and (b) preserves the duality between non-degenerate double quasi-Poisson structures and quasi-bisymplectic structures. As an application we obtain that Van den Bergh's Poisson structures on the moduli spaces of representations of deformed multiplicative preprojective algebras coincide with the ones induced by the 2-Calabi-Yau structures on (dg-versions of) these algebras.
This is a survey on Drinfeld associators and their generalizations, where we focus on operadic aspects.
We study truncated gauge-orbits through principal parts of irregular-singular connection germs, in the untwisted/unramified setting: for any connected complex reductive structure group $G$, in the general multilevel case. In particular, we compute the stabilisers of the formal normal forms using filtrations of Levi root systems, showing that they are connected. When the residue is semisimple we then stratify the space of orbits by the conjugacy class of the stabilisers, i.e., by quotients of root-valuation strata; the dense stratum corresponds to the generic setting of isomonodromic deformations, \`a la Jimbo--Miwa--Ueno. Then we adapt a result of Alekseev--Lachowska to deformation-quantise nongeneric orbits. The $\ast$-product involves affine-Lie-algebra modules, extending: (i) the parabolic Verma modules (in the case of regular singularities); and (ii) the `singularity' modules of F.--R. (in the case of generic irregular singularities). They contain Whittaker vectors for the Gaiotto--Teschner/Bonelli--Maruyoshi--Tanzini Virasoro pairs in irregular Liouville conformal field theory, and they provide all the quotients obtained by leaving the aforementioned dense strata. We also construct Shapovalov forms for the corresponding representations of truncated-current Lie algebras, which enter into the category $\mathcal O$ of Chaffe--Topley; and we state a sharp irreducibility criterion. Finally, we use these representations to construct vector bundles of genus-zero vacua/covacua, equipped with flat connections \`a la Knizhnik--Zamolodchikov/Reshetikhin.
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks. Derived stacks are the main players in derived geometry, the purpose of which is to deal with singular spaces, while symplectic structures are an essential ingredient of the geometric formalism of classical mechanics and classical field theory. In addition to providing an overview of a relatively young field of research, we provide a case study on Casson's invariant.
We prove that the derived critical locus of a $G$-invariant function $S:X\to\mathbb{A}^1$ carries a shifted moment map, and that its derived symplectic reduction is the derived critical locus of the induced function $S_{red}:X/G\to\mathbb{A}^1$ on the orbit stack. We also provide a relative version of this result, and show that derived symplectic reduction commutes with derived lagrangian intersections.
In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes.
We construct a family of oriented extended topological field theories using the AKSZ construction in derived algebraic geometry, which can be viewed as an algebraic and topological version of the classical AKSZ field theories that occur in physics. These have as their targets higher categories of symplectic derived stacks, with higher morphisms given by iterated Lagrangian correspondences. We define these, as well as analogous higher categories of oriented derived stacks and iterated oriented cospans, and prove that all objects are fully dualizable. Then we set up a functorial version of the AKSZ construction, first implemented in this context by Pantev-Toën-Vaquié-Vezzosi, and show that it induces a family of symmetric monoidal functors from oriented stacks to symplectic stacks. Finally, we construct forgetful functors from the unoriented bordism $(\infty,n)$-category to cospans of spaces, and from the oriented bordism $(\infty,n)$-category to cospans of spaces equipped with an orientation; the latter combines with the AKSZ functors by viewing spaces as constant stacks, giving the desired field theories.
A theorem of Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. We prove a generalization of this correspondence, relating formal moduli problems parametrized by algebras over a Koszul operad to algebras over its Koszul dual operad. In particular, when the Lie algebra associated to a deformation problem is induced from a pre-Lie structure it corresponds to a permutative formal moduli problem. As another example we obtain a correspondence between operadic formal moduli problems and augmented operads.
We show that there is an equivalence of $\infty$-categories between Lie algebroids and certain kinds of curved Lie algebras. For this we develop a method to study the $\infty$-category of curved Lie algebras using the homotopy theory of algebras over a complete operad.
We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of $Γ$-structured elliptic curves with marked points, where $Γ=\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$, and $M,N\geq1$ are two integers. It restricts to a flat connection on $Γ$-twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subgroups of the pure braid group on the torus. We show that the universal ellipsitomic KZB connection realizes as the usual KZB connection associated with elliptic dynamical $r$-matrices with spectral parameter, and finally, also produces representations of cyclotomic Cherednik algebras.
In this paper we identify the cotangent to the derived stack of representations of a quiver $Q$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with $Q$. More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane.
We develop a notion of ellipsitomic associators by means of operad theory. We take this opportunity to review the operadic point-of-view on Drinfeld associators and to provide such an operadic approach for elliptic associators too. We then show that ellipsitomic associators do exist, using the monodromy of the universal ellipsitomic KZB connection, that we introduced in a previous work. We finally relate the KZB ellipsitomic associator to certain Eisenstein series associated with congruence subgroups of $SL_2(\mathbb{Z})$, and to twisted elliptic multiple zeta values.
This is a companion paper to "Ellipsitomic associators". We provide a (m)operadic description of Enriquez's torsor of cyclotomic associators, as well as of its associated cyclotomic Grothendieck-Teichm\"uller groups.