Higher structures in Algebra and Topology – SAT

This project proposes first to enrich one step further the theory of higher structures with new models, like a suitable notion of symmetric operad up to homotopy and a good notion of infinity-cooperad. Then it aims at applying these new methods to the mathematical study of field theories, like E_n-algebras, factorization algebras, topological field theories, cohomological field theories, vertex algebras and conformal field theories.

* Homotopy theory of symmetric operads over any ring. Construction of a cofibrant replacement functor, which applied to the operad of commutative algebras, provides us with an explicit E_infinity-operad over any ring. [Dehling--Vallette]

* Construction of a homotopy nerve functor from coloured unital homotopy operads and infinity-operads. [Le Grignou]

* Introduction et development of higher category of iterated cobordisms which should be used in Baez--Dolan cobordism hypothesis. [Calaque--Scheimbauer]

* Construction of shifted symplectic structures on the level of the derived stacks of applications under boundary conditions which are mandatory in most of applications in quantum field theories. [Calaque]

* Ultimate form of the homotopy trivialisation of the circle action, that is optimal form of the d-dbar lemma of rational homotopy theory. [Dotsenko--Shadrin--Vallette]

* Structure of homotopy cohomological field theory on the De Rham cohomology of Poisson manifolds: new invariants which keep faithfully track of the homotopy type of the Batalin--Vilkovisky structure of the De Rham algebra. [Dotsenko--Shadrin--Vallette]

* Non-commutative theory of Batalin--Vilkovisky algebras, cohomological field theories and little disks operads in terms of toric varieties associated to the representations with integer coordinates of the associera due to Loday.

New foundational developments of high algebra directed toward the resolution of problems in the topological study of field theories.

D. Calaque, T. Willwacher, Triviality of the higher formality theorem, Proceedings of the AMS 143 (2015), 5181-5193.

D. Calaque, Lagrangian structures on mapping stacks and semi-classical TFTs, in Stacks and Categories in Geometry, Topology, and Algebra, 1-23, Contemp. Math., 643, Amer. Math. Soc., (2015).

D. Calaque, F. Naef, A trace formula for the quantization of coadjoint orbits, Int. Math. Res. Not. IMRN (2015), no. 21, 11236-11252.

C. Cazanave, Appendix of the article of B. Farb and J. Wolfson, Topology and arithmetic of resultants, II: the resultant = 1 hypersurface, (2015), to appear in Algebraic Geometry, 4 pages , arXiv: 1507.01283

B. Le Grignou, From homotopy operads to infinity-operads, (December 2014), to appear in Journal of Noncommutative Geometry, 36 pages, arXiv:1412.4968

C. Dupont, B. Vallette, Brown's moduli spaces of curves and the gravity operad, (2015), to appear in Geometry & Topology, 26 pages, arXiv:1509.08840

V. Dotsenko, S. Shadrin, B. Vallette, PreLie deformation theory, Moscow Mathematical Journal, Volume 16, Issue 3 (2016) 505-543.

V. Dotsenko, S. Shadrin, B. Vallette, Givental Action and Trivialisation of Circle Action, Journal de l’École polytechnique – Mathématiques, 2 (2015), 213-246.

V. Dotsenko, S. Shadrin, B. Vallette, De Rham cohomology and homotopy Frobenius manifolds, Journal of the European Mathematical Society, Volume 17, Issue 2 (2015), 535-547.

The present project is a program of fundamental research in mathematics, more precisely in algebra and topology.

The theory of categories has proven to be a valuable algebraic tool to state and to organize the results in many areas in mathematics. In the same way, the notion of an operad, which emerged from the study of iterated loop spaces in the early 1970’s, now plays a structural role by organizing the many operations with several inputs acting on the various objects in algebra, geometry, topology and mathematical physics.

At the end of the last century, it appeared clearly that the introduction of higher algebraic notions was necessary in order to encode higher structures naturally ap- pearing in mathematical problems. Homotopy theory provides us with natural phenomena whose description requires higher maps, that is some notion of higher category. In the same way, the algebraic notion of an operad, while already used to describe homotopy algebras, was proved to be a too strict notion, e.g. for the purposes of deformation theory.

After a long period of research in algebra and topology, the theory of higher structures gave rise recently to radically new and manageable notions, like infinity-categories, homotopy al- gebras and homotopy operads, which make possible the study of some unsolved questions like formality theorems (Kontsevich) and cobordism hypothesis (Lurie), for instance.

In an attempt to provide well established mathematical definitions to objects studied in field theory in physics, several renowned mathematicians introduced the notions of topological field theories, cohomological field theories and conformal field theories using categorical or operadic structures on geometrical objects: smooth manifolds with boundary (Atiyah), algebraic curves with marked points (Kontsevich–Manin), and Riemann surfaces with holomorphic holes (Segal) respectively.

This project consists in new foundational developments of high algebra di- rected toward the resolution of problems in the topological study of field theories.

It proposes first to enrich one step further the theory of higher structures with new models, like a suitable notion of symmetric operad up to homotopy and a good notion of infinity- cooperad. Then it aims at applying these new methods to the mathematical study of field theories, like En-algebras, factorization algebras, topological field theories, cohomological field theories, vertex algebras and conformal field theories.

This ANR JCJC proposal gathers a group of young researchers, including Ph.D. students. It aims at creating a new transversal team of research in algebra and topology in the South part of France, where some of the proposed topics of study, like factorization algebras and extended field theories, are not yet represented.