Deformation theory of cohomological field theories-Reference-Cited by-同舟云学术

Deformation theory of cohomological field theories

Published:2024-01-30 Issue:0 Volume:0 Page:
ISSN:0075-4102
Container-title:Journal für die reine und angewandte Mathematik (Crelles Journal)
language:en
Short-container-title:

Author:

Dotsenko Vladimir¹,Shadrin Sergey²,Vaintrob Arkady³,Vallette Bruno⁴

Affiliation:

1. Institut de Recherche Mathématique Avancée , UMR 7501 , Université de Strasbourg et CNRS , 7 rue René-Descartes, 67000 Strasbourg , France

2. Korteweg-de Vries Institute for Mathematics , University of Amsterdam , P. O. Box 94248, 1090 GE Amsterdam , The Netherlands

3. Department of Mathematics , University of Oregon , OR 97403 , Eugene , USA

4. Laboratoire de Géométrie, Analyse et Applications , LAGA, CNRS, UMR 7539 , Université Sorbonne Paris Nord , 93430 , Villetaneuse , France

Abstract

Abstract We develop the deformation theory of cohomological field theories (in short, CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduce two new natural extensions of the notion of a CohFT: homotopy (necessary algebraic toolkit to develop chain-level Gromov–Witten invariants) and quantum (with examples found in the works of Buryak and Rossi on integrable systems). The universal group of symmetries of morphisms of modular operads, based on Kontsevich’s graph complex, is shown to be trivial. Using the tautological rings on moduli spaces of curves, we introduce a natural enrichment of Kontsevich’s graph complex. This leads to universal groups of non-trivial symmetries of both homotopy and quantum CohFTs, which, in the latter case, is shown to contain both the prounipotent Grothendieck–Teichmüller group and the Givental group.

Publisher

Walter de Gruyter GmbH

Link

https://www.degruyter.com/document/doi/10.1515/crelle-2023-0098/pdf

Reference66 articles.

1. J. Alm and D. Petersen, Brown’s dihedral moduli space and freedom of the gravity operad, Ann. Sc. Éc. Norm. Supér. (4) 50 (2017), no. 5, 1081–1122.

2. E. Arbarello and M. Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996), no. 4, 705–749.

3. J. C. Baez and J. Dolan, Higher-dimensional algebra. III. n-categories and the algebra of opetopes, Adv. Math. 135 (1998), no. 2, 145–206.

4. S. Barannikov, Modular operads and Batalin–Vilkovisky geometry, Int. Math. Res. Not. IMRN 2007 (2007), no. 19, Article ID rnm075.

5. M. Batanin, J. Kock and M. Weber, Regular patterns, substitudes, Feynman categories and operads, Theory Appl. Categ. 33 (2018), 148–192.