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We develop the deformation theory of cohomological field theories (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: homotopical (necessary to structure chain-level Gromov--Witten invariants) and quantum (with examples found in the works of Buryak--Rossi on integrable systems). We introduce a new version of Kontsevichs graph complex, enriched with tautological classes on the moduli spaces of stable curves. We use it to study a new universal deformation group which acts naturally on the moduli spaces of quantum homotopy CohFTs, by methods due to Merkulov--Willwacher. This group is shown to contain both the prounipotent Grothendieck--Teichmuller group and the Givental group.
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We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-disks, framed little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, coming from the fact that they admit several equivalent interpretations: as the toric varieties of Lodays realisations of the associahedra, as the brick manifolds recently defined by Escobar, and as the De Concini-Procesi wonderful models for certain subspace arrangements.
A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.
In his 2011 paper, Teleman proved that a cohomological field theory on the moduli space $overline{mathcal{M}}_{g,n}$ of stable complex curves is uniquely determined by its restriction to the smooth part $mathcal{M}_{g,n}$, provided that the underlying Frobenius algebra is semisimple. This leads to a classification of all semisimple cohomological field theories. The present paper, the outcome of the authors masters thesis, presents Telemans proof following his original paper. The author claims no originality: the main motivation has been to keep the exposition as complete and self-contained as possible.
The main purpose of this paper is a mathematical construction of a non-perturbative deformation of a two-dimensional conformal field theory. We introduce a notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of a full vertex algebra which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformation is expressed as a double coset of an orthogonal group, a quotient of an orthogonal Grassmannian. As an application, we consider a deformation of chiral conformal field theories, vertex operator algebras. A current-current deformation of a vertex operator algebra may produce new vertex operator algebras. We give a formula for counting the number of the isomorphic classes of vertex operator algebras obtained in this way. We demonstrate it for some holomorphic vertex operator algebra of central charge $24$.
We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L infinity algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toen--Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as Andre--Quillen cohomology.
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