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Register a new userToric varieties of Lodays associahedra and noncommutative cohomological field theories
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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.
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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.
We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety $X$ with branching set the invariant divisor under the action of $(mathbb{C}^*)^n$. This is the proalgebraic toric-completion $X_{mathbb{Q}}$ of $X$. The ramification over the invariant divisor and the singular invariant divisors of $X$ impose topological constraints on the automorphisms of $X_{mathbb{Q}}$. Considering this proalgebraic space as the toric functor on the adelic complex plane multiplicative semigroup, we calculate its automorphic group. Moreover we show that its vector bundle category is the direct limit of the respective categories of the finite toric varieties coverings defining the proalgebraic toric-completion.
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.
We revisit the question of microcausality violations in quantum field theory on noncommutative spacetime, taking $O(x)=:phistarphi:(x)$ as a sample observable. Using methods of the theory of distributions, we precisely describe the support properties of the commutator [O(x),O(y)] and prove that, in the case of space-space noncommutativity, it does not vanish at spacelike separation in the noncommuting directions. However, the matrix elements of this commutator exhibit a rapid falloff along an arbitrary spacelike direction irrespective of the type of noncommutativity. We also consider the star commutator for this observable and show that it fails to vanish even at spacelike separation in the commuting directions and completely violates causality. We conclude with a brief discussion about the modified Wightman functions which are vacuum expectation values of the star products of fields at different spacetime points.
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