Toric varieties of Loday's associahedra and noncommutative cohomological field theories (1510.03261v2)

Abstract: 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 Loday's realisations of the associahedra, as the brick manifolds recently defined by Escobar, and as the De Concini-Procesi wonderful models for certain subspace arrangements.

• The paper introduces noncommutative cohomological field theories (ncCohFTs) as a novel extension, defining them via the Koszul dual of noncommutative gravity operads.
• It demonstrates that toric varieties from Loday’s associahedra serve as brick manifolds, establishing a geometric framework for noncommutative Deligne-Mumford spaces.
• The paper employs Givental group action and formality theorems to derive rich deformation properties, linking algebraic topology with computational algebraic geometry.

Noncommutative Cohomological Field Theories and Toric Varieties of Loday’s Associahedra

The paper of operads has long provided a profound and versatile framework across algebraic topology, category theory, and beyond. This paper addresses a novel conjugation of noncommutative cohomological field theories (ncCohFTs) with the geometric realisation through toric varieties, focusing on structures associated to Loday's associahedra. The authors, Dotsoenko, Shadrin, and Vallette, explore the intricate algebraic structures and suggest a newfound interpretation for these operads.

Key Results and Claims

1. Noncommutative Cohomological Field Theories: The authors introduce the concept of ncCohFTs, which, in essence, abstracts cohomological field theories to a noncommutative setting. These theories operate over a graded vector space by exploring the combined algebra and geometry realms, introducing an ncHyperCom operad, representing the Koszul dual of the operad of noncommutative gravity algebras.
2. Toric Varieties and Brick Manifolds: The paper further details how the toric varieties associated with Loday’s realisation of associahedra can be understood as brick manifolds. This connects the geometric framework with algebraic structures by demonstrating the equivalence of certain toric models and noncommutative braid arrangements, helping define the concept of "noncommutative Deligne-Mumford spaces."
3. Givental Group Action: Employing the intersection theory model, the research examines the application of Givental action on ncCohFTs. Here, the elaboration includes actions on a Lie algebra, leading to a rich symphony of deformations within the field theories.
4. Formality Theorems: The formality of the operads AsS1, AsS1 ⋊ S, and the ns brick operad were shown using theories pertaining to topological operads, providing a substantial underpinning for high-level algebraic manipulations within the provided framework.

Theoretical and Practical Implications

On a theoretical level, these results underline the significance of operads in abstract algebra, feeding into broader theories concerning homotopy and deformation. The posited ncCohFTs pave pathways for considering field theories in noncommutative geometries, which could find applications in topological recursion relations and intersection theories in higher-dimensional algebraic varieties.

Practically, the establishment of toric varieties of Loday’s associahedra and their classification as noncommutative geometrical moduli spaces offer insights into many areas requiring extensive computational algebraic geometry. The formalism introduced could greatly facilitate the construction of explicit models possessing these novel properties, opening up possible endeavors in computational topology and algebraic stacks.

Speculations on Future Developments

As research gravitates further into understanding the components and implications of noncommutative projective geometry, the current paper sets foundations that might navigate future studies towards extending these constructions over broader classes of operads. This could include exploring further applications of ncHyperCom in graphics both from theoretical physics and higher category theory plots, destined for complex systems.

Another future trajectory might pivot towards establishing more generalized models of ncCohFTs using categorical approaches. Interactions between operads with braided and monoidal categories may further illuminate new algebraic properties of ncCohFTs which could hold quantum mechanical or quantum information theoretic significance.

Conclusion

In summation, this paper advances the understanding of noncommutative field theories and their associated toric geometries. While grounded in deep algebraic and topological methods, the work convincingly applies complex theoretical constructs to elucidate noncommutative analogues, significantly broadening the scope of algebraic operads. The collaboration between algebraists and geometers on such initiatives will continually sculpt the contour of mathematical landscapes.