代数量子场论中的Operads、同伦论与高阶范畴

算子，算子范畴与团块复形

Operads, operadic categories and the blob complex

本文证明拓扑量子场论中的Morrison-Walker团块复形是由场和局部关系组成的算子的算子bar分解，并发展了一元算子范畴理论。

Type II Superstring Field Theory: Geometric Approach and Operadic Description

We outline the construction of type II superstring field theory leading to a geometric and algebraic BV master equation, analogous to Zwiebach's construction for the bosonic string. The construction uses the small Hilbert space. Elementary vertices of the non-polynomial action are described with the help of a properly formulated minimal area problem. They give rise to an infinite tower of superstring field products defining a $\mathcal{N}=1$ generalization of a loop homotopy Lie algebra, the genus zero part generalizing a homotopy Lie algebra. Finally, we give an operadic interpretation of the…

Involutive categories, colored $\ast$-operads and quantum field theory

Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field) theory, we develop the involutive analogs of colored operads and their algebras, named colored $\ast$-operads and $\ast$-algebras. Central to the definition of colored $\ast$-operads is the involutive monoidal category of symmetric sequences, which we obtain from a general product-exponential $2$-adjunction whose right adjoint forms involutive functor categorie…

Homotopical Quantum Field Theory

Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. In this monograph, using a new coend definition of the Boardman-Vogt construction of a colored operad, we define homotopy algebraic quantum field theories and homotopy prefactorization algebras and investigate their homotopy coherent structures. Homotopy coherent diagrams, homotopy inverses, A-infinity-algebras, E-infinity-algebras, and E-infinity-modules arise naturally in this context. In particular, each homotopy algebraic quantum field theory has the structure of a homotopy …

Homotopy morphisms between convolution homotopy Lie algebras

In previous works by the authors, a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a homotopy Lie algebra structure. We build on this result by using a more general notion of $\infty$-morphism between (co)algebras over a (co)operad associated to a twisting morphism, and show that this bifunctor can be extended to take such $\infty$-morphisms in either one of its two slots. We also provide a counterexample proving that it cannot be coherently…

操作素算术用于更高色-动能对偶性

Operadic Calculus for Higher Colour-Kinematics Duality

本文提出了一种同调coexact BV代数的概念，用于研究色-动能对偶性及更高阶结构。

$\n\mathbb{C}^d$ 中点构型模空间的 Operad 再探

Operads of moduli spaces of points in $\mathbb{C}^d$ revisited

研究射影空间构型模空间同调上的 operad，用生成元和关系刻画，证明其同伦 Koszul 性及 Givental 型作用，并揭示其与框架化圆盘 operad 同伦商的关系，为 $d=1$ 情形提供新组合视角。