Higher Operations in Perturbation Theory

Abstract

We discuss the role of formal deformation theory in quantum field theories and present various ``higher operations'' which control their deformations, (generalized) OPEs, and anomalies. Particular attention is paid to holomorphic-topological theories where we systematically describe and regularize the Feynman diagrams which compute these higher operations in free and perturbative scenarios, including examples with defects. We prove geometrically that the resulting higher operations satisfy expected ``quadratic axioms,'' which can be interpreted physically as a form of Wess-Zumino consistency condition for BRST symmetry. We discuss a higher-dimensional analogue of Kontsevich's formality theorem, which proves the absence of perturbative corrections in TQFTs with two or more topological directions. We discuss at some length the relation of our work to the theory of factorization algebras and provide an introduction to the subject for physicists.

Figures (12)

• Figure 1: Left, a cycle in the configuration space $\mathrm{Conf}_3(\mathbb{R}^2)$ corresponding to a 2-bracket of $\mathcal{O}_1$ and $\mathcal{O}_2$ concatenated with a 2-bracket with $\mathcal{O}_3$. Right, a cycle in the same configuration space of 3-points corresponding to a 2-bracket of $\mathcal{O}_2$ and $\mathcal{O}_3$; $\mathcal{O}_1$ is fixed "at infinity."
• Figure 2: We give the first four 3-Laman graphs (up to 2 loops). In general, for $n$-Laman graphs, there will only be one $0$-loop graph given by a single edge, one $1$-loop graph given by an $(n+1)$ gon, and $(n-1)$$2$-loop graphs obtained by sewing an $(n+1)$-gon to an $(n+k)$-gon along $k$ consecutive edges, for $k=1,\dots, n-1$.
• Figure 3: Left, an almost $3$-Laman graph $\Gamma$ of degree $1$. A square $3$-Laman subgraph $\Gamma[S]$ is marked in red. If the red square is shrunk down to a point $p_0$, the remaining graph $\Gamma(S)$ is also $3$-Laman.
• Figure 4: The quadratic identity for $3$-Laman graphs implied by the almost $3$-Laman square-pentagon graph in Figure \ref{['fig:shrinkingGraph']}. We label all of the vertices and edges following the ordering conventions implied by Figures \ref{['fig:canLabLaman']} and \ref{['fig:shrinkingGraph']}. We employ here the shorthand $\lambda_{i_1 + \dots + i_n} := \sum_{j=1}^n \lambda_{i_j}$.
• Figure 5: Degree $1$ almost 1-Laman graphs have three vertices, labeled as $v_1$, $v_2$, $v_3$, with $N_{12}$, $N_{13}$ and $N_{23}$ edges. Note, when there is only one spacetime dimension, i.e. $(H,T)=(0,1)$, these graphs should be viewed as one dimensional, not stretching into a second direction.

Theorems & Definitions (9)

• Definition 1: Operad
• Definition 2: Operad Morphism
• Definition 3: Algebra over an Operad
• Definition 4: Symmetric Operad
• Definition 5: Module over an Operad
• Definition 6: Coloured Operad
• Definition 7: Little $d$-Disks Operad, $E_d$ Operad
• Definition 8: Disjoint Sets Operad, $\mathrm{Disj}_M$
• Definition 9: Prefactorization Algebra