Secondary products in supersymmetric field theory

TL;DR

This work identifies and analyzes higher, secondary operations in the operator algebras of cohomological TQFTs, showing that local operators carry a shifted Poisson ($P_d$) structure and that extended operators (notably line operators) acquire richer, higher disc structures. Central to the framework is topological descent, which defines a degree-(1−d) bracket that acts derivatively on the primary product and satisfies Jacobi; in odd dimensions this yields a genuine Poisson bracket, while in even dimensions one obtains a Gerstenhaber structure. The Omega-background is shown to induce deformation quantization of these algebras by encoding equivariant homology on configuration spaces, with explicit realizations in Rozansky–Witten theory and in 4d GL twists related to geometric Langlands. The paper works out concrete examples in the 2d B-model, 3d RW twists, and 4d gauge theories, clarifying how secondary operations on local and line operators measure noncommutativity of categorical structures (e.g., the spherical category) and connecting to deeper phenomena like Ngô actions and boundary conditions under S-duality. These insights provide a unified, topological mechanism for quantization and deformation of operator algebras, with potential applications to deformations, higher-form symmetries, and holomorphic twists.

Abstract

The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect to the primary product. We describe the mathematical structures involved and illustrate this general phenomenon in a range of physical examples arising from supersymmetric field theories in spacetime dimension two, three, and four. In the Rozansky-Witten twist of three-dimensional N=4 theories, this gives an intrinsic realization of the holomorphic symplectic structure of the moduli space of vacua. We further give a simple mathematical derivation of the assertion that introducing an Omega-background precisely deformation quantizes this structure. We then study the secondary product structure of extended operators, which subsumes that of local operators but is often much richer. We calculate interesting cases of secondary brackets of line operators in Rozansky-Witten theories and in four-dimensional N=4 super Yang-Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.

Figures (14)

• Figure 1: An illustration of an element of ${\mathrm {Op}}_\delta$, coming from a multi-point insertion of ordinary local operators and a compact loop operator, all supported in the ball $B_\delta$.
• Figure 2: A point of the connected space $\mathcal{C}_2(B)$ for $d = 2$.
• Figure 3: Left: a point of the component $\mathcal{C}_{2,a}(B)$. Right: a point of the component $\mathcal{C}_{2,b}(B)$.
• Figure 4: Connectedness of $\mathcal{C}_3(B)$ leads to associativity of $*$.
• Figure 5: Construction of the product operator \ref{['eq:descent-product-op']}: the local operator ${\mathcal{O}}_1$ is placed at $x$, and the $(d-1)$-form descendant of the local operator ${\mathcal{O}}_2$ is integrated over a surrounding sphere. The large ball $B$ is not shown explicitly.

Theorems & Definitions (3)

• Proposition 9.1
• Theorem 9.2
• Claim 9.3