[Paper Review] A mathematical foundation of Rozansky-Witten theory

Kwokwai Chan, Naichung Conan Leung+1 more|arXiv (Cornell University)|Feb 12, 2015

Homotopy and Cohomology in Algebraic Topology33 references3 citations

TL;DR

This paper provides a rigorous mathematical construction of Rozansky-Witten's 3-dimensional sigma-model as a perturbative quantum field theory using Costello's BV formalism and configuration space techniques. It establishes that the cohomology of local quantum observables on a genus $g$ handlebody is isomorphic to $H^*(X, (\wedge^*T_X)^{\otimes g})$, and proves the partition function equals the Rozansky-Witten invariants, thereby giving a mathematically sound foundation for these invariants.

ABSTRACT

We give a mathematically rigorous construction of Rozansky-Witten's 3-dimensional $\sigma$-model as a perturbative quantum field theory (QFT) by applying Costello's approach using the Batalin-Vilkovisky (BV) formalism. The quantization of our model is obtained via the technique of configuration spaces. We also investigate the observable theory following the work of Costello-Gwilliam. In particular, we show that the cohomology of local quantum observables on a genus $g$ handle body is given by $H^*(X,(\wedge^*T_X)^{\otimes g})$, where $X$ is the target hyperk\ahler manifold. We further give a mathematical definition of the partition function and prove that it coincides with the Rozansky-Witten invariants.

Motivation & Objective

• To provide a mathematically rigorous formulation of Rozansky-Witten's 3D sigma-model as a perturbative QFT.
• To apply Costello's BV formalism and configuration space methods to quantize the model.
• To characterize the cohomology of local quantum observables on a genus $g$ handlebody.
• To define the partition function mathematically and show its equivalence to Rozansky-Witten invariants.

Proposed method

• Utilizes Costello's approach to perturbative QFT based on the Batalin-Vilkovisky (BV) formalism.
• Applies configuration space techniques to perform the quantization of the sigma-model.
• Constructs the quantum observables algebraically using the BV complex and homological perturbation theory.
• Analyzes the cohomology of local observables on a genus $g$ handlebody via sheaf cohomology on the target hyperkähler manifold $X$.
• Defines the partition function through the path integral in the BV framework.
• Establishes the coincidence of the partition function with known Rozansky-Witten invariants via cohomological comparison.

Experimental results

Research questions

• RQ1 How can Rozansky-Witten theory be rigorously formulated as a perturbative quantum field theory?
• RQ2 What is the precise mathematical structure of the cohomology of local quantum observables on a genus $g$ handlebody in this model?
• RQ3 How does the partition function of the constructed QFT relate to the Rozansky-Witten invariants?
• RQ4 Can the configuration space method be effectively applied to quantize this 3D sigma-model within the BV formalism?
• RQ5 What role does the hyperkähler structure of the target manifold $X$ play in determining the observable cohomology?

Key findings

• The cohomology of local quantum observables on a genus $g$ handlebody is isomorphic to $H^*(X, (\wedge^*T_X)^{\otimes g})$, providing a precise characterization of the observable algebra.
• The partition function of the constructed QFT is mathematically defined and proven to coincide with the Rozansky-Witten invariants.
• The quantization procedure via configuration spaces yields a well-defined perturbative QFT structure within the BV formalism.
• The construction establishes a rigorous link between physical quantum field theory and topological invariants in 3D.
• The method successfully extends Costello's framework to a class of 3D sigma-models with hyperkähler target spaces.
• The results confirm the mathematical consistency and physical relevance of Rozansky-Witten invariants through a systematic QFT derivation.