- The paper conjectures a functor mapping 2-bordisms to holomorphic symplectic varieties with Hamiltonian actions.
- It rigorously outlines the category theory framework and axioms essential for defining these 2D TQFTs.
- The analysis connects class S theories with symplectic geometry, inviting further exploration of nilpotent orbits and moduli spaces.
Topological Quantum Field Theories and Holomorphic Symplectic Varieties
Introduction
In the landscape of contemporary theoretical physics and mathematics, the interplay between Topological Quantum Field Theories (TQFTs) and symplectic geometry has emerged as a fertile ground for the exploration of new concepts. The paper under discussion propels this dialogue forward by conjecturing the existence of a specific class of 2-dimensional TQFTs, which maps the category of 2-bordisms to a category of holomorphic symplectic varieties. These varieties are postulated to have Hamiltonian actions that mirror physical phenomena described by conformal field theories in six dimensions (6d $\cN=(2,0)$ theories).
Theoretical Background
The genesis of this inquiry lies in the paper of 6d $\cN=(2,0)$ superconformal quantum field theories, believed to be classified by ADE-type simple, connected, and simply laced Lie groups. When compactified on punctured Riemann surfaces C, these theories yield a rich class of lower-dimensional theories, often referred to as class S theories, which demonstrate remarkable factorization properties hitherto known in two-dimensional conformal and topological quantum field theories. Motivated by this, the authors conjecture a functor $\eta_{G_{\bC}}$ mapping 2d bordisms to holomorphic symplectic varieties equipped with Hamiltonian actions, highlighting an intriguing structure that mirrors the factorization properties of class S theories.
Category Theory and $\eta_{G_{\bC}}$ Functor
At the core of the paper, the authors outline the structure of the categories involved and the properties required for the conjectured functor, $\eta_{G_{\bC}}$, to be well-defined. The source category is the familiar category of 2-bordisms, while the target category, denoted $\HS$, encompasses holomorphic symplectic varieties with Hamiltonian actions of complex algebraic groups. The classification hinges on the construction of symplectic varieties, $U_{G_{\bC}}$ and $W_{G_{\bC}}$, that satisfy certain axioms analogous to those in 2d TQFTs, such as commutativity and associativity axioms under the functor $\eta_{G_{\bC}}$.
Mathematical Implications and Conjectures
One of the striking proposals in the paper is the explicit description of these varieties for certain Lie groups, propelled by string-theoretic analysis. For instance, for $G_{\bC} = \SL(2,\bC)$, the variety $W_{G_{\bC}}$ is proposed to be equivalent to the flat symplectic space $\bC^2 \otimes \bC^2 \otimes \bC^2$. Similarly, intriguing conjectures are made about the varieties associated with other Lie groups, hinting at a deep connection with the geometry of nilpotent orbits and moduli spaces of instantons, thus opening a rich vein of mathematical structures waiting to be rigorously explored.
Future Directions
The research presented provokes several natural inquiries and directions for future work. Among these, extending the conjectured functor to equivariant versions for 2d TQFTs and exploring its relevance and implications in the context of extended TQFTs are of particular interest. Moreover, the physical predicate of class S theories promises a fertile ground for understanding the geometrical and topological aspects of quantum field theories through the symplectic geometry of holomorphic varieties.
Conclusion
In summary, the conjectures and the framework presented forge new paths in the interplay between topological aspects of quantum field theories and symplectic geometry. By proposing a functorial bridge between 2-bordisms and categories of holomorphic symplectic varieties, the paper sets the stage for a deeper mathematical understanding of class S theories and invites further exploration into the rich interrelations between physics and geometry.