On 2d TQFTs whose values are holomorphic symplectic varieties
Gregory W. Moore, Yuji Tachikawa
TL;DR
The paper proposes a conjectural framework for a 2D topological quantum field theory whose values are holomorphic symplectic varieties, realized as a functor $\eta_{G_{\mathbb{C}}}$ from the 2D bordism category to a holomorphic-symplectic target category. Built from physics of six-dimensional $\mathcal{N}=(2,0)$ theories of type $G$, the construction encodes gluing via holomorphic symplectic quotients and organizes data through basic objects $U_{G_{\mathbb{C}}}$ and $W_{G_{\mathbb{C}}}$ tied to Nahm-moduli and Slodowy slices, with explicit examples for $A_1$ and $A_2$ aligning with minimal nilpotent orbits in $SO(8,\mathbb{C})$ and $E_6(\mathbb{C})$. The work catalogs a rich set of conjectures and properties for general $G_{\mathbb{C}}$, including moment-map relations across triplets, realizations as symplectic vector spaces, and connections to instanton moduli spaces, while addressing the hyperkähler nature of Higgs branches and proposing an area-dependent refinement to obtain a genuine TFT. It also outlines extensions to equivariant and fully extended TFT frameworks, suggesting deep links to Langlands duality and domain-wall physics. Overall, the paper lays out a coherent program to axiomatize and rigorously construct a 2D TFT valued in holomorphic symplectic varieties rooted in the physics of class $S$ theories, with concrete low-rank examples and broad conjectural structure guiding future mathematics.
Abstract
For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the diagonal action of G. We describe various properties of eta_G obtained via string-theoretic analysis. Mathematicians are urged to construct eta_G rigorously.