The partition bundle of type A_{N-1} (2, 0) theory

TL;DR

The paper develops a geometric framework for the generalized partition data Z of the 6D (2,0) theory by encoding dependence on six-manifold data through the intermediate Jacobian $T$ and a hermitian line bundle $\mathcal{L}$ with curvature $\omega$. It shows that Z takes values in $V\simeq H^{0}(T,\mathcal{L}^{N})$, with dimension $N^{n}$, and that the full structure reorganizes into a partition bundle $E$ obtained by pulling back a holomorphic, hermitian bundle over a Siegel-space parameterization of complex structures. By constructing a holomorphic frame from theta functions and analyzing modular transformations via ${\rm Sp}_{2n}(\mathbb{Z})$, the work links the generalized partition data to Siegel modular geometry and a flat vector-bundle component, while illustrating the dependence on spin structures in the $T^{6}$ case. This provides a concrete, modular, and geometric perspective on the elusive partition function-like object in the (2,0) theory with potential connections to holography and topological field theories.

Abstract

Six-dimensional (2, 0) theory can be defined on a large class of six-manifolds endowed with some additional topological and geometric data (i.e. an orientation, a spin structure, a conformal structure, and an R-symmetry bundle with connection). We discuss the nature of the object that generalizes the partition function of a more conventional quantum theory. This object takes its values in a certain complex vector space, which fits together into the total space of a complex vector bundle (the `partition bundle') as the data on the six-manifold is varied in its infinite-dimensional parameter space. In this context, an important role is played by the middle-dimensional intermediate Jacobian of the six-manifold endowed with some additional data (i.e. a symplectic structure, a quadratic form, and a complex structure). We define a certain hermitian vector bundle over this finite-dimensional parameter space. The partition bundle is then given by the pullback of the latter bundle by the map from the parameter space related to the six-manifold to the parameter space related to the intermediate Jacobian.