IIB flux non-commutativity and the global structure of field theories

TL;DR

The work shows that the ambiguous global structure of six-dimensional $(2,0)$ theories, and their 4d/N=4 descendants, can be understood from boundary data of RR fluxes in Type IIB string theory on ALE spaces, where torsion induces non-commuting flux operators. By formulating a Heisenberg algebra of RR fluxes and computing the relevant K-theory torsion and linking pairings, the authors derive how maximal isotropic polarizations specify boundary conditions that reproduce known 4d global form classifications (Aharony–Seiberg–Tachikawa) and determine fractional instanton data via the torsion linking form. The framework unifies 6d $(2,0)$ theories and 4d$\\mathcal{N}=4$ theories, clarifying how self-dual boundary conditions yield modular-invariant partition data on special manifolds (e.g., K3) and how duality defects affect global structure. It also provides a natural route to 2d/4d correspondences through Hecke transforms and suggests broad generalizations to other geometries and lower-supersymmetry settings, with potential connections to anomaly theories in non-compact spacetimes.

Abstract

We discuss the origin of the choice of global structure for six dimensional $(2,0)$ theories and their compactifications in terms of their realization from IIB string theory on ALE spaces. We find that the ambiguity in the choice of global structure on the field theory side can be traced back to a subtle effect that needs to be taken into account when specifying boundary conditions at infinity in the IIB orbifold, namely the known non-commutativity of RR fluxes in spaces with torsion. As an example, we show how the classification of $\mathcal{N}=4$ theories by Aharony, Seiberg and Tachikawa can be understood in terms of choices of boundary conditions for RR fields in IIB. Along the way we encounter a formula for the fractional instanton number of $\mathcal{N}=4$ ADE theories in terms of the torsional linking pairing for rational homology spheres. We also consider six-dimensional $(1,0)$ theories, clarifying the rules for determining commutators of flux operators for discrete 2-form symmetries. Finally, we analyze the issue of global structure for four dimensional theories in the presence of duality defects.

Figures (7)

• Figure 1: Dynkin diagram for $D_n$.
• Figure 2: The generalized geometries considered in the text. Nodes denote two-cycles, a line connecting two nodes indicates that the cycles intersect each other transversely, and a number next to the node denotes (minus) its self-intersection.
• Figure 3: Maximal isotropic subspace of $H^1(T^2;\mathbb{Z}_3)=\mathbb{Z}_3\oplus\mathbb{Z}_3$ with respect to the perfect pairing (\ref{['eqn:ZNperfectpairing']}). We have labelled the possibilities using the nomenclature of Aharony:2013hda. Each filled dot corresponds to a genuine line operator.
• Figure 4: \ref{['sfig:flux-diff-a']} The fractional instanton number can be viewed as the anomaly coming from a large diffeomorphism in the presence background torsion flux. At the level of the fluxes, this can be implemented by the insertion of suitable operators in the anomaly theory. \ref{['sfig:flux-diff-b']} Gluing two copies of the configuration giving the anomaly to two configurations without flux we obtain the anomaly theory with four operator insertions. Bringing the operators together we obtain the commutator, a c-number.
• Figure 5: The paths that implement the monodromies $S$ and $T$ at the point $p$ in the presence of the degenerations of the elliptic fibration discussed in the text. The dotted lines indicate branch cuts. The monodromies indicated in the text are obtained by crossing the branch cut counterclockwise.