Topological duality twist and brane instantons in F-theory

TL;DR

The paper introduces a topological duality twist (TDT) that extends the standard topological twist by incorporating SL(2,$\mathbb{Z}$) dualities to allow a holomorphic, space-dependent coupling $\tau(z)$ on a Kähler surface. It develops both the local spectrum and the global structure of a U(1) N=4 SYM under TDT, revealing a network of duality walls (3d Chern–Simons–like terms) and chiral defects (2d theories) that end walls and cancel anomalies to preserve twisted supersymmetry; it then embeds this framework in F-theory, including axionic moduli and bulk gauge fields, to study Euclidean D3-brane instantons. An explicit Hirzebruch surface example and Sen limit demonstrate how wall networks and monodromies arise and how the partition function becomes independent of the bulk Kähler moduli up to a topological factor, linking to non-perturbative F-theory corrections. The work lays groundwork for quantization, anomaly analysis, and extending to higher-rank gauge groups, with potential applications to brane instanton effects in F-theory compactifications.

Abstract

A variant of the topological twist, involving SL(2,Z) dualities and hence named topological duality twist, is introduced and explicitly applied to describe a U(1) N=4 super Yang-Mills theory on a Kaehler space with holomorphically space-dependent coupling. Three-dimensional duality walls and two-dimensional chiral theories naturally enter the formulation of the duality twisted theory. Appropriately generalized, this theory is relevant for the study of Euclidean D3-brane instantons in F-theory compactifications. Some of its properties and implications are discussed.

Figures (7)

• Figure 1: A local region of $S$ constituted by two patches $\Sigma$ and $\Sigma'$ divided by a $\gamma$ wall ${\cal B}$. Two of the directions parallel to ${\cal B}$ are suppressed.
• Figure 2: A local deformation of a $\gamma$ wall from ${\cal B}$ (blue) to ${\cal B}'$ (violet). It corresponds to performing a $\gamma^{-1}$ duality in the region $\Sigma$ surrounded by ${\cal B}$ and ${\cal B}'$.
• Figure 3: A junctions of three duality walls which separates the three patches $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$ and meets at the two-dimensional space ${\cal C}$. The picture shows a two-dimensional slice transversal to ${\cal C}$.
• Figure 4: Two dimensional slice, transversal to a chiral defect ${\cal C}$, of a patch $\Sigma$ surrounding ${\cal C}$. As we encircle ${\cal C}$ anticlockwise, $\tau$ undergoes a monodromy $\tau\rightarrow \tau+1$. Hence the wall ${\cal B}$ is associated to the $T^{-1}$ duality which brings $\tau$ back to its original value.
• Figure 5: A $T$-conjugated $\gamma$-monodromy, with $\gamma=M T M^{-1}$, can be locally described as a $T$-monodromy by appropriately choosing the duality walls. The $\gamma^{-1}$ wall ${\cal B}$ on the left is substituted by a junction of the three $\gamma^{-1}$, $T^{-1}$ and $M$ walls (${\cal B}$, ${\cal B}'$ and ${\cal B}"$ respectively) on the right.