An emergent higher-form symmetry from type IIB superstring theory
Naoto Kan, Masashi Kawahira, Hiroki Wada
TL;DR
The paper addresses how emergent higher-form symmetries arise in type IIB string theory from the gauged $SL(2,\mathbb{Z})$ duality. It develops a framework based on quantum (duality) symmetries and constructs explicit topological operators using the discriminant $\Delta(\tau)$ within an F-theory setup, identifying ${\mathbb Z}_{12}^{[8]}$ as the solitonic symmetry associated with $7$-branes. By examining extensions of the duality group to Mp$(2,\mathbb{Z})$, GL$(2,\mathbb{Z})$, and Pin$^+(GL(2,\mathbb{Z}))=GL^{+}(2,\mathbb{Z})$, it reveals a richer ${\mathbb Z}_{24}^{[8]}$ structure intertwined with worldsheet orientation reversal, forming a higher-group symmetry ${\mathbb Z}_{24}^{[8]}\rtimes ({\mathbb Z}_{2}^{[0]})_{\Omega}$. The work clarifies how low-energy effective theories encode topological data and heavy degrees of freedom, and discusses subtle distinctions between string theory and field theory in the presence of topological phases, with potential extensions to Type 0B and phenomenological implications.
Abstract
We investigate a higher-form symmetry in type IIB superstring theory, which possesses an ${\rm SL}(2,\mathbb{Z})$ symmetry. From the point of view of the low-energy effective field theory, the ${\rm SL}(2,\mathbb{Z})$ symmetry is treated as a gauge symmetry. Hence, an $8$-form global symmetry $\mathbb{Z}_{12}^{[8]}$ emerges as a quantum symmetry. In this paper, we present an explicit construction of the topological operator associated with the $\mathbb{Z}_{12}^{[8]}$ symmetry. In this construction, the discriminant $Δ(τ)$ plays a central role. As a result, it becomes manifest that $\mathbb{Z}_{12}^{[8]}$ is the solitonic symmetry of $7$-branes. Furthermore, taking into account the extensions of the duality group, we also discuss what global symmetries emerge when considering not ${\rm SL}(2,\mathbb{Z})$ but ${\rm Mp}(2,\mathbb{Z})$, ${\rm GL}(2,\mathbb{Z})$, and ${\rm Pin}^+({\rm GL}(2,\mathbb{Z}))={\rm GL}^+(2,\mathbb{Z})$.