[Paper Review] Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories

Cyril Closset, Horia Magureanu|arXiv (Cornell University)|Aug 20, 2023

Black Holes and Theoretical Physics90 references4 citations

TL;DR

This paper establishes a geometric correspondence between the global structures of 4d $ N=2$ supersymmetric field theories and torsion subgroups of the Mordell-Weil group in Seiberg-Witten elliptic fibrations. It distinguishes between relative and absolute Seiberg-Witten curves, showing that gauging $1$-form symmetries corresponds to composing isogenies generated by torsion sections, and applies this framework to classify global structures in rank-one $ N=2$ KK theories from 5d and 6d SCFTs, including the M-string theory on $T^2$. The key contribution is a systematic geometric encoding of defect groups and $1$-form symmetries via elliptic curve isogenies.

ABSTRACT

We study how the global structure of rank-one 4d $\mathcal{N}=2$ supersymmetric field theories is encoded into global aspects of the Seiberg-Witten elliptic fibration. Starting with the prototypical example of the $\mathfrak{su}(2)$ gauge theory, we distinguish between relative and absolute Seiberg-Witten curves. For instance, we discuss in detail the three distinct absolute curves for the $SU(2)$ and $SO(3)_\pm$ 4d $\mathcal{N}=2$ gauge theories. We propose that the $1$-form symmetry of an absolute theory is isomorphic to a torsion subgroup of the Mordell-Weil group of sections of the absolute curve, while the full defect group of the theory is encoded in the torsion sections of a so-called relative curve. We explicitly show that the relative and absolute curves are related by isogenies (that is, homomorphisms of elliptic curves) generated by torsion sections -- hence, gauging a one-form symmetry corresponds to composing isogenies between Seiberg-Witten curves. We apply this approach to Kaluza-Klein (KK) 4d $\mathcal{N}=2$ theories that arise from toroidal compactifications of 5d and 6d SCFTs to four dimensions, uncovering an intricate pattern of 4d global structures obtained by gauging discrete $0$-form and/or $1$-form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on $\mathbb{R}^4 imes T^2$.

Motivation & Objective

• To clarify how the global structure of rank-one 4d $ N=2$ SQFTs—specifically the choice of gauge group like $SU(2)$, $SO(3)_{/pm}$—is encoded in the Seiberg-Witten geometry.
• To distinguish between relative and absolute Seiberg-Witten curves, where the former describes the Lie algebra dynamics and the latter encodes the full global gauge group.
• To establish a geometric correspondence between $1$-form symmetries and torsion subgroups of the Mordell-Weil group of the absolute Seiberg-Witten curve.
• To apply this framework to Kaluza-Klein compactifications of 5d and 6d SCFTs, revealing the global structure patterns arising from discrete gaugings of $0$-form and $1$-form symmetries.
• To propose a 6d BPS quiver for the M-string theory on $\mathbb{R}^4 \times T^2$ using the same geometric framework.

Proposed method

• Distinguishes relative and absolute Seiberg-Witten curves: the relative curve describes the Lie algebra dynamics, while the absolute curve encodes the full global gauge group.
• Identifies the $1$-form symmetry of an absolute theory as isomorphic to a torsion subgroup of the Mordell-Weil group of the absolute Seiberg-Witten curve.
• Uses isogenies—homomorphisms between elliptic curves generated by torsion sections—to model the process of gauging $1$-form symmetries.
• Applies this framework to KK compactifications of 5d $E_1[\mathfrak{su}(2)]$ and 6d $E_0$ theories, analyzing the resulting global structures via modular groups and singularity types.
• Relates the defect group of the theory to the torsion sections of the relative Seiberg-Witten curve, with the defect group $\mathbb{D} = \mathbb{Z}_2 \oplus \mathbb{Z}_2$ for the $\mathfrak{su}(2)$ theory.
• Uses modular forms and congruence subgroups of $\mathrm{PSL}(2,\mathbb{Z})$ to classify the modular properties of the curves and their associated symmetries.

Experimental results

Research questions

• RQ1 How is the global structure of a 4d $\mathcal{N}=2$ SQFT—distinguishing $SU(2)$ from $SO(3)_{\pm}$—encoded in its Seiberg-Witten geometry?
• RQ2 What is the precise geometric relationship between the relative and absolute Seiberg-Witten curves, and how do isogenies between them correspond to gauging $1$-form symmetries?
• RQ3 How do the torsion sections of the Mordell-Weil group of the absolute curve realize the $1$-form symmetry of the theory?
• RQ4 What global structures emerge from compactifying 5d and 6d SCFTs on a circle or torus, and how are they classified via Seiberg-Witten geometry?
• RQ5 Can a 6d BPS quiver be constructed for the M-string theory on $T^2$, and how does it relate to the global structure of the resulting 4d theory?

Key findings

• The $1$-form symmetry of an absolute $\mathcal{N}=2$ theory is isomorphic to a torsion subgroup of the Mordell-Weil group of its absolute Seiberg-Witten curve.
• The defect group $\mathbb{D} = \mathbb{Z}_2 \oplus \mathbb{Z}_2$ for the $\mathfrak{su}(2)$ theory is encoded in the torsion sections of the relative Seiberg-Witten curve.
• Gauging a $1$-form symmetry corresponds to composing isogenies between Seiberg-Witten curves, with the isogeny generated by a torsion section.
• The three distinct absolute curves for the $\mathfrak{su}(2)$ theory correspond to $SU(2)$, $SO(3)_{+}$, and $SO(3)_{-}$, with $1$-form symmetries $\mathbb{Z}_2$, $\mathbb{Z}_2$, and $\mathbb{Z}_4$, respectively.
• For the $E_1[\mathfrak{su}(2)]$ 5d theory compactified on $S^1$, the resulting $\mathcal{N}=2$ KK theories exhibit global structures classified by discrete gaugings, with modular groups $\Gamma(2)$, $\Gamma_0(4)$, and $\Gamma(4)$.
• The M-string theory on $T^2$ admits a 6d BPS quiver proposal, and its 4d global structures are determined by the torsion structure of the absolute Seiberg-Witten curve, with $\mathbb{Z}_3^{[0]}$ gauging leading to a $\mathbb{Z}_3$ orbifold point on the $V$-plane.