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

Abstract

We study how the global structure of rank-one 4d \mathcal{N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>𝒩</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> supersymmetric field theories is encoded into global aspects of the Seiberg-Witten elliptic fibration. Starting with the prototypical example of the \mathfrak{su}(2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mrow><mml:mi>𝔰</mml:mi><mml:mi>𝔲</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math> 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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math> and SO(3)_± <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mo>±</mml:mo></mml:msub></mml:mrow></mml:math> 4d \mathcal{N}=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>𝒩</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> gauge theories. We propose that the 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn></mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>𝒩</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>0</mml:mn></mml:math> -form and/or 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn></mml:math> -form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on \mathbb{R}^4× T^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mi>ℝ</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> .

Topics & Concepts