Liberating confinement from Lagrangians: 1-form symmetries and lines in 4d N=1 from 6d N=(2,0)

<jats:p>We study confinement in 4d <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=1</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theories obtained by deforming 4d <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=2</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theories of Class S. We argue that confinement in a vacuum of the <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=1</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theory is encoded in the 1-cycles of the associated <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=1</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> curve. This curve is the spectral cover associated to a generalized Hitchin system describing the profiles of two Higgs fields over the Riemann surface upon which the 6d <jats:inline-formula><jats:alternatives><jats:tex-math>(2,0)</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theory is compactified. Using our method, we reproduce the expected properties of confinement in various classic examples, such as 4d <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=1</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> pure Super-Yang-Mills theory and the Cachazo-Seiberg-Witten setup. More generally, this work can be viewed as providing tools for probing confinement in non-Lagrangian <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=1</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theories, which we illustrate by constructing an infinite class of non-Lagrangian <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=1</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theories that contain confining vacua. The simplest model in this class is an <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=1</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> deformation of the <jats:inline-formula><jats:alternatives><jats:tex-math>\mathcal{N}=2</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle mathvariant="script"><mml:mi>𝒩</mml:mi></mml:mstyle><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theory obtained by gauging <jats:inline-formula><jats:alternatives><jats:tex-math>SU(3)^3</jats:tex-math><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:mo stretchy="false" form="prefix">(</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mo stretchy="false" form="postfix">)</mml:mo><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> flavor symmetry of the <jats:inline-formula><jats:alternatives><jats:tex-math>E_6</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>E</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> Minahan-Nemeschansky theory. </jats:p>