1-form symmetry, isolated $$ \mathcal{N} $$ = 2 SCFTs, and Calabi-Yau threefolds
Matthew Buican, Hongliang Jiang
Abstract
A bstract We systematically study 4D $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 superconformal field theories (SCFTs) that can be constructed via type IIB string theory on isolated hypersurface singularities (IHSs) embedded in ℂ 4 . We show that if a theory in this class has no $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2-preserving exactly marginal deformation (i.e., the theory is isolated as an $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 SCFT), then it has no 1-form symmetry. This situation is somewhat reminiscent of 1-form symmetry and decomposition in 2D quantum field theory. Moreover, our result suggests that, for theories arising from IHSs, 1-form symmetries originate from gauge groups (with vanishing beta functions). One corollary of our discussion is that there is no 1-form symmetry in IHS theories that have all Coulomb branch chiral ring generators of scaling dimension less than two. In terms of the a and c central charges, this condition implies that IHS theories satisfying $$ a<\frac{1}{24}\left(15r+2f\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> <mml:mo><</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>24</mml:mn> </mml:mfrac> <mml:mfenced> <mml:mrow> <mml:mn>15</mml:mn> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>f</mml:mi> </mml:mrow> </mml:mfenced> </mml:math> and $$ c<\frac{1}{6}\left(3r+f\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> <mml:mo><</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>6</mml:mn> </mml:mfrac> <mml:mfenced> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:mfenced> </mml:math> (where r is the complex dimension of the Coulomb branch, and f is the rank of the continuous 0-form flavor symmetry) have no 1-form symmetry. After reviewing the 1-form symmetries of other classes of theories, we are motivated to conjecture that general interacting 4D $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 SCFTs with all Coulomb branch chiral ring generators of dimension less than two have no 1-form symmetry.